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Proofs and Refutations: The Logic of Mathematical Discovery First Edition

by Imre Lakatos (Editor), John Worrall (Editor), Elie Zahar (Editor)
4.5 4.5 out of 5 stars 71 ratings
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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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  1. ISBN-10
    0521290384
  2. ISBN-13
    978-0521290388
  3. Edition
    First Edition
  4. Publisher
    Cambridge University Press
  5. Publication date
    January 1, 1976
  6. Part of series
    Cambridge Philosophy Classics
  7. Language
    English
  8. Dimensions
    5.5 x 0.75 x 8.5 inches
  9. Print length
    188 pages
  10. See all details
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Editorial Reviews

Review

'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

Book Description

Proofs and Refutations is for those interested in the methodology, philosophy and history of mathematics.

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 ‏ : ‎ 8.2 ounces
  • Dimensions ‏ : ‎ 5.5 x 0.75 x 8.5 inches
  • Customer Reviews:
    4.5 4.5 out of 5 stars 71 ratings

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Imre Lakatos

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Customer reviews

4.5 out of 5 stars
4.5 out of 5
71 global ratings

Top reviews from the United States

William Oliver
5.0 out of 5 stars If you are looking for a book on practical mathematical foundations, this is the book for you.
Reviewed in the United States on January 24, 2018
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I have never written a review for a book on amazon before this point, but I felt this book really deserves 5 stars, and I wanted to address the criticisms of the book directly. I believe many of the criticisms are a result of a misunderstanding of this book's goals. Specifically, the comparison to Polya's books are somewhat unfounded, because I believe the books have different goals. The goal of Polya's books are to provide a general method for finding solutions to unsolved problems. The goal of this book, on the other hand, is to explain the purpose and meaning of a proof once we already have it.

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.
14 people found this helpful
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Michael Demkowicz
5.0 out of 5 stars the heuristic of mathematical discovery
Reviewed in the United States on March 9, 2006
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In a footnote to chapter 2 (much of the content of "Proofs and Refutations" is in the footnotes) Lakatos writes: "Until the seventeenth century, Euclidians approved the Platonic method of analysis as the method of heuristic; later they replaced it by the stroke of luck and/or genius." That stroke of luck and/or genius is a slight of hand that hides much of the story of the unfolding of mathematical research.

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.
14 people found this helpful
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Reader7
5.0 out of 5 stars A Math Mind for Philosophy
Reviewed in the United States on July 30, 2014
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This book is a classic, and I am so happy to finally have my own copy. Lakatos was one of those rare minds in philosophy who influenced generations of people in multiple disciplines.
2 people found this helpful
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jmmephd
4.0 out of 5 stars Classic. Timeless.
Reviewed in the United States on February 4, 2013
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An important read for anyone studying the philosophy of mathematics. Also interesting for mathematics educators. Classic. Timeless. Not too long.
3 people found this helpful
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K. in Texas
5.0 out of 5 stars Excellent book showing that doing mathematics is a complicated process ...
Reviewed in the United States on June 11, 2017
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Excellent book showing that doing mathematics is a complicated process they goes far beyond just discovering results of some realm.
One person found this helpful
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Harvey
3.0 out of 5 stars Too specific
Reviewed in the United States on July 4, 2019
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I was hoping to read something about the general logic of mathematical discovery, as promised in the subtitle.
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.
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Richard
5.0 out of 5 stars An excellent, intriguing read.
Reviewed in the United States on February 10, 2014
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Not only is this a very interesting argument about the nature of mathematical proof, it is a highly entertaining read. A classic of late 20th century mathematical philosophy.
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Amazon Customer
5.0 out of 5 stars A beautifully readable story with very little maths in it - a real classic
Reviewed in the United Kingdom on February 23, 2014
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This is a story book. It chronicles a series of high school lessons about crystal shapes called polyhedra. The class spend the whole book in a lively, wryly humorous and immensely readable dialogue with their teacher, trying to find a mathematical definition that makes sense. The reader is drawn in to the characters, their arguments and the way the plot develops.

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.
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Gaetano
5.0 out of 5 stars un classico
Reviewed in Italy on April 26, 2013
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Si tratta di un testo classico sui fondamenti della matematica. Indispensabile per chi vuole comprendere i meccanismi più profondi della creazione matematica.
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Willy Van den Driessche
4.0 out of 5 stars Illustrating mathematical discovery
Reviewed in France on October 25, 2010
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This is a book about mathematics in the sense that it does not really prove theorems but rather it talks about what it is like to make a theorem.
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.
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CAGEN
5.0 out of 5 stars Great book, great seller.
Reviewed in the United Kingdom on January 1, 2018
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Great book, great seller. Found this very interesting to read as a History and Philosophy of Science student.
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andrew berkeley
5.0 out of 5 stars Five Stars
Reviewed in the United Kingdom on March 12, 2017
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excellent
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