Articles
Essays, notes and commentary. Newest first, nothing filed into categories — the kind of piece is noted above each title.
The definition always comes last
Every definition in mathematics is an answer to a question that was asked first. We teach them in the opposite order and then wonder why students experience the subject as a list of arbitrary rules to be endured. This is the long version of the argument the whole project rests on.
What I got wrong about compactness for two years
I could state the definition, prove the standard theorems, and had no idea what the thing was for. These are the notes from the week that fixed it, including the wrong pictures I was carrying around and the example that finally dislodged them.
Rigour is a destination, not a filter
There is a style of teaching that treats the difficulty of the presentation as a measure of seriousness. It selects for tolerance rather than ability, and it has cost mathematics more good people than any other habit in the discipline.
Cantor's diagonal is not a trick
It is usually presented as a clever device, which makes it memorable and forgettable at the same time. Read as the only available response to a specific impossibility, it stops being clever and becomes obvious — which is what a good proof should feel like afterwards.
Reading Rudin, chapter three, slowly
A page-by-page record of working through the chapter on sequences, with the exercises I could not do at first and what eventually unlocked them. Kept partly as a study log and partly as evidence that this is what the process actually looks like.
Why I am writing Fermat from the beginning
The proof of Fermat's Last Theorem is routinely described as inaccessible, which is true of the final argument and false of nearly everything leading to it. An account of what the book is trying to do and why the first chapter is about creativity rather than number theory.
On a certain kind of popular mathematics
Two recent books, both excellent at the picture and both unwilling to follow it into a proof. The gap between them and a textbook is the space this project is trying to occupy, so it is worth saying precisely where they stop.
The unfinished square
A short note on the mark used throughout this site: the proof box drawn with one side missing, and why a closed argument is the least interesting stage of a mathematical idea's life.