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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To create defutations most apt theorem statement, the proof is examined for ‘hidden assumptions’, ‘domain of applicability’, and even for sources of definitions. No trivia or quizzes yet. What Lakatos shows you is that math is not the rigid formalistic refutationd 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.
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Theorems begin as mere conjectures, whose proofs are informal and whose terms are vaguely defined. The difference between man and animals is thus a matter of degree and not of kind.
The book is structured as a philosophical dialogue. 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. The gist of it is that non-obvious mathematical concepts and definitions emerge through the process of refuting proposed proofs by exhibiting counter-examples. But back to Lakatos. Jun 30, Kelly John Rose rated it it was amazing. 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.
Just a moment while we sign you in to your Goodreads account. This poverty of rewards is the explicit claim of Kline, whom I had read years before coming across Lakatos. How we “monster-bar” by ptoofs that an exception to the rule is irrelevant or worse “proves the rule.
Math as evolving social construct. Indeed the distinctive feature of Lakatos’ work is to eefutations the rigorists with their own tools including their tedious “microanalysis.
Many are apt to shy away from it due to its apparent levity and lack of rigor. The very idea of mathematical truth and refutatios changing notions of rigour and proof are all discussed with stunning clarity.
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 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 refutationz bracketing.
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.
And it is presented in the form of an entertaining and even suspenseful narrative. Jul 09, Devi rated it it was amazing Shelves: Portions of Proofs and Refutations were required reading for one of my classes for my master’s degree, but I liked it enough that I finished it after the course was completed. Many of you, I’m guessing, have some math problems. Feb 05, Julian rated it really liked it Shelves: For this reason, Oroofs argues, teachers and textbooks must provide a heuristic presentation reftations the arguments and the proofs; the ontogenesis of mathematical discovery does not proceed through an anx ‘definition, theorem, proof’ style.
Proofs and Refutations: The Logic of Mathematical Discovery
I really enjoyed wrestling with the idea that “proofs” can not be the perfect ideal that mathematics and mathematicians should strive for. And like Otis, it appears that, by taking Popper’s argument too far, Lakatos incurred the disapproval, if not emnity, of the former.
William rated it it was amazing Jan 10, In the first, Refhtations gives examples of the heuristic process in mathematical discovery. But I warn you, it’s a slow go itself. Jul 08, Vasil Kolev rated it it was amazing Shelves: 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: Thus the old proofs are seen as ‘obviously’ assuming a ‘hidden lemma’.
Overall pretty readable for what it is – will revisit again someday.
Proofs and Refutations – Wikipedia
A line I thought was pretty interesting is the following: 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 adn from a position of respect and distance. Anv and refutations is set as a dialog between students and teacher, where the teacher slowly goes through teaching a proof while students, representing famous mathematicians pipe in with conjecture and counter points.
The book has been translated into more than 15 languages worldwide, including Chinese, Korean, Serbo-Croat and Turkish, and went into its second Chinese edition in