Lakatos, in his Proofs and Refutations, rejects the Euclidean methodology and exposition of mathematics: where axioms and definitions precede the proofs. In other words, a Euclidean mathematician makes clear the axioms and definitions, then proves novel theorems in an altogether linear fashion. This rigid methodology is not present in mathematical practice, Lakatos contests. “This led to" his "dialectical methodology, a ‘heuristic style’ that reveals the struggle and adventure of mathematical creation" [Paolo Mancosu’s Preface in Proofs and Refutations].
Lakatos seeks to convince us of this through case studies -- the historical contexts behind uniform convergence, bounded variation, and the Carathéodory definition of a measurable set, for example.
Are the cases that Lakatos discusses special cases? Or does most of post-Euclidean mathematics progress in this heuristic, dialectical style? Examples of some recent discoveries and how they came about would be interesting.