## Proofs and Refutations: The Logic of Mathematical DiscoveryProofs 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. |

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### Contents

CHAPTER 1 A Problem and a Conjecture | 6 |

2 A Proof | 7 |

3 Criticism of the Proof by Counterexamples which are Local hut not Global | 10 |

4 Criticism of the Conjecture by Global Counterexamples | 13 |

b Rejection of the counterexample The method of monsterbarring | 14 |

c Improving the conjecture by exceptionbarring methods Piecemeal exclusions Strategic withdrawal or playing for safety | 24 |

d The method of monsteradjustment | 30 |

e Improving the conjecture by the method of lemmaincorporation Proof generated theorem versus naive conjecture | 33 |

b Proofgenerated versus naive concepts Theoretical versus naive classification | 88 |

c Logical and heuristic refutations revisited | 92 |

d Theoretical versus naive conceptstretching Continuous versus critical growth | 93 |

e The limits of the increase in content Theoretical versus naive refutations | 96 |

9 How Criticism may turn Mathematical Truth into Logical Truth | 99 |

b Mitigated conceptstretching may turn mathematical truth into logical truth | 102 |

Editors Introduction | 106 |

2 Another Proof of the Conjecture | 116 |

5 Criticism of the ProofAnalysis by Counterexamples which are Global but not Local The Problem of Rigour | 42 |

b Hidden lemmas | 43 |

c The method of proof and refutations | 47 |

d Proof versus proofanalysis The relativisation of the concepts of theorem and rigour in proofanalysis | 50 |

6 Return to Criticism of the Proof by Counterexamples which are Local hut not Global The Problem of Content | 57 |

b Drive towards final proofs and corresponding sufficient and necessary conditions | 63 |

c Different proofs yield different theorems | 65 |

7 The Problem of Content Revisited | 66 |

b Induction as the basis of the method of proofs and refutations | 68 |

c Deductive guessing versus naive guessing | 70 |

d Increasing content by deductive guessing | 76 |

e Logical versus heuristic counterexamples | 82 |

a Refutation by conceptstretching A reappraisal of monsterbarring and of the concepts of error and refutation | 83 |

3 Some Doubts about the Finality of the Proof Translation Procedure and the Essentialist versus the Nominalist Approach to Definitions | 119 |

Another CaseStudy in the Method of Proofs and Refutations | 127 |

2 Seidels Proof and the ProofGenerated Concept of Uniform Convergence | 131 |

3 Abels ExceptionBarring Method | 133 |

4 Obstacles in the Way of the Discovery of the Method of ProofAnalysis | 136 |

The Deductive versus the Heuristic Approach | 142 |

2 The Heuristic Approach ProofGenerated Concepts | 144 |

b Bounded variation | 146 |

c The Caratheodory definition of measurable set | 152 |

155 | |

167 | |

170 | |

### Other editions - View all

Proofs and Refutations: The Logic of Mathematical Discovery Imre Lakatos,John Worrall,Elie Zahar Limited preview - 1976 |

### Common terms and phrases

Abel Alpha analysis argument axioms Beta boundary bounded variation Cauchy Cauchy's proof certainly concept-stretching continuous functions convex polyhedra counter criticism cube cylinder deductive guessing define definition Delta Dirichlet discovery domain edges empty set Epsilon Euclidean Euler characteristic Euler theorem Euler's theorem Eulerian Eulerian polyhedra examples exception-barring method exceptions fact false footnote formalist formula Fourier's Gamma generalisation global counterexamples heptahedron Hessel heuristic hidden lemmas inductive infallibilist interpretation intuition Kappa Lakatos Lambda lemma-incorporation Lhuilier logic mathe mathematical proof mathematicians mathematics matics method of proofs monster monster-barring naive conjecture number of vertices Omega original conjecture perfectly known philosophy of mathematics picture-frame plane Poinsot Polya polygons polyhedra are Eulerian polyhedron primitive conjecture problem proof-analysis proof-generated concept proofs and refutations proposition prove rigour ringshaped faces Sigma simply-connected star-polyhedra stretching Teacher Theta thought-experiment tions translation triangles trivial true truth tunnels uniform convergence V—E+F validity vertex Zeta

### Popular passages

Page 4 - Kant: the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty. . . . The formalist philosophy of mathematics has very deep roots. It is the latest link in the long chain of dogmatist philosophies of mathematics. For more than 2,000 years there has been an argument between dogmatists and sceptics. In this great debate, mathematics has been...