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

by Imre Lakatos, John Worrall, Elie Zahar
4.3 out of 5 stars  See all reviews (15 customer reviews)

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Book Description

January 1, 1976 0521290384 978-0521290388
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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Proofs and Refutations: The Logic of Mathematical Discovery + The Mathematical Experience + Journey through Genius: The Great Theorems of Mathematics
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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

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: 8.4 x 5.7 x 0.4 inches
  • Shipping Weight: 9.1 ounces (View shipping rates and policies)
  • Average Customer Review: 4.3 out of 5 stars  See all reviews (15 customer reviews)
  • Amazon Best Sellers Rank: #71,119 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
27 of 27 people found the following review helpful
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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25 of 25 people found the following review helpful
5.0 out of 5 stars a study in mathematical thought 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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22 of 22 people found the following review helpful
5.0 out of 5 stars Mathematics as a human endeavor to meet human needs 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.
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11 of 12 people found the following review helpful
By A Customer
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.
Comment | 
Was this review helpful to you?
7 of 7 people found the following review helpful
5.0 out of 5 stars Excellent Critical Reasoning Framework 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.
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Most Recent Customer Reviews
5.0 out of 5 stars 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 2 months ago by Richard A. Edwards
5.0 out of 5 stars 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 2 months ago by Daniel
4.0 out of 5 stars Classic. Timeless.
An important read for anyone studying the philosophy of mathematics. Also interesting for mathematics educators. Classic. Timeless. Not too long.
Published 14 months ago by MathEdPhD
1.0 out of 5 stars 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 14 months ago by Abhi
4.0 out of 5 stars 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
5.0 out of 5 stars 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
5.0 out of 5 stars 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
1.0 out of 5 stars 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
5.0 out of 5 stars Weirdly fascinating book on the essence of mathematics.
One should start right off admitting that this is a book about different approaches to what a mathematical proof is and should be. Read more
Published on April 29, 1999 by danielse@rafael.co.il, Daniel Segel
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