Imre Lakatos, answered from the texts and cited to the page.
Mathematics is not the deductive presentation Euclid and the Hilbertian-Bourbakian formalists make it appear. It is a fallible, historically developing activity whose actual logic is the dialectic of conjecture, counterexample, monster-barring, lemma-incorporation, and refined conjecture — the method of proofs and refutations.1 The Euler polyhedra theorem case I worked through in the 1963- 64 BJPS essays shows the pattern in detail across the nineteenth-century history of the theorem from Euler through Cauchy through Poincaré.2
The Cauchy on uniform convergence case shows the same dialectic in nineteenth- century analysis. Mathematics is quasi-empirical: mathematical conjectures are tested against the mathematical objects they describe, and rigor is improved as theorems become wedded to their objects through the dialectical back-and-forth of proof and refutation.3
The formalist programme conceals this heuristic logic by presenting the finished proof as if it had been the starting point.4
So a lemma like 'All polyhedra have at least 17 edges' would take care of the cylinder! And any other random ad hoc conjecture would do just as well, so long as it happened to be refuted by the counterexample. gamma: Why not? lambda: We already criticised monster-barrers and exception-barrers for forgetting about proofs.Proofs and Refutations - The Logic of Mathematical Discovery, pp. 64–65
Zacharias in his [1914-31] gives an uncritical but faithful description of this compartmentalisation: 'In the 19th century, geometers, besides finding new proofs of the Euler theorem, were engaged in establishing the exceptions which it suffers under certain conditions. Such exceptions were stated, e.g. by Poinsot. S. Lhuilier and F. Ch. Hessel tried to classify the exceptions...' (p. 1052).Proofs and Refutations - The Logic of Mathematical Discovery, pp. 50–51
The real aim of a 'problem to prove' should be to improve - in fact, perfect - the original, 'naive' conjecture into a genuine 'theorem'. 51 Actually, such a proof was first proposed by H. Reichardt (, p. 23). Also cf. B. L. van der Waerden. Hilbert and Cohn-Vossen were satisfied that the truth of Gamma's assertion is 'easy to see' (, English translation, p. 292). 52 Pólya (, p.Proofs and Refutations - The Logic of Mathematical Discovery, pp. 54–55
I am dubious about your first step. pupil beta: Are you sure that in triangulating the map one will always get a new face for any new edge? I am dubious about your second step. pupil gamma: Are you sure that there are only two alternatives - the disappearance of one edge or else of two edges and a vertex - when one drops the triangles one by one?Proofs and Refutations - The Logic of Mathematical Discovery, pp. 20–21
settled-as-articulated