Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...) Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

Imre Lakatos' conception of the history of science is explicated with the purpose of replying to criticism leveled against it by Thomas Kuhn, Ian Hacking, and others. Kuhn's primary argument is that the historian's internal—external distinction is methodologically superior to Lakatos' because it is "independent" of an analysis of rationality. That distinction, however, appears to be a normative one, harboring an implicit and unarticulated appeal to rationality, despite Kuhn's claims to the contrary. Lakatos' history, by contrast, is clearly the history (...) of a normatively defined discipline; of science and not scientists and their activities. How such history can be written, the historiographic and critical tools available for its construction, and its importance as history, are considered in detail. In an afterword, the prevalence of Lakatos' treatment of history in philosophical discussion is indicated: A related approach is shown to arise in social contract theory. (shrink)

This paper discusses the connection between the actual history of mathematics and Lakatos's philosophy of mathematics, in three parts. The first points to studies by Lakatos and others which support his conception of mathematics and its history. In the second I suggest that the apparent poverty of Lakatosian examples may be due to the way in which the history of mathematics is usually written. The third part argues that Lakatos is right to hold philosophy accountable to history, even if Lakatos's (...) own view of mathematics fails that test. (shrink)

It is shown how the historiographic purport of Lakatosian methodology of mathematics is structured on the theme of analysis and synthesis. This theme is explored and extended to the revolutionary phase around 1800. On the basis of this historical investigation it is argued that major innovations, crucial to the appraisal of mathematical progress, defy reconstruction as irreducibly rational processes and should instead essentially be understood as processes of social-cognitive interaction. A model of conceptual change is developed whose essential ingredients are (...) the variability of rational responses to new intellectual and practical challenges arising in the cultural environment of mathematics, and the shifting selective pressure of society. The resulting view of mathematical development is compared with Kuhn's theory of scientific paradigms in the light of some personal communications. (shrink)

In this paper I explore possibilities of bringing post-positivist philosophies of empirical science to bear on the dynamics of mathematical development. This is done by way of a convergent accommodation of a mathematical version of Lakatos's methodology of research programmes, and a version of Kuhn's account of scientific change that is made applicable to mathematics by cleansing it of all references to the psychology of perception. The resulting view is argued in the light of two case histories of radical conceptual (...) innovations. (shrink)

Kitcher and Aspray distinguish a mainstream tradition in the philosophy of mathematics concerned with foundationalist epistemology, and a ‘maverick’ or naturalistic tradition, originating with Lakatos. My claim is that if the consequences of Lakatos's contribution are fully worked out, no less than a radical reconceptualization of the philosophy of mathematics is necessitated, including history, methodology and a fallibilist epistemology as central to the field. In the paper an interpretation of Lakatos's philosophy of mathematics is offered, followed by some critical discussion, (...) and an extension to a social constructivist position (which might well have been unacceptable to Lakatos). (shrink)

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume II presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues. Imre Lakatos had an influence (...) out of all proportion to the length of his philosophical career. This collection exhibits and confirms the originality, range and the essential unity of his work. It demonstrates too the force and spirit he brought to every issue with which he engaged, from his most abstract mathematical work to his passionate 'Letter to the director of the LSE'. Lakatos' ideas are now the focus of widespread and increasing interest, and these volumes should make possible for the first time their study as a whole and their proper assessment. (shrink)