Patrick Stevens

Claude knows who you are

Sat, 18 Apr 2026 09:41:00 +0100

Claude Opus 4.7 is solidly superhuman at stylometric identification of authors.

Another trip to Tromsø

Sun, 05 Apr 2026 11:42:00 +0000

Travelogue of another trip to Norway.

Fun LLM prompt: where are my mental models defective?

Mon, 02 Mar 2026 22:05:00 +0000

A fun prompt to use with a powerful LLM, to learn about something in great detail.

LLM arithmetic

Sat, 13 Dec 2025 09:41:00 +0000

I keep seeing people going on about the 3.9 - 3.11 = 0.79 thing, but we already know why they do that and how to avoid it!

Robocop: automated code review

Mon, 10 Nov 2025 17:25:00 +0000

I wrote a basic LLM-based code review tool that’s distressingly effective.

LLMs sanding off the edges

Mon, 06 Oct 2025 22:59:00 +0100

I’m a bit concerned about a failure mode that is increasingly arising with LLMs: the divorce between how something looks, and its quality.

Boolean blindness

Fri, 29 Aug 2025 10:23:00 +0100

Why booleans are an advanced technique that should be used with care.

Some Python surprises

Sat, 21 Jun 2025 00:00:00 +0100

I am forever astonished that people describe Python as a simple language. Here are some of the things I found very surprising about it.

Things I've learned about the .NET runtime

Fri, 20 Jun 2025 00:00:00 +0100

While writing an MSIL interpreter, I discovered a bunch of unexpected things about the .NET runtime.

Announcing WoofWare.Expect

Tue, 17 Jun 2025 00:00:00 +0100

A basic but functional expect-testing framework for F#.

LLM effect on my programming (2025 edition)

Mon, 16 Jun 2025 09:28:00 +0100

Since my last post about this in 2024-03, LLMs have become a distinctly positive addition to my ability.

A Bitcoin analogy for Multiple Drafts theory of consciousness

Sat, 08 Mar 2025 13:30:00 +0000

I’ve never actually seen anyone write down precisely the way I think about consciousness, so here it is: it’s (gasp!) like a blockchain.

NativeAOT in .NET 9 and Nixpkgs/darwin

Thu, 28 Nov 2024 20:10:00 +0000

How to get a NativeAOT build using nixpkgs

Mary's Room

Fri, 25 Oct 2024 23:30:00 +0100

Last weekend I was in a discussion ostensibly about whether the mind is nonphysical. Much of the discussion ended up as a rather polite shouting match about Mary’s Room. Here is what I actually believe about it.

My reading list

Fri, 27 Sep 2024 19:31:58 +0100

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.

Unhinged rant about software

Fri, 27 Sep 2024 18:14:00 +0100

Modern software practices and their sadness.

Lessons from a massage course

Sat, 07 Sep 2024 11:47:00 +0100

I went to a one-day intro to massage taster course, and it was fun and interesting!

WoofWare.Myriad.Plugins learns to parse args

Mon, 26 Aug 2024 12:26:00 +0100

My F# source generators have some new features, including an argument parser.

New GitHub workflows

Thu, 15 Aug 2024 17:51:00 +0100

I’ve made a GitHub workflow to assert that all required GitHub checks are complete, and one to publish and attest NuGet packages.

Learning plan for "Program Equilibrium in the Prisoner's Dilemma"

Thu, 25 Jul 2024 19:48:00 +0100

The questions whose answers I don’t know, and the things I intend to learn, on the way to understanding a paper.

Code having "the right philosophy"

Wed, 24 Jul 2024 17:38:00 +0100

Those who would give up essential safety, to purchase a little temporary simplicity, deserve (and will get) neither safety nor simplicity.

The phrase "I Notice That I Am Confused"

Wed, 01 May 2024 11:38:00 +0100

To me it’s got a specific meaning, but I’ve seen it used much more generally, and I think its meaning should not be polluted.

Games

Sun, 28 Apr 2024 20:51:00 +0100

This page holds an incomplete list of games I have played.

Video games

In progress

ibb and obb. Fun two-player co-op, neat and slightly mind-bending physics.
It Takes Two. Amusing but not worth the price, I think. This is more like a constant stream of “oh, what funny one-off mechanic have they implemented next”.

