Description
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.
Chapter
1 A Problem and a Conjecture
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
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
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 tum mathematical truth into logical truth
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
c. The Carathéodory definition of measurable set
Proofs and Refutations
：The Logic of Mathematical Discovery
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