Table Of Content

Cover Page: i Half title Page: i Title page Page: iii Imprints page Page: iv Contents Page: v Preface to this edition Page: vii Editors’ preface Page: ix Acknowledgments Page: xii Author’s introduction Page: 1 Chapter 1 Page: 6 1. A problem and a conjecture Page: 6 2. A proof Page: 7 3. Criticism of the proof by counterexamples which are local but not global Page: 10 4. Criticism of the conjecture by global counterexamples Page: 13 (a) Rejection of the conjecture. The method of surrender Page: 13 (b) Rejection of the counterexample. The method of monster-barring Page: 15 (c) Improving the conjecture by exception-barring methods. Piecemeal exclusions. Strategic withdrawal or playing for safety Page: 26 (d) The method of monster-adjustment Page: 32 (e) Improving the conjecture by the method of lemma-incorporation. Proof generated theorem versus naive conjecture Page: 35 5. Criticism of the proof-analysis by counterexamples which are global but not local. The problem of rigour Page: 45 (a) Monster-barring in defence of the theorem Page: 45 (b) Hidden lemmas Page: 45 (c) The method of proof and refutations Page: 50 (d) Proof versus proof-analysis. The relativisation of the concepts of theorem and rigour in proof-analysis. Page: 53 6. Return to criticism of the proof by counterexamples which are local but not global. The problem of content Page: 60 (a) Increasing content by deeper proofs Page: 60 (b) Drive towards final proofs and corresponding sufficient and necessary conditions Page: 67 (c) Different proofs yield different theorems Page: 69 7. The problem of content revisited Page: 70 (a) The naiveté of the naive conjecture Page: 70 (b) Induction as the basis of the method of proofs and refutations Page: 72 (c) Deductive guessing versus naive guessing Page: 74 (d) Increasing content by deductive guessing Page: 81 (e) Logical versus heuristic counterexamples. Page: 87 8. Concept-formation Page: 89 (a) Refutation by concept-stretching. A reappraisal of monster-barring – and of the concepts of error and refutation Page: 89 (b) Proof–generated versus naive concepts. Theoretical versus naive classification Page: 92 (c) Logical and heuristic refutations revisited Page: 98 (d) Theoretical versus naive concept-stretching. Continuous versus critical growth Page: 99 (e) The limits of the increase in content. Theoretical versus naive refutations Page: 101 9. How criticism may turn mathematical truth into logical truth Page: 105 (a) Unlimited concept-stretching destroys meaning and truth Page: 105 (b) Mitigated concept-stretching may turn mathematical truth into logical truth Page: 108 Chapter 2 Page: 112 Editors’ introduction Page: 112 1. Translation of the conjecture into the ‘perfectly known’ terms of vector algebra. The problem of translation Page: 112 2. Another proof of the conjecture Page: 123 3· Some doubts about the finality of the proof. Translation procedure and the essentialist versus the nominalist approach to definitions Page: 126 Appendices Page: 134 Appendix 1 Another case-study in the method of proofs and refutations Page: 135 1. Cauchy’s defence of the ‘principle of continuity’ Page: 135 2. Seidel’s proof and the proof-generated concept of uniform convergence Page: 139 3· Abel’s exception-barring method Page: 141 4· Obstacles in the way of the discovery of the method of proof-analysis Page: 144 Appendix 2 The deductivist versus the heuristic approach Page: 151 1. The deductivist approach Page: 151 2. The heuristic approach. Proof-generated concepts Page: 153 (a) Uniform convergence Page: 153 (b) Bounded variation Page: 155 (c) The Carathéodory definition of measurable set Page: 162 Bibliography Page: 164 Index of names Page: 175 Index of subjects Page: 178

Description:

Imre Lakatos's Proofs and Refutations is an enduring classic, which has never lost its relevance. Taking the form of a dialogue between a teacher and some students, the book considers various solutions to mathematical problems and, in the process, raises important questions about the nature of mathematical discovery and methodology. Lakatos shows that mathematics grows through a process of improvement by attempts at proofs and critiques of these attempts, and his work continues to inspire mathematicians and philosophers aspiring to develop a philosophy of mathematics that accounts for both the static and the dynamic complexity of mathematical practice. With a specially commissioned Preface written by Paolo Mancosu, this book has been revived for a new generation of readers.