From 1f3e10710615402f4b636bb02e38a1c0dfaa4796 Mon Sep 17 00:00:00 2001
From: Smaug123 <patrick+github@patrickstevens.co.uk>
Date: Sat, 30 Sep 2023 18:30:52 +0100
Subject: [PATCH] Import Hugo

---
 README.md                                     |    36 +-
 flake.lock                                    |    17 -
 flake.nix                                     |     7 +-
 hugo/.gitignore                               |    13 +
 hugo/assets/css/fontawesome.css               |    74 +
 hugo/assets/css/sidenote.css                  |    62 +
 hugo/assets/css/syntax.css                    |    59 +
 hugo/config.toml                              |   111 +
 hugo/content/ILAS/index.md                    |     8 +
 hugo/content/about-this-site/index.md         |    35 +
 hugo/content/about/index.md                   |    52 +
 hugo/content/anki-decks/index.md              |    18 +
 ...2015-08-19-category-theory-introduction.md |    18 +
 .../awodey/2015-08-19-what-is-a-category.md   |    57 +
 .../2015-08-20-new-categories-from-old.md     |    71 +
 ...5-08-21-free-categories-and-foundations.md |    56 +
 hugo/content/awodey/2015-09-02-epis-monos.md  |    60 +
 ...2015-09-02-initial-generalised-elements.md |    57 +
 .../2015-09-08-products-in-category-theory.md |    49 +
 .../2015-09-10-homsets-and-exercises.md       |    63 +
 .../2015-09-15-duality-in-category-theory.md  |    59 +
 hugo/content/awodey/2015-09-16-equalisers.md  |    53 +
 .../awodey/2015-09-19-duality-exercises.md    |    66 +
 .../awodey/2015-09-19-groups-in-categories.md |    57 +
 .../awodey/2015-09-22-limits-and-pullbacks.md |    50 +
 ...15-09-22-properties-of-pullbacks-limits.md |    61 +
 .../awodey/2015-09-23-properties-of-limits.md |    71 +
 ...5-09-29-exponentials-in-category-theory.md |    51 +
 .../awodey/2015-09-29-limit-exercises.md      |    83 +
 .../awodey/2015-09-30-heyting-algebras.md     |    49 +
 hugo/content/comparisons/index.md             |    82 +
 hugo/content/films/index.md                   |   130 +
 hugo/content/lifehacks/index.md               |    29 +
 .../posts/2013-06-26-cucats-puzzlehunt.md     |    27 +
 hugo/content/posts/2013-06-26-first-post.md   |    16 +
 .../posts/2013-06-26-sylow-theorems.md        |   110 +
 ...13-07-03-in-which-i-augment-the-lexicon.md |    27 +
 ...13-07-04-cambridge-undergrad-maths-tips.md |    31 +
 ...mbridge-vocab-a-guide-for-the-mystified.md |    23 +
 .../posts/2013-07-07-mundane-magics.md        |    24 +
 ...-07-08-an-obvious-improvement-to-tennis.md |    40 +
 ...013-07-09-stumbled-across-9th-july-2013.md |    32 +
 ...-07-10-imre-leader-appreciation-society.md |    15 +
 ...-framework-for-discussing-pricelessness.md |    36 +
 ...13-07-13-stumbled-across-13th-july-2013.md |    24 +
 ...erequisites-for-hypothetical-situations.md |    83 +
 ...e-multiple-drafts-view-of-consciousness.md |    48 +
 ...-objection-to-the-one-logical-leap-view.md |    80 +
 .../posts/2013-07-21-on-shakespeare.md        |    58 +
 .../2013-07-22-the-orbitstabiliser-theorem.md |    28 +
 ...13-07-24-stumbled-across-24th-july-2013.md |    25 +
 hugo/content/posts/2013-07-25-metathought.md  |    42 +
 ...2013-07-29-stumbled-across-29-july-2013.md |    21 +
 ...-30-on-to-do-lists-as-direction-in-life.md |    35 +
 .../posts/2013-08-04-new-computer-setup.md    |    60 +
 ...013-08-04-stumbled-across-4-august-2013.md |    25 +
 ...-08-11-stumbled-across-11th-august-2013.md |    24 +
 .../posts/2013-08-18-thinking-styles.md       |    37 +
 .../2013-08-21-my-experiences-with-flow.md    |    30 +
 .../2013-08-22-how-to-punt-in-cambridge.md    |    43 +
 ...13-08-24-stumbled-across-24-august-2013.md |    24 +
 .../posts/2013-08-26-topology-made-simple.md  |    43 +
 ...-31-slightly-silly-sylow-pseudo-sonnets.md |   171 +
 ...-13-stumbled-across-14th-september-2013.md |    29 +
 ...3-09-21-how-to-prove-that-you-are-a-god.md |    38 +
 hugo/content/posts/2013-10-10-plot-armour.md  |    51 +
 .../posts/2013-10-11-meaning-what-you-say.md  |    34 +
 .../2013-10-13-training-away-mental-bias.md   |    46 +
 hugo/content/posts/2013-10-20-the-ravenous.md |    56 +
 ...2013-10-24-how-to-do-analysis-questions.md |   115 +
 .../2013-11-07-my-quest-for-a-new-phone.md    |    96 +
 .../2013-11-12-markov-chain-card-trick.md     |    59 +
 ...013-11-23-the-jean-paul-sartre-cookbook.md |    41 +
 .../posts/2013-12-14-the-training-game.md     |    24 +
 ...-explanations-of-the-monty-hall-problem.md |    40 +
 .../posts/2013-12-30-smartphone-charter.md    |    30 +
 hugo/content/posts/2014-01-02-the-creation.md |    33 +
 ...2014-01-12-denouement-of-myst-iii-exile.md |    53 +
 ...ional-programming-syntax-of-mathematica.md |   115 +
 .../posts/2014-01-28-writing-essays.md        |    60 +
 ...age-rage-against-the-poets-hardest-sell.md |    38 +
 ...014-03-20-a-roundup-of-some-board-games.md |    62 +
 ...iscover-the-contraction-mapping-theorem.md |    72 +
 ...4-04-discovering-a-proof-of-heine-borel.md |    68 +
 .../2014-04-07-useful-conformal-mappings.md   |    44 +
 .../2014-04-15-sample-topology-question.md    |    53 +
 .../2014-04-17-cayley-hamilton-theorem.md     |    58 +
 ...-04-26-sequentially-compact-iff-compact.md |   122 +
 ...ng-a-proof-of-sylvesters-law-of-inertia.md |    66 +
 ...t-symmetric-matrices-are-diagonalisable.md |    28 +
 .../2014-06-25-possible-cons-of-Soylent.md    |    29 +
 .../2014-07-13-solvability-of-nonograms.md    |    48 +
 ...2014-07-15-what-maths-does-to-the-brain.md |    47 +
 .../posts/2014-07-19-music-practice.md        |    38 +
 .../content/posts/2014-07-21-perfect-pitch.md |    48 +
 hugo/content/posts/2014-08-19-parables.md     |    34 +
 .../2014-08-26-python-script-shadowing.md     |    26 +
 .../2014-09-09-sum-of-two-squares-theorem.md  |    85 +
 .../posts/2014-12-02-christmas-carols.md      |    43 +
 ...-12-09-film-recommendation-interstellar.md |    25 +
 .../content/posts/2014-12-19-matrix-puzzle.md |    91 +
 .../2014-12-23-latin-translation-tips.md      |    42 +
 .../posts/2015-01-29-motivational-learning.md |    28 +
 hugo/content/posts/2015-08-19-awodey.md       |    14 +
 .../2015-08-21-proof-by-contradiction.md      |    33 +
 hugo/content/posts/2015-09-25-lottery-odds.md |    36 +
 .../posts/2015-11-12-eilenberg-moore.md       |    50 +
 .../posts/2015-11-28-my-first-forcing.md      |    61 +
 ...5-12-24-general-adjoint-functor-theorem.md |    20 +
 .../posts/2015-12-31-monadicity-theorems.md   |    16 +
 .../2016-01-01-multiplicative-determinant.md  |    20 +
 .../2016-01-26-representable-functors.md      |    17 +
 .../2016-02-05-friedberg-muchnik-theorem.md   |    17 +
 .../posts/2016-03-03-a-certain-limit.md       |    67 +
 .../posts/2016-03-28-clojure-exercism.md      |   173 +
 ...16-04-08-another-monty-hall-explanation.md |    52 +
 .../2016-04-13-independence-of-choice.md      |    59 +
 .../posts/2016-04-21-modular-machines.md      |    32 +
 .../posts/2016-04-27-tennenbaums-theorem.md   |    18 +
 .../2016-05-25-finitistic-reducibility.md     |    68 +
 .../posts/2016-06-13-the-use-of-jargon.md     |    67 +
 .../posts/2016-06-15-part-iii-essay.md        |    20 +
 .../content/posts/2016-08-05-be-a-beginner.md |    45 +
 .../content/posts/2016-08-07-a-free-market.md |    50 +
 .../posts/2016-08-10-reinvent-maths.md        |    34 +
 .../posts/2016-12-31-complex-infinity.md      |    32 +
 .../posts/2017-02-14-cauchy-schwarz-proof.md  |    17 +
 .../posts/2017-03-14-maths-olympiad.md        |    25 +
 .../posts/2017-11-05-abuse-of-notation.md     |    38 +
 .../content/posts/2018-02-03-epsilon-delta.md |    26 +
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 .../content/posts/2018-06-02-json-comments.md |    25 +
 .../2018-07-21-dependent-types-overview.md    |   185 +
 hugo/content/posts/2018-09-10-don-syme.md     |   105 +
 .../posts/2018-09-20-mathematics-teaching.md  |    24 +
 hugo/content/posts/2019-01-16-mystery-hunt.md |    19 +
 hugo/content/posts/2019-12-27-cooking.md      |    18 +
 hugo/content/posts/2020-01-16-mystery-hunt.md |    57 +
 .../2020-01-25-mit-mystery-hunt-photos.md     |    12 +
 .../posts/2020-03-04-defunctionalisation.md   |   394 +
 .../content/posts/2020-04-05-static-config.md |    22 +
 .../content/posts/2020-07-22-tailrecursion.md |   229 +
 .../posts/2020-08-02-uncountability.md        |    25 +
 .../posts/2020-10-10-null-and-option.md       |    23 +
 hugo/content/posts/2020-10-18-gravity.md      |    27 +
 .../content/posts/2020-10-23-anki-learning.md |   217 +
 .../posts/2020-11-21-christmas-dinner.md      |   113 +
 .../posts/2020-12-26-christmas-quiz.md        |    25 +
 hugo/content/posts/2020-12-28-a-bug-in-git.md |    85 +
 hugo/content/posts/2021-01-18-find-the-bug.md |    58 +
 .../posts/2021-01-21-evaluation-checklist.md  |    82 +
 .../posts/2021-01-30-cyclic-dependencies.md   |    28 +
 .../2021-02-03-senior-engineer-syllabus.md    |    45 +
 .../2021-02-06-property-based-metatesting.md  |   267 +
 .../posts/2021-02-20-in-praise-of-dry-run.md  |   197 +
 ...021-03-17-rewriting-technical-interview.md |   392 +
 .../2021-03-22-continuation-passing-style.md  |   571 +
 hugo/content/posts/2021-10-19-crates.md       |   298 +
 hugo/content/posts/2021-10-22-argument.md     |    58 +
 hugo/content/posts/2021-10-25-avoid-rm-rf.md  |    63 +
 .../posts/2022-05-15-water-filtration.md      |    26 +
 .../posts/2023-07-12-transformer-questions.md |   107 +
 hugo/content/posts/2023-09-06-git-showcase.md |    89 +
 .../posts/2023-09-28-sum-of-two-squares.md    |   165 +
 hugo/content/reading-list/index.md            |   155 +
 hugo/content/restaurants/index.md             |    22 +
 hugo/content/top-posts/index.md               |    39 +
 hugo/i18n/en.toml                             |     6 +
 hugo/layouts/shortcodes/baseurl.html          |     5 +
 hugo/layouts/shortcodes/foldergallery.html    |    18 +
 hugo/layouts/shortcodes/side.html             |     7 +
 hugo/static/CNAME                             |     1 +
 hugo/static/MysteryHunt2019/PictureBook.nb    | 25225 ++++++++++++++++
 hugo/static/MysteryHunt2019/PictureBook.pdf   |   Bin 0 -> 1185830 bytes
 hugo/static/MysteryHunt2020/BestSongEver.nb   |   343 +
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 250 files changed, 59591 insertions(+), 48 deletions(-)
 create mode 100644 hugo/.gitignore
 create mode 100644 hugo/assets/css/fontawesome.css
 create mode 100644 hugo/assets/css/sidenote.css
 create mode 100644 hugo/assets/css/syntax.css
 create mode 100644 hugo/config.toml
 create mode 100644 hugo/content/ILAS/index.md
 create mode 100755 hugo/content/about-this-site/index.md
 create mode 100644 hugo/content/about/index.md
 create mode 100755 hugo/content/anki-decks/index.md
 create mode 100644 hugo/content/awodey/2015-08-19-category-theory-introduction.md
 create mode 100644 hugo/content/awodey/2015-08-19-what-is-a-category.md
 create mode 100644 hugo/content/awodey/2015-08-20-new-categories-from-old.md
 create mode 100644 hugo/content/awodey/2015-08-21-free-categories-and-foundations.md
 create mode 100644 hugo/content/awodey/2015-09-02-epis-monos.md
 create mode 100644 hugo/content/awodey/2015-09-02-initial-generalised-elements.md
 create mode 100644 hugo/content/awodey/2015-09-08-products-in-category-theory.md
 create mode 100644 hugo/content/awodey/2015-09-10-homsets-and-exercises.md
 create mode 100644 hugo/content/awodey/2015-09-15-duality-in-category-theory.md
 create mode 100644 hugo/content/awodey/2015-09-16-equalisers.md
 create mode 100644 hugo/content/awodey/2015-09-19-duality-exercises.md
 create mode 100644 hugo/content/awodey/2015-09-19-groups-in-categories.md
 create mode 100644 hugo/content/awodey/2015-09-22-limits-and-pullbacks.md
 create mode 100644 hugo/content/awodey/2015-09-22-properties-of-pullbacks-limits.md
 create mode 100644 hugo/content/awodey/2015-09-23-properties-of-limits.md
 create mode 100644 hugo/content/awodey/2015-09-29-exponentials-in-category-theory.md
 create mode 100644 hugo/content/awodey/2015-09-29-limit-exercises.md
 create mode 100644 hugo/content/awodey/2015-09-30-heyting-algebras.md
 create mode 100644 hugo/content/comparisons/index.md
 create mode 100644 hugo/content/films/index.md
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 create mode 100644 hugo/content/posts/2013-06-26-cucats-puzzlehunt.md
 create mode 100644 hugo/content/posts/2013-06-26-first-post.md
 create mode 100644 hugo/content/posts/2013-06-26-sylow-theorems.md
 create mode 100644 hugo/content/posts/2013-07-03-in-which-i-augment-the-lexicon.md
 create mode 100644 hugo/content/posts/2013-07-04-cambridge-undergrad-maths-tips.md
 create mode 100644 hugo/content/posts/2013-07-06-cambridge-vocab-a-guide-for-the-mystified.md
 create mode 100644 hugo/content/posts/2013-07-07-mundane-magics.md
 create mode 100644 hugo/content/posts/2013-07-08-an-obvious-improvement-to-tennis.md
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 create mode 100644 hugo/content/posts/2013-07-12-a-framework-for-discussing-pricelessness.md
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 create mode 100644 hugo/content/posts/2013-07-21-on-shakespeare.md
 create mode 100644 hugo/content/posts/2013-07-22-the-orbitstabiliser-theorem.md
 create mode 100644 hugo/content/posts/2013-07-24-stumbled-across-24th-july-2013.md
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 create mode 100644 hugo/content/posts/2013-08-04-stumbled-across-4-august-2013.md
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 create mode 100644 hugo/content/posts/2013-08-18-thinking-styles.md
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 create mode 100644 hugo/content/posts/2013-08-22-how-to-punt-in-cambridge.md
 create mode 100644 hugo/content/posts/2013-08-24-stumbled-across-24-august-2013.md
 create mode 100644 hugo/content/posts/2013-08-26-topology-made-simple.md
 create mode 100644 hugo/content/posts/2013-08-31-slightly-silly-sylow-pseudo-sonnets.md
 create mode 100644 hugo/content/posts/2013-09-13-stumbled-across-14th-september-2013.md
 create mode 100644 hugo/content/posts/2013-09-21-how-to-prove-that-you-are-a-god.md
 create mode 100644 hugo/content/posts/2013-10-10-plot-armour.md
 create mode 100644 hugo/content/posts/2013-10-11-meaning-what-you-say.md
 create mode 100644 hugo/content/posts/2013-10-13-training-away-mental-bias.md
 create mode 100644 hugo/content/posts/2013-10-20-the-ravenous.md
 create mode 100644 hugo/content/posts/2013-10-24-how-to-do-analysis-questions.md
 create mode 100644 hugo/content/posts/2013-11-07-my-quest-for-a-new-phone.md
 create mode 100644 hugo/content/posts/2013-11-12-markov-chain-card-trick.md
 create mode 100644 hugo/content/posts/2013-11-23-the-jean-paul-sartre-cookbook.md
 create mode 100644 hugo/content/posts/2013-12-14-the-training-game.md
 create mode 100644 hugo/content/posts/2013-12-22-three-explanations-of-the-monty-hall-problem.md
 create mode 100644 hugo/content/posts/2013-12-30-smartphone-charter.md
 create mode 100644 hugo/content/posts/2014-01-02-the-creation.md
 create mode 100644 hugo/content/posts/2014-01-12-denouement-of-myst-iii-exile.md
 create mode 100644 hugo/content/posts/2014-01-24-introduction-to-functional-programming-syntax-of-mathematica.md
 create mode 100644 hugo/content/posts/2014-01-28-writing-essays.md
 create mode 100644 hugo/content/posts/2014-02-16-rage-rage-against-the-poets-hardest-sell.md
 create mode 100644 hugo/content/posts/2014-03-20-a-roundup-of-some-board-games.md
 create mode 100644 hugo/content/posts/2014-03-30-how-to-discover-the-contraction-mapping-theorem.md
 create mode 100644 hugo/content/posts/2014-04-04-discovering-a-proof-of-heine-borel.md
 create mode 100644 hugo/content/posts/2014-04-07-useful-conformal-mappings.md
 create mode 100644 hugo/content/posts/2014-04-15-sample-topology-question.md
 create mode 100644 hugo/content/posts/2014-04-17-cayley-hamilton-theorem.md
 create mode 100644 hugo/content/posts/2014-04-26-sequentially-compact-iff-compact.md
 create mode 100644 hugo/content/posts/2014-05-03-discovering-a-proof-of-sylvesters-law-of-inertia.md
 create mode 100644 hugo/content/posts/2014-05-26-proof-that-symmetric-matrices-are-diagonalisable.md
 create mode 100644 hugo/content/posts/2014-06-25-possible-cons-of-Soylent.md
 create mode 100644 hugo/content/posts/2014-07-13-solvability-of-nonograms.md
 create mode 100644 hugo/content/posts/2014-07-15-what-maths-does-to-the-brain.md
 create mode 100644 hugo/content/posts/2014-07-19-music-practice.md
 create mode 100644 hugo/content/posts/2014-07-21-perfect-pitch.md
 create mode 100644 hugo/content/posts/2014-08-19-parables.md
 create mode 100644 hugo/content/posts/2014-08-26-python-script-shadowing.md
 create mode 100644 hugo/content/posts/2014-09-09-sum-of-two-squares-theorem.md
 create mode 100644 hugo/content/posts/2014-12-02-christmas-carols.md
 create mode 100644 hugo/content/posts/2014-12-09-film-recommendation-interstellar.md
 create mode 100644 hugo/content/posts/2014-12-19-matrix-puzzle.md
 create mode 100644 hugo/content/posts/2014-12-23-latin-translation-tips.md
 create mode 100644 hugo/content/posts/2015-01-29-motivational-learning.md
 create mode 100644 hugo/content/posts/2015-08-19-awodey.md
 create mode 100644 hugo/content/posts/2015-08-21-proof-by-contradiction.md
 create mode 100644 hugo/content/posts/2015-09-25-lottery-odds.md
 create mode 100644 hugo/content/posts/2015-11-12-eilenberg-moore.md
 create mode 100644 hugo/content/posts/2015-11-28-my-first-forcing.md
 create mode 100644 hugo/content/posts/2015-12-24-general-adjoint-functor-theorem.md
 create mode 100644 hugo/content/posts/2015-12-31-monadicity-theorems.md
 create mode 100644 hugo/content/posts/2016-01-01-multiplicative-determinant.md
 create mode 100644 hugo/content/posts/2016-01-26-representable-functors.md
 create mode 100644 hugo/content/posts/2016-02-05-friedberg-muchnik-theorem.md
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 create mode 100644 hugo/content/posts/2016-03-28-clojure-exercism.md
 create mode 100644 hugo/content/posts/2016-04-08-another-monty-hall-explanation.md
 create mode 100644 hugo/content/posts/2016-04-13-independence-of-choice.md
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 create mode 100644 hugo/content/posts/2016-04-27-tennenbaums-theorem.md
 create mode 100644 hugo/content/posts/2016-05-25-finitistic-reducibility.md
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 create mode 100644 hugo/content/posts/2016-06-15-part-iii-essay.md
 create mode 100644 hugo/content/posts/2016-08-05-be-a-beginner.md
 create mode 100644 hugo/content/posts/2016-08-07-a-free-market.md
 create mode 100644 hugo/content/posts/2016-08-10-reinvent-maths.md
 create mode 100644 hugo/content/posts/2016-12-31-complex-infinity.md
 create mode 100644 hugo/content/posts/2017-02-14-cauchy-schwarz-proof.md
 create mode 100644 hugo/content/posts/2017-03-14-maths-olympiad.md
 create mode 100644 hugo/content/posts/2017-11-05-abuse-of-notation.md
 create mode 100644 hugo/content/posts/2018-02-03-epsilon-delta.md
 create mode 100644 hugo/content/posts/2018-04-08-kinds-of-number.md
 create mode 100644 hugo/content/posts/2018-06-02-json-comments.md
 create mode 100644 hugo/content/posts/2018-07-21-dependent-types-overview.md
 create mode 100644 hugo/content/posts/2018-09-10-don-syme.md
 create mode 100644 hugo/content/posts/2018-09-20-mathematics-teaching.md
 create mode 100644 hugo/content/posts/2019-01-16-mystery-hunt.md
 create mode 100644 hugo/content/posts/2019-12-27-cooking.md
 create mode 100644 hugo/content/posts/2020-01-16-mystery-hunt.md
 create mode 100644 hugo/content/posts/2020-01-25-mit-mystery-hunt-photos.md
 create mode 100644 hugo/content/posts/2020-03-04-defunctionalisation.md
 create mode 100644 hugo/content/posts/2020-04-05-static-config.md
 create mode 100644 hugo/content/posts/2020-07-22-tailrecursion.md
 create mode 100644 hugo/content/posts/2020-08-02-uncountability.md
 create mode 100644 hugo/content/posts/2020-10-10-null-and-option.md
 create mode 100644 hugo/content/posts/2020-10-18-gravity.md
 create mode 100644 hugo/content/posts/2020-10-23-anki-learning.md
 create mode 100644 hugo/content/posts/2020-11-21-christmas-dinner.md
 create mode 100644 hugo/content/posts/2020-12-26-christmas-quiz.md
 create mode 100644 hugo/content/posts/2020-12-28-a-bug-in-git.md
 create mode 100644 hugo/content/posts/2021-01-18-find-the-bug.md
 create mode 100644 hugo/content/posts/2021-01-21-evaluation-checklist.md
 create mode 100644 hugo/content/posts/2021-01-30-cyclic-dependencies.md
 create mode 100644 hugo/content/posts/2021-02-03-senior-engineer-syllabus.md
 create mode 100644 hugo/content/posts/2021-02-06-property-based-metatesting.md
 create mode 100644 hugo/content/posts/2021-02-20-in-praise-of-dry-run.md
 create mode 100644 hugo/content/posts/2021-03-17-rewriting-technical-interview.md
 create mode 100644 hugo/content/posts/2021-03-22-continuation-passing-style.md
 create mode 100644 hugo/content/posts/2021-10-19-crates.md
 create mode 100644 hugo/content/posts/2021-10-22-argument.md
 create mode 100644 hugo/content/posts/2021-10-25-avoid-rm-rf.md
 create mode 100644 hugo/content/posts/2022-05-15-water-filtration.md
 create mode 100644 hugo/content/posts/2023-07-12-transformer-questions.md
 create mode 100644 hugo/content/posts/2023-09-06-git-showcase.md
 create mode 100644 hugo/content/posts/2023-09-28-sum-of-two-squares.md
 create mode 100755 hugo/content/reading-list/index.md
 create mode 100644 hugo/content/restaurants/index.md
 create mode 100644 hugo/content/top-posts/index.md
 create mode 100644 hugo/i18n/en.toml
 create mode 100644 hugo/layouts/shortcodes/baseurl.html
 create mode 100644 hugo/layouts/shortcodes/foldergallery.html
 create mode 100644 hugo/layouts/shortcodes/side.html
 create mode 100644 hugo/static/CNAME
 create mode 100644 hugo/static/MysteryHunt2019/PictureBook.nb
 create mode 100644 hugo/static/MysteryHunt2019/PictureBook.pdf
 create mode 100644 hugo/static/MysteryHunt2020/BestSongEver.nb
 create mode 100644 hugo/static/MysteryHunt2020/BestSongEver.pdf
 create mode 100644 hugo/static/MysteryHunt2020/Story.nb
 create mode 100644 hugo/static/MysteryHunt2020/Story.pdf
 create mode 100644 hugo/static/MysteryHunt2020/Z3Interop.wl
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 create mode 100644 hugo/static/MysteryHunt2020/nauseator.pdf
 create mode 100644 hugo/static/MysteryHunt2020/nauseator.svg
 create mode 100644 hugo/static/MysteryHunt2020/nauseator.xlsx
 create mode 100644 hugo/static/MysteryHunt2020/turtle.nb
 create mode 100644 hugo/static/MysteryHunt2020/turtle.pdf
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 create mode 100644 hugo/static/ProjectEuler/10.pdf
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 create mode 100644 hugo/static/ProjectEuler/index.html
 create mode 100644 hugo/static/archive/2015IntroToCategoryTheory.pdf
 create mode 100644 hugo/static/cucats/Puzzlehunt2013/KeyboardCat.nb
 create mode 100644 hugo/static/cucats/Puzzlehunt2013/TheChase.nb
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 create mode 100644 hugo/static/favicons/favicon.png
 create mode 100644 hugo/themes/anatole/.gitattributes
 create mode 100644 hugo/themes/anatole/.gitignore
 create mode 100644 hugo/themes/anatole/LICENSE
 create mode 100644 hugo/themes/anatole/README.md
 create mode 100644 hugo/themes/anatole/archetypes/post.md
 create mode 100644 hugo/themes/anatole/assets/css/style.css
 create mode 100644 hugo/themes/anatole/assets/js/anatole-header.js
 create mode 100644 hugo/themes/anatole/assets/js/medium-zoom.js
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 create mode 100644 hugo/themes/anatole/layouts/partials/medium-zoom.html
 create mode 100644 hugo/themes/anatole/layouts/partials/navbar.html
 create mode 100644 hugo/themes/anatole/layouts/partials/sidebar.html
 create mode 100644 hugo/themes/anatole/layouts/partials/taxonomy/categories.html
 create mode 100644 hugo/themes/anatole/layouts/partials/taxonomy/tags.html
 create mode 100644 hugo/themes/anatole/theme.toml

diff --git a/README.md b/README.md
index 06c57bb..7b9ed94 100644
--- a/README.md
+++ b/README.md
@@ -7,33 +7,13 @@ Every part of it is terrible.
 Some of it is unmaintained.
 You have been warned.
 
-## Checking out
-
-The builder is intended to `git subtree` to separate the mechanics of building from the content to be built, though you could simply inline your own content into the repo at the specified locations, or use submodules if you don't care about Nix support.
-
-The subtree checkout is automated using `./checkout.sh`, which you should update to point to your own content.
-After `./checkout.sh` has executed, you should be on a new branch called `working`, in which those remote repos have been read in appropriately as subtrees.
-When you want to commit changes to this repo, you need to cherry-pick them out onto `main` or similar, so as not to introduce the subtrees to the history of `main`; the `working` branch is created so that it's harder to get confused this way.
-At some point, I may write helper scripts to make the workflows sane.
-
 ## How to use
 
-I currently maintain two distinct build pipelines that should do the same thing.
-I'm in the process of standardising them as much as possible so that the choice boils down to "use Docker or use Nix to get my dependencies in place", with all the actual scripts shared.
+`nix build`.
 
-* A Nix flake (`./flake.nix`). Invoke using a plain old `nix build` to get the rendered site symlinked to `./result`.
-* A shell script (`./build.sh`) which runs a collection of pipelines in Docker images. Invoke using `sh ./build.sh` to get the rendered site in `./public`.
+### The `pdfs` flake
 
-The repository is intended to contain subtrees (see "Checking out" above) which refer to example content:
-
-* `hugo`, which refers to a [Hugo](https://github.com/gohugoio/hugo) static site directory, no tweaks required.
-* `pdfs`, which must contain a collection of TeX files and a text file `pdf-targets.txt`.
-* `images`, which must contain a collection of folders containing image files, and a text file `image-targets.txt`.
-* `meta`, which contains some amount of miscellaneous metadata.
-
-### The `pdfs` folder
-
-The `pdfs` folder is expected to contain a structure such as the following:
+The `pdfs` flake is expected to output a structure such as the following:
 
 ```
 file1.tex
@@ -60,9 +40,9 @@ static/Quux/file2.tex
 static/Quux/file2.pdf
 ```
 
-### The `images` folder
+### The `images` flake
 
-The `images` folder is expected to contain a structure such as the following:
+The `images` flake is expected to output a structure such as the following:
 
 ```
 FolderName/image1.jpg
@@ -107,3 +87,9 @@ However, in the immediate future I intend adding support for the following:
 
 There is a work-in-progress linting script, which is not currently included in the Nix build.
 It is intended to be run after `./build.sh`, and it runs a number of checks on the rendered output, such as ensuring that all HTML is syntactically valid.
+
+## License
+
+Code from the Anatole theme is MIT-licenced, and there's a copy next to it.
+The content of this website does not yet have a licence, because I haven't thought that far ahead: all rights reserved, you can `git clone` the repository from GitHub, but nothing else.
+Contact me if you want to use it for some reason.
diff --git a/flake.lock b/flake.lock
index 32e7208..b6cca1d 100644
--- a/flake.lock
+++ b/flake.lock
@@ -42,22 +42,6 @@
         "type": "github"
       }
     },
-    "content-source": {
-      "flake": false,
-      "locked": {
-        "lastModified": 1696092055,
-        "narHash": "sha256-CmQ0pcr0yiDQypcvFJ1jCviGgZCJ4Zw0t1JooX5LshM=",
-        "owner": "Smaug123",
-        "repo": "static-site-content",
-        "rev": "56b120fccfea31c0f761b4ebd68aa9af7d8d40e2",
-        "type": "github"
-      },
-      "original": {
-        "owner": "Smaug123",
-        "repo": "static-site-content",
-        "type": "github"
-      }
-    },
     "extra-content": {
       "flake": false,
       "locked": {
@@ -247,7 +231,6 @@
     "root": {
       "inputs": {
         "anki-decks": "anki-decks",
-        "content-source": "content-source",
         "extra-content": "extra-content",
         "flake-utils": "flake-utils_2",
         "images": "images",
diff --git a/flake.nix b/flake.nix
index 0b239f1..d1c2b1e 100644
--- a/flake.nix
+++ b/flake.nix
@@ -25,10 +25,6 @@
       url = "github:Smaug123/anki-decks";
       inputs.flake-utils.follows = "flake-utils";
     };
-    content-source = {
-      url = "github:Smaug123/static-site-content";
-      flake = false;
-    };
   };
 
   outputs = {
@@ -39,7 +35,6 @@
     images,
     pdfs,
     anki-decks,
-    content-source,
     extra-content,
     scripts,
   }:
@@ -91,7 +86,7 @@
           pname = "patrickstevens.co.uk";
           version = "0.1.0";
 
-          src = content-source;
+          src = ./hugo;
 
           buildInputs = [
             pkgs.hugo
diff --git a/hugo/.gitignore b/hugo/.gitignore
new file mode 100644
index 0000000..46a2051
--- /dev/null
+++ b/hugo/.gitignore
@@ -0,0 +1,13 @@
+/result
+public/
+.ionide/
+static/images/galleries
+images/**/*-thumb.jpg
+
+static/misc/**/*.tex
+static/misc/**/*.pdf
+
+.DS_Store
+
+.idea/
+.hugo_build.lock
diff --git a/hugo/assets/css/fontawesome.css b/hugo/assets/css/fontawesome.css
new file mode 100644
index 0000000..b24f753
--- /dev/null
+++ b/hugo/assets/css/fontawesome.css
@@ -0,0 +1,74 @@
+.fa,.fab,.fad,.fal,.far,.fas {
+    -moz-osx-font-smoothing: grayscale;
+    -webkit-font-smoothing: antialiased;
+    display: inline-block;
+    font-style: normal;
+    font-variant: normal;
+    text-rendering: auto;
+    line-height: 1
+}
+.fa-2x {
+    font-size: 2em
+}
+
+.fa-github:before {
+    content: "\f09b"
+}
+
+.fa-stack-exchange:before {
+    content: "\f18d"
+}
+
+.fa-envelope:before {
+    content: "\f0e0"
+}
+
+.fa-linkedin:before {
+    content: "\f08c"
+}
+
+.fa-calendar-day:before {
+    content: "\f783"
+}
+
+@font-face {
+    font-family:"Font Awesome 5 Brands";
+    font-style:normal;
+    font-weight:400;
+    font-display:block;
+    src:url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-brands-400.eot);
+    src:url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-brands-400.eot?#iefix) format("embedded-opentype"),
+    url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-brands-400.woff2) format("woff2"),
+    url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-brands-400.woff) format("woff"),
+    url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-brands-400.ttf) format("truetype"),
+    url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-brands-400.svg#fontawesome) format("svg")
+   }
+
+@font-face {
+    font-family: "Font Awesome 5 Free";
+    font-style: normal;
+    font-weight: 900;
+    font-display: block;
+    src: url('https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-solid-900.eot');
+    src: url('https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-solid-900.eot?#iefix') format("embedded-opentype"),
+    url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-solid-900.woff2) format("woff2"),
+    url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-solid-900.woff) format("woff"),
+    url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-solid-900.ttf) format("truetype"),
+    url(https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/webfonts/fa-solid-900.svg#fontawesome) format("svg")
+}
+
+.fab {
+    font-family: "Font Awesome 5 Brands"
+}
+
+.fab,.far {
+    font-weight: 400
+}
+
+.fa,.far,.fas {
+    font-family: "Font Awesome 5 Free"
+}
+
+.fa,.fas {
+    font-weight: 900
+}
diff --git a/hugo/assets/css/sidenote.css b/hugo/assets/css/sidenote.css
new file mode 100644
index 0000000..0a37dc0
--- /dev/null
+++ b/hugo/assets/css/sidenote.css
@@ -0,0 +1,62 @@
+span .sidenote {
+    display: inline;
+}
+
+.sidenote:hover .sidenote-label {
+    background-color: #36e281;
+    color:#fff
+}
+
+.sidenote:hover .sidenote-content {
+    border: .2rem dashed;
+    padding: .875rem;
+    border-color:#36e281
+}
+
+.sidenote-label {
+    border-bottom:.2rem dashed #36e281
+}
+
+.sidenote-checkbox {
+    display:none
+}
+
+.sidenote-content {
+    display: block;
+    position: absolute;
+    box-sizing: border-box;
+    border: .075rem solid #bfbfbf;
+    border-radius: .2rem;
+    margin-top: -1.5rem;
+    padding: 1rem;
+    text-align:left
+}
+
+.sidenote-content.sidenote-right {
+    right: 0;
+    width: 16%;
+}
+
+@media screen and (max-width: 78.5rem) {
+    .sidenote-content.sidenote-right {
+        display:none
+    }
+}
+
+.sidenote-delimiter {
+    display:none
+}
+
+@media screen and (max-width: 78.5rem) {
+    .sidenote-content.sidenote-right {
+        position: static;
+        margin-top: 1rem;
+        margin-bottom: 1rem;
+        width: 100%;
+        margin-right:0
+    }
+
+    .sidenote-checkbox:checked ~ .sidenote-content.sidenote-right {
+        display: block
+    }
+}
\ No newline at end of file
diff --git a/hugo/assets/css/syntax.css b/hugo/assets/css/syntax.css
new file mode 100644
index 0000000..350286e
--- /dev/null
+++ b/hugo/assets/css/syntax.css
@@ -0,0 +1,59 @@
+/* Background */ .chroma { color: #f8f8f2; background-color: #272822 }
+/* Error */ .chroma .err { color: #960050; background-color: #1e0010 }
+/* LineTableTD */ .chroma .lntd { vertical-align: top; padding: 0; margin: 0; border: 0; }
+/* LineTable */ .chroma .lntable { border-spacing: 0; padding: 0; margin: 0; border: 0; width: auto; overflow: auto; display: block; }
+/* LineHighlight */ .chroma .hl { display: block; width: 100%;background-color: #ffffcc }
+/* LineNumbersTable */ .chroma .lnt { margin-right: 0.4em; padding: 0 0.4em 0 0.4em;color: #7f7f7f }
+/* LineNumbers */ .chroma .ln { margin-right: 0.4em; padding: 0 0.4em 0 0.4em;color: #7f7f7f }
+/* Keyword */ .chroma .k { color: #66d9ef }
+/* KeywordConstant */ .chroma .kc { color: #66d9ef }
+/* KeywordDeclaration */ .chroma .kd { color: #66d9ef }
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diff --git a/hugo/config.toml b/hugo/config.toml
new file mode 100644
index 0000000..ce3a426
--- /dev/null
+++ b/hugo/config.toml
@@ -0,0 +1,111 @@
+baseURL = "/"
+disablePathToLower = true
+languageCode = "en-gb"
+title = "Patrick Stevens"
+theme = "anatole"
+buildFuture = false
+enableEmoji = true
+paginate = 20
+
+[params]
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+title = "Patrick Stevens"
+author = "Patrick Stevens"
+customCss = ["css/syntax.css", "css/fontawesome.css"]
+favicon = "favicons/"
+
+[params.math]
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+
+[[params.socialIcons]]
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+title = "GitHub"
+url = "https://github.com/Smaug123/"
+
+[[params.socialIcons]]
+icon = "fab fa-stack-exchange"
+title = "Stack Exchange"
+url = "https://math.stackexchange.com/users/259262/patrick-stevens"
+
+[[params.socialIcons]]
+icon = "fas fa-envelope"
+title = "e-mail"
+url = "mailto:patrick+sidebar@patrickstevens.co.uk"
+
+[[params.socialIcons]]
+icon = "fab fa-linkedin"
+title = "LinkedIn"
+url = "https://www.linkedin.com/in/patrick-stevens-2846017b/"
+
+[markup.highlight]
+  codeFences = true
+  guessSyntax = true
+  hl_Lines = ""
+  lineNoStart = 1
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+pygmentsCodefences = true
+
+[menu]
+  [[menu.main]]
+    name = "Home"
+    identifier = "home"
+    url = "/"
+  [[menu.main]]
+    name = "Posts"
+    identifier = "posts"
+    url = "/posts"
+  [[menu.main]]
+    name = "About Me"
+    identifier = "about-me"
+    url = "/about"
+  [[menu.main]]
+    name = "About This Site"
+    identifier = "about-this-site"
+    url = "/about-this-site"
+  [[menu.main]]
+    name = "Top Posts"
+    identifier = "top-posts"
+    url = "/top-posts"
+  [[menu.main]]
+    name = "Reading List"
+    identifier = "reading-list"
+    url = "/reading-list"
+  [[menu.main]]
+    name = "Film List"
+    identifier = "films"
+    url = "/films"
+  [[menu.main]]
+    name = "Lifehacks"
+    identifier = "lifehacks"
+    url = "/lifehacks"
+
+[frontmatter]
+date = ["date", "publishDate", "lastmod"]
+lastmod = ["lastmod", "date", "publishDate"]
+publishDate = ["publishDate", "date"]
+expiryDate = ["expiryDate"]
+
+[outputFormats]
+  [outputFormats.Markdown]
+    baseName = "markdown"
+    isPlainText = true
+    mediaType = "text/markdown"
+
+[mediaTypes]
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+    suffixes = ['md']
+
+[outputs]
+post = ['HTML', 'markdown']
+page = ['HTML', 'markdown']
+
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+[languages]
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diff --git a/hugo/content/ILAS/index.md b/hugo/content/ILAS/index.md
new file mode 100644
index 0000000..7d0ee74
--- /dev/null
+++ b/hugo/content/ILAS/index.md
@@ -0,0 +1,8 @@
+---
+lastmod: "2023-09-09T23:30:00.0000000+01:00"
+title: Imre Leader Appreciation Society
+author: patrick
+layout: page
+---
+
+I used to maintain an archive of the Imre Leader Appreciation Society for posterity through WebCitation, but WebCitation itself is now dead, so here I simply link to [Konrad Dąbrowski's capture][https://www.konraddabrowski.co.uk/ilas/index.html).
diff --git a/hugo/content/about-this-site/index.md b/hugo/content/about-this-site/index.md
new file mode 100755
index 0000000..78cb80e
--- /dev/null
+++ b/hugo/content/about-this-site/index.md
@@ -0,0 +1,35 @@
+---
+lastmod: "2022-07-31T20:16:44.0000000+01:00"
+title: About this website
+author: patrick
+layout: page
+---
+
+This website has been around in one form or another since June 26th, 2013.
+
+The website is hosted on [DigitalOcean] and is served statically by [NGINX].
+[Cloudflare] is sitting between my DigitalOcean droplet and you.
+Your HTTPS connection is secure to Cloudflare, and secure from Cloudflare to the droplet.
+
+The rendering engines are [Hugo] for the site, [pdftex] for PDFs, and [ImageMagick] to create image thumbnails.
+The Hugo theme is [Anatole] with a variety of modifications, most notably to remove most uses of JavaScript and to incorporate [Danila Fedore's sidenotes](https://danilafe.com/blog/sidenotes/) ([archive](https://web.archive.org/web/20210116232126/https://danilafe.com/blog/sidenotes/)).
+Mathematical notation in HTML is rendered by [KaTeX].
+
+You can access the TeX source of any PDFs I authored in TeX, by replacing the ".pdf" extension with ".tex".
+
+The infrastructure for this website is defined and managed by [Pulumi]; you can see it [on GitHub](https://github.com/Smaug123/PulumiConfig/).
+That repository also specifies the [Nix] configuration for the server.
+
+ [static]: https://en.wikipedia.org/wiki/Static_web_page
+ [GitHub Pages]: https://pages.github.com
+ [Hugo]: https://gohugo.io/
+ [Wordpress]: https://wordpress.org
+ [Anatole]: https://themes.gohugo.io/anatole/
+ [CloudFlare]: https://www.cloudflare.com
+ [DigitalOcean]: https://www.digitalocean.com
+ [NGINX]: https://www.nginx.com/
+ [pdftex]: https://www.tug.org/applications/pdftex/
+ [ImageMagick]: https://imagemagick.org/index.php
+ [KaTeX]: https://katex.org/
+ [Pulumi]: https://www.pulumi.com
+ [Nix]: https://nixos.org/
diff --git a/hugo/content/about/index.md b/hugo/content/about/index.md
new file mode 100644
index 0000000..998d033
--- /dev/null
+++ b/hugo/content/about/index.md
@@ -0,0 +1,52 @@
+---
+lastmod: "2022-08-20T14:18:00.0000000+01:00"
+title: About me
+author: patrick
+layout: page
+sidenotes: true
+---
+I am Patrick Stevens, a software engineer based in London, England.
+I completed my BA+MMath at the University of Cambridge.
+
+Social media accounts:
+
+* [Github][Github: Smaug123].
+* [Hacker News][Hacker News: Smaug123].
+* [Email](mailto:patrick+sidebar@patrickstevens.co.uk).
+* [LinkedIn][LinkedIn] (used almost never).
+* [Twitter][Twitter: smaug12345] (used almost never). My handle is @smaug12345.
+
+I am very interested in maths and puzzle-solving.
+For instance:
+
+* I have one of the [top twenty answers on the Maths StackExchange](https://math.stackexchange.com/questions/1681993/why-is-1-frac11-frac11-ldots-not-real/1682008#1682008) by upvotes.
+* On [Hacker.org], I am [laz0r][Hacker.org: laz0r], one of the top 50 users, although I have run out of low-hanging fruit on that site and I haven't returned to it for a while.
+* For three years running, I have participated in-person in the [MIT Mystery Hunt](https://en.wikipedia.org/wiki/MIT_Mystery_Hunt), solving with Team Palindrome; if interested, see [captain's write-up from 2019](https://www.ericberlin.com/2019/01/23/mystery-hunt-2019/) and [from 2020](https://www.ericberlin.com/2020/01/22/a-really-absurdly-long-post-about-the-mit-mystery-hunt/). In 2021, [we won](https://www.ericberlin.com/2021/01/19/my-mystery-hunt-2021-wrapup/).
+* I have solved {{< side right project-euler "a number">}}<img src="/images/AboutMe/project_euler.png" alt="Badge indicating at least 185 problems solved on Project Euler"/>{{< /side >}} of [Project Euler] problems.
+
+Languages:
+
+* F#: this is my day job.
+* Mathematica (recreationally); it's a lovely Lisp-ish thing. At one point, I was very active on the [Mathematica StackExchange].
+* Python: better-than-code-monkey experience. The open-source [Sextant] and [Endroid] are in Python; I contributed to both of these in my dim and distant past.
+* C#: my day job interacts moderately frequently with C# code.
+* Delphi: the language I learned first.
+* Agda: I'm [playing around][Agda] with this one at the moment.
+
+I have an interest in rationality, philosophy, lifehacking, and the links between these. I like to play around with words and constrained writing (such as poetry).
+
+[Twitter: smaug12345]: https://twitter.com/smaug12345 "My Twitter account"
+[Github: Smaug123]: https://github.com/Smaug123/ "Patrick Stevens Github account"
+[Hacker News: Smaug123]: https://news.ycombinator.com/user?id=Smaug123
+[Public key: Patrick Stevens]: https://keybase.io/patrickstevens
+[Project Euler]: https://projecteuler.net/
+[Hacker.org]: http://www.hacker.org "Hacker.org"
+[Hacker.org: laz0r]: http://www.hacker.org/forum/profile.php?mode=viewprofile&#38;u=13437 "My Hacker.org profile"
+[Agda]: {{< ref "2018-07-21-dependent-types-overview" >}}
+[GitHub Page]: https://pages.github.com
+[Endroid]: https://launchpad.net/endroid
+[Launchpad]: https://launchpad.net/~patrickas
+[Sextant]: https://launchpad.net/ensoft-sextant
+[Maths StackExchange]: https://math.stackexchange.com
+[Mathematica StackExchange]: https://mathematica.stackexchange.com/users/30771/patrick-stevens
+[LinkedIn]: https://www.linkedin.com/in/patrick-stevens-2846017b/
diff --git a/hugo/content/anki-decks/index.md b/hugo/content/anki-decks/index.md
new file mode 100755
index 0000000..e5cea1b
--- /dev/null
+++ b/hugo/content/anki-decks/index.md
@@ -0,0 +1,18 @@
+---
+lastmod: "2023-09-08T19:22:27.0000000+01:00"
+title: Anki decks
+author: patrick
+layout: page
+comments: true
+---
+
+I have deleted almost all of the Anki decks on this page, because I think they would do more harm than good.
+They were made during a time when I didn't really know how to use Anki appropriately.
+Any remaining decks here are CC-BY-SA.
+
+*   [Geography]. You can filter out the `london-tube` tag if you like, or `world-capitals`, or `american-geography`.
+
+<a href="http://creativecommons.org/licenses/by-sa/3.0/deed.en_GB" rel="license"><img style="border-width: 0;" alt="Creative Commons Licence" src="https://licensebuttons.net/l/by-sa/3.0/88x31.png" /></a>
+This work by <a href="/anki-decks" rel="cc:attributionURL">Patrick Stevens</a> is licensed under a <a href="http://creativecommons.org/licenses/by-sa/3.0/deed.en_GB" rel="license">Creative Commons Attribution-ShareAlike 3.0 Unported License</a>.
+
+[Geography]: /AnkiDecks/Geography.apkg
\ No newline at end of file
diff --git a/hugo/content/awodey/2015-08-19-category-theory-introduction.md b/hugo/content/awodey/2015-08-19-category-theory-introduction.md
new file mode 100644
index 0000000..75be1b4
--- /dev/null
+++ b/hugo/content/awodey/2015-08-19-category-theory-introduction.md
@@ -0,0 +1,18 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-08-19T00:00:00Z"
+aliases:
+- /categorytheory/category-theory-introduction/
+- /category-theory-introduction/
+title: Category Theory introduction
+---
+
+The next few posts will be following me on my journey through the book [Category Theory], by Steve Awodey. I’m using the second edition, if anyone wants to join me. I will read the book and make notes here as I go along: doing the exercises (if they seem interesting enough, I’ll post them up here), coming up with my own intuition pumps, and generally writing down my thought processes. The idea is to see how a fledgling mathematician studies a text, and to record my thoughts so I can refresh my memory more easily in future.
+
+As I go, I’m also creating an Anki deck of the definitions, by the way, although that might not appear on this site.
+
+[Category Theory]: http://ukcatalogue.oup.com/product/9780199237180.do
diff --git a/hugo/content/awodey/2015-08-19-what-is-a-category.md b/hugo/content/awodey/2015-08-19-what-is-a-category.md
new file mode 100644
index 0000000..4cc89e1
--- /dev/null
+++ b/hugo/content/awodey/2015-08-19-what-is-a-category.md
@@ -0,0 +1,57 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-08-19T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/what-is-a-category/
+- /what-is-a-category/
+title: What is a Category?
+---
+
+This post will cover the initial "examples" section of [Category Theory]. Because there aren't really very deep concepts in this section, this is probably a less interesting post to read than the others in this series.
+
+The introduction lasts until the bottom of page 4, which is where a *category* is defined. I read the definition in a kind of blank haze, not really taking it in, but I was reassured by the line "we will have plenty of examples very soon". On re-reading the definition, I've summarised it into "objects, arrows which go from object to object, associative compositions of arrows, identity arrows which compose in the obvious way". That's a very general definition, as the text points out, so I'm just going to wait until the examples before trying to understand this properly.
+
+The first example is the category of sets, with arrows being the functions between sets. That destroys my nice idea that "a category can be represented by a [simple (directed) graph][simple graph] together with a single identity arrow on each node": indeed, there are lots and lots of functions between the same two sets, and indeed more than one arrow \\(A \to A\\). I'll relax my mental idea into "directed multigraph".
+
+Then there's the category of finite sets. I'll just check that's a category - oh, it's actually really obvious and there's not really anything to check.
+
+Then the category of sets with injective functions. The "is this a category" check is done in the text.
+
+What about surjective functions? The composition of surjective functions is surjective, and the identity function is surjective, so that does also form a category.
+
+The first exercise in the text is where the arrows are \\(f: A \to B\\) such that \\(f^{-1}(b) \subset A\\) has at most two elements. (A moment of confusion before I realise that this is almost the definition of "injective".) That's clearly not a category: the composition of two of those might fail to satisfy the property. For instance, \\(f: \{0, 1, 2, 3 \} \to \{0, 1\}\\) the "is my input odd" function, and \\(g: \{0, 1\} \to \{0\}\\) the constant function; the composition of these is the constant zero function which is four-to-one.
+
+Now comes the category of posets with monotone functions. Not much comes to mind about that.
+
+The category of sets with binary relations as the arrows is one that is less intuitive for me, mainly because I'm still not used to thinking of relations \\(\sim\\) (such that \\(x \in X\\) may \\(\sim y \in Y\\)) as subsets of \\(X \times Y\\). The identity arrow is easy enough: it's the obvious "equality" relation that \\(a \sim a\\) only. The composition is a little less obvious: \\(a (S \circ R) c\\) iff there is \\(b\\) such that \\(a S b\\) and \\(b R c\\). Can I come up with an example of that? Let \\(S = \ \leq\\) on \\(\mathbb{R}\\), and \\(R = \ \geq\\). Then \\(S \circ R\\) is just the "everything is related" relation, since we may either let \\(b=a\\) or \\(b=c\\) depending on whether \\(a \leq c\\) or \\(a \geq c\\). OK, I'm a bit happier about that. It's easy to show that we have a category.
+
+Then comes a matrices example (which I've simplified from the textual example), where the objects are natural numbers - possibly repeated - and the arrows are integer matrices of the right dimensions that matrix multiplication is defined. I thought that was a pretty neat example.
+
+Finite categories: the book gives the definitions of \\(0\\), \\(1\\), \\(2\\) and \\(3\\). There's an obvious way to extend this to higher natural numbers. The section about "we may specify a finite category by just writing down a directed graph and making sure the arrows work" rings a strong bell with [free group]s, and indeed, the book calls them "free categories".
+
+Now we come to the definition of a "functor", which I immediately parse as a "category homomorphism" and move on. (Questions which come to mind: are any of the above categories related by some functor? I don't care much about that for the moment.)
+
+Preorders form a category which is drawn in almost exactly the same way as the Hasse diagram for a partial order (omitting identity arrows). That's a category in which the arrows are representing relation rather than domain/codomain.
+
+The topological-space example I skipped because I didn't know what a \\(T_0\\) space was. (However, the specialisation ordering I did observe to be trivial on sufficiently separated spaces.)
+
+Example from the category of proofs in a particular deductive system: the identity arrow \\(1_{\phi}\\) should be the trivial deduction \\(\phi\\) from premise \\(\phi\\). Very neat. It rings a bell from what I've heard of the [Curry-Howard isomorphism], and indeed the next example makes me think even more strongly of that.
+
+Discrete category on a set: yep, checks out. I should verify that they are posets, which they are: the poset with order relation "almost nothing is comparable".
+
+Monoids: oh dear, this example looks long. OK, I know what a monoid is ("group without inverses"), but how is it a category? Little mental shift of gear to thinking of elements as arrows, and it all becomes clear. The "free category" relations from earlier, then, correspond to the "free group" relations on the generators. I check that the set of functions \\(X \to X\\) actually a monoid, which it is. It seems easier to view it as a subcategory of the category of sets; and lo and behold the next paragraph points this out. We get to the bit about "monoid homomorphisms" - yes, they are indeed functors, which is not at all unexpected given that my understanding of "functor" is "category homomorphism", and monoids are categories.
+
+## Summary
+This is actually the second time I've read this section - the first time was before I had the idea of blogging my progress - and now I think I've got a good feel for what a category is. The next section is titled "Isomorphisms", which should give me a better idea of which categories are "the same". I noticed that the integers (when implemented as categories) seem to form a preorder, and indeed a poset; this corresponds nicely with their implementation as finite ordinals, with \\(3 = \{2\}\\) and so forth. I like seeing things crop up in different implementations all over the place like that.
+
+
+[Category Theory]: http://ukcatalogue.oup.com/product/9780199237180.do
+[simple graph]: https://en.wikipedia.org/wiki/Graph_(mathematics)#Simple_graph
+[free group]: https://en.wikipedia.org/wiki/Free_group
+[Hasse diagram]: https://en.wikipedia.org/wiki/Hasse_diagram
+[Curry-Howard isomorphism]: https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence
diff --git a/hugo/content/awodey/2015-08-20-new-categories-from-old.md b/hugo/content/awodey/2015-08-20-new-categories-from-old.md
new file mode 100644
index 0000000..25f6357
--- /dev/null
+++ b/hugo/content/awodey/2015-08-20-new-categories-from-old.md
@@ -0,0 +1,71 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-08-20T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/new-categories-from-old/
+- /new-categories-from-old/
+title: New categories from old
+---
+
+Here, I will be going through the Isomorphisms and Constructions sections of Awodey's Category Theory - pages 12 through 17.
+
+The first definition here is that of an isomorphism within a category. I notice that it corresponds with the usual definition of an isomorphism, but it's not phrased in exactly the same way. Up til now, "isomorphism" has strictly meant "bijective homomorphism". Are these two notions secretly the same? They can't be, because arrows aren't necessarily homomorphisms. Let's proceed with this slightly unfamiliar definition: it is an "arrow which is invertible on either side by the same inverse". The book asks us to prove that inverses are unique - that's easy by the usual group-inverses proof, which only really requires associativity.
+
+I need to be careful to remember that isomorphisms (as defined here) aren't between categories, but between members of a category. That is, they're not functors but arrows. (Though of course an arrow may represent a functor, but that's beside the point.)
+
+Now comes a paragraph about abstract definitions, which basically crystallises my thoughts that isomorphism is a more general form of "bijective homomorphism" which works in all categories. The example from the poset category with monotone functions as arrows is something I'm going to have to get my head around. Here goes.
+
+What does the category-theoretic definition of an isomorphism look like in the category of posets? It's a monotone function which has a monotone inverse. (Ah, that's more like the definition I remember: "a homeomorphism is a continuous function with a continuous inverse".) How is that different from "bijective homomorphism"? We'll want a monotone function which has an inverse which is not monotone. The standard topological spaces example was on an arbitrary space, between the discrete topology and the indiscrete topology. One direction is continuous and the other is not. Can I quickly turn that into a poset example? The obvious way to go would be on the same set from "nothing is related" to "total order". Definitely order-preserving: if \\(x < y\\) then \\(f(x) < f(y)\\) is vacuously true; definitely invertible; definitely not what we want an isomorphism to look like. I think I've got my head around the difference now.
+
+In the case of a monoid (viewed as a category), "only the abstract definition makes sense". Is that true? Firstly, what does the abstract definition look like? In a group, all elements are isomorphisms. If we take the monoid \\((\mathbb{Z} \cup \{ \infty \}, +)\\), the arrow \\(\infty: G \to G\\) is not an isomorphism because it has no inverse. That seems fine. Can I make sense of the idea of a monoid element being a "bijective homomorphism"? I could make the element act on the monoid by left multiplication, and I don't see anything wrong with that at the moment. I moved on at the time, but asked someone a bit later about this. The answer is that there are some categories which can't be viewed concretely at all, so the idea of "an arrow is a function" can't be made to make sense in some categories.
+
+Definition of a group is next; I definitely understand that, and I discovered for myself that a group has all its arrows as isomorphisms. I'll skip the bit about some examples of groups, because I know it, and go to the definition of a group homorphism. That bit is clear too, so on to Cayley's Theorem.
+
+The proof which appears here is basically the same as the one I was taught: show that action-on-the-left gives us a way to turn \\(G\\) into a permutation group on itself.
+
+The warning is interesting, and I hadn't noticed the feature it points out. I'll think about that a bit further. OK, it doesn't actually seem to be that problematic to understand, but definitely important to keep my thinking type-checked.
+
+Theorem 1.6. This looks important. We instantiate objects by their collection of incoming arrows, and instantiate arrows by functions which "represent" an arrow in the same way as the regular representation does in groups. Actually, that doesn't seem particularly important: it's just saying "we can instantiate categories whose arrows form a set". Maybe the Remark will clear things up. It's basically saying by analogy that "there's nothing special about permutation groups, since all groups may be viewed as permutation groups, so stop thinking about them in that way please". I think I'll wait until the discussion of terminal objects before I try and get my head around the true interpretation of a concrete category.
+
+Now the New Categories From Old section. The product looks easy enough, and its two projections are natural. The dual category likewise is pretty obvious, and makes the dual vector spaces idea much neater.
+
+The arrow category takes me a while to get my head around. The composition operation clearly does compose arrows correctly. What does the arrow category of the integer category \\(3\\) look like? Let's call the objects of \\(3\\) by the names \\(a, b, c\\). Then the arrow category has six objects (three identity and three non-identity arrows). We can find all the commutative squares by brute force, which I did on paper: there are \\(3^4\\) squares, but anything with \\(c\\) in the top left corner must be the identity arrow on the arrow \\(c \to c\\). That narrows it down enough for me to do this by hand. We end up with \\(a \to a\\) being connected to every arrow; \\(a \to b\\) connected to every arrow except \\(a \to a\\); \\(a \to c\\) connected only to \\(a \to c, b \to c, c \to c\\); \\(b \to b\\) connected to \\(b \to b, b \to c, c \to c\\); \\(b \to c\\) connected to \\(b \to c, c \to c\\); and \\(c \to c\\) connected to \\(c \to c\\). That is, if we omit the identity arrows, we obtain the following Hasse diagram.
+
+![Arrow category of 3][arrow]
+
+I don't think that was very enlightening. Motto: arrow categories aren't obviously anything in particular. What about the forgetful functors specified by taking the codomain or the domain? I'm happy that those are both functors, having stared at my diagram.
+
+Now comes the slice category. I've read this over once and got absolutely nowhere, so let's try again more carefully. The objects I can deal with: any arrow which goes into \\(C\\). The arrows? I'll do this with the category \\(3\\) again. If we slice on \\(a\\) then the only object is the identity arrow, and the only arrow is another identity. If we slice on \\(b\\) then there are two objects: \\(a \to b\\) and \\(b \to b\\). (Just quickly went back to the definition of a category, to check that \\((a \to b) \circ (b \to b)\\) isn't another arrow; in general it could be, but there isn't in this category.) Then in the slice category, there's an arrow \\((a \to b) \to (a \to b)\\) - namely the \\(C\\)-arrow \\(a \to a\\) - and an arrow \\((a \to b) \to (b \to b)\\) - namely the \\(C\\)-arrow \\(a \to b\\). We also have \\(b \to b\\)'s identity arrow. Therefore, we have recovered the category \\(2\\). That gives me intuition about what the identity arrows in the slice category look like.
+
+I don't think I've got any more intuition here. I'll briefly move on to the bit about the functor which forgets the sliced object. Certainly I agree that the given functor behaves correctly on objects. Does it behave on arrows? Yes, that's obvious from the syntactic definition, but I'm not certain I grok it. (I notice at this point that the functor is not necessarily surjective, as the \\(3\\) example above shows.)
+
+If I understand the composition law, then I should understand the arrows, so I'll aim for that instead. The composition law is clear from the book's diagram, on page 16: just add another triangle joined along edge \\(f'\\) to make a bigger supertriangle. OK, now I'm happier about the arrows in the slice category: they really are just arrows in the original category, and they join two slice-category objects (that is, arrows in \\(C\\)) if the two objects form a commutative triangle. This is actually a lot like the arrow category, by the looks of it.
+
+What about this composition functor? It lets us slice out on a different vertex by "changing the worldview", viewing everything through the lens of a particular arrow. I'm happy enough with that as a concept, although I recognise that my "understanding that this is a functor" is purely syntactic. Hopefully I'll get used to this with time.
+
+"The whole slicing construction is a functor". Yes, OK, that follows from the existence of the composition functor. I repeat that I'm understanding this at the surface level only, and I don't really grok any of it.
+
+What happens if we slice out a group (viewed as a category) by its only object? Then we get a category which has objects {the elements of the group}, and arrows \\(g \to h\\) given by \\((h^{-1} \circ g) \in C\\). That seems to have taken the group and told us how all its elements are related, which is mildly interesting.
+
+I verify that the slice category of a poset category is the "principal ideal" as stated, and note with relief that we will see more examples soon.
+
+The coslice category: that's obviously just the dual of the slice category.
+
+The category of pointed sets: yep, it's a category. I really don't' understand the isomorphism with the coslice category on sets. I can just about see it syntactically, but this is going to need a lot more work. I spent about ten minutes trying to work out what this really meant.
+
+## Summary
+
+I'm happy with some of these constructions, but I'll need a lot more work on others. I'll do these constructions on some more categories and see what happens.
+
+After composing this post, I asked someone for intuition, and got the reply:
+
+"The coslice category has objects which may be viewed as pairs \\((A, f)\\), where \\(f:\{ * \} \to A\\). So \\(f\\) is exactly a choice of element in \\(A\\). And the morphisms are maps such that the triangle commutes, i.e. the element "chosen" by \\(f\\) is the same as the one "chosen" by \\(f'\\)."
+
+I think this has cleared things up, but time will tell.
+
+[arrow]: {{< baseurl >}}images/CategoryTheorySketches/ArrowCategoryOf3.jpg
diff --git a/hugo/content/awodey/2015-08-21-free-categories-and-foundations.md b/hugo/content/awodey/2015-08-21-free-categories-and-foundations.md
new file mode 100644
index 0000000..eac8a14
--- /dev/null
+++ b/hugo/content/awodey/2015-08-21-free-categories-and-foundations.md
@@ -0,0 +1,56 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-08-21T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/free-categories-and-foundations/
+- /free-categories-and-foundations/
+title: Free categories and foundations
+---
+
+Here, I will be going through the Free Categories and Foundations sections of Awodey's Category Theory - pages 18 through 25.
+
+The definition of a free monoid is basically the same as that of a free group. However, I skim past and see the word "functor" appearing in the "no noise" statement, so I'll actually read this section properly.
+Everything is familiar up until the definition of the universal mapping property. One bit confuses me for a moment - "every monoid has an underlying set, and every monoid homomorphism an underlying function; this is a functor" - until I realise that by "this", Awodey means "this construction" rather than "this underlying function".
+
+Now comes the Universal Mapping Property of free monoids. This is a painful definition - I've spent fifteen minutes trying and failing to understand it - so I'll skip past it and come back when I've read some more.
+
+Proposition 1.9: this is a proof in an area where I'm wrestling to keep everything in my mind at once, so I'll just prove the proposition myself, using Awodey to take short-cuts. Let \\(i: A \to \vert A^* \vert\\) be defined by inclusion - that is, taking \\(a\\) to the single-character word \\(a\\). Let \\(N\\) be a monoid and \\(f: A \to \vert N \vert\\). Define \\(\bar{f}: A^* \to N\\) as stated; it's clearly a homomorphism. It has \\(\vert \bar{f} \vert  \circ i = f\\): indeed, \\(\vert \bar{f} \vert \circ i(a) = f(a)\\) manifestly. The homomorphism is unique, as Awodey proves at the end. Very well: I'm satisfied that \\(A^*\\) has the UMP of the free monoid on \\(A\\).
+
+Apparently the UMP captures "no junk and no noise". What Awodey says is plausible to me in that it hits the right words on the mark scheme, but the definition of the UMP is just too abstract. I'll try and break it into parts.
+
+"There is a function \\(i : A \to \vert M(A) \vert\\)." That bit's fine: it's saying that the inclusion exists. "Words are built up from the set in some way."
+
+"Given any monoid \\(N\\) and any function \\(f: A \to \vert N \vert \\), there is a monoid homomorphism \\(\bar{f} : M(A) \to N\\) such that \\(\vert \bar{f} \vert  \circ i = f\\)." The final equality is saying "we may represent \\(f\\) instead by first including into the free group, then applying some analog of \\(f\\)". Makes sense: "if we know where members of the free group go, then we definitely know where the generators go".
+
+"Moreover, \\(\bar{f}\\) is unique." Well, if it weren't unique, we would have a choice of places to send a word in the free group, even if we knew where all the generators went.
+
+I think I understand it better now. Still not on a particularly intuitive level, but now I'm convinced by Awodey's explanation.
+
+Let's move on to Proposition 1.10, that the free monoid is determined uniquely up to isomorphism. That seems plain enough, on a syntactic level.
+
+The bit about graphs is clear, but now there's another UMP to worry about. (Ah, I'm starting to understand that a UMP is a class of property, not just one particular property. Presumably there's one for lots of different structures.) The forgetful functor from Cat to Graphs is fine; the "different point of view" of a graph homomorphism makes me stop. Let's break down that diagram more carefully.
+
+\\(i: C_0 \to C_1\\) is indeed a valid map: we may view the identity arrow operation as taking an object to its associated identity. The codomain and domain functions do indeed take arrows to objects. The composition operation takes pairs of arrows (which have the right codomain/domain) to single arrows. OK, that's not too scary a diagram, and I agree that a functor is as claimed.
+
+After the same process of thought, I agree with the formulation of Graphs; and then I get to the description of the forgetful function Cat to Graphs. That is immediately comprehensible, and my first thought is that I don't know why Awodey didn't just come out with it straight away.
+
+"Similarly for functors…" - this bit is "easier to demonstrate with chalk", but I'll just go back and do it mentally. It works out in the obvious way.
+
+Finally, our second universal mapping property, this time the free category generated by a graph. Armed with the (meagre) intuition from the free-monoid UMP, this is easier to understand. "We may include the graph into the free category, and given somewhere to map the generators, there is a unique way to determine where elements of the free category go". I had one of those rare moments of "I know exactly what is going on here", which is hopefully a good sign.
+
+I'm intuitively happy with the examples given in the epilogue. If I were less lazy, I'd check from the UMP that the examples worked (that is, show that categories so defined were unique, and that the free catgory satisfied the UMP).
+
+Page 24 (on foundations) is familiar to me. I note the definition of "small" and "large" categories - natural enough. The definition of "locally small" looks a bit frightening at first, but on second glance it really is just what you'd expect "locally small" to mean. What would it mean for \\(Cat\\), the collection of small categories, to not be locally small? There would have to be two small categories such that the collection of functors between them was not a set. But the two categories are small, so they are sets, and there is a set of all functions between two sets. (However, the category of locally small categories would not be locally small: pick a non-small member \\(C\\), and define a functor \\(1 \to C\\) which selects an element. There are non-set-many of these.)
+
+Finally, the warning that "concrete" is not "small". Once given the example of the poset category \\(\mathbb{R}\\), I'm satisfied.
+
+# Summary
+
+I took a few days to understand this section, not working at it very hard but just letting it trickle in when the mood took me. It was massively more difficult than the previous sections, but I think I've got my head around the universal mapping properties described. I don't know whether I could come up with them myself to describe other free objects, but I could certainly give it a go.
+
+The exercises at the end of this chapter will be the true test of understanding.
diff --git a/hugo/content/awodey/2015-09-02-epis-monos.md b/hugo/content/awodey/2015-09-02-epis-monos.md
new file mode 100644
index 0000000..a9144ce
--- /dev/null
+++ b/hugo/content/awodey/2015-09-02-epis-monos.md
@@ -0,0 +1,60 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-02T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/epis-monos/
+- /epis-monos/
+title: Epis and monos
+---
+
+This post is on pages 29 through 33 of Awodey. It took me a while to do this, because I was basically on holiday for the past week.
+
+The definition of a [mono] and an [epi] seems at first glance to be basically the same thing as "injection" and "surjection". A mono is \\(f: A \to B\\) such that for all \\(g, h: C \to A\\), if \\(fg = fh\\) then \\(g=h\\). Indeed, if we take this in the category of sets, and let \\(g, h: \{1 \} \to A\\) ("picking out an element"), we precisely have the definition of "injection". An epi is \\(f: A \to B\\) such that for all \\(i, j:B \to D\\), if \\(if = jf\\) then \\(i=j\\). Again, in the category of sets, let \\(i, j: B \to \{1\}\\); then… ah. \\(if = jf\\) and \\(i=j\\) always, because there's only one function to the one-point set from a given set. I may have to rethink the "surjection" bit.
+
+Then there's Proposition 2.2, which I'm happy I've just basically proved anyway, so I skim it.
+
+Example 2.3: "monos are often injective homomorphisms". I glance through the example as preparation for going through it with pencil and paper, and see "this follows from the presence of objects like the free monoid \\(M(1)\\)", which is extremely interesting. Now I'll go back through the example properly.
+
+Suppose \\(h: M \to N\\) is monic. For any two distinct ways of selecting an element of the monoid's underlying set, we can lift those selections into mappings on the free monoid \\(M(1) \to M\\); they are distinct by the UMP. Applying \\(h\\) then takes the mappings into \\(N\\), maintaining distinctness by monicity; then the UMP lets us drag the mappings back into the sets, making selections from \\(1 \to \vert N \vert\\). The converse is quite clear.
+
+So it is clear where we needed the free monoid and its UMP: it was to give us a way to pass from talking about monoids to talking about sets, and back.
+
+Example 2.4: every arrow in a poset category is both monic and epic. An arrow \\(f: A \to B\\) is monic iff for all \\(g, h: C \to A\\), \\(f g = f h \Rightarrow g = h\\). That is, to abuse notation horribly, \\(a \leq b\\) is monic iff \\(c \leq a \leq b, c \leq a \leq b \Rightarrow ((c \leq a) = (c \leq a))\\). Ah, it's clear why all arrows are monic: it's because there is at most one arrow between \\(A, B\\), so two arrows with the same codomain and domain must be the same. The same reasoning works for "the arrows are epic".
+
+"Dually to the foregoing, the epis in the category of sets are the surjective functions". This is the bit from earlier I had to rethink. OK, let's take \\(f: A \to B\\) an epi in the category of sets. Let \\(i, j: B \to C\\), for some set \\(C\\). (Hopefully it'll become clear what \\(C\\) is to be.) Then \\(i f = j f\\) implies \\(i = j\\); we want to show that \\(f\\) hits every element of \\(B\\), so suppose it didn't hit \\(b\\). Then when we take the compositions \\(if, jf\\), we see that \\(i, j\\) never are asked about \\(b \in B\\), so in fact we are free to choose \\(i, j\\) to differ. That means we just need to pick \\(C\\) to be a set with more than one element. OK, that's much easier, although it's not quite clear to me how this is "dually".
+
+Then the example of the inclusion map \\(i\\) of the monoid \\(\mathbb{N} \cup \{ 0 \}\\) into the monoid \\(\mathbb{Z}\\). We're going to prove it's epic, so I'll try that before reading the proof. Let \\(g, h: \mathbb{Z} \to M\\) for some monoid \\(M\\); we want to show that \\(g i = h i \Rightarrow g = h\\). Indeed, suppose \\(g i = h i\\), but \\(g \not = h\\): that is, there is some \\(z \in \mathbb{Z}\\) such that \\(g(z) \not = h(z)\\). Since \\(gi = hi\\), we must have that \\(i\\) does not hit \\(z\\): that is, \\(z < 0\\). But \\(gi(-z) = hi(-z)\\) and so \\(g(-z) = h(-z)\\); whence \\(g(0) = g(z)+g(-z) \not = h(z)+h(-z) = h(0)\\). That is, \\(g, h\\) differ in the image of the unit. That is a contradiction because a homomorphism of monoids has a defined place to send the unit.
+
+Looking back over the proof in the book, it's basically the same. Awodey specialises to \\(-1\\) first.
+
+Proposition 2.6: every iso is monic and epic. I can't help but see the diagram when I read this, but I'll try and ignore it so I can prove it myself. Recall that an iso is an arrow such that there is an "inverse arrow". Let \\(f: A \to B\\) be an iso, and \\(i, j: B \to C\\) such that \\(if = jf\\). Then we may post-compose by \\(f\\)'s inverse - ah, it's clear now that this will work both forwards and backwards. This is exactly analogous to saying "we may left- or right-cancel in a group", and now I come to think of it, "epis are about right-cancelling" is something I just skipped over in the book.
+
+I'm happy with "every mono-epi is iso in the category of sets", since we've already proved that the injections are precisely the monos, and the epis are precisely the surjections.
+
+Now, the definition of a split mono/epi. That seems fine - it's a weaker form of mono/epi. "Functors preserve identities" does indeed mean that they preserve split epis and split monos, clearly, because a split epi comes in a pair with a split mono.
+
+The forgetful functor Mon to Set does not preserve the epi \\(\mathbb{N} \to \mathbb{Z}\\): we want to show that the inclusion of \\(\mathbb{N} \to \mathbb{Z}\\) (as sets) is not surjective. Oh, that's trivially obvious.
+
+In Sets, every mono splits except the empty ones: yes, we already have a theorem that injections have left inverses. "Every epi splits" is the categorical axiom of choice: we already have a theorem that "surjections have right inverses" is equivalent to AC, so I'm happy with this bit.
+
+Now the definition of a projective object. It's basically saying "arrows from this object may be pulled back through epis". A projective object "has a more free structure"? I don't really understand what that's saying, so I'll just accept the words and move on.
+
+All sets are projective because of the axiom of choice? Fix set \\(P\\); we want to show that for any function \\(f: P \to X\\) and any surjection \\(e: E \to X\\), there is \\(\bar{f}: P \to E\\) with \\(e \circ \bar{f} = f\\). We have (by Choice) that \\(e\\) splits: there is a right inverse \\(e^{-1}\\) such that \\(e \circ e^{-1} = 1_X\\). Define \\(\bar{f} = e^{-1} \circ f\\) and we're done.
+
+Any retract of a projective object is itself projective: I absolutely have to draw a diagram here. After a bit of confusion over left-composition happening as you go further to the right along the arrows, I spit out an answer.
+
+![Retract of a projective object is projective][retract]
+
+# Summary
+
+This section was more definitional than idea-heavy, so I think I've got my head around it for now. I do still need to practise my fluency with converting compositions of arrows on the diagrams into composition of arrows as algebraically notated - I still have to keep careful track of domain and codomain to make sure I don't get confused.
+
+[mono]: https://en.wikipedia.org/wiki/Monomorphism
+[epi]: https://en.wikipedia.org/wiki/Epimorphism
+
+[retract]: {{< baseurl >}}images/CategoryTheorySketches/RetractOfProjectiveIsProjective.jpg
diff --git a/hugo/content/awodey/2015-09-02-initial-generalised-elements.md b/hugo/content/awodey/2015-09-02-initial-generalised-elements.md
new file mode 100644
index 0000000..6077aed
--- /dev/null
+++ b/hugo/content/awodey/2015-09-02-initial-generalised-elements.md
@@ -0,0 +1,57 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-02T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/initial-generalised-elements/
+- /initial-generalised-elements/
+title: Initial, terminal, and generalised elements
+---
+
+This is pages 33 to 38 of Awodey.
+
+This bit looks really cool. A categorical way of expressing "this set has one element only": a terminal object. We have more examples of UMPs - these aren't quite of the same form as the previous ones.
+
+The proof that initial objects are unique up to unique isomorphism is easy - no need for me even to consider the diagram. On to the huge list of examples.
+
+Sets example: agreed. I actually asked about this (the fact that Set is not isomorphic to its dual) on Stack Exchange, and got basically this answer. Just a quick check that the one-point sets are indeed unique up to unique isomorphism, which they are.
+
+The category 0 is definitely initial in Cat; I agree that 1 is also terminal.
+
+In Groups: the initial objects are those for which there is precisely one homomorphism between it and any other group. Such a group needs to be the trivial group, since if \\(G\\) contains any other element, we can send \\(G\\) to \\(G\\) in a non-identifying way by sending every element to its inverse. The terminal objects: again that's just the trivial group, because for any other group \\(G\\), we can take two different homomorphisms into \\(G \times G\\), by projection onto the first or second coordinates. In Rings, on the other hand, I agree that \\(\mathbb{Z}\\) is initial: the unit has to go somewhere, and that determines the image of all of \\(\mathbb{Z}\\).
+
+Boolean algebras are something I ought to have met before in Part II Logic and Sets, but it was not lectured. I think I'll come back to this if it becomes important, because I feel like I have a good idea for the moment of what an initial/terminal object are.
+
+Posets: an object is indeed initial iff it is the least element. We have that initial elements are isomorphic up to unique isomorphism. What does that mean here? It means there is a unique arrow which has an inverse between these two elements. That is, it means the two elements are comparable and equal (by \\(a \leq b, b \leq a \Rightarrow a=b\\)). We therefore require there to be a *single* least element, if it is to be initial. What about the poset consisting of two identical copies of \\(\mathbb{N}\\), the elements of each copy incomparable to those of the other? There is no arrow from the 1 in the first \\(\mathbb{N}\\) into any element of the second \\(\mathbb{N}\\), so I'm happy that this is indeed not initial.
+
+Identity arrow is terminal in the slice category: everything has a unique morphism into this arrow, yes, because there is always a single commutative triangle between an arrow into \\(X\\) and the identity arrow on \\(X\\).
+
+Generalised elements, now. Hopefully this will be about ways of saying categorically that "this set has three elements", in the same way as "this set is terminal" was a categorical way of identifying a set with one element.
+
+"A set \\(A\\) has an arrow \\(f\\) into the initial object \\(A \to 0\\) just if it is itself initial." An initial object, remember, is one which has exactly one arrow into every other object, so it must have an arrow into \\(A\\); but the composition of \\(f\\) with that arrow must then be the identity on \\(0\\), since there is only one arrow \\(0 \to 0\\). Therefore \\(A, 0\\) are isomorphic and hence both initial.
+
+In monoids and groups, every object has a unique arrow to the initial object - that's trivial, since there is only one object. Unless it means objects in the category of monoids? The unique initial object is the trivial group, and it's also terminal. That makes more sense.
+
+Curses, I'm actually going to have to understand Boolean algebras now. I'll flick back to the definition and try to understand example 4 above. The definition looks an awful lot like the definitions of intersection and union, so I think I'll just think of them in that way. What's a filter? It's what we get when we infect some sets with filterness, and filterness propagates to "parents" and to "children of two parents" (intersections). An ultrafilter then is a filter where adding any other set infects everything.
+
+A filter \\(F\\) on \\(B\\) is an ultrafilter iff for every \\(b \in B\\), either \\(b \in F\\) or \\(b^C \in F\\) but not both: if \\(b \in F\\) then \\(b^C\\) can't be in \\(F\\) because then the empty set (that is, the intersection) is in the filter, so the filter is "everything". If \\(b \not \in F\\) then unless \\(b^C \in F\\), we could add \\(b\\) to \\(F\\) to obtain a strictly larger filter which still isn't everything, since \\(b^C\\) is still not in the augmented filter.
+
+Then I agree with the following stuff about "ultrafilters correspond to maps \\(B \to 2\\)". Not much more I can find to say there immediately.
+
+Ring homomorphisms \\(p\\) from ring \\(A\\) into the initial ring \\(\mathbb{Z}\\) correspond with prime ideals: yep, since \\(p^{-1}(0)\\) is an ideal of \\(A\\) (being the kernel of \\(p\\)), which is prime because we quotient by it to get a Euclidean domain \\(\mathbb{Z}\\).
+
+From arrows from initial objects to arrows from terminal objects. The definition of a point of object \\(A\\) is a natural one, as is the warning that objects are not necessarily determined by points (this is in the case that structural information is bound up in the arrows, like in a monoid viewed as a category). How many points does a Boolean algebra have? The terminal Boolean algebra is \\( \{ 0 \} \\); an arrow from \\(\{0\}\\) to a Boolean algebra must only ever pick out the \\(0\\) element, because the arrows must preserve the zero. That is, Boolean algebras have only one point.
+
+"Generalised elements" is therefore a way of trying to capture all the information, which the terminal object does not necessarily. The example which follows is a summary of this idea. There is something there to prove: that \\(f = g\\) iff \\(fx = gx\\) for all arrows \\(x\\). This leaves me stuck for a bit - I'm reviewing possible ways to prove that two arrows are the same, but the only ways I can think of require some kind of invertibility. What does it even mean for two arrows to be equal? At this point I got horribly confused and asked StackExchange, where I was told that I don't need to worry about that - just let \\(x\\) be the identity arrow. (By the way, it seems that equality of arrows is in the first-order logic sense here.)
+
+Example 2.13: aha, a way of showing categories are not isomorphic. Always handy to have ways of doing this. The number of \\(\mathbf{2}\\)-elements from \\(\{0 \leq 1 \}\\) to \\(\{x \leq y, x \leq z \}\\): \\(0\\) may map to \\(x\\), then \\(1\\) may map to \\(x\\) \\(y\\) or \\(z\\), or \\(0\\) may map to \\(y\\) or \\(z\\), when \\(1\\) must map to the same, producing five such 2-elements. I'm not sure I see why this is invariant, but on the next page I see that will be explained, and it all seems quite satisfactory.
+
+Example 2.14: ah, the "figures of shape \\(T\\) in \\(A\\)" interpretation makes it actually intuitive why the number of \\(2\\)-elements of the posets above are what they are. The arrows from the free monoid on one generator suffice to distinguish homomorphisms? That is, if we know where all \\(\mathbb{N}\\)-shapes go from \\(M\\), can we entirely determine the homomorphism? Yes, we can. If we have access to the elements of the monoid, we can do better (by simply specifying the image of each element), but of course we don't have the elements.
+
+# Summary
+
+I might need a bit more exposure to these ideas before I understand them properly, but I suspect the exercises at the end of this chapter will help with that. This feels like the first really categorical thing that has happened: ways of cheating so that we can consider the elements of structures without actually needing any elements.
diff --git a/hugo/content/awodey/2015-09-08-products-in-category-theory.md b/hugo/content/awodey/2015-09-08-products-in-category-theory.md
new file mode 100644
index 0000000..de9f08c
--- /dev/null
+++ b/hugo/content/awodey/2015-09-08-products-in-category-theory.md
@@ -0,0 +1,49 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-08T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/products/
+- /products-in-category-theory/
+title: Products in category theory
+---
+
+This is on pages 38 through 48 of Awodey. I've been looking forward to it, because products are things which come up all over the place and I'd heard that they are one of the main examples of a categorical idea.
+
+I skim over the definition of the product in the category of sets, and go straight to the general definition. It seems natural enough: the product is defined uniquely such that given any generalised element of the product, it projects in a unique way to corresponding elements of the children.
+
+Proving that products are unique up to isomorphism presumably goes in the same way as the other UMP-proofs have gone. I draw out the general diagram, then because we need to show isomorphism of two objects, I replace the "free" (unspecified) test object with one of the two objects of interest. Then the uniqueness conditions make everything fall out correctly. Moral: if we have a mapping property along the lines of "we can find unique arrows to make this diagram commute" then everything is easy.
+
+![Diagrams for the UMP of the product][UMP of product]
+
+Then we introduce some notation for the product. "A pair of objects may have many different products in a category". Yes, I can see why that's plausible, because we could define \\(\langle a, b \rangle\\) to be the ordered pair \\((b, a)\\), for instance, without changing any of the properties we're interested in.
+
+"Something is gained by considering arrows out of products": I'm aware of currying, which when Awodey points it out, makes me think nothing is really gained after all. I think I'll wait for Chapter 6 before I pass judgement on that.
+
+Now for a huge list of examples. First there are two definitions of "ordered pair", which I called earlier (though not in this exact form). Then we see the usual products of structured sets, with which I'm already very familiar.
+
+I'll verify the UMP for the product of two categories: let \\(x_1: X \to C, x_2: X \to D\\) be generalised elements. We want there to be a unique arrow \\(u : X \to (C \times D)\\) with \\(p_1 u = x_1, p_2 u = x_2\\), where \\(p_1, p_2\\) are the projection functors. Certainly there is an arrow given by stitching \\(x_1, x_2\\) together componentwise; is there another? Clearly not. Suppose \\(u_2\\) were another arrow \\(u: X \to (C \times D)\\). If \\(u_2(x) = (c, d)\\) then \\(p_1 u_2(x) = c\\) by the UMP, and \\(u_2\\) is therefore specified on all generalised elements already. That argument is not very formal, and I don't really see how to formalise it properly.
+
+The product of two groups according to this product construction is then self-evidently the product group we know and love. The product of two posets is also manifestly a poset, being a category where any pair of objects has at most one arrow between them. (Indeed, if there were two, we could project down to one of the posets to obtain two arrows between two elements.)
+
+The greatest lower bound example takes me a while to get my head around. The UMP for the product says, "define \\(p \times q\\) such that for all \\(x\\), if \\(a \leq p \times q, b \leq p \times q\\), and \\(a \leq x, b \leq x\\), then \\(x \leq p \times q\\)". That is indeed the greatest lower bound, but it took me ages to work this out.
+
+I work through the topological spaces example without thinking too hard about it. It's not clear to me that Awodey has proved that the uniqueness part of the UMP is satisfied, but I'll just accept it and move on.
+
+Type theory example: I've already met the lambda calculus, though never studied it in any detail. I skim over this, pausing at the equation "\\(\lambda x . c x = c\\) if no \\(x\\) is in \\(c\\)" - is this a typo for \\(\lambda x . c = c\\)? No, stupid of me - \\(c\\) represents a function, and the function \\(x \mapsto c(x)\\) is the same as the function \\(c\\). Then the category of types is indeed a category, and I'm happy with the proof that it has products. This time Awodey does certainly verify the uniqueness part of the UMP, by simply expanding everything and reducing it again.
+
+A long remark on the Curry-Howard correspondence. Clearly the product here is conjunction - skimming down I see that Awodey says it is indeed a product (or, at least, that there is a functor from types to proofs in which products have conjunctions as their images). Very pretty.
+
+"Categories with products": supposing every pair of objects has a product, we define a functor taking every pair to its product. That's intuitive in the sense of "structured sets", since I'm very familiar with that product construction. What does it mean in the poset case? Recall that the product was the greatest lower bound. A poset where every pair of elements has a greatest lower bound is actually a totally ordered set, and the greatest lower bound is the least of the two elements, so that also makes sense. I think I'll skip over the UMPs for \\(n\\)-ary products, but the idea of a terminal object as a nullary product is pretty neat. So that's why the empty product of real numbers is 1.
+
+
+
+# Summary
+
+As seems to be a general theme, I understand the syntax of products, and I can recognise some of them when they turn up, but have no real intuition for how they work. There will be more examples at the end of the chapter, which should clear things up a bit.
+
+[UMP of product]: {{< baseurl >}}images/CategoryTheorySketches/UMPofProduct.jpg
diff --git a/hugo/content/awodey/2015-09-10-homsets-and-exercises.md b/hugo/content/awodey/2015-09-10-homsets-and-exercises.md
new file mode 100644
index 0000000..c564df6
--- /dev/null
+++ b/hugo/content/awodey/2015-09-10-homsets-and-exercises.md
@@ -0,0 +1,63 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-10T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/homsets-and-exercises/
+- /homsets-and-exercises/
+title: Hom-sets and exercises
+---
+
+This is on pages 48 through 52 of Awodey, covering the hom-sets section and the exercises at the end of Chapter 2. Only eight more chapters after this, and I imagine they'll be more difficult - I should probably step up the speed at which I'm doing this.
+
+Awodey assumes we are working with locally small categories - recall that such categories have "given any two objects, there is a bona fide set of all arrows between those objects". That is, all the hom-sets are really sets.
+
+We see the idea that any arrow induces a function on the hom-sets by composing on the left. Awodey doesn't mention currying here, but that seems to be the same phenomenon. Why is the map \\(\phi(g): g \mapsto (f \mapsto g \circ f)\\) a functor from \\(C\\) to the category of sets? I agree with the step-by-step proof Awodey gives, but I don't really have intuition for it. It feels a bit misleading to me that this is thought of as a functor into the category of sets, because that category contains many many more things than we're actually interested in. It's like saying \\(f: \mathbb{N} \to \mathbb{R}\\) by \\(n \mapsto n\\), when you only ever are interested in the fact that \\(f\\) takes integer values. I'm sure it'll become more natural later when we look at representable functors.
+
+Then an alternative explanation of the product construction, as a way of exploding an arrow \\(X \to P\\) into two child arrows \\(X \to A, X \to B\\). A diagram is a product iff that explosion is always an isomorphism. Then a functor preserves binary products if it… preserves binary products. I had to draw out a diagram to convince myself that \\(F\\) preserves products iff \\(F(A \times B) \cong FA \times FB\\) canonically, but I'm satisfied with it.
+
+
+# Exercises
+
+Exercises 1 and 2 I've [done already][epis-monos]. The uniqueness of inverses is easy by the usual group-theoretic argument: \\(fg = f g'\\) means \\(gfg = gf g'\\), so \\(g = g'\\) by cancelling the \\(g f = 1\\) term.
+
+The composition of isos is an iso: easy, since \\(f^{-1} \circ g^{-1} = (g \circ f)^{-1}\\). \\(g \circ f\\) monic implies \\(f\\) monic and \\(g\\) epic: follows immediately by just writing out the definitions. The counterexample to "\\(g \circ f\\) monic implies \\(g\\) monic" can be found in the category of sets: we want an injective composition where the second function is not injective. Easy: take \\(\{1 \} \to \mathbb{N}\\) and then \\(\mathbb{N} \to \{ 1 \}\\). The composition is the identity, but the second function is very non-injective.
+
+Exercise 5: a) and d) are equivalent by definition of "iso" and "split mono/epi". Isos are monic and epic, as we've already seen in the text (because we can cancel \\(f\\) in \\(x f = x' f\\), for instance), so we have that a) implies b) and c). If \\(f\\) is a mono and a split epi, then it has a right-inverse \\(g\\) such that \\(fg = 1\\); we claim that \\(g\\) is also a left-inverse. Indeed, \\(f g f = f 1\\) so \\(g f = 1\\) by mono-ness. Therefore b) implies a). Likewise c) implies a).
+
+Exercise 6: Let \\(h: G \to H\\) be a monic graph hom. Let \\(v_1, v_2: 1 \to G\\) be homs from the graph with one vertex and no edges. Then \\(h v_1 = h v_2\\) implies \\(v_1 = v_2\\), so in fact \\(h\\) is injective. Likewise with edges, using the graph with one edge and two vertices, and the graph with one edge and one vertex. Conversely, suppose \\(h: G \to H\\) is not monic. Then there are \\(v_1: F_1 \to G, v_2: F_2 \to G\\) with \\(h v_1 = h v_2\\) but \\(v_1 \not = v_2\\). Since \\(h v_1 = h v_2\\), we must have that "their types match": \\(F_1 = F_2\\). We will denote that by \\(F\\). Then there is some vertex or edge on which \\(v_1\\) and \\(v_2\\) have different effects. If it's a vertex: then \\(v_1(v) \not = v_2(v)\\) for that vertex \\(v\\), but \\(h v_1 (v) = h v_2(v)\\), so \\(h\\) can't be injective. Likewise if it's an edge.
+
+Exercises 7 and 8 I've [done already][epis-monos].
+
+Exercise 9: the epis among posets are the surjections-on-elements. Let \\(f: P \to Q\\) be an epi of posets, so \\(x f = y f\\) implies \\(x = y\\). Suppose \\(f\\) is not surjective, so there is \\(q \in Q\\) it doesn't hit. Then let \\(x, y: Q \to \{ 1, 2 \}\\), disagreeing at \\(q\\). We have \\(x f = y f\\) so \\(x, y\\) must agree at \\(q\\). This is a contradiction. Conversely, any surjection-on-elements is an epi, because if \\(x(q) \not = y(q)\\) then we may write \\(q = f(p)\\) for some \\(p\\), whence \\(x f(p) \not = y f(p)\\). The one-element poset is projective: let \\(s: X \to \{1\}\\) be an epi (surjective), and \\(\phi: P \to \{ 1 \}\\). Then \\(X\\) has an element, \\(u\\) say, since \\(s\\) is surjective. Then we may lift \\(\phi\\) over \\(s\\) by letting \\(\bar{\phi}: p \mapsto u\\), so that the composite \\(s \circ \bar{\phi} = \phi\\). (Quick check in my mind that this works for \\(P\\) the empty poset - it does.)
+
+Exercise 10: Sets (implemented as discrete posets) are projective in the category of posets: the one-element poset is projective, and retracts of projective objects are projective. Let \\(A\\) be an arbitrary discrete poset. Define \\(r: 1 \to A\\) by selecting an element, and \\(s: A \to \{1\}\\). Then \\(A\\) is a retraction of \\(B\\), so is projective. Afterwards, I looked in the solutions, and Awodey's proof is much more concrete than this. I [asked on Stack Exchange][SE question] whether my proof was valid, and the great Qiaochu Yuan himself pointed out that I had mixed up what "retract" meant, and had actually showed that \\(\{1\}\\) was a retract of \\(A\\). Back to the drawing board.
+
+Exercise 10 revisited: Take a discrete poset \\(P\\), and let \\(f: X \to P\\) be an epi - that is, surjection. Let \\(A\\) be a poset and \\(\phi: A \to P\\) an arrow (monotone map). For each \\(a \in A\\) we have \\(\phi(a)\\) appearing in some form in \\(X\\); pick any inverse image \\(x_a\\) such that \\(f(x_a) = \phi(a)\\). I claim that the function \\(a \mapsto x_a\\) is monotone (whence we're done). Indeed, if \\(a \leq b\\) then \\(\phi(a) \leq \phi(b)\\) so \\(f(x_a) \leq f(x_b)\\) so \\(x_a \leq x_b\\) because \\(f, \phi\\) are monotone.
+
+Example of a non-projective poset: let \\(A = P\\) be the poset \\(0 \leq 1 \leq 2\\), and let \\(i:A \to P\\) the inclusion. Let \\(E\\) be the poset \\(0 \leq 2, 1 \leq 2\\), with its obvious inclusion as the epi. Then \\(i\\) doesn't lift across that epi, because \\(0_A\\) must map to \\(0_E\\) and \\(1_A\\) to \\(1_E\\), but \\(0 \leq_A 1\\) and \\(0 \not \leq_E 1\\).
+
+Now, all projective posets are discrete: suppose the comparison \\(a < b\\) exists in the poset \\(P\\), and let \\(X\\) be \\(P\\) but where we break that comparison. Let the epi \\(X \to P\\) be the obvious inclusion. Then the inclusion \\(\text{id}: P \to P\\) doesn't lift across \\(X\\).
+
+Exercise 11: Of course, the first thing is a diagram. An initial object in \\(A-\mathbf{Mon}\\) is \\((I, i)\\) such that there is precisely one arrow from \\((I, i)\\) to any other object: that is, precisely one commutative triangle exists. A free monoid \\(M(A)\\) on \\(A\\) is such that there is \\(j: A \to \vert M(A) \vert\\), and for any function \\(f: A \to \vert N \vert\\) there is a unique monoid hom \\(\bar{f}: M(A) \to N\\) with \\(\vert \bar{f} \vert \circ j = f\\). If \\((I, i)\\) is initial, it is therefore clear that \\(I\\) has the UMP of the free monoid on \\(A\\), just by looking at the diagram. Initial objects are unique up to isomorphism, and free monoids are too, so we automatically have the converse.
+
+Exercise 12 I did in my head to my satisfaction while I was following the text.
+
+Exercise 13: I wrote out several lines for this, amounting to showing that the unique \\(x: (A \times B) \times C \to A \times (B \times C)\\) guaranteed by the UMP of \\(A \times (B \times C)\\) is in fact an iso. The symbol shunting isn't very enlightening, so I won't reproduce it here.
+
+Exercise 14: the UMP for an \\(I\\)-indexed product should be: \\(P\\) with arrows \\(\{ (p_i: P \to A_i) : i \in I \}\\) is a product iff for every object \\(X\\) with collections \\(\{ (x_i: X \to A_i) : i \in I \}\\) of arrows, there is a unique \\(x : X \to P\\) with \\(p_i \circ x = x_i\\) for each \\(i \in I\\). Then in the category of sets, the product of \\(X\\) over \\(i \in I\\) satisfies that for all \\(T\\) with \\( \{ (t_i: T \to X): i \in I \}\\) arrows, there is a unique \\(t: T \to P\\) with \\(p_i \circ t = t_i\\). If we let \\(P = \{ f: I \to X \} = X^I\\), we do get this result: let \\(t(\tau) : i \mapsto t_i(\tau)\\). This works if \\(p_i \circ (\tau \mapsto (i \mapsto t_i(\tau))) = (\tau \mapsto t_i(\tau))\\), so we just need to define the projection \\(p_i: X^I \to X\\) by \\(p_i(i \mapsto x) = x\\). I think that makes sense.
+
+Exercise 15: I first draw a diagram. \\(\mathbb{C}_{A, B}\\) has a terminal object iff there is some \\((X, x_1, x_2)\\) such that for all \\((Y, y_1, y_2)\\), there is precisely one arrow \\((Y, y_1, y_2) \to (X, x_1, x_2)\\). \\(A\\) and \\(B\\) have a product in \\(\mathbb{C}\\) iff there is \\(P\\) and \\(p_1: P \to A, p_2: P \to B\\) such that for every \\(x_1: X \to A, x_2: X \to B\\) there is unique \\(x: X \to P\\) with the appropriate diagram commuting. If we let \\((Y, y_1, y_2) = (P, p_1, p_2)\\) then it becomes clear that if \\(A, B\\) have a product then \\(\mathbb{C}_{A, B}\\) has a terminal object - namely \\((Y, y_1, y_2)\\). Conversely, if \\(\mathbb{C}_{A, B}\\) has a terminal object \\((Y, y_1, y_2)\\), then our unique arrow \\(x: X \to Y\\) in \\(\mathbb{C}_{A, B}\\) corresponds to a unique product arrow in \\(\mathbb{C}\\), so the UMP for products is satisfied.
+
+Exercise 16: Is this really as easy as it looks? The product functor takes \\(a: A, b: B \mapsto \langle a, b \rangle : A \times B\\). Maybe I've misunderstood something, but I can't see that it's any harder than that. There's a functor \\(X \mapsto (A \to X)\\), given by coslicing out by \\(A\\). I've squinted at the answers Awodey supplies, and this isn't an exercise he gives. I'll just shut my eyes and pretend this exercise didn't exist.
+
+Exercise 17: The given morphism is indeed monic, because \\(1_A x = 1_A y\\) implies \\(x = y\\), and \\(\Gamma(f)x = \Gamma(f)y\\) implies \\(1_A x = 1_A y\\) because of the projection we may perform on the pair \\(\langle 1_A, f \rangle\\). \\(\Gamma\\) is a functor from sets to relations, clearly, but we've already done that in Section 1 question 1b).
+
+Exercise 18: It would really help if Awodey had told us what a representable functor was, rather than just giving an example. Is he asking us to show that "the representable functor of Mon is the forgetful functor"? I'm going to hope that I can just drop Mon in for the category C in section 2.7. If we let \\(A\\) be the trivial monoid, then \\(\text{Hom}(A, -)\\) is a functor taking a monad \\(M\\) to its set of underlying elements (each identified with a different hom \\(\{ 1 \} \to M\\)) - but hang on, there's only one such hom, so that line is nonsense. It would work in Sets, but not in Mon. We need \\(\text{Hom}(M, N)\\) to be isomorphic in some way to the set \\(\vert N \vert\\), and I just don't see how that's possible. Infuriatingly, this exercise doesn't have a solution in the answers section. I ended up looking this up, and the trick is to pick \\(M = \mathbb{N}\\). Then the homomorphisms \\(\phi: \mathbb{N} \to N\\) precisely determine elements of \\(N\\), by \\( \phi(1)\\). So that proves the result. Why did I not think of \\(\mathbb{N}\\) instead of \\(\{ 1 \}\\)? Probably just lack of experience.
+
+[epis-monos]: {% post_url 2015-09-02-epis-monos %}
+[SE question]: http://math.stackexchange.com/q/1429746/259262
diff --git a/hugo/content/awodey/2015-09-15-duality-in-category-theory.md b/hugo/content/awodey/2015-09-15-duality-in-category-theory.md
new file mode 100644
index 0000000..9c01581
--- /dev/null
+++ b/hugo/content/awodey/2015-09-15-duality-in-category-theory.md
@@ -0,0 +1,59 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-15T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/duality/
+- /duality-in-category-theory/
+title: Duality in category theory
+---
+
+I don't have strong preconceptions about this chapter. The previous chapter I knew would contain general constructions, and I was looking forwards to that, but this one is more unfamiliar to me. I'll be doing pages 53 through 61 of Awodey here - coproducts.
+
+The first bits are stuff I recognise from when I flicked through Categories for the Working Mathematician, I think. Or something. Anyway, I recognise the notion of formal duality and the very-closely-related semantic duality. (Like the difference between "semantic truth" and "syntactic truth" in first-order logic.) It's probably a horrible sin to say it, but both of these are just obvious, once they've been pointed out.
+
+Now the definition of a coproduct. The notation \\(A+B\\) is extremely suggestive, and I'd have preferred to try and work out what the coproduct was without that hint. \\(z_1: A \to Z\\) and \\(z_2: B \to Z\\) are "ways of selecting \\(A\\)- and \\(B\\)-shaped subsets of any object" (yes, that's not fully general, but for intuition I'll pretend I'm in a concrete category). So for any \\(Z\\), and for any way of selecting an \\(A\\)-shaped and a \\(B\\)-shaped subset of \\(Z\\), we can find a unique way of selecting an \\(A+B\\)-shaped subset according to the commuting-diagram condition. I'm still a bit unclear as to what that all means, so I whizz down to the Sets example below.
+
+In Sets, if we can find an \\(A\\)-shaped subset of some set \\(Z\\), and a \\(B\\)-shaped subset, then we can find a subset which is shaped like the disjoint union of \\(A\\) and \\(B\\) in a unique way. (Note that our arrows need not be injective, which is why the \\(A+B\\)-shaped subset exists. For instance, if \\(A = \{1\}, B = \{1\}\\), and our \\(A\\)-shaped subset and \\(B\\)-shaped subset of \\(\{a,b \}\\) were both \\(\{a\}\\), then the \\(A+B\\)-shaped subset would be simply \\(\{a \}\\). Both selections of shape end up pointing at the same element.)
+
+This leads me to wonder: what about in the category of sets with injections as arrows? Now it seems that the coproduct is only defined on disjoint sets, because the arrows \\(z_1, z_2\\) which pick out \\(A\\)- and \\(B\\)-shaped subsets now need to have distinct images in \\(Z\\) so that the coproduct may pick out an \\(A \cup B\\)-shaped subset.
+
+The free-monoids coproduct: given any "co-test object" \\(N\\), and any two monoid homomorphisms selecting subsets of \\(N\\) corresponding to the shapes of \\(M(A)\\) and \\(M(B)\\), there should be a natural way to define a shape corresponding to some kind of union. The shape of \\(M(A)\\) corresponds exactly to "where we send the generators", so we can see intuitively that \\(M(A) + M(B) = M(A+B)\\). This is very much not a proof, and I'll make sure to check the diagrammatic proof from the book first; that proof is fine with me. "Forgetful functor preserves products -> a structure-imposing functor preserves coproducts" has a certain appeal to it, but I don't quickly see a sense in which the structure can be imposed in general.
+
+Coproduct of two topological spaces: given a co-test topological space \\(X\\), and two continuous functions into \\(X\\) which pick out subspaces of shape \\(A\\) and \\(B\\), we want to find a space \\(P\\) such that for all \\(A\\)- and \\(B\\)-shape subspaces of \\(P\\), there is a unique \\(P\\)-shaped subspace of \\(X\\) composed of the same shapes as the \\(A\\)- and \\(B\\)-subspaces. Then it's fairly clear that \\(P\\) should be the disjoint union of \\(A\\) and \\(B\\) (compare with the fact that the forgetful functor to Set again yields the correct Set coproduct), but what topology? Surely it should be the "product" given by sets of the form (open in \\(A\\), open in \\(B\\)), since \\(A\\)-shaped subspaces of this will map directly into \\(A\\)-shaped subspaces of the co-test space, etc.
+
+Coproducts, therefore, are a way of putting two things next to each other, and this is pointed out in the next paragraph, where the coproduct of two posets is the "disjoint union" of them. The coproduct of two rooted posets is what I'd have guessed, as well, given that we need to make sure the coproduct is also rooted.
+
+Coproduct of two elements in a poset: that's easy by duality, since the opposite category of a poset is just the same poset with the opposite ordering. The product is the greatest lower bound, so the coproduct must be the least upper bound. How does this square with the idea of "put the two elements side by side"? This category is not concrete, so we need to work out what we mean by "an element of shape \\(A\\)". Since an arrow \\(A \to X\\) is precisely the fact that \\(A \leq X\\), we have that for every element \\(y\\) of the poset, all elements which compare less than or equal to that element have "images of shape \\(y\\)" in \\(X\\). Therefore, the coproduct condition says "for every co-test object \\(X\\), for every pair of images of shape \\(A, B\\) in \\(X\\), the there is an image of shape \\(A+B\\) in \\(X\\) which restricts to the images of shape \\(A\\) and \\(B\\) respectively". With a bit of mental gymnastics, that does correspond to \\(A+B\\) being the least upper bound.
+
+Coproduct of two formulae in the category of proofs: an arrow from one formula to another is a deduction. An "image of shape \\(A\\) in \\(X\\)" - an arrow \\(A \to X\\) - is therefore a statement that we can deduce \\(X\\) from \\(A\\). We want a formula \\(A+B\\) such that for any co-test formula \\(X\\), and for any images of \\(A, B\\) in \\(X\\), there is a unique image of \\(A+B\\) in \\(X\\) which respects the shapes of \\(A\\) and \\(B\\) in \\(A+B\\). Hang on - at this point I realise that the opposite category of the category of proofs is the "category of negated proofs", and the "opposite category" functor is simply taking "not" of everything. That's because the contrapositive of a statement is equivalent to the statement. Therefore since the product is the "and", the coproduct should be the "or" (which is the composition of "not-and-not", or "dual-product-dual). I'll keep going anyway.
+
+We need to be able to prove \\(A+B\\) from \\(A\\), and to prove \\(A+B\\) from \\(B\\). That's already mighty suggestive. Moreover, if there's a proof of \\(X\\) from \\(A\\), there needs to be a unique corresponding proof of \\(X\\) from \\(A+B\\). That's enough for my intuition to say "this is OR, beyond all reasonable doubt".
+
+I now look at the book's explanation of this. Of course, I omitted to perform an actual proof that OR formed the coproduct, and that bites me here: identical arrows must yield identical proofs, but any proof which goes via "a OR b" must be different from one which ignores b. Memo: need to prove the intuitions.
+
+Coproduct of two monoids. Ah, this is a cunning idea, viewing a monoid as a sub-monoid of its free monoid. We already know how to take the coproduct of two free monoids, and we can do the equiv-rel trick that worked with the category of proofs above. Is it possible that in general we do coproducts by moving into a free construction and then quotienting back down? I'm struggling to see how free posets might work, so I'll shelve that idea for now.
+
+I went to sleep in between the previous paragraph and this one, so I'm now in a position to write out a proper proof that the coproduct of two monoids is as stated. I did it without prompting in a very concrete way: given a word('s equivalence class) in \\(M(\vert A \vert + \vert B \vert)\\), and two maps \\(z_1: A \to N\\) and \\(z_2: B \to N\\), we send the letter \\(a \in \vert A \vert\\) to \\(z_1(a)\\), etc. The book gives a more abstract way of doing it. I don't feel like I could come up with that myself in a hurry without a better categorical idea of "quotient by an equivalence relation". At least this way gave me a good feel for why we needed to do the quotient: otherwise our \\(\phi: a \mapsto z_1(a)\\) could have been replaced by \\(a \mapsto u_A z_1(a)\\). The map is unique in this setting. Indeed, suppose \\(\phi([w]) \not = \phi_2([w])\\) for some \\(w\\). We may wlog that \\(w\\) is just one character long, since any longer and we could use that \\(\phi, \phi_2\\) are "homomorphic" to find a character where they differed. (That's where we need that we're working with equivalence classes.) Wlog \\([w] = [w_1]\\). Then \\(\phi([w_1]) \not = \phi_2([w_1])\\); but that means \\(z_1(w_1) \not = z_1(w_1)\\) because the map \\(\phi_2\\) also needs to commute with \\(z_1\\).
+
+I make sure to note that the forgetful functor Mon to Sets doesn't preserve coproducts.
+
+Aha! An example I've seen recently in a different context. (Oops, I've glanced to the bottom of the page, Proposition 3.11. I'll wait til I actually get there.)
+
+I'm confused by the "in effect we have pairs of elements" idea. What about a word like \\(a_1 a_2 b_1 b_2 b_3\\)? Then we don't get a pair of elements containing \\(b_3\\). Ah, I see - Awodey is implicitly pairing with \\(0_A\\) in that example. I'd have preferred to have that spelled out. Now I do see that the underlying set of the coproduct should be the same as that of the product, and that the given coproduct is indeed a coproduct.
+
+Now my "aha" moment from earlier. I've seen this fact referenced [a few days ago][stack exchange] on StackExchange. I can follow the proof, and I see where it relies on abelian-ness, but don't really see much of an idea behind it. The obvious arrows \\(A \to A\\) and \\(B \to B\\) are picked, but it seems to be something of a trick to pick the zero homomorphism \\(A \to B\\). In hindsight, it's the only homomorphism we could have picked, but it would have taken me a while to think of it.
+
+I skim over the bit about abelian categories, and over the various dual notions to products (like "coproducts are unique up to isomorphism", "the empty coproduct is an initial object" etc).
+
+# Summary
+
+This was a bit less intuitive than the idea of the product. Instead of having "finding \\(Z\\)-shaped things in \\(A\\) and \\(B\\) means we can find a \\(Z\\)-shaped thing in the product", we have "finding \\(A\\)- and \\(B\\)-shaped things in \\(Z\\) means we can find a coproduct-shaped thing in \\(Z\\) too", but it took me a while to fix this in my mind, and it still seems to me a little less easy for something to be a coproduct: we've been bothering with equivalence classes more.
+
+[semantic truth]: https://en.wikipedia.org/wiki/Semantic_theory_of_truth
+[syntactic truth]: https://en.wikipedia.org/wiki/Logical_consequence
+[stack exchange]: https://math.stackexchange.com/a/1430755/259262
diff --git a/hugo/content/awodey/2015-09-16-equalisers.md b/hugo/content/awodey/2015-09-16-equalisers.md
new file mode 100644
index 0000000..cc4b54d
--- /dev/null
+++ b/hugo/content/awodey/2015-09-16-equalisers.md
@@ -0,0 +1,53 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-16T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/equalisers/
+- /equalisers/
+title: Equalisers
+---
+
+This is pages 62 through 71 of Awodey, on [equalisers] and coequalisers.
+
+The first paragraph is really quite exciting. I can see that there would be a common generalisation of kernels and varieties - they're the same idea that lets us find complementary functions and particular integrals of linear differential equations, for instance. But the axiom of separation ("subset selection") as well? Now that's intriguing.
+
+We are given the definition of an equaliser: given a pair of arrows with the same domain and codomain, it's an arrow \\(e\\) which may feed into that domain to make the two arrows be "the same according to \\(e\\)".
+
+Let's see the examples of \\(f, g: \mathbb{R}^2 \to \mathbb{R}\\) with \\(f(x, y) = x^2+y^2, g(x,y) = 1\\). I'll try and find the equaliser (in Top) myself. It'll be a topological space \\(E\\) and a continuous function \\(e: E \to \mathbb{R}^2\\) such that \\(f \circ e = g \circ e\\). That is, such that \\(f \circ e = 1\\). That makes it easy: were it not for the "universal" property, \\(E\\) could be anything which has a continuous function mapping it into the unit circle in \\(\mathbb{R}^2\\), and \\(e\\) would be that mapping. (I'm beginning to see where the axiom of subset selection might come in.) But if we took the space \\(E = \{ (1, 0) \}\\) and the inclusion mapping as \\(e\\), this would fail the uniqueness property because there's more than one way we can continuously map a single point into that unit circle. In order to make sure everything is specified uniquely, we'll want \\(E\\) to be the entire unit circle and its inherited topology. Ah, Awodey points out that in this case, the work is easy because the inclusion is monic and so uniqueness is automatic.
+
+Let's do the same thing for Set. The equaliser of \\(f, g: A \to B\\) is a function \\(e: E \to A\\) such that \\(f \circ e = g \circ e\\). We need to make sure \\(f, g\\) only ever see the elements where they're equal after the \\(e\\)-filter has been applied to them, so \\(e\\) must only map into the set \\(\{a \in A : f(a) = g(a) \}\\). It should be easy to show that the equaliser is actually that set with the obvious inclusion into \\(A\\), and I look at the book to see that it is indeed so.
+
+"Every subset is an equaliser" is therefore true, and the characteristic function is indeed the obvious way to go about it. Huh - the axiom of subset selection has just fallen out, stating that there is an inverse to the characteristic function. Magic. Then \\(\text{Hom}(A, 2) \cong \mathbb{P}(A)\\), which we already knew because to specify an element of the power-set is precisely to specify which elements of \\(A\\) are included.
+
+Equalisers are monic: well, the diagram certainly looks like being the right shape, and it's intuitive for Sets: if \\(E \to A\\) weren't injective, then we could choose more than one way of mapping \\(Z \to E \to A \to B\\). The proof in general mimics the Sets example.
+
+Then a blurb on how equalisers can often be made as "restrict the sets and inherit the structure". That's a nice rule of thumb. Awodey points out the "kernel of homomorphism" interpretation, which I'd already pattern-matched in the first paragraph. The equaliser is basically specifying an equivalence class under an equiv rel.
+
+Hah, I wrote that before seeing that a coequaliser is a generalisation of "take the quotient by an equiv rel". Makes sense: if the equaliser is an equivalence class, it seems reasonable for its dual to be a quotient. I skip past the definition of an equivalence relation, because I already know it. What does the definition of a coequaliser really mean? It's an arrow \\(q: B \to Q\\) such that once we've let \\(f, g\\) do their thing, we can apply \\(q\\) to make it look like they've done the *same* thing. It's the other way round from the equaliser, which restricted \\(f, g\\) so that they could only do the same thing. I can see why this is like taking a quotient, and the next example makes that very clear.
+
+Coproduct of two rooted posets: we quotient by the appropriate equivalence relation. That is, we co-equalise using \\(\{ 0 \}\\) and its obvious inclusions into the two posets. I draw out the diagram and after some wrestling I convince myself that the rooted-posets coproduct is as stated. I'm still getting used to this diagram-chasing. I'll wait til the exercises to do the Top example.
+
+Presentations of algebras. I've never seen a demonstration that all groups can be obtained as presentations of free groups, I think, although it's fairly clear that it can be done (just specify every single relation that can possibly hold - in effect writing out the Cayley table). I would prefer it if Awodey defined \\(F(1)\\) explicitly, since it takes me a moment to realise it's the free algebra on one generator. Then \\(F(3) = F(x(1), y(1), z(1))\\). We then perform the next coequaliser. Awodey is again confusing me a bit, and I have to stop and work out that by \\(q(y^2)\\), he means \\(q \circ (1 \mapsto y^2)\\), and by \\(q(1)\\) he means \\(q \circ (1 \mapsto 1)\\). It's obvious what the intent is - chaining together these coequalisers in the obvious way. Each coequaliser doesn't significantly change the structure of the free group, so each coequaliser can be applied in turn, using the inherited structure where necessary. However, this is a bit of a confusing write-up.
+
+"The steps can be done simultaneously": oh dear. It looks like we should be able to do this construction sequentially all the time, and that is conceptually easier, but I'll try and understand the all-at-once construction anyway. Firstly, we define \\(F(2)\\) because we want to equalise two 2-tuples. (It would be \\(F(3)\\) if we had three constraints and so wanted to equalise two 3-tuples.) My instinct would have been to do this with the algebra product instead of the algebra coproduct - using \\(F(2) = F(1) \times F(1)\\). I asked a friend about this. The intuition is apparently that the product is good for imposing multiple conditions at the same time, while what we really want is a way to impose one of a number of conditions. The coproduct (by way of the direct sum) has the notion of "one of a number of things, but not necessarily all of them together".
+
+I draw out the diagram again the next day. This time it makes a bit more sense that we need the coproduct: it's because we need a way of getting from \\(F(1)\\) to \\(F(3)\\), and if we used the product we'd have all our arrows ending at \\(F(1)\\) rather than originating at \\(F(1)\\). I can see why the UMP guarantees the uniqueness of the algebra given by these generators and relations now.
+
+On to the specialisation to monoids. The construction is… odd, so I'll try and forget about it for a couple of minutes and then do it myself. We want to construct the free group on all the generators we have available - that is, all the monoid elements - so we're going to need a functor \\(T\\) (to use Awodey's notation) taking \\(N \to M(\vert N \vert)\\). Then we're also going to need a way to specify all the different relations in the monoid. We can do that by specifying a mapping taking a word \\(x_1, x_2, \dots, x_n\\) to the monoid product \\(x_1 x_2 \dots x_n\\). Write \\(C\\) for that multiplication ("C" for "collapsing") functor \\(T(N) \to N\\). Aha: our restrictions become of the form \\(C(x_1, x_2) = x_3\\), representing the equation \\(x_1 x_2 = x_3\\).
+
+Very well: we're going to need our left-hand sides to be going from \\(T^2 N \to T N\\), and our right-hand sides likewise. Then we'll coequalise them. Let \\(f: T^2 N \to T N\\) taking a word of words to its corresponding word of products, and let \\(g: T^2 N \to T N\\) taking a word of words to the product of products. Wait, that's got the wrong codomain. Let \\(g: T^2 N \to T N\\) taking a word of words to the corresponding word of letters. That's better: we have basically provided a list of equivalences between \\((x_1, x_2)\\) and \\(x_1 x_2\\).
+
+Finally, we take the coequaliser \\(e\\) of \\(f\\) and \\(g\\), and hope and pray that \\(N\\) (our original monoid) has the UMP for the resulting object. I remember from the proof in the book that we should first show that the coequaliser arrow is the operation "take the product of the word". (In hindsight, that's a great place to start. It's harder to deal with a function without knowing what it is.) Certainly the "take the product of the word" function does what we want, but does it actually satisfy the UMP? Drawing out a diagram convinces me that it does: any \\(\phi\\) which doesn't care how the letters are grouped between words, descends uniquely to a map from the monoid of all words.
+
+The coequaliser arrow therefore definitely does go into \\(N\\), and we can identify \\(N\\) with the coequaliser in the obvious way by including from the coequaliser (which still technically has the structure of a free group).
+
+# Summary
+
+This is yet another thing I've started to get a feel for, but not really understood. I now know what coequalisers and equalisers are for, and the utility of the "duals" idea. The exercises will certainly be helpful.
+
+[equalisers]: https://en.wikipedia.org/wiki/Equaliser_(mathematics)#In_category_theory
diff --git a/hugo/content/awodey/2015-09-19-duality-exercises.md b/hugo/content/awodey/2015-09-19-duality-exercises.md
new file mode 100644
index 0000000..763b7fc
--- /dev/null
+++ b/hugo/content/awodey/2015-09-19-duality-exercises.md
@@ -0,0 +1,66 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-19T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/duality-exercises/
+- /duality-exercises/
+title: Duality exercises
+---
+
+Exercise 1 is easy: at the end of Chapter 2 the corresponding products statement was proved, and the obvious dual statement turns out to be this one.
+
+Exercise 2 falls out of the appropriate diagram, whose upper triangle is irrelevant.
+
+![Free monoid functor preserves coproducts][ex 2]
+
+Exercise 3 I've [already proved][duality] - search on "sleep".
+
+Exercise 4: Let \\(\pi_1: \mathbb{P}(A + B) \to \mathbb{P}(A)\\) be given by \\(\pi_1(S) = S \cap A\\), and \\(\pi_2: \mathbb{P}(A+B) \to \mathbb{P}(B)\\) likewise by \\(S \mapsto S \cap B\\). Claim: this has the UMP of the product of \\(\mathbb{P}(A)\\) and \\(\mathbb{P}(B)\\). Indeed, if \\(z_1: Z \to \mathbb{P}(A)\\) and \\(z_2: Z \to \mathbb{P}(B)\\) are given, then \\(z: Z \to \mathbb{P}(A + B)\\) is specified uniquely by \\(S \mapsto z_1(S) \cup z_2(S)\\) (taking the disjoint union).
+
+Exercise 5: Let the coproduct of \\(A, B\\) be their disjunction. Then the "coproduct" property is saying "if we can prove \\(Z\\) from \\(A\\) and from \\(B\\), then we can prove it from \\(A \vee B\\)", which is clearly true. The uniqueness of proofs is sort of obvious, but I don't see how to prove it - I'm not at all used to the syntax of natural deduction. I look at the answer, which makes everything clear, although I still don't know if I could reproduce it. I understand its spirit, but not the mechanics of how to work in the category of proofs.
+
+Exercise 6: we need that for any two monoid homomorphisms \\(f, g: A \to B\\) there is a monoid \\(E\\) and a monoid homomorphism \\(e: E \to A\\) universal with \\(f e = g e\\). Certainly there is a monoid hom \\(e: E \to A\\) with that property (namely the trivial hom), so we just need to find one that is "big enough". Let \\(E\\) be the subset of \\(A\\) on which \\(f = g\\), which is nonempty because they must be equal on \\(1_A\\). I claim that it is a monoid with \\(A\\)'s operation. Indeed, if \\(f(a) = g(a)\\) and \\(f(b) = g(b)\\) then \\(f(ab) = f(a) f(b) = g(a) g(b) = g(ab)\\). This also works with abelian groups - and apparently groups as well.
+
+Finally we need that this structure satisfies the universal property. Let \\(Z\\) be a monoid with hom \\(h: Z \to A\\), such that \\(f h = g h\\). We want a hom \\(\bar{h} : Z \to E\\) with \\(e \bar{h} = h\\). But if \\(f h = g h\\) then we must have the image of \\(h\\) being in \\(E\\), so we can just take \\(\bar{h}\\) to be the inclusion. This reasoning works for abelian groups too. We relied on Mon having a terminal element and monoids being well-pointed.
