I can also recommend Proofs and Refutations by Imre Lakatos. It's an unusually easy read for a philosophy book (fewer than 200 pages and written as a sort of S

I can also recommend Proofs and Refutations by Imre Lakatos. It's an unusually easy read for a philosophy book (fewer than 200 pages and written as a sort of Socratic dialogue) and engages with some very deep problems related to the nature of mathematical proofs.