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Proofs and Refutations: The Logic of Mathematical Discovery (Cambridge Philosophy Classics) Paperback – October 15, 2015
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"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 1963-4 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."
"How is mathematics really done, and - once done - how should it be presented? Imre Lakatos had some very strong opinions about this. The current book, based on his PhD work under George Polya, is a classic book on the subject. It is often characterized as a work in the philosophy of mathematics, and it is that - and more. The argument, presented in several forms, is that mathematical philosophy should address the way that mathematics is done, not just the way it is often packaged for delivery."
William J. Satzer, MAA Reviews
Imre Lakatos's influential and enduring work on the nature of mathematic discovery and development continues to be relevant to philosophers of mathematics. Including a specially commissioned preface written by Paolo Mancosu, and presented in a fresh twenty-first-century series livery, it is now available for a new generation of readers.
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To be clear, I am interested only thought process involved in problem solving. Many websites recommend this book to illuminate the reader as to how you can learn the skill of mathematical investigation. This book may be good for other reasons but to understand the process of mathematical discovery, this book is quite useless. If you are looking for something outside the realm of problem solving, please ignore this review.
Edit on 06/24/2013: I have read a lot of books on solving tough math problems. I am only rating this book relative to the rest of them that I have read. Also, I am rating this book based on how helpful it is for the subject of "Logic of Mathematical Discovery". I was misguided into thinking that he will actually help the reader to become a better "proof maker". He has attempted to walk the student through the thought process involved in proof making by providing a dialogue between students and the teacher. It was not useful at all. Polya, on the other hand, in his series of books on problem solving, speaks of ways to think in order to solve the problems. People say that his approach is that of heuristics and that won't work all the time. In his book called "Mathematical Discovery", he does speak of mental strategies to some extent but there is another book written by Paul Zeitz which is at a higher level and speaks of mental strategies used to solve very tough problems.
In summary, there are much better books that you can invest your time and money on. Many of the examples he has used are used by others as well for the purpose of illustrating a point. I think it came from Polya's work originally since his is the oldest of all these books. I highly recommend Polya's book over this book. Of course, there are more advanced books if you can digest Polya's books starting from "How to Solve it" and ending in "Mathematical Discovery Vol II".
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 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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(a) "Under the present dominance of formalism, ...Read more