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Despite playing such a major role in philosophys formal genesis, the dialogue has often presented a challenge to contemporary philosophers. Many are apt to shy away from it due to its apparent levity and lack of rigor. However, the dialogue possesses significant didactic and autotelic advantages. At its best, it can reveal without effort the dialectic manner in which knowledge and disciplines develop. This way, the reader has a chance to experience the process. Proofs and Refutations is a paragon of dialogical philosophy. Using just a few historical case studies, the book presents a powerful rebuttal of the formalist characterization of mathematics as an additive process in which absolute truth is gradually arrived at through infallible deductions. The logic of discovery, he claims, is a much messier affair. Theorems begin as mere conjectures, whose proofs are informal and whose terms are vaguely defined. It is only through a dialectical process, which Lakatos dubs the method of proofs and refutations, that mathematicians finally arrive at the subtle definitions and absolute theorems that they later end up taking so much for granted. Lakatos contrasts the formalist method of approaching mathematical history against his own, consciously heuristic approach. Instead of treating definitions as if they have been conjured up by divine insight to allow the mathematician to deduce theorems from the bottom up, the heuristic approach recognizes the very top down aspect of performing mathematics, by which definitions develop as a consequence of the refinement of proofs and their related concepts. Ultimately, the naive conjecture (the top) is where the mathematician begins, and it is only after the process of proofs and refutations has finalized that we are even prepared to present mathematics as beginning from first principles and flourishing therefrom.

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