Imre Lakatos (,Philosophy of Science: Popper and Lakatos, lecture on the philosophy of science of Karl Popper and Imre Lakatos, delivered to master's students at the University of Sussex in November 2014. ; ; 9 November 1922 – 2 February 1974) was a Hungarian-born philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its "methodology of proofs and refutations" in its pre-axiomatic stages of development, and also for introducing the concept of the "research programme" in his methodology of scientific research programmes.
Life Lakatos was born Imre Lipsitz to a well-to-do Hungarian-Jewish family in Debrecen, Hungary, in 1922. He received a degree in mathematics, physics, and philosophy from the University of Debrecen in 1944. In March 1944, the Germans invaded Hungary, and Lakatos, along with Éva Révész, his then-girlfriend and subsequent wife, formed a Marxist resistance group. In May of that year, the group was joined by Éva Izsák, a 19-year-old Jewish antifascist activist. Lakatos, considering that there was a risk that she would be captured and forced to betray them, decided that her duty to the group was to commit suicide. Subsequently, a member of the group took her to Debrecen and gave her cyanide.
During the occupation, Lakatos avoided Nazi persecution of Jews by changing his surname to Molnár. His mother and grandmother were murdered in Auschwitz. He changed his surname once again to Lakatos (Locksmith) in honor of Géza Lakatos.
After the war, from 1947, he worked as a senior official in the Hungarian Ministry of Education. He also continued his education with a PhD at Debrecen University (awarded in 1947) and also attended György Lukács's weekly Wednesday afternoon private seminars. He also studied at the Moscow State University under the supervision of Sofya Yanovskaya in 1949. When he returned, however, he found himself on the losing side of internal arguments within the Hungarian communist party and was imprisoned on charges of revisionism from 1950 to 1953. More of Lakatos's activities in Hungary after World War II have recently become known. In fact, Lakatos was a hardline Stalinist and, despite his young age, had an important role between 1945 and 1950 (his own arrest and jailing) in building up the Communist rule, especially in cultural life and academia, in Hungary.Bandy 2010.
After his release, Lakatos returned to academic life, doing mathematical research and translating George Pólya's How to Solve It into Hungarian. Still nominally a communist, his political views had shifted markedly, and he was involved with at least one dissident student group in the lead-up to the 1956 Hungarian Revolution.
After the Soviet Union invaded Hungary in November 1956, Lakatos fled to Vienna and later reached England. He lived there for the rest of his life, however, he never achieved British citizenship.György Kampis, L. Kvasz, Michael Stöltzner (eds.), Appraising Lakatos: Mathematics, Methodology, and the Man, Springer, 2013, p. 296. He received a PhD in philosophy in 1961 from the University of Cambridge; his doctoral thesis was entitled Essays in the Logic of Mathematical Discovery, and his doctoral advisor was R. B. Braithwaite. The book Proofs and Refutations: The Logic of Mathematical Discovery, published after his death, is based on this work.
In 1960, he was appointed to a position in the London School of Economics (LSE), where he wrote on the philosophy of mathematics and the philosophy of science. The LSE philosophy of science department at that time included Karl Popper, Joseph Agassi and J. O. Wisdom.. It was Agassi who first introduced Lakatos to Popper under the rubric of his applying a fallibilist methodology of conjectures and refutations to mathematics in his Cambridge PhD thesis.
With co-editor Alan Musgrave, he edited the often cited Criticism and the Growth of Knowledge, the Proceedings of the International Colloquium in the Philosophy of Science, London, 1965. Published in 1970, the 1965 Colloquium included well-known speakers delivering papers in response to Thomas Kuhn's The Structure of Scientific Revolutions.
In January 1971, he became editor of the British Journal for the Philosophy of Science, which J. O. Wisdom had built up before departing in 1965, and he continued as editor until his death in 1974,See Lakatos's 5 Jan 1971 letter to Paul Feyerabend pp. 233–234 in Motterlini's 1999 For and Against Method. after which it was then edited jointly for many years by his LSE colleagues John W. N. Watkins and John Worrall, Lakatos's ex-research assistant.
Lakatos and his colleague Spiro Latsis had organized an international conference in Greece in 1975, and it went ahead despite his death. It was devoted entirely to historical case studies in Lakatos's methodology of research programmes in the physical sciences and economics. These case studies covered cases such as Einstein's relativity programme, Fresnel's wave theory of light and neoclassical economics, and were published by Cambridge University Press in two separate volumes in 1976, one devoted to physical sciences and Lakatos's general programme for rewriting the history of science, with a concluding critique by his great friend Paul Feyerabend, and the other devoted to economics.These were respectively Method and Appraisal in the Physical Sciences: The Critical Background to Modern Science 1800–1905 by Colin Howson (ed.) and Method and Appraisal in Economics by Spiro J. Latsis (ed.).
He remained at LSE until his sudden death in 1974 of a heart attack at the age of 51. The Lakatos Award was set up by the school in his memory. His last lectures along with parts of his correspondence with Paul Feyerabend have been published in For and Against Method.
Philosophical work Philosophy of mathematics
Lakatos's philosophy of mathematics was inspired by both Hegel's and 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 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 (). 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 counterexamples. 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.
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 axioms were tautological, i.e. logically true.)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".
Lakatos proposed an account of mathematical knowledge based on the idea of heuristics. 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 experiments" 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 proofs are valid and which are not. Therefore, he fundamentally disagreed with the "formalist" conception of proof that prevailed in Frege's and 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 Poincaré (1893)Poincaré, H. (1893). "Sur la Généralisation d'un Théorème d'Euler relatif aux Polyèdres", Comptes Rendus des Séances de l'Académie des Sciences, 117 p. 144, as cited in Lakatos, Worrall and Zahar, p. 162. 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.Lakatos, Worrall and Zahar (1976), Proofs and Refutations , 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, rewrites Euler's conjecture into a tautology of vector algebra.
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.
Research programmes