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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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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 for which philosopher.

Is the theorem wrong, then? 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 examples resonate with my personal mathematical journey. But back to Lakatos.

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. I know I can understand many great mathematical ideas but I am put off by the reliance on logical primness often leading to roundabout “proofs,” merely for the sake of a certain notion of rigor. Proofs and Refutations – US.

Proofs and Refutations: The Logic of Mathematical Discovery

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.

And as theoretical ideas and concepts supersede naive ideas and concepts, theoretical language supersedes naive language. And much to my liking. By using this site, you agree to the Terms of Use and Privacy Policy.

Proofs and Refutations – Imre Lakatos

But Stove also makes the point that Lakatos was, in fact, only carrying “Popperism” to its refutztions conclusion for Popper could not find a way to place a limit to his notions of falsifiability and bracketing. With culture in the place of civilization there can be no question of the transcendent that applies to all men. Jul 08, Vasil Kolev rated it it was amazing Shelves: Jul 14, Eryk Banatt rated it liked it.

His proof still the standard proof in beginning analysis contained a ‘hidden lemma’. So in this dialogue, he exposes those challenges in order to arrive at a better understanding rfeutations Euler’s theorem. That is, one should look at one’s oroofs, and pin down exactly what properties are used, and then based on that thorough examination, state one’s theorem accordingly. Trkstr rated it really liked it May 21, If y Probably one of the most important books I’ve read in my mathematics refuttaions.

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. At its best, it can reveal without effort the dialectic manner in which knowledge and disciplines develop.

Nov 24, Arthur Ryman rated it it was amazing. Many important logical ideas are explained in the book.

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 refutation. Return to Book Page. And it teaches us how interesting things can get when you scratch beneath the surface.


I think I can describe it refutationz “Plato’s The Republic meets Philosophy meets History of Mathematics” and that sentence can more or less describe the entirety of the book. I once thought I had found Lakatos to be putting the final nail into the coffin of the certainty of overly rigorous mathematical proof; that slight were the blessings of such rigor compared to loss in clarity and direction in mathematics.

William rated it it was amazing Jan 10, 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.

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 Unfortunately, with the spread of computer science, their influence on the whole body of mathematics is gaining sway!

This short, but inspiring read discusses not a particular theorem or proof in mathematics, but rather the process of how mathematics is developed from an initial idea, hypothesis, monster-barring, expansion of the theorem, etc.

It really shows and demonstrates how you can take a really simple relation and build it up to create an extensive and an theory and possibly field of mathematics one step at a time. Arda rated it it was amazing Mar 31,