Proofs and Refutations: The Logic of Mathematical DiscoveryProofs 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. 
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Review: Proofs and Refutations: The Logic of Mathematical Discovery
User Review  Robb Seaton  GoodreadsBegins strong with a deconstruction of the Euler characteristic, but soon gets bogged down in philosophy, along with a troubling amount of relativism, although I'm not entirely clear about what Lakatos intends when he writes about truth, certainty, and progress. Read full review
Review: Proofs and Refutations: The Logic of Mathematical Discovery
User Review  David  GoodreadsIt is common for people starting out in Mathematics, by the time they've mastered Euclidean Geometry or some other first rigorous branch, to believe in its complete infallibility. If something is ... Read full review