Proofs and Refutations: The Logic of Mathematical Discovery

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