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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Jun 23, J. 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.
It is only through a refutationd 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. Trying to meet all your book preview and review needs. 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.
Proof 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 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 refutayions.
This is an excellent, though very difficult, read.
The very idea of mathematical truth and the changing notions of rigour and proof are all discussed with stunning clarity. This way, the reader has a chance to experience the process.
Proofs and Refutations – UK. Thus the old proofs are seen as ‘obviously’ assuming a ‘hidden lemma’. Is the theorem wrong, then? 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 refurations thinking to the specific example of mathematics. In contrast most mathematical papers and textbooks present the final, polished product prokfs the style of Euclid’s Elements, leaving the reader wondering how the author came up with them.
A central theme is that definitions are not carved in stone, but often have to be patched up in the lakatow of later insights, in particular failed proofs. This page was last edited on 28 Februaryat 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.
An enjoyable dialogue examining when the demonstrable is or isn’t the “de-monsterable”. Unfortunately, he choose Popper as his model. If something is mathematically proven we know beyond any shadow of a doubt that it is true because it follows from elementary axioms.
Preview — Proofs and Refutations by Imre Lakatos. I have a background in set theory or axiomatics, and so the material in this book initially appeared quite shocking to me. Probably one prpofs the most important books I’ve read in my mathematics career. Today all we have is culture and that allows no judgment as to progress of mankind–except as an outworking of an all-encompassing statism. And much to my liking.
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.
Both of these This is a frequently cited work in the philosophy of mathematics.
For this reason, Lakatos argues, teachers and textbooks must provide a heuristic presentation behind the arguments and the proofs; the ontogenesis of mathematical discovery does not proceed through an arbitrary ‘definition, theorem, proof’ style. The book is profoundly deep, in a philosophical way, and it was not too difficult, which is probably why I enjoyed it so much. Imre Lakatos has written a highly readable book that ought to be read and re-read, to remind current and future generations of mathematicians that mathematics is not a quest for knowledge with an actual end, but shared cultural, even psychological, human activity.
This poverty of rewards is the explicit claim of Kline, whom I had read years before coming across Lakatos. While their dispute is ultimately intellectual for the most part the personal tensions also realistically make themselves felt.
Proofs and Refutations: The Logic of Mathematical Discovery
Lakqtos is this destruction, not irrefutability as Popper claims, that has lead to the ascendancy of bogus ideas such as Marxism, feminism and, lately, deconstructionism.
Here is Lakatos talking about the formalists, “Formalism denies the status of mathematics to most of what has been commonly understood to be mathematics, and can say nothing about its growth. 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 rrfutations the theorem.
This gives mathematics a somewhat experimental flavour. 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. Or perhaps they do for “We might be more interested in this proposition if we really understood just why the Riemann — Stieltjes integrable functions are so important.
To the critics that say such a textbook would be too long, he replies: This book is warmly recommended to anyone who does mathematics, is interested in philosophy of mathematics lkaatos science or simply enjoys a well-written dialogue about redutations questions.
Proofs and Refutations – Imre Lakatos
This deserves a higher rating, but the math was beyond my meager understanding so I struggled a bit. Lakatos also displays a fine wit, and an elegant writing style.
In best mathematical fashion each line builds on the previous, with all the lajatos trimmed away. Lakatos argues that this view misses quite a lot of how mathematical ideas historically have emerged.
Mar 18, Arron rated it it was amazing Shelves: I rated this alkatos 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. How we “monster-bar” by claiming that an exception to the rule proods irrelevant or worse “proves the rule. Books by Imre Lakatos. 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.