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

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

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

A quick overview of dependent types.

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

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

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

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

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

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”.

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

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

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

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

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).

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

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

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.

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.

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.