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Top positive review
3 people found this helpful
5.0 out of 5 starsIf you are looking for a book on practical mathematical foundations, this is the book for you.
ByWilliam Oliveron January 24, 2018
I have never written a review for a book on amazon before this point, but I felt this book really deserves 5 stars, and I wanted to address the criticisms of the book directly. I believe many of the criticisms are a result of a misunderstanding of this book's goals. Specifically, the comparison to Polya's books are somewhat unfounded, because I believe the books have different goals. The goal of Polya's books are to provide a general method for finding solutions to unsolved problems. The goal of this book, on the other hand, is to explain the purpose and meaning of a proof once we already have it.
This book answers the questions "How can we be sure a formal proof is correct?" and "How can we be sure it actually proves what we intuitively intended?", and it does so better than anything else I have ever read. As a result, this is a book more about mathematical philosophy than mathematical technique.
If you are someone who has trouble reading or writing proofs because you keep thinking of weird edge cases and have to verify that the proof handles all of them, or you have frequent existential crises about how written mathematical symbols (which are just symbols and syntax) can be shown to say anything about reality, this is the book for you.
This book answers the questions "How can we be sure a formal proof is correct?" and "How can we be sure it actually proves what we intuitively intended?", and it does so better than anything else I have ever read. As a result, this is a book more about mathematical philosophy than mathematical technique.
If you are someone who has trouble reading or writing proofs because you keep thinking of weird edge cases and have to verify that the proof handles all of them, or you have frequent existential crises about how written mathematical symbols (which are just symbols and syntax) can be shown to say anything about reality, this is the book for you.
Top critical review
10 people found this helpful
1.0 out of 5 starsUseless book
ByAbhion February 3, 2013
Simply drop it. Polya's Mathematics and plausible reasoning is the right book. I have been reading books to understand conjecture making. This one turned out to be useless.
To be clear, I am interested only thought process involved in problem solving. Many websites recommend this book to illuminate the reader as to how you can learn the skill of mathematical investigation. This book may be good for other reasons but to understand the process of mathematical discovery, this book is quite useless. If you are looking for something outside the realm of problem solving, please ignore this review.
Edit on 06/24/2013: I have read a lot of books on solving tough math problems. I am only rating this book relative to the rest of them that I have read. Also, I am rating this book based on how helpful it is for the subject of "Logic of Mathematical Discovery". I was misguided into thinking that he will actually help the reader to become a better "proof maker". He has attempted to walk the student through the thought process involved in proof making by providing a dialogue between students and the teacher. It was not useful at all. Polya, on the other hand, in his series of books on problem solving, speaks of ways to think in order to solve the problems. People say that his approach is that of heuristics and that won't work all the time. In his book called "Mathematical Discovery", he does speak of mental strategies to some extent but there is another book written by Paul Zeitz which is at a higher level and speaks of mental strategies used to solve very tough problems.
In summary, there are much better books that you can invest your time and money on. Many of the examples he has used are used by others as well for the purpose of illustrating a point. I think it came from Polya's work originally since his is the oldest of all these books. I highly recommend Polya's book over this book. Of course, there are more advanced books if you can digest Polya's books starting from "How to Solve it" and ending in "Mathematical Discovery Vol II".
To be clear, I am interested only thought process involved in problem solving. Many websites recommend this book to illuminate the reader as to how you can learn the skill of mathematical investigation. This book may be good for other reasons but to understand the process of mathematical discovery, this book is quite useless. If you are looking for something outside the realm of problem solving, please ignore this review.
Edit on 06/24/2013: I have read a lot of books on solving tough math problems. I am only rating this book relative to the rest of them that I have read. Also, I am rating this book based on how helpful it is for the subject of "Logic of Mathematical Discovery". I was misguided into thinking that he will actually help the reader to become a better "proof maker". He has attempted to walk the student through the thought process involved in proof making by providing a dialogue between students and the teacher. It was not useful at all. Polya, on the other hand, in his series of books on problem solving, speaks of ways to think in order to solve the problems. People say that his approach is that of heuristics and that won't work all the time. In his book called "Mathematical Discovery", he does speak of mental strategies to some extent but there is another book written by Paul Zeitz which is at a higher level and speaks of mental strategies used to solve very tough problems.