+
+Finite products: we just need to check binary products and the existence of an initial object. Initial objects are easy: the trivial monoid/group is initial. Binary products: the componentwise direct product satisfies the UMP for the product, since if \\(z_1: Z \to A, z_2: Z \to B\\) then take \\(z: Z \to A \times B\\) by \\(z(y) = \langle z_1(y), z_2(y) \rangle\\). This is obviously homomorphic, while the projections make sure it is unique.
+
+Exercise  7 falls out of another diagram. The (1) label refers to arrows forced by the first step of the argument; the (2) label to the arrow forced by the (1) arrows.
+
+![Coproduct of projectives is projective][ex 7]
+
+Exercise 8: an injective object is \\(I\\) such that for any \\(X, E\\) with arrows \\(h: X \to I, m: X \to E\\) with \\(m\\) monic, there is \\(\bar{h}: E \to I\\) with \\(\bar{h} m = h\\). Let \\(P, Q\\) be posets, and let \\(f: P \to Q\\) be monic. Then for any points \\(x, y: \{ 1 \} \to P\\) we have \\(fx = fy \Rightarrow x=y\\), so \\(f\\) is injective. Conversely, if \\(f\\) is not monic then we can find \\(a: A \to P, b: B \to P\\) with \\(fa = fb\\) but \\(a \not = b\\). This means \\(A = B\\) because the arrows \\(fa, fb\\) agree on their domain; so we have \\(a, b: A \to P\\) and \\(x \in A\\) with \\(a(x) \not = b(x)\\). But \\(f a(x) = f b(x)\\), so we have \\(f\\) not injective.
+
+Now, a non-injective poset: we want to set up a situation where we force some extra structure on \\(X\\). If \\(I\\) is has two distinct nontrivial chunks which have no elements comparable between the chunks, then \\(I\\) is not injective. Indeed, let \\(X = I\\). Then the inclusion \\(X \to I\\) does not lift across the map which sends one chunk "on top of" the other: say one chunk is \\(\{a \leq b \}\\) and the other \\(\{c \leq d\}\\), then the map would have image \\(a \leq b \leq c \leq d\\).
+
+What about an injective poset? The dual of "posets" is "posets", so we can just take the dual of any projective poset - for instance, any discrete poset. Anything well-ordered will also do, suggests my intuition, but I looked it up and apparently the injective posets are exactly the complete lattices. Therefore a wellordering will almost never do. I couldn't see why \\(\omega\\) failed to be injective, so I asked a question on Stack Exchange; midway through, I [realised why][SE].
+
+Exercise 9: \\(\bar{h}\\) is obviously a homomorphism. Indeed, \\(\bar{h}(a) \bar{h}(b) = h i(a) h i(b) = h(i(a) i(b))\\) because \\(h\\) is a homomorphism. But \\(i(a)\\) is the wordification of the letter \\(a\\), and \\(i(b)\\) likewise of \\(b\\), so we have \\(i(a) i(b)\\) is the word \\((a, b)\\), which is itself the inclusion of the product \\(ab\\).
+
+Exercise 10: Functors preserve the structure of diagrams, so we just need to show that that the unique arrow guaranteed by the coequaliser UMP corresponds to a *unique* arrow in Sets. We need to show that given a function \\(\vert M \vert \to \vert N \vert\\) there is only one possible homomorphism \\(M \to N\\) which forgetful-functors down to it. But a homomorphism \\(M \to N\\) does specify where every single set element in \\(\vert M \vert\\) goes, so uniqueness is indeed preserved.
+
+Exercise 11: Let \\(R\\) be the smallest equiv rel on \\(B\\) with \\(f(x) \sim g(x)\\) for all \\(x \in A\\). Claim: the projection \\(\pi: B \to B/R\\) is a coequaliser of \\(f, g: A \to B\\). Indeed, let \\(C\\) be another set, with a function \\(c: B \to C\\). Then there is a unique function \\(q: B/R \to C\\) with \\(q \pi = c\\): namely, \\(q([b]) = c(b)\\). This is well-defined because if \\(b \sim b'\\) then \\(c(b') = q([b']) = q([b]) = c(b)\\).
+
+Exercise 12 I've [already done][duality] - search on "wrestling", though I didn't write this up.
+
+Exercise 13: I left this question to the end and couldn't be bothered to decipher the notation.
+
+Exercise 14: The equaliser of \\(f p_1\\) and \\(f p_2\\) is universal \\(e: E \to A \times A\\) such that \\(f p_1 e = f p_2 e\\). Let \\(E = \{ (a, b) \in A \times A : f(a) = f(b) \}\\) and \\(e\\) the inclusion. It is an equivalence relation manifestly: if \\(f(a) = f(b)\\) and \\(f(b) = f(c)\\) then \\(f(a) = f(c)\\), and so on.
+
+The kernel of \\(\pi: A \mapsto A/R\\), the quotient by an equiv rel \\(R\\), is \\(\{ (a, b) \in A \times A : \pi(a) = \pi(b) \}\\). This is obviously \\(R\\), since \\(a \sim b\\) iff \\(\pi(a) = \pi(b)\\). That's what it means to take the quotient.
+
+The coequaliser of the two projections \\(R \to A\\) is the quotient of \\(A\\) by the equiv rel generated by the pairs \\(\langle \pi_1(x), \pi_2(x) \rangle\\), as in exercise 11. This is precisely the specified quotient.
+
+The final part of the exercise is a simple summary of the preceding parts.
+
+Exercise 15 is more of a "check you follow this construction" than an actual exercise. I do follow it.
+
+[duality]: {{< ref "2015-09-15-duality-in-category-theory" >}}
+[ex 2]: /images/CategoryTheorySketches/FreeMonoidFunctorPreservesCoproducts.jpg
+[ex 7]: /images/CategoryTheorySketches/CoproductOfProjectivesIsProjective.jpg
+[SE]: https://math.stackexchange.com/a/1442264/259262
diff --git a/hugo/content/awodey/2015-09-19-groups-in-categories.md b/hugo/content/awodey/2015-09-19-groups-in-categories.md
new file mode 100644
index 0000000..d4c91bf
--- /dev/null
+++ b/hugo/content/awodey/2015-09-19-groups-in-categories.md
@@ -0,0 +1,57 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-19T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/groups-in-categories/
+- /groups-in-categories/
+title: Groups in categories
+---
+
+I go into this chapter hoping that it will be on things I already know about group theory. This post will be on pages 75 through 85 of Awodey.
+
+I already know about groups over sets, but this looks like they can be made more generally over other categories. It is clear that we will need to consider only categories with finite products, because the notion of a binary operation requires us to work on pairs of elements.
+
+The definition of a group is the obvious one: an object \\(G\\) with an "inverses" arrow \\(i: G \to G\\), a "multiplication" arrow \\(m: G \times G \to G\\) and a "unit" arrow \\(u: 1 \to G\\), such that \\(m\\) is associative in the obvious way, \\(u\\) is a unit for \\(m\\), and \\(i\\) is an inverse with respect to \\(m\\) - drawing out the appropriate diagrams.
+
+The definition of a homomorphism is likewise very familiar, and the examples which follow are very clear. (The operations are arrows, so they must preserve structure.)
+
+Example 4.4 is a group in the category of groups. I remember having proved Proposition 4.5 on an example sheet somewhere, but it wasn't indicated there that it was anything particularly important. I've only glanced over the construction of a group in the category of groups, so I'll try and work out what it is myself. A group in the category of groups is a group \\(G\\) together with its self-product \\(G \times G\\), and associative homomorphism \\(m: G \times G \to G\\), and \\(u: \{ 1 \} \to G\\), and \\(i: G \to G\\) which acts as an inverse for \\(m\\). This is still a bit nonspecific, so can we say anything about \\(m\\)? It must preserve the group structure on \\(((G, \cdot), m, i)\\), and we know \\(\cdot\\) preserves the group structure on \\((G, \cdot)\\). Is there perhaps a way to get them to play nicely together?
+
+I'll write \\(\times\\) as a shorthand for \\(m\\). Then \\(m(a \cdot b, c \cdot d) = m(a, c) \cdot m(b, d)\\) because \\(m\\) is a homomorphism \\(G \times G \to G\\). Letting \\(a = 1_G, d = 1_G\\) yields \\(m(b, c) = c \cdot b\\). Letting \\(b=1_G, c=1_G\\) yields \\(m(a, d) = a \cdot d\\). Therefore in fact \\(m\\) is the group operation on \\(G\\), and \\(G\\) is also abelian. (I won't bother with the converse, since on looking, the book says it's easy.)
+
+A strict monoidal category is a monoid in Cat. Dear heavens, this is confusingly general. I'll have to go through the examples Awodey gives.
+
+The operation of taking products and coproducts (that is, the meet/join operations) does indeed satisfy the criterion - ah, I move down and see that Awodey points out that these only hold up to isomorphism, not equality, so this isn't "strict". In posets, though, there's at most one arrow between any two objects, so we really do have equality.
+
+A discrete monoidal category is a standard Set-monoid: I can see that each Set-monoid is a discrete monoidal category. How about the converse? Yep, that's fine as long as we're talking about locally small categories. (I briefly got confused between the morphisms and the \\(\otimes\\) operation, but that's cleared up now.)
+
+A strict non-poset monoidal category is the finite ordinals: since no arrow between two different ordinals has an inverse, we must have that objects are unique not just up to isomorphism, but in a more specific sense. This again lets us say that this is a strict monoidal category.
+
+I'll leave that and hope for the best. Next is the category of groups, and we see the familiar equivalence between kernels of homomorphisms and normal subgroups. There's also this idea of "the equaliser is the subgroup; the coequaliser is the quotient" from earlier. I prove the coequaliser statement myself without looking at the proof - it's not hard, and it just involves showing that for \\(H\\) normal in \\(G\\), if \\(k: G \to K\\) is such that \\(k i = k u\\), then \\(k\\) is constant on \\(H\\) and so descends to the quotient \\(G/H\\). The category-theoretic statement about coequalisers is much more fearsome than the concrete group-theoretic one!
+
+I'm very familiar with these results, so having done one of them (the coproducts one), I skip through to the first one I don't know, which is Corollary 4.11. Actually this is the First Isomorphism Theorem in disguise. I whizz down to the exercises and see that the cokernel construction is an exercise, so I'll leave it til then (I'd like to avoid fragmenting them, and also I can't be bothered at the moment).
+
+Section 4.3: groups as categories. Groups certainly are categories - that's how I defined them in my Anki card for the category theory deck. A functor is therefore clearly a group hom, as Awodey says.
+
+Ah, that's cool. Functors from a group (viewed as a category) to any category form "representations" of that group. Elements of \\(G\\) become automorphisms of an object in \\(C\\). In the case of the functor into the category of finite-dimensional vector spaces with linear maps, we can have \\(G\\) appearing as the automorphism group of a wide variety of different objects: for instance, \\(C_5\\) acts on \\(\mathbb{C}^1\\), or on \\(\mathbb{C}^2\\) or on…
+
+In the case of the functor into the category of sets, it's most natural to identify \\(G\\) with a subgroup of some permutation group and to make \\(G\\) act on the appropriate set; in fact this looks like the only way of describing such a functor, since every group is unique up to isomorphism, so corresponds to only one "distinct" permutation-subgroup.
+
+Now we see the definition of a congruence on a category. It's easy to see that this is the equivalence relation we get by identifying all arrows which go to and from the same place, or an equiv rel with more classes than that.
+
+The congruence category uses some rather strange notation. What even is \\(C_0\\)? Surely it must be the set of objects, and \\(C_1\\) the set of arrows, but that isn't notation I remember from earlier in the book. Once that's settled, the definitions become easy: the congruence category is "the thing on which we need to take the quotient" in order to get the quotient by the congruence. It is the category where the morphisms are instead "congruent pairs of arrows" in the original category, and the composition is well-defined because \\(f' f\\) and \\(g' g\\) are congruent if \\(f', f\\) and \\(g', g\\) are.
+
+There are indeed two projection functors, because we're working on a category which has "pairs of arrows" as its morphisms; then the coequaliser of those two is the desired quotient. That seems fine.
+
+We then construct the "kernel of a functor \\(F\\)" in an analogous way to groups: two arrows are \\(F\\)-congruent iff \\(F\\) treats them in the same way, and we define the quotient category to be universal such that for any congruence \\(\sim\\), \\(F\\) descends to the quotient by \\(\sim\\) iff \\(\sim\\) is a sub-congruence of \\(\sim_F\\). (I had to sleep on this one, but I think I understand it now.)
+
+Finally, picking \\(\sim\\) to equal \\(\sim_F\\) gives that all functors descend to their quotients, where the descent is bijective on objects, surjective on hom-sets, and the descended map is injective on hom-sets.
+
+# Summary
+
+This section was an interesting one, but it took me a while to get the hang of it. I'm used to all of this in a concrete setting; seeing it in the abstract makes everything quite difficult. I'm going back to the section on hom-sets now, because the last paragraph is not intuitive at all to me, and I feel it ought to be.
diff --git a/hugo/content/awodey/2015-09-22-limits-and-pullbacks.md b/hugo/content/awodey/2015-09-22-limits-and-pullbacks.md
new file mode 100644
index 0000000..07e0137
--- /dev/null
+++ b/hugo/content/awodey/2015-09-22-limits-and-pullbacks.md
@@ -0,0 +1,50 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-22T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/limits-and-pullbacks/
+- /limits-and-pullbacks/
+title: Limits and pullbacks
+---
+
+I'm going to skip pages 85 through 88 of Awodey for the moment, because time is starting to get short and I want to make sure I'm doing stuff which is relevant to the Part III course on category theory. Therefore, I'll skip straight to Chapter 5, pages 89 through 95. (There's not really a nice way to break this chapter up into small chunks, because the next many pages are on pullbacks.)
+
+We have indeed seen that every subset of a set is an equaliser: just define two functions which agree on that subset and nowhere else. (The indicator function on the subset, and the constant-1 function, for example.) A mono is a generalised subset: well, we have that arrows are generalised elements, so can we make a mono represent a collection of generalised elements? Yes, sort of: given any generalised element which is "in the subset" - that is, on which the equaliser-functions agree - that element lifts over the mono, so can be interpreted as an element of the mono. It's a bit dubious, but it'll do.
+
+The idea of "an injective function which is isomorphic onto its image" comes up quite often, so the next chunk is quite familiar. Then the collection of subobjects of \\(X\\) is just the slice \\(\mathbf{C}/X\\), and the morphisms are the same as in the slice category: commuting triangles.
+
+Because our arrows are monic, we can have at most one way to complete any given commuting triangle, so we get the natural idea of "there is exactly one natural inclusion map from a subset to its parent set". Finally, we define what it means for two objects to be "the same object" in this setting: namely, each includes into the other. (Remark 5.2 describes the process of quotienting out those objects which are "the same" in this sense, and points out that in Set, each subobject is isomorphic only to itself.)
+
+We then see that subobjects of subobjects of \\(X\\) are subobjects of \\(X\\), because the composition of monic things is monic. We therefore have a way of including subobjects of subobjects of \\(X\\) into \\(X\\), and that lets us define the obvious membership relation.
+
+The final example in this section is that of the equaliser, which is actually a subobject consisting of generalised elements which \\(f, g\\) view as being the same. I follow this construction as symbols, but as ever, I don't really have an intuition for it. I'll accept that and move on.
+
+Pullbacks next. A pullback is a universal way of completing a square. My first thought on seeing the definition is that this is an awful lot like a product: given \\(f: A \to C, g: B \to C\\) we seek a product of \\(A\\) and \\(B\\) such that the projection diagram commutes with \\(f\\) and \\(g\\) in the right way. However, products are unique up to isomorphism, so there is "only one" product anyway: we can't just look for one which behaves in the right way, can we?
+
+I'm going to have to try and get this in Sets. Let \\(A = \{ 1, 2 \}\\), \\(B = \{4, 5 \}\\), \\(C = \{1, 2, 4, 5 \}\\) and \\(f, g\\) the inclusions. Then the pullback \\(P\\) must be the empty set - ah, this is the intersection operation Awodey mentioned earlier, and I sense an equaliser going on here. What about \\(A = \{1, 2, 4 \}\\) instead? Then we need \\(P\\) to be \\(\{4\}\\) only.
+
+Ah, I understand my confusion. Products are indeed unique - but they are universal: they are the most general kind of thing which satisfies the UMP of the product. There are other things which satisfy the "UMP-without-the-U" of the product: the statement of the UMP but without the word "unique". We want to pick the most general one of those which satisfies a certain property. So a product is just a pullback where \\(C\\) is initial, for instance.
+
+Proposition 5.5 is a description of the pullback as an equaliser. I knew there would be something like this! Without looking at the proof, I can tell it'll revolve around the fact that equalisers are monic (that'll be the step which guarantees uniqueness). The proof follows just by drawing out the diagram, really.
+
+![Pullback exists if equalisers and products do][pullback exists]
+
+Now comes a demonstration that inverse images are a kind of pullback. I don't see a way to understand this intuitively enough that I could reproduce it - the idea is simple but very much counter to my intuitions. I'll just plough on.
+
+In a pullback of \\(f: A \to B, m: M \to B\\), if \\(m\\) is monic then its parallel arrow \\(m'\\) is: that follows from another diagram.
+
+![Monic implies parallel arrow is monic in a pullback][monic]
+
+# Summary
+
+I get the impression that the idea of a limit is a very general one, of which presumably pullbacks are a specific example - I can't think of something which generalises the idea of "inverse image" off the top of my head. We're going to have six more pages on pullbacks, and then the idea of a limit will be introduced. (This chapter is rather long.)
+
+I do like the way Awodey is doing this: give examples of specific constructions, and then show how they may be unified. I glanced down to the blurb at the start of the "limits" section, and saw that another such unification is about to take place. I'm looking forwards to that.
+
+[pullback exists]: {{< baseurl >}}images/CategoryTheorySketches/PullbackExistsWithEqualisers.jpg
+[monic]: {{< baseurl >}}images/CategoryTheorySketches/ParallelArrowInPullbackIsMonic.jpg
diff --git a/hugo/content/awodey/2015-09-22-properties-of-pullbacks-limits.md b/hugo/content/awodey/2015-09-22-properties-of-pullbacks-limits.md
new file mode 100644
index 0000000..091f22a
--- /dev/null
+++ b/hugo/content/awodey/2015-09-22-properties-of-pullbacks-limits.md
@@ -0,0 +1,61 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-22T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/properties-of-pullbacks-limits/
+- /properties-of-pullbacks-limits/
+title: Limits and pullbacks 2
+---
+
+We have just had the definition of a pullback; now in Awodey pages 95 through 100 we'll see some more about them, and after that we'll get the more general unifying idea of the limit in pages 100 through 105.
+
+Lemma 5.8 states that a certain commuting diagram is a pullback. The proof is by "diagram chase", and I can see why - my proof goes along the lines of gesturing several times at various parts of the diagram. Then the corollary takes me a moment to get my head around, but then I turn my head sideways and it pops out of the diagram. If you push the \\(h'\\) line into the page in Lemma 5.8, and rotate the diagram by ninety degrees, you end up with the diagram of Corollary 5.9; then part 2 of Lemma 5.8 is the corollary.
+
+The operation of pullback is a functor. Given a "base" arrow \\(h\\), we may define a functor which takes an arrow \\(f\\) and pulls back along \\(f, h\\). It seems very plausible, but it takes me a while of staring at the diagrams before it makes sense. In particular, the diagram in the book doesn't do a good job of splitting up the two statements which are proved: namely that \\(h^* 1_X = 1_{X'}\\) and \\(1_{X'} = 1_{h^* X}\\).
+
+The corollary is that \\(f^{-1}\\) is a functor, which follows because the operation "pull things back along \\(f\\)" is a functor. Then we get that \\(f^{-1}\\) descends to the quotient by equivalence. This is all a set of symbols which I barely understand, so I have a break and the go back over the whole thing again.
+
+Pullback is a functor. Fine. Then \\(f^{-1}: \text{Sub}(B) \to \text{Sub}(A)\\) - which is defined as the pullback of an inclusion and \\(f\\) - must also be a functor, because it is exactly the operation "take a certain pullback". The statement that \\(M \subseteq N \Rightarrow f^{-1}(M) \subseteq f^{-1}(N)\\) is just the statement that "if we apply the pullback by \\(f\\) and an inclusion, the relation \\(\subseteq\\) is preserved", which is true because pullback is a functor. (Recall that \\(M \subseteq N\\) iff there is \\(g: M \to N\\) with the triangle \\(m: M \to Z, n: N \to Z\\) commuting. We're working throughout with subobjects of the object \\(Z\\).)
+
+Now we do have \\(M \equiv N\\) implies \\(f^{-1}(M) \equiv f^{-1}(N)\\) - recall that \\(M \equiv N\\) iff both are \\(\subseteq\\) each other - so \\(f^{-1}\\) is constant on equivalence classes and so descends to the quotient. That's a bit clearer now.
+
+Phew, a concrete example is coming up: a pullback in Sets. I draw out the general diagram first, then write in the assumptions we make, and end up with a diagram a lot like the one in the book, except that I've labelled the unlabelled arrow "inclusion".
+
+Ah, I'm starting to get this. The operation "take inverses" is a function which takes one "major" argument \\(f: A \to B\\), and one "minor" argument \\(M\\) (from which we extract the corresponding subset-arrow \\(m: M \to B\\)). The output is the pullback diagram, which may be interpreted as just the pullback object from those two arrows.
+
+Once I've realised that the operation "take inverses" is as above, the top of the following page (p99) becomes trivially obvious, although I still have to do some mental work to do the interpretation in terms of substituting a term \\(f\\) for a variable \\(f\\) in function \\(\phi\\). It seems like a very complicated way of saying something very simple.
+
+Then we see the naturality of the isomorphism \\(2^A \cong \mathbb{P}(A)\\). First, am I convinced we've even shown that there is an isomorphism? Certainly each function \\(A \to 2\\) corresponds (by inverses) to a unique member of \\(\mathbb{P}(A)\\), while each member of \\(\mathbb{P}(A)\\) corresponds to a unique member of \\(2^A\\) given by the characteristic function. Now, does the naturality diagram really commute? Yes, that's what happened above: \\(f^{-1}(V_{\phi}) = V_{\phi f}\\).
+
+This section has one final example: reindexing an indexed family of sets. The definition of \\(p\\) is fine; then we pull it back along \\(\alpha\\). I need to check that the guessed pullback object is indeed a pullback, for which I need a diagram. The required property eludes me completely until I realise that the topmost arrow of Awodey's diagram is in fact the identity; then the UMP falls out easily.
+
+Section 5.4 is entitled "limits", and it promises to unify pretty much everything we've already seen. Recall the theorem that if a category has pullbacks and a terminal object if has finite products and equalisers, because we may take the equaliser of the product to obtain a pullback, and we may perform the empty product to obtain a terminal object. Now Awodey proves the converse: constructing the product as a pullback from a terminal object, and constructing the equaliser as a pullback of the identity pair of arrows with the pair \\(\langle f, g \rangle\\) we want to equalise.
+
+Define a "diagram of type \\(\mathbf{J}\\) in \\(\mathbf{C}\\)" in the way you'd hope: since an arrow \\(X \to Y\\) is thought of as a shape-\\(X\\) subset of \\(Y\\), we should consider a shape-\\(\mathbf{J}\\) "subset" of \\(\mathbf{C}\\) to be a functor \\(\mathbf{J} \to \mathbf{C}\\).
+
+Define a *cone* to the diagram \\(D\\) as - well, the name is quite suggestive. Fix a base object \\(C\\) of \\(\mathbf{C}\\), and then the subcategory of all the arrows \\(C \to D_j\\) forms a category of shape \\(\mathbf{J}\\) in \\(\mathbf{C}\\), all linked to this base \\(C\\). (Of course, we insist that the arrows of this cone commute with the base object.)
+
+A morphism of cones behaves in the obvious way: send the base point to its new position, and send each arrow to its new arrow. (We keep the \\(D_j\\) the same, because we need to preserve the diagram; we're only changing the position of the apex of the cone.)
+
+Finally, the definition of a limit! It's a terminal object in the category of cones on a given diagram. All cones have exactly one arrow going into this cone (if it exists). The "closest cone to the diagram" idea is a nice one, and I can see how this links with the idea of a universal mapping property. The UMPs we've seen up to now are of the form "draw this diagram, and select the closest object that fulfils it" - how neat. This immediately covers the product, pullback and equaliser examples; from the empty diagram, there is precisely one cone for each object (namely "pick a vertex, and have no maps at all"), so the category of cones is just the original category, so the limit is a terminal object.
+
+Now, a theorem on an equivalent condition for having all finite limits. If a category has all finite limits, then it trivially has all finite products and equalisers, because they're limits. Therefore we need to show that if a category has all finite products and equalisers, then we can build any limit. The proof will have to start by fixing some finite category \\(\mathbf{J}\\) and considering some fixed diagram of shape \\(\mathbf{J}\\) in \\(\mathbf{C}\\). Construct the cone category. We're going to have to manufacture the limit somehow, given that we have finite products and equalisers. At this point I look in the book and it tells me that the first step is to consider the product of all the objects in the diagram. OK, that is a cone-shape - it has the right arrows. Could it be a limit? We'd need that for any other cone \\(X\\), there was a unique arrow \\(X \to \prod D_i\\) commuting with the projections. That doesn't actually hold, though: consider \\(D_1, D_2\\) as our diagram, and take \\(D_1 \times D_2\\) as the product. Then \\(D_1 \times \{ \langle x, x \rangle : x \in D_2 \}\\) doesn't have a unique arrow into \\(D_1 \times D_2\\), because we could take either the second or the third projection, so we want to equalise out by such manipulations.
+
+Ugh, I just don't see how to do this. I'll have to look at the book again. The construction is quite complicated: we take the product over all the possible arrows (ways to get to) \\(D_j\\) from any object \\(D_i\\), and we'll equalise out the different ways to get to each object. This becomes much clearer from a diagram, where it actually looks like the only possible way to do it: basically list all the different ways to get from A to B, and equalise out by "viewing them all as being the same way".
+
+![Equalisers to make limits][equalisers to make limits]
+
+Once that's done, the rest is bookkeeping to check that we've actually made a cone, and that the cone is a limit, by showing that "cone" is precisely "thing which satisfies the equaliser diagram"; then the fact that we made a limit falls straight out of the uniqueness part of the UMP.
+
+The final bit on "we didn't use the finiteness condition" is clear, and the dual bit is clear (though I have not much idea about what a colimit or a cocone is). Presumably we'll see some examples of colimits later, but I imagine the coequaliser and coproduct are examples.
+
+# Summary
+
+This section was really neat. Quite hard to understand - took a lot of time and effort to get the pullbacks idea - but the feeling of unification was great fun. Next up will be "preservation of limits" and colimits, and after that will come some exercises (which I think are sorely needed). Then the next chapter is on another kind of construction which is not a limit, and then the really meaty sections which Awodey has called "higher category theory" and which occupy a large chunk of the Part III introductory category theory course.
+
+[equalisers to make limits]: {{< baseurl >}}images/CategoryTheorySketches/EqualisersToMakeLimits.jpg
diff --git a/hugo/content/awodey/2015-09-23-properties-of-limits.md b/hugo/content/awodey/2015-09-23-properties-of-limits.md
new file mode 100644
index 0000000..fa03cab
--- /dev/null
+++ b/hugo/content/awodey/2015-09-23-properties-of-limits.md
@@ -0,0 +1,71 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-23T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/properties-of-limits/
+- /properties-of-categorical-limits/
+title: Properties of categorical limits
+---
+
+We've seen how limits are formed, and that they exist iff products and equalisers do. Now we get to see about continuous functions and colimits, pages 105 through 114 of Awodey.
+
+The definition of a continuous functor is obvious in hindsight given the real-valued version: it "preserves all limits", where "preserves a particular limit" means the obvious thing that limits of cones of the given shape remain limits when the functor is applied.
+
+The example is the representable functor, taking any arrow in category \\(\mathbf{C}\\) to its corresponding "apply me on the left!" arrow in Sets. That is basically to the relevant commutative triangle in \\(\mathbf{C}\\). I hope the following proof will help me understand the representable functors more clearly.
+
+Representable functors preserve all limits: we need to preserve all products and all equalisers. Awodey shows the empty product first, which is clear: the terminal object goes to the terminal object. Then an arbitrary product \\(\prod_{i \in I} X_i\\) gets sent to \\(\text{Hom}(C, \prod_i X_i)\\), which is itself a product because \\(f: C \to \prod_i X_i)\\) corresponds exactly with \\(\{ f_i: C \to X_i \mid i \in I\}\\). (Indeed, the projections give \\(f \mapsto \{ f_i \mid i \in I\}\\); conversely, the UMP of the product gives a unique \\(f\\) for the collection \\(\{ f_i \mid i \in I \}\\).)
+
+This has given me the intuition that "the representable functor preserves all the structure" in the sense that the diagrams will look the same before and after having done the functor.
+
+Equalisers are the other thing to show, and that falls out of the definition in a completely impenetrable way. I can't distill that into "the representable functor preserves all the structure" so easily.
+
+Then the definition of a contravariant functor. I've heard the terms "covariant" and "contravariant" before, several times, when people talk about tensors and general relativity and electromagnetism, but I could never understand what was meant by them. This definition is clearer: a functor which reverses input arrows with respect to the objects. Operations like \\(f \mapsto f^{-1}\\) would be contravariant, for instance.
+
+The representable functor \\(\text{Hom}_{\mathbf{C}} ( -, C) : \mathbf{C}^{\text{op}} \to \mathbf{Sets}\\) is certainly contravariant, taking \\(A\\) to \\(\text{Hom}(A, C)\\) and an arrow \\(f: B \to A\\) to \\(f^* : \text{Hom}(A, C) \to \text{Hom}(B, C)\\) by \\((a \mapsto g(a)) \mapsto (b \mapsto g(f(b)))\\). The contravariant functor reverses the order of arrows in its argument; it takes arrows to co-arrows, so it should take colimits to co-colimits, or limits. I need to keep in mind this example, to avoid the intuition that "functors take things to things and cothings to cothings": if the functor is covariant, it flips the co-ness of its input.
+
+Example: a coproduct is a colimit, so \\(\text{Hom}_{\mathbf{C}} ( - , C)\\) should take the coproduct to a product. That might be why we had \\(\mathbb{P}(A+B) \cong \mathbb{P}(A) \times \mathbb{P}(B)\\) as Boolean algebras: the functor \\(\mathbb{P}\\) might be contravariant. What does it do to the arrow \\(B \to A\\)? Recall that an arrow in the category of Boolean algebras (interpreted as posets) is an order-preserving map. Huh, not contravariant after all: the \\(\mathbb{P}\\) functor seems covariant to me. There must be some other reason; [it turns out][SE] that I'm mixing up two different functors, one of which is covariant and takes sets to sets, and one of which is contravariant and takes sets to Boolean algebras.
+
+"The ultrafilters in a coproduct of Boolean algebras correspond to pairs of ultrafilters": recall that the functor \\(\text{Ult}: \mathbf{BA}^{\text{op}} \to \mathbf{Sets}\\) takes an ultrafilter to the corresponding set of indicator functions picking out whether a given subset is in the filter, and an arrow \\(f: B \to A\\) of ultrafilters to the arrow \\(\text{Ult}(f): \text{Ult}(A) \to \text{Ult}(B)\\) by \\(\text{Ult}(f)(1_U) = 1_U \circ f\\), and so it is representable. (I barely remember this. I think I deferred properly thinking about representable functors until Awodey covered them properly.) At least once we've proved that, we do get "ultrafilters in the coproduct correspond to pairs of ultrafilters", by the iso in the previous paragraph.
+
+The exponent law is much easier - it follows immediately from the same iso.
+
+(Oh, by the way, we have that limits are unique up to unique isomorphism, because they may be formed from products and equalisers which are themselves unique up to unique isomorphism.)
+
+Next section: colimits. The construction of the co-pullback (that is, pushout) is dual to that of the pullback: take the coproduct and then coequalise across the two sides of the square. So the coproduct of two rooted posets would be the pushout of the two "pick out the root" functions: let \\(A = \{ 0 \}\\), and \\(B, C\\) be rooted posets with roots \\(0_B, 0_C\\). Then the pushout of \\(f: A \to B\\) by \\(f(0) = 0_B\\) and \\(g: A \to C\\) by \\(g(0) = 0_C\\) is just the coproduct of the two rooted posets.
+
+Ugh, a geometrical example next. Actually, this is fairly neat: the coproduct of two discs, but where we view two points as being the same if they are both images of the inclusion. That's just two circles glued together on the boundary, which is topologically the same as a sphere. In the next lower dimension, we want to take two intervals, glued together at their endpoints, making a circle.
+
+Then the definition of a colimit, which is the obvious dual to that of a limit. I skip through to the "direct limit" idea, where the colimit is taken over a linearly ordered indexing category. I can immediately see that this might be associated with the idea of a limit in \\(\mathbb{R}\\), but I'll save that until after the worked example, which is the direct limit of groups.
+
+The colimit setup is all pretty obvious in retrospect, but I didn't try and come up with it myself. (The exercises will show whether it really is obvious!) The colimiting object does exist because coproducts and coequalisers do, and we can construct it as the coproduct followed by a certain coequaliser - namely, the one where "following a path through the sequence, then going out to the colimit, is the same as just going straight to the colimit". That is, such that \\(p_n g_{n-1} g_{n-2} \dots g_i = p_i\\), where the \\(p_i: G_i \to L\\) are the maps into the colimit. The equivalence relation whose quotient we take, is therefore: if \\(x \in G_n, y \in G_m\\), then \\(x \sim y\\) iff there is some \\(k\\) such that if we follow along the homomorphisms starting from \\(x\\) and \\(y\\), we eventually hit a common element. (Indeed, if there existed elements \\(x, y\\) which didn't have this property, then \\(p_m g_{m-1} \dots g_n(x) \not = p_n(x)\\).) I think I've got that.
+
+The operations are the obvious ones, and we've made a kind of "infinite intersection" of these groups, where the maps \\(u_n: G_n \to G_{\infty}\\) are the "inclusions". Universality is inherited from Sets, so as long as the limiting structure obeys the group axioms, we have indeed ended up with a colimit.
+
+What does it mean, then, for functor \\(F: \mathbf{C} \to \mathbf{D}\\) to "create limits of type \\(\mathbf{J}\\)"? For each diagram \\(D\\) in \\(\mathbf{D}\\) of type \\(\mathbf{J}\\), and each limit of that diagram, there is a unique cone in \\(\mathbf{C}\\) which is sent to \\(D\\) by \\(F\\), and moreover that cone is itself a limit.
+
+In the example above, \\(F\\) is the forgetful functor Groups to Sets, \\(\mathbf{J}\\) is the ordinal category \\(\omega\\). For each diagram \\(D\\) in Sets of type \\(\omega\\), the colimit of the diagram is given by taking the coproduct of all the \\(D_i\\), and identifying \\(x_n \sim g_n(x_n)\\) (where \\(g_n: D_n \to D_{n+1}\\) is the arrow in \\(D\\) corresponding to the arrow in \\(\omega\\) from \\(n\\) to \\(n+1\\)). Then we can pull this back through the forgetful functor to obtain a corresponding cocone in Groups, and we can check that it's still a colimit. That is, \\(F\\) creates \\(\omega\\)-colimits.
+
+Why does it create all limits? Take a diagram \\(C: \mathbf{J} \to \mathbf{Groups}\\) and limit \\(p_j: L \to U C_j\\) in \\(\mathbf{Sets}\\). Then we need a unique Groups-cone which is a limit for \\(C\\). The Set-limit can be assigned a group structure, apparently. It's obvious how to do that in the case that the limit was an ordinal - it's the same as we saw above - but in general…
+
+I'll leave that for the moment, because I want to get on to adjoints sooner rather than later (they're apparently treated very early in the Part III course).
+
+The idea behind the cumulative hierarchy construction is clear in the light of the \\(\omega\\) example above, and this makes it immediately obvious that each \\(V_{\alpha}\\) is transitive. The construction of the colimit is the obvious one (although I keep having to convince myself that it is indeed a colimit, rather than a limit).
+
+What does it mean to have all colimits of type \\(\omega\\)? A diagram of \\(\omega\\)-shape is an \\(\omega\\)-chain. A colimit of that chain would compare bigger than all the elements of the chain (that's "there is an arrow \\(n \to \omega\\)" - that is, "it is a cocone"), and would have the property that if \\(n \leq x\\) for all \\(n\\) then \\(\omega \leq x\\) (that's "all other cocones have a map into the colimit"). The colimit is a "least upper bound" for the specified chain. A monotone map is called continuous if it maintains this kind of least upper bound.
+
+Then we have a restated version of the theorem that "an order-preserving map on a complete poset has a fixed point", which I remember from Part II Logic and Sets. The proof here is very different, though. I follow it through, doing pretty natural things, until "The last step follows because the first term \\(d_0 = 0\\) of the sequence is trivial". Does it actually make a difference? If we remove the first element of the chain, I think it couldn't possibly alter anything in this case, even if the first element were not trivial, because we've already taken the quotient identifying "things which are eventually equal".
+
+I was a little confused by the statement of the theorem. "Of course \\(h\\) has a least fixed point, because it's well-ordered" was my thought, but obviously that's nonsense because \\([0,1]\\) is not well-ordered. So there is some work to do here, although it's easy work.
+
+The final example seems almost trivial when it's spelled out, but I would never have come up with it myself. Basically saying that "you need to check that your proposed colimit object actually exists", and if it doesn't, you might have to add things to your colimit until it starts existing". I don't know how common a problem this turns out to be in practice, but the dual says that we can't assume naive limits exist either.
+
+# Summary
+
+This was another rather difficult section. Fortunately the exercises come next, and that should help a lot. I've dropped behind a bit on my Anki deck, and need to Ankify the colimits section.
+
+[SE]: http://math.stackexchange.com/a/1448655/259262
diff --git a/hugo/content/awodey/2015-09-29-exponentials-in-category-theory.md b/hugo/content/awodey/2015-09-29-exponentials-in-category-theory.md
new file mode 100644
index 0000000..793459d
--- /dev/null
+++ b/hugo/content/awodey/2015-09-29-exponentials-in-category-theory.md
@@ -0,0 +1,51 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-29T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/exponentials/
+- /exponentials-in-category-theory/
+title: Exponentials in category theory
+---
+
+Now we come to Chapter 6 of Awodey, on exponentials, pages 119 through 128. Supposedly, this represents a kind of universal property which is not of the form "for every arrow which makes this diagram commute, that arrow factors through this one".
+
+First, we define the currying of a function \\(f: A \times B \to C\\), producting a function \\(f(a) : B \to C\\) - that is, a function \\(f(a) \in C^B\\). That is, we view \\(f: A \to C^B\\), defining an isomorphism of homsets \\(\text{Hom}_{\mathbf{Sets}}(A \times B, C)\\) to \\(\text{Hom}_{\mathbf{Sets}}(A, C^B)\\).
+
+Now, we try to generalise this construction, by generalising the "currying" construct to allow for more kinds of evaluation. We just need a way to take \\(C^B \times B \to C\\) in a universal way. The resulting diagram is perhaps not something I could have come up with, but it is extremely reminiscent of the UMP of the free monoid.
+
+The general definition of an exponential is then "a way of currying", defined in terms of "a way of evaluating". We get some terminology - the "evaluation" is the way of evaluating, and the "transpose" of an arrow is the curried form. We can also define the transpose of a curried arrow, by giving it a way of evaluating on any input; the UMP tells us that if we transpose twice, we recover the original arrow; therefore, the "curry me" operation is an isomorphism between \\(\text{Hom}_{\mathbf{C}}(A \times B, C)\\) and \\(\text{Hom}_{\mathbf{C}}(A, C^B)\\). (This is all probably very harmful, thinking of this in terms of currying, but so far I think it is helping.)
+
+A category is then Cartesian closed if it has all finite products and exponentials. That is, if we can define multi-variable functions which curry. (Yes, arrows are usually not functions. This is for my beginner's intuition.)
+
+Then Example 6.4, showing that the product of two posets is a poset, and defining the exponential to be the Sets-exponential but with the pointwise ordering on arrows. There is work to do to show that the evaluation is an arrow and that the transpose of an arrow is an arrow.
+
+Restricting to \\(\omega\\)CPOs, we still need to show that \\(Q^P\\) is an \\(\omega\\)CPO. Indeed, given an \\(\omega\\)-chain in \\(Q^P\\), we need to find an upper bound in \\(Q^P\\). Say the chain was \\(f_1, f_2, \dots\\). Then for each \\(p\\), the chain with members \\(f_i(p)\\) has a least upper bound \\(f(p)\\). This defines an order-preserving function because if \\(p \leq q\\) then each \\(f_i(p) \leq f_i(q)\\), and weak inequalities respect the limiting operation. Therefore our prospective exponential is in fact in the category.
+
+\\(\epsilon\\) needs to be \\(\omega\\)-continuous: it needs to respect least upper bounds. Let \\((f_i, p_i)\\) be an \\(\omega\\)-chain in \\(Q^P \times P\\). (I'll take it as read that products exist.) We need that evaluating the least upper bound, \\(\epsilon(f, p)\\), yields the limit of \\(\epsilon(f_i, p_i)\\). This follows from the lemma that if the LUB of \\((f_i)\\) is \\(f\\), and of \\((p_i)\\) is \\(p\\), then the least upper bound of \\((f_i, p_i)\\) is \\((f, p)\\) (which is true: it is an upper bound, while any other upper bound is bigger than it). Then \\(\epsilon(f, p) = f(p)\\) while \\(\epsilon(f_i, p_i) = f_i(p_i)\\), so we do get the result: each \\(f_i(p_i) \leq f(p)\\) because \\(f_i(p_i) \leq f(p_i) \leq f(p)\\), while any other upper bound \\(g\\) would have all \\(f_i(p_i) \leq g\\) so (fixing \\(j\\)) all \\(f_j(p_i) \leq g\\), so all \\(f_j(p) \leq g\\), so (releasing \\(j\\)) \\(f(p) \leq g\\).
+
+Finally, the transpose of an \\(\omega\\)-continuous function needs to be \\(\omega\\)-continuous: let \\(f: A \times B \to C\\) be \\(\omega\\)-continuous. Its transpose is \\(\bar{f}: A \to C^B\\) given by \\(\epsilon \circ (\bar{f} \times 1_B) = f\\). If \\(\bar{f}\\) weren't \\(\omega\\)-continuous, there would be a witness sequence \\((a_i)\\) which had \\(\lim \bar{f}(a_i) \not = \bar{f}(\lim a_i)\\); plugging this into the definition of \\(\bar{f}\\) gives that \\((a_i)\\) is a witness against the \\(\omega\\)-continuity of \\(f\\). Contradiction.
+
+And now for something completely different: an exponential with more structure than previously. I just check the definition of the product graph, because I don't think we had it in our Graph Theory course; it seems to be the obvious one, taking pairs of vertices and corresponding pairs of edges. Then the exponential graph. This is defined as to have vertices "set-exponential of the vertices", and an edge between \\(\phi: G \to H\\), \\(\psi: G \to H\\) is a \\(e(G)\\)-indexed collection of edges in \\(H\\) which have "the source is where \\(\phi\\) takes the corresponding \\(G\\)-source" and "the target is where \\(\psi\\) takes the corresponding \\(G\\)-target". It's a way of embedding \\(G\\) into \\(H\\) along \\(\phi\\) and \\(\psi\\).
+
+The evaluation is the obvious one given those structures, and the transpose of a map is the curried version of that map. The different thing about this system is the fact that our maps have to have two parts (one for vertices and one for edges).
+
+"Basic facts about exponentials". The transpose of evaluation, without looking at the rest of the page: \\(\epsilon: B^A \times A \to B\\) must transpose to \\(\bar{\epsilon}: B^A \to B^A\\) with \\(\epsilon \circ (\bar{\epsilon} \times 1_{A}) = \epsilon\\). If \\(\epsilon\\) were monic, we could say that \\(\bar{\epsilon} = 1_{B^A}\\) immediately, but it's not monic. Ah, but we do have that \\(\bar{\epsilon}\\) is uniquely specified by the UMP, so it must be \\(1_{B^A}\\) after all. Maybe that'll help me remember things, if nothing else.
+
+A proof that "exponentiation by a fixed object" is a functor: it starts in Set, which makes me worry that representable functors are going to be involved again (because we seem to be able to cast many things as Set-based things). Onwards: currying is certainly functorial in Set because application of functions is associative, and because we check that the identity curries in the right way.
+
+In general, the definition of the exponential of an arrow \\(\beta: B \to C\\) is the obvious one: there's only one way to make an element of \\(C^A\\) given one in \\(B^A\\) and a map \\(\beta : B \to C\\), and that's to "evaluate at \\(a\\), then do \\(\beta\\)". This method does keep the identity map as an identity: \\(1: B \to B\\) causes \\(f: A \to B\\) to become \\(f: A \to B\\), of course. It respects composition by just writing a couple of lines of symbol-manipulation.
+
+Finally, the transpose of \\(1_{A \times B}\\), which is a map \\(\eta: A \to (A \times B)^B\\). This takes a value \\(a\\) and returns a function \\(b \mapsto (a, b)\\). Then some symbol shunting gives \\(\bar{f} = f^A \circ \eta\\).
+
+![Calculating the exponential][exponential]
+
+# Summary
+
+This section is the one I've thought most concretely about so far. That's probably something I'll have to unlearn. It's useful already being familiar with currying; this chapter would have been a lot harder without already having that intuition.
+
+[exponential]: {{< baseurl >}}images/CategoryTheorySketches/ExponentialEvaluation.jpg
diff --git a/hugo/content/awodey/2015-09-29-limit-exercises.md b/hugo/content/awodey/2015-09-29-limit-exercises.md
new file mode 100644
index 0000000..dfdea99
--- /dev/null
+++ b/hugo/content/awodey/2015-09-29-limit-exercises.md
@@ -0,0 +1,83 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-29T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/limit-exercises/
+- /limit-exercises/
+title: Limits exercises
+---
+
+These are located on pages 114 through 118 of Awodey.
+
+Exercise 1 follows by just drawing out the diagrams for the product and the pullback: they end up being the same diagram and the same UMP.
+
+Exercise 2 a): \\(m\\) is monic iff \\(mx = my \Rightarrow x=y\\); the diagram is a pullback iff for all \\(x: A \to M\\) and \\(y: A \to M\\) with \\(m x = m y\\), have \\(z: A \to M\\) such that \\(my = mz = mz = mx\\).
+
+Exercise 2 b): We can draw the cube line by line, checking that each pullback arrow exists, and ending up with a diagram.
+
+![Pullback cube][pullback]
+
+We still need the pullback of the pullback square to be a pullback square. If we can prove that \\(P\\) forms a pullback of \\(f \circ f^{-1}(\alpha), \beta\\) then we're done by the two-pullbacks lemma using the square with downward-arrow \\(f^{-1}(\beta): f^{-1}(B) \to Y\\). But it is: if we pull back the "diagonal square" \\(A \times_X B \to X\\) and \\(f\\), then we do get \\(P\\), and so all the commutative properties hold.
+
+Exercise 2 c): this follows by drawing out the diagram. We pull back the "\\(m\\) is monic" square along \\(f\\) to obtain the "\\(m'\\) is monic" square; this is a pullback because of the "\\(f, m\\) pull back to \\(m'\\)" square.
+
+Exercise 3: Let \\(x', y': R \to M'\\) with \\(m' x' = m' y'\\). Then \\(f m' x' = f m' y'\\); while labelling the unlabelled arrow in Awodey's diagram \\(\alpha\\), have \\(m \alpha x' = m \alpha y'\\) because the diagram commutes. But by monicness of \\(m\\), have \\(\alpha x' = \alpha y'\\). By the UMP of the pullback, there is a unique arrow \\(r\\) which arises \\(R \to M\\) such that \\(\alpha r = \alpha x'\\) and \\(m' r = m' x'\\), and so \\(r=x\\). Likewise \\(r=y\\) (since \\(\alpha y' = \alpha x'\\) and \\(m' x' = m' y'\\)). Hence \\(x=y\\).
+
+Exercise 4: One direction is easy. Suppose \\(z \in_A M \Rightarrow z \in_A N\\). Let \\(z = m: M \to A\\). Then \\(M \in_A N\\) so \\(M \subseteq N\\).
+
+Conversely, suppose \\(M \subseteq N\\) by means of \\(f: M \to N\\), and \\(z: Z \to A\\) gives \\(Z \in_A M\\). Then \\(z\\) lifts to \\(fz: Z \to N\\), and the entire diagram commutes as required.
+
+Exercise 5 is apparently a duplicate of Exercise 4.
+
+Exercise 6 is very similar in shape to some things we've already proved. Let \\(z: Z \to A\\) be such that \\(fz = gz\\). We need to find \\(\bar{z}: Z \to E\\) such that \\(e \bar{z} = z\\). Since \\(fz = gz\\), the arrow \\(Z \to B \times B\\) by \\(\langle f, g \rangle \circ z\\) is equal to the arrow \\(Z \to B \times B\\) given by \\(\langle 1_B, 1_B \rangle \circ f \circ z\\); so by the UMP of the pullback, there is \\(\bar{z}: Z \to E\\) with \\(e\bar{z} = z\\). That's all we needed.
+
+Exercise 7: we need to show that \\(\text{Hom}_{\mathbf{C}}(C, L)\\) is a limit for \\(\text{Hom}_{\mathbf{C}}(C, \cdot) \circ D = \text{Hom}_{\mathbf{C}}(C, D): \mathbf{J} \to \mathbf{Sets}\\). Equivalently, we need to show that the representable functor preserves products and equalisers, so let \\(p_1: P \to A, p_2: P \to B\\) be a product in \\(\mathbf{C}\\). I claim that \\(p_1' : \text{Hom}_{\mathbf{C}}(C, P) \to \text{Hom}_{\mathbf{C}}(C, A)\\) by \\(p_1': f \mapsto p_1 f\\), and likewise \\(p_2': f \mapsto p_2 f\\), form a product. Indeed, let \\(x_1: X \to \text{Hom}_{\mathbf{C}}(C, A)\\) and \\(x_2: X \to \text{Hom}_{\mathbf{C}}(C, B)\\). Then \\(\langle x_1(z), x_2(z) \rangle\\) is of the form \\(\langle C \to A, C \to B \rangle\\) for all \\(z \in X\\), so there is a unique corresponding \\(C \to P\\) for each \\(z \in X\\). This therefore constructs a product.
+
+Now the equalisers part. Let \\(e: E \to A\\) equalise \\(f, g: A \to B\\), and write \\(f^*, g^*\\) for the images of \\(f, g\\) under the representable functor. Let \\(x: X \to \text{Hom}_{\mathbf{C}}(C, A)\\) be such that \\(f^* x = g^* x\\). We need to lift \\(x\\) over \\(e^*\\). For each \\(z \in X\\), we have \\(x(z): C \to A\\) an arrow in \\(\mathbf{C}\\); this has \\(f \circ x(z) = g \circ x(z)\\), so \\(x(z)\\) lifts to unique \\(\overline{x(z)}: C \to E\\). This specifies a unique morphism \\(X \to \text{Hom}_{\mathbf{C}}(C, E)\\) as required.
+
+Exercise 8: It seems intuitive that partial maps should define a category. However, let's go for it. There is an identity arrow - namely, the pair \\((\vert id_A \vert, A)\\). This does behave as the identity, because the pullback of the identity with anything gives that anything. The composition of arrows is evidently an arrow (because the composition of monos is monic). We just need associativity of composition, which comes out of drawing the diagrams of what happens when we do the triple composition in the two available ways. We can complete each of the two diagrams using the two pullbacks lemma, as in the picture.
+
+![Partial maps associative][partial]
+
+The map \\(\mathbf{C} \to \mathbf{Par}(\mathbf{C})\\) given by \\((f: A \to B) \mapsto (\vert f \vert, A)\\) is a functor: it respects the identity arrow by inspection, while composition is respected by just looking at the diagram. It is clearly the identity on objects, by definition of the partial-maps category.
+
+![Partial maps functor is a functor][Partial maps functor]
+
+Exercise 9: Diagrams is a category: identity arrows are just identity arrows from the parent category; the composition of commutative squares is itself a commutative square (well, rectangle); composing with the identity arrow doesn't change anything. Taking the vertex objects of limits does determine a functor: it takes the identity arrows to identity arrows because taking a diagram to itself means taking its unique limit vertex to itself. It respects domains/codomains, because… well, it just does: if \\(f: D_1 \to D_2\\) in Diagrams, then \\(\lim f\\) is uniquely specified to go from limit-vertex 1 to limit-vertex 2. (By the way, the intuition for what an arrow in this category is, is the placing of one diagram above another with linking arrows between the objects.) Better justification: there is a unique morphism between the limit vertices, because we can use the arrow to determine a collection of morphisms from one limit vertex to the other making \\(D_1\\) into a cone for \\(D_2\\).
+
+The last part follows because \\(\mathbf{Diagrams}(I, \mathbf{Sets})\\) is isomorphic to \\(\mathbf{Sets}^I\\). Sets has all limits, so the theorem holds, and hence there is a product functor. This seems a little nonrigorous, but I can't put my finger on why.
+
+Exercise 10: we've already seen this. I'll state it anyway. The copullback of arrows \\(f: A \to B\\) and \\(g: A \to C\\) is the universal \\(P\\) and arrows \\(p_1: B \to P, p_2: C \to P\\) such that for any \\(b: B \to Z, c: C \to Z\\) with \\(cg = bf\\), there is a unique \\(p: P \to Z\\) with \\(p p_1 = b, p p_2 = c\\), as in the diagram.
+
+![Definition of a pushout][pushout]
+
+The construction of a pushout with coequalisers and coproducts is done by taking the coproduct of \\(B\\) and \\(C\\), and coequalising the two sides of the square.
+
+Exercise 11: To show that the diagram is an equaliser, we need to show that any \\(z: Z \to \mathbb{P}(X)\\), which causes the two \\(\mathbb{P}(r_i): \mathbb{P}(X) \to \mathbb{P}(R)\\) to be equal, factors through \\(\mathbb{P}(q): \mathbb{P}(Q) \to \mathbb{P}(X)\\). Any \\(z: Z \to \mathbb{P}(X)\\) is a selection of subsets of \\(X\\) for each element of \\(Z\\); the condition that it equalises \\(\mathbb{P}(r_1), \mathbb{P}(r_2)\\) is exactly the same as saying that if we take the \\(r_1\\)-inverse image and the \\(r_2\\)-inverse image of the result, then we get the same subset of \\(R\\). Can we make it assign an indicator function on \\(Q\\)? We're going to have to prove that \\(z: Z \to \mathbb{P}(X)\\) maps only into unions of equivalence classes, and then the map will descend. 
+
+OK, we have "for each element of \\(Z\\), we pick out a subset of \\(X\\) which has the property that finding everything which that subset twiddles on the left, we get the same set as everything which that subset twiddles on the right". Suppose an element \\(a\\) is in the image of \\(z \in Z\\). Then we must have the entire equivalence class of \\(a\\) in the image set, because \\(\mathbb{P}(r_1)(\{ a \}) = \{ (a, x) \mid x \sim a \}\\) but \\(\mathbb{P}(r_2)(\{ a \}) = \{ (x, a) \mid x \sim a \}\\). These can't be equal unless the only thing in the equivalence class is \\(a\\). The reasoning generalises for when more than one thing is in the image set, by taking appropriate unions. Therefore the map does descend.
+
+Exercise 12: the limit is such that for any cone, there is a unique way to factor the cone through the limit. What is a cone? It's a way of identifying a subshape of every element of the sequence, such that all other subshapes also appear in this limit subshape. But the only shape in \\([0]\\) is \\([0]\\), so the limit must be isomorphic to \\([0]\\).
+
+The colimit must be \\(\omega\\). Indeed, a cocone is precisely an identification of a subset which contains an \\(\omega\\)-wellordered subset, and the colimit is the smallest \\(\omega\\)-well-ordered subset.
+
+Exercise 13 a): The limit of \\(M_0 \to M_1 \to \dots\\) is just \\(M_0\\) - same reasoning as Exercise 12 - so it's an abelian group. It seems like the colimit should also be abelian. Let \\(C\\) be the colimit, and let \\(x, y \in C\\). I claim that there is some \\(n\\) such that \\(x, y \in M_n\\), whence we're done because \\(M_n\\) is abelian. (Strictly, I claim that there is \\(n\\) and \\(\alpha, \beta\\) such that \\(i_n(\alpha) = x, i_n(\beta) = y\\), where \\(i_n\\) is the inclusion.) It's enough to show that there is \\(m\\) and \\(n\\) such that \\(x \in M_m, y \in M_n\\), because then the maximum of \\(m, n\\) would do. If there weren't such an \\(m\\) for \\(x\\), we could take the cocone \\(C \setminus \{ x \}\\), and this would fail to factor through \\(C\\).
+
+I then had a clever but sadly bogus idea: the second diagram is the same as the first but in the opposite category. Therefore by duality, we have that colimits <-> limits, so the limits and colimits are indeed abelian. This is bogus because the opposite category of Monoids is not Monoids, so we're not working the right category any more.
+
+Let's go back to the beginning. The colimit is \\(N_0\\) by the same reasoning that made the limit of the \\(M_i\\) sequence be \\(M_0\\). That means it's an abelian group. Taking the limit of \\(N_0 \gets N_1 \gets \dots\\): our limit is a shape \\(L\\) which is in \\(N_0\\), which is itself an image of \\(N_1\\), which… This is a kind of generalised intersection, and the (infinite) intersection of abelian groups is an abelian group, so the intuition should be that the limit is also an abelian group.
+
+Some on Stack Exchange [gave a cunning way to continue][SE], considering involutions \\(x \mapsto x^{-1}\\). I don't know if I'd ever have come up with that.
+
+Exercise 13 b): now they are all finite groups. The limit of the \\(M_i\\) is \\(M_0\\), so this certainly has the "all elements have orders" property. The colimit of the \\(N_i\\) is \\(N_0\\), so likewise. The colimit \\(M\\) of the \\(M_i\\): every element \\(x\\) appears in some \\(M_x\\) (and all later ones) as above, and it must have an order in those groups, so it has an order in \\(M\\) too (indeed, each \\(M_i\\) is a subgroup of \\(M\\)). The limit of the \\(N_i\\): what about \\(C_2 \gets C_2^2 \gets C_2^3 \gets \dots\\), each arrow being the quotient by the first coordinate? No, the limit of that is \\(C_2^{\mathbb{N}}\\) in which every element has order 2. If we use \\(C_{n!}\\) instead? Ugh, I'm confused. I'll leave this for the moment and try to press on. If it becomes vital to understand limits in great detail in the time left before my course starts, I'll come back to this.
+
+[pullback]: /images/CategoryTheorySketches/PullbackCube.jpg
+[Partial maps functor]: /images/CategoryTheorySketches/PartialMapsFunctor.jpg
+[partial]: /images/CategoryTheorySketches/PartialMapAssociative.jpg
+[pushout]: /images/CategoryTheorySketches/PushoutDefinition.jpg
+[SE]: https://math.stackexchange.com/a/1454266/259262
diff --git a/hugo/content/awodey/2015-09-30-heyting-algebras.md b/hugo/content/awodey/2015-09-30-heyting-algebras.md
new file mode 100644
index 0000000..18b203d
--- /dev/null
+++ b/hugo/content/awodey/2015-09-30-heyting-algebras.md
@@ -0,0 +1,49 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-09-30T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/heyting-algebras/
+- /heyting-algebras/
+title: Heyting algebras
+---
+
+Now that we've had the definition of an exponential, we move on to the Heyting algebra, pages 129 through 131 of Awodey. This is still in the "exponentials" chapter. I stop shortly after the definition of a Heyting algebra, so as to move on to the more general stuff which is more relevant to the Part III course.
+
+The first thing to come is the definition of an exponential \\(b^a\\) in a Boolean algebra \\(B\\) (regarded as a poset category). Without looking at the definition, I draw out a picture. We need to find \\(c^b\\) and \\(\epsilon: c^b \times b \to c\\) such that for all \\(f: a \times b \to c\\) there is \\(\bar{f}: a \to c^b\\) unique with \\(\epsilon \circ (\bar{f} \times 1_b) = f\\).
+
+The first thing to note is that arrows are already unique if they exist, because we are in a poset category, so we don't have to worry about uniqueness of \\(\bar{f}\\). Then note that \\(f: a \times b \to c\\) is nothing more nor less than the statement that \\(a \times b \leq c\\) - that is, that the greatest lower bound of \\(a\\) and \\(b\\) is \\(\leq c\\), or that \\(c\\) is not a lower bound for both \\(a\\) and \\(b\\) simultaneously (assuming \\(a \times b \not = c\\)). The definition of \\(\bar{f}\\) is precisely the statement that \\(a \leq c^b\\), and \\(\epsilon\\) says precisely that the GLB of \\(c^b\\) and \\(b\\) is \\(\leq c\\).
+
+In order to piece this together, we're going to want to know what the product of two arrows looks like. We're in a poset category, so it comes from "propagating the two arrows downwards until they hit a common basepoint, and taking that arrow": it is the arrow between the GLB of the domains and the GLB of the codomains. Therefore the product arrow \\(\bar{f} \times 1_B\\) is the arrow between the GLB of \\(a, b\\) and the GLB of \\(c^b, b\\).
+
+![Product of arrows][arrow product]
+
+Therefore the following picture is justified.
+
+![Exponential in boolean category][exponential]
+
+What could \\(c^b\\) be? If we let \\(f\\) be the arrow \\(\text{GLB}(c^b, b) \to c^b\\), then \\(\bar{f} = f\\), and \\(\bar{f} \times 1_b\\) is the identity arrow on that GLB. I don't know if this is helping, and I'm forced to look at the book.
+
+The book gives \\(c^b\\) as \\((\neg b \vee c)\\), the LUB of \\(\neg b\\) and \\(c\\). This certainly does have an appropriate evaluation arrow and it is an exponential (having worked through the lines in the book), but I really don't see how one could have come up with that.
+
+A Heyting algebra has finite intersections, unions and exponentials (where \\(a \Rightarrow b\\) is defined such that \\(x \leq (a \Rightarrow b)\\) iff \\((x \wedge a) \leq b\\)). What does this exponential really mean? In a Boolean algebra, it's an object which has as its subsets precisely those things which intersect with \\(a\\) to give a subset of \\(b\\). I can draw that in terms of a Venn diagram.
+
+The distributive property holds, as I write out myself given the first line.
+
+Now the definition of a complete poset (which I already know as "all subsets have a least upper bound"). Why is completeness equivalent to cocompleteness? In a Boolean algebra, this is easy because "join" is "complement-meet-complement". Actually, I'm now a bit confused: \\(\omega\\), the first infinite well-ordering, is not complete as a poset, but it certainly looks cocomplete. I check the definition of "complete" again to see if I'm going mad, and I see that it's "all limits exist", not just "\\(\omega\\)-limits exist". But then why does the book say "a poset is complete if it is so as a category - that is, if it has all set-indexed meets"? OK, \\(\omega\\) has a meet - namely \\(0\\) - but for it to have a join, we need \\(a \in \omega\\) such that for any \\(c \in \omega\\), have all elements of \\(\omega\\) are \\(\leq c\\) iff \\(a \leq c\\). Since \\(c+1 \not \leq c\\), we must have \\(a \not \leq c\\): that is, \\(a\\) is bigger than all members of \\(\omega\\). Therefore \\(\omega \subseteq \omega\\) doesn't have a join. Can we find a corresponding subset of \\(\omega\\) without a meet? No: the meet of any subset of a well-ordered set is just the least element. I'm horribly confused, so I've asked on [Stack Exchange]; the reply came that the corresponding meetless subset is \\(\emptyset\\), which I forgot to consider.
+
+OK, let's try again. Suppose our poset has a meetless subset \\((a_i)\\) - that is, one which doesn't have a greatest lower bound. Remember, our poset might not have a terminal object, so actually we might have to change this into a proof by contradiction rather than contrapositive: let's assume all subsets have joins, so in particular there is a terminal object (the empty join). I would love to say "Then the corresponding complement of \\(\{ a_i \}\\) has no join, because its least upper bound is a greatest lower bound for \\(\{ a_i \}\\)", but \\(\{ 1 \} \subset \omega\\) has \\(1\\) as its LUB, but its complement has \\(0\\) as its GLB. However, what I could say is "Let \\(\{ b_i \}\\) be the set of elements which are less than every element of \\(a_i\\). This doesn't have a least upper bound, because that would be a GLB of \\(a_i\\)." That's better.
+
+The power set algebra is certainly a complete Heyting algebra, as I mentioned above with the Venn diagram, or by Awodey's reasoning with the distributive law. The statement that Heyting algebras correspond to intuitionistic propositional calculus (where excluded middle may not apply) is pretty neat, but I'm afraid I'm still a bit lost.
+
+The next section is on propositional calculus, where Awodey provides a set of axioms for intuitionistic logic.
+
+At this point, I was told that exponentials don't really turn up in the Part III course, and since my aim here is to get an advantage in terms of the course, I'm skipping to the next chapter.
+
+[arrow product]: {{< baseurl >}}images/CategoryTheorySketches/ArrowProduct.jpg
+[exponential]: {{< baseurl >}}images/CategoryTheorySketches/ExponentialInBooleanAlgebra.jpg
+[Stack Exchange]: http://math.stackexchange.com/q/1459373/259262
diff --git a/hugo/content/comparisons/index.md b/hugo/content/comparisons/index.md
new file mode 100644
index 0000000..4d53802
--- /dev/null
+++ b/hugo/content/comparisons/index.md
@@ -0,0 +1,82 @@
+---
+lastmod: "2021-05-21T18:23:48.0000000+01:00"
+title: Product comparisons
+author: patrick
+layout: page
+---
+
+This page is an ongoing history of product comparisons, made informally but blindly.
+
+# Fig rolls
+
+Compared Boland with McVities, 2022-04-18.
+It was pretty obvious to me in the blind testing that the McVities ones were the big-brand ones, but Boland were much nicer.
+Tasted more of fig, and had less pastry around them.
+
+# Miso soup
+
+I'm still on a quest for the perfect miso soup; I remember one that was really great from my childhood, and I don't know which it was.
+
+Clearspring Organic Japanese Brown White miso soup is really insipid.
+Their "hearty red" tastes dark and kind of a bit unpleasant to me.
+Their "mellow white with tofu and green onions" seemed oddly vegetal on the first tasting, but on re-tasting seems good. One to compare with the others. Not mind-blowing.
+Their "instant miso soup with sea vegetables" is a bit darker than the "mellow white", also perfectly adequate; should taste-test against the others. Not mind-blowing.
+
+## Clearspring Organic White Miso Instant Soup Paste vs Itsu Miso'easy Traditional Miso
+
+These were quite different from each other.
+Itsu was powerful; Clearspring was subtle.
+Initially I thought that Itsu won by a country mile, but actually I could imagine a mood in which Clearspring wins; it's not totally clear cut.
+Itsu is still the victor.
+
+## Yutaka
+
+This was actually pretty good, though I didn't have it in a comparison with any other.
+I'll buy this again.
+
+# Houmous
+
+A standard useful thing to have around when you need calories.
+
+If you want to push the boat out, Natoora Spring Herb Houmous is very nice but rather expensive.
+
+## Ocado own-brand vs Tesco own-brand
+
+There's no contest here.
+Ocado tastes of tahini; Tesco is just a bit insipid.
+Ocado is the clear winner.
+
+# Baked beans
+
+Heinz beats Sainsbury's own brand by a country mile.
+It's not remotely close.
+Sainsbury's is sickly-sweet.
+
+Marks and Spencer beans are perfectly fine. I didn't try them side-by-side with Heinz, but I'd say they're just as good.
+
+# Meat replacements
+
+Beyond Burgers are pretty good, honestly, although they are much better when cooked correctly (i.e not overdone).
+Honest Burger on Tottenham Court Road did them well; Neat Burger also do them well.
+
+I was unable to tell the difference between Impossible (in Boston, 2020) and real meat.
+However, according to the person I was with, it was extremely obvious which was which.
+So your mileage probably will vary.
+
+# Tinned tuna
+
+Sainsbury's Taste the Difference Albacore tuna is really good: meaty chunks, not flakes.
+Similarly Waitrose own-brand tuna.
+
+Princes is clearly cheap and flaky.
+
+Marks and Spencer Tuna Steaks are somewhere in between - not as good as Waitrose or the Taste the Difference Sainsbury's, but much better than Princes.
+
+# Nutella
+
+The "cocoa" one is much less sweet, though I'm not sure I'd call the taste "chocolate" - it just tastes less nutty.
+I think I prefer the normal one.
+
+# Tinned tomatoes
+
+Marks and Spencer own-brand are, bizarrely, not very full of tomatoes. I tried making the Dishoom Chicken Ruby recipe with them and it took an extra tin again from when I made the same thing with Waitrose own-brand.
diff --git a/hugo/content/films/index.md b/hugo/content/films/index.md
new file mode 100644
index 0000000..0bf4aed
--- /dev/null
+++ b/hugo/content/films/index.md
@@ -0,0 +1,130 @@
+---
+lastmod: "2023-09-02T13:30:00.0000000+01:00"
+title: Films
+author: patrick
+layout: page
+---
+
+This page holds a list of films I have watched, spoiler-free, starting from 9th January 2015.
+
+*  [A Haunting in Venice](https://www.imdb.com/title/tt22687790/): Well, I really enjoyed this, and I think I was surrounded by heathens in the cinema. I successfully called precisely none of the plot, and it all tied up so neatly. Ariadne Oliver will always be Zoë Wanamaker to me, but I believed Kenneth Branagh. Top-tier Poirot.
+
+*  [Oppenheimer](https://en.wikipedia.org/wiki/Oppenheimer_(film)): Brilliant. It was a little too long, but I couldn't pick out anything to take away. Great acting, great filming, a bit harrowing.
+
+*  [Barbie](https://en.wikipedia.org/wiki/Barbie_(film)): Pretty good. I did feel a bit like I was having messages shoved down my throat - so many different messages! - the whole way through. Enjoyable watch, though.
+
+*  [Man of Steel]: Not much to say here, really. A film with no apparent merit at all, and so long, too! There was one single cute moment, where Superman gets thrown into a sign saying "106 Days Without an Accident", knocking off the digits 1 and 6 to produce "0 Days Without an Accident" as he destroys the building site.
+
+*  [Pride and Prejudice and Zombies]: We watched this because we wanted a zom-rom-com. While perhaps not strictly a rom-com, it did not disappoint. Funny throughout. While I can never tell the Bennets apart at the best of times, it doesn't really matter.
+
+  * [Bohemian Rhapsody]: I was glued to this the whole way through. It was touching in all the right places, heroic in all the right places, and generally just a great introduction to some history I actually knew surprisingly little about. Would watch again.
+
+  * [Good Omens][Good Omens Wikipedia]: The book is laugh-out-loud funny, and the series (written by Neil Gaiman) did not disappoint. Very true to the book, many excellent actors acting excellently.
+
+  * [Star Wars: The Last Jedi][TLJ]: I know this has sparked some mixed reviews, but I absolutely loved this. There is a particular scene (spoiler-free words to identify the scene: light speed, silence, escape) which was an absolute treat, but every scene with a lightsaber or with Rey experiencing the Force is great. I could watch this again and again.
+
+  * [Arrival][Arrival IMDB]: I loved this. It felt like it was trying to be cerebral, and it largely succeeded. The end dragged on a bit, and there was a little bit of rabbit-out-of-a-hattery, but some great ideas. Could have done with a bit more exploring of the language.
+
+  * Netflix's House of Cards (not technically a film but a TV series): I loved the whole series. I really understand the motivation of most of the characters; they are unabashedly evil but I can see nuanced reasons for *why* they're evil. I described this series as "the Count of Monte Cristo, but more so", which I think is fair.
+
+  * [TRON: Legacy][Tron Legacy IMDB]: Ehhhh. Flashy graphics, minimal and predictable plot. Computers! Technology! Quantum teleportation and genetic algorithms! Perhaps I'm just getting cynical in my old age. At least it's better than Hackers (below).
+
+  * [Suicide Squad][Suicide Squad IMDB]: This was mildly amusing, though the plot was a bit simple and predictable. With some minor changes, it could have been a much more interesting film. Flashy, anyway.
+
+  * [Metropolis][Metropolis IMDB]: I can see why this film is considered iconic. Its depiction of technology must be well ahead of its time, and I think it's a good film. It does an excellent job of displaying peril and similar. The problem is that it's very long for my modern-day attention span.
+
+  * [Ender's Game][Ender's Game IMDB]: Ehhhh. It was watchable. The book was substantially better, although the book wasn't any better than "moderately good". Don't bother with this film.
+
+  * [Eddie the Eagle][Eddie the Eagle IMDB]: I read a review of this film, calling it "saccharine". While everyone else seemed to love it, I thought it was watchable but too sickly.
+
+  * [Donnie Darko][Donnie Darko IMDB]: Given that I'd heard Primer was "Donnie Darko for grown-ups", I was expecting a bit more from this film. It was pretty good anyway, but somewhat less satisfying than I'd hoped. A watchable sort-of-horror film.
+
+  * [Children of Men][Children of Men IMDB]: The premise of this film is very promising, but to be honest, it was mainly just used to set up a fairly run-of-the-mill action movie. Not a bad action movie, but there are certainly better ones (such as any of the Bourne series).
+
+  * [Hackers][Hackers IMDB]: Good grief. This is literally the worst film I have ever seen. Apparently this was Angelina Jolie's first film. I'm shocked her career took off, after this. Such realistic. Many technical. Very hackers. Wow.
+
+  * [V for Vendetta][V IMDB]: Wow. This is possibly my favourite film ever. All the Alan Moore-related films I've seen have been excellent, but this is even better than Watchmen, and I was not expecting any film to achieve that.
+
+  * [Memento][Memento IMDB]: I loved this film. It's a kind of thriller, about someone who has no ability to lay down new memories. I don't want to say much more about it, for spoiler purposes, but it was excellent. Contains several members of the cast of The Matrix.
+
+  * [Life of Pi][Life of Pi IMDB]: a very pretty film, with some gently funny bits, but if I'd paid to see it in a cinema I think I'd have been a bit disappointed. A film to go to sleep to, as a friend described it.
+
+  * [Star Wars VII][Star Wars VII IMDB]: this is one of two films I've ever seen after which the audience applauded; the other was [The King's Speech][King's Speech IMDB]. Good fun, and not the disappointment I was expecting.
+
+*  [Airplane!][Airplane IMDB]: brilliant and extremely random comedy. Many very quotable lines.
+
+*  [Some Like It Hot][SLIH IMDB]: one of the funniest sort-of-rom-coms ever made. Falls into the "we watch this when ill". It contains a cross-dressing Tony Curtis. What more is there to want from a film?
+
+*  [My Favourite Wife][My Favourite Wife IMDB]: another happy Cary Grant film. Not much more to say, really. Great fun. A bit less funny than Bringing Up Baby (below).
+
+*  [Bringing Up Baby][Bringing Up Baby IMDB]: very laugh-out-loud comedy with Cary Grant and Katharine Hepburn. It's one of the films my family watches if any of us is ill - quiet but hilarious.
+
+*  [The Importance of Being Earnest][Earnest IMDB]: extremely funny, very British, and has all the right people in it. Great film for cheering-up.
+
+*  [The Lion King][Lion King IMDB]: Good heavens, this was boring. Scar was the only remotely good character, and he got nowhere near enough screen time. Otherwise, just a deathly dull film.
+
+* [Inside Out][Inside Out IMDB]: sweet adventure film centred on the personified emotions of a girl who is undergoing a major life event. A [feels]y film; nearly everyone I know loved it, and I certainly did.
+
+* [Mad Max: Fury Road][Mad Max IMDB]: mildly entertaining action/post-apocalyptic film, which was basically just one very long action scene. I could have done with about an hour less of this film.
+
+* [Avengers: Age of Ultron][Ultron IMDB]: flashy and pretty fun action film, lots of nods to earlier films in the series, but a bit less plotty than I'd hoped. Even the usual suspension of disbelief wasn't quite enough. You get a bit tired of whizz-bang effects after having them nonstop for two hours.
+
+* [Kingsman][Kingsman IMDB]: very fun action film which I'd happily see again. Colin Firth is particularly good in his role.
+
+*  [The Borrowers][Borrowers IMDB]: I loved the book when I was much smaller, but sadly even the presence of Stephen Fry and Victoria Wood isn't enough to make up for the mess they made of the plot. Arriety was made into a complete idiot who did pretty much exactly the wrong thing at every turn.
+
+*  [Interstellar][Interstellar IMDB]: as of this writing, [my favourite film][Interstellar review] in its genre (disaster movie with space/time shenanigans, I suppose).
+
+*  [Inception][Inception IMDB]: kind of predictable, and while people said things like "it's a game of chess" and "mind-bending", it really wasn't. Not a bad film per se, but Predestination and Primer (below) are just better.
+
+*  [Predestination][Predestination IMDB]: I called the plot from about halfway through, but lots of other people said they didn't and it was all quite mysterious to them. Great film for those who want a more interesting Inception.
+
+*  [Primer][Primer IMDB]: mind-boggling time-travel film whose events I still don't properly understand. One of the most cerebral films I've ever seen.
+
+*  [Now You See Me][NYSM IMDB]: magic-tricks/heist film which really didn't live up to its trailer. I predicted essentially the entire plot from about ten minutes in, and the film was miles too long for the amount of interesting stuff which happened in it.
+
+*  [The Illusionist][Illusionist IMDB]: film with magic and political intrigue, which I really enjoyed.
+
+*  [Limitless][Limitless IMDB]: fun, if non-cerebral, film about someone who becomes much more intelligent than normal. Much better than Now You See Me, which I think was aiming for the same general effect.
+
+[Tron Legacy IMDB]: https://www.imdb.com/title/tt1104001
+[Metropolis IMDB]: https://www.imdb.com/title/tt0017136
+[Suicide Squad IMDB]: https://www.imdb.com/title/tt1386697/
+[Hackers IMDB]: https://www.imdb.com/title/tt0113243/
+[V IMDB]: https://www.imdb.com/title/tt0434409/
+[Memento IMDB]: https://www.imdb.com/title/tt0209144/
+[Interstellar IMDB]: https://www.imdb.com/title/tt0816692/
+[Predestination IMDB]: https://www.imdb.com/title/tt2397535/
+[Primer IMDB]: https://www.imdb.com/title/tt0390384
+[Inception IMDB]: https://www.imdb.com/title/tt1375666/
+[NYSM IMDB]: https://www.imdb.com/title/tt1670345/
+[Illusionist IMDB]: https://www.imdb.com/title/tt0443543/
+[Limitless IMDB]: https://www.imdb.com/title/tt1219289/
+[Borrowers IMDB]: https://www.imdb.com/title/tt1975269/
+[Kingsman IMDB]: https://www.imdb.com/title/tt2802144/
+[Ultron IMDB]: https://www.imdb.com/title/tt2395427/
+[Inside Out IMDB]: https://www.imdb.com/title/tt2096673/
+[Mad Max IMDB]: https://www.imdb.com/title/tt1392190/
+[Lion King IMDB]: https://www.imdb.com/title/tt0110357
+[Earnest IMDB]: https://www.imdb.com/title/tt0278500
+[Bringing Up Baby IMDB]: https://www.imdb.com/title/tt0029947/
+[My Favourite Wife IMDB]: https://www.imdb.com/title/tt0029284/
+[SLIH IMDB]: https://www.imdb.com/title/tt0053291/
+[Airplane IMDB]: https://www.imdb.com/title/tt0080339/
+[Star Wars VII IMDB]: https://www.imdb.com/title/tt2488496/
+[King's Speech IMDB]: https://www.imdb.com/title/tt1504320
+[Life of Pi IMDB]: https://www.imdb.com/title/tt0454876/
+[Donnie Darko IMDB]: https://www.imdb.com/title/tt0246578
+[Children of Men IMDB]: https://www.imdb.com/title/tt0206634
+[Eddie the Eagle IMDB]: https://www.imdb.com/title/tt1083452
+[Ender's Game IMDB]: https://www.imdb.com/title/tt1731141/
+[Arrival IMDB]: https://www.imdb.com/title/tt2543164/
+[TLJ]: https://www.imdb.com/title/tt2527336/
+[Good Omens Wikipedia]: https://en.wikipedia.org/wiki/Good_Omens_(TV_series)
+[Bohemian Rhapsody]: https://en.wikipedia.org/wiki/Bohemian_Rhapsody_(film)
+[Pride and Prejudice and Zombies]: https://www.imdb.com/title/tt1374989/
+[Man of Steel]: https://www.imdb.com/title/tt0770828/
+
+[Interstellar review]: {{< ref "2014-12-09-film-recommendation-interstellar" >}}
+
+[feels]: https://knowyourmeme.com/memes/feels
diff --git a/hugo/content/lifehacks/index.md b/hugo/content/lifehacks/index.md
new file mode 100755
index 0000000..4cbf06a
--- /dev/null
+++ b/hugo/content/lifehacks/index.md
@@ -0,0 +1,29 @@
+---
+lastmod: "2022-12-31T23:21:44.0000000+00:00"
+title: Lifehacks
+author: patrick
+layout: page
+---
+If I ever become rich and famous, I'm sure I'll be besieged with requests for "how to do better in life". I hereby head such requests off at the pass, by providing a list of [lifehacks] I am either using or considering the use of.
+
+*   For learning smallish but numerous facts (such as a list of theorems), I use [Anki], which is a [spaced-repetition] learning system, allowing you to enter flashcards and have them shown to you regularly. The time between repetitions of a certain flashcard changes, depending on how well you've been doing on that flashcard - so marking your performance on a particular card as "easy" rather than "hard" tells Anki that you don't want to see that card for a while. It's a bit like the antithesis of cramming, where you see the material exactly once and use it a short time later; Anki is designed for reviewing the material many times (at an optimal spacing) for recall whenever you need it. The idea is to make use of the [spacing effect] - an extremely powerful memory technique that is currently ignored by almost all methods of formal teaching ([Memrise] is a notable exception; I used Memrise until I used Anki).
+*   A surprisingly good way of making myself work when I'm feeling unmotivated is to gather a few like-minded friends and to work in absolute silence with them (possibly on completely unrelated topics). Oddly, I'd not thought of it until reading <a title="Co-working LessWrong post" href="http://lesswrong.com/lw/gwo/coworking_collaboration_to_combat_akrasia/" target="_blank">a LessWrong post on the subject</a>. There's a kind of "all in this together" feeling, as well as the public commitment effect.
+*   I use [f.lux], an application which dims and tints red the computer screen after dusk. I have no idea whether or not it has any effect on wakefulness at night (that is, whether or not being bathed in a standard blue glow keeps me awake), but it certainly feels nicer on the eye.
+*   I am currently in the middle of learning [Dvorak], which is a keyboard layout (QWERTY is the usual one) that is supposedly easier on the hands than QWERTY. It puts vowels all together in easy-to-reach places, and the most common consonants in easy places such that words tend to be made of letters which lie in different hands. (In QWERTY, for example, the word "the" is oddly hard to type, for such a common word - all the characters are away from the home row - but in Dvorak it's just a simple flourish from right to left on the home row.) A friend tells me that [Colemak] is better than Dvorak, but I'd already half-learnt Dvorak by the time ey told me this, and Dvorak interfered heavily with my attempts to learn Colemak. It appears to be much of a muchness, anyway - both are considerably better than QWERTY.
+*   I don't know if it qualifies as a lifehack - more of a biohack or something - but [lucid dreaming] is really cool, and it doesn't take an enormous amount of commitment to learn to do (it just requires the setting up of a few habits throughout the day).
+*   Something I'm strongly considering when it's set up and running properly is [Soylent], a food substitute being developed by an engineer, which is nutritionally complete and satisfying. As of this writing, the creator has been on it for three and a half months without ill effects (once he'd sorted out the balance, anyway - he discovered a sulphur deficiency at the start of the third month, which is a very hard deficiency to give someone normally!) Currently, someone I know is using [the Exante diet][1] (the presence of a link is not necessarily an endorsement), which consists of similar but very low-calorie meal replacements; this person has been a very interesting source of information on replacing meals in this way. Their main objection to the diet seems to be the monotony, but supposedly Soylent is bland enough not to suffer from this (I could eat bread until the cows came home, for instance, but not chocolate). The Soylent Corp. says that Soylent will get cheaper as the company is set up and grows. (The creator wrote a response, but the link is now dead.)
+*   How to get up in the morning: because it's really quite hard to motivate yourself to move, I count down from 10 to 1, with the resolution that on the count of 1 I will get up. It's much easier to motivate yourself to count down from 10 than it is to move your entire body somewhere uncomfortable, and once I'm counting, consistency pressure is enough to make me follow through. I'm careful with this technique - I never use it on anything I'm not absolutely certain to do. It might pollute the technique irreparably if I had an excuse that "oh, once I did this and it didn't work!".
+
+More to come.
+
+ [1]: https://www.exantediet.com/ "Exante diet"
+ [lifehacks]: https://en.wikipedia.org/wiki/Life_hack
+ [spaced-repetition]: https://en.wikipedia.org/wiki/Spaced_repetition
+ [Anki]: http://ankisrs.net/
+ [spacing effect]: https://en.wikipedia.org/wiki/Spacing_effect
+ [Memrise]: https://www.memrise.com/
+ [f.lux]: https://justgetflux.com/
+ [Dvorak]: https://en.wikipedia.org/wiki/Dvorak_Simplified_Keyboard
+ [Colemak]: http://colemak.com/
+ [lucid dreaming]: https://en.wikipedia.org/wiki/Lucid_dream
+ [Soylent]: https://soylent.com
diff --git a/hugo/content/posts/2013-06-26-cucats-puzzlehunt.md b/hugo/content/posts/2013-06-26-cucats-puzzlehunt.md
new file mode 100644
index 0000000..e5e042f
--- /dev/null
+++ b/hugo/content/posts/2013-06-26-cucats-puzzlehunt.md
@@ -0,0 +1,27 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-06-26T00:00:00Z"
+aliases:
+- /archives/10/index.html
+- /wordpress/archives/10/index.html
+- /uncategorized/cucats-puzzlehunt/
+title: CUCaTS Puzzlehunt
+---
+At the end of last (that is, Lent 2012-2013) term at Cambridge, I took part in the [Cambridge University Computing and Technology Society](https://cucats.org) [Puzzlehunt](https://cucats.org/puzzlehunt) (for some reason, as of this writing, they haven't yet updated that page for this year's Puzzlehunt, but last year's is up there). A short summary: the Puzzlehunt is a treasure hunt around Cambridge, crossed with a whole bunch of online computing-based puzzles. It's very difficult, and it lasts for twenty-four hours.
+
+It was great fun, and while my team was hampered considerably by the fact that (having found out about the event only a day in advance) we had all planned various May Week celebrations to coincide with the first five hours or so of the twenty-four hour competition, we still gave it a good shot and came fifth of about nine, as far as I remember. (Team G, for the win!)
+
+For possibly the first time ever, I adopted a sensible strategy of separating the programs I wrote for each puzzle, and saving them as I went. This means I have a record of my attempts at each puzzle - they're all in the form of [Mathematica](https://www.wolfram.com) notebooks.
+
+My attempts are *extremely* rough-and-ready, being thrown together in the shortest time possible.
+
+Mathematica Notebook files (.nb) can be read through the Wolfram CDF Player, which can be installed free from [the Wolfram website](https://www.wolfram.com/player "Wolfram CDF player page"); the plugin is quite large, so I can release them as PDFs instead if anyone wants. (Using the CDF player gives syntax highlighting and interactivity, not that many of these files will be interactive, because they were made so quickly.)
+
+*   [Keyboard Cat](/cucats/Puzzlehunt2013/KeyboardCat.nb)
+*   [The Chase](/cucats/Puzzlehunt2013/TheChase.nb)
+
+More to follow, when I've put a bit of explanatory commentary in them.
diff --git a/hugo/content/posts/2013-06-26-first-post.md b/hugo/content/posts/2013-06-26-first-post.md
new file mode 100644
index 0000000..37b8e1e
--- /dev/null
+++ b/hugo/content/posts/2013-06-26-first-post.md
@@ -0,0 +1,16 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+date: "2013-06-26T00:00:00Z"
+aliases:
+- /archives/4/index.html
+- /wordpress/archives/4/index.html
+- /uncategorized/first-post/
+- /first-post/
+title: First post
+---
+Hello all!
+
+In the spirit of shouting into an echoing void, this is my first post, testing whether the setup works. Some content will probably turn up soon.
diff --git a/hugo/content/posts/2013-06-26-sylow-theorems.md b/hugo/content/posts/2013-06-26-sylow-theorems.md
new file mode 100644
index 0000000..88b3d96
--- /dev/null
+++ b/hugo/content/posts/2013-06-26-sylow-theorems.md
@@ -0,0 +1,110 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2013-06-26T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/sylow-theorems/
+- /sylow-theorems/
+title: Sylow theorems
+summary: "A fairly long and winding way through a proof of the three Sylow theorems."
+---
+(This post is mostly to set up a kind of structure for the website; in particular, to be the first in a series of posts summarising some mathematical results I stumble across.)
+
+EDIT: There is now [an Anki deck](/AnkiDecks/SylowTheoremsProof.apkg) of this proof, and a [collection of poems][sylow sonnets] summarising it.
+
+In Part IB of the Mathematical Tripos (that is, second-year material), there is a course called Groups, Rings and Modules. I took it in the academic year 2012-2013, when it was lectured by [Imre Leader](https://en.wikipedia.org/wiki/Imre_Leader). He told us that there were three main proofs of the [Sylow theorems](https://en.wikipedia.org/wiki/Sylow_theorems), two of which were horrible and one of which was nice; he presented the "nice" one. At the time, I thought this was the most beautiful proof of anything I'd ever seen, although other people have told me it's a disgusting proof.
+
+# Theorem - the Sylow Theorems
+
+Let \\(G\\) be a group, of order \\(p^k m\\) for some prime \\(p\\), where the [HCF](https://en.wikipedia.org/wiki/Greatest_common_divisor) \\((p,m) = 1\\). Then:
+
+1.  There is a subgroup \\(H\\) of \\(G\\), of order \\(p^k\\) (a Sylow p-subgroup);
+2.  All such subgroups are conjugate to each other;
+3.  The number of such subgroups, \\(n_p\\), satisfies \\(n_p \equiv 1 \pmod p\\) and \\(n_p \mid m\\).
+
+# Proof
+
+The proof goes as follows: pick a p-subgroup \\(P\\) to be of maximal size; then introduce its normaliser \\(N\\), and show that the orbit of \\(P\\) under the conjugation action when \\(P\\) acts on itself is precisely the set of Sylow p-subgroups.
+
+## First Sylow theorem
+
+The proof starts out in a natural way, by naming a subgroup \\(P\\) of order \\(p^a\\) for some \\(a\\). Such a subgroup certainly exists, by [Cauchy's Theorem](https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(group_theory)) (which has \\(a=1\\)). If we select \\(a\\) to be maximal, then we wish to show that \\(a=k\\), or equivalently (which seems even easier) that \\(\dfrac{ \vert G \vert }{ \vert P \vert }\\) is not a multiple of \\(p\\).
+
+Now, how do we show that \\(\dfrac{ \vert G \vert }{ \vert P \vert }\\) is not a multiple of \\(p\\)? Well, we don't know anything about such a quotient unless \\(P\\) is normal in \\(G\\). But we can't guarantee this - so let's introduce a subgroup, \\(N\\), in which \\(P\\) is normal. The natural one to pick, because we're trying to make the subgroup as big as possible, is the [normaliser](https://en.wikipedia.org/wiki/Centralizer_and_normalizer) \\(N(P)\\) - that is, \\(\{g : g P g^{-1} = P\}\\), or \\(Stab_G(P)\\) under the conjugation action. This is the largest subgroup of \\(G\\) in which \\(P\\) is normal.
+
+Then we want to show that \\(\dfrac{ \vert G \vert }{ \vert N \vert } \times \dfrac{ \vert N \vert }{ \vert P \vert }\\) is not a multiple of \\(p\\); this is true if and only if neither of the multiplicands is divisible by \\(p\\).
+
+### The second multiplicand
+
+It looks like it will be easier to start with the second multiplicand, because it's got a really really obvious interpretation.
+
+We want to show that \\(\dfrac{ \vert N \vert }{ \vert P \vert }\\) is not a multiple of \\(p\\). Now, from the [First Isomorphism Theorem](https://en.wikipedia.org/wiki/Centralizer_and_normalizer) we have \\(\dfrac{ \vert N \vert }{ \vert P \vert } =  \vert \dfrac{N}{P} \vert \\).
+
+Suppose \\( \vert \dfrac{N}{P} \vert  \equiv 0 \pmod p\\). Then by Cauchy's Theorem, there is an element \\(h \in \dfrac{N}{P}\\) such that the [order](https://en.wikipedia.org/wiki/Order_(group_theory)) \\(o(h) = p\\); let \\(H = \langle h \rangle\\), the group generated by \\(h\\). But we got to this quotient group \\(\dfrac{N}{P}\\) by applying the projection map \\(\pi : N \rightarrow \dfrac{N}{P}\\), so what happens when we "un-quotient" (that is, apply \\(\pi^{-1}\\))? We have \\(\pi^{-1}(H)\\) has order \\( \vert H \vert   \vert P \vert \\), because \\(\pi\\) was a \\( \vert P \vert \\)-to-one mapping, and so \\(\pi^{-1}(H) \leq P\\) has order \\(p  \vert P \vert \\). This is a contradiction.
+
+Hence \\( \vert \dfrac{N}{P} \vert  \not \equiv 0 \pmod p\\).
+
+### The first multiplicand
+
+The first multiplicand, \\(\dfrac{ \vert G \vert }{ \vert N \vert }\\): this is the number of conjugates of \\(P\\), by the [Orbit-Stabiliser Theorem](https://en.wikipedia.org/wiki/Orbit_stabiliser_theorem#Orbit-stabilizer_theorem_and_Burnside.27s_lemma) (by using the conjugation action: the stabiliser is \\(N\\); while the orbit of \\(P\\) is simply the set of conjugate subgroups). We want to show that this is not divisible by \\(p\\). We can do much more with the conjugates themselves, so let \\(X = \{gPg^{-1}, g \in G\}\\).
+
+We would like to show that \\( \vert X \vert  \not \equiv 0 \pmod p\\). This expression rings a bell - we've seen it before, as a key idea in the [class equation](https://en.wikipedia.org/wiki/Conjugacy_class#Conjugacy_class_equation). In order to use the class equation, we need to act on \\(X\\). There are only three groups we've met so far: \\(N\\), \\(P\\) and \\(G\\). The group we haven't yet used is \\(P\\), and it's a [p-group](https://en.wikipedia.org/wiki/P-group) (and we know a bit about actions of p-groups). What's the only obvious action to use? It has to be conjugation.
+
+Let \\(P\\) act on \\(X\\) by conjugation. Since the orbits partition the set \\(X\\) and have order dividing \\( \vert P \vert \\), the order of each orbit is one of \\(1, p, p^2, \dots , p^a =  \vert P \vert \\). \\(P\\) is clearly in an orbit all of its own (since \\(p P p^{-1} \in P\\) for every \\(p \in P\\)). What we really want is for \\(P = e P e^{-1}\\) to be the only conjugate of \\(P\\) which is in its own orbit, because then we have \\( \vert X \vert  \equiv 1 \pmod p\\) (since the orbits partition the set).
+
+Suppose we have \\(g\\) such that \\(g P g^{-1}\\) is in an orbit of size 1. Then \\(p g P g^{-1} p^{-1} = g P g^{-1}\\) for all \\(p \in P\\), and so (by conjugating with \\(g^{-1}\\)) we have \\(g^{-1} p g P g^{-1} p^{-1} g = P\\), and so \\(g^{-1} p g\\) stabilises \\(P\\) and so is in \\(N\\). So \\(g^{-1} P g\\) is contained within \\(N\\).
+
+Now, we know that \\(g^{-1} P g\\) is contained within \\(N\\), so we can now use functions defined on \\(N\\). We have that \\(\pi : N \rightarrow \dfrac{N}{P}\\) (the quotient map) is a homomorphism with kernel \\(P\\). That is, \\(\pi(P) = \{e\}\\). Hence considering \\(\pi(g^{-1} P g) = \pi(g^{-1}) \pi(P) \pi(g)\\) because \\(\pi\\) is a homomorphism; but \\(\pi(P) = \{e\}\\) so this expression is just \\(\{\pi(g^{-1}) \pi(g)\} = \{\pi(g^{-1} g)\} = \{e\}\\).
+
+Hence \\(g^{-1} P g\\) is contained in the kernel of \\(\pi\\). But it's also the same size as \\(P\\) which is itself the kernel of \\(\pi\\). Hence \\(g^{-1} P g = P\\).
+
+So there is only one orbit of size \\(1\\), and hence because orbits partition the set, \\(\dfrac{ \vert G \vert }{ \vert N \vert }\\) is not divisible by \\(p\\).
+
+This concludes the proof of the first Sylow theorem.
+
+## Second Sylow theorem
+
+Given a Sylow p-subgroup \\(Q\\) of \\(G\\), we want to show that it is conjugate to \\(P\\).
+
+Use \\(X\\) as before, the set of \\(\{g P g^{-1}\, g \in G \}\\). In the first theorem, we had \\(P\\) acting on \\(X\\); now let's use \\(Q\\) in the same way. We want to show that there is some \\(g \in G\\) such that \\(g^{-1} Q g = P\\), or equivalently that \\(Q \in X\\).
+
+Let \\(Q\\) act on \\(X\\) by conjugation. We have that \\( \vert X \vert \\) is not a multiple of \\(p\\) by the earlier part, but \\(X\\) is a union of orbits which are of size \\(p^s\\) for some \\(s\\). Hence there is a \\(g \in G\\) such that \\(\{g P g^{-1} \}\\) is the entire orbit of \\(P\\) when \\(Q\\) acts on that conjugate. (That is, there is \\(g \in G\\) such that \\(q g P g^{-1} q^{-1} = g P g^{-1}\\) for all \\(q \in Q\\).) Hence, as before, all elements of \\(g^{-1} Q g\\) fix \\(P\\) under conjugation, and hence \\(g^{-1} Q g \subset N\\).
+
+Now, \\(g^{-1} Q g \subset N\\) so we can apply the projection map \\(\pi\\) to it. We show that \\(\pi(g^{-1} Q g) = \{e\}\\). Indeed, suppose it isn't. Then \\(H = \pi(g^{-1} Q g)\\) is a non-trivial subgroup of \\(\dfrac{N}{P}\\), because \\(g^{-1} Q g\\) was a subgroup of \\(N\\). It has order dividing that of \\(g^{-1} Q g\\), because applying a homomorphism to a subgroup yields a subgroup of order dividing that of the original - and so its order is a multiple of \\(p\\). Also, its order divides that of \\(\dfrac{N}{P}\\), by Lagrange, because it's a subgroup of \\(\dfrac{N}{P}\\) - and this is not a multiple of \\(p\\). But now we have a multiple of \\(p\\) which divides a non-multiple of \\(p\\) - contradiction.
+
+Then \\(\{e\} = \pi(g^{-1} Q g) = \pi(g^{-1}) \pi(Q) \pi(g)\\); and hence we must have \\(\pi(Q) = \{e\}\\). So \\(g^{-1} Q g \subset \mathrm{Ker}(\pi)\\) and hence \\(g^{-1} Q g = P\\).
+
+This concludes the proof of the second Sylow theorem.
+
+## Third Sylow theorem
+
+We now want to show that the number \\(n_p\\) of Sylow p-subgroups is \\(1 \pmod p\\) and divides \\(m\\).
+
+We certainly have that \\(n_p =  \vert X \vert \\), because every Sylow p-subgroup is a conjugate of \\(P\\), but also every conjugate of \\(P\\) (that is, every member of \\(X\\)) is itself a subgroup of \\(G\\), and has the same size as \\(P\\), so is also a Sylow p-subgroup. Hence, just as before, \\(n_p \equiv 1 \pmod p\\).
+
+Also, \\(n_p\\) is the size of an orbit under conjugation, and hence by the Orbit/Stabiliser Theorem, it divides \\( \vert G \vert  = p^a m\\); but \\(n_p\\) does not have a factor of \\(p\\), so it must divide \\(m\\).
+
+This concludes the proof of the third Sylow theorem.
+
+# Summary
+
+So the proof went as follows:
+
+1.  We're looking for information about Sylow p-subgroups, so we pick the maximum possible p-subgroup and hope that it's a Sylow one.
+2.  How do we know whether this p-group is Sylow? If \\(\dfrac{ \vert G \vert }{ \vert P \vert }\\) is not divisible by \\(p\\).
+3.  What can we do with a quotient? Not much, but we *can* use a quotient of a normal subgroup. We can't guarantee that \\(P\\) is normal in \\(G\\), so we split up the fraction into \\(\dfrac{ \vert G \vert }{ \vert N \vert }\\) and \\(\dfrac{ \vert N \vert }{ \vert P \vert }\\).
+4.  What's a good normal subgroup to use? We have a choice. We'll go for the normaliser \\(N = N(P)\\), because that gives a nice interpretation to \\(\dfrac{ \vert G \vert }{ \vert N \vert }\\). (But otherwise, this step seems a bit arbitrary to me.)
+5.  Now we'll go for \\(\dfrac{ \vert N \vert }{ \vert P \vert }\\); this is definitely something to do with the quotient group \\(\dfrac{N}{P}\\). Let's imagine its size were divisible by \\(p\\); then we can use Cauchy on \\(\dfrac{N}{P}\\) and get a contradiction on moving back to \\(N\\).
+6.  Let's now consider \\(\dfrac{ \vert G \vert }{ \vert N \vert }\\); the normaliser is something to do with conjugates, so we'll consider the conjugation action. Happily, this expression then becomes the size of the orbit of \\(P\\) under the conjugation action; call that orbit \\(X\\).
+7.  We need \\( \vert X \vert  \not \equiv 0 \pmod p\\). Remember the class equation; we want to act on \\(X\\) using a p-group. \\(P\\) is such a p-group, so we'll let \\(P\\) act on \\(X\\). The only natural action to use is conjugation. We know straight away that \\(P\\) is in an orbit all to itself; we need it to be the only one.
+8.  Name a different conjugate of \\(P\\); call it \\(g P g^{-1}\\). We need this to be exactly \\(P\\). It's got the right size already, so we just need it to be contained in \\(P\\). Here a leap of faith: what's special about \\(P\\)? It's the kernel of a homomorphism \\(\pi: N \rightarrow \dfrac{N}{P}\\) (because it's a normal subgroup of \\(N\\)). So, after proving that \\(\pi\\) is defined on what we want to give as its arguments (that is, after showing that \\(g P g^{-1}\\) is contained in \\(N\\), or equivalently that all elements of \\(g P g^{-1}\\) stabilise \\(P\\) under conjugation), consider \\(\pi(g^{-1} P g)\\). This is clearly \\(\{e\}\\), and hence \\(g^{-1} P g\\) is in the kernel of \\(\pi\\), and hence is a subset of \\(P\\), as required.
+9.  Now the second theorem: all the Sylow p-subgroups need to be conjugate. Name a Sylow p-subgroup \\(Q\\), and have it act on \\(X\\) as above. Then in exactly the same way as in step 7, since \\( \vert X \vert \\) is not a multiple of \\(p\\), we have that there is some \\(h \in G\\) such that \\(\{h P h^{-1}\}\\) is an entire orbit under conjugation by \\(Q\\).
+10. Exactly as in step 8, a conjugate \\(h P h^{-1}\\) is on its own in an orbit, so it is fixed under conjugation by every element in \\(h^{-1} Q h\\). Hence \\(H = h^{-1} Q h\\) is contained within \\(N\\) and we can use \\(\pi\\). Suppose that \\(H\\) is not fully contained in the kernel of \\(\pi\\); then applying \\(\pi\\) to it gives us a subgroup, which must have prime power order (from the fact that \\(h^{-1} Q h\\) had prime power order); it also has order dividing that of \\(\dfrac{N}{P}\\), which is not a multiple of \\(p\\): contradiction.
+11. \\(H\\), a conjugate of \\(Q\\), is hence contained in the kernel of \\(\pi\\). Then since it is of the same size as the kernel, it must be the kernel, but that is \\(P\\).
+12. Now the third theorem: we've just shown that \\(X\\) is precisely the set of Sylow p-subgroups, so \\( \vert X \vert  \equiv 1 \pmod p\\) is just what we want (but we've already shown it back in step 8); and since it is also precisely an orbit when \\(G\\) acts on \\(P\\) by conjugation, it must have order dividing that of \\(G\\).
+
+ [sylow sonnets]: {{< ref "2013-08-31-slightly-silly-sylow-pseudo-sonnets" >}}
diff --git a/hugo/content/posts/2013-07-03-in-which-i-augment-the-lexicon.md b/hugo/content/posts/2013-07-03-in-which-i-augment-the-lexicon.md
new file mode 100644
index 0000000..45fcc8b
--- /dev/null
+++ b/hugo/content/posts/2013-07-03-in-which-i-augment-the-lexicon.md
@@ -0,0 +1,27 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-07-03T00:00:00Z"
+aliases:
+- /uncategorized/in-which-i-augment-the-lexicon/
+- /in-which-i-augment-the-lexicon/
+title: In which I augment the lexicon
+summary: "A few dubiously-real words which I think should be more widely used."
+---
+(This is my first post written in Dvorak; accordingly, it is a bit shorter than I would like, since I am very slow at it. [Tsuyoku naritai](http://lesswrong.com/lw/h8/tsuyoku_naritai_i_want_to_become_stronger/ "I want to become stronger"), and all that.)
+
+A really nice website I've come across in my wanderings is [Pretty Rational], a growing collection of pithy quotes about rationality, illustrated by one Katie Hartman.
+
+![Reality provides us with facts so romantic that imagination itself could add nothing to them][reality]
+
+This particular Jules Verne quote is expounded upon in [a LessWrong post](http://lesswrong.com/lw/or/joy_in_the_merely_real/ "Joy in the Merely Real"), as so many things are, but I can't help noticing that the source of the quote doesn't seem to appear on the Internet. If anyone knows where the quote appears, please let me know! It may turn out to be another Einsteinism - a word I hereby coin to mean "something misattributed to a(n) historical figure whom we think of as wise" - but the quote itself would be undiminished.
+
+Another niche in the language is "[evilogue](http://www.cracked.com/article_18798_6-words-that-need-to-be-invented-5Bcomic5D.html "Evilogue")" - don't click any links on that page, as Cracked is the third-hardest website on the Internet to escape, after [TV Tropes](http://tvtropes.org) and the [SCP wiki](http://scp-wiki.net). An evilogue is claimed in a situation in which someone has asked you for your opinion of (for example) a company, and you hate that company without at this time being able to recall any specific evidence. Then you may state that you have an evilogue, meaning that if ey wants you to, you will find the evidence you were referring to, at your leisure. (Beware, of course, of being unduly influenced by your past opinion - if in the course of your research you find your concerns to be unjustified, do tell the other person and update accordingly. You shouldn't be looking for new evidence, but finding the evidence you used originally.)
+
+My final bestowal on the English language (for the moment) is the word "yop", being a "yes" in response to a negative question. When asked "So I'm not the Pope after all?", the correct answer for most people would be "No" (you're not the Pope); the answer to "So I'm not sentient after all?" would usually (but not necessarily, according to [John Searle](https://en.wikipedia.org/wiki/Philosophical_zombie "P-zombie")) be "Yop" (you are sentient). This avoids the needless ambiguity of "Yes" or the prolixity of "No [or Yes], you are sentient".
+
+[Pretty Rational]: https://web.archive.org/web/20151016143228/http://prettyrational.com/
+[reality]: https://web.archive.org/web/20141012121546/http://prettyrational.com/wp-content/uploads/2013/06/PrettyRational_Reality.jpg
diff --git a/hugo/content/posts/2013-07-04-cambridge-undergrad-maths-tips.md b/hugo/content/posts/2013-07-04-cambridge-undergrad-maths-tips.md
new file mode 100644
index 0000000..45a91e7
--- /dev/null
+++ b/hugo/content/posts/2013-07-04-cambridge-undergrad-maths-tips.md
@@ -0,0 +1,31 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-07-04T00:00:00Z"
+aliases:
+- /uncategorized/cambridge-undergrad-maths-tips/
+- /cambridge-undergrad-maths-tips/
+title: Cambridge undergrad maths tips
+---
+I wrote this when I was excessively bored during exam term of my first year. It may grow as I get better at working (I'm something of a [revisionist](https://en.wikipedia.org/wiki/Ministry_of_Truth)). The advice is entirely Cambridge-based; a lot of it probably applies to other places with minor alterations. Most of this comes from personal experience.
+
+During a supervision, your supervisor will be writing all the time. As soon as you leave the supervision, mark the sheets that are particularly important in some obvious way (eg. by colouring in the corner). That way, when you're frantically flicking through the notes at the end of the year, you'll see where the information you need is. By "most important", I mean the places where the supervisor explains something fundamental to many questions, rather than the ins and outs of one particular question.
+
+Use Anki during the course - after each lecture, add the key factoids to your Anki deck for that course. (It's a bit annoying to do at the time, but it's seriously *so* much easier this way when it comes to the end of the year.) Try and get into the habit of doing some Anki every day. Remember that Anki does LaTeX!
+
+If there's anything you don't understand, email your supervisor quickly. Some of the supervisors are absolutely brilliant at replying to emails; but all of them will reply eventually. If you have a DoS [director of studies] who is all-powerful (my first-year DoS was head of almost everything in the maths department), most of your requests can be granted (even possibly to the extent of shuffling lecture times around at the start of the year, if given plenty of warning and a *very* good reason).
+
+It's going to feel weird at first, but you almost certainly aren't the best in the year - you're likely to be average. (That's what "average" is usually taken to *mean* - pun not originally intended.) This means that the lecturer probably isn't interested in hearing your pedantry or requests for rigour during the lecture. It's the supervisor's job to clear up points that you didn't understand. If everyone you've spoken to doesn't understand something from the lecture, then it might indeed be the lecturer's fault; in that case, email the lecturer, or go down to speak to them at the end of the lecture. If you notice something wrong that the lecturer's written, then unless you're absolutely sure the lecturer's made a mistake, check with the person next to you before calling it out. Protocol is to wait for a brief pause in speech before shouting "Should *this bit* be *this* instead of *what's on the board*?" - try and be as specific as you can, saying (for instance) "In your statement of Theorem 16, the first line says "f is differentiable" - should that be "g is differentiable"?". Most people are not specific when they spot a problem, and it makes it much harder for the lecturer to diagnose the problem if they don't know exactly where the problem is.
+
+Don't let your sleep cycle get too out-of-sync. It's absolutely fine (after the first couple of weeks of the term, anyway) to go to bed at whatever time you're tired - in my experience, everyone else is also tired and welcomes the chance to sleep. This is put on hold during the first couple of weeks of the term, because that's when everyone's all excited to be there and there's not too much work.
+
+If you have anything impairing your work that your DoS could conceivably help with, raise it as soon as you can. The earlier your DoS knows about it, the earlier something can be done, and your DOS is paid to worry about this sort of thing.
+
+If both you and a friend are having trouble working, go together to the library and work next to each other. You might find it helpful to view it as a competition between you and them, or as a "suffering in comradeship" kind of thing. Maintain an absolute rule of "no talking to each other", though. Schedule a break every 45mins or so, go outside and stretch your legs, and at the start of each 45min block you can ask each other about things you got stuck with on the previous 45min block.
+
+You will not be able to do every question on the example sheets [problem sheets you do as homework] easily. You're expected to have a good go at them all, but not to complete them (that would be a bonus). For those questions you can't do, pretend you are in an interview: write down your thought processes, what you've tried and why it failed. Pretend you're trying to appear really intelligent and solution-seeking in front of a prospective employer.
+
+In your answers, use lots of words; your answer should not just be a list of equations, but a coherent argument. It's a hundred times easier to mark if you explain every step properly, and it means you can go back over it at revision time; it's not that hard to do at the time, too. If you find that you pick up your work before a supervision and have no idea what you're wittering on about, you need to make your answers clearer.
diff --git a/hugo/content/posts/2013-07-06-cambridge-vocab-a-guide-for-the-mystified.md b/hugo/content/posts/2013-07-06-cambridge-vocab-a-guide-for-the-mystified.md
new file mode 100644
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--- /dev/null
+++ b/hugo/content/posts/2013-07-06-cambridge-vocab-a-guide-for-the-mystified.md
@@ -0,0 +1,23 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-07-06T00:00:00Z"
+aliases:
+- /uncategorized/cambridge-vocab-a-guide-for-the-mystified/
+- /cambridge-vocab-a-guide-for-the-mystified/
+title: Cambridge vocab - a guide for the mystified
+---
+There is an awfully large collection of confusing words you will encounter on first coming to study at Cambridge. You pick them up really quickly in the natural run of things, but I thought perhaps a mini-dictionary might be helpful. The list is alphabetised (if I'm competent enough, anyway) and may, like so many of my writings, grow. Apologies for my crude attempts at pronunciations for the non-obvious words, but it's very hard to find someone who can read [IPA](https://en.wikipedia.org/wiki/IPA).
+
+*   **Boatie** - one of the many people who row. Rowing is a very big thing at Cambridge, and some people are extremely dedicated to it (to the extent of getting up at six in the morning to train).
+*   **Formal** - a contraction of "**formal hall**", this is an event in which you are served a three-course meal in college. Exceptions in the number of courses may apply between colleges, but I'm not aware of any non-three-coursers; exceptions also apply on special occasions, so the Jesus Christmas formal had seven courses (if I recall correctly). Usually you would wear a suit or mid-scale posh dress, with gown. Most formals start and end with a Latin grace. This is probably the closest experience to Hogwarts that Cambridge has to offer. You would almost always go to formal with other people you know (booking en-masse), as a celebration (such as for birthdays).
+*   **Mathmo** - a mathematician. The word is used to refer both to maths students, and also (less commonly) to people who may not be studying maths but who share the mildly Aspergers-y traits of stereotypical mathematicians. The word is very adaptable - so, for instance, a Trinity mathmo might be referred to as a **Trinmo**, a mathmo who enjoys applied courses rather than pure courses might be referred to as an **appliedmo**, and so forth. It can also (in some circles) be femininised as **mathma**.
+*   **Muso** - a music student.
+*   **Natsci** (pron. "nat-ski") - a contraction of Natural Sciences, the subject studied by anyone who wishes to study a scientific subject. People studying (say) Biology would apply for Natsci, and then specialise later through judicious choice of courses. The Natscis are broadly subdivided into **Physnatscis** and **Bionatscis**. Also refers to Natsci students.
+*   **Pennying** - a drinking game (in the loosest sense of the word "game", even for drinking games) fairly common across the UK, as far as I can tell. To my knowledge, the rules differ between Oxford, Durham and Cambridge; I present the Cambridge rules. If your drink is sitting on a surface, without your hand being contact with the glass, anyone else (though decorum dictates that this may only be done by people who are themselves drinking alcohol) is at liberty to drop a penny into the glass, whereupon you are honour-bound to down the drink. "An empty glass is a full glass" - that is, if an empty glass is pennied, you must fill your glass with drink and then down it. For this reason, it is wise to keep some liquid in your glass at all times. If you catch the penny in your teeth as you finish your drink, the pennier must down eir drink in turn. A "double penny" occurs when two people penny the same drink; in this situation, the second pennier must down eir drink, and the one who is pennied does not have to do so.
+*   **Staircase** - this is the generic term for where students live if they are in college - the very vertical equivalent of a block of flats. They are essentially the same as dormitories, and usually have their own kitchen(s). A house owned by the college and used as accommodation can be referred to as an **external staircase**.
+*   **Swap** - this is a sort of cross between a party and a speed-dating event. Usually they take the form of a formal (see above) or a trip to a local curry-house. They are designed to get lots of people who share an interest, or some sort of connection, to get to know each other very quickly. The Christ's College hockey team might **swap with** the Jesus hockey team, for example, meaning that the teams go to a formal (or curry-house) and have a meal. Swaps are usually pretty ad-hoc; they are planned entirely by the people who are swapping.
+*   **Tripos** (pron. "try-poss") - the [Wikipedia article](https://en.wikipedia.org/wiki/Tripos "Tripos Wikipedia page") says it all, really, but this is the term used to refer to a course of study (the Mathematical Tripos, or the Historical Tripos, for example).
diff --git a/hugo/content/posts/2013-07-07-mundane-magics.md b/hugo/content/posts/2013-07-07-mundane-magics.md
new file mode 100644
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--- /dev/null
+++ b/hugo/content/posts/2013-07-07-mundane-magics.md
@@ -0,0 +1,24 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- psychology
+comments: true
+date: "2013-07-07T00:00:00Z"
+aliases:
+- /psychology/mundane-magics/
+- /mundane-magics/
+title: Mundane magics
+---
+I have stumbled across a LessWrong post on the importance of [seeing what is real for just how cool it is](http://lesswrong.com/lw/ve/mundane_magic/ "LessWrong post on Mundane Magic"). It lists such examples as:
+
+*   *Vibratory Telepathy*.  By transmitting invisible vibrations through the very air itself, two users of this ability can *share thoughts*.  As a result, Vibratory Telepaths can form emotional bonds much deeper than those possible to other primates.
+*   *Psychometric Tracery*.  By tracing small fine lines on a surface, the Psychometric Tracer can leave impressions of emotions, history, knowledge, even the structure of other spells.  This is a higher level than Vibratory Telepathy as a Psychometric Tracer can share the thoughts of long-dead Tracers who lived thousands of years earlier.  By reading one Tracery and inscribing another simultaneously, Tracers can duplicate Tracings; and these replicated Tracings can even contain the detailed pattern of other spells and magics.  Thus, the Tracers wield almost unimaginable power as magicians; but Tracers can get in trouble trying to use complicated Traceries that they could not have Traced themselves.
+
+I thought I would give a few more. First, I hereby rename *The Eye* (as that post's author names this ability) to *Force Perception*, and I dub a user of any of these magics a Mage.
+
+*   *Modular Incarnation*. An extremely powerful technique that allows enormous flexibility of function, Modular Incarnation is a method of creating superstructures out of tiny Modules, each specialised for a specific task. Out of a single generic Module, a huge variety of specialised Modules can be created, which together can be assembled into structures which can channel various other magics, including Modular Incarnation itself. Thus can an Incarnator increase eir abilities by leaps and bounds from the moment of the birth of eir Incarnatory power. The Incarnator must be wary of this ability, for in its nigh-unimaginable power lies the danger of upsetting the balance of the Modules: an Incarnator can become overrun by eir own frantically replicating Modules, the tide of which is as yet very hard to stem, even using the greatest achieved extent of the Ultimate Power.
+*   *Elemental Shielding.* Users of this passive ability are granted a flexible, regenerating defence against fire, earth, air and water. It also gives the user a constant diagnostic of eir surroundings, allowing the Shielder to understand what adjustments to make to eir environment *without even thinking about it*.
+*   *Infiltration Adaptation.* One of the most successful forms of attack that can be made on a Mage is the insertion of weapons so small that even the greatest of Force Perceptors cannot detect them. These weapons are a perversion of Modular Incarnation, and as such have the potential to be immensely powerful, but users of the Infiltration Adaptation ability can detect and neutralise them by creating a defence consisting of many thousands of Modules, each tailored to be highly effective against a single weapon that was once used against the Mage. In this way, each unsuccessful attack strengthens the Mage: after only a short period to gather eir strength, the Mage recovers, usually with no discernible damage dealt.
+*   *The Web of Pure Extraction*. Among the many ways to apply the Ultimate Power, the WPE may be one of its purest instantiations. Thousands of Extractors have together spent thousands of years in building a magnificent edifice which lies just outside this world, intersecting everywhere yet nowhere tangible. Through this power, Extractors can predict with staggering accuracy the outcomes of events happening at all scales, from the level of the fabric of reality itself up to levels encompassing all that is known to exist, and even further. Extractors can use the structure already created to solve problems that no other art can; and the structure is so well integrated with itself that particularly strong Extractors may use parts of the structure to affect other parts which lesser Extractors deem totally unrelated.
+*   *The Web of Mental Distribution*. Closely related in structure to the Web of Pure Extraction (to the extent that its name even derives from the WPE), the WMD represents the culmination of decades of work to integrate the arts of Psychometric Tracery with the Force. Through use of an abstracted version of Psychometric Tracery, users of the WMD may share thoughts across enormous distances and times, connecting all Distributors to better fuel the Ultimate Power.
diff --git a/hugo/content/posts/2013-07-08-an-obvious-improvement-to-tennis.md b/hugo/content/posts/2013-07-08-an-obvious-improvement-to-tennis.md
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--- /dev/null
+++ b/hugo/content/posts/2013-07-08-an-obvious-improvement-to-tennis.md
@@ -0,0 +1,40 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-07-08T00:00:00Z"
+aliases:
+- /uncategorized/an-obvious-improvement-to-tennis/
+- /an-obvious-improvement-to-tennis/
+title: An obvious improvement to tennis
+---
+So yesterday the [Wimbledon tennis tournament](https://en.wikipedia.org/wiki/The_Championships,_Wimbledon) was decided. The system for verifying whether the tennis ball is out or not (and hence whether play for the point stops or continues) on the main courts is as follows:
+
+1.  The ball lands.
+2.  The linesperson keeping charge of the line nearest to the landing point of the ball works out whether the ball landed inside or outside the region demarcated by the line.
+3.  The umpire decides whether or not to overrule the linesperson's decision.
+4.  The [Hawkeye](https://en.wikipedia.org/wiki/Hawk-Eye) ball-tracking system determines whether the ball landed inside or outside the region demarcated by the line.
+5.  If either player disagrees with the official decision (that is, if the linesperson called "out" when the player thought the ball was in, or the linesperson was silent when the player thought the ball was out, or if the umpire overruled a decision that the player thinks was correct) then that player informs the umpire that ey wishes to "challenge" the linesperson. In this instance, the Hawkeye reading is consulted (and the ball's trajectory slowly animated on a big screen, for added tension) and regarded as definitive.
+
+The problem I have with this system is the process of "challenging". Each player starts out with a challenge count of three. If a player makes a challenge, and Hawkeye contradicts the official call, then the challenge count is maintained at its current level. If a player makes a challenge, and Hawkeye agrees with the official call, then the challenge count for that player is decremented. A player cannot challenge if eir challenge count is 0. On entering a tie-break, each player's challenge count is incremented.
+
+This resulted in an unhappy event in the last match of the Wimbledon tournament. The player who went on to lose ([Novak Djokovic](https://en.wikipedia.org/wiki/Novak_Djokovic)) used up his three permitted challenges in unsuccessful attempts to overrule the official rulings. Then in one particularly close game, Djokovic was denied a point when his opponent's shot was deemed to be "in". He became angry (displaying the unfortunate tendency of professional sports players to throw temper tantrums at the drop of a hat) and shouted at the umpire that the call should be overruled. He had no challenges remaining, and so could not force the official decision to be reassessed; I suspect his attitude very much did not help to press his case at this point. Later, the commentators showed the Hawkeye ruling to the TV broadcast; the opponent's shot was in fact "out", and Djokovic was vindicated. As I say, he went on to lose (pretty comprehensively, I gather, although I didn't really pay attention); it is conceivable, though admittedly unlikely, that this dispute cost Djokovic the match.
+
+My question is this: why do we rely on linespeople to do that which is done better by Hawkeye?
+
+Would it not be massively more sensible if the linespeople were allowed to do exactly what they normally do (as a salve to those who do not wish to sully the tradition), but the umpire were provided with Hawkeye's ruling after every point so that ey could overrule as necessary? This changes nothing except the umpire's ability to carry out a task ey already has to do. Of course, in instances where Hawkeye is unavailable, such as on the lower courts at Wimbledon, nothing need change.
+
+Hawkeye supposedly has an average error of 3.6mm, roughly equivalent to the fluff on the ball. I propose that the umpire should be provided with the possible error along with Hawkeye's decision, and that it should be down to eir judgement which verdict to accept in such tight cases that Hawkeye might have made an error. (I would suggest defaulting to the normal method of judgement in that case - that is, "continue as if Hawkeye had not been invented".)
+
+The only reason that I can think of to limit the number of allowable challenges is to prevent time being wasted in the administrative process. However, umpire overrules (and challenges themselves) happen rarely enough that I think the following procedure would be quite sufficient:
+
+1.  The ball lands.
+2.  The linesperson keeping charge of the line nearest to the landing point of the ball works out whether the ball landed "in" or "out".
+3.  Hawkeye determines whether the ball landed "in" or "out".
+4.  The umpire reads Hawkeye's decision off a screen.
+5.  The umpire decides whether or not to overrule the official call.
+6.  If the umpire decides to overrule the call, the ball's trajectory is animated slowly on a big screen.
