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.
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.
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.
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.
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.
*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.
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.
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.
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.
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.
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.
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.
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.
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.
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.