Proofs and Refutations: The Logic of Mathematical Discovery

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Cambridge University Press, Jan 1, 1976 - Philosophy - 174 pages
21 Reviews
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.
  

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Review: Proofs and Refutations: The Logic of Mathematical Discovery

User Review  - Christin Bailey - Goodreads

This book ruined my life. Read full review

Review: Proofs and Refutations: The Logic of Mathematical Discovery

User Review  - Goodreads

This book ruined my life. Read full review

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
Bibliograpy
155
Indexes of Names
167
Indexes of Subjects
170
Copyright

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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...

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