+
+Now, this does (of course) do nothing to resolve the problem of conflicting verdicts during a very fast rally - the umpire cannot concentrate on both the game and the Hawkeye reading at the same time. But then there's no existing solution to that problem anyway, and I do not propose to resolve this problem at the current time.
diff --git a/hugo/content/posts/2013-07-09-stumbled-across-9th-july-2013.md b/hugo/content/posts/2013-07-09-stumbled-across-9th-july-2013.md
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--- /dev/null
+++ b/hugo/content/posts/2013-07-09-stumbled-across-9th-july-2013.md
@@ -0,0 +1,32 @@
+---
+lastmod: "2022-08-21T10:39:44.0000000+01:00"
+author: patrick
+categories:
+- stumbled_across
+comments: true
+date: "2013-07-09T00:00:00Z"
+aliases:
+- /stumbled_across/stumbled-across-9th-july-2013/
+- /stumbled-across-9th-july-2013/
+title: Stumbled across 9th July 2013
+---
+Being bored over the summer holiday, I decided that I would document the cool things I ran across on the Internet. Over the last week, there have been many of these. If I see anything particularly amazing, it'll go in one of these aggregation posts.
+
+*   Neurons are surprisingly beautiful: <http://blog.eyewire.org/gallery/image-gallery/>
+*   A rather neat and very short story: <https://qntm.org/timeloop>
+*   A *bit* less short but just as good a short story: <https://qntm.org/responsibility>
+*   A rant with which students can all identify, in The Cambridge Student magazine: now lost from the Internet.
+*   An Easter Island word "tingo" means "to borrow objects from a friend’s house one by one until there are none left": <link to the Internet Archive>("http://web.archive.org/web/20100516040410/http://blog.web-translations.com/2008/12/toujours-tingo-words-that-dont-exist-in-english/)
+*   Musings on free will: <http://www.mit.edu/people/dpolicar/writing/prose/text/godTaoist.html>
+*   A thing that I just have to share again: <http://nextbigfuture.com/2013/06/technical-hurdles-have-been-overcome.html>
+*   The human brain is a really weird piece of kit: <http://lesswrong.com/lw/20/the_apologist_and_the_revolutionary/>
+*   We *have* to make one of these at some point: <http://www.pimpthatsnack.com/project/302/1">
+*   This is quite soothing in a weird kind of way: <https://thingsfittingperfectlyintothings.tumblr.com/>
+*   It is possible to be deficient in arsenic. (Link to the Soylent Discourse forum is permanently defunct.)
+*   A really useful website for when you don't want to have to spin up Wolfram|Alpha to work out time differences: <http://everytimezone.com/>
+*   Why never to talk to the police (seriously, never talk to the police): <https://www.youtube.com/watch?v=6wXkI4t7nuc>
+*   A fascinating book about the power of positive and negative reinforcement, and why they're often done wrongly: [Don’t Shoot the Dog]
+*   The Church of England really took its time, but at last they've done it: <https://www.bbc.co.uk/news/uk-23215388>
+*   The Hawkeye Initiative, for the liberation of women in comics: <http://thehawkeyeinitiative.com/>
+
+[Don’t Shoot the Dog]: https://web.archive.org/web/20130206170903/http://www.papagalibg.com/FilesStore/karen_pryor_-_don_t_shoot_the_dog.pdf
diff --git a/hugo/content/posts/2013-07-10-imre-leader-appreciation-society.md b/hugo/content/posts/2013-07-10-imre-leader-appreciation-society.md
new file mode 100644
index 0000000..5cf842a
--- /dev/null
+++ b/hugo/content/posts/2013-07-10-imre-leader-appreciation-society.md
@@ -0,0 +1,15 @@
+---
+lastmod: "2022-08-21T11:04:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-07-10T00:00:00Z"
+aliases:
+- /uncategorized/imre-leader-appreciation-society/
+- /imre-leader-appreciation-society/
+title: Imre Leader Appreciation Society
+---
+There was once a small website devoted to noting the more interesting quotes from our more idiosyncratic lecturers.
+It sadly vanished from the web, although after some detective work, I found a copy floating around on one of Amazon's servers.
+I stored them for posterity using the archival service WebCitation, which is itself now dead, so instead I shall link to [Konrad Dąbrowski's capture](https://www.konraddabrowski.co.uk/ilas/index.html).
diff --git a/hugo/content/posts/2013-07-12-a-framework-for-discussing-pricelessness.md b/hugo/content/posts/2013-07-12-a-framework-for-discussing-pricelessness.md
new file mode 100644
index 0000000..8614bc6
--- /dev/null
+++ b/hugo/content/posts/2013-07-12-a-framework-for-discussing-pricelessness.md
@@ -0,0 +1,36 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- philosophy
+comments: true
+date: "2013-07-12T00:00:00Z"
+aliases:
+- /philosophy/a-framework-for-discussing-pricelessness/
+- /a-framework-for-discussing-pricelessness/
+title: A framework for discussing "pricelessness"
+---
+Sometimes some people argue that certain things are "priceless" - that is, worth an infinite amount of money to them. I posit that what this really means is that it would take work and uncomfortable imagination to evaluate the worth of that thing to them.
+
+The example that triggered this framework was my evaluation of how much my sense of smell was worth to me. (It was late at night and I couldn't get to sleep, so I just let my mind wander around for a bit.) I was unable to quantify the amount I would pay to keep my sense of smell, but it is certainly finite, as the following thought experiment demonstrates.
+
+Suppose that you are the Master [hmm, no gender-neutral version of that word exists, as far as I know] of the Universe. For the purposes of this discussion, humans haven't explored the rest of space, and so while you are the Master of everything, you don't actually know what the "everything" is - but it doesn't really matter to you, because there's so much you can do on Earth. Perhaps you'll branch out later. In the absence of your commands, the world ticks over much as it normally does, but if you want anything at all, you can issue a demand, and it will be met as soon as possible, by the people best-suited to dealing with it. You could, for instance, insist on being given a project to work on, which will lie within your range of abilities but will be nice and challenging, and will take you at least a week but less than a year. (This allows you to prevent yourself from becoming [a mere wanting-thing](http://lesswrong.com/lw/ww/high_challenge/ "LessWrong page on High Challenge"), if you don't want to be one of those.)
+
+The penalty for abdication is pretty severe. You were elected Master of the Universe because you are the single person best suited to the role; no-one else can come close to your suitability, so to make sure you never abdicate, it is enshrined in immutable law (the only thing you can't change, in fact) that were you to abdicate, you would have everything taken from you, and would be dumped penniless without a single possession (including clothes) in the centre of London (or substitute place where it's really hard to get started in life). After all, reasoned the lawmakers, why on earth would you want to retire?
+
+Now suppose that you are kidnapped, entirely by surprise, by a mad scientist. Ey says to you:
+
+> I want to be Master of the Universe. If you don't elect me MotU, I will in my anger take away your sense of smell - but of course I don't have the power to take the Mastery of the Universe from you, so you'll still be MotU. But I am a merciful mad scientist, so I will give you this device that hooks straight into your brain and tells you what you would be smelling if you still had a sense of smell. That way you'll know whether your toast is burning - you just won't have the [quale], and I am so cunning that it will be beyond the ken of mortals to replace the quale. I will be so depressed that I will retire to the Bahamas [[capital Nassau](/anki-decks "My Anki decks, including capitals of the world")] and never trouble you again.  
+> If you hand the Mastery of the Universe to me, I will be ever so grateful - I will leave you with your sense of smell. But the penalty for abdication is pretty severe, as you know.  
+> Make your choice.
+
+Of course, assume the [least convenient possible universe] when considering a thought experiment - for instance, assume that the smelling-device is no better and no worse than your nose at detecting chemicals, so that it is not an improvement to what is currently your sense of smell; assume that you never bothered to change the dictionary so that the penalty for abdication as outlined in law would no longer be what it says on the tin, etc.
+
+In this thought experiment, it's a one-off: you lose your sense of smell and keep Mastery of the Universe, or you become absolutely nothing and keep your sense of smell. (A variant might be that the mad scientist replaces your senses one by one until you give up the Mastery.)
+
+I rather suspect I would forgo the qualia associated with smell, in order to keep my Mastery of the Universe. This imposes an upper limit on the value of the qualia associated with my sense of smell - and hence my sense of smell cannot be priceless to me.
+
+This framework is very flexible - it adapts to thinking about essentially anything. You may, of course, feel that you would give up the Mastery in order to retain your sense of smell; in that case, the thought experiment has given a lower limit, and your sense of smell could still be priceless, but at least you've actually thought about it.
+
+[least convenient possible universe]: http://lesswrong.com/lw/2k/the_least_convenient_possible_world/
+[quale]: https://en.wikipedia.org/wiki/Qualia
diff --git a/hugo/content/posts/2013-07-13-stumbled-across-13th-july-2013.md b/hugo/content/posts/2013-07-13-stumbled-across-13th-july-2013.md
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@@ -0,0 +1,24 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stumbled_across
+comments: true
+date: "2013-07-13T00:00:00Z"
+aliases:
+- /stumbled_across/stumbled-across-13th-july-2013/
+- /stumbled-across-13th-july-2013/
+title: Stumbled across 13th July 2013
+---
+*   This is really quite heartwarming: <http://www.reddit.com/r/Random_Acts_Of_Pizza/>
+*   Interesting article on current trends in fiction: <http://www.locusmag.com/Perspectives/2013/01/david-brin-our-favorite-cliche-a-world-filled-with-idiots-orwhy-films-and-novels-routinely-depict-society-and-its-citizens-as-fools/>
+*   A ridiculous reason for a rocket to explode: <https://arstechnica.com/science/2013/07/parts-installed-upside-down-caused-last-weeks-russian-rocket-to-explode/>
+*   A very information-dense way of storing data long-term: <https://phys.org/news/2013-07-5d-optical-memory-glass-evidence.html> (compare <http://rosettaproject.org/disk/technology/> which is much less information-dense but much more easily decoded in the event of being discovered after the collapse of civilisation)
+*   A cool thing to do with a Raspberry Pi and a microwave: <http://madebynathan.com/2013/07/10/raspberry-pi-powered-microwave/>
+*   I really want one of these - I think I might order one: <http://www.kickstarter.com/projects/cloud-guys/plug-the-brain-of-your-devices> (also, the word "plug" is insanely wonderful when spoken in a French accent)
+*   An interesting idea for making the world a better place: <http://web.archive.org/web/20130713135924/http://simulacrum.cc/2013/07/10/three-trends-that-push-us-towards-an-unconditional-basic-income/>
+*   A look at how to infer causality or not, as the case may be, depending on the data: <http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/>
+*   I hope they get to producing this quickly: <http://technabob.com/blog/2013/05/22/lumigrids-bike-leds/>
+*   Thank goodness for that - regular expressions are the most unreadable things ever: <https://phys.org/news/2013-07-ordinary-language.html>
+*   Something else I would do if I had eternity to play with: <https://www.youtube.com/watch?v=voB6WiP83NU>
+*   Glass ceiling issues: <http://whatwouldkingleonidasdo.tumblr.com/post/54989171152/how-i-discovered-gender-discrimination>
diff --git a/hugo/content/posts/2013-07-14-prerequisites-for-hypothetical-situations.md b/hugo/content/posts/2013-07-14-prerequisites-for-hypothetical-situations.md
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--- /dev/null
+++ b/hugo/content/posts/2013-07-14-prerequisites-for-hypothetical-situations.md
@@ -0,0 +1,83 @@
+---
+lastmod: "2022-08-21T10:47:44.0000000+01:00"
+author: patrick
+categories:
+- philosophy
+- psychology
+comments: true
+date: "2013-07-14T00:00:00Z"
+aliases:
+- /philosophy/psychology/prerequisites-for-hypothetical-situations/
+- /prerequisites-for-hypothetical-situations/
+title: Prerequisites for hypothetical situations
+---
+Usually when I discover (or, more rarely, think up) a thought experiment about a moral point, and discuss it with an arbitrary person whom I will (for convenience) call Kim, the conversation usually goes like this:
+
+> Me: {Interesting scenario} - what do you think?
+> 
+> Kim: I would just {avoids point of scenario by nitpicking}
+> 
+> Me: You know what I meant. {applies easy fix to scenario to prevent nitpick}
+> 
+> Kim: Well then, I'd {avoids point of scenario by raising unrelated moral issue}
+> 
+> Me: That's not the point. The point is {point} - let's say I constructed the scenario to make {moral issue} not an issue.
+> 
+> Kim: Hmm. {avoids point again}
+
+And so on, and so on.
+
+Now I have a platform on which to present the prerequisites for using hypothetical situations as aids to moral understanding.
+
+# Logical rudeness
+
+I have read two excellent pieces about logical rudeness - one [by a Peter Suber][logical rudeness], and one on [LessWrong][lw logical rudeness].
+Logical rudeness is a term used to denote a whole variety of techniques used to *appear to win arguments*, rather than to *address the issues at hand*.
+I can't offhand think of a way to improve Eliezer Yudkowsky's explanation on the LessWrong page I linked, so I won't elaborate on it.
+
+The main way people are logically rude with moral dilemmas [I suffered a little dilemma here myself, wondering whether to sound pretentious by pluralising as "dilemmata"] is in working out lots of ways in which your hypothetical situation could, in fact, not be about the point you want it to be about.
+A paraphrased real-life example that actually happened to me:
+
+> Me: \<explains the [torture vs. dust specks] moral problem\>
+> 
+> Kim: But how can you possibly even contemplate torturing a person! You're an evil person!
+> 
+> Me: I would contemplate torturing a person if it would avert some greater harm, yes. That's not to say I would torture a person.
+> 
+> Kim: But torture! Evil!
+
+This example shows Kim latching on to an emotional part of the hypothetical situation, and using it to launch an [ad hominem].
+This is not only logically rude (I could have outlined any scenario at all, and included the word "torture", and got the same result; Kim ignores the effort I put in to the explanation) but also verges on the socially rude.
+(In the actual situation in which this happened, I lost my temper, I am ashamed to say; the discussion, which was between about ten people, quickly turned into what was essentially a shouting match, that was only dissolved when some of us insisted on watching the latest episode of Doctor Who.)  
+The key way to avoid this is to make sure that you never stop yourself considering something, and never condemn others for considering something.
+It's a moral dilemma - you're meant to feel uncomfortable while thinking about it.
+You shouldn't be afraid just to think something, and it takes some time and effort to learn [not to avoid uncomfortable thoughts](http://lesswrong.com/lw/21b/ugh_fields/ "Ugh Fields LessWrong post").
+(Obviously, speaking those uncomfortable thoughts is certainly something to consider avoiding.)
+
+#  The Least Convenient Possible World
+
+The other major way people avoid grappling with moral dilemmas is to say, "But your hypothetical situation doesn't actually work, because of \<this objection\>."
+It's a very natural thing to do.
+My major inspiration on this is the LessWrong post on [considering the least convenient possible world](http://lesswrong.com/lw/2k/the_least_convenient_possible_world/) during debates.
+(As an aside, I'm not sure whether to use the word "argument", "debate" or "discussion" - an argument is a pointless thing, while a debate is something you enter with the aim of winning.
+Neither of these is what I am actually talking about, but the word "discussion" is becoming a little monotonous.)
+
+The usual situation: it's perfectly obvious to you (or at least would become so after five minutes of thought) what the flaws are in the presentation of the hypothetical situation, and it is probably abundantly clear that those flaws could be fixed, but because you want to *win the argument* rather than *address the moral issue*, you point out the flaws and waste the time of all concerned.
+
+However, the aim of a moral discussion is not to prove yourself to be a better arguer, but to discover what your thoughts are on an issue you've never really seen before. If you are going to point out the flaws in the given situation, at least do so while presenting a solution. My usual tactic when someone (let's make it Kim again) presents me with a moral dilemma is to begin the discussion with something like:
+
+> I presume we can ignore \<this flaw\>? I could fix it with \<very brief explanation of fix\>.
+
+Invariably Kim will reply with something along the lines of "Yeah, that's what I meant" - and that is the signal for "I am trying to discuss a moral problem, not to construct a watertight scenario." If Kim were instead to respond with "Hmm, I hadn't considered that…", then that would be an indication that ey was looking for the implementation flaws in the situation ey had outlined. Then and only then would I generate more such flaws.
+
+I'm not holding myself up to be a paragon of hypothetical-considerators, but I like to think that I'm a bit better at it than most people are. My overarching rule is:
+
+> If either party in a discussion has become angry, you have failed.
+
+Of course, [some people][trolls] just enter into arguments in order to make you or them angry (after all, it's quite fun to be angry about something that doesn't matter) - but if you actually want a fruitful discussion, avoid inflaming people.
+
+[trolls]: https://en.wikipedia.org/wiki/Trolling
+[ad hominem]: https://yourlogicalfallacyis.com/ad-hominem
+[torture vs. dust specks]: http://lesswrong.com/lw/kn/torture_vs_dust_specks/
+[logical rudeness]: https://dash.harvard.edu/bitstream/handle/1/4317660/suber_rudeness.html
+[lw logical rudeness]: http://lesswrong.com/lw/1p1/logical_rudeness/
diff --git a/hugo/content/posts/2013-07-14-the-multiple-drafts-view-of-consciousness.md b/hugo/content/posts/2013-07-14-the-multiple-drafts-view-of-consciousness.md
new file mode 100644
index 0000000..b5daf26
--- /dev/null
+++ b/hugo/content/posts/2013-07-14-the-multiple-drafts-view-of-consciousness.md
@@ -0,0 +1,48 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- philosophy
+- psychology
+comments: true
+date: "2013-07-14T00:00:00Z"
+aliases:
+- /philosophy/psychology/the-multiple-drafts-view-of-consciousness/
+- the-multiple-drafts-view-of-consciousness/
+title: The Multiple Drafts view of consciousness
+---
+I've been reading one of [Daniel Dennett's](https://en.wikipedia.org/wiki/Daniel_Dennett) books, *Consciousness Explained*. Aside from the fact that the author has an incredible beard and is therefore correct on all matters, he can also write a very cogent book. In *Consciousness Explained*, Dennett outlines what he calls the Multiple Drafts approach to explaining consciousness; this blog post is my attempt to summarise that view in a couple of short analogies.
+
+Dennett starts off by providing evidence that our time-perception is somewhat malleable: we can interpret [two dots of different colours][colour phi] (appearing separated by a short distance in time and space) as a single moving dot that changes colour abruptly at some point. The key puzzle here is that we perceive the colour to have changed *before* seeing the second coloured dot. Dennett then outlines what seem to be the two mainstream points of view on how this happens.
+
+*   The Orwellian view: that at the time of perception, we saw exactly what happened, and then we edit this after the fact to reflect a more logical sequence of events (à la [Minitrue]);
+*   The Stalinist view: that the information is edited before even making its way into the consciousness.
+
+Dennett points out that both of these options implicitly assert the existence of a "Cartesian Theatre" - a place where consciousness is experienced as information is gathered. In particular, the Stalinist view requires consciousness to be experienced after sufficient time has passed for some decisions to be made. By the way, in arguing against this supposition, Dennett doesn't mention that there is precedent for this kind of behaviour in the reflex action, which we explicitly only realise we have made after it has happened; but it's a minor point, since there are sound physiological reasons for why the reflex action doesn't come under conscious control (the signal for action never actually enters the brain, but is headed off at the brain stem). He then gives a third possible view - the Multiple Drafts model. In each of the next two analogies, I will liken the consciousness to a general in war, making decisions based on reports from the battlefield. In fact, Dennett argues that since the Cartesian Theatre does not exist (that is, consciousness isn't something that is recorded and played back to some internal watcher), this type of analogy is deeply flawed, and the third analogy will contain an appropriate adjustment.  
+Central to the analogies are two reports in particular:
+
+1.  "At location X at time 15:00:00, M happened", analogous to the report-to-the-consciousness "My hand tells me that I drew near to a source of intense heat at time \___";
+2.  "At location X at time 15:00:02, N happened", analogous to the report-to-the-consciousness "My eyes tell me that I touched the hot plate at time \___".
+
+We consider the case that report 2 arrives before report 1 (even though report 2 describes events which occurred later than report 1) - this is quite conceivable given the distance that messages must travel in the nervous system. (Please ignore the fact that this particular effect is probably going to work in reverse for this particular example, the eyes being closer to the brain than the hand - and assume that every decision is made in the brain, so that reflexes don't happen. It's harder than you might think to come up with something sufficiently urgent that isn't made as a reflex!)
+
+# The Stalinist analogy
+
+In this version of events, the reports come in from the battlefield, and flow through the general's underlings. The underlings see that the reports are in the wrong order, and switch them round so that they are in the right order, before presenting them to the general in the order {2,1} to consider; they also decide that there is a missing piece of information [corresponding to the "change-in-colour-of-dot" situation, but that doesn't fit with this analogy] between reports 2 and 1, so they insert it. The general acts on the augmented reports, and they are then sent off to be filed away for future reference.
+
+# The Orwellian analogy
+
+In this version of events, the reports come in from the battlefield, but the underlings don't correct the order of the reports, so the general sees {1,2}. The general acts on the reports once they've both been received, noticing that some information seems to be missing and adding it in, and sends them off to be filed. The archivist sees that they are in the wrong order, and switches them round just before filing them.
+
+# The Multiple Drafts analogy
+
+In this version of events, the reports come in one by one from the battlefield, but there is no general - just a room full of underlings. The first report (which records a later event) comes in, and the underlings all update their states-of-mind accordingly. Then the second report (which records an earlier event) comes in; the underlings nearest the door update soonest, and the report makes its way around the room from underling to underling. The underlings act on the reports (Multiple Drafts doesn't address how this happens - for the purposes of this analogy, let it be by everyone shouting at once, and the majority view prevails). As time goes on, more reports flood in, but eventually every underling has received reports 1 and 2 (this may happen before or after the action based on those reports is taken), and the archivist-underling files what ey thinks happened.
+
+Under Multiple Drafts, then, there is never a "point at which information enters the consciousness", but rather a "time interval in which information is making its way around the consciousness". The name of the model comes from an analogy to writing a summary of the events - starting from report 1, a summary is written; then report 2 is added, it progresses around the consciousness, and wherever it arrives, the summary is updated to reflect the new information. Thus there are *multiple drafts* of the summary at once. When the information is fully incorporated (that is, consensus has been reached on what the summary should contain), the consciousness is free to store the consensus draft in memory for future reference. Note that this could happen some time after the events described in the summary - Dennett is careful to separate "what happened" from "how the consciousness stores what happened".
+
+The reason Multiple Drafts is so attractive is that there is no experimental way to differentiate between Orwellian and Stalinist. Either way, the subject of an experiment will report the same thing, so it is strange to draw a distinction between these two possible methods. Having noticed that Orwellian and Stalinist are indistinguishable, the natural question is "why do we think they are different?" - and it turns out that the only real reason is that we think there is a centre of consciousness, through which the information must flow. Only under that interpretation is there a difference between amending-before-consideration and amending-after-consideration. So we relax the assumption of a centre of consciousness, and we end up with a "smear" of time during which information is incorporated, rather than an absolute time of perception. (This is borne out by experiment, by the way - we are very flexible when it comes to simultaneous perception.) This idea makes sense - we don't perceive space absolutely, and can happily work with receiving information about space at smeared-out times, adding more information to the model as we find out more. I nudge the table-leg with my foot, someone reacts, I am swinging my foot to kick it again, but just when I can no longer stop the kick I realise that it was in fact a human leg, the other person glares at me - my perception of the layout of space below the table has developed as new information came in, but out of sync with the information itself (the information bunched up and all came along at once). There is no particular reason why our time-perception should be any different.
+
+The book is an excellent one, very coherently written - this blog post doesn't really do it justice (although that's the point of this blog - to get me practised at writing). As of this writing, I am only half-way through the book, but it is shaping up well.
+
+[colour phi]: https://en.wikipedia.org/wiki/Color_Phi_phenomenon
+[Minitrue]: https://en.wikipedia.org/wiki/Minitrue "Ministry of Truth"
diff --git a/hugo/content/posts/2013-07-18-my-objection-to-the-one-logical-leap-view.md b/hugo/content/posts/2013-07-18-my-objection-to-the-one-logical-leap-view.md
new file mode 100644
index 0000000..e325bbf
--- /dev/null
+++ b/hugo/content/posts/2013-07-18-my-objection-to-the-one-logical-leap-view.md
@@ -0,0 +1,80 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- philosophy
+- psychology
+comments: true
+date: "2013-07-18T00:00:00Z"
+math: true
+aliases:
+- /philosophy/psychology/my-objection-to-the-one-logical-leap-view/
+- /my-objection-to-the-one-logical-leap-view/
+title: My objection to the One Logical Leap view
+---
+A large chunk of the reason why changing someone's mind is so difficult is the fact that our deeply-held beliefs seem so obviously true to us, and we find it hard to understand why those beliefs aren't obvious to others. Example:
+
+> A: A god exists - look around you; everything you see is so obviously created, not stumbled upon!
+> B: No, that's rubbish - look around you, everything you see is easily explained by understood processes!
+
+The basic problem here is that B *sees things differently* to A. Everything that B sees is automatically interpreted through the prism of "process that is understood", and that is a really hard thing to convey to A. The same evidence is spun in two totally different ways, and yet people argue as if the other party were only *one logical leap* away from coming round to their point of view. Weirdly, I can't find a source for that - I have heard the phrase before. In lieu of a source, I will (very briefly) summarise that viewpoint. I was under the impression that it is often trumpeted around that at the heart of every argument is one logical leap (OLL) that made that argument significantly different from the opposition, and that if one could only convey why that point was so important, then one could sway everyone to that point of view. (As I say, I can no longer find anyone saying this.) There is implicit evidence that people think in this way - the fact that when people are arguing earnestly with each other, each in an actual attempt to change the other's mind, they usually repeat the same argument again and again, as if that were simply a killer blow. Now, quite aside from the obvious symmetry here (both sides feel that their point is the one thing that just needs to be understood), there is a deeper point to be drawn about how we think, that exposes the OLL as a fallacy.
+
+# Ideas are not atomic
+
+There are precious few ideas that are what I call "atomic". An atomic idea is one that most observers will experience in essentially the same way. The idea that \\(1+1=2\\) is the closest I can get to an atomic idea - most people, I suspect, know that numbers can be added, and for small numbers, I posit that we all think of addition in the same way. Certainly the concept of "addition" is very much observer-dependent, in that a mathematician will probably have a very different view of addition to, say, a painter - but we have all been so well drilled that \\(1+1=2\\) that I suspect we all view it (not "addition", but "\\(1+1=2\\)") in the same way - as an isolated fact. By the way, the main difference in the concept of "addition" generally is, I think, that for a mathematician, addition is a small part of a much larger edifice (involving the Peano axioms and so forth), whereas I have met many people to whom "maths" is merely a collection of isolated computational techniques, for whom addition is simply an extra tool. Most ideas are not like \\(1+1=2\\). If you were to get me to [free-associate](https://en.wikipedia.org/wiki/Free_association_%28psychology%29) on the word "death", for instance, my immediate reaction would probably be "bad, get rid of it". If you were to get J. K. Rowling to do the same, you'd probably get "inevitable, must reconcile with". (I base this on the final book of the Harry Potter series, in which a major theme is the portrayal of death as "the next great adventure".) "Death" is a concept which varies heavily from person to person - it is *not atomic*. In order to change someone's view of death, it is likely that (for most people) a large reshuffling of the worldview would have to take place - for me, you would probably have to do one of the following:
+
+*   weaken my "human life is to be desired" axiom (in the process drastically altering my aesthetic principles);
+
+*   prove to me that there was something desirable after death (in the process weakening my ultra-materialistic worldview);
+
+*   show me that there would be horrific consequences to the prolonging of life (but that wouldn't change my view that "it would be better if we could get rid of death").
+
+Common to the two options that are actually changing my mind (the first and second) is the requirement that you break down a key part of my worldview. I think that this is why opinions are so hard to change - because they so quickly become very heavily bound up with the entire worldview. Few ideas have sufficient force to alter my entire model of the world (although they do exist: for me, one such idea was [Cached Thoughts](http://lesswrong.com/lw/k5/cached_thoughts/)). The "one logical leap" in an argument is merely the global interface of a particularly large chunk of world-model - the tip of an iceberg.
+
+At this point, I will explicitly attempt to do what I have been claiming is the impossible - to convey my worldview to you. I attempt this in order to show just how much worldview sits behind my simple opinion on the topic that "the One Logical Leap does not exist" - and how much harder than it first appears it would be to change my mind on it. I very much doubt that I will succeed in explaining my mind to the extent that I would like, for reasons explained throughout this post. Anyway, I took fifteen minutes of introspection, and here (hopefully in a reasonably logical order) are the major areas of worldview on which this article rests. I will refer to these bullet points throughout the article, and will leave out my views on mathematics and death (which have already been mentioned, but are not central to my argument). It should go without saying that these are incomplete generalisations.
+
+1.  Thought is computation, mainly oriented around pattern-matching and caching
+
+2.  There is a correct answer to essentially every question - we just don't have the computational power (in our brains or otherwise), or are not using it correctly, to answer them
+
+3.  The process of speech is literally the process of sharing thoughts (to a lesser extent, my [non-rent-paying](http://lesswrong.com/lw/i3/making_beliefs_pay_rent_in_anticipated_experiences/) belief that the mind is an entity that is distributed across multiple brains, as Hofstadter outlines in his book *I Am a Strange Loop*)
+
+4.  There are low- and high-bandwidth ways to share thoughts (one-way blog posts are not high on the list of effective thought-sharing means), but we only really use low-bandwidth ones
+
+5.  The mind is a vast collection of models of the world, constantly reaching consensus to provide a single contiguous model
+
+6.  Humans are very bad at evaluating new ideas, and most of the thought happens below the level of consciousness
+
+7.  For most people, argument is a battle to prove yourself right
+
+Of course, my statement of these aspects of my worldview is really inadequate - a soundbite summary of a seething mass of thought (viewpoints 4 and 5). As an exercise to the unusually-interested reader, you might find it interesting to go back and see where these views appeared implicitly up to now. I have tried to make this post as non-circular as I can, but it was harder than I expected to express that "the worldview is tightly bound up and hard to express". Now that I have these viewpoints explicitly labelled, I can outline my argument properly.
+
+*   People think in "one logical leap" terms - that is, they believe that B is only one short step of understanding away from coming round to A's viewpoint
+
+*   Worldviews are very hard (and therefore slow) to convey, because although we can share thoughts (viewpoint 3), we can't do it anywhere near fast enough (viewpoint 4) to put across what we call "a viewpoint" but is really a truly massive edifice (viewpoint 5)
+
+*   People therefore receive novel thoughts slowly enough that they have time to pattern-match some standard answers to them (viewpoint 1), and thereby avoid dealing with the "logical leap" the other party is trying to convey
+
+*   Unless A is very careful, B will interpret A's argument as an attack on B's own worldview (viewpoint 7) and is thus incentivised to find objections
+
+*   Hence, it is extremely hard to change a worldview.
+
+*   What I think of as "an obvious idea" is only obvious to me because that's how my pattern-matcher works (viewpoints 1 and 5)
+
+*   To change your pattern-matcher sufficiently to view my idea as "obvious" is to alter your worldview
+
+*   Therefore, my "obvious idea" is outlandish to you, unless our worldviews are sufficiently well-aligned already.
+
+An atomic idea, of course, doesn't suffer from this problem - it is seen by everyone in the same way, so it can just be packaged up and spoken, and understood as it was intended. Now, a single idea can be so powerful that it reshapes my worldview; or many different small, nearly-but-not-quite-atomic ideas relating to the same worldview can be presented, with my worldview adapting to accommodate these (viewpoints 1, 5 and 6); or I suppose I could maintain some kind of cognitive dissonance where an idea doesn't fit on top of my regular worldview, but I don't really count this as a good solution. But when conveying ideas, people don't use this fact-which-is-so-obvious-to-me, that it is hard to persuade people because the task is huge. (I hasten to add, by way of example, that it only became obvious after the considerable change in worldview brought about by reading most of LessWrong.) People almost invariably present to me a single idea without the supporting worldview (and of course I include myself as "people", but I do try to ameliorate the effect) - and then the idea has no worldview to slot into when it arrives in my brain, so I unconsciously and consciously find ways to reject it. To defeat this effect is the essence of a pretty big chunk of rationality - learning to recognise when you're automatically rejecting an argument, and stopping yourself - but that's a post for another time.
+
+# What to do with this information
+
+You may well say, "That's all very well, but what difference does it make?" - and that's a very natural question to ask, because (in my experience) the balance of probability suggests that you don't have the worldview which would make it obvious (and after all, you're reading this blog post to learn about my worldview). Over so short a period as the last few months, it has become a part of my worldview that people really don't like to evaluate arguments - probably because to do so means the other person has "won", as there's a reasonably effective social norm against adapting your view in response to evidence or argument. There are two very simply-stated ways you can use what I've been trying to convey of my worldview:
+
+*   Notice when you're rejecting an idea on worldview-incompatibility grounds, so that you can actually think about it rather than letting your already-stored model of the world decide for you
+
+*   When you are trying to convey a point, and you have the luxury of time, give much more justification than you think should be necessary, and remember, when the other person is obtusely refusing to absorb your idea, that it's probably because you aren't conveying enough background-idea. Yes, it may be very hard and time-consuming to do enough of this - but at least it stops you from unconsciously or consciously thinking that the only reason the other person isn't taking in your idea is that ey's stupid - and in my experience, that seems to be a major driving force behind the development of discussions into proper angry rows.
+
+# Post Scriptum
+
+This is far from the most coherent work to flow from my pen, I realise - it's an oddly hard thing to construct a cogent discussion on, because the argument is itself that the argument is hard to put across. My usual structure for an argument would be something like "Statement of argument, evidence, expound", but here my evidence is part of the statement of the argument, and my "expound" is also "evidence". That throws the whole structure into disarray. Ah well - let it stand as a key example in favour of the argument.
diff --git a/hugo/content/posts/2013-07-21-on-shakespeare.md b/hugo/content/posts/2013-07-21-on-shakespeare.md
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@@ -0,0 +1,58 @@
+---
+lastmod: "2022-08-21T10:51:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-07-21T00:00:00Z"
+aliases:
+- /uncategorized/on-shakespeare/
+- /on-shakespeare/
+title: On Shakespeare
+---
+I've now seen two Shakespeare plays at the [Globe](https://en.wikipedia.org/wiki/Shakespeare%27s_Globe) - once in person, to see A Midsummer Night's Dream, and once with a one-year-and-eighty-mile gap between viewing and performance (through the [Globe On Screen](https://www.dramaonlinelibrary.com/shakespeares-globe-on-screen) project), to see Twelfth Night.
+Both times the plays were excellent.
+Both were comedies, and both were laugh-out-loud funny.
+
+The performance of Twelfth Night, then, was beamed into a local-ish cinema for our viewing pleasure.
+(Definitely more comfortable than the seating at the Globe, although I am reliably informed that if you go to the Globe, you really have to be a groundling, standing at the front next to the stage, in order to get the proper experience.)
+My seat was next to those of some young-ish children.
+The result of taking several young children to a three-hour performance of a play which isn't in Modern English was predictable, but it got me thinking.
+(Bear with me - this will become relevant.)
+
+I said that both the plays were laugh-out-loud funny.
+I'll be talking about Twelfth Night, as that's the one I remember best (since it happened in the past week).
+In fact, it started out pretty dull - I was completely lost for the first five minutes while some Count or other pontificated about how much he loved a reclusive Lady.
+I was only able to get the gist of what he was saying, through snatching out some words every now and again.
+However, as soon as the Count got off-stage, the play picked up immensely, and became properly funny.
+It was really noticeable that Shakespeare was writing in two different registers - the posh one, with the Count wittering on in soliloquy, was all but incomprehensible to me, while the standard register, in which everyone else spoke, was pretty much just English.
+
+It is also really hard to grasp the nature of the humour of Shakespeare just from reading the plays.
+Once they are being performed, however, it becomes immediately obvious that every other line is an innuendo of some sort.
+Even while the Count is talking, Shakespeare gives him double-entendres ("How will she love, when the rich golden shaft/ Hath kill'd the flock of all affections else/ That live in her…") - we are clearly meant to be laughing at him, such a serious character accidentally making ribald puns - and once the silly characters come on, the humour just gets coarser.
+You don't see that from the script unless you're actually looking for it - but actors can make so much more of it, with their freedom to move around and inflect.
+In fact, with the exception of the wordplay of the Fool and the plot-based shenanigans (twins being mistaken for one another, and so on), I would say that well over half the humour in Twelfth Night is sexual in nature.
+
+Cue smooth segue to the English National Curriculum, which seems desperate to get children learning Shakespeare.
+Thankfully, [Michael Gove](https://en.wikipedia.org/wiki/Michael_Gove) doesn't seem to have gone sufficiently mad as to insist on its teaching in primary school (that is, from the age of 4 to 11), but before his reforms take place in 2014, it is/was required (link now dead) that pupils be taught on at least one Shakespeare play in Key Stage 3 (that is, aged 11 to 14).
+I can't find information about the draft 2014 curriculum for Key Stage 3, but I'm sure it appears in there too, given Gove's attitudes to pedagogy.
+
+I just don't understand why pupils are taught Shakespeare at such a young age.
+I speak for myself here, but 14 is really not old enough to understand the main source of humour (innuendo) in Shakespeare plays.
+Shakespeare's comedies are full of it - essentially all non-plot-based humour is sexual in nature.
+It is bizarre that pupils who are too young to understand the humour would be taught to analyse it.
+The language is difficult enough to read (another hurdle that simply goes away when it's acted properly), but the plays are simply horrendously drab unless you are able to grasp the humour - when you remove three-quarters of the humour from a comedy, what is left?
+
+Aside from the fact that such young children can't really understand the humour, it's also difficult for a teacher to teach, unless that teacher is one of a very unusual breed who can talk to eir pupils candidly about anything at all without it feeling awkward.
+Most teachers would find it much easier simply to ignore the double-entendres in the first place - I know that when I was taught A Midsummer Night's Dream in Year 6 (aged 10-11), my teacher focused entirely on plot, but the plot of AMND is nothing special.
+The same happened when I was subsequently taught AMND in Year 8 (aged 12-13) - even worse, we were shown a film adaptation that was just not funny.
+(This may be that I was too young to be amused by Shakespeare-humour, but I actually think the film didn't portray it at all.)
+
+So we have this strange situation of young children being taught centuries-old plays, of which they understand neither the content nor the syntax.
+There is absolutely no reason for a pupil to find Shakespeare relevant or useful in any way, taught like this.
+It's a shame, because the simple fact that "Twelfth Night is laugh-out-loud funny" is enough to tell me that Shakespeare is relevant.
+There are a couple of interesting historical notes to be gleaned from it - for instance, the treatment of the puritanical Malvolio, the only character not to receive a happy ending (aside from the pirate and the Fool), seems to show that people really liked to put down killjoys back then, in contrast to our view now (I find Malvolio's plight rather sad, and so does everyone else I've spoken to).
+But that's not really why I think Shakespeare is relevant - I think his plays are relevant in much the same way that I think the Marx Brothers' films are.
+They are really entertaining plays.
+Humour seems not to have changed very much over the last few centuries.
+Taking pupils at a young age, and turning them off good plays which are part of our cultural heritage, is something of a travesty.
diff --git a/hugo/content/posts/2013-07-22-the-orbitstabiliser-theorem.md b/hugo/content/posts/2013-07-22-the-orbitstabiliser-theorem.md
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index 0000000..38f4d51
--- /dev/null
+++ b/hugo/content/posts/2013-07-22-the-orbitstabiliser-theorem.md
@@ -0,0 +1,28 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2013-07-22T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/the-orbitstabiliser-theorem/
+- /the-orbitstabiliser-theorem/
+title: The Orbit/Stabiliser Theorem
+---
+The Orbit/Stabiliser Theorem is a simple theorem in group theory. Thanks to [Tim Gowers](https://gowers.wordpress.com/2011/11/09/group-actions-ii-the-orbit-stabilizer-theorem/) for the proof I outline here - I find it much more intuitive than the proof that was presented in lectures, and it involves equivalence relations (which I think are wonderful things).
+
+Theorem: \\(\vert \{g(x), g \in G\} \vert \times \vert \{g \in G: g(x) = x\} \vert = \vert G \vert\\).
+
+Proof: We fix an element \\(x \in G\\), and define two equivalence relations: \\(g \sim h\\) iff \\(g(x) = h(x)\\), and \\(g \cdot h\\) if \\(h^{-1} g \in \text{Stab}_G(x)\\), where \\(\text{Stab}_G(k) = \{g \in G: g(k) = k\}\\).
+
+Now, these are the same relation (we will check that they are indeed equivalence relations - don't worry!). This is because \\(g \sim h \iff g(x) = h(x) \iff h^{-1}g(x) = x \iff h^{-1}g \in \text{Stab}_G(x) \iff g \cdot h\\).
+
+And \\(\sim\\) is an equivalence relation, almost trivially: it is reflexive since \\(g \sim g \iff g(x) = g(x)\\) is obviously true; it is symmetric, since \\(g \sim h \iff g(x) = h(x) \iff h(x) = g(x) \iff h \sim g\\); it is transitive similarly.
+
+Now, it is clear that the number of equivalence classes of \\(\sim\\) is just the size of the orbit \\(\{g(x), g \in G \}\\), because for each equivalence class there is one member of the orbit (with \\([g]\\) representing \\(g(x)\\)), and for each member of the orbit there is one equivalence class (with \\(g(x)\\) being represented solely by \\([g]\\)).
+
+It is also clear that the size of the stabiliser \\(\text{Stab}_G(x)\\) is just the size of an equivalence class \\([g]\\) of \\(\cdot\\), since for each member \\(s\\) of the stabiliser, we have that \\(g \cdot (g s)\\) so \\(\vert [g] \vert \geq  \vert \text{Stab}_G(x) \vert"\\), while for each for each member \\(h\\) of \\([g]\\) we have that \\(h^{-1}g \in \text{Stab}_G(x)\\) by definition of \\(\cdot\\) - but all these \\(h^{-1}g\\) are different (because otherwise we could cancel a \\(g\\)) so \\(\vert [g] \vert \leq \vert \text{Stab}_G(x) \vert\\).
+
+And the equivalence classes of \\(\sim \ = \cdot\\) partition the set \\(G\\), so (size of equivalence class) times (number of equivalence classes) is just \\( \vert G \vert\\) - but this is exactly what we required.
diff --git a/hugo/content/posts/2013-07-24-stumbled-across-24th-july-2013.md b/hugo/content/posts/2013-07-24-stumbled-across-24th-july-2013.md
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--- /dev/null
+++ b/hugo/content/posts/2013-07-24-stumbled-across-24th-july-2013.md
@@ -0,0 +1,25 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stumbled_across
+comments: true
+date: "2013-07-24T00:00:00Z"
+aliases:
+- /stumbled_across/stumbled-across/
+- /stumbled-across/
+title: Stumbled across 24th July 2013
+---
+
+*   This is something I will try at some point, probably when I get back to uni: <http://www.bulletproofexec.com/how-to-make-your-coffee-bulletproof-and-your-morning-too/>
+*   This was fun: <http://www.sporcle.com/games/Government_Agent/true-or-false-logic-quiz>
+*   Hah - stupid copyright owners: <https://torrentfreak.com/hbo-wants-google-to-censor-hbo-com-130203/>
+*   The government's got around to allowing the testing of driverless cars: <https://arstechnica.com/tech-policy/2013/07/uk-govt-approves-autonomous-cars-on-public-roads-before-years-end/>
+*   An insightful comic about getting to sleep: <https://abstrusegoose.com/523>
+*   Roll on the cheap and easy satellites: <http://www.guardian.co.uk/science/across-the-universe/2013/jul/17/sabre-rocket-engine-reaction-skylon>
+*   A bunch of interesting sciency things, including a new application of zapping current through the brain: <https://arstechnica.com/science/2013/07/weird-science-always-runs-current-through-its-brain-before-speed-dating/>
+*   At last! <http://coderinaworldofcode.blogspot.co.uk/2013/07/50-shades-of-grey-made-illegal-in-uk.html>
+*   I didn't see this at the time - consider my faith in humanity restored: <https://www.bbc.co.uk/news/world-asia-pacific-13598607>
+*   Excellent essay on why it's hard to prohibit same-sex marriage: [cached][gender and same-sex marriage]
+
+[gender and same-sex marriage]: http://web.archive.org/web/20140723074138/http://linuxmafia.com/faq/Essays/marriage.html
diff --git a/hugo/content/posts/2013-07-25-metathought.md b/hugo/content/posts/2013-07-25-metathought.md
new file mode 100644
index 0000000..b582f85
--- /dev/null
+++ b/hugo/content/posts/2013-07-25-metathought.md
@@ -0,0 +1,42 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- psychology
+comments: true
+date: "2013-07-25T00:00:00Z"
+aliases:
+- /psychology/metathought/
+- /metathought/
+title: Metathought
+sidenotes: true
+---
+
+I have recently discovered the game of [Agricola](https://en.wikipedia.org/wiki/Agricola_%28board_game%29), a board game involving using resources (family members, stone, etc) to build a thriving farm.
+The game is turn-based, with the possible actions each turn being severely limited.
+This makes the game be in large part about optimising under constraint (the foundation of any good game).
+However, during gameplay I also detected a certain resonance between Agricola and the game of [Magic: The Gathering](https://en.wikipedia.org/wiki/Magic_the_gathering), beyond the usual "constrained optimisation" theme.
+While I was playing Agricola, there was a kind of niggle in the back of my mind, telling me that "ooh, this is like Magic".
+
+I notice a similar affinity when reading essentially anything by [Douglas R Hofstadter](https://en.wikipedia.org/wiki/Doug_Hofstadter), an author [famed](https://xkcd.com/917/ "xkcd I'm So Meta") for his "metaness".
+That is, when reading a good Hofstadter piece, I get a similar niggle (considerably weaker than the Magic-Agricola one) telling me that "ooh, this is a bit like Magic".
+Hofstadter invents puns and connections which feel so natural that you'd be forgiven for thinking that he had invented English specially for the purpose, were it not for the fact that his book [Gödel, Escher, Bach](https://en.wikipedia.org/wiki/Godel_escher_bach) was translated (I am told) with the same level of scintillation into at least {{< side right lang "eight other languages." >}}French, German, Spanish, Chinese, Swedish, Dutch, Italian and Russian, according to the bottom of <a href="http://tal.forum2.org/geb">Tal Cohen's review</a> (<a href="http://web.archive.org/web/20220428223304/http://tal.forum2.org/geb">cached</a>).{{< /side >}}
+This leads me to wonder whether what I'm really noticing is not just constrained optimisation, but "metathought" - thought on a higher level of abstraction to the usual.
+With Hofstadter, it's on the level of words as well as of the symbols of thought that the words invoke; with Magic, it's thinking about plans and strategies involving the other player(s) and the interactions between their cards and mine; with Agricola it's thinking about the aims of the other player(s) and how best to compete for the limited actions available to us both.
+I note that I don't feel the resonance with chess - archetypical of "deterministic games", where you know exactly what moves are available to both sides - so the resonance is not a marker for "putting myself in others' shoes".
+Rather, it seems to be a marker of *interaction* - between players' plans, or between words and meaning, and so on.
+
+Closely linked to this is the related concept of [introspection](http://lesswrong.com/lw/6p6/the_limits_of_introspection/). It's a [well-researched](https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect "Dunning-Kruger effect Wikipedia page") [fact](https://en.wikipedia.org/wiki/Four_stages_of_competence "Four stages of competence Wikipedia page") that people are ([in general](http://lesswrong.com/lw/1xh/living_luminously/ "Luminosity LessWrong page")) [bad](http://lesswrong.com/lw/i4/belief_in_belief/ "Belief in belief LessWrong page") at introspection (hence the existence of [behaviourism](https://en.wikipedia.org/wiki/Behaviorism) and [heterophenomenology](https://en.wikipedia.org/wiki/Heterophenomenology)). I've trained myself over the last year or so to be much better at introspection than I was [^uncertain] - I notice myself shying away from thinking things, I recognise when there's a specific thing I can't be bothered to think about, and so forth. Of course, I am (as yet) imperfect, but I am [trying not to be](http://lesswrong.com/lw/h8/tsuyoku_naritai_i_want_to_become_stronger/ "I want to become stronger").
+
+How is this, as I have claimed, "closely linked"? I am slowly forming the opinion that it takes a reasonably good level of introspective ability just to be able to notice resonances between things. [^general] I am waiting for experimental evidence on this (and it is possible that my subjects are reading this blog, so I won't say what the tests are). {{< side right interrelation "However, I've noticed these resonances myself to a greater degree since learning introspection.">}}A possible explanation is that I've just been doing more interrelated things recently, so I would be very likely to spot more interrelations.{{< /side >}} The feeling of "affinity" between things is very difficult for me to describe - it's kind of a shade of extra interpretation laid on a concept, but it's not linked to any of the commonly-recognised senses, so English isn't very well set up to define it - but the feeling is very weak. I sometimes think of it as making an extra brushstroke on a watercolour - the added colour is there, but it's very slight - perhaps slight enough as to go unnoticed by someone who is not in the habit of noticing eir thoughts. It also feels like an area of light (in both senses - "not dark" and "not heavy") at the (literal) back of my mind. (Ah, how difficult to describe qualia accurately!) However it feels to me, it is my experience that people very rarely claim that one activity is similar to another in some abstract way (as I do with Magic and Gödel, Escher, Bach) - this may be because I don't notice it when they so claim, or that they never so claim because no-one else ever so claims, or that they never so claim because they don't notice the resonance, or that the resonance isn't actually there and I'm delusional (although in this instance that seems a bit unlikely, if I say so myself).
+
+ Why do I think that what I'm noticing is "metathought" rather than merely "constrained optimisation"? Well, I very rarely feel the resonance, and I'm always solving constrained optimisation problems without feeling the resonance (how succinct can I make this post, how many chocolates can I get away with eating…) so I suspect that it's not just the optimisation aspect. The only other link I have come up with at the moment is metathought. Magic, in particular, has the potential for very complicated interactions involving thinking hard about which strategies will be successful and when exactly to do things; Hofstadter's punning is ridiculously meta anyway; while Agricola is heavily based on working out what the opponents will be doing and taking that into account (that is, it requires *reflection*, a key component of metathought), while juggling your own strategies. I note for completeness that I read Gödel, Escher, Bach well before I discovered the game of Magic, and I didn't feel the resonance with GEB on first playing Magic - it was only once I'd played Magic that I started feeling the resonance. Alternatively put, I feel the "resonance with Magic", rather than "resonance with things in the class to which Magic belongs".
+
+I get slight shades of the same resonance when solving crosswords, and maybe even sometimes when proving mathematical statements - but take this with a pinch of salt, because I've had time to create a pattern for "I feel this resonance when…", and it's much easier to fill that pattern than to actually work out whether I do feel that resonance. I explicitly noticed it and noted it to myself when playing Agricola and when reading the Ricercar from Gödel, Escher, Bach - any other examples are potentially suspect, now that I've thought the concept through, because the feeling of resonance is so weak compared to the thought "If my hypothesis is correct, I should feel resonance now". (I came to this realisation while writing this paragraph.) It would appear that I may have accidentally corrupted my ability to feel this "resonance" in weak cases. Unfortunately, this makes it very hard to provide further tests: in particular, I need cases when I would predict feeling resonance but in fact do not.
+
+Anyway, I hypothesised that "resonance" is only felt by people who naturally or artificially have good introspection. I would be very interested to hear of evidence on this point - if you feel (with some kind of justification) that you have unusually good introspection, or if you think you have felt the kind of resonance that I describe (of course, my description was poor!), do let me know - I don't know which way causality runs, if any, and I would like to know whether it's just some oddity of my own, or whether It's A Thing that no-one bothers to mention for some reason.
+
+
+[^general]: The resonance which is the main subject of this post is a single instance of a more general class of relations - for instance, there is a different kind of resonance between Scrabble and [Countdown](https://en.wikipedia.org/wiki/Countdown_%28game_show%29).
+
+[^uncertain]: Or at least, I hope I have - it certainly feels like it's working, but then again, it would probably feel like that if I were getting *worse* at introspection, because I'd be getting worse at telling whether I was getting better or not.
diff --git a/hugo/content/posts/2013-07-29-stumbled-across-29-july-2013.md b/hugo/content/posts/2013-07-29-stumbled-across-29-july-2013.md
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--- /dev/null
+++ b/hugo/content/posts/2013-07-29-stumbled-across-29-july-2013.md
@@ -0,0 +1,21 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stumbled_across
+comments: true
+date: "2013-07-29T00:00:00Z"
+aliases:
+- /stumbled_across/stumbled-across-2/
+- /stumbled-across-29-july-2013/
+title: Stumbled across 29th July 2013
+---
+*   Hehe: <http://www.pixartheory.com/>
+*   Wow - light trapped for a full minute: <http://www.newscientist.com/article/dn23925-light-completely-stopped-for-a-recordbreaking-minute.html>
+*   The importance of a consistent utility function: <http://lesswrong.com/lw/my/the_allais_paradox/>
+*   Obama promised to be friendly to whistleblowers, and has quietly removed said promise: <http://sunlightfoundation.com/blog/2013/07/25/obama-promises-disappear-from-web/>
+*   I wholeheartedly agree with this site: <https://abandonmatlab.wordpress.com/>
+*   Good post on belief-in-belief: <http://web.archive.org/web/20120629012114/http://stairs.umd.edu/236/meta-atheism.html>
+*   Huh. A strange system, the US medical system: <http://stallman.org/articles/asked_to_lie.html>
+*   Very much this - about how the media has lost the plot about PRISMgate: <http://m.guardiannews.com/technology/2013/jul/28/edward-snowden-death-of-internet>
+*   Aaand my faith in humanity is once again shattered: <http://i.imgur.com/mdfvFA6.jpg>
diff --git a/hugo/content/posts/2013-07-30-on-to-do-lists-as-direction-in-life.md b/hugo/content/posts/2013-07-30-on-to-do-lists-as-direction-in-life.md
new file mode 100644
index 0000000..752b0e1
--- /dev/null
+++ b/hugo/content/posts/2013-07-30-on-to-do-lists-as-direction-in-life.md
@@ -0,0 +1,35 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-07-30T00:00:00Z"
+aliases:
+- /uncategorized/on-to-do-lists-as-direction-in-life/
+- /on-to-do-lists-as-direction-in-life/
+title: On to-do lists as direction in life
+---
+[Getting Things Done](https://en.wikipedia.org/wiki/Getting_Things_Done) has gathered something of a [cult following](http://web.archive.org/web/20130428015707/http://www.wired.com/techbiz/people/magazine/15-10/ff_allen? "Wired article on GTD") [archived due to [link rot][1]] since its inception. As a way of getting things done, it's pretty good - separate tasks out into small bits on your to-do list so that you have mental room free to consider the bigger picture. However, there's a certain aspect of to-do lists that I've not really seen mentioned before, and which I find to be really helpful.
+
+My to-do list takes up a large amount of space on one of my virtual desktops (specifically, on [Dashboard](https://en.wikipedia.org/wiki/Dashboard_%28Mac_OS%29)). It consists of a large number of short-term goals, with some longer-term goals and a couple of very long-term goals mixed in. Sample:
+
+> Library books: Flow, The Mind's I, Consciousness Explained
+>
+> Go and see the [Aurora](https://en.wikipedia.org/wiki/Aurora_borealis)
+>
+> See how many [taste buds](https://en.wikipedia.org/wiki/Supertaster "Supertaster") I have
+>
+> Update list of books on blog
+
+There are very long-term goals like seeing the Aurora (which I intend doing during the next solar maximum in seven years or so), some goals which can be accomplished very quickly (like seeing whether I am officially a supertaster), an ongoing task (updating the blog) and a list of the library books I have out at the moment.
+
+The reason I like this arrangement so much is that it doesn't make you feel bad to see a wall full of to-do items that you've not done. Because a fair few of the goals are so long-term, I expect to see lots of items on the list, so I don't get the sinking feeling when I see everything I have left to do. It also feels really good to tick off a long-term goal (my most recent being "Get a [Kindle](https://en.wikipedia.org/wiki/Amazon_Kindle)"), and it feels better than it otherwise would to tick off a short-term goal, since it is surrounded by things that I know won't get ticked off for a while, so it feels (by association) like a bigger accomplishment.
+
+It also means that I should never forget to do something big that I want to do. So often, I hear people say "I wish I could… before I die", or similar. Now I have a system for recording all these things that cross my mind, so I will eventually get round to doing them. (I should note that on a fairly regular basis, I read through the whole list and work out which items are feasible right now - hopefully this will mitigate the "that's a long-term goal, ignore it" effect.) My goal to "play in the [Tallis Fantasia](https://en.wikipedia.org/wiki/Fantasia_on_a_Theme_by_Thomas_Tallis)" is one such entry.
+
+I think that this kind of method of writing down goals could be used to create some sort of life direction. I've seen services into which you enter your long-term goals, and then when you complete one, you tell the system and you gain "experience points", levelling up after reaching a certain threshold of points. I like this idea, but I postulate that it encourages thinking of long-term goals as different things to short-term goals, and that this is not necessarily desirable. A goal is a goal; some are big-impact long-term things, some are big-impact short-term things, and so on; the system seems to create an artificial distinction between short-term and long-term. My system, in its simplicity, avoids this distinction. I can see a pattern of goals that reflects my future life; to get a bit soppy about it, I can see a much clearer "direction" this way, listing internships, the research I want to do for interest, a certain walk that is strongly recommended from Cambridge to Grantchester, and so on. The lack of "levels of abstraction", I think, makes it much easier to do long-term things that I would otherwise put off.
+
+I now get to tick something else off the list - hooray! I hope something comes along soon to replace it.
+
+ [1]: https://en.wikipedia.org/wiki/Link_rot "Link rot Wikipedia page"
diff --git a/hugo/content/posts/2013-08-04-new-computer-setup.md b/hugo/content/posts/2013-08-04-new-computer-setup.md
new file mode 100644
index 0000000..a5bab01
--- /dev/null
+++ b/hugo/content/posts/2013-08-04-new-computer-setup.md
@@ -0,0 +1,60 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-08-04T00:00:00Z"
+aliases:
+- /uncategorized/new-computer-setup/
+- /new-computer-setup/
+title: New computer setup
+---
+
+*Editor's note: this is a snapshot of life in 2013-08-04. My setup has changed substantially since then.*
+
+In case I ever have to get a new computer (or, indeed, in case anyone else is interested), I hereby present the (updating) list of applications and so forth that I would immediately install to get a computer up to usability.
+
+*   Browser: [Firefox] with [Ghostery], [HTTPS Everywhere], and [NoScript] (and remember to turn on Do Not Track…)
+*   Mail client: [Thunderbird] with [Enigmail]
+*   Messaging client: [Adium] on Mac, and possibly [Pidgin] for others - I've never used a non-Mac chat client. Beware: as of this writing, Pidgin stores passwords in plain text, so don't save passwords in Pidgin.
+*   Encryption: GPG ([Windows][GPG Windows], [Mac][GPG Mac], [Linux][GPG Linux])
+*   Text editor: Vim
+*   Memory training: [Anki]
+*   Movie viewing: [VLC]
+*   Screen colour muter: [f.lux]
+*   Backup software: [CrashPlan] - but I also keep local backups using whatever built-in automated backup utility the OS provides
+*   FTP client: [FileZilla], or [Cyberduck] on a Mac
+*   Syncing: [Dropbox] (but I want to get rid of this, because of privacy concerns)
+*   Computational software: [Mathematica]
+*   Music: [iTunes] (but I want to switch this for something not-Apple, and it has no Linux version)
+*   Gaming: [Steam]
+*   RSS reader: Currently, my RSS feed is presented in-browser, at [NewsBlur].
+
+
+[Firefox]: https://www.mozilla.org/en-US/firefox/new/
+[Thunderbird]: https://www.mozilla.org/en-US/thunderbird/
+
+[Ghostery]: https://www.ghostery.com/
+[HTTPS Everywhere]: https://www.eff.org/https-everywhere
+[NoScript]: https://addons.mozilla.org/en-US/firefox/addon/noscript/
+[Enigmail]: http://www.enigmail.net/home/index.php
+
+[Dropbox]: https://www.dropbox.com/
+[Mathematica]: https://www.wolfram.com
+[iTunes]: https://www.apple.com/itunes/
+[Steam]: https://store.steampowered.com/
+[Anki]: http://ankisrs.net/
+[NewsBlur]: https://www.newsblur.com
+[FileZilla]: https://filezilla-project.org/
+[Cyberduck]: http://cyberduck.io/
+[CrashPlan]: https://www.crashplan.com/
+
+[f.lux]: http://stereopsis.com/flux/
+[VLC]: https://videolan.org/vlc/
+[Notepad++]: http://notepad-plus-plus.org/
+[Pidgin]: https://www.pidgin.im/
+[Adium]: https://adium.im/
+[GPG Linux]: https://gnupg.org/
+[GPG Mac]: https://gpgtools.org/
+[GPG Windows]: http://www.gpg4win.org/
diff --git a/hugo/content/posts/2013-08-04-stumbled-across-4-august-2013.md b/hugo/content/posts/2013-08-04-stumbled-across-4-august-2013.md
new file mode 100644
index 0000000..2ce554a
--- /dev/null
+++ b/hugo/content/posts/2013-08-04-stumbled-across-4-august-2013.md
@@ -0,0 +1,25 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stumbled_across
+comments: true
+date: "2013-08-04T00:00:00Z"
+aliases:
+- /stumbled_across/stumbled-across-3/
+- /stumbled-across-4-august-2013/
+title: Stumbled across 4th August 2013
+---
+*   An ad developer has misgivings: <http://seriouspony.com/blog/2013/7/24/your-app-makes-me-fat>
+*   Hint for dealing with some automated phone helplines - swear at them and they'll put you through to a human: <https://www.vice.com/en/article/z444dx/if-you-swear-at-apple-s-automated-customer-service-they-ll-put-you-through-to-a-human>
+*   The future is coming: <http://www.extremetech.com/extreme/162678-harvard-creates-brain-to-brain-interface-allows-humans-to-control-other-animals-with-thoughts-alone>
+*   A large collection of replacements for various PRISM-vulnerable services: <https://prism-break.org/>
+*   Some people think in a really rather interesting way: <https://www.schneier.com/blog/archives/2005/02/smart_water.html>
+*   The joys of a memoryless distribution: <http://io9.com/the-quantum-zeno-effect-actually-does-stop-the-world-977909459>
+*   An impressive photograph: [largest photo cached]
+*   A fair chunk of the "1910's predicted Year 2000 technologies" has been invented: <http://www.sadanduseless.com/2011/03/world-in-2000/>
+*   A sweet video about Street View: <http://vimeo.com/32397612>
+*   How to enable encryption in your emails using [GPG][GPG]: <http://arstechnica.com/security/2013/06/encrypted-e-mail-how-much-annoyance-will-you-tolerate-to-keep-the-nsa-away/>
+
+[GPG]: https://en.wikipedia.org/wiki/GNU_Privacy_Guard
+[largest photo cached]: https://web.archive.org/web/20130814173950/http://www.oddly-even.com/2013/07/31/the-largest-photo-ever-taken-of-tokyo-is-zoomable-and-it-is-glorious/
diff --git a/hugo/content/posts/2013-08-11-stumbled-across-11th-august-2013.md b/hugo/content/posts/2013-08-11-stumbled-across-11th-august-2013.md
new file mode 100644
index 0000000..c825cbe
--- /dev/null
+++ b/hugo/content/posts/2013-08-11-stumbled-across-11th-august-2013.md
@@ -0,0 +1,24 @@
+---
+lastmod: "2022-08-21T11:10:44.0000000+01:00"
+author: patrick
+categories:
+- stumbled_across
+comments: true
+date: "2013-08-11T00:00:00Z"
+aliases:
+- /stumbled_across/stumbled-across-11th-august-2013/
+- /stumbled-across-11th-august-2013/
+title: Stumbled across 11th August 2013
+---
+*   A thousand times this (EDIT 2022: the link is dead and I have no idea what I was referring to).
+*   A possible fix for the "[economic problem][1] of democracy": <https://mason.gmu.edu/~rhanson/futarchy.html>
+*   A fascinating look at privacy online, how we're not built for privacy, and how tribal cultures attain privacy: <https://aeon.co/essays/facebook-s-privacy-settings-aren-t-the-problem-ours-are/>
+*   I'm all for healthy competition and so forth, but do we really want such massive phones? <https://arstechnica.com/gadgets/2013/08/the-smallest-new-android-phone-you-can-buy-isnt-small-at-all/>
+*   This is the kind of thing that I never quite have the courage or the morals to do: <http://web.archive.org/web/20130809202515/http://www.minyanville.com/business-news/editors-pick/articles/A-Russian-Bank-Is-Sued-for/8/7/2013/id/51205>
+*   This is an excellent summary for why I'm trying to find a good Gmail replacement: <https://ar.al/notes/schnail-mail-free-real-mail-for-life/>
+*   A guide for dealing with introverts (not that many of my friends need it - perhaps that's why they're my friends): <http://laughingsquid.com/how-to-live-with-introverts/>
+*   I didn't know this was such a wide-spread problem: <http://www.coding2learn.org/blog/2013/07/29/kids-cant-use-computers/>
+*   I agree with this article on the state of maths teaching entirely - I had some excellent teachers, but I could see from the textbooks how it was designed to be taught: <http://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf>
+*   How is it that Scandinavia manages to be so nice all the time?! <https://www.bbc.co.uk/news/world-europe-23655675>
+
+ [1]: https://en.wikipedia.org/wiki/Criticism_of_democracy#Economic_criticisms
diff --git a/hugo/content/posts/2013-08-18-thinking-styles.md b/hugo/content/posts/2013-08-18-thinking-styles.md
new file mode 100644
index 0000000..b9fa7aa
--- /dev/null
+++ b/hugo/content/posts/2013-08-18-thinking-styles.md
@@ -0,0 +1,37 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- psychology
+comments: true
+date: "2013-08-18T00:00:00Z"
+math: true
+aliases:
+- /psychology/thinking-styles/
+- /thinking-styles/
+title: Thinking styles
+---
+All the way back into primary school (ages 4 to 11 years old, in case a non-Brit is reading this), we have been told repeatedly that "people learn things in different ways". There were two years in primary school when I had a teacher who was very into [Six Thinking Hats](https://en.wikipedia.org/wiki/Six_Thinking_Hats) (leading to the worst outbreak of headlice I've ever encountered) and [mind maps](https://en.wikipedia.org/wiki/Mind_map). I never understood mind maps, and whenever we were told to create a mind map, I'd make mine as linear and boxy as possible, out of simple frustration with the pointless task of making a picture of something that I already had perfectly well-set-out in my mind. I quickly learnt to correlate "making a mind map" with "being slow and inefficient at thinking". (This was back when my memory was still exceptionally good, so I wasn't really learning much at school - having read, and therefore memorised, a good children's encyclopaedia was enough for me - and hence relative to me, pretty much everyone else was slow and inefficient, because I'd already learnt the material.)
+
+It's only now that I've realised that perhaps some people actually do think in a way that makes mind maps helpful. I'm not bad at spatial visualisation (not great, but not totally inept), but I don't think in pictures at all. Apparently, [about 3% of people](https://www.lesswrong.com/posts/baTWMegR42PAsH9qJ/generalizing-from-one-example) [sorry, the source for the statistic wasn't given on that page] simply do not have mental images - I don't fall into that 3%, but a close family member tells me ey does - ey cannot make sense of pictures at all without translating them into words. (Possibly a genetic bias? At least another two close family members are very visual indeed.) Ey told me that the world is really not set up for people who can't visualise: whenever you say you don't understand something, the default response is apparently to say exactly the same thing again, but accompanied by a picture - completely useless for a non-visualiser. I've never noticed this before, and a quick memory trawl is inconclusive, but I will certainly keep a look out for it and against it.
+
+A prime example (no pun intended) of an extraneous visual approach to something was the multiplication of two two-digit numbers by using the fact that \\((a+b)(c+d)=ac+bc+ad+bd\\). As an example, I'll take the numbers 35 and 27. The method involved drawing a box of (nominal) side lengths 27x35, and drawing two lines to divide the sides (nominally) into 20+7 and 30+5. Then in each of the four sub-boxes thus created, you had to write the area of that sub-box (that is, calculate \\(30 \times 20\\), \\(30 \times 7\\), \\(5 \times 20\\), \\(5 \times 7\\)) and then add them all up to get the total area. This method seemed like an enormous waste of time and space to me; I had already learnt to multiply arbitrary numbers together through the [Kumon](https://en.wikipedia.org/wiki/Kumon) program by using the standard [long multiplication](https://en.wikipedia.org/wiki/Multiplication_algorithm#Long_multiplication), and to have to learn a method that was about ten times slower and used four times more paper seemed immensely wasteful. I formed the opinion that the reason people were bad at multiplication was that they were being told to use these useless methods that no-one in their right mind could possibly understand. The [Generalising from One Example](http://lesswrong.com/lw/dr/generalizing_from_one_example/) LessWrong post contains an extremely relevant passage:
+
+> I only really discovered this in my last job as a school teacher. There's a lot of data on teaching methods that students enjoy and learn from. I had some of these methods...inflicted...on me during my school days, and I had no intention of abusing my own students in the same way. And when I tried the sorts of really creative stuff I would have loved as a student...it fell completely flat. What ended up working? Something pretty close to the teaching methods I'd hated as a kid. Oh. Well. Now I know why people use them so much. And here I'd gone through life thinking my teachers were just inexplicably bad at what they did, never figuring out that I was just the odd outlier who couldn't be reached by this sort of stuff.
+
+And it's only very recently that it occurred to me that this is quite possibly exactly my experience. The visual techniques simply work for other people.
+
+Another example (again from arithmetic) is the [number line][1] (and the closely related and suggestively named [real line][2]). A large chunk of the first few years at primary school was devoted to learning to count and add (pretty tedious stuff, especially if you already knew how to count and add!). One of the key methods used was the number line - so, for instance, to work out \\(8-3\\), you had to count forward 8 and go back 3. I hated this method - again, it wasted time (why not just go forward 5?) and space (draw out a line? no thanks!). Apparently there was a study done on an untouched-by-society tribe, and it turns out that viewing numbers spatially is not inbuilt in humans. [^study] Maybe I was just unusually unable to learn this view of numbers.
+
+Over the last few years, however, most noticeably as I have come to learn more maths, I have started to rely on pictures considerably more than I used to. I discovered the memory technique of "imagine a picture, the more ridiculous the better" to link two concepts (that's how I'm learning the capitals of the world - Luanda is the capital of Angola, which I remember as a [footballer](https://en.wikipedia.org/wiki/Soccer) scoring a GOAL [Angola] by kicking the ball into a [LOO](https://en.wikipedia.org/wiki/Toilet "Toilet") which is sitting between the goalposts), and I have used it to learn a variety of things. In the topic of [analysis](https://en.wikipedia.org/wiki/Mathematical_analysis), I rely on pictures as a guide to intuition - the statement that "for every \\(\epsilon > 0\\), there is a \\(\delta > 0\\) such that for all \\(y\\) where \\(\vert y<x \vert < \delta\\), we have \\( \vert f(y)-f(x) \vert < \epsilon\\) is pretty horrific to grok, but a [simple picture](https://en.wikipedia.org/wiki/File:Example_of_continuous_function.png "Continuous function example") is enough for me to understand it. However, I don't use pictures except as intuition-pumps - they're often too unreliable to use as proofs, and I don't (yet) think sufficiently pictorially to do this.
+
+I also have an unusually wide vocabulary. While I used to find it facile to augment the erudition of my verbiage, I am now afflicted by a slowness of thought that makes it something of an effort. It feels perhaps as if my facility with words has decreased as my mental use of pictures has increased (though this is hard to gauge, because I also know that my mental faculties in general have declined precipitously since perhaps the age of 13). It is possible that I was forced by my brain to become excellent with words, as a substitute for thinking in pictures (since words are the most general means we have of representing thought-symbols, though pictures may be higher-fidelity).
+
+I also wonder whether gesturing during speech has an association with picture-representation. As far as I am aware, I don't gesture very much when speaking; I know that the close-family-member-who-doesn't-think-in-pictures does not gesture at all. However, the greatest gesturer I know certainly thinks in pictures; and [Tom Körner](https://en.wikipedia.org/wiki/Tom_Korner) gestures vigorously while lecturing proofs, while also using pictures very frequently as intuition guides. I don't have very much data on this, though, and I would be interested to add more data points.
+
+Anyway, the message to take away is that people don't all think in the same way - if you find yourself trying to communicate something to someone who seems to be refusing obstinately to understand, do consider the possibility that you're not approaching the explanation in a way that the other person can comprehend.
+
+[^study]: I saw this study over a year ago, and I can't find it now. I thought it was reported on [Ars Technica](https://arstechnica.com), but apparently not.
+
+ [1]: https://en.wikipedia.org/wiki/Number_line "Number line Wikipedia page"
+ [2]: https://en.wikipedia.org/wiki/Real_line "Real line Wikipedia page"
diff --git a/hugo/content/posts/2013-08-21-my-experiences-with-flow.md b/hugo/content/posts/2013-08-21-my-experiences-with-flow.md
new file mode 100644
index 0000000..14f4162
--- /dev/null
+++ b/hugo/content/posts/2013-08-21-my-experiences-with-flow.md
@@ -0,0 +1,30 @@
+---
+lastmod: "2022-12-31T23:46:44.0000000+00:00"
+author: patrick
+categories:
+- psychology
+comments: true
+date: "2013-08-21T00:00:00Z"
+aliases:
+- /psychology/flow/
+- /my-experiences-with-flow/
+title: My experiences with flow
+---
+I'm in the middle of reading [Flow](https://en.wikipedia.org/wiki/Flow_(psychology)), by [Mihály Csíkszentmihályi][1], and so far, I love it. It describes the "[flow state](https://en.wikipedia.org/wiki/Flow_%28psychology%29)" of consciousness, that state of "everything is irrelevant except for the task at hand" in which time flies past without your noticing, and you don't notice hunger or thirst or people moving around you. Flow can be induced when performing a difficult task which lies within your abilities, where immediate feedback is provided. I, at least, feel characteristically exhausted after coming out of a long period of flow - but it's a good kind of mental exhaustion, much as the tiredness after a long swim is a good kind of physical exhaustion (in contrast to tiredness-after-a-long-day-of-doing-nothing, which feels sort of lazier and unwholesome). The Wikipedia page is a good enough explanation of flow that I will not describe it further here.
+
+I myself have experienced flow when playing violin in the orchestra, playing the piano solo, playing chess, doing maths, doing sudoku-type puzzles, reading (fiction and non-fiction), doing exams, and programming. These are the examples that come to mind during five minutes' probing of my mind - there may very well be more.
+
+I never experienced flow during a music exam (unfortunately) - this is probably because during an exam, good results are *required*, and this creates a lot of pressure to perform well. As an added source of worry during music exams, you are performing alone, doing something where it is excruciatingly obvious when you get it wrong. If the pressure to do well is too great, it is very hard to achieve flow; Csíkszentmihályi would claim that this is because "the self, in the form of worry and fear, is too strongly asserted when under pressure, and flow requires losing the sense of self". I hold this to be a fringe benefit to the modular system of exams - if you mess up one module, it's not the end of the world.
+
+I no longer experience flow during chess, because I have had several years of hiatus since being fairly good at chess - time enough to forget many of the patterns that I used to be able to see effortlessly - and so the game has become a little too difficult for me to attain flow. However, now that I am playing casually again, I hope that it might become easier to achieve flow during chess. Maths is the main source of flow in my life at the moment. I measure how long I have spent doing a particular problem by how many sheets of paper I have used, and I always end up having used far more sheets that I feel there could possibly have been time for. I also notice flow-tiredness pretty often after doing a long bout of maths, which is a fair indication that I have been in flow. It is from maths that I have come to think that I can kick flow up a notch if required, because I can sit a maths exam, completing (to a better standard than normal) questions that feel harder than those I do for practice, and come out of the exam absolutely exhausted - with flow-tiredness times two. However, the experience feels much the same as "normal flow" at the time - that is, it doesn't really feel like anything, and the problem I'm solving is all there is. It's quite hard to perform accurate introspection on a state of mind in which introspection is entirely halted! I have to rely on evidence from outside the experience: the fact that I am much more tired and have completed more than I would have done in "normal flow". I would be interested to know if anyone else has what they perceive to be two levels of flow. As further evidence in favour of "multiple levels of flow", it is apparently possible for groups to achieve flow together, working on the same task - I struggle to come up with an activity in which the delay of talking is not a huge barrier to flow. (I suppose that if a group can get to the stage where its members are passing around information before it is requested, by anticipating the questions, then it could be possible.) It seems much more likely that a "lesser level" of flow could be achieved in a group, during which interruptions are not so disruptive; I cannot imagine my maths-flow or anything like it surviving with people's talking around me and my having to reply (with anything more sophisticated than "mmm").  
+I go into flow with maths exams ([A-level][2], [BMO][3], [Tripos][4], and the corresponding mock exams), but also in certain other subjects. When writing an essay for my A-level Latin exam, I went into flow. Interestingly, I never went into flow when writing practice essays for Latin - it took the structure and importance of the actual exam to get me into flow. I'm looking for ways to get more flow; I remember that I used to go into flow all the time when I was younger, through doing [sudoku][5] and related puzzles, so this is perhaps an avenue of exploration.
+
+I like the contrast that Csíkszentmihályi gives between "pleasure" and "enjoyment". "Pleasure" is your standard run-of-the-mill emotion, something that hasn't got a strong thought component involved. It is (of course) not a bad thing, but it is somehow less wholesome than "enjoyment", which is the pleasure obtained from investing a large amount of mental energy in something. Watching television is usually pleasurable (assuming it isn't boring, anyway), while reading a really good book is enjoyable - possibly because of the work involved in constructing a mental model, which elevates the act of reading from "pleasure" to "enjoyment". Thinking on the distinction, I realised that in fact most of the things I do as a distraction are not enjoyable, and many are not even pleasurable - forthwith, I am deleting almost all the iOS games I have on my iPod, because their only purpose (from my perspective) is to be used as a time-sink, rather than to provide pleasure. I might take up crosswords again - while not necessarily an easy flow-inducing task (because the crossword is split up into many small but very difficult tasks, rather than small and reasonable ones), it is at least enjoyable.
+
+The book Flow describes many different activities that can induce flow in a skilled practitioner - rock climbing, ice skating, even assembly-line work - so it is not a phenomenon that is confined to mental pursuits. (The skew of this blog post is just because I have never achieved proficiency in any physical activity.) I would be interested to know what activities induce flow in my readers.
+
+ [1]: https://en.wikipedia.org/wiki/Mihaly_Csikszentmihalyi "Mihaly Csikszentmihalyi"
+ [2]: https://en.wikipedia.org/wiki/A-level "A-levels Wikipedia page"
+ [3]: https://en.wikipedia.org/wiki/British_Mathematical_Olympiad "BMO Wikipedia page"
+ [4]: https://en.wikipedia.org/wiki/Mathematical_Tripos "Mathematical Tripos Wikipedia page "
+ [5]: https://en.wikipedia.org/wiki/Sudoku "Sudoku Wikipedia page"
diff --git a/hugo/content/posts/2013-08-22-how-to-punt-in-cambridge.md b/hugo/content/posts/2013-08-22-how-to-punt-in-cambridge.md
new file mode 100644
index 0000000..46b9621
--- /dev/null
+++ b/hugo/content/posts/2013-08-22-how-to-punt-in-cambridge.md
@@ -0,0 +1,43 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-08-22T00:00:00Z"
+aliases:
+- /uncategorized/how-to-punt-in-cambridge/
+- /how-to-punt-in-cambridge/
+title: How to punt in Cambridge
+---
+[When in Cambridge][1]…
+
+The river is always full of beginners and professional puntists. The beginners veer all over the place, getting very wet, while the professionals zip between them, somehow managing to avoid collision by the width of an otter's hair. The worst attempt by a beginner I've ever seen at punting was an attempt to use the pole rather like an oar, without ever touching the bottom of the river with it. This patent perplexity pertaining to the point of the punt provoked a pertinent post.
+
+Step 1: hire a punt. They are usually hired hourly, with a three-minute grace at returning-time. [Scudamore's][2] is the usual punt-hire company for the public; they have punts stationed on the "town" side of Magdalene College (under the bridge there), and another set stationed on the river past Queen's College (or approaching from Pembroke College, cross the road into Mill Lane and go down to the river; you're pretty much there). If you're a student, they charge £16.50/hr, which is absurd. (I have not yet tried any of the following student alternatives.) Nominally, Trinity College does punts at [£4/hr][3] for Trinity members and £12/hr for other students. Magdalene does punts at what may still be £5/hr for Magdalene students, but I believe this is only available during the Easter term (that is, the summer term). Clare College has punts which may or may not be free to Clare students. St John's College has punts for John's students at [£4/hr][5] (also, bizarrely, discount rates for the British Antarctic Survey). Other options may exist, but I don't know about them.
+
+Step 2: ensure that the punt hire people have given you a pole for punting with, and you might want a paddle too, just in case. I have myself accidentally left the pole anchored in the river-bed, and narrowly avoided being left clinging onto it for dear life (à la [Three Men in a Boat][6]) - in this situation, a paddle is invaluable.
+
+Step 3: note your starting time, for payment purposes, and calculate the time you need to be back by. You should probably remember to turn back after you have got a little more than halfway to this end time (as it's always faster once you've got into the swing of punting).
+
+Step 4: the designated first puntist takes the pole and puts it into the water (remembering to keep hold of one end). The water end often has a hook or some other distinguishing feature on it. Then ey walks with the pole to the end of the punt, which will be flat and elevated. This end is the back of the punt; passengers sit in front of you.
+
+Step 5: passengers embark, keeping their hands inside the punt (not gripping the edge, as this can lead to horrific injury).
+
+Step 6: you will be released, or you will have to arrange your own release, by dint of untying whatever ropes attach you to the shore. The puntist immerses the pole until its bottom hits the floor, and then uses this as a lever to push the Earth away from the punt. In this manner, you can rotate the Earth beneath you until you are in the middle of the river. Do try to avoid doing this while people are on top of the Eiffel tower, or they might fall off due to the sudden movement of the Earth. Also avoid hitting other punts.
+
+Step 7: Now that you are thoroughly in reverse, you will need to apply some sort of forward propulsion to avoid ploughing into the other bank of the river and thereby annoying the John's porters and clogging up the river. Do not do this by putting the pole directly behind the punt, as this could cause you to be levered straight off when the punt continues to go backwards. Instead, put the pole behind and to one side of the back of the punt, and push away from you, so that the back of the punt swings away from the pole (and hence the front of the punt swings towards the pole). It will take a lot of pressure to alter course, but once you start changing direction, it is oddly difficult to stop - so do this step as gradually as seems prudent.
+
+Step 8: You should now be parallel with the river, or at least pointing in some direction approximating "water". To go forwards, put the pole directly behind the punt and push against the river bed with it. Steering is accomplished by dint of waving the pole behind you (still trailing in the water, but no longer touching the bottom) - a long stick of wood/metal suffices as a pretty good rudder. I find it helpful to use my own body as a fulcrum. You will find that it is possible to make very large course adjustments very quickly if you apply enough force to the pole, but again remember that changes of direction are hard to start and easy to keep going, so be gentle. Whenever you need more speed, repeat the "push against river bed" manouevre. If you're going quickly, you need to do this quickly, as otherwise the punt will be racing away from the point at which you put the pole in. If you're under a bridge, remember that you won't have enough height available to do this.
+
+For some reason, courtesy dictates that you keep to the right-hand side of the river in the direction you are going. Why it's not the left is beyond me.
+
+Step 9: If you find that the punt is not coming out of the river bed, apply a twisting motion while pulling quite hard. It should come suddenly free (and may overbalance you, so keep your weight low); if it does not, you should let go so that you are not left clinging onto it while the punt drifts away from you. Judicious use of the paddle will get you back to retrieve it.
+
+And that's it. You are now a master puntist. Go forth and wreak havoc upon the river.
+
+ [1]: https://en.wiktionary.org/wiki/when_in_Rome,_do_as_the_Romans_do "When in Rome…"
+ [2]: http://www.scudamores.com/ "Scudamore"
+ [3]: http://www.trin.cam.ac.uk/index.php?pageid=664 "Trinity punting scheme"
+ [5]: http://www.joh.cam.ac.uk/punt-society "St John's punts"
+ [6]: /reading-list "Things Everyone Should Read"
diff --git a/hugo/content/posts/2013-08-24-stumbled-across-24-august-2013.md b/hugo/content/posts/2013-08-24-stumbled-across-24-august-2013.md
new file mode 100644
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--- /dev/null
+++ b/hugo/content/posts/2013-08-24-stumbled-across-24-august-2013.md
@@ -0,0 +1,24 @@
+---
+lastmod: "2022-08-21T11:30:44.0000000+01:00"
+author: patrick
+categories:
+- stumbled_across
+comments: true
+date: "2013-08-24T00:00:00Z"
+aliases:
+- /stumbled_across/stumbled-across-4/
+- /stumbled-across-24-august-2013/
+title: Stumbled across 24th August 2013
+---
+*   The much-vaunted Hyperloop looks really cool, if it could ever be built: <https://arstechnica.com/business/2013/08/hyperloop-a-theoretical-760-mph-transit-system-made-of-sun-air-and-magnets/>
+*   But it may be a bit too half-baked: <https://pedestrianobservations.com/2013/08/13/loopy-ideas-are-fine-if-youre-an-entrepreneur/>
+*   I love a good visualisation: <http://nickolaylamm.com/art-for-clients/what-if-you-could-see-wifi/>
+*   I laughed pretty much constantly through this piece of bureaucracy-hacking: <http://www.slate.com/articles/life/culturebox/2013/08/the_kindly_brontosaurus_the_amazing_prehistoric_posture_that_will_get_you.html>
+*   This is a problem with the Internet of Things as well as with mind-computer interfaces: <http://www.extremetech.com/extreme/134682-hackers-backdoor-the-human-brain-successfully-extract-sensitive-data>
+*   Wow - it's possible to represent words as vectors so that *vector('Paris') - vector('France') + vector('Italy')* results in a vector that is very close to *vector('Rome')*: <https://code.google.com/p/word2vec/>
+*   Let there be food: <http://web.archive.org/web/20150219023941/http://thisiswhyyourefat.kinja.com/cadbury-creme-eggs-benedict-230182670>
+*   One of the manifold reasons why the USA's [TSA][1] should be scrapped: <https://varnull.adityamukerjee.net/2013/08/22/dont-fly-during-ramadan>
+*   An excellent witty dialogue between some experts in their respective fields: <http://mathwithbaddrawings.com/2013/08/21/five-math-experts-split-the-check/>
+*   How to disagree correctly: <http://www.paulgraham.com/disagree.html>
+
+ [1]: https://en.wikipedia.org/wiki/Transportation_Security_Administration "TSA Wikipedia page"
diff --git a/hugo/content/posts/2013-08-26-topology-made-simple.md b/hugo/content/posts/2013-08-26-topology-made-simple.md
new file mode 100644
index 0000000..3747ae2
--- /dev/null
+++ b/hugo/content/posts/2013-08-26-topology-made-simple.md
@@ -0,0 +1,43 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2013-08-26T00:00:00Z"
+math: true
+aliases:
+- /wordpress/archives/364/index.html
+- /mathematical_summary/topology-made-simple/
+- /topology-made-simple/
+title: Topology made simple
+---
+I've been learning some basic [topology][1] over the last couple of months, and it strikes me that there are some *very* confusing names for things. Here I present an approach that hopefully avoids confusing terminology.
+
+We define a **topology** \\(\tau\\) on a set \\(X\\) to be a collection of sets such that: for every pair of sets \\(x,y \in \tau\\), we have that \\(x \cap y \in \tau\\); \\(\phi\\) the empty set and \\(X\\) are both in \\(\tau\\); for every \\(x \in \tau\\) we have that \\(x \subset X\\); and that \\(\displaystyle \cup_{\alpha} x_{\alpha}\\) is in \\(\tau\\) if all the \\(x_{\alpha}\\) are in \\(\tau\\). (That is: \\(\tau\\) contains the empty set and the entire set; sets in \\(\tau\\) are subsets of \\(X\\); not-necessarily-countable unions of sets in \\(\tau\\) are in \\(\tau\\); and finite intersections of sets in \\(\tau\\) are in \\(\tau\\).) We then say that \\((X, \tau)\\) is a **topological space**.
+
+If a set \\(x\\) is in \\(\tau\\), then we say that \\(x\\) is **fibble**. On the other hand, if \\(x^{\mathsf{c}}\\) (the complement of \\(x\\)) is in \\(\tau\\), then we say that \\(x\\) is **gobble**.
+
+We define a **metric space** \\((X,d)\\) to be a set \\(X\\) together with a "distance" function \\(d: X \to \mathbb{R}\\) such that: \\(d(x,y)=0\\) iff \\(x=y\\); \\(d(x,y)=d(y,x)\\); and \\(d(x,y)+d(y,z) \geq d(x,z)\\). (That is, "the distance between two points is 0 iff they're the same point; distance is the same if we reverse as if we go forward; and if we take a detour then the distance is greater".)
+
+We then define a **fiball** \\(B(x,\delta )\\) to be "a set for which every \\(y \in X\\) is within \\(\delta\\) of \\(x\\)" - that is, \\(\{ y \in X: d(x,y)<\delta \}\\).
+
+It turns out that we can create (or **induce**) a topology out of a metric space, by considering the fiballs. Let \\(x \in \tau\\) iff \\(x\\) is a union (not necessarily countable) of fiballs in the metric space. We can see that this is a topology, because unions of (things which are unions of fiballs) are unions of fiballs; the empty set is the union of no fiballs; the entire set \\(X\\) is the union of all possible fiballs; and it can be checked that intersections behave as required (although that takes a tiny bit of work).
+
+Now we see why fiballs are called "fiballs" - because in the induced topology, fiballs are fibble.
+
+We can define a **gobball** in the same way, by making the weak inequality strict in the definition of the fiball. And it can be verified that gobballs are gobble.
+
+We can keep going with these definitions - a **continuous function** between two topological spaces \\(f: (X, \tau) \to (Y, \sigma)\\) is defined to be one such that if \\(y \subset Y\\) is fibble in \\(Y\\), then \\(f^{-1}(y)\\) is fibble in \\(X\\), and so forth.
+
+Eventually we come to the reason that I've used the words "fibble" and "gobble". Consider the metric \\(d: \mathbb{R} \to \mathbb{R}\\) given by \\(d(x,y) = \vert x-y \vert\\). It can easily be checked that \\((\mathbb{R},d)\\) is a metric space, and so it induces a topology on \\(\mathbb{R}\\). What is the fiball \\(B(x,\delta)\\)? It is precisely the set of points which are within \\(\delta\\) of \\(x\\) - that is, the open interval \\((x-\delta, x+\delta)\\). So we know that open intervals are fibble. Note also that \\((1,2) \cup (3,4)\\) is fibble, but is not an open interval. All well and good.
+
+But now consider a different topology on \\(\mathbb{R}\\). Let \\(x\\) be fibble if it is a union of half-open intervals \\([a,b)\\). It can be checked that this is a topology. Now the set \\([1,2) \cup [3,4)\\) is fibble, and note that it is not an open interval. We can see that \\((1,2)\\) is still fibble (it's the union of the fibble sets \\([x, 2)\\) for \\(1<x<1.1\\), for example).
+
+And consider a third, final topology on\\(\mathbb{R}\\). Let \\(x\\) be fibble iff \\(x\\) is \\(\mathbb{R}\\) or the empty set. We can easily see that this is a topology. Now no open interval is fibble.
+
+The problem is that in standard notation, fibble sets are referred to as **open**. It's all fine when you have that open intervals are open in the usual topology, but we can construct a topology in which there is an open set which is not an open interval, and we can construct a topology where no open intervals are open. What madness is this? Why not have a different word, because the meaning is different?!
+
+When I am Master of the Universe, I will reform topology so that it makes sense.
+
+ [1]: https://en.wikipedia.org/wiki/Topology "Topology Wikipedia page"
diff --git a/hugo/content/posts/2013-08-31-slightly-silly-sylow-pseudo-sonnets.md b/hugo/content/posts/2013-08-31-slightly-silly-sylow-pseudo-sonnets.md
new file mode 100644
index 0000000..23e4a86
--- /dev/null
+++ b/hugo/content/posts/2013-08-31-slightly-silly-sylow-pseudo-sonnets.md
@@ -0,0 +1,171 @@
+---
+lastmod: "2022-01-01T22:20:19.0000000+00:00"
+author: patrick
+categories:
+- creative
+- mathematical_summary
+comments: true
+date: "2013-08-31T00:00:00Z"
+math: true
+aliases:
+- /wordpress/archives/379/index.html
+- /creative/mathematical_summary/slightly-silly-sylow-pseudo-sonnets/index.html
+- /slightly-silly-sylow-pseudo-sonnets/index.html
+title: Slightly silly Sylow pseudo-sonnets
+---
+This is a collection of poems which together prove the [Sylow theorems][1].
+
+# Notes on pronunciation
+
+* Pronounce \\( \vert P \vert \\) as "mod P", \\(a/b\\) or \\(\dfrac{a}{b}\\) as "a on b", and \\(=\\) as "equals".
+* \\(a^b\\) for positive integer \\(b\\) is pronounced "a to the b".
+* \\(g^{-1}\\) is pronounced "gee inverse".
+* "Sylow" is pronounced "see-lov", for the purposes of these poems.
+* \\(p\\) and \\(P\\) and \\(n_p\\) are different entities, so they're allowed to rhyme.
+
+# [Monorhymic][4] Motivation [^notsonnet]
+
+Suppose we have a finite group called \\(G\\).  
+This group has size \\(m\\) times a power of \\(p\\).  
+We choose \\(m\\) to have coprimality:  
+the power of \\(p\\)'s the biggest we can see.  
+Then One: a subgroup of that size do we  
+assert exists. And Two: such subgroups be  
+all conjugate. And \\(m\\)'s nought mod \\(n_p\\),  
+while \\(n_p = 1 \pmod{p}\\); that's Three.
+
+# Theorem One
+
+## Little [Lemmarick][5]
+
+*Subtitle: "The size of the normaliser \\(N\\) of a maximal \\(p\\)-subgroup \\(P\\) has \\(N/P\\) coprime to \\(p\\)"*
+
+There was a \\(p\\)-subgroup of \\(G\\)  
+(by Cauchy). The largest was \\(P\\).  
+Let \\(N\\) normalise,  
+Take \\(\dfrac{N}{P}\\)'s size,  
+Suppose that it's zero mod \\(p\\).
+
+---
+
+Now \\(\dfrac{N}{P}\\) also has some  
+p-subgroup (by Cauchy); take one.  
+Take it un-projected,  
+\\(P\\)'s most big? Corrected!  
+We've found one sized \\(p \vert P \vert \\): done.
+
+## Introductory Interlude (to the tune of "[Jerusalem](https://en.wikipedia.org/wiki/Jerusalem_%28hymn%29)")
+
+*Subtitle: "\\(\{P\}\\) is an orbit of size \\(1\\) under the conjugation action of \\(P\\) on the set of \\(G\\)-conjugates of \\(P\\)"*
+
+Let \\(X\\) be \\(P\\)'s orbit under \\(G\\)  
+Acting by conjuga-ti-on.  
+Mod \\(G\\) o'er \\(N\\)'s the size of \\(X\\)  
+The Orbit/Stabiliser's done.  
+And in its turn, \\(P\\) acts on \\(X\\)  
+By conjugating, as before,  
+Then \\(P\\) is certainly all alone:  
+Its orbit is itself, no more.  
+
+---
+
+Let \\(gPg^{-1}\\) be alone,  
+\\(P\\) stabilises it, and hence  
+\\(pgPg^{-1}p^{-1}\\)  
+Is \\(gPg^{-1}\\) - from whence  
+We conjugate by \\(g^{-1}\\):  
+\\(g^{-1}Pg\\) fixes \\(P\\).  
+\\(g^{-1}Pg\\) is in \\(N\\),  
+so \\(\pi\\) applies. From this, we'll see:
+
+## [Cinquain][6] Claim [^cinquain]
+
+*Subtitle: "\\(\{P\}\\) is the only orbit of size \\(1\\)"*
+
+A claim:  
+\\(\pi(g^{-1}Pg)\\) is \\({1}\\).  
+Call it \\(K\\). If false, \\(p\\)  
+divides \\( \vert K \vert \\),  
+as \\(\pi\\)  
+a hom [^hom].  
+Also, \\( \vert K \vert \\)  
+divides \\( \vert N/P \vert \\)  
+(Lagrange). Then Lemmarick proves: \\(K\\)  
+Is \\({1}\\).
+
+## [Trochaic Tetrameter][7] Tying Together [^rhyme]
+
+*Subtitle: "\\(\{P\}\\) is Sylow, since \\(G/N\\) has size coprime to \\(p\\)"*
+
+\\(\pi\\) has kernel \\(P\\) - but also  
+\\(K\\) is \\({1}\\), so lies inside it.  
+\\(P\\) contains \\(g^{-1}Pg\\);  
+Both have size \\(p^a\\). So
+since they're finite, they're the same set.  
+Any set alone in orbit  
+must be \\(P\\). The class equation  
+Tells us \\( \vert G \vert / \vert N \vert \\) is  
+Just precisely \\(1 \pmod{p}\\). Then  
+\\( \vert G \vert / \vert P \vert \\) is not a  
+multiple of \\(p\\) because it's  
+\\( \vert \dfrac{N}{P} \vert \\) multiplied by  
+\\(\dfrac{ \vert G \vert }{ \vert N \vert }\\) and \\(p\\) can't  
+possibly divide those two. So  
+Maximal the power of \\(p\\) is:  
+\\(P\\)'s a Sylow \\(p\\)-subgroup.  
+
+# Theorem Two - Quad-[quatrain][8] [^quatrain]
+
+A Sylow \\(p\\)-subgroup let \\(Q\\) be:  
+a subgroup, size \\(p^a\\).  
+Because it's the same size as was \\(P\\),  
+it acts on \\(X\\) in the same way.
+
+---
+
+Mod \\(p\\), we have \\( \vert X \vert \\) is \\(1\\) -  
+the orbits of \\(Q\\) will divide it;  
+Now invoke the class equation:  
+an orbit, size \\(1\\), lies inside it.  
+
+---
+
+We dub this one \\(gPg^{-1}\\),  
+then \\(g^{-1}Qg\\)'s in \\(N\\).  
+Projection works just as well in verse:  
+\\(\pi(g^{-1}Qg)\\) is \\({1}\\).
+
+---
+
+The previous poem's our saviour:  
+\\(g^{-1}Qg\\) is in \\(P\\).  
+The Pigeonhole tells its behaviour:  
+that \\(P\\) is \\(g^{-1}Qg\\).
+
+# Theorem Three - Hindmost [Haiku][9] [^haiku]
+
+\\( \vert X \vert \\): \\(1 \pmod{p}\\)  
+Orbit \\(X\\) divides \\(G\\)'s size:  
+We have proved the Third.
+
+[^notsonnet]: This is not a sonnet - it is six lines too short, and is monorhymic rather than following a more varied rhyme scheme. I started out intending it to be a sonnet, but all the rhymes for "p", "G" and so forth were irresistible. "Power" is a monosyllable.
+
+[^cinquain]: I use a form of reverse cinquain, with syllable count 2,8,6,4,2,2,4,6,8,2.
+
+[^hom]: "Hom", of course, is short for "homomorphism". Imre Leader used it all the time, so I took it to be legitimate.
+
+[^rhyme]: This section is unrhymed; although Shakespeare rhymes his tetrameter, Longfellow doesn't. The strong iambic nature of English makes enjambement very natural to write when you're constrained to trochees, so I have just gone with the flow.
+
+[^quatrain]: Quatrains have a variety of allowable rhyme schemes, but I plumped for ABAB for the sake of variety. Yes, "N" rhymes with "one". For the purposes of scansion, pronounce each line as the first line of a limerick, with an optional weak syllable at the end if necessary.
+
+[^haiku]: I know that a haiku should mention a season, etc - but that is a constraint I am willing to relax. Gareth pointed out that if "sum" and "size" were synonymous, then " \|X\| : 1 (mod p)/Orbit X divides G's sum/A proof of the Third" would mention the season "sum-A".
+
+ [1]: {{< ref "2013-06-26-sylow-theorems" >}}
+ [2]: http://tartarus.org/gareth/
+ [3]: http://mmeblair.tumblr.com/post/61532912275/carnival-of-mathematics-102-my-summation-of-other
+ [4]: https://en.wikipedia.org/wiki/Monorhyme
+ [5]: https://en.wikipedia.org/wiki/Limerick_%28poetry%29
+ [6]: https://en.wikipedia.org/wiki/Cinquain
+ [7]: https://en.wikipedia.org/wiki/Trochaic_tetrameter
+ [8]: https://en.wikipedia.org/wiki/Quatrain
+ [9]: https://en.wikipedia.org/wiki/Haiku
diff --git a/hugo/content/posts/2013-09-13-stumbled-across-14th-september-2013.md b/hugo/content/posts/2013-09-13-stumbled-across-14th-september-2013.md
new file mode 100644
index 0000000..81b6f5e
--- /dev/null
+++ b/hugo/content/posts/2013-09-13-stumbled-across-14th-september-2013.md
@@ -0,0 +1,29 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stumbled_across
+comments: true
+date: "2013-09-13T00:00:00Z"
+aliases:
+- /stumbled_across/stumbled-across-14th-september-2013/
+- /stumbled-across-14th-september-2013/
+title: Stumbled across 14th September 2013
+---
+*  On the merits of silence (I wholeheartedly agree): <http://www.nytimes.com/2013/08/25/opinion/sunday/im-thinking-please-be-quiet.html>
+
+*  Given the previous results on humans' sense of physical location, I'm not particularly surprised that you can make yourself identify your body as being somewhere other than where it really is: <http://www.psychologicalscience.org/index.php/news/releases/visualized-heartbeat-can-trigger-out-of-body-experience.html>
+
+*  Aaand the future arrives: <http://www.washington.edu/news/2013/08/27/researcher-controls-colleagues-motions-in-1st-human-brain-to-brain-interface>
+
+*  Another reason why Finland is amazing: <http://web.archive.org/web/20150302131427/http://neomam.com/blog/there-is-no-homework-in-finland>
+
+*  A thought-provoking story: [WebCite version](http://web.archive.org/web/20010802144026/http://www.tor.com/72ltrs.html)
+
+*  On the "mundane magics" kind of lines: <http://i.imgur.com/hINj1xf.png>
+
+*  Not sure what to make of this - I actually can't remember who narrated Paddington in the audio-books of my youth: <http://www.bbc.co.uk/news/entertainment-arts-24077834>
+
+*  This links in heavily with the thesis of the book [Flow](https://en.wikipedia.org/wiki/Mihaly_Csikszentmihalyi#Flow), which I'm reading at the moment: <https://www.washingtonpost.com/news/wonk/wp/2013/09/13/being-poor-changes-your-thinking-about-everything/>
+
+This is the first post that I'm syndicating to social media. I hope it works. (If anyone has any ideas about what to syndicate - for instance, that Stumbled Across posts should not, or especially should, or things like that - then do let me know.)
diff --git a/hugo/content/posts/2013-09-21-how-to-prove-that-you-are-a-god.md b/hugo/content/posts/2013-09-21-how-to-prove-that-you-are-a-god.md
new file mode 100644
index 0000000..f8a9ff5
--- /dev/null
+++ b/hugo/content/posts/2013-09-21-how-to-prove-that-you-are-a-god.md
@@ -0,0 +1,38 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-09-21T00:00:00Z"
+aliases:
+- /uncategorized/how-to-prove-that-you-are-a-god/
+- /how-to-prove-that-you-are-a-god/
+title: How to prove that you are a god
+---
+I came across an interesting question while reading the blog of [Scott Aaronson][1] today. The question was as follows:
+
+> In the world of the colour-blind, how could I prove that I could see colour?
+
+I'm presuming, to make the discussion more life-like and less cheaty, that this civilisation hasn't discovered that light comes in wavelengths, or that it has but it can't distinguish very well between wavelengths (so that all coloured light falls into the same bucket of 100nm to 1000nm, for instance). The challenge is to design an experimental protocol to confirm or deny that I have access to information that the colour-blind do not. This question is much harder than the corresponding question in the world of the blind, because having vision tells you so much more than having colour vision (simply set up a flag two miles away, have someone raise it at a random time, note down the time you saw it raised, and compare notes).
+
+Oh. That's unfortunate. This protocol works perfectly well to determine colour, too - I just need to provide two flags of different colour, present them for inspection so that the experimenters verify that they look identical to them, mark the base of the flagpoles A and B in some way that the colour-blind can detect (etching?), note down which colour corresponds to which flag, walk a hundred metres, have the experimenter wave one of the flags at random, write down which flag was waved, repeat to taste.
+
+How about a proof that I could hear when no-one else knew what hearing was? I would need to find something that I could hear that no-one else could detect - perhaps the dropping of a vase fifty metres away - and, while blindfolded, raise my hand when I heard the vase drop. I would, of course, have to remember to explain that there could be a time delay over long distances.
+
+Stereo sound (the ability to detect where something is by the sound it makes)? I shut my eyes, someone walks around me and claps once; I point to that person.
+
+Smell? Easy - simply uncork a test tube of water or hydrogen sulphide. I can identify which was used.
+
+Taste? Again, we could dissolve sugar and salt separately into water.
+
+[Proprioception][2]? It seems odd to me that any physical being could have managed to evolve language and not proprioception, but I could at least demonstrate the ability to exercise fine control over my body by pulling the spring of a [Newton meter][3] with a toe, finger, mouth, etc. This should be good evidence that I know how much strength I am exerting. I could also do this blind (although my results would be more rough, because I'm not a good proprioceptor), and I could also not see the scale of the Newton meter. To test awareness of where my body parts are, I could place both hands behind my back and have someone move one of them (with me blindfolded); I would then touch the moved hand with my other hand.
+
+Language (the fact that "the sounds I am making convey information")? This would require two people who spoke the same language, of course. We could be placed in separate booths, with a row of pictures in front of us. Someone would point to a picture in one booth, the person in the relevant booth would describe it, the other person would point to the corresponding picture.
+
+This discussion turned out to be less interesting than I would have liked. Anyway, it would appear to imply that if someone did indeed have extra senses, that person would easily be able to convince me of this fact. For instance, in the world of the colour-blind, I would present two items, saying "These items differ in a property which I can sense and which you cannot; show me one and I will tell you which it was". If the experiment were repeated and I were consistently able to say which item was shown, then I think this should count as proof that I can see colour. Of course, any limitations on my power ("I can't necessarily distinguish between any two items, but I can distinguish between these two", or "I can only distinguish between two items when it's a full moon in three days and when I've received a blood sacrifice and when the experimenter has sufficient faith in my abilities") should be declared up front, so that they can't be used to explain away failure (so, for instance, we could in advance find someone very credulous). Parallels to the [Randi prize][4] fully intended.
+
+ [1]: http://www.scottaaronson.com
+ [2]: https://en.wikipedia.org/wiki/Proprioception
+ [3]: https://en.wikipedia.org/wiki/Spring_scale
+ [4]: http://www.skepdic.com/randi.html
diff --git a/hugo/content/posts/2013-10-10-plot-armour.md b/hugo/content/posts/2013-10-10-plot-armour.md
new file mode 100644
index 0000000..5ab2e25
--- /dev/null
+++ b/hugo/content/posts/2013-10-10-plot-armour.md
@@ -0,0 +1,51 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- creative
+comments: true
+date: "2013-10-10T00:00:00Z"
+aliases:
+- /creative/plot-armour/
+- /plot-armour/
+title: Plot Armour
+---
+*Wherein I dabble in parodic fiction. The title refers to the TV Tropes page on [Plot Armour][1], but don't follow that link unless you first resolve not to click on any links on that page. TV Tropes is the hardest extant website from which to escape.*
+
+Jim, third-in-command of the Watchers, ducked behind the Warlord's force-field, desperately trying to catch his breath in the face of an inexorable onslaught. His attackers, the hundred-strong members of the Hourglass Collective, had never been defeated in pitched battle. As testament to their ability, two thousand of the finest troops the Watchers had to offer stood motionless around him, suspended in time; even now, even with five of the most experienced Watchers still fighting, the Hourglass forces were calmly and efficiently slitting the throats of the frozen soldiers. Skilled in cultivating terror, they were working in from afar, and it looked to Jim as though he would have to endure another half-hour of helplessness before they got to him at last. Jim and the Warlord had only survived this far by virtue of an accidental and uncontrollable burst of power from the Founder of the Watchers, released at a fortuitous moment to counter the time-suspension channelled by the Hourglass. That had given the Warlord time to protect five people, before the Founder had collapsed.
+
+Sophia, the Second Vigilant, most powerful of the Watchers, the Founder's first recruit, was still fighting. She had been the recipient of the Warlord's first force-field, naturally, and she was using her borrowed time well. Jim was recovering, moving nearer to her, and her power waxed correspondingly: he was exerting his power to heal her and to fuel her efforts. She began to glow, first dimly but soon as bright as the moon and then as the sun on a cloudy day, and as her light fell on the ranks of the Hourglass, all movement across the battlefield stopped. Sophia gently closed her eyes; in response, threads of light began to take shape around the Hourglass, weaving a net to contain the enemy. Too slowly: a pulse of power blasted forth from the Collective, tearing through the weave and ripping away the Warlord's force-fields. Sophia teetered on her feet, her power spent, but Jim was too far away, having been frozen in place by the calm Sophia had laid on the battlefield. She fell even as he ran towards her, his healing power growing as he did so, but he was too late to stop her from falling unconscious.
+
+Three remaining Watchers, against a hundred of the Hourglass. The Warlord had used everything he had to create his force-fields. Jim had no offensive abilities at all. That left Christine, who (as the recipient of the Warlord's final, weakest, force-field) had been badly affected by the Collective's retaliation. Even with Jim's presence already staunching her head wound, her skill of intuition was still very much off-kilter. Her mind was sluggish, the chains of correlation and causation drifting to her as through treacle.
+
+After far too long, the first key insight came to her.
+
+"Warlord! Jim! Do you remember anything at all from before you threw up the force-fields?" she whispered to him, with as little voice as she could manage. Even that would have been audible to some of the far-away Hourglass, such was the eerie silence over the battlefield.
+
+Her two comrades stared at her in confusion for ten seconds. At last, the Warlord's furrowed brow cleared, and he announced proudly that he could recall the whole series of events in perfect detail. Jim nodded along.
+
+Christine closed her eyes. She, too, could now remember the assassination, the declarations of war, the summoning of the Watchers, and the start of the battle. Odd - but the exchange slipped from her mind as she made another connection.
+
+"Since when could anyone stop time?! How can the Hourglass possibly have the power to suspend an entire army? How come you can heal us, Jim? These aren't normal things for humans to be able to do!"
+
+At the edge of her mind, she could feel an explanation forming, but she was thoroughly spooked by now, and she squashed the nascent reasoning. The final piece clicked into place.
+
+"Jim - say something. Anything - recite the first ten digits of pi, in as normal a voice as you can," she ordered.
+
+"Three point one four one five nine two six…" Jim recited. He was beyond thinking that Christine was being weird, requiring the value of pi with an army advancing upon them - he had long since learnt to go with her requests for information, as you could never tell which piece of data would cause everything to make sense to her superhuman intuitive powers.
+
+"No - can you say it more *normally*?" Christine clarified, cutting him off.
+
+In a monotone, Jim reeled off "Three point one four one five nine two six…"
+
+That was all the confirmation Christine's inductive power needed.
+
+"OK. This will be a shock to you both, Warlord, Jim, but we're in a story. We're fictional. This situation we're in makes no sense at all. We had no backstory until I explicitly requested it, and it took a little while to come to us. And no-one seems to be capable of just *saying* something! Every time, we're ordering, or clarifying, or reeling things off, but never *saying*! We are fictional, and our author is not particularly competent to boot. That gives us a way out of our conveniently dramatic Dire Straits.
+
+"Author! We're three of the most powerful members of the Watchers, and we've been the entire focus of this short story. There are no other plausible protagonists. You must find a way for us to survive, or else the story ends and you will have wasted all this time on another creative endeavour that came to nothing!"
+
+The Hourglass were approaching faster now, provoked by Christine's loud outburst. Only thirty feet away, then twenty, then ten.
+
+The front runner drew a dagger, and slit Jim's throat, then the Warlord's, then Christine's.
+
+ [1]: http://tvtropes.org/pmwiki/pmwiki.php/Main/PlotArmor
diff --git a/hugo/content/posts/2013-10-11-meaning-what-you-say.md b/hugo/content/posts/2013-10-11-meaning-what-you-say.md
new file mode 100644
index 0000000..964beab
--- /dev/null
+++ b/hugo/content/posts/2013-10-11-meaning-what-you-say.md
@@ -0,0 +1,34 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-10-11T00:00:00Z"
+aliases:
+- /uncategorized/meaning-what-you-say/
+- /meaning-what-you-say/
+title: Meaning what you say
+---
+In conversation with (say, for the purposes of propagating a sterotype) humanities students, I am often struck by how imprecisely language is used, and how much confusion arises therefrom. A case in point:
+
+> A: I think that froogles should be sprogged!
+> 
+> B: Sprogging froogles would make the bimmers go plog.
+> 
+> A: But I use froogles all the time - I don't care about the bimmers! Why are you so caught up on the plogging of bimmers?
+
+Here, we see Person A espousing a view, Person B contributing a fact, and Person A responding as if the fact were an attack. This happens *all the time*, and it's not just that I'm incapable of sounding non-threatening, because Person A-style people seem to respond in the same way whoever's acting as Person B. It may well be an excellent tactic during a competitive debate, because a Person A-style response makes you sound impassioned and commanding. However, when it comes to attempting to divine truth, it's thoroughly detrimental. Person B has to spend the next few sentences saying that ey's not attacking A [^pun] - and that's time wasted which could have been spent discussing bimmer-plogging and its relevance.
+
+As a mathematician, you quickly learn to be able to shift into a state of mind in which you mean exactly what you say, and no more. Without this skill, I suspect it is very hard to be a mathematician. Imagine I, as Person B, said "10 is not a multiple of 3", and you (as person A) replied, "But 10 is a multiple of 2, and you didn't mention that!" You would be laughed out of the room, because it is simply taken for granted that I didn't mean to say anything beyond "10 is not a multiple of 3".
+
+Similarly, as a truth-finder (as opposed to debater), I should have the freedom to say "If the cinema were closed, it would very likely have little to no impact on your life" without my interlocutor assuming that I mean "The cinema should be closed" or "The cinema will be closed" or "You are a moron".[^cinema] Fine, in a competitive debate, no holds are barred, but in real life we should be trying to find truth, and it's much harder to do that if you have to keep clarifying every statement. "If the cinema were to be closed (and it might not be), then it would very likely have little to no impact on your life, but I'm not saying that its overall cultural value shouldn't mean that the cinema ought to stay" is considerably less easy to read and write. I, as its author, have limited room in my memory to store all the little dangly bits of sentence that I intend to include. You, as its reader, have limited room in your memory, some of which is taken up in holding irrelevant points like "Patrick is not arguing with you".
+
+Once Person A responds in that way, it becomes much harder for Person B to maintain a calm fact-finding frame of mind. It flashes through B's mind, "Person A has just attacked me! I must defend myself!", even if ey is trying as hard as possible to be balanced and to think clearly. [^experience] It clouds the rest of the discussion.
+
+Essentially, what I want is for everyone to receive training in meaning exactly what you say, and in understanding exactly what is said. I find that it adds greatly to pretty much every conversation if all parties are able to switch into this mode as necessary (to resolve some particular question of fact, for instance). I recognise that my causing offence to another person is always a failing on my part, but it is a lot of work to maintain the context of "I must de-offendify every factual statement". If nothing else, it's just one more thing I have to remember to do once I've realised what words are coming out of my mouth. Fine for normal conversation, since so much of that is based around appearing as un-offensive [^inoffensive] as possible, but it is a great burden when you're attempting to perform a distributed computation (namely, using two or more brains to discover whether a statement is true or false).
+
+[^pun]: See what I did there? "Ey, A"?
+[^cinema]: This example comes from a discussion about a certain news story about anti-monopoly laws in Cambridge cinemas (entitled "Cambridge set to lose Cineworld or Arts Picturehouse following Competition Commission ruling", of Cambridge News, published 2013-10-08 and now defunct). In fact, in this example, the cinema will be *sold on*, not closed - another reason for clear mathematical thinking in distinguishing "if X then Y" from "X is true". 
+[^experience]: I am extrapolating from my own experience here - I am not yet well-enough practised at adopting the frame of mind that "the other person is only attacking me because ey doesn't know better - it doesn't really count".
+[^inoffensive]: Not inoffensive, which is something a little different.
diff --git a/hugo/content/posts/2013-10-13-training-away-mental-bias.md b/hugo/content/posts/2013-10-13-training-away-mental-bias.md
new file mode 100644
index 0000000..cadb376
--- /dev/null
+++ b/hugo/content/posts/2013-10-13-training-away-mental-bias.md
@@ -0,0 +1,46 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- psychology
+comments: true
+date: "2013-10-13T00:00:00Z"
+aliases:
+- /psychology/training-away-mental-bias/
+- /training-away-mental-bias/
+title: Training away mental bias
+---
+*In which I recount an experiment I have been performing. Please be aware that in this article I am in "[meaning what I say][1]" mode.*
+
+For the past year or so, I have been consciously trying to identify and counteract places in the "natural", everyday use of language in which gender bias is implicitly assumed to be correct. The kind of thing I mean is:
+
+> A: I called the plumber.
+> 
+> B: And what did he say?
+
+I have also been keeping tabs on the way the word "man" creeps into occupations and so forth:
+
+> We are looking for a new chairman for the society.
+
+Specifically, I did the following to counter this culturally-imposed tendency:
+
+1.  I switched to using gender-neutral pronouns in my writing (although more recently, I have reverted to "he-or-she" in conversation)
+2.  I formed a habit of noticing whenever I thought the words "she" or "he", and checking whether I actually knew the gender of the person in question
+3.  If for some reason I need to invent a person, and gender-neutral won't do, I flip a coin to determine that character's gender (which will, in theory, completely eliminate gender bias in my characters)
+
+I think that I have succeeded in correcting the bias, at least partially. A week ago, I was even caught by surprise when someone referred to an electrician of unspecified gender as "he" - I had to backtrack mentally and work out whether I'd missed the specification of eir gender, before I realised that this was simply the usual bias being demonstrated by other people. In much the same way, it would surprise me for an electrician to be *assumed* to be called Fred [^coinflips] ("I phoned for an electrician to fix our wiring. Fred was amazing."), or to be gluten-intolerant ("I phoned for an electrician to fix our wiring, but of course he couldn't eat the cheesecake I'd made for him."), or to be particularly tall ("I phoned for an electrician to fix our wiring, but she had the obvious trouble moving around in the attic.").
+
+This is not to proselytise - I've never pointed out people's gender bias unless they've specifically asked for me to do something similar, because in my experience (sample size of 1, when I explained the problem as I see it to someone) people get annoyed and call me a "feminist". [^slight] I don't understand why people would get annoyed that I would like women and men to be treated equally, and the issue is further obscured by the labelling of my views as "feminism". Not that I am against feminism particularly, but using the word "feminist" just clouds the issue. In the same way, calling yourself a "liberal" encompasses an enormous range of policies, and I may not agree with every single one to identify myself as a liberal. For me to be a "feminist" could be interpreted by some as "this person wishes for men to be replaced entirely by women", rather than the interpretation I would prefer (namely, "this person wishes for males to be treated as fairly as females in all things"). Classifying your argument immediately makes everything harder for all parties, as it then sets up a pressure to remain consistent with the entire category you have given, rather than with what you intended to convey. Silly example: "I'm a utilitarian" vs "I think people should act so as to maximise the happiness of people around them".
+
+Given that the person-who-got-annoyed in question was female, I don't think I have the right to overrule her position. [^parallel]
+
+Anyway, I think that my attempt to realign my thoughts so that the implicit, unannounced anti-female bias is less pronounced has been a success. I do not claim perfection, of course, and I will keep going with my new habits (they're a part of me now, so it's easier to keep going than not). I have no real-world outcomes to measure, apart from being acutely aware of everyone else's bias [^mean] - I intend at some point to take a test to give me a quantitative answer, but at present I can't find the particular test I have in mind. [^test] I hope these habits are having a good effect on my thinking.
+
+
+[^coinflips]: I'm flipping coins for the genders in this paragraph, too.
+[^slight]: As if that were some kind of horrendous slight upon my good name.
+[^parallel]: Although I can't help drawing a parallel with extremely wealthy people claiming that "money is unimportant"; it feels as bogus as a philosopher claiming that "truth is relative" - which is simply asking for you not to believe em.
+[^mean]: Again, I mean what I say: I do not necessarily mean that "I am unbiased" or "I am unaware of my own bias".
+[^test]: Its protocol is to flash up pairs of words like \{"good","female"\} or \{"uncle, male"\}, whereupon the testee presses a button for "related" and a different button for "unrelated". The idea is that it is easy to determine whether \{"uncle","male"\} are related, but if bias is present then \{"competent","female"\} will be harder (and hence slower) to determine than \{"competent", "male"\}, because we are so used to thinking of males implicitly as more competent.
+
+[1]: {% post_url 2013-10-11-meaning-what-you-say %}
diff --git a/hugo/content/posts/2013-10-20-the-ravenous.md b/hugo/content/posts/2013-10-20-the-ravenous.md
new file mode 100644
index 0000000..4aa3a78
--- /dev/null
+++ b/hugo/content/posts/2013-10-20-the-ravenous.md
@@ -0,0 +1,56 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- creative
+comments: true
+date: "2013-10-20T00:00:00Z"
+aliases:
+- /creative/the-ravenous/
+- /the-ravenous/
+title: The Ravenous
+sidenotes: true
+---
+[Once upon a midnight dreary][1], while I pondered, weak and weary,  
+I required a snack to feed me. Reaching in the kitchen drawer -  
+With the scissors, cut the wrapping, I revealed a jar of tapen-  
+Ade of olives. Gently snapping, snapping off the lid, I saw:  
+Lines of mouldy olive scored the tapenade. The lid I saw  
+Speckled with each mocking spore.
+
+How the pangs of hunger rumbled while I cursed the jar I'd fumbled;  
+Indistinct, I faintly mumbled, "May this torture last no more!"  
+Suddenly I saw the bread bin; eagerly towards it edging,  
+Bravely to my stomach pledging, pledging food would be in store.  
+Opening that sacred vessel, only crumbs were left in store.  
+Savagely the bag I tore.
+
+Now my thoughts turned to basmati; I would make a dish quite hearty,  
+And my shattered brain was party to such plans of starch galore.  
+Trembling I imagined sauces rich in spice and such resources,  
+Gripped by these enchanting forces, opened I the cupboard door.  
+Slavering, excitement mounting, opened I the cupboard door;  
+Rice stocks were exceeding poor.
+
+How my stomach needed filling. Dreams of pancakes gently grilling  
+Served to give me eager willingness to find a bag of {{< side right flour-footnote `flour.`>}} Pronounced "floor". {{< /side >}}  
+Happily it was not lacking. Took the eggs out from their packing,  
+Fetched a bowl, and in it cracking, cracking eggs so batter'd pour.  
+Tipped the milk (blue top, full-fat) in, mixing up so batter'd pour.  
+Sugar I could not ignore.
+
+Took out oil, and put the gas on. Measured out a goodly ration,  
+Ladled it in practised fashion, spread it thin, my movements sure.  
+Round the edges batter bubbled, far too quiet. The heat I doubled;  
+Soon I'd be no longer troubled: hunger'd bother me no more.  
+Oh, to be no longer troubled, hunger both'ring me no more.  
+Crêpes: a food which I adore.
+
+Tested I the pancake, dipping fish-slice in to start its flipping;  
+Grabbed the pan, towards me tipping. "Now be cooked!" I did implore.  
+In my eagerness to turn it (lest I tarry and I burn it)  
+With such horror I discern it: I had dropped it to the floor.  
+Ah, with terror I discern that I had dropped it to the floor.  
+Quoth the pancake: "Nevermore."
+
+ [1]: https://en.wikipedia.org/wiki/The_Raven "The Raven Wikipedia page"
diff --git a/hugo/content/posts/2013-10-24-how-to-do-analysis-questions.md b/hugo/content/posts/2013-10-24-how-to-do-analysis-questions.md
new file mode 100644
index 0000000..3fe711b
--- /dev/null
+++ b/hugo/content/posts/2013-10-24-how-to-do-analysis-questions.md
@@ -0,0 +1,115 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+- proof_discovery
+comments: true
+date: "2013-10-24T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/how-to-do-analysis-questions/
+- /how-to-do-analysis-questions/
+title: How to do Analysis questions
+---
+This post is for posterity, made shortly after [Dr Paul Russell][1] lectured Analysis II in Part IB of the Maths Tripos at Cambridge. In particular, he demonstrated a way of doing certain basic questions. It may be useful to people who are only just starting the study of analysis and/or who are doing example sheets in it.
