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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A number of mathematics teachers have implemented Lakatos’ method proofa proofs and refutations in the classroom, when teaching other mathematical topics. Teaching it this way, then, is a mistake.
Proofs and Refutations – Imre Lakatos
His proof still the standard proof in beginning analysis contained a ‘hidden lemma’. Science and math make progress by conjectures leading to proofs which are refuted with counterexamples. The philosophy was good though.
The cool part of this part of this passage is the idea that statements have different consistency values depending on the language in which you talk about them – you have certain things that might be true in a naive language i.
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 lakato place a limit lkaatos his notions of falsifiability and bracketing.
If something is mathematically proven we know beyond any shadow of a doubt that it is true because it follows from elementary axioms. 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. 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.
In fact, the definitions themselves have become more generally encompassing without this fact being consciously realized by the mathematicians working with the new definitions. The most important lesson from this book is the idea of proof-based theorems. A fairly simple mathematical concept is used as an example: The students put forward attempts of proofs that correspond to the historical development of Euler’s conjecture about polyhedra.
Both of these This is a frequently cited work in the philosophy of mathematics.
Arda rated it it was amazing Mar 31, A novel introduction to the philosophy of mathematics, mostly in the form of a discussion between a group of students and their teacher.
The book is structured as a philosophical dialogue.
See 1 question about Proofs and Refutations…. Theorems begin as mere conjectures, whose proofs are informal and whose terms are vaguely defined. To the critics that say such a textbook would be too long, he replies: We also see how reutations 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 refutatikns the theorem.
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 The discovery led to the definitional distinction between ‘point-wise convergence’ and ‘uniform convergence’.
One particularly enlightening application of this ‘proof-first’ method comes via the proof of Cauchy that the limit of a sequence of continuous functions is continuous. William rated it it was amazing Jan 10, That is, the proof always takes precedence. The difference between man and animals is thus a matter of degree and not of kind.
As an enthusiastic but relatively feeble intellect–at least by refutatikns standards of today’s annd modern university wizards–I felt cheated.
Proofs and Refutations: The Logic of Mathematical Discovery
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 stude Although I appreciates Lakatos’ classroom discussion style as original I had a hard time keeping up with the development of the conversation and keeping the original question in mind. Strongly invoking Popper both in its title and subtitle echoing Popper’s Conjectures and Refutations and The Logic of Scientific DiscoveryLakatos applies much of the master’s thinking to the specific example of mathematics.
Oct 22, Andrew ptoofs it Shelves: And it teaches us how interesting things can get when you scratch lkatos the surface.
It reminds me of Ernest Mach’s “Science of Mechanics”–the latter is not in the form of a dialogue. And indeed, that is one of the main messages to take away from reading this book. It was a little dry at times but the dialogue was very interesti I picked this up seeing it on a list of Robb Seaton’s favorite books”. Overall pretty readable for what it is – will revisit again someday. Though the book is written as a narrative, an actual method of investigation, that of “proofs and refutations”, is developed.