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# Proofs and Refutations: The Logic of Mathematical Discovery

by
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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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"Proofs and Refutations" is a par ...more

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 ...more

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 ...more

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 ...more

*for a mathematics book*...more

*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*(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 ...more

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 ...more

*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 ...more

*Automathography*and Davis's The Mathematical Experience; we'll see.

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 ...more

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 ...more

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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.”
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“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.”
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