Imre Lakatos , Lakatos Imre. 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.

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Goodreads helps you keep track of books you want to read. Want to Read saving…. Want to Read Currently Reading Read. Other editions. Enlarge cover. Error rating book. Refresh and try again. Open Preview See a Problem? Details if other :. Thanks for telling us about the problem. Return to Book Page.

Preview — Proofs and Refutations by Imre Lakatos. John Worrall Editor. Elie Zahar Editor. 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. Get A Copy. Paperback , pages. Published January 1st by Cambridge University Press. More Details Original Title.

Other Editions Friend Reviews. To see what your friends thought of this book, please sign up. To ask other readers questions about Proofs and Refutations , please sign up. What is the them of this book? See 1 question about Proofs and Refutations…. Lists with This Book. Community Reviews. Showing Average rating 4. Rating details. More filters. Sort order. Aug 11, Ben Labe rated it it was amazing. Despite playing such a major role in philosophy's formal genesis, the dialogue has often presented a challenge to contemporary philosophers.

Many are apt to shy away from it due to its apparent levity and lack of rigor. However, the dialogue possesses significant didactic and autotelic advantages.

At its best, it can reveal without effort the dialectic manner in which knowledge and disciplines develop. This way, the reader has a chance to experience the process. Using just a few historical case studies, the book presents a powerful rebuttal of the formalist characterization of mathematics as an additive process in which absolute truth is gradually arrived at through infallible deductions.

The "logic of discovery," he claims, is a much messier affair. Theorems begin as mere conjectures, whose proofs are informal and whose terms are vaguely defined. 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.

Lakatos contrasts the formalist method of approaching mathematical history against his own, consciously "heuristic" approach. Instead of treating definitions as if they have been conjured up by divine insight to allow the mathematician to deduce theorems from the bottom up, the heuristic approach recognizes the very top down aspect of performing mathematics, by which definitions develop as a consequence of the refinement of proofs and their related concepts.

Ultimately, the naive conjecture the top is where the mathematician begins, and it is only after the process of "proofs and refutations" has finalized that we are even prepared to present mathematics as beginning from first principles and flourishing therefrom. View 2 comments. Mar 08, Michael Nielsen rated it it was amazing Shelves: favorites.

Radically changed my idea of what mathematical definitions and proofs are, and where they come from. In particular, Lakatos convincingly refutes the idea that definitions come before theorems and proofs as often seems the case. Rather, they arise out of repeated back-and-forth interplay between conjectures and proof-ideas. That's a pretty abstract- and weird-sounding review. The book itself is incredibly readable, incredibly fun, and by the end will if you're anything like me have caused an e Radically changed my idea of what mathematical definitions and proofs are, and where they come from.

The book itself is incredibly readable, incredibly fun, and by the end will if you're anything like me have caused an earthquake in your worldview. So ignore the weirdness of my last paragraph, and just go read the book.

It's amazing. Jul 09, Devi rated it it was amazing Shelves: non-fiction. It is common for people starting out in Mathematics, by the time they've mastered Euclidean Geometry or some other first rigorous branch, to believe in its complete infallibility.

If something is mathematically proven we know beyond any shadow of a doubt that it is true because it follows from elementary axioms. Lakatos argues that this view misses quite a lot of how mathematical ideas historically have emerged.

His main argument takes the form of a dialogue between a number of students and a te It is common for people starting out in Mathematics, by the time they've mastered Euclidean Geometry or some other first rigorous branch, to believe in its complete infallibility. His main argument takes the form of a dialogue between a number of students and a teacher.

The dialogue itself is very witty and entertaining to read. The students put forward attempts of proofs that correspond to the historical development of Euler's conjecture about polyhedra. We see how new definitions emerge, like simply connected, from the nature of the naive, but incomplete, proofs of the conjecture. We also see how generally it is the refutations, the counterexamples, that help us in the development by forcing us to specify more conditions in the theorems, using more specific definitions and hint at further developments of the theorem.

