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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If y Probably one of the most important books I’ve read in my mathematics career. Lakatos contrasts the formalist method of approaching mathematical history against his own, consciously “heuristic” approach. Nov 24, Arthur Ryman rated it it was amazing.
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. Anyways, the bad is so negligible, I would say that if you like philosophy and if you like math, AND if you want a new perspective — read this very readable book.
Sep 30, Robb Seaton rated it it was amazing Shelves: Retrieved from ” https: In this view of things, the theorem statement becomes secondary to the proof idea, which then takes precedence as the most important part of the mathematical work. Naive conjectures and naive concepts are superseded by improved conjectures theorems and concepts proof-generated or theoretical concepts growing out of the method of proofs and refutations.
Many of you, I’m guessing, have some math problems. His main argument takes the form of a dialogue between a number of students and a teacher.
Proofs and Refutations: The Logic of Mathematical Discovery
Mar 24, Conrad rated it it was amazing Shelves: Jun 13, Douglas rated it it was amazing Shelves: May 12, Ari rated it really liked it. The book includes two appendices. Lakatos argues that this view misses quite a lot of how mathematical ideas historically have emerged.
But back to Lakatos. This book is warmly recommended to anyone who does mathematics, is interested in philosophy of mathematics or science or simply enjoys a well-written dialogue about philosophical questions.
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.
Then the conjectures can be modified and tightened up to make theories. Unfortunately, he choose Popper as his model. And Lakatos knows the history of eulers theorem, presents it as a classroom discussion making us realize that nothing is ever static in mathematics.
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.
Teaching it this way, then, is a mistake. Hence when I put quotes around the word proof, as I just did, I was following Lakatos.
Proofs and Refutations – Imre Lakatos
What seems relatively straightforward is in fact a complex and convoluted problem, and as the various opinions regarding proper approaches are voiced the characters also grow richer.
To see what your friends thought of this book, please sign up. This is a frequently cited work in the philosophy of mathematics. 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. Surprisingly interesting, like Wittgenstein if he wrote in a human fashion, and longer than one would think possible given how straightforward amd problem initially appears. 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.
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”. Jul 09, Devi rated it it was amazing Shelves: Indeed the distinctive feature of Lakatos’ work is to rffutations the rigorists with refuttaions own tools including their tedious “microanalysis. If something is mathematically proven we know beyond any shadow of a doubt that it is true because it follows from elementary axioms.
I believe Lakatos’ basic diagnosis is essentially correct. While their dispute is ultimately intellectual for the most part the personal tensions also realistically make themselves felt. And it teaches us how interesting things can get when you scratch beneath the surface.
If you are going into mathematics at a University level, I would highly recommend this book. The complete review ‘s Review:.