+
+The first example sheet of an Analysis course will usually be full of questions designed to get you up and running with the basic definitions. For instance, one question from the first example sheet of Analysis II this year is as follows:
+
+> Show that if \\((f_n)\\) is a sequence of uniformly continuous real functions on \\(\mathbb{R}\\), and if \\(f_n \to f\\) uniformly, then \\(f\\) is uniformly continuous.
+
+This is one of those questions which only exists to make sure that you know what "uniformly continuous" and "converges uniformly" mean.
+
+How do we solve this question? The key with a definitions-question is to avoid employing the brain wherever possible. So the first step is to define \\((f_n)\\) and \\(f\\), and to write down everything we know about them:
+
+*   Let \\((f_n)\\) be a sequence of uniformly continuous real functions on \\(\mathbb{R}\\), and \\(f\\) a real function on \\(\mathbb{R}\\), such that \\(f_n \to f\\) uniformly.
+*   Since each \\(f_n\\) is uniformly continuous, we have that for all \\(n\\), we have for every \\(\epsilon\\) there is a \\(\delta\\) such that for all \\(x,y\\) with \\(\vert y-x \vert < \delta\\), it is true that \\(\vert f_n(y)-f_n(x) \vert < \epsilon\\).
+*   Since \\(f_n \to f\\) uniformly, we have that for all \\(\epsilon\\), there exists \\(N\\) such that for all \\(n \geq N\\), for every \\(x\\) we have \\(\vert f_n(x)-f(x) \vert < \epsilon\\).
+
+Now, what do we want to prove?
+
+*   [Don't write this down yet - this line goes at the end of the proof!] Therefore, for every \\(\epsilon\\) there is a \\(\delta\\) such that for all \\(x,y\\) with \\(\vert x-y  < \delta\\), \\(\vert f(y)-f(x) \vert < \epsilon\\). Hence \\(f\\) is uniformly continuous.
+
+So what can we get from what we know? Everything we know is about "for all \\(\epsilon\\)". So we fix an arbitrary \\(\epsilon\\). If we can prove something that is true for this \\(\epsilon\\), with no further assumptions, then we are done for all \\(\epsilon\\).
+
+*   Fix arbitrary \\(\epsilon\\) greater than \\(0\\).
+
+Now what do we know?
+
+*   Since each \\(f_n\\) is uniformly continuous, we have that for all \\(n\\), there is a \\(\delta\\) such that for all \\(x,y\\) with \\(\vert y-x \vert < \delta\\), it is true that \\(\vert f_n(y)-f_n(x) \vert < \epsilon\\).
+*   Since \\(f_n \to f\\) uniformly, we have that there exists \\(N\\) such that for all \\(n \geq N\\), for every \\(x\\) we have \\(\vert f_n(x)-f(x)\vert < \epsilon\\).
+*   \\(\epsilon > 0\\).
+
+Aha! Now we have a definite something existing (namely, the \\(N\\) in the second condition). Let's fix it into existence.
+
+*   Let \\(N\\) be such that for all \\(n \geq N\\), for every \\(x\\) we have \\(\vert f_n(x)-f(x)\vert < \epsilon\\).
+
+What do we know?
+
+*   Since each \\(f_n\\) is uniformly continuous, we have that for all \\(n\\), there is a \\(\delta\\) such that for all \\(x,y\\) with \\(\vert y-x \vert < \delta\\), it is true that \\(\vert f_n(y)-f_n(x)\vert < \epsilon\\).
+*   Since \\(f_n \to f\\) uniformly, we have that for all \\(n \geq N\\), for every \\(x\\) we have \\(\vert f_n(x)-f(x)\vert < \epsilon\\).
+*   \\(\epsilon > 0\\), and \\(N\\) is an integer.
+
+Now, we have two "for all"s competing with each other. The more specific is the second one, so we'll fix that into existence.
+
+*   Fix arbitrary \\(n\\) greater than or equal to \\(N\\).
+
+What do we know?
+
+*   Since each \\(f_n\\) is uniformly continuous, we have that for all \\(n\\), there is a \\(\delta\\) such that for all \\(x,y\\) with \\(\vert y-x \vert < \delta\\), it is true that \\(\vert f_n(y)-f_n(x)\vert < \epsilon\\).
+*   Since \\(f_n \to f\\) uniformly, we have that for every \\(x\\), \\(\vert f_n(x)-f(x)\vert < \epsilon\\).
+*   \\(\epsilon > 0\\), and \\(N\\) is an integer, and \\(n \geq N\\).
+
+Now we have a choice of "for all"s again, but this time they aren't "talking about the same thing" (last time, both were integers referring to which \\(f_n\\) we were talking about; this time, one is an integer and one is an arbitrary real). However, now we have \\(n \geq N\\) which we can talk about; let's wring more information out of it, by using the "uniformly continuous" bit.
+
+*   Since each \\(f_n\\) is uniformly continuous, there is a \\(\delta\\) such that for all \\(x,y\\) with \\(\vert y-x\vert < \delta\\), it is true that \\(\vert f_n(y)-f_n(x)\vert < \epsilon\\).
+*   Since \\(f_n \to f\\) uniformly, we have that for every \\(x\\), \\(\vert f_n(x)-f(x)\vert < \epsilon\\).
+*   \\(\epsilon > 0\\), and \\(N\\) is an integer, and \\(n \geq N\\).
+
+Aha - another "there exists" condition (on \\(\delta\\)). Let's fix it.
+
+*   Fix \\(\delta\\) such that for all \\(x,y\\) with \\(\vert y-x\vert < \delta\\), it is true that \\(\vert f_n(y)-f_n(x)\vert < \epsilon\\).
+
+What do we know?
+
+*   Since each \\(f_n\\) is uniformly continuous, for all \\(x,y\\) with \\(\vert y-x\vert < \delta\\), it is true that \\(\vert f_n(y)-f_n(x) \vert < \epsilon\\).
+*   Since \\(f_n \to f\\) uniformly, we have that for every \\(x\\), \\(\vert f_n(x)-f(x)\vert < \epsilon\\).
+*   \\(\epsilon > 0\\), and \\(N\\) is an integer, and \\(n \geq N\\), and \\(\delta > 0\\).
+
+Two more "for all" conditions. Let's fix them into existence:
+
+*   Let \\(x\\) be an arbitrary real, and let \\(y\\) be such that \\(\vert y-x\vert < \delta\\).
+
+What do we know?
+
+*   Since each \\(f_n\\) is uniformly continuous, \\(\vert f_n(y)-f_n(x) \vert < \epsilon\\).
+*   Since \\(f_n \to f\\) uniformly, we have that \\(\vert f_n(x)-f(x) \vert < \epsilon\\).
+*   \\(\epsilon > 0\\), and \\(N\\) is an integer, and \\(n \geq N\\), and \\(\delta > 0\\), and \\(x\\) is real, and \\(\vert y-x\vert < \delta\\).
+
+Now the conditions are really small things. It's kind of unclear how to proceed from here, so let's look at what we wanted to prove again:
+
+> For every \\(\epsilon\\) there is a \\(\delta\\) such that for all \\(x,y\\) with \\(\vert x-y\vert < \delta\\), \\(\vert f(y)-f(x)\vert < \epsilon\\).
+
+Applying what we know, this becomes:
+
+*   [to be proved] For all \\(y\\) with \\(\vert x-y\vert < \delta\\), \\(\vert f(y)-f(x)\vert < \epsilon\\).
+
+Aha! We have already got something to do with \\(y\\) (namely that \\(\vert f_n(y)-f_n(x)\vert < \epsilon\\)), and we have something to do with \\(f(x)\\) (namely that \\(\vert f_n(x)-f(x)\vert < \epsilon\\)). Hence \\(\vert f_n(y)-f_n(x)\vert + \vert f_n(x)-f(x)\vert < 2\epsilon\\), and the triangle inequality gives us that \\(\vert f_n(y)-f(x)\vert < 2\epsilon\\). Eek - we need to turn that \\(f_n(y)\\) into an \\(f(y)\\). We have no way of doing that, so we must have missed out some information somewhere. Backtracking, the nearest-to-the-end bit of missed out information was when we fixed \\(x, y\\). We threw away information in "for every \\(x\\), \\(\vert f_n(x)-f(x)\vert < \epsilon\\)" when we fixed \\(x\\) - it applies to \\(y\\) too. So we'll add a new statement to the "what do we know?" list:
+
+*   \\(\vert f_n(y)-f(x)\vert < 2\epsilo\\)n
+*   \\(\vert f_n(y)-f(y)\vert < \epsilon\\).
+*   \\(\epsilon > 0\\), and \\(N\\) is an integer, and \\(n \geq N\\), and \\(\delta > 0\\), and \\(x\\) is real, and \\(\vert y-x \vert < \delta\\).
+
+And now it just drops out of the triangle inequality that \\( \vert f(y)-f(x) \vert < 3 \epsilon\\).
+
+Now, \\(\epsilon\\) was arbitrary, \\(N\\) was dictated by the conditions, \\(n \geq N\\) was arbitrary, \\(\delta\\) was dictated by the conditions, \\(x\\) was arbitrary, \\(y\\) was arbitrary subject to \\(\vert y-x \vert < \delta\\).
+
+Hence we have proved that for every \\(\epsilon\\) there exists \\(N\\) such that for all \\(n \geq N\\) there is a \\(\delta\\) such that for all \\(x\\), for all \\(y\\) with \\(\vert y-x\vert  < \delta\\), \\(\vert f(y)-f(x)\vert  < 3\epsilon\\).
+
+We can clean this statement up. Notice that neither \\(n\\) nor \\(N\\) was involved in the final expression, so we can simply get rid of them to obtain:
+
+> For every \\(\epsilon\\) there is a \\(\delta\\) such that for all \\(x\\), for all \\(y\\) with \\(\vert y-x\vert  < \delta\\), \\(\vert f(y)-f(x) \vert < 3\epsilon\\).
+
+From this, it is easy to obtain the required result. We want to turn \\(3 \epsilon\\) into \\(\epsilon\\) - but that's fine, because the expression holds for every \\(\epsilon\\), so in particular if we fix \\(\epsilon\\) then it holds for \\(\dfrac{\epsilon}{3}\\). We'll just use the \\(\delta\\) from that \\(\dfrac{\epsilon}{3}\\) instead. This gives us that \\(f\\) is uniformly continuous, as required, and without actually engaging the brain except to carry out the algorithm of "write down what we know; if there exists something, fix it, and repeat; if for all something, then fix an arbitrary one, and repeat; if we're stuck, go back through, looking to see if we missed out any information during a fixing-arbitrary-for-all phase" and to carry out the algorithm of "when the information we have is simple enough, compare terms from what we know with the expression that we want to show; use the triangle inequality to get them in there".
+
+ [1]: https://www.dpmms.cam.ac.uk/~par31 "Paul Russell"
diff --git a/hugo/content/posts/2013-11-07-my-quest-for-a-new-phone.md b/hugo/content/posts/2013-11-07-my-quest-for-a-new-phone.md
new file mode 100644
index 0000000..4358cfd
--- /dev/null
+++ b/hugo/content/posts/2013-11-07-my-quest-for-a-new-phone.md
@@ -0,0 +1,96 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-11-07T00:00:00Z"
+aliases:
+- /uncategorized/my-quest-for-a-new-phone/
+- /my-quest-for-a-new-phone/
+title: My quest for a new phone
+---
+*This post is unfinished, and may never be finished - I have decided that the Nexus 5 is sufficiently cheap, nice-looking and future-proof to outweigh the boredom of continuing the research here, especially given that such research by necessity has a very short lifespan. I am one of those people who hates shopping with a fiery passion.*
+
+My current phone is a five-year-old [Nokia 1680]. It has recently developed a disturbing tendency to turn off when I'm not watching it.
+This puts me in the market for a new phone. Having looked over the Internet for guides to which phone to buy, I've become lost in the swamp of information, so I am using this post to order my thoughts.
+
+# My current phone usage
+
+I use my phone pretty rarely. It has a camera, but I have only used it once ever (and that picture was so blurry that it doesn't really count). It has a colour screen, which I would happily forgo if it made the battery life better. The battery lasts about a week between charges at my current usage level. I have made about five calls on it in the past year, and sent a few hundred texts. The £20 of credit I gave it about four months ago is now down to £2.50, but I used it unusually often to make calls (four of the five calls I mentioned were in that period). The phone can connect to the Internet, but I have never used it thusly, because interacting with web pages would be too painful on that screen and with those buttons. My current tariff is pay-as-you-go, with Tesco Mobile, on a plan that doesn't seem to exist any more (4p per text, and some unspecified amount for calls).
+
+# Projected phone usage
+
+I have two main options available.
+
+*   Buy a dumbphone
+*   Buy a smartphone.
+
+These options greatly affect the way I would use the phone. For a dumbphone, I would use it much as I use my current phone: for rare calls and for less-rare texts. For a smartphone, I would branch out considerably, into using it for calendar syncing, to-do lists, GPS/maps/directions, on-the-go information, computation and so forth. I would not use it for games (because they are simply a waste of time that I could be using to [become more awesome][2], and because they aren't fun anyway). I don't see myself using it as a camera, either. I will not be installing social media apps on a smartphone, because I hate it when people use them in front of me, and because I categorically do not want to become one of these people who incessantly posts about what food they had this morning. I reserve public self-broadcasting platforms for those things which I think could be important or interesting to many people (and I amend the "what-I-post" category in response to feedback), or which I'm proud of having created, and it's much harder to find/make these things on a phone screen than on a computer with keyboard and big screen.
+
+# Requirements for a dumbphone
+
+*   Cheap - I do not want to spend more than £50 on a dumbphone
+*   Long battery life
+*   No need for a camera or a colour screen - [eInk] sounds ideal
+*   No need for Internet access
+
+# Requirements for a smartphone
+
+*   Calendar syncing (I could host a [CalDAV] server on this website, so interoperability should be easy)
+*   To-do list syncing (I have switched to [Workflowy] for to-do lists, and that can be accessed in-browser, so it only needs a web browser)
+*   Preferably maps/GPS
+*   Smooth user experience (I want to feel like I'm controlling [JARVIS])
+*   Cheaper is better -  I do not want to spend more than £400 on a smartphone
+*   Preferably [libre][7] and more preferably secure/NSA-proof, although this is not paramount
+*   At least a four-inch screen, preferably larger (up to a maximum of six inches)
+
+# Dumbphone research
+
+It would appear that very few purely eInk phones have ever been created. There are a few dual-screen [LCD and eInk phones][8], but they are primarily smartphones; what I want from eInk is more like a [Kindle] turned into a phone. The [eInk page on phones][10] demonstrates three phones, but they are either dual-screen or truly dreadful ([as in][11], only [two lines of text][12] can appear on the screen at once). It looks like eInk is a no-go.
+
+I am reduced to looking for dumbphones without a camera, colour screen or Internet access.
+
+# Smartphone research
+
+## Operating system
+
+There are two main OSs in use: [Android] and [iOS]. I say this because [Windows Phone] OS is ugly enough to flout the JARVIS requirement, and [Blackberry] phones… hmm. My cached thoughts on Blackberry phones run along the lines of "don't like them, uncool" more than anything else. I find myself generating excuses not to include them in this list, even though I don't actually know much about them. Better put them in.
+
+### iOS
+
+The only phone devices which run iOS are Apple's iPhones. With an education discount, the only model I can buy new within my £400 limit is the [iPhone 4S] (at £349). This model has access to [Siri] (the Apple personal assistant).
+
+Apple offers a "[refurbished and clearance][19]" store, but they do not offer iPhones through this.
+
+### Android
+
+Android phones are very widely available. Because there is such a huge choice of phones already, I will make the simplifying assumption that I only want a phone which runs [Android 4.4 "KitKat"][KitKat] (the latest version of Android, as of this writing).
+
+### Blackberry
+
+It turns out that only two Blackberry phones have full-size touchscreens. The JARVIS criterion is failed for screens which are too small to fit reasonable amounts of text on, which leaves only the [Z30][21] and [Z10][22]. However, from what [I've seen][23], the Blackberry OS is kind of uglier than Android or iOS. For the sake of simplifying the discussion, I will go with my cached self and rule out Blackberry.
+
+ [Nokia 1680]: https://en.wikipedia.org/wiki/Nokia_1680_classic "Nokia 1680 Wikipedia page"
+ [2]: http://lesswrong.com/lw/iri/how_to_become_a_1000_year_old_vampire/ "Thousand year old vampire LessWrong page"
+ [eInk]: http://www.eink.com "eInk"
+ [calDAV]: https://en.wikipedia.org/wiki/CalDAV "CalDAV Wikipedia page"
+ [Workflowy]: https://workflowy.com "Workflowy"
+ [JARVIS]: https://www.youtube.com/watch?v=D156TfHpE1Q "JARVIS Youtube video"
+ [7]: https://en.wikipedia.org/wiki/Gratis_versus_libre "Free-as-in-freedom Wikipedia page"
+ [8]: http://gizmodo.com/5967746/this-dual-lcd-and-e-ink-phone-will-be-available-in-2013 "LCD/eInk phone example"
+ [Kindle]: https://en.wikipedia.org/wiki/Amazon_Kindle "Amazon Kindle"
+ [10]: http://web.archive.org/web/20130718152515/http://www.eink.com/customer_showcase_cell_phones.html "eInk phones showcase"
+ [11]: https://en.wikipedia.org/wiki/Motorola_Fone "Motofone eInk phone"
+ [12]: https://en.wikipedia.org/wiki/Motofone_f3#Display_technology "Motofone F3 Wikipedia page"
+ [Android]: https://en.wikipedia.org/wiki/Android_OS "Android Wikipedia page"
+ [iOS]: https://en.wikipedia.org/wiki/IOS "iOS Wikipedia page"
+ [Windows Phone]: https://en.wikipedia.org/wiki/Windows_Phone_8 "Windows Phone Wikipedia page"
+ [Blackberry]: http://uk.blackberry.com/smartphones.html "Blackberry phone"
+ [iPhone 4S]: https://en.wikipedia.org/wiki/Iphone_4s "iPhone 4S Wikipedia page"
+ [Siri]: https://en.wikipedia.org/wiki/Siri "Siri Wikipedia page"
+ [19]: http://store.apple.com/uk/browse/home/specialdeals "Apple Refurbished store"
+ [KitKat]: https://www.android.com/kitkat/ "Android KitKat"
+ [21]: http://uk.blackberry.com/smartphones/blackberry-z30.html "Blackberry Z30"
+ [22]: http://uk.blackberry.com/smartphones/blackberry-z10 "Blackberry Z10"
+ [23]: http://www.youtube.com/watch?v=nyjMVJ3ISDQ "Blackberry Z30 Youtube video"
diff --git a/hugo/content/posts/2013-11-12-markov-chain-card-trick.md b/hugo/content/posts/2013-11-12-markov-chain-card-trick.md
new file mode 100644
index 0000000..bf81fb9
--- /dev/null
+++ b/hugo/content/posts/2013-11-12-markov-chain-card-trick.md
@@ -0,0 +1,59 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2013-11-12T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/markov-chain-card-trick/
+- /markov-chain-card-trick/
+title: Markov Chain card trick
+---
+In my latest lecture on [Markov Chains][1] in Part IB of the Mathematical Tripos, our lecturer showed us a very nice little application of the theorem that "if a discrete-time chain is aperiodic, irreducible and positive-recurrent, then there is an invariant distribution to which the chain tends as time increases". In particular, let \\(X\\) be a Markov chain on a state space consisting of "the value of a card revealed from a deck of cards", where aces count 1 and picture cards count 10. Let \\(P\\) be randomly chosen from the range \\(1 \dots 5\\), and let \\(X_0 = P\\). Proceed as follows: define \\(X_n\\) as "the value of the \\(\sum_{i=0}^{n-1} X_i\\)-th card". Stop when the newest \\(X_n\\) would be greater than \\(52\\).
+
+That is, I shuffle a pack of cards, and you select one of the first five at random. I then deal out the rest of the cards in order; you hop through the cards as they are revealed. For instance, if the deck looked like \\(\{5,4,9,10,1,2,6,8,8,3, \dots \}\\) and you picked \\(2\\) as your starting value, then your list of numbers would look like \\(\{4, 2,8, \dots \}\\) (moving forward four cards, then two, then eight, and so on). We keep going until I run out of cards to deal out, at which point I triumphantly announce the value of the card which you last remembered.
+
+How is this done? The point is that we are both walking along the same Markov chain, just from different starting positions. As soon as we both hit the same card, we are locked together for all time, and it is simply a matter of ensuring that we hit the same card at some point. But this is precisely what the quoted theorem tells us: if we go on for long enough, we will fall into the same distribution, and hence will likely hit the same card as each other at some point. I ran some simulations to determine the probability with which we end on the same value. The code is kind of dirty, for which I apologise - it was thrown together quickly, and is written in the [write-only][2] [Mathematica][3]. We first assume that all picture cards are 10s, and that aces are 1s.
+
+    nums = Flatten[{ConstantArray[Range[1, 10], 4], ConstantArray[10, 12]}];
+
+The following function runs one simulation using each of the supplied starting indices, using the given order of cards:
+
+    test\[perm\_, startPos\_List] := ({Length[#[[1]]], #[[2]]} &@ NestWhile[{#[[1]\]\[[#[[2\]] + 1 ;;]], #\[[1]\]\[[#[[ 2\]]]]} &, {perm, #}, Length[#[[1]]] #[[2]] &]) & /@ startPos
+
+It is astonishingly illegible. Read it as: "For each starting position supplied: start off with the input permutation and starting position. While the starting position is a valid position of the list (so it is less than or equal to the length of the list), set the starting position to the value of the card at that starting position, and set the list of cards to be everything after that position. Repeat until we've run out of cards. Then output the length of the remaining list of cards [and hence, indirectly, the final position we hit], and the last value we remembered."
+
+The following line of code runs a hundred thousand simulations with a random order of cards each time:
+
+    True/(False + True) /. Rule @@@ Tally[ Function[{inputStartPos}, #[[1, 1]] == #[[2, 1]] &@ test[RandomSample[nums, Length@nums], inputStartPos]] /@ RandomChoice[Range[4], {100000, 2}]] // N
+
+Again, it is illegible. Read it as "We're going to want the proportion of good results to all results, where "good" is defined as follows: call a run "good" if we stopped at the same card at the end. Do that for a hundred thousand different pairs of random starting points less than \\(6\\), and tally them all up. Give me a numerical answer at the end, not a fraction." This program output 0.76764 - that is, there is a better-than-three-quarters chance of "winning" in this variant, where we insist that players pick one of the first five cards to start with, and where we don't care that queens, kings, jacks and tens are all different.
+
+In order to try and be a bit more clever, I used a simple [Bayesian update technique][4] to try and get the confidence of the answer. Performing 5000 trials and updating from a prior of "uniformly likely that the required probability is any \\(\dfrac{n}{5000}\\) for integer \\(n\\)", I got the following PDF:
+
+![Bayesian updated probability]({{< baseurl >}}images/MarkovChainCardTrick/MarkovChainBayesPlot.jpg "Probability after Bayesian updates")
+
+This has mean 0.756297 and standard deviation 0.00606961.
+
+What if we want a different range of starting values? The following table gives the mean and standard deviation of \\(p\\) for different ranges of allowed starting cards.
+
+N=1: {0.999884, 0.000191799} [the true value is, of course, 1]  
+N=2: {0.840064, 0.0051822}  
+N=3: {0.805078, 0.0056006}  
+N=5: {0.756897, 0.00606454}  
+N=10: {0.69912, 0.00648421}
+
+How about if we make 10s different from picture cards? Let's make jacks 11, queens 12 and kings 13:
+
+N=2: {0.834066, 0.00525959}  
+N=5: {0.716913, 0.0063691}  
+N=10: {0.673331, 0.0066306}
+
+So your odds of winning are still pretty good, even if we insist that all cards are different (ignoring suit).
+
+ [1]: http://www.statslab.cam.ac.uk/~grg/teaching/markovc.html "Markov Chains course page"
+ [2]: https://en.wikipedia.org/wiki/Write-only_language "Write-only language Wikipedia page"
+ [3]: https://www.wolfram.com "Wolfram Mathematica"
+ [4]: https://web.archive.org/web/20131019220645/http://www.databozo.com/2013/09/15/Bayesian_updating_of_probability_distributions.html "Bayesian updating"
diff --git a/hugo/content/posts/2013-11-23-the-jean-paul-sartre-cookbook.md b/hugo/content/posts/2013-11-23-the-jean-paul-sartre-cookbook.md
new file mode 100644
index 0000000..049c644
--- /dev/null
+++ b/hugo/content/posts/2013-11-23-the-jean-paul-sartre-cookbook.md
@@ -0,0 +1,41 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-11-23T00:00:00Z"
+aliases:
+- uncategorized/the-jean-paul-sartre-cookbook/
+title: The Jean-Paul Sartre Cookbook
+---
+> Many thanks to the [Guru Bursill-Hall][1] for bringing this tract to my attention through his weekly History of Maths bulletins. It was originally written in 1987 by Marty Smith, according to the Internet.
+
+# The Jean-Paul Sartre Cookbook
+
+**October 3.**   Spoke with Camus today about my cookbook. Though he has never actually eaten, he gave me much encouragement. I rushed home immediately to begin work. How excited I am! I have begun my formula for a Denver omelet.
+
+**October 4.**  Still working on the omelet. There have been stumbling blocks. I keep creating omelets one after another, like soldiers marching into the sea, but each one seems empty, hollow, like stone. I want to create an omelet that expresses the meaninglessness of existence, and instead they taste like cheese. I look at them on the plate, but they do not look back. Tried eating them with the lights off. It did not help. Malraux suggested paprika.
+
+**October 6.**  I have realized that the traditional omelet form (eggs and cheese) is bourgeois. Today I tried making one out of cigarette, some coffee, and four tiny stones. I fed it to Malraux, who puked. I am encouraged, but my journey is still long.
+
+**October 10.** I find myself trying ever more radical interpretations of traditional dishes, in an effort to somehow express the void I feel so acutely. Today I tried this recipe:
+
+> **Tuna Casserole**
+> Ingredients: 1 large casserole dish.
+>
+> Place the casserole dish in a cold oven. Place a chair facing the oven and sit in it forever. Think about how hungry you are. When night falls, do not turn on the light.
+
+While a void is expressed in this recipe, I am struck by its inapplicability to the bourgeois lifestyle. How can the eater recognize that the food denied him is a tuna casserole and not some other dish? I am becoming more and more frustrated.
+
+**October 25.** I have been forced to abandon the project of producing an entire cookbook. Rather, I now seek a single recipe which will, by itself, embody the plight of man in a world ruled by an unfeeling God, as well as providing the eater with at least one ingredient from each of the four basic food groups.
+
+To this end, I purchased six hundred pounds of foodstuffs from the corner grocery and locked myself in the kitchen,refusing to admit anyone. After several weeks of work, I produced a recipe calling for two eggs, half a cup of flour, four tons of beef, and a leek. While this is a start, I am afraid I still have much work ahead.
+
+**November 15.** Today I made a Black Forest cake out of five pounds of cherries and a live beaver, challenging the very definition of the word cake. I was very pleased. Malraux said he admired it greatly, but could not stay for dessert. Still, I feel that this may be most profound achievement yet, and have resolved to enter it in the Betty Crocker Bake-Off.
+
+**November 30.** Today was the day of the Bake-Off. Alas, things did not go as I had hoped. During the judging, the beaver became agitated and bit Betty Crocker on the wrist. The beaver's powerful jaws are capable of felling blue spruce in less than ten minutes and proved, needless to say, more than a match for the tender limbs of America's favourite homemaker. I only got third place. Moreover, I am now the subject of a rather nasty lawsuit.
+
+**December 1.** I have been gaining twenty-five pounds a week for two months, and I am now experiencing light tides. It is stupid to be so fat. My pain and ultimate solitude are still as authentic as they were when I was thin, but seem to impress girls far less. From now on, I will live on cigarettes and black coffee.
+
+ [1]: http://web.archive.org/web/20201113203936/https://www.dpmms.cam.ac.uk/~piers/ "Guru Piers Bursill-Hall"
diff --git a/hugo/content/posts/2013-12-14-the-training-game.md b/hugo/content/posts/2013-12-14-the-training-game.md
new file mode 100644
index 0000000..0e251d8
--- /dev/null
+++ b/hugo/content/posts/2013-12-14-the-training-game.md
@@ -0,0 +1,24 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-12-14T00:00:00Z"
+aliases:
+- /uncategorized/the-training-game/
+- /the-training-game/
+title: The Training Game
+---
+The book Don't Shoot the Dog, by Karen Pryor, contains a simple exercise in demonstrating [clicker training][1]. This is a very successful technique used to produce behaviour in animals: having first associated the sound of a click with the reward of attention or food, one can then use the click as an immediate substitute for the reward (so that one can train more complicated, time-critical actions through positive reinforcement; a click is instant, but food or attention requires the trainer approaching the trainee). The demonstration exercise involves a person designated the Trainer, and a person designated the Trainee. The trainer has a goal in mind, but cannot communicate that goal to the trainee; the only interaction allowed is a click when the trainee is doing something vaguely correct. As an example, the trainee can be made to move towards a light switch by dint of a click when ey is pointing towards the switch, then a click when ey moves in that direction (ignoring any attempts to move in a different direction); the trainer then draws attention to the general area of the light by clicking whenever the trainee looks in the right direction, and then for any hand movement, then for hand movement in the direction of the light switch. This kind of incremental reinforcement can be used to achieve all sorts of interesting behaviour. (I seem to remember, from Don't Shoot the Dog, that it has been used in chickens to make them do hundred-step dances, although I may have mis-remembered that.)
+
+The exercise, then, demonstrates the power of reinforcement to produce order from chaos. With one trainer and several trainees, I would imagine that the problem becomes harder, but not insurmountably so (click when the person whose attention you need moves - it would take a while, but eventually I think I could train individual behaviour out of the group).
+
+But what about one trainee and several trainers? Imagine a scenario in which a single trainee is in a room alone, with the clicks of two trainers coming through the door in such a way that the trainee can hear only a single click. No matter which of the trainers produced it, the trainee can't tell the difference between different trainers' commands. The two trainers have competing goals (or the same goals?), and they perform the above clicker-training procedure. Would any useful behaviour result? I can imagine that an animal would get hopelessly confused by the competing goals, but a human might be able to get some kind of result. (We must assume in the contradictory case that the trainers have among their goals that "progress towards the opposing goal should be minimised"; that prevents them from teaming up to, say, perform the two goals sequentially.)
+
+Imagine that one trainer aims to make the trainee do the [Macarena][2], while the other trainer wishes the trainee to assume the [lotus position][3]. The goals are contradictory. I would imagine that the trainee would receive reinforcement towards being low down (in order to sit), as well as for standing straight and still (the starting position for the Macarena). I suspect that the trainee would infer some completely unrelated behaviour. I don't know if there's an official name for "excessively powerful inference" - [pareidolia][4] (the tendency to see faces in random settings) is a related phenomenon, and might cover this. I would be interested to know what behaviour would result from this kind of stimulus. Perhaps an experiment is in order (or, if you are also interested, do convey your results to me).
+
+ [1]: https://en.wikipedia.org/wiki/Clicker_training "Clicker training Wikipedia page"
+ [2]: https://en.wikipedia.org/wiki/Macarena_%28song%29 "Macarena Wikipedia page"
+ [3]: https://en.wikipedia.org/wiki/Lotus_position "Lotus position Wikipedia page"
+ [4]: https://en.wikipedia.org/wiki/Pareidolia "Pareidolia Wikipedia page"
diff --git a/hugo/content/posts/2013-12-22-three-explanations-of-the-monty-hall-problem.md b/hugo/content/posts/2013-12-22-three-explanations-of-the-monty-hall-problem.md
new file mode 100644
index 0000000..613bd71
--- /dev/null
+++ b/hugo/content/posts/2013-12-22-three-explanations-of-the-monty-hall-problem.md
@@ -0,0 +1,40 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2013-12-22T00:00:00Z"
+aliases:
+- /mathematical_summary/three-explanations-of-the-monty-hall-problem/
+- /three-explanations-of-the-monty-hall-problem/
+title: Three explanations of the Monty Hall Problem
+---
+Earlier today, I had a rather depressing conversation with several people, in which it was revealed to me that many people will attempt to argue against the dictates of mathematical and empirical fact in the instance of the [Monty Hall Problem][1]. I present a version of the problem which is slightly simpler than the usual statement (I have replaced goats with empty rooms).
+
+> Monty Hall is a game show presenter. He shows you three doors; behind one of the three is a car, and the other two hide empty rooms. You have a free choice: you pick one of the doors. Monty Hall then opens a door which you did not pick, which he knows is an empty-room door. Then he gives you the choice: out of the two doors remaining, you may switch your choice to the other door, or stick with the one you first picked. You will get whatever is behind the door you end up with. You want to pick the car; do you stick with your first choice, or do you switch to the other door?
+
+The solution is that you should switch. I present three explanations for why this is true, each of which makes it obvious to me in a different way. They may not help.
+
+# Different worlds
+
+Imagine three possible worlds: you pick a door, and the car is behind the first, second or third door. These choices are equally likely: the position of the car is randomly chosen by Monty Hall beforehand. Hence there are three possible worlds that I could find myself in. Let's suppose I picked door 1; it doesn't matter.
+
+*   In the "I pick door 1, car in 1" world, if I switch my door, I lose; if I keep my door, I win.
+*   In the "I pick door 1, car in 2" world, if I switch my door, I win; if I keep my door, I lose.
+
+The "I pick door 1, car in 3" world is identical to the previous one.
+
+That is, in two cases out of three, switching wins for me. That means switching is better than sticking: I win in two-thirds of the worlds if I switch, and I only win in a third of the worlds if I stick. (This is the brute-force approach to understanding the problem.)
+
+# Extra information
+
+Let's suppose we pick a door, and then Monty Hall reveals a false door. Of course, when I picked my door, I had a 1/3 chance of having picked the car, and that probability is unchanged when Monty reveals the false door. However, if I switch, only then am I given a chance to use the information that Monty has provided to me. Only if I switch am I able to use the fact that only two doors remain (one of them hiding nothing, and one of them hiding a car) - that makes my chance of winning a car 1/2 if I switch (actually, it's 2/3 if we condition correctly, but that's not instantly obvious and this is an informal explanation), but only 1/3 if I stick. This means it's better for me to switch. Essentially, I'm restarting the game if I switch, because nothing was special about my original choice so I can discard it without changing anything. If I switch, I discard my original choice, changing nothing, and re-pick from the improved game with one fewer door. (This is the information-theoretic approach of incorporating new information.)
+
+The same idea can be seen if we think of the question in a slightly different way. Once you've picked a door, and before Monty Hall opens any door, Monty asks you, "Would you like to look behind the door you picked, or behind the two doors you didn't pick?" If you reply "my door, please", that's the same as sticking with your original choice: Monty opens an empty-room door (changing nothing; after all, you know Monty will do this before you even start the game) and then your original door. If you reply "the other two, please", Monty opens an empty-room door and then the other door. (That's the same as switching choices.) Essentially, Monty is giving you a choice of two doors in the second case, and only one door in the first. The reason that his opening an empty-room door changes something in this case, is because we might as well consider it as "Monty opens the other two doors simultaneously": you get a 2-in-3 chance this time, since Monty's opening two of the three doors.
+
+# Extreme problem
+
+Consider the phrasing of the problem as "You pick a door. If you picked the car, Monty Hall opens every door except the one you picked, and one random empty-room door. If you picked an empty room, Monty Hall opens every door except the one you picked, and the car." Now, this exactly reflects the original problem, but is amenable to extension in the following way. Instead of having one car and two empty rooms, have one car and a hundred empty rooms. Now, when you pick, Monty Hall opens every door except for the one you picked, and one other. You started with a one-in-101 chance of having picked the car. At the end, Monty Hall has left only two doors. The probability that you originally picked the car is very low (1 in 101). But if we switch, we suddenly see that Monty Hall has removed almost all of the chaff that caused us to have only a 1 in 101 chance originally. Now it's just obvious that we have a 1 in 2 chance of picking the car if we re-pick from the game in its new state. The only way we have of re-picking is to switch doors.
+
+ [1]: https://en.wikipedia.org/wiki/Monty_hall_problem "Monty Hall problem Wikipedia page"
diff --git a/hugo/content/posts/2013-12-30-smartphone-charter.md b/hugo/content/posts/2013-12-30-smartphone-charter.md
new file mode 100644
index 0000000..1837855
--- /dev/null
+++ b/hugo/content/posts/2013-12-30-smartphone-charter.md
@@ -0,0 +1,30 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2013-12-30T00:00:00Z"
+aliases:
+- /uncategorized/smartphone-charter/
+- /smartphone-charter/
+title: Smartphone Charter
+---
+I am shortly to receive a new [Nexus 5][1]. I am determined not to become a smartphone zombie, and so I hereby commit to the following Charter.
+
+*   I will keep my phone free of social networking apps, and I will ensure that I do not know the passwords to access their web interfaces. While they can be really quite handy, they are usually simply a distraction. People are used to the fact that I am present on the Internet only when I have my computer with me; there's no need for that to change.
+*   I will only look at text messages when I'm not talking to someone already.
+*   I will never look at [reddit][2] or [Hacker News][3] or suchlike on my phone, unless there is no-one else around. Similarly, I will not access my news feeds from my phone. It's far too easy to waste time and attention on them, when such attention is expected from the people I'm with.
+*   If I am doing something on my phone, and someone asks me to stop, I will do one of the following (with number 1 being heavily preferred, and number 3 only in emergency):
+    1.  I will stop using my phone within ten seconds
+    2.  I will explain what I am doing, and ask permission to continue
+    3.  I will explain what I am doing (or say that an explanation will be forthcoming as soon as possible), and continue.
+*   I will keep my phone out of reach of my bed when I go to sleep. It's easy to become lost in the Internet, especially when you're tired and not really concentrating.
+*   I will be able to access emails on my phone, but I will set it up so that it only checks manually.
+*   I will not install games on my phone. It's not there as "something to keep me entertained when I'm bored" but as "something to be useful when needed", and in my experience, games seem to intrude.
+
+If I break any of these, you're allowed to get annoyed with me. (The converse is false in general.)
+
+ [1]: https://en.wikipedia.org/wiki/Nexus_5 "Nexus 5 Wikipedia page"
+ [2]: http://www.reddit.com/ "reddit"
+ [3]: https://news.ycombinator.com "Hacker News"
diff --git a/hugo/content/posts/2014-01-02-the-creation.md b/hugo/content/posts/2014-01-02-the-creation.md
new file mode 100644
index 0000000..4cd2ff3
--- /dev/null
+++ b/hugo/content/posts/2014-01-02-the-creation.md
@@ -0,0 +1,33 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- creative
+comments: true
+date: "2014-01-02T00:00:00Z"
+aliases:
+- /creative/the-creation/
+- /the-creation/
+title: The Creation
+---
+Once upon a time, before this bountiful age of Matter and Light, there was only the Fell. A single being, surrounded by Chaos, content to remain alone forever (for it did not know what a "friend" was). It had not the power to shape the Chaos; neither had it the inclination, for it needed nothing and had no desires. For seething unchanging aeons, it persisted.
+
+Then Chaos bore new fruit. A single electron, a point source of *charge*. The *electric field* thereby induced resonated throughout all of Chaos, propagating yet further, every second by the same amount; and so the Fell recognised *distance*. The Fell experienced *curiosity* then: for an electromagnetic field was entirely a novel sensation to it. The place it inhabited was changed, from isotropic to merely *spherically symmetric*: now the Fell identified *direction*. It began to *move towards* the point charge, first *slowly*, and then *faster*, until its *velocity* approached that of the electric field itself. All this was for to discover the nature of the descendant of Chaos.
+
+As the Fell approached the electron, its existence became *threatened*: as a simple pattern in Chaos, it could exist indefinitely, but approaching a source of electric charge was a new disturbance, one which the pattern had not been purposelessly selected to overcome. And it recoiled from the intrusion with great force, the influence of the electron growing with the square of the Fell's distance from it, much faster than was comfortable.
+
+But the pattern that was the Fell was changed by the charge, and the charge was changed by the pattern. The same perturbations that had caused the first electron were still latent in the Chaos, and the Fell's scramble to escape the charge was enough to revive them. A second electron emerged, accompanied by a single *photon*.
+
+Now there was unbound *energy* in Chaos. Before the Fell could even begin to *react*, Chaos began to resonate, shuffling, its patterns collapsing into such regularity that a great explosion of *matter* emerged. At the speed of light, *things* emerged, a great array of *muons*, *quarks* and their ilk. The Fell could but race away from the catastrophe; most of it was shorn away in that first burst of creation, before it could flee. And so it continued to exist.
+
+*Gradually*, the flurry of *order* was calmed. Chaos is infinite, unquenchable, and the energy which the Fell unwittingly brought into existence was but finite. At the boundary of the sphere of roiling matter did the Fell rest, recovering itself, painstakingly forging its old patterns anew from the Chaos. It felt the unconstrained resonance of the matter, and so could it know what was *happening* in this new world.
+
+And indeed it came to pass that the *Universe* settled down, protected from Chaos by its sheer radius. *Gravity*, not present in the isotropic Chaos, was very much a factor in the Universe, and things came together to form new patterns. With nothing better to *do*, the Fell learnt to peer into the Universe, *polling* it with the gentlest bursts of electromagnetism to discern what new *wonders* occurred. (The Fell grew larger and larger, forcing its pattern onto Chaos, to keep and examine this new information.) It learnt to send information into the Universe by gently affecting the boundary, and eventually it occurred to the Fell to *create* something. It planned and tweaked, and when it was satisfied, it chose a star and a newly-made planet, and altered it subtly.
+
+It came to pass that self-replicating structures emerged on that planet. With startling speed, they became better-adapted to their environment. The Fell's usual languid pace of existence was not enough to keep up with the rapidity of the changes, so it began to poll for information much more frequently. It felt *tenderness* for what it had wrought, and it tried to keep that planet from harm.
+
+And the changes accelerated, faster and faster: an *exponential* with no apparent end. The Fell struggled to keep up, polling yet faster; its error rate was low, but with so many polls occurring, every so often it misjudged and sent a beam of energy that was so powerful that it affected the planet's star itself, causing plasma to gout out of it.
+
+Reptiles had emerged before the Fell realised how quickly the changes were now happening. It stretched itself to its limit, polling more and more frequently until it could go no faster, desperate to document everything. It had no energy spare to protect the planet, and it came to pass that a very large chunk of rock hit, causing the destruction of the incumbent life; and so mammals emerged, followed in short order by primates and then humans.
+
+Therefore, send not to know for whom the Fell polls - it polls for thee.
diff --git a/hugo/content/posts/2014-01-12-denouement-of-myst-iii-exile.md b/hugo/content/posts/2014-01-12-denouement-of-myst-iii-exile.md
new file mode 100644
index 0000000..4be6996
--- /dev/null
+++ b/hugo/content/posts/2014-01-12-denouement-of-myst-iii-exile.md
@@ -0,0 +1,53 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2014-01-12T00:00:00Z"
+aliases:
+- /uncategorized/denouement-of-myst-iii-exile/
+- /denouement-of-myst-iii-exile/
+title: 'Denouement of Myst III: Exile'
+---
+A long time ago, in a galaxy far far away, I completed [Myst III: Exile][1]. It's a stupendously good puzzle game. For some reason, it popped into my mind again a couple of days ago. This post contains very hefty spoilers for that game (it will completely ruin the ending - I will be discussing information-exchange protocols which are key to completing it), so if you're ever going to play it, don't read this post yet. It's a brilliant game - I highly recommend it.
+
+The spoilers start here. Weak spoilers are first, so that you have time to stop reading if your eyes are accidentally moving downwards. After those weak spoilers will come a discussion of the final puzzle of the game. If you're familiar with the general Myst universe, skip the next two paragraphs; start at the sentence "Proper spoilers start here".
+
+The series of Myst games revolves around the concept of a *Linking Book*, a means of moving from world to world (these worlds are called "Ages") by touching the front page of certain books. Each book is a link to its ultimate *Descriptive Book*, which was at some point written by a Writer, and which describes an Age in such detail that the Age actually comes to exist. The act of touching the front page of a Linking Book or the Descriptive Book causes you to be transported into the Age described by the corresponding Descriptive Book. (By the way, you can't bring the Linking Book with you into the Age it links to.) The destruction of the Descriptive Book of an Age causes the Age to be destroyed.
+
+Atrus is a master Writer. After the destruction of his civilisation (the D'ni), he writes a new Age, called Releeshan, in which the remnants of the D'ni can start afresh. Myst III: Exile starts with you ("the Stranger", since you are never named or depicted in any way, in any of the Myst series) being invited to explore Releeshan yourself with Atrus for the first time; he shows you the Descriptive Book, which you will use to enter Releeshan. However, just as you're about to enter, a person, Saveedro, appears, starts a fire, and grabs the Descriptive Book, before linking back out. The book he used falls to the floor, and you rashly follow him. So the events of the game begin. (It turns out that the fire burns the Linking Book, so you're conveniently on your own.)
+
+Proper spoilers start here; the next couple of paragraphs describe the set-up of the final puzzle. At the end of the game, you've tracked down Saveedro. It turns out that for plotty reasons, he hates Atrus's sons with a fiery passion (this is elaborated on in games 1 and 4), and this was his way of getting back at Atrus. (He intended Atrus to follow him, not you, the Stranger.) He also wanted to show Atrus the consequences of his sons' actions. To that end, he has caused you to end up in the Age of Narayan, Saveedro's home Age, which used to be vibrantly natural but was ruined by Atrus's sons. Saveedro's home, which is where you have ended up, is shielded off from the rest of Narayan, and Saveedro desperately wants to get out into Narayan proper.
+
+Saveedro's home is divided into two chunks, which we will consider to be in concentric circles, with an impenetrable shield between them, and an impenetrable shield surrounding the whole set-up. (That's why Saveedro can't escape: he is stuck behind two shields, unable to get through even one.) Linking to Narayan takes you to the inner chunk. From there, you can turn the power on to a device which has enough power to inhibit one of the shields, but not both. Being a friend of Atrus, you (naturally) have access to his journal, which gives you the key insights necessary to activate this device. The device can switch between inhibiting either of the two shields, but not both. The mechanism to control which shield is inhibited is inside both the concentric circles. (That is, while the inner shield is raised, you can't access it from the outside-circle.)
+
+One can leave the house only if one is in the outside-circle and the outer shield is inhibited. However, one person alone can't do this: after activating the inhibitor, one can cause the inner shield to be inhibited and the outer shield to remain, and thereby one can get into the outside-circle; or one can cause the outer shield to be inhibited and the inner shield to remain. There is a single small passage between inner and outer circles, but it's not big enough for you to get through.
+
+Saveedro still holds Releeshan, which it is your objective to retain. In his home, you have found a Linking Book that will return you to Atrus's home, Tomahna. You want to obtain Releeshan and return it to Atrus. Saveedro wants to escape his house. You start this scenario with you in the inner circle next to the inhibitor's controls, and Saveedro standing next to the outer shield, which is currently raised, and the inner shield is inhibited.
+
+The official solutions are as follows (least-optimal to optimal in order, so you can think about the puzzle if you want to):
+
+1.  Leave immediately, using the Tomahna Linking Book. (Then Saveedro follows you through the book you leave behind, and kills you.)
+2.  Release Saveedro immediately. (Then he destroys Releeshan, because he is still angry with Atrus. Many years of Atrus's life's work are now gone.)
+3.  Cut the power to the inhibitor, thereby raising both shields. Then Saveedro collapses, having had freedom waved in front of him and snatched away (this time, he is trapped away from all his belongings, which are in the inner circle.) He hands you Releeshan through the small passage, and pleads with you to let him out. You have several choices: you can link back to Atrus (you end the game remorseful that Saveedro is alone); you can turn power back onto the inhibitor (then Saveedro runs back and kills you); or you can turn the controls on the inhibitor and then power it back up (then Saveedro gets out of his house, and salutes you as he leaves towards the houses in the distance), before linking back to Atrus. This last ending is optimal.
+
+# The actual point of this post
+
+I was wondering whether there is a better solution, in the sense that "we don't have to cause Saveedro unnecessary anguish, and/or we can complete the scenario without requiring anyone to trust anyone else". (Recall that the solutions officially require Saveedro to be provoked into trusting you.) What we need is some sort of mechanism or box to hold Releeshan, which is open if and only if the outer shield is inhibited. Then Saveedro can go into the inner circle with you (inner shield inhibited), switch the inhibitor (opening the box), and place the book inside it (after verifying that it does indeed open under and only under those conditions). Then he switches the inhibitor (closing the box, and allowing him through to the outer circle), goes through to the outer circle, you switch the inhibitor (opening the box and allowing him to escape), take Releeshan, and link away. He cannot return to kill you (because the inner shield is up).
+
+Possible failure modes: you could shut off power to the inhibitor, thereby causing Saveedro to be trapped and you to have the unopenable box. This is obviously non-optimal, but it is precluded in-game by the fact that the controller of power to the inhibitor is located quite far from the inhibitor itself, so Saveedro has ample time to get back into the inner circle and kill you. You could link straight out, which leads to the exact scenario portrayed in-game. Saveedro could simply destroy Releeshan before putting it in the box (but then you will never release him).
+
+I can see parallels between this kind of scenario and the creation of currencies like Bitcoin. There are some pretty impressive protocols to allow parties to spend money without being able to spend the same money twice, and so forth. What I really want is an information-based solution to this Myst problem.
+
+I have heard of information-swapping protocols, to ensure that two parties swapping information will not do so asymmetrically, even if one party is evil with respect to the other. That sounds perfect for this.
+
+New plan: create a box with a long combination lock, and modify the inhibitor so that it requires the same long combination to open it. Place Releeshan in the box. Each pick digits for the combination in turn, without the other knowing the digits. Lock the box using that combination, and lock the inhibitor in the "inhibiting the inside" position. Then Saveedro goes into the outer circle, taking the box; you go with him, and ensure that it is anchored firmly in place, before going back. Saveedro starts reading out his first digit. You punch it in to the inhibitor, and he punches it into the box. You read out your first digit; then repeat. If at any point you stop entering digits into the inhibitor, or do so incorrectly, Saveedro simply stops reading out his numbers, and you can't get to Releeshan. (This is why Saveedro needs to have some numbers at all.) Similarly, Saveedro can't stop punching the digits into the safe, or else you will not release him.
+
+Now, as you get nearer to the end, you might be able to stop entering the digits and just brute-force the box open. Then Saveedro can come and kill you. Alternatively, Saveedro can stop reading his digits; but that doesn't serve him any purpose, for you know the code up to the point where he stopped reading his digits, and he's back where he started. (We will assume that there is a way to ensure that the same combination has been set for the box and the inhibitor.) Now, Saveedro has read all his digits; there are four left for you to enter. You don't read them out, but punch them in to the inhibitor and release Saveedro. Then you re-set the inhibitor and collect Releeshan, since you have the complete code to open the box.
+
+If you turn off power to the inhibitor, Saveedro will simply never give you the code to access Releeshan, so that option is ruled out on the grounds that an approach exists which doesn't require you to trust each other but still lets you both get what you want.
+
+Can anyone see any failure modes I've missed, or any simplifications? It probably works fine with just five numbers (one picked by Saveedro, and four by you), but I wanted to include a way to exchange arbitrary information.
+
+ [1]: https://en.wikipedia.org/wiki/Myst_III:_Exile "Myst III: Exile Wikipedia page"
diff --git a/hugo/content/posts/2014-01-24-introduction-to-functional-programming-syntax-of-mathematica.md b/hugo/content/posts/2014-01-24-introduction-to-functional-programming-syntax-of-mathematica.md
new file mode 100644
index 0000000..28ebe0c
--- /dev/null
+++ b/hugo/content/posts/2014-01-24-introduction-to-functional-programming-syntax-of-mathematica.md
@@ -0,0 +1,115 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- programming
+comments: true
+date: "2014-01-24T00:00:00Z"
+math: true
+aliases:
+- /uncategorized/introduction-to-functional-programming-syntax-of-mathematica/
+- /introduction-to-functional-programming-syntax-of-mathematica/
+title: Introduction to functional programming syntax of Mathematica
+---
+Recently, I was browsing the [Wolfram Community][1] forum, and I came across the following question:
+
+> What are the symbols @, #, / in Mathematica?
+
+I remember that grasping the basics of functional programming took me quite a lot of mental effort (well worth it, I think!) so here is my attempt at a guide to the process.
+
+In Mathematica, there are only two things you can work with: the Symbol and the Atom. There is only one way to combine these things: you can provide them as arguments to each other. We denote "\\(x\\) with arguments \\(y\\) and \\(z\\)" by "`x[y,z]`".
+
+What is an Atom? As the name suggests, it is something indivisible, like the number 2 or the string "Hello!". So that the language isn't too complicated to implement, we mean "indivisible without any further work" - so the number 15 is "divisible" (in the sense that it's 3x5), but not in our sense: it takes work to find the divisors of a number. Similarly, the string "Hello!" is "divisible" into characters, but that again takes work.
+
+A Symbol is something which we, as a programmer, tell Mathematica to give meaning to. We also tell it under what circumstances that Symbol has meaning. For instance, I might say to Mathematica, "In future, when I ask you the Symbol $MachinePrecision, you will pretend I said instead the Atom 15.9546." Something else I might say to Mathematica is, "In future, when I ask you for the Symbol Plus, combined with the arguments 1 and 2, you will pretend I said instead the Atom 3."
+
+In Mathematica's syntax, we write the above as:
+
+    $MachinePrecision = 15.9546;
+    Plus[1, 2] = 3;
+
+(The semicolons prevent Mathematica from printing the value we gave. Without the semicolons, it would print out 15.9546 and 3. In fact, the semicolons are a shorthand for the Symbol CompoundExpression, but that's not important.)
+
+Furthermore, we can ask Mathematica, "In future, when I ask you for Plus combined with zero and any other argument x, return that argument x". In Mathematica's syntax, that is:
+
+`Plus[0, Pattern[x, Blank[]] ] := x`
+
+More compactly:
+
+`Plus[0, x_] := x`
+
+Now, we have had to be careful. Mathematica needs a way of distinguishing the Symbol `x` from the "free argument" `x`. We want the "free argument" - that is, we want to be able to supply any argument we like, and just temporarily call it x. We do that using the Pattern symbol, better recognised as `x_` . I won't go into how Pattern works in terms of the Symbol/Atom idea, but just recognise that `x_` *matches* things, rather than *being* a thing.
+
+Now, we'll assume that there is already a "plus one" method - that Mathematica already knows how to do `Plus[1, x_]`. Let's also assume that it knows what `Plus[-1, x_]` is (not hard to do, in principle, once we know `Plus[1, x_]`). Then we can define Plus over the positive integers:
+
+`Plus[x_, y_] := Plus[Plus[-1, x], Plus[1, y]]`
+
+And so forth. This is how we build up functions out of Symbols and Atoms.
+
+Now, there is a shorthand for `f[x]`. We can instead write `f@x`. This means exactly the same thing.
+
+A really important Symbol is `List`. `List[x, y, z]` (or, in shorthand, `{x, y, z}`) represents a collection of things. There's nothing "special" about `List` - it's interpreted in exactly the same way as everything else - but it's a convenient, conventional way to lump several things together. (It would all have worked in exactly the same way if the creators of the language had decided that Yerpik would be the symbol that represented a generic collection; even `Plus` could be used this way, if we made sure to tell Mathematica that "Plus" should not be evaluated in the usual way. You could even use the number 2 as the list-indicating symbol, or even use it as `Plus` usually is used, leading to expressions like `2[5,6] == 11`.) We can define functions like `Length[list_]`, so `Length[{1, 2, 3}]` is just 3.
+
+Since everything is essentially function application ("apply a symbol to an expression"), we might explore ways to apply several functions at once, or to apply a function to several different parts of an expression. It turns out that a really useful thing to do is to be able to apply a function to all the inside bits of a List. We call this "mapping":
+
+`Map[f, {a, b, c}] == {f[a], f[b], f[c]}`
+
+More generally, `Map[f, s[a1, a2, … ]] == s[f[a1], f[a2], …]`, but we use `List` instead of `s` for convenience. There is a shorthand, reminiscent of the `f@x` notation: we use `f /@ {a, b, c}` to denote "mapping".
+
+It's all very well to want to map a function across the arguments to a symbol (let's call that symbol, which has those arguments, the Head of an expression, so `Head[f[x,y]]` is just `f`), but what about if we want to apply the function *to the Head symbol*? Actually, this turns out to be quite rare (the function is `Operate[p, f[x,y]]` to give `(p[f])[x,y]` ), but it's much more common to want to replace the Head completely. For instance, we might want to supply a List as arguments to a function, as follows:
+
+`f[x_, y_] := x + y^2`
+
+How would we get `f` to act on the List `{5, 6}`? We can't just say `f[{5, 6}]` because f requires two inputs, not the one that is `List[5, 6]`. Mathematica's syntax is that instead of `f@{5,6}`, we use `f@@{5, 6}`. This is shorthand for `Apply[f, {5,6}]`, and it returns `f[5, 6]`, which is 41.
+
+More generally, `f@@g[x, y] == f[x, y]`. (Note, however, that Mathematica evaluates things as much as possible before doing these transformations, so `f@@Plus[5,6]` doesn't give you `f[5,6]` but `f@@11`, an expression which makes no sense. Mathematica's convention is that Atoms don't really have a Head, so replacing the Head with `f` does nothing; hence `f@@11` will return 11.)
+
+Particularly in conjunction with `Map`, it can be useful to Apply a function not to an expression, but to the arguments of the expression. That is, given a List `{{1, 2}, {3, 4}}`, which is `{List[1, 2], List[3, 4]}`, we might want to output `{f[1, 2], f[3, 4]}`. We do this with the shorthand `f@@@{{1, 2}, {3, 4}}`, which is really `Apply[f, {{1, 2}, {3, 4}}, 2]`. This situation might arise if we wanted to "transpose" two strings "ab" and "cd" to return "ac" and "bd" (imagine writing the strings out in a table, and reading the answer down the columns instead of across the rows). We could use `StringJoin@@@Transpose@Map[Characters, {"ab", "cd"}]`. Indeed, what does this expression do? The first thing that will actually change when it is evaluated is `Map[Characters, {"ab", "cd"}]`. This will return `{{"a", "b"}, {"c", "d"}}`. Then Transpose sees that new list, and flips things round to `{{"a", "c"}, {"b", "d"}}`, which is `{List["a", "c"], List["b", "d"]}`. Then `StringJoin` is asked not to hit the outer `List`, or even to hit the inner Lists, but to *replace* the List head on the inner Lists: the expression becomes `{StringJoin["a", "c"], StringJoin["b", "d"]}`, or `{"ac", "bd"}`.
+
+Now, it's all very well to have functions that work like this. But what if we wanted to take the second character of a string? There's a function for that - `StringTake` - but it needs arguments. We could define a new function `takeSecondChars[str_] := StringTake[str, {2}]`, but that's unwieldy if we only want this function once - and what about if we wanted the third character instead, the next time?
+
+There is a really useful way to define functions without names. Unsurprisingly, they look like:
+
+`Function[{x, y, …}, …]`
+
+So in the above example, we'd have `Function[{str}, StringTake[str, {2}]]`. And then to map it across a list would look like:
+
+`Function[{str}, StringTake[str, {2}]] /@ {"str1", "str2", "str3"}`
+
+We can also apply it to a string: `Function[{str}, StringTake[str, {2}]]["string"]`, or `Function[{str}, StringTake[str, {2}]]@"string"`.
+
+There's a really compact shorthand. Instead of `Function[{args}, body]` we use `(body)&`. We don't even bother naming the arguments; we use the `Slot[i]` function to get the `i`th argument. `Slot[i]` is more neatly written as `#i`, while just the `#` symbol is interpreted as `#1`.
+
+Hence our function becomes `StringTake[#, {2}]&`, and its mapping looks like:
+
+`StringTake[#, {2}]& /@ {"str1", "str2", "str3"}`
+
+It takes some getting used to, but after a while it becomes extremely natural. In my most recent coursework project, there are almost no programs I wrote which don't use this syntax, even though the coursework is aimed at the language Matlab which is almost the antithesis of this idea of "symbols with arguments". Once you become able to see problems in this way - mapping small functions over expressions, and so forth - you start seeing it everywhere. The idea is about sixty years old - it's the principle of Lisp - and it's ridiculously powerful. Since functions are just expressions, you can use them to alter themselves. For instance, memoisation is trivial:
+
+    fibonacci[n_] := (fibonacci[n] = fibonacci[n-1] + fibonacci[n-2])
+    fibonacci[1] = 1;
+
+That is, "Whenever I ask you for fibonacci[n], you will set the value of fibonacci[n] to be the sum of the two previous values." Note that this is "set the value of fibonacci[n] to be", not "return" - this is a permanent change (well, as permanent as the Mathematica session), and it means that the value of fibonacci[36] is instantly available forever after once you've calculated it once.
+
+You can also get some crazy things with Slot notation, because `#0` (which is `Slot[0]`) represents *the function itself*. Off the top of my head, an example is:
+
+    (Boole[# < 10] #0[# + 1] + #) &[1]
+
+This generates the tenth triangle number. (The function `Boole[arg]` returns 1 if arg is `True`, and 0 otherwise.) This is because the function evaluates to exactly its input unless that input is less than 10; in that case, the function evaluates to (its input, plus "this function evaluated at input+1"). Recursively expanded, it is `f[x_] := If[x < 10, f[x+1]+x, x]`, evaluated at the input 1. It gets quite mind-bending quite quickly, and I don't think I have ever used `#0` in earnest. Another example I came up with quickly was:
+
+    If[Cos[#] == #, #, #0[Cos[#]]] &[1.]
+
+This finds a fixed point of the function Cos, starting at the initial input 1. (It has to be a numerical input, otherwise Mathematica will just keep going forever with better and better symbolic expressions for this fixed point, like `Cos[Cos[Cos[1]]]`. It rightly recognises that, for instance, `Cos[Cos[Cos[1]]]` is not equal to `Cos[Cos[Cos[Cos[1]]]]`, so it never stops.)
+
+The last really useful piece of shorthand I can think of at the moment is // which is another way to apply functions.  
+Instead of `f@x`, we can use `x//f` . This has the benefit of making it a bit clearer what is actually contentful, and what is just afterthoughts, because the functions which are evaluated last actually appear at the end:
+
+`CharacterRange["a","z"] // StringJoin`
+
+Of course, the usual function notation can be used:
+
+`1 // (Boole[# < 10] #0[#+1] + # &)`
+
+Phew, that was a whistlestop tour in rather more words than I had hoped - turns out there are far more Mathematica concepts that I've internalised than I had thought, all of which are really quite fundamental and indispensable. I understand much better why people say Mathematica has a steep learning curve, and why it is derided as a "write-only language" - that final example is ridiculous!
+
+ [1]: http://community.wolfram.com
diff --git a/hugo/content/posts/2014-01-28-writing-essays.md b/hugo/content/posts/2014-01-28-writing-essays.md
new file mode 100644
index 0000000..c79b16b
--- /dev/null
+++ b/hugo/content/posts/2014-01-28-writing-essays.md
@@ -0,0 +1,60 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2014-01-28T00:00:00Z"
+aliases:
+- /uncategorized/writing-essays/
+- /writing-essays/
+title: Writing essays
+---
+The aim of this post is twofold: to find out whether a certain mental habit of mine is common, and to draw parallels between that habit and the writing of essays.
+
+I don't know whether this is common or not, but when I'm feeling particularly not-alert (for instance, when I'm nearly asleep, or while I'm doing routine tasks like cooking), I sometimes accidentally latch onto a topic and mentally explain it to myself, as if I were teaching it to the Ancient Greeks (who, naturally, speak English). As an example, last night's topic of discourse was "the composition of soil", in which I "talked" about soil, in a manner roughly according to the following diagram. It is laid out so as to display roughly what occurred to me, and the order in which it occurred to me to "say" it.
+
+The contents of soil
+
+*  Soil contains fungi
+
+* *  what is a fungus?
+
+*  Soil contains fungi, lots and lots, which contributes to
+
+*  we eat fungi
+
+* *  we don't just eat mushrooms, we also are starting to eat <a title="Quorn Wikipedia page" href="https://en.wikipedia.org/wiki/Quorn">Quorn</a> etc
+
+*  we eat fungi - more specifically, the reproductive organs of the mycelium
+
+* *  what is a mycelium? it's a web that can span large areas
+
+* * *  "fairy circles" - mycelium is why mushrooms often appear in arcs, because the mushrooms - the reproductive organs - appear at the periphery of the web
+
+*  Soil contains fungi, lots and lots, which contributes to its taste
+
+* *  I once accidentally ate some of a mouldy slice of bread, and it tasted just like soil
+
+* * *  the mould looks the same as the mould which you get in damp areas of a house
+
+* *  you can actually see something which is closely related to a mycelium on mouldy bread - webs of fungus
+
+*  we eat fungi - Quorn, for instance, something similar to which was eaten during the first world war in Germany because of famine
+
+* *  Quorn can be made in huge vats, 250 kg can be made using the same resources as would make a kilogram of chicken
+
+* * *  chicken is the most-eaten meat in the world, and our treatment of them can be horrible
+
+* * *  there are environmental problems associated with using the resources that could produce a quarter of a ton of food to instead produce a kilogram of chicken
+
+You get the idea. I'm essentially doing a depth-first search of my internal knowledge-base, starting from a particular place. When I feel that a topic is getting too big to include (for instance, I stop after "environmental problems" because that leads to a very large nexus of topics in my knowledge-base), I stop the search and backtrack. When I feel that a fact is particularly interesting but doesn't have too much relevant content after it, I stop (for instance, the "fairy circles" fact, which could lead to a digression on myths and legends, but I deem that too big a logical leap).
+
+This rings a bell with [an essay Paul Graham wrote][1] about essays, and more strongly with an anecdote which (infuriatingly) I can't find or recall the name of, by a teacher of English. As an exercise, he (I think it was "he") set a student an exercise to "write an essay about your home town". She looked at him blankly, and so he refined it to "write an essay about the High Street of your home town". The process continued until it got to "write 500 words about the top-left brick in the front of the bank on the High Street of your home town". The student left, almost in tears. The next day, she returned with five thousand words of essay, and said that "once I got started, I just didn't stop".
+
+What I am doing is very close to this view of writing an essay. If I kept going long enough (in an awake state), I would presumably hit areas I don't know much about (for instance, how is it that there is a kind of mushroom that can punch through tarmac? Hydraulics, I know, but that's a [stop sign][2].) That's where the research would start, and where I would start discovering new things - and that's where Paul Graham's view of writing an essay would happen.
+
+This very post was written somewhat in this manner, but to save space and time, I made the knowledge-tree much smaller. (Alternative ending to that sentence: "I destroyed my time machine and burned all my papers".) Naturally, when actually formulating an essay from such a tree, it is important only to keep that which is interesting and/or useful, and it is necessary to restrict the output to a reasonable length. A blog format, in particular, prefers shorter pieces, so maybe I should**—**
+
+ [1]: http://www.paulgraham.com/essay.html "Paul Graham on essays"
+ [2]: http://lesswrong.com/lw/it/semantic_stopsigns/ "Semantic Stop-signs LessWrong page"
diff --git a/hugo/content/posts/2014-02-16-rage-rage-against-the-poets-hardest-sell.md b/hugo/content/posts/2014-02-16-rage-rage-against-the-poets-hardest-sell.md
new file mode 100644
index 0000000..51903f6
--- /dev/null
+++ b/hugo/content/posts/2014-02-16-rage-rage-against-the-poets-hardest-sell.md
@@ -0,0 +1,38 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- creative
+comments: true
+date: "2014-02-16T00:00:00Z"
+aliases:
+- /creative/rage-rage-against-the-poets-hardest-sell/
+- /rage-rage-against-the-poets-hardest-sell/
+title: Rage, rage against the poet’s hardest sell
+---
+I feel that I can write a sonnet well.  
+While sonnets are an easy thing to spout,  
+It’s really hard to write a villanelle.
+
+By rhyming, any story I can tell:  
+in couplets, rhyme and rhythm evens out.  
+I feel that I can write a sonnet well.
+
+But alternately-structured verse is hell.  
+The poet struggles, juggles words about:  
+It’s really hard to write a villanelle.
+
+Enthusiasm’s difficult to quell.  
+An acolyte of Shakespeare, I’m devout:  
+I feel that I can write a sonnet well.
+
+But triplets are a task on which I dwell,  
+I’m running out of rhymes, without a doubt.  
+It’s really hard to write a villanelle.
+
+For sonnets, you don’t have to be [Kal-El][1]  
+to make a super stanza just work out.  
+I feel that I can write a sonnet well;  
+It’s really hard to write a villanelle.
+
+ [1]: https://en.wikipedia.org/wiki/Superman "Superman Wikipedia page"
diff --git a/hugo/content/posts/2014-03-20-a-roundup-of-some-board-games.md b/hugo/content/posts/2014-03-20-a-roundup-of-some-board-games.md
new file mode 100644
index 0000000..a566d17
--- /dev/null
+++ b/hugo/content/posts/2014-03-20-a-roundup-of-some-board-games.md
@@ -0,0 +1,62 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2014-03-20T00:00:00Z"
+aliases:
+- /uncategorized/a-roundup-of-some-board-games/
+- /a-roundup-of-some-board-games/
+title: A roundup of some board games
+---
+It has been commented to me that it's quite hard to find out (on the Internet) what different games involve. For instance, [Agricola][1] is a game about farming (and that's easy to find out), but what you actually do while playing it is not easy to discover. Here, then, is a brief overview of some games.
+
+# Agricola
+
+[Agricola][2] is a game in which you control a farm, and are aiming to make your farm thrive. It is a multiplayer game (for two to five) divided into turns. During each turn, you can make several actions (the number of actions you can make is determined by the number of people you have on your farm; you start out with two, and some actions increase the number of people you have). The actions are shared between all players - that is, if I make an action, you may not make that same action this turn. There is no other inter-player interaction - no attacking or anything, and you all have your own farm to manage. Your aim is to use actions to gather resources, build and extend your house, and plough fields; at the end of the game (after fourteen rounds, which is about forty minutes) everyone scores their own farm according to a set checklist, and the winner is the one who has the most prosperous farm.
+
+# Settlers of Catan
+
+[Catan][3] is a game in which you are trying to build up your civilisation essentially from scratch. It is multiplayer (two to four), and is divided into turns. The game is played on a common board, which you gradually populate with your own settlements, cities and roads, while attempting to make sure that other people can't foil your plans with their own building. (Once something is built, it can't be un-built, so the game is competitive only in a strategic sense, not a combat sense.) You aim to gather resources (which you can trade freely with opponents) so as to build more such trappings (your settlements and cities gain resources according to dice rolls), and the winner is the first to reach a certain size of civilisation. Games last about 45mins.
+
+# Diplomacy
+
+[Diplomacy][4] is a game almost entirely down to how well you can connive with and against opponents. It takes place in turns, but actions happen essentially simultaneously in a turn; the real action happens in between turns, when you go and plot with other people. Games take many hours, and are very multiplayer (eight or so, I think, is normal). Your aim is to take over the world, which you can only feasibly do by persuading people both to assist you and to foil your opponents' attempts.
+
+# Dominion
+
+[Dominion][5] is a two-to-four player deck-building game. You aim to have acquired the most Victory cards by the end of the game. Turns involve playing cards you have already acquired, and acquiring more cards; the cards you acquire become part of an "economy" that is almost never subtracted from, but you may only use a small subset of your cards during any one turn. It is somewhat like Magic: the Gathering (below), but restricted so that cards only modify the structure of your turn and allow you to draw more cards. (There are some "attack" cards, but I find them not to be conducive to fun play.)
+
+# Magic: the Gathering
+
+[Magic][6] is a rather different game to those listed above. It is a collector's game: you acquire cards over your lifetime, although some formats involve getting a random selection of cards and doing the best you can with those. It is multiplayer (two players is common, but it goes arbitrarily high). The format is turn-based - each turn is subdivided - but the key point is that cards can do pretty much anything to the game. Win conditions can be altered, turns can be prevented, cards can be renamed, all as the result of card effects. Your aim is to win the game, which is usually done by taking the opponent's life total down to 0 or by forcing them to draw a card when they have no cards left to draw (that is, after they have already drawn all of their cards). Many other win conditions exist - [one card][7] causes you to win if you have a certain ridiculous number of cards; [one card][8]'s active effect is that a target opponent loses the game; [one card][9] causes you to win if nothing happens for a while; and so forth. A complete list can be seen on the [Gatherer card search facility][10] (you might want to search for "wins the game" as well as the default "win the game").
+
+That description of win conditions was intended to convey how complicated the game can become. It is not for the faint-hearted - it takes a while to get to grips with the myriad mechanics.
+
+# Mafia/Werewolf/Avalon/The Resistance
+
+These two games ([Mafia][11] and Werewolf are isomorphic games, as are Avalon and [The Resistance][12]) are both highly-multiplayer games (up to ten people) in which there are two teams: a team of innocents and a team of hidden spies (I'll refer to those as the Mafia). The job of the former team is to unmask the spies; the latter team usually wins by remaining undetected.
+
+In Mafia, the game revolves around group voting to "kill" people (and the first person killed will have a much less interesting game!). The innocents naturally want to kill the Mafia; the Mafia want to kill all the innocents. The Mafia get an extra turn in between every group vote, in which they can elect among themselves to kill someone. (That is, every normal turn, two people die.) The innocents win if all the Mafia are dead and an innocent is still alive; the Mafia win if they kill all the innocents. There are some extra roles to complicate things, but those are the basics.
+
+Avalon is slightly different - nobody dies at any point. In a round, someone chooses a team of people to go Questing, and the group votes on whether to allow that team to Quest. If the vote comes out negative, the next person chooses a team, and so on. This repeats until a team is approved; then if the team contains a Mafia member, the mission can be failed by that member (secretly: no-one finds out who the Mafia was). Otherwise, it succeeds. The aim is to accumulate failed/succeeded missions (depending on if you are Mafia/innocent).
+
+# Dixit
+
+[Dixit][13] is a [Keynesian beauty contest][14] style game. Everyone gets cards with pictures on them, and each round, a different Storyteller describes one of their cards. Then everyone puts one of their cards into a pile, and the cards are all placed in the centre of the table. Everyone then guesses which card was the Storyteller's card, based on their description. Points are allocated to any player whose card was guessed (that is, a player who put in a card which matches the description enough for someone to mistake it for the Storyteller's card). The Storyteller gets points *unless* everyone or no-one guessed correctly. (That is, the description must not be so obscure that no-one gets the answer right, and it must not be so obvious that everyone does.)
+
+ [1]: https://en.wikipedia.org/wiki/Agricola_(board_game) "Agricola Wikipedia page"
+ [2]: https://en.wikipedia.org/wiki/Agricola_(board_game) "Agricola (Wikipedia page)"
+ [3]: https://en.wikipedia.org/wiki/Catan "Settlers of Catan Wikipedia page"
+ [4]: https://en.wikipedia.org/wiki/Diplomacy_(board_game) "Diplomacy Wikipedia page"
+ [5]: https://en.wikipedia.org/wiki/Dominion_(game) "Dominion Wikipedia page"
+ [6]: https://en.wikipedia.org/wiki/Magic:_The_Gathering "Magic: the Gathering Wikipedia page"
+ [7]: https://gatherer.wizards.com/Pages/Card/Details.aspx?multiverseid=288878 "Battle of Wits Magic card"
+ [8]: https://gatherer.wizards.com/Pages/Card/Details.aspx?multiverseid=288992 "Door to Nothingness Magic card"
+ [9]: https://gatherer.wizards.com/Pages/Card/Details.aspx?multiverseid=265418 "Azor's Elocutors Magic card"
+ [10]: https://gatherer.wizards.com/Pages/Search/Default.aspx?text=+[win]+[the]+[game] "Gatherer"
+ [11]: https://en.wikipedia.org/wiki/Mafia_(party_game) "Mafia Wikipedia page"
+ [12]: https://en.wikipedia.org/wiki/The_Resistance_(party_game) "The Resistance Wikipedia page"
+ [13]: https://en.wikipedia.org/wiki/Dixit_(card_game) "Dixit Wikipedia page"
+ [14]: https://en.wikipedia.org/wiki/Keynesian_beauty_contest "Keynesian beauty contest"
diff --git a/hugo/content/posts/2014-03-30-how-to-discover-the-contraction-mapping-theorem.md b/hugo/content/posts/2014-03-30-how-to-discover-the-contraction-mapping-theorem.md
new file mode 100644
index 0000000..ba1fa47
--- /dev/null
+++ b/hugo/content/posts/2014-03-30-how-to-discover-the-contraction-mapping-theorem.md
@@ -0,0 +1,72 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+- proof_discovery
+comments: true
+date: "2014-03-30T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/proof_discovery/how-to-discover-the-contraction-mapping-theorem/
+- /how-to-discover-the-contraction-mapping-theorem/
+title: How to discover the Contraction Mapping Theorem
+---
+A little while ago I set myself the exercise of stating and proving the [Contraction Mapping Theorem][1]. It turned out that I mis-stated it in three different aspects ("contraction", "non-empty" and "complete"), but I was able to correct the statement because there were several points in the proof where it was very natural to do a certain thing (and where that thing turned out to rely on a correct statement of the theorem).
+
+Here, then, is how you might go about discovering it from the point of having a definition of a [Lipschitz function][2] on a metric space \\((X, d)\\) (that is, a function \\(f\\) for which there exists \\(\lambda \in \mathbb{R}^{>0}\\) such that for all \\(x, y \in X\\), \\(d(f(x),f(y)) \leq \lambda d(x,y)\\)). We'll aim for a statement describing the fixed points of such a function.
+
+## Define the terms
+
+What is a "fixed point"? There's nowhere obvious to start other than working out what we mean by one of these. Well, what we mean is "a point \\(x \in X\\) such that \\(f(x) = x\\)". We'll also define \\((X, d)\\) to be an arbitrary metric space, and \\(f\\) an arbitrary Lipschitz function on that space with Lipschitz constant \\(\lambda\\).
+
+## How might we proceed?
+
+We're looking for a fixed point. We have a Lipschitz function (that is, one which "draws points together", in the sense that two points which are originally \\(\delta\\) apart end up \\(\lambda \delta\\) apart, or closer, after \\(f\\) is applied to them). That suggests the idea of starting out with two arbitrary points, repeatedly pulling them closer together with \\(f\\), and seeing where we end up. Actually, on second thoughts, we can dispense with one of the arbitrary points, because we can make another point given our arbitrary \\(x\\) - namely \\(f(x)\\).
+
+## What did we assume?
+
+So far, we've made a (silly) assumption: that the space \\(X\\) is not empty, because we've just picked a point in it. In order to use this "\\(f\\) draws points together", we're going to want \\(\lambda < 1\\), otherwise it's actually blowing them outwards.
+
+## How might we proceed?
+
+We have two points, \\(x\\) and \\(f(x)\\). We want to pull them together using \\(f\\), so it's natural to keep applying \\(f\\) to them. So that we can have access to all these values, we'll define a sequence \\(z_i = f(z_{i-1})\\) and \\(z_0 = x\\). What we really want is for this sequence to converge to the fixed point (after all, if we're drawing the points together to some limit, we'd imagine that the limit of the sequence is a local accumulator in some sense).
+
+Now, we know nothing about this metric space, and we know nothing about the limit of the sequence. There's a key thing we do in analysis if we want a limit of a sequence but know nothing about it: we show that it is [Cauchy][3]. In order to use this, though, we'll need to suppose that the metric space is complete (so that Cauchy sequences converge).
+
+Then we want to show that this sequence \\(z_i\\) is Cauchy. That is, we want \\(d(z_i,z_j) \to 0\\) as \\(i,j \to \infty\\) independently of each other, which means that for all \\(\epsilon > 0\\) there exists \\(N \in \mathbb{N}\\) such that for all \\(i, j > N\\), \\(d(f^i(x), f^j(x)) < \epsilon\\).
+
+Aha - we have \\(d(f^i(z), f^j(z))\\). We know \\(f\\) is Lipschitz, so ([wlog][4] \\(i \leq j\\)) this is \\(d(f^i(z), f^j(z)) \leq \lambda^i d(x, f^{j-i}(x))\\). It would be very convenient if the \\(d\\) expression were bounded, because then as \\(i \to \infty\\), the \\(\lambda^i\\) will take care of the rest (since \\(\lambda < 1\\)).
+
+But what else do we know about \\(d\\)? We're going to need something to bound \\(d(x, f^{j-i}(x))\\), but we don't know anything about this expression - we only know about \\(d(z_i, f(z_i)) \leq \lambda d(z_{i-1}, z_i)\\), by the Lipschitzness of \\(f\\). But in fact we can make \\(d(x, f^{j-i}(x))\\) in terms of those: \\(d\\) is a metric, which means that it obeys the triangle inequality.
+
+Hence \\(\displaystyle d(x, f^{j-i}(x)) \leq d(x, f(x)) + d(f(x), f^{j-i}(x)) \leq \dots \leq \sum_{k=1}^{j-i} d(z_{k-1}, z_k)\\). This we can bound: it's \\(\displaystyle \leq \sum_{k=1}^{j-i} \lambda^{k-1} d(z_0, z_1) = d(z_0, z_1) \sum_{k=1}^{j-i} \lambda^{k-1}\\). And, joy of joys, this sum is bounded, because the infinite sum \\(\displaystyle \sum_{k=1}^{\infty} \lambda^{k-1} = \dfrac{1}{1-\lambda}\\).
+
+Hence \\(d(z_i, z_j) < \lambda^i d(z_0, z_1) \dfrac{1}{1-\lambda}\\). This goes to \\(0\\) as \\(i \to \infty\\), so the sequence \\(z_i\\) is Cauchy.
+
+## What did we assume?
+
+In this section, we assumed that the space was complete.
+
+## Summary
+
+So far, we have shown that the sequence \\(f^i(x)\\) is Cauchy, so it converges to a limit. We'll call the limit \\(L\\): so we have \\(f^i(x) \to L\\) as \\(i \to \infty\\).
+
+## What next?
+
+It feels like we're very close to a result now. What we really want is for \\(L\\) to be a fixed point: we need \\(f(L) = L\\). Equivalently, we need \\(f(\lim z_i) \to \lim z_i\\); but \\(z_i = f(z_{i-1})\\), so this is \\(f(\lim z_i) \to \lim f(z_i)\\). This will be trivial if \\(f\\) is continuous. But \\(f\\) is Lipschitz, so it is uniformly continuous and hence continuous (this is a really simple lemma).
+
+That is, \\(L\\) is a fixed point of \\(f\\): we have proved that \\(f\\) has a fixed point.
+
+## Extension
+
+But we don't have to stop there - if we're drawing points together using \\(f\\), and we end up at a fixed point, surely there can't be two fixed points (since if there were, \\(f\\) would draw them together). Let's aim to prove that \\(f\\)'s fixed point is unique, by supposing that \\(L_1, L_2\\) are fixed points. Then \\(d(L_1, L_2) = d(f(L_1), f(L_2))\\), because \\(L_1, L_2\\) are fixed points, and then \\(d(f(L_1), f(L_2)) \leq \lambda d(L_1, L_2) < d(L_1, L_2)\\), contradiction.
+
+## Summary
+
+We have shown that there exists a unique fixed point \\(L\\) of a Lipschitz function \\(f\\) with Lipschitz constant \\(\lambda < 1\\) on a non-empty complete metric space. Moreover, we have shown that \\(f^(i)(x) \to L\\) for all \\(x\\), because we can perform this same construction of \\(z_i\\) starting from any point \\(x\\). Even more, we have shown that convergence is geometrically fast (by the \\(\lambda^i\\) term). This is a really strong theorem, and all I needed to remember in order to construct it was that Lipschitz functions were important and that we were looking for information about fixed points. (I didn't look up anything during the proof - I checked my statement of it afterwards, and it turned out to be correct. I didn't change anything after I finished it.)
+
+ [1]: https://en.wikipedia.org/wiki/Contraction_mapping_theorem "Contraction Mapping Theorem Wikipedia page"
+ [2]: https://en.wikipedia.org/wiki/Lipschitz_function "Lipschitz function Wikipedia page"
+ [3]: https://en.wikipedia.org/wiki/Cauchy_sequence "Cauchy sequence Wikipedia page"
+ [4]: https://en.wikipedia.org/wiki/Wlog "Wlog Wikipedia page"
diff --git a/hugo/content/posts/2014-04-04-discovering-a-proof-of-heine-borel.md b/hugo/content/posts/2014-04-04-discovering-a-proof-of-heine-borel.md
new file mode 100644
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--- /dev/null
+++ b/hugo/content/posts/2014-04-04-discovering-a-proof-of-heine-borel.md
@@ -0,0 +1,68 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+- proof_discovery
+comments: true
+date: "2014-04-04T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/proof_discovery/discovering-a-proof-of-heine-borel/
+- /discovering-a-proof-of-heine-borel/
+title: Discovering a proof of Heine-Borel
+---
+I'm running through my Analysis proofs, trying to work out which ones are genuinely hard and which follow straightforwardly from my general knowledge base. I don't find the [Heine-Borel Theorem][1] "easy" enough that I can even forget its statement and still prove it (like [I can with the Contraction Mapping Theorem][2]), but it turns out to be easy in the sense that it follows simply from all the theorems I already know. Here, then, is my attempt to discover a proof of the theorem, using as a guide all the results I know but can't necessarily prove without lots of effort.
+
+# Statement of the theorem
+
+The Heine-Borel Theorem states that a subset of \\(\mathbb{R}^n\\) is compact if and only if it is closed and bounded.
+
+# First direction
+
+One direction looks easy - if we assume our set is not closed or not bounded, it should be simple to show that it is not compact, using an argument based on the fact that \\((0,1]\\) is not compact and \\([0, \infty)\\) is not compact. Both of those I know how to prove.
+
+## Assume not closed
+
+If the set \\(S\\) is not closed, the only thing we can do is take a sequence \\((x_n)_{n \geq 1}\\) tending to a limit \\(x\\) which is not in \\(S\\). From this, we need to create an open cover of \\(S\\) which has no finite subcover.
+
+In one dimension, this is easy because we can just take a ball around each \\(x_i\\), each ball overlapping by a tiny bit with the next. Clearly since any finite cover must include each \\(x_i\\), it must also include those balls, whence it must include an infinite number of balls (contradiction). However, in more dimensions this is not so obvious, because we don't have this handy "next ball" concept. What was really key in that 1D example was that the balls around each \\(x_i\\) didn't overlap to the extent that a ball contained more than one \\(x_i\\), and that no ball got near the forbidden limit point. (There was always "room to keep going" - in the \\((0,1]\\) example, taking the sets \\((\dfrac{1}{n+1}, \dfrac{1}{n})\\) and filling in some tiny balls around each \\(\dfrac{1}{n}\\), every set is some finite distance away from \\(0\\).)
+
+In more dimensions, if we create an open cover of \\(S\\) such that no set gets near the limit point \\(x\\) - that is, such that each set in the cover has some neighbourhood of \\(x\\) which it doesn't encroach upon - then any finite cover must also have some neighbourhood of \\(x\\) which it doesn't encroach upon. (A finite collection of things which don't get close to 0 must also not get close to 0.) Hence, because we have a sequence tending to \\(x\\) in \\(S\\), which \*does\* get close to \\(x\\), one of the \\(x_i\\) can't be included in our finite cover. That contradicts compactness.
+
+## Assume not bounded
+
+Remember our key example here was \\([0, \infty)\\). Since our set isn't bounded, we can take a sequence in it getting arbitrarily far out from \\(0\\) (that is, for every \\(n\\) there is \\(x_n\\) such that \\(\vert x_n \vert \geq n\\)). But then the easiest cover to use is just the set of balls centred on \\(0\\) with radius \\(n\\); this is an infinite cover, but there is no finite subcover because if we ever stop, there's an \\(x_n\\) we've missed.
+
+# The other direction
+
+Here's the bit that looks harder, because we're taking any closed bounded set and showing a strong property of it. Remember, though, that we have in fact proved this in 1D already: we proved the Bolzano-Weierstrass property of the reals, and it is a fact (although I don't remember how to prove it) that sequential compactness implies compactness. Let's see if we can make that proof work. (The proof I know goes along the lines of "fix an infinite sequence in an interval; keep halving the interval; there's an infinite subsequence in one of the halves; repeat".)
+
+Firstly, we're faced with an arbitrary closed bounded set. With the not-closed or not-bounded sets it was easier - we had somewhere to start from. We're going to need to make the problem simpler, because closed bounded sets can look really quite odd. The simplest possible closed bounded set is the closed ball centred on the origin of radius \\(r\\), but that's not great for halving. What we can halve is a box \\([-r, r]^n\\) - that's the second-simplest possible closed bounded set (arguably the most simple).
+
+Take an open cover of the box, and assume for contradiction that it has no finite subcover. Divide the box up into \\(2^n\\) smaller boxes by cutting halfway along each side. One of these boxes must have no finite subcover of the original cover (otherwise they'd all have finite subcovers, so we could union them all together to get a finite subcover of the big box), so we can repeat on that box. Inductively we obtain a sequence of nested boxes, none of which has a finite subcover in the original cover, and they are boxes of side length \\(r \times 2^{-n}\\).
+
+What do we know about these nested boxes? In 1D, the proof then went "our infinite sequence therefore has a limit": there was a point which lay in every box. We'd love that to be true here: an infinite sequence of closed nested boxes must have non-empty intersection. Fortunately, that's easy to prove: take a sequence \\(z_n\\) such that each \\(z_n\\) lies in the \\(n\\)th box but not the \\(n+1\\)th. This sequence tends to a limit, because it's clearly Cauchy; we'll show that the limit lies in every box. Indeed, we know that the boxes are closed, so the sequence \\(z_n, z_{n+1}, z_{n+2}, \dots \to z\\) tells us that \\(z\\) lies in box \\(n\\) for every \\(n\\), so there is no \\(n\\) such that \\(z\\) is not in the \\(n\\)th box, and hence \\(z\\) is in every box.
+
+Now, we have our concentric boxes homing in on \\(z\\), and \\(z\\) lies in all of these boxes. Moreover, the boxes get smaller and smaller, quite rapidly, and each of them requires an infinite number of sets from our original cover in order to cover it. But where is \\(z\\)? \\(z\\) lies in some set \\(U\\) in the original cover; \\(U\\) is some finite size, so it must cover one of the boxes completely, because the sizes of the boxes goes to zero. Formally, \\(U\\) contains some ball \\(B_z(\epsilon)\\); for all \\(\epsilon\\) there is \\(n\\) such that the \\(n\\)th box lies wholly in \\(B_z(\epsilon)\\); hence \\(U\\) contains the \\(n\\)th box, for some \\(n\\).
+
+This contradicts the fact that the \\(n\\)th box requires an infinite cover of open sets - we've done it in just one!
+
+Hence all boxes are compact.
+
+## Dealing with all possible closed bounded sets
+
+We've dealt with the easiest kind of closed bounded sets. How can we transform any other closed bounded set into one of these? We can't do that necessarily - closed sets aren't necessarily unions of closed boxes - but what we can say is that all closed bounded sets are contained some closed box. (Indeed, all bounded sets are.) It would be great if a closed subset \\(C\\) of a compact set \\(X\\) were compact.
+
+That's easy, though - if we have an open cover of \\(C\\), we can make an open cover of \\(X\\) by just adding \\(U = \mathbb{R}^n - C\\) to the cover. (That extra set is open, being the complement of a closed set.) Then this has a finite subcover, by compactness of \\(X\\); that subcover probably contains \\(U\\), but if it does, just throw it out and we've got a finite subcover of \\(C\\). Hence \\(C\\) is compact.
+
+# Summary
+
+We proved it as follows:
+
+1.  Show the easier direction: assume not closed, make sequence tending to point not in set, define open cover such that no set individually gets close to that point; any finite subcover doesn't get close to that point, so the sequence can't be in the finite subcover. Assume not bounded, then use the nested balls centred on the origin.
+2.  Do the easiest case of boxes, by taking an open cover with no finite subcover, repeatedly dividing up the box to get a sequence of nested boxes each with no finite subcover; there is a point in every box (by defining a sequence of points, one in each box, which must tend to a limit); that point is in some open set in the original cover, and eventually the boxes get small enough that a box is entirely contained within an open set - contradiction.
+3.  Do the harder cases of boxes, by showing that a closed subset of a compact set is compact (by taking an open cover, extending it to a cover of the big set, and compactly taking a finite subcover, which turns back into a finite subcover of the small set).
+
+ [1]: https://en.wikipedia.org/wiki/Heine-Borel_theorem "Heine-Borel theorem"
+ [2]: {% post_url 2014-03-30-how-to-discover-the-contraction-mapping-theorem %}
diff --git a/hugo/content/posts/2014-04-07-useful-conformal-mappings.md b/hugo/content/posts/2014-04-07-useful-conformal-mappings.md
new file mode 100644
index 0000000..4df55a7
--- /dev/null
+++ b/hugo/content/posts/2014-04-07-useful-conformal-mappings.md
@@ -0,0 +1,44 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2014-04-07T00:00:00Z"
+math: true
+aliases:
+- /uncategorized/useful-conformal-mappings/
+- /useful-conformal-mappings/
+title: Useful conformal mappings
+---
+This post is to be a list of conformal mappings, so that I can get better at answering questions like "Find a conformal mapping from \<this domain\> to \<this domain\>". The following Mathematica code is rough-and-ready, but it is designed to demonstrate where a given region goes under a given transformation.
+
+    whereRegionGoes[f_, pred_, xrange_, yrange_] := 
+
+    whereRegionGoes[f, pred, xrange, yrange] = 
+     With[{xlist = Join[{x}, xrange], ylist = Join[{y}, yrange]},
+      ListPlot[
+       Transpose@
+        Through[{Re, Im}[
+         f /@ (#[[1]] + #[[2]] I & /@ 
+          Select[Flatten[Table[{x, y}, xlist, ylist], 1], 
+           With[{z = #[[1]] + I #[[2]]}, pred[z]] &])]]]]
+
+*   Möbius maps - these are of the form \\(z \mapsto \dfrac{az+b}{c z+d}\\). They keep circles and lines as circles and lines, so they are extremely useful when mapping a disc to a half-plane. A map is defined entirely by how it acts on any three points: there is a unique Möbius map taking any three points to any three points (and hence any circle/line to circle/line). (Some of the following are Möbius maps.)
+*   To take the unit disc to the upper half plane, \\(z \mapsto \dfrac{z-i}{i z-1\\)}
+*   To take the upper half plane to the unit disc, \\(z \mapsto \dfrac{z-i}{z+i}\\) (the [Cayley transform][1])
+*   To rotate by 90 degrees about the origin, \\(z \mapsto i \\)z
+*   To translate by \\(a\\), \\(z \mapsto a+\\)z
+*   To scale by factor \\(a \in \mathbb{R}\\) from the origin, \\(z \mapsto a \\)z
+*   \\(z \mapsto exp(z)\\) takes a vertical strip to an annulus - but note that it is not bijective, because its domain is simply connected while its range is not.
+*   \\(z \mapsto exp(z)\\) takes a horizontal strip, width \\(\pi\\) centred on \\(\mathbb{R}\\) onto the right-half-plane.
+
+## Maps which might not be conformal
+
+These maps are useful but we can only use them when the domain doesn't include a point where \\(f'(z) = 0\\) (as that would stop the map from being conformal).
+
+*   To "broaden" a wedge symmetric about the real axis pointing rightwards, \\(z \mapsto z^\\)2
+*   To take a half-strip \\(Re(z) > 0, 0 < Im(z) < \dfrac{\pi}{2}\\) to the top-right quadrant: \\(z \mapsto \sinh(z\\))
+*   to take a half-strip \\(Im(z) > 0, -\frac{\pi}{2} < Re(z) < \frac{\pi}{2}\\) to the upper half plane, \\(z \mapsto \sin(z\\))
+
+ [1]: https://en.wikipedia.org/wiki/Cayley_transform#Conformal_map "Cayley transform Wikipedia page"
diff --git a/hugo/content/posts/2014-04-15-sample-topology-question.md b/hugo/content/posts/2014-04-15-sample-topology-question.md
new file mode 100644
index 0000000..41bcf3c
--- /dev/null
+++ b/hugo/content/posts/2014-04-15-sample-topology-question.md
@@ -0,0 +1,53 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+- proof_discovery
+comments: true
+date: "2014-04-15T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/sample-topology-question/
+- /sample-topology-question/
+title: Sample topology question
+---
+As part of the recent series on how I approach maths problems, I give another one here (question 14 on the Maths Tripos IB 2007 paper 4). The question is:
+
+> Show that a compact metric space has a countable dense subset.
+
+This is intuitively clear if we go by our favourite examples of metric spaces (namely \\(\mathbb{R}^n\\), the discrete metric and the indiscrete metric). Indeed, in \\(\mathbb{R}^n\\), which isn't even compact, we have the rationals (so the theorem doesn't give a necessary condition, only a sufficient one); in the indiscrete metric, any singleton \\(\{x \}\\) is dense (since the only closed non-empty set is the whole space); in the discrete metric, where every set is open, we can't possibly be compact unless the space is finite, so that's why the theorem doesn't hold for a topology with so many sets.
+
+However, there are some really weird metric spaces out there, and if there's one thing I've learnt about topology it's that intuition-by-examples is an extremely bad way to prove things, although it's often a good way to work out *how* to prove something.
+
+Right. Our metric space could be really odd - it might be massively uncountable or something - so that means we're going to have to build our dense subset anew for each metric space. (It's like trying to find a good diet for your pet - the possible pets are so diverse that one diet won't fit all, so we have to find the right diet for each pet individually.) The "countable" bit can only come in from the rationals or naturals - it can't pop out of the metric space itself, because we have no idea how huge the metric space might be.
+
+That's all I can come up with for meta-reasoning at the moment. Let's find an example to guide intuition. By far the simplest is \\([a,b] \subset \mathbb{R}\\), whose dense subset is \\(\mathbb{Q} \cap [a,b]\\).
+
+My first thought is to make a dense subset by grabbing an arbitrary point \\(x\\) and then taking one point \\(x_p\\) such that \\(d(x_p, x) = p\\) for all rational \\(p\\). That definitely works for \\([a,b]\\), but actually it clearly fails in \\(\mathbb{R}^2\\) - what if we happened to pick our points so they all lay on the same line? They'd be dense along that line, but not anywhere else in the set. It's going to be a lot of work to fix this in \\(\mathbb{R}^2\\) without using special properties of \\(\mathbb{R}\\), so I'll abandon that line of thought.
+
+Nothing obvious has come of the "density" part of the statement. Let's move on to the other bit - we know our metric space is compact (or, equivalently, that any open cover has a finite subcover). That means we're going to want to create an open cover. Because our metric space might be so odd, the only obvious cover to take is one consisting of a ball around every point. (Those balls might all be different sizes, of course.) That's the only way to make sure that we have actually included our entire space in the cover.
+
+Compactness then gives us that there is a finite subcover of this cover of balls. That's not going to get us very far if we require a countable number of points, though. Where might we get a *point* rather than an open set (after all, compactness is all about sets, not points)? The only possible place is as the centre of some ball. Aha - we need to create a countable number of points, each of which lies at the centre of some ball. Equivalently, we want a countable number of balls.
+
+OK, we can create hugely many balls to cover the set (wrap every point in a ball), and we can turn that into finitely many balls to cover the set (by compactness). How can we get countably many? Obviously not from the "hugely many" directly, because it might be very very uncountable - but we can make countable from finite, by taking a countable union. That is, we're going to need a countable union of {finitely many balls which cover the set}.
+
+The simplest way I can create that countable union is to make every ball the same size (\\(\frac{1}{n}\\)), and use the cover \\(B_{\frac{1}{n}}\\) consisting of a \\(\frac{1}{n}\\)-ball around every point. We use compactness to turn that into \\(C_{\frac{1}{n}}\\) a collection of finitely many balls (which covers the entire space), and consider the union of all these \\(C_{\frac{1}{n}}\\).
+
+This has given us a countable collection of points \\(\cup_{n \geq 1} \cup_{j = 1}^{i_n} P_{\frac{1}{n},j}\\) (namely, the centres of the balls; notationally, \\(P_{\frac{1}{n}, j}\\) refers to the centre of the \\(j\\)th element of \\(C_{\frac{1}{n}}\\)). Now, we want that set to be dense - we need the closure to be the entire space. What would it mean if the closure weren't the entire space? There would be a point which was in the space but not the closure.
+
+At this point, I move back to the \\(\mathbb{R}^2\\) intuition-guide. I have drawn a mental picture of \\([0,1] \times [0,1]\\) with a countable collection of balls covering it, with a single point not in the closure of the set of centres. Aha, something is not right here - how can a point manage not to be in the closure of that set, unless it is outside the cover?
+
+Suppose \\(x\\) does not lie in \\(\text{cl}(\cup_n P_{\frac{1}{n}})\\) - that is, \\(x\\) is outside the closure. Then \\(x\\) lies in an open set - namely the complement of the closure - so there is an open ball \\(B_{\epsilon}(x)\\) which lies outside the closure. I can feel that we're going to use \\(\frac{1}{n}\\)-ness at some point, because that's how we defined our cover, so let's make \\(\epsilon = \frac{1}{m}\\) for some \\(m\\) (which we can do - if our original \\(\epsilon\\) didn't work, make it smaller until it is the reciprocal of an integer).
+
+Then we have a radius-\\(\frac{1}{m}\\) ball which doesn't lie inside the closure. That doesn't bode well for \\(C_{\frac{1}{m}}\\) being a cover, but it's just possible that the balls may sit next to each other in some way that makes it work (that's how vague my thoughts are, not just my incompetence at communication). For safety, let's consider \\(C_{\frac{1}{2m}}\\) instead.
+
+Then we have \\(x \in B_{\frac{1}{2m}}(k)\\) for some \\(k \in P_{\frac{1}{2m}}\\), because \\(C_\frac{1}{2m}\\) was a cover so \\(x\\) does lie in a ball in that cover; pick \\(k\\) to be the cetntre of that ball. In particular, \\(k\\) lies at most \\(\frac{1}{2m}\\) away from \\(x\\), and \\(k\\) lies both in \\(B_{\frac{1}{m}}(x)\\) (which is not in the closure) and in \\(P\\) (which is in the closure). This is a contradiction - we've found a point which is both in and not in the closure.
+
+Hence we must have the closure being the entire space, which means our countable collection of points is dense.
+
+# Summary
+
+I started off by thinking about the problem - working out roughly how I might be able to attack it, and deciding that it was too general for clever tricks to work. I then constructed an intuition-guide example, and worked off that, but decided that the line of attack suggested by my example would be very hard in general.
+
+Having exhausted one of the parts of the theorem's statement, I moved to the other, and followed my nose. The problem was so general that there were only a few possible places we could acquire a countable collection of points; compactness suggested using balls around every point in the space, to get a finite cover of balls. From finite we can create countable by just taking a union, so I made the finite covers more formal (giving the balls a particular size) and took the union of all of them. That naturally gives a countable set of points (the centres of the balls); in the spirit of "do as little work as possible", I set out to prove that this set was dense. Assuming the contrary made it obvious from my intuition-picture that the set was indeed dense.
diff --git a/hugo/content/posts/2014-04-17-cayley-hamilton-theorem.md b/hugo/content/posts/2014-04-17-cayley-hamilton-theorem.md
new file mode 100644
index 0000000..fc202aa
--- /dev/null
+++ b/hugo/content/posts/2014-04-17-cayley-hamilton-theorem.md
@@ -0,0 +1,58 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2014-04-17T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/cayley-hamilton-theorem/
+- /cayley-hamilton-theorem/
+title: Cayley-Hamilton theorem
+---
+This is to detail a much easier proof (at least, I find it so) of [Cayley-Hamilton][1] than the ones which appear on the Wikipedia page. It only applies in the case of complex vector spaces; most of the post is taken up with a proof of a lemma about complex matrices that is very useful in many contexts.
+
+The idea is as follows: given an arbitrary square matrix, upper-triangularise it (looking at it in basis \\(B\\)). Then consider how \\(A-\lambda I\\) acts on the vectors of \\(B\\); in particular, how it deals with the subspace spanned by \\(b_1, \dots, b_i\\).
+
+# Lemma: upper-triangulation
+
+> Given a square matrix \\(A\\), there is a basis with respect to which \\(A\\) is upper-triangular.
+
+Proof: by induction. It's obviously true for \\(1 \times 1\\) matrices, as they're already triangular. Now, let's take an arbitrary \\(n \times n\\) matrix \\(A\\). We want to make it upper-triangular. In particular, thinking about the top-left element, we need \\(A\\) to have an eigenvector (since if \\(A\\) is upper-triangular with respect to basis \\(B\\), then \\(A(b_1) = \lambda b_1\\), where \\(\lambda\\) is the top-left element). OK, let's grab an eigenvector \\(v_1\\) with eigenvalue \\(\lambda\\).
+
+We'd love to be done by induction at this point - if we extend our eigenvector to a basis, that extension itself forms a smaller space, on which \\(A\\) is upper-triangulable. We have that every subspace has a complement, so let's call pick a complement of \\(\text{span}(v_1)\\) and call it \\(W\\).
+
+Now, we want \\(A\\) to be upper-triangulable on \\(W\\). It makes sense, then, to restrict it to \\(W\\) - we'll call the restriction \\(\tilde{A}\\), and that's a linear map from \\(W\\) to \\(V\\). Our inductive hypothesis requires a square matrix, so we need to throw out one of the rows of this linear map - but in order that we're working with an endomorphism (rather than just a linear map) we need \\(A\\)'s domain to be \\(W\\). That means we have to throw out the top row as well - that is, we compose with \\(\pi\\) the projection map onto \\(W\\).
+
+Then \\(\pi \cdot \tilde{A}\\) is \\((n-1)\times(n-1)\\), and so we can induct to state that there is a basis of \\(W \leq V\\) with respect to which \\(\pi \cdot \tilde{A}\\) is upper-triangular. Let's take that basis of \\(W\\) as our extension to \\(v_1\\), to make a basis of \\(V\\). (These are \\(n-1\\) length-\\(n\\) vectors.)
+
+Then we construct \\(A\\)'s matrix as \\(A(v_1), A(v_2), \dots, A(v_n)\\). (That's how we construct a matrix for a map in a basis: state where the basis vectors go under the map.)
+
+Now, with respect to this basis \\(v_1, \dots, v_n\\), what does \\(A\\) look like? Certainly \\(A(v_1) = \lambda v_1\\) by definition. \\(\pi(A(v_2)) = \pi(\tilde{A}(v_2))\\) because \\(\tilde{A}\\) acts just the same as \\(A\\) on \\(W\\); by upper-triangularity of \\(\pi \cdot \tilde{A}\\), we have that \\(\pi \cdot \tilde{A}(\pi(v_2)) = k v_2\\) for some \\(k\\). The first element (the \\(v_1\\) coefficient) of \\(A(v_2)\\), who knows? (We threw that information away by taking \\(\pi\\).) But that doesn't matter - we're looking for upper-triangulability rather than diagonalisability, so we're allowed to have spare elements sitting at the top of the matrix.
+
+And so forth: \\(A\\) is upper-triangular with respect to some basis.
+
+## Note
+
+Remember that we threw out some information by projecting onto \\(W\\). If it turned out that we didn't throw out any information - if it turned out that if we could always "fill in with zeros" - then we'd find that we'd constructed a basis of eigenvectors, and that the matrix was diagonalisable. (This is how the two ideas are related.)
+
+# Theorem
+
+Recall the statement of the theorem:
+
+> Every square matrix satisfies its characteristic polynomial.
+
+Now, this would be absolutely trivial if our matrix \\(A\\) were diagonalisable - just look at it in a basis with respect to which \\(A\\) is diagonal (recalling that change-of-basis doesn't change characteristic polynomial), and we end up with \\(n\\) simultaneous equations which are conveniently decoupled from each other (by virtue of the fact that \\(A\\) is diagonal).
+
+We can't assume diagonalisability - but we've shown that there is something nearly as good, namely upper-triangulability. Let's assume (by picking an appropriate basis) that \\(A\\) is upper-triangular. Now, let's say the characteristic polynomial is \\(\chi(x) = (x - \lambda_1)(x-\lambda_2) \dots (x-\lambda_n)\\). What does \\(\chi(A)\\) do to the basis vectors?
+
+Well, let's consider the first basis vector, \\(e_1\\). We have that \\(A(e_1) = \lambda_1 e_i\\) because \\(A\\) is upper-triangular with top-left element \\(\lambda_1\\), so we have \\((A-\lambda_1 I)(e_1) = 0\\). If we look at the characteristic polynomial as \\((x-\lambda_n)\dots (x-\lambda_1)\\), then, we see that \\(\chi(A)(e_1) = 0\\).
+
+What about the second basis vector? \\(A(e_2) = k e_1 + \lambda_2 e_2\\); so \\((A - \lambda_2 I)(e_2) = k e_1\\). We've pulled the \\(2\\)nd basis vector into an earlier-considered subspace, and happily we can kill it by applying \\((A-\lambda_1 I)\\). That is, \\(\chi(A)(e_2) = (A-\lambda_n I)\dots (A-\lambda_1 I)(A-\lambda_2 I)(e_2) = (A-\lambda_n I)\dots (A-\lambda_1 I) (k e_1) = 0\\).
+
+Keep going: the final case is the \\(n\\)th basis vector, \\(e_n\\). \\(A-\lambda_n I\\) has a zero in the bottom-right entry, and is upper-triangular, so it must take \\(e_n\\) to the subspace spanned by \\(e_1, \dots, e_{n-1}\\). Hence \\((A-\lambda_1 I)\dots (A-\lambda_n I)(e_n) = 0\\).
+
+Since \\(\chi(A)\\) is zero on a basis, it must be zero on the whole space, and that is what we wanted to prove.
+
+ [1]: https://en.wikipedia.org/wiki/Cayley-Hamilton_theorem "Cayley-Hamilton theorem"
diff --git a/hugo/content/posts/2014-04-26-sequentially-compact-iff-compact.md b/hugo/content/posts/2014-04-26-sequentially-compact-iff-compact.md
new file mode 100644
index 0000000..8bc6ff0
--- /dev/null
+++ b/hugo/content/posts/2014-04-26-sequentially-compact-iff-compact.md
@@ -0,0 +1,122 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+- proof_discovery
+comments: true
+date: "2014-04-26T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/proof_discovery/sequentially-compact-iff-compact/
+- /sequentially-compact-iff-compact/
+title: Sequentially compact iff compact
+---
+[Prof Körner][1] told us during the [IB Metric and Topological Spaces][2] course that the real meat of the course (indeed, its hardest theorem) was "a metric space is sequentially compact iff it is compact". At the moment, all I remember of this result is that one direction requires Lebesgue's lemma (whose statement I don't remember) and that the other direction is quite easy. I'm going to try and discover a proof - I'll be honest when I have to look things up.
+
+# Easy direction
+
+I don't remember which direction was the easy one. However, I do know that in Analysis we prove very early on that closed intervals are sequentially compact (that is, they have the Bolzano-Weierstrass theorem), so I'm going to guess that that's the easy direction.
+
+## Thought process
+
+Suppose the space is compact. (Then for every open cover there is a finite subcover.) We want to show that every sequence has a convergent subsequence, so of course we'll try a proof by contradiction, because the statement is so general.
+
+Suppose the sequence \\(x_n\\) has no convergent subsequence. That is, no subsequence of \\(x_n\\) converges to \\(y\\), for any \\(y\\). We're aiming for some kind of open cover, and we're in a very general kind of metric space, so we're going to have to generate our cover by considering balls around every point.
+
+What does it mean for every subsequence of \\(x_n\\) not to converge to \\(y\\)? It means that for every ball around \\(y\\), and for every subsequence, we can find arbitrarily many \\(x_n\\) in the subsequence such that \\(x_n\\) is outside that ball. My first thought is that we've made a sequence which might be useful - the \\(x_n\\) outside balls of radius \\(\delta_i\\) for \\(i = \frac{1}{m}\\) - but it's not obvious whether that will in fact be useful, because all we know about this sequence is that it doesn't get near a particular point.
+
+OK, let's look at the "for every \\(y\\)" bit, because that's bound to be where our cover comes from. We're going to want a ball around each \\(y\\), so let's say the ball is of radius \\(\delta_y\\). (We'll delay stating what \\(\delta_y\\) actually is in value, because I have no idea what it's going to be.) Ah, then we know that for every subsequence, there are infinitely many \\(x_i\\) which lie outside the ball \\(B(y, \delta_y)\\).
+
+What does our finite subcover look like? It's a finite collection (say, \\(k\\) many) of balls, and we know that there are infinitely many \\(x_i\\) in any subsequence such that the \\(x_i\\) are outside a given one of those balls. But this is a contradiction: take the subsequence of \\(x_n\\) such that all of the \\(x_i\\) in the subsequence lie outside ball 1. Then take a subsequence of that such that all the elements lie outside ball 2. Repeat: eventually we end up with a subsequence of \\(x_n\\) such that all the elements lie in ball \\(k\\). This subsequence converges.
+
+## Proof
+
+Suppose \\((X,d)\\) is a compact metric space, and take a sequence \\(x_n\\) in \\(X\\). We show that there exists \\(y \in X\\) such that there is a subsequence \\(z_i\\) of the \\(x_n\\) such that \\(z_i \to y\\).
+
+Indeed, if the sequence \\(x_n\\) gets arbitrarily close to \\(y\\) then there is a subsequence of \\(x_n\\) tending to \\(y\\) (namely, let \\(\epsilon_m = \frac{1}{m}\\); then pick \\(x_{n_m}\\) such that \\(d(x_{n_m}, y) < \epsilon_m\\)), so it is enough to show that there is some \\(y\\) such that the sequence \\(x_n\\) gets arbitrarily close to \\(y\\).
+
+We show that this is true. Indeed, suppose not. Then for all \\(y\\) there exists \\(\delta_y\\) such that \\(x_n\\) never gets within \\(\delta_y\\) of \\(y\\) (for all \\(n > N\\), some \\(N\\) - the sequence might have started at \\(y\\), but we know it never returns after some point). Take a cover consisting of those \\(B(y, \delta_y)\\); by compactness, there is a finite subcover.
+
+Now, we have that for the \\(i\\)th ball in the cover, there exists \\(N_i\\) such that \\(x_n\\) never gets into the \\(i\\)th ball for \\(n > N_i\\); but there are only finitely many balls, so \\(x_n\\) never gets into any of the balls for \\(n > N = \text{max}(N_i)\\). But the finite collection of balls is a cover. That is, no \\(x_n\\) is in \\(X\\), for \\(n > N\\) - contradiction.
+
+## Postscript
+
+That did indeed turn out to be the easier direction, then.
+
+# Hard direction
+
+I'm not even going to begin attempting to find out what Lebesgue's lemma is on my own, so I'll just look it up and state it.
+
+> For a sequentially compact metric space \\((X, d)\\), and an open cover \\(U_{\alpha}\\), we have that there exists \\(\delta\\) such that for all \\(x \in X\\), there exists \\(\alpha_x\\) such that \\(B(x, \delta) \subset U_{\alpha_x}\\).
+
+That is, "given any open cover, we can find a ball-width such that for every point, a ball of that width lies entirely in some set in the cover". It feels kind of related to Hausdorffness - while "metric spaces are Hausdorff" guarantees that we can wrap distinct points in non-overlapping balls, Lebesgue's lemma tells us that if our distinct points are not covered by the same set then we can separate them while remaining in those different sets in the cover.
+
+OK, let's go for a proof of this.
+
+## Proving Lebesgue's lemma
+
+Well, where can we start? To actually produce such a \\(\delta\\), it looks like we'd need to take some kind of minimum, and that would require a finite cover (which is assuming compactness). So that's not a good place to start.
+
+If we don't know where to start, we contradict. Suppose there is no \\(\delta\\) such that for all \\(x \in X\\) there exists \\(\alpha_x\\) such that \\(B(x, \delta) \subset U_{\alpha}\\). That is, for every \\(\delta\\) there exists \\(x \in X\\) such that for all open sets in the cover, \\(B(x, \delta) \not \subset U_{\alpha}\\).
+
+We're in a sequentially compact space - we need a sequence, so that it can have a convergent subsequence. Mindlessly (nearly literally - I'm exhausted at the moment, having had an unusually long supervision since proving the easier direction), I'll take \\(\delta_n = \frac{1}{n}\\) and create a sequence \\(x_n\\) such that \\(B(x_n, \frac{1}{n})\\) is not wholly contained in any set of the cover. Then the \\(x_n\\) has a convergent subsequence \\(x_{n_i} \to x\\), say.
+
+Picture pause. We've got our \\(x_{n_i}\\) tending to \\(x\\), with ever-decreasing balls around them. It seems sensible that at some point (since the position of the balls, the centre \\(x_{n_i}\\), is hardly changing, while the radius is getting smaller) the balls will get so small that they start being contained in some cover-set.
+
+That's actually so close to a proof that I'll write it up formally from this point.
+
+### Proof
+
+Let \\((X, d)\\) be a sequentially compact metric space, and let \\(U_\alpha\\) be a cover (ranging \\(\alpha\\) over some indexing set). Assume for contradiction that for every \\(\delta\\) there exists \\(x \in X\\) such that for all \\(\alpha\\), \\(B(x, \delta) \not \subset U_{\alpha}\\).
+
+Specialise to the sequence \\(\delta_n = \frac{1}{n}\\), and let \\(x_n\\) be the corresponding \\(x \in X\\). Then by sequential compactness, there exists a subsequence \\(x_{n_i}\\) tending to some \\(x\\).
+
+Now, \\(B(x_{n_i}, \frac{1}{n_i}) \not \subset U_{\alpha}\\) for any \\(\alpha\\). Also, because each \\(U_{\alpha}\\) is open, we have that for every \\(\alpha\\) such that \\(x \in U_{\alpha}\\) there exists \\(\epsilon_{\alpha}\\) such that \\(B(x, \epsilon_{\alpha})\\) is wholly contained within \\(U_{\alpha}\\).
+
+Fix some \\(\alpha\\) such that \\(x \in U_{\alpha}\\), and let \\(\epsilon = \epsilon_{\alpha}\\). Take \\(n_i\\) such that \\(d(x_{n_i}, x) < \frac{\epsilon}{2}\\) (possible, because \\(x_{n_i} \to x\\)). We have \\(B(x_{n_i}, \frac{1}{n_i})\\) entirely contained in \\(B(x, \epsilon)\\), because any point in the former ball is at most \\(\frac{1}{n_i}\\) away from \\(x_{n_i}\\), which is itself at most \\(\frac{\epsilon}{2}\\) away from \\(x\\); hence any point in \\(B(x_{n_i}, \frac{1}{n_i})\\) is at most \\(\frac{1}{n_i} +\frac{\epsilon}{2}\\) away from \\(x\\). Picking \\(n_i > \frac{2}{\epsilon}\\) (as well as such that \\(d(x_{n_i}, x) < \frac{\epsilon}{2}\\)) ensures that \\(\frac{1}{n_i} +\frac{\epsilon}{2} < \epsilon\\).
+
+But this is a contradiction: we have a ball entirely contained in some \\(U_{\alpha}\\) - namely \\(B(x, \epsilon)\\) - which contains a ball which is not entirely contained in \\(U_{\alpha}\\) - namely \\(B(x_{n_i}, \frac{1}{n_i})\\).
+
+## Proving the main theorem
+
+OK, what do we have? We have that any open cover of a sequentially compact space allows us to draw a ball of *predetermined width* around each point, such that every ball is contained entirely in a set from the cover.
+
+What do we want? We want every open cover of a sequentially compact space to have a finite subcover. [^when]
+
+OK, let's do the only possible thing and take an open cover of a sequentially compact space. We might be able to build a finite subcover because of our predetermined-width balls, but I want a picture first.
+
+### Pictures (feel free to skip)
+
+Let's use \\([0, 1]\\) and the cover \\([0, \frac{1}{5}), (\frac{1}{n+2}, \frac{1}{n}), (\frac13, 1]\\) where \\(n \geq 2\\), and let's suppose \\(\delta = \frac17\\). (This clearly works as a \\(\delta\\) in Lebesgue's lemma.) Then a \\(\frac17\\)-ball around any point remains in some set of the cover. The reason we have a finite subcover in this case is that the sets in the cover get smaller, so eventually we can just discard the ones which are too small to contain a \\(\frac17\\)-ball. It turns out that wasn't a great intuition guide - metric spaces can be a lot odder than that.
+
+We want a space where the "balls get smaller" argument fails. Let's use \\(\mathbb{R} \cup \{ \infty \}\\) under the usual metric, and the cover \\((n-\frac34, n+\frac34)\\) along with some ball around \\(\infty\\). The reason this one works is because the ball around infinity makes sure we can throw out most of the sets of the cover, because they are contained in the ball around infinity. (A suitable \\(\delta\\) is \\(\frac14\\).)
+
+### End of pictures
+
+Hmm, I don't think I can easily come up with an example which explains exactly why the theorem is true. I slept on this, and got no further, so I looked up the next step: assume that it is not possible to cover the space with a finite number of \\(B(x_i, \delta)\\). (This should perhaps have been suggested to me by my finite examples, in hindsight.) It turns out that this step makes it really easy.
+
+Then for all finite sequences \\((x_i)_{i=1}^n\\), there is a point \\(x_{n+1}\\) such that \\(x_{n+1}\\) is not in the cover; this forms a sequence which must have a convergent subsequence. Because the covering-balls are all of fixed width \\(\delta\\), we must have that eventually the points in the subsequence draw together enough to sit in the same ball.
+
+## Proof
+
+Suppose \\((X, d)\\) is a sequentially compact metric space which is not compact, and fix an arbitrary open cover \\(U_{\alpha}\\) such that there is no finite subcover. Then by Lebesgue's lemma, there is \\(\delta\\) such that for all \\(x \in X\\), there is \\(\alpha_x\\) such that \\(B(x, \delta) \subset U_{\alpha}\\).
+
+Now, if it were possible to cover \\(X\\) with a finite number of \\(B(x_i, \delta)\\) then we would have a finite subcover (namely, \\(U_{\alpha_{x_i}}\\) for each \\(i\\)). Hence it is impossible to cover \\(X\\) with a finite number of \\(B(x_i, \delta)\\). Take a sequence \\((x_n)_{n=1}^{\infty}\\) such that \\(x_i\\) does not lie in any \\(B(x_j, \delta)\\) for \\(j < i\\) (and where \\(x_1\\) is arbitrary). Then there is a convergent subsequence \\(x_{n_i} \to x\\), say; wlog let \\(n_i = i\\), for ease of notation (so the original sequence converged).
+
+But this contradicts the requirement that \\(x_i\\) always lies outside \\(B(x_j, \delta)\\) for \\(j < i\\): indeed, \\(d(x_i, x_j) < \delta\\) for sufficiently large \\(i, j\\), since convergent sequences are Cauchy.
+
+Hence \\((X, d)\\) is compact.
+
+# Postscript
+
+Ouch, that took a long time. There were three key ideas I ended up using.
+
+1.  One direction is so easy that it's one of the first theorems we prove in Analysis.
+2.  Lebesgue's lemma.
+3.  Contradict ALL the things. (Every single major step in either direction of the proof is a contradiction, and everything just falls out.)
+
+[^when]: When do we want it? Now!
+
+ [1]: https://en.wikipedia.org/wiki/Tom_Körner "Prof Körner Wikipedia page"
+ [2]: https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/ "Met+Top"
diff --git a/hugo/content/posts/2014-05-03-discovering-a-proof-of-sylvesters-law-of-inertia.md b/hugo/content/posts/2014-05-03-discovering-a-proof-of-sylvesters-law-of-inertia.md
new file mode 100644
index 0000000..b24bbf1
--- /dev/null
+++ b/hugo/content/posts/2014-05-03-discovering-a-proof-of-sylvesters-law-of-inertia.md
@@ -0,0 +1,66 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+- proof_discovery
+comments: true
+date: "2014-05-03T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/proof_discovery/discovering-a-proof-of-sylvesters-law-of-inertia/
+- /discovering-a-proof-of-sylvesters-law-of-inertia/
+title: Discovering a proof of Sylvester's Law of Inertia
+---
+*This is part of what has become a series on discovering some fairly basic mathematical results, and/or discovering their proofs. It's mostly intended so that I start finding the results intuitive - having once found a proof myself, I hope to be able to reproduce it without too much effort in the exam.*
+
+# Statement of the theorem
+
+[Sylvester's Law of Inertia][1] states that given a quadratic form \\(A\\) on a real finite-dimensional vector space \\(V\\), there is a diagonal matrix \\(D\\), with entries \\(( 1_1,1_2,\dots,1_p, -1_1, -1_2, \dots, -1_q, 0,0,\dots,0 )\\), to which \\(A\\) is congruent; moreover, \\(p\\) and \\(q\\) are the same however we transform \\(A\\) into this diagonal form.
