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Proofs and Refutations: The Logic of Mathematical Discovery
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Proofs and Refutations: The Logic of Mathematical Discovery

4.25  ·  Rating details ·  640 ratings  ·  34 reviews
A novel introduction to the philosophy of mathematics, mostly in the form of a discussion between a group of students and their teacher. It combats the positivist picture and develops a much richer, more dramatic progression.
Paperback, 188 pages
Published January 1st 1976 by Cambridge University Press
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4.25  · 
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 ·  640 ratings  ·  34 reviews

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Ben Labe
Aug 11, 2011 rated it it was amazing
Despite playing such a major role in philosophy's formal genesis, the dialogue has often presented a challenge to contemporary philosophers. Many are apt to shy away from it due to its apparent levity and lack of rigor. However, the dialogue possesses significant didactic and autotelic advantages. At its best, it can reveal without effort the dialectic manner in which knowledge and disciplines develop. This way, the reader has a chance to experience the process.

"Proofs and Refutations" is a par
Michael Nielsen
Mar 08, 2019 rated it it was amazing
Shelves: favorites
Radically changed my idea of what mathematical definitions and proofs are, and where they come from. In particular, Lakatos convincingly refutes the idea that definitions come before theorems and proofs (as often seems the case). Rather, they arise out of repeated back-and-forth interplay between conjectures and proof-ideas.

That's a pretty abstract- and weird-sounding review. The book itself is incredibly readable, incredibly fun, and by the end will (if you're anything like me) have caused an e
Jul 09, 2013 rated it it was amazing
Shelves: non-fiction
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. If something is mathematically proven we know beyond any shadow of a doubt that it is true because it follows from elementary axioms. Lakatos argues that this view misses quite a lot of how mathematical ideas historically have emerged. His main argument takes the form of a dialogue between a number of students and a te ...more
Jul 16, 2012 rated it really liked it
Surprisingly interesting, like Wittgenstein if he wrote in a human fashion, and longer than one would think possible given how straightforward the problem initially appears.
Mar 24, 2007 rated it it was amazing
By far one of the best philosophical texts I've read. It takes a theory about the sides of a polyhedron by Euler and uses dialogue form to show how the methods of inquiry of a handful of different theoreticians fall apart when attempting to prove or disprove the proposition. 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 ...more
Jun 13, 2016 rated it it was amazing
This is an excellent, though very difficult, read. It reminds me of Ernest Mach's "Science of Mechanics"--the latter is not in the form of a dialogue.

Having heard Lakatos speak I can see how the book's dialogue format fits in with his style which is to the point and voluble. He makes you think about the nature of proof, kind of along the lines of the great Morris Kline--still an occasional presence during my graduate school days at New York University--and who's wonderful book, "Mathematics and
Many of you, I'm guessing, have some math problems. 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.

What Lakatos shows you is that math is not the rigid formalistic system you may conceive of, but something far more fluid, something prone to frequent revision, something that must always have its underpinnings challenged in order to reach mathematical t
Jan 07, 2013 rated it really liked it
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 book. Portions of Proofs and Refutations were required reading for one of my classes for my master's degree, but I liked it enough that I finished it after the course was completed. I really enjoyed wrestling with the idea that "proofs" can not be the perfect ideal that mathematics and mathematicians should strive for. Lakatos argues that proof ...more
May 29, 2013 rated it it was amazing
Math as evolving social construct. Truth itself evolves. And Lakatos knows the history of eulers theorem, presents it as a classroom discussion making us realize that nothing is ever static in mathematics.
Vasil Kolev
Jul 08, 2018 rated it it was amazing
Shelves: science, math
I would have to reread this some day. This book describes a lot of what I found missing while studying mathematics in the university, mainly the reasoning for the way proofs were, and the overall reasoning for the definitions and terms used. The book looks into those from the purely mathematical standpoint, and shows that they can be a lot easier to grasp and understand.

(the other part, the actual usage of most of it, doesn't seem to exist in a single book, but in bits and pieces in the actual a
Jul 14, 2015 rated it it was amazing
To quote Northrop Frye, we go see MacBeth to learn what it feels like for a man to gain a kingdom but lose his soul. We develop mathematical definitions, examples, theorems, and proofs to meet human needs through heuristics. We assume, incorrectly that mathematics are solid continents of rules and facts, but what we observe are loosely connected archipelagos of calibrated and stable forms where those islands are in constant risk of being retaken by the sea. Proof and refutations is set as a dial ...more
Keith Peters
Jan 21, 2019 rated it really liked it
Shelves: kindle, science
I got into an argument recently with a PhD about the I infallibility of math. My argument came down a bit on the human endeavor at the heart of math. This book is a bit awkward and suffers from the hubris I see in basically all philosophy texts but covers my concerns well.
Jun 23, 2014 rated it liked it
Recommends it for: mathematicians
A book about the meaning and philosophy of mathematical proofs.

The most important lesson from this book is the idea of proof-based theorems. That is, one should look at one's proof, and pin down exactly what properties are used, and then based on that thorough examination, state one's theorem accordingly. 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.

To create the most apt theorem
Eryk Banatt
Jul 14, 2018 rated it liked it
I picked this up seeing it on a list of Robb Seaton's favorite books". I think I can describe it as "Plato's The Republic meets Philosophy meets History of Mathematics" and that sentence can more or less describe the entirety of the book.