This book is warmly recommended to anyone who does mathematics, is interested in philosophy of mathematics or science or simply enjoys a well-written dialogue about philosophical questions. The mathematics is generally except in the appendices about analysis quite elementary and doesn't require any prior knowledge, though it will feel more familiar if you have some experience with mathematical proofs.

Jul 16, Gwern rated it really liked it. Surprisingly interesting, like Wittgenstein if he wrote in a human fashion, and longer than one would think possible given how straightforward the problem initially appears. Oct 22, Andrew added it Shelves: analytic-philosophy , philosophy. Many of you, I'm guessing, have some math problems. You didn't do so hot in higher-level math, are more comfortable with the subjectivity of the written word, and view the process of mathematical discovery from a position of respect and distance.

What Lakatos shows you is that math is not the rigid formalistic system you may conceive of, but something far more fluid, something prone to frequent revision, something that must always have its underpinnings challenged in order to reach mathematical t Many of you, I'm guessing, have some math problems. What Lakatos shows you is that math is not the rigid formalistic system you may conceive of, but something far more fluid, something prone to frequent revision, something that must always have its underpinnings challenged in order to reach mathematical truth.

So in this dialogue, he exposes those challenges in order to arrive at a better understanding of Euler's theorem. What's important here, for the non-mathematically inclined, is to understand how we apply those same formalisms to our day-to-day thought. How we "monster-bar" by claiming that an exception to the rule is irrelevant or worse "proves the rule. And it teaches us how interesting things can get when you scratch beneath the surface.

Shit, I think I might get a tattoo of that ferocious "urchin" on the book cover. Mar 24, Conrad rated it it was amazing Shelves: philosophy , masterpieces , owned , in-storage.

By far one of the best philosophical texts I've read. 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. I've never gotten past Algebra II, and I still understood most of the book, though to be sure I missed out on the bits of calculus here and there, and didn't know enough about math to discern which dialogue participant stood By far one of the best philosophical texts I've read.

I've never gotten past Algebra II, and I still understood most of the book, though to be sure I missed out on the bits of calculus here and there, and didn't know enough about math to discern which dialogue participant stood for which philosopher. Definitely worthwhile. Jun 13, Douglas rated it it was amazing Shelves: mathematics , philosophy. This is an excellent, though very difficult, read.

It reminds me of Ernest Mach's "Science of Mechanics"--the latter is not in the form of a dialogue. Having heard Lakatos speak I can see how the book's dialogue format fits in with his style which is to the point and voluble. 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.

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 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.

But back to Lakatos. 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. This poverty of rewards is the explicit claim of Kline, whom I had read years before coming across Lakatos. Both men believed that claims by its proponents to the contrary, rigor was more obfuscation than clarification.

Indeed the distinctive feature of Lakatos' work is to skewer the rigorists with their own tools including their tedious "microanalysis. Such a view fit in with my own frustration over rigorism which diverts the student from the rich meat of mathematical ideas towards the details of the implements by which it is to be served.

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## Imre Lakatos

Imre Lakatos — was a Hungarian-born philosopher of mathematics and science who rose to prominence in Britain, having fled his native land in when the Hungarian Uprising was suppressed by Soviet tanks. Despite the star-studded array of academic lords and knights who were willing to testify on his behalf, neither MI5 nor the Special Branch seem to have trusted him, and no less a person than Roy Jenkins, the then Home Secretary, signed off on the refusal to naturalize him. See Bandy ch. According to Google Scholar, by the 25 th of January , that is, just twenty-five days into the new year, thirty-three papers had been published citing Lakatos in that year alone , a citation rate of over one paper per day. Introductory texts on the Philosophy of Science typically include substantial sections on Lakatos, some admiring, some critical, and many an admixture of the two see for example Chalmers and Godfrey-Smith The premier prize for the best book in the Philosophy of Science funded by the foundation of a wealthy and academically distinguished disciple, Spiro Latsis is named in his honour. Moreover, Lakatos is one of those philosophers whose influence extends well beyond the confines of academic philosophy.

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## Proofs and Refutations : The Logic of Mathematical Discovery

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