+
+# Proof
+
+The very first thing we need to know is that \\(A\\) is diagonalisable. (If it isn't diagonalisable, we don't have a hope of getting into this nice form.) We know of a few classes of diagonalisable matrices - symmetric, Hermitian, etc. All we know about \\(A\\) is that it is a real quadratic form. What does that mean? It means that \\(A(x) = x^T A x\\) if we move into some coordinate system; transposing gives us that \\(A(x)^T = x^T A^T x\\), but the left-hand-side is scalar so is symmetric, whence \\(A = A^T\\) (because \\(x\\) was arbitrary). Hence \\(A\\) has a symmetric matrix and so is diagonalisable: there is an orthogonal matrix \\(P\\) such that \\(P^{-1}AP = D\\), where \\(D\\) is diagonal. (Recall that a matrix \\(M\\) is orthogonal if it satisfies \\(M^{-1} = M^T\\).)
+
+Now we might as well consider \\(D\\) in diagonal form. Some of the elements are positive, some negative, and some zero - it's easy to transform \\(D\\) so that the positive ones are all together, the negative ones are all together and the zeros are all together, by swapping basis vectors. (For instance, if we want to swap diagonal elements in positions \\((i,i), (j,j)\\), just swap \\(e_i, e_j\\).) Now we can scale every diagonal element down to \\(1\\), by scaling the basis vectors - if we scale \\(e_i\\) by \\(\sqrt{ \vert A_{i,i} \vert }\\), calling the resulting basis \\(f_i\\) we'll get \\(A(f_i) =  \vert A_{i,i} \vert  A(e_i) = A_{i,i} e_i\\) as required. (The squaring comes from the fact that \\(A\\) is a \*quadratic\* form, so \\(A(a x) = a^2 A(x)\\).)
+
+Hence we've got \\(A\\) into the right form. But how do we show that the number of positive and negative elements is an invariant?
+
+## Positive bit
+
+All I remember from the notes is that there's something to do with positive definite subspaces. It turns out that's a really big hint, and I haven't been able to think up how you might discover it. Sadly, I'll just continue as if I'd thought it up for myself rather than remembering it.
+
+The following section was my first attempt. My supervisor then told me that it's a bit inaccurate (and some of it doesn't make sense). In particular, I talk about the dimension of \\(V \backslash P\\) for \\(P\\) a subspace of \\(V\\) - but \\(V \backslash P\\) isn't even a space (it doesn't contain \\(0\\)). During the supervision I attempted to refine it by using \\(P^C\\) the complement of \\(P\\) in \\(V\\), but even that is vague, not least because complements aren't unique.
+
+### Original attempt
+
+We have a subspace \\(P\\) on which \\(A\\) is positive definite - namely, make \\(A\\) diagonal and then take the first \\(p\\) basis vectors. (Remember, positive definite iff \\(A(x, x) > 0\\) unless \\(x = 0\\); but \\(A(x,x) > 0\\) for \\(x \in P\\) because \\(x^T A x\\) is a sum of positive things.) Similarly, we have a subspace \\(Q\\) on which \\(A\\) is negative semi-definite (namely "everything which isn't in \\(P\\)"). Then what we want is: for any other diagonal form of \\(A\\), there is the same number of 1s on the diagonal, and the same number of -1s, and the same number of 0s. That is, we want to ensure that just by changing basis, we can't alter the size of the subspace on which \\(A\\) is positive-definite.
+
+We'll show that for any subspace \\(R\\) on which \\(A\\) is positive-definite, we must have \\(\dim(R) \leq \dim(P)\\). Indeed, let's take \\(R\\) on which \\(A\\) is positive definite. The easiest way to ensure that its dimension is less than that of \\(P\\) is to show that it's contained in \\(P\\). Now, that might be hard - we don't know anything about what's in \\(R\\) - but we might do better in showing that nothing in \\(R\\) is also in \\(V \backslash P\\), because we know \\(A\\) is negative semi-definite on \\(V \backslash P\\), and that's inherently in tension with the positive-definiteness on \\(R\\).
+
+Suppose \\(r \not \in P\\) and \\(r \in R\\). Then \\(A(r,r) \leq 0\\) (by the first condition) and \\(A(r,r) > 0\\) (by the second condition, since \\(R\\) is positive-definite) - contradiction.
+
+That was quick - we showed, for all subspaces \\(R\\) on which \\(A\\) is positive-definite, that \\(\dim(R) \leq \dim(P)\\).
+
+### Supervisor-vetted version
+
+We have a subspace \\(P\\) on which \\(A\\) is positive-definite - namely, make \\(A\\) diagonal and take the first \\(p\\) basis vectors. We'll call the set of basis vectors \\(\{e_1, \dots, e_n \}\\); then \\(P\\) is spanned by \\(\{e_1, \dots, e_p \}\\).
+
+Now, let's take any subspace \\(\tilde{P}\\) on which \\(A\\) is positive-definite. We want \\(\dim(\tilde{P}) \leq \dim(P)\\); to that end, take \\(N\\) spanned by \\(\{e_{p+1}, \dots, e_n \}\\). We show that \\(\tilde{P} \cap N = \{0\}\\). Indeed, if \\(r \in\tilde{P} \cap N\\), with \\(r \not = 0\\), then:
+
+*   \\(r \in \tilde{P}\\) so \\(A(r,r) > \\)0
+*   \\(r \in N\\) so \\(A(r,r) \leq \\)0
+
+But this is a contradiction. Hence \\(\tilde{P} \cap N\\) is the zero space, and so \\(\dim(\tilde{P}) \leq \dim(P)\\) because \\(\dim(P) + \dim(N) = n\\) while \\(\dim(\tilde{P}) + \dim(N) \leq n\\).
+
+### Commentary
+
+Notice that my original version is conceptually quite close to correct: "take something in a positive-definite space, show that it can't be in the negative-semi-definite bit and hence must be in \\(P\\)". I was careless in not checking that what I had written made sense. I am slightly surprised that no alarm bells were triggered by my using \\(V \backslash P\\) as a space - I hope that now my background mental checks will come to include this idea of "make sure that when you transform objects, you retain their properties".
+
+### Completion (original and hopefully correct)
+
+Identically we can show that for all subspaces \\(Q\\) on which \\(A\\) is negative-definite, that \\(\dim(Q) \leq \dim(N)\\) (with \\(N\\) defined analogously to \\(P\\) but with negative-definiteness instead of positive-definiteness). And we already know that congruence preserves matrix rank (because matrix rank is a property of the eigenvalues, and basis change in this way only alters eigenvalues by multiples of squares), so we have that the number of zeros in any diagonal representation of \\(A\\) is the same.
+
+Hence in any diagonal representation of \\(A\\) with \\(p', q', z'\\) the number of \\(1, -1, 0\\) respectively on the diagonal, we need \\(p' \leq p, q' \leq q, z' = z\\) - but because the diagonal is the same size on each matrix (since the matrices don't change dimension), we must have equality throughout.
+
+ [1]: https://en.wikipedia.org/wiki/Sylvester's_Law_of_Inertia "Sylvester's law of inertia Wikipedia page"
diff --git a/hugo/content/posts/2014-05-26-proof-that-symmetric-matrices-are-diagonalisable.md b/hugo/content/posts/2014-05-26-proof-that-symmetric-matrices-are-diagonalisable.md
new file mode 100644
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--- /dev/null
+++ b/hugo/content/posts/2014-05-26-proof-that-symmetric-matrices-are-diagonalisable.md
@@ -0,0 +1,28 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2014-05-26T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/proof-that-symmetric-matrices-are-diagonalisable/
+- /proof-that-symmetric-matrices-are-diagonalisable/
+title: Proof that symmetric matrices are diagonalisable
+---
+This comes up quite frequently, but I've been stuck for an easy memory-friendly way to do this. I trawled through the 1A Vectors and Matrices course notes, and found the following mechanical proof. (It's not a discovery-proof - I looked it up.)
+
+## Lemma
+
+Let \\(A\\) be a symmetric matrix. Then any eigenvectors corresponding to different eigenvalues are orthonormal. (This is a very standard fact that is probably hammered very hard into your head if you have ever studied maths post-secondary-school.) The proof of this is of the "write it down, and you can't help proving it" variety:
+
+Suppose \\(\lambda, \mu\\) are different eigenvalues of \\(A\\), corresponding to eigenvectors \\(x, y\\). Then \\(Ax = \lambda x\\), \\(A y = \mu y\\). Hence (transposing the first equation) \\(x^T A^T = \lambda x^T\\); the left hand side is \\(x^T A\\). Hence \\(x^T A y = \lambda x^T y\\); but \\(A y = \mu y\\) so this is \\(x^T \mu y = \lambda x^T y\\). Since \\(\lambda \not = \mu\\), this means \\(x^T y = 0\\).
+
+## Theorem
+
+Now, suppose \\(A\\) has eigenvalues \\(\lambda_1, \dots, \lambda_n\\). They might not be distinct; take the ones which are, \\(\lambda_1, \dots, \lambda_r\\). Then extend this to a basis of \\(\mathbb{R}^n\\), and orthonormalise that basis using the [Gram-Schmidt process][1]. (This can be proved - it's tedious but not hard, as long as you remember what the Gram-Schmidt process is, and I think it's safe to assume.) With respect to this basis, \\(A\\) is a matrix which is diagonal in the first \\(r\\) entries. Moreover, we are performing an orthonormal change of basis, and conjugation by orthogonal matrices preserves the property of "symmetricness" (proof: \\((P^T A P)^T = P^T A^T P = P^T A P\\)), so the \\(r+1\\)th to \\(n\\)th row/column block is symmetric. It is also real (because we have performed a conjugation by a real matrix). And we have that the first \\(r\\) columns of \\(P^T A P\\) are filled with zeros below the diagonal (being the image of eigenvectors), so \\(P^T A P\\) is also filled with zeros in the first \\(r\\) rows above the diagonal, because it is a symmetric matrix.
+
+Now by induction, that sub-matrix \\(A_{r,r} \dots A_{n,n}\\) is diagonalisable by an orthogonal matrix. Hence we are done: all symmetric matrices are diagonalisable by an orthogonal change of basis. (The eigenvectors produced by the inductive step must be orthogonal to the ones we've already found, because they fall in a subspace which is orthogonal to that of the one we already found.)
+
+ [1]: https://en.wikipedia.org/wiki/Gram-Schmidt_process "Gram-Schmidt process Wikipedia page"
diff --git a/hugo/content/posts/2014-06-25-possible-cons-of-Soylent.md b/hugo/content/posts/2014-06-25-possible-cons-of-Soylent.md
new file mode 100644
index 0000000..546f6f4
--- /dev/null
+++ b/hugo/content/posts/2014-06-25-possible-cons-of-Soylent.md
@@ -0,0 +1,29 @@
+---
+lastmod: "2022-12-31T23:21:00.0000000+00:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2014-06-25T00:00:00Z"
+aliases:
+- /uncategorized/possible-cons-of-Soylent/
+- /possible-cons-of-Soylent/
+title: Possible cons of Soylent
+---
+
+I have seen many glowing reviews of [Soylent](https://soylent.com), and many vitriolic [naturalistic](https://en.wikipedia.org/wiki/Appeal_to_nature) arguments against it. What I have not really seen is a proper collection of credible reasons why you might not want to try Soylent (that is, reasons which do not boil down to "it’s not natural, therefore Soylent is bad" or "food is great, therefore Soylent is bad").
+
+This page used to contain citations in the form of links to the Soylent Discourse forum at `discourse.soylent.com`.
+However, that site is now defunct.
+
+* *Soylent is untested.* Indeed, there are apparently trials being run (there was originally a link to a post from the founder of Soylent, but the link is dead), but I have not seen any data coming out of them (or indeed any evidence of a trial, other than the founder’s word). It is perfectly plausible that Soylent misses out something important - [lycopene](https://en.wikipedia.org/wiki/Lycopene), for instance, may turn out to be highly beneficial. Of course, various fast-foody diets don’t contain lycopene or whatever anyway. The current fact that no-one has become ill (apart from a well-known and easily-fixed sodium problem) in a diet-related way from Soylent is insufficient as evidence that Soylent is safe.
+
+* *Soylent is even more addictive than whole food.* People often report that Soylent makes them feel really really good for a few days, before they adjust to their new level of wellbeing and "good" becomes "normal". Then returning to whole food causes them to feel sluggish and generally not very well. On the other hand, some report that whole food becomes extra-tasty, so perhaps it’s a balancing act - switching from Soylent to a good diet may be important.
+
+* *You hate the idea/you find cooking too fun.* Fine, don’t eat it.
+
+* *It’s effort to test and tune your home-made recipe.* Everyone is different, and you might need to make up for pre-existing deficiencies or whatever. As much as the DIY community and [Rosa Labs](http://www.rosalabs.com/) would like it, one size does not fit all, and it might take a while to find out what you need.
+
+* *There are side-effects of adjusting to Soylent.* People usually report gas when starting a soylent, and sometimes it doesn’t seem to settle down. It seems to be unclear why this issue is experienced, too. There are other symptoms, like headaches (which are apparently usually down to having not enough sodium or not enough water) and bloatedness (which is apparently solved by not drinking the Soylent so quickly).
+
+* *Expense.* There are some DIY recipes which are very expensive. This is often because protein is dear, and low-carb soylents are to be mostly protein and fat by necessity. Too high a fat content is unpalatable, so the expensive protein makes up the calories.
diff --git a/hugo/content/posts/2014-07-13-solvability-of-nonograms.md b/hugo/content/posts/2014-07-13-solvability-of-nonograms.md
new file mode 100644
index 0000000..f86ea1e
--- /dev/null
+++ b/hugo/content/posts/2014-07-13-solvability-of-nonograms.md
@@ -0,0 +1,48 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2014-07-13T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/nonograms/
+- /solvability-of-nonograms/
+title: Solvability of nonograms
+---
+Recently, a friend re-introduced me to the joys of the [nonogram] (variously known as "hanjie" or "griddler"). I was first shown these about ten years ago, I think, because they appeared in [The Times]. When The Times stopped printing them, I forgot about them for a long time, until two years ago, or thereabouts, I tried these on [a website][griddlers.net]. I find the process much more satisfying on paper with a pencil than on computer, so I gave them up again and forgot about them again.
+
+Anyway, the thought occurred to me: is a given griddler always solvable, and is it solvable uniquely? That is, given a grid and the edge entries, is it always a valid puzzle?
+
+Notation: we will say that a given *solved grid* has an *edge-set* consisting of the numbers we would see if we were about to start solving the nonogram. We say that an edge-set *applies to* a solved grid if that edge-set is consistent with the solved grid. (For instance, the empty edge-set doesn't apply to any solved grid apart from the zero-size grid.)
+
+Then our question has become: is there in some way a bijection between (edge-sets) and (solved grids)?
+
+# Existence of edge-sets
+
+We can trivially describe any solved grid by an edge-set and a grid size: simply write down the grid size of the solved grid, and write down the obvious edge-set. (We do need the grid size to be specified, because given an edge-set which applies to a solved grid, we can create a new grid to which that edge-set applies by simply appending a blank row to the solved grid.)
+
+# Uniqueness of edge-sets
+
+Is there an obvious reason why we could never have two different edge-sets applying to the same solved grid? It seems intuitively clear that a given solved grid can only have the obvious edge-set (namely, the one we get by writing down the blocks in each row and column in the obvious way). Is this rigorous as a proof? Yes: suppose that we had two edge-sets describing the same solved grid, and (wlog) the sets differ in the first row. In fact, let us wlog that our solved grid is only one row long.
+
+* If one edge-set is empty, we're done: because the two edge-sets are not the same, that means the other edge-set is non-empty, and so under the first edge-set the solved grid is empty, while under the second the solved grid is nonempty.
+* If both edge-sets are non-empty: suppose the first starts with the number \\(a\\), and the second with the number \\(b\\). Then we have some number of blank squares, and then \\(a\\) filled-in squares (by edge-set 1) and also \\(b\\) filled-in squares (by edge-set 2); hence \\(a=b\\), because our solved grid is fixed.
+
+# Existence of solutions
+Must a solution exist for a given grid size and edge-set? Is it possible to create a nonogram with no solution? One strategy for proving this might be to count the number of allowable edge-sets and to count the number of allowable solved grids (the latter problem is extremely easy if we consider a grid as being a binary number whose bits are laid out in a rectangle), because we have that any two finite sets of the same size must biject. However, the former problem sounds very hard.
+
+On second thoughts (read: I slept on this), it's blindingly obvious that there is a grid with no solution - namely, the one-by-one grid with edge-set "1 as column heading, 0 as row heading". So there certainly are edge-sets which don't have a solution grid.
+
+# Uniqueness of solutions
+OK, if we don't always have solvability, how about the "easy puzzle-setting property": that a given edge-set and grid-size cannot have two solved grids to which the edge-set applies? If this were true, it would make generating puzzles extremely easy: simply draw out a solved grid, write down its edge-set (which is unique, as shown above), and set that edge-set and grid-size as the puzzle, without fear that someone could sit down and solve the puzzle validly to get a different grid to your solution.
+
+On the same second thoughts as the 'existence of solutions' thoughts, it's clear that the 2-by-2 grid with a diagonal black stripe has two solutions - namely, send the stripe top-left to bottom-right, or top-right to bottom-left. Curses.
+
+# Summary
+Every solved grid has an edge-set, which is unique to that grid. However, not all edge-sets are solvable, and we don't have uniqueness of solutions. That was much less interesting than I had hoped.
+
+[nonogram]: https://en.wikipedia.org/wiki/Nonogram
+[griddlers.net]: https://www.griddlers.net/home
+[The Times]: http://www.thetimes.co.uk/tto/news/
diff --git a/hugo/content/posts/2014-07-15-what-maths-does-to-the-brain.md b/hugo/content/posts/2014-07-15-what-maths-does-to-the-brain.md
new file mode 100644
index 0000000..023446b
--- /dev/null
+++ b/hugo/content/posts/2014-07-15-what-maths-does-to-the-brain.md
@@ -0,0 +1,47 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- psychology
+comments: true
+date: "2014-07-15T00:00:00Z"
+disqus: true
+math: true
+aliases:
+- /psychology/what-maths-does-to-the-brain/
+- /what-maths-does-to-the-brain/
+title: What maths does to the brain
+---
+
+In my activities on [The Student Room], a student forum, someone (let's call em Entity, because I like that word) recently asked me about the following question.
+
+>Isaac places some counters onto the squares of an 8 by 8 chessboard so that there is at most one counter in each of the 64 squares. Determine, with justification, the maximum number that he can place without having five or more counters in the same row, or in the same column, or on either of the two long diagonals.
+
+You might like to have a think about it before I give first the answer Entity gave, and my commentary on it.
+
+I paraphrase Entity's answer:
+
+>The maximum is 32, because the maximum along each row is 4 and so having 33 counters means having more than one row being full. Moreover, I have found a pattern which satisfies the 32 requirement. Hence we have shown that the correct answer is at most and at least 32, so it must be 32.
+
+I'm going to assume that the 32-pattern is correct, because I wasn't shown the purported answer. What interested me was that my mind immediately pointed out internally that we have made an unproved claim. Again, you might like to think what the unproved claim might be - it's completely trivial to prove, but I found it fascinating. It'll come in the next paragraph.
+
+The unproved claim is "having 33 counters means having more than one row being full". There are a couple of trivial proofs:
+
+* \\(\frac{33}{64} > \frac{1}{2}\\) is the proportion of the board which is becountered, and the mean of eight quantities (the proportion of counters in each row) which are all less than or equal to a half cannot itself be greater than a half. Hence at least one of the eight quantities is greater than a half (that is, a row has more than four counters in).
+* The [pigeonhole principle] gives the result directly in a similar way (33 pigeons into eight holes means one hole has more than four pigeons).
+
+However, my mind flagged this claim automatically as something that wasn't necessarily obvious. It turned out to be trivial, but it is an example of a step which is in general not true. For instance:
+
+> Consider the natural numbers (greater than 0). The set of even numbers takes up half the space. Now if we remove the number 2 from the set of even numbers, we have the collection still taking up half the space, but now it's a smaller set - it's missing an element. Conundrum.
+
+Here, we used very similar reasoning ("removing something from a set makes it take up proportionally less space") but got nonsense, ultimately because the pigeonhole principle doesn't apply to infinite sets.
+
+I think what I did here was recognise a general pattern, but I struggle to work out what that pattern might be. The closest I've come is "if one property of a structure holds, then an obviously related property of that structure holds", because I'm pretty sure my thought wasn't triggered by the need for the pigeonhole principle. (In that case, the pattern would have been "if we fill up some slots, then some subset of the slots must be full", which is much more specific and trivial than I feel my reaction was. It felt like a specific instance of a very general check.)
+
+A similar pattern which is much more concrete is the distinction between "if" and "only if" and "if and only if". A mathematician trains emself early on to not get confused between these. It doesn't take too long before you simply stop having the mental architecture that lets you make a mistake like "all odd squares are squares of odd numbers. Indeed, if n is odd then n^2 is odd. QED" unless the structures you're working with are quite a bit more complicated. Of course, my mental checks can be overwhelmed by complexity, and I have certainly proved the wrong direction of a problem many times, but in everyday conversation and in simpler mathematical problems, it becomes not only easy but automatic to distinguish between "implies" and "is implied by".
+
+It feels vaguely similar to some of the filters I've installed in myself for other reasons. For instance, earlier today I was asked which of five leaflets looked best. I had already seen one of them before, and my first reaction (before any other) was "I've seen this one before, so I'm likely to think it looks better". I have a few anti-bias systems like these, and I have no idea whether they're useful or not, but I can certainly feel them going sometimes, without any input from myself.
+
+
+[The Student Room]: http://www.thestudentroom.co.uk
+[pigeonhole principle]: https://en.wikipedia.org/wiki/Pigeonhole_principle
diff --git a/hugo/content/posts/2014-07-19-music-practice.md b/hugo/content/posts/2014-07-19-music-practice.md
new file mode 100644
index 0000000..62d2029
--- /dev/null
+++ b/hugo/content/posts/2014-07-19-music-practice.md
@@ -0,0 +1,38 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2014-07-19T00:00:00Z"
+aliases:
+- /uncategorized/music-practice/
+- /music-practice/
+title: Music practice
+---
+
+A couple of weeks ago, someone opined to me that there was a type of person who was just able to sit down and play at the piano, without sheet music.
+
+I, myself, am capable of playing [precisely one piece][Jarrod Radnich PotC] inexpertly, from memory, at the piano. (My rendering of that piece is *nowhere* near the arranger's standard.) I can play nothing else without sheet music. I very much think that this is the natural state for essentially every musician who has not spent thousands upon thousands of hours practising in a general way. That is, almost no-one can naturally sit down and play a piece from memory without a lot of work beforehand, and almost no-one can improvise well without a great deal of effort directed either at learning how to improvise, or at learning generally the mechanics of playing.
+
+# How technical practice helps
+The syllabus for [ABRSM exams] contains a large body of scale-work, arpeggios and related patterns. There is a reason these are so heavily featured that one cannot attain a Distinction grade without them: because while they may not help much in learning pieces up to Grade 8, they really are useful beyond then. Someone who can sit down in front of an unseen piece and note automatically that "the left-hand is an Alberti bass in F#-major" is at a distinct advantage compared to someone who has never practised F#-major arpeggios, because the latter person has to read each note in both hands; the former can concentrate almost solely on the right hand's melody line. An impossible piece can become quite do-able if you can reduce the left-hand's job to that of repeating a memorised action.
+
+# How general performance practice helps
+Because the same patterns occur so often in music, there is essentially no upper limit on how many useful actions there are to memorise. Most phrases of music end in one of a couple of [recognised ways][cadence] (cadences), and a given cadence doesn't vary that much in its presentation. Someone well-practised could quite conceivably only need to read three-quarters of a piece, knowing that the remaining quarter is already-familiar cadences.
+
+And, of course, it is hard to practise actual pieces without coming across cadences - they show up so regularly. By just performing general practice of a wide range of pieces, you naturally come to be able to play cadences without much thought. If you devote effort to learning particular chordal patterns, this process becomes even easier.
+
+If you play the piano with any level of seriousness, you have probably played a [fugue] at some point. A fugue is a piece of music mainly characterised by a single melody which is repeated at various pitches, and around which a richly textured harmony is built. The idea is to bat this theme between several 'voices' (for instance, a fugue might have four voices, two played in the left hand and two in the right, analogously to a four-part choir), with each voice either playing the theme or embellishing upon it. It's kind of like a more complicated and interesting canon. The key point is that fugues are all very similar in style, and if you have the skill of playing a single tune more than once simultaneously, in different voices and offset from each other, then you can pretty much play a fugue. That skill comes with practice.
+
+Anyone who has sat a music exam knows how important it is to be able to recover from mistakes. The best way to recover from a mistake would be to improvise something that sounds plausible (ideally the original piece!) until you picked up the thread again. This, too, comes with practice: I have noticed myself that I have over time got substantially better at ignoring mistakes I make during a performance. Every so often, if I'm caught out while playing a piece I know well, I can just about invent a semi-plausible bar to fill in the gap before I recover. (If nothing else, I might be able to play the right chords in an unexpected [inversion] or something.) I understand that this skill has pretty much no upper bound.
+
+# Summary
+The point is, then, that it is probable that sheer mind-numbing amounts of practice is what makes people able to sit down and play. Certainly some may require less practice than others, but anyone who can play at the drop of a hat has probably practised an awful lot to get like that. I certainly know of no counterexamples.
+
+
+[Jarrod Radnich PotC]: https://www.youtube.com/watch?v=n4JD-3-UAzM
+[ABRSM exams]: https://en.wikipedia.org/wiki/ABRSM#Practical_exams
+[cadence]: https://en.wikipedia.org/wiki/Cadence_(music)
+[fugue]: https://en.wikipedia.org/wiki/Fugue
+[inversion]: https://en.wikipedia.org/wiki/Chord_inversion#Chords
diff --git a/hugo/content/posts/2014-07-21-perfect-pitch.md b/hugo/content/posts/2014-07-21-perfect-pitch.md
new file mode 100644
index 0000000..fa87ea0
--- /dev/null
+++ b/hugo/content/posts/2014-07-21-perfect-pitch.md
@@ -0,0 +1,48 @@
+---
+lastmod: "2022-08-21T12:09:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2014-07-21T00:00:00Z"
+aliases:
+- /uncategorized/perfect-pitch/
+- /perfect-pitch/
+title: Perfect pitch
+---
+
+I have a limited form of [perfect (absolute) pitch][perfect pitch], which I am sometimes asked about. Often it's the same questions, so here they are. No doubt people with better perfect pitch than mine will be annoyed at this impudent upstart claiming the ability, but perfect pitch comes on a spectrum anyway. Apparently some people can identify notes to within the nearest fifth of a semitone, while some can only identify the semitone closest to the note. I am a bit further towards the "tone-deaf" end of that spectrum.
+
+# References for notes
+
+Anyway, I have been able to sing a [concert A] without reference since about the age of 12, I think, on account of having learnt the violin since much younger than then. From then, until about the age of 15, I kind of accumulated more notes I could use as references (A because it's concert tuning; E because it's the start of [Für Elise]; D because it's the start of [Pachelbel's Canon] and of the Libera Me from [Fauré's Requiem]). Annoyingly, these were all notes we tune violin strings to; it's very easy to find the four notes of a violin given an A, because we hear that sound every time we start playing the violin. Eventually I picked up an unreliable B-flat (from [a rather rousing Christmas carol][This Little Babe]), which I always had to cross-check with the A.
+
+Then I started noticing that the F which lies at the bottom of my vocal range had a very distinctive feel on the piano. Not a particular piano - just that I could recognise that F when played on the piano. Similarly, middle C started to feel like a C. I came to be able to reproduce the C vocally, by imagining pressing the middle-C key and singing the note that it played.
+
+That is, I could identify notes ACDEF B-flat. More tentatively, I could identify G as being kind of weedy and characterless (as opposed to the rich understated heroism of F - sounds silly, but I can find no other way to describe it off the top of my head).
+
+I still have trouble with most accidentals (that is, flats and sharps), although I've just now realised that I can do F-sharp from [Tim Minchin]'s excellent [song of the same name][F Sharp] and I can do D-sharp from the start of [Chopin]'s [Nocturne in B][Chopin Nocturne in B]. So it's really just C-sharp, F-sharp, A-flat and B that I don't have references for. I can identify the white notes (except B, which feels a bit like a chameleon, could be either a C or a B-flat) on a piano by sound, and I can identify all the notes by producing them, or producing the next-door note, and comparing with what I heard.
+
+Having said that, I'm significantly slower and less accurate when there is background noise - particularly tuned background noise. It feels like my internal scale is fuzzy and easily subject to external influence.
+
+# FAQ
+*Have you always had it?* No, I picked it up mid-to-late secondary school. Also, my ability depends on having been playing music recently (by "recently" I mean "in the last week or so"). If the last few weeks have been musicless, I become much slower and less accurate.
+
+*What's it like to have it?* No different than otherwise, for the most part. It doesn't get in the way unless I ask for it, with some exceptions. In particular, I usually listen to a piece of music without noticing the notes, although I am not that fast at identifying most notes, so they might well pass me by before I have a chance to decide what they are. The individual letters of a text don't bother you.
+
+I said "exceptions": I am quite sensitive to instruments being out of tune. I don't know whether I'm much more sensitive than other people in this area - maybe they're all being polite in pretending not to notice. After a few minutes to get used to the pitch, it usually swamps my absolute representation of notes, and then I stop noticing out-of-tuneness (because I no longer have a reliable baseline).
+
+*Can you distinguish sound better than normal?* Apparently so, but I don't think it's caused by my perfect pitch. On a now-defunct online test, I scored in the 87th percentile of test takers, reliably distinguishing between 0.75 hertz around 500 hertz. I imagine that's to do with musical training.
+
+
+[perfect pitch]: https://en.wikipedia.org/wiki/Perfect_pitch
+[concert A]: https://en.wikipedia.org/wiki/Concert_A
+[Für Elise]: https://en.wikipedia.org/wiki/Fur_Elise
+[Pachelbel's Canon]: https://en.wikipedia.org/wiki/Pachelbel%27s_Canon
+[This Little Babe]: https://www.youtube.com/watch?v=BTyIP7m8Btg
+[Fauré's Requiem]: https://en.wikipedia.org/wiki/Faur%C3%A9_Requiem
+[F Sharp]: https://www.youtube.com/watch?v=5Ju8Wxmrk3s
+[Tim Minchin]: https://en.wikipedia.org/wiki/Tim_Minchin
+[Chopin Nocturne in B]: https://www.youtube.com/watch?v=BhIP4hDBp-E
+[Chopin]: https://en.wikipedia.org/wiki/Chopin
+[ear test]: http://tonometric.com/adaptivepitch/
diff --git a/hugo/content/posts/2014-08-19-parables.md b/hugo/content/posts/2014-08-19-parables.md
new file mode 100644
index 0000000..43cf0cd
--- /dev/null
+++ b/hugo/content/posts/2014-08-19-parables.md
@@ -0,0 +1,34 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- creative
+comments: true
+date: "2014-08-19T00:00:00Z"
+aliases:
+- /creative/parables/
+- /parables/
+title: Parables, chapter 1, verses 1-10
+---
+
+One day, a group of investors came to [Bezos] in the Temple and begged of him, "You are known throughout the land for your wisdom. Please tell us: what lessons did you learn early in life, which we have not yet learnt?"
+
+Bezos replied thus.
+
+"When I was but a child, when I had not yet seen seven summers, I discovered that my teacher had a bountiful store of chocolates hidden in the stationery cupboard. Being of an enterprising frame of mind, I proceeded to eat one of them every day for a week." For he was mindful of the need to preserve the source of good things.
+
+"The next Monday, the teacher took me aside, and asked me whether I had been eating the chocolates. I replied that I had no idea who had been eating the chocolates, and expressed astonishment that indeed there were free chocolates to be had so near to my place of work." He knew that the key to deceit was remembering what you *should* know, as a cover for what you *did* know.
+
+"But the teacher was wise beyond my years. Ey said to me, 'I saw you take chocolates last Friday!' And to prove it, ey brandished the selfsame wrapper I had carelessly discarded." And even these decades later, a tear ran down Bezos's cheek, that his scheme had failed in so predictable a manner.
+
+"I realised that now was the time for the truth. I explained myself: 'I am sorry, O teacher, that I allowed you to discover my scheme. I understand now that you become suspicious after only four repetitions of a deception, and not the five I thought were safe. In future, I shall be more careful.' I was a simple mind then, and believed that it was right to tell the truth. I wished to be held accountable for my lies." One of the investors nodded sympathetically.
+
+"To my surprise, the teacher flew into a rage. I was put into detention. That day I learnt that while the truth should set you free, this only holds up to the point of maintaining your societal role." He knew now that truth is secondary, when one is an underling.
+
+"I saw an opportunity to prevent further suffering. 'I see you are attempting to negatively reinforce me against telling the truth and explaining my actions. I have learnt my lesson - you need not apply further reinforcement. I shall remember this.'"
+
+"And that was the day I was expelled from my school, and was left to forge my own path."
+
+One's prescribed roles should not confine behaviour overmuch. That way lies stagnation and inactivity.
+
+[Bezos]: https://en.wikipedia.org/wiki/Jeff_Bezos
diff --git a/hugo/content/posts/2014-08-26-python-script-shadowing.md b/hugo/content/posts/2014-08-26-python-script-shadowing.md
new file mode 100644
index 0000000..06dbf5e
--- /dev/null
+++ b/hugo/content/posts/2014-08-26-python-script-shadowing.md
@@ -0,0 +1,26 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- programming
+comments: true
+date: "2014-08-26T00:00:00Z"
+aliases:
+- /uncategorized/python-script-shadowing/
+title: Python, script shadowing
+---
+
+*A very brief post about the solution to a problem I came across in Python.*
+
+In the course of my work on [Sextant] (specifically the project to add support for accessing a [Neo4j] instance by SSH), I ran into a problem whose nature is explained [here][Name shadowing trap] as the Name Shadowing Trap. Essentially, in a project whose root directory contains a `bin/executable.py` script, which is intended as a thin wrapper to the module `executable`, you can't `import executable`, because the `bin/executable.py` shadows the module `executable`.
+
+The particular example I had was a wrapper called `sextant.py`, which needed to `import sextant` somewhere in the code. There was no guarantee that the wrapper script would be located in a predictable place relative to the module, because `pip` has a lot of liberty about where it puts various files during a package installation. I really didn't want to mess with the PythonPath if at all possible; a maybe-workable solution might have been to alter the PythonPath to put the module `sextant` at the front temporarily, so that its import would take precedence over that of `sextant.py`, but it seemed like a dirty way to do it.
+
+No workaround was listed, other than to rename the script. A brief Google didn't give me anything more useful. Eventually, I asked someone in person, and ey told me to get rid of the `.py` from the end of the script name. That stops Python from recognising it as a script (for the purposes of `import`). As long as you have the right [shebang] at the top of the script, though, and its permissions are set to be executable, you can still run it.
+
+(Keywords in the hope that Google might direct people to this page if they have the same problem: Python shadow module script same name.)
+
+[Sextant]: https://launchpad.net/ensoft-sextant
+[Neo4j]: https://neo4j.com
+[Name shadowing trap]: http://python-notes.curiousefficiency.org/en/latest/python_concepts/import_traps.html
+[shebang]: https://en.wikipedia.org/wiki/Shebang_(Unix)
diff --git a/hugo/content/posts/2014-09-09-sum-of-two-squares-theorem.md b/hugo/content/posts/2014-09-09-sum-of-two-squares-theorem.md
new file mode 100644
index 0000000..5750627
--- /dev/null
+++ b/hugo/content/posts/2014-09-09-sum-of-two-squares-theorem.md
@@ -0,0 +1,85 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2014-09-09T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/sum-of-two-squares-theorem/
+- /sum-of-two-squares-theorem/
+title: Sum-of-two-squares theorem
+---
+
+*Wherein I detail the most beautiful proof of a theorem I've ever seen, in a bite-size form suitable for an Anki deck. I attach the [Anki deck], which contains the bulleted lines of this post as flashcards.*
+
+# Statement
+There's no particularly nice way to motivate this in this context, I'm afraid, so we'll just dive in. I have found this method extremely hard to motivate - a few of the steps are a glorious magic.
+
+* \\(n\\) is a sum of two squares iff in the prime factorisation of \\(n\\), primes 3 mod 4 appear only to even powers.
+
+# Proof
+We're going to need a few background results.
+
+## Background
+* \\(\mathbb{Z}[i]\\), the ring of [Gaussian integers], is a UFD.
+* In a UFD, [irreducible]s are [prime].
+* \\(-1\\) is square mod \\(p\\) iff \\(p\\) is not 3 mod 4.
+
+Additionally, we'll call a number which is the sum of two squares a **nice** number.
+
+## First implication: if primes 3 mod 4 appear only to even powers…
+We prove the result first for the primes, and will then show that niceness is preserved on taking products.
+
+
+
+* Let \\(p=2\\). Then \\(p\\) is trivially the sum of two squares: it is \\(1+1\\).
+* Let \\(p\\) be 1 mod 4.
+* Then modulo \\(p\\), we have \\(-1\\) is square.
+* That is, there is \\(n \in \mathbb{N}\\) such that \\(x^2 + 1 = n p\\).
+* That is, there is \\(n \in \mathbb{N}\\) such that \\((x+i)(x-i) = n p\\).
+* \\(p\\) divides \\((x+i)(x-i)\\), but it does not divide either of the two multiplicands (since it does not divide their imaginary parts).
+* Therefore \\(p\\) is not prime in the complex integers.
+* Since \\(\mathbb{Z}[i]\\) is a UFD, \\(p\\) is not irreducible in the complex integers.
+* Hence there exist non-invertible \\(a, b \in \mathbb{Z}[i]\\) such that \\(a b = p\\).
+* Taking norms, \\(N(p) = N(ab)\\).
+* Since the norm is multiplicative, \\(N(p) = N(a) N(b)\\).
+* \\(N(p) = p^2\\), so \\(p^2 = N(a) N(b)\\).
+* Neither \\(a\\) nor \\(b\\) was invertible, so neither of them has norm 1 (since in \\(Z[i]\\), having norm 1 is equivalent to being invertible).
+* Hence wlog \\(N(a)\\) is exactly \\(p\\), since the product of two numbers being \\(p^2\\) means either one of them is 1 or they are both \\(p\\).
+* Let \\(a = u+iv\\). Then \\(N(a) = u^2 + v^2 = p\\), which was what we needed.
+
+Next, we need to take care of this "even powers" business:
+
+* \\(p^2\\) is a sum of two squares if \\(p\\) is 3 mod 4: indeed, it is \\(0 + p^2\\).
+
+All we now need is for niceness to be preserved under multiplication. (Recall \\(w^*\\) denotes the conjugate of \\(w\\).)
+
+* Let \\(x, y\\) be the sum of two squares each, \\(x_1^2 + x_2^2\\) and \\(y_1^2 + y_2^2\\).
+* Then \\(x = (x_1 + i x_2)(x_1 - i x_2)\\), and similarly for \\(y\\).
+* Then \\(x y = (x_1 + i x_2)(x_1 - i x_2)(y_1 + i y_2)(y_1 - i y_2)\\).
+* So \\(x y = w w^*\\), where \\(w = (x_1 + i x_2)(y_1 + i y_2)\\).
+* Hence \\(x y = N(w)\\), so is a sum of two squares (since norms are precisely sums of two squares).
+
+Together, this is enough to prove the first direction of the theorem.
+
+## Second implication: if \\(n\\) is the sum of two squares…
+We'll suppose that \\(n = x^2 + y^2\\) has a prime factor which is 3 mod 4, and show that it divides both \\(x\\) and \\(y\\).
+
+* Let \\(n = x^2 + y^2\\) have prime factor \\(p\\) which is 3 mod 4.
+* Then taken mod \\(p\\), we have \\(x^2 + y^2 = 0\\).
+* That is, \\(x^2 = - y^2\\).
+* If \\(y\\) is not zero mod \\(p\\), it is invertible.
+* That is, \\((x y^{-1})^2 = -1\\).
+* This contradicts that \\(p\\) is 3 mod 4 (since \\(-1\\) is not square mod \\(p\\)). So \\(y\\) is divisible by \\(p\\).
+* Symmetrically, \\(x\\) is divisible by \\(p\\).
+* Hence \\(p^2\\) divides \\(n\\), so we can divide through by it and repeat inductively.
+
+That ends the proof. Its beauty lies in the way it regards sums of two squares as norms of complex integers, and dances into and out of \\(\mathbb{C}\\), \\(\mathbb{Z}[i]\\) and \\(\mathbb{Z}\\) where necessary.
+
+[Gaussian integers]: https://en.wikipedia.org/wiki/Gaussian_integers
+[UFD]: https://en.wikipedia.org/wiki/Unique_factorization_domain
+[irreducible]: https://en.wikipedia.org/wiki/Irreducible_element
+[prime]: https://en.wikipedia.org/wiki/Prime_element
+[Anki deck]: {{< baseurl >}}AnkiDecks/SumOfTwoSquaresTheorem.apkg
diff --git a/hugo/content/posts/2014-12-02-christmas-carols.md b/hugo/content/posts/2014-12-02-christmas-carols.md
new file mode 100644
index 0000000..1cbd678
--- /dev/null
+++ b/hugo/content/posts/2014-12-02-christmas-carols.md
@@ -0,0 +1,43 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2014-12-02T00:00:00Z"
+aliases:
+- /uncategorized/christmas-carols/
+- /christmas-carols/
+title: Christmas carols
+---
+
+In which I provide my favourite carols and my favourite renditions of them. 
+
+In no particular order, except that 1) must be at the start and 9) at the end.
+
+1) [Once in Royal David's City][Once]. Always opens the Festival of Nine Lessons and Carols. Has the same problem as 9) in that the only nice recordings seem to have congregations in, but I suppose that's all part of it.
+
+2) [The Three Kings]. My favourite. This performance (King's College) has a soloist who is a bit strident, I think, but all the other ones I've listened to are even stridenter.
+
+3) O Holy Night. My second-favourite. It took until 2017 before I found a recording I liked: it's by the Elora Festival Singers. (Pavarotti is a bit forceful. Most of the recordings appear to be soloists only, singing in very American voices. I want a SATB choir with soloist(s) and, if there must be accompaniment, organ. The soloist(s) must be reverent rather than joyful, and the choir must be singing the standard chordal patterns rather than funky modern ones. There's a version done by Libera which almost passes muster, but it's not SATB and it is accompanied by lighthearted orchestra. It's a solemn piece.)
+
+4) [This Little Babe]. I don't usually like Britten, but this one is too rousing. I had trouble finding a good version of this, but these people nailed it.
+
+5) [In Dulci Jubilo]. King's College does it perfectly.
+
+6) [In the Bleak Midwinter][Bleak] (Darke's setting). I'm sensing a theme with the King's choir.
+
+7) [It Came Upon the Midnight Clear][Midnight Clear]. This performance is beautifully smooth.
+
+8) [This Is the Truth Sent From Above]. Vaughan Williams had to make it into the list.
+
+9) [Hark, the Herald Angels Sing][Hark]. Have to end a carol service with that. Wow, there are some bad arrangements of this out there (Mormon Tabernacle Choir, I'm looking at you, and Pentatonix, which would be so nice if they didn't sing with such weirdly non-British vowel sounds). I still haven't found one in which there isn't a congregation.
+
+[Once]: https://www.youtube.com/watch?v=NMGMV-fujUY
+[The Three Kings]: https://www.youtube.com/watch?v=HIedUioo_Jk
+[This Little Babe]: https://www.youtube.com/watch?v=aPnP5zzHJoQ
+[In Dulci Jubilo]: https://www.youtube.com/watch?v=iXze_TLUTqM
+[Bleak]: https://www.youtube.com/watch?v=GPpy3XSk6c0
+[Midnight Clear]: https://www.youtube.com/watch?v=rSn0_Zj6gjQ
+[This Is the Truth Sent From Above]: https://www.youtube.com/watch?v=5M_8vjqWYmM
+[Hark]: https://www.youtube.com/watch?v=A_iLXNSIaYc
diff --git a/hugo/content/posts/2014-12-09-film-recommendation-interstellar.md b/hugo/content/posts/2014-12-09-film-recommendation-interstellar.md
new file mode 100644
index 0000000..3f6a5d7
--- /dev/null
+++ b/hugo/content/posts/2014-12-09-film-recommendation-interstellar.md
@@ -0,0 +1,25 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+comments: true
+date: "2014-12-09T00:00:00Z"
+aliases:
+- /uncategorized/film-recommendation-interstellar/
+title: Film recommendation, Interstellar
+---
+
+I’ve just come back from seeing [Interstellar], a film of peril and physics. This post will be spoiler-free except for sections which are in [rot13].
+
+I thought the film was excellent. My previous favourite film in its genre was [Sunshine], but this beats it in many ways, chiefly that the physics portrayed in Interstellar - relativity, primarily - is not so wrong that it’s immediately implausible. Indeed, some physics-driven plot twists (such as *gvqny sbeprf arne n oynpx ubyr*) I called in advance, which is a testament to how closely the film matched my physical expectations. My stomach nearly dropped out when the characters realised what relativity meant for them.
+
+This is one of few films whose outcome was truly tense and uncertain for me. Characters were reasonably well-developed, and Michael Caine was in it. Good long story, told at the right pace, and there weren’t too many concessions made to the plot. (By which I mean, it felt like things often happened as they would in real life, rather than just to make a good story, and I had genuine feelings of empathic frustration when reality intervened in the plot.)
+
+The film lasted perhaps seven minutes too long, in my opinion. *V gubhtug vg fubhyq unir raqrq jvgu gur cebgntbavfg qlvat bhgfvqr Fnghea, naq uhznavgl'f shgher hapregnva ohg thnenagrrq gb pbagnva tbqubbq.* I think it’s made to cater to USA audiences rather than British ones; we Brits tend to like emotions to be portrayed with subtlety in films. There were several places I thought the ending was going to be very different: *gung Pbbcre jbhyq qvr ba gur sebmra cynarg; gung gurl jbhyq fynz vagb gur oynpx ubyr naq qvr; gung Zhecul'f oebgure jbhyq xvyy Zhecul jura fur oenaqvfurq gur jngpu*. My favourite ending would simply have been the film without its last scene.
+
+Additionally, a little too much was made of *ybir genafpraqf gvzr naq fcnpr*: while I can believe one irrational person saying this, it stretches the imagination for an entire team of scientists to think it.
+
+I should stress that those are pretty much my only problems with this film, and they’re all pretty minor. I loved the soundtrack; the visual effects were astonishing (vaguely reminiscent of 2001: A Space Odyssey). I’d go so far as to say that this film is beautiful, not just in a visual sense but in an arty sense: its spirit is pure, or something like that. Very much worth the price of entry, at a little under £3/hr.
+
+[Interstellar]: https://en.wikipedia.org/wiki/Interstellar_(film)
+[rot13]: https://rot13.com/
+[Sunshine]: https://en.wikipedia.org/wiki/Sunshine_(2007_film)
diff --git a/hugo/content/posts/2014-12-19-matrix-puzzle.md b/hugo/content/posts/2014-12-19-matrix-puzzle.md
new file mode 100644
index 0000000..23efd84
--- /dev/null
+++ b/hugo/content/posts/2014-12-19-matrix-puzzle.md
@@ -0,0 +1,91 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2014-12-19T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/matrix-puzzle/
+title: Matrix puzzle
+---
+
+I recently saw a problem from an Indian maths olympiad:
+
+> There is a square arrangement made out of n elements on each side (n^2 elements total). You can put assign a value of +1 or -1 to any element. A function f is defined as the sum of the products of the elements of each row, over all rows and g is defined as the sum of the product of elements of each column, over all columns. Prove that, for n being an odd number, f(x)+g(x) can never be 0.
+
+There is a very quick solution, similar in flavour to that [famous dominoes puzzle][Mutilated chessboard]. However, I didn’t come up with it immediately, and my investigation led down an interesting route.
+
+Preliminary observations
+===========
+
+It is easy to see that given a matrix of \\(1\\) and \\(-1\\), we have \\(f, g\\) unchanged on reordering rows and columns, and on taking the transpose. This leads to a very useful lemma: \\(f, g\\) are unchanged if we negate the corners of a rectangle in the matrix.
+
+The idea then occurs: perhaps there is a [normal form] of some kind?
+
+Specification of normal form
+========
+
+Given any four -1’s laid out at the corners of a rectangle, we may flip them all into 1’s without changing \\(f, g\\). Similarly, given any three -1’s on the corners of a rectangle, where the fourth corner is 1, we may flip to get a rectangle with one -1 and three 1’s.
+
+Repeat this procedure until there are no rectangles with three or more corners -1. (Note that we might get a different answer depending on the order we do this in!) A Mathematica procedure to do this (expressed in a very disgusting way) is as follows.
+{% raw %}
+    internalReduce[mat_] :=  Module[{m = mat},
+  Do[If[(i != k && j != l) &&
+     Count[Extract[m, {{i, j}, {k, j}, {i, l}, {k, l}}], -1] >
+      2, {m[[i, j]], m[[i, l]], m[[k, j]],
+       m[[k, l]]} = -{m[[i, j]], m[[i, l]], m[[k, j]], m[[k, l]]};
+    ], {i, 1, Length[mat]}, {j, 1, Length[mat]}, {k, 1,
+    Length[mat]}, {l, 1, Length[mat]}];
+    m]
+    reduce[mat_] := FixedPoint[internalReduce, mat]
+{% endraw %}
+
+Notice that columns which contain more than one -1 must not overlap, in the sense that no two columns with more than one -1 may have a -1 in the same row. Indeed, if they did, we’d have a submatrix somewhere of the form {{-1, -1}, {1, -1}}, which contradicts the “we’ve finished flipping” condition. Hence we may rearrange rows so that all -1’s appear together in contiguous columns.
+
+We may then rearrange columns so that reading from the left, we see successive columns with decreasingly many -1’s. Rearrange rows again so that they appear stacked on top of each other.
+
+![example of reduced matrix][reduced matrix]
+
+We’ve ended up with a normal form: columns of -1’s, diagonally adjoined to each other, followed by rows of -1’s. (The following Mathematica code relies on the fact that SortBy is a stable sort.)
+
+`normalform[mat_] := SortBy[Transpose@SortBy[Transpose@reduce[mat], -Count[#, -1] &], Count[#, -1] /. {0 -> Infinity} &]`
+
+We haven’t shown that it’s unique yet, and indeed it’s not. As a counterexample, {{-1,1,1,1,1}, {-1,1,1,1,1}, {1,-1,1,1,1}, {1,-1,1,1,1}, {1,1,1,1,1}} is transformed into {{-1,1,1,1,1}, {-1,1,1,1,1}, {-1,1,1,1,1}, {-1,1,1,1,1},{1,1,1,1,1}} by a rectangle-flip.
+
+This suggests a further improvement to the normal form: by flipping in this way, we may insist that any column of -1’s, other than the first, must contain only one -1. Indeed, if it contained two or more, we would flip two of them into the first column, rearrange so that all columns were contiguous -1’s again, and repeat.
+
+What does our matrix look like now? It’s a column of -1, followed by some diagonal -1’s, followed by a row of -1. We’ll call this the canonical form, although I’ve still not shown uniqueness.
+
+![example of matrix in canonical form][canonical matrix]
+
+Restatement of problem
+========
+
+The problem then becomes: given a matrix in canonical form, show that \\(f+g\\) cannot be 0.
+
+Notice that if the long column is \\(r\\) long, and there are \\(s\\) diagonal -1’s, and the long row is \\(t\\) long, and the matrix is \\(n \times n\\), then \\(f = -r-s+(-1)^t + (n-s-r-1)\\), \\(g = -t-s+(-1)^r + (n-s-t-1)\\).
+
+Hence \\(f+g = 2n - 2(r+2s+t+1) + (-1)^r + (-1)^t\\).
+
+Any choice of \\(r, s, t, n\\) with \\(r+s+1 \leq n; s+t+1 \leq n; r, t>1\\) yields a valid matrix. We therefore need to show that for all \\(r, s, t, n\\) we have \\(2(n-r-2s-t-1) + (-1)^r + (-1)^t \not = 0\\).
+
+Solution
+=======
+
+Reducing this mod 4, it is enough to show that \\(2(n-r-t-1) + (-1)^r + (-1)^t \not \equiv 0 \pmod{4}\\). But we can easily case-bash the four cases which arise depending on the odd-even parity of \\(r, t\\), to see that in all four cases, the congruence does indeed not hold.
+
+* \\(r, t\\) even: \\(2(n-1) + 2 = 2n\\), but since \\(n\\) is odd, this is not \\(0 \pmod{4}\\).
+* \\(r\\) even, \\(t\\) odd: \\(2n - 1\\), since \\(t-1\\) is even so \\(2(t-1)\\) is a multiple of 4. \\(2n-1\\) isn’t even even, let alone divisible by \\(4\\).
+* \\(r, t\\) odd: \\(2(n-1) + 2 = 2n\\) which is again not \\(0 \pmod{4}\\).
+
+Summary
+=======
+
+Once we had this canonical form, it was easy to find \\(f, g\\) and therefore analyse the behaviour of \\(f+g\\). Next steps: prove that canonical forms are unique (perhaps using the fact that \\(f, g\\) are invariant across forms, and showing a result along the lines that any two canonical forms with the same \\(f, g\\) must be equivalent). I won’t do that now.
+
+[Mutilated chessboard]: https://en.wikipedia.org/wiki/Mutilated_chessboard_problem
+[normal form]: https://en.wikipedia.org/wiki/Canonical_form
+[reduced matrix]: {{< baseurl >}}images/Matrices/matrix_reduced.jpg
+[canonical matrix]: {{< baseurl >}}images/Matrices/matrix_canonical.jpg
diff --git a/hugo/content/posts/2014-12-23-latin-translation-tips.md b/hugo/content/posts/2014-12-23-latin-translation-tips.md
new file mode 100644
index 0000000..af024a8
--- /dev/null
+++ b/hugo/content/posts/2014-12-23-latin-translation-tips.md
@@ -0,0 +1,42 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2014-12-23T00:00:00Z"
+aliases:
+- /uncategorized/latin-translation-tips/
+- /latin-translation-tips/
+title: Latin translation tips
+---
+
+I'm clearing out my computer, and found a file which may as well be here.
+
+Chunking:
+----
+
+1. The first thing to do is to run through the sentence, identifying the verbs and anything that looks like it might be a verb (even in a strange form, like “passus” or “ascendere”).
+2. Run through a second time, looking for structures like “ut + subjunctive” and “non solum… sed etiam…” - if a verb you spotted is in an odd form, this is when you look quickly for why it’s in that form.
+3. Look for any subordinate clauses (like “dixit Caecilius, qui in horto laborabat…”) 
+4. If you see an adjective-looking thing, it probably has to go with a noun.
+5. With that in mind, chunk the text, remembering that two verbs in the same chunk is unlikely unless one is something like “dixit” or “poterat”, which can modify another verb. Remember that chunks shouldn’t be too long, but lots of really short words together might not count against the length limit. Try reading out each chunk - rhythm takes time to learn to grasp, but it might help you.
+
+Once the text is chunked:
+----
+
+1. Remember that your chunking is probably wrong somewhere, but also is probably broadly right. 
+2. In each chunk, if there’s a nominative and a verb then try and translate those first. Then think about what the verb “expects”; if the verb is looking for an accusative, find an accusative, while if it’s looking for a dative, find a dative. For example, “docet” = “he teaches” is looking for an accusative, while “trahet” = “he drags” is looking both for an accusative (“he drags something”) and possibly a dative (“he drags something somewhere”).
+3. If it looks like a jumble of words, identify the case of everything (in poetry, it can help if you scan the text) - this should tell you what goes with what. Don’t be too fussy about getting the right case, though - I’d be happy with “dative or ablative”, most of the time, because that’s usually clear from context - as long as you have the right case among your options!
+
+Guessing vocab:
+----
+
+Try and work out what the principal parts of a verb are. The English word from a given Latin one almost always comes from the past passive participle (the fourth principal part), by adding “tion” instead of “us”: “passus” -> “passion” [a bit misleading if you don’t know about the Passion of the Christ, because it means “suffering”], “traho” -> “tractus” -> “traction”; it actually means “drag”.
+How to guess the principal parts is the kind of thing you learn with time, but as a general rule, “t” -> “s” (as in “patior passus”) and almost everything else goes to “ct”: “pingere pictus” from which “depiction” so “painting”, “facere factus” from which “manufaction” which isn’t really English but tells you it means “making”, etc.
+
+General:
+----
+
+* If you see lots and lots of things in the same case, ask yourself whether they all go together, or whether there’s some reason that more things than usual should be in that case. Usually it’ll be the former, with the major exception being “que” = “and”. (eg. Caesarem Brutumque - Caesar isn’t described by the word “Brutus”, but they’re both affected by the same verb.)
+* Don’t be afraid to amend your earlier translation, if something becomes clarified by later text. Keep looking at the English you get, to make sure you’re on track; while you’re working, it’s better to leave something blank than to get it wrong, so don’t guess too early. Once you’ve gone over the whole thing, or you’ve got to a point where everything afterwards is impossible without help from earlier, then you can guess. (And, of course, leave nothing blank when you hand it in!) If you do amend the translation, score out the old one with a thin line - don’t scribble it out - because then the examiners might take pity on you if it turned out to be right the first time after all. If you make a significant amendment (you find out that Brutus is actually doing what you thought Caesar was doing, for example) then you should reread the whole translation; check that the new interpretation isn’t just impossible from what Latin has come earlier, and check whether earlier parts make more sense under the new interpretation.
diff --git a/hugo/content/posts/2015-01-29-motivational-learning.md b/hugo/content/posts/2015-01-29-motivational-learning.md
new file mode 100644
index 0000000..6055f17
--- /dev/null
+++ b/hugo/content/posts/2015-01-29-motivational-learning.md
@@ -0,0 +1,28 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2015-01-29T00:00:00Z"
+aliases:
+- /uncategorized/motivational-learning/
+- /motivational-learning/
+title: Motivational learning
+---
+
+*In which I am a wizard.*
+
+Sometimes as a student, the work piles up and I start to think "I'll never finish this". It becomes easy to think that there's no point in working because the work will never be over. When that happens to me, I imagine that my course is magic/alchemy/something with flashy special effects. I'm going through the Wizardry Academy, and I'll graduate able to manipulate the four elements. Even if I'm not the best in the year at it, I'm still able to *manipulate the elements*, and if I work at it, I'll be able to manipulate them better and in flashier ways - that's not something most people can do!
+
+I tend not to take this analogy very far. It's usually enough just for me to pretend I'm [Kvothe] for a moment, and I'm all motivated again. However, the trick kind of works for specific topics, too. At the moment, for instance, I need to know how to classify the [representations] of a group, per [Slate Star Codex's article][extreme mnemonics].
+
+An arcanist who is working with minerals needs to know lots of properties of those minerals, and is greatly advantaged by performing certain rituals to divine the Affinities of a metal. As you know, metals are nothing more nor less than a physical embodiment of a collection of Aspects, and you get a different kind of metal for each Aspect that has gone into its construction. All metals have an Affinity with Nothing - that's just standard Elemental Theory. Metals only have a certain number of Affinities, too, and it turns out to be a fact that each Affinity corresponds exactly with a purity band of the metal, and you can see which purity band goes with an Affinity if you look at the Affinity through a Tracer. (On that note, recall from the first Alchemy course you ever took that there is a ritual we can perform to extract a particular Aspect already present in a metal. Purity bands are what we call the product of that ritual, and represent a distilled Aspect which is still related to the original metal.)
+
+A mineral is an algebraic structure; a metal, a finite group. An Aspect is a group element, and so if we have different generators for the group, we get a different group. An Affinity of a group is a complex irreducible representation. All finite groups have the trivial representation, as is standard Representation Theory. Finite groups only have a certain number of irreducible complex representations, and they are in bijection with the conjugacy classes of the group. (If you apply the trace operator to a representation, you obtain a character.) From any first course in group theory, we can extract the conjugacy class of an element of a group, and it is those conjugacy classes which are in bijection with the the characters.
+
+It's paraphrased a bit, and my notation is a bit sloppy, but it certainly sounds more interesting than representation theory.
+
+[Kvothe]: https://en.wikipedia.org/wiki/The_Kingkiller_Chronicle
+[representations]: https://en.wikipedia.org/wiki/Group_representation
+[extreme mnemonics]: https://slatestarcodex.com/2013/08/14/extreme-mnemonics/
diff --git a/hugo/content/posts/2015-08-19-awodey.md b/hugo/content/posts/2015-08-19-awodey.md
new file mode 100644
index 0000000..227dc4e
--- /dev/null
+++ b/hugo/content/posts/2015-08-19-awodey.md
@@ -0,0 +1,14 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+title: Sequence on Awodey's Category Theory
+author: patrick
+layout: page
+date: "2015-08-19T00:00:00Z"
+comments: true
+---
+
+In the summer of 2015, I worked through Awodey's [Category Theory][book], and I produced [a large collection of posts][posts] as I tried to understand its contents.
+These posts are probably not of much interest to anyone who is just looking for something to read, so they're siloed off.
+
+[book]: https://global.oup.com/ukhe/product/category-theory-9780199237180
+[posts]: /awodey
diff --git a/hugo/content/posts/2015-08-21-proof-by-contradiction.md b/hugo/content/posts/2015-08-21-proof-by-contradiction.md
new file mode 100644
index 0000000..3c9d7ed
--- /dev/null
+++ b/hugo/content/posts/2015-08-21-proof-by-contradiction.md
@@ -0,0 +1,33 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2015-08-21T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/proof-by-contradiction/
+- /proof-by-contradiction/
+title: Proof by contradiction
+summary: Here I explain proof by contradiction so that anyone who has ever done a sudoku and seen algebra may understand it.
+---
+
+Here I explain proof by contradiction so that anyone who has ever done a [sudoku] and seen algebra may understand it.
+
+Imagine you are doing a sudoku, and you have narrowed a particular cell down to being either a 1 or a 3. You're not sure which it is, so you do the "guess and see" approach: you guess it's a 1. That forces this other cell to be an 8, this one to be a 5, and then - oh no! That one over there has to be a 7, but there's already a 7 in its row! That means we have to backtrack: our first guess of 1 was wrong, so it has to be a 3 after all.
+
+That was a proof by contradiction that the cell was a 3.
+
+Now I present the standard proof that \\(\sqrt{2}\\) is not [expressible as a fraction][rational] \\(\frac{p}{q}\\) where \\(p, q\\) are whole numbers.
+
+Analogy: "the cell was a 1" corresponds to "\\(\sqrt{2}\\) is fraction-expressible". "The cell was a 3" corresponds to "\\(\sqrt{2}\\) is not fraction-expressible".
+
+Suppose \\(\sqrt{2}\\) were fraction-expressible. Then we could write it explicitly as \\(\sqrt{2} = \frac{p}{q}\\), and we can insist that \\(q > 0\\): if it's negative, we can move the negative up to the \\(p\\). If we clear denominators, we get \\(q \sqrt{2} = p\\); then square both sides, to get \\(2 q^2 = p^2\\).
+
+But now think about how many times 2 divides the left-hand side and the right-hand side. 2 divides a square an even number of times, if it divides it at all (because any square which is divisible by 2 is also divisible by 4, so we can pair off the 2-factors). So 2 must divide \\(q^2\\) an even number of times, and hence the left-hand side an odd number of times (because that's \\(2 \times q^2\\)). It divides the right-hand side an even number of times. So the number of times 2 divides \\(p^2\\) is both odd and even. No number is both odd and even!
+
+We've done the equivalent of finding a 7 appearing twice in a single row. We have to backtrack and conclude that the starting cell was a 3 after all: \\(\sqrt{2}\\) is not fraction-expressible.
+
+[sudoku]: https://en.wikipedia.org/wiki/Sudoku
+[rational]: https://en.wikipedia.org/wiki/Rational_number
diff --git a/hugo/content/posts/2015-09-25-lottery-odds.md b/hugo/content/posts/2015-09-25-lottery-odds.md
new file mode 100644
index 0000000..479b14a
--- /dev/null
+++ b/hugo/content/posts/2015-09-25-lottery-odds.md
@@ -0,0 +1,36 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2015-09-25T00:00:00Z"
+redirect-from:
+- /mathematical_summary/lottery-odds/
+- /lottery-odds/
+title: Lottery odds
+summary:
+    It has been proposed to me that if one is to play the National Lottery, one should be sure to select one's own numbers instead of allowing the machine to select them for you. This is not an optimal strategy.
+---
+
+It has been proposed to me that if one is to play the National Lottery, one should be sure to select one's own numbers instead of allowing the machine to select them for you.
+
+To summarise and slightly simplify the Lottery: at some point during the week, the entrant picks six distinct numbers from 1 to 49 inclusive, and buys a ticket with those numbers on. There is also the option to let the ticket vending machine choose numbers at random, instead of having you choose them. Then on Wednesday evening, six numbers are selected from 1 to 49 on live TV by a process which is as near to true random as we can get while still retaining drama. If all six of your numbers match all six of the prize numbers, you win a prize. (In the actual game, there are also smaller prizes for matching fewer numbers, and so on.)
+
+The argument goes as follows: if you let the vending machine decide your numbers, you have the square of the probability of winning. (That is, a much smaller chance.) Indeed, in order to win, the vending machine first needs to select the winning numbers, and then the TV machine also does.
+
+This is, of course, a confusion of the probability of A given B, and the probability of A and B. What was calculated was the probability that the vending machine picks the six given numbers and the TV picks the six given numbers. What is actually required is the probability that the TV picks the six given numbers given that the vending machine also did.
+
+By the way, "A and B" is definitely distinct from "A given B": in a population, the probability that a person is both Albert Einstein and a man is rather low, but the probability that a given person is a man given that they are Albert Einstein is 1.
+
+The first way to make the lottery more intuitive is to note that we could have conducted the lottery so that we already drew the TV's winning numbers, in secret, before you bought your ticket. Only on buying it do you find out whether you've won or not. Now we are simply trying to match six specific numbers by buying our ticket (although we do not know what they are in advance, we do know they are fixed), and the vending machine can guess exactly as well as I can. By analogy: the TV person flips a coin, and then tells you that you will win if you can guess what the outcome of the coin flip was. It's obvious that you'll win half the time if you pick heads, and half the time if you pick tails, and you won't do any better than the vending machine if you guess. Now, instead, let's say that you pick your heads/tails option first, and then the TV person flips a coin. Nothing has changed except the order in which we do things, and the machine will still do just as well as you. (Analogy, of course, is that selecting the six numbers you want to win is the same as selecting the head/tail option you want to win.)
+
+That is, the bogus argument of the third paragraph is not time independent. If you simply shuffle some of the stages of the lottery around, even though this should have no effect on the outcome, the bogus argument says the outcome should be different.
+
+The second way is let's say I'm in competition with you to win most money on the lottery. I'm going to pick the "vending machine" option. You claim I'm thirteen million times less likely to win when the vending machine has picked my numbers, so you surely won't object if I change the lottery slightly so that if I choose the "vending machine" option, it picks two sets of six numbers and enters me for them both simultaneously. That doubles my winning chance, but it's still a damn sight worse than the penalty of thirteen million times I incurred by picking the "vending machine" option. You likewise won't mind if I change the lottery so that the "vending machine" chance picks ten sets of numbers. A hundred. Thirteen million, which brings me into parity with your lottery: according to you, we're now equally likely to win. But wait - now them machine has picked every combination. I win if any combination wins! And I'm still… just as likely as you to win? Come back to me when you're winning every time and we can rethink.
+
+The third way to make the distinction more intuitive is to make everything much smaller. Let's say I just need to pick one number, and the TV picks one number, each out of 3 instead of 50. Now, the cases in which I win are precisely {(1,1), (2,2), (3,3)}, where (a,b) means "I picked number a, and the machine picked number b". The cases in which I do not win are precisely {(1,2), (1,3), (2,1), (2,3), (3,1), (3,2)}. All of these are equally likely - (1,1) is exactly as likely as (1,2), because if I sneakily relabeled the TV's lottery balls by swapping 1 for 2 then that should have no effect on the outcomes - so my chance of winning is 3/9, or 1/3. This is independent of the means I used to pick my choice, because there is exactly one winning outcome for each of my possible choices. The situation is completely symmetrical: relabelling all the choices doesn't change anything. If it helps, we could think of the option "let the vending machine decide" as "I choose the number 1. Now I let the vending machine apply some scrambling operation I don't know, and it will spit out the number I'll end up using." This doesn't change any of the probabilities, because the statement of the problem is completely independent of what labels appear on the choices (as long as they're all different).
+
+I fear that my third way might require more maths than most people have - the idea of symmetry isn't exactly common.
+
+Anyway, everyone should agree that the lottery is a bad investment if your intention is only to gain money out of it. (Aside from anything else, if you stood to gain anything from playing the lottery, then by symmetry so must everyone else, so the lottery itself must stand to lose. There's simply nowhere else the gain could come from. The lottery would be closed down immediately if it made a loss.)
diff --git a/hugo/content/posts/2015-11-12-eilenberg-moore.md b/hugo/content/posts/2015-11-12-eilenberg-moore.md
new file mode 100644
index 0000000..e61637c
--- /dev/null
+++ b/hugo/content/posts/2015-11-12-eilenberg-moore.md
@@ -0,0 +1,50 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- awodey
+comments: true
+date: "2015-11-12T00:00:00Z"
+math: true
+aliases:
+- /categorytheory/eilenberg-moore/
+- /eilenberg-moore/
+title: Eilenberg-Moore
+summary: As an exercise in understanding the definitions involved, I find the Eilenberg-Moore category of a certain functor.
+---
+
+During my attempts to understand the fearsomely difficult Part III course "[Introduction to Category Theory][course]" by PT Johnstone, I came across the monadicity of the power-set functor \\(\mathbf{Sets} \to \mathbf{Sets}\\). The monad is given by the triple \\((\mathbb{P}, \eta_A: A \to \mathbb{P}(A), \mu_A: \mathbb{PP}(A) \to \mathbb{P}(A))\\), where \\(\eta_A: a \mapsto \{ a \}\\), and \\(\mu_A\\) is the union operator. So \\(\mu_A(\{ \{1, 2 \}, \{3\} \}) = \{1,2,3 \}\\).
+
+It's easy enough to check that this is a monad. We have a theorem saying that every monad has an associated "[Eilenberg-Moore]" category - the category of algebras over that monad. What, then, is the E-M category for this monad?
+
+Recall: an algebra over the monad is a pair \\((A, \alpha)\\) where \\(A\\) is a set and \\(\alpha: \mathbb{P}(A) \to A\\), such that the following two diagrams commute. (That is, \\(\alpha\\) here is an operation which takes a collection of elements of \\(A\\), and returns an element of \\(A\\).)
+
+![Power-set monad algebra diagram][PowersetMonad]
+
+Aha! The second diagram says that the operation \\(\alpha\\) is "massively associative": however we group up terms and successively apply \\(\alpha\\) to them, we'll come up with the same answer. Mathematica calls this attribute "[Flat]"ness, when applied to finite sets only.
+
+Moreover, it doesn't matter what order we feed the elements in to \\(\alpha\\), since it works only on sets and not on ordered sets. So \\(\alpha\\) is effectively commutative. (Mathematica calls this "[Orderless]".)
+
+The first diagram says that \\(\alpha\\) applied to a singleton is just the contained element. Mathematica calls this attribute "[OneIdentity]".
+
+Finally, \\(\alpha(a, a) = \alpha(a)\\), because \\(\alpha\\) is implemented by looking at a set of inputs.
+
+So what is an algebra over this monad? It's a set equipped with an infinitarily-Flat, OneIdentity, commutative operation which ignores repeated arguments. If we forgot that "repeated arguments" requirement, we could use any finite set with any commutative monoid structure; the nonnegative reals with infinity, as a monoid, with addition; and so on. However, this way we're reduced to monoids which have an operation such that \\(a+a = a\\). That's not many monoids.
+
+What operations do work this way? The [Flatten]-followed-by-[Sort] operation in Mathematica obeys this, if the underlying set \\(A\\) is a power-set of a well-ordered set. The union operation also works, if the underlying set is a complete poset - so the power-set example is subsumed in that.
+
+Have we by some miracle got every algebra? If we have an arbitrary algebra \\((A, \alpha)\\), we want to define a complete poset which has \\(\alpha\\) acting as the union. So we need some ordering on \\(A\\); and if \\(x \leq y\\), we need \\(\alpha(\{x, y\}) = y\\). That looks like a fair enough definition to me. It turns out that this definition just works.
+
+So the Eilenberg-Moore category of the covariant power-set functor is just the category of complete posets.
+
+(Subsequently, I looked up the definition of "complete poset", and it turns out I mean "complete lattice". I've already identified the need for unions of all sets to exist, so this is just a terminology issue. A complete poset only has sups of directed sequences. A complete lattice has all sups.)
+
+
+[course]: /archive/2015IntroToCategoryTheory.pdf
+[Eilenberg-Moore]: https://ncatlab.org/nlab/show/Eilenberg-Moore+category
+[PowersetMonad]: {{< baseurl >}}images/CategoryTheorySketches/PowersetMonad.jpg
+[OneIdentity]: https://reference.wolfram.com/language/ref/OneIdentity.html
+[Orderless]: https://reference.wolfram.com/language/ref/Orderless.html
+[Flat]: https://reference.wolfram.com/language/ref/Flat.html
+[Flatten]: https://reference.wolfram.com/language/ref/Flatten.html
+[Sort]: https://reference.wolfram.com/language/ref/Sort.html
diff --git a/hugo/content/posts/2015-11-28-my-first-forcing.md b/hugo/content/posts/2015-11-28-my-first-forcing.md
new file mode 100644
index 0000000..227edd5
--- /dev/null
+++ b/hugo/content/posts/2015-11-28-my-first-forcing.md
@@ -0,0 +1,61 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2015-11-28T00:00:00Z"
+math: true
+aliases:
+- /mathematical_summary/my-first-forcing/
+- /my-first-forcing/
+title: My First Forcing
+summary:
+    In the Part III Topics in Set Theory course, we have used forcing to show the consistency of the Continuum Hypothesis, and we are about to show the consistency of its negation. I don't really grok forcing at the moment, so I thought I would go through an example.
+---
+
+In the Part III Topics in Set Theory course, we have used [forcing] to show the consistency of the [Continuum Hypothesis][CH], and we are about to show the consistency of its negation. I don't really grok forcing at the moment, so I thought I would go through an example.
+
+A forcing is just a quasiorder, so I'll pick a nice one: \\(\mathbb{N}\\), with the usual order. Let's go through some terminology: condition \\(p \in \mathbb{N}\\) is stronger than condition \\(q \in \mathbb{N}\\) (according to my course's convention) iff \\(q \leq p\\). All conditions are compatible, because for every pair of conditions there is a condition stronger than both of them.
+
+The dense subsets of this forcing are precisely the unbounded ones: that is, the infinite ones.
+
+The directed subsets are precisely all subsets, because there is always a natural bigger-than-or-equal-to any two specified naturals. The downward-closed subsets are the initial segments.
+
+The generic set existence theorem is in this case satisfied trivially by \\(G = \mathbb{N}\\), which is generic relative to any collection of dense subsets, and which contains any specified element.
+
+The sets which are \\(\mathbb{P}\\)-generic over \\(\mathbb{M}\\) (any model which contains \\(\mathbb{N}\\)) are those initial segments of \\(\mathbb{N}\\) which intersect every dense set: that is, the only \\(\mathbb{P}\\)-generic set over \\(\mathbb{M}\\) is \\(\mathbb{N}\\) itself.
+
+\\(\mathbb{P}\\) is not separative, because it's total so every pair of elements is compatible. That means our forcing isn't guaranteed to add any elements. Let's plough on anyway.
+
+What are the \\(\mathbb{P}\\)-names of rank \\(0\\)? The empty set is the only such name.
+
+What are the \\(\mathbb{P}\\)-names of rank \\(1\\)? They are all of the form \\(\tau = \{ (n_i, \sigma_i) : n_i \in \mathbb{N}, \sigma_i = \emptyset, i < i_0 \in \text{Ord} \}\\): that is, \\(\{ (n_i, \emptyset): n_i \in \mathbb{N} \}\\). Hence the \\(\mathbb{P}\\)-names of rank \\(1\\) are in one-to-one correspondence with the subsets of \\(\mathbb{N}\\), and subset \\(N\\) is taken to \\(\{ (n, \emptyset) : n \in N \}\\).
+
+What are the \\(\mathbb{P}\\)-names of rank \\(2\\)? They are of the form \\(\tau = \{ (n_i, \sigma_i): n_i \in \mathbb{N}, (\sigma_i = \emptyset) \vee (\sigma_i = N \subseteq \mathbb{N}) \}\\), where I'm abusing notation and identifying the subset of \\(\mathbb{N}\\) with its corresponding \\(\mathbb{P}\\)-name of rank \\(1\\). (This isn't a horrible abuse, because \\(\emptyset\\) means the same thing in the two contexts.) That is, it's basically an arbitrary relation between naturals and subsets of naturals.
+
+The ones of rank \\(3\\), after some mental gymnastics, turn out effectively to be arbitrary relations between pairs of naturals and subsets of naturals; and those of rank \\(n\\) are arbitrary relations between \\(n-1\\)-tuples of naturals and subsets of naturals.
+
+The ones of rank \\(\omega\\) look like being relations between \\(\omega\\)-indexed tuples of naturals and subsets of naturals, and so on. I'm willing to proceed on the assumption that they are.
+
+On to the interpretation. We can interpret with respect to any set \\(G \subseteq \mathbb{N}\\), although most of our theorems only really talk about when \\(G\\) is \\(\mathbb{P}\\)-generic: that is, when it is \\(\mathbb{N}\\) itself.
+
+The interpretation of anything of rank \\(0\\) is, of course, the empty set. If we take anything of rank \\(1\\) - that is, a subset of the naturals - its interpretation is either the empty set (if \\(G\\) doesn't intersect the subset) or the set containing the empty set (if \\(G\\) does intersect the subset).
+
+Let \\(\sim\\) be a relation between the naturals and subsets of the naturals: that is, a name of rank \\(2\\). Then the interpretation is \\(\{ \sigma_G: (\exists p \in G: p \sim \sigma) \}\\). That is, for everything in \\(G\\), take everything it twiddles, and interpret that (producing the empty set if \\(G\\) doesn't intersect the twiddled thing, or \\(\{ \emptyset \}\\) if it does). Hence we produce the empty set if nothing in \\(G\\) twiddles anything; we get \\(\{ \emptyset \}\\) if everything in \\(G\\) only twiddles things which intersect \\(G\\); and \\(\{ \{ \emptyset \}, \emptyset \}\\) if {something in \\(G\\) twiddles something which intersects \\(G\\), and something in \\(G\\) twiddles something which is disjoint from \\(G\\)}.
+
+Repeating, it looks like we're building the ordinals, and with the right choice of \\(\mathbb{P}\\)-name, we'll get every ordinal for most choices of \\(G\\) (including the only generic one, \\(\mathbb{N}\\)).
+
+I'm struggling to think why the entire class of ordinals isn't in this extension. If we started from a countable transitive model, there's a theorem which says that not only have we gained no new ordinals, but we still remain countable. So perhaps we've only actually generated the ordinals up to the Hartogs ordinal of the CTM (that is, \\(\omega_1\\)).
+
+Let's move into \\(\mathbb{M}\\). As far as \\(\mathbb{M}\\) is concerned, we've just verified the existence of the von Neumann hierarchy (that is, we can show that every subset of every ordinal is present as an interpretation), so our forcing hasn't added anything at all. Aha, I've got it! Every \\(\mathbb{P}\\)-name lives in \\(\mathbb{M}\\), and so there are only countably many of those, but \\(\mathbb{M}\\) thinks that lots of those \\(\mathbb{P}\\)-names are different, though they are actually (from our outside, $V$, perspective) the same. \\(\mathbb{M}\\) doesn't have enough power to show they're the same. Therefore, from \\(\mathbb{M}\\)'s point of view, every ordinal really does exist. The previous paragraph was all backwards: our interpretations contain every ordinal because \\(\mathbb{M}\\) thinks there is every ordinal represented among the \\(\mathbb{P}\\)-names, even though to us with the super-strong large cardinal axiom that "the CTM isn't everything" fundamentally is, there's only countably many of those names.
+
+Are there indeed countably many of those names, to us in \\(V\\)? There must be, because we're in a CTM. Indeed, if we go up to \\(\alpha = \omega_1\\), we are attempting to talk about \\(V\\)-uncountable families of elements drawn from this countable model, so actually there aren't any \\(\mathbb{P}\\)-names of rank \\(\omega_1\\).
+
+OK. The above all goes to show that if we force our CTM by \\(\mathbb{N}\\), we don't get anything new. (And this doesn't contradict our theorem that if \\(\mathbb{P}\\) is separative, then we do get something new, because \\(\mathbb{N}\\) is not separative.) Hooray! I feel like I've just cast my first spell with a shiny new magic wand, examined what the spell did, and discovered that it did nothing more than check that magic was still working today.
+
+Next time, I'll try a separative forcing, so I'm guaranteed something new.
+
+[forcing]: https://en.wikipedia.org/wiki/Forcing_(mathematics)
+[CH]: https://en.wikipedia.org/wiki/Continuum_hypothesis
+[quasiorder]: https://en.wikipedia.org/wiki/Preorder
diff --git a/hugo/content/posts/2015-12-24-general-adjoint-functor-theorem.md b/hugo/content/posts/2015-12-24-general-adjoint-functor-theorem.md
new file mode 100644
index 0000000..d4101d8
--- /dev/null
+++ b/hugo/content/posts/2015-12-24-general-adjoint-functor-theorem.md
@@ -0,0 +1,20 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- categorytheory
+comments: true
+date: "2015-12-24T00:00:00Z"
+aliases:
+- /categorytheory/general-adjoint-functor-theorem/
+- /general-adjoint-functor-theorem/
+title: General Adjoint Functor Theorem
+---
+
+Just a post to draw attention to [my new article][article] about the [General Adjoint Functor Theorem][GAFT].
+It's a motivation of the GAFT and its proof.
+I've never seen it motivated in this way, and it's actually quite a natural theorem.
+I haven't managed to motivate the Special Adjoint Functor Theorem at all, although I'm told that it's natural if you know Stone-Cech compactification.
+
+[article]: /misc/AdjointFunctorTheorems/AdjointFunctorTheorems.pdf
+[GAFT]: https://ncatlab.org/nlab/show/adjoint+functor+theorem
diff --git a/hugo/content/posts/2015-12-31-monadicity-theorems.md b/hugo/content/posts/2015-12-31-monadicity-theorems.md
new file mode 100644
index 0000000..f10959c
--- /dev/null
+++ b/hugo/content/posts/2015-12-31-monadicity-theorems.md
@@ -0,0 +1,16 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- categorytheory
+comments: true
+date: "2015-12-31T00:00:00Z"
+aliases:
+- /categorytheory/monadicity-theorems/
+- /monadicity-theorems/
+title: Monadicity Theorems
+---
+
+Another short post to highlight the existence of [an article about the Monadicity Theorems][mts], in which I prove one direction of both the Crude and Precise versions. Comments and corrections would be very much appreciated, because there is an awful lot of work involved in proving those theorems. It would be good to know of any parts where the argument is unclear, unmotivated, too long-winded, or wrong.
+
+[mts]: /misc/MonadicityTheorems/MonadicityTheorems.pdf
diff --git a/hugo/content/posts/2016-01-01-multiplicative-determinant.md b/hugo/content/posts/2016-01-01-multiplicative-determinant.md
new file mode 100644
index 0000000..f73de87
--- /dev/null
+++ b/hugo/content/posts/2016-01-01-multiplicative-determinant.md
@@ -0,0 +1,20 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2016-01-01T00:00:00Z"
+aliases:
+- /mathematical_summary/multiplicative-determinant/
+- /multiplicative-determinant/
+title: Multiplicative determinant
+---
+
+I'm clearing out my desktop again, and found [this document on the multiplicativity of the
+determinant][doc], which I wrote in 2014. It might as well be up here.
+
+I should note that this document contains no motivation of any kind. It is simply an
+exercise in symbol-shunting, and it has no clever ideas in it.
+
+[doc]: /misc/MultiplicativeDetProof/MultiplicativeDetProof.pdf
diff --git a/hugo/content/posts/2016-01-26-representable-functors.md b/hugo/content/posts/2016-01-26-representable-functors.md
new file mode 100644
index 0000000..0c08ba6
--- /dev/null
+++ b/hugo/content/posts/2016-01-26-representable-functors.md
@@ -0,0 +1,17 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- categorytheory
+comments: true
+date: "2016-01-26T00:00:00Z"
+aliases:
+- /categorytheory/representable-functors/
+- /representable-functors/
+title: Representable functors
+---
+
+Just a post to draw attention to [my new article][article] about representable functors and their links to adjoint functors.
+It's very short, but it gives a reason for being interested in representable functors: they are basically "those with left adjoints", up to minor quibbles.
+
+[article]: /misc/RepresentableFunctors/RepresentableFunctors.pdf
diff --git a/hugo/content/posts/2016-02-05-friedberg-muchnik-theorem.md b/hugo/content/posts/2016-02-05-friedberg-muchnik-theorem.md
new file mode 100644
index 0000000..78aeecb
--- /dev/null
+++ b/hugo/content/posts/2016-02-05-friedberg-muchnik-theorem.md
@@ -0,0 +1,17 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2016-02-05T00:00:00Z"
+aliases:
+- /mathematical_summary/friedberg-muchnik-theorem/
+- /friedberg-muchnik-theorem/
+title: Friedberg-Muchnik theorem
+---
+
+Another short post to point out [my new article on the Friedberg-Muchnik theorem][FM], a theorem from computability theory. It uses what is known officially as a finite injury priority method, and the proof is cribbed entirely from [Dr Thomas Forster][tf].
+
+[FM]: /misc/FriedbergMuchnik/FriedbergMuchnik.pdf
+[tf]: https://www.dpmms.cam.ac.uk/~tf/
diff --git a/hugo/content/posts/2016-03-03-a-certain-limit.md b/hugo/content/posts/2016-03-03-a-certain-limit.md
new file mode 100644
index 0000000..977ac75
--- /dev/null
+++ b/hugo/content/posts/2016-03-03-a-certain-limit.md
@@ -0,0 +1,67 @@
+---
+lastmod: "2021-10-25T23:24:01.0000000+01:00"
+author: patrick
+categories:
+- stack-exchange
+comments: true
+math: true
+date: "2016-03-03T00:00:00Z"
+title: Why do we get complex numbers in a certain expression?
+summary: Answering the question, "Why does a continued fraction containing only 1, subtraction, and division result in one of two complex numbers?".
+---
+
+*This is my answer to the same [question posed on the Mathematics Stack Exchange](https://math.stackexchange.com/q/1681993/259262). It is therefore licenced under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).*
+
+# Question
+
+So we all know that the continued fraction containing all 1s...
+
+$$
+x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}}
+$$
+
+yields the golden ratio \\(x = \phi\\), which can easily be proven by rewriting it as \\(x = 1 + \dfrac{1}{x}\\), solving the resulting quadratic equation and assuming that a continued fraction that only contains additions will give a positive number.
+
+Now, a friend asked me what would happen if we replaced all additions with subtractions:
+
+$$
+x = 1 - \frac{1}{1 - \frac{1}{1 - \ldots}}
+$$
+
+I thought "oh cool, I know how to solve this...":
+
+$$
+x = 1 - \frac{1}{x}
+$$
+
+$$
+x^2 - x + 1 = 0
+$$
+
+And voila, I get...
+
+$$ x \in \{e^{i\pi/3}, e^{-i\pi/3} \} $$
+
+Ummm... why does a continued fraction containing only 1s, subtraction and division result in one of two complex (as opposed to real) numbers?
+
+(I have a feeling this is something like the \\(\sum_i (-1)^i\\) thing, that the infinite continued fraction isn't well-defined unless we can express it as the limit of a converging series, because the truncated fractions \\(1 - \frac{1}{1-1}\\) etc. aren't well-defined, but I thought I'd ask for a well-founded answer. Even if this is the case, do the two complex numbers have any "meaning"?)
+
+# Answer
+
+You're attempting to take a limit.
+
+$$
+x_{n+1} = 1-\frac{1}{x_n}
+$$
+
+This recurrence actually never converges, from any real starting point.
+Indeed, \\(x_2 = 1-\frac{1}{x_1}; \\ x_3 = 1-\frac{1}{1-1/x_1} = 1-\frac{x_1}{x_1-1} = \frac{1}{1-x_1}; \\ x_4 = x_1\\)
+
+So the sequence is periodic with period 3.
+Therefore it converges if and only if it is constant; but the only way it could be constant is, as you say, if \\(x_1\\) is one of the two complex numbers you found.
+
+Therefore, what you have is actually basically a proof by contradiction that the sequence doesn't converge when you consider it over the reals.
+
+However, you have found exactly the two values for which the iteration does converge; that is their significance.
+
+Alternatively viewed, the map \\(z \mapsto 1-\frac{1}{z}\\) is a certain transformation of the complex plane, which has precisely two fixed points. You might find it an interesting exercise to work out what that map does to the complex plane, and examine in particular what it does to points on the real line.
diff --git a/hugo/content/posts/2016-03-28-clojure-exercism.md b/hugo/content/posts/2016-03-28-clojure-exercism.md
new file mode 100644
index 0000000..7e964ee
--- /dev/null
+++ b/hugo/content/posts/2016-03-28-clojure-exercism.md
@@ -0,0 +1,173 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- programming
+comments: true
+date: "2016-03-28T00:00:00Z"
+aliases:
+- /clojure-exercism/
+title: Clojure and Exercism
+summary:
+    I've been trying to learn Clojure through Exercism, a programming exercises tool.
+    It took me an hour to get Hello, World! up and running, so I thought I'd document how it's done.
+    I'm using Leiningen on Mac OS 10.11.4.
+---
+
+I've been trying to learn [Clojure] (a LISP) through [Exercism], a programming exercises tool.
+It took me an hour to get Hello, World! up and running, so I thought I'd document how it's done.
+I'm using [Leiningen] on Mac OS 10.11.4.
+
+The [Installing Clojure page] on Exercism details how to install Leiningen; that part is easy.
+Installing `exercism` is likewise easy, so we run `exercism fetch clojure hello-world`.
+
+And then we enter a world of pain.
+
+`exercism` downloads a project structure:
+
+    hello-world/
+    -- project.clj
+    -- README.md
+    -- test/
+       -- hello_world_test.clj
+
+The README helpfully tells us what Hello, World! is, and a specification for the answer.
+How are we to come up with our answer?
+`lein` gives access to a REPL we can use to write an answer, but there's no indication of
+where to put our files so that `lein` can see them.
+
+Let's run `lein test` to see what `lein` complains about.
+
+    Exception in thread "main" java.io.FileNotFoundException:
+    Could not locate hello_world__init.class or hello_world.clj on classpath.
+    Please check that namespaces with dashes use underscores in the Clojure file name.,
+    compiling:(hello_world_test.clj:1:1)
+
+Fine. It's looking for `hello_world.clj`. Let's make one!
+
+I've put the following in `hello-world/hello_world.clj`:
+
+    (defn hello
+     []
+     "Hello, World!"
+     [name]
+     (str "Hello, " name "!"))
+
+    (defn main- [& _] (println "Hello!"))
+
+`lein test` fails again, with the same error.
+
+Do we get any hints from the test file?
+It starts with a namespace declaration:
+
+    (ns hello-world-test
+     (:require [clojure.test :refer [deftest is]]
+               hello-world))
+
+We're going to want a `hello-world` namespace, so let's put that at the top of our `hello_world.clj`.
+
+    (ns hello-world)
+
+Still fails with the same error.
+OK, the thing that is telling `lein` what to do must be `project.clj`, and it turns out to contain the following:
+
+    (defproject hello-world "0.1.0-SNAPSHOT"
+     :description "hello-world exercise."
+     :url "https://github.com/exercism/xclojure/tree/master/exercises/hello-world"
+     :dependencies [[org.clojure/clojure "1.8.0"]])
+
+None of that tells `lein` where to look for the source file.
+If we make a new `lein` project somewhere, let's see what the project file is supposed to look like.
+
+Go to a temporary directory and use `lein new app newproj`.
+The source tree looks like:
+
+    newproj/
+    -- CHANGELOG.md
+    -- LICENSE.md
+    -- README.md
+    -- doc/
+       -- intro.md
+    -- project.clj
+    -- resources/
+    -- src/
+       -- newproj/
+          -- core.clj
+    -- test/
+       -- newproj/
+          -- core_test.clj
+
+And `project.clj` looks like:
+
+    (defproject newproj "0.1.0-SNAPSHOT"
+     :description "FIXME: write description"
+     :url "http://example.com/FIXME"
+     :license {:name "Eclipse Public License"
+               :url "http://www.eclipse.org/legal/epl-v10.html"}
+     :dependencies [[org.clojure/clojure "1.8.0"]]
+     :main ^:skip-aot newproj.core
+     :target-path "target/%s"
+     :profiles {:uberjar {:aot :all}})
+
+The only interesting thing there seems to be `:main ^:skip-aot newproj.core`.
+Let's try putting `:main ^:skip-aot hello-world` into our own `project.clj`.
+
+`lein test` continues to fail with the same error.
+Looking up `:skip-aot`, it just tells `lein` to skip Ahead-Of-Time compilation, which isn't what we want.
+
+With a heavy heart, then, let's restructure `hello-world` so it looks exactly like `newproj`:
+
+    hello-world/
+    -- README.md
+    -- project.clj
+    -- src/
+       -- hello_world/
+          -- hello_world.clj
+    -- test/
+       -- hello_world/
+          -- hello_world_test.clj
+
+Miraculous! We have a different error!
+
+    Exception in thread "main" java.io.FileNotFoundException:
+    Could not locate hello_world_test__init.class or hello_world_test.clj on classpath.
+
+I think this might be a back-step, because beforehand it was at least finding the test file.
+I get the same error if I navigate into the test folder and run `lein test`.
+And if we try `lein run`, we get the original error:
+
+    Exception in thread "main" java.io.FileNotFoundException:
+    Could not locate hello_world__init.class or hello_world.clj on classpath.
+
+From the [Leiningen documentation]:
+
+> The `src/my_stuff/core.clj` file corresponds to the `my-stuff.core` namespace.
+
+That would imply that our source file corresponds to the `hello-world.hello-world` namespace.
+Let's try flattening out the structure a bit, and returning the `hello_world_test.clj` to where at least
+`lein` recognised it:
+
+    hello-world/
+    -- README.md
+    -- project.clj
+    -- src/
+       -- hello_world.clj
+    -- test/
+       -- hello_world_test.clj
+
+And it works! Woohoo!
+(Well, the tests fail, but that's because I'm new to Clojure and missed out a bunch of parentheses.)
+
+The final contents of `src/hello_world.clj`, causing the tests to pass, were:
+
+    (ns hello-world)
+
+    (defn hello
+     ([] "Hello, World!")
+     ([namevar] (str "Hello, " namevar "!")))
+
+[Clojure]: https://clojure.org/
+[Exercism]: https://exercism.io/
+[Installing Clojure page]: https://exercism.io/languages/clojure
+[Leiningen]: https://leiningen.org
+[Leiningen documentation]: https://github.com/technomancy/leiningen/blob/stable/doc/TUTORIAL.md#creating-a-project
diff --git a/hugo/content/posts/2016-04-08-another-monty-hall-explanation.md b/hugo/content/posts/2016-04-08-another-monty-hall-explanation.md
new file mode 100644
index 0000000..beca258
--- /dev/null
+++ b/hugo/content/posts/2016-04-08-another-monty-hall-explanation.md
@@ -0,0 +1,52 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2016-04-08T00:00:00Z"
+aliases:
+- /another-monty-hall-explanation/
+title: Another Monty Hall explanation
+---
+
+Recall the [Monty Hall problem]: the host, Monty Hall, shows you three doors, named A, B and C.
+You are assured that behind one of the doors is a car, and behind the two others there is a goat each.
+You want the car.
+You pick a door, and Monty Hall opens one of the two doors you didn't pick that he knows contains a goat.
+He offers you the chance to switch guesses from the door you first picked to the one remaining door.
+Should you switch or stick?
+
+I'll slightly reframe the problem: let's pretend you are playing cooperatively with Monty Hall, where he
+knows the layouts and he is trying to open two goat-doors, and you're trying for the car; you're not allowed to communicate.
+The game is (noting the distinction between "picking" a door - i.e. announcing your intention to open it - and opening it):
+
+* You pick a door;
+* Monty Hall opens a door you didn't pick;
+* You open a door Monty Hall didn't just pick;
+* Monty Hall opens the remaining door.
+
+(The problem is the same: in standard Monty Hall, you win if and only if you open the car door and Monty Hall opens two goat doors.
+Let's say Monty Hall really likes goats, and not inquire further.)
+
+You pick a door, B say. Monty Hall now opens a goat-door, C say,
+because he knows the layouts and can pick one with a goat behind.
+
+At this point, you know Monty Hall *decided not to open* door A.
+Why would he not have chosen door A?
+It's either because he chose randomly between his available goaty options A and C,
+or because he knew A had a car behind so he was choosing the only goat door available to him.
+(Remember, Monty Hall wants to find goats.)
+
+If he chose randomly, you're better off sticking, because that means you have the car.
+But if he *actively refused* door A (which can only happen because it had a car behind), that means you need to switch to door A.
+
+He chose randomly with probability 1/3 (because he chose randomly if, and only if, you originally picked the car).
+He actively refused door A with probability 2/3, therefore.
+
+So with 2/3 probability, you're in the case that means you guarantee yourself a car if you switch.
+With 1/3 probability, you're in the case that means you guarantee yourself a car if you stick.
+
+So you should switch.
+
+[Monty Hall problem]: {{< ref "2013-12-22-three-explanations-of-the-monty-hall-problem" >}}
diff --git a/hugo/content/posts/2016-04-13-independence-of-choice.md b/hugo/content/posts/2016-04-13-independence-of-choice.md
new file mode 100644
index 0000000..aee64d5
--- /dev/null
+++ b/hugo/content/posts/2016-04-13-independence-of-choice.md
@@ -0,0 +1,59 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2016-04-13T00:00:00Z"
+math: true
+aliases:
+- /independence-of-choice/
+title: Independence of the Axiom of Choice (for programmers)
+summary: So you've heard that the Axiom of Choice is magical and special and unprovable and independent of set theory, and you're here to work out what that means.
+---
+
+So you've heard that the Axiom of Choice is magical and special and unprovable and independent of set theory,
+and you're here to work out what that means.
+Let's not get too hung up on what the Axiom of Choice (or "AC") actually is, because you probably don't care.
+Let's instead discuss what it means for something to be "independent".
+
+Often I hear the layperson say things like "AC is unprovable".
+This is true in a sense, but it's misleading.
+
+Take an object \\(n\\) of the type "integer" - so \\(5\\), \\(-100\\), that kind of thing.
+Here is what I will call the Positivity Hypothesis (or "PH"):
+
+> \\(n\\) is (strictly) greater than \\(0\\).
+
+Of course, depending on how we chose \\(n\\), PH might be true or it might be false, although it can't be both.
+So, while maths might let us prove which of PH or not-PH holds for our given \\(n\\),
+maths will emphatically not let us prove that PH is always true, and it will not let us prove that PH is always false.
+(Maths would be stupid if it did that, because PH is neither always true nor always false.
+The integers \\(5\\) and \\(-100\\) witness that PH can be true and can be false respectively.)
+
+Therefore PH is independent of integer theory.
+It's not magic: there is no god-given reason why PH mysteriously resists all efforts to prove it.
+It's simply not always true, but it's not always false either.
+
+Let's go back to the Axiom of Choice.
+
+The usual system of set theory (which is used as a foundation for all of maths) is a collection of nine axioms,
+together comprising what is known as ZF.
+(If we add Choice to that collection as a tenth axiom, we obtain the set theory called ZFC.)
+In the "integers" analogy above, "the integer type" plays the role of ZF.
+
+Now, just as we may pick an object of type "integer", we may pick a set-theory of type "ZF".
+A "set theory of type ZF" is my informal phrasing for what is usually called "a model of ZF".
+(I'm eliding the question of the consistency of ZF, and I'll just assume it's consistent.)
+In the "integers" analogy, the number \\(5\\) plays the role of one of these set theories,
+as does the number \\(-100\\).
+We can ask of this set theory whether it obeys AC (for which we substituted PH in the analogy).
+
+And it turns out that for some models of set theory, AC holds, and for some models, it doesn't.
+It's quite hard to describe models of set theory, because set theory supports so much complexity;
+the integers are much easier to specify.
+However, if you want the names of two models: in the model which contains precisely the "constructible sets", AC holds, while in Solovay's model, AC fails.
+
+That's all there is to it.
+Maths won't let us prove AC, because it's not true of every set theory of the type "ZF".
+Maths won't let us prove AC is false, because there are some set theories of the type "ZF" in which it is true.
diff --git a/hugo/content/posts/2016-04-21-modular-machines.md b/hugo/content/posts/2016-04-21-modular-machines.md
new file mode 100644
index 0000000..4f7473c
--- /dev/null
+++ b/hugo/content/posts/2016-04-21-modular-machines.md
@@ -0,0 +1,32 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2016-04-21T00:00:00Z"
+aliases:
+- /modular-machines/
+title: Modular machines
+---
+
+I've written [a blurb][MM] about what a modular machine is (namely, another Turing-equivalent form of computing machine),
+and how a Turing machine may be simulated in one.
+(In fact, that blurb now contains an overview of how we may use modular machines to produce a group with insoluble word problem,
+and how to use them to embed a recursively presented group into a finitely presented one.)
+
+A modular machine is like a slightly more complicated version of a Turing machine, but it has the advantage
+that it is easier to embed a modular machine into a group than it is to embed a Turing machine directly into a group.
+We can use this embedding to show that there is a group with unsolvable word problem:
+solving the word problem would correspond to determining whether a certain Turing machine halted.
+
+This is as part of my revision process for the Part III course on "Infinite Groups and Decision Problems".
+It's probably more comprehensible if you already know what a modular machine is.
+Below are some notes which are handwritten, because I needed to draw pictures easily; the linked notes are typeset but might be less legible.
+
+![Notes1]
+![Notes2]
+
+[MM]: /misc/ModularMachines/EmbedMMIntoTuringMachine.pdf
+[Notes1]: /images/ModularMachines/ModularMachines1.jpg
+[Notes2]: /images/ModularMachines/ModularMachines2.jpg
diff --git a/hugo/content/posts/2016-04-27-tennenbaums-theorem.md b/hugo/content/posts/2016-04-27-tennenbaums-theorem.md
new file mode 100644
index 0000000..95ecd71
--- /dev/null
+++ b/hugo/content/posts/2016-04-27-tennenbaums-theorem.md
@@ -0,0 +1,18 @@
+---
+lastmod: "2020-11-07T15:42:41.0000000+00:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2016-04-27T00:00:00Z"
+aliases:
+- /tennenbaums-theorem/
+title: Tennenbaum's theorem
+---
+
+Most recent exposition: [an article][tennenbaum] on [Tennenbaum's Theorem].
+Comments welcome.
+The proof is cribbed from Dr Thomas Forster, but his notes only sketched the fairly crucial last step, on account of the notes not yet being complete.
+
+[tennenbaum]: /misc/Tennenbaum/Tennenbaum.pdf
+[Tennenbaum's Theorem]: https://en.wikipedia.org/wiki/Tennenbaum%27s_theorem
diff --git a/hugo/content/posts/2016-05-25-finitistic-reducibility.md b/hugo/content/posts/2016-05-25-finitistic-reducibility.md
new file mode 100644
index 0000000..7f84d1f
--- /dev/null
+++ b/hugo/content/posts/2016-05-25-finitistic-reducibility.md
@@ -0,0 +1,68 @@
+---
+lastmod: "2021-09-12T22:50:36.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2016-05-25T00:00:00Z"
+math: true
+aliases:
+- /finitistic-reducibility/
+title: Finitistic reducibility
+summary: A quick overview of the definition of the mathematical concept of finitistic reducibility.
+---
+
+There is a [Hacker News thread][HN] at the moment about [an article on Quanta][quanta]
+which describes a paper which claims to prove that Ramsey's theorem for pairs is finitistically reducible.
+That thread contains lots of people being a bit confused about what this means.
+I wrote a comment which I hope is elucidating; this is that comment.
+
+It is a fact of mathematics that there are some statements which are solely about finite objects,
+but to prove them requires reasoning about an infinite object.
+The [TREE function]'s well-definedness is one of them.
+For a more accessible example than TREE, I think the [Ackermann function] falls into this category.
+The Ackermann function \\(A(n+1, m+1) = A(n, A(n+1, m))\\) is well-defined for all \\(n\\) and \\(m\\)
+(we prove this by induction over \\(\mathbb{N} \times \mathbb{N}\\)),
+but the proof relies on considering the [lexicographic order][lex] on \\(\mathbb{N} \times \mathbb{N}\\)
+which is inherently infinite.
+(I'm not totally certain that all proofs of Ackermann's well-definedness rely on an infinite object,
+but the only proof known to me does.)
+Ackermann's function itself is in some sense a "finite" object,
+but the proof of its well-definedness is in some sense "infinite".
+
+Whatever the status of my conjecture that "you can't prove that Ackermann's function is well-defined without considering an infinite object",
+it is [certainly a fact][ack not primrec] that Ackermann is not [primitive-recursive],
+and "primitive-recursive functions" corresponds to the lowest level of the five "mysterious levels" the article talks about.
+
+There are some mathematicians ("finitists") who don't believe that any infinite objects exist.
+Such mathematicians will reject any proof that relies on an infinite object,
+so their mathematics is necessarily less wide-ranging than the usual version.
+Any result that shows that more things are finitistically true is good,
+because it means the finitists get to use these facts the rest of us were already happy about.
+
+So the analogy is as follows.
+Imagine that we knew of this "infinitary" proof that Ackermann is well-defined,
+but we hadn't proved that no "finitary" proof exists.
+(So finitists are not happy to use Ackermann, because it might not actually be well-defined according to them:
+any known proof requires dealing with an infinite object.)
+Now, this paper comes along and proves that actually a finitary proof exists.
+Suddenly the finitists are happy to use the Ackermann function.
+
+Similarly, in real life, most mathematicians were quite happy to use \\(R_2^2\\) to reason about finite objects,
+but the finitists rejected such proofs.
+Now, because of the paper, it turns out that the finitists are allowed to use \\(R_2^2\\) after all,
+because there is a purely finitistic reason why \\(R_2^2\\) is true.
+
+The actual definition of TREE is a bit too long for me to explain here,
+but it is an example of a function like Ackermann, which is well-defined,
+but in fact if you're not allowed to consider infinite objects during the proof then it is provably impossible to prove that TREE is well-defined.
+So the statement "TREE is well-defined" is, in some sense, "less constructive" or "more infinitary" than \\(R_2^2\\).
+
+
+[HN]: https://news.ycombinator.com/item?id=11763080
+[quanta]: https://www.quantamagazine.org/mathematicians-bridge-finite-infinite-divide-20160524
+[TREE function]: https://en.wikipedia.org/wiki/Kruskal's_tree_theorem
+[Ackermann function]: https://en.wikipedia.org/wiki/Ackermann_function
+[lex]: https://en.wikipedia.org/wiki/Lexicographical_order
+[primitive-recursive]: https://en.wikipedia.org/wiki/Primitive_recursive_function
+[ack not primrec]: http://planetmath.org/ackermannfunctionisnotprimitiverecursive
diff --git a/hugo/content/posts/2016-06-13-the-use-of-jargon.md b/hugo/content/posts/2016-06-13-the-use-of-jargon.md
new file mode 100644
index 0000000..116fcf4
--- /dev/null
+++ b/hugo/content/posts/2016-06-13-the-use-of-jargon.md
@@ -0,0 +1,67 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- psychology
+comments: true
+date: "2016-06-13T00:00:00Z"
+aliases:
+- /the-use-of-jargon/
+title: The use of jargon
+summary: "Why jargon is a really useful thing to have and use."
+---
+
+I was recently having a late-night argument with someone about the following thesis:
+
+> If you can't explain something in a simple way, you don't understand it.
+
+They were using this to argue something like the following:
+
+> Jargon is unhelpful because it sets a very high barrier for entry into any field.
+
+My reply, as something of a mathematician, is as follows.
+
+While there are certainly more accessible parts of physics and maths which can be well-explained by analogies and imprecise language
+(and, indeed, we often use them to students, and Brian Cox tries to use them in e.g. documentaries),
+it has led to the horrible nightmare which is everyone thinking wrongly that they understand quantum mechanics [QM] because they heard some cool analogies.
+QM has very little in common with its analogies;
+the analogies are basically just there to give an idea that "things are weird, classical intuition will fail".
+It's the flip side to "if you use abstruse language then you create an environment where you must pass the initiation tests to take part":
+
+> If you use imprecise language then you create an environment where everyone thinks they understand but they're all wrong.
+
+Both approaches have merits, and boringly the correct answer is probably "use a mixture of the two, with the ratio depending on appropriateness to the subject".
+However, physics is increasingly a subfield of maths since the advent of QM and general relativity (which are purely mathematical frameworks),
+and in maths we find the precise language *extremely* important because we strive for total rigour in this, the only subject where it's actually possible.
+Most people start doing maths without access to the language,
+and they often find lots of interesting stuff
+([Ramanujan] is a particular example of such a mathematician,
+who did a lot of great work before ever interacting with Western mathematicians),
+but once you know the language, the language creates a framework which goes some way to guaranteeing the correctness of your results and which can help you spot connections/see more patterns.
+
+From a maths point of view, documentaries are there to get people interested in playing around for themselves,
+rather than to actually impart mathematical knowledge.
+In an ideal world, I think we'd let people discover loads of maths on their own,
+and then show them the precise framework and language it fits into,
+but there just isn't time,
+so we teach it by shoving the framework down students' throats until they either give up maths or become divinely inspired and start playing with it for themselves.
+Additionally a lot of the maths I study [though this might be historical accident,
+derived from our tradition of using jargon] consists of the study of objects which have very few properties, so they defy analogy.
+
+Sometimes it turns out that a certain collection of "very few properties",
+like the collection by which we define the objects we call [groups],
+happen to capture a certain intuition
+(in this case, the idea of "symmetry" [turns out in a deep way][Cayley's theorem] to be precisely captured by groups).
+However, that seems like being the exception rather than the rule,
+and a general collection of "few properties" won't have a neat accessible analogy that anyone has been able to find.
+Especially when you study metamathematics, as well,
+some very deep theorems turn out to hinge on *exactly* what you mean by "the integers" or "the real numbers" or whatever.
+In such fringe cases it is absolutely necessary to be totally precise that we mean "the integers" in a specific technical sense rather than "the integers" as a fuzzy concept,
+or else one will almost certainly go wrong.
+
+So there are definitely cases where the "stupid jargon" is necessary to maintain clarity of thought.
+(Some such theorems do actually impinge on reality, too! Usually via computer science.)
+
+[Ramanujan]: https://en.wikipedia.org/wiki/Srinivasa_Ramanujan
+[groups]: https://en.wikipedia.org/wiki/Group_(mathematics)
+[Cayley's theorem]: https://arbital.com/p/cayley_theorem_symmetric_groups/
diff --git a/hugo/content/posts/2016-06-15-part-iii-essay.md b/hugo/content/posts/2016-06-15-part-iii-essay.md
new file mode 100644
index 0000000..ef4f909
--- /dev/null
+++ b/hugo/content/posts/2016-06-15-part-iii-essay.md
@@ -0,0 +1,20 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematical_summary
+comments: true
+date: "2016-06-15T00:00:00Z"
+aliases:
+- /part-iii-essay/
+title: Part III essay
+---
+
+Now that my time in [Part III] is over, I feel justified in releasing [my essay],
+which is on the subject of [Non-standard Analysis].
+It was supervised by Dr Thomas Forster
+(to whom I owe many thanks for exposing me to such an interesting subject, and for agreeing to supervise the essay).
+
+[Part III]: https://en.wikipedia.org/wiki/Part_III_of_the_Mathematical_Tripos
+[Non-standard Analysis]: https://en.wikipedia.org/wiki/Non-standard_analysis
+[my essay]: https://www.patrickstevens.co.uk/misc/NonstandardAnalysis/NonstandardAnalysisPartIII.pdf
diff --git a/hugo/content/posts/2016-08-05-be-a-beginner.md b/hugo/content/posts/2016-08-05-be-a-beginner.md
new file mode 100644
index 0000000..d0d31eb
--- /dev/null
+++ b/hugo/content/posts/2016-08-05-be-a-beginner.md
@@ -0,0 +1,45 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+comments: true
+date: "2016-08-05T00:00:00Z"
+aliases:
+- /be-a-beginner/
+title: Be a Beginner
+summary: Being a beginner at something is great, especially if it's something that humans are built for.
+---
+
+TL;DR: Being a beginner at something is great, especially if it's something that humans are built for.
+
+Humans are, among other things, [persistence hunters].
+That means one of the ways we are adapted to catch prey is by the brutest of brute-force techniques:
+on foot, we follow a large animal as it runs, until it sits down and dies of exhaustion,
+whereupon we eat it.
+We're pretty slow, but we can run for hours in the full heat of the day
+(we're unreasonably effective at regulating our own body temperature)
+and we just don't stop.
+One of the adaptations by which the human body is built is the ability to run at a constant speed for a long time.
+This art is, of course, increasingly unnecessary,
+as we have supplanted it with tools wrought of pure intellect (agriculture and so forth);
+but the underlying mechanisms are still there in [most of] our bodies.
+
+If you start something as a beginner, you make extremely rapid progress.
+The general effect has a name: the [Pareto principle],
+which is a rule of thumb which states that 80% of the effects come from 20% of the causes.
+If you just learn the most basic 20% of something,
+that often gets you 80% of the total possible effects.
+Beginners improve rapidly in most human endeavours.
+
+I started running using the NHS [Couch to 5k] programme, about nine weeks ago.
+In that time, I have gone from being able to run fitfully for about thirty seconds before having to stop and breathe,
+to being able to run for thirty minutes and only stopping because that's when the timer finished.
+It wasn't particularly fun, but it's always satisfying to improve rapidly at something,
+and it is certainly better to be able to run for half an hour than not to be able to run at all.
+(I had a similar experience with lifting weights, a year and a half ago, except I actually find that fun.)
+
+This post is to recommend being a beginner every so often,
+and specifically to point to the Couch to 5k programme for those who don't currently do things that involve running.
+
+[persistence hunters]: https://en.wikipedia.org/wiki/Persistence_hunting
+[Pareto principle]: https://en.wikipedia.org/wiki/Pareto_principle
+[Couch to 5k]: https://www.nhs.uk/live-well/exercise/couch-to-5k-week-by-week
diff --git a/hugo/content/posts/2016-08-07-a-free-market.md b/hugo/content/posts/2016-08-07-a-free-market.md
new file mode 100644
index 0000000..7f9c9f4
--- /dev/null
+++ b/hugo/content/posts/2016-08-07-a-free-market.md
@@ -0,0 +1,50 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- creative
+- fiction
+comments: true
+date: "2016-08-07T00:00:00Z"
+aliases:
+- /a-free-market/
+title: A Free Market
+summary: The story of Martin's search for a kaki fruit.
+---
+
+Martin was walking through the farmers' market.
+He had scored off nearly everything on his shopping list, but one item stubbornly remained:
+he needed some kaki fruit for a new sorbet recipe he wanted to try out.
+
+High and low he searched,
+weaving in and out of the stalls,
+but his mission proved… well, let us say that it was not successful.
+
+Finally, he thought to give up and place his problem into better hands than his own.
+He forged towards the market's finest attraction,
+the Personal Shopper ("Guaranteed to find your stuff!").
+Her name was Posy,
+and she had been a fixture here for the last twenty years:
+that was when she first noticed the curious way that no-one could ever find quite what they wanted at the weirdly inefficient market.
+Posy was uncannily good at navigating the cobbled rows between the stalls,
+and had an unerring eye for picking out exactly what the customer required.
+
+Martin poured out his problems.
+"Please! I need your help to find a kaki fruit. The recipe will be ruined without it."
+
+Posy smiled, assumed a look of determination, and forged off,
+leaving Martin to scurry behind her as she ducked first left,
+then left again, then (for some reason) a third and a fourth time.
+After what had to be the eighth or ninth left turn through the higgledy-piggledy stalls,
+with Martin hopelessly lost,
+she stopped in front of a little tent whose sign read
+"Children educated and tutored in etiquette: inquire within".
+She raised the entrance flap, and an elderly lady emerged.
+
+Angry, baffled and confused, Martin raised his voice,
+ignoring the proper-and-prim-looking lady from the tent.
+"Why haven't you found me a kaki fruit?
+I thought you knew this market like the back of your hand!"
+
+"Haven't you heard?" said Posy incredulously.
+"It's better to ask for a governess than seek persimmons."
diff --git a/hugo/content/posts/2016-08-10-reinvent-maths.md b/hugo/content/posts/2016-08-10-reinvent-maths.md
new file mode 100644
index 0000000..e6cd89e
--- /dev/null
+++ b/hugo/content/posts/2016-08-10-reinvent-maths.md
@@ -0,0 +1,34 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stack-exchange
+comments: true
+date: "2016-08-10T00:00:00Z"
+title: How far back does mathematical understanding go?
+summary: Answering the question, "how far back in time would maths be understandable to a modern mathematician?".
+---
+
+*This is my answer to the same [question posed on the WorldBuilding Stack Exchange](https://worldbuilding.stackexchange.com/q/51166/13796). It is therefore licenced under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).*
+
+# Question
+
+How far could a mathematician go back in time, and have to spend as less time as possible in relearning stuff?
+
+Background: The main character has realised that he can travel back in time voluntarily, and he wishes to travel back in time to a time-period where he can participate in the beginning of maths, but without relearning as much as possible.
+
+Magic: To make things clear, I'll add this in. The magic allows him to communicate in the time-periods language easily. He can understand it effortlessly, and it stops the other people from asking him very incriminating questions (like where are you from, etc). They simply think he is a travelling scholar and leave it at that. (It stops them from digging to deeply, even if he does not know what they think is common sense.) They also have given him food and a place to stay.
+
+# Answer
+
+It strongly depends which area of maths you're talking about.
+
+* Category theory is basically new, so before the 1950s or so, it just didn't exist in anything like its modern form.
+* Combinatorics has been around for a long time, but before Erdös it looked very different.
+* Before Newton and Leibniz, the notion of calculus wasn't very clear, and its notation would make it very difficult for us modern-day people to work with.
+* Before Cauchy, they didn't really have what we would refer to as a "rigorous" foundation of analysis, and the relevant language changed substantially since Cauchy to take into account the new approach to rigour.
+* There was a time, even some point after the Renaissance IIRC, when mathematicians were still not really sold on this whole "rigour" thing, and the art of defining things crisply so as to deduce (nearly) incontrovertible stuff about them. The entire mindset of mathematics is different now.
+
+A first-year undergraduate going back before Newton could, if their ideas were taken seriously, revolutionise multiple areas of maths simply because we now know (and take for granted) the correct ways of thinking about certain fields of study.
+Conversely, of course, the first-year undergraduate would have a hard time following the maths of the day, because the technical language and frameworks are so strongly unfamiliar.
+The only frameworks I can think of which haven't changed much post-Renaissance are Euclidean geometry and arithmetic, though of course geometry and number theory have advanced substantially since then.
diff --git a/hugo/content/posts/2016-12-31-complex-infinity.md b/hugo/content/posts/2016-12-31-complex-infinity.md
new file mode 100644
index 0000000..841005b
--- /dev/null
+++ b/hugo/content/posts/2016-12-31-complex-infinity.md
@@ -0,0 +1,32 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stack-exchange
+comments: true
+date: "2016-12-31T00:00:00Z"
+title: What does Mathematica mean by ComplexInfinity?
+summary: Answering the question, "Why does WolframAlpha say that a quantity is ComplexInfinity?".
+---
+
+*This is my answer to the same [question posed on the Mathematics Stack Exchange](https://math.stackexchange.com/q/2078754/259262). It is therefore licenced under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).*
+
+# Question
+
+When entered into [Wolfram|Alpha](https://www.wolframalpha.com/), \\(\infty^{\infty}\\) results in "complex infinity".
+Why?
+
+# Answer
+
+WA's `ComplexInfinity` is the same as Mathematica's: it represents a complex "number" which has infinite magnitude but unknown or nonexistent phase.
+One can use `DirectedInfinity` to specify the phase of an infinite quantity, if it approaches infinity in a certain direction.
+The standard `Infinity` is the special case of phase `0`.
+Note that `Infinity` is different from `Indeterminate` (which would be the output of e.g. `0/0`).
+
+Some elucidating examples:
+
+* `0/0` returns `Indeterminate`, since (for instance) the limit may be approached as \\(\frac{1/n}{1/n}\\) or \\(\frac{2/n}{2/n}\\), resulting in two different real numbers.
+* `1/0` returns `ComplexInfinity`, since (for instance) the limit may be approached as \\(\frac{1}{-1/n}\\) or as \\(\frac{1}{1/n}\\), but every possible way of approaching the limit gives an infinite answer.
+* `Abs[1/0]` returns `Infinity`, since the limit is guaranteed to be infinite and approached along the real line in the positive direction.
+
+In your particular example, you get `ComplexInfinity` because the infinite limit may be approached as (e.g.) \\(n^n\\) or as \\(n^{n+i}\\).
diff --git a/hugo/content/posts/2017-02-14-cauchy-schwarz-proof.md b/hugo/content/posts/2017-02-14-cauchy-schwarz-proof.md
new file mode 100644
index 0000000..a242ebb
--- /dev/null
+++ b/hugo/content/posts/2017-02-14-cauchy-schwarz-proof.md
@@ -0,0 +1,17 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mathematics
+comments: true
+date: "2017-02-14T00:00:00Z"
+aliases:
+- /cauchy-schwarz-proof/
+title: Proof of Cauchy-Schwarz
+---
+
+This is just a link to a [beautiful proof][proof] of the [Cauchy-Schwarz inequality][CS].
+There are a number of elegant proofs, but this is by far my favourite, because (as pointed out in the paper) it "builds itself".
+
+[CS]: https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality
+[proof]: http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/New%20Problems/CSNewProof/CauchySchwarzInequalityProof.pdf
diff --git a/hugo/content/posts/2017-03-14-maths-olympiad.md b/hugo/content/posts/2017-03-14-maths-olympiad.md
new file mode 100644
index 0000000..574f2da
--- /dev/null
+++ b/hugo/content/posts/2017-03-14-maths-olympiad.md
@@ -0,0 +1,25 @@
+---
+lastmod: "2021-01-24T12:53:36.0000000+00:00"
+author: patrick
+categories:
+- stack-exchange
+comments: true
+date: "2017-03-14T00:00:00Z"
+title: The relationship between the IMO and research mathematics
+summary: Answering the question, "does the International Maths Olympiad help research mathematics?".
+---
+
+*This is my answer to the same [question posed on the Academia Stack Exchange](https://academia.stackexchange.com/q/86451/51909). It is therefore licenced under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).*
+
+# Question
+
+I was reading a note of Hojoo Lee on inequality which is written for International Math Olympiad (IMO) participants. Although he writes that “target readers are challenging high schools students and undergraduate students“, it appears to be quite advanced.
+
+It occurred to me to ask, do these IMO problems contribute towards research work in math? Do these math notes/books give good overview for research work?
+
+# Answer
+
+I think of Olympiad problems more as "parlour tricks".
+They're really difficult, and it's super-impressive if someone's good at them, but the skills are very different to the skills you need in research.
+As a big example of a difference: the Olympiad rewards quick accurate leaps of reasoning, because you're under such time pressure.
+Research rewards long-term grit and persistence through blind alleys and repeated failure.
diff --git a/hugo/content/posts/2017-11-05-abuse-of-notation.md b/hugo/content/posts/2017-11-05-abuse-of-notation.md
new file mode 100644
index 0000000..db288ec
--- /dev/null
+++ b/hugo/content/posts/2017-11-05-abuse-of-notation.md
@@ -0,0 +1,38 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stack-exchange
+comments: true
+date: "2017-11-05T00:00:00Z"
+title: Abuse of notation in function application
+summary: Answering the question, "Are these examples of abuses of notation?".
+---
+
+*This is my answer to the same [question posed on the Mathematics Stack Exchange](https://math.stackexchange.com/q/2505777/259262). It is therefore licenced under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).*
+
+# Question
+
+I have often seen notation like this:
+
+> Let \\(f : \mathbb{R}^2 \to \mathbb{R}\\) be defined by \\(f(x, y) = x^2 + 83xy + y^7\\).
+
+How does this make any sense?