Not in progress

Chants of Sennaar. An extremely impressive 81% of players have the Steam achievement for leaving the first game area, which I think tells you how gripping this game is. It’s basically a particular kind of IQ test, but it’s really fun. Reminiscent of Return of the Obra Dinn. I could have done with a little more variation in the languages; there are a few neat bits, but there was a broad language design space they mostly didn’t touch.
Baldur’s Gate 3 (hundreds of hours). Glorious game, aesthetically beautiful, absolutely gigantic amount of content, and sometimes succeeds at making you empathise with the characters. Having played this, I wouldn’t really want to go back to playing a DnD-style RPG any other way: all the bookkeeping happens in the background.
Factorio, and Dyson Sphere Program (both hundreds of hours). Grouped together because they are basically the same game. I have banned myself from playing this genre of game (I gave myself RSI from it for a while). Between the two, DSP is much prettier and somewhat more forgiving, but also it has the annoying mechanic of laying your factory out on spheres. “Cannot place blueprint across tropic lines” just feels bad; you really want your factory layout not to depend on where it is situated.
Outer Wilds (about half an hour). Everyone raves about this game, but it makes me travel-sick very fast. No further comment.
Human Resource Corporation, and 7 Billion Humans (25 hours total). As I’ve got older, games which are purely programming have become less appealing (that is, after all, my day job). These ones are visually pretty, but I have no idea how they’ll play for someone who is not a programmer.
Among Us (70 hours). I love this game (multiplayer social deduction), and am quite sad that everyone stopped playing it after the pandemic. Your first few games will be extremely hard if you’re an impostor, because your plausibility relies on being able to make up what you were doing where (which means you need to know the maps), but once you’ve got over that it’s just really fun.
Antichamber (2 hours). This makes me motion-sick really quickly, and I have only played it in ten-minute bursts. I imagine this would be very interesting if I could play it.
Baba Is You (68 hours). Cute puzzle game, with some extremely difficult puzzles. Mind-blowingly beautiful mechanics as you get towards the end. Tier 1. I played this with my housemates on a big screen and it was a great way to play!
Braid (9 hours). Lovely puzzle game, innovative, coherent; a classic Jonathan Blow game. See qntm’s write-up.
Classic Sudoku (135 hours) and Killer Sudoku (107 hours), by the Cracking the Cryptic team. A bunch of these puzzles are just really neat! A few are slogs, but many of them are cute, and I got the impression at the time that it might be quite a good introduction to advanced Sudoku techniques. The over-100-hours is because I have done each puzzle twice.
Death and Taxes (2 hours). Meh. Three years after playing it I can’t really remember it at all.
Disco Elysium (17 hours). Some of this game is pretty funny. I haven’t completed it, because it was getting to be a bit of a slog.
Dota 2 (thousands of hours). Deeply addictive. You will spend the first thousand hours or so completely sucking at this game, and improvement comes from recognising that you suck (thanks Purge for the framing on this). I stopped playing this game many years ago when I discovered that I didn’t actually enjoy it. My mechanical skill was pretty mediocre, so my specialist skill, inasmuch as I had one, was to bind together a team and keep morale up. A team of five coherent players will beat a team of five individuals any day of the week, and a team of five players will beat a team of four players plus a griefer, so an extremely valuable role is “keep your team feeling like a unit”. But it grinds away at the soul, and eventually I realised it isn’t fun.
Elsinore (7 hours). Intriguing and cute puzzle game: you’re playing Ophelia, caught in a time loop for several critical days in the setting of Hamlet. You gradually learn what happens when, and how you can act to change what happens. I haven’t finished this because it had a bit too much of the Shlemiel the Painter about it: as you solve milestones, you keep revealing more, but have to keep retracing the same steps anyway.
The Zachtronics games (Exapunks, TIS-100, Shenzhen I/O, Opus Magnum, MOLEK-SYNTEZ; excluding Infinifactory; a couple of hours). Again, like Human Resource Corporation, these games are purely programming, and I’m getting too old for that sort of thing to be a fun way to relax.
The House of da Vinci (and 2 and 3; 20 hours). These are all a bit too point-and-click linear for my liking.
Inscryption (15 hours). Interesting card game mechanic; intriguing metagame.
Islands of Insight (55 hours). Loads and loads of small puzzles, many of them Nikoli-like. Some are just quite annoying (“go to this place and hunt around”). Some are deeply absorbing. Very pretty aesthetics.
The Witness (55 hours). Good lord is this game amazing. It’s a superstimulus for the sense of discovering an insight. The game mechanics are all taught entirely without words, which is a brilliant trick if you can manage it.
The Looker (2 hours). Parody of The Witness. Some of it laugh-out-loud funny.
Loop Hero (25 hours, not all of which was me). Very playable roguelike.
Obduction (7 hours). I distinctly remember that when I was much younger I enjoyed Myst and friends. This was back in the days when I had the patience to read The Lord of the Rings, for example. Nowadays I suspect I just don’t have the attention span and/or patience. Obduction in particular requires a lot of walking around.
Offworld Trading Company (30 hours). I am terrible at this game, and find it deeply stressful, but it’s extremely compelling. The market mechanic is either very innovative or very innovatively phrased.
Oxygen Not Included (8 hours). I feel like I’m missing some fundamental part of this game. It feels like it should be some massive Factorio-like infinite world automation thing, but in practice I ended up doing an awful lot of micromanagement. Probably a skill issue.
Patrick’s Parabox (10 hours). Cute puzzle game. Mind-bending, innovative. Really good.
Portal (and 2, and the Reloaded mod; 35 hours in total). Classics, of course.
Return of the Obra Dinn (12 hours). Aesthetically beautiful game. When the main mechanic was introduced, I sat back in awe. You gradually get to have quite a good mental map of what’s going on here. Highly recommended.
Slay the Spire (hundreds of hours). Deep gameplay (extremely deep - check out Jorbs’s streams!). One of those games where it takes a while to realise just how bad at it you are; all your losses feel like luck, but actually this game is at least 80% skill.
Roguebook (23 hours). A variant of Slay the Spire. Aesthetically quite pleasing, but I didn’t find it as gripping as StS.
Subnautica (26 hours). Personally I am a wimp and found quite a bit of this game primally terrifying; so much so that I ended up watching a stream of the game past a certain point. I don’t much like the choice to phrase the story in terms of time pressure, when everything is actually triggered by reaching certain milestones. It’s pretty, though, and if I weren’t a wuss I’d definitely have played it to the end.
SUPERHOT (3 hours). This is the first-person shooter for the person like me who is terrible at first-person shooters! It’s more like a puzzle game than a shooter, but for the first time it made me feel like I had a game in the shooter genre that I could play!
Taiji (12 hours). This is a puzzle game much like The Witness but a bit worse (but still a great game!).
The Talos Principle (and 2; 35 hours total). Puzzle games with a skin of basic philosophy. Aesthetically beautiful; some very hard puzzles.
Torment: Tides of Numenera (40 hours, because I’ve played it twice). If you’re going to play an RPG, play Baldur’s Gate 3 instead. Torment is much less pretty and considerably smaller, but I did have fun playing it.
Understand (3 hours). Tiny puzzle game. Almost precisely an IQ test. Great fun.
We Were Here (and Too, and Together; a few hours total). Two-player co-op games. The in-game chat is really janky, so we just used Discord. I feel like these were nearly great games, but something just felt missing. A Baldur’s Gate 3 campaign completely supplanted this.

