MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Here's my favorite example. Imre Lakatos' book Proofs and Refutations contains a very long dialogue between a teacher and pupils who debate what are good definitions of polyhedra, with respect to a claimed proof that $V-E+F=2$ is true for polyhedra. It's common that a good definition (or reformulation) of a concept can help yield proofs of theorems, and this book promotes the "dual" view that a proof of a theorem can lead to a good definition in hindsight.
Here's my favorite example. Imre Lakatos' book Proofs and Refutations contains a very long dialogue between a teacher and pupils who debate what are good definitions of polyhedra, with respect to a claimed proof that $V-E+F=2$ is true for polyhedra. It's common that a good definition (or reformulation) of a concept can help yield proofs of theorems, and this book promotes the "dual" view that a proof of a theorem can lead to a good definition in hindsight.