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Proofs and Refutations: The Logic of Mathematical Discovery First Edition
- ISBN-100521290384
- ISBN-13978-0521290388
- EditionFirst Edition
- PublisherCambridge University Press
- Publication dateJanuary 1, 1976
- LanguageEnglish
- Dimensions5.5 x 0.75 x 8.5 inches
- Print length188 pages
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Editorial Reviews
Review
'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
Book Description
Book Description
Product details
- Publisher : Cambridge University Press; First Edition (January 1, 1976)
- Language : English
- Paperback : 188 pages
- ISBN-10 : 0521290384
- ISBN-13 : 978-0521290388
- Item Weight : 8.8 ounces
- Dimensions : 5.5 x 0.75 x 8.5 inches
- Best Sellers Rank: #2,440,071 in Books (See Top 100 in Books)
- #1,137 in Mathematical Logic
- #5,893 in Philosophy (Books)
- #8,554 in History & Philosophy of Science (Books)
- Customer Reviews:
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This book answers the questions "How can we be sure a formal proof which is formally correct, is also intuitively 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.
Instead it talked entirely about ONE proof, in an unhelpful dialogue format which only obscured the underlying points.
Perhaps if I were more clever I could have generated a general logic from that, but if I were that clever, I wouldn't have needed the book.
Top reviews from other countries
Along the way you find that you are discovering how to do maths. I don't mean the nuts-and-bolts, how to do the boring stuff like equations, but the interesting things that make you wonder, like why, whether, what is it anyway and, most significant of all, how to do better next time. Oh, and what a polyhedron is not, too - you'll be amazed at how many lousy definitions mathematicians have nailed their flag to over the years. But that's the point Lakatos is making, maths is not some dead library catalogue that grows and grows, it is forever being torn up, burned and rewritten better than ever.
The most frightening thing about this book is its title. Get past that and you are in for a truly enjoyable ride, with no more visible maths than a few simple and well-illustrated geometrical ideas to absorb.
There are some rather more heavyweight appendices, but you don't have to plough through those unless you want to.
Lakatos' book is a real classic and serves as an excellent introduction to the philosophy of mathematics.
The book starts with Euler's polyhedron formula. Then it goes on to search exceptions to this formula,
leading to a tighter and tighter definition of a good polyhedron. The process is inspiring and you should try to find exceptions yourself instead of just reading them in the book. Each time you think the definition is waterproof, there is still some other clever monster construction that does not fit the bill. It shows how easy it is to think you have a general theorem and the value of having counterexamples.
I love this book and would probably recommend it. The only thing against it is the somewhat dated style.