Board games

(This section is basically a placeholder for now.)

The Yoneda lemma

Sat, 13 Apr 2024 16:58:00 +0100

Another attempt to explain that the Yoneda lemma is actually intuitive.

ChatGPT's effect on my programming

Wed, 27 Mar 2024 12:01:00 +0000

After a decent while programming with ChatGPT, I’m not sure it’s even a net positive on my ability.

YAML is not a superset of JSON

Thu, 14 Mar 2024 00:12:00 +0000

All the reasons I know for why YAML is not a superset of JSON.

Trip to Tromsø

Wed, 13 Mar 2024 12:18:00 +0000

Travelogue of my trip to Norway.

Why does no-confusion use equality rather than a recursive call?

Fri, 01 Mar 2024 13:51:00 +0000

A question about the definition of a no-confusion type.

Starting a suspended process

Fri, 19 Jan 2024 18:51:00 +0000

How to start a process into a suspended state on Linux

Announcing WoofWare.Myriad.Plugins

Sun, 31 Dec 2023 13:42:00 +0000

Some F# source generators to solve common problems I have.

iOS interface

Sun, 31 Dec 2023 12:49:00 +0000

A bunch of ways the iOS user interface is bad, and some undiscoverable features.

Squashed stacked PRs workflow

Wed, 18 Oct 2023 20:36:00 +0100

How to handle stacked pull requests in a repository which requires squashing all history on merge.

Raymond Smullyan chess problem walkthrough

Fri, 06 Oct 2023 20:53:00 +0100

The thought process behind solving a particular Raymond Smullyan chess retrograde analysis puzzle.

Nix fireside chat outline

Thu, 05 Oct 2023 20:00:00 +0100

A talk I gave at work about the Nix build system.

Property-based testing introduction

Sun, 01 Oct 2023 20:53:00 +0100

A talk I gave at work introducing property-based testing and then giving some more advanced techniques.

The tiny proof that primes 1 mod 4 are sums of two squares

Thu, 28 Sep 2023 00:15:00 +0000

Exploding the incredibly terse proof into a bunch of exposition.

Imre Leader Appreciation Society

Sat, 09 Sep 2023 23:30:00 +0100

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).

Anki decks

Fri, 08 Sep 2023 19:22:27 +0100

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 (the deck has a misleading name; it’s actually a general Geography deck). You can filter out the london-tube tag if you like, or world-capitals, or american-geography.

This work by Patrick Stevens is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

Notes for a Git fireside chat

Wed, 06 Sep 2023 13:51:00 +0000

A syllabus for a fireside chat to give at work, on Git

Films

Sat, 02 Sep 2023 13:30:00 +0100

This page holds a list of films I have watched, spoiler-free, starting from 9th January 2015.

Project Hail Mary: Great film. Visually beautiful; tense where appropriate; I was absolutely surrounded by people crying. Strongly recommended; I saw this on the BFI IMAX screen, which I think was a good choice.
Dungeons and Dragons: Honour Among Thieves: Really rather good! As they say, “Consider this fan serviced”. Light-hearted, plenty of flashy callouts.

Questions I had about transformers

Wed, 12 Jul 2023 00:00:00 +0000

Some basic questions I had about transformers, and their possible answers.

