- Paperback: 188 pages
- Publisher: Cambridge University Press; 1st edition (January 1, 1976)
- Language: English
- ISBN-10: 0521290384
- ISBN-13: 978-0521290388
- Product Dimensions: 5.4 x 0.5 x 8.5 inches
- Shipping Weight: 9.1 ounces (View shipping rates and policies)
- Average Customer Review: 18 customer reviews
- Amazon Best Sellers Rank: #1,083,524 in Books (See Top 100 in Books)
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Proofs and Refutations: The Logic of Mathematical Discovery 1st Edition
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'For anyone interested in mathematics who has not encountered the work of the late Imre Lakatos before, this book is a treasure; and those who know well the famous dialogue, first published in 1963-64 in the British Journal for the Philosophy of Science, that forms the greater part of this book, will be eager to read the supplementary material ... the book, as it stands, is rich and stimulating, and, unlike most writings on the philosophy of mathematics, succeeds in making excellent use of detailed observations about mathematics as it is actually practised.' Michael Dummett, Nature
'The whole book, as well as being a delightful read, is of immense value to anyone concerned with mathematical education at any level.' C. W. Kilmister, The Times Higher Education Supplement
'In this book the late Imre Lakatos explores 'the logic of discovery' and 'the logic of justification' as applied to mathematics ... The arguments presented are deep ... but the author's lucid literary style greatly facilitates their comprehension ... The book is destined to become a classic. It should be read by all those who would understand more about the nature of mathematics, of how it is created and how it might best be taught.' Education
A novel introduction to the philosophy of mathematics, mostly in the form of a discussion between a group of students and their teacher. It combats the positivist picture and develops a much richer, more dramatic progression.
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This book answers the questions "How can we be sure a formal proof is correct?" and "How can we be sure it actually proves what we intuitively intended?", and it does so better than anything else I have ever read. As a result, this is a book more about mathematical philosophy than mathematical technique.
If you are someone who has trouble reading or writing proofs because you keep thinking of weird edge cases and have to verify that the proof handles all of them, or you have frequent existential crises about how written mathematical symbols (which are just symbols and syntax) can be shown to say anything about reality, this is the book for you.
In "Proofs and Refutations," Lakatos illustrates how a single mathematical theorem developed from a naive conjecture to its present (far more sophisticated) form through a gruelling process of criticism by counterexamples and subsequent improvements. Lakatos manages to seemlessly narrate over a century of mathematical work by adopting a quasi-Platonic dialogue form (inspired by Galileo's "Dialogues"?), which he thoroughly backs up with hard historical evidence in the voluminous footnotes. The story he tells explores the clumsy and halting heuristic processes by which mathematical knowledge is created: the very process so carfully hidden from view in most mathematics textbooks!
The participants of Lakatos' dialogue argue over questions like "when is something proved?", "what is a trivial vs. severe counterexample?", "must you state all your assumptions or can some be thought of as implicit?", "in the end, what has been proved?",etc.. The answers to these questions change as the theorem under consideration is successively seen in a new light. Throughout, Lakatos is at pains to point out that the different perspectives adopted by his characters are representative of viewpoints that were once taken by the heroes of mathematics.
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(a) "Under the present dominance of formalism, ...Read more