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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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.
The gist of it is that non-obvious mathematical concepts and definitions emerge through the process of refuting proposed proofs by exhibiting counter-examples.
The dialogue is fairly natural as natural as is possible, given the maths that make up much of itand through the use of verbatim quotes and his varied subjects he has created a fine work. Preview — Proofs and Refutations by Imre Lakatos. I have studied Hegel for quite some time now, but Lakatos’ book introduced me to a new side of the dialectical method — yes, this book will teach you the method of “Proofs and Refutations” which is, a dialectical method of mathematical discovery.
I would recommend it to anyone with an interest in mathematics and philosophy. Feb 05, Julian rated it really liked it Shelves: I would like to give this book a 4.
It combats the positivist picture and develops a much richer, more dramatic progression. A finely pakatos, well-argued book, it is exemplary in its succinct and elegant presentation. 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.
At some parts of the book, the amount of prerequisite mathematical knowledge is small, then suddenly takes a giant leap into undefined but commonly known in advanced mathematics literatureso it can be a little difficult. Lakatos attacks the formalist philosophy of maths, which disagrees with the history of mathematical discovery. The complete review ‘s Review:. Lakatos contrasts the formalist method of approaching mathematical history against his own, consciously “heuristic” approach. The book has lakats translated into more than 15 languages worldwide, including Chinese, Korean, Serbo-Croat and Turkish, and went into its second Chinese edition in As an enthusiastic but relatively feeble intellect–at least by the standards of today’s ultra-competitive modern university wizards–I felt cheated.
To see what your friends thought of this book, please sign up. The counter-examples are then analyzed and new concepts are identified. The idea that the definition creates the mathematical meaning is a another powerful one, and I think it would be interesting to do an activity where students could come up with initial definitions and then try to rewrite them to make them more broad or more narrow.
Oct 22, Andrew added it Shelves: Lakatos uses the form to dramatic effect. Unfortunately, he choose Popper as his model. Definitely required reading for mathematicians and philosophers of mathematics. 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.
Apr 02, Jonathan rated it it was amazing. One particularly enlightening application of proocs ‘proof-first’ method comes via the proof of Cauchy that the limit of a sequence of continuous functions is continuous. It is this destruction, not irrefutability as Popper claims, that has lead to the ascendancy of bogus ideas such as Marxism, feminism and, lately, deconstructionism.
Definitions stretch as the history of mathematics rolls on; quite often slowly, and imperceptibly, so that when old theorems are seen in the light of the new stretched definitions, suddenly the proof is seen to be false, or to assume a ‘hidden lemma’. In this view of things, the theorem statement becomes secondary to the proof idea, which then takes precedence as lakattos most important part of the mathematical work.
Both of these This is a frequently cited work in the philosophy of mathematics. This is a frequently cited work in the philosophy of mathematics. The additional essays included here another case-study of the xnd idea, and a comparison of The Deductivist versus the Heuristic Approach offer more insight into Lakatos’ philosophy and are welcome appendices.
View all 4 comments. Mar 18, Arron rated it it was amazing Shelves: The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron. There are no discussion topics on this book yet. I really enjoyed wrestling with the idea refutatioons “proofs” can not be the perfect ideal that mathematics and mathematicians should strive for. But Stove also makes the point that Lakatos was, in fact, only carrying “Popperism” to its logical conclusion for Popper could not find a way to place a limit to his notions of falsifiability and bracketing.
Shit, I think I might get a tattoo of that ferocious “urchin” on the book cover. To ask other readers questions about Proofs and Refutationsplease sign up. But I warn you, it’s a slow go itself. However, the dialogue possesses significant didactic and autotelic advantages.
I think I can describe it as “Plato’s The Republic meets Philosophy meets History of Mathematics” and prolfs sentence can more or less describe the entirety of the book. Trivia About Proofs and Refuta Thanks for telling us about the problem.
While their dispute is ultimately intellectual for the most part the personal tensions also realistically make themselves felt. This way, the reader has a chance to experience the process. Mar 12, Samuel Fout rated it it was amazing. And this is why, even though I recommend Lakatos’ book, ultimately I must back away from it. I know I can understand many great mathematical ideas but I am put off by the reliance on logical primness often leading to roundabout “proofs,” merely for the sake of a certain notion of rigor.
What is the them of this book?