Proofs and Refutations

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 ...

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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.

Contents v
Preface to this edition vii
Editors' preface ix
Author's introduction 1
Chapter 1 6
1 A problem and a conjecture 6
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Contents v
Preface to this edition vii
Editors' preface ix
Author's introduction 1
Chapter 1 6
1 A problem and a conjecture 6
2 A proof 7
3 Criticism of the proof by counterexamples which are local but not global 10
4 Criticism of the conjecture by global counterexamples 14
a. Rejection of the conjecture. The method of surrender 14
b. Rejection of the counterexample. The method of monster-barring 15
c. Improving the conjecture by exception-barring methods. Piecemeal exclusions. Strategic withdrawal or playing for safety 26
d. The method of monster-adjustment 32
e. Improving the conjecture by the method of lemma-incorporation. Proof-generated theorem versus naive conjecture 35
5 Criticism of the proof-analysis by counterexamples which are global but not local. The problem of rigour 45
a. Monster-barring in defence of the theorem 45
b. Hidden lemmas 46
c. The method of proof and refutations 50
d. Proof versus proof-analysis. The relativisation of the concepts of theorem and rigour in proof-analysis 53
6 Return to criticism of the proof by counterexamples which are local but not global. The problem of content 60
a. Increasing content by deeper proofs 60
b. Drive towards final proofs and corresponding sufficient and necessary conditions 67
c. Different proofs yield different theorems 69
7 The problem of content revisited 70
a. The naiveté of the naive conjecture 70
b. Induction as the basis of the method of proofs and refutations 72
c. Deductive guessing versus naive guessing 74
d. Increasing content by deductive guessing 81
e. Logical versus heuristic counterexamples 87
8 Concept-formation 89
a. Refutation by concept-stretching. A reappraisal of monster-barring – and of the concepts of error and refutation 89
b. Proof-generated versus naive concepts. Theoretical versus naive classification 93
c. Logical and heuristic refutations revisited 98
d. Theoretical versus naive concept-stretching. Continuous versus critical growth 99
e. The limits of the increase in content. Theoretical versus naive refutations 101
9 How criticism may turn mathematical truth into logical truth 105
a. Unlimited concept-stretching destroys meaning and truth 105
b. Mitigated concept-stretching may turn mathematical truth into logical truth 108
Chapter 2 112
Editors' introduction 112
1 Translation of the conjecture into the 'perfectly known' terms of vector algebra. The problem of translation 112
2 Another proof of the conjecture 123
3 Some doubts about the finality of the proof. Translation procedure and the essentialist versus the nominalist approach to definitions 126
Appendix 1 Another case-study in the method of proofs and refutations 135
1 Cauchy's defence of the 'principle of continuity' 135
2 Seidel's proof and the proof-generated concept of uniform convergence 139
3 Abel's exception-barring method 141
4 Obstacles in the way of the discovery of the method of proof-analysis 144
Appendix 2 The deductivist versus the heuristic approach 151
1 The deductivist approach 151
2 The heuristic approach. Proof-generated concepts 153
a. Uniform convergence 153
b. Bounded variation 155
c. The Carathéodory definition of measurable set 162
Bibliography 164
Index of names 175
Index of subjects 178
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