Editing Imre Lakatos (section)

=== Philosophy of mathematics ===
{{main|Proofs and Refutations}}

Lakatos's philosophy of mathematics was inspired by both [[Georg Hegel|Hegel]]'s and [[Karl Marx|Marx]]'s [[dialectic]], by [[Karl Popper]]'s theory of knowledge, and by the work of mathematician [[George Pólya]].

The 1976 book ''Proofs and Refutations'' is based on the first three chapters of his 1961 four-chapter doctoral thesis ''Essays in the Logic of Mathematical Discovery''. But its first chapter is Lakatos's own revision of its chapter&nbsp;1 that was first published as ''Proofs and Refutations'' in four parts in 1963–64 in the ''British Journal for the Philosophy of Science''. It is largely taken up by a fictional [[dialogue]] set in a mathematics class. The students are attempting to prove the formula for the [[Euler characteristic]] in [[algebraic topology]], which is a [[theorem]] about the properties of [[polyhedra]], namely that for all polyhedra the number of their vertices ''V'' minus the number of their edges ''E'' plus the number of their faces ''F'' is 2 ({{nobr|''V'' − ''E'' + ''F'' {{=}} 2}}). The dialogue is meant to represent the actual series of attempted proofs that mathematicians historically offered for the [[conjecture]], only to be repeatedly refuted by [[counterexample]]s. Often the students paraphrase famous mathematicians such as [[Cauchy]], as noted in Lakatos's extensive footnotes.

Lakatos termed the polyhedral counterexamples to Euler's formula ''monsters'' and distinguished three ways of handling these objects: Firstly, ''monster-barring'', by which means the theorem in question could not be applied to such objects. Secondly, ''monster-adjustment'', whereby by making a re-appraisal of the ''monster'' it could be ''made'' to obey the proposed theorem. Thirdly, ''exception handling'', a further distinct process. These distinct strategies have been taken up in qualitative physics, where the terminology of ''monsters'' has been applied to apparent counterexamples, and the techniques of ''monster-barring'' and ''monster-adjustment'' recognized as approaches to the refinement of the analysis of a physical issue.<ref>{{cite web |title=Lakatosian Monsters |url=http://harveycohen.net/dragons/Lakatosian_Monsters.htm |access-date=18 January 2015}}</ref>

What Lakatos tried to establish was that no theorem of [[informal mathematics]] is final or perfect. This means that we should not think that a theorem is ultimately true, only that no [[counterexample]] has yet been found. Once a counterexample is found, we adjust the theorem, possibly extending the domain of its validity. This is a continuous way our knowledge accumulates, through the logic and process of proofs and refutations. (If axioms are given for a branch of mathematics, however, Lakatos claimed that proofs from those [[axiom]]s were [[Tautology (logic)|tautological]], i.e. [[logical truth|logically true]].)<ref>See, for instance, Lakatos's ''A renaissance of empiricism in the recent philosophy of mathematics'', section 2, in which he defines a Euclidean system to be one consisting of all logical deductions from an initial set of axioms and writes that "a Euclidean system may be claimed to be true".</ref>

Lakatos proposed an account of mathematical knowledge based on the idea of [[heuristic]]s. In ''Proofs and Refutations'' the concept of "heuristic" was not well developed, although Lakatos gave several basic rules for finding proofs and counterexamples to conjectures. He thought that mathematical "[[thought experiment]]s" are a valid way to discover mathematical conjectures and proofs, and sometimes called his philosophy "quasi-[[empiricism]]".

However, he also conceived of the mathematical community as carrying on a kind of dialectic to decide which [[mathematical proof]]s are [[Validity (logic)|valid]] and which are not. Therefore, he fundamentally disagreed with the "[[formalism (mathematics)|formalist]]" conception of proof that prevailed in [[Gottlob Frege|Frege]]'s and [[Bertrand Russell|Russell]]'s [[logicism]], which defines proof simply in terms of ''formal'' validity.

On its first publication as an article in the ''British Journal for the Philosophy of Science'' in 1963–64, ''Proofs and Refutations'' became highly influential on new work in the philosophy of mathematics, although few agreed with Lakatos's strong disapproval of formal proof. Before his death he had been planning to return to the philosophy of mathematics and apply his theory of research programmes to it. Lakatos, Worrall and Zahar use [[Henri Poincaré|Poincaré]] (1893)<ref>Poincaré, H. (1893). "Sur la Généralisation d'un Théorème d'Euler relatif aux Polyèdres", ''Comptes Redus des Séances de l'Académie des Sciences'', '''117''' p. 144, as cited in Lakatos, Worrall and Zahar, p. 162.</ref> to answer one of the major problems perceived by critics, namely that the pattern of mathematical research depicted in ''Proofs and Refutations'' does not faithfully represent most of the actual activity of contemporary mathematicians.<ref>Lakatos, Worrall and Zahar (1976), ''Proofs and Refutations'' {{ISBN|0-521-21078-X}}, pp. 106–126, note that Poincaré's formal proof (1899) "Complèment à l'Analysis Situs", ''Rediconti del Circolo Matematico di Palermo'', '''13''', pp. 285–343, [[Rewriting|rewrite]]s Euler's conjecture into a [[tautology (logic)|tautology]] of vector algebra.</ref>

==== Cauchy and uniform convergence ====
In a 1966 text ''Cauchy and the continuum'', Lakatos re-examines the history of the calculus, with special regard to [[Augustin-Louis Cauchy]] and the concept of uniform convergence, in the light of [[non-standard analysis]]. Lakatos is concerned that historians of mathematics should not judge the evolution of mathematics in terms of currently fashionable theories. As an illustration, he examines Cauchy's proof that the sum of a series of continuous functions is itself continuous. Lakatos is critical of those who would see Cauchy's proof, with its failure to make explicit a suitable convergence hypothesis, merely as an inadequate approach to Weierstrassian analysis. Lakatos sees in such an approach a failure to realize that Cauchy's concept of the continuum differed from currently dominant views.