An ongoing list of the concepts, methods, and heuristics used by the characters of Proofs and Refutations: The Logic of Mathematical Discovery, by Imre Lakatos. This is what I would have found useful the first time reading Proofs and Refutations, and I hope you find it useful too.

Chapter 1, §4 focuses on methods for responding to global counterexamples of a proof.

An ongoing list of the concepts, methods, and heuristics used by the characters of Proofs and Refutations: The Logic of Mathematical Discovery, by Imre Lakatos. This is what I would have found useful the first time reading Proofs and Refutations, and I hope you find it useful too.

Chapter 1, §§1-3 focus on a proposed proof for the Descartes-Euler conjecture (the Cauchy proof) and types of counterexamples.