Proofs and refutations
by Imre Lakatos, John Worrall, Elie Zahar · first published 1976
76% recommend22% neutral2% critical
Why HN recommends it
A short Socratic dialogue following one theorem, the Euler characteristic of polyhedra, through repeated counterexamples and revised definitions. It shows how proof and discovery work; suits curious non-specialists.
Summarised by Claude from recommending comments
Proofs and Refutations is for those interested in the methodology, philosophy and history of mathematics.
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I would go with "Proofs and Refutations: The Logic of Mathematical" by Lakatos, its a dialog-form history of defining Euler's formula and it shows the human side of mathematics as a science. I think its great as math has this unique status…
on “Ask HN: Which book can attract anyone towards your field of study?” ↗…delete/clear cells at all from my jupyter notebooks. :) It is always informative/fun to refer to them at a later point in time to relive that problem solving experience. PS: In case you haven't, you should try Proofs and Refutations by Lakatos. Edit: Fixed name of the book mentioned and added some text.
on “In Search of Proofs from “The Book”” ↗Funny related reading: Imre Lakatos - Proofs and refutations
on “Intuition in Mathematics (2012) [pdf]” ↗I'm guessing you have not read Proofs and Refutations by Imre Lakatos. It is worthwhile. Also if you read on the history of mathematics, the various back and forth foundational issues raised by infinitesmals, Fourier series, proof by contradiction and…
on “Just 11% of 53 cancer research papers were reproducible” ↗"Proofs and Refutations" is a fantastic book, I really enjoy that style. If someone is looking for a more lightweight introduction to Lakatos I can highly recommend "For and Against Method" which outlines his "arguments" with…
on “Great books about mathematics” ↗…History of Mathematics - William Dunham - Journey Through Genius - Philip J. Davi & Reuben Hersh - The Mathematical Experience - Martin Aigner & Günter M. Ziegler - Proofs from the Book - Imre Lakatos - Proofs and Refutations - Robert M. Young - Excursions in Calculus: An Interplay of the Continuous and the Discrete - Courant & Robbins - What is Mathematics? - George Pólya - How to Solve It - Morris Kline - Kline’s Mathematics:…
on “Ask HN: How or where to begin learning mathematics from first principles?” ↗Read 'Proofs and Refutations, The Logic of Mathematical Discovery' by Imre Lakatos. That may provide a good answer to your question.
on “The Man Who Knew Infinity: A Life of the Genius Ramanujan [pdf]” ↗…Thomas Kuhn's The Structure of Scientific Revolutions, Imre Lakatos' Falsification and the Methodology of Scientific Research Programmes (great short read, btw, applying those ideas to maths: Lakatos' Proofs and Refutations), and Feyerabend's Against Method. Chalmers then covers Bayesianism, new experimentalism, the nature of scientific laws, and scientific realism. They've all pointed out problems with Popper's demarcation…
on “Philosophy Has Made Plenty of Progress” ↗I wrote my final year thesis on Lakatos’s book “Proofs and Refutations”[0]. It’s a short, mind-blowing book that answers the question: “Is mathematics discovered or invented?” You may think the abstract structures that you learn about are fixed fundamental truths, but just…
on “Imre Lakatos and the philosophy of bad science” ↗This is also the example used in Imre Lakatos fascinating essay 'Proofs and Refutations', about the nature of proof. https://math.berkeley.edu/~kpmann/Lakatos.pdf
on “Twenty Proofs of Euler's Formula: V-E+F=2” ↗…pan-universal. what is or isn't considered math, or more specifically what kinds of reasoning are considered convincing/sound/etc have evolved a lot over history, and is continuing to do so. lakatos' 'proofs and refutations' is a nice short introduction to some of the issues involved here. long story short, what it really means to say 'math is universal' is actually pretty hard to pin down precisely enough to make anything…
on “Is the Schrödinger Equation True?” ↗Whenever I want to think about "what does it mean to prove something?" I dig into my copy of Imre Lakatos's "Proofs and Refutations", a Socratic dialogue-style story where a bunch of aspiring theorem-provers try to prove a particular theorem about the Euler characteristic of polyhedra and discuss what they are actually doing and why…
on “Why Mathematical Proof Is a Social Compact” ↗…accepted proofs are wrong, only that we've drawn a few black balls from the urn of double-checking-established-proofs already, and so we should not assume that we won't find more. Imre Lakatos' essay Proofs and Refutations is a fun read on the subject. A simple deductive proof of 2+2=4 (say in Peano Arithmetic) is clear enough that I'd give it a very small chance of being wrong, easily less than 10^-9 but certainly not…
on “Against Bayesianism – David Deutsch” ↗For an excellent treatment of how doing mathematics is a subtle and dynamic interplay of the "Greek" and "Babylonian" approaches I recommend Imre Lakatos' "Proofs and Refutations" (http://books.google.ca/books?id=EjQqJT4Z-VoC&lpg=PP1&...). It's presented as a Socratic dialog and is simply fantastic.
on “Feynman the Babylonian” ↗…Philosophical Investigations, the second book of Wittgenstein, it's dense, and obscure to some extent, but it's really eye-opener, some ideas that influenced the last century are introduced there. - Proofs and Refutations, a posthumous work by Imre Lakatos, it defied much of my established view on how Mathematics works. - Thus Spoke Zarathustra, a powerful book, it's dense and you can only drag many conclusions from it…
on “Ask HN: Best Philosophical Books?” ↗