Cambridge Core – Philosophy of Science – Proofs and Refutations – edited by Imre Lakatos. PROOFS AND REFUTATIONS. ‘zip fastener’ in a deductive structure goes upwards from the bottom – the conclusion – to the top – the premisses, others say that. I. LAKATOS. 6 7. The Problem of Content Revisited. (a) The naivet6 of the naive conjecture. (b) Induction as the basis of the method of proofs and refutations.
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Retrieved from ” https: To see what your friends thought of this book, please sign up. We also see how generally it is the refutations, the counterexamples, that help us in the development by forcing us proofss specify more conditions in the theorems, using more specific definitions and hint at further developments of the theorem.
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Proofs and Refutations: The Logic of Mathematical Discovery
It is only through a dialectical process, which Lakatos dubs the method of “proofs and refutations,” that mathematicians finally arrive at the subtle definitions and absolute theorems that they later end up taking so much for granted. Instead I follow–and point the reader towards–a wonderful essay by the little-known Australian philosopher, David Stove, entitled, “Cole Porter and Karl Popper: And like Otis, it appears that, by taking Popper’s argument too far, Lakatos incurred the proogs, if not emnity, of the former.
Theorems begin as mere conjectures, whose proofs are informal and whose terms are vaguely defined. 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 verificationamong others.
Relatively short, it is also a very dense book, with hardly a wasted word. Thanks for telling us about the problem. Despite playing such a major role anv philosophy’s formal genesis, the dialogue has often presented a challenge to contemporary philosophers. Trkstr rated it really liked refutationz May 21, He makes you think about the nature of proof, kind rdfutations along the lines of the great Morris Kline–still an occasional presence during my graduate school days at New York University–and lzkatos wonderful book, “Mathematics and the Loss of Certainty” reinvigorated my love for mathematics; because it showed mathematics didn’t have to be presented in the dry theorem-lemma-proof style that has had it in a strangle hold since the 20th century predominance of the rigorists called formalists by Lakatos.
Nevertheless, I can name a few lessons learned.
The dialogue itself is very witty and entertaining to read. To create the most apt theorem statement, the proof is examined for ‘hidden assumptions’, ‘domain of applicability’, and even for sources of definitions. His main argument takes the form of a dialogue between a number of students and a teacher. Stove attempts to show how this has lead to what he calls irrationalism; by which he means the destruction of the intellect.
The book is profoundly deep, in a philosophical way, and it was not too difficult, which is probably why I enjoyed it so much. Paperbackpages. Definitely required reading for mathematicians and philosophers of mathematics. This poverty of rewards is the explicit claim of Kline, whom I had read years before coming across Lakatos.
If you are going into mathematics at a University level, I would highly recommend this book. Indeed the distinctive feature of Lakatos’ work is to skewer the rigorists with their own tools including their tedious “microanalysis.
It takes a theory about the sides of a polyhedron by Euler and uses dialogue form to show how the methods of inquiry of a handful of different theoreticians fall apart when attempting to prove or disprove the proposition. The additional essays included here another case-study of the proofs-and-refutations idea, and a comparison of The Deductivist versus the Heuristic Approach offer more insight into Lakatos’ philosophy and are welcome appendices. Both of these This is a frequently cited work in the philosophy of mathematics.
Proofs and Refutations: The Logic of Mathematical Discovery by Imre Lakatos
E prima di tutto il confronto: While their dispute is ultimately intellectual for the most part the personal tensions also realistically make themselves felt. Out of this Lakatos has fashioned an extremely effective essay explaining much about mathematics and refutatiobs methods. Math as evolving social construct.
George rated it it was amazing Feb 02, The students, named rwfutations 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.
A fairly simple mathematical concept is used as an example: The philosophy was good though. In the first, Lakatos gives examples of the heuristic process in mathematical discovery.
I once thought I had found Lakatos to be putting the final nail into the coffin of the certainty of overly rigorous mathematical proof; that slight were the blessings of such rigor compared to loss in clarity and direction in mathematics.
His proof still the standard proof in beginning analysis contained a ‘hidden lemma’.
Proofs and Refutations – Imre Lakatos
He makes you think about the nature of proof, kind of along the lines of the great Morris Kline–still an occasional presence during my graduate school days at New York University–and who’s wonderful book, “Mathematics and This is an excellent, though very difficult, read.
Lakatos argues that proof I rated this book 4 stars but it would be more accurate to call it 4 stars out of 5 for a mathematics book or for a school book or for a required reading book.
Jul 14, Jake rated it it was amazing. It does seem that the prevailing belief that we cannot really know anything–that there is uncertainty even in mathematical proof–has something to do with the loss of confidence in Western civilization itself; that the return to verifiability from falsifiability would herald a return to the old confidence in not only Western civilization but the idea of civilization itself. Return to Book Page.
Both men believed that claims by its proponents to the contrary, rigor was more obfuscation than clarification.
Is the theorem wrong, then? We develop mathematical definitions, examples, theorems, and proofs to meet human needs through heuristics. Jun 30, Kelly John Rose rated it it was amazing. So in this dialogue, he exposes those challenges in order to arrive at a better understanding of Euler’s theorem.
The most important lesson from this book is the idea of proof-based theorems. Published January 1st by Cambridge Refutayions Press. Trying to meet all your book preview and review needs.