+If the domain is \\(\mathbb{R}^2\\) then \\(f\\) should be mapping individual tuples.
+
+Also when speaking of algebraic structures why do people constantly interchange the carrier set with the algebraic structure itself?
+For example you might see someone write this:
+
+> Given any field \\(\mathbb{F}\\) take those elements in our field \\(a \in \mathbb{F}\\) that satisfy the equation \\(a^8 = a\\).
+
+How does this make any sense?
+If \\(\mathbb{F}\\) is a field then it is a tuple equipped with two binary operations and corresponding identity elements all of which satisfy a variety of axioms.
+
+# Answer
+
+The example you've given of a function is not an abuse. \\(x\\) is instead shorthand for \\(\pi_1(t)\\) and \\(y\\) is shorthand for \\(\pi_2(t)\\) and \\((x, y)\\) is shorthand for \\(t\\).
+
+\\(g \in G\\) is a very minor abuse, yes.
+"A group \\(G\\) is a set \\(G\\) endowed with some operations" is a slight abuse, but one which will never be misinterpreted.
+It is done this way to avoid the proliferation of unnecessary and confusing symbols.
+For the same reason, we use the symbol \\(+\\) to refer to the three different operations of addition of integers, rationals, and reals.
diff --git a/hugo/content/posts/2018-02-03-epsilon-delta.md b/hugo/content/posts/2018-02-03-epsilon-delta.md
new file mode 100644
index 0000000..5c1f985
--- /dev/null
+++ b/hugo/content/posts/2018-02-03-epsilon-delta.md
@@ -0,0 +1,26 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stack-exchange
+comments: true
+date: "2018-02-03T00:00:00Z"
+title: Infinitesimals as an idea that took a long time
+summary: Answering the question, "Which mathematical ideas took a long time to define rigorously?".
+---
+
+*This is my answer to the same [question posed on the Mathematics Stack Exchange](https://math.stackexchange.com/q/2633847/259262). It is therefore licenced under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).*
+
+# Question
+
+It often happens in mathematics that the answer to a problem is "known" long before anybody knows how to prove it. (Some examples of contemporary interest are among the Millennium Prize problems: E.g. Yang-Mills existence is widely believed to be true based on ideas from physics, and the Riemann hypothesis is widely believed to be true because it would be an awful shame if it wasn't. Another good example is Schramm–Loewner evolution, where again the answer was anticipated by ideas from physics.)
+
+More rare are the instances where an abstract mathematical "idea" floats around for many years before even a rigorous definition or interpretation can be developed to describe the idea. An example of this is umbral calculus, where a mysterious technique for proving properties of certain sequences existed for over a century before anybody understood why the technique worked, in a rigorous way.
+
+I find these instances of mathematical ideas without rigorous interpretation fascinating, because they seem to often lead to the development of radically new branches of mathematics. What are further examples of this type?
+
+# Answer
+
+Following from the continuity example, in which the epsilon-delta formulation eventually became ubiquitous, I submit the notion of the infinitesimal. It took until Robinson in the 1950s and early 60s before we had "the right construction" of infinitesimals via ultrapowers, in a way that made infinitesimal manipulation fully rigorous as a way of dealing with the reals. They were a very useful tool for centuries before then, with (e.g.) Cauchy using them regularly, attempting to formalise them but not succeeding, and with Leibniz's calculus being defined entirely in terms of infinitesimals.
+
+Of course, there are other systems which contain infinitesimals - for example, the field of formal Laurent series, in which the variable may be viewed as an infinitesimal - but e.g. the infinitesimal \\(x\\) doesn't have a square root in this system, so it's not ideal as a place in which to do analysis.
diff --git a/hugo/content/posts/2018-04-08-kinds-of-number.md b/hugo/content/posts/2018-04-08-kinds-of-number.md
new file mode 100644
index 0000000..84fd955
--- /dev/null
+++ b/hugo/content/posts/2018-04-08-kinds-of-number.md
@@ -0,0 +1,25 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stack-exchange
+comments: true
+date: "2018-04-08T00:00:00Z"
+title: What is lost when we move between number systems?
+summary: Answering the question, "What is lost when we move from the reals to the complex numbers?".
+---
+
+*This is my answer to the same [question posed on the Mathematics Stack Exchange](https://math.stackexchange.com/q/2728317/259262). It is therefore licenced under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).*
+
+# Question
+
+As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?
+
+# Answer
+
+The most important ones as I see it:
+
+* Naturals to integers: lose well-orderedness, gain "abelian group" (and, indeed, "ring").
+* Integers to rationals: lose discreteness, gain "field".
+* Rationals to reals: lose countability, gain "Cauchy-complete".
+* Reals to complexes: lose a compatible total order, gain the Fundamental Theorem of Algebra.
diff --git a/hugo/content/posts/2018-06-02-json-comments.md b/hugo/content/posts/2018-06-02-json-comments.md
new file mode 100644
index 0000000..0b8f9f0
--- /dev/null
+++ b/hugo/content/posts/2018-06-02-json-comments.md
@@ -0,0 +1,25 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- hacker-news
+- programming
+comments: true
+date: "2018-06-02T00:00:00Z"
+title: JSON comments (a note from Hacker News)
+summary: "A quick note from Hacker News about why the comment-handling situation in JSON is bad."
+---
+
+In response to [a linkpost](https://news.ycombinator.com/item?id=17358103) to [an article about how YAML isn't perfect](https://arp242.net/weblog/yaml_probably_not_so_great_after_all.html), [user jiveturkey](https://news.ycombinator.com/user?id=jiveturkey) [commented with confusion](https://news.ycombinator.com/item?id=17359727):
+
+> > JSON doesn't support comments
+
+> eh?
+>
+> `{ "firstName": "John", "lastName": "Smith", "comment": "foo", }`
+>
+> I know it isn't the same as `#comments`, but who cares really.
+
+[I replied](https://news.ycombinator.com/item?id=17359800):
+
+> The trouble there is that your comments come in-band. What if you're trying to serialise something and you don't have the power to insist that it's not a dictionary with "comment" as a key?
diff --git a/hugo/content/posts/2018-07-21-dependent-types-overview.md b/hugo/content/posts/2018-07-21-dependent-types-overview.md
new file mode 100644
index 0000000..5c117c6
--- /dev/null
+++ b/hugo/content/posts/2018-07-21-dependent-types-overview.md
@@ -0,0 +1,185 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- programming
+- mathematics
+comments: true
+date: "2018-07-21T00:00:00Z"
+aliases:
+- /dependent-types-overview/
+title: Dependent types overview
+summary: "A quick overview of dependent types."
+---
+
+# Proving things in Agda, part 1: what is dependent typing?
+
+[Agda] is a [dependently-typed] programming language which I've been investigating over the last couple of months, inspired by Conor McBride's [CS410] lecture series.
+Being dependently-typed, its type system is powerful enough to encode mathematical truth, and you can use the type system to verify proofs of mathematical statements, as well as to almost completely obviate the need for tests by having the compiler verify almost any property of your program.
+This post is an overview of what that means.
+
+Before you read any of the Agda code that lives in [my Agda repository][GitHub], please keep in mind that I'm an Agda novice who is exploring.
+I make no claims that any of this code is any good; only that it is correct.
+I'm also not interested in performance, since I'm using it as a proof environment rather than as a source of runnable programs; while all of the code is runnable, I have not optimised it at all.
+We shall see that the mere existence of these programs is enough to constitute mathematical proof.
+
+## What is a type system?
+
+I think of a type system as one or both of two things.
+
+* A way of informing the compiler that certain objects are supposed to match up in certain ways, such that this information may vanish at runtime but allows the compiler to help you when you're writing the program.
+* A way of ensuring at runtime that you don't perform nonsensical operations on objects that don't support those operations.
+
+For example, the language Python has a type system which is "dynamic": you don't specify the type of an object while you're writing the program, so the compiler can't really use type information to help you.
+The language F# has a "static" type system: you specify the type of every object up front, while you're writing the program, so the compiler has more opportunities to tell whether you've told your program to do something inconsistent.
+
+From now on, I'll focus on the first kind of type system (i.e. on type systems where you specify types while you're writing the program, so the compiler can help you).
+
+## What can a type system do for you?
+
+Any Python programmer has probably encountered a certain extremely common bug: since strings are iterable, it's all too easy to iterate accidentally over a single string when you intended to iterate over a list of strings.
+A baby example, in which the bug is very obvious, is as follows:
+
+{{< highlight python >}}
+stringsList = ["hello", "world"]
+for ch in stringsList[0]: # Oops - stringsList[0], not stringsList
+    print(ch)
+# Expected: "hello" and then "world"
+# Actual: "h", then "e", then "l", then "l", then "o"
+{{< / highlight >}}
+
+Python's dynamic typing means that you often can't find out that you've iterated over the wrong thing until you come to run the program and discover that it blows up.
+(It doesn't help that there's no such thing as a character in Python; only a string of length 1.)
+
+In F#, this is a class of bug that never makes it to runtime, because you know the type of every variable up front.
+
+{{< highlight fsharp >}}
+let stringsList = [ "hello" ; "world" ]
+stringsList.[0]
+|> List.map (printfn "%s") // doesn't compile!
+{{< / highlight >}}
+
+`List.map` can't take a string as an argument.
+Even if it could, `printfn "%s"` can't take a character as an argument.
+
+The type system has protected you from this particular bug.
+So far, so familiar.
+
+## Dependent types?
+
+In most common type systems, you're restricted to declaring that any particular object inhabits one of some fixed collection of types, or inhabits a type that is built out of those.
+(For example, `string` or `int` or `List<Map<int, Set<char>>>`).
+This deprives you of some power: there may be things you know about your program, which you proved to your own satisfaction when you wrote it, but which you have been unable to tell the compiler because the language's type system was not expressive enough.
+
+For example, the following (very inefficient) program computes highest common factors of integers:
+
+{{< highlight fsharp >}}
+let rec euclid a b =
+    if a < b then euclid b a
+    elif a = b then a
+    else euclid (a - b) b
+{{< / highlight >}}
+
+The F# compiler can tell me that I haven't made any of a collection of stupid errors - it will stop me from trying to compute the highest common factor of `"hello"` and `[1 ; 2]`, for example, and it tells me that I've given `euclid` the right number of arguments in the recursive call.
+However, in order to know that my function really does compute highest common factors, you need to execute tests.
+(Did you spot the bug in the program? The compiler certainly didn't, because F# doesn't let me inform the compiler that the program is meant to be computing highest common factors, so the compiler doesn't know what to check for.)
+
+But a *dependently-typed* language has the expressive power to lift values themselves, and therefore much more general statements about values, into the type system.
+A dependently-typed language can define a function whose type signature contains terms which depend on other arguments to the function, and can thereby express restrictions on the possible arguments that the function can take *which are enforced at compile-time*.
+Much like F# can prevent you from applying `euclid` to a string, so a dependently-typed language can prevent you from applying `euclid` to a nonpositive number, and can thereby protect you from one bug which is present in the `euclid` above (namely, that the function does not terminate under some conditions, e.g. when `b = 0`).
+Several examples of this will appear later.
+
+Moreover, while any particular non-dependently typed language could in theory have been designed to contain a type for the strictly positive integers (and thereby protect you from that bug, converting it into a compile error), only a dependently-typed language will allow you to construct arbitrary restrictions which the language inventors didn't think of in advance.
+
+In theory, you can use a dependent type system to encode *any* information about the output of your program, in such a way that the compiler knows so much that it will refuse to compile any program that fails to adhere to the specification you gave.
+You can rule out *any* output-based bug if you take enough time over the program specification.
+(Strictly, you need to assume that the compiler is bug-free, and that the implementation of the computer you're running the program on is bug-free, and so forth. No compiler will protect you in full generality from cosmic rays flipping bits in memory while your program is executing.)
+
+## Propositions as types
+
+Health warning: this concept is one that takes a while to grok, so I will not devote much time to it.
+For me, it took several hours of example sheets followed by a very clear explanation from my Logic and Sets supervisor.
+
+The trick for using a dependently-typed language to encode properties of your program is to use this magical ability to raise values up into the type level.
+The most fundamental expression of this concept is the "equality type": for every value `a`, there is a type `=a= : T -> T` (where `T` is the universe of all possible values and types).
+A member of `=a= b` is an object that somehow expresses the notion that `a = b`.
+(Much in the same way as how a member of `int` somehow captures an object that is an integer, so a member of `=a= b` somehow captures a proof that `a = b`.)
+
+There are several possible ways one could implement this type.
+For example, one could define `=a= b` to be the type of all proofs that `a = b` in a mathematical sense; then there are a couple of ways to generate members of `=a= b` (for example, "prepend a proof that `a = c` to a proof that `c = b`"), and there might be many different proofs that `a = b`.
+But in any sane world, no matter what the other ways are to create a proof that `a = b`, there is certainly a canonical thing that `a` is equal to: namely, `a` itself!
+So `=a= a` is a type that should definitely have an inhabitant no matter what system we're working in: namely, the inhabitant that expresses the fact that `a = a` canonically.
+
+In fact, Agda goes further than this and asserts that this is the *only* inhabitant of that type, and moreover that there is no other way to create a member of `=a= b` than to identify `b` with `a` and to use the canonical `a = a`.
+In Agda, if you've proved that two things are equal, then they are literally identical; if you want anything looser, like the notion of "isomorphic", or if you want to capture the idea that you might be able to prove equality in more than one way, then you'll have to define primitives for that yourself.
+
+## A real example
+
+The upshot of the above is that you may insist, in the functions that you define, that something is equal to something else.
+This opens the door to expressing in the type system, for example, that an integer can be decomposed into a modulus and a remainder in a certain way.
+In Agda (here, this is in my [EuclideanAlgorithm.agda]):
+
+```agda
+record divisionAlgResult (a : ℕ) (b : ℕ) : Set where
+    field
+    quot : ℕ
+    rem : ℕ
+    pr : a *N quot +N rem ≡ b
+    remIsSmall : (rem <N a) || (a ≡ 0)
+```
+
+(Here, `*N` is the operation of multiplication on the natural numbers, `<N` is the type of proofs that the left-hand is less than the right-hand as natural numbers, and `+N` is the addition of naturals.
+Also `||` is the "or" type which expresses that a particular one of the left-hand or the right-hand operand is inhabited.)
+
+This is a declaration of a record type with four fields: a quotient and a remainder (which are naturals), together with a proof object witnessing that `a * quot + rem = b`, and a proof object witnessing that the remainder is smaller than `a` (and hence that `rem` really is the value of `b` modulo `a`).
+
+An inhabitant of this record type is simply a proof that `b` can be decomposed into a quotient and a remainder on division by `a`.
+
+Observe that this doesn't actually construct the record for you; it's a declaration of a type, not a construction of an object of that type.
+It still remains for the programmer to provide a function `(a : ℕ) -> (b : ℕ) -> divisionAlgResult a b` and thereby show that the decomposition is always possible.
+Note also that the function signature I just specified is itself an example of lifting values into the type system: the return type `divisionAlgResult a b` depends on the input values `a` and `b`.
+This reflects the fact that one may construct proofs *about specific objects*: a `divisionAlgResult a b` is a proof of a certain property of specific integers `a` and `b` (namely "we can find `b % a` and `b / a`").
+
+Crucially, if I can ever get my hands on a `divisionAlgResult a b`, then I know incontrovertibly (in a way that is guaranteed at compile-time) that the `rem` field and the `quot` field together satisfy `a * quot + rem = b`.
+There is no way to make one of these objects without supplying a proof that the quotient and the remainder behave in this way.
+That means, for example, that there is no possible off-by-one error: I cannot somehow accidentally use the fact `a * quot + rem = b + 1` in a context where I intended to use `a * quot + rem = b`, because the compiler will notice that the thing I am trying to use has type `pr : a *N quot +N rem ≡ b +N 1` instead of `pr : a *N quot +N rem ≡ b`.
+The potential off-by-one error has become a type error, caught at compile time, rather than a runtime error that could only be caught by tests.
+
+## Example: the Hello World of dependently typed languages
+
+The canonical intro to a dependently-typed language is a program `rev` which reverses lists, together with a proof that applying `rev` twice is just the same as the identity function.
+
+In [my own Agda library][GitHub], this is the function `rev` in [Lists/Reversal/Reversal.agda], and the proof is named `revRevIsId`.
+I shall remove some of the arguments which are only there for technical reasons related to preventing Russell's paradox, and give its signature:
+`revRevIsId : {A : Set} → (l : List A) → (rev (rev l) ≡ l)`.
+That is, whenever we have a type (or, in Agda's language, `Set`) called `A`, we can take a list `l` of things of type `A`, and produce a proof that reversing `l` twice yields a list which is identical to `l`.
+
+If I had made any mistakes in the definition of `rev`, then the compiler would have caught them insofar as those mistakes prevented `rev (rev l)` from being identical to `l` for all `l`.
+There is no need to test the `rev` function for this property: I have proved that it holds, and the compiler has verified my proof.
+If it were ever false, there would be an error in my proof, and the compiler would have been unable to compile the function `revRevIsId` that embodies that proof.
+
+One could, if desired, run my function `revRevIsId` explicitly on some list.
+Then one would obtain at runtime a proof that `rev (rev l)` is equal to `l`.
+But I certainly don't expect anyone to run this function; merely the fact that one *could* run it, and if one did then it would always produce an equality, is enough to guarantee the correctness of `rev` in this aspect.
+
+Ultimately my proof uses a few intermediate facts, which I had to prove first:
+
+* Concatenating the empty list to the end of `l` yields `l`;
+* Concatenation of lists is associative;
+* We can pull the head off a concatenated list in the obvious way (this is hard to say in any kind of slick way, but I named the theorem `canMovePrepend` in [Lists.agda]);
+* `rev (l ++ m)` is equal to `(rev m) ++ (rev l)`.
+
+I proved each of these facts, then converted my proof into a program that could be used in theory on any given list to produce objects that witnessed the truth of each fact when specialised to that list.
+
+# Next steps: mathematics
+
+This post has been a whistlestop tour of why one might be interested in dependent types, and how in principle they can be used to write proofs of correctness instead of having to rely on tests.
+The next post will look in more depth at how Agda's type system can be used to check mathematical proofs, and will go into some detail about how "(constructive) proof" and "program" are really the same thing.
+Ultimately we will produce a program/proof that is on the one hand a proof of the Fundamental Theorem of Arithmetic (roughly, "every natural decomposes uniquely into prime factors"), and is on the other hand a fully-verified program that factorises naturals with no tests required.
+
+[Agda]: https://en.wikipedia.org/wiki/Agda_(programming_language)
+[dependently-typed]: https://en.wikipedia.org/wiki/Dependent_type
+[CS410]: https://www.youtube.com/watch?v=O4oczQry9Jw
+[Lists/Reversal/Reversal.agda]: https://github.com/Smaug123/agdaproofs/blob/master/Lists/Reversal/Reversal.agda
+[GitHub]: https://github.com/Smaug123/agdaproofs
+[EuclideanAlgorithm.agda]: https://github.com/Smaug123/agdaproofs/blob/cab004f6d84dfd13a12ca1e73a68aed23d42a348/Numbers/Naturals/EuclideanAlgorithm.agda
diff --git a/hugo/content/posts/2018-09-10-don-syme.md b/hugo/content/posts/2018-09-10-don-syme.md
new file mode 100644
index 0000000..de7eb8f
--- /dev/null
+++ b/hugo/content/posts/2018-09-10-don-syme.md
@@ -0,0 +1,105 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- programming
+comments: true
+date: "2018-09-10T00:00:00Z"
+title: Chatting with Don Syme about the F# compiler
+summary: "Notes from a chat with the creator of F#, about how to contribute to the compiler."
+---
+
+In 2018, my team at G-Research had a fruitful meeting with Don Syme about how to contribute to the F# compiler.
+I made rough notes at the time.
+They may be out of date by now.
+
+# Organisational structure
+
+* *Suggestions* are for debate; Don goes through every three months or so, and approves or rejects or leaves open.
+* *RFCs* are for things which are approved in principle - but there's no guarantee that anyone will commit time to them.
+* Once they're approved, they kick around the RFC folder until they're implemented, at which point they get bucketed into the appropriate historical record of F# version.
+
+RFCs act as primary documentation, since the technical specification of F# is pretty hard to read.
+It should therefore record drawbacks and alternatives that were considered.
+
+The process for getting a feature to "approved in principle" is simply to ask Don.
+
+Features must be complete when they're pushed - no partial features that will be extended later.
+(This is so that we resolve feature interactions up front.)
+
+# Why does F# not have functors?
+
+F# doesn't have a functor system because the process of taking functors *out* of code once they're in is very hard.
+Don wants to add features in such a way that it's not too hard to unwind a decision to start using them.
+
+# Working on the compiler
+
+Sidetrack: he said that when working on the compiler, all the tools basically needed to be the same versions; is that some kind of design decision?
+Is it going to become less true?
+I didn't ask this.
+
+You can always push stuff up to a branch on the FSharp repo to run the CI and verify that existing F# code won't be broken by your addition to the language.
+
+Basically everything happens in the typechecker.
+"Pretty much every time we try to do things earlier, we regret it."
+This is because of the language machinery, which is wedded heavily to the actual code rather than to any other representation.
+
+# Specific notes on adding syntactic sugar for `fun x -> x.Prop`
+
+During typechecking we would expand the `_.Prop` syntax.
+Would want to expand to `fun (obj : ^T) -> (^T : (member Prop : 'U) obj)`.
+This would add a constraint during typechecking, that would be solved automatically.
+But this approach doesn't extend to `.Method` instead of `.Property` - since generics can't be placed on that member.
+
+A naive find-replace of "replace `.Prop` syntactic sugar with `fun x -> x.Prop`" fails because of the example of `.Method(2)` - this can't easily be converted to `fun x -> x.Method(2)`.
+
+There is no ability to have "or"-based constraints.
+
+Implementation strategy for the "replace `.x` with `fun i -> i.x`" feature idea:
+1. Find how `_.S` is parsed currently (hint: it's not).
+2. Alter parser so that `_.S` is actually parsed to an alternative.
+3. Look in the typechecker where that AST gets resolved.
+4. Expand the typechecker and try again.
+(Keep track of where our operator's syntax may legitimately be used, by checking the `pars.fsy` parse file.)
+
+Pitfall to remember: the artificial name `x` in `fun x -> x.S` needs to not be reported in debug symbols etc.
+Specifically during typechecking, Don hasn't made the decision whether it would be typecheck-then-replace, or replace-then-typecheck.
+
+# The F# parser
+
+Useful flags in tooling exist.
+
+* `fsc --tokenize`, which tells us that `_.Length` e.g. doesn't have an expression form: the tokeniser just sees `UNDERSCORE`.
+* `fsc --parseonly` to run only the parsing stage.
+
+Specifically to see what happens with our proposed underscore syntax, go to `pars.fsy` to see where the underscore parse token appears, and `lex.fsl` to see what it lexes to.
+We discover that `_` is a keyword in the language (see `lexhelp.fs`), and it's declared in `pars.fsy`.
+Find usages of it in `pars.fsy`.
+Seems reasonable to parse `_` as an identifier (alternative: create a whole new syntax node).
+
+The parser defines a grammar with various types of expression. Atomic (e.g. `1`, `true`, `(blah)`, `1.0`); other expressions (`1+1`, etc).
+Atomic expressions have known-in-advance delimitation in the code - e.g. bracketed expressions.
+Things which will be parsed along similar code paths to our proposed underscore syntax: `base.blah`, `global.System.Blah`.
+
+# How the typechecker works
+
+The typechecker will first typecheck the thing on the left, and then has a chain of qualifications afterwards (i.e. the type gets refined as we shove more on the right).
+E.g. `_.Name.Name2.Name3` is typechecked as "`_`" and then refined with "`.Name`", and so on.
+Type-checking and name resolution are interleaved.
+
+At the end, after typecheck, `FindUnsolved.fs` infers the rest of the types.
+
+# How to go about implementing applicative computation expression builders
+This is [issue #579](https://github.com/fsharp/fslang-suggestions/issues/579).
+*Editor's note: this is now implemented as of F# 5.*
+
+Tomas Petricek had a wide-ranging change ("joinads") which would have encompassed this feature plus many more, but it was rejected, because it tried to change the semantics of pattern matching.
+
+Question of whether a builder should have a flag to indicate "I follow the applicative laws" - e.g. for optimising when you could use "map" rather than "bind then return".
+Answer: yes, this would be good - for backwards compat, even if for no other reason.
+This would probably be a new attribute on the map.
+
+# An annoyance: not being able to have a crate returning `unit`
+
+We could go ahead and fix this by using the `void` return type on `type C () = member x.M() : unit`, and using the `unit` return type on everything else.
+(Our current workaround is to define a type `FakeUnit` which we return instead of `unit`.)
diff --git a/hugo/content/posts/2018-09-20-mathematics-teaching.md b/hugo/content/posts/2018-09-20-mathematics-teaching.md
new file mode 100644
index 0000000..360f0db
--- /dev/null
+++ b/hugo/content/posts/2018-09-20-mathematics-teaching.md
@@ -0,0 +1,24 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- hacker-news
+comments: true
+date: "2019-09-20T00:00:00Z"
+title: The stages of mathematics teaching (a note from Hacker News)
+summary: "A quick note from Hacker News about the various stages of learning and teaching in mathematics."
+---
+
+In response to [a linkpost](https://news.ycombinator.com/item?id=18034714) to [an article about the disputed nature of a proof of the ABC conjecture](https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/), [user barbecue_sauce](https://news.ycombinator.com/user?id=barbecue_sauce) [commented with surprise](https://news.ycombinator.com/item?id=18035656):
+
+> You have an MSc in mathematics but don't consider yourself an expert?
+
+[I replied](https://news.ycombinator.com/item?id=18035956):
+
+> Where high-school mathematics was analogous to learning to form letters with a pencil, undergraduate BSc mathematics is analogous to learning to form words and short sentences.
+>
+> By the end of your BSc in the analogy, you can look at a sentence and recognise at least what sort of genre it might be part of, and you can write down many of the most common words as well as a number of rehearsed useful sentences.
+>
+> An MSc is like reading some examples of long sentences and short paragraphs, and being shown some long paragraphs or even chapters of books (which you have no hope of understanding, but you try, and the experience is salutary). If you're lucky, you did a research project in which you perhaps rewrote a certain very specific short paragraph in your own words.
+>
+> The job of a research mathematician is analogously then to read and write books.
diff --git a/hugo/content/posts/2019-01-16-mystery-hunt.md b/hugo/content/posts/2019-01-16-mystery-hunt.md
new file mode 100644
index 0000000..e8738a9
--- /dev/null
+++ b/hugo/content/posts/2019-01-16-mystery-hunt.md
@@ -0,0 +1,19 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mystery-hunt
+- programming
+comments: true
+date: "2019-01-16T00:00:00Z"
+title: MIT Mystery Hunt 2019 answers
+summary: "A couple of solution documents I made during the progress of the 2019 MIT Mystery Hunt."
+---
+
+# Picture Book
+
+The [Picture Book](https://www.mit.edu/~puzzle/2020/puzzle/turtle/) puzzle was an image of James Joyce, in which information was hidden.
+See [PictureBook.nb] for a Mathematica notebook, or [PictureBook.pdf] for a statically rendered pdf.
+
+[PictureBook.nb]: /MysteryHunt2019/PictureBook.nb
+[PictureBook.pdf]: /MysteryHunt2019/PictureBook.pdf
diff --git a/hugo/content/posts/2019-12-27-cooking.md b/hugo/content/posts/2019-12-27-cooking.md
new file mode 100644
index 0000000..174dbd3
--- /dev/null
+++ b/hugo/content/posts/2019-12-27-cooking.md
@@ -0,0 +1,18 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- hacker-news
+comments: true
+date: "2019-12-27T00:00:00Z"
+title: Teaching how to cook (a note from Hacker News)
+summary: "A quick note from Hacker News about my thoughts on teaching cookery."
+---
+
+In response to [a linkpost](https://news.ycombinator.com/item?id=21884927) to [an article about how it's important to be precise when baking](https://theoutline.com/post/8307/follow-recipe-baking-thanksgiving), [I commented](https://news.ycombinator.com/item?id=21890535):
+
+> Much of the art of (non-professional) cooking, in my view, lies in knowing what must be rigid and what need not be. Baking is a science, relying on precise chemical reactions that must happen within pretty tightly defined windows. Making a curry is not a science, in that you can get a good curry by just making it up on the spot, tasting as you go, and fixing it if it needs fixing. (Maybe following a recipe precisely will get you the best results, but you can get most of the way there with just a few rough signposts; nothing will go disastrously wrong if you shake things up a bit.)
+>
+> One of the things I found most annoying about watching someone trying to teach someone else how to cook (while at uni) was that the teacher never pointed out when something didn't matter. Whether it was "now you put in 1.5tsp of salt to the sauce" or "now you put in 1.5tsp of yeast to the dough", the method was presented as unyielding. In fact the sauce can be made in infinite variety, while the dough must be made just so. Know when to yield to the expert, and know when not to bother.
+
+Other commenters noted that things are usually a bit more flexible than I made out, but the overall point stands.
diff --git a/hugo/content/posts/2020-01-16-mystery-hunt.md b/hugo/content/posts/2020-01-16-mystery-hunt.md
new file mode 100644
index 0000000..8fd1f4b
--- /dev/null
+++ b/hugo/content/posts/2020-01-16-mystery-hunt.md
@@ -0,0 +1,57 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mystery-hunt
+- programming
+comments: true
+date: "2020-01-16T00:00:00Z"
+title: MIT Mystery Hunt 2020 answers
+summary: "A couple of solution documents I made during the progress of the 2020 MIT Mystery Hunt."
+---
+
+# Turtle
+
+The [Turtle](https://www.mit.edu/~puzzle/2020/puzzle/turtle/) puzzle was a collection of Logo programs which ultimately spelled out words.
+See [turtle.nb] for a Mathematica notebook, or [turtle.pdf] for a statically rendered pdf.
+
+# Best Song Ever
+
+The [Best Song Ever](https://www.mit.edu/~puzzle/2020/puzzle/best_song/) puzzle was a .wav file of a recording of Disney's "It's A Small World".
+Its data segment encoded a .jpg file which suggested how to proceed.
+See [BestSongEver.nb] for a Mathematica notebook which processed the data segment (it depends on [tiny_planet.wav]), or [BestSongEver.pdf] for a statically rendered PDF; see [out.jpg] for the extracted JPG.
+
+# Story
+
+The [Story](https://www.mit.edu/~puzzle/2020/puzzle/story/) puzzle was a collection of rather incoherent paragraphs.
+Embedded in the page source was also a collection of very large integers, which turned out to have binary expansions that were of exactly the same length as their corresponding paragraph.
+Selecting the letters from the paragraph corresponding to 0-valued bits of the integers yielded a message which told us how to proceed.
+
+See [Story.nb] for a Mathematica notebook which performs that extraction, or [Story.pdf] for a statically rendered PDF.
+
+# Nauseator
+
+The [Nauseator](https://www.mit.edu/~puzzle/2020/puzzle/nauseator/) puzzle started with a coloured nonogram.
+See [Nauseator.nb] for a Mathematica notebook to parse out the data into structured form and shell out to `z3` to solve it, or [Nauseator.pdf] for a statically rendered PDF.
+(The PDF has a very blurry image at the end, for some reason; see [the actual image] for full resolution.)
+The input spreadsheet is [nauseator.xlsx].
+
+The notebook relies on [Z3Interop.wl] (static: [Z3Interop.pdf]), an extremely janky DSL to express SAT problems to send to the [Z3] SAT solver.
+See the application to Sudoku and to nonograms, at [Z3Interop.nb] (static: [Z3Interop.pdf]).
+
+[turtle.nb]: /MysteryHunt2020/turtle.nb
+[turtle.pdf]: /MysteryHunt2020/turtle.pdf
+[BestSongEver.nb]: /MysteryHunt2020/BestSongEver.nb
+[BestSongEver.pdf]: /MysteryHunt2020/BestSongEver.pdf
+[tiny_planet.wav]: https://puzzles.mit.edu/2020/puzzle/best_song/tiny_planet.wav
+[out.jpg]: /images/MysteryHunt2020/out.jpg
+[Story.nb]: /MysteryHunt2020/Story.nb
+[Story.pdf]: /MysteryHunt2020/Story.pdf
+[Nauseator.nb]: /MysteryHunt2020/nauseator.nb
+[Nauseator.pdf]: /MysteryHunt2020/nauseator.pdf
+[Nauseator.xlsx]: /MysteryHunt2020/nauseator.xlsx
+[the actual image]: /MysteryHunt2020/nauseator.svg
+[Z3Interop.wl]: /MysteryHunt2020/Z3Interop.wl
+[Z3Interop.nb]: /MysteryHunt2020/z3interop.nb
+[Z3Interop.pdf]: /MysteryHunt2020/z3interop.pdf
+[Z3]: https://github.com/Z3Prover/z3
diff --git a/hugo/content/posts/2020-01-25-mit-mystery-hunt-photos.md b/hugo/content/posts/2020-01-25-mit-mystery-hunt-photos.md
new file mode 100644
index 0000000..ae3a533
--- /dev/null
+++ b/hugo/content/posts/2020-01-25-mit-mystery-hunt-photos.md
@@ -0,0 +1,12 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- mystery-hunt
+date: "2020-01-25T00:00:00Z"
+title: MIT Mystery Hunt photos
+summary: "Photos from the Mystery Hunt."
+gallery: true
+---
+
+{{< foldergallery src="images/galleries/Mit2020/" >}}
diff --git a/hugo/content/posts/2020-03-04-defunctionalisation.md b/hugo/content/posts/2020-03-04-defunctionalisation.md
new file mode 100644
index 0000000..a2f5a40
--- /dev/null
+++ b/hugo/content/posts/2020-03-04-defunctionalisation.md
@@ -0,0 +1,394 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- programming
+- g-research
+- hacker-news
+comments: true
+date: "2020-03-04T00:00:00Z"
+aliases:
+- /defunctionalisation/
+title: Defunctionalisation
+summary: "An underappreciated tool for writing good software."
+---
+
+# Defunctionalisation: an underappreciated tool for writing good software
+
+*Cross-posted at the [G-Research official blog](https://www.gresearch.co.uk/article/defunctionalisation/); comments at [Hacker News](https://news.ycombinator.com/item?id=22525038).*
+
+I work in an engineering team at G-Research dedicated to creating performant and correct libraries for our quant researchers to use. In the process of achieving this goal, we have settled on a number of patterns to make it easier to write code that is both correct and amenable to optimisation, and I’d like to explain one such pattern in this blog post. My personal motto distilled from these patterns is “load as much information as you can into the type system, so that the computer can tell you what to write”, and this post can be viewed as exploring an aspect of that philosophy.
+
+We have talked about "initial algebras" [at some length](https://github.com/nickcowle/Talks/blob/3babf57d01db905ef057624250543753acfde4cb/Initial%20Algebras%20for%20the%20Uninitiated.pdf); this post is to discuss *defunctionalisation* as a complementary technique.
+Defunctionalisation is a well-known technique among compiler writers, but I consider it to be under-appreciated as a tool for writing good software more generally.
+It pairs up particularly well with the "initial algebra" pattern.
+
+# What problem can defunctionalisation help to solve?
+
+Recall that the initial algebra pattern, in a nutshell, is "don't manipulate *things*, but instead manipulate *descriptions of those things*".
+An ideal program written using the initial algebra pattern is essentially a "compilation pipeline" of successive refinements to these descriptions, until finally the descriptions contain all the information required to produce a performant program; whereupon it is easy to construct that program.
+More philosophically, "don't throw away information; instead, refine the information as much as possible, and only turn it into an actual program as late as possible".
+
+This pattern, used correctly, has a number of advantages, all related to each other.
+
+* The "compilation pipeline" of successively refined descriptions is very inspectable.
+At any point during this "compilation", you can explore the current structure of the descriptions you have made so far.
+* Keeping as much information as possible means you have more scope to optimise.
+For example, if your library takes as input a description of what the user wants to do, you have complete freedom to notice any redundancy or inefficiency in the user's description before you ever start producing any concrete output.
+* Such programs are very easy to test.
+They are naturally modular in structure (as implied by the pipeline metaphor).
+
+But while the initial algebra pattern helps a lot in writing correct software that can also be made performant, it is still very possible to write in such a way as to nullify the benefits above; and this is where defunctionalisation comes in.
+
+In brief, "defunctionalisation" means "converting programs into specifications of programs, and only later on interpreting those specifications back into programs".
+Defunctionalisation is to *program structure* as the initial algebra is to *datatype structure*; a defunctionalised program can be thought of as a program which first describes what it will do, and then does it.
+This might seem like needless complexity, but in fact it can be extraordinarily helpful to separate the logic of your programs in this way.
+
+## How could the initial algebra pattern be misused, and how does defunctionalisation help?
+
+We will discuss the three benefits of the initial algebra pattern listed above, and we will indicate how these benefits can be subverted by careless design.
+The first two benefits will use the same example; the third ("testability") is best demonstrated by adding a little more complexity, so will use a different example.
+
+It's true that both these examples are somewhat contrived; they are designed to highlight specific principles without any of the distractions that might make the code a little more realistic.
+Rest assured that the principles continue to hold in real code.
+
+### Inspectability
+
+The first of the benefits listed above was "inspectability".
+
+For this section and the next, let us imagine the guts of a very basic calculator application written in F#, expressed in the initial algebra style.
+Perhaps this calculator has a graphical user interface with a `+` button and a unary `-` button, and the GUI constructs an expression which represents the user's input before that input is handed off to a library for processing.
+The library will take an `Expr` as its input, defined below, and it will interpret that input into an `int`, thereby performing the calculation specified.
+
+```fsharp
+type Expr =
+  | Const of int
+  | CustomOneArg of (int -> int) * Expr
+  | CustomTwoArgs of (int -> int -> int) * Expr * Expr
+
+module Expr =
+  let rec interpret (e : Expr) : int =
+    match e with
+    | Const a -> a
+    | CustomOneArg (f, a) -> f (interpret a)
+    | CustomTwoArgs (f, a, b) -> f (interpret a) (interpret b)
+```
+
+So the `+` button on this calculator produces an `Expr` which looks like a `CustomTwoArgs ((+), a, b)`; the `-` button produces a `CustomOneArg ((fun x -> -x), a)`.
+
+Now, what happens if you were to pause in the middle of execution, just after the user's input has been parsed into an `Expr` but before we go on to call `interpret`?
+Well, in your hand, perhaps you have a `CustomTwoArgs (closure<49>, (Const 3), (CustomOneArg(closure<103>, Const 9)))`.
+But has this actually helped you?
+
+The resolution to this problem is very likely obvious.
+
+```fsharp
+type OneArgFunction =
+  | Negate
+  | Custom of (int -> int)
+
+type TwoArgFunction =
+  | Add
+  | Subtract
+  | Custom of (int -> int -> int)
+
+type Expr =
+  | Const of int
+  | OneArg of OneArgFunction * Expr
+  | TwoArg of TwoArgFunction * Expr * Expr
+
+module OneArgFunction =
+  let interpret (f : OneArgFunction) : int -> int =
+    match f with
+    | Negate -> (~-)
+    | Custom f -> f
+
+module TwoArgFunction =
+  let interpret (f : TwoArgFunction) : int -> int -> int =
+    match f with
+    | Add -> (+)
+    | Subtract -> (-)
+    | Custom f -> f
+
+module Expr =
+  let rec interpret (e : Expr) : int =
+    match e with
+    | Const a -> a
+    | OneArg (f, x) ->
+      (OneArgFunction.interpret f) (interpret x)
+    | TwoArg (f, x, y) ->
+      (TwoArgFunction.interpret f) (interpret x) (interpret y)
+```
+
+Hook up the `+` button so that it produces the `Sum` union case, and the `-` button so that it produces the `Negate` union case.
+
+Now it's very likely that the debugger tells you that we are currently processing a `Sum (Const 3, Negate (Const 9))`.
+Much better than the previous `CustomTwoArgs` confusion!
+
+The trick was to *describe what we wanted to do* wherever possible, and then implement that description.
+Notice the kinship with the initial algebra pattern: in our first attempt, we were careful not to throw away the user's input (after all, we could have hooked it up so that `3+3` was directly `6`, rather than `CustomTwoArgs((+), Const 3, Const 3)`), but it was only when we defunctionalised the operations that the power of the initial algebra really came into its own.
+
+Note further that there is no loss of flexibility - we've still retained the `CustomOneArg` and `CustomTwoArgs` cases.
+(Perhaps we do need this: perhaps the calculator library needs to take arbitrary functions from its consumers, and we genuinely might not be able to describe in advance what the user wants to do.)
+This goes to show that defunctionalisation can often be done incrementally - you can get significant benefits without completely rearchitecting your program.
+If after several rounds of defunctionalisation it turns out that the `CustomOneArg` and `CustomTwoArgs` cases are no longer required, then we can simply delete them and end up with a fully defunctionalised datatype; but if they are required, we can just keep them around and still retain many of the benefits from the rest of the program's defunctionalisation.
+
+Remember, this particular instance of the problem is small, contrived, and designed to be easy to understand.
+The principle does still hold in real life, and indeed one of our most recent internal projects has seen great structural and performance improvements after a couple of rounds of defunctionalising its core in precisely the incremental "identify the things you want to use this library for, and special-case them with unique descriptions" way described above.
+
+### Scope for optimisation
+
+The second of the benefits listed above was "keeping as much information as possible, leaving scope to optimise".
+
+Initial algebras naturally don't lose much information; but it's quite possible to lose the information you need very early in the refinement pipeline.
+In the toy calculator example, the original attempt (without `Sum` or `Negate` as distinct cases) made certain optimisations completely impossible.
+
+For example, it's true that `-(-a) = a` for every integer `a`, so we might wish to optimise `Negate (Negate e)` down to just `e` for any expression `e`.
+But if all we have is `OneArgFunction.Custom (_, OneArgFunction.Custom (_, e))`, then we simply can't do it.
+Despite the use of an initial algebra, we've still lost too much information because of being insufficiently defunctionalised.
+
+### Testability
+
+The third of the benefits was "testability".
+This one is easiest to demonstrate, and it's probably the most tangible way I use these patterns day-to-day.
+
+Consider a simple, contrived, F# program that has some validation requirements, and then does one of two different things depending on whether its input was negative.
+
+```fsharp
+let writeSmallOddToFile (input : int) : unit =
+    if input % 2 = 0 then
+        failwith "Input did not pass validation: input cannot be even."
+    if input > 100 then
+        failwith "Input did not pass validation: input cannot be over 100."
+
+    if input < 0 then
+        printfn "Number was negative: %i" input
+    else
+        System.IO.WriteToFile("output.txt", sprintf "%i" input)
+```
+
+What about when you want to test this?
+It's easy: simply catch and handle the exception in your tests, and mock out the file system.
+(Of course, the function signature will need to be altered to take a file system as input.)
+
+But there is a dual approach: using defunctionalisation to push the work forward, into the future.
+Instead of handing the file system dependency *down* into `writeSmallOddToFile` from its caller, we have `writeSmallOddToFile` hand its desired actions *up* into its caller.
+
+The following example is written rather in longhand to demonstrate the general pattern.
+
+```fsharp
+type ValidationError =
+  | Even of int
+  | TooBig of int
+
+[<RequireQualifiedAccess>]
+module ValidationError =
+  let toString (error : ValidationError) =
+    match error with
+    | Even i -> sprintf "Input did not pass validation: input cannot be even."
+    | TooBig i -> sprintf "Input did not pass validation: input cannot be over 100."
+
+type WriteInstruction =
+  | Print of string
+  | WriteFile of FilePath * string
+
+[<RequireQualifiedAccess>]
+module WriteInstruction =
+  let execute (instruction : WriteInstruction) : unit =
+    match instruction with
+    | Print s -> printfn s
+    | WriteFile (fp, s) -> System.IO.WriteToFile (fp, s)
+
+let describe (input : int) : Result<WriteInstruction, ValidationError> =
+  if input % 2 = 0 then
+    Error (Even input)
+  else if input > 100 then
+    Error (TooBig input)
+  else
+    if input < 0 then
+      Print (sprintf "Number was negative: %i" input)
+    else
+      WriteFile (System.IO.FilePath "output.txt", sprintf "%i" input)
+    |> Ok
+
+let writeSmallOddToFile (input : int) : unit =
+  match describe input with
+  | Error e -> failwith (ValidationError.toString e)
+  | Ok instruction -> WriteInstruction.execute instruction
+```
+
+Now it is incredibly simple to test the logic of the `describe` function: there are no dependencies to mock, so we can simply call the function precisely as it will be called in production.
+The `WriteInstruction.execute` function has similarly been stripped to the bare bones; to test this function, it will be necessary to mock out dependencies (though because the dependencies have been forced as far forward as possible, they are much easier to mock).
+Succinctly: *the logic of deciding how to do things* is completely decoupled from *the logic of deciding what to do*, and this separation makes it easier both to test *what the program thinks it should do* and *how the program thinks it should do it*.
+
+If a more end-to-end test is required, a function `WriteInstruction -> unit` could be injected into `writeSmallOddToFile`, to be used instead of `WriteInstruction.execute`.
+Note that this looks very similar to the object-orientation-inspired idea of mocking out the filesystem; but mocking an entire filesystem is very much overkill here, giving the caller much less of an idea of how the filesystem dependency is going to be used.
+By instead having `describe` pass its intended actions *up* to its caller, and separating the interpretation of those actions in `execute`, the requirements are made clear.
+
+# How can you start defunctionalising your programs?
+
+One of the great benefits of defunctionalisation is that it can be introduced gradually to an existing system.
+When a data structure contains functions that you (the creator of a library) have put there, you can always pull off the following manoeuvre.
+
+Imagine we have a discriminated union called `FunctionalType`, which expresses several possible ways of getting information from the outside world for use in subsequent processing.
+One of these ways is an `InternetRequest`, which takes an integer (say, `3`), then requests `http://wellknown/3.txt` and checks if that page exists.
+
+```fsharp
+[<Obsolete "Use Defunctionalising instead.">]
+type FunctionalType =
+  /// Requests document '3.txt', say, under a well-known URL.
+  /// Returns true if that document exists.
+  | InternetRequest of (int -> bool)
+  /// Some other possible things we might want to do
+  | AnotherCase of (string -> float)
+  | AThirdCase of (unit -> float)
+
+type Defunctionalising =
+  | InternetRequest of Uri
+  | AnotherCase of (string -> float)
+  | AThirdCase of (unit -> float)
+```
+
+Imagine the call site used to look like the following (where I have assumed access to some helper functions in a hypothetical `Uri` module, rather than writing out a complete implementation):
+
+```fsharp
+let makeFunctionalType () : FunctionalType =
+  let remoteFileExists (i : int) =
+    let document : string option =
+      Uri.append (Uri.make "http://wellknown") (sprintf "%i.txt" i)
+      |> Uri.fetch
+    match document with
+    | None -> false
+    | Some _ -> true
+  FunctionalType.InternetRequest remoteFileExists
+
+let doStuff (f : FunctionalType) (i : int) =
+  match f with
+  | FunctionalType.InternetRequest (f : int -> bool) ->
+    if f i then 0.0 else 1.0
+  | AnotherCase (f : string -> float) ->
+    f "hello!"
+  | AThirdCase (f : unit -> float) ->
+    f ()
+```
+
+Then in the new world, it could look like this:
+
+```fsharp
+let makeDefunctionalising () : Defunctionalising =
+  Uri.make ("http://wellknown")
+  |> Defunctionalising.InternetRequest
+
+let doStuff (f : Defunctionalising) (i : int) =
+  match f with
+  | Defunctionalising.InternetRequest (u : Uri) ->
+    let document =
+      Uri.append u (sprintf "%i.txt" i)
+      |> Uri.fetch
+    match document with
+    | None -> false
+    | Some _ -> true
+    |> fun result -> if result then 0.0 else 1.0
+  | Defunctionalising.AnotherCase (f : string -> float) ->
+    // This is unchanged: we haven't defunctionalised it yet
+    f "hello!"
+  | Defunctionalising.AThirdCase (f : unit -> float) ->
+    f ()
+```
+
+This hasn't actually decreased the complexity: all the same code is present in the old world as in the new world, though some of it has shuffled around.
+However, this new world enjoys some of the benefits discussed above, and without the need to make sweeping changes to the entire program: we have defunctionalised only one case where it seemed easy to do so.
+
+Of course, if we would like to keep the `doStuff` function short, then we can extract the rearranged code out into its own “interpreting” function with type signature `Uri -> (int -> bool)`.
+
+## What about more complex dependencies?
+
+Sometimes there may be interdependencies knitted into the structure of your code, such that several DU cases have to be migrated together; but even this is not an insurmountable obstacle.
+By creating a structure that holds the dependencies, you can make sure you have them whenever you need them; and once all the relevant DU cases have been "nearly defunctionalised" so that they consist of a description and a blob of dependencies, then you can start to move the dependencies out again.
+
+```fsharp
+type Dependencies = Dependencies of (int ref) * OtherThings
+
+type Defunctionalising =
+  /// The InternetRequest case is completely defunctionalised, as above.
+  | InternetRequest of Uri
+  /// The AnotherCase case is not completely defunctionalised, but is "nearly" so:
+  /// it no longer contains a function string -> float, but instead a structure
+  /// that contains all the information that would be required to create that function.
+  | AnotherCase of Dependencies
+  | AThirdCase of Dependencies * bool
+```
+
+Now, when we come to use the `Defunctionalising` type, we can recreate the functions that were previously contained within the `FunctionalType`'s cases, even if there was lots of information shared between those cases.
+In this example, there is some shared mutable state (the `int ref` of the `Dependencies` object), which is why we didn't want to migrate the two cases independently.
+But we can still move the construction of those functions forward, into the future, constructing them only at the point where they are needed and hence obtaining many of the benefits of defunctionalisation:
+
+```fsharp
+let doStuff (f : Defunctionalising) (i : int) =
+  match f with
+  | Defunctionalising.InternetRequest (u : Uri) ->
+    // We already saw this defunctionalisation, so I won't repeat it
+    failwith "etc."
+  | Defunctionalising.AnotherCase (Dependencies (sharedState, otherThings)) ->
+    let f (s : string) : float =
+      sharedState := float (String.length s)
+      sharedState
+    f "hello!"
+  | Defunctionalising.AThirdCase (Dependencies (sharedState, otherThings), b : bool) ->
+    let f () : float =
+      sharedState := sharedState + 1.
+      sharedState
+    f ()
+```
+
+The idea is that instead of having a function which [closes over] its dependencies implicitly, we bring those arguments out so they are explicitly represented in the defunctionalised data structure.
+Then we can get rid of the function itself from our data type, and rehydrate the function later on.
+
+Once we have manipulated the data structures into this more cleanly separated form (that is, "a blob of stuff the function needs" and "the input to the function"), we could simply stop and consider our job to be done.
+It's almost certainly produced a more inspectable, more testable data structure, with nice handy labels for everything that the debugger would previously have had to its best at disentangling for us (in showing all the variables that had been closed over).
+Or we could go on and see how far forward in the "compilation pipeline" we can push the computations that create the `Dependencies` object.
+
+For example, we could discover that the `sharedState` was actually only ever mutated by the functions contained in these two discriminated-union cases, and that it was safe to create this dependency later, omitting it from the `Dependencies` object and therefore making the code simpler (since we can guarantee that nothing else is mutating that shared state):
+
+```fsharp
+type Defunctionalising =
+  /// The InternetRequest case is completely defunctionalised, as above.
+  | InternetRequest of Uri
+  /// Even more "nearly" defunctionalised: we have successfully extracted the int ref
+  /// from the `Dependencies` object that used to be part of these cases
+  | AnotherCase of OtherThings
+  | AThirdCase of OtherThings * bool
+
+let doStuff (f : Defunctionalising) (i : int) =
+  // Now it's only `doStuff` which can be mutating this state!
+  let mutable sharedMutableState = 0.
+  match f with
+  | Defunctionalising.InternetRequest (u : Uri) ->
+    // We already saw this defunctionalisation, so I won't repeat it
+    failwith "etc."
+  | Defunctionalising.AnotherCase (deps : OtherThings) ->
+    let f (s : string) : float =
+      sharedMutableState <- float (String.length s)
+      sharedMutableState
+    f "hello!"
+  | Defunctionalising.AThirdCase (deps : OtherThings, b : bool) ->
+    let f () : float =
+      sharedMutableState <- sharedMutableState + 1.
+      sharedMutableState
+    f ()
+```
+
+It's entirely up to you to decide how far along the defunctionalisation path you wish to go.
+
+# Conclusion
+
+It really is very hard to express in words how beautiful a "compilation pipeline" of initial algebras can be.
+Functional programming is in some sense "programming via data", and while the initial algebra pattern lets you see what you're going to do, it is only when you add in defunctionalisation to the "compilation pipeline" that the program can truly fall out into layer after layer of pure data transformations.
+
+Defunctionalisation makes it easier to track the dependencies of your code, and it can simultaneously make your code more amenable to optimisation and more debuggable.
+Indeed, internally we have used this extra power to help create visualisations of the various stages of "compilation" underlying a particular one of our projects (since at any stage of "compilation" of an initial algebra, you can always choose to interpret your data structure not as a specification of a computation, but as a specification of a visualisation!).
+If you have never tried working while you have good visualisations of the internals of your programs, I strongly recommend giving it a try; it is mind-expanding.
+
+Be on the lookout for opportunities to throw away less information, and the data structures will reward you.
+
+[closes over]: https://en.wikipedia.org/wiki/Closure_(computer_programming)
diff --git a/hugo/content/posts/2020-04-05-static-config.md b/hugo/content/posts/2020-04-05-static-config.md
new file mode 100644
index 0000000..e463679
--- /dev/null
+++ b/hugo/content/posts/2020-04-05-static-config.md
@@ -0,0 +1,22 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- hacker-news
+- programming
+comments: true
+date: "2020-04-05T00:00:00Z"
+title: Static config (a note from Hacker News)
+summary: "A quick note from Hacker News about my preference for static config rather than dynamic."
+---
+
+Hacker News user [amelius](https://news.ycombinator.com/user?id=amelius) [commented](https://news.ycombinator.com/item?id=22788623) on a [linkpost](https://news.ycombinator.com/item?id=22787332) to [an article about how configuration languages are all terrible](https://beepb00p.xyz/configs-suck.html):
+
+> I don't agree. Configuration files tell a program what to do. You want expressive power there. Telling a program what to do merely through values only makes things more indirect.
+> To give an example: you can get into the situation that some configuration values are only valid when configuration value X is Y, and otherwise other configuration values are valid. What better way to model this than through an if-statement in a programming language? This makes it immediately apparent which settings have effect.
+
+[I responded](https://news.ycombinator.com/item?id=22788713):
+
+> I am involuntarily howling internally in anguish, and if the feeling could speak, it would be screaming "[parse, don't validate](https://lexi-lambda.github.io/blog/2019/11/05/parse-don-t-validate/)".
+> If some config values are valid when X is true, and other config values are valid when X is false, then use a different schema for the two cases. This is what discriminated unions are for: for representing conditionals statically, inspectably, serialisably!
+> If some config values are valid when X is 0.339, and some are valid when X is 0.340, and some are valid when X is 0.341, and so on, then… I don't know how to help, and maybe I must just avert my eyes in shame as I implement the dynamic logic. (But in that case it seems a bit odd to say you're "telling the program what to do" with this configuration; I'd say you're bolting on a little extra program at the start.)
diff --git a/hugo/content/posts/2020-07-22-tailrecursion.md b/hugo/content/posts/2020-07-22-tailrecursion.md
new file mode 100644
index 0000000..ac9af49
--- /dev/null
+++ b/hugo/content/posts/2020-07-22-tailrecursion.md
@@ -0,0 +1,229 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- programming
+- g-research
+comments: true
+date: "2020-07-22T00:00:00Z"
+aliases:
+- /tail-recursion-for-loops/
+title: In favour of recursive functions, not imperative constructs, to make loops
+summary: "How to write loops immutably and safely."
+---
+
+*Cross-posted at the [G-Research official blog](https://www.gresearch.co.uk/article/in-favour-of-recursive-functions-not-imperative-constructs-to-make-loops/)*
+
+## Setting the scene
+
+It's a fresh new day at work, and you want to find the dot product of two arrays (that is, the sum of the product of the pairs of elements).
+You're using your favourite language, [F#](https://en.wikipedia.org/wiki/F_Sharp_(programming_language)).
+You're allergic to `Seq.zip` (maybe you're terrified of virtual calls or something), and `Array.zip` just allocates too much memory.
+Also, you want to make sure you can bail out of the iteration early; what if someone tells you at the last minute that you have to stop if you see the number `100`?
+
+Cracking your knuckles, the following pours from your fingertips:
+
+```fsharp
+type Index = int
+
+let truncatedDotProduct
+    (shouldContinue : Index -> int -> int -> bool)
+    (xs : int array)
+    (ys : int array)
+    : int
+    =
+    let mutable ans = 0
+    let mutable (i : Index) = 0
+    while (shouldContinue i xs.[i] ys.[i]) do
+        ans <- ans + (xs.[i] * ys.[i])
+    ans
+```
+
+Easy, of course!
+And then you write the unit tests and discover how truly exquisite is the infinite loop you have made.
+
+With a gentle sigh of one who has arisen bright and refreshed only to discover the computer in a sullen and recalcitrant mood, you insert the line that makes everything work.
+
+```fsharp
+type Index = int
+
+let truncatedDotProduct
+    (shouldContinue : Index -> int -> int -> bool)
+    (xs : int array)
+    (ys : int array)
+    : int
+    =
+    let mutable ans = 0
+    let mutable (i : Index) = 0
+    while (shouldContinue i xs.[i] ys.[i]) do
+        ans <- ans + (xs.[i] * ys.[i])
+        // How weary, stale, flat, and unprofitable
+        // seem to me all the uses of this world
+        i <- i + 1
+    ans
+```
+
+Now the tests pass, and you can stride forward into a glorious day of productivity.
+
+But wouldn't it have been lovely if this mistake hadn't happened?
+
+## Going loopy
+
+The mistake was that we introduced a piece of state, but never changed it.
+A sufficiently advanced compiler can warn us about a variable that is set but never used - but the problem still remains in general.
+(What if we had printed the iteration variable, and therefore used it? Or what if we had a slightly more complicated computation for which we legitimately performed some mutation to the state in our loop body, but still forgot to do the final increment?)
+
+With a small change to the shape of this function, we can make the bug so obvious that we would never dream of making it.
+
+```fsharp
+type Index = int
+
+let truncatedDotProduct
+    (shouldContinue : Index -> int -> int -> bool)
+    (xs : int array)
+    (ys : int array)
+    : int
+    =
+    let rec go (ans : int) (i : Index) =
+        if shouldContinue i xs.[i] ys.[i] then
+            go (ans + (xs.[i] * ys.[i])) (i + 1)
+        else
+            ans
+
+    go 0 0
+```
+
+Now we are forced to declare explicitly that the index in the next iteration will be `i + 1`.
+Merely omitting this decision is not an option any more, because the compiler will force us to give a second argument to `go` whenever we invoke it.
+
+As an added bonus, the termination condition of the loop has been tidied up; I would argue that it's now easier to see that if `shouldContinue` fails, we return the value we'd got up to *before* the current index.
+(Admittedly, that one is more of a matter of taste; maybe you personally find it easier to understand the termination condition of the explicit `while` loop.)
+
+And in a language like F#, we don't even lose any performance using this approach.
+A language with [tail-call optimisation](https://en.wikipedia.org/wiki/Tail_call) will optimise away the recursive call into an imperative loop behind the scenes; in fact, the recursive and the imperative code may well boil down to exactly the same machine code.
+(By contrast, some languages like Python lack tail-call optimisation, so in Python this approach is liable to end in a stack overflow.)
+
+## Backing off
+
+For another prime example, consider a famous saying:
+
+> If at first you don't succeed, try, try again.
+
+What is less often noted is that one should wait some period of time in between each retry.
+
+Of course, there are libraries which do this for you ([Polly], for example).
+But for the sake of argument, let's say we want to write a simple exponential backoff ourselves.
+(We'll mock out the `sleep` function, to allow us to test the logic more easily; it'll probably be given as `System.Threading.Thread.Sleep` at the call site, but nobody wants to be calling `Thread.Sleep` in a unit test.)
+
+```fsharp
+open System
+
+let doThingRetryingWithBackoff<'err>
+    (sleep : TimeSpan -> unit)
+    (maxRetries : int)
+    (backoff : TimeSpan)
+    (doThing : unit -> Result<unit, 'err>)
+    : Result<unit, 'err>
+    =
+    let mutable retries = 0
+    let mutable lastError = None
+    let mutable isDone = false
+    while (not isDone) && retries < maxRetries do
+        match doThing () with
+        | Ok () -> isDone <- true
+        | Error e ->
+            lastError <- Some e
+            retries <- retries + 1
+            if retries < maxRetries then
+                sleep (backoff * (2. ** float retries))
+
+    if (not isDone) && retries = maxRetries then
+        Error (Option.get lastError)
+    else Ok ()
+```
+
+Easy enough, but let's give it the recursive treatment:
+
+```fsharp
+open System
+
+let doThingRetryingWithBackoff<'err>
+    (sleep : TimeSpan -> unit)
+    (maxRetries : int)
+    (backoff : TimeSpan)
+    (doThing : unit -> Result<unit, 'err>)
+    : Result<unit, 'err>
+    =
+    let rec go (retriesPerformed : int) =
+        match doThing () with
+        | Ok () -> Ok ()
+        | Error e ->
+            let newRetries = retriesPerformed + 1
+            if newRetries >= maxRetries then
+                Error e
+            else
+                sleep (backoff * (2. ** float newRetries))
+                go newRetries
+
+    go 0
+```
+
+No need for the sad `isDone` sentinel here!
+Since looping was an active choice (an explicit call to `go`), we were free simply to choose not to loop at any point, and specifically at the point where `doThing` had returned successfully.
+
+## The barest essentials
+
+Sometimes we don't even need any state at all!
+
+```fsharp
+open System
+
+let pollInfinitely<'a>
+    (sleep : TimeSpan -> unit)
+    (poll : unit -> 'a)
+    (doThings : 'a -> unit)
+    : unit
+    =
+    while true do
+        let values = poll ()
+        doThings values
+        sleep (TimeSpan.FromSeconds 5.)
+```
+
+Or, in the shiny new world which is free of `while`:
+
+```fsharp
+open System
+
+let pollInfinitely<'a>
+    (sleep : TimeSpan -> unit)
+    (poll : unit -> 'a)
+    (doThings : 'a -> unit)
+    : unit
+    =
+    let rec go () =
+        let values = poll ()
+        doThings values
+        sleep (TimeSpan.FromSeconds 5.)
+        go ()
+
+    go ()
+```
+
+This example looks too simple to be of any use, and maybe indeed it is; and of course in real life there is `System.Timers.Timer` which solves this problem better.
+(This is only a toy example.)
+
+But I would argue that once you're familiar with the general pattern, this is a natural way to express the loop.
+Moreover, it is extremely extensible: if we decide we don't actually want to loop forever, it's trivial to add extra state by inserting it as parameters to `go`, and the compiler will try its best to help us use it correctly.
+(By contrast, with the `while` loop and mutable state, we are on our own: the compiler can't know what we had in mind for the state.
+It will freely let us either modify the state when we didn't want to, or fail to modify it when we did want to.)
+
+## Winding up
+
+Though it may look very odd at first sight, it soon becomes quite natural to think about loops in this recursive way.
+It allows you to express more of the problem in the type system, helping the compiler to point out when you've done things like "adding a new piece of state but forgetting to use it everywhere you wanted it".
+It also makes the act of looping into a conscious choice at each stage; if you've ever longed for the good old days of `goto` to break out of loops, this actually gives you back something of that flavour, but much more safe!
+
+Making the loop state immutable makes it easier to reason about the invariants and behaviour of the loop, and helps you write safer programs fast.
+
+[Polly]: https://github.com/App-vNext/Polly
diff --git a/hugo/content/posts/2020-08-02-uncountability.md b/hugo/content/posts/2020-08-02-uncountability.md
new file mode 100644
index 0000000..6c7f8ca
--- /dev/null
+++ b/hugo/content/posts/2020-08-02-uncountability.md
@@ -0,0 +1,25 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- hacker-news
+- mathematical_summary
+comments: true
+date: "2020-08-02T00:00:00Z"
+title: The uncountability of the reals (a note from Hacker News)
+summary: "A quick note from Hacker News about a beautiful proof of the uncountability of the reals."
+---
+
+[My barely-related response](https://news.ycombinator.com/item?id=24030441) to a [linkpost](https://news.ycombinator.com/item?id=24029791) on Hacker News to [a paper by Chaitin](https://arxiv.org/abs/math/0411418).
+
+The link is now https://web.archive.org/web/20120915113514/http://people.math.gatech.edu/~mbaker/pdf/mmrealgame.pdf .
+
+> I know it's completely unrelated to the article, but I would like to take a moment to plug an absolutely beautiful proof of the uncountability of [0,1], which I first saw in `http://people.math.gatech.edu/~mbaker/pdf/realgame.pdf` .
+
+> Briefly: Alice and Bob play a game. Alice starts at 0, Bob starts at 1, and they alternate taking turns (starting with Alice), each picking a number between Alice and Bob's current numbers. (So start with A:0, B:1, then A:0.5, B:0.75, A:0.6, … is an example of the start of a valid sequence of plays.) We fix a subset S of [0,1] in advance, and Alice will win if at the end of all time the sequence of numbers she has picked converges to a number in S; Bob wins otherwise. (Alice's sequence does converge: it's increasing and bounded above by 1.)
+>
+> It's obvious that if S = [0,1] then Alice wins no matter what strategy either of them uses: a convergent sequence drawn from [0,1] must converge to something in [0,1].
+>
+> Also, if S = (s1, s2, …) is countable then Bob has a winning strategy: at move n, pick s_n if possible, and otherwise make any legal move. (Think for a couple of minutes to see why this is true: if Bob couldn't pick s_n at time n, then either Alice has already picked a number bigger, in which case she can't ever get back down near s_n again, or Bob has already picked a number b which is smaller, in which case she is blocked off from reaching s_n because she can't get past b.)
+>
+> So if [0,1] is countable then Alice must win no matter what either of them does, but Bob has a winning strategy; contradiction.
diff --git a/hugo/content/posts/2020-10-10-null-and-option.md b/hugo/content/posts/2020-10-10-null-and-option.md
new file mode 100644
index 0000000..df41b94
--- /dev/null
+++ b/hugo/content/posts/2020-10-10-null-and-option.md
@@ -0,0 +1,23 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- hacker-news
+- programming
+comments: true
+date: "2020-10-10T00:00:00Z"
+title: Nulls and options (a note from Hacker News)
+summary: "A quick note from Hacker News about why we want optional types but why 'null' is unintuitive."
+---
+
+A [question by user gitgud](https://news.ycombinator.com/item?id=24737961) on Hacker News:
+
+> > "4. Objects must always be initialized to a valid state. Not doing so is a compile-time error."
+>
+> So this essentially means there can never be a NULL object assigned to anything... how is this possible? is it avoided by throwing exceptions instead of returning NULL?
+
+[My response](https://news.ycombinator.com/item?id=24738020):
+
+> If you forget everything you ever learned about programming, and then I ask you for an integer, you're never in a million years going to reply "null"; the existence of "null" is something you had to learn. A language could simply not have a concept of "null". Even an OO language could have no concept of "null".
+> In practice, it's useful to be able to answer "I'm sorry, Dave, I can't do that". Some languages do that by silently and implicitly allowing this special value "null" in a return type. Some languages allow you to throw an exception (again without ever having declared that this is something you might do). But some languages take one of those approaches and make it explicit: in the two cases, you have the "option"/"maybe" type, or checked exceptions. Either of these explicit methods let a function declare that it may fail, and moreover the language forces you to handle the possibility of failure in any function which may fail.
+> The "reasonability" comes from the fact that the language forces you to be explicit about the possibility of failure, rather than implicitly allowing all functions to fail.
diff --git a/hugo/content/posts/2020-10-18-gravity.md b/hugo/content/posts/2020-10-18-gravity.md
new file mode 100644
index 0000000..e82f9f7
--- /dev/null
+++ b/hugo/content/posts/2020-10-18-gravity.md
@@ -0,0 +1,27 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- hacker-news
+comments: true
+date: "2020-10-18T00:00:00Z"
+title: General relativity (a note from Hacker News)
+summary: "A quick note from Hacker News about a neat fact from general relativity."
+---
+
+[My barely-related response](https://news.ycombinator.com/item?id=24821415) to a [linkpost](https://news.ycombinator.com/item?id=24821141) on Hacker News to [Tim Hutton's website](https://timhutton.github.io/GravityIsNotAForce/).
+
+> I heard an interesting question at one point: "how come, when you throw a ball up on Earth, the parabola is so strongly curved? Spacetime is nearly flat, so how can a straight line become such a steep parabola?"
+>
+> I'll answer this question as I understand it, but I only took four lectures of General Relativity before I gave it up in favour of computability and logic, so if there is a more intuitive and/or less wrong answer out there, please correct me.
+>
+> Intuitive answer: the curve is indeed very gentle, and (e.g.) light will be deflected only very slightly by the curvature; but the ball is moving for a couple of seconds, and that's an eternity. On human scales, the time dimension is much "bigger" than the space dimensions (we're quite big in the time dimension and quite small in the spatial dimensions); the ball moves only a small distance through space but a very large distance through time, amounting to a big distance in spacetime, and so the slight curvature has a bigger effect than you might expect.
+
+Below, [user vbezhenar](https://news.ycombinator.com/user?id=vbezhenar) [asked](https://news.ycombinator.com/item?id=24821492):
+
+> Are we moving through time with constant speed? Or we're constantly accelerating through time?
+
+[My response](https://news.ycombinator.com/item?id=24821520):
+
+> Under special relativity, everyone and everything moves at a constant speed `c` through spacetime. If you feel like you're not moving, it's because all your speed is being put towards travelling faster through time. Conversely, if you manage to move very fast through space, the world around you will appear to speed up, because you've had to trade off some of your forward travel through time so as to travel in space; the rest of the world is moving forward in time faster than you are.
+> So you can change your acceleration through the time dimension of spacetime, by dint of changing your acceleration in the spatial ones.
diff --git a/hugo/content/posts/2020-10-23-anki-learning.md b/hugo/content/posts/2020-10-23-anki-learning.md
new file mode 100644
index 0000000..8d55719
--- /dev/null
+++ b/hugo/content/posts/2020-10-23-anki-learning.md
@@ -0,0 +1,217 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- g-research
+- hacker-news
+comments: true
+date: "2020-10-23T00:00:00Z"
+math: true
+aliases:
+- /anki-learning-superpower-computer-science/
+title: Anki as Learning Superpower
+summary: "Some techniques to help you get more out of Anki."
+---
+
+# Anki as Learning Superpower: computer science edition
+
+*Cross-posted at the [G-Research official blog](https://www.gresearch.co.uk/article/anki-as-learning-superpower-computer-science-edition/); comments at [Hacker News](https://news.ycombinator.com/item?id=24878171).*
+
+TL;DR: If you aren't using [Anki] or one of its siblings to remember what you learn, you are missing out on literal superpowers.
+There are specific techniques which I've found helpful for using it to learn maths and computer science in particular.
+
+This post will discuss extremely briefly what Anki is, but is mainly aimed at helping the student of maths or computer science to squeeze out some extra droplets of productivity from it.
+If you aren't already familiar with Anki, go and read Michael Nielsen's post [Augmenting Long-Term Memory] as well or instead of this one; if you're short of time, the main value of the post to someone new to Anki is in the first one-eighth or so, up to but not including "Using Anki to thoroughly read a research paper in an unfamiliar field".
+That essay gives a broad-strokes motivation for why one might want to use Anki at all. (Spoiler: it gives you a superhuman memory!)
+
+## What is Anki?
+
+[Anki] is a [spaced repetition system](https://en.wikipedia.org/wiki/Spaced_repetition): essentially a program that shows you flashcards at intervals tailored to help you remember their contents.
+Such systems are based on the observation that while you can't stop memories from decaying, repeated exposure makes the decay exponentially slower.
+So while you need to rehearse a fact frequently when you first come to learn it, reviews can become very sparse very quickly without losing much of your hard-won memory.
+
+Anki is only one such system; other implementations such as SuperMemo and Mnemosyne exist.
+There are also applications using the same principles for more specific goals; for example, Memrise is designed for language learning.
+This post is phrased in terms of Anki, because for historical reasons that's the one I use.
+
+# Acknowledgements and further reading
+
+I've placed this section at the start, because I believe that spaced repetition is genuinely a superpower that anyone can acquire.
+If I can't hold your interest in this post, I encourage you to glance over these other resources in the hope that one of them does the trick!
+
+* [Augmenting Long-Term Memory] - a brief introduction to various memory systems with a particular focus on Anki. The money quote: "Anki makes memory a choice". If you choose to remember something, you can.
+
+* [Quantum Country](https://quantum.country) - an introduction to quantum computation with integrated spaced-repetition flashcards, which comes with emailed reminders to use the site at intervals.
+Even if you have no interest in quantum computation, I recommend skimming at least the first bits of the site, so that you can see what really good flashcards based on a text can look like. (Note that some of its cards are not ideal; the site does sometimes fall into the "yes/no" pattern, with flashcards whose answer is a bald "yes" or "no". Cards of this form aren't as good as cards which get you to reproduce actually content-based parts of a fact.)
+
+* [The Twenty Rules of Formulating Knowledge in Learning](https://www.supermemo.com/en/archives1990-2015/articles/20rules) - the creator of the spaced-repetition system SuperMemo discusses how to formulate knowledge so that it sticks.
+Again, even if you don't use spaced repetition, definitely read this; the advice is absolutely foundational to my understanding of how to learn.
+A nontrivial amount of this very blog post is essentially specialising some of these rules to the case of maths and computer science.
+
+* [Reddit: Medical School Anki](https://www.reddit.com/r/medicalschoolanki/), a subreddit with almost 55,000 subscribers at the time of writing. This link is here just to demonstrate that there are fields (such as medicine) in which Anki is very popular and nowhere near as niche as it is in most areas.
+
+# General guidance: break things up, then break them up more
+
+Absolutely crucial to the effective use of Anki for even a moderate collection of cards is that answering a card should be a single mental motion.
+
+What does this mean?
+We'll demonstrate with some positive and negative examples.
+
+* "What is the capital of France?" is probably answered with a single mental motion: it's just "Paris", obtained by a simple lookup in your memory.
+* "What is 367 times 399?" is many mental motions unless you are really unusually good at mental arithmetic. Almost everyone has to carry out some algorithm to get an answer to this question.
+* "What is the composition of air?" is probably many mental motions: for example, a lookup of "which gases are in air?" and a number of motions like "what proportion of air is this specific gas?". Or it could be the various mental motions of "Which gas is the nth common in air, and what is its proportion in air?" as n varies.
+* "What is the most common gas in air, and what is its proportion?" is very possibly a single mental motion despite the multiple components in its answer. For me, the knowledge that "nitrogen is the most common gas in air" is intimately bound up with "78% of air is nitrogen"; to answer one question is to access the same mental lump of storage as is required to answer the other question. Of course, for someone else, this may be two distinct facts and two distinct memory lookups.
+
+A card that is answered with one mental motion may be multiple motions for someone else; and a question answered with many mental motions may become one motion after you learn some additional fact or part of the model that ties the facts together.
+So it's helpful if you personally keep track of what you're doing to answer a given card, so that you can recognise some of this structure in your own mind.
+
+Any card which is multiple mental motions should be broken down into smaller chunks.
+
+If you're doing too much work to answer a card, it's important to notice, so that you can break the card down.
+And don't worry that you might "break down a card too much"; if the cards are easy, Anki will exponentially decrease how often it shows them to you.
+The cost of creating a card that's too easy is almost entirely in the time it takes you to create the card in the first place;
+a card that's too difficult will sap your time and energy again and again.
+
+# Specific guidance for maths: conditions on statements
+
+As an example, recall the [ratio test for convergence], likely to be covered in any first course in analysis:
+
+> Let \\(\sum_{n=1}^{\infty} a_n\\) be a complex series, with each \\(a_i\\) nonzero.
+> Suppose that there is \\(a\\) such that \\(\vert \frac{a_{n+1}}{a_n}\vert \to a\\).
+> Then, if \\(a < 1\\) then the series converges absolutely; if \\(a > 1\\) then the series diverges.
+
+(We'll discuss its proof in a later section.)
+
+This statement has a number of moving parts, and is definitely much too large to appear as a single card.
+(Remember the rule of thumb: answering a single card should be a single mental motion.)
+How could we formulate this statement in an Anki-friendly way?
+
+My own personal Anki collection contains the following cards.
+(Anki does allow LaTeX, though I often optimise for speed of card creation and just use plain text, however ugly it ends up.
+As long as it's easily legible, it doesn't need to be pretty.)
+
+* What conditions exist on the ratio test? (answer: nonzero, sequence of ratios has a limit)
+* When does the ratio test let us conclude something? (answer: successive ratios converge to something not equal to 1)
+* Which cases exist on the ratio test? (answer: ratios converge to less than, or greater than, 1)
+* What can we say from the ratio test if the ratios converge to something less than 1? (answer: converges absolutely)
+* What can we say from the ratio test if the ratios converge to something greater than 1? (answer: diverges)
+* What can we say from the ratio test if the ratios converge to 1? (answer: nothing)
+* Give an example where the ratio test is inconclusive. (answer: [harmonic series])
+
+All these cards are very simple; perhaps they might look laughably simple.
+This is fine! Remember that Anki will rapidly stop showing you cards you already know.
+Crucially each card is really a single mental motion to answer.
+
+Note also that information is duplicated between cards.
+For example, there are two cards saying in slightly different ways that the ratio test talks about when successive ratios converge to something not equal to 1.
+Again, it's fine to repeat information, because cards are cheap.
+Here, the information is duplicated to better facilitate the links between the cards: one card is a general "when can we conclude anything", and one card is the same information rephrased as "which cases do I need to consider to produce more information here".
+Without the additional rephrasing, there's more of a risk of developing "isolated" clusters of cards; the aim is that given just the prompt "ratio test", I can reproduce the *entire* cluster without thinking any further.
+
+And don't forget to store key examples and counterexamples!
+
+# Specific guidance for maths/computer science: conditions as part of definitions
+
+Remembering a definition is rather like remembering the statement of a theorem.
+Without further ado, here is my definition of a [red-black tree]:
+
+* What, imprecisely, are the colouring rules on a red-black tree? ("global; local; base-case")
+* What are the leaves of a red-black tree? ("null, black")
+* What is the base-case colouring rule of a red-black tree? ("leaves are black")
+* What is the local colouring rule of a red-black tree? ("red node => black children")
+* What is the global colouring rule of a red-black tree? ("black depth is well defined")
+* What is the black depth of a node in a red-black tree? ("the number of black nodes encountered on a path from that node down to a leaf")
+
+Again, there is some duplication (two cards note that the leaves of a red-black tree are coloured black), and all the cards are extremely simple.
+The point is to make sure both that the actual *content* is remembered, but also that the content is interlinked enough that you can reproduce the entire definition from the single prompt "red-black tree".
+Moreover, one card is explicitly structural and carries no actual content as such: the card that lists the three types of colouring rule.
+This card is there solely to allow me to remember which *cards* are relevant.
+
+Naively, one might create instead a single card, "what is a red-black tree?".
+But in my experience, this is too big.
+If you're reviewing even 30 cards a day, and they're all as big as "what is a red-black tree?", you may very well get bored or start spending too long on reviews.
+I personally review about 85 cards per day, all of them small; whenever I have a couple of minutes spare, I'll do a few cards.
+If many of your cards are big, reviewing becomes an awful chore and it becomes very hard to make yourself put in even the small amount of time that's necessary to stay on top of it.
+
+# Specific guidance for maths: remember waypoints, not proofs
+
+It *is* possible to rote-learn text using Anki.
+For example, I've learnt the first four chapters of the [Tao Te Ching] as 90 cards, each having the same structure: the front face is line 43 of the text, say, and the back face is line 44.
+However, this is a task for which human memory is quite poorly suited: almost everyone learns best through association, creating large networks of interrelated facts that together form a coherent world-model, each fact boosting the memory of the others.
+
+> To learn anything at all, it is better first to understand it, however vaguely.
+> Don't jump straight to memorisation without first having the skeleton of a model on which to hang the facts you're memorising.
+
+For much, much more on the nature of understanding proofs and how to use Anki, see the excellent blog post [Seeing Through a Piece of Mathematics](http://cognitivemedium.com/srs-mathematics) by Michael Nielsen.
+Nielsen walks through a simple problem and discusses the possible ways one might understand it, and how one might use Anki to encode one's understanding.
+
+So let's say you've understood a proof, and you now want to use Anki to immortalise the proof in your memory.
+It may feel quite natural to try and rote-learn the proof; but this is a big waste of energy, since unstructured text does not play to the mind's strengths.
+Instead, construct a map through the proof, placing waypoints as you go, then learn the waypoints.
+Remember also that you can use cards which encode *structure* rather than *content*, if necessary, as discussed in the "what is a red-black tree?" example.
+
+## Simple example: the ratio test
+
+We defined the ratio test above; let's see how we might formulate its proof in an Anki-friendly way.
+
+Here is one proof of the ratio test, copied almost verbatim from my university lecture notes.
+It's not important to understand this fully, or even in any detail at all; I'm using it only as an example, for conversion into Anki cards.
+
+> There are two cases: the sequence \\(\vert \frac{a_{n+1}}{a_n}\vert \\) tends to \\(a < 1\\) as \\(n \to \infty\\), or to \\(a > 1\\).
+>
+> Suppose first that \\(a > 1\\). Then there is some \\(M\\) such that beyond \\(a_M\\), all \\(a_{n+1}\\) are greater in absolute value than their \\(a_n\\). Therefore the sequence cannot tend to \\(0\\), so the series cannot converge.
+>
+> Suppose therefore that \\(a < 1\\). Pick any \\(r\\) between \\(a\\) and \\(1\\); then there is some \\(M\\) such that beyond \\(a_M\\), all \\(\vert \frac{a_{n+1}}{a_n}\vert  < r\\). So beyond \\(a_M\\), all \\(\vert a_n\vert \\) are less than \\(r^{n-M} \vert a_M\vert \\); but \\(\sum_{n=M}^{\infty} \vert a_M\vert  r^{n-M}\\) is a geometric series which converges, and so by the comparison test, \\(\sum_{n=M}^{\infty} \vert a_n\vert \\) converges. Since initial terms do not affect convergence, \\(\sum_{n=1}^{\infty} a_n\\) must converge absolutely. This completes the proof.
+
+For me, the cards are as follows:
+
+* What is the first step of the proof of the ratio test? (split into \\(a < 1\\) and \\(a > 1\\))
+* What is the key idea in the proof of the ratio test, \\(a > 1\\)? (terms don't tend to 0)
+* What is the key idea in the proof of the ratio test, \\(a < 1\\)? (pick \\(a < r < 1\\), comparison test after two adjustments)
+* What is the first adjustment we make in the proof of the ratio test, \\(a < 1\\)? (ignore initial terms so all ratios < r)
+* What is the second adjustment we make in the proof of the ratio test, \\(a < 1\\)? (divide out first term so all terms < r^n)
+* What do we compare against in the proof of the ratio test, \\(a < 1\\)? (\\(r^n\\))
+
+Using this, I can construct the following proof, which is in a little less detail than the original but which I could easily flesh out if required.
+
+> Let the absolute value of the limit of \\(\frac{a_{n+1}}{a_n}\\) be \\(a\\). We split into two cases: \\(a < 1\\) and \\(a > 1\\).
+>
+> If \\(a > 1\\), the series is a summation of terms which do not tend to \\(0\\) (indeed, beyond a certain point the terms become strictly increasing in modulus), so it cannot converge.
+>
+> If \\(a < 1\\), select \\(r\\) between \\(a\\) and \\(1\\). Then because initial terms don't affect convergence, we can omit initial terms so that without loss of generality all \\(\frac{a_{n+1}}{a_n}\\) are less than \\(r\\) in modulus. Then we may divide out by \\(a_1\\) so that without loss of generality every \\(\vert a_n\vert  \leq r^n\\). Finally, the comparison test gives the required result.
+
+Perhaps there are better ways to formulate these cards; it's never ideal to have to remember rather arbitrary lists such as "what is the first adjustment" and "what is the second adjustment".
+But the "first and second adjustments" cards are safe for me to remember, because I remember them together spatially:
+the first is a feeling of motion, translating the series to the left, and the second is a feeling of shrinking/squashing, dividing out the first term.
+Possibly different people will find different and more personally effective ways to factor out the various components.
+
+Notice that these cards encode hints rather than formal proof.
+The aim is to set up the right structure, rather than to rehearse a proof verbatim.
+Even if you're studying for an exam, you still should trust yourself to be able to do things without having explicitly memorised them!
+
+# General guidance: prioritise ruthlessly, learn to recognise bad cards
+
+The usual rule of thumb is that you spend 50% of your time on the hardest 5% of your flashcards.
+A spaced repetition system will show you easy cards at exponentially decreasing frequency, so you'll rapidly stop reviewing your easiest cards.
+But some cards just stubbornly refuse to be learned, so Anki will show them to you again and again in the hope that you'll get it this time (until eventually it gives up, marking the card automatically as a "leech", and stops ever showing it to you).
+
+There will be cards that behave this way; that's just life.
+Do keep an eye out for them; try and learn to recognise the mental state that is "urrgh, not this card again".
+When you find a card that is showing up repeatedly, or that it's consistently an unpleasant effort to answer, or which you keep getting wrong:
+
+1. Ditch the card! Do you really need to learn this fact anyway? There are so many other things out there to learn; do you have a good reason to spend your time on this one? Or might you be able to derive the contents of this card if necessary without even trying to memorise it?
+2. Split up the card. Is it too big? Can you reduce the mental effort required to learn the card?
+3. Make better mnemonics. Do you need to construct a different mental scene describing the card, or some clever phrase that sticks in your mind, or set something to music?
+4. Revisit the basics. Are you getting this card wrong because your mental model for the entire subject is not sufficiently well-founded?
+
+# Conclusion
+
+I've left it to other people to persuade you to *use* Anki, but this was a whip around a few things to keep in mind so that you can get a bit more out of your Anki usage, especially when it comes to learning more complex technical concepts where it's not so obvious how to create the cards.
+The art of Anki is easiest to learn by example; hopefully this has provided a couple more demonstrations to help fill out the arsenal.
+
+[harmonic series]: https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)
+[red-black tree]: https://en.wikipedia.org/wiki/Red-black_tree
+[Tao Te Ching]: https://en.wikipedia.org/wiki/Tao_Te_Ching
+[ratio test for convergence]: https://en.wikipedia.org/wiki/Ratio_test
+[Anki]: https://apps.ankiweb.net
+[Augmenting Long-Term Memory]: http://augmentingcognition.com/ltm.html
diff --git a/hugo/content/posts/2020-11-21-christmas-dinner.md b/hugo/content/posts/2020-11-21-christmas-dinner.md
new file mode 100644
index 0000000..cf34fa3
--- /dev/null
+++ b/hugo/content/posts/2020-11-21-christmas-dinner.md
@@ -0,0 +1,113 @@
+---
+lastmod: "2020-11-21T13:39:08.0000000+00:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2020-11-21T00:00:00Z"
+title: Christmas dinner notes
+summary: "My notes on the production of a Christmas dinner."
+---
+
+For the past several years, I have hosted a Christmas dinner in November (according to Cambridge tradition).
+These are my notes.
+
+Page references are to Rose Elliot's New Complete Vegetarian, and items are listed roughly in order of when you can make them.
+
+Changelog:
+
+* In 2019 I started tracking whether I had bought the right amount of food for 11 people.
+
+# Executive summary
+Turkey, nut roast, onion gravy, Dublin vegetables, roast potatoes, sage & onion stuffing, cranberry sauce, bread sauce, sweetcorn, red cabbage, pigs in blankets.
+Followed by Christmas pudding, brandy butter, and bought custard.
+
+# Cranberry sauce
+Page 65.
+225g cranberries, 175g caster sugar.
+Make entirely in advance.
+
+2019: bought 300g cranberries, made about twice as much as was eaten on the day.
+
+# Bread sauce
+Page 65.
+275ml milk, 3 cloves, 2 large slices of bread, 15g butter, 1 onion, 1 bay leaf, nutmeg.
+Make entirely in advance.
+
+2019: Made double the recipe and this was about right for 11 people.
+
+# Red cabbage
+Page 133.
+1kg cabbage, and whatever else you want to throw in.
+Prepare the day before and just slow-cook for hours on the day.
+
+2019: used 1kg cabbage, it was nearly all eaten - this was the right quantity.
+
+# Roast potatoes
+Page 131.
+2.5kg Maris Piper potatoes easily serves 12 people, ~6tbsp plain flour, olive oil.
+Parboil the previous night and store in water.
+Roast for about an hour after they're parboiled.
+
+## Alternative if you don't have space: potatoes Dauphinoise
+For two years I did not have enough cooking space to accommodate roast potatoes, so made potatoes Dauphinoise or boulangère.
+Page 129.
+Remember to layer salt and thyme in with the onions. Slow-cook, and add in loads of double cream an hour before serving.
+
+2019: bought 1kg potatoes and 600ml double cream, this was about right.
+
+## Nut roast
+Page 171.
+The following ingredients are for double the recipe, and serves 12.
+130g butter, olive oil, 2 large onions, 450g cashews, 400g breadcrumbs, 2 lemons' worth of juice, 8tbsp chopped flat-leaf parsley.
+Prepared entirely in advance, baked on the day. Requires a full hour of baking, but tin can sit on top of turkey tray so will fit.
+
+2019: bought 60g parsley and made about 3x the recipe; this was nearly double what was required for 11 people.
+
+## Sage and onion stuffing
+Make in advance, and cook in oven for any time between 30min and an hour.
+Put in as soon as turkey comes out to sit.
+
+Add half the packet of sage to the onion and butter mixture, with pepper, and mix.
+Then mix with breadcrumbs.
+Add salt gradually, in case you decide it doesn't need so much after all, and more sage as you go, if needed.
+Too much sage is chemical; not enough is disappointing.
+Taste as you go.
+
+2019: made with 2.5x800g loaves of bread, which was more than twice as much as required.
+
+## Creamed style sweetcorn
+This one is inherited from my grandparents, who lived for a while in America and must have picked this up there.
+Pour out of tin, and microwave before serving.
+
+## Red onion gravy
+Page 66.
+2 onions, olive oil, 50g plain flour, 2 garlic cloves.
+Make the day before and microwave before serving.
+
+2019: made two saucepansful (the red Le Creuset ones); this was about twice as much as required.
+2018: ran out of this serving eleven people with double the quantity in the recipe.
+
+## Mashed carrots and parsnips (Dublin vegetables)
+Mashing is because I have never lived anywhere with room to roast them.
+
+1kg of each.
+At least 45min, probably an hour, roasting.
+Boil the day before until soft, then mash with butter and put in fridge.
+Microwave on the day.
+
+## Pigs in blankets
+Put in as soon as turkey comes out.
+
+2019: bought 16 chipolatas, ended up with three spare.
+
+## Turkey
+Start defrosting on early afternoon of day before, put in fridge to finish defrosting overnight, get out of fridge first thing in the morning.
+
+2019: bought 2x "medium serves 4 minimum 2.2kg", this was much, much more than required. We ate a little over 2kg.
+
+## Christmas pudding
+
+And brandy butter, custard.
+
+2019: bought about 900g of Christmas pudding, this was about right or a tiny bit under. Bought 200g brandy butter, 1l custard; this was twice as much custard and brandy butter as required.
diff --git a/hugo/content/posts/2020-12-26-christmas-quiz.md b/hugo/content/posts/2020-12-26-christmas-quiz.md
new file mode 100644
index 0000000..c674428
--- /dev/null
+++ b/hugo/content/posts/2020-12-26-christmas-quiz.md
@@ -0,0 +1,25 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2020-12-26T00:00:00Z"
+title: Christmas quiz
+summary: "A round of questions I wrote for a Christmas quiz."
+---
+
+# General knowledge
+
+Answers are given in [ROT13](https://rot13.com/).
+
+* What comes fourth? Nougat and caramel; soil; goddess of love; (yvdhvq zrgny; guvf vf gur cynargf tbvat vajneqf)
+* What are the flavours of a Neapolitan ice cream? (inavyyn fgenjoreel pubpbyngr)
+* There are plenty of very early people like Kayumarth of Aryans, or Adam, who might or might not have existed. The earliest specific person who definitely existed and whose name we know is from where? His name is Narmer. (ur jnf na Rtlcgvna cunenbu)
+* The first Poirot book is “The Mysterious Affair at Styles”. How did its murder take place? (cbvfbavat jvgu fgelpuavar)
+* Half a point for each Tate Museum you can name (oevgnva yvirecbby zbqrea fg virf)
+* Which is the only letter worth five points in Scrabble? (x nf va xvyb)
+* When Fay Wray was told she would act alongside the “tallest, darkest leading man in Hollywood”, she thought this meant Cary Grant. Who was it? (xvat xbat)
+* What is the most sold flavour of Walker’s crisps? (purrfr naq bavba)
+* Daft Punk “Technologic” starts with the first two lines “Buy it, use it, break it, fix it, trash it, change it, mail, upgrade it”. In the next six lines, name any two of the other things you do to it. (“Punetr vg, cbvag vg, mbbz vg, cerff vg, fanc vg, jbex vg, dhvpx, renfr vg, jevgr vg, phg vg, cnfgr vg, fnir vg, ybnq vg, purpx vg, dhvpx, erjevgr vg, cyht vg, cynl vg, ohea vg, evc vg, qent naq qebc vg, mvc, hamvc vg”)
+* Who played Jim Hacker in “Yes, Minister”? (Cnhy Rqqvatgba)
diff --git a/hugo/content/posts/2020-12-28-a-bug-in-git.md b/hugo/content/posts/2020-12-28-a-bug-in-git.md
new file mode 100644
index 0000000..7dd3deb
--- /dev/null
+++ b/hugo/content/posts/2020-12-28-a-bug-in-git.md
@@ -0,0 +1,85 @@
+---
+lastmod: "2021-02-07T08:50:46.0000000+00:00"
+author: patrick
+categories:
+- programming
+comments: true
+date: "2020-12-28T00:00:00Z"
+title: A bug in Git
+summary: "A bug I found and reported in Git."
+---
+
+*This is a copy of [a mail](https://lore.kernel.org/git/31599b45-cf4e-be77-22bb-8fa03f0a52d6@patrickstevens.co.uk/) I sent to the git mailing list.*
+
+Below the fold is a `git bugreport`-generated report of a bug with `git apply --cached --reject`, which I have reproduced in three different environments.
+Summary: we do not correctly stage the removal of a file if there is also an unrelated change that cannot be applied.
+
+I don't think this behaviour is intended; in the report I give a couple of variations which correctly do what I expected, and this one breaks the semantics I expect, given the behaviour of those variations.
+I have not tried to find the source of the bug.
+
+Thanks,
+
+Patrick Stevens
+
+---
+
+Thank you for filling out a Git bug report!
+Please answer the following questions to help us understand your issue.
+
+What did you do before the bug happened? (Steps to reproduce your issue)
+
+Complete reproduction instructions:
+
+```bash
+mkdir badrepo && cd badrepo && git init
+echo "start" >> file.txt && git add file.txt && git commit -m "Initial
+commit"
+git tag initialcommit
+echo "another" >> another.txt && git add another.txt && git commit -m
+"Something else"
+git tag commit2
+git rm another.txt && git rm file.txt && git commit -m "Remove both"
+git tag commit3
+git checkout initialcommit
+git diff commit2 commit3 | git apply --reject --cached
+```
+
+What did you expect to happen? (Expected behavior)
+
+We should be left detached at tag `initialcommit`, with the working copy clean (i.e. containing only `file.txt`), and with the deletion of `file.txt` staged.
+
+What happened instead? (Actual behavior)
+
+The final `git apply` correctly leaves us detached at tag `initialcommit`, with a clean working copy, but incorrectly the deletion of `file.txt` is not staged: the index is also clean.
+
+What's different between what you expected and what actually happened?
+
+The file `file.txt` should have had its deletion staged, because this is part of the diff which could apply cleanly.
+
+Anything else you want to add:
+
+If I delete the `--cached` from the last line, the right thing happens: the working copy has `file.txt` deleted in the working copy but the deletion is not staged.
+If instead I remove the `git rm another.txt`, then similarly the right thing happens: now we end up with the deletion of `file.txt` staged (but the working copy is correctly unchanged).
+
+I have reproduced this using git version 2.24.1.windows.2, as well as 2.29.2 built on Windows using the WSL and invoked through WSL, as well as from the Mac HomeBrew install where I produced this bug report.
+
+Please review the rest of the bug report below.
+You can delete any lines you don't wish to share.
+
+[System Info]
+git version:
+git version 2.29.2
+cpu: x86_64
+no commit associated with this build
+sizeof-long: 8
+sizeof-size_t: 8
+shell-path: /bin/sh
+uname: Darwin 18.7.0 Darwin Kernel Version 18.7.0: Mon Aug 31 20:53:32
+PDT 2020; root:xnu-4903.278.44~1/RELEASE_X86_64 x86_64
+compiler info: clang: 11.0.0 (clang-1100.0.33.17)
+libc info: no libc information available
+$SHELL (typically, interactive shell): /bin/zsh
+
+
+[Enabled Hooks]
+
diff --git a/hugo/content/posts/2021-01-18-find-the-bug.md b/hugo/content/posts/2021-01-18-find-the-bug.md
new file mode 100644
index 0000000..8d14824
--- /dev/null
+++ b/hugo/content/posts/2021-01-18-find-the-bug.md
@@ -0,0 +1,58 @@
+---
+lastmod: "2021-06-05T13:10:16.0000000+01:00"
+author: patrick
+categories:
+- programming
+comments: true
+date: "2021-01-18T00:00:00Z"
+title: Find the Bug, C# edition
+summary: "A cute little exercise in bug-spotting."
+---
+
+Thanks to [Joe Roberts](https://github.com/Joey9801/) for showing me the code which I've rendered below as an exercise in bug-spotting.
+
+The following C# defines a class with a private nullable int field `_data`, initialised to `null`, and a public int property `Data`, which assumes the value of `_data` if `_data` has a value, or sets `_data` to `3` and returns `3` otherwise.
+
+```csharp
+class Foo {
+  private int? _data = null;
+  public int Data => _data ??= 3;
+}
+
+Foo f;
+// more code here...
+Console.WriteLine(f.Data);
+```
+
+It is very likely that this code contains a bug. What is the probable bug?
+
+# Hint, which you will certainly need
+
+C# compiles down to Microsoft Intermediate Language ("IL") bytecode.
+The example above compiles to essentially the following:
+
+```
+int32 local0;
+local0 = this._data.GetValueOrDefault();
+if (this._data.HasValue)
+    goto label1;
+local0 = 3;
+this._data = new Nullable<int32>(local0);
+stack.push(local0);
+goto label2;
+
+label1:
+    stack.push(local0);
+label2:
+    return stack.pop();
+```
+
+# Hint 2
+In [ROT13](https://rot13.com/):
+
+Gurer'f n enpr pbaqvgvba - jung unccraf va n zhygv-guernqrq pbagrkg?
+
+# Answer
+In ROT13:
+
+Rkrphgvba ortvaf, naq gur ybpny inevnoyr vf nffvtarq gur qrsnhyg inyhr mreb. Gur guernq vf fhfcraqrq. N arj guernq nfxf sbe gur inyhr bs Qngn, naq pbeerpgyl trgf gur inyhr guerr. Rkrphgvba bs gur bevtvany guernq erfhzrf; qngn abj unf n inyhr, fb jr whzc gb gur svefg ynory, naq erghea gur qrsnhyg inyhr bs na vag, anzryl mreb.
diff --git a/hugo/content/posts/2021-01-21-evaluation-checklist.md b/hugo/content/posts/2021-01-21-evaluation-checklist.md
new file mode 100644
index 0000000..d76fe9b
--- /dev/null
+++ b/hugo/content/posts/2021-01-21-evaluation-checklist.md
@@ -0,0 +1,82 @@
+---
+lastmod: "2021-01-22T22:52:50.0000000+00:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2021-01-21T00:00:00Z"
+title: An incomplete life evaluation checklist
+summary: "A checklist I have used during my regular six-monthly life review."
+---
+
+I'm clearing up my desktop, and found this, a checklist I've used during my six-monthly life review.
+Note that I believe [Alex Vermeer](https://alexvermeer.com/8760hours/) has the state of the art on this topic; use his guide rather than this one.
+
+# Jobs
+
+## Introspect
+
+This section is not only to determine whether I'm happy at the moment, but is also a more generally useful tool to work out what my day-to-day preferences are for any job.
+
+* Do I enjoy my job?
+  * If not: what specifically is bad about it? What are the things that make me not want to go into work? Are there any good features of my job? Can any of the things that make me not want to go into work be mitigated in the short term? What lessons should I learn from disliking this job?
+  * If so: what features of the job are good? Are there any features which are bad - on the days when I don't want to go into work, why is that? Can those be mitigated? What lessons should I learn from liking this job?
+
+Specific ideas: what is management like? How are the people/atmosphere? Does it give me what I want from a work social life? Is it satisfying in a big-picture sort of way?
+
+## Evaluate
+
+* What are my goals in life? Have they changed from last time?
+* Is my job fulfilling those goals?
+* Are my aptitudes different to last time - what have I gained, what has atrophied? Should I update based on that?
+* What jobs would, in an ideal world, contribute to my goals the most?
+* Can I pursue those jobs?
+* Are there specific skills I should work at to prevent atrophy?
+* Are there any specific skills I could acquire (to open more doors or to go into some specific job)?
+
+## Consolidate
+
+* Update my master CV.
+* Review last time's criteria for success/failure.
+
+## Project
+
+* Make a plan for the next six months.
+  * What does a successful next-six-months look like? (Be specific and measurable if possible.)
+  * What does a failed next-six-months look like?
+* Schedule the next session for six month's time.
+
+# Free time
+
+* What do I spend my free time doing? Do I enjoy it? (Try and guess e.g. how much time is spent mindlessly poking the Internet.) Is it improving me? How much of my free time do I want to be spent improving myself?
+* What do I want to do with my free time?
+* Why is "what do I want to do" not the same as "what I do"? Is any of the slack avoidable? Set calendar reminders to act on things where appropriate.
+
+# Wellbeing
+
+* Are there any physical conditions I have been suffering from but ignoring?
+* Are there any mental conditions I have been suffering from but ignoring?
+* Are there any cheap interventions or experiments I can perform that might improve my wellbeing?
+
+# Financial
+
+## Collect
+
+* What are my assets, and where are they stored?
+* Are any of my assets volatile?
+* What did I spend my money on in the last six months?
+
+## Assess
+
+* Am I happy with the volatility of my assets? Do I have enough liquid emergency fund?
+* Did my assets return what I expected of them? If not, why not? (E.g. introductory bonus rates on bank accounts)
+* Are any of my assets in inefficient places? (E.g. interest rates which are beaten by other accounts)
+* Are there outgoings which are systematically too high (e.g. bills, coffees/takeaways)?
+
+## Plan
+
+* Where specifically should I move my assets?
+* Are my savings goals still the same?
+* Roughly how much do I expect to save over the coming six months?
+* Are there any supplier changes I should make e.g. energy providers?
+* Are there any lifestyle changes I should make e.g. not having coffees?
diff --git a/hugo/content/posts/2021-01-30-cyclic-dependencies.md b/hugo/content/posts/2021-01-30-cyclic-dependencies.md
new file mode 100644
index 0000000..68b36a7
--- /dev/null
+++ b/hugo/content/posts/2021-01-30-cyclic-dependencies.md
@@ -0,0 +1,28 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- programming
+- hacker-news
+comments: true
+date: "2021-01-30T00:00:00Z"
+title: Cyclic dependencies (a note from Hacker News)
+summary: "A quick note from Hacker News about how to model in F# something that might look like cyclic dependencies."
+---
+
+In response to [a comment by bob1029 on Hacker News](https://news.ycombinator.com/item?id=25970583), which ran as follows.
+
+> Language constraints aside, the real world is not something that can be cleanly modeled without the notion of circular dependencies between things. Not very many real, practical activities can be truly isolated from other closely-related activities and wrapped up in some leak-proof contract.
+>
+> Consider briefly the domain model of a bank. Customers rely on Accounts (I.e. have one or many). Accounts rely on Customers (i.e. have one or many). This is a simple kind of test you can apply to your language and architecture to see if you have what it takes to attack the problem domain. Most approaches lauded on HN would be a messy clusterfuck when attempting to deal with this. Now, if I can simply call CustomerService from AccountService and vice versa, there is no frustration anymore. This is the power of reflection. It certainly has other caveats, but there are advantages when it is used responsibly.
+>
+> If you want to understand why functional-only business applications are not taking the world by storm, this is the reason. If it weren't for a few "messy" concepts like reflection, we would never get anything done. Having 1 rigid graph of functions and a global ordering of how we have to compile these things... My co workers would laugh me off the conference call if I proposed these constraints be imposed upon us today.
+
+[I reply](https://news.ycombinator.com/item?id=25970628):
+
+> In F#, you would naturally approach this by defining a `Customer` independent of the `Account` (e.g. just containing a name and address), and an `Account` independent of the `Customer` (e.g. just containing an ID), and then a `Bank` which is a mapping of `Account` to `Set<Customer>`. What you see as a cyclic dependency, I see as a data type that you haven't reified.
+
+Other commenters note that indeed this is precisely how SQL would model the domain without cyclic dependencies.
+And [JackFr](https://news.ycombinator.com/user?id=JackFr) gave a [rather nice soundbite](https://news.ycombinator.com/item?id=25970879):
+
+> Not really rocket science but some times your model is telling you something if you’re willing to listen.
diff --git a/hugo/content/posts/2021-02-03-senior-engineer-syllabus.md b/hugo/content/posts/2021-02-03-senior-engineer-syllabus.md
new file mode 100644
index 0000000..e2169a3
--- /dev/null
+++ b/hugo/content/posts/2021-02-03-senior-engineer-syllabus.md
@@ -0,0 +1,45 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- programming
+comments: true
+date: "2021-02-03T00:00:00Z"
+title: Software engineer syllabus
+summary: "An incomplete but growing list of things I would make mandatory reading if I were building a software engineering syllabus."
+---
+
+This is an ad-hoc list of posts or books that would form part of my software engineering syllabus if I were to make one.
+The list grows when I remember something I've read before or discover something new.
+
+# Source control
+
+* [Pro Git](https://git-scm.com/book/en/v2).
+
+# Testing
+
+* Property-based testing. Since many people start with Python, I'd start with the [Hypothesis docs](https://hypothesis.readthedocs.io/en/latest/).
+
+# Design
+
+* [Domain-driven design](https://fsharpforfunandprofit.com/series/designing-with-types.html).
+
+# People skills
+
+* Turn The Ship Around, and its sequel, Leadership Is Language, by David Marquet.
+
+# Documentation
+
+* Look at some examples of truly first-class documentation. https://reference.wolfram.com/language/ref/Graph.html is interactive documentation (if you have Mathematica) that is absolutely superlative. From another angle, the [Rust book](https://doc.rust-lang.org/book/) is widely agreed to be excellent.
+* The nature of documentation: https://documentation.divio.com/
+
+# Things everyone always gets wrong
+
+Things are in this section to make sure that you recognise that they are *hard*.
+
+* [Unicode](https://www.joelonsoftware.com/2003/10/08/the-absolute-minimum-every-software-developer-absolutely-positively-must-know-about-unicode-and-character-sets-no-excuses/). The article is from 2003, but the only thing which has aged is that everyone agrees now that UTF8 is the correct way to go (it was pretty new at the time of the article).
+* [Time](https://infiniteundo.com/post/25326999628/falsehoods-programmers-believe-about-time)
+* [Names](https://www.kalzumeus.com/2010/06/17/falsehoods-programmers-believe-about-names/)
+* [Addresses](https://www.mjt.me.uk/posts/falsehoods-programmers-believe-about-addresses/)
+* [Cryptography](https://security.stackexchange.com/a/18198)
+* [Package managers](https://medium.com/@sdboyer/so-you-want-to-write-a-package-manager-4ae9c17d9527)
diff --git a/hugo/content/posts/2021-02-06-property-based-metatesting.md b/hugo/content/posts/2021-02-06-property-based-metatesting.md
new file mode 100644
index 0000000..e1a66a4
--- /dev/null
+++ b/hugo/content/posts/2021-02-06-property-based-metatesting.md
@@ -0,0 +1,267 @@
+---
+lastmod: "2021-06-01T19:45:48.0000000+01:00"
+author: patrick
+categories:
+- programming
+- g-research
+comments: true
+date: "2021-05-27T00:00:00Z"
+title: Metatesting your property-based tests
+summary: "A rather little-known but very easy and high-reward way to sanity-check your property-based tests."
+math: true
+---
+
+*Cross-posted at the [G-Research official blog](https://www.gresearch.co.uk/article/metatesting-your-property-based-tests/)*.
+
+Property-based testing is a great technique for getting the computer to do much of the drudgery when you're writing tests.
+But it's pretty easy to accidentally test much less than you thought.
+Here, we'll go over an extremely easy technique you can use to automatically test your own test coverage.
+
+This post is written in [F#], an impure functional-first language.
+F#'s impurity is actually what makes this technique so easy; in a pure language like Haskell, the same goal has to be achieved using more machinery.
+
+# What is property-based testing?
+
+"Property-based testing" refers to any method of testing which focuses around defining *properties that your programs obey* rather than *specific test cases your programs satisfy*.
+That is, the programmer writes down some kind of general desired fact about the program, and the computer creates test cases to try and disprove that fact.
+
+The first and most popular implementation of a property-based testing library is Haskell's [QuickCheck].
+QuickCheck has many ports into other languages, and the one we'll be using is [FsCheck] for F#.
+
+## Example
+
+Here's a little bare-bones implementation of an "interval set": a set of integers, but stored in a space-efficient way if there are many consecutive runs of integers in the set.
+
+The defining example is as follows.
+
+```fsharp
+type IntervalSet = private | IntervalSet of (int * int) list
+
+IntervalSet [ (1, 4) ; (8, 10) ]
+|> IntervalSet.toSet
+|> shouldEqual (Set.ofList [ 1 ; 2 ; 3 ; 4 ; 8 ; 9 ; 10 ])
+```
+
+(I've used a `list` here rather than a `Set` as the underlying data structure, just to make the code a bit shorter.)
+
+Let's say we want to test a small public API.
+
+```fsharp
+module IntervalSet =
+    val empty : IntervalSet
+    val add : int -> IntervalSet -> IntervalSet
+    val contains : int -> IntervalSet -> bool
+    val ofList : int list -> IntervalSet
+```
+
+Here's a small, trivial implementation - which does have a correctness bug, since this is a post about testing!
+To make the code smaller, it also has efficiency bugs: for example, it doesn't even try to join intervals together if they grow to overlap each other, and it's also possible for the same integer to appear twice in the interval-set.
+
+```fsharp
+type IntervalSet = private | IntervalSet of (int * int) list
+module IntervalSet =
+    let empty = IntervalSet []
+    let contains (a : int) (IntervalSet intervals) =
+        intervals
+        |> List.exists (fun (min, max) -> min <= a && a <= max)
+
+    let rec private add' (a : int) (intervals : (int * int) list) =
+        match intervals with
+        | [] -> [ (a, a) ]
+        | (min, max) :: rest ->
+            if min <= a && a <= max then
+                // No need to add, because it's already there
+                intervals
+            elif min - 1 = a then
+                // Augment this interval to contain the new number
+                (a, max) :: rest
+            elif max + 1 = a then
+                // Augment this interval to contain the new number
+                (min, a) :: rest
+            else
+                // Can't add it to this interval; recurse.
+                add' a rest
+
+    let add (a : int) (IntervalSet intervals) =
+        add' a intervals
+        |> IntervalSet
+
+    let ofList (numbers : int list) : IntervalSet =
+        numbers
+        |> List.fold
+            (fun set i -> add i set)
+            empty
+```
+
+Here are some tests, which all pass, so we commit the code and move on with our lives.
+
+```fsharp
+IntervalSet.ofList [ 3 ; 4 ]
+|> IntervalSet.contains 5
+|> shouldEqual false
+
+IntervalSet.ofList [ 3 ; 5 ]
+|> IntervalSet.contains 5
+|> shouldEqual true
+
+IntervalSet.ofList [ 3 ; 4 ; 5 ]
+|> IntervalSet.contains 5
+|> shouldEqual true
+```
+
+Then eventually someone comes along and points out that the following test fails!
+```fsharp
+IntervalSet.ofList [ 0 ; 2 ]
+|> IntervalSet.contains 0
+|> shouldEqual true
+```
+
+Property-based testing is an answer to this problem of incomplete test coverage.
+It allows us to write more comprehensive tests with less drudgery: we specify directly the properties we wish to hold, rather than listing specific examples of those properties.
+
+The obvious property we want here is that checking a list of integers for containment should be the same as testing the interval set for containment.
+
+```fsharp
+let property (ints : int list) (toCheck : int) : bool =
+    IntervalSet.ofList ints
+    |> IntervalSet.contains toCheck
+    |> (=) (List.contains toCheck ints)
+```
+
+And FsCheck can check this for us:
+
+```fsharp
+open FsCheck
+
+[<Test>]
+let ``Test by comparing to a list`` () =
+    Check.QuickThrowOnFailure property
+```
+
+After one run, I get this output:
+
+```
+Falsifiable, after 8 tests (4 shrinks) (StdGen (476991543,296861283)):
+Original:
+[-3; -1; 0; 0; 0; 0]
+-3
+Shrunk:
+[-3; 0]
+-3
+```
+
+FsCheck has found a failing case and has even found another one which is nice and small for us.
+
+## The bug
+
+By the way, the bug was in the recursion for `add'`:
+
+```fsharp
+let rec private add' (a : int) (intervals : (int * int) list) =
+    match intervals with
+    | [] -> [ (a, a) ]
+    | (min, max) :: rest ->
+        if min <= a && a <= max then
+            // No need to add, because it's already there
+            intervals
+        elif min - 1 = a then
+            // Augment this interval to contain the new number
+            (a, max) :: rest
+        elif max + 1 = a then
+            // Augment this interval to contain the new number
+            (min, a) :: rest
+        else
+            // Can't add it to this interval; recurse.
+            // *** This line has changed! ***
+            (min, max) :: add' a rest
+```
+
+# The silent difficulty with property-based testing
+
+How do we know we ever hit any interesting behaviour?
+In the example above, our property is of the form "do something, then check whether the result is equal to a `List.contains` call".
+But what if that `List.contains` never outputs `true`?
+Then we're just generating lots of uninteresting tests which all look like "is `59` in the set `{1, 2, 3}`?", and we're never actually testing the important cases!
+
+Happily, though, F# is impure, so we can actually *assert* whether this is happening extremely easily.
+
+```fsharp
+let property
+    (positiveCount : int ref) (negativeCount : int ref)
+    (ints : int list)
+    (toCheck : int)
+    : bool
+    =
+    let contains = List.contains toCheck ints
+    if contains then incr positives else incr negatives
+    IntervalSet.ofList ints
+    |> IntervalSet.contains toCheck
+    |> (=) contains
+```
+
+Now, the call site looks like this:
+
+```fsharp
+[<Test>]
+let ``Test with instrumentation`` () =
+    let pos = ref 0
+    let neg = ref 0
+    Check.QuickThrowOnFailure (property pos neg)
+
+    pos.Value |> shouldBeGreaterThan 0
+    neg.Value |> shouldBeGreaterThan 0
+
+    // And test the balance of cases, for good measure
+    (float pos.Value) / (float pos.Value + float neg.Value)
+    |> shouldBeGreaterThan 0.1
+```
+
+We're now *asserting at test time* that at least 10% of our cases are indeed the "list does contain" case.
+
+And in fact this test now becomes flaky!
+
+```
+Expected: 0.1
+Actual: 0.08
+
+at FsUnitTyped.TopLevelOperators.shouldBeGreaterThan
+```
+
+We have successfully turned our own uncertainty into a flaky test.
+
+# What to do about it
+
+We need to fix the flaky test.
+
+One way to do this is to manipulate the generators FsCheck is using to construct test cases.
+We can make a generator of `int list * int`, where the list contains the `int` about 50% of the time.
+(I won't go into how to do this now; the [FsCheck documentation on Test Data](https://fscheck.github.io/FsCheck/TestData.html) is pretty comprehensive.)
+Then when we use this new generator, we’re extremely likely to pass the “more than 10% of our generated test cases are of the positive case” check.
+Indeed, we’re taking many independently-sampled cases (by default, FsCheck generates 100 of them), and we’ve tweaked the distribution so that these samples are of the positive case with 50% probability, and to pass the check we only need 10% of them to be testing the positive case; so the probability of this test failing after the generator is set up correctly is actually around \\(1.5 \times 10^{-17}\\) (being \\(\mathbb{P}(X \le 10)\\) when \\(X\\) follows the binomial distribution on 100 trials with probability \\(\frac{1}{2}\\) of success).
+
+Depending on how many property-based tests you have, and how frequently they are run, you may want to make spurious failure less likely even than this; you can do so by skewing the generator even further towards positive cases, or by increasing the number of trials FsCheck runs.
+
+# How would you approach this in Haskell?
+
+Haskell is a pure language, so you can't just do the cheap answer we did here of mutating some external state from within the property.
+Instead, QuickCheck comes with the function [`checkCoverage`](https://hackage.haskell.org/package/QuickCheck-2.14.2/docs/Test-QuickCheck.html#v:checkCoverage), which you can set up with conditions you want to assert about test distributions.
+Note that `checkCoverage` is smarter than our cheap solution, though, because it does statistical tests to try and minimise the chance of spurious failures; above, I just did the calculation to discover the probability of spurious failure, but it's certainly not always so easy.
+
+# FsCheck's in-built support
+
+FsCheck does have some methods (documented as [FsCheck: Observing Test Case Distribution](https://fscheck.github.io/FsCheck/Properties.html#Observing-Test-Case-Distribution)) which let you find things out about test distributions.
+For example, `Prop.trivial` allows you to mark an outcome as being "trivial" or as falling into one or more categories, and you can then get FsCheck to tell you the distribution of test cases across the various categories.
+However, I honestly think it's easier and more readable to do it yourself using the cheap method from this post.
+
+# Conclusion
+
+When I implemented automated coverage checks this way in one of our internal projects, I discovered several parts of our API surface that just weren't tested at all by property-based tests.
+It was trivial to fix up the test generator so that those parts were hit, but if we hadn't put in this automated instrumentation, we would probably never have found out until our users started submitting bug reports about those under-tested parts of our API.
+
+If you're using property-based testing (which you should be!), you should pay attention to the distribution of test cases you're getting from the generator library.
+Check that the tests are actually as comprehensive as you hoped.
+It's extremely easy to do this in F#, and it takes no knowledge of the FsCheck library to do it; anyone who speaks only a little F# can come in and add this metatesting to an existing property-based test suite, even if they know nothing about how FsCheck works.
+
+[F#]: https://en.wikipedia.org/wiki/F_Sharp_(programming_language)
+[QuickCheck]: https://en.wikipedia.org/wiki/QuickCheck
+[FsCheck]: https://github.com/fscheck/FsCheck
diff --git a/hugo/content/posts/2021-02-20-in-praise-of-dry-run.md b/hugo/content/posts/2021-02-20-in-praise-of-dry-run.md
new file mode 100644
index 0000000..8169db1
--- /dev/null
+++ b/hugo/content/posts/2021-02-20-in-praise-of-dry-run.md
@@ -0,0 +1,197 @@
+---
+lastmod: "2021-05-24T20:49:46.0000000+01:00"
+author: patrick
+categories:
+- programming
+- g-research
+comments: true
+date: "2021-05-21T00:00:00Z"
+title: In praise of `--dry-run`
+summary: "First-class support for a --dry-run mode makes a tool more maintainable, more testable, and more user-friendly."
+---
+
+*Cross-posted at the [G-Research official blog](https://www.gresearch.co.uk/article/in-praise-of-dry-run); comments at [Hacker News](https://news.ycombinator.com/item?id=27263136).*
+
+One part of my job as a software engineer is writing tools.
+
+Something I always want to see in a tool which does anything non-trivial is a `--dry-run` mode.
+To be able to know what you're about to do, before you do it, is a great and wondrous thing, helpful to the novice and the experienced user alike.
+But it's sadly often the case that a tool's `--dry-run` is an unloved second-class citizen, prone to drifting out of date or failing to provide all the information you need.
+
+There's a simple method that keeps you honest, ensures that your `--dry-run` has all the information the user wants to see, and (an unexpected bonus!) helps make the entire tool more maintainable: use the `--dry-run` code path as an input to the "execute" code path.
+
+# The idea
+
+If you have a separate `--dry-run` code path which is to any extent independent of the main execution flow, then you've immediately opted into the huge problem that documentation also tends to suffer: there's now a way to get your explanations out of sync with what actually happens.
+
+So how do we solve this in general?
+The best answer the world has seen so far is that we need a machine to somehow check whether we've stayed in sync.
+
+## Synchronisation through testing?
+
+One way a machine can check things by testing.
+For example, the Wolfram Language [is documented](https://reference.wolfram.com/language/) almost entirely through interactive notebooks written in the Wolfram Language itself, and the documentation itself [forms a huge test suite](https://reference.wolfram.com/language/tutorial/TheInternalsOfTheWolframSystem.html#3925) that is evaluated as part of the release pipeline, ensuring that a large number of documentation problems actually turn automatically into *blockers* for release.
+
+## Synchronisation through types
+
+But in many languages (such as [F#], the language I use most and the language this post is written in), we have a simpler and more lightweight way to machine-check things: we have the type system.
+We can use the type system to keep our `--dry-run` and our "execute" flow in sync, by *artificially introducing an API boundary down the middle of our tool*: a boundary which the compiler can check.
+
+The `--dry-run` mode produces a strongly-typed output which is essentially a declaration of what actions we want to carry out.
+The "execute" mode consumes the `--dry-run` output as its own input.
+
+This way, we're forced to stay relatively honest.
+If the "execute" flow changes to require different information, you have to make sure it gets that information in the first place, and the easiest way you can do that is by making sure `--dry-run` outputs it.
+
+# An explicit example
+
+Let's take an incredibly simple tool: a command-line utility we'll call `delete`, which takes a collection of [globs](https://en.wikipedia.org/wiki/Glob_(programming)) and deletes every file matching one or more of those globs.
+Obviously this isn't something you'd write a tool for in practice, since `rm` exists, but it will demonstrate the point.
+
+In F#, one way you might do this could be:
+
+```fsharp
+type Glob = Glob of string
+module Glob =
+    let paths (Glob g) : FileInfo list = failwith "(implementation here)"
+
+let deleteFiles (paths : Glob list) : unit =
+    paths
+    |> List.iter (fun glob ->
+        Glob.paths glob
+        |> List.iter (fun fileInfo -> fileInfo.Delete ())
+    )
+```
+
+Now, there are a couple of ways you could make a `--dry-run` for this tool.
+One way (which is perfectly fine but is not what I'm trying to show here) would be to inject a dependency:
+
+```fsharp
+let deleteFiles<'ret> (toDo : FileInfo -> 'ret) (paths : Glob list) : 'ret list =
+    paths
+    |> List.collect (fun glob ->
+        Glob.paths glob
+        |> List.map toDo
+    )
+```
+
+Then you'd implement `--dry-run` by passing some variant on `println` as the `toDo` parameter, and you'd implement the real thing by passing `fun fileInfo -> fileInfo.Delete ()`.
+
+But another way to do it, demonstrating the API-boundary technique, is to split up `deleteFiles` into a `gather : Glob list -> FileInfo list`, and an `execute : FileInfo list -> unit`:
+
+```fsharp
+let gather (paths : Glob list) : FileInfo list =
+    paths
+    |> List.collect Glob.paths
+
+let execute (files : FileInfo list) : unit =
+    files
+    |> List.iter (fun fileInfo -> fileInfo.Delete ())
+```
+
+In this particular example, there's not much difference between the two styles.
+But as the tool gets bigger and does more things which are less trivial, it becomes more and more attractive to use the second style and bundle up the `execute` logic together.
+If nothing else, we want to avoid duplicating work, and the first (dependency-injection) style doesn't make it obvious how.
+
+Of course, there's a judgement you need to make about where to draw the boundary: what is "information retrieval" and what is "execution"?
+In this example, `execute` is just a sequence of `doSomething` applied to various parts of its input, but in more complex cases, it's up to you exactly what you consider to be "input" and what you consider merely to be "intermediate processing during execution".
+But the point is that there's *some* hard boundary drawn between the "find out what you're going to do" phase and the "do it" phase.
+
+## Finishing the example
+
+In real life, you're probably doing something a bit less trivial than just deleting a bunch of files, so you'll have a correspondingly more complex domain model.
