Proofs and Refutations

Proof and Refutations is a book by the philosopher Imre Lakatos expounding his view of the progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students. A central theme is that definitions are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proofs. This gives mathematics a somewhat experimental flavour.