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.

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.

Recall the Monty Hall problem: the host, Monty Hall, shows you three doors, named A, B and C. You are assured that behind one of the doors is a car, and behind the two others there is a goat each. You want the car. You pick a door, and Monty Hall opens one of the two doors you didn’t pick that he knows contains a goat. He offers you the chance to switch guesses from the door you first picked to the one remaining door. Should you switch or stick?

I’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.

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

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.

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.

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.

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.

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.

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.

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.

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

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.

In which I am a wizard.

Sometimes as a student, the work piles up and I start to think “I’ll never finish this”. It becomes easy to think that there’s no point in working because the work will never be over. When that happens to me, I imagine that my course is magic/alchemy/something with flashy special effects. I’m going through the Wizardry Academy, and I’ll graduate able to manipulate the four elements. Even if I’m not the best in the year at it, I’m still able to manipulate the elements, and if I work at it, I’ll be able to manipulate them better and in flashier ways - that’s not something most people can do!

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

Chunking:

1. The first thing to do is to run through the sentence, identifying the verbs and anything that looks like it might be a verb (even in a strange form, like “passus” or “ascendere”).
2. Run through a second time, looking for structures like “ut + subjunctive” and “non solum… sed etiam…” - if a verb you spotted is in an odd form, this is when you look quickly for why it’s in that form.
3. Look for any subordinate clauses (like “dixit Caecilius, qui in horto laborabat…”)
4. If you see an adjective-looking thing, it probably has to go with a noun.
5. With that in mind, chunk the text, remembering that two verbs in the same chunk is unlikely unless one is something like “dixit” or “poterat”, which can modify another verb. Remember that chunks shouldn’t be too long, but lots of really short words together might not count against the length limit. Try reading out each chunk - rhythm takes time to learn to grasp, but it might help you.

Once the text is chunked:

1. Remember that your chunking is probably wrong somewhere, but also is probably broadly right.
2. In each chunk, if there’s a nominative and a verb then try and translate those first. Then think about what the verb “expects”; if the verb is looking for an accusative, find an accusative, while if it’s looking for a dative, find a dative. For example, “docet” = “he teaches” is looking for an accusative, while “trahet” = “he drags” is looking both for an accusative (“he drags something”) and possibly a dative (“he drags something somewhere”).
3. If it looks like a jumble of words, identify the case of everything (in poetry, it can help if you scan the text) - this should tell you what goes with what. Don’t be too fussy about getting the right case, though - I’d be happy with “dative or ablative”, most of the time, because that’s usually clear from context - as long as you have the right case among your options!

Guessing vocab:

Try and work out what the principal parts of a verb are. The English word from a given Latin one almost always comes from the past passive participle (the fourth principal part), by adding “tion” instead of “us”: “passus” -> “passion” [a bit misleading if you don’t know about the Passion of the Christ, because it means “suffering”], “traho” -> “tractus” -> “traction”; it actually means “drag”. How to guess the principal parts is the kind of thing you learn with time, but as a general rule, “t” -> “s” (as in “patior passus”) and almost everything else goes to “ct”: “pingere pictus” from which “depiction” so “painting”, “facere factus” from which “manufaction” which isn’t really English but tells you it means “making”, etc.

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.

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.

In which I provide my favourite carols and my favourite renditions of them.

In no particular order, except that 1) must be at the start and 9) at the end.

1. Once in Royal David’s City. Always opens the Festival of Nine Lessons and Carols. Has the same problem as 9) in that the only nice recordings seem to have congregations in, but I suppose that’s all part of it.

2. The Three Kings. My favourite. This performance (King’s College) has a soloist who is a bit strident, I think, but all the other ones I’ve listened to are even stridenter.