[mherdeg]

Whenever I want to think about "what does it mean to prove something?" I dig into my copy of Imre Lakatos's "Proofs and Refutations", a Socratic dialogue-style story where a bunch of aspiring theorem-provers try to prove a particular theorem about the Euler characteristic of polyhedra and discuss what they are actually doing and why it works or doesn't.

I originally picked up the book because I was trying to understand Euler's polyhedral formula better -- which in retrospect is kind of like reading Zen and the Art of Motorcycle Maintenance because you wanted to fix a motorcycle.

I'm not wired for pure math -- I loved real analysis then quit while I was ahead. Still it's fun to pretend sometimes, and Lakatos does a great job of making you feel like you're learning inside knowledge about what mathematicians do. He introduces fun concepts like "monster-barring" (the way people sometimes carve out special cases in a proof when they encounter counterexamples).

I've never made it the whole way through, but I like to go back every few months and absorb a little more.

edit to add: I just now skimmed the author's Wikipedia entry and the Stanford Encyclopedia of Philosophy's story about the author's interaction with someone named Éva Izsák and I have a ton of questions.

[theresistor]

I had to read that for a first-semester discrete math course in undergrad. It was... challenging.