Lifehacks

Sat, 31 Dec 2022 23:21:44 +0000

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 LessWrong post on the subject. 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 (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.

About me

Sat, 20 Aug 2022 14:18:00 +0100

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.
Hacker News.
Email.
LinkedIn (used almost never).
Twitter (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 by upvotes.
On Hacker.org, I am 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, solving with Team Palindrome; if interested, see captain’s write-up from 2019 and from 2020. In 2021, we won.
I have solved a number of Project Euler problems.

Languages:

About this website

Sun, 31 Jul 2022 20:16:44 +0100

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 (archive). Mathematical notation in HTML is rendered by KaTeX.

The water filtration industry has played us for absolute fools

Sun, 15 May 2022 00:00:00 +0000

My Brita water filter does not in fact purify.

Don't supply `-f` to `rm` unless you know you need it

Mon, 25 Oct 2021 00:00:00 +0000

A vignette on the theme of ‘do not allow yourself to get into the habit of supplying the -f flag to rm’.

Argument in a high-trust environment

Thu, 21 Oct 2021 00:00:00 +0000

I tend to argue things in a particular way because I’m in a high-trust environment.

Crates (existentials in F#)

Tue, 19 Oct 2021 00:00:00 +0000

An introduction to the crate pattern for representing existential quantification in F#.

Restaurants

Sun, 12 Sep 2021 22:47:44 +0100

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.

Metatesting your property-based tests

Thu, 27 May 2021 00:00:00 +0000

A rather little-known but very easy and high-reward way to sanity-check your property-based tests.

Product comparisons

Fri, 21 May 2021 18:23:48 +0100

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.

In praise of `--dry-run`

Fri, 21 May 2021 00:00:00 +0000

First-class support for a –dry-run mode makes a tool more maintainable, more testable, and more user-friendly.

Continuation-passing style

Mon, 22 Mar 2021 00:00:00 +0000

Motivating the technique of continuation-passing style, by looking at recursive functions.

Rewriting the Technical Interview, in Mathematica

Wed, 17 Mar 2021 00:00:00 +0000

An exploration into reaching into the internals of Mathematica to natively evaluate C code.

Software engineer syllabus

Wed, 03 Feb 2021 00:00:00 +0000

An incomplete but growing list of things I would make mandatory reading if I were building a software engineering syllabus.

Cyclic dependencies (a note from Hacker News)

Sat, 30 Jan 2021 00:00:00 +0000

A quick note from Hacker News about how to model in F# something that might look like cyclic dependencies.

An incomplete life evaluation checklist

Thu, 21 Jan 2021 00:00:00 +0000

A checklist I have used during my regular six-monthly life review.

Find the Bug, C# edition

Mon, 18 Jan 2021 00:00:00 +0000

A cute little exercise in bug-spotting.

A bug in Git

Mon, 28 Dec 2020 00:00:00 +0000

A bug I found and reported in Git.

Christmas quiz

Sat, 26 Dec 2020 00:00:00 +0000

A round of questions I wrote for a Christmas quiz.

Christmas dinner notes

Sat, 21 Nov 2020 00:00:00 +0000

My notes on the production of a Christmas dinner.

Anki as Learning Superpower

Fri, 23 Oct 2020 00:00:00 +0000

Some techniques to help you get more out of Anki.

General relativity (a note from Hacker News)

Sun, 18 Oct 2020 00:00:00 +0000

A quick note from Hacker News about a neat fact from general relativity.

Nulls and options (a note from Hacker News)

Sat, 10 Oct 2020 00:00:00 +0000

A quick note from Hacker News about why we want optional types but why ’null’ is unintuitive.

The uncountability of the reals (a note from Hacker News)

Sun, 02 Aug 2020 00:00:00 +0000

A quick note from Hacker News about a beautiful proof of the uncountability of the reals.

In favour of recursive functions, not imperative constructs, to make loops

Wed, 22 Jul 2020 00:00:00 +0000

How to write loops immutably and safely.

Static config (a note from Hacker News)

Sun, 05 Apr 2020 00:00:00 +0000

A quick note from Hacker News about my preference for static config rather than dynamic.

Defunctionalisation

Wed, 04 Mar 2020 00:00:00 +0000

An underappreciated tool for writing good software.

MIT Mystery Hunt photos

Sat, 25 Jan 2020 00:00:00 +0000

Photos from the Mystery Hunt.

MIT Mystery Hunt 2020 answers

Thu, 16 Jan 2020 00:00:00 +0000

A couple of solution documents I made during the progress of the 2020 MIT Mystery Hunt.

Teaching how to cook (a note from Hacker News)

Fri, 27 Dec 2019 00:00:00 +0000

A quick note from Hacker News about my thoughts on teaching cookery.

The stages of mathematics teaching (a note from Hacker News)

Fri, 20 Sep 2019 00:00:00 +0000

A quick note from Hacker News about the various stages of learning and teaching in mathematics.