+So naturally you'd introduce a type for the instructions which we previously had implicitly as a `FileInfo list`.
+
+```fsharp
+type Instructions =
+    {
+        PathsToDelete : FileInfo list
+    }
+
+module Instructions =
+    let toString (instructions : Instructions) : string = failwith "(implementation here)"
+```
+
+Then your main program is simply a matter of the following:
+
+```fsharp
+type DryRunMode = | Dry | Wet
+
+module ArgParse =
+    let extractGlobs (args : string[]) : Glob [] =
+        failwith "(implementation here)"
+    let dryRun (args : string[]) : DryRunMode =
+        failwith "(implementation here)"
+
+let main (args : string[]) : int =
+    let inputGlobs = ArgParse.parseInputGlobs args
+    let instructions = gather inputGlobs
+
+    match ArgParse.dryRun args with
+    | DryRunMode.Dry ->
+        printfn "Would execute the following:"
+        printfn "%s" (Instructions.toString instructions)
+        printfn "Halting due to --dry-run."
+        0
+    | DryRunMode.Wet ->
+        printfn "Executing the following:"
+        printfn "%s" (Instructions.toString instructions)
+        match execute instructions with
+        | Result.Error err ->
+            eprintfn "%O" err
+            1
+        | Result.Ok output ->
+            printfn "%O" output
+            0
+```
+
+(Of course, you probably want to be using some more resilient logging than merely `printfn`, you want to be using an actual argument parser rather than my hand-rolled `ArgParse` module, you probably want different return codes based on the different error cases, and so on.)
+
+# An additional benefit: librarification
+
+This "work out what you're going to do, and then do it" idiom has an extra architectural benefit: it really pushes you to make your tool's functionality into a library.
+If you've got a function which works out what to do, and a function which does it, then you've got two ready-made units of compilation to pull out into the library; additionally (in a statically-typed language) you've got some ready-made types (the `ProgramInstructions`) which serve as places from which to hang docstrings.
+
+In general, a tool which can be used as a library is much more helpful than a tool which can only be used from the command line, and I try to do this with every tool I write - sometimes unlocking unexpected use cases months down the line, when I look at a problem and think "Oh, I've solved that problem already in this other tool - let me just import it"!
+
+If I can import the "work out what to do" component of my tool, then even better, because now I'm not restricted to solving only the problem I originally wrote the tool to deal with.
+Instead, my tool's library can help me solve any problem for which it's useful to know what the tool knows.
+
+By the way, turning a tool into a library is really helpful whatever language you're using.
+For example, in Python, there are (often) no static types, but it's still great to be able to import the logic of a tool, even if you're not getting any additional static guarantees by making the `gather`/`execute` split.
+
+# Why the technique works especially well in F#
+
+F# is a strongly- and statically-typed language, so the technique works particularly well here.
+The API boundary down the middle of our tool is necessarily expressed in terms of some static type: the `--dry-run` mode is really a function like `gather : ProgramArguments -> ProgramInstructions`, and the "execute" mode is a function like `execute : ProgramInstructions -> Result<_, _>`.
+
+We have been forced to model our domain explicitly, by reifying the `ProgramInstructions` type which might otherwise have been implicit (if the tool were all knitted together as a single `doIt : ProgramArguments -> Result<_, _>`).
+
+This is very similar to the idea of "defunctionalisation", by the way, which [we've discussed before](https://www.gresearch.co.uk/article/defunctionalisation/) (see [the canonical post](http://www.pathsensitive.com/2019/07/the-best-refactoring-youve-never-heard.html) for a comprehensive introduction).
+We get the same benefits, and our code is:
+
+* more readable, because of the explicit domain modelling;
+* more easily testable, because we can now test the "information discovery" process independently of the "program execution" process;
+* more amenable to optimisation, since (for example) the "execute" phase might be able to realise it can pipeline together two of the things it's being asked to do. It's easier to make this kind of deduction when all your instructions are laid out explicitly.
+
+# Case study
+
+All three of these benefits played out recently for me personally.
+Some months before, I'd written an internal tool (now used multiple times a day by dozens of developers and quants, as well as at least one automated pipeline) using these principles, and it subsequently came to light that one of the tool's core dependencies actually couldn't do what it said on the tin safely.
+To work around this, I had to completely rewrite the part of the tool that used this dependency in that particular unsafe way.
+
+But because I'd used the API-boundary technique to structure the tool:
+
+* The tool's statement of intent (the `--dry-run`) was completely unchanged, so immediately the scope of the rewrite was cut by about a third: it was obvious from the type signatures that I wouldn't need to touch the `gather : ProgramArguments -> ProgramInstructions` phase of the program at all.
+* A whole raft of tests didn't need to change, because they were testing the `gather` function rather than the `execute` function. It was an unbelievable relief to know that I still had a really solid part of the system to rely on.
+* The rewrite was simpler, because of the additional constraints imposed by the fact that my rewritten `execute : ProgramInstructions -> Result<_, _>` could only use the `ProgramInstructions` it was given. I'd essentially already done the work months ago to shrink down the space of possibilities so that the shape of the rewrite was very nearly forced.
+* It was extremely easy to test the new `execute` functionality. In some cases, I didn't actually have to go through the effort of setting up all the preconditions for my test cases, because the pre-existing tests for `gather` still proved we could correctly create `ProgramInstructions`. Instead, sometimes, I could simply manually construct a `ProgramInstructions` that would test the `execute` paths, without having to set up the environment and go through the `gather` step first.
+* I was also able to optimise away some of the possible instructions by noticing when our core dependency gave us the tools to do some subset of them all in one operation. I may very well not have noticed this if the domain model wasn't sitting there in front of me.
+
+# Conclusion
+
+A `--dry-run` mode is really helpful for the new user who is playing around with your tool, as well as the experienced user who is trying to explore some edge case or who is unsure if their environment is right for the tool to do what they expect.
+You can help make your `--dry-run` truly meaningful by *depending* on it.
+This leads to a more modular architecture, with all its benefits.
+
+[F#]: https://en.wikipedia.org/wiki/F_Sharp_(programming_language)
diff --git a/hugo/content/posts/2021-03-17-rewriting-technical-interview.md b/hugo/content/posts/2021-03-17-rewriting-technical-interview.md
new file mode 100644
index 0000000..9227fc8
--- /dev/null
+++ b/hugo/content/posts/2021-03-17-rewriting-technical-interview.md
@@ -0,0 +1,392 @@
+---
+lastmod: "2021-03-27T14:34:02.0000000+00:00"
+author: patrick
+categories:
+- programming
+- g-research
+comments: true
+date: "2021-03-17T00:00:00Z"
+title: Rewriting the Technical Interview, in Mathematica
+summary: "An exploration into reaching into the internals of Mathematica to natively evaluate C code."
+---
+
+*Cross-posted at the [G-Research official blog](https://www.gresearch.co.uk/article/rewriting-the-technical-interview-in-mathematica/).*
+
+*Mandatory prerequisite reading for this post: Kyle Kingsbury's ["Technical Interview"](https://aphyr.com/tags/Interviews) series. Don't read further until you've read from at least "[Reversing the Technical Interview](https://aphyr.com/posts/340-reversing-the-technical-interview)" through to the fourth interview, "Rewriting the Technical Interview". I promise that it will be worth your time.*
+
+In Kyle Kingsbury's excellent [Rewriting the Technical Interview](https://aphyr.com/posts/353-rewriting-the-technical-interview), our protagonist solves [FizzBuzz](https://en.wikipedia.org/wiki/Fizz_buzz) in Clojure by copy-pasting a solution written in C, and then building a term-rewriting system in Clojure to evaluate the C code directly.
+
+The reason this approach was so worthy of inclusion in the "Technical Interview" series is because Clojure is manifestly unsuited to approaching the problem in this way.
+Much of the piece is devoted to building a term-rewriting system, even before getting to the meaty work of constructing the appropriate macros so that Clojure accepts the input C expression in a form which Clojure can manipulate.
+But an [interesting question on Hacker News](https://news.ycombinator.com/item?id=24489112) arose: could this be done easily in a language which *already* supports term rewriting?
+
+## A very quick intro to Mathematica
+
+[Wolfram Mathematica] is an implementation of the [Wolfram Language], a language which is rather unusually based on [M-expressions][M-expression] (as opposed to the standard [S-expressions][S-expression] of LISP).
+Usefully, it has excellent term-rewriting capabilities, and idiomatic Mathematica code often makes use of the function [`Replace`](https://reference.wolfram.com/language/ref/Replace.html) and its friends to traverse the form of an expression and apply replacement rules at various points.
+This machinery comes out-of-the-box and could be used to replace the first half of Rewriting the Technical Interview if the language of choice were the Wolfram Language rather than Clojure.
+
+However, a fundamental and unusual design feature of the Wolfram Language goes much further.
+In fact, functions are usually *defined* as replacement rules.
+To "define a function" is (more or less) to add one or more rules to a kind of global replacement table.
+Whenever Mathematica encounters an expression which has a form matching the structure of a rule in its global replacement table, it replaces that expression with the right-hand-side of that rule.
+This is *precisely* the functionality required to solve FizzBuzz using the insane method of the original post.
+
+### Brief example
+
+We can "define the function \\(f(x) = x^2\\)" by setting up a global replacement rule as follows:
+```mathematica
+f[x_] := Power[x, 2]
+```
+Whenever Mathematica encounters the symbol `f` applied to a single argument (here bound to the name `x`), it will replace the matched expression with `Power[x, 2]`.
+
+For example, the expression `g[f[x, 3], f[6], f[y]]` (where `g` has not yet had any replacement rules set up for it) is evaluated in several steps:
+
+* Mathematica walks the expression to discover all the sub-expressions. Most of them clearly have no replacement rules set up for them, so are left untouched (for example, `x`, `3`, `y`).
+* `f[6]` and `f[y]` match the structure of the left-hand-side of the rule `f[x_] := Power[x, 2]`, so they are evaluated, into `Power[6, 2]` and `Power[y, 2]` respectively.
+* `Power[6, 2]` matches a built-in rule and is evaluated to `36`. `Power[y, 2]` matches no built-in rules (and no user-defined rules) so is left unevaluated.
+* `f[x, 3]` does not match the structure of the left-hand-side of any known rule. It has head `f`, but there are two arguments (`x` and `3`), whereas the pattern `f[x_]` strictly matches only the head `f` applied to *one* argument. So `f[x, 3]` is left unevaluated.
+
+The final result is `g[f[x, 3], 36, Power[y, 2]]`.
+
+# Solving the problem
+
+We want to be able to accept the following expression as valid Wolfram Language code.
+
+```c
+for (i = 1 ; i < 101 ; i++) {
+    if (i % 15 == 0) {
+        println ("FizzBuzz");
+    } else if (i % 3 == 0) {
+        println ("Fizz");
+    } else if (i % 5 == 0) {
+        println ("Buzz");
+    } else {
+        println (i);
+    };
+}
+```
+
+This code is very much not valid, but we wish to set up Mathematica so that it recognises it as valid code (without having to put the code into a string or anything; just a direct paste into a notebook cell).
+
+To do this, we need to know at a high level what happens when we evaluate a cell in Mathematica's notebook interface.
+Mathematica's documentation, as ever, is exemplary for this purpose, and there are a number of tutorials (starting with [Low-Level Notebook Structure](http://reference.wolfram.com/language/guide/LowLevelNotebookStructure.html)) which point the reader towards the necessary symbols.
+
+In brief, then, Mathematica proceeds as follows.
+We will expand on all the magic here very soon.
+
+* Tokenise the input.
+* Perform some magic to group terms together in what is referred to as "[boxes](http://reference.wolfram.com/language/ref/MakeBoxes.html)".
+* Apply the head `MakeExpression` to the boxed terms, constructing a syntax tree ready for user-defined (and built-in) replacement rules to apply. `MakeExpression` has many built-in replacement rules in Mathematica, allowing it to parse all valid Wolfram Language code; a failure of any existing replacement rule to match indicates a syntax error. We will be solving the problem by augmenting these rules to expand the range of syntax Mathematica thinks is valid.
+
+## A simple example
+
+Suppose we have already entered the `f[x_] := Power[x, 2]` rule above into Mathematica, so that from now on, expressions of the form `f[x_]` will be replaced with `Power[x, 2]` during evaluation.
+
+How, then, does Mathematica process the input `g[f[x, 3], f[6], f[y]]` in a new notebook cell?
+
+First, it tokenises (basically [lexing]) and groups terms into "boxes".
+A box is a group of tokens which are part of the same sub-expression; the construction of boxes is an intermediate stage of Mathematica's parsing pipeline.
+
+We can observe one possible collection of boxes that represents the expression `g[f[x, 3], f[6], f[y]]` by using the function [`MakeBoxes`](http://reference.wolfram.com/language/ref/MakeBoxes.html).
+
+```mathematica
+> MakeBoxes[g[f[x, 3], f[6], f[y]]]
+
+[Out]: RowBox[{
+  "g",
+  "[",
+  RowBox[{RowBox[{"f", "[", RowBox[{"x", ",", "3"}], "]"}],
+  ",",
+  RowBox[{"f", "[", "6", "]"}],
+  ",",
+  RowBox[{"f", "[", "y", "]"}]}],
+  "]"
+}]
+```
+
+Notice how Mathematica has already grouped terms which belong together according to its internal grouping rules.
+For example, the three tokens `x` `,` `3` have been put together in their own box, because Mathematica saw them together inside square brackets.
+This is going to be very helpful for us, because Mathematica has already made sane decisions about what sub-expressions we need to parse; these decisions turn out to be fully compatible with those we will require for the subset of C we intend to "interpret".
+
+### Aside if you're confused about evaluation at this point
+You can, and should, ignore this section unless it occurred to you to wonder how `MakeBoxes` managed to see the expression `f[6]` rather than the evaluated `36`.
+
+Normally, `f[6]` inside an expression will get turned into `Power[6, 2]` and then into `36` before the parent expression even gets a chance to see the presence of `f`.
+However, `MakeBoxes` has a certain *attribute* placed upon it, which exempts its argument from evaluation.
+This is not magic, and you can do it to any head (i.e. any function) you define yourself; the attribute is called [`HoldAllComplete`](http://reference.wolfram.com/language/ref/HoldAllComplete.html).
+This means we really are seeing the result of `MakeBoxes` on the expression `g[f[x, 3], f[6], f[y]]` rather than on the evaluated `g[f[x, 3], 36, Power[y, 2]]`.
+Contrast this with how almost every other head behaves: they *will* evaluate their input before their own replacement rules apply.
+
+For example, the head `Print`:
+
+```mathematica
+> Print[g[f[x, 3], f[6], f[y]]]
+
+g[f[x,3],36,y^2]
+[Out]: Null
+```
+(The printed output contains the syntactic sugar `y^2` which is exactly the same as `Power[y, 2]`.)
+
+Or the head `List`:
+```mathematica
+> List[g[f[x, 3], f[6], f[y]]]
+
+[Out]: {f[x, 3], 36, Power[y, 2]}
+```
+
+### What does Mathematica do with the parsed tokens?
+
+Recall that we entered the text `g[f[x, 3], f[6], f[y]]` into a notebook cell, whereupon Mathematica lexed and then parsed the input into a pile of nested `RowBox`es.
+(We used the function `MakeBoxes` to discover one sequence of boxes that corresponds to this expression; other such sequences do exist, but it so happens that we have seen the exact same form that Mathematica itself creates when parsing the raw notebook input of this expression.)
+
+Now that we have our input in the form of nested boxes, Mathematica applies the function [`MakeExpression`](http://reference.wolfram.com/language/ref/MakeExpression.html) to the boxes.
+This is the function that parses the stream of grouped tokens, and it has many built-in rules allowing it to parse all valid Wolfram Language syntax.
+
+For example, the built-in rules assigned to `MakeExpression` take the following:
+
+```mathematica
+RowBox[{
+  "g",
+  "[",
+  RowBox[{RowBox[{"f", "[", RowBox[{"x", ",", "3"}], "]"}],
+  ",",
+  RowBox[{"f", "[", "6", "]"}],
+  ",",
+  RowBox[{"f", "[", "y", "]"}]}],
+  "]"
+}]
+```
+
+And they transform it into this:
+
+```mathematica
+g[f[x, 3], f[6], f[y]]
+```
+
+That is how Mathematica obtains its in-memory expression from the input we entered.
+
+### What does it do with the parsed expression?
+
+Now that Mathematica has parsed our text into an actual M-expression, it proceeds to apply evaluation rules.
+We've already seen how this works, and how it comes up with the final answer `g[f[x, 3], 36, Power[y, 2]]`.
+
+But if we want to match how Rewriting the Technical Interview worked, we need to do all our evaluation *before* this point, at parse time.
+So we'll have to wind back a bit and consider how we will get computation to happen *after* we've obtained the `RowBox` expression, during `MakeExpression`.
+
+# Reaching into the parser
+
+The simple and beautiful insight is that `MakeExpression` is a function *exactly* like any other.
+So, for example, we can supply our own replacement rules which will operate according to Mathematica's usual rule-precedence logic, which approximately boils down to "first apply the rules which match most precisely".
+
+To see how this logic works, consider the following two replacement rules associated with the head `h`, effectively defining a function called `h`.
+
+```mathematica
+h[x_] := x ^ 2
+h[3] := 0
+```
+
+(Recall the built-in infix syntactic sugar `x ^ 2` for `Power[x, 2]`.)
+
+If the argument is a symbol `y` Mathematica doesn't recognise, it will evaluate `h[y]` into `Power[y, 2]` by matching the first rule.
+But it will evaluate the verbatim expression `h[3]` into `0` rather than `9`, because the rule `h[3] := 0` is "more specific" than the other matching rule `h[x_] := x ^ 2`, so the "more specific" rule will be used preferentially wherever it matches.
+There is a precise notion of what it means for one rule to be "more specific" than another, but in practice it almost always just aligns with intuition and you don't have to worry about the details.
+
+The upshot is that if we create our own specific rules for `MakeExpression`, they will match with higher precedence than the built-in general ones, and so our own rules will be applied.
+The difficult part will be discovering which rules we need to add to `MakeExpression`.
+
+## Aside on the nature of LISP
+
+This solution method, "reach down into the parser to adjust how it works", mirrors a beautiful aspect of many LISPs, in which very little is hidden from you and everything is modifiable.
+The Wolfram Language is not strictly speaking a LISP (it's rather more like the macro system of a LISP, based around rewriting terms), but it shares many similarities in spirit.
+
+Much of the implementation of Mathematica is hidden from the user, because it's proprietary software, but like a LISP, the language is built around the idea of [homoiconicity] - that is, "code is data which you can manipulate just like any other data".
+In Kingsbury's original post (written in Clojure, a genuine LISP), much time was spent building macros which parse the C code into a Clojure data structure.
+Here, though, we will cut out one step: we won't reify the C code as a data structure, but instead will skip straight to evaluating it during parse.
+In adhering strictly to the principle of homoiconicity by building an explicit representation of the C code, Kingsbury's method is in some sense "pure" and it certainly allows more complex operations such as inspecting the code structure; but it's a lot less pragmatic when all you want to do is run some C, side-effects and all!
+
+## A simple example
+
+The Wolfram Language does not recognise the infix operator `%`, which is the modular-arithmetic operator in C.
+In fact, `%` has a totally different meaning in the Wolfram Language: it refers to the output of the most recently evaluated expression (which, in Mathematica, is the most recently evaluated notebook cell).
+Naturally, we have to change this if we wish to evaluate FizzBuzz in C at parse time; and this is one of the simplest places to start.
+
+We'll need a quick overview of the "blank pattern" syntax of the Wolfram Language before we start.
+A blank is simply one or more holes in an expression; when Mathematica tries to match the pattern, either it will succeed (and fill in the blank), or it will fail.
+It's just like a capturing group in a regex.
+
+* A blank whose name ends with exactly one underscore (`x_`) matches exactly one thing. `f[x_]` does not match `f[1, 2]` or `f[]`, but it does match `f[3]` or `f[y]`.
+* A blank whose name ends with exactly two underscores (`x__`) matches a sequence of at least one thing. `f[x__]` does not match `f[]`, but it matches any other expression whose head is `f`. Similarly, `f[x__, 3, y__]` matches any expression whose head is `f` with some arguments, then a `3`, then some more arguments. During a successful match, the name `x` gets locally assigned with the first arguments up to the `3`, and the name `y` gets locally assigned with the last arguments after the `3`.
+* A blank whose name ends with exactly three underscores (`x___`) matches any (possibly empty) sequence. `f[x___]` matches any expression whose head is `f`. `f[x___, 3, y___]` matches, for example, `f[3]`, `f[3, 5, 6]`, `f[1, 2, 3]`, and `f[1, 3, 5, 6]`. It matches `f[3, 3]` in several different ways (so you have to be quite careful when defining patterns with the more permissive blanks, to make sure you know which one you're going to get and what `x` and `y` will end up as!).
+
+There are more ways to specify patterns and blanks in more complex forms, but this is all we will need.
+
+Now we have the tools required to turn `%` into an infix operator:
+
+```mathematica
+MakeExpression[RowBox[{x__, "%", y__}], form_] :=
+  MakeExpression[
+    RowBox[{"Mod", "[", RowBox[{x, ",", y}], "]"}],
+    form
+  ]
+```
+
+(Ignore the `form` argument to `MakeExpression`. There are multiple different "box forms" corresponding to the different layouts of text that Mathematica will recognise, such as "traditional form" where Mathematica tries to parse ordinary mathematical text; here, for convenience, we're saying our logic will apply no matter which form is in use.)
+
+This creates a new replacement rule for `MakeExpression`.
+Whenever `MakeExpression` is called on any `RowBox` which contains some sequence of symbols (bound to the name `x`), then the single-character string `"%"`, then another sequence of symbols (bound to the name `y`), we swap out the expression it's parsing with one we know will parse into Mathematica's `Mod` function applied to the two arguments.
+
+Note that we've basically just rewritten the stream of tokens; there's no fancy logic taking place, and no real computation occurring at parse-time.
+
+## Printing
+
+The C function `println` will work in a very similar way, except this time we need to take care of some brackets ourselves, because the Wolfram Language uses square brackets whereas C uses round brackets for function application.
+
+```mathematica
+MakeExpression[RowBox[{"println", RowBox[{"(", xs___, ")"}]}], form_] :=
+  MakeExpression[RowBox[{"Print", "[", xs, "]"}], form]
+```
+
+## If-then-else
+
+We can handle if-then-else in a slightly different way.
+
+Aside: a lot of trial and error went into discovering exactly the form of the left-hand-side here!
+It's not obvious by eye how a string will be lexed; I found the lex by matching on very general structures like `RowBox[{"if", rest__}]` and judiciously using `Print` statements to discover the form of `rest`.
+Of course, if you get this wrong, then your code is a syntax error (because `MakeExpression`'s built-in rules don't match, because it's not valid Wolfram Language code).
+
+```mathematica
+MakeExpression[
+ RowBox[
+   {
+     "if",
+     RowBox[{"(", xs__, ")"}],
+     RowBox[{"{", ys___, "}"}],
+     "else",
+     zs___
+   }
+ ],
+ form_
+] :=
+ If[
+   ReleaseHold[MakeExpression[xs, form]],
+   MakeExpression[ys, form],
+   MakeExpression[RowBox[{zs}], form]
+ ]
+```
+
+This is somewhat different in structure to the previous examples, because now we're doing *actual computation* at parse time.
+
+The left-hand side is much as we have seen before: we match a certain lexical structure, binding subexpressions to the names `xs`, `ys`, and `zs`.
+However, the right-hand side is very different.
+
+Instead of replacing the expression with a statically-known form, we are dynamically chosing which form to recurse into.
+More concretely:
+
+* Use `MakeExpression` to parse the expression `xs`, resulting in a Mathematica expression like `Equal[x, 3]` (if, for example, `xs` referred to the token stream for the input `x == 3`).
+* Call `ReleaseHold` to cause Mathematica to *evaluate* the parsed expression. This results in a Mathematica boolean `True` or `False`.
+* Depending on the Mathematica boolean, proceed to parsing either the `ys` or the `zs`.
+
+Notice how we've performed some actual computation in Mathematica *during expression parse*, while deciding what to parse next.
+This is vital when `ys` and `zs` contain side-effects (as they are bound to do, since C is an imperative language); we have to avoid parsing the code we aren't going to execute, because we'll be performing side-effects at parse-time.
+
+## For-loop
+
+Now that we've seen the main idea from `if`/`then`/`else`, we can do the same for `for`.
+
+```mathematica
+MakeExpression[
+  RowBox[{
+    "for",
+    RowBox[{"(", RowBox[{init__, ";", cond__, ";", incr__}], ")"}],
+    RowBox[{"{", expr__, "}"}]
+  }],
+  form_
+] :=
+  Hold[
+    For[
+      ReleaseHold[MakeExpression[init, form]],
+      ReleaseHold[MakeExpression[cond, form]],
+      ReleaseHold[MakeExpression[incr, form]],
+      ReleaseHold[MakeExpression[expr, form]]
+    ]
+  ]
+```
+
+This is pretty much the same as in the `if` case: simply use a lot of trial and error to discover how Mathematica lexes the input stream in this particular case of the C source, then extract the desired parts of the token stream, and translate them into Mathematica's own `For` construct (which has basically the same form as C's, because the Wolfram Language's designers wanted to make it a bit easier for people to migrate to Mathematica from C).
+
+An oddity: note here that we have wrapped the entire thing in `Hold`, to *prevent* immediate evaluation; `Hold` is one way you can do quoting of expressions in Mathematica.
+Without this `Hold`, you'll find a syntax error ("more input is needed") when you evaluate the C code cell.
+
+In all our previous examples, we've ultimately been descending into another `MakeExpression` call.
+This for-loop is the first time we have not done so (though we could have done it that way with `printf` earlier).
+In fact, `MakeExpression` is always assumed to return a Wolfram Language expression held unevaluated, so that the result of parsing is an *unevaluated* expression.
+Since we've always been descending into `MakeExpression` calls, we've never broken that property.
+If we omit the `Hold` here, then `For` is just a normal Mathematica `For`, so it returns nothing; and in particular it fails to return an unevaluated expression.
+That breaks the contract `MakeExpression` implicitly has with the notebook interface, so whenever we come to use our new syntax `for (init; cond; incr) { expr }`, the parser returns nothing at all, and the notebook interface interprets this as a syntax error.
+
+The `if` case was different, of course, because in both its branches, it always returns a `MakeExpression` output, which is already an unevaluated expression.
+
+# The entire program
+
+In two cells:
+
+```mathematica
+MakeExpression[RowBox[{x__, "%", y__}], form_] := MakeExpression[RowBox[{"Mod", "[", RowBox[{x, ",", y}], "]"}], form]
+
+MakeExpression[RowBox[{"println", RowBox[{"(", xs___, ")"}]}], form_] := MakeExpression[RowBox[{"Print", "[", xs, "]"}], form]
+
+MakeExpression[RowBox[{"if", RowBox[{"(", xs__, ")"}], RowBox[{"{", ys___, "}"}], "else", zs___}], form_] :=
+ If[ReleaseHold[MakeExpression[xs, form]], MakeExpression[ys, form], MakeExpression[RowBox[{zs}], form]]
+
+MakeExpression[RowBox[{"for", RowBox[{"(", RowBox[{init__, ";", cond__, ";", incr__}], ")"}], RowBox[{"{", expr__, "}"}]}], form_] :=
+ Hold[For[ReleaseHold@MakeExpression[init, form],
+  ReleaseHold@MakeExpression[cond, form],
+  ReleaseHold@MakeExpression[incr, form],
+  ReleaseHold@MakeExpression[expr, form]]]
+```
+
+```mathematica
+for (i = 1 ; i < 101 ; i++) {
+    if (i % 15 == 0) {
+        println ("FizzBuzz");
+    } else if (i % 3 == 0) {
+        println ("Fizz");
+    } else if (i % 5 == 0) {
+        println ("Buzz");
+    } else {
+        println (i);
+    };
+}
+```
+
+Naturally, Mathematica's syntax highlighting and pretty-printing will go *absolutely nuts* on the second cell; but evaluating the cell will cause the appropriate FizzBuzz output to appear (and nothing else to appear!), and does not even throw a syntax error.
+
+# Remark on the Clojure original
+
+The original Clojure program from Kingsbury's post, by the way, cheats with its semicolons.
+The semicolon used to delimit the `for` initialisation/condition/increment, and the semicolon used to denote the end of the `if`/`else` chain, are both actually the very similar-looking character Unicode U+037E "GREEK QUESTION MARK".
+(The article gently calls this out, referring to the construction as "not exactly evil".)
+
+This makes it easier for the hand-rolled Clojure term rewriter to distinguish terms which belong to different sub-expressions, but in fact it makes the Mathematica version *much* harder.
+The syntax of C and its operators is sufficiently close to that of the Wolfram Language that we can simply rely on the built-in lexer to give us the right tokens and the built-in parser to group terms for us.
+Mathematica parses sufficiently correctly for our FizzBuzz purposes.
+(Notice, for example, that we got the `==` token for free with no further work: it is the same token, with the same infix-ness, both in C and in the Wolfram Language.)
+But using a character that is not a true semicolon *really* confuses the parser, and we end up with a stream of tokens grouped wildly differently into boxes, making it very hard to untangle into a useful expression.
+
+# Possible extensions
+
+The evaluator I've outlined here is hyper-focused on solving one specific problem.
+It's not truly parsing the input according to anything remotely resembling a C standard.
+It would be possible to do it correctly, though, if we spent a lot more effort in crafting more specific and more general rules!
+
+For example, Kingsbury (the original author) was worried about being able to distinguish between `f(a, b)` and `f((a, b))`.
+The program I've outlined here makes no attempt to do this - it is not necessary if we only want to solve FizzBuzz - but it is possible, because parentheses appear in the lexed stream.
+We don't lose any information in the lexing or parsing up to the point of calling `MakeExpression`, so in principle we could perform arbitrary analysis and transformation on the C code.
+
+[lexing]: https://en.wikipedia.org/wiki/Lexical_analysis
+[M-expression]: https://en.wikipedia.org/wiki/M-expression
+[S-expression]: https://en.wikipedia.org/wiki/S-expression
+[Wolfram Mathematica]: https://en.wikipedia.org/wiki/Wolfram_Mathematica
+[Wolfram Language]: https://en.wikipedia.org/wiki/Wolfram_Language
+[homoiconicity]: https://en.wikipedia.org/wiki/Homoiconicity
diff --git a/hugo/content/posts/2021-03-22-continuation-passing-style.md b/hugo/content/posts/2021-03-22-continuation-passing-style.md
new file mode 100644
index 0000000..0732818
--- /dev/null
+++ b/hugo/content/posts/2021-03-22-continuation-passing-style.md
@@ -0,0 +1,571 @@
+---
+lastmod: "2021-03-22T18:21:49.0000000+00:00"
+author: patrick
+categories:
+- programming
+- g-research
+comments: true
+date: "2021-03-22T00:00:00Z"
+title: Continuation-passing style
+summary: "Motivating the technique of continuation-passing style, by looking at recursive functions."
+---
+
+*Cross-posted at the [G-Research official blog](https://www.gresearch.co.uk/article/continuation-passing-style/).*
+
+In a previous post, [Chris Arnott examined a few different techniques to do recursion][Chris] in F#.
+Here, we will expand on a particular one of those techniques (continuation-passing style, or "CPS") from a slightly different angle.
+CPS is one of those topics which you can only really learn by soaking in various different explanations and by giving it a go until it clicks.
+
+In brief, we'll motivate continuation-passing style by thinking about how to deal with stack overflows in recursive functions.
+
+# A quick tail-recursion refresher
+
+I've previously [discussed] how to write loops using immutable state.
+Here is a pair of equivalent functions, presented without comment to jog your memory.
+
+```fsharp
+let tryFindMax (xs : 'a list) : 'a option =
+    let rec go (biggest : 'a) (xs : 'a list) : 'a =
+        match xs with
+        | [] -> biggest
+        | x :: xs -> go (max biggest x) xs
+
+    match xs with
+    | [] -> None
+    | x :: xs -> Some (go x xs)
+```
+
+```fsharp
+let tryFindMax' (xs : 'a list) : 'a option =
+    match xs with
+    | [] -> None
+    | x :: xs ->
+        let mutable biggest = x
+        let mutable xs = xs
+        while not (List.isEmpty xs) do
+            biggest <- max biggest (List.head xs)
+            xs <- List.tail xs
+        Some biggest
+```
+
+We won't discuss this further, except to note three things:
+
+* we've avoided using `List.fold`, purely for pedagogical reasons (because `fold` sidesteps all the issues we'll discuss in this post, being essentially equivalent to the "accumulator" recursion technique Chris discusses);
+* a sufficiently smart compiler can mechanically transform the former into the latter;
+* the latter phrasing obviously does not grow the stack, and so a sufficiently smart compiler can transparently rewrite so that the former does not grow the stack either.
+
+(The F# compiler is indeed sufficiently smart, although in some cases you have to compile explicitly with optimisations enabled before it will perform this transformation.)
+
+# The motivation for continuation-passing style
+
+We start with a recursive function, expressed in F#.
+Here, we'll generate all the permutations of a list.
+For example, there are six permutations of `[1;2;3]`; two such permutations are `[1;2;3]` and `[2;1;3]`.
+
+The algorithm will be a very naive recursion: the permutations of `[1;2;3]` are precisely the permutations of `[2;3]` but where we have inserted the number `1` into every possible position.
+Concretely:
+
+* `[2;3]` yields `[1;2;3]`, `[2;1;3]`, `[2;3;1]` by inserting `1` into each possible slot;
+* `[3;2]` yields `[1;3;2]`, `[3;1;2]`, `[3;2;1]` similarly.
+
+This gives us the required six permutations.
+
+Firstly we'll consider the "insert `1` into every possible position" helper function.
+
+```fsharp
+let rec insertions (y : 'a) (xs : 'a list) : 'a list list =
+    match xs with
+    | [] ->
+        // There is only one place to insert `y` into the empty list.
+        List.singleton [ y ]
+    | x :: xs ->
+        // Either we insert `y` at the beginning, or we don't.
+        (y :: x :: xs)
+        ::
+        (insertions y xs |> List.map (fun subResult -> x :: subResult))
+```
+
+This is all very well, but it will stack overflow if `xs` is too big.
+Make sure you understand why: it's because the program needs to remember the entire context in which it's constructing the answer, for the entire time in which it's constructing the answer.
+
+This is ultimately because the function is not tail-recursive, so the compiler is unable to optimise it into a simple loop.
+
+You may want to meditate on how we might solve the problem in principle.
+The stack is running out of space; how can we fix this?
+
+Some answers are as follows.
+
+## Make the stack bigger?
+
+This works, but does not scale indefinitely; you need to know up front what your recursion limit is going to be.
+We won't consider this solution further.
+
+## Use less space?
+
+The following phrasing, where we use a seq instead of a list, could allow a sufficiently smart compiler to throw away almost all the context at every stage.
+But in fact the F# compiler is not sufficiently smart, even in Release mode with optimisations turned on (and I'm not aware of a language with a compiler that will do this in general).
+This still overflows the stack.
+
+```fsharp
+let rec insertions (y : 'a) (xs : 'a list) : 'a list seq =
+    match xs with
+    | [] ->
+        // There is only one place to insert `y` into the empty list
+        Seq.singleton [y]
+    | x :: xs ->
+        // Either we insert `y` at the beginning, or we don't.
+        seq {
+            yield y :: x :: xs
+            yield! (insertions y xs |> Seq.map (fun subResult -> x :: subResult))
+        }
+```
+
+## Use the same amount of space, but put it on the heap
+
+*This* is the magic thought that leads us to a way of doing arbitrary recursion without stack overflow.
+
+One possibility would be to construct our own stack object on the heap, and manipulate it as if we were an interpreter.
+This technique works, but we won't pursue it further.
+It implies a lot of manual bookkeeping, and (all else being equal) it would be nice to make use of the existing F# compiler to do the heavy lifting for us.
+
+Instead, after sitting with a cup of hot chocolate by the fire for a couple of hours and thinking very hard, we could use the fact that F# is a language with good functional support to capture the intent of what we're trying to do.
+The stack holds information about what to do after a recursive call has terminated; but in a functional language, how do we usually express the notion of "what to do now"?
+Answer: that's simply a function call.
+
+So how can we express "what to do after the recursion has finished" as a function call, using the heap instead of the stack?
+The answer is to *pass in the entirety of the rest of the program as a function*, so that we can call it after our recursive computation has finished.
+In more technical terms, we *pass in a continuation*, and we have written a function *in continuation-passing style*.
+
+```fsharp
+let insertions
+    (y : 'a) (xs : 'a list)
+    (andThen : 'a list list -> 'ret)
+    : 'ret
+    =
+    failwith "implement me!"
+```
+
+An example call site is:
+
+```fsharp
+// Old world, without continuation-passing style
+let y = insertions 4 [ 1 ; 2 ; 3 ]
+printfn "%+A" y
+
+// Rephrased in the new world
+insertions 4 [ 1 ; 2 ; 3 ] (fun y -> printfn "%+A" y)
+// or, succinctly:
+insertions 4 [ 1 ; 2 ; 3 ] (printfn "%+A")
+```
+
+By the way, in real life, it's rather inconvenient to use this phrasing, so we'll often define a little helper function that moves us back into "normal" style:
+
+```fsharp
+let insertions' i xs = insertions i xs (fun ret -> ret) // or just `id`
+
+let y = insertions' 4 [1 ; 2 ; 3]
+printfn "%+A" y
+```
+
+You can think of this helper function as saying "rather than pushing the rest of the program into `insertions`, I will define a little sub-program that simply returns the computed answer, and use the result of that instead".
+
+Now all we have to do is implement the function `insertions` so that it doesn't use any further stack space!
+
+# Implementing the continuation-passing recursion
+
+The easiest way to avoid using any stack space is to make sure that all our recursions are tail-recursions.
+Let's start coding, and see what goes wrong and where we need to fix it.
+
+```fsharp
+let rec insertions
+    (y : 'a) (xs : 'a list)
+    (andThen : 'a list list -> 'ret)
+    : 'ret
+    =
+    match xs with
+    | [] ->
+        // This one's easy; we already know how to compute the value
+        // with no recursion at all, so just call `andThen` (i.e.
+        // continue with the rest of the program).
+        andThen (List.singleton [ y ])
+    | x :: xs ->
+        // This one's not so clear.
+        failwith "implement me!"
+```
+
+There's no way we can possibly create a value of type `'ret` except by calling `andThen` or `insertions`.
+But to help us decide which to use, remember that the whole point of this method is to use only tail recursion.
+We're building up the knowledge of where to go not on the stack, but as explicit closures (which will each be named `andThen` at some point during our repeated recursive calls to `insertions`).
+
+Just to show what goes wrong if we decide not to make our recursive call straight away, let's try making a call to `andThen` here.
+
+```fsharp
+let rec insertions
+    (y : 'a) (xs : 'a list)
+    (andThen : 'a list list -> 'ret)
+    : 'ret
+    =
+    match xs with
+    | [] ->
+        andThen (List.singleton [ y ])
+    | x :: xs ->
+        failwith<'a list list> "what goes here?"
+        |> andThen
+```
+
+We need to make an `'a list list`.
+But there's no way to get hold of one that has a hope of containing what we want!
+All we have access to is an `'a` and an `'a list`, as well as a way of making `'ret`s (by recursive calls to `insertions`).
+
+Because we started out by making a call to `andThen`, we immediately cut off all our avenues for meaningfully recursing.
+
+So instead of calling `andThen`, we have to backtrack: we call `insertions` instead.
+Note that this is a tail call, so the compiler will optimise it away into a loop, using no additional stack space.
+Using the types to guide us, we obtain the following:
+
+```fsharp
+let rec insertions
+    (y : 'a) (xs : 'a list)
+    (andThen : 'a list list -> 'ret)
+    : 'ret
+    =
+    match xs with
+    | [] ->
+        andThen (List.singleton [ y ])
+    | x :: xs ->
+        fun (result : 'a list list) ->
+            failwith<'ret> "TODO"
+        |> insertions (failwith<'a> "TODO") (failwith<'a list> "TODO")
+```
+
+Now, we can dispense with a couple of these TODOs straight away.
+The recursive call to `insertions` is to insert `y` into the remaining `xs`; there's only one possible way the last line could look if we want to preserve the intent of the recursive call.
+
+```fsharp
+let rec insertions
+    (y : 'a) (xs : 'a list)
+    (andThen : 'a list list -> 'ret)
+    : 'ret
+    =
+    match xs with
+    | [] ->
+        andThen (List.singleton [ y ])
+    | x :: xs ->
+        fun (result : 'a list list) ->
+            failwith<'ret> "TODO"
+        |> insertions y xs
+```
+
+So there's just one `failwith` left to fill, where we need to come up with a `'ret`.
+In hand, we have `result`, the result of the recursive call.
+(Remember, we're constructing a lambda which answers the question "what will we do after the recursive call has terminated?".)
+
+We probably don't want to make any *further* recursive calls after we've made one recursive call - certainly in the original naive phrasing we started with, we didn't have to recurse twice - so there's only one possible way left that we could get a `'ret`.
+
+```fsharp
+let rec insertions
+    (y : 'a) (xs : 'a list)
+    (andThen : 'a list list -> 'ret)
+    : 'ret
+    =
+    match xs with
+    | [] ->
+        andThen (List.singleton [ y ])
+    | x :: xs ->
+        fun (result : 'a list list) ->
+            failwith<'a list list> "TODO"
+            |> andThen
+        |> insertions y xs
+```
+
+Now the body of our lambda says "when you've got a result from your recursive call, do some processing which I've `failwith`ed out, and then go on to do the rest of the program with it".
+Going back to our naive phrasing from the very beginning, we used the following to construct a final answer:
+
+```fsharp
+(y :: x :: xs)
+::
+(insertions y xs |> List.map (fun subResult -> x :: subResult))
+```
+
+But now we've already *got* the result of the recursive call, because we're in a context where we're continuing with that answer (we're constructing a closure which tells us how to continue).
+In the context of this closure, that result is labelled with the name `result`.
+So the following snippet is what we want:
+
+```fsharp
+(y :: x :: xs)
+::
+(result |> List.map (fun subResult -> x :: subResult))
+```
+
+That is, the final version of the function is as follows:
+
+```fsharp
+let rec insertions
+    (y : 'a) (xs : 'a list)
+    (andThen : 'a list list -> 'ret)
+    : 'ret
+    =
+    match xs with
+    | [] ->
+        andThen (List.singleton [ y ])
+    | x :: xs ->
+        fun (result : 'a list list) ->
+            (
+                (y :: x :: xs)
+                ::
+                (result |> List.map (fun subResult -> x :: subResult))
+            )
+            |> andThen
+        |> insertions y xs
+```
+
+Sure, it might use a lot of heap space, and it's miles slower than a version written with an explicit accumulator or with mutable state - but it won't overflow the stack, and the method we used to generate this function was actually very mechanical!
+
+As a final note, you can recover a non-stack-overflowing function with the original type signature as follows:
+
+```fsharp
+let insertions' (y : 'a) (xs : 'a list) : 'a list list =
+    insertions y xs id
+```
+
+Recall that this is essentially "enter a little sub-program which will do nothing more than return the computed value", so that instead of having to express the rest of our program in the `andThen` argument to `insertions`, we can just assign a variable and proceed as in any other F# program:
+
+```fsharp
+// Slightly confusing and rather strange to read
+insertions 4 [1 ; 2 ; 3] (fun inserted ->
+    // proceed with more stuff here
+)
+
+// Much more natural F#
+let inserted : int list list = insertions' 4 [ 1 ; 2 ; 3 ]
+// proceed with more stuff here
+```
+
+# But what about recursing multiple times?
+
+The previous example worked because we were able to tail-call our single recursion, so we didn't need any stack space.
+But what about when we write the function `permutations`, which constructs all the permutations of an input?
+
+```fsharp
+let insertions (x : 'a) (xs : 'a list) : 'a list list =
+    failwith "We did this earlier"
+
+let rec permutations (xs : 'a list) : 'a list list =
+    match xs with
+    | [] -> [[]]
+    | x1 :: xs ->
+        permutations xs |> List.collect (insertions x1)
+```
+
+This is all fine, and we could use our previous methods to protect us from stack overflow in the recursive call to `permutations`.
+But to do so, we would throw away our previous hard work; we'd treat `insertions` purely as an ordinary function `'a -> 'a list -> 'a list list`, without using any of the hard-won CPS machinery with which we implemented `insertions'`.
+Then we'd go and implement *yet more* machinery in defining a continuation-passing version of `permutations`.
+This is a little warning sign.
+
+In fact, if you have to recurse *twice* rather than merely once, and you can't work out how to express the problem using an explicit accumulator or mutable state, then you really do need something more.
+(That's not the case for this problem, but the example is still complex enough that we will see how to develop the machinery to solve it.)
+If there are many recursive calls, there's no possible way that *two* recursive calls could both be the last thing the function does, so there's no way the simple tail-call optimisation could take place!
+Given a collection of recursive calls, we need to sequence them to be made one at a time.
+
+The machinery we develop to do this will allow us to express `permutations` in terms of the continuation-passing-style `insertions`.
+This particular example doesn't need the machinery, and we are not developing it to solve the general "multiple recursive calls" problem here - there is only one recursive call, after all - but in fact the machinery generalises instantly to that problem too.
+
+We will sequence the many continuation-passing calls to `insertions` in such a way that `permutations` can be expressed neatly in continuation-passing style.
+
+Remember, a recursive call in continuation-passing style essentially results in a `('result -> 'ret) -> 'ret` function.
+(Imagine partially applying the `insertions` function above. Alternatively, look at the continuation we constructed before passing it into a recursive call to `insertions`.)
+So our putative "sequencing" function will take a list of `('result -> 'ret) -> 'ret`, and return a CPS-style `'result list` (i.e. `('result list -> 'ret) -> 'ret`).
+That is, we seek a function of the following type signature:
+
+```fsharp
+let rec sequence
+    (results : (('result -> 'ret) -> 'ret) list)
+    (andThen : 'result list -> 'ret)
+    : 'ret
+    =
+    failwith "implement me!"
+```
+
+Parenthetical aside: sometimes we might use the following type alias to neaten up the signatures, where a `Continuation<'a, 'ret>` is simply "an `'a`, but in continuation-passing style".
+
+```fsharp
+type Continuation<'a, 'ret> = ('a -> 'ret) -> 'ret
+```
+
+Anyway, once we've written out the type signature of `sequence`, there are basically only two possible ways to make the types line up while still making all the recursive calls we will need to make.
+It's an excellent exercise to try and do this yourself; the experience may well feel like flailing around trying to slot differently-shaped bits of a jigsaw together.
+[Chris] presented one of the ways; I'll present the other here, because it fits better with the method used above.
+
+```fsharp
+let rec sequence
+    (results : (('result -> 'ret) -> 'ret) list)
+    (andThen : 'result list -> 'ret)
+    : 'ret
+    =
+    match results with
+    | [] -> andThen []
+    | andThenInner :: andThenInners ->
+        fun (results : 'result list) ->
+            fun (result : 'result) ->
+                result :: results
+                |> andThen
+            |> andThenInner
+        |> sequence andThenInners
+```
+
+## What does `Continuation.sequence` do?
+
+It's all very well to have written out a function with the right type signature, but what does it actually do?
+
+Just like in the other examples of a tail-recursive CPS function, the call to `sequence` essentially iterates over the input list constructing an ever larger closure on the heap.
+The actual contents of this closure don't matter until the input list runs out; we're simply spinning round in a loop, building a bigger and bigger closure while truncating the input list one at a time.
+No meaningful computation is taking place at all: we're just building the instructions in memory that we are *going* to carry out, in the form of an enormous closure.
+
+But when the input list *does* run out, we evaluate this enormous closure on the input `[]`.
+That is, we evaluate `andThenInner (fun result -> andThen (result :: []))`, where `andThenInner` is the most-recently-seen (i.e. the last) element of the original list `results` of continuation-passing-style inputs, and `andThen` is the enormous closure but with one layer unpeeled (because we applied it to `[]` in the previous step).
+
+That is, we evaluate the last element of the list, and then inject it into the enormous closure, effectively leaving us with the enormous closure applied to `results.Last :: []` (where we have used an imaginary `list.Last` notation that doesn't actually exist in F#).
+
+The computation unspools in exactly the same way through the input list, at each stage unwrapping one layer of the enormous closure and injecting the value represented by the previous element of the input list of continuations.
+
+At the very end, the enormous closure has been unwrapped all the way down to the original `andThen` that was passed in at the very start; no longer does the binding `andThen` correspond to a closure that we created during the course of our recursion.
+This is the final escape hatch: the evaluation is complete, and `andThen` tells us to continue execution in user code.
+
+By the way, this demonstrates that the *last* element of the input list is the one which executes *earliest*.
+While the order of elements as presented to the user's continuation is the same as the order in which they appear in the input list of continuations, the order in which the result list is *generated* is from the end backwards.
+If all functions are side-effect-free, this doesn't matter at all, but if your continuations are side-effectful then you should keep this at the back of your mind.
+The clue in the name ("sequence") indicates that the continuations are being, well, put into sequence; but you should know what order the sequence is being computed in.
+
+# Apply `Continuation.sequence` to compute permutations
+
+Recall our definition of `permutations` from earlier:
+
+```fsharp
+let rec permutations (xs : 'a list) : 'a list list =
+    match xs with
+    | [] -> [[]]
+    | x1 :: xs ->
+        permutations xs
+        |> List.collect (insertions x1)
+```
+
+To translate this into continuation-passing style, it needs the following type signature:
+
+```fsharp
+let rec permutations (xs : 'a list) (andThen : 'a list list -> 'ret) : 'ret =
+    failwith "implement me!"
+```
+
+Now, we certainly *could* do this without `Continuation.sequence`.
+We already have an implementation of `insertions` which uses continuation-passing style to make itself tail-recursive, after all.
+
+```fsharp
+let rec permutations' (xs : 'a list) (andThen : 'a list list -> 'ret) : 'ret =
+    match xs with
+    | [] -> andThen [[]]
+    | x :: xs ->
+        fun (remainder : 'a list list) ->
+            remainder
+            |> List.collect (fun i -> insertions x i id)
+            |> andThen
+        |> permutations' xs
+```
+
+However, it's a bit weird to jump in and out of CPS like this.
+There's nothing wrong with it, but for the exercise, could we use `insertions x i` in continuation-passing style rather than by forcing its execution with `id`?
+
+This is why we need `Continuation.sequence`.
+
+Start in the only possible way we can absolutely guarantee tail recursion: by writing down the recursive call to `permutations'`.
+
+```fsharp
+let rec permutations' (xs : 'a list) (andThen : 'a list list -> 'ret) : 'ret =
+    match xs with
+    | [] -> andThen [[]]
+    | x :: xs ->
+        fun (recurseResult : 'a list list) ->
+            failwith<'ret> "now implement me"
+        |> permutations' xs
+```
+
+What do we have in scope to fill the `failwith`?
+It has to be either `andThen`, or a call to `Continuation.sequence`, since we probably don't want to do *another* call to `permutations'`.
+
+If we consider for a moment what would happen if we tried `andThen`:
+```fsharp
+let rec permutations' (xs : 'a list) (andThen : 'a list list -> 'ret) : 'ret =
+    match xs with
+    | [] -> andThen [[]]
+    | x :: xs ->
+        fun (recurseResult : 'a list list) ->
+            failwith<'a list list> "implement this?"
+            |> andThen
+        |> permutations' xs
+```
+This is how we recover the solution further above, which I described as "a bit weird": we're going to have to jump out of CPS to get an `'a list list` from `insertions`.
+
+So we'll go with `Continuation.sequence` instead, so that we can collect results from multiple CPS calls to `insertions` - but note that this introduces a free type parameter, expressed with `_`:
+
+```fsharp
+let rec permutations' (xs : 'a list) (andThen : 'a list list -> 'ret) : 'ret =
+    match xs with
+    | [] -> andThen [[]]
+    | x :: xs ->
+        fun (recurseResult : 'a list list) ->
+            sequence
+                (failwith<((_ -> 'ret) -> 'ret) list> "implement")
+                (failwith<_ list -> 'ret> "implement")
+        |> permutations' xs
+```
+
+And since it's multiple calls to `insertions` we wish to collect together, the type of those holes is decided:
+
+```fsharp
+let rec permutations' (xs : 'a list) (andThen : 'a list list -> 'ret) : 'ret =
+    match xs with
+    | [] -> andThen [[]]
+    | x :: xs ->
+        fun (recurseResult : 'a list list) ->
+            let conts : (('a list list -> 'ret) -> 'ret) list =
+                List.map (insertions x) recurseResult
+            sequence
+                conts
+                (failwith<('a list list) list -> 'ret> "implement")
+        |> permutations' xs
+```
+
+Notice how the original `List.collect` was forced to become a `List.map` here: we were unable to immediately concatenate the results of the various calls to `insertions`.
+In the original non-CPS style, our `'a list list list` could become `'a list list` by implicitly squashing together the outermost pair of `list` with a call to `List.collect`; but now there is a CPS `(_ -> 'ret) -> 'ret` in the way.
+So we'll have to `List.concat` explicitly after sequencing these recursive calls together.
+
+```fsharp
+let rec permutations' (xs : 'a list) (andThen : 'a list list -> 'ret) : 'ret =
+    match xs with
+    | [] -> andThen [[]]
+    | x :: xs ->
+        fun (recurseResult : 'a list list) ->
+            let conts = List.map (insertions x) recurseResult
+            sequence conts (List.concat >> andThen)
+        |> permutations' xs
+```
+
+Voilà!
+
+# Exercises
+
+* Easy: define the type `Cont<'a, 'ret> = ('a -> 'ret) -> 'ret`, and implement `Continuation.bind : ('a -> Cont<'b, 'ret>) -> Cont<'a, 'ret> -> Cont<'b, 'ret>`. You've essentially defined the Continuation monad. Note that `Continuation.sequence` is the standard notion of `sequence` for a traversable, applied to the functor `List`.
+* Moderately difficult: Make sure you understand, in the definition of `permutations'`, when each part executes relative to each other part. To really test your bookkeeping: in what order are the permutations created, and in what order are they output?
+* Difficult: Check Chris's definition of `Continuation.sequence`, which is different from the one presented here. How are the two definitions different? In what order do they evaluate, and in what order do they return?
+Hint: Try it out using the following!
+
+```fsharp
+let mutable j = 0
+sequence [for _ in 1..100 do yield (fun andThen -> andThen (j <- j + 1 ; j))] (printfn "%+A")
+```
+
+* Moderately difficult: construct a "seq-safe" version of `Continuation.sequence`, with type signature `(inputs : (('result -> 'ret) -> 'ret) seq) -> (andThen : 'result list -> 'ret) -> 'ret`, where we have replaced the inputs with a `seq` rather than a `list`. This version must enumerate the input sequence only once; if you ever call `Seq.head`, for example, you've already enumerated the input sequence up to the first element, and you've run out of enumeration budget! This would matter a lot if, for example, the sequence opens a database connection or mutates a stream.
+* Very difficult: is it possible to construct a lazy version of `Continuation.sequence`, with type signature `(inputs : (('result -> 'ret) -> 'ret) seq) -> (andThen : 'result seq -> 'ret) -> 'ret`, where we have replaced the `list`s with `seq`s, which only enumerates as much of the input sequence `inputs` as `andThen` asks for?
+
+
+[Chris]: https://www.gresearch.co.uk/article/advanced-recursion-techniques-in-f/
+[discussed]: {{< ref "2020-07-22-tailrecursion" >}}
diff --git a/hugo/content/posts/2021-10-19-crates.md b/hugo/content/posts/2021-10-19-crates.md
new file mode 100644
index 0000000..2018cfa
--- /dev/null
+++ b/hugo/content/posts/2021-10-19-crates.md
@@ -0,0 +1,298 @@
+---
+lastmod: "2021-10-23T18:57:58.0000000+01:00"
+author: patrick
+categories:
+- programming
+comments: true
+date: "2021-10-19T00:00:00Z"
+title: Crates (existentials in F#)
+summary: "An introduction to the crate pattern for representing existential quantification in F#."
+---
+
+The usual blog post introducing crates in F# is [G-Research's](https://www.gresearch.co.uk/article/squeezing-more-out-of-the-f-type-system-introducing-crates/).
+However, I believe that post approaches things from a much more abstract perspective than is useful.
+Instead, I will approach them by starting from the concrete.
+
+(The crate pattern is also called [skolemization](https://en.wikipedia.org/wiki/Skolem_normal_form), by the way.)
+
+# The problem crates solve
+
+Say we want to represent lists in some kind of [defunctionalised]({{< ref "2020-03-04-defunctionalisation" >}}) way.
+
+```fsharp
+type ListBuilder<'a> =
+    | OfList of 'a list
+    | Cons of 'a * 'a ListBuilder
+    | Concat of 'a ListBuilder list
+
+module ListBuilder =
+    let toList (lb : 'a ListBuilder) : 'a list =
+        failwith "implement"
+```
+
+Because we don't need to reify each of the individual input lists inside a `ListBuilder`, in principle the following could be a much more efficient way to construct the specified list (assuming we define appropriate module methods to deal with currying):
+
+```fsharp
+[
+    ListBuilder.ofList [ 1 ; 2 ; 3 ]
+    ListBuilder.cons 3 [ 3 ; 6 ; 8 ]
+    ListBuilder.concat [ ListBuilder.ofList [ 1 ; 3 ] ; ListBuilder.ofList [ 10 ; 12 ] ]
+]
+|> ListBuilder.concat
+|> ListBuilder.toList
+```
+For example, if we did this purely in `List`-space, we'd have to reify the concatenated `[ 1 ; 3 ; 10 ; 12 ]`, whereas in principle we could avoid doing that here: we stand a chance of directly emitting the single output list using the compiler's built-in magical mutable-tail cons lists which are very fast.
+
+This is all very well, but what about the following extension to the API?
+
+```fsharp
+type ListBuilder<'a> =
+    | OfList of 'a list
+    | Cons of 'a * 'a ListBuilder
+    | Concat of 'a ListBuilder list
+    | Map of ('b -> 'a) * 'b ListBuilder
+```
+
+This won't compile, because we've got this type parameter `'b` which we're not generic on.
+(The structure I've written down is essentially trying to be a [generalised algebraic datatype](https://en.wikipedia.org/wiki/Generalized_algebraic_data_type).)
+
+# How do we approach this kind of problem?
+
+Let's drop down a level of abstraction.
+
+To represent sum types in F#, it's very easy:
+
+```fsharp
+type MySumType =
+    | Foo of int
+    | Bar of string
+
+let toString (input : MySumType) : string =
+    match input with
+    // hooray, the language lets us do our desired operation on the object
+    | Foo i -> sprintf "%i" i
+    | Bar s -> s
+```
+
+But C# doesn't let you do this! (Well, in ten years C# will have grown to subsume all other programming languages. But as of this writing, it can't.)
+How do you approach exactly the same problem in C#?
+
+The [canonical trick](https://blog.ploeh.dk/2018/06/25/visitor-as-a-sum-type/) when your language can't represent the decomposition of an object is to *seal up the object, but teach it how to accept operations from the outside*.
+We can't take the object to the operation, so we have to take the operation to the object.
+(This is the "[visitor pattern](https://en.wikipedia.org/wiki/Visitor_pattern)".)
+
+I'll present the solution in F# restricted to concepts that are present in C#.
+
+```fsharp
+type IMySumType =
+    abstract Match<'ret> : (int -> 'ret) -> (string -> 'ret) -> 'ret
+
+type FooCase (i : int) =
+    interface IMySumType with
+        member _.Match<'ret> (f : int -> 'ret) (_ : string -> 'ret) =
+            f i
+
+type BarCase (s : string) =
+    interface IMySumType with
+        member _.Match<'ret> (_ : int -> 'ret) (g : string -> 'ret) =
+            g s
+
+let toString (input : IMySumType) : string =
+    input.Match (sprintf "%i") id
+```
+
+Notice what we've done: we have in some sense "taught the discriminated union how to perform operations on itself", to compensate for the fact that the language can't express the notion "open up the object to see what's inside".
+
+If you wanted to be *really* object-oriented, you could make an object that represents the operation, and rejig so that `Match` now takes an Operation instead:
+
+```fsharp
+type Operation<'ret> =
+    abstract OperateOnFoo : int -> 'ret
+    abstract OperateOnBar : string -> 'ret
+
+type IMySumType =
+    abstract Match<'ret> : Operation<'ret> -> 'ret
+
+type FooCase (i : int) =
+    interface IMySumType with
+        member _.Match<'ret> (operation : Operation<'ret>) : 'ret =
+            operation.OperateOnFoo i
+
+type BarCase (s : string) =
+    interface IMySumType with
+        member _.Match<'ret> (operation : Operation<'ret>) : 'ret =
+            operation.OperateOnBar s
+
+let toString =
+    { new Operation<string> with
+        member _.OperateOnFoo (i : int) = sprintf "%i" i
+        member _.OperateOnBar (s : string) = s
+    }
+```
+
+# A little more general
+
+OK, how about the following, where we're now generic?
+
+```fsharp
+type 'a MySumType =
+    | Foo of int
+    | Bar of 'a
+
+let toInt<'a> (input : MySumType<'a>) : int =
+    match input with
+    | Foo i -> i
+    | Bar a ->
+        // what can go here?
+        failwith "implement"
+```
+
+Well, the only way this code can make sense is if `toInt` also takes an `'a -> int` function:
+
+```fsharp
+type 'a MySumType =
+    | Foo of int
+    | Bar of 'a
+
+let toInt<'a> (barCase : 'a -> int) (input : MySumType<'a>) : int =
+    match input with
+    | Foo i -> i
+    | Bar a -> barCase a
+```
+
+How might we do this in C#? Here's exactly the same transformation as before:
+
+```fsharp
+type IMySumType<'a> =
+    abstract Match<'ret> : (int -> 'ret) -> ('a -> 'ret) -> 'ret
+
+type FooCase<'a> (i : int) =
+    interface IMySumType<'a> with
+        member _.Match<'ret> (f : int -> 'ret) (_ : 'a -> 'ret) =
+            f i
+
+type BarCase<'a> (a : 'a) =
+    interface IMySumType<'a> with
+        member _.Match<'ret> (_ : int -> 'ret) (g : 'a -> 'ret) =
+            g a
+
+let toInt<'a> (barCase : 'a -> int) (input : IMySumType<'a>) : int =
+    input.Match id barCase
+```
+
+Again, for the really object-oriented inclined:
+
+```fsharp
+type Operation<'a, 'ret> =
+    abstract OperateOnFoo : int -> 'ret
+    abstract OperateOnBar : 'a -> 'ret
+
+type IMySumType<'a> =
+    abstract Match<'ret> : Operation<'a, 'ret> -> 'ret
+
+type FooCase<'a> (i : int) =
+    interface IMySumType<'a> with
+        member _.Match operation =
+            operation.OperateOnFoo i
+
+type BarCase<'a> (a : 'a) =
+    interface IMySumType<'a> with
+        member _.Match operation =
+            operation.OperateOnBar a
+
+
+let toInt<'a> (barCase : 'a -> int) =
+    { new Operation<'a, int> with
+        member _.OperateOnFoo i = i
+        member _.OperateOnBar a = barCase a
+    }
+```
+
+# How this applies to the Map case
+
+Recall the problematic code:
+
+```fsharp
+type ListBuilder<'a> =
+    | OfList of 'a list
+    | Cons of 'a * 'a ListBuilder
+    | Concat of 'a ListBuilder list
+    // This is unrepresentable
+    | Map of ('b -> 'a) * 'b ListBuilder
+```
+
+Well, the language doesn't give us the tools to "open up" the contents of the Map case.
+So let's pull the same visiting trick, starting from the top:
+
+```fsharp
+type IListBuilder<'a> =
+    // This doesn't compile either
+    abstract Match<'ret> : ('a list -> 'ret) -> ('a -> 'a IListBuilder -> 'ret) -> ('a IListBuilder list -> 'ret) -> (('b -> 'a) -> 'b IListBuilder -> 'ret) -> 'ret
+```
+
+Well, this looks ghastly and also it doesn't compile (because again there's that `'b` we can't place anywhere), but happily if we try and make it look nicer by moving to the "really object-oriented" world, *both* those problems go away!
+
+```fsharp
+type Operation<'a, 'ret> =
+    abstract OperateOnOfList : 'a list -> 'ret
+    abstract OperateOnCons : 'a -> 'a IListBuilder -> 'ret
+    abstract OperateOnConcat : 'a IListBuilder list -> 'ret
+    // Ta-da!
+    abstract OperateOnMap<'b> : ('b -> 'a) -> 'b IListBuilder -> 'ret
+
+and IListBuilder<'a> =
+    abstract Match<'ret> : Operation<'a, 'ret> -> 'ret
+
+type OfListCase<'a> (xs : 'a list) =
+    interface IListBuilder<'a> with
+        member _.Match op = op.OperateOnOfList xs
+
+type OfConsCase<'a> (x : 'a, xs : 'a IListBuilder) =
+    interface IListBuilder<'a> with
+        member _.Match op = op.OperateOnCons x xs
+
+type OfConcat<'a> (xs : 'a IListBuilder list) =
+    interface IListBuilder<'a> with
+        member _.Match op = op.OperateOnConcat xs
+
+type OfMap<'a, 'b> (f : 'b -> 'a, xs : 'b IListBuilder) =
+    interface IListBuilder<'a> with
+        member _.Match op = op.OperateOnMap f xs
+```
+
+Now, it's a little upsetting that the user can't operate on a nice `OfList` case without making one of our nonsense interfaces.
+It would be handy if we could retain the terse F# pattern match in all the cases where the language can represent it.
+
+In the following, I just deleted three cases from `Operation`, and three implementors of `IListBuilder`, and created a discriminated union `ListBuilder` which is just "an ordinary ListBuilder, but with an extra case where we have to go through the indirection nonsense".
+
+```fsharp
+type Operation<'a, 'ret> =
+    abstract OperateOnMap<'b> : ('b -> 'a) -> 'b ListBuilder -> 'ret
+
+and IListBuilder<'a> =
+    abstract Match<'ret> : Operation<'a, 'ret> -> 'ret
+
+and ListBuilder<'a> =
+    | OfList of 'a list
+    | OfCons of 'a * 'a ListBuilder
+    | OfConcat of 'a ListBuilder list
+    | OfMap of 'a IListBuilder
+```
+
+I'll just rename this a bit:
+
+```fsharp
+type MapCaseEvaluator<'a, 'ret> =
+    abstract Eval<'b> : ('b -> 'a) -> 'b ListBuilder -> 'ret
+
+and MapCaseCrate<'a> =
+    abstract Apply<'ret> : MapCaseEvaluator<'a, 'ret> -> 'ret
+
+and ListBuilder<'a> =
+    | OfList of 'a list
+    | OfCons of 'a * 'a list
+    | OfConcat of 'a ListBuilder list
+    | OfMap of 'a MapCaseCrate
+```
+
+And that is a crate being used as part of the implementation of a GADT.
diff --git a/hugo/content/posts/2021-10-22-argument.md b/hugo/content/posts/2021-10-22-argument.md
new file mode 100644
index 0000000..0b18356
--- /dev/null
+++ b/hugo/content/posts/2021-10-22-argument.md
@@ -0,0 +1,58 @@
+---
+lastmod: "2021-10-24T21:14:55.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+comments: true
+date: "2021-10-21T00:00:00Z"
+title: Argument in a high-trust environment
+summary: "I tend to argue things in a particular way because I'm in a high-trust environment."
+---
+
+This post describes a pattern I tend to follow (unconsciously) when it comes to arguing things, and suggests cases where this pattern may work very poorly.
+
+# The method, in one line
+
+I often begin an argument by making extremely strong claims, and then trying to guide the collective mind of the arguers to show me why my claim is false.
+
+# A concrete example
+
+A couple of days ago, I made the claim to some of my team at work that "functional programming is more intuitive than imperative programming".
+My state of mind before I made this claim was a tentative suspicion that this was idealistic and not necessarily true, but no concrete reasons why it would be false, and a preference on ideological and tribal grounds for it to be true.
+
+Then someone replied to the effect that my assertion was false.
+(Hooray! We're about to have a fun argument.)
+
+There was some back-and-forth in which we determined a bit more closely what my claim was and what the counterclaim was; we ended up with something more like "someone who has never programmed before, and has never encountered the imperative programming model before, will find functional programming easier to pick up, and this is for fundamental reasons to do with how people think".
+
+Then I started trying to construct counterfactuals and hypotheticals to describe the state of the world in the case that the hypothesis was true or false, and I came up with recipe books as an example of an everyday thing that people are generally expected to be able to use.
+
+We had a bit of a discussion about whether recipe books appear to be "functional" or "procedural", and eventually I was persuaded that they are primarily procedural.
+I tried to think what a recipe book would look like if it came from a civilisation of beings who naturally thought using the functional paradigm, and the result appeared to be quite different from what we see in recipe books.
+
+So I am now decently convinced that people naturally think procedurally.
+
+# Why does this method work?
+
+The method works in a high-trust environment where everyone understands that everyone else is operating in good faith and can have their opinions changed by data.
+It's basically a clumsy and inconsistent way of finding [cruxes](https://www.lesswrong.com/posts/hNztRARB52Riy36Kz/the-basic-double-crux-pattern) in the LessWrong sense: we are essentially binary-chopping through the space of hypotheses, finding the minimal places where we differ on our opinion.
+
+# Why is the method a bad one in general?
+
+Well, it is entirely phrased around an adversarial conversation, which might be mind-killing in itself!
+The method may well tend to get people digging their heels in.
+
+Whenever I remember, I'll try and practise the double-crux method in a way that's closer to its original formulation, because it should be possible to get the benefits without being so adversarial.
+
+# Under what specific conditions does the method perform particularly badly?
+
+One easy place where the method totally fails is if there's a big power imbalance among the people in the conversation, such that some people feel they can't contradict others or point out obvious flaws.
+It also fails when people aren't used to a culture of calling people out when they say things that are false.
+
+# Appendix: what would a functional-paradigm recipe book look like?
+
+It seems to me that a functional-paradigm recipe book would be phrased less in terms of the instructions you perform to get from one state to the next, and more in terms of descriptions of the states and the possible transitions between them.
+Recipe books currently appear essentially as an imperative list of instructions; a functional recipe book might appear more like a graph of the intermediate states you expect to reach during the recipe.
+
+I think recipe books might be better if they took some inspiration from this!
+A recipe book which was more explicit about the dependency order of the steps (rather than cleaving strictly to the "linear sequence of instructions" idiom), and which described each state thoroughly (perhaps with pictures), might well be easier for a beginner to follow/understand/learn from.
diff --git a/hugo/content/posts/2021-10-25-avoid-rm-rf.md b/hugo/content/posts/2021-10-25-avoid-rm-rf.md
new file mode 100644
index 0000000..947d2b3
--- /dev/null
+++ b/hugo/content/posts/2021-10-25-avoid-rm-rf.md
@@ -0,0 +1,63 @@
+---
+lastmod: "2021-10-25T23:41:14.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+date: "2021-10-25T00:00:00Z"
+sidenotes: true
+title: Don't supply `-f` to `rm` unless you know you need it
+summary: "A vignette on the theme of 'do not allow yourself to get into the habit of supplying the `-f` flag to `rm`'."
+---
+
+```bash
+➜  hugo git:(master) ✗ ./build.sh
+Building thumbnails.
+Building LaTeX.
+Building site.
+Start building sites …
+
+                   | EN
+-------------------+------
+  Pages            | 196
+  Paginator pages  |   6
+  Non-page files   |   0
+  Static files     | 371
+  Processed images |   0
+  Aliases          | 208
+  Sitemaps         |   1
+  Cleaned          |   0
+
+Total in 45838ms
+Linting.
+
+➜ chrome --disable-web-security --user-data-dir=~/.chrome
+```
+
+{{< side right time-note "(checks local render of website)" >}}Yes, you read that timing correctly. Docker for Mac is truly astonishingly slow.{{< /side >}}
+
+```bash
+➜  hugo git:(master) ✗ git status
+On branch master
+Your branch is up to date with 'alligator/master'.
+
+Changes not staged for commit:
+  (use "git add <file>..." to update what will be committed)
+  (use "git restore <file>..." to discard changes in working directory)
+        modified:   build.sh
+        modified:   content/posts/2016-03-03-a-certain-limit.md
+        modified:   layouts/partials/math.html
+
+Untracked files:
+  (use "git add <file>..." to include in what will be committed)
+        lint.sh
+        ~/
+
+no changes added to commit (use "git add" and/or "git commit -a")
+
+➜  hugo git:(master) ✗ rm -r ~
+rm: /Users/Patrick/Music: Permission denied
+^C
+^C
+^C
+^C
+```
diff --git a/hugo/content/posts/2022-05-15-water-filtration.md b/hugo/content/posts/2022-05-15-water-filtration.md
new file mode 100644
index 0000000..2b097a5
--- /dev/null
+++ b/hugo/content/posts/2022-05-15-water-filtration.md
@@ -0,0 +1,26 @@
+---
+lastmod: "2022-05-15T11:00:00.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+date: "2022-05-15T00:00:00Z"
+title: The water filtration industry has played us for absolute fools
+summary: "My Brita water filter does not in fact purify."
+gallery: true
+---
+
+I wanted to know whether my Brita water filter ("MAXTRA") actually purifies water.
+The manufacturer [advertises](https://www.brita.co.uk/water-filter-system/water-filter-jugs#to-filtration) it as:
+
+1. removing large particles;
+2. removing metals via ion exchange;
+3. removing chemicals like chlorine.
+
+So I took a fresh cartridge, rinsed it with water following its directions, and put it through two filtration cycles of plain water.
+Then I emptied the jug, made one mug of strong cherry-flavoured squash, and poured the squash into the filter jug.
+
+I defy you to tell the difference between the following three pictures, which are the result of the first, second, and third successive filter runs respectively.
+
+{{< foldergallery src="images/galleries/WaterFiltration/" >}}
+
+The real mug was me all along!
\ No newline at end of file
diff --git a/hugo/content/posts/2023-07-12-transformer-questions.md b/hugo/content/posts/2023-07-12-transformer-questions.md
new file mode 100644
index 0000000..ed5bb3c
--- /dev/null
+++ b/hugo/content/posts/2023-07-12-transformer-questions.md
@@ -0,0 +1,107 @@
+---
+lastmod: "2023-07-14T20:00:00.0000000+01:00"
+author: patrick
+categories:
+- uncategorized
+date: "2023-07-12T00:00:00Z"
+title: Questions I had about transformers
+summary: "Some basic questions I had about transformers, and their possible answers."
+math: true
+---
+
+I was learning about transformers, variously through [Neel Nanda's video on transformers](https://www.youtube.com/watch?v=bOYE6E8JrtU) and his [video about A Mathematical Framework for Transformer Circuits](https://www.youtube.com/watch?v=KV5gbOmHbjU).
+Here are several questions I had throughout.
+
+## Prediction of previous tokens
+
+In a decoder-only transformer applied to a stream of text, the final output (after all layers have been computed) is a vector of logits, one for each element of the context window, and the "prediction" of the transformer is the token indicated by the logit that is at the final place of the context window.
+Does that mean its "prediction" for the *previous* elements of the context window is always a trivial "this particular token, with extremely high probability", because that particular element of the context window was already in the stream before we started to predict?
+
+ChatGPT4 did me wrong and said that this was correct.
+In fact, it is not: this is what the *masked* part of "masked self-attention" is for, to prevent predictions earlier in the sequence from using information from later in the sequence.
+The prediction at any given token is the prediction using all information available up to immediately before the point of that token, but no further.
+
+## Ordering
+
+What gives the residual stream any ordering properties?
+Why can we view it as being somehow ordered, when interactions with it are all linear?
+
+Answer: positional encoding.
+(This is also answered during the "A Mathematical Framework" video.)
+
+## Strong reliance on specific paths
+
+Neel said:
+
+> An important property of the transformer is that the model functionality arises as a sum of paths through the residual stream.
+> This is important because we should expect a lot of the model's behaviour to be localised: there will be some path through the model, going through some heads and some MLPs to the output, and we might predict that in general some paths matter but most paths don't.
+> And in practice this seems to be basically true."
+
+But I don't see why we should predict that.
+Why shouldn't the model's behaviour arise through a roughly-equally-weighted combination of all the paths?
+
+Answer: there's no inherent reason why this should be true, but in practice the model tends to learn better by having individual heads focusing on specific features and compositions thereof.
+(I suspect this is also related to the problem that superposition solves: there's only a limited amount of information you're able to extract at once from the residual stream, because the stream relies heavily on superposition to fit the data in. So your features are inherently limited to operating on "single cross-sections" of the superposed data at once.)
+
+## Attention is limited
+
+Why do attention heads learn to attend only to specific parts of the input, rather than as much of the input as is available at once?
+
+Answer: specialisation is just more efficient in practice, and also each head is kind of small and can't attend to everything at once, even if it can in principle access most of the data to some degree.
+
+## Dimensions
+
+In GPT2-small, the residual stream is of width 768.
+Does that mean the residual stream is essentially a tensor, with first dimension 768 and second dimension about 50,000 (i.e. the size of the token dictionary)?
+
+Answer: no.
+The initial embedding layer converts the input token stream (that is, a collection of 50,000-length vectors) into the residual stream (that is, a collection of context-window-many vectors of size model-width=768).
+The number 50,000 then has no further part in things until the projection back into token-space at the very end.
+
+## Is the "context window" the same as the "sequence length"?
+
+Answer: yes.
+
+## Adjusting the context window
+
+Do we have to fix the context size up front, or is it easy to adjust after the main body of training?
+The release of "increased context window" versions of the GPT models suggests that it's cheap to adjust, but I don't see how.
+
+Answer: I was really confused about this, and my best answer was "no, you really can't adjust this, they're just training new models".
+ChatGPT4 said that that was correct, but I am reliably informed that it was lying.
+I am assured that the answer to this question will be very obvious once I grok what an attention head is.
+
+## Why not concatenate things to the residual stream?
+
+Surely if the residual stream is under immense compression pressure (leading to superposition), we could do massively better by just making the residual stream even slightly higher-dimensional?
+What considerations go into picking the size of the residual stream?
+
+Relatedly, Neel Nanda said we elementwise-add rather than concat positional information to the input "because the residual stream is shared memory and likely under significant superposition".
+Is that a response to "why don't we have a specific dedicated area for position that never changes" (i.e. "because it's inefficient, the model wants to use any space available so it's inefficient to carve out some space which can't change")?
+
+## Why do the same prediction problem at every position?
+
+Is there some structural reason why we predict text *at each position*, rather than doing something else and only caring about the prediction at the last token?
+
+Answer (I think): I had a longer explanation of this question, but I think I answered it in the process of writing it.
+The loss function we use for text generation is "perplexity", which structurally does mean "how far is each predicted token from what actually happened".
+
+## Why can't an MLP layer move information between tokens?
+
+"An attention head is the only part of a transformer which can move information between positions."
+But why can't an MLP layer do this?
+
+## Hold on, the attention pattern is an activation and not a parameter?
+
+Don't we learn up front how much attention a head is going to pay to each component of the input?
+
+Answer: I think this was bad exposition on Neel's part.
+The attention pattern \\(A\\) is computed by converting \\(x \mapsto x^T W x\\) to a probability distribution with softmax.
+That is, it is an activation (because it depends on \\(x\\)), but it is learned (because it contains a parameter \\(W\\)).
+
+## Is an "attention-only" model one which doesn't even have any MLP layers, just attention heads?
+
+## Why "Q-K" at all?
+
+It just collapses to one matrix, doesn't it?
+So why do we separate "keys" from "queries" at all?
\ No newline at end of file
diff --git a/hugo/content/posts/2023-09-06-git-showcase.md b/hugo/content/posts/2023-09-06-git-showcase.md
new file mode 100644
index 0000000..be3f74e
--- /dev/null
+++ b/hugo/content/posts/2023-09-06-git-showcase.md
@@ -0,0 +1,89 @@
+---
+lastmod: "2023-09-06T13:51:00.0000000+01:00"
+author: patrick
+categories:
+- programming
+date: "2023-09-06T13:51:00Z"
+title: Notes for a Git fireside chat
+summary: "A syllabus for a fireside chat to give at work, on Git"
+---
+
+This is simply an outline, with no actual content.
+
+# How Git works
+
+* Recommendation to read Pro Git
+* Fundamental model: the object database
+  * Blobs, trees, tags, commits
+* Extra metadata
+  * Branches
+  * Lightweight tags
+  * HEAD (and other symbolic refs)
+  * Remotes
+* Mention of performance optimisations such as packfiles
+* Merge algorithms
+  * recursive
+  * ort
+  * ours
+* Useful commands
+  * `git rev-parse`
+  * `git merge-base`
+  * `git symbolic-ref`
+  * `git clone --depth=1`
+  * `git commit --amend`
+  * `git grep` (although ripgrep is generally faster anyway)
+  * `git log -p` (for patch) and `-S function_name`
+  * `git blame`
+* Brief mention of what a submodule is
+* Brief mention of what `git rebase` is
+* How-tos
+  * Stacked PRs in a squash-merge-to-`main` world
+* Case studies
+  * `$INTERNAL_WORKSPACE_TOOL push-files`
+  * my internal Git-related dotfiles
+  * [git-appraise](https://github.com/google/git-appraise)
+
+
+# Problems with Git, and why Pijul is interesting
+
+## Merging
+
+Git works on snapshots => merging is impossible in principle.
+Here's the textbook example:
+
+Start:
+```
+A
+B
+```
+
+Branch 1:
+```
+A
+X
+B
+```
+
+Branch 2:
+```
+G
+A
+B
+```
+then
+```
+A
+B
+G
+A
+B
+```
+
+When merging branch 1 and branch 2, the snapshot-based merge algorithm can't know which AB pair the X was added in between, so it must choose arbitrarily.
+
+
+## Conflicts
+
+Conflicts are not modelled at all: a conflict is only a property of the working tree, and you can't "commit it".
+That means *you can never tell Git how to resolve a conflict* in such a way that it can reliably resolve the same conflict again in the future.
+(`git rerere` is a hack that in my experience frequently does not do the right thing.)
\ No newline at end of file
diff --git a/hugo/content/posts/2023-09-28-sum-of-two-squares.md b/hugo/content/posts/2023-09-28-sum-of-two-squares.md
new file mode 100644
index 0000000..7178ca7
--- /dev/null
+++ b/hugo/content/posts/2023-09-28-sum-of-two-squares.md
@@ -0,0 +1,165 @@
+---
+lastmod: "2023-09-28T00:15:00.0000000+01:00"
+author: patrick
+categories:
+- programming
+date: "2023-09-28T00:15:00Z"
+title: The tiny proof that primes 1 mod 4 are sums of two squares
+summary: "Exploding the incredibly terse proof into a bunch of exposition."
+math: true
+sidenotes: true
+---
+
+Recall the incredibly terse proof by Zagier that primes \\(1 \pmod{4}\\) can be expressed as sums of two squares:
+
+> The {{< side right involution "involution">}}An involution is a function which is its own inverse.{{< /side >}} on the finite set \\(S = \\{(x, y, z) \in \mathbb{N}^3 : x^2 + 4yz = p \\}\\) defined by:
+>
+> \\(
+(x, y, z) \mapsto
+\begin{cases}
+(x+2z, z, y-x-z) & \text{if $x < y-z$} \\\\
+(2y-x, y, x-y+z) & \text{if $y-z < x < 2y$} \\\\
+(x-2y, x-y+z, y) & \text{if $x > 2y$}
+\end{cases}
+\\)
+>
+> has exactly one fixed point, so \\(|S|\\) is odd and the involution defined by \\((x,y,z) \mapsto (x, z, y)\\) also has a fixed point.
+
+This proof is really quite beautiful, and I want to wave at how one might come up with it.
+
+# Core idea
+
+The point is to look at the question of finding a solution to \\(x^2 + y^2 = p\\), and to note that this can be rephrased as finding a fixed point of a certain function.
+Moreover, if we build the function to have the property that it is an *involution* (that is, it is inverse to itself), then we get access to a wildly non-constructive but correspondingly perhaps-easier method to show that there is a fixed point.
+We then use exactly the same method in reverse, coupled with a different choice of involution which is absolute witchcraft, and the answer pops out.
+
+## Core idea, but concrete
+
+An involution on a set can be viewed as a way to pair up elements of that set.
+For example, \\(x \mapsto -x\\) pairs up the reals, each one being paired with its negation.
+
+Some elements, though, have a "degenerate" pairing formed this way, where we pair up the elemet with itself.
+(In the negations example, there's exactly one degenerate element which gets paired up with itself, namely \\(0\\).)
+To be precise, an involution pairs up each element with exactly one other element, *except* for the fixed points of that involution.
+The fixed points of an involution are precisely those which aren't in a pair but instead stand alone.
+
+{{< side right aside "This can also be explained with excessive machinery." >}}
+The involution is a permutation of the set.
+It's self-inverse, so the subgroup of the symmetric group generated by that permutation has size 2.
+Therefore the orbits of the set under the usual action of permutations on sets are of size 2 or 1, by the orbit-stabiliser theorem.
+{{< /side >}}
+
+So to show that an involution has a fixed point, it's enough to show that the set it's permuting has an odd number of elements.
+Indeed, if the set has an odd number of elements, then we can't possibly pair all up (that's what it means for a number to be odd!), so in particular no involution can possibly pair them all up, and in even more particular our *chosen* involution can't pair them all up.
+
+Moreover, to show that a set has an odd number of elements, it's enough to find *some* involution on that set which has an odd number of fixed points.
+In greater detail: if an involution has an odd number of fixed points, then the set consists of "everything which is paired with something" (necessarily there are evenly many of these, because we constructed it out of pairs), together with "everything which doesn't get paired with something" (i.e. "the fixed points", which we've just assumed is a set of odd cardinality); and even plus odd is odd.
+
+The upshot is: we can show that one particular involution on a set has a fixed point by showing that *some other involution of our choice* on that set has an odd number of fixed points.
+(Then the set itself must have odd cardinality, so our original involution must have a fixed point because all involutions on that set must.)
+
+# The involution that would witness \\(p\\) being a sum of two squares
+
+We want \\(x^2 + y^2 = p\\).
+
+Since \\(p\\) was assumed to be odd, we know by parity that exactly one of \\(x\\) and \\(y\\) is even.
+So (since squaring and square-rooting preserves parity) we could instead assume \\(y\\) is even and divide through to write the equation as \\(x^2 + 4y^2 = p\\).
+
+The magic rabbit out of the hat is to introduce a third variable \\(z\\), so that we're not finding solutions to \\(x^2 + 4y^2 = p\\) but instead \\(x^2 + 4yz = p\\) subject to the constraint that \\(y=z\\).
+Then that's just the same as asking for a fixed point of the involution \\((x, y, z) \mapsto (x, z, y)\\) where the involution is defined on a set whose members are all solutions to \\(x^2 + 4yz = p\\).
+
+This involution is super general: it's an involution with only very weak restrictions on its domain (that is, you need to be able to swap \\(y\\) with \\(z\\), but nothing else matters).
+To make that involution useful for our problem, we've additionally assumed that it's defined on a subset of \\(\mathbb{Z}^3\\), so that our fixed point is giving us *integer* solutions to the equation.
+
+The other half of the proof requires defining an involution on that same subset with an odd number of fixed points, and that's not necessarily easy; we'll later find it helpful to restrict the domain further so that *that* involution has fewer possible fixed points.
+
+# The involution that would witness \\(|S|\\) being odd
+
+Now we've defined \\(S = \{(x, y, z) : x^2 + 4yz = p\}\\) - which took the magic idea of introducing the third variable.
+So far, we've also hand-waved around the exact domain because we don't want to paint ourselves into a corner: we need all the flexibility we can get to define an involution that has an odd number of fixed points (at which point we'll be done with the entire proof).
+
+In fact, we'll make it conceptually easier for ourselves and go for an involution that has *exactly one* fixed point.
+
+## Choosing the fixed point
+
+How can we guarantee exactly one fixed point?
+Well, since \\(p\\) is \\(1 \pmod{4}\\), and we need *some* way for our proof to fail when \\(p\\) violates the hypotheses (otherwise we might prove too much and show the falsehood that *every* number is a sum of two squares!), it's not too much of a stretch to declare that our fixed point is going to be \\((x, y, z) = \left(1, 1, \frac{p-1}{4} \right)\\).
+(I chose that because it's definitely in \\(S\\), so the line of the proof that's going to fail when \\(p\\) violates the hypothesis is the line that says "this proposed fixed point is in \\(S\\)".)
+
+Now we "just" need to build our involution of some subset of \\(S\\), in such a way that the above \\((x, y, z)\\) is a fixed point, and in such a way that there are no other fixed points!
+(This isn't necessarily as hard as it sounds: there's a lot of freedom, especially because we've now listed literally all the requirements we're imposing on the involution.)
+
+## Possible functions which stay in \\(S\\)
+We can make this a fixed point with something like \\((x, y, z) \mapsto (y, x, z)\\), but we do need to make sure that we stay within \\(S\\), and we don't want to use up too much freedom by defining \\(S\\) so early to be some perverse subset of what we've currently got.
+
+Let's just hope for the best, and define \\(y \mapsto y\\).
+(We could also have tried keeping the \\(x\\) coordinate unchanged. I haven't checked happens when you try that.)
+
+Then \\((x, y, z) \mapsto (f(x,y,z), y, g(x,y,z))\\), and also \\(f(x,y,z)^2 + 4 y g(x,y,z) = p\\) (which is \\(x^2 + 4yz\\)).
+
+How can we make the left and right hand sides look more alike?
+\\(f\\) is definitely going to want to contain an \\(x\\) component so that we get the squared terms to simplify, so say \\(f(x,y,z) = \pm x + f'(x, y, z)\\).
+Expanding, \\(x^2 \pm 2 x f' + f'^2 + 4yg = x^2 + 4yz\\); so \\(\pm 2xf' + f'^2 + 4yg = 4yz\\).
+
+This is screaming at us to factor out at least a \\(2\\), so let \\(f' := 2 f'' \\); then \\(\pm xf'' + f''^2 + yg = yz\\).
+Similarly, we now want to factor out at least a \\(y\\), so let \\(f'' = y \hat{f}\\); then \\(\pm x \hat{f} + y \hat{f}^2 + g = z\\) (assuming \\(y\\) is nonzero, but we can assume whatever we want when we're playing around, as long as we check our final answer works at the end).
+
+To summarise, our involution so far looks like \\((x, y, z) \mapsto (\pm x + 2y \hat{f}, y, z \mp x \hat{f} - y \hat{f}^2)\\).
+All of this was done essentially without loss of generality: any generality we've lost can be made up for in \\(\hat{f}\\).
+
+## Making sure this has the right fixed point
+We haven't yet done anything to make this be an involution - everything so far was aimed at making sure we stayed within \\(S\\).
+We also haven't yet made sure that \\(\left(1, 1, \frac{p-1}{4}\right)\\) is a fixed point, so let's plug those in (writing \\(k\\) for the fraction): \\((1, 1, k) = (\pm 1 + 2 \hat{f}(1, 1, k), 1, k \mp \hat{f}(1, 1, k) - \hat{f}(1, 1, k)^2)\\).
+
+Looking at the \\(x\\)-coordinate, we see that either \\(\hat{f} = 0\\) or \\(\hat{f} = 1\\) at these coordinates.
+Moreover, if we look at what it means to have an involution, we end up with a very complicated constraint on what \\(\hat{f}\\) must be at certain points, but actually that condition is trivial if \\(\hat{f} = 0\\) always, or if \\(\hat{f} = 1\\) always and the \\(\pm\\) is the negative case.
+Before we tie ourselves completely in knots, it's worth checking whether either of those will do.
+
+Firstly, if \\(\hat{f} = 0\\) then our involution is \\((x, y, z) \mapsto (\pm x, y, z)\\).
+We'll discard the positive case straight away because that's got all the fixed points in the world.
+In the negative case: this certainly is an involution, and it's indeed well-defined as a function \\(S \to S\\), but it's not got \\((1, 1, k)\\) as a fixed point.
+
+So we try \\(\hat{f} = 1\\).
+Then the involution is \\((x, y, z) \mapsto (2y-x, y, z + x - y)\\).
+
+## Checking for any other fixed points
+
+We constructed this to be an involution, but we should sanity-check at this point, because some of my reasoning was a bit sloppy (e.g. I didn't check that its image was a subset of \\(S\\)!).
+
+* If \\((x, y, z) \in S\\), then we need \\((2y-x, y, z+x-y) \in S\\). This follows from some handle-cranking algebra.
+* It's got a fixed point at \\((1, 1, k)\\) for any \\(k\\).
+
+But in fact its fixed points are precisely when \\(x=y\\), which might be a bit of a problem, because we wanted to ensure there was *exactly one* fixed point.
+So far, we've got fixed points for all \\(x, z\\) such that \\(x^2 + 4xz = p\\).
+
+That's not such a problem, though.
+The left-hand side is divisible by \\(x\\), so (since the right-hand side is prime) we have \\(x = 0\\) (impossible because \\(p\\) is positive), or \\(x = 1\\) (in which case there's exactly one solution for \\(z\\)), or \\(x = p\\) (which we can rule out by decreeing that \\(x,y,z\\) all be positive, whereupon this choice would make \\(z\\) negative).
+
+## Adjusting for the further restriction we just applied
+
+We're actually kind of done!
+The only problem is that we ruled out \\(x=p\\) by decreeing that the members of \\(S\\) were nonnegative, which means our involution now no longer maps into \\(S\\).
+So we just need to carve up the involution so that it does what we've just described wherever that lands in \\(S\\), and otherwise we define it to be anything that retains the "involution" property but has no fixed points.
+
+When does our involution escape \\(S\\)?
+That's when \\(2y-x < 0\\) or \\(z+x-y < 0\\).
+We can do whatever we want to make sure we don't leave any fixed points in our new definitions in those ranges; one way is just to send the points from one case into the other case, and vice versa.
+
+If \\(2y < x\\), then, we'll be sending \\((x,y,z)\\) to some new values such that \\(x < y-z\\), and vice versa.
+We also need to preserve the property that \\(x^2 + 4yz = p\\), and we need to stay nonnegative.
+
+For the first case, \\(x\\) is very big, so let's just translate it down to be small enough: \\(x \mapsto x - 2y\\).
+(It's got to stay positive, so this is the obvious thing to try: make it as close as possible to being negative.)
+Then we have \\((x-2y)^2 + 4 \hat{y} \hat{z} = p\\), which after a litle algebra (using that \\(x^2 + 4yz = p\\)) simplifies by Mathematica's `FullSimplify` to \\(\hat{y} \hat{z} = y (x - y + z) \\).
+That suggests two possible ways to define the unknown \\(\hat{y}, \hat{z}\\), and once we've done that, it only remains to find the right one of those two ways!
+
+# Summary
+
+There was only really one magic idea: we can express \\(p\\) as a sum of two squares if we can find a fixed point of a certain involution.
+I don't have a good reason why one might not consider switching out \\(x^2 = xz\\) instead to get an involution. (I haven't tried it, and maybe it actually works!)
+"Could this be a fixed point of an involution?" is the sort of magic which you could have in the general box of tricks: it doesn't strike me as being particularly specialised, it just happens that this is a problem on which it works beautifully.
+
+The rest comes from following your nose, and in fact the situation turns out to be very welcoming:
+
+* There was actually an involution with the right properties that kept one coordinate constant.
+* It could easily have been that a much more complicated \\(\hat{f}\\) was needed.
diff --git a/hugo/content/reading-list/index.md b/hugo/content/reading-list/index.md
new file mode 100755
index 0000000..879850e
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+++ b/hugo/content/reading-list/index.md
@@ -0,0 +1,155 @@
+---
+lastmod: "2023-09-02T13:31:58.0000000+01:00"
+title: My reading list
+author: patrick
+layout: page
+---
+For a film list, see [films].
+
+This page holds a list of the books I am reading, and a list of books I have read (starting from the 8th July 2013, which is when the list started). I start with high hopes of keeping the list updated in real time, but of course this plan may go off the rails. There is also a list of things I very strongly recommend reading - they changed my thought patterns extensively, and/or are amazing books.
+
+# Things you should read
+*   [Don't Shoot the Dog] - Nonfiction book on animal training and operant conditioning: the art of getting obedience (from animals and humans alike) without using punishment. It has clarified my understanding of a whole variety of phenomena related to motivation and so on.
+*   [Inadequate Equilibria](https://equilibriabook.com/), by Eliezer Yudkowsky. Fascinating look at why everything is broken and nobody can fix anything. I'm a sucker for ideas presented as Socratic dialogue.
+*   [A Person Paper on Purity in Language][Person Paper], by Douglas Hofstadter (linked to the Wayback Machine version) - it is pretty shocking to realise just how much the English language discriminates against females, and how routine it is. Whether or not you think it's an issue, this is an excellent satire.
+*   [Gödel, Escher, Bach], by Douglas Hofstadter (an incredible book on pretty much everything - possibly the most meta thing I've ever read). Revisiting it now, I realise that I already know quite a lot of the mathematical content from different sources, but then to the extent that I specialised in anything during Part III, I specialised in logic.
+*   The [LessWrong] entry on [Cached Thoughts] - of the many fascinating LessWrong entries, this is probably the one that had the most profound and immediate effect on my thinking. I don't know if that effect was just a culmination of my previous readings, but this was the moment that I really got the idea that "human thinking was not designed, it's a bit of a miracle that we can think at all, and there are steps we can take to make our thinking less sloppy".
+*   [Harry Potter and the Methods of Rationality][HPMoR] - it's a very long fanfic about what might happen if Harry came from a different intellectual background and if every character were actually *trying* at life. It successfully manages to find coherent explanations for many of the random inconsistencies of the original. Give it a try - if you don't like it by the time we reach Hogwarts, feel free to stop. It's got its own subreddit, of which the most pertinent post is probably [what HPMoR is about][HPMoR subreddit]. People either find HPMoR horribly dull or absolutely incredible, as far as I can tell.
+*   Another LessWrong entry, this one on [effective altruism][EA LessWrong] ([factual summary][EA Wikipedia] at Wikipedia) - the holy grail of charitable giving is surely to get maximum bang for your buck. (On a vaguely related note, a link that for ethical reasons doesn't make it onto the list but seems worryingly insightful is a post from some random blog describing a horrible unit of currency, the "dead child", being the amount of money required to save the life of one child's life through charity - it makes very uncomfortable reading, so consider yourself duly warned that you might not want to read it. The idea is to make numbers like "£250,000 spent on a luxury dog kennel" make *sense* to us. The link is [this post on a horrible unit of currency][dead child].)
+*   [The Name of the Wind] by Patrick Rothfuss is the fantasy book with the best story I've ever read. It's the start of a trilogy (the Kingkiller Chronicle) and, if you like fantasy (and maybe even if you don't), is a stupendously well-told book. I gave it to a friend who professes never to read, and I didn't see em again for another week. It's not uniformly liked, though: another friend described it as an obvious self-insert wish-fulfillment fantasy from beginning to end.
+*   [Three Men in a Boat] - the funniest book I've read. After the first two pages, I can barely move after having dissolved in laughter, and it just gets better. I was banned from reading this in the presence of other people, probably because I was having too much fun.
+*   [Midnight's Children] - justly famous mystical fiction, set in twentieth century India. Wonderful book, beautiful turns of phrase and awe-inspiring plot.
+*   [Flowers for Algernon] - possibly the saddest book I've ever read, and at least one other person has agreed with me.
+*   [Turn The Ship Around!] - book on the nature of leadership. Central thesis: if a great commander leaves a ship and the ship falls apart straight away, in what sense is it reasonable to say they were a "great" commander? Its teachings are weirdly similar, in fact to [Never Split the Difference], though they're approaching two very different problems.
+*   The Debt to Pleasure, by John Lanchester - fiction/cookbook. One of the most erudite books I've read; this book really benefits from being read on a Kindle, with its inbuilt dictionary. A joy to read, excellent characterisation of the narrator. I'd advise not Googling the book before you read it (and ideally don't even read the blurb); there are spoilers to be found, and I think the best experience of this book would be to go in completely blind. Do read it, though; it has many beautiful turns of phrase, and is just generally extremely well written.
+
+[Gödel, Escher, Bach]: https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach
+[LessWrong]: http://www.lesswrong.com
+[Cached Thoughts]: http://lesswrong.com/lw/k5/cached_thoughts/
+[HPMoR]: http://hpmor.com
+[HPMoR subreddit]: https://www.reddit.com/r/HPMOR/comments/1h5yru/no_spoilers_so_what_exactly_is_harry_potter_and/
+[dead child]: http://web.archive.org/web/20150228045100/www.raikoth.net/deadchild.html "Horrible unit of currency"
+[The Name of the Wind]: http://www.patrickrothfuss.com/content/books.asp
+[Three Men in a Boat]: https://en.wikipedia.org/wiki/Three_Men_in_a_Boat
+[EA Wikipedia]: https://en.wikipedia.org/wiki/Effective_altruism "Effective altruism"
+[EA LessWrong]: http://lesswrong.com/lw/3gj/efficient_charity_do_unto_others/ "Effective altruism on LessWrong"
+[Person Paper]: http://web.archive.org/web/20130624210555/www.cs.virginia.edu/~evans/cs655/readings/purity.html "Person Paper on Purity in Language"
+[Midnight's Children]: https://en.wikipedia.org/wiki/Midnight's_Children
+[Flowers for Algernon]: https://en.wikipedia.org/wiki/Flowers_for_Algernon
+[Turn The Ship Around!]: https://davidmarquet.com/turn-the-ship-around-book/
+
+# Currently reading
+
+*   In Praise of Shadows, by Junichiro Tanizaki
+* Building Secure and Reliable Systems, by the Google SRE team. This is genuinely a textbook, so it's quite slow going. A lot of this is *very* Google-centric, where it's assumed that everything is a microservice and any given query to a service will require hundreds of RPC calls. However, the general lessons seem to be valuable.
+
+# Bought and ready to read
+
+*   The Attention Merchants, by Tim Wu
+*   American Gods, by Neil Gaiman
+
+# To read
+
+*   More from Less, by Andrew McAfee
+*   Superforecasting, by Philip Tetlock
+*   [A Formal Solution to the Grain of Truth problem](http://www.auai.org/uai2016/proceedings/papers/87.pdf), a paper from MIRI
+*   [Cubical Agda](http://www.cse.chalmers.se/~abela/types19-vezzosi.pdf), a paper describing an extension to Agda which supplies univalence and higher inductive types
+*   Linear Algebra Done Right, by Sheldon Axler (non-fiction, apparently the definitive textbook on linear algebra)
+*   Reinventing Discovery, by Michael Nielsen (non-fiction: how modern-day collaboration could remake science)
+*   The Strategy of Conflict, by Tom Schelling (book about game theory by a professional game theorist)
+*   Slaughterhouse Five, by Kurt Vonnegut (depressing fiction)
+*   The Quincunx, by Charles Palliser (Wilkie Collins-esque Dickensian fiction)
+*   Genesis, by Bernard Beckett (future ?dystopian fiction)
+*   Small Gods, by Terry Pratchett (fiction)
+*   Linked, by Albert Lazlo Barabassi (study of networks)
+*   Libriomancer, by Jim C Hines (fantasy fiction)
+*   Breakfast of Champions, by Kurt Vonnegut (fiction of some sort)
+*   Nature via Nurture, by Matt Ridley (study of interaction of genes and environment and development)
+*   The Culture series, by Iain M Banks
+*   A Place of Greater Safety, by Hilary Mantel
+*   So Good They Can't Ignore You, by Cal Newport
+*   84, Charing Cross Road, by Helene Hanff
+*   Black Box Thinking, by Matthew Syed
+*   Impro, by Keith Johnstone
+
+# Read some of and then put down
+
+*   [Legal Systems Very Different From Ours](http://web.archive.org/web/20220806200919/http://www.daviddfriedman.com/Legal%20Systems/LegalSystemsContents.htm): it's all in the title, really. Case studies of different legal systems, by the economist David Friedman. I read the first third or so of this, and it was a very interesting set of things to think about. How can a civilisation run itself without our systems of law courts? There have been many answers through the centuries.
+*   [Extraordinary Popular Delusions and the Madness of Crowds](https://en.wikipedia.org/wiki/Extraordinary_Popular_Delusions_and_the_Madness_of_Crowds), by Charles Mackay - nonfiction about the ways in which the crowd can get things wrong. This book is very, very long; I made it through the first half of the first volume before giving up. A pretty eye-opening book, though: Mackay's description (from the year 1841!) of the South Sea Bubble is almost indistinguishable from the recent cryptocurrency crazes, right down to an extensive list of some of the crazy companies that were eventually made illegal ("for improving the art of making soap", "for a wheel of perpetual motion", "for extracting silver from lead", "for a grand American fishery") - all very reminiscent of the various shitcoins.
+*   [Quantum Computing for the Very Curious](https://quantum.country/qcvc): essay to teach the fundamentals of quantum computing, with embedded spaced repetition cards.
+
+# Have read
+
+*   The Remains of the Day, by Kazuo Ishiguro. Short, sweet, and tragic. Simple throwaway lines become retrospectively rich in meaning as you read on. Spoiler in rot13: gur haeryvnoyr aneengbe tvirf guvf fgenatr frafr bs grafvba guebhtubhg.
+*   A Pocketful of Rye, by Agatha Christie. One of the Marples. Christie is generally excellent light reading, perhaps partly because her writing is all so familiar. This particular book I first read as an early teenager, but recently reread as part of a marathon in which we watched two adaptations and listened to a radio version; the exercise was actually quite interesting in seeing what the various adapters (and actors) chose to pull out.
+*   The Hornblower series, by CS Forester. I read some of these over and over as a child, and they bear rereading as an adult. The later books become less interesting to me, though, as Hornblower is promoted and gets further from the action.
+*   Station Eleven, by Emily St. John Mandel.
+*   Sea of Tranquility, by Emily St. John Mandel. Short and enjoyable mystery book set across pretty large tracts of space and time. I agree with a GoodReads reviewer, though, that Cloud Atlas does a very similar thing but much better.
+*   Cloud Atlas, by David Mitchell. Beautiful book. Each story is gripping in its own way. Imaginative.
+*   The Constant Rabbit, by Jasper Fforde. I love Fforde's stuff, but this particular book actually felt quite clumsy. The Fforde whimsy was generally ham-fistedly in service of a specific set of politics, which meant the overall sense was not "oh, that's clever" but "and this is where I'm meant to sneer at this group". Sadly not recommended.
+*   The Colour out of Space, by HP Lovecraft. A short fiction horror story which was indeed genuinely unsettling!
+*   Boyfriend Material, by Alexis Hall (and its sequel, Husband Material). I enjoyed this romcom, which describes two very different personalities trying to make it work. Laugh-out-loud funny in several places. Good light reading.
+*   The Little Learner, by Friedman and Mendhekar. A textbook introducing neural networks, with the rather unusual choice of Scheme as the programming language (as with the rest of the Little series). Sadly my main beef with machine learning frameworks in general is that they're all so weakly typed, and the choice of Scheme in a textbook really makes this problem much worse: as a student, you absolutely have to know what type everything has, so that you can understand it! I was struck by how tight a path this book treads, in that it teaches *precisely* what is necessary, in *precisely* the right order, to get to a very specific outcome (being able to train a network to predict the [iris dataset](https://en.wikipedia.org/wiki/Iris_flower_data_set)).
+*   The SRE handbook, by the Google Site Reliability Engineering team. Super interesting. A bunch of this stuff is slowly seeping into how I think about software, with the main obvious effect on my work being that I am now much more strongly inclined to notice when I'm doing a task that could in principle be automated. However, if you're not Google, some of this stuff is probably not aimed at you; they have unique problems. I think this is worth reading for any software engineer.
+*   Leadership Is Language, by David Marquet (sequel to Turn The Ship Around, discussed above). Writing this review some time after reading the book, I don't actually retain anything specific from it, but I have certainly cargo-culted some of what I know was in it - e.g. "the way you phrase things can have a strong effect on the responses you get, so think about what you're implicitly telling people when you choose to phrase something a particular way". I suspect this was worth reading.
+*   The Elephant In The Brain, by Robin Hanson. This is one of those books that colours everything you think about afterwards. The premise is that all aspects of humans can be explained, and more specifically that disparate things like art and modern medicine are in large part about social signalling. Very mind-expanding, very cynical-economist. Recommended.
+*   The Making Of The Atomic Bomb, by Richard Rhodes (non-fiction, about what it says on the tin). Absolutely brilliant book. Strong layman's introduction to the nuclear physics relevant to building the bomb, and plenty of humanising stories. I was not expecting to read so many pages of _Mein Kampf_ quoted in this book (I think most readers probably accept the "Hitler evil" premise). There was a chapter devoted to eyewitness accounts of the destruction of Hiroshima, which was literally sickening to read. Highly recommended.
+*   If on a winter's night a traveller, by Italo Calvino. I couldn't really put this down, but I also wouldn't describe it as "gripping". There are some lovely vignettes in here, and many of the stories I'd love to read more of. Recommended.
+*   Infinite Jest, by David Foster Wallace. This was an infuriating book. It's so, so long, and it's so nearly excellent throughout, but honestly I got to the end and thought it wasn't worth the time I'd put into it. Some lovely bits of wordplay, a fairly good plot, excellent characterisation; if it were half the length this would be a snap recommendation.
+*   To Be Taught If Fortunate, by Becky Chambers. Basically four vignettes about extrasolar worlds, each very different. Its core premise is rather different from any sci-fi I have ever read before, and the whole book is gentle.
+*   The Third Policeman, by Flann O'Brien. Bizarre book, some quite funny bits; I think the author may have read a lot of Salman Rushdie before writing this.
+*   The Hair-Carpet Weavers, by Andreas Eschbach, translated from German (Die Haarteppichknüpfer) by Doryl Jensen. An astonishing book. It starts out small, then zooms out again and again into an epic tale. Whenever you think it's going to play to some standard trope, it doesn't. Kind of jaw-dropping, and the mounting horror as you work out what's going on.
+*   A Gentleman in Moscow, by Amor Towles. A lovely book, with an air of The Grand Budapest Hotel. Witty, erudite without being too exclusive, and quite gripping.
+*   Neuromancer, by William Gibson. Honestly I think this book has suffered somewhat from being a genre-defining work: it's been copied and knocked off so many times that it feels a bit derivative now. Interestingly, my memory of this book is quite strongly "oh, it's like the film of Blade Runner".
+*   [Book Review: Progress and Poverty](https://astralcodexten.substack.com/p/your-book-review-progress-and-poverty), by Lars Doucet - non-fiction, a summary of Henry George's "Progress and Poverty". Very clear and readable intro to Georgism, the obviously correct economic policy.
+*   [Never Split The Difference], by Chris Voss - non-fiction, about negotiation. Fascinating read. Lots of stuff which is obvious in hindsight but was totally new to me. I think this book has contributed to my understanding of how people work, and I should probably reread it periodically.
+*   [The World of Null-A](https://en.wikipedia.org/wiki/The_World_of_Null-A), by A. E. van Vogt - science-fiction from 1948. I believe this book spawned the phrase "the map is not the territory". It was clearly not modern science-fiction: while I can certainly forgive anachronisms due to failure to predict the future (difficult by all accounts!), I believe the simple art of writing has advanced considerably since this was written. Parts of it are clumsy and rushed. The overall premise, though, I enjoyed a lot, and the glimpses into a different way of thinking based on a different philosophy of mind.
+*   [The Full Stack of Society](https://www.conradbastable.com/essays/the-full-stack-of-society-can-you-make-a-whole-society-wealthier-full-version), by Conrad Bastable - a moderately easy-going intro to economics and how to recognise certain general classes of method of wealth-extraction. I found this absolutely fascinating, with a new insight for me every couple of paragraphs; my economics background is pretty lacking (at the time of writing), and I'm sure he's not saying anything new or groundbreaking, but as an introduction for the novice this appears to be excellent.
+*   Thief of Time, by Terry Pratchett - one of the more standalone books of the Discworld; I usually recommend newcomers to the Discworld to start with this one. I first started reading Pratchett when I was probably a bit too young to understand him, but much as Pullman does with the Northern Lights trilogy, Pratchett is masterful in writing books that have real appeal to all audiences. The experience of reading Pratchett now is vastly richer.
+*   Why We Sleep, by Matthew Walker - nonfiction pop-science classic about the nature of sleep. This book reads like Walker has bought into some sort of cult; whether or not he's right, his bright-eyed enthusiasm reads as being naive, and much of what he says pattern-matches into the same class of statements as has caused the replication crisis in psychology. Everywhere you look in this book, you can see motivated reasoning and tortured post-hoc rationalisation. I'm aware that sleep is extremely important, but he could have spared us the improbable evo-psych justifications and the meaningless analogies.
+*   [Unsong], by Scott Alexander - silly fiction, full of puns, lots of Jewish lore. Very amusing; probably insufferable if you're not into puns.
+*   Some of the Mistborn books, by Brandon Sanderson - fantasy about a magical system. Sanderson is great at coming up with magical systems, but as I've got older I've stopped thinking his writing is particularly good.
+*   Watchmen, by Alan Moore et al - graphic novel about superheroes/masked vigilantes. The film is one of my favourite films, and it turns out to have been very true to the book. I think the book is better after having seen the film: I prefer the portrayal of Doctor Manhattan in the film, and there is a certain wildly silly plot device in the book which was replaced with a much more sensible one in the film. The book contains many little bits which made me go "ooh, that was good".
+*   The Signal and the Noise, by Nate Silver - non-fiction about what the founder of [FiveThirtyEight] has learnt about making predictions. Basically espouses Bayesian reasoning, so its ultimate message is not really new to me. It talks a *lot* about predictions in the context of American sports; I care not one iota about baseball, basketball or American football, so I just skimmed those chapters.
+*   Proofs and Refutations, by Imre Lakatos - examining how/whether proofs can be considered to prove things, and how finding counterexamples can change proofs. I've been told this book is required reading for students of the philosophy of mathematics, but I didn't find it particularly interesting. Perhaps I'm a barbarian, because everyone seems to think it's great.
+*   The Northern Lights trilogy, by Philip Pullman - brilliant books for both children and adults. I read these as a small-ish child and they were great stories; I've just reread them and they have become even better. Excellent.
+*   The Nursery Crime series, by Jasper Fforde - even better than the Thursday Next books (below), taking place in a fictionalised city of Reading where the local population includes nursery-rhyme and other oral-tradition characters.
+*   The Thursday Next series, by Jasper Fforde - absolutely hilarious fiction, some of it taking place within the BookWorld (the world of the written word). Meta ensues, and also lots and lots of references to other fiction, all written beautifully and very funnily.
+*   Lectures on the Hyperreals: an introduction to nonstandard analysis, by Robert Goldblatt - mathematical textbook, very readable as textbooks go, with an excellent introduction to hyperreals and some interesting applications. I've looked at a few presentations of nonstandard analysis, and this is by far the best I've read. It introduces everything neatly and in a good order, and has sufficient exercises.
+*   [Crystal: Society][Crystal], by Max Harms - fiction about the escape of an artificial intelligence. It's mostly plausible, and it features some genuinely alien aliens. Fun to read, and I couldn't put it down at the start (when it was all based in the same laboratory); as it got more complicated and the robot began moving around and interacting in different scenarios, I got a bit less interested. The exploration of the aliens picked it right back up, though.
+*   Dictator, by Robert Harris - great conclusion to the fictional-autobiographical trilogy following Cicero's life through the eyes of his secretary. All three books are excellent, and I learnt nearly everything I know about Roman culture from these books.
+*   [Rationality: from AI to Zombies], by Eliezer Yudkowsky - compendium of posts from Less Wrong about the art of rationality. I don't know what this would be like as an introduction, but it's designed for that purpose, and I certainly re-learnt a lot from the compendium that I'd forgotten from the individual posts.
+*   [Pact], by wildbow - fictional web serial about a certain system of magic. I preferred Worm (see below); Pact and Worm are both very long, so I'd recommend reading Worm over Pact, because time is actually a factor when reading these. It's less clear to the reader how the rules of the Pact universe govern the actions of the characters, which is probably why Worm wins.
+*   Category Theory, by Steve Awodey - non-fiction, "gentle" introduction to category theory, which covers about a third of the Part III course at Cambridge. For a maths book, it takes rather little effort to read.
+*   Bad Science, by Ben Goldacre - excellent layperson's introduction to trial design and how medical matters are misreported in the media. This is one of the books which really hammers home the principle that "you know how any news story about your area of knowledge is completely laughable? well, why do you trust the rest of the news?" It's a little out-of-date now, but for the best of reasons: the transgressions of some of the most topical chapters have been partially fixed in real life.
+*   Candide, by Voltaire (in English translation by Tobias Smollett) - mildly amusing throughout, with a couple of laugh-out-loud moments. The end is rather too overtly moralising, and I much preferred the first half of the book. Many of the references have dated badly, and without the footnotes I wouldn't have understood the point of large tracts of it.
+*   The Metamorphosis, by Franz Kafka (in English translation by David Wyllie) - tragic in several ways, and sometimes scary. Rather short; if it were longer, I might not say it was worth reading. I think Flowers for Algernon is better at doing what this book tries to do.
+*   The Count of Monte Cristo, by Alexandre Dumas (in English translation; I don't know who translated it, but it's the Project Gutenberg version) - fictional story of vengeance served cold. One of my favourite TV series is a French version of this with Gérard Depardieu, but I gave up on the book halfway through because it was just getting more and more boring. The TV series sets everything up beautifully all the time, revealing only parts of the plans as they become important; the book has basically everything going on in parallel, which takes away much of the magic. The characters are just tedious in the book, lacking in much personality at all.
+*   Operation Mincemeat, by Ben Macintyre (non-fiction about World War II) - gripping, reads like it's nearly a fiction book, some nice turns of phrase.
+*   The Gospel of Loki, by Joanne Harris (Norse mythology) - entertaining retelling of the Norse mythos, centred on Loki. Pretty self-aware, beautiful ending, and it's very fun even if you know the mythology.
+*   The Slow Regard of Silent Things, by Patrick Rothfuss (kind of fantasy, but mostly a vignette about a character) - a very pretty book, following Auri, a minor character in the Kingkiller Chronicles. It's short, gentle, and sweet; it bears rereading.
+*   Queen Lucia, by E F Benson (not sure how to describe it, other than "comedy of manners") - funny and gentle, lots of characters socially manipulating others. Also all the rest of the Mapp and Lucia series. They're all great fun. The kind of book I might read when ill.
+*   Grimus, by Salman Rushdie - very odd, slightly dreamlike book, full of puns and the sort of beautiful writing for which I loved Midnight's Children. Enjoyed it a lot, though it wasn't as good as Midnight's Children.
+*   [Superintelligence: Paths, Dangers, Strategies][Superintelligence], by Nick Bostrom (non-fiction treatment of the social and ethical implications of artificial superintelligence) - very well-written and persuasive about the potential dangers of naively-implemented artificial intelligence. Definitely worth a read.
+*   The Bourne Identity, by Robert Ludlum (thriller fiction) - this is like the film but with about a hundred more plot points. It's fairly well (but not astonishingly well) written - it's certainly gripping, but some of Bourne's panic scenes come across a little clumsy. There is also quite a lot of implausible exposition at various points - people are far too willing to talk about things they have no reason (other than plot) to discuss. Good read, though.
+*   Midnight's Children, by Salman Rushdie (mystical fiction) - absolutely excellent book, quite worthy of winning the Best of Booker prize. It is full of truly beautiful phrases, and every so often I literally sat back in awe at what I had just read. The first book of the three which make up Midnight's Children is extremely gripping in a wonderful-story kind of way, the second is a little less so, but the third is gripping in a riveting-plot way. I really, really recommend this book, and I think it's a travesty that students sometimes have to study it.
+*   Thinking: Fast and Slow, by Daniel Kahneman (cognitive science) - I already knew quite a lot of the content of this book (from my perusal of [LessWrong], for instance), but it was a good refresher. If you haven't had much exposure to rationality and cognitive bias, this book is extremely strongly recommended.
+*   Ender's Game, by Orson Scott Card (science-fiction with an overpowered main character) - interesting, but the character doesn't actually face many challenges. The protagonist of a book should have some difficulty overcoming problems - while Ender is nearly killed by his exertions, there is no particular adversary.
+*   Flow, by Mihály Csíkszentmihályi (cognitive science) - excellent book, although if you are already familiar with the [flow phenomenon][1], you probably won't learn much. The book sometimes comes across as a bit New Agey.
+*   [Worm] (online fiction about superheroes), by wildbow. Excellent superhero story, with characters who actually make sense and do sensible things. A huge variety of fascinating superpowers. Well worth reading, but it's very long and hard to put down, so choose your starting time well! I've scraped it into a PDF for Kindle reading, but the author has specifically requested that ebooks not be distributed, so I will only even consider giving it to you if you are personally a friend of mine.
+*   Consciousness Explained, by Daniel Dennet (explaining the Multiple Drafts view of consciousness) - very interesting book, thought-provoking, and I've absorbed some of its contents into my world-view.
+*   The Mind's I, by Douglas Hofstadter and Daniel Dennet (various musings on consciousness and the nature of the self) - this is a collection of pieces by other people, with commentary by Hofstadter and Dennet, which wasn't quite what I expected. I'd read several of the pieces before, although Hofstadter sparkled just as he usually does. I got bored after about the tenth excerpt, but certainly up to the eighth my attention was held.
+
+ [1]: https://en.wikipedia.org/wiki/Flow_%28psychology%29 "Flow Wikipedia page"
+ [Worm]: https://parahumans.wordpress.com/ "Worm (fiction)"
+ [Pact]: https://pactwebserial.wordpress.com/
+ [Don't Shoot the Dog]: https://www.goodreads.com/book/show/31052.Don_t_Shoot_the_Dog_
+ [LessWrong]: http://lesswrong.com
+ [Superintelligence]: https://en.wikipedia.org/wiki/Superintelligence_paths_dangers_strategies
+ [Rationality: from AI to Zombies]: https://intelligence.org/rationality-ai-zombies/
+ [Crystal]: http://crystal.raelifin.com/
+ [FiveThirtyEight]: https://fivethirtyeight.com
+ [Unsong]: http://unsongbook.com/
+ [Never Split the Difference]: https://www.goodreads.com/book/show/26156469-never-split-the-difference
+
+[films]: {{< baseurl >}}films
diff --git a/hugo/content/restaurants/index.md b/hugo/content/restaurants/index.md
new file mode 100644
index 0000000..fb63c52
--- /dev/null
+++ b/hugo/content/restaurants/index.md
@@ -0,0 +1,22 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+title: Restaurants
+author: patrick
+layout: page
+---
+
+This page is a list of restaurants I know to be good, or which I intend to try.
+
+# London
+
+* Dishoom (Bombay/Irani). For good reason are they something of a meme; I might actually agree that it's worth queueing outside to get in. There's almost always room for breakfast, and their breakfast is excellent (bacon and egg naan roll, chai). My favourite dishes thus far are probably the black lentil dahl and the ruby chicken.
+* Shoryu (ramen). Solid, middlingly-expensive chain. Everything they do is about equally good.
+* Taro (Japanese). On the cheaper end. The chicken teri don is a *lot* of food (if you double the chicken, it's considerably more than I could eat) and is really rather good. Bento boxes also good, but really the chicken teriyaki can't be beaten on price-to-calories.
+* Hawksmoor (steak). Very expensive. I'm not a steak person, but their fish menu is also excellent. No vegan offering, if I recall correctly.
+* Casa do Frango (Portuguese chicken; basically an upmarket Nando's). Located very near Borough Market. Quite a bit nicer than Nando's, and only modestly more expensive. The pastel de nata is *very* expensive but sublime. Vegan/vegetarian options very limited.
+* Applebee's Fish (fish). Located in Borough Market. We encountered no duds on their menu and said we'd go again.
+* Aster (Scandi/French). Located near Victoria. Pretty expensive, but we had a great birthday meal here.
