Proofs and Refutations: The Logic of Mathematical Discovery [Paperback]

Imre Lakatos (Editor), John Worrall (Editor), Elie Zahar (Editor)

Book Description

ISBN-10: 0521290384 | ISBN-13: 978-0521290388 | Publication Date: January 1, 1976

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.

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

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.

Product Details

• Paperback: 188 pages
• Publisher: Cambridge University Press (January 1, 1976)
• Language: English
• ISBN-10: 0521290384
• ISBN-13: 978-0521290388
• Product Dimensions: 8.3 x 5.5 x 0.7 inches
• Shipping Weight: 9.1 ounces (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (11 customer reviews)
• Amazon Best Sellers Rank: #314,965 in Books (See Top 100 in Books)

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27 of 27 people found the following review helpful:

5.0 out of 5 stars The fundamental work on what mathematics really does, May 22, 2000

By 

Stavros Macrakis (Cambridge, MA, USA) - See all my reviews
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This review is from: Proofs and Refutations: The Logic of Mathematical Discovery (Paperback)

Definitions, examples, theorems, proofs -- they all seem so inevitable. But how did they come to be that way? What is the role of counterexamples? Why are some definitions so peculiar? What good are proofs?

In this brilliant and deep -- yet easy to read -- book, Lakatos shows how mathematicians explore concepts; how their ideas can develop over time; and how misleading the "textbook" presentation of math really is.

Fascinating for anyone who has seen mathematical proofs (even high-school Euclidean geometry) and essential for anyone studying mathematics at any level.

If you'd like to read more discussion about Lakatos and the intellectual context of P&R, you'll be interested in Brendan Larvor's "Lakatos: An Introduction".

25 of 25 people found the following review helpful:

5.0 out of 5 stars a study in mathematical thought, October 2, 2001

By 

Vin de Silva (Claremont, California USA) - See all my reviews
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This review is from: Proofs and Refutations: The Logic of Mathematical Discovery (Paperback)

I want to add a few words to the brief comment by the reader in Monroe (who gave this book one star). I tend to agree that "Proofs and Refutations" isn't a primer in mathematical proof-writing; it's certainly not a textbook for beginning mathematicians wanting to know how to practice their craft.

However, for those readers (including beginning mathematicians) who are interested in the broader picture, who are interested in the nature of mathematical proof, then Lakatos is essential reading. The examples chosen are vivid, and there is a rich sense of historical context. The dramatised setting (with Teacher and students Alpha, Beta, Gamma, etc) is handled skilfully. Now and then, a foolish-seeming comment from one of the students has a footnote tagged to it; more often than not, that student is standing in for Euler, Cauchy, Poincare or some other great mathematician from a past era, closely paraphrasing actual remarks made by them. That in some ways is the most important lesson I learned from this book; "obvious" now doesn't mean obvious then, even to the greatest intellects of the time.

Although "Proofs and Refuatations" is an easy book to begin reading, it is not an easy book per se. I have returned to it repeatedly over the last ten years, and I always learn something new. The text matures with the reader.

19 of 19 people found the following review helpful:

5.0 out of 5 stars Mathematics as a human endeavor to meet human needs, March 18, 2001

By 

Tom Gray (Fort-Coulonge, Quebec Canada) - See all my reviews

This review is from: Proofs and Refutations: The Logic of Mathematical Discovery (Paperback)

I would recommend that anyone interested in mathermaics or indeed anyone interested in human activities read Imre Lakatos's seminal book 'Proofs and Refutations: The Logic of Mathematical Discovery'.

Lakatos direcctly makes the distinction between formal and informal mathematics. Formal mathematics is contained in the proofs published in mathematical journals. Informal mathematics are the strategies that working mathemeticians use to make their work a useful exercise in mathematical discovery.

The proof provided for the four colour theorm which was derved in the 1970's relied heavily on the sue of computers and brute force technqiues. It was extremely cotroversial not because it was invalid but because of the issues which Lakatos so clearly describes in this book.It was undoubtedly a valid formal proof. However it did nothing to advance the cause of mathematics beyond this.

The reason that Lakatos equates proofs and refutation in his title is his contention that it is the refutations that are developed that show mathematicians the deficiencies and indeed teh possibilites in their theories. A refutation does not necessarily discredit a theory. Instead it provides insights to the theory's limitations and possibiliites for future development. It is their attempts to deal with unwanted and unexpected refutations - to preserve a valuable theory in the face of imperfect axioms and proof methods - that teach mathemeticians the true depths of their conceptions and to point the way to new and deeper ones.

Lakatos shows this by an account of the historical development of the concept of proof in mathematics and by showing in historical detail how certain valuable 'proofs' were preserved in the face of refutation. To this point Lakatos shows that the 'proofs' of the truth of Euler's number are no proofs at all. The great mathemetician Euler noticed that for any regular polyhedron the formula V-E+F=2 holds where V is the number of vertexes, E is the number of edges and F is the number of faces. Euler's and his successors proofs fall before any number of counterexamples. Does this prove that the theorem is 'incorrect?' Or does it mean as mathemetician's actions show that they thought it meant was that their concept of what constituted a regular polyhedron was deficient. Lakatos shows how these conceptions were modified over a couple of hundred years as counterexample after counterexample were faced.

These counterexamples all made mathematics stronger by deepening the conception of what polyhedra really are and by discovering new classes of them. In the end Euler's formula turned out not to have a proof but to be in effect a tautology. It is true for the regular polyhedra for which it is true by the definition of what constitutes a polyhedron. It is true because human mathematicians in order to make progress need it to be true.

The computer proof of the four color theorem was a triumph of formal mathematics. Its critics complained and if interpreted according to what Lakatos wrote in this book, they complained because it defeated the progress of informal mathematics.

Mathematical proofs are useful tools. The tell us what we need to know. Formal mathematics is about finding them. Informal mathematics is about making them useful. Mathematics is not some Platonian ideal divorced from humanity, painting, poetry ... It is a human endeavor to meet human needs.