# Proofs and Refutations: The Logic of Mathematical Discovery

Cambridge University Press, 1976 - Philosophy - 174 pages
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.

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### Contents

 CHAPTER 1 A Problem and a Conjecture 6 2 A Proof 7 3 Criticism of the Proof by Counterexamples which are Local hut not Global 10 4 Criticism of the Conjecture by Global Counterexamples 13 b Rejection of the counterexample The method of monsterbarring 14 c Improving the conjecture by exceptionbarring methods Piecemeal exclusions Strategic withdrawal or playing for safety 24 d The method of monsteradjustment 30 e Improving the conjecture by the method of lemmaincorporation Proof generated theorem versus naive conjecture 33
 b Proofgenerated versus naive concepts Theoretical versus naive classification 88 c Logical and heuristic refutations revisited 92 d Theoretical versus naive conceptstretching Continuous versus critical growth 93 e The limits of the increase in content Theoretical versus naive refutations 96 9 How Criticism may turn Mathematical Truth into Logical Truth 99 b Mitigated conceptstretching may turn mathematical truth into logical truth 102 Editors Introduction 106 2 Another Proof of the Conjecture 116

 5 Criticism of the ProofAnalysis by Counterexamples which are Global but not Local The Problem of Rigour 42 b Hidden lemmas 43 c The method of proof and refutations 47 d Proof versus proofanalysis The relativisation of the concepts of theorem and rigour in proofanalysis 50 6 Return to Criticism of the Proof by Counterexamples which are Local hut not Global The Problem of Content 57 b Drive towards final proofs and corresponding sufficient and necessary conditions 63 c Different proofs yield different theorems 65 7 The Problem of Content Revisited 66 b Induction as the basis of the method of proofs and refutations 68 c Deductive guessing versus naive guessing 70 d Increasing content by deductive guessing 76 e Logical versus heuristic counterexamples 82 a Refutation by conceptstretching A reappraisal of monsterbarring and of the concepts of error and refutation 83
 3 Some Doubts about the Finality of the Proof Translation Procedure and the Essentialist versus the Nominalist Approach to Definitions 119 Another CaseStudy in the Method of Proofs and Refutations 127 2 Seidels Proof and the ProofGenerated Concept of Uniform Convergence 131 3 Abels ExceptionBarring Method 133 4 Obstacles in the Way of the Discovery of the Method of ProofAnalysis 136 The Deductive versus the Heuristic Approach 142 2 The Heuristic Approach ProofGenerated Concepts 144 b Bounded variation 146 c The Caratheodory definition of measurable set 152 Bibliograpy 155 Indexes of Names 167 Indexes of Subjects 170 Copyright

### Popular passages

Page 4 - Kant: the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty. . . . The formalist philosophy of mathematics has very deep roots. It is the latest link in the long chain of dogmatist philosophies of mathematics. For more than 2,000 years there has been an argument between dogmatists and sceptics. In this great debate, mathematics has been...