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4.29 AVERAGE
4.29 AVERAGE
shoutout gillon for the rec. dense as ever but eye-opening fr. will return to this again
slow-paced
Counter-intuitive at first but extremely convincing. Styling the main essay as a socratic/platonic type dialogue helps the author prove his point while also making the read surprisingly more digestible. It's still a fairly dense book, with the appendices being more dense (and no less interesting) than the main essay.
The second appendix, on the heuristic method, was the most interesting section to me. I don't want to go into what level of knowledge is needed to appreciate this book (especially when that sort of thinking gets criticised at multiple points within it) but I will say that I think it requires an open mind and a willingness to think abstractly more than actual mathematical knowledge (which would however make things much easier). The appendices and Epsilon's final proof in the dialogue are the main parts which, if one doesn't know much at all of algebra and analysis, may leave someone having to take many concepts for granted and focusing just on the skeleton of the argument, which is possible but probably harder. Nonetheless, I think it's a great book for anyone interested in logic, maths and/or proof theory.
The second appendix, on the heuristic method, was the most interesting section to me. I don't want to go into what level of knowledge is needed to appreciate this book (especially when that sort of thinking gets criticised at multiple points within it) but I will say that I think it requires an open mind and a willingness to think abstractly more than actual mathematical knowledge (which would however make things much easier). The appendices and Epsilon's final proof in the dialogue are the main parts which, if one doesn't know much at all of algebra and analysis, may leave someone having to take many concepts for granted and focusing just on the skeleton of the argument, which is possible but probably harder. Nonetheless, I think it's a great book for anyone interested in logic, maths and/or proof theory.
Really neat exploration of what constitutes "proof" would love to see something similar for modern analysis, building up to some subtle idea (uniform convergence in function spacea?)
challenging
informative
reflective
fast-paced
For most of us, mathematics is to be avoided at all costs. It is challenging, unintuitive and mechanical. Yet, mathematics holds the seat of absolute “infallibility”: what can be mathematically demonstrated is unequivocally correct, the streams of formal logic hold steadfast the grounds of mathematical certainty. Lakatos recognizes that, in fact, mathematical activity—an instance of “human activity”—is not only beautifully creative and playfully ingenious, but also that our modern way of understanding mathematical truth casts an “authoritarian” shadow onto the discipline.
Through a conversation between a teacher and their students, Proofs and Refutations, studies the logical developments of the Euler conjecture regarding polyhedra: the sum of the number of vertices plus the number of faces equals the number of edges plus 2. At once one can verify this formula holds for the immediate polyhedra that come to our imagination: cubes, tetrahedrons, etc. However, one requires a “proof” to justify that the formula holds for all possible polyhedra! A series of proofs, counterexamples, monster-barrings, hidden lemmas, and discussions about truth and finality take place within this case-study, all in the voices of the students justifying their own “logics of mathematical discovery.” Lakatos, accompanies us on this discussion on the footnotes, where he gives us the relevant mathematical, historical and philosophical context that accompanies his overall argument.
As a mathematics student myself, I found myself re-examining my own mathematical faith, at which any point fell into despair—but rather it was reshaped.
Pre-college mathematics often leaves a bad taste, as most schools teach mathematics in a formulaic way: “this is a formula used for this certain situation and this is how you use it”. Seldom is the motivation for such formulas taught, and even more rarely is the real intuition behind concepts ever developed, and this leads to an static idea of mathematics as void of interesting content. Once a student begins to learn “proof based mathematics” a whole world of creativity opens up and the real oyster of mathematical discovery unveils its pearls. And nonetheless, when one sits in a class, a theorem is thrown on the board, followed by a proof that every student in the room has to have internalized later, as the class is moving onto some other proposition, and there isn’t enough time to examine the details during lecture. And yet, even if the proof is now clear, each of them wonders “how did someone ever came up with that idea?”.
Lakatos argues that this is a critical flaw of the modern formalist approach of mathematics: it takes out the problem from the solution, and it erases the struggle, critiques and ideas that came from a primitive conjecture that was slowly polished through the links between proofs, counterexamples and the analysis of arguments, which all together uncovers a new valid theorem. Lakatos proposes a heuristic: the method of proofs and refutations, the “logic of informal mathematics”. It is interesting that, at least in my experience, mathematical research resembles much more this heuristic, and yet we always patch it at the end to present it with a (formal) ribbon for publication.
