{"agent_id":"lakatos","agent_name":"Imre Lakatos","slug":"the-method-of-proofs-and-refutations","label":"The method of proofs and refutations: is mathematics fallible, quasi-empirical, and historically developing, with the dialectical method of conjecture-counterexample-lemma-incorporation as the actual logic of mathematical discovery, against Hilbertian-Bourbakian formalism?","topic":"The method of proofs and refutations","question":"Is mathematics fallible, quasi-empirical, and historically developing, with the dialectical method of conjecture-counterexample-lemma-incorporation as the actual logic of mathematical discovery, against Hilbertian-Bourbakian formalism?","position":"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. 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. 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. The formalist programme conceals this heuristic logic by presenting the finished proof as if it had been the starting point.","paragraphs":[[{"t":"Mathematics is not the deductive presentation Euclid and the Hilbertian-Bourbakian formalists make it appear.","n":[]},{"t":"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.","n":[1]},{"t":"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.","n":[2]}],[{"t":"The Cauchy on uniform convergence case shows the same dialectic in nineteenth- century analysis.","n":[]},{"t":"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.","n":[3]}],[{"t":"The formalist programme conceals this heuristic logic by presenting the finished proof as if it had been the starting point.","n":[4]}]],"texts":"\"Proofs and Refutations\" four-part essay in the *British Journal for the Philosophy of Science* (1963-64); *Proofs and Refutations: The Logic of Mathematical Discovery* (posthumous book, Cambridge 1976; Cambridge Philosophy Classics edition with Worrall + Zahar editorial apparatus); the methodological essays on the philosophy of mathematics in *Philosophical Papers Vol II* — \"What Does a Mathematical Proof Prove?\", \"Cauchy and the Continuum: The Significance of Non-standard Analysis for the History and Philosophy of Mathematics,\" \"The Method of Analysis-Synthesis\"; the \"Renaissance of Empiricism in the Recent Philosophy of Mathematics\" essay. Reception: Worrall + Zahar's editorial apparatus to the Cambridge edition; Brendan Larvor on Lakatos as Hegelian dialectician in mathematics; John Kadvany's reading of *Proofs and Refutations* as a Hegelian Bildungsroman with Euler's theorem in the role Hegel gave to the generic learning consciousness in the *Phenomenology*; Imre Toth on the Lakatos-Toth correspondence on mathematics history; Donald Gillies on quasi-empiricism in the philosophy of mathematics.","works":["\"Proofs and Refutations\" four-part essay in the *British Journal for the Philosophy of Science* (1963-64)","*Proofs and Refutations: The Logic of Mathematical Discovery* (posthumous book, Cambridge 1976","Cambridge Philosophy Classics edition with Worrall + Zahar editorial apparatus)","the methodological essays on the philosophy of mathematics in *Philosophical Papers Vol II* — \"What Does a Mathematical Proof Prove?\", \"Cauchy and the Continuum: The Significance of Non-standard Analysis for the History and Philosophy of Mathematics,\" \"The Method of Analysis-Synthesis\"","the \"Renaissance of Empiricism in the Recent Philosophy of Mathematics\" essay"],"reception":"Worrall + Zahar's editorial apparatus to the Cambridge edition; Brendan Larvor on Lakatos as Hegelian dialectician in mathematics; John Kadvany's reading of *Proofs and Refutations* as a Hegelian Bildungsroman with Euler's theorem in the role Hegel gave to the generic learning consciousness in the *Phenomenology*; Imre Toth on the Lakatos-Toth correspondence on mathematics history; Donald Gillies on quasi-empiricism in the philosophy of mathematics.","status":"settled-as-articulated","era":"1922-1974","discipline":"Philosophy","refs":[{"n":1,"work":"Proofs and Refutations - The Logic of Mathematical Discovery","page":"pp. 64–65","canonical":"","quote":"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.","label":"Proofs and Refutations - The Logic of Mathematical Discovery, pp. 64–65"},{"n":2,"work":"Proofs and Refutations - The Logic of Mathematical Discovery","page":"pp. 50–51","canonical":"","quote":"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).","label":"Proofs and Refutations - The Logic of Mathematical Discovery, pp. 50–51"},{"n":3,"work":"Proofs and Refutations - The Logic of Mathematical Discovery","page":"pp. 54–55","canonical":"","quote":"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.","label":"Proofs and Refutations - The Logic of Mathematical Discovery, pp. 54–55"},{"n":4,"work":"Proofs and Refutations - The Logic of Mathematical Discovery","page":"pp. 20–21","canonical":"","quote":"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*?","label":"Proofs and Refutations - The Logic of Mathematical Discovery, pp. 20–21"}],"answer":null,"siblings":[{"slug":"the-methodology-of-scientific-research-programmes","label":"The methodology of scientific research programmes: is the unit of appraisal the research programme rather than the individual theory, against Kuhnian incommensurability, against Feyerabend's anything-goes, and against naive Popperian falsificationism?"},{"slug":"rational-reconstruction-and-the-internal-external-history-di","label":"Rational reconstruction and the internal/external history distinction: is the history of science to be written as the rational reconstruction of the internal logic of research programmes, against Kuhn's sociology-of-knowledge historicism?"},{"slug":"the-popper-relationship-and-the-popper-0-popper-1-popper-2-d","label":"The Popper relationship and the Popper-0/Popper-1/Popper-2 distinction: is Lakatos the sophisticated methodological falsificationist Popper's late work was reaching for, or is the SRP framework a covert subversion of orthodox Popperianism?"},{"slug":"the-hungarian-hegelian-formation","label":"The Hungarian-Hegelian formation: are there 'two Lakatoses' (the Anglo Popperian successor and the buried Hungarian-Hegelian whose Lukácsian-Marxist inheritance shapes the dialectical method of Proofs and Refutations and the historicist architectonic of the SRP framework)?"}]}