Proofs and Refutations: The Logic of Mathematical Discovery 1st Edition

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

16 customer reviews

ISBN-13: 978-0521290388

ISBN-10: 0521290384

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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.

Proofs and Refutations: The Logic of Mathematical Discovery

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Product Details

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
Average Customer Review: 4.4 out of 5 stars See all reviews (16 customer reviews)
Amazon Best Sellers Rank: #687,501 in Books (See Top 100 in Books)
- #263 in Books > Science & Math > Mathematics > Pure Mathematics > Logic
- #3279 in Books > Science & Math > History & Philosophy
- #4045 in Books > Textbooks > Humanities > Philosophy

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Customer Reviews

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28 of 28 people found the following review helpful

a study in mathematical thought

By Vin de Silva on October 2, 2001

Format: 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.

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28 of 28 people found the following review helpful

The fundamental work on what mathematics really does

By Stavros Macrakis on May 22, 2000

Format: 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".

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22 of 22 people found the following review helpful

Mathematics as a human endeavor to meet human needs

By Tom Gray on March 18, 2001

Format: 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.Read more ›

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11 of 12 people found the following review helpful

The fundamental work on what mathematics really does.

By A Customer on August 6, 1996

Format: 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.

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7 of 7 people found the following review helpful

Excellent Critical Reasoning Framework

By A Customer on September 26, 2001

Format: Paperback

As a lay reader of mathematics, I am prone to read for more for analogy and thought methods instead of, for example, the real implications of variations on Eulers Formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges.
Displaying solid content with artful execution, this book interested me in both the math of the thing and the acompanying thought processes.
Content: This book has near-poetic density and elegance in arguing a non-linear approach to mathematical development and, for me, to just plain thinking. Our tendency (as born worshippers of linearity and causality) is to discover a brick for the building then immediately look for the next to stack on top. Lakatos contends that PERHAPS you have discovered a brick worthy of the building, now let's see what truly objective tests we will put to this brick and before giving it a final stamp of approval. It seems obvious to say "always question", but the exercise in this book will take you through the process and show you what you may take for granted in this simple concept. For example, do you observe HOW you question? See his discussion throughout on global vs. local counterexamples, just as a start.
Execution of the text: This is the beautiful part. Mr. Lakatos has written this book as theater: characters with definite identities, plot, drama. The narrative flows in the voices of students and a professor who proves to be a sound moderator, intervening at timely points, i.e. those where questions may be crystallized or thoughts prodded to that point. This is where learning takes place, in a heated, moderated debate over Euler's formula.Read more ›

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Most Recent Customer Reviews

A Math Mind for Philosophy

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... Read more

Published 16 months ago by Reader7

An excellent, intriguing read.

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.

Published 22 months ago by Richard

My Favorite Book on Mathematical Philosophy

This is a book that really gets into how mathematics is done, instead of just focusing on what mathematics "is" or "should be. Read more

Published 22 months ago by Amazon Customer

Classic. Timeless.

An important read for anyone studying the philosophy of mathematics. Also interesting for mathematics educators. Classic. Timeless. Not too long.

Published on February 4, 2013 by MathEdPhD

Useless book

Simply drop it. Polya's Mathematics and plausible reasoning is the right book. I have been reading books to understand conjecture making. This one turned out to be useless. Read more

Published on February 3, 2013 by Abhi

Largely trivial

Lakatos' motives for writing this book seem to have been:
(a) "Under the present dominance of formalism, ... Read more

Published on February 8, 2008 by Viktor Blasjo

the heuristic of mathematical discovery

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... Read more

Published on March 8, 2006 by Michael Demkowicz

nice reading for the general public

Very nice book if you are in high school or in college and would like to see how mathematics evolves. Read more

Published on November 20, 2003

excentric prof's coffee chat

i strongly disadvise this book as a learning tool. "students" of mathematics who wish to explore the nature of "proofs" should definitely pass this one up. Read more

Published on July 13, 2000