I've a (possibly unfortunate) soft spot in my heart for informal reasoning in maths. Like don't get me wrong being correct and being demonstrably so is absolutely necessary. I'd never tell an undergrad to only show the vibes portion of a proof. But the so-called post-rigour mindset (don't like that word, also not a Terry Tao fan) has such a wonderful appeal.
Being one with the currents of reason as they take shape not merely within formulae and proofs but within groups of people, among hundreds and hundreds of proofs, within bodies of work. The stuff that takes you a year to learn to do in 5 mins. Dunking your head in the clear stream in which god fishes for theorems, oxygen be damned.
This is one of the things that bothers me so much about the use of LLMs for proofs (even Lean checked proofs). They output five hundred pages of stuff that no one has the time to read, is often subtly wrong (and Lean isn't the be-all end-all of proof correctness either but that's a separate can of worms), and takes from us (in the literal sense of prevents the author from doing themselves) the part of maths that I love most: the musing! The pontificating. The imagination! The imagination that comes from working through shit yourself and letting the waves take you somewhere you never saw coming! The informal reasoning that can only come out of a deep understanding of technicalities.
To me it's what animates a good talk. That the speaker has drowned themselves in that stream and come back with a parable. The best works of mathematics are emotional. They make an argument for what *should* be and *then* make an argument for why it *is*. It's a profoundly unscientific way to think, and it intersects an unfortunate amount with most LLMs' core capabilities. But it's what I love about maths, and it connects me to other people who love it.
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