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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Review: Proofs and Refutations: The Logic of Mathematical Discovery
User Review  Christin Bailey  GoodreadsThis book ruined my life. Read full review
Review: Proofs and Refutations: The Logic of Mathematical Discovery
User Review  GoodreadsThis 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 
155  
167  
170  
Common terms and phrases
Abel Alpha analysis argument axioms Beta boundary bounded variation Cauchy Cauchy's proof certainly conceptstretching 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 exceptionbarring 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 lemmaincorporation Lhuilier logic mathe mathematical proof mathematicians mathematics matics method of proofs monster monsterbarring naive conjecture number of vertices Omega original conjecture perfectly known philosophy of mathematics pictureframe plane Poinsot Polya polygons polyhedra are Eulerian polyhedron primitive conjecture problem proofanalysis proofgenerated concept proofs and refutations proposition prove rigour ringshaped faces Sigma simplyconnected starpolyhedra stretching Teacher Theta thoughtexperiment 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...