MIT Mystery Hunt 2019 answers

Wed, 16 Jan 2019 00:00:00 +0000

A couple of solution documents I made during the progress of the 2019 MIT Mystery Hunt.

Chatting with Don Syme about the F# compiler

Mon, 10 Sep 2018 00:00:00 +0000

Notes from a chat with the creator of F#, about how to contribute to the compiler.

Dependent types overview

Sat, 21 Jul 2018 00:00:00 +0000

A quick overview of dependent types.

JSON comments (a note from Hacker News)

Sat, 02 Jun 2018 00:00:00 +0000

A quick note from Hacker News about why the comment-handling situation in JSON is bad.

What is lost when we move between number systems?

Sun, 08 Apr 2018 00:00:00 +0000

Answering the question, “What is lost when we move from the reals to the complex numbers?”.

Infinitesimals as an idea that took a long time

Sat, 03 Feb 2018 00:00:00 +0000

Answering the question, “Which mathematical ideas took a long time to define rigorously?”.

Abuse of notation in function application

Sun, 05 Nov 2017 00:00:00 +0000

Answering the question, “Are these examples of abuses of notation?”.

The relationship between the IMO and research mathematics

Tue, 14 Mar 2017 00:00:00 +0000

Answering the question, “does the International Maths Olympiad help research mathematics?”.

Proof of Cauchy-Schwarz

Tue, 14 Feb 2017 00:00:00 +0000

This is just a link to a beautiful proof of the Cauchy-Schwarz inequality. There are a number of elegant proofs, but this is by far my favourite, because (as pointed out in the paper) it “builds itself”.

What does Mathematica mean by ComplexInfinity?

Sat, 31 Dec 2016 00:00:00 +0000

Answering the question, “Why does WolframAlpha say that a quantity is ComplexInfinity?”.

How far back does mathematical understanding go?

Wed, 10 Aug 2016 00:00:00 +0000

Answering the question, “how far back in time would maths be understandable to a modern mathematician?”.

A Free Market

Sun, 07 Aug 2016 00:00:00 +0000

The story of Martin’s search for a kaki fruit.

Be a Beginner

Fri, 05 Aug 2016 00:00:00 +0000

Being a beginner at something is great, especially if it’s something that humans are built for.

Part III essay

Wed, 15 Jun 2016 00:00:00 +0000

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).

The use of jargon

Mon, 13 Jun 2016 00:00:00 +0000

Why jargon is a really useful thing to have and use.

Finitistic reducibility

Wed, 25 May 2016 00:00:00 +0000

A quick overview of the definition of the mathematical concept of finitistic reducibility.

Tennenbaum's theorem

Wed, 27 Apr 2016 00:00:00 +0000

Most recent exposition: an article 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.

Modular machines

Thu, 21 Apr 2016 00:00:00 +0000

I’ve written a blurb 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.

Independence of the Axiom of Choice (for programmers)

Wed, 13 Apr 2016 00:00:00 +0000

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.

Another Monty Hall explanation

Fri, 08 Apr 2016 00:00:00 +0000

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?

Clojure and Exercism

Mon, 28 Mar 2016 00:00:00 +0000

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.

Why do we get complex numbers in a certain expression?

Thu, 03 Mar 2016 00:00:00 +0000

Answering the question, “Why does a continued fraction containing only 1, subtraction, and division result in one of two complex numbers?”.

Friedberg-Muchnik theorem

Fri, 05 Feb 2016 00:00:00 +0000

Another short post to point out my new article on the Friedberg-Muchnik theorem, 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.

Representable functors

Tue, 26 Jan 2016 00:00:00 +0000

Just a post to draw attention to my new 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.

Multiplicative determinant

Fri, 01 Jan 2016 00:00:00 +0000

I’m clearing out my desktop again, and found this document on the multiplicativity of the determinant, 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.

Monadicity Theorems

Thu, 31 Dec 2015 00:00:00 +0000

Another short post to highlight the existence of an article about the Monadicity Theorems, 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.

General Adjoint Functor Theorem

Thu, 24 Dec 2015 00:00:00 +0000

Just a post to draw attention to my new article about the General Adjoint Functor Theorem. 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.

My First Forcing

Sat, 28 Nov 2015 00:00:00 +0000

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.

Eilenberg-Moore

Thu, 12 Nov 2015 00:00:00 +0000

As an exercise in understanding the definitions involved, I find the Eilenberg-Moore category of a certain functor.

Heyting algebras

Wed, 30 Sep 2015 00:00:00 +0000

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 \(\epsilon \circ (\bar{f} \times 1_b) = f\).

Exponentials in category theory

Tue, 29 Sep 2015 00:00:00 +0000

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 \(\text{Hom}{\mathbf{Sets}}(A, C^B)\).

Limits exercises

Tue, 29 Sep 2015 00:00:00 +0000

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 \(my = mz = mz = mx\).

Lottery odds

Fri, 25 Sep 2015 00:00:00 +0000

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.

