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
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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 |
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| 102 | |
Editors Introduction | 106 |
2 Another Proof of the Conjecture | 116 |
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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 |
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 7 - V - E + F = 2, where V is the number of vertices, E the number of edges and F the number of faces.
Page 89 - I have already said somewhere that mathematics is the art of giving the same name to different things.
Page 29 - I am sure that it is not altogether misleading. If we were to push it to its extreme we should be led to a rather paradoxical conclusion; that there is, strictly, no such thing as mathematical proof; that we can, in the last analysis, do nothing but point; that proofs are what Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils.
Page 9 - I propose to retain the time-honored technical term "proof" for a thought-experiment — or "quasi-experiment" — which suggests a decomposition of the original conjecture into subconjectures or lemmas, thus embedding it in a possibly quite distant body of knowledge. Our "proof," for instance, has embedded the original conjecture — about crystals, or, say, solids — in the theory of rubber sheets. Descartes or Euler, the fathers of the original conjecture, certainly did...
Page 22 - TEACHER: Counterexamples 2a and 2b." DELTA: I admire your perverted imagination, but of course I did not mean that any system of polygons is a polyhedron. By polyhedron I meant a system of polygons arranged in such a way that (1) exactly two polygons meet at every edge and (2) it is possible to get from the inside of any polygon to the inside of any other polygon by a route which never crosses any edge at a vertex.
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...
Page 7 - Let us imagine the polyhedron to be hollow. with a surface made of thin rubber. If we cut out one of the faces we can stretch the remaining surface flat on the blackboard, without tearing it. The faces and edges will be deformed, the edges may become curved, but V and E will not alter, so that if and only if V...
Page 142 - This style starts with a painstakingly stated list of axioms, lemmas and/or definitions. The axioms and definitions frequently look artificial and mystifyingly complicated. One is never told how these complications arose. The list of axioms and definitions is followed by the carefully worded theorems. These are loaded with heavy-going conditions; it seems unlikely that anyone should ever have guessed them. The theorem is followed by the proof. (P&R, p. 142) This 'deductivist...
Page 27 - And if no exception occur from phenomena, the conclusion may be pronounced generally. But if at any time afterwards any exception shall occur from experiments, it may then begin to be pronounced with such exceptions as occur.
Page 142 - Deductivist style hides the struggle, hides the adventure. The whole story vanishes, the successive tentative formulations of the theorem in the course of the proof-procedure are doomed to oblivion while the end result is exalted into sacred infallibility.
References to this book
Bibliographic information
| Title | Proofs and Refutations: The Logic of Mathematical Discovery |
| Author | Imre Lakatos |
| Editor | Imre Lakatos |
| Edition | illustrated, reprint |
| Publisher | Cambridge University Press, 1976 |
| ISBN | 0521290384, 9780521290388 |
| Length | 174 pages |
| Subjects | › Philosophy / Logic Science / Philosophy & Social Aspects |
| Export Citation | BiBTeX EndNote RefMan |