I will admit that the book was a bit challenging for me, and I suspect I will revisit this book when I get a bit better at math, but for what it was I think it was quite readable and I enjoyed it. It was a little dry at times but the dialogue was very interesti
Aleks Veselovsky
Jan 28, 2012 rated it liked it
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. Nevertheless, I can name a few lessons learned. I think that the use of counterexamples is underutilized in the classroom and Lakatos shows how useful it can be. 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 ...more
Nick Black
Apr 15, 2009 marked it as to-read
Recommended to Nick by: Conrad
Amazon third-party 2009-04-15. I'm excited about this one, riding in as it does on a ringing recommendation of Conrad's (although I'm a bit puzzled by his tagging of House of Leaves with "masterpieces"). Looks to contain echoes of Halmos's Automathography and Davis's The Mathematical Experience; we'll see.
Kelly John Rose
Jun 30, 2013 rated it it was amazing
Probably one of the most important books I've read in my mathematics career. 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 interesting theory (and possibly) field of mathematics one step at a time.

If y
Apr 02, 2015 rated it it was amazing
The very idea of mathematical truth and the changing notions of rigour and proof are all discussed with stunning clarity. 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.
Aug 25, 2014 rated it liked it  ·  review of another edition
This deserves a higher rating, but the math was beyond my meager understanding so I struggled a bit. The philosophy was good though. Science and math make progress by conjectures leading to proofs which are refuted with counterexamples. Then the conjectures can be modified and tightened up to make theories. Written in Socratic dialogue.
Robb Seaton
Begins strong with a deconstruction of the Euler characteristic, but soon gets bogged down in philosophy, along with a troubling amount of relativism, although I'm not entirely clear about what Lakatos intends when he writes about truth, certainty, and progress.

Mar 18, 2013 rated it it was amazing
Shelves: math, philosophy
This book is a wonderful blend of philosophy, history, pedagogy, and interesting mathematics. Worth multiple readings.
Philip Naw
Jan 01, 2018 rated it it was amazing
I would like to give this book a 4.5/5 but I rounded up and gave it a 5/5.

The good:
I have a background in set theory or axiomatics, and so the material in this book initially appeared quite shocking to me. 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. The book is profoundly deep, in a philosophic
Arthur Ryman
Nov 24, 2018 rated it it was amazing
This is a frequently cited work in the philosophy of mathematics. The gist of it is that non-obvious mathematical concepts and definitions emerge through the process of refuting proposed proofs by exhibiting counter-examples. The counter-examples are then analyzed and new concepts are identified. The two main examples given are 1) the definition of a polyhedron in the context of Euler's formula V-E+F=2, and 2) the definition of uniform convergence in the context of Fourier series. Both of these ...more
Mar 26, 2019 rated it it was amazing
An entertaining read about how proofs are formed, broken down, and reformed by way of Socratic dialogue. The second section involving the vector algebraic proof of Epsilon was harder for me to follow, but the insights about mathematical thinking (and what it means to prove something with any certainty) are sound. I expect I will have to come back and reread this one someday.
Apr 06, 2019 rated it really liked it
Fantastic - such depth of research and knowledge into the ins and outs of the history of just one theorem. And making it a dialogue among students was brilliant - so much more fun and interesting than it would have been interesting. And very valuable to learn something realistic about how math (or any similar endeavor) actually works. I would love more like this.
John Lepp
Jan 12, 2019 rated it really liked it
Honestly, I prefer it to MSRP.
Mugizi Rwebangira
Mar 31, 2019 rated it really liked it
I should probably read this again, I am pretty sure I didn't totally get it.
Sep 22, 2017 rated it it was amazing  ·  review of another edition
Ci sono quegli autori di cui vorresti disperatamente leggere qualcosa perché ti intrigano i concetti che intendono esprimere e di cui però non riesci a trovare alcuna opera: è il caso di Imre Lakatos, pilastro della filosofia della matematica e della scienza del secolo scorso la cui fama ha probabilmente risentito del recente scritto di Januaria Piromallo, la quale ne mette in luce l'oscuro passato. Tuttavia, "il filosofo bastardo" quale è stato definito dalla stessa Piromallo in un commento al ...more
Jul 15, 2017 rated it really liked it
Shelves: philosophy
"We might be more interested in this proposition if we really understood just why the Riemann – Stieltjes integrable functions are so important. [Walter] Rudin does not even mention the intuitively most obvious concept of integrability, namely Cauchy-integrability, criticism of which led to Riemann-integrability. So now we have got a theorem in which two mystical concepts, bounded variation and Riemann-integrability, occur. But two mysteries do not add up to understanding. Or perhaps they do for ...more
May 12, 2012 rated it really liked it
Lovely and thought-provoking. The book is structured as a philosophical dialogue. Whenever one of the characters says something flowery and absurd, there's a little footnote to something almost identical said by Poincare or Dedekind or some other prominent mathematician.
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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 the 'research programme' in his methodology of scientific research programmes.
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“Женщина с ребенком во чреве не может быть контрапримером для тезиса, что люди имеют одну голову.” 4 likes
“Гамма. Я думаю, что если мы хотим изучить что-нибудь действительно глубоко, то нам нужно исследовать это не в его «нормальном», правильном, обычном виде, но в его критическом положении, в лихорадке и страсти. Если вы хотите узнать нормальное здоровое тело, то изучайте его, когда оно в ненормальном положении, когда оно болеет. Если вы хотите знать функции, то изучайте их странности. Если вы хотите познать обычные многогранники, то изучайте их причудливые обрамления. Вот только так можно внести математический анализ в самое сердце вещей. Но если даже в основе вы правы, разве вы не видите бесплодия вашего метода ad hoc? Если вы хотите провести пограничную линию между контрапримерами и монстрами, то этого нельзя сделать в припадках и срывах.” 2 likes
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