In summary, there are much better books that you can invest your time and money on. Many of the examples he has used are used by others as well for the purpose of illustrating a point. I think it came from Polya's work originally since his is the oldest of all these books. I highly recommend Polya's book over this book. Of course, there are more advanced books if you can digest Polya's books starting from "How to Solve it" and ending in "Mathematical Discovery Vol II".
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ByMichael Demkowiczon March 8, 2006
In a footnote to chapter 2 (much of the content of "Proofs and Refutations" is in the footnotes) Lakatos writes: "Until the seventeenth century, Euclidians approved the Platonic method of analysis as the method of heuristic; later they replaced it by the stroke of luck and/or genius." That stroke of luck and/or genius is a slight of hand that hides much of the story of the unfolding of mathematical research.
In "Proofs and Refutations," Lakatos illustrates how a single mathematical theorem developed from a naive conjecture to its present (far more sophisticated) form through a gruelling process of criticism by counterexamples and subsequent improvements. Lakatos manages to seemlessly narrate over a century of mathematical work by adopting a quasi-Platonic dialogue form (inspired by Galileo's "Dialogues"?), which he thoroughly backs up with hard historical evidence in the voluminous footnotes. The story he tells explores the clumsy and halting heuristic processes by which mathematical knowledge is created: the very process so carfully hidden from view in most mathematics textbooks!
The participants of Lakatos' dialogue argue over questions like "when is something proved?", "what is a trivial vs. severe counterexample?", "must you state all your assumptions or can some be thought of as implicit?", "in the end, what has been proved?",etc.. The answers to these questions change as the theorem under consideration is successively seen in a new light. Throughout, Lakatos is at pains to point out that the different perspectives adopted by his characters are representative of viewpoints that were once taken by the heroes of mathematics.
In "Proofs and Refutations," Lakatos illustrates how a single mathematical theorem developed from a naive conjecture to its present (far more sophisticated) form through a gruelling process of criticism by counterexamples and subsequent improvements. Lakatos manages to seemlessly narrate over a century of mathematical work by adopting a quasi-Platonic dialogue form (inspired by Galileo's "Dialogues"?), which he thoroughly backs up with hard historical evidence in the voluminous footnotes. The story he tells explores the clumsy and halting heuristic processes by which mathematical knowledge is created: the very process so carfully hidden from view in most mathematics textbooks!
The participants of Lakatos' dialogue argue over questions like "when is something proved?", "what is a trivial vs. severe counterexample?", "must you state all your assumptions or can some be thought of as implicit?", "in the end, what has been proved?",etc.. The answers to these questions change as the theorem under consideration is successively seen in a new light. Throughout, Lakatos is at pains to point out that the different perspectives adopted by his characters are representative of viewpoints that were once taken by the heroes of mathematics.
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ByTom Grayon March 18, 2001
I would recommend that anyone interested in mathermaics or indeed anyone interested in human activities read Imre Lakatos's seminal book 'Proofs and Refutations: The Logic of Mathematical Discovery'.
Lakatos direcctly makes the distinction between formal and informal mathematics. Formal mathematics is contained in the proofs published in mathematical journals. Informal mathematics are the strategies that working mathemeticians use to make their work a useful exercise in mathematical discovery.
The proof provided for the four colour theorm which was derved in the 1970's relied heavily on the sue of computers and brute force technqiues. It was extremely cotroversial not because it was invalid but because of the issues which Lakatos so clearly describes in this book.It was undoubtedly a valid formal proof. However it did nothing to advance the cause of mathematics beyond this.
The reason that Lakatos equates proofs and refutation in his title is his contention that it is the refutations that are developed that show mathematicians the deficiencies and indeed teh possibilites in their theories. A refutation does not necessarily discredit a theory. Instead it provides insights to the theory's limitations and possibiliites for future development. It is their attempts to deal with unwanted and unexpected refutations - to preserve a valuable theory in the face of imperfect axioms and proof methods - that teach mathemeticians the true depths of their conceptions and to point the way to new and deeper ones.
