Michael Nielsen's Reviews > Proofs and Refutations: The Logic of Mathematical Discovery

Radically changed my idea of what mathematical definitions and proofs are, and where they come from. In particular, Lakatos convincingly refutes the idea that definitions come before theorems and proofs (as often seems the case). Rather, they arise out of repeated back-and-forth interplay between conjectures and proof-ideas.

That's a pretty abstract- and weird-sounding review. The book itself is incredibly readable, incredibly fun, and by the end will (if you're anything like me) have caused an earthquake in your worldview. So ignore the weirdness of my last paragraph, and just go read the book. It's amazing.