Now, I don’t claim to be a smart person, and certainly a lot of Lakatos points flew over my head so this is mostly a surface level review, but this book was a challenging and rewarding read. The ways that Lakatos explains the mathematics is very approachable, at least for most of the book (basic linear algebra is a requirement for Chapter 2 and some basic analysis for the appendices), and the proofs he discusses are very beautiful pieces of mathematics. If anything the book is heavier on the philosophy side, and I think some familiarity with Hegel and Popper is a going to help—although absolutely not necessary, as I myself haven’t read them. It seems Lakatos also changed some of his ideas through time (the text was originally part of his PhD thesis), and unfortunately he passed away before he fully updated the manuscript for publication in this format. The editors—John Worrall and Elie Zahar, both friend of Lakatos—were very respectful of their friends work, and their insights on the evolution of Lakatos’s thoughts was very appreciated.
This is a book that I would recommend to mathematicians and mathematics students (and in fact I deem it of great importance that they should read it), for it challenges the way we present mathematics, and the way we think about the developments of our discipline. I think this is also a fantastic read for anyone interested in philosophy and logic, in particular with an eye towards methodology of science and discovery.
Through a conversation between a teacher and their students, Proofs and Refutations, studies the logical developments of the Euler conjecture regarding polyhedra: the sum of the number of vertices plus the number of faces equals the number of edges plus 2. At once one can verify this formula holds for the immediate polyhedra that come to our imagination: cubes, tetrahedrons, etc. However, one requires a “proof” to justify that the formula holds for all possible polyhedra! A series of proofs, counterexamples, monster-barrings, hidden lemmas, and discussions about truth and finality take place within this case-study, all in the voices of the students justifying their own “logics of mathematical discovery.” Lakatos, accompanies us on this discussion on the footnotes, where he gives us the relevant mathematical, historical and philosophical context that accompanies his overall argument.
As a mathematics student myself, I found myself re-examining my own mathematical faith, at which any point fell into despair—but rather it was reshaped.
Pre-college mathematics often leaves a bad taste, as most schools teach mathematics in a formulaic way: “this is a formula used for this certain situation and this is how you use it”. Seldom is the motivation for such formulas taught, and even more rarely is the real intuition behind concepts ever developed, and this leads to an static idea of mathematics as void of interesting content. Once a student begins to learn “proof based mathematics” a whole world of creativity opens up and the real oyster of mathematical discovery unveils its pearls. And nonetheless, when one sits in a class, a theorem is thrown on the board, followed by a proof that every student in the room has to have internalized later, as the class is moving onto some other proposition, and there isn’t enough time to examine the details during lecture. And yet, even if the proof is now clear, each of them wonders “how did someone ever came up with that idea?”.
Lakatos argues that this is a critical flaw of the modern formalist approach of mathematics: it takes out the problem from the solution, and it erases the struggle, critiques and ideas that came from a primitive conjecture that was slowly polished through the links between proofs, counterexamples and the analysis of arguments, which all together uncovers a new valid theorem. Lakatos proposes a heuristic: the method of proofs and refutations, the “logic of informal mathematics”. It is interesting that, at least in my experience, mathematical research resembles much more this heuristic, and yet we always patch it at the end to present it with a (formal) ribbon for publication.
Now, I don’t claim to be a smart person, and certainly a lot of Lakatos points flew over my head so this is mostly a surface level review, but this book was a challenging and rewarding read. The ways that Lakatos explains the mathematics is very approachable, at least for most of the book (basic linear algebra is a requirement for Chapter 2 and some basic analysis for the appendices), and the proofs he discusses are very beautiful pieces of mathematics. If anything the book is heavier on the philosophy side, and I think some familiarity with Hegel and Popper is a going to help—although absolutely not necessary, as I myself haven’t read them. It seems Lakatos also changed some of his ideas through time (the text was originally part of his PhD thesis), and unfortunately he passed away before he fully updated the manuscript for publication in this format. The editors—John Worrall and Elie Zahar, both friend of Lakatos—were very respectful of their friends work, and their insights on the evolution of Lakatos’s thoughts was very appreciated.
This is a book that I would recommend to mathematicians and mathematics students (and in fact I deem it of great importance that they should read it), for it challenges the way we present mathematics, and the way we think about the developments of our discipline. I think this is also a fantastic read for anyone interested in philosophy and logic, in particular with an eye towards methodology of science and discovery.
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.
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.
Charmigt format, med en pedagogisk uppbyggnad av hur ett bevis skapades inom proofs and refutations-metodiken. Den saknade viss analys tyckte jag dock, jag upplevde det som att författaren hade sin åsikt färdig innan han påbörjade boken.
Interesting. Although some of the math went over my head
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 proofs are either far too limited to be of any use, or else they invariable let in some "monsters". I can see my self re-reading this book in the future, but I would not recommend it to anyone in my social circle. I would recommend it to anyone with an interest in mathematics and philosophy.