Imre Lakatos (1922–1974)

Indeholder "Paolo Mancosu: Preface to this edition", "Editor's preface", "Acknowledgments", "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".

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