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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See 1 question about Proofs and Refutations…. The polyhedron-example that is used has, in fact, a long and storied past, and Lakatos uses this to keep the example from being simply an abstract one — the book allows one to see the historical progression of maths, and to hear the echoes of the voices of past mathematicians that grappled with the same question. Philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its ‘methodology of proofs and refutations’ in its pre-axiomatic stages of development, and also for introducing the concept of refutatiobs ‘research programme’ in his methodology of scientific research programmes.
Want to Read Currently Reading Read. A novel introduction to the philosophy of mathematics, mostly in the form of a discussion between a group of students and their teacher.
Proofs and Refutations – Imre Lakatos
As an enthusiastic but relatively feeble intellect–at least by the standards of today’s ultra-competitive modern university wizards–I felt cheated. The book is profoundly deep, in a philosophic I would like to give this book a 4. If you are going into mathematics at a University level, I would highly recommend this book. The dialogue is fairly natural as natural as is prpofs, 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.
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Proofs and Refutations: The Logic of Mathematical Discovery by Imre Lakatos
I am not a philosopher and so I make no pretense to speak authoritatively about this. Just a moment while we sign you in to your Goodreads account. I would have to reread this some day.
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. 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.
The gist of it is that non-obvious mathematical concepts and definitions emerge through the process of refuting proposed proofs by exhibiting counter-examples. 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.
The dialogue itself is very witty and entertaining to read. 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.
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.
The counter-examples are then analyzed and new concepts are identified. In contrast most mathematical papers and textbooks present the final, polished product in the style of Euclid’s Elements, leaving the reader wondering how the author came up with them.
And it teaches us how oroofs things can get when you scratch beneath the surface. I really enjoyed wrestling with the idea that “proofs” can not be the perfect ideal that mathematics and mathematicians should strive for. Although Anr 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. Despite playing such a major role in philosophy’s formal genesis, the dialogue has often presented a challenge to contemporary philosophers.
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 reuftations conversation and keeping the original question in mind. But Stove also makes the point that Lakatos was, in fact, only carrying “Popperism” to its logical conclusion for Popper could not find a lakatoa to place a limit to his notions of falsifiability and bracketing.
Thanks for telling us about the problem. I think I can describe it as refutayions The Republic meets Philosophy meets History of Mathematics” and that sentence can proofz or less describe the entirety of the book.
A finely written, well-argued book, it is exemplary in its succinct and elegant presentation. 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.
Proofs and Refutations: The Logic of Mathematical Discovery
To create the most apt theorem statement, the proof is examined for ‘hidden assumptions’, ‘domain of applicability’, and even for sources of definitions. Lakatos goes through great pains, to succinctly convey the broad perspective of the students Euler, Cauchy, Poincare, etc. 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.
The students, named after 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. This deserves a higher rating, but the math was beyond my meager understanding so I struggled a bit.
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.
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. Nevertheless, I can name a few lessons learned. His main argument takes the form of a dialogue between a number of students and a teacher.
And it is presented in the form of an entertaining and even suspenseful narrative. Probably one of the most important books I’ve read in my mathematics career.
The book looks into those from the purely mathematical standpoint, and shows that they can be a lot easier to grasp and understand. For example, the difference between a counterexample to a lemma a so-called ‘local counterexample’ and a counterexample to the specific conjecture under attack a ‘global counterexample’ to the Euler characteristic, in anx case is discussed. 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 refutatins I can see my self re-reading this book in the future, but I would not recommend it to anyone in my porofs circle.
An enjoyable dialogue examining when the demonstrable is or isn’t the “de-monsterable”. A fairly simple mathematical concept is used as an example: Of course I would. Lakatos argues for a different kind of textbook, one that uses heuristic style.