Lakatos shows this by an account of the historical development of the concept of proof in mathematics and by showing in historical detail how certain valuable 'proofs' were preserved in the face of refutation. To this point Lakatos shows that the 'proofs' of the truth of Euler's number are no proofs at all. The great mathemetician Euler noticed that for any regular polyhedron the formula V-E+F=2 holds where V is the number of vertexes, E is the number of edges and F is the number of faces. Euler's and his successors proofs fall before any number of counterexamples. Does this prove that the theorem is 'incorrect?' Or does it mean as mathemetician's actions show that they thought it meant was that their concept of what constituted a regular polyhedron was deficient. Lakatos shows how these conceptions were modified over a couple of hundred years as counterexample after counterexample were faced.
These counterexamples all made mathematics stronger by deepening the conception of what polyhedra really are and by discovering new classes of them. In the end Euler's formula turned out not to have a proof but to be in effect a tautology. It is true for the regular polyhedra for which it is true by the definition of what constitutes a polyhedron. It is true because human mathematicians in order to make progress need it to be true.
The computer proof of the four color theorem was a triumph of formal mathematics. Its critics complained and if interpreted according to what Lakatos wrote in this book, they complained because it defeated the progress of informal mathematics.
Mathematical proofs are useful tools. The tell us what we need to know. Formal mathematics is about finding them. Informal mathematics is about making them useful. Mathematics is not some Platonian ideal divorced from humanity, painting, poetry ... It is a human endeavor to meet human needs.
Lakatos direcctly makes the distinction between formal and informal mathematics. Formal mathematics is contained in the proofs published in mathematical journals. Informal mathematics are the strategies that working mathemeticians use to make their work a useful exercise in mathematical discovery.
The proof provided for the four colour theorm which was derved in the 1970's relied heavily on the sue of computers and brute force technqiues. It was extremely cotroversial not because it was invalid but because of the issues which Lakatos so clearly describes in this book.It was undoubtedly a valid formal proof. However it did nothing to advance the cause of mathematics beyond this.
The reason that Lakatos equates proofs and refutation in his title is his contention that it is the refutations that are developed that show mathematicians the deficiencies and indeed teh possibilites in their theories. A refutation does not necessarily discredit a theory. Instead it provides insights to the theory's limitations and possibiliites for future development. It is their attempts to deal with unwanted and unexpected refutations - to preserve a valuable theory in the face of imperfect axioms and proof methods - that teach mathemeticians the true depths of their conceptions and to point the way to new and deeper ones.
Lakatos shows this by an account of the historical development of the concept of proof in mathematics and by showing in historical detail how certain valuable 'proofs' were preserved in the face of refutation. To this point Lakatos shows that the 'proofs' of the truth of Euler's number are no proofs at all. The great mathemetician Euler noticed that for any regular polyhedron the formula V-E+F=2 holds where V is the number of vertexes, E is the number of edges and F is the number of faces. Euler's and his successors proofs fall before any number of counterexamples. Does this prove that the theorem is 'incorrect?' Or does it mean as mathemetician's actions show that they thought it meant was that their concept of what constituted a regular polyhedron was deficient. Lakatos shows how these conceptions were modified over a couple of hundred years as counterexample after counterexample were faced.
These counterexamples all made mathematics stronger by deepening the conception of what polyhedra really are and by discovering new classes of them. In the end Euler's formula turned out not to have a proof but to be in effect a tautology. It is true for the regular polyhedra for which it is true by the definition of what constitutes a polyhedron. It is true because human mathematicians in order to make progress need it to be true.
The computer proof of the four color theorem was a triumph of formal mathematics. Its critics complained and if interpreted according to what Lakatos wrote in this book, they complained because it defeated the progress of informal mathematics.
Mathematical proofs are useful tools. The tell us what we need to know. Formal mathematics is about finding them. Informal mathematics is about making them useful. Mathematics is not some Platonian ideal divorced from humanity, painting, poetry ... It is a human endeavor to meet human needs.
ByVin de Silvaon October 2, 2001
I want to add a few words to the brief comment by the reader in Monroe (who gave this book one star). I tend to agree that "Proofs and Refutations" isn't a primer in mathematical proof-writing; it's certainly not a textbook for beginning mathematicians wanting to know how to practice their craft.
