### Paris in the Spring

#### Posted by David Corfield

I’m just back from a couple of days of philosophy of mathematics workshops in Paris, which turns out to be one of the liveliest places on the planet for the subject. Had I realised, I might have extended my trip by a day to take in the first screening of Edward Frenkel’s Rites of Love and Math.

I was invited to contribute to the second day on the theme of the ‘complexity of proof’ by Andrew Arana. I’m rather doubtful that one can give anything resembling a measure of the complexity of a proof, although I did hear some interesting ideas from Paul-André Melliès and Alessandra Carbone who are looking to assign invariants to formal proofs, group theoretic ones for the latter.

But the gap between proof theory and real mathematics yawns rather wide, so I thought to look more closely at the combinations of ‘ideas’ which appear in a proof. As Michael Harris pointed out in his talk, when praising mathematicians’ work in a letter of recommendation, one opts for the word ‘deep’ over ‘complex’, the latter suggesting an unnecessary intricacy. He is surely right then that a treatment of mathematical depth is of central importance. Depth may indicate novel and powerful combinations of ideas.

My approach then was to consider proofs at a high level of description as a collection of ideas and their relations. For example, Tom Leinster gave us a nice sketch of the proof that the probability that a randomly dropped needle will land across a crack between boards whose width is the length of the needle is $2/\pi$. The proof can be boiled down to

Needle can only cross once. Probability = Expectation. Expectation for needle depends linearly on length, since expectation of sum = sum of expectations. Expectation for noodle (floppy needle) depends similarly linearly on length. Expectation for unit diameter circle (circumference $\pi$) is 2.

I wonder then how far one could get in sketching the gist of any proof in terms of the relations between combinations of ideas drawn from a certain stock, which includes:

expectation, sum, measure, obstruction, completion, extension/lift, descent, dimension, representation, subobject, quotient, cover, dual, identity, nilpotent, idempotent, invariant, cohomology, orientation, generator, kernel, factor, bounded, projection, convex, anomaly, ergodic, entropy, random, compact, splitting, section, linear, approximation…

This leads me to a couple of questions,

- Have I bundled together ideas of different kinds? If so, how should they be separated?
- Why do some ideas lend themselves so readily to category theoretic description, e.g., section and completion, while others, e.g., approximation, ergodic, seem not to?

On another matter, I was trying to expound James’ point from back here that Cauchy’s theorem fails in the topos of sets equipped with an automorphism, when a question revealed that while I can see how to define the order of an element, I didn’t know how to define the order of a group. Presumably, James meant us to take the order of $\mathbb{Z}/3 \mathbb{Z}$ with its nontrivial automorphism to be 3, but how do you define the order of a group in that setting when there’s only one map in the topos from the terminal object?

## Re: Paris in the Spring

Regarding your last question, it seems to me that really the right thing to ask is whether the statement of Cauchy’s theorem is valid in the internal logic of the topos in question. And if that’s the question, then I don’t even think that $\mathbb{Z}/3\mathbb{Z}$ with its nontrivial automorphism counts as a finite set in that internal logic, so I wouldn’t expect Cauchy’s theorem to apply to it. The “finite sets” in $Set^{B Z}$ are the “externally” finite sets equipped with

identityautomorphisms only.