Properties of categorical limits

Wed, 23 Sep 2015 00:00:00 +0000

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.

Limits and pullbacks

Tue, 22 Sep 2015 00:00:00 +0000

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.)

Limits and pullbacks 2

Tue, 22 Sep 2015 00:00:00 +0000

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.

Duality exercises

Sat, 19 Sep 2015 00:00:00 +0000

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.

Exercise 3 I’ve already proved - 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).

Groups in categories

Sat, 19 Sep 2015 00:00:00 +0000

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.

Equalisers

Wed, 16 Sep 2015 00:00:00 +0000

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 \(e\)”.

Duality in category theory

Tue, 15 Sep 2015 00:00:00 +0000

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.

Hom-sets and exercises

Thu, 10 Sep 2015 00:00:00 +0000

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.

Products in category theory

Tue, 08 Sep 2015 00:00:00 +0000

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.

Epis and monos

Wed, 02 Sep 2015 00:00:00 +0000

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.

Initial, terminal, and generalised elements

Wed, 02 Sep 2015 00:00:00 +0000

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.

Free categories and foundations

Fri, 21 Aug 2015 00:00:00 +0000

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”.

Proof by contradiction

Fri, 21 Aug 2015 00:00:00 +0000

Here I explain proof by contradiction so that anyone who has ever done a sudoku and seen algebra may understand it.

New categories from old

Thu, 20 Aug 2015 00:00:00 +0000

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.

Category Theory introduction

Wed, 19 Aug 2015 00:00:00 +0000

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.

Sequence on Awodey's Category Theory

Wed, 19 Aug 2015 00:00:00 +0000

In the summer of 2015, I worked through Awodey’s Category Theory, and I produced a large collection of 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.

What is a Category?

Wed, 19 Aug 2015 00:00:00 +0000

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.

Motivational learning

Thu, 29 Jan 2015 00:00:00 +0000

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!

Latin translation tips

Tue, 23 Dec 2014 00:00:00 +0000

I’m clearing out my computer, and found a file which may as well be here.

Chunking:

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”).
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.
Look for any subordinate clauses (like “dixit Caecilius, qui in horto laborabat…”)
If you see an adjective-looking thing, it probably has to go with a noun.
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:

Remember that your chunking is probably wrong somewhere, but also is probably broadly right.
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”).
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.

Matrix puzzle

Fri, 19 Dec 2014 00:00:00 +0000

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.

Film recommendation, Interstellar

Tue, 09 Dec 2014 00:00:00 +0000

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.

Christmas carols

Tue, 02 Dec 2014 00:00:00 +0000

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.

Once in Royal David’s City. 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.
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.

Sum-of-two-squares theorem

Tue, 09 Sep 2014 00:00:00 +0000

*Wherein I detail the most beautiful proof of a theorem I’ve ever seen, in a bite-size form suitable for an Anki deck.

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.

Python, script shadowing

Tue, 26 Aug 2014 00:00:00 +0000

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 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.

Parables, chapter 1, verses 1-10

Tue, 19 Aug 2014 00:00:00 +0000

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.

Perfect pitch

Mon, 21 Jul 2014 00:00:00 +0000

I have a limited form of perfect (absolute) 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.

Music practice

Sat, 19 Jul 2014 00:00:00 +0000

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 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.

What maths does to the brain

Tue, 15 Jul 2014 00:00:00 +0000

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.

Solvability of nonograms

Sun, 13 Jul 2014 00:00:00 +0000

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. 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.

Possible cons of Soylent

Wed, 25 Jun 2014 00:00:00 +0000

I have seen many glowing reviews of Soylent, and many vitriolic naturalistic 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.

Proof that symmetric matrices are diagonalisable

Mon, 26 May 2014 00:00:00 +0000

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:

Discovering a proof of Sylvester's Law of Inertia

Sat, 03 May 2014 00:00:00 +0000

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 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.

Sequentially compact iff compact

Sat, 26 Apr 2014 00:00:00 +0000

Prof Körner told us during the IB Metric and Topological Spaces 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.

Cayley-Hamilton theorem

Thu, 17 Apr 2014 00:00:00 +0000

This is to detail a much easier proof (at least, I find it so) of Cayley-Hamilton 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 \(b_1, \dots, b_i\).

Sample topology question

Tue, 15 Apr 2014 00:00:00 +0000

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.

Useful conformal mappings

Mon, 07 Apr 2014 00:00:00 +0000

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)
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).

Discovering a proof of Heine-Borel

Fri, 04 Apr 2014 00:00:00 +0000

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 “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.

How to discover the Contraction Mapping Theorem

Sun, 30 Mar 2014 00:00:00 +0000

A little while ago I set myself the exercise of stating and proving the Contraction Mapping Theorem. 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).

A roundup of some board games

Thu, 20 Mar 2014 00:00:00 +0000

It has been commented to me that it’s quite hard to find out (on the Internet) what different games involve. For instance, Agricola 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 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.

Rage, rage against the poet’s hardest sell

Sun, 16 Feb 2014 00:00:00 +0000

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.