However, for those readers (including beginning mathematicians) who are interested in the broader picture, who are interested in the nature of mathematical proof, then Lakatos is essential reading. The examples chosen are vivid, and there is a rich sense of historical context. The dramatised setting (with Teacher and students Alpha, Beta, Gamma, etc) is handled skilfully. Now and then, a foolish-seeming comment from one of the students has a footnote tagged to it; more often than not, that student is standing in for Euler, Cauchy, Poincare or some other great mathematician from a past era, closely paraphrasing actual remarks made by them. That in some ways is the most important lesson I learned from this book; "obvious" now doesn't mean obvious then, even to the greatest intellects of the time.
Although "Proofs and Refuatations" is an easy book to begin reading, it is not an easy book per se. I have returned to it repeatedly over the last ten years, and I always learn something new. The text matures with the reader.
However, for those readers (including beginning mathematicians) who are interested in the broader picture, who are interested in the nature of mathematical proof, then Lakatos is essential reading. The examples chosen are vivid, and there is a rich sense of historical context. The dramatised setting (with Teacher and students Alpha, Beta, Gamma, etc) is handled skilfully. Now and then, a foolish-seeming comment from one of the students has a footnote tagged to it; more often than not, that student is standing in for Euler, Cauchy, Poincare or some other great mathematician from a past era, closely paraphrasing actual remarks made by them. That in some ways is the most important lesson I learned from this book; "obvious" now doesn't mean obvious then, even to the greatest intellects of the time.
Although "Proofs and Refuatations" is an easy book to begin reading, it is not an easy book per se. I have returned to it repeatedly over the last ten years, and I always learn something new. The text matures with the reader.
ByViktor Blasjoon February 8, 2008
Lakatos' motives for writing this book seem to have been:
(a) "Under the present dominance of formalism, ... the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty." (p. 2)
(b) "present mathematical and scientific education is a hotbed of authoritarianism and is the worst enemy of independent and critical thought" (pp. 142-143)
I passionately agree, but still found the actual book quite bland. It consists in a fairly amusing, semi-historical dialogue on Euler's formula V-E+F=2, intended to illustrate the very trivial thesis that creative mathematics is based on informal reasoning, heuristics, conjectures, counterexamples, etc., while also noting some general patterns of thought within this framework. Illustrations of similar patters in the history of the foundations of the calculus are also pointed out briefly; e.g., "the exception-barring method," is exemplified by Abel's reaction to his discovery of counterexamples such as sin(x)+sin(2x)/2+sin(3x)/3+... to Cauchy's theorem that the limit function of a convergent series of continuous functions is always continuous: "His response to these counterexamples is to start guessing: 'What is the safe domain of Cauchy's theorem?' ... All the known exceptions to this basic continuity principle were trigonometrical series so he proposed to withdraw analysis to within the safe boundaries of power series, thus leaving behind Fourier's cherished trigonometrical series as an uncontrollable jungle." (p. 133).
(a) "Under the present dominance of formalism, ... the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty." (p. 2)
(b) "present mathematical and scientific education is a hotbed of authoritarianism and is the worst enemy of independent and critical thought" (pp. 142-143)
I passionately agree, but still found the actual book quite bland. It consists in a fairly amusing, semi-historical dialogue on Euler's formula V-E+F=2, intended to illustrate the very trivial thesis that creative mathematics is based on informal reasoning, heuristics, conjectures, counterexamples, etc., while also noting some general patterns of thought within this framework. Illustrations of similar patters in the history of the foundations of the calculus are also pointed out briefly; e.g., "the exception-barring method," is exemplified by Abel's reaction to his discovery of counterexamples such as sin(x)+sin(2x)/2+sin(3x)/3+... to Cauchy's theorem that the limit function of a convergent series of continuous functions is always continuous: "His response to these counterexamples is to start guessing: 'What is the safe domain of Cauchy's theorem?' ... All the known exceptions to this basic continuity principle were trigonometrical series so he proposed to withdraw analysis to within the safe boundaries of power series, thus leaving behind Fourier's cherished trigonometrical series as an uncontrollable jungle." (p. 133).
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