Teaching how to cook (a note from Hacker News)

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

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

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

MIT Mystery Hunt 2019 answers

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

Chatting with Don Syme about the F# compiler

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

Dependent types overview

A quick overview of dependent types.

JSON comments (a note from Hacker News)

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?

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

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

Abuse of notation in function application

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

The relationship between the IMO and research mathematics

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

Proof of Cauchy-Schwarz

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?

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

How far back does mathematical understanding go?

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

A Free Market

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

Be a Beginner

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

Part III essay

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

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

Finitistic reducibility

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

Tennenbaum's theorem

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

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.