Writing essays

Tue, 28 Jan 2014 00:00:00 +0000

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.

Introduction to functional programming syntax of Mathematica

Fri, 24 Jan 2014 00:00:00 +0000

Recently, I was browsing the Wolfram Community 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]”.

Denouement of Myst III: Exile

Sun, 12 Jan 2014 00:00:00 +0000

A long time ago, in a galaxy far far away, I completed Myst III: Exile. 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 Creation

Thu, 02 Jan 2014 00:00:00 +0000

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.

Smartphone Charter

Mon, 30 Dec 2013 00:00:00 +0000

I am shortly to receive a new Nexus 5. 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 or Hacker News 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.)

Three explanations of the Monty Hall Problem

Sun, 22 Dec 2013 00:00:00 +0000

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. 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 Training Game

Sat, 14 Dec 2013 00:00:00 +0000

The book Don’t Shoot the Dog, by Karen Pryor, contains a simple exercise in demonstrating clicker training. 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 Jean-Paul Sartre Cookbook

Sat, 23 Nov 2013 00:00:00 +0000

Many thanks to the Guru Bursill-Hall 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.

Markov Chain card trick

Tue, 12 Nov 2013 00:00:00 +0000

In my latest lecture on Markov Chains 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 \(52\).

My quest for a new phone

Thu, 07 Nov 2013 00:00:00 +0000

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.

How to do Analysis questions

Thu, 24 Oct 2013 00:00:00 +0000

This post is for posterity, made shortly after Dr Paul Russell 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:

The Ravenous

Sun, 20 Oct 2013 00:00:00 +0000

Once upon a midnight dreary, 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.

Training away mental bias

Sun, 13 Oct 2013 00:00:00 +0000

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.

Meaning what you say

Fri, 11 Oct 2013 00:00:00 +0000

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?

Plot Armour

Thu, 10 Oct 2013 00:00:00 +0000

Wherein I dabble in parodic fiction. The title refers to the TV Tropes page on Plot Armour, 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.

How to prove that you are a god

Sat, 21 Sep 2013 00:00:00 +0000

I came across an interesting question while reading the blog of Scott Aaronson 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).

Stumbled across 14th September 2013

Fri, 13 Sep 2013 00:00:00 +0000

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
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

Slightly silly Sylow pseudo-sonnets

Sat, 31 Aug 2013 00:00:00 +0000

This is a collection of poems which together prove the Sylow theorems.

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 Motivation ¹

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.

Topology made simple

Mon, 26 Aug 2013 00:00:00 +0000

I’ve been learning some basic topology 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.

Stumbled across 24th August 2013

Sat, 24 Aug 2013 00:00:00 +0000

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 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

How to punt in Cambridge

Thu, 22 Aug 2013 00:00:00 +0000

When in Cambridge…

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.

My experiences with flow

Wed, 21 Aug 2013 00:00:00 +0000

I’m in the middle of reading Flow, by Mihály Csíkszentmihályi, and so far, I love it. It describes the “flow state” 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.

Thinking styles

Sun, 18 Aug 2013 00:00:00 +0000

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 (leading to the worst outbreak of headlice I’ve ever encountered) and mind maps. 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.)

Stumbled across 11th August 2013

Sun, 11 Aug 2013 00:00:00 +0000

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 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

New computer setup

Sun, 04 Aug 2013 00:00:00 +0000

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, Mac, 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.

Stumbled across 4th August 2013

Sun, 04 Aug 2013 00:00:00 +0000

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: http://arstechnica.com/security/2013/06/encrypted-e-mail-how-much-annoyance-will-you-tolerate-to-keep-the-nsa-away/

On to-do lists as direction in life

Tue, 30 Jul 2013 00:00:00 +0000

Getting Things Done has gathered something of a cult following [archived due to link rot] 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.

Stumbled across 29th July 2013

Mon, 29 Jul 2013 00:00:00 +0000

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

Metathought

Thu, 25 Jul 2013 00:00:00 +0000

I have recently discovered the game of Agricola, 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, 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”.

Stumbled across 24th July 2013

Wed, 24 Jul 2013 00:00:00 +0000

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

The Orbit/Stabiliser Theorem

Mon, 22 Jul 2013 00:00:00 +0000

The Orbit/Stabiliser Theorem is a simple theorem in group theory. Thanks to Tim Gowers 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 \(\text{Stab}_G(k) = {g \in G: g(k) = k}\).

On Shakespeare

Sun, 21 Jul 2013 00:00:00 +0000

I’ve now seen two Shakespeare plays at the 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 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.)

My objection to the One Logical Leap view

Thu, 18 Jul 2013 00:00:00 +0000

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!

Prerequisites for hypothetical situations

Sun, 14 Jul 2013 00:00:00 +0000

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}

The Multiple Drafts view of consciousness

Sun, 14 Jul 2013 00:00:00 +0000

I’ve been reading one of Daniel Dennett’s 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 (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.