+
+## Not tried
+
+* Heddon Yokocho (ramen). Jay Rayner recommended; better than Shoryu and same price; near Regent Street.
diff --git a/hugo/content/top-posts/index.md b/hugo/content/top-posts/index.md
new file mode 100644
index 0000000..2d40288
--- /dev/null
+++ b/hugo/content/top-posts/index.md
@@ -0,0 +1,39 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+title: Top Posts
+author: patrick
+layout: page
+---
+
+The generally accessible posts of which I am most proud on this website are:
+
+* [Anki as Learning Superpower] (how to squeeze a bit more out of your Anki usage)
+* [The Ravenous] (parody of [The Raven])
+* [Thinking Styles] (about how people think differently to each other)
+* [On Shakespeare] (the foolishness of teaching Shakespeare in school to pupils of a young age)
+
+The mathematical posts:
+
+* [Slightly Silly Sylow Pseudo-Sonnets][Sylow sonnets] (poetical proof of the Sylow Theorems)
+* [General Adjoint Functor Theorem][GAFT] (exposition on the GAFT and how it is actually a very natural theorem)
+
+The computer-science posts:
+
+* [Defunctionalisation] (on an underappreciated method of writing software)
+* [Continuation-passing style] (on motivating a technique in functional programming)
+* [Rewriting the Technical Interview, in Mathematica] (digging into the internals of Mathematica's parser)
+* [In praise of `--dry-run`][dryrun] (a technique for writing better tools)
+* [Metatesting your property-based tests] (an easy technique to improve your property-based tests)
+
+[The Ravenous]: {{< ref "2013-10-20-the-ravenous" >}}
+[The Raven]: https://en.wikipedia.org/wiki/The_Raven "The Raven Wikipedia page"
+[Sylow sonnets]: {{< ref "2013-08-31-slightly-silly-sylow-pseudo-sonnets" >}}
+[Thinking Styles]: {{< ref "2013-08-18-thinking-styles" >}}
+[On Shakespeare]: {{< ref "2013-07-21-on-shakespeare" >}}
+[GAFT]: {{< ref "2015-12-24-general-adjoint-functor-theorem" >}}
+[Defunctionalisation]: {{< ref "2020-03-04-defunctionalisation" >}}
+[Anki as Learning Superpower]: {{< ref "2020-10-23-anki-learning" >}}
+[Continuation-passing style]: {{< ref "2021-03-22-continuation-passing-style" >}}
+[Rewriting the Technical Interview, in Mathematica]: {{< ref "2021-03-17-rewriting-technical-interview" >}}
+[dryrun]: {{< ref "2021-02-20-in-praise-of-dry-run" >}}
+[Metatesting your property-based tests]: {{< ref "2021-02-06-property-based-metatesting" >}}
diff --git a/hugo/i18n/en.toml b/hugo/i18n/en.toml
new file mode 100644
index 0000000..c3b8d26
--- /dev/null
+++ b/hugo/i18n/en.toml
@@ -0,0 +1,6 @@
+[page_not_found]
+  other = "Page not found"
+[page_does_not_exist]
+  other = "Page does not exist"
+[head_back]
+  other = "<a href='{{.Site.BaseURL}}'>Head back home</a>"
\ No newline at end of file
diff --git a/hugo/layouts/shortcodes/baseurl.html b/hugo/layouts/shortcodes/baseurl.html
new file mode 100644
index 0000000..7a1f8be
--- /dev/null
+++ b/hugo/layouts/shortcodes/baseurl.html
@@ -0,0 +1,5 @@
+{{- if strings.HasSuffix .Page.Site.BaseURL "/" -}}
+    {{ .Page.Site.BaseURL }}
+{{- else -}}
+    {{ printf "%s/" .Page.Site.BaseURL }}
+{{- end -}}
\ No newline at end of file
diff --git a/hugo/layouts/shortcodes/foldergallery.html b/hugo/layouts/shortcodes/foldergallery.html
new file mode 100644
index 0000000..2baff43
--- /dev/null
+++ b/hugo/layouts/shortcodes/foldergallery.html
@@ -0,0 +1,18 @@
+<div class="gallery">
+  {{ $base := .Get "src" }}
+  {{ $path := print "static/" (.Get "src") }}
+
+  {{ range (readDir $path) }}
+    {{- $thumbext := "-thumb" }}
+    {{- $isthumb := .Name | findRE ($thumbext | printf "%s\\.") }}<!-- is the current file a thumbnail image? -->
+    {{- $isimg := lower .Name | findRE "\\.(gif|jpg|jpeg|tiff|png|bmp)" }}<!-- is the current file an image? -->
+    {{- if and $isimg (not $isthumb) }}
+    {{- $thumb := .Name | replaceRE "(\\.)" ($thumbext | printf "%s.") }}
+    {{- $alttextfile := .Name | replaceRE "(\\..+)" ".md" }}
+    {{- $alttext := printf "%s/%s" $path $alttextfile | os.ReadFile }}
+      <a data-fancybox="gallery" href="/{{ $base }}/{{ .Name }}">
+          <img src="/{{ $base  }}/{{ $thumb }}" width=100 height=100 alt={{ $.Page.RenderString $alttext }}>  <br/>
+      </a>
+  {{- end }}
+{{ end }}
+</div>
\ No newline at end of file
diff --git a/hugo/layouts/shortcodes/side.html b/hugo/layouts/shortcodes/side.html
new file mode 100644
index 0000000..5fdebda
--- /dev/null
+++ b/hugo/layouts/shortcodes/side.html
@@ -0,0 +1,7 @@
+<span class="sidenote">
+    <label class="sidenote-label" for="{{ .Get 1 }}"> {{ .Get 2 }}</label>
+    <input class="sidenote-checkbox" type="checkbox" id="{{ .Get 1 }}" />
+    <span class="sidenote-content sidenote-{{ .Get 0 }}">
+    {{ .Inner }}
+    </span>
+</span>
\ No newline at end of file
diff --git a/hugo/static/CNAME b/hugo/static/CNAME
new file mode 100644
index 0000000..15a05eb
--- /dev/null
+++ b/hugo/static/CNAME
@@ -0,0 +1 @@
+www.patrickstevens.co.uk
\ No newline at end of file
diff --git a/hugo/static/MysteryHunt2019/PictureBook.nb b/hugo/static/MysteryHunt2019/PictureBook.nb
new file mode 100644
index 0000000..7837327
--- /dev/null
+++ b/hugo/static/MysteryHunt2019/PictureBook.nb
@@ -0,0 +1,25225 @@
+(* Content-type: application/vnd.wolfram.mathematica *)
+
+(*** Wolfram Notebook File ***)
+(* http://www.wolfram.com/nb *)
+
+(* CreatedBy='Mathematica 12.0' *)
+
+(*CacheID: 234*)
+(* Internal cache information:
+NotebookFileLineBreakTest
+NotebookFileLineBreakTest
+NotebookDataPosition[       158,          7]
+NotebookDataLength[   1497841,      25217]
+NotebookOptionsPosition[   1492269,      25124]
+NotebookOutlinePosition[   1492607,      25139]
+CellTagsIndexPosition[   1492564,      25136]
+WindowFrame->Normal*)
+
+(* Beginning of Notebook Content *)
+Notebook[{
+Cell[BoxData[
+ RowBox[{
+  RowBox[{"img", "=", 
+   RowBox[{
+   "Import", "[", 
+    "\"\<https://web.mit.edu/puzzle/www/2019/assets/puzzles/picture_book/\
+image.png\>\"", "]"}]}], ";"}]], "Input",
+ CellChangeTimes->{{3.7882411776002073`*^9, 3.788241178964216*^9}, {
+  3.788245392803112*^9, 3.788245395636941*^9}},
+ CellLabel->"In[1]:=",ExpressionUUID->"9f5bbdd0-e526-45f0-8d98-429fbe918bbd"],
+
+Cell[CellGroupData[{
+
+Cell[BoxData["img"], "Input",
+ CellChangeTimes->{{3.822638079980654*^9, 3.822638080129547*^9}},
+ CellLabel->"In[34]:=",ExpressionUUID->"f6620b4c-97fb-4916-a3be-2c32439e8df7"],
+
+Cell[BoxData[
+ GraphicsBox[
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diff --git a/hugo/static/MysteryHunt2020/Story.nb b/hugo/static/MysteryHunt2020/Story.nb
new file mode 100644
index 0000000..3367c6c
--- /dev/null
+++ b/hugo/static/MysteryHunt2020/Story.nb
@@ -0,0 +1,406 @@
+(* Content-type: application/vnd.wolfram.mathematica *)
+
+(*** Wolfram Notebook File ***)
+(* http://www.wolfram.com/nb *)
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+
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+WindowFrame->Normal*)
+
+(* Beginning of Notebook Content *)
+Notebook[{
+Cell["Lots of data contained in the following un-Open cell.", "Text",
+ CellChangeTimes->{{3.822501195789774*^9, 
+  3.822501206196813*^9}},ExpressionUUID->"e5aca7b5-45b8-4062-aff1-\
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+Cell[BoxData[
+ RowBox[{
+  RowBox[{"s", "=", 
+   RowBox[{"{", 
+    RowBox[{
+    "\"\<I still remember the first time I ever visited the house. My friend \
+Emmett - from the Jazz lounge where I played sometimes Friday nights - had \
+lived there for years and invited me over for house dinner and to watch a \
+movie with him. When I arrived, he gave me a tour through the maze of rooms \
+and muralled hallways, even the treehouse out back with a drawbridge and a \
+fire pole from the third floor to the first. Just about every room in the \
+house (and there were a lot!) had a story attached to it, which made the tour \
+both really interesting and really long. When we finally heard the shouts for \
+dinnertime, I served myself the lentil curry, roasted brussels sprouts, and \
+quinoa, and wandered toward the tables. \\\"Sit over here!\\\" a guy wearing \
+an xkcd shirt called to me. I squeezed in beside him as he explained, \\\"we \
+call this the inductive table. If it can fit n people, it can fit n+1.\\\" \
+The conversation over the next half hour covered a Martian rover, raising \
+chickens, and principal ideal domains...I kind of felt like I'd known these \
+people my whole life. \\\"I wish I lived here,\\\" I said offhandedly as I \
+gathered my empty plate to wash. \\\"You totally should!\\\" said xkcd-guy \
+(whose name, it turned out, was Oliver). And I got to thinking. Much later, \
+in the darkness of a starless night as I walked home, the idea of moving in \
+seemed completely natural. I turned back to the house to see the rainbow \
+\\\"you are welcome here\\\" sign (on which someone had scrawled \\\"straight \
+people welcome on alternate Thursdays\\\") and the flower bed containing a \
+pinwheel masquerading as a pansy. I was sold.\>\"", ",", 
+     "\"\<The day I moved in, I was welcomed by a girl with lavender hair and \
+a pocket protector who gave me the unofficial tour. She pointed at a whip on \
+a hook by the door. \\\"That's in case of burglars and the like. Alice found \
+it at Goodwill along with a pair of handcuffs and a leather thong after this \
+sex dungeon went out of business.\\\" Next, she pointed at an alcove with a \
+public computer. \\\"That's to look up directions or to check facts on \
+Wikipedia if you get in an argument with someone and have to prove you're \
+right.\\\" I was entranced by the fractal screensaver that kept drawing \
+itself in exponentially multicolored smaller iterations until it was too \
+small to detect, then zoom in and keep going. The girl followed my stare. \
+\\\"Oh yeah, Ethan programmed the screensaver,\\\" she said, \\\"he could \
+keep a genius occupied for hours with math questions.\\\" As she led me up \
+the stairs, the wood floors creaked beneath our feet. \\\"It's an old house, \
+but it's totally safe,\\\" she told me. She tilted her head to the side. \
+\\\"I think,\\\" she added. \\\"And this is you!\\\" She dramatically opened \
+the door to my new room, which was super-duper tiny, but there was this prism \
+hung in the window and the late afternoon sun made shards of light dance \
+around the room. I looked at the rainbow lights hopping across the wood floor \
+and I was home. I moved the desk to the center of the room, directly below \
+the skylight in the ceiling, so I would feel like a princess in a spotlight \
+while doing psets, and plunked myself down on the chair. \\\"Seems like \
+you're all good here,\\\" she said with a smile, and it was only as she \
+turned to go that I noticed the tiny Rubik's cube earring in the loeb of her \
+left ear. Definitely at home.\>\"", ",", "\[IndentingNewLine]", 
+     "\"\<When Emmett and Savanah (also from the Jazz lounge) came to visit \
+the next week, I took them on a tour of my own through the maze of staircases \
+and rooms. \\\"This place is really cool,\\\" Savanah gushed, investigating \
+our hallway murals. \\\"It has so much charact -\\\" she stopped short and \
+stared open-mouthed at a guy dissecting an octopus and wearing exactly \
+nothing. I tugged at her elbow and led her through our common room and \
+towards our central staircase. \\\"That's Jacob,\\\" I explained. But I guess \
+that wasn't much of an explanation because she still looked like she was in \
+shock. \\\"He's course 7. He's doing his senior thesis on sea animals or \
+something.\\\" Savanah looked at me like I was crazy. \\\"But he wasn't \
+wearing anything!\\\" she exclaimed. \\\"Oh right. We're clothing-optional \
+here.\\\" She stared at me. \\\"It's really great when you forget to bring \
+your towel with you to the shower.\\\" Light from the skylight fell across \
+her skeptical face. How was it that in just a week I'd come to feel like I'd \
+lived here my whole life?\>\"", ",", 
+     "\"\<Sleep deprivation being the drug of choice in my life at the time, \
+it wasn't unusual for me to wake up disoriented with the imprint of my laptop \
+keyboard on my cheek and late afternoon sunlight streaming in through the \
+skylight above my desk. Such was my half-awake and dazed state the day I \
+walked outside and saw the pirate ship in the backyard, like a beacon \
+vertically aligned with the shed. I kind of blinked for a minute, because I \
+swear the yard was empty that morning and now there was this sailboat in it, \
+complete with a skull-and-crossbones flag. \\\"Ahoy matey!\\\" Oliver called \
+to me as he took a running start, leapt into the boat, and struck a pose. \
+\\\"Ahoy?\\\" I called back, a little confused by where this was going with \
+no water. The compass he held in his hand glinted in the sun. \\\"What's a \
+pirate's favorite mathematical field?\\\" he asked me. I stared at him. \
+\\\"Arrrr!\\\" called Ethan as he hurdled himself from \
+I'm-not-even-sure-where into the boat and gave Oliver a high five.\>\"", ",", 
+     "\[IndentingNewLine]", 
+     "\"\<By the end of the day, the boat, which Oliver had completely fixed \
+up from an old crumbling dinghy, was ready for its big debut in open water. \
+We got it loaded onto a makeshift cart-thingy and about a dozen of us walked \
+it down to the river for moonlight sail. Ethan and Oliver eased it into the \
+water and hopped on board, while the rest of us split into twos and threes \
+and boarded other sailboats at the dock. Soon we were all out on the river, \
+gliding peacefully through the crisp night air and gazing at Orion in the \
+southeastern skies. That is, until Alice and Jacob slid in close to Ethan and \
+Oliver and splashed them with a bucket of river water. Within minutes, all \
+five of our boats were in a water war, each of us attempting to get close \
+enough to another to throw water at one another and then sail quickly away. \
+We were all relatively evenly matched and relatively evenly soaked until the \
+pirate ship started pirating. Ryan and I watched it coming towards us and \
+prepared our bucket for attack. But just as it got within feet of our boat, \
+and before we could even lift the bucket, Ethan leapt off his vessel and \
+landed beside me. He grabbed hold of the sail of our boat with each hand and \
+leaned back with all his weight. Within moments I was in the river, coughing \
+and spluttering and looking around for our capsized dinghy. It only took me a \
+few seconds to locate it and swim towards it, and as I grabbed hold of the \
+side and began trying to right it with Ryan's help, I saw a dripping wet \
+Ethan grinning and waving at us beneath his black flag as he sailed away. My \
+teeth were chattering when we all walked home together about an hour later, \
+and I was desperately looking forward to the toastie warmth of home and a \
+promised fire in the fireplace. The rainbow sign on the door could not have \
+come soon enough. By the time I was dry and changed into fresh clothes, the \
+fire was crackling and half a dozen people gathered around it singing. I \
+joined in \\\"Maxwell's Silver Hammer\\\", warmed by the music of this cozy \
+little group gathered in the darkness lit only by the twisting flames. \
+\\\"Anyone want cookies?\\\" Amy asked as she came in from the kitchen with a \
+heavenly smelling tray, and my evening was complete.\>\"", ",", 
+     "\[IndentingNewLine]", 
+     "\"\<Despite all the warm-fuzzies of the fireplace and singing and \
+kumbaya and all that, Ethan was not off the hook. Oh no. I mean, it was all \
+in good fun, and yeah it was a pretty badass move, but Ryan and I knew we had \
+to plot a comeback. The idea came to me when Ryan sent out an email asking \
+all the cooks to send him their menus, since he was in charge of the shopping \
+(usually he tried to do his house job as early as possible in the week, but \
+he'd been hosed.) See, I knew that while Ethan loved broccoli and cauliflower \
+for their repetitive geometry, he hated tropical fruit. So I just suggested \
+we replace at least one requested item from each day's menu with a tropical \
+fruit as a little surprise. I figured the cooks would be totally down to \
+experiment with how to incorporate an unexpected ingredient...and I happen to \
+love papaya. By the time the last menu came in that evening, we only had \
+about a half hour before the grocery store closed for the night and in the \
+rush to go, I tripped over the flower bed outside the front door. By the time \
+we had scooped the soil back in, we had twenty minutes left. We literally ran \
+to the store, breathlessly grabbed a cart, and practically sailed through the \
+maze of aisles, trying to find each one of the dozens of items listed, not \
+always very helpfully (like seriously, somebody had written \\\"granola from \
+a brand that's called nature something or maybe something nature and I think \
+the package is red but it might be green\\\"). Anyway, by 10:59 our mission \
+was accomplished: all weekly food had been bought plus two bags of fruit. The \
+next day, Priya looked at the twenty mangoes we had bought her instead of her \
+requested bell peppers, read the note we had enclosed with it, and laughed. \
+We all enjoyed the mango pilaf that night...except Ethan. But the best part \
+was the look on his face when he tried the dragon fruit pudding (and the rush \
+he was in to rinse his mouth after he spit it out). Priceless.\>\"", ",", 
+     "\[IndentingNewLine]", 
+     "\"\<Ethan prepared his response that night. He crept out to the back \
+yard in the darkness with a pineapple in tow. He had decorated it to look \
+like a face with wild carrot-spirals for hair and a superman cape. He placed \
+his artwork at the end of the bow with a sign that said, \\\"I may not be \
+delicious but I am the captain of the pirate ship and I contain the Fibonacci \
+sequence. Mwahahaha.\\\"\>\"", ",", "\[IndentingNewLine]", 
+     "\"\<I noticed Captain Pineapple the following morning as I went out for \
+a run. Several of my housemates were already outside doing pushups and \
+sweating profusely. \\\"We're doing the Murph,\\\" Amanda gasped as she \
+rested for a moment. \\\"It's a CrossFit workout...next exercise is 300 \
+squats; want to join?\\\" she asked, with a can-do attitude. I watched Carlos \
+collapse into the grass and roll onto his back. \\\"I'm good,\\\" I said and \
+took off toward the Esplanade. The rush of blood felt good throughout my body \
+after a long week of psets and the early morning world seemed exceptionally \
+alive. I watched the T glide across the Longfellow bridge like a caterpillar \
+on ice skates and thought about all the people inside going places, thinking \
+thoughts, experiencing experiences...it just kind of suddenly felt like I was \
+this simultaneously insignificant and all-consuming being, like everything I \
+did and said was the entirety of my own life but it was all a speck of dust \
+in the world at large. I gazed at the sparkling river in the morning sunlight \
+that just days ago I had fallen into in the darkness of night, and I planned \
+my next move. As soon as I got home, I went to the fridge and took out all \
+the dragon fruit left over from the pudding. I cut it up and arranged the \
+pieces on the seat of the boat, forming the word, \\\"truce?\\\"\>\"", ",", 
+     "\[IndentingNewLine]", 
+     "\"\<After that, the pirate ship moved to the shed. In the years to \
+follow, it would go on many more journeys, but this is the tale we still tell \
+first to curious visitors. I had a professor that semester that always used \
+to say, \\\"this isn't fun-and-games topology; this is point-set \
+topology!\\\" And I guess I could say the same thing about that house. I mean \
+sure, it was fun, and sure there were games, but it wasn't just fun-and-games \
+coop life, and over time I would discover layers of depth in this group of \
+people that I hadn't noticed on first meeting. My housemates were the compass \
+that got me through each day that year. And sure, it was kind of annoying to \
+find that the Coke someone had left open on the side counter was still there \
+and still open 10 days later. And then to discover a lightened ring in the \
+wood of the countertop 14 days after that, and the Coke still there and still \
+open. And then to discover 9 days after that that was an intentional \
+experiment, not to see if it would grow mold (which it did!) but to see if \
+someone would take the initiative to throw it away (which obviously no one \
+did, but honestly we were just trying to be respectful of someone else's \
+drink/experiment!). And yeah, it's kind of fun to take a break at 3am from my \
+automorphic forms project and hear someone tell the story of the superman \
+Fibonacci fruit (and have someone else argue that it wasn't superman \
+Fibonacci fruit, it was electric bluegaloo Fibonacci fruit). But it wasn't \
+just about the games and it wasn't just about the pranks. It was about living \
+a life worth living with people worth living it with. And peace and love and \
+all that hippie stuff. So yeah, that's my story. The end.\>\""}], 
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diff --git a/hugo/static/MysteryHunt2020/Z3Interop.wl b/hugo/static/MysteryHunt2020/Z3Interop.wl
new file mode 100644
index 0000000..7ed0f6c
--- /dev/null
+++ b/hugo/static/MysteryHunt2020/Z3Interop.wl
@@ -0,0 +1,98 @@
+(* ::Package:: *)
+
+BeginPackage["Z3Interop`"]
+
+
+Assertion::usage="Assertion[condition] is a symbolic representation of the assertion that the given condition holds.";
+Declare::usage="Declare[var_Symbol, Integer] is a symbolic representation of the integer variable named var.";
+CheckSat::usage="CheckSat is a symbolic representation of the command to check satisfiability at the current point in the model specification.";
+GetModel::usage="GetModel is a symbolic representation of the command to print out a satisfying model.";
+
+getDefinitions::usage="getDefinitions extracts the integer variables from a Z3 output and converts them into Wolfram Language expressions."
+toString::usage="toString[symbols_, expr] converts a line of Z3 program (given as a specification either as a Wolfram Language expression or in terms of Z3Interop primitives) into a string.";
+
+
+Begin["`Private`"]
+
+
+toString[symbols_, Equal[i_,j_]]:=StringJoin["(= ",toString[symbols, i]," ",toString[symbols,j],")"]
+toString[symbols_, Greater[i_,j_]]:=StringJoin["(> ",toString[symbols,i]," ",toString[symbols,j],")"]
+toString[symbols_, Less[i_,j_]]:=StringJoin["(< ",toString[symbols,i]," ",toString[symbols,j],")"]
+toString[symbols_, GreaterEqual[i_,j_]]:=StringJoin["(>= ",toString[symbols,i]," ",toString[symbols,j],")"]
+toString[symbols_, LessEqual[i_,j_]]:=StringJoin["(<= ",toString[symbols,i]," ",toString[symbols,j],")"]
+toString[symbols_, Or[a_,b_]]:=StringJoin["(or ",toString[symbols,a]," ",toString[symbols,b],")"]
+toString[symbols_, Xor[a_,b_]]:=StringJoin["(xor ",toString[symbols,a]," ",toString[symbols,b],")"]
+toString[symbols_, And[a_,b_]]:=StringJoin["(and ",toString[symbols,a]," ",toString[symbols,b],")"]
+toString[symbols_, Not[a_]]:=StringJoin["(not ", toString[symbols,a], ")"]
+toString[symbols_, Unequal[a_, b_]]:=StringJoin["(not ",toString[symbols,a==b],")"]
+toString[symbols_, i_Integer]:=ToString[i]
+toString[symbols_, a_+b_]:=StringJoin["(+ ",toString[a]," ",toString[b],")"]
+toString[symbols_, Integer]="Int";
+toString[symbols_, Declare[var_,type_]]:=StringJoin["(declare-fun ",toString[symbols,var]," () ",toString[symbols,type],")"]
+toString[symbols_, Assertion[expr_]]:=StringJoin["(assert ",toString[symbols,expr],")"]
+toString[symbols_, CheckSat]:="(check-sat)"
+toString[symbols_, GetModel]:="(get-model)"
+toString[symbols_, h_[tail___]] /; MemberQ[First/@symbols,h]:=StringJoin[h/.symbols, "_", Riffle[toString[symbols, #]&/@{tail},"_"]]
+
+
+r=RegularExpression["\\(define-fun (.+?) \\(\\) Int\\s+([0-9]*)\\)"];
+getDefinitions[symbols_, output_]:=(First[#][Sequence@@(FromDigits[#,10]&/@Most[Rest[#]])]->FromDigits@Last[#])&/@(MapAt[#/.(Reverse/@symbols)&,StringSplit[#,"_"],1]&/@StringCases[StringReplace[output,"\n"->""],r->"$1_$2"]);
+
+
+End[]
+
+
+Protect[Assertion];
+Protect[Declare];
+Protect[CheckSat];
+Protect[GetModel];
+
+
+Begin["`Nonogram`"]
+
+
+gapsToConstraints::usage="gapsToConstraints[column_List, columnIndex_Integer, totalNumberOfRows_Integer, variableHead_Symbol] converts the data from the given column (which is column number `columnIndex`) into a set of constraints on that column (without reference to any rows). The same is true interchanging the notion of 'row' and 'column'.";
+constrainedCells::usage="constrainedCells[rowGap_Symbol, colGap_Symbol, cellVariable_Symbol, rows_, cols_, constraintsFromRows_, constraintsFromCols_,colourMapping_] uses the data from `gapsToConstraints` applied to the rows and columns, and produces the additional constraint which tie the rows and columns together. The output omits the original input constraints. `rows` and `cols` should be the actual data of the shape {{{index, colour}, {index, colour}}}.";
+
+
+Begin["`Private`"]
+
+
+Z3Interop`Nonogram`gapsToConstraints[column_,colNum_Integer,totalRows_Integer,variable_Symbol]:=
+With[{lhs=MapIndexed[variable[colNum,First@#2]+First@#1&,column]},
+With[{rhs=Append[variable[colNum,#+1]&/@Most@Range[Length@lhs],totalRows+1]},
+With[{operations=Append[If[#[[1]]===#[[2]],Less,LessEqual]&/@Partition[Last/@column,2,1],LessEqual]},
+With[{middle=#[[1]][#[[2]],#[[3]]]&/@Transpose[{operations,lhs,rhs}]},
+Join[{1<=variable[colNum,1]},middle]]]]]
+
+
+Z3Interop`Nonogram`constrainedCells[rowGap_Symbol, colGap_Symbol, cell_Symbol, rowsIn_, colsIn_, constrainedRows_, constrainedColumns_,mapping_] :=
+Join[
+(* Constraints on the rows *)
+Flatten[Table[
+Xor@@Prepend[
+(rowGap[row,#]<=col&&col<rowGap[row,#]+rowsIn[[row,#,1]] && cell[row,col]==rowsIn[[row,#,2]])&/@Range[Length@constrainedRows[[row]]-1],
+And@@Prepend[rowGap[row,#]>col||col>=rowGap[row,#]+rowsIn[[row,#,1]] &/@Range[Length@constrainedRows[[row]]-1],cell[row,col]==White]
+],
+{row,1,Length@rowsIn},{col,1,Length@colsIn}],1],
+(* Constraints on the columns *)
+Flatten[Table[
+Xor@@Prepend[
+(colGap[col,#]<=row&&row<colGap[col,#]+colsIn[[col,#,1]] && cell[row,col]==colsIn[[col,#,2]])&/@Range[Length@constrainedColumns[[col]]-1],
+And@@Prepend[colGap[col,#]>row||row>=colGap[col,#]+colsIn[[col,#,1]] &/@Range[Length@constrainedColumns[[col]]-1],cell[row,col]==White]
+],
+{row,1,Length@rowsIn},{col,1,Length@colsIn}],1]
+]/.mapping;
+
+
+End[]
+
+
+Protect[gapsToConstraints];
+Protect[constrainedCells];
+
+
+End[]
+
+
+EndPackage[]
diff --git a/hugo/static/MysteryHunt2020/nauseator.nb b/hugo/static/MysteryHunt2020/nauseator.nb
new file mode 100644
index 0000000..6cf72a4
--- /dev/null
+++ b/hugo/static/MysteryHunt2020/nauseator.nb
@@ -0,0 +1,1451 @@
+(* Content-type: application/vnd.wolfram.mathematica *)
+
+(*** Wolfram Notebook File ***)
+(* http://www.wolfram.com/nb *)
+
+(* CreatedBy='Mathematica 12.0' *)
+
+(*CacheID: 234*)
+(* Internal cache information:
+NotebookFileLineBreakTest
+NotebookFileLineBreakTest
+NotebookDataPosition[       158,          7]
+NotebookDataLength[     57800,       1443]
+NotebookOptionsPosition[     53068,       1366]
+NotebookOutlinePosition[     53529,       1384]
+CellTagsIndexPosition[     53486,       1381]
+WindowFrame->Normal*)
+
+(* Beginning of Notebook Content *)
+Notebook[{
+Cell[BoxData[
+ RowBox[{"Get", "@", 
+  RowBox[{"FileNameJoin", "[", 
+   RowBox[{"{", 
+    RowBox[{
+     RowBox[{"DirectoryName", "@", 
+      RowBox[{"NotebookFileName", "[", "]"}]}], ",", "\"\<Z3Interop.wl\>\""}],
+     "}"}], "]"}]}]], "Input",
+ InitializationCell->True,
+ CellChangeTimes->{{3.82254947495747*^9, 3.8225495093988934`*^9}},
+ CellLabel->"In[75]:=",ExpressionUUID->"3c1daa83-8c0a-4e99-ad7f-c3f04877f88a"],
+
+Cell[BoxData[
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+ CellLabel->"In[2]:=",ExpressionUUID->"7775dbb6-7ac4-457a-96d6-a9ae875b828b"],
+
+Cell[CellGroupData[{
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+  3.822505625838307*^9}},ExpressionUUID->"02135865-b6e5-4cad-ad4b-\
+5778f2375c71"],
+
+Cell["\<\
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+\>", "Text",
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+  3.822504539065948*^9}},ExpressionUUID->"54930d51-3a8d-4d53-bf35-\
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+   RowBox[{"fixColour", "[", "\"\<FFFF0000\>\"", "]"}], "=", "Red"}], 
+  ";"}], "\[IndentingNewLine]", 
+ RowBox[{
+  RowBox[{
+   RowBox[{"fixColour", "[", "\"\<FFFFFF00\>\"", "]"}], "=", "Yellow"}], 
+  ";"}], "\[IndentingNewLine]", 
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+   RowBox[{"Throw", "[", "x", "]"}]}], ";"}]}], "Input",
+ CellChangeTimes->{{3.822503800728251*^9, 3.822503887208557*^9}, {
+  3.8225039516130342`*^9, 3.822503977819137*^9}, {3.822552940100294*^9, 
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+Cell[BoxData[
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diff --git a/hugo/static/MysteryHunt2020/turtle.nb b/hugo/static/MysteryHunt2020/turtle.nb
new file mode 100644
index 0000000..b2b6f22
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+++ b/hugo/static/MysteryHunt2020/turtle.nb
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diff --git a/hugo/static/MysteryHunt2020/z3interop.nb b/hugo/static/MysteryHunt2020/z3interop.nb
new file mode 100644
index 0000000..f58262c
--- /dev/null
+++ b/hugo/static/MysteryHunt2020/z3interop.nb
@@ -0,0 +1,2917 @@
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+                "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+         DefaultBaseStyle->{},
+         Evaluator->Automatic,
+         Method->"Preemptive"],
+        RGBColor[1, 0, 0],
+        Editable->False,
+        Selectable->False]}], "}"}], ",", 
+     RowBox[{"{", 
+      RowBox[{"2", ",", 
+       InterpretationBox[
+        ButtonBox[
+         TooltipBox[
+          GraphicsBox[{
+            {GrayLevel[0], RectangleBox[{0, 0}]}, 
+            {GrayLevel[0], RectangleBox[{1, -1}]}, 
+            {GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}},
+           AspectRatio->1,
+           DefaultBaseStyle->"ColorSwatchGraphics",
+           Frame->True,
+           FrameStyle->GrayLevel[0.],
+           FrameTicks->None,
+           
+           ImageSize->
+            Dynamic[{
+             Automatic, 1.35 CurrentValue["FontCapHeight"]/
+              AbsoluteCurrentValue[Magnification]}],
+           PlotRangePadding->None],
+          StyleBox[
+           RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]],
+         Appearance->None,
+         BaseStyle->{},
+         BaselinePosition->Baseline,
+         ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+           If[
+            Not[
+             AbsoluteCurrentValue["Deployed"]], 
+            SelectionMove[Typeset`box$, All, Expression]; 
+            FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+            FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; 
+            FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+            MathLink`CallFrontEnd[
+              FrontEnd`AttachCell[Typeset`box$, 
+               FrontEndResource["GrayLevelColorValueSelector"], {
+               0, {Left, Bottom}}, {Left, Top}, 
+               "ClosingActions" -> {
+                "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+         DefaultBaseStyle->{},
+         Evaluator->Automatic,
+         Method->"Preemptive"],
+        GrayLevel[0],
+        Editable->False,
+        Selectable->False]}], "}"}], ",", 
+     RowBox[{"{", 
+      RowBox[{"2", ",", 
+       InterpretationBox[
+        ButtonBox[
+         TooltipBox[
+          GraphicsBox[{
+            {GrayLevel[0], RectangleBox[{0, 0}]}, 
+            {GrayLevel[0], RectangleBox[{1, -1}]}, 
+            {RGBColor[0, 0, 1], RectangleBox[{0, -1}, {2, 1}]}},
+           AspectRatio->1,
+           DefaultBaseStyle->"ColorSwatchGraphics",
+           Frame->True,
+           FrameStyle->RGBColor[0., 0., 0.6666666666666666],
+           FrameTicks->None,
+           
+           ImageSize->
+            Dynamic[{
+             Automatic, 1.35 CurrentValue["FontCapHeight"]/
+              AbsoluteCurrentValue[Magnification]}],
+           PlotRangePadding->None],
+          StyleBox[
+           RowBox[{"RGBColor", "[", 
+             RowBox[{"0", ",", "0", ",", "1"}], "]"}], NumberMarks -> False]],
+         
+         Appearance->None,
+         BaseStyle->{},
+         BaselinePosition->Baseline,
+         ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+           If[
+            Not[
+             AbsoluteCurrentValue["Deployed"]], 
+            SelectionMove[Typeset`box$, All, Expression]; 
+            FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+            FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0, 0, 1]; 
+            FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+            MathLink`CallFrontEnd[
+              FrontEnd`AttachCell[Typeset`box$, 
+               FrontEndResource["RGBColorValueSelector"], {
+               0, {Left, Bottom}}, {Left, Top}, 
+               "ClosingActions" -> {
+                "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+         DefaultBaseStyle->{},
+         Evaluator->Automatic,
+         Method->"Preemptive"],
+        RGBColor[0, 0, 1],
+        Editable->False,
+        Selectable->False]}], "}"}], ",", 
+     RowBox[{"{", 
+      RowBox[{"5", ",", 
+       InterpretationBox[
+        ButtonBox[
+         TooltipBox[
+          GraphicsBox[{
+            {GrayLevel[0], RectangleBox[{0, 0}]}, 
+            {GrayLevel[0], RectangleBox[{1, -1}]}, 
+            {RGBColor[0, 0, 1], RectangleBox[{0, -1}, {2, 1}]}},
+           AspectRatio->1,
+           DefaultBaseStyle->"ColorSwatchGraphics",
+           Frame->True,
+           FrameStyle->RGBColor[0., 0., 0.6666666666666666],
+           FrameTicks->None,
+           
+           ImageSize->
+            Dynamic[{
+             Automatic, 1.35 CurrentValue["FontCapHeight"]/
+              AbsoluteCurrentValue[Magnification]}],
+           PlotRangePadding->None],
+          StyleBox[
+           RowBox[{"RGBColor", "[", 
+             RowBox[{"0", ",", "0", ",", "1"}], "]"}], NumberMarks -> False]],
+         
+         Appearance->None,
+         BaseStyle->{},
+         BaselinePosition->Baseline,
+         ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+           If[
+            Not[
+             AbsoluteCurrentValue["Deployed"]], 
+            SelectionMove[Typeset`box$, All, Expression]; 
+            FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+            FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0, 0, 1]; 
+            FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+            MathLink`CallFrontEnd[
+              FrontEnd`AttachCell[Typeset`box$, 
+               FrontEndResource["RGBColorValueSelector"], {
+               0, {Left, Bottom}}, {Left, Top}, 
+               "ClosingActions" -> {
+                "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+         DefaultBaseStyle->{},
+         Evaluator->Automatic,
+         Method->"Preemptive"],
+        RGBColor[0, 0, 1],
+        Editable->False,
+        Selectable->False]}], "}"}], ",", 
+     RowBox[{"{", 
+      RowBox[{"1", ",", 
+       InterpretationBox[
+        ButtonBox[
+         TooltipBox[
+          GraphicsBox[{
+            {GrayLevel[0], RectangleBox[{0, 0}]}, 
+            {GrayLevel[0], RectangleBox[{1, -1}]}, 
+            {RGBColor[1, 0, 0], RectangleBox[{0, -1}, {2, 1}]}},
+           AspectRatio->1,
+           DefaultBaseStyle->"ColorSwatchGraphics",
+           Frame->True,
+           FrameStyle->RGBColor[0.6666666666666666, 0., 0.],
+           FrameTicks->None,
+           
+           ImageSize->
+            Dynamic[{
+             Automatic, 1.35 CurrentValue["FontCapHeight"]/
+              AbsoluteCurrentValue[Magnification]}],
+           PlotRangePadding->None],
+          StyleBox[
+           RowBox[{"RGBColor", "[", 
+             RowBox[{"1", ",", "0", ",", "0"}], "]"}], NumberMarks -> False]],
+         
+         Appearance->None,
+         BaseStyle->{},
+         BaselinePosition->Baseline,
+         ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+           If[
+            Not[
+             AbsoluteCurrentValue["Deployed"]], 
+            SelectionMove[Typeset`box$, All, Expression]; 
+            FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+            FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[1, 0, 0]; 
+            FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+            MathLink`CallFrontEnd[
+              FrontEnd`AttachCell[Typeset`box$, 
+               FrontEndResource["RGBColorValueSelector"], {
+               0, {Left, Bottom}}, {Left, Top}, 
+               "ClosingActions" -> {
+                "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+         DefaultBaseStyle->{},
+         Evaluator->Automatic,
+         Method->"Preemptive"],
+        RGBColor[1, 0, 0],
+        Editable->False,
+        Selectable->False]}], "}"}], ",", 
+     RowBox[{"{", 
+      RowBox[{"4", ",", 
+       InterpretationBox[
+        ButtonBox[
+         TooltipBox[
+          GraphicsBox[{
+            {GrayLevel[0], RectangleBox[{0, 0}]}, 
+            {GrayLevel[0], RectangleBox[{1, -1}]}, 
+            {RGBColor[1, 0, 0], RectangleBox[{0, -1}, {2, 1}]}},
+           AspectRatio->1,
+           DefaultBaseStyle->"ColorSwatchGraphics",
+           Frame->True,
+           FrameStyle->RGBColor[0.6666666666666666, 0., 0.],
+           FrameTicks->None,
+           
+           ImageSize->
+            Dynamic[{
+             Automatic, 1.35 CurrentValue["FontCapHeight"]/
+              AbsoluteCurrentValue[Magnification]}],
+           PlotRangePadding->None],
+          StyleBox[
+           RowBox[{"RGBColor", "[", 
+             RowBox[{"1", ",", "0", ",", "0"}], "]"}], NumberMarks -> False]],
+         
+         Appearance->None,
+         BaseStyle->{},
+         BaselinePosition->Baseline,
+         ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+           If[
+            Not[
+             AbsoluteCurrentValue["Deployed"]], 
+            SelectionMove[Typeset`box$, All, Expression]; 
+            FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+            FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[1, 0, 0]; 
+            FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+            MathLink`CallFrontEnd[
+              FrontEnd`AttachCell[Typeset`box$, 
+               FrontEndResource["RGBColorValueSelector"], {
+               0, {Left, Bottom}}, {Left, Top}, 
+               "ClosingActions" -> {
+                "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+         DefaultBaseStyle->{},
+         Evaluator->Automatic,
+         Method->"Preemptive"],
+        RGBColor[1, 0, 0],
+        Editable->False,
+        Selectable->False]}], "}"}], ",", 
+     RowBox[{"{", 
+      RowBox[{"4", ",", 
+       InterpretationBox[
+        ButtonBox[
+         TooltipBox[
+          GraphicsBox[{
+            {GrayLevel[0], RectangleBox[{0, 0}]}, 
+            {GrayLevel[0], RectangleBox[{1, -1}]}, 
+            {RGBColor[1, 0, 0], RectangleBox[{0, -1}, {2, 1}]}},
+           AspectRatio->1,
+           DefaultBaseStyle->"ColorSwatchGraphics",
+           Frame->True,
+           FrameStyle->RGBColor[0.6666666666666666, 0., 0.],
+           FrameTicks->None,
+           
+           ImageSize->
+            Dynamic[{
+             Automatic, 1.35 CurrentValue["FontCapHeight"]/
+              AbsoluteCurrentValue[Magnification]}],
+           PlotRangePadding->None],
+          StyleBox[
+           RowBox[{"RGBColor", "[", 
+             RowBox[{"1", ",", "0", ",", "0"}], "]"}], NumberMarks -> False]],
+         
+         Appearance->None,
+         BaseStyle->{},
+         BaselinePosition->Baseline,
+         ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+           If[
+            Not[
+             AbsoluteCurrentValue["Deployed"]], 
+            SelectionMove[Typeset`box$, All, Expression]; 
+            FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+            FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[1, 0, 0]; 
+            FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+            MathLink`CallFrontEnd[
+              FrontEnd`AttachCell[Typeset`box$, 
+               FrontEndResource["RGBColorValueSelector"], {
+               0, {Left, Bottom}}, {Left, Top}, 
+               "ClosingActions" -> {
+                "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+         DefaultBaseStyle->{},
+         Evaluator->Automatic,
+         Method->"Preemptive"],
+        RGBColor[1, 0, 0],
+        Editable->False,
+        Selectable->False]}], "}"}], ",", 
+     RowBox[{"{", 
+      RowBox[{"12", ",", 
+       InterpretationBox[
+        ButtonBox[
+         TooltipBox[
+          GraphicsBox[{
+            {GrayLevel[0], RectangleBox[{0, 0}]}, 
+            {GrayLevel[0], RectangleBox[{1, -1}]}, 
+            {GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}},
+           AspectRatio->1,
+           DefaultBaseStyle->"ColorSwatchGraphics",
+           Frame->True,
+           FrameStyle->GrayLevel[0.],
+           FrameTicks->None,
+           
+           ImageSize->
+            Dynamic[{
+             Automatic, 1.35 CurrentValue["FontCapHeight"]/
+              AbsoluteCurrentValue[Magnification]}],
+           PlotRangePadding->None],
+          StyleBox[
+           RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]],
+         Appearance->None,
+         BaseStyle->{},
+         BaselinePosition->Baseline,
+         ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+           If[
+            Not[
+             AbsoluteCurrentValue["Deployed"]], 
+            SelectionMove[Typeset`box$, All, Expression]; 
+            FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+            FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; 
+            FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+            MathLink`CallFrontEnd[
+              FrontEnd`AttachCell[Typeset`box$, 
+               FrontEndResource["GrayLevelColorValueSelector"], {
+               0, {Left, Bottom}}, {Left, Top}, 
+               "ClosingActions" -> {
+                "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+         DefaultBaseStyle->{},
+         Evaluator->Automatic,
+         Method->"Preemptive"],
+        GrayLevel[0],
+        Editable->False,
+        Selectable->False]}], "}"}]}], "}"}]}], ";"}]], "Input",
+ CellChangeTimes->{{3.822548903710017*^9, 3.82254890451241*^9}, {
+  3.8225515791443157`*^9, 3.822551579952506*^9}},
+ CellLabel->"In[7]:=",ExpressionUUID->"081cfc5e-5d5a-4391-b143-93cdddea7504"],
+
+Cell["\<\
+Find the positions of colour changes x1,x2,x3,x4,\[Ellipsis] such that:
+1<=x1
+x1+4<=x2
+x2+2<=x3
+x3+2<x4
+x4+5<=x5
+x5+1<x6
+x6+4<x7
+x7+4<=x8
+x8+12<=#rows\
+\>", "Text",
+ CellChangeTimes->{{3.7883347723337708`*^9, 3.788334867311164*^9}, {
+   3.8225407857864428`*^9, 3.822540789481996*^9}, 3.8225421024396*^9, {
+   3.822545965661354*^9, 
+   3.822545973469831*^9}},ExpressionUUID->"aa91654c-6ec8-477c-b2b8-\
+e4f1bd9156ea"],
+
+Cell["\<\
+We represent this as the list of left-hand sides (x1+4, x2+2, \[Ellipsis], \
+x8+12), right-hand sides (x2, x3, \[Ellipsis], x9), and operations (<=, <=, \
+<, \[Ellipsis]), then prepend the first line and append the last.\
+\>", "Text",
+ CellChangeTimes->{{3.822542603335876*^9, 
+  3.8225426393920116`*^9}},ExpressionUUID->"2b7b53b0-0e04-44ac-9201-\
+6100ed0fcc0f"],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"gapsToConstraints", "[", 
+  RowBox[{"col", ",", "1", ",", "60", ",", "colGap"}], "]"}]], "Input",
+ CellChangeTimes->{{3.822542658635066*^9, 3.822542667656575*^9}, 
+   3.8225460347447777`*^9, {3.822551443401033*^9, 3.8225514507274446`*^9}, {
+   3.822551585243217*^9, 3.82255159216156*^9}, 3.8225516429649343`*^9, {
+   3.822551811290702*^9, 3.822551829202127*^9}, {3.8225518945045357`*^9, 
+   3.8225519686303368`*^9}},
+ CellLabel->"In[9]:=",ExpressionUUID->"3bf729b4-46c8-4e1a-96bd-4753e38e461b"],
+
+Cell[BoxData[
+ RowBox[{"{", 
+  RowBox[{
+   RowBox[{"1", "\[LessEqual]", 
+    RowBox[{"colGap", "[", 
+     RowBox[{"1", ",", "1"}], "]"}]}], ",", 
+   RowBox[{
+    RowBox[{"4", "+", 
+     RowBox[{"colGap", "[", 
+      RowBox[{"1", ",", "1"}], "]"}]}], "\[LessEqual]", 
+    RowBox[{"colGap", "[", 
+     RowBox[{"1", ",", "2"}], "]"}]}], ",", 
+   RowBox[{
+    RowBox[{"2", "+", 
+     RowBox[{"colGap", "[", 
+      RowBox[{"1", ",", "2"}], "]"}]}], "\[LessEqual]", 
+    RowBox[{"colGap", "[", 
+     RowBox[{"1", ",", "3"}], "]"}]}], ",", 
+   RowBox[{
+    RowBox[{"2", "+", 
+     RowBox[{"colGap", "[", 
+      RowBox[{"1", ",", "3"}], "]"}]}], "<", 
+    RowBox[{"colGap", "[", 
+     RowBox[{"1", ",", "4"}], "]"}]}], ",", 
+   RowBox[{
+    RowBox[{"5", "+", 
+     RowBox[{"colGap", "[", 
+      RowBox[{"1", ",", "4"}], "]"}]}], "\[LessEqual]", 
+    RowBox[{"colGap", "[", 
+     RowBox[{"1", ",", "5"}], "]"}]}], ",", 
+   RowBox[{
+    RowBox[{"1", "+", 
+     RowBox[{"colGap", "[", 
+      RowBox[{"1", ",", "5"}], "]"}]}], "<", 
+    RowBox[{"colGap", "[", 
+     RowBox[{"1", ",", "6"}], "]"}]}], ",", 
+   RowBox[{
+    RowBox[{"4", "+", 
+     RowBox[{"colGap", "[", 
+      RowBox[{"1", ",", "6"}], "]"}]}], "<", 
+    RowBox[{"colGap", "[", 
+     RowBox[{"1", ",", "7"}], "]"}]}], ",", 
+   RowBox[{
+    RowBox[{"4", "+", 
+     RowBox[{"colGap", "[", 
+      RowBox[{"1", ",", "7"}], "]"}]}], "\[LessEqual]", 
+    RowBox[{"colGap", "[", 
+     RowBox[{"1", ",", "8"}], "]"}]}], ",", 
+   RowBox[{
+    RowBox[{"12", "+", 
+     RowBox[{"colGap", "[", 
+      RowBox[{"1", ",", "8"}], "]"}]}], "\[LessEqual]", "61"}]}], 
+  "}"}]], "Output",
+ CellChangeTimes->{
+  3.8225426679409018`*^9, 3.8225460376953783`*^9, {3.822546067794279*^9, 
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+   3.8225514509951763`*^9}, {3.822551585402956*^9, 3.822551607610095*^9}, {
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+   3.822551918670422*^9}, {3.8225519487021513`*^9, 3.822551968945219*^9}, 
+   3.822552073176504*^9},
+ CellLabel->"Out[9]=",ExpressionUUID->"5a00bdce-b583-4188-90ce-b44cd2c866d4"]
+}, Open  ]],
+
+Cell[BoxData[
+ RowBox[{
+  RowBox[{"constrainedColumns", ":=", 
+   RowBox[{"MapIndexed", "[", 
+    RowBox[{
+     RowBox[{
+      RowBox[{"gapsToConstraints", "[", 
+       RowBox[{"#1", ",", 
+        RowBox[{"First", "@", "#2"}], ",", 
+        RowBox[{"Length", "@", "rowsIn"}], ",", "colGap"}], "]"}], "&"}], ",",
+      "colsIn"}], "]"}]}], ";"}]], "Input",
+ CellChangeTimes->{{3.822542276174329*^9, 3.8225423268775177`*^9}, 
+   3.822546145935195*^9},
+ CellLabel->"In[10]:=",ExpressionUUID->"233d1ff7-031b-4ad0-999c-c79ffef67cbd"],
+
+Cell[BoxData[
+ RowBox[{
+  RowBox[{"constrainedRows", ":=", 
+   RowBox[{"MapIndexed", "[", 
+    RowBox[{
+     RowBox[{
+      RowBox[{"gapsToConstraints", "[", 
+       RowBox[{"#1", ",", 
+        RowBox[{"First", "@", "#2"}], ",", 
+        RowBox[{"Length", "@", "colsIn"}], ",", "rowGap"}], "]"}], "&"}], ",",
+      "rowsIn"}], "]"}]}], ";"}]], "Input",
+ CellChangeTimes->{{3.822542338936245*^9, 3.82254235153629*^9}, 
+   3.822546147070763*^9},
+ CellLabel->"In[11]:=",ExpressionUUID->"f708add8-2e46-4ba2-abae-b327ee76fabf"],
+
+Cell["\<\
+And tie together the row and column constraints.
+The cell at index {row, column} is of colour \
+colsIn\[LeftDoubleBracket]column, i, 2\[RightDoubleBracket] if colGap[column, \
+i] <= row < colGap[column, i]+colsIn\[LeftDoubleBracket]column,i,1\
+\[RightDoubleBracket], and of colour white otherwise.
+Similarly for the rows.\
+\>", "Text",
+ CellChangeTimes->{{3.822542410601672*^9, 3.8225424178323603`*^9}, {
+   3.8225427665073023`*^9, 3.822542767901464*^9}, {3.8225432294539423`*^9, 
+   3.822543248110524*^9}, {3.8225433595858517`*^9, 3.822543619335075*^9}, {
+   3.82254389249254*^9, 3.822543938002324*^9}, 3.822544695943574*^9, {
+   3.822548621431777*^9, 3.822548647592437*^9}, {3.8225487431283493`*^9, 
+   3.8225487667290583`*^9}},ExpressionUUID->"77deb73a-f5f7-46c6-86d4-\
+60fecdb64e5b"],
+
+Cell[BoxData[
+ RowBox[{
+  RowBox[{"additionalConstraints", ":=", 
+   RowBox[{"constrainedCells", "[", 
+    RowBox[{
+    "rowGap", ",", "colGap", ",", "cell", ",", "rowsIn", ",", "colsIn", ",", 
+     "constrainedRows", ",", "constrainedColumns", ",", "mapping"}], "]"}]}], 
+  ";"}]], "Input",
+ CellChangeTimes->{{3.822551299141808*^9, 3.8225513326405487`*^9}, {
+  3.8225522315043573`*^9, 3.8225522323102083`*^9}},
+ CellLabel->"In[26]:=",ExpressionUUID->"34d3d4e1-69c4-4677-9d9f-76c813fd905c"],
+
+Cell["Form the program:", "Text",
+ CellChangeTimes->{{3.822543172308032*^9, 3.8225431759837637`*^9}, {
+  3.8225432065952673`*^9, 
+  3.8225432068193693`*^9}},ExpressionUUID->"0d554cf3-6796-49cf-9639-\
+f2e39f1d1c0d"],
+
+Cell[BoxData[
+ RowBox[{
+  RowBox[{"vars", ":=", 
+   RowBox[{"DeleteDuplicates", "@", 
+    RowBox[{"Flatten", "@", 
+     RowBox[{"{", 
+      RowBox[{
+       RowBox[{"Cases", "[", 
+        RowBox[{
+         RowBox[{"{", 
+          RowBox[{
+          "constrainedCells", ",", "constrainedColumns", ",", 
+           "constrainedRows"}], "}"}], ",", 
+         RowBox[{"colGap", "[", 
+          RowBox[{"_", ",", "_"}], "]"}], ",", "Infinity"}], "]"}], ",", 
+       RowBox[{"Cases", "[", 
+        RowBox[{
+         RowBox[{"{", 
+          RowBox[{
+          "additionalConstraints", ",", "constrainedColumns", ",", 
+           "constrainedRows"}], "}"}], ",", 
+         RowBox[{"rowGap", "[", 
+          RowBox[{"_", ",", "_"}], "]"}], ",", "Infinity"}], "]"}], ",", 
+       RowBox[{"Cases", "[", 
+        RowBox[{
+         RowBox[{"{", 
+          RowBox[{
+          "additionalConstraints", ",", "constrainedColumns", ",", 
+           "constrainedRows"}], "}"}], ",", 
+         RowBox[{"cell", "[", 
+          RowBox[{"_", ",", "_"}], "]"}], ",", "Infinity"}], "]"}]}], 
+      "}"}]}]}]}], ";"}]], "Input",
+ CellChangeTimes->{{3.788337270244377*^9, 3.7883374021686573`*^9}, {
+  3.822551373702321*^9, 3.822551378429165*^9}},
+ CellLabel->"In[13]:=",ExpressionUUID->"6483cafd-b7c8-40ae-931e-03d3228c2c56"],
+
+Cell[BoxData[
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+  RowBox[{"Assertion", "/@", 
+   RowBox[{"Flatten", "[", 
+    RowBox[{"{", 
+     RowBox[{
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+      "constrainedRows"}], "}"}], "]"}]}]}]], "Input",
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+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[1],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[1], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.6666666666666666],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "1", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[1]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[1],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[0],
+      Editable->False,
+      Selectable->False]},
+    {
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[0],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[0],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[0],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[1], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.6666666666666666],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "1", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[1]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[1],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[0],
+      Editable->False,
+      Selectable->False]},
+    {
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[0],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[1], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.6666666666666666],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "1", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[1]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[1],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[1], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.6666666666666666],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "1", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[1]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[1],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[1], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.6666666666666666],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "1", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[1]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[1],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[1], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.6666666666666666],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "1", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[1]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[1],
+      Editable->False,
+      Selectable->False]},
+    {
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[1], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.6666666666666666],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "1", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[1]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[1],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[1], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.6666666666666666],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "1", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[1]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
+      GrayLevel[1],
+      Editable->False,
+      Selectable->False], 
+     InterpretationBox[
+      ButtonBox[
+       TooltipBox[
+        GraphicsBox[{
+          {GrayLevel[0], RectangleBox[{0, 0}]}, 
+          {GrayLevel[0], RectangleBox[{1, -1}]}, 
+          {GrayLevel[0], RectangleBox[{0, -1}, {2, 1}]}},
+         AspectRatio->1,
+         DefaultBaseStyle->"ColorSwatchGraphics",
+         Frame->True,
+         FrameStyle->GrayLevel[0.],
+         FrameTicks->None,
+         ImageSize->
+          Dynamic[{
+           Automatic, 1.35 CurrentValue["FontCapHeight"]/AbsoluteCurrentValue[
+            Magnification]}],
+         PlotRangePadding->None],
+        StyleBox[
+         RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> False]],
+       Appearance->None,
+       BaseStyle->{},
+       BaselinePosition->Baseline,
+       ButtonFunction:>With[{Typeset`box$ = EvaluationBox[]}, 
+         If[
+          Not[
+           AbsoluteCurrentValue["Deployed"]], 
+          SelectionMove[Typeset`box$, All, Expression]; 
+          FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
+          FrontEnd`Private`$ColorSelectorInitialColor = GrayLevel[0]; 
+          FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
+          MathLink`CallFrontEnd[
+            FrontEnd`AttachCell[Typeset`box$, 
+             FrontEndResource["GrayLevelColorValueSelector"], {
+             0, {Left, Bottom}}, {Left, Top}, 
+             "ClosingActions" -> {
+              "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]],
+       DefaultBaseStyle->{},
+       Evaluator->Automatic,
+       Method->"Preemptive"],
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diff --git a/hugo/static/ProjectEuler/index.html b/hugo/static/ProjectEuler/index.html
new file mode 100644
index 0000000..4cc275f
--- /dev/null
+++ b/hugo/static/ProjectEuler/index.html
@@ -0,0 +1,5 @@
+<!DOCTYPE html>
+<html>
+<head><title>Project Euler work</title></head>
+<body>This folder contains some of my work on <a href="https://projecteuler.net">Project Euler</a>. Append "problem.pdf" to the directory, so Problem 1 can be accessed at www.patrickstevens.co.uk/ProjectEuler/1.pdf .</body>
+</html>
diff --git a/hugo/static/archive/2015IntroToCategoryTheory.pdf b/hugo/static/archive/2015IntroToCategoryTheory.pdf
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diff --git a/hugo/static/cucats/Puzzlehunt2013/KeyboardCat.nb b/hugo/static/cucats/Puzzlehunt2013/KeyboardCat.nb
new file mode 100644
index 0000000..2519f15
--- /dev/null
+++ b/hugo/static/cucats/Puzzlehunt2013/KeyboardCat.nb
@@ -0,0 +1,5798 @@
+(* Content-type: application/vnd.wolfram.mathematica *)
+
+(*** Wolfram Notebook File ***)
+(* http://www.wolfram.com/nb *)
+
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+
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+CellTagsIndexPosition[    329550,       5746]
+WindowFrame->Normal*)
+
+(* Beginning of Notebook Content *)
+Notebook[{
+Cell["\<\
+This challenge was a column of keyboard pictures (like \
+\[OpenCurlyDoubleQuote]img\[CloseCurlyDoubleQuote] below), each presented \
+with one of two cat pictures. One cat picture was Dvorak with cat-ears, and \
+one was the Keyboard Cat. \
+\>", "Text",
+ CellChangeTimes->{{3.5812686464719677`*^9, 3.581268727540566*^9}}],
+
+Cell["\<\
+The pictures which accompanied the keyboards:\
+\>", "Text",
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+  3.581268701267626*^9, 3.5812687014923563`*^9}}],
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+       "], {{0, 152}, {230, 0}}, {0, 255},
+       ColorFunction->RGBColor],
+      BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
+      Selectable->False],
+     BaseStyle->"ImageGraphics",
+     ImageSize->{299.13671875, Automatic},
+     ImageSizeRaw->{230, 152},
+     PlotRange->{{0, 230}, {0, 152}}]}]}], ";"}]], "Input",
+ CellChangeTimes->{{3.58029131060008*^9, 3.580291311102138*^9}, {
+  3.580291544702215*^9, 3.580291546345409*^9}}],
+
+Cell["\<\
+The keyboard pictures represented binary (black key is a 0, and no black key \
+is a 1) for ASCII.
+I use this reference image to work out the columns in which we can check for \
+a black pixel. It turns out that the appropriate values are \
+{30,50,80,100,130,150,172,195}.\
+\>", "Text",
+ CellChangeTimes->{{3.581268252452532*^9, 3.5812682812890472`*^9}, {
+  3.581268734140572*^9, 3.581268768050536*^9}}],
+
+Cell[BoxData[
+ RowBox[{
+  RowBox[{"binFromImg", "[", "img_", "]"}], ":=", 
+  RowBox[{
+   RowBox[{"FromDigits", "[", 
+    RowBox[{
+     RowBox[{"1", "-", 
+      RowBox[{"Flatten", "[", 
+       RowBox[{"ImageData", "/@", 
+        RowBox[{"(", 
+         RowBox[{
+          RowBox[{
+           RowBox[{"ImageTake", "[", 
+            RowBox[{
+             RowBox[{"Binarize", "@", "img"}], ",", 
+             RowBox[{"{", "10", "}"}], ",", 
+             RowBox[{"{", "#", "}"}]}], "]"}], "&"}], "/@", 
+          RowBox[{"{", 
+           RowBox[{
+           "30", ",", "50", ",", "80", ",", "100", ",", "130", ",", "150", 
+            ",", "172", ",", "195"}], "}"}]}], ")"}]}], "]"}]}], ",", "2"}], 
+    "]"}], "//", "FromCharacterCode"}]}]], "Input",
+ CellChangeTimes->{{3.5802914122725773`*^9, 3.580291602194956*^9}, {
+  3.580291734088152*^9, 3.580291748637876*^9}, {3.581268298037182*^9, 
+  3.581268299448848*^9}}],
+
+Cell["\<\
+I then dumped the web page into a folder on my Desktop; thanks to the puzzle \
+creators, the relevant pictures were precisely the PNG files.\
+\>", "Text",
+ CellChangeTimes->{{3.5812683117521143`*^9, 3.5812683719728737`*^9}}],
+
+Cell[BoxData[
+ RowBox[{
+  RowBox[{"imgsList", "=", 
+   RowBox[{
+    RowBox[{
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diff --git a/hugo/themes/anatole/.gitattributes b/hugo/themes/anatole/.gitattributes
new file mode 100644
index 0000000..dfe0770
--- /dev/null
+++ b/hugo/themes/anatole/.gitattributes
@@ -0,0 +1,2 @@
+# Auto detect text files and perform LF normalization
+* text=auto
diff --git a/hugo/themes/anatole/.gitignore b/hugo/themes/anatole/.gitignore
new file mode 100644
index 0000000..ee4d895
--- /dev/null
+++ b/hugo/themes/anatole/.gitignore
@@ -0,0 +1,4 @@
+**/themes/
+demo/
+.hugo/*
+!.hugo/version
\ No newline at end of file
diff --git a/hugo/themes/anatole/LICENSE b/hugo/themes/anatole/LICENSE
new file mode 100644
index 0000000..635021b
--- /dev/null
+++ b/hugo/themes/anatole/LICENSE
@@ -0,0 +1,20 @@
+The MIT License (MIT)
+
+Copyright (c) 2020 lxndrblz
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal in
+the Software without restriction, including without limitation the rights to
+use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
+the Software, and to permit persons to whom the Software is furnished to do so,
+subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS
+FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR
+COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER
+IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
diff --git a/hugo/themes/anatole/README.md b/hugo/themes/anatole/README.md
new file mode 100644
index 0000000..e6a1ec2
--- /dev/null
+++ b/hugo/themes/anatole/README.md
@@ -0,0 +1,3 @@
+This is a copy of [Anatole](https://github.com/lxndrblz/anatole/) by [Alexander Bilz](https://github.com/lxndrblz), with adjustments by Patrick Stevens.
+It is substantially the original, which is licensed under the MIT licence, a copy of which is beside this README.
+I license the edits I have made under the MIT licence too.
\ No newline at end of file
diff --git a/hugo/themes/anatole/archetypes/post.md b/hugo/themes/anatole/archetypes/post.md
new file mode 100644
index 0000000..607aa3f
--- /dev/null
+++ b/hugo/themes/anatole/archetypes/post.md
@@ -0,0 +1,8 @@
+---
+title: "{{ replace .Name "-" " " | title }}"
+date: {{ .Date }}
+Description: ""
+Tags: []
+Categories: []
+DisableComments: false
+---
diff --git a/hugo/themes/anatole/assets/css/style.css b/hugo/themes/anatole/assets/css/style.css
new file mode 100644
index 0000000..ede8b2f
--- /dev/null
+++ b/hugo/themes/anatole/assets/css/style.css
@@ -0,0 +1,918 @@
+@charset "UTF-8";
+
+:root {
+    --bg-color: #fff;
+    --secondary-bg-color: #eeeeee;
+    --heading-color: #5f5f5f;
+    --body-color: rgba(0, 0, 0, 0.7);
+    --post-color: rgba(0, 0, 0, 0.44);
+    --border-color: rgba(0, 0, 0, 0.15);
+    --pre-bg-color: #f9f9fd;
+    --nav-text-color: #5a5a5a;
+    --tag-color: #424242;
+    --blockquote-text-color: #858585;
+    --blockquote-border-color: #dfe2e5;
+    scroll-padding-top: 100px;
+}
+
+html[data-theme='dark'] {
+    --bg-color: #292a2d;
+    --secondary-bg-color: #2c2d32;
+    --heading-color: rgb(169, 169, 179);
+    --body-color: rgb(169, 169, 179);
+    --post-color: rgba(0, 0, 0, 0.44);
+    --border-color: rgb(38, 38, 38);
+    --pre-bg-color: rgb(33, 33, 45);
+    --nav-text-color: rgb(191, 191, 191);
+    --tag-color: rgb(191, 191, 191);
+    --blockquote-text-color: #808080;
+    --blockquote-border-color: #424242;
+}
+
+html {
+    background-color: var(--bg-color);
+    -webkit-font-smoothing: antialiased;
+}
+
+body {
+    color: var(--body-color);
+    font-family: 'Verdana', sans-serif;
+    font-size: 15px;
+    width: 100%;
+    margin: 0 auto 30px auto;
+    background-color: var(--bg-color);
+}
+
+p {
+    line-height: 1.9em;
+    font-weight: 400;
+    font-size: 14px;
+}
+
+a {
+    text-decoration: none;
+}
+
+blockquote {
+    padding: 0 1em;
+    border-left: .25em solid var(--blockquote-border-color);
+    color: var(--blockquote-text-color);
+}
+
+.category {
+    padding: 4px 6px;
+    border-radius: 3px;
+    color: var(--tag-color) !important;
+    background-color: var(--secondary-bg-color);
+    border: 1px solid var(--border-color);
+}
+
+.tag::before {
+    content: "#";
+    opacity: .5;
+}
+
+.tag,
+.category {
+    display: inline-block;
+    font-size: 15px;
+    line-height: 1;
+    margin: 5px 8px 5px 0;
+}
+
+pre {
+    background-color: var(--pre-bg-color);
+    padding: 5px;
+    display: block;
+    overflow-x: auto;
+}
+
+.info i {
+    opacity: 0.5;
+    margin-right: 5px;
+}
+
+a:link,
+a:visited {
+    opacity: 1;
+    color: var(--tag-color);
+}
+
+a:hover,
+a:active {
+    color: #2660ab;
+}
+
+/*basic styles ends*/
+
+.main {
+    width: 80%;
+    height: auto;
+    float: right;
+}
+
+.content {
+    height: auto;
+    float: left;
+    width: 80%;
+    margin-top: 60px;
+}
+
+.page-top {
+    width: 80%;
+    position: fixed;
+    right: 0;
+    z-index: 3;
+    background-color: var(--bg-color);
+    height: 60px;
+    border-bottom: 1px solid var(--border-color);
+}
+
+.page-top .nav {
+    list-style: none;
+    float: left;
+    font-size: 12px;
+    width: calc(100% - 30px);
+    margin: 0;
+    padding-left: 15px;
+}
+
+.page-top .nav li {
+    position: relative;
+    display: initial;
+    padding-right: 20px;
+    padding-bottom: 10px;
+}
+
+.page-top .nav a {
+    color: var(--nav-text-color);
+}
+
+.page-top .nav a:hover {
+    color: #2660ab;
+}
+
+.page-top .nav a.current {
+    color: var(--nav-text-color);
+    padding-bottom: 6px;
+    border-bottom: 1px solid var(--nav-text-color);
+}
+
+.theme-switch-item {
+    position: relative;
+    float: right;
+    padding-top: 11px;
+    font-size: 2em;
+    margin-top: -12px;
+}
+
+.page-top .information {
+    float: right;
+    padding-top: 12px;
+    padding-right: 20px;
+}
+
+.page-top .information .avatar {
+    float: right;
+}
+
+.page-top .information .avatar img {
+    width: 32px;
+    height: 32px;
+    border-radius: 300px;
+}
+
+.page-top .information .back_btn {
+    float: left;
+    padding-top: 5px;
+    margin-right: -10px;
+}
+
+.page-top .information .back_btn li {
+    display: initial;
+    padding-right: 40px;
+}
+
+.sidebar {
+    width: 20%;
+    float: left;
+    -webkit-background-size: cover;
+    background-size: cover;
+    background-color: var(--bg-color);
+    height: 100%;
+    top: 0;
+    left: 0;
+    position: fixed;
+    z-index: 4;
+    border-right: 1px solid var(--border-color);
+    display: flex;
+    flex-direction: column;
+    min-height: 100%;
+}
+
+.sidebar .logo-title {
+    text-align: center;
+    padding-top: 240px;
+    flex: 1;
+}
+
+.sidebar .logo-title .description {
+    font-size: 14px;
+}
+
+.sidebar .logo-title .logo {
+    margin: 0 auto;
+}
+
+.sidebar .logo-title .title img {
+    width: 127px;
+    height: 127px;
+    border-radius: 50%;
+}
+
+.sidebar .logo-title .title h3 {
+    text-transform: uppercase;
+    font-size: 2rem;
+    font-weight: bold;
+    letter-spacing: 2px;
+    line-height: 1;
+}
+
+.sidebar .logo-title .title a {
+    text-decoration: none;
+    color: var(--heading-color);
+    font-size: 2rem;
+    font-weight: bold;
+}
+
+.sidebar .social-links {
+    list-style: none;
+    padding: 0;
+    font-size: 14px;
+    text-align: center;
+    flex: 7;
+}
+
+.sidebar .social-links span {
+    margin-right: 3px;
+}
+
+.sidebar .social-links td {
+    display: inline;
+    padding: 0 4px;
+    line-height: 0;
+}
+
+.sidebar .social-links a {
+    color: var(--heading-color);
+}
+
+.sidebar .social-links a:hover {
+    color: #2660ab;
+}
+
+.post {
+    background-color: var(--bg-color);
+    margin: 30px;
+}
+
+.post .post-title h1 {
+    text-transform: uppercase;
+    font-size: 30px;
+    letter-spacing: 5px;
+    line-height: 1;
+}
+
+.post .post-title h2 {
+    text-transform: uppercase;
+    letter-spacing: 1px;
+    font-size: 28px;
+    line-height: 1;
+    font-weight: 600;
+    color: var(--heading-color);
+}
+
+.post .post-title h3 {
+    text-transform: uppercase;
+    letter-spacing: 1px;
+    line-height: 1;
+    font-weight: 600;
+    /* color: #464646; */
+    color: var(--heading-color);
+    font-size: 22px;
+    margin: 0;
+}
+
+.post .post-title a {
+    text-decoration: none;
+    letter-spacing: 1px;
+    color: var(--heading-color);
+}
+
+.post .post-title a:hover {
+    text-decoration: underline;
+}
+
+.post .post-content a {
+    text-decoration: none;
+    letter-spacing: 1px;
+    color: #2660ab;
+}
+
+.post .post-content a:hover {
+    color: #2F69B3;
+}
+
+.post .post-content h3 {
+    font-size: 22px;
+    font-weight: 600;
+}
+
+.post .post-content h4 {
+    /* color: var(--heading-color); */
+    font-size: 16px;
+}
+
+.post .post-content img {
+    max-width: 100%;
+}
+
+.post .post-footer {
+    padding: 0 0 10px 0;
+    border-bottom: 1px solid var(--border-color);
+}
+
+.post .post-footer .meta {
+    max-width: 100%;
+    display: flex;
+    color: #bbbbbb;
+}
+
+.post .post-footer .meta .info {
+    float: left;
+    font-size: 12px;
+    margin-bottom: 1em;
+    color: var(--body-color);
+}
+
+.post .post-footer .info .separator a {
+    margin-right: 0.2em;
+}
+
+.post .post-footer .meta .info .date {
+    margin-right: 10px;
+    margin-left: 5px
+}
+
+.info {
+    margin: 1em;
+}
+
+.post .post-footer .meta a {
+    text-decoration: none;
+    color: var(--body-color);
+    padding-right: 10px;
+}
+
+.post .post-footer .meta a:hover {
+    color: #2660ab;
+}
+
+.post .post-footer .meta i {
+    margin-right: 6px;
+}
+
+.post .post-footer .tags {
+    padding-bottom: 15px;
+    font-size: 13px;
+}
+
+.post .post-footer .tags ul {
+    list-style-type: none;
+    display: inline;
+    margin: 0;
+    padding: 0;
+}
+
+.post .post-footer .tags ul li {
+    list-style-type: none;
+    margin: 0;
+    padding-right: 5px;
+    display: inline;
+}
+
+.post .post-footer .tags a {
+    text-decoration: none;
+    color: var(--post-color);
+    font-weight: 400;
+}
+
+.post .post-footer .tags a:hover {
+    text-decoration: none;
+}
+
+.pagination {
+    margin: 30px;
+    padding: 0px 0 56px 0;
+    text-align: center;
+}
+
+.pagination ul {
+    list-style: none;
+    margin: 0;
+    padding: 0;
+    height: 13px;
+}
+
+.pagination ul li {
+    margin: 0 2px 0 2px;
+    display: inline;
+    line-height: 1;
+}
+
+.pagination ul li a {
+    text-decoration: none;
+    color: var(--body-color);
+}
+
+.pagination .pre {
+    float: left;
+}
+
+.pagination .next {
+    float: right;
+}
+
+.like-reblog-buttons {
+    float: right;
+}
+
+.like-button {
+    float: right;
+    padding: 0 0 0 10px;
+}
+
+.reblog-button {
+    float: right;
+    padding: 0;
+}
+
+#install-btn {
+    position: fixed;
+    bottom: 0px;
+    right: 6px;
+}
+
+#disqus_thread {
+    margin: 30px;
+    border-bottom: 1px solid var(--border-color);
+}
+
+.footer {
+    clear: both;
+    text-align: center;
+    font-size: 10px;
+    margin: 0 auto;
+    bottom: 0;
+    width: 100%;
+    padding-bottom: 20px;
+    flex: 0;
+    position: relative;
+}
+
+.footer a {
+    color: #A6A6A6;
+}
+
+.footer a:hover {
+    color: #2660ab;
+}
+
+/*for archive*/
+.archive {
+    width: 100%;
+}
+
+.list-with-title {
+    font-size: 14px;
+    margin: 30px;
+    padding: 0;
+}
+
+.list-with-title li {
+    list-style-type: none;
+    padding: 0;
+}
+
+.list-with-title .listing-title {
+    font-size: 24px;
+    color: #666666;
+    font-weight: 600;
+    line-height: 2.2em;
+}
+
+.list-with-title .listing {
+    padding: 0;
+}
+
+.list-with-title .listing .listing-post {
+    padding-bottom: 5px;
+}
+
+.list-with-title .listing .listing-post .post-time {
+    float: right;
+    color: #C5C5C5;
+}
+
+.list-with-title .listing .listing-post a {
+    color: #8F8F8F;
+}
+
+.list-with-title .listing .listing-post a:hover {
+    color: #2660ab;
+}
+
+/* share */
+.share {
+    margin: 0px 30px;
+    display: inline-flex;
+}
+
+.evernote {
+    width: 32px;
+    height: 32px;
+    border-radius: 300px;
+    background-color: #3E3E3E;
+    margin-right: 5px;
+}
+
+.evernote a {
+    color: #fff;
+    padding: 11px;
+    font-size: 12px;
+}
+
+.evernote a:hover {
+    color: #ED6243;
+    padding: 11px;
+}
+
+.weibo {
+    width: 32px;
+    height: 32px;
+    border-radius: 300px;
+    background-color: #ED6243;
+    margin-right: 5px;
+}
+
+.weibo a {
+    color: #fff;
+    padding: 9px;
+}
+
+.weibo a:hover {
+    color: #BD4226;
+}
+
+.twitter {
+    width: 32px;
+    height: 32px;
+    border-radius: 300px;
+    background-color: #59C0FD;
+    margin-right: 5px;
+}
+
+.twitter a {
+    color: #fff;
+    padding: 9px;
+}
+
+.twitter a:hover {
+    color: #4B9ECE;
+}
+
+/* about */
+.about {
+    margin: 30px;
+}
+
+.about h3 {
+    font-size: 22px;
+}
+
+/* links*/
+.links {
+    margin: 30px;
+}
+
+.links h3 {
+    font-size: 22px;
+}
+
+.links a {
+    cursor: pointer;
+}
+
+/* Comments */
+.comment-count {
+    color: #666;
+}
+
+.tab-community {
+    color: #666;
+}
+
+.read_more {
+    font-size: 14px;
+}
+
+.back-button {
+    padding-top: 30px;
+    max-width: 100px;
+    padding-left: 40px;
+    float: left;
+}
+
+/* Facebook Comments */
+#fb_comments_container {
+    margin: 30px;
+}
+
+/* Buttons */
+a.btn {
+    color: #868686;
+    font-weight: 400;
+}
+
+.btn {
+    display: inline-block;
+    position: relative;
+    outline: 0;
+    color: var(--post-color);
+    background: transparent;
+    font-size: 14px;
+    text-align: center;
+    text-decoration: none;
+    cursor: pointer;
+    border: 1px solid var(--border-color);
+    white-space: nowrap;
+    font-weight: 400;
+    font-style: normal;
+    border-radius: 999em;
+}
+
+.btn:hover {
+    display: inline-block;
+    position: relative;
+    outline: 0px;
+    color: #464545;
+    background: transparent;
+    font-size: 14px;
+    text-align: center;
+    text-decoration: none;
+    cursor: pointer;
+    border: 1px solid #464545;
+    white-space: nowrap;
+    font-weight: 400;
+    font-style: normal;
+    border-radius: 999em;
+}
+
+[role="back"] {
+    padding: 0.5em 1.25em;
+    line-height: 1.666em;
+}
+
+[role="home"] {
+    padding: 0.5em 1.25em;
+    line-height: 1.666em;
+}
+
+[role="navigation"] {
+    padding: 0.5em 1.25em;
+    line-height: 1.666em;
+}
+
+[role="tags"] {
+    padding: 6px 12px;
+}
+
+/* In full-screen mode, the navbar toggle checkbox has no use; don't display it */
+.navbar-toggle {
+  display:none
+}
+
+#navbar-toggle-cbox {
+  opacity: 0;
+  float: none;
+  position: absolute;
+}
+
+/* Menu */
+.menu {
+    float: right;
+    padding-top: 30px;
+}
+
+.menu .btn-down {
+    margin: 0px;
+}
+
+.menu .btn-down li {
+    list-style: none;
+    width: 100px;
+}
+
+.menu .btn-down li a {
+    display: inline-block;
+    position: relative;
+    padding: 0.5em 1.25em;
+    outline: 0;
+    color: var(--post-color);
+    background: transparent;
+    font-size: 14px;
+    text-align: center;
+    text-decoration: none;
+    cursor: pointer;
+    border: 1px solid var(--border-color);
+    white-space: nowrap;
+    font-weight: 400;
+    font-style: normal;
+    border-radius: 999em;
+    margin-top: 5px;
+}
+
+.menu .btn-down li a:hover {
+    position: relative;
+    padding: 0.5em 1.25em;
+    outline: 0;
+    color: #fff;
+    background: #3CBD10;
+    font-size: 14px;
+    text-align: center;
+    text-decoration: none;
+    cursor: pointer;
+    border: 1px solid rgba(0, 0, 0, 0.15);
+    white-space: nowrap;
+    font-weight: 400;
+    font-style: normal;
+    border-radius: 999em;
+    margin-top: 5px;
+}
+
+.menu .btn-down div {
+    position: absolute;
+    visibility: hidden;
+    width: 100px;
+    float: right;
+}
+
+.page_404 {
+    text-align: center;
+    padding-top: 50px;
+}
+
+@media screen and (max-width: 960px),
+print {
+    .sidebar {
+        width: 100%;
+        position: absolute;
+        border-right: none;
+        z-index: 1;
+    }
+
+    .sidebar .logo-title {
+        padding-top: 120px;
+    }
+
+    .sidebar .logo-title .title img {
+        width: 100px;
+        height: 100px;
+    }
+
+    .sidebar .logo-title .title h3 {
+        font-size: 20px;
+    }
+
+    .main {
+        width: 100%;
+        height: auto;
+        float: right;
+    }
+
+    .page-top {
+        width: 100%;
+    }
+
+    .post-title h3 {
+        line-height: 1.6;
+    }
+
+    .content {
+        margin-top: 420px;
+        width: 100%;
+        z-index: 2;
+        position: absolute;
+    }
+
+    .footer {
+        display: none;
+    }
+
+    .share {
+        display: grid;
+    }
+
+    .navbar-toggle {
+        display: inline-block;
+        padding: 11px 10px;
+    }
+
+    label.navbar-toggle {
+        position: fixed;
+        padding-top: 18px;
+    }
+
+    #navbar-toggle-cbox:focus ~ label span {
+        display: inline-block;
+        outline :1px dashed cyan;
+    }
+
+    #navbar-toggle-cbox:not(:checked) ~ .nav {
+        display: none;
+    }
+
+    .page-top .nav {
+        background-color: var(--secondary-bg-color);
+        box-shadow: 0 8px 16px rgba(10, 10, 10, .1);
+        padding: .5rem 0;
+        width: 100%;
+        margin-top: 0px;
+    }
+
+    .page-top .nav li {
+        flex-grow: 0;
+        flex-shrink: 0;
+        display: block;
+        line-height: 2.5;
+        padding: .5rem .75rem;
+        position: relative;
+        text-transform: uppercase;
+        text-align: center;
+        font-size: 1.3em;
+    }
+
+    .page-top .nav a.current {
+        border-bottom: none;
+    }
+}
+
+/* Medium zoom */
+.medium-zoom-overlay {
+    position: fixed;
+    top: 0;
+    right: 0;
+    bottom: 0;
+    left: 0;
+    opacity: 0;
+    transition: opacity 300ms;
+    will-change: opacity;
+    background: var(--bg-color);
+}
+
+.medium-zoom--opened .medium-zoom-overlay {
+    cursor: pointer;
+    cursor: zoom-out;
+    opacity: 1;
+}
+
+.medium-zoom-image {
+    cursor: pointer;
+    cursor: zoom-in;
+    /*
+      The `transition` is marked as "!important" for the animation to happen
+      even though it's overriden by another inline `transition` style attribute.
+      This is problematic with frameworks that generate inline styles on their
+      images (e.g. Gatsby).
+      See https://github.com/francoischalifour/medium-zoom/issues/110
+     */
+    transition: transform 300ms cubic-bezier(0.2, 0, 0.2, 1) !important;
+    z-index: 100;
+}
+
+.medium-zoom-image--hidden {
+    visibility: hidden;
+}
+
+.medium-zoom-image--opened {
+    position: relative;
+    cursor: pointer;
+    cursor: zoom-out;
+    will-change: transform;
+}
+
+@media print {
+	.page-top {
+		display: none;
+	}
+}
\ No newline at end of file
diff --git a/hugo/themes/anatole/assets/js/anatole-header.js b/hugo/themes/anatole/assets/js/anatole-header.js
new file mode 100644
index 0000000..ddc0c21
--- /dev/null
+++ b/hugo/themes/anatole/assets/js/anatole-header.js
@@ -0,0 +1,51 @@
+// initialize default value
+function getTheme() {
+    return localStorage.getItem('theme') ? localStorage.getItem('theme') : null;
+}
+
+function setTheme(style) {
+    document.documentElement.setAttribute('data-theme', style);
+    localStorage.setItem('theme', style);
+}
+
+function init() {
+    // initialize default value
+    var theme = getTheme();
+
+    // check if a prefered color theme is set for users that have never been to our site
+    const userPrefersDark = window.matchMedia && window.matchMedia('(prefers-color-scheme: dark)').matches;
+    if (theme === null) {
+        if (userPrefersDark) {
+            setTheme('dark');
+        } else if (!document.documentElement.getAttribute('data-theme')) {
+            setTheme('light');
+        } else {
+            setTheme(document.documentElement.getAttribute('data-theme'));
+        }
+    } else {
+        // load a stored theme
+        if (theme == 'light') {
+            document.documentElement.setAttribute('data-theme', 'light');
+        } else {
+            document.documentElement.setAttribute('data-theme', 'dark');
+        }
+    }
+}
+
+
+// switch themes
+function switchTheme(e) {
+    var theme = getTheme();
+    if (theme == 'light') {
+        setTheme('dark');
+    } else {
+        setTheme('light');
+    }
+}
+
+document.addEventListener('DOMContentLoaded', function () {
+    var themeSwitcher = document.querySelector('.theme-switch');
+    themeSwitcher.addEventListener('click', switchTheme, false);
+}, false);
+
+init();
diff --git a/hugo/themes/anatole/assets/js/medium-zoom.js b/hugo/themes/anatole/assets/js/medium-zoom.js
new file mode 100644
index 0000000..a190f27
--- /dev/null
+++ b/hugo/themes/anatole/assets/js/medium-zoom.js
@@ -0,0 +1,458 @@
+/*! medium-zoom 1.0.5 | MIT License | https://github.com/francoischalifour/medium-zoom */
+(function (global, factory) {
+    typeof exports === "object" && typeof module !== "undefined" ? module.exports = factory() : typeof define === "function" && define.amd ? define(factory) : (global = global || self,
+        global.mediumZoom = factory());
+})(this, function () {
+    "use strict";
+    var _extends = Object.assign || function (target) {
+        for (var i = 1; i < arguments.length; i++) {
+            var source = arguments[i];
+            for (var key in source) {
+                if (Object.prototype.hasOwnProperty.call(source, key)) {
+                    target[key] = source[key];
+                }
+            }
+        }
+        return target;
+    };
+    var isSupported = function isSupported(node) {
+        return node.tagName === "IMG";
+    };
+    var isNodeList = function isNodeList(selector) {
+        return NodeList.prototype.isPrototypeOf(selector);
+    };
+    var isNode = function isNode(selector) {
+        return selector && selector.nodeType === 1;
+    };
+    var isSvg = function isSvg(image) {
+        var source = image.currentSrc || image.src;
+        return source.substr(-4).toLowerCase() === ".svg";
+    };
+    var getImagesFromSelector = function getImagesFromSelector(selector) {
+        try {
+            if (Array.isArray(selector)) {
+                return selector.filter(isSupported);
+            }
+            if (isNodeList(selector)) {
+                return [].slice.call(selector).filter(isSupported);
+            }
+            if (isNode(selector)) {
+                return [selector].filter(isSupported);
+            }
+            if (typeof selector === "string") {
+                return [].slice.call(document.querySelectorAll(selector)).filter(isSupported);
+            }
+            return [];
+        } catch (err) {
+            throw new TypeError("The provided selector is invalid.\n" + "Expects a CSS selector, a Node element, a NodeList or an array.\n" + "See: https://github.com/francoischalifour/medium-zoom");
+        }
+    };
+    var createOverlay = function createOverlay() {
+        var overlay = document.createElement("div");
+        overlay.classList.add("medium-zoom-overlay");
+        return overlay;
+    };
+    var cloneTarget = function cloneTarget(template) {
+        var _template$getBounding = template.getBoundingClientRect(), top = _template$getBounding.top,
+            left = _template$getBounding.left, width = _template$getBounding.width,
+            height = _template$getBounding.height;
+        var clone = template.cloneNode();
+        var scrollTop = window.pageYOffset || document.documentElement.scrollTop || document.body.scrollTop || 0;
+        var scrollLeft = window.pageXOffset || document.documentElement.scrollLeft || document.body.scrollLeft || 0;
+        clone.removeAttribute("id");
+        clone.style.position = "absolute";
+        clone.style.top = top + scrollTop + "px";
+        clone.style.left = left + scrollLeft + "px";
+        clone.style.width = width + "px";
+        clone.style.height = height + "px";
+        clone.style.transform = "";
+        return clone;
+    };
+    var createCustomEvent = function createCustomEvent(type, params) {
+        var eventParams = _extends({
+            bubbles: false,
+            cancelable: false,
+            detail: undefined
+        }, params);
+        if (typeof window.CustomEvent === "function") {
+            return new CustomEvent(type, eventParams);
+        }
+        var customEvent = document.createEvent("CustomEvent");
+        customEvent.initCustomEvent(type, eventParams.bubbles, eventParams.cancelable, eventParams.detail);
+        return customEvent;
+    };
+    var mediumZoom = function mediumZoom(selector) {
+        var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {};
+        var Promise = window.Promise || function Promise(fn) {
+            function noop() {
+            }
+
+            fn(noop, noop);
+        };
+        var _handleClick = function _handleClick(event) {
+            var target = event.target;
+            if (target === overlay) {
+                close();
+                return;
+            }
+            if (images.indexOf(target) === -1) {
+                return;
+            }
+            toggle({
+                target: target
+            });
+        };
+        var _handleScroll = function _handleScroll() {
+            if (isAnimating || !active.original) {
+                return;
+            }
+            var currentScroll = window.pageYOffset || document.documentElement.scrollTop || document.body.scrollTop || 0;
+            if (Math.abs(scrollTop - currentScroll) > zoomOptions.scrollOffset) {
+                setTimeout(close, 150);
+            }
+        };
+        var _handleKeyUp = function _handleKeyUp(event) {
+            var key = event.key || event.keyCode;
+            if (key === "Escape" || key === "Esc" || key === 27) {
+                close();
+            }
+        };
+        var update = function update() {
+            var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};
+            var newOptions = options;
+            if (options.container && options.container instanceof Object) {
+                newOptions.container = _extends({}, zoomOptions.container, options.container);
+            }
+            if (options.template) {
+                var template = isNode(options.template) ? options.template : document.querySelector(options.template);
+                newOptions.template = template;
+            }
+            zoomOptions = _extends({}, zoomOptions, newOptions);
+            images.forEach(function (image) {
+                image.dispatchEvent(createCustomEvent("medium-zoom:update", {
+                    detail: {
+                        zoom: zoom
+                    }
+                }));
+            });
+            return zoom;
+        };
+        var clone = function clone() {
+            var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};
+            return mediumZoom(_extends({}, zoomOptions, options));
+        };
+        var attach = function attach() {
+            for (var _len = arguments.length, selectors = Array(_len), _key = 0; _key < _len; _key++) {
+                selectors[_key] = arguments[_key];
+            }
+            var newImages = selectors.reduce(function (imagesAccumulator, currentSelector) {
+                return [].concat(imagesAccumulator, getImagesFromSelector(currentSelector));
+            }, []);
+            newImages.filter(function (newImage) {
+                return images.indexOf(newImage) === -1;
+            }).forEach(function (newImage) {
+                images.push(newImage);
+                newImage.classList.add("medium-zoom-image");
+            });
+            eventListeners.forEach(function (_ref) {
+                var type = _ref.type, listener = _ref.listener, options = _ref.options;
+                newImages.forEach(function (image) {
+                    image.addEventListener(type, listener, options);
+                });
+            });
+            return zoom;
+        };
+        var detach = function detach() {
+            for (var _len2 = arguments.length, selectors = Array(_len2), _key2 = 0; _key2 < _len2; _key2++) {
+                selectors[_key2] = arguments[_key2];
+            }
+            if (active.zoomed) {
+                close();
+            }
+            var imagesToDetach = selectors.length > 0 ? selectors.reduce(function (imagesAccumulator, currentSelector) {
+                return [].concat(imagesAccumulator, getImagesFromSelector(currentSelector));
+            }, []) : images;
+            imagesToDetach.forEach(function (image) {
+                image.classList.remove("medium-zoom-image");
+                image.dispatchEvent(createCustomEvent("medium-zoom:detach", {
+                    detail: {
+                        zoom: zoom
+                    }
+                }));
+            });
+            images = images.filter(function (image) {
+                return imagesToDetach.indexOf(image) === -1;
+            });
+            return zoom;
+        };
+        var on = function on(type, listener) {
+            var options = arguments.length > 2 && arguments[2] !== undefined ? arguments[2] : {};
+            images.forEach(function (image) {
+                image.addEventListener("medium-zoom:" + type, listener, options);
+            });
+            eventListeners.push({
+                type: "medium-zoom:" + type,
+                listener: listener,
+                options: options
+            });
+            return zoom;
+        };
+        var off = function off(type, listener) {
+            var options = arguments.length > 2 && arguments[2] !== undefined ? arguments[2] : {};
+            images.forEach(function (image) {
+                image.removeEventListener("medium-zoom:" + type, listener, options);
+            });
+            eventListeners = eventListeners.filter(function (eventListener) {
+                return !(eventListener.type === "medium-zoom:" + type && eventListener.listener.toString() === listener.toString());
+            });
+            return zoom;
+        };
+        var open = function open() {
+            var _ref2 = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {}, target = _ref2.target;
+            var _animate = function _animate() {
+                var container = {
+                    width: document.documentElement.clientWidth,
+                    height: document.documentElement.clientHeight,
+                    left: 0,
+                    top: 0,
+                    right: 0,
+                    bottom: 0
+                };
+                var viewportWidth = void 0;
+                var viewportHeight = void 0;
+                if (zoomOptions.container) {
+                    if (zoomOptions.container instanceof Object) {
+                        container = _extends({}, container, zoomOptions.container);
+                        viewportWidth = container.width - container.left - container.right - zoomOptions.margin * 2;
+                        viewportHeight = container.height - container.top - container.bottom - zoomOptions.margin * 2;
+                    } else {
+                        var zoomContainer = isNode(zoomOptions.container) ? zoomOptions.container : document.querySelector(zoomOptions.container);
+                        var _zoomContainer$getBou = zoomContainer.getBoundingClientRect(),
+                            _width = _zoomContainer$getBou.width, _height = _zoomContainer$getBou.height,
+                            _left = _zoomContainer$getBou.left, _top = _zoomContainer$getBou.top;
+                        container = _extends({}, container, {
+                            width: _width,
+                            height: _height,
+                            left: _left,
+                            top: _top
+                        });
+                    }
+                }
+                viewportWidth = viewportWidth || container.width - zoomOptions.margin * 2;
+                viewportHeight = viewportHeight || container.height - zoomOptions.margin * 2;
+                var zoomTarget = active.zoomedHd || active.original;
+                var naturalWidth = isSvg(zoomTarget) ? viewportWidth : zoomTarget.naturalWidth || viewportWidth;
+                var naturalHeight = isSvg(zoomTarget) ? viewportHeight : zoomTarget.naturalHeight || viewportHeight;
+                var _zoomTarget$getBoundi = zoomTarget.getBoundingClientRect(), top = _zoomTarget$getBoundi.top,
+                    left = _zoomTarget$getBoundi.left, width = _zoomTarget$getBoundi.width,
+                    height = _zoomTarget$getBoundi.height;
+                var scaleX = Math.min(naturalWidth, viewportWidth) / width;
+                var scaleY = Math.min(naturalHeight, viewportHeight) / height;
+                var scale = Math.min(scaleX, scaleY);
+                var translateX = (-left + (viewportWidth - width) / 2 + zoomOptions.margin + container.left) / scale;
+                var translateY = (-top + (viewportHeight - height) / 2 + zoomOptions.margin + container.top) / scale;
+                var transform = "scale(" + scale + ") translate3d(" + translateX + "px, " + translateY + "px, 0)";
+                active.zoomed.style.transform = transform;
+                if (active.zoomedHd) {
+                    active.zoomedHd.style.transform = transform;
+                }
+            };
+            return new Promise(function (resolve) {
+                if (target && images.indexOf(target) === -1) {
+                    resolve(zoom);
+                    return;
+                }
+                var _handleOpenEnd = function _handleOpenEnd() {
+                    isAnimating = false;
+                    active.zoomed.removeEventListener("transitionend", _handleOpenEnd);
+                    active.original.dispatchEvent(createCustomEvent("medium-zoom:opened", {
+                        detail: {
+                            zoom: zoom
+                        }
+                    }));
+                    resolve(zoom);
+                };
+                if (active.zoomed) {
+                    resolve(zoom);
+                    return;
+                }
+                if (target) {
+                    active.original = target;
+                } else if (images.length > 0) {
+                    var _images = images;
+                    active.original = _images[0];
+                } else {
+                    resolve(zoom);
+                    return;
+                }
+                active.original.dispatchEvent(createCustomEvent("medium-zoom:open", {
+                    detail: {
+                        zoom: zoom
+                    }
+                }));
+                scrollTop = window.pageYOffset || document.documentElement.scrollTop || document.body.scrollTop || 0;
+                isAnimating = true;
+                active.zoomed = cloneTarget(active.original);
+                document.body.appendChild(overlay);
+                if (zoomOptions.template) {
+                    var template = isNode(zoomOptions.template) ? zoomOptions.template : document.querySelector(zoomOptions.template);
+                    active.template = document.createElement("div");
+                    active.template.appendChild(template.content.cloneNode(true));
+                    document.body.appendChild(active.template);
+                }
+                document.body.appendChild(active.zoomed);
+                window.requestAnimationFrame(function () {
+                    document.body.classList.add("medium-zoom--opened");
+                });
+                active.original.classList.add("medium-zoom-image--hidden");
+                active.zoomed.classList.add("medium-zoom-image--opened");
+                active.zoomed.addEventListener("click", close);
+                active.zoomed.addEventListener("transitionend", _handleOpenEnd);
+                if (active.original.getAttribute("data-zoom-src")) {
+                    active.zoomedHd = active.zoomed.cloneNode();
+                    active.zoomedHd.removeAttribute("srcset");
+                    active.zoomedHd.removeAttribute("sizes");
+                    active.zoomedHd.src = active.zoomed.getAttribute("data-zoom-src");
+                    active.zoomedHd.onerror = function () {
+                        clearInterval(getZoomTargetSize);
+                        console.warn("Unable to reach the zoom image target " + active.zoomedHd.src);
+                        active.zoomedHd = null;
+                        _animate();
+                    };
+                    var getZoomTargetSize = setInterval(function () {
+                        if (active.zoomedHd.complete) {
+                            clearInterval(getZoomTargetSize);
+                            active.zoomedHd.classList.add("medium-zoom-image--opened");
+                            active.zoomedHd.addEventListener("click", close);
+                            document.body.appendChild(active.zoomedHd);
+                            _animate();
+                        }
+                    }, 10);
+                } else if (active.original.hasAttribute("srcset")) {
+                    active.zoomedHd = active.zoomed.cloneNode();
+                    active.zoomedHd.removeAttribute("sizes");
+                    var loadEventListener = active.zoomedHd.addEventListener("load", function () {
+                        active.zoomedHd.removeEventListener("load", loadEventListener);
+                        active.zoomedHd.classList.add("medium-zoom-image--opened");
+                        active.zoomedHd.addEventListener("click", close);
+                        document.body.appendChild(active.zoomedHd);
+                        _animate();
+                    });
+                } else {
+                    _animate();
+                }
+            });
+        };
+        var close = function close() {
+            return new Promise(function (resolve) {
+                if (isAnimating || !active.original) {
+                    resolve(zoom);
+                    return;
+                }
+                var _handleCloseEnd = function _handleCloseEnd() {
+                    active.original.classList.remove("medium-zoom-image--hidden");
+                    document.body.removeChild(active.zoomed);
+                    if (active.zoomedHd) {
+                        document.body.removeChild(active.zoomedHd);
+                    }
+                    document.body.removeChild(overlay);
+                    active.zoomed.classList.remove("medium-zoom-image--opened");
+                    if (active.template) {
+                        document.body.removeChild(active.template);
+                    }
+                    isAnimating = false;
+                    active.zoomed.removeEventListener("transitionend", _handleCloseEnd);
+                    active.original.dispatchEvent(createCustomEvent("medium-zoom:closed", {
+                        detail: {
+                            zoom: zoom
+                        }
+                    }));
+                    active.original = null;
+                    active.zoomed = null;
+                    active.zoomedHd = null;
+                    active.template = null;
+                    resolve(zoom);
+                };
+                isAnimating = true;
+                document.body.classList.remove("medium-zoom--opened");
+                active.zoomed.style.transform = "";
+                if (active.zoomedHd) {
+                    active.zoomedHd.style.transform = "";
+                }
+                if (active.template) {
+                    active.template.style.transition = "opacity 150ms";
+                    active.template.style.opacity = 0;
+                }
+                active.original.dispatchEvent(createCustomEvent("medium-zoom:close", {
+                    detail: {
+                        zoom: zoom
+                    }
+                }));
+                active.zoomed.addEventListener("transitionend", _handleCloseEnd);
+            });
+        };
+        var toggle = function toggle() {
+            var _ref3 = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {}, target = _ref3.target;
+            if (active.original) {
+                return close();
+            }
+            return open({
+                target: target
+            });
+        };
+        var getOptions = function getOptions() {
+            return zoomOptions;
+        };
+        var getImages = function getImages() {
+            return images;
+        };
+        var getZoomedImage = function getZoomedImage() {
+            return active.original;
+        };
+        var images = [];
+        var eventListeners = [];
+        var isAnimating = false;
+        var scrollTop = 0;
+        var zoomOptions = options;
+        var active = {
+            original: null,
+            zoomed: null,
+            zoomedHd: null,
+            template: null
+        };
+        if (Object.prototype.toString.call(selector) === "[object Object]") {
+            zoomOptions = selector;
+        } else if (selector || typeof selector === "string") {
+            attach(selector);
+        }
+        zoomOptions = _extends({
+            margin: 0,
+            scrollOffset: 40,
+            container: null,
+            template: null
+        }, zoomOptions);
+        var overlay = createOverlay();
+        document.addEventListener("click", _handleClick);
+        document.addEventListener("keyup", _handleKeyUp);
+        document.addEventListener("scroll", _handleScroll);
+        window.addEventListener("resize", close);
+        var zoom = {
+            open: open,
+            close: close,
+            toggle: toggle,
+            update: update,
+            clone: clone,
+            attach: attach,
+            detach: detach,
+            on: on,
+            off: off,
+            getOptions: getOptions,
+            getImages: getImages,
+            getZoomedImage: getZoomedImage
+        };
+        return zoom;
+    };
+    return mediumZoom;
+});
+
+mediumZoom(document.querySelectorAll('div.post-content img'));
diff --git a/hugo/themes/anatole/layouts/404.html b/hugo/themes/anatole/layouts/404.html
new file mode 100644
index 0000000..57fa172
--- /dev/null
+++ b/hugo/themes/anatole/layouts/404.html
@@ -0,0 +1,8 @@
+{{ define "main" }}
+    <div class="page_404">
+        <h1>404</h1>
+        <h2>{{ i18n "page_not_found" }}</h2>
+        <p>{{ i18n "page_does_not_exist" }}</p>
+        <p>{{ i18n "head_back" . | safeHTML }}</p>
+    </div>
+{{ end }}
diff --git a/hugo/themes/anatole/layouts/_default/baseof.html b/hugo/themes/anatole/layouts/_default/baseof.html
new file mode 100644
index 0000000..ed42060
--- /dev/null
+++ b/hugo/themes/anatole/layouts/_default/baseof.html
@@ -0,0 +1,21 @@
+<!DOCTYPE html>
+<html lang="{{- site.Language.Lang -}}" data-theme="{{- .Site.Params.displayMode -}}">
+{{- partial "head.html" . -}}
+<body>
+
+<div>
+{{ partial "navbar.html" . }}
+<div class="main">
+    <div class="autopagerize_page_element">
+        <div class="content">
+            {{- block "main" . }}{{- end }}
+        </div>
+    </div>
+</div>
+{{- partial "sidebar.html" . -}}
+</div>
+
+{{- partial "footer.html" . -}}
+</body>
+
+</html>
diff --git a/hugo/themes/anatole/layouts/_default/list.html b/hugo/themes/anatole/layouts/_default/list.html
new file mode 100644
index 0000000..ddf7749
--- /dev/null
+++ b/hugo/themes/anatole/layouts/_default/list.html
@@ -0,0 +1,18 @@
+{{ define "main" }}
+    <div class="archive animated fadeInDown">
+        <ul class="list-with-title">
+            {{ range .Data.Pages.GroupByDate "2006" }}
+                <div class="listing-title">{{ .Key }}</div>
+                {{ range .Pages }}
+                    <ul class="listing">
+                        <div class="listing-item">
+                            <div class="listing-post"><a href="{{ .Permalink }}" title="{{ .Title }}">{{ .Title }}</a>
+                                <div class="post-time"><span class="date">{{.Date.Format "Jan 2" }}</span></div>
+                            </div>
+                        </div>
+                    </ul>
+                {{ end }}
+            {{ end }}
+        </ul>
+    </div>
+{{ end }}
diff --git a/hugo/themes/anatole/layouts/_default/single.html b/hugo/themes/anatole/layouts/_default/single.html
new file mode 100644
index 0000000..5c034ea
--- /dev/null
+++ b/hugo/themes/anatole/layouts/_default/single.html
@@ -0,0 +1,42 @@
+{{ define "main" }}
+    <div class="post animated fadeInDown">
+        <div class="post-content">
+
+            <div class="post-title">
+                <h3>{{ .Title }}</h3>
+                {{ if eq .Type "post"}}
+                    <div class="info">
+                        <em class="fas fa-calendar-day"></em>
+                        <span class="date">{{ .Date.Format "Mon, Jan 2, 2006" }}</span>
+                        <em class="fas fa-stopwatch"></em>
+                        <span class="reading-time">{{ i18n "reading_time" .ReadingTime }}</span>
+                    </div>
+                {{ end }}
+            </div>
+
+            {{ .Content }}
+        </div>
+        <div class="post-footer">
+            <div class="info">
+                {{ if not .PublishDate.IsZero }} Published: <span class="date"> {{ .PublishDate.Format "2006-01-02" }}</span><br /> {{end}}
+                {{ if not .Lastmod.IsZero }} Last modified: <span class="date"> {{ .Lastmod.Format "2006-01-02 15:04" }}</span><br /> {{end}}
+                {{ .WordCount }} words<br />
+                {{ with .Page.Params.Categories }}{{ partial "taxonomy/categories.html" . }}{{ end }}
+                {{ with .Page.Params.Tags }}{{ partial "taxonomy/tags.html" . }}{{ end }}
+            </div>
+            <div class="feedback-link">
+                <a href=https://docs.google.com/forms/d/e/1FAIpQLSfBIFO50asRr6QmzRzCmtiRBXrImIraOnHuum9I798pSW_pjA/viewform>Submit anonymous feedback</a>
+            </div>
+        </div>
+
+        {{ if eq .Type "post"}}
+            {{ if .Site.DisqusShortname -}}
+                <div id="fb_comments_container">
+                    <h2>{{ i18n "comments" }}</h2>
+                    {{ template "_internal/disqus.html" . }}
+                </div>
+            {{- end }}
+        {{ end }}
+    </div>
+
+{{ end }}
diff --git a/hugo/themes/anatole/layouts/_default/single.md b/hugo/themes/anatole/layouts/_default/single.md
new file mode 100644
index 0000000..26c9c57
--- /dev/null
+++ b/hugo/themes/anatole/layouts/_default/single.md
@@ -0,0 +1,3 @@
+{{ define "main" }}
+    {{.Content}}
+{{ end }}
\ No newline at end of file
diff --git a/hugo/themes/anatole/layouts/index.html b/hugo/themes/anatole/layouts/index.html
new file mode 100644
index 0000000..dfcecdd
--- /dev/null
+++ b/hugo/themes/anatole/layouts/index.html
@@ -0,0 +1,30 @@
+{{ define "main" }}
+    {{ $paginator := .Paginate (where .Site.RegularPages "Type" "in" .Site.Params.mainSections) }}
+    {{ range $paginator.Pages }}
+        <div class="post animated fadeInDown">
+            <div class="post-title">
+                <h3><a href="{{ .RelPermalink }}">{{ .Title }}</a>
+                </h3>
+            </div>
+            <div class="post-content">
+                <div class="p_part"><p>{{ .Summary }}</p></div>
+            </div>
+            <div class="post-footer">
+                <div class="meta">
+                    <div class="info">
+                        <em class="fas fa-calendar-day">&nbsp;</em><span class="date">{{ .Date.Format "2006-01-02" }}</span>
+                        {{ with .Params.tags }}
+                            {{- range $index, $el := . -}}
+                                <a class="tag"
+                                   href="{{ ( printf "tags/%s/" ( . | urlize ) ) | relLangURL }}">{{ . }}</a>
+                            {{- end -}}
+                        {{ end }}
+                    </div>
+                </div>
+            </div>
+        </div>
+    {{ end }}
+    <div class="pagination">
+        {{ template "_internal/pagination.html" . }}
+    </div>
+{{ end }}
diff --git a/hugo/themes/anatole/layouts/partials/footer.html b/hugo/themes/anatole/layouts/partials/footer.html
new file mode 100644
index 0000000..b12a157
--- /dev/null
+++ b/hugo/themes/anatole/layouts/partials/footer.html
@@ -0,0 +1,2 @@
+{{if .Params.gallery }} {{- partial "medium-zoom.html" . -}} {{- partial "gallery.html" . -}}{{ end }}
+{{if .Params.math }} {{- partial "math.html" . -}} {{ end }}
\ No newline at end of file
diff --git a/hugo/themes/anatole/layouts/partials/gallery.html b/hugo/themes/anatole/layouts/partials/gallery.html
new file mode 100644
index 0000000..841ce6f
--- /dev/null
+++ b/hugo/themes/anatole/layouts/partials/gallery.html
@@ -0,0 +1,17 @@
+{{- if (isset .Params "gallery") -}}
+    <script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"
+            integrity="sha384-tsQFqpEReu7ZLhBV2VZlAu7zcOV+rXbYlF2cqB8txI/8aZajjp4Bqd+V6D5IgvKT"
+            crossorigin="anonymous"></script>
+    {{- $url := "https://cdnjs.cloudflare.com/ajax/libs/fancybox/3.4.0/jquery.fancybox.min.js" -}}
+    {{- $hash := "sha384-fPPwDm9Mfp/tF9MgxjA4B33PPUlNmlXdP3oKSJgdAywm+cfAEYllAIRl/D3SBlpc" -}}
+    <script defer
+            src="{{- $url -}}"
+            integrity="{{- $hash -}}"
+            crossorigin="anonymous"></script>
+    {{- $url := "https://cdnjs.cloudflare.com/ajax/libs/fancybox/3.4.0/jquery.fancybox.min.css" -}}
+    {{- $hash := "sha384-Ewh0nI9dXkThksh2nOTHmr8wAk5UlhiE2FiGSuZSnYkQ/G1W5wPuXGBnXodqD8GC" -}}
+    <link rel="stylesheet"
+          href="{{- $url -}}"
+          integrity="{{- $hash -}}"
+          crossorigin="anonymous" />
+{{- end -}}
\ No newline at end of file
diff --git a/hugo/themes/anatole/layouts/partials/head.html b/hugo/themes/anatole/layouts/partials/head.html
new file mode 100644
index 0000000..696b57a
--- /dev/null
+++ b/hugo/themes/anatole/layouts/partials/head.html
@@ -0,0 +1,84 @@
+<head>
+    <title> {{.Site.Params.author}}{{ with .Title }} | {{ . }}{{ end }} </title>
+
+    <!-- Meta -->
+    <meta charset="utf-8">
+    {{- hugo.Generator -}}
+    <meta name="viewport" content="width=device-width,initial-scale=1,viewport-fit=cover">
+    <meta name="description" content="{{ .Site.Params.description }}">
+    {{- if .Site.Params.googleSiteVerify }}
+    <meta name="google-site-verification" content="{{ .Site.Params.googleSiteVerify }}">
+    {{- end -}}
+
+    <!-- CSS -->
+    {{ $style := resources.Get "css/style.css" | resources.Minify | resources.Fingerprint }}
+    <link rel="stylesheet"
+          href="{{ $style.Permalink }}"
+          integrity="{{ $style.Data.Integrity }}"
+          crossorigin="anonymous"
+          type="text/css">
+    {{ if .Params.sidenotes }}
+    {{ $style := resources.Get "css/sidenote.css" | resources.Minify | resources.Fingerprint }}
+    <link rel="stylesheet"
+          href="{{ $style.Permalink }}"
+          integrity="{{ $style.Data.Integrity }}"
+          crossorigin="anonymous"
+          type="text/css">
+    {{ end }}
+    {{- $css := "" -}}
+    {{- range .Site.Params.customCss -}}
+    {{ $css := resources.Get . | fingerprint }}
+    <link rel="stylesheet"
+          href="{{ $css.Permalink }}"
+          integrity="{{ $css.Data.Integrity }}"
+          crossorigin="anonymous"
+          type="text/css">
+    {{- end -}}
+    <!-- Desired parts of fontawesome have been extracted to fontawesome.css
+    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.1/css/all.min.css"
+    integrity="sha512-+4zCK9k+qNFUR5X+cKL9EIR+ZOhtIloNl9GIKS57V1MyNsYpYcUrUeQc9vNfzsWfV28IaLL3i96P9sdNyeRssA=="
+    crossorigin="anonymous" />-->
+
+    <!-- Favicons -->
+    <link rel="shortcut icon" href="{{ .Site.Params.favicon | absURL }}favicon.ico" type="image/x-icon">
+
+    <link rel="canonical" href="{{ .Permalink }}">
+
+    <!-- RSS -->
+    {{ with .OutputFormats.Get "rss" -}}
+        {{ printf `<link rel="%s" type="%s" href="%s" title="%s" />` .Rel .MediaType.Type .Permalink $.Site.Title | safeHTML }}
+    {{ end -}}
+
+    <!-- JavaScript -->
+    {{ $anatole := resources.Get "js/anatole-header.js" }}
+    {{ $secureJS := $anatole |  resources.Minify | resources.Fingerprint }}
+    <script src="{{ $secureJS.Permalink }}"
+            integrity="{{ $secureJS.Data.Integrity }}"
+            crossorigin="anonymous"></script>
+
+    {{- $js := "" -}}
+    {{- range .Site.Params.customJs -}}
+        {{- if or (in . "http://") (in . "https://") -}}
+            <script src="{{ . | relURL }}"></script>
+        {{- else -}}
+            {{- $customJS := resources.Get . -}}
+            {{- if $customJS -}}
+                {{- if eq $js "" -}}
+                    {{- $js = $customJS -}}
+                {{- else -}}
+                    {{- $js = slice $js $customJS | resources.Concat "js/custom.js" -}}
+                {{- end -}}
+            {{- end -}}
+        {{- end -}}
+    {{- end -}}
+
+    {{- if ne $js "" -}}
+    {{- $secureJS := $js |  resources.Minify | resources.Fingerprint -}}
+    <script src="{{ $secureJS.Permalink }}"
+            integrity="{{ $secureJS.Data.Integrity }}"
+            crossorigin="anonymous"></script>
+    {{- end -}}
+
+    <!-- Twitter Cards -->
+    {{ template "_internal/twitter_cards.html" . }}
+</head>
diff --git a/hugo/themes/anatole/layouts/partials/math.html b/hugo/themes/anatole/layouts/partials/math.html
new file mode 100644
index 0000000..6d1d98f
--- /dev/null
+++ b/hugo/themes/anatole/layouts/partials/math.html
@@ -0,0 +1,79 @@
+{{- if or (eq site.Params.math true) (eq .Params.math true) -}}
+    {{- $use := "katex" -}}
+
+    {{- with site.Params.math -}}
+        {{- if and (isset . "use") (eq (.use | lower) "mathjax") -}}
+            {{- $use = "mathjax" -}}
+        {{- else if and (isset . "use") (eq (.use | lower) "katex-css") -}}
+            {{- $use = "katex-css" -}}
+        {{- end -}}
+    {{- end -}}
+
+    {{- if eq $use "mathjax" -}}
+        {{- $url := "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-AMS-MML_HTMLorMML" -}}
+        {{- $hash := "sha384-e/4/LvThKH1gwzXhdbY2AsjR3rm7LHWyhIG5C0jiRfn8AN2eTN5ILeztWw0H9jmN" -}}
+        <script defer
+                src="{{- $url -}}"
+                integrity="{{- $hash -}}"
+                crossorigin="anonymous"></script>
+        <script type="text/x-mathjax-config">
+            MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], ['\\(','\\)']] } });
+        </script>
+    {{- else if eq $use "katex" -}}
+        {{- $url := "https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.css" -}}
+        {{- $hash := "sha384-zB1R0rpPzHqg7Kpt0Aljp8JPLqbXI3bhnPWROx27a9N0Ll6ZP/+DiW/UqRcLbRjq" -}}
+        <link rel="preload"
+              as="script"
+              href="{{- $url -}}"
+              integrity="{{- $hash -}}"
+              crossorigin="anonymous" />
+        <noscript><link rel="stylesheet"
+              href="{{- $url -}}"
+              integrity="{{- $hash -}}"
+              crossorigin="anonymous" />
+        </noscript>
+
+        {{- $url := "https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.js" -}}
+        {{- $hash := "sha384-y23I5Q6l+B6vatafAwxRu/0oK/79VlbSz7Q9aiSZUvyWYIYsd+qj+o24G5ZU2zJz" -}}
+        <script defer
+                src="{{- $url -}}"
+                integrity="{{- $hash -}}"
+                crossorigin="anonymous"></script>
+
+        {{- $url := "https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/contrib/auto-render.min.js" -}}
+        {{- $hash := "sha384-kWPLUVMOks5AQFrykwIup5lo0m3iMkkHrD0uJ4H5cjeGihAutqP0yW0J6dpFiVkI" -}}
+        <script defer
+                src="{{- $url -}}"
+                integrity="{{- $hash -}}"
+                crossorigin="anonymous"
+                onload="renderMathInElement(document.body);"></script>
+    {{- else if eq $use "katex-css" -}}
+        {{ $css := resources.Get "katex.min.css" | resources.Minify | resources.Fingerprint }}
+        <link rel="stylesheet"
+              href="{{ $css.Permalink }}"
+              integrity="{{ $css.Data.Integrity }}"
+              crossorigin="anonymous" />
+        {{ $katexjs := resources.Get "katex.min.js" | resources.Minify | resources.Fingerprint }}
+        <script defer
+                src="{{ $katexjs.Permalink }}"
+                integrity="{{ $katexjs.Data.Integrity }}"
+                crossorigin="anonymous"></script>
+        {{ $katexrender := resources.Get "contrib/auto-render.min.js" | resources.Minify | resources.Fingerprint }}
+        <script defer
+                src="{{ $katexrender.Permalink }}"
+                integrity="{{ $katexrender.Data.Integrity }}"
+                crossorigin="anonymous"
+                onload="renderMathInElement(document.body);"></script>
+        {{ $fonts := readDir "themes/anatole/assets/fonts" }}
+        {{- range $fonts -}}
+            {{- if strings.HasSuffix .Name ".woff2" -}}
+                {{ $font := resources.Get (printf "fonts/%s" .Name) }}
+                <link rel="preload"
+                    href="{{ $font.Permalink }}"
+                    as="font"
+                    type="font/woff2"
+                    crossorigin="anonymous" />
+            {{- end -}}
+        {{- end -}}
+    {{- end -}}
+{{- end -}}
diff --git a/hugo/themes/anatole/layouts/partials/medium-zoom.html b/hugo/themes/anatole/layouts/partials/medium-zoom.html
new file mode 100644
index 0000000..bc43a7c
--- /dev/null
+++ b/hugo/themes/anatole/layouts/partials/medium-zoom.html
@@ -0,0 +1,5 @@
+{{ $js := resources.Get "js/medium-zoom.js" }}
+{{ $secureJS := $js |  resources.Minify | resources.Fingerprint }}
+<script src="{{ $secureJS.Permalink }}"
+        integrity="{{ $secureJS.Data.Integrity }}"
+        crossorigin="anonymous"></script>
\ No newline at end of file
diff --git a/hugo/themes/anatole/layouts/partials/navbar.html b/hugo/themes/anatole/layouts/partials/navbar.html
new file mode 100644
index 0000000..fb894e5
--- /dev/null
+++ b/hugo/themes/anatole/layouts/partials/navbar.html
@@ -0,0 +1,25 @@
+<div class="page-top animated fadeInDown">
+    <div class="theme-switch-item">
+        <button class="theme-switch" title="Switch Theme">
+            <i>Lighting</i>
+        </button>
+    </div>
+    <input type="checkbox" class="navbar-toggle" tabindex="0" id="navbar-toggle-cbox" />
+    <label class="navbar-toggle" for="navbar-toggle-cbox"><span>Toggle navigation menu</span></label>
+    <ul class="nav">
+        <label for="navbar-toggle-cbox" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false" aria-controls="navbar-toggle-cbox">
+            &nbsp;
+        </label>
+        {{ $url := .RelPermalink }}
+        {{ range .Site.Menus.main }}
+            {{ $active := eq $url .URL }}
+            <li><a {{if $active }}class="current"{{end}}
+                href="{{ .URL }}"
+                        {{ if strings.HasPrefix .URL "http" }}
+                            target="_blank"
+                            rel="noopener noreferrer"
+                        {{ end }}
+                title="{{ .Title }}">{{ .Name }}</a></li>
+        {{ end }}
+    </ul>
+</div>
\ No newline at end of file
diff --git a/hugo/themes/anatole/layouts/partials/sidebar.html b/hugo/themes/anatole/layouts/partials/sidebar.html
new file mode 100644
index 0000000..147aa83
--- /dev/null
+++ b/hugo/themes/anatole/layouts/partials/sidebar.html
@@ -0,0 +1,27 @@
+<div class="sidebar animated fadeInDown">
+    <div class="logo-title">
+        <div class="title">
+            <img src="{{ printf "%s.png" .Site.Params.profilePicture | absURL }}" srcset="{{ printf "%s-small.png" .Site.Params.profilePicture | absURL }} 337w, {{ printf "%s.png" .Site.Params.profilePicture | absURL }} 1020w" sizes="337px, 1020px" alt="profile picture" width="127" height="127" />
+            <h3 title=""><a href="/">{{ .Site.Params.Title }}</a></h3>
+            {{ if .Site.Params.description }}
+            <div class="description">
+                <p>{{ .Site.Params.description }}</p>
+            </div>
+            {{ end }}
+        </div>
+    </div>
+    <table class="social-links">
+        <tr>
+        {{ range $item := .Site.Params.socialIcons }}
+            <td>
+                <a href="{{ $item.url }}" rel="me" aria-label="{{ $item.title }}">
+                    <span class="{{ $item.icon }} fa-2x" aria-hidden="true">&nbsp;</span>
+                </a>
+            </td>
+        {{ end }}
+        </tr>
+    </table>
+    <div class="footer">
+        <div class="by_farbox">&copy; {{ .Site.Params.author }} {{ now.Format "2006"}} </div>
+    </div>
+</div>
diff --git a/hugo/themes/anatole/layouts/partials/taxonomy/categories.html b/hugo/themes/anatole/layouts/partials/taxonomy/categories.html
new file mode 100644
index 0000000..fd4988d
--- /dev/null
+++ b/hugo/themes/anatole/layouts/partials/taxonomy/categories.html
@@ -0,0 +1,5 @@
+<span class="separator">
+{{- range $index, $el := . -}}
+    <a class="category" href="{{ ( printf "categories/%s/" ( . | urlize ) ) | relLangURL }}">{{ . }}</a>
+{{- end -}}
+</span>
diff --git a/hugo/themes/anatole/layouts/partials/taxonomy/tags.html b/hugo/themes/anatole/layouts/partials/taxonomy/tags.html
new file mode 100644
index 0000000..86c0673
--- /dev/null
+++ b/hugo/themes/anatole/layouts/partials/taxonomy/tags.html
@@ -0,0 +1,5 @@
+<span class="separator">
+{{- range $index, $el := . -}}
+    <a class="tag" href="{{ ( printf "tags/%s/" ( . | urlize ) ) | relLangURL }}">{{ . }}</a>
+{{- end -}}
+</span>
diff --git a/hugo/themes/anatole/theme.toml b/hugo/themes/anatole/theme.toml
new file mode 100644
index 0000000..6e15b08
--- /dev/null
+++ b/hugo/themes/anatole/theme.toml
@@ -0,0 +1,21 @@
+# theme.toml template for a Hugo theme
+# See https://github.com/gohugoio/hugoThemes#themetoml for an example
+
+name = "Anatole"
+license = "MIT"
+licenselink = "https://github.com/lxndrblz/anatole/blob/master/LICENSE"
+description = "Anatole is a beautiful minimalist two-column hugo theme based on farbox-theme-Anatole."
+homepage = "https://github.com/lxndrblz/anatole/"
+tags = ["blog", "responsive", "clean", "minimalist", "minimal", "personal", "dark"]
+features = ["blog", "seo", "responsive", "mobile", "disqus", "fontawesome", "math"]
+min_version = "0.53.0"
+
+[author]
+  name = "Alexander Bilz"
+  homepage = "https://alexbilz.com"
+
+# If porting an existing theme
+[original]
+  author = "Cai Cai"
+  homepage = "https://www.caicai.me/"
+  repo = "https://github.com/hi-caicai/farbox-theme-Anatole"