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Proofs and Refutations
by
Imre Lakatos
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 The Logic of Mathematical Discovery
 Edited by John Worrall and Elie Zahar
 This is a revision of the essay Proofs and Refutations that had appeared in The British Journal for the Philosophy of Science in 19634. Lakatos was working on the longplanned book version at the time of his death.
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Our Assessment:
A : very well done survey of philosophical and practical issues dealing with mathematics.
See our review for fuller assessment.
Opinions:
 "Lakatos's contribution to the philosophy of mathematics was, to put it simply, definitive: the subject will never be the same again. (...) Lakatos made us think instead about what most research mathematicians do. He wrote an amazing philosophical dialogue around the proof of a seemingly elementary but astonishingly deep geometrical idea pioneered by Euler. It is a work of art  I rank it right up there with the dialogues composed by Hume or Berkeley or Plato. He made us see a theorem, a mathematical fact, coming into being before our eyes."  Ian Hacking, in a review of For and Against Method by Lakatos/Feyerabend (see our review), in the London Review of Books (20/1/2000)
 "Proofs and Refutations has the impact it does because Lakatos knows the mathematics  or as much of it as he chooses to know  and can take the reader through it. The question is how he came to jettison so much of his intellectual past, and to believe himself a friend of reason, while making his living out of a project fundamentally irrational."  James Franklin, in a nonreview of For and Against Method by Lakatos/Feyerabend (see our review), in The New Criterion (5/2000)
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The complete review's Review:
Lakatos' didactic text, the title essay which makes up the bulk of this book, is presented in the form of a discussion between a teacher and a number of students.
Lakatos uses the form to dramatic effect.
The students, named after letters of the Greek alphabet, represent a broad spectrum of viewpoints that can be held about the issues at hand, all engaged in argument with their mentor.
Out of this Lakatos has fashioned an extremely effective essay explaining much about mathematics and its methods.
And it is presented in the form of an entertaining (and even suspenseful) narrative.
Strongly invoking Popper both in its title and subtitle (echoing Popper's Conjectures and Refutations and The Logic of Scientific Discovery), Lakatos applies much of the master's thinking to the specific example of mathematics.
A fairly simple mathematical concept is used as an example: anyone who knows what a polyhedron is should be able to follow the bulk of the arguments (those whose mathematical literacy does not extend this far will probably have difficulties with the book).
One of the issues is, in fact, the definition of a polyhedron, as well as the difference between Eulerian and nonEulerian polyhedra.
Taking the apparently simple problem before the class the teacher shows how many difficulties there in fact are  from that of proof to definition to verification, among others.
The possible approaches to advancing mathematical concepts are gone over, cleverly introduced in examples (and undermined in counterexamples).
The polyhedronexample that is used has, in fact, a long and storied past, and Lakatos uses this to keep the example from being simply an abstract one  the book allows one to see the historical progression of maths, and to hear the echoes of the voices of past mathematicians that grappled with the same question.
Most remarkable is the narrative drive behind the argument.
What seems relatively straightforward is in fact a complex and convoluted problem, and as the various opinions regarding proper approaches are voiced the characters also grow richer.
While their dispute is ultimately intellectual (for the most part) the personal tensions also realistically make themselves felt.
Lakatos also displays a fine wit, and an elegant writing style.
The dialogue is fairly natural (as natural as is possible, given the maths that make up much of it), and through the use of verbatim quotes and his varied subjects he has created a fine work.
Relatively short, it is also a very dense book, with hardly a wasted word.
In best mathematical fashion each line builds on the previous, with all the fat trimmed away.
Even (or perhaps: especially) the footnotes are a mine of information.
Lakatos himself did not finish the preparations to publish his essay in book form, but his editors have done a fine job.
The additional essays included here (another casestudy of the proofsandrefutations idea, and a comparison of The Deductivist versus the Heuristic Approach) offer more insight into Lakatos' philosophy and are welcome appendices.
An important look at the history and philosophy of maths (a field not quite as esoteric as one might imagine) this book is certainly recommended to all who are involved with mathematics, as well as all historians and philosophers of science.
A finely written, wellargued book, it is exemplary in its succinct and elegant presentation.
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About the Author:
Imre Lakatos was born in Hungary as Imre Lipsitz in 1922.
Active in the Communist Party in Hungary after World War II he worked in the Ministry of Education.
He earned his Ph.D from Debrecen University in 1947.
Expelled from the Communist Party in 1950, he was interned for three years.
He fled Hungary in 1956, and was awarded a Rockefeller Fellowship to study at Cambridge, where he completed another Ph.D.
He became a lecturer at the London School of Economics where Karl Popper was a great influence on him.
Lakatos died in 1974.
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