Proofs and Refutations: The Logic of Mathematical DiscoveryImre Lakatos, John Worrall, Elie Zahar Cambridge University Press, 1 sty 1976 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. |
Spis treści
Criticism of the Proof by Counterexamples which are Local but not Global | |
Criticism of the ProofAnalysis by Counterexamples which are Global but | |
Return to Criticism of the Proof by Counterexamples which are Local but | |
b Induction as the basis of the method of proofs and refutations | |
e Logical versus heuristic counterexamples | |
How Criticism may turn Mathematical Truth into Logical Truth | |
CHAPTER 2 | |
Editors Introduction | |
APPENDIX 1 | |
APPENDIX 2 | |
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accept according admit ALPHA already analysis apply argument BETA boundary bounded Cauchy Cauchy’s certainly claim closed completely concept conjecture continuous convergence convex counterexamples course criticism cube deductive define definition DELTA Dirichlet discovered discovery discussion domain edges EPSILON Euler Eulerian example exceptions explain faces fact false formula Fourier’s functions GAMMA give global guessing heuristic hidden idea improved increase inductive instance interesting interpretation KAPPA knowledge known language lemma logic look mathematicians mathematics meaning method mind monster naive naive conjecture never OMEGA ordinary original perfectly plane polygons polyhedra polyhedron possible present problem proof proof-analysis proofs and refutations proposition prove reason refutations remove replace rigour Rule seems SIGMA simple starts stretching TEACHER theorem theory thought translation triangles trivial true truth turn V—E+F I validity vertices ZETA