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Proofs and Refutations: The Logic of Mathematical Discovery by John Worrall,Elie Zahar,Imre Lakatos
  • Author: John Worrall,Elie Zahar,Imre Lakatos
  • Title: Proofs and Refutations: The Logic of Mathematical Discovery
  • Size PDF ver: 1424 kb
  • Size ePUB ver: 1859 kb
  • Size Fb2 ver: 1648 kb
  • ISBN: 052121078X
  • ISBN13: 978-0521210782
  • Pages: 186
  • Other formats: azw mbr mobi lit
  • Category: Science & Math
  • Subcat: Mathematics
  • Language: English
  • Rating: 4.2 of 5
  • Votes: 842
  • Publisher: Cambridge University Press; 1 edition (September 24, 1976)
  • Hardcover: Here
Proofs and Refutations: The Logic of Mathematical Discovery
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.

Reviews num: (7)

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 is 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.
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.
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.
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.
Very good text that gives many refutations to traditional proofs in science.
An important read for anyone studying the philosophy of mathematics. Also interesting for mathematics educators. Classic. Timeless. Not too long.
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.

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