Stumbled across 13th July 2013

Sat, 13 Jul 2013 00:00:00 +0000

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

A framework for discussing "pricelessness"

Fri, 12 Jul 2013 00:00:00 +0000

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.

Imre Leader Appreciation Society

Wed, 10 Jul 2013 00:00:00 +0000

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.

Stumbled across 9th July 2013

Tue, 09 Jul 2013 00:00:00 +0000

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
A rather neat and very short story, Time Loop by qntm
A bit less short but just as good a short story: I don’t know, Timmy, being God is a big responsibility by qntm
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
Musings on free will: Is God a Taoist?
A thing that I just have to share again: Technical hurdles have been overcome for the first human head transplant
The human brain is a really weird piece of kit: The Apologist and the Revolutionary
We have to make one of these at some point: the Creme de la Creme Egg
This is quite soothing in a weird kind of way: Things Fitting Perfectly Into Things
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: Every Time Zone
Why never to talk to the police (seriously, never talk to the police): Don’t Talk to the Police
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: Church of England makes Chichester child abuse apology
The Hawkeye Initiative, for the liberation of women in comics: The Hawkeye Initiative

An obvious improvement to tennis

Mon, 08 Jul 2013 00:00:00 +0000

So yesterday the Wimbledon tennis tournament 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:

The ball lands.
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.
The umpire decides whether or not to overrule the linesperson’s decision.
The Hawkeye ball-tracking system determines whether the ball landed inside or outside the region demarcated by the line.
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.

Mundane magics

Sun, 07 Jul 2013 00:00:00 +0000

I have stumbled across a LessWrong post on the importance of seeing what is real for just how cool it is. 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.

Cambridge vocab - a guide for the mystified

Sat, 06 Jul 2013 00:00:00 +0000

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.

Cambridge undergrad maths tips

Thu, 04 Jul 2013 00:00:00 +0000

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). 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.

In which I augment the lexicon

Wed, 03 Jul 2013 00:00:00 +0000

A few dubiously-real words which I think should be more widely used.

CUCaTS Puzzlehunt

Wed, 26 Jun 2013 00:00:00 +0000

At the end of last (that is, Lent 2012-2013) term at Cambridge, I took part in the Cambridge University Computing and Technology Society 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.

First post

Wed, 26 Jun 2013 00:00:00 +0000

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.

Sylow theorems

Wed, 26 Jun 2013 00:00:00 +0000

A fairly long and winding way through a proof of the three Sylow theorems.

Mon, 01 Jan 0001 00:00:00 +0000

Redirecting…

Click here if you are not redirected.

Patrick Stevens

Claude knows who you are

Another trip to Tromsø

Fun LLM prompt: where are my mental models defective?

LLM arithmetic

Robocop: automated code review

LLMs sanding off the edges

Boolean blindness

Some Python surprises

Things I've learned about the .NET runtime

Announcing WoofWare.Expect

LLM effect on my programming (2025 edition)

A Bitcoin analogy for Multiple Drafts theory of consciousness

NativeAOT in .NET 9 and Nixpkgs/darwin

Mary's Room

My reading list

Unhinged rant about software

Lessons from a massage course

WoofWare.Myriad.Plugins learns to parse args

New GitHub workflows

Learning plan for "Program Equilibrium in the Prisoner's Dilemma"

Code having "the right philosophy"

The phrase "I Notice That I Am Confused"

Games

Video games

In progress

Not in progress

Board games

The Yoneda lemma

ChatGPT's effect on my programming

YAML is not a superset of JSON

Trip to Tromsø

Why does no-confusion use equality rather than a recursive call?

Starting a suspended process

Announcing WoofWare.Myriad.Plugins

iOS interface

Squashed stacked PRs workflow

Raymond Smullyan chess problem walkthrough

Nix fireside chat outline

Property-based testing introduction

The tiny proof that primes 1 mod 4 are sums of two squares

Imre Leader Appreciation Society

Anki decks

Notes for a Git fireside chat

Films

Questions I had about transformers

Lifehacks

About me

About this website

The water filtration industry has played us for absolute fools

Don't supply `-f` to `rm` unless you know you need it

Argument in a high-trust environment

Crates (existentials in F#)

Restaurants

London

Not tried

Top Posts

Metatesting your property-based tests

Product comparisons

Fig rolls

Miso soup

In praise of `--dry-run`

Continuation-passing style

Rewriting the Technical Interview, in Mathematica

Software engineer syllabus

Cyclic dependencies (a note from Hacker News)

An incomplete life evaluation checklist

Find the Bug, C# edition

A bug in Git

Christmas quiz

Christmas dinner notes

Anki as Learning Superpower

General relativity (a note from Hacker News)

Nulls and options (a note from Hacker News)

The uncountability of the reals (a note from Hacker News)

In favour of recursive functions, not imperative constructs, to make loops

Static config (a note from Hacker News)

Defunctionalisation

MIT Mystery Hunt photos

MIT Mystery Hunt 2020 answers