Independence of the Axiom of Choice (for programmers)

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

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.

Clojure and Exercism

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?

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

Friedberg-Muchnik theorem

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

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

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

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

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

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.


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

Lottery odds

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.

Proof by contradiction

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

Sequence on Awodey's Category Theory

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.

Motivational learning

In which I am a wizard. Sometimes as a student, the work piles up and I start to think “I’ll never finish this”. It becomes easy to think that there’s no point in working because the work will never be over. When that happens to me, I imagine that my course is magic/alchemy/something with flashy special effects. I’m going through the Wizardry Academy, and I’ll graduate able to manipulate the four elements.

Latin translation tips

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.

Matrix puzzle

I recently saw a problem from an Indian maths olympiad: There is a square arrangement made out of n elements on each side (n^2 elements total). You can put assign a value of +1 or -1 to any element. A function f is defined as the sum of the products of the elements of each row, over all rows and g is defined as the sum of the product of elements of each column, over all columns.

Film recommendation, Interstellar

I’ve just come back from seeing Interstellar, a film of peril and physics. This post will be spoiler-free except for sections which are in rot13. I thought the film was excellent. My previous favourite film in its genre was Sunshine, but this beats it in many ways, chiefly that the physics portrayed in Interstellar - relativity, primarily - is not so wrong that it’s immediately implausible. Indeed, some physics-driven plot twists (such as gvqny sbeprf arne n oynpx ubyr) I called in advance, which is a testament to how closely the film matched my physical expectations.

Christmas carols

In which I provide my favourite carols and my favourite renditions of them. In no particular order, except that 1) must be at the start and 9) at the end. 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.

Sum-of-two-squares theorem

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