Customer Reviews

Proofs and Refutations: The Logic of Mathematical Discovery

5 star:		(9)
4 star:		(1)
3 star:		(0)
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1 star:		(1)

 

 

Definitions, examples, theorems, proofs -- they all seem so inevitable. But how did they come to be that way? What is the role of counterexamples? Why are some definitions so peculiar? What good are proofs?

In this brilliant and deep -- yet easy to read -- book, Lakatos shows how mathematicians explore concepts; how their ideas can develop over time; and how misleading the "textbook" presentation of math really is.

Fascinating for anyone who has seen mathematical proofs (even high-school Euclidean geometry) and essential for anyone studying mathematics at any level.

If you'd like to read more discussion about Lakatos and the intellectual context of P&R, you'll be interested in Brendan Larvor's "Lakatos: An Introduction".

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.

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.

Definitions, examples, theorems, proofs -- they all seem so inevitable. But how did they come to be that way? What is the role of counterexamples? Why are some definitions so peculiar? What good are proofs?

In this brilliant and deep -- yet easy to read -- book, Lakatos shows how mathematicians explore concepts; how their ideas can develop over time; and how misleading the "textbook" presentation of math really is.

Fascinating for anyone who has seen mathematical proofs (even high-school Euclidean geometry) and essential for anyone studying mathematics at any level.

As a lay reader of mathematics, I am prone to read for more for analogy and thought methods instead of, for example, the real implications of variations on Eulers Formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges.

Displaying solid content with artful execution, this book interested me in both the math of the thing and the acompanying thought processes.

Content: This book has near-poetic density and elegance in arguing a non-linear approach to mathematical development and, for me, to just plain thinking. Our tendency (as born worshippers of linearity and causality) is to discover a brick for the building then immediately look for the next to stack on top. Lakatos contends that PERHAPS you have discovered a brick worthy of the building, now let's see what truly objective tests we will put to this brick and before giving it a final stamp of approval. It seems obvious to say "always question", but the exercise in this book will take you through the process and show you what you may take for granted in this simple concept. For example, do you observe HOW you question? See his discussion throughout on global vs. local counterexamples, just as a start.

Execution of the text: This is the beautiful part. Mr. Lakatos has written this book as theater: characters with definite identities, plot, drama. The narrative flows in the voices of students and a professor who proves to be a sound moderator, intervening at timely points, i.e. those where questions may be crystallized or thoughts prodded to that point. This is where learning takes place, in a heated, moderated debate over Euler's formula. What was most interesting to me about this method was that it lent itself easily to isolating a particular thread of discussion. I literally chose certain characters to research from beginning to end in order to follow the evolution or confirmation of their thinking.

You emerge with a good framework that makes this book excellent reference material for problem-solving.

One last, but important note. This book will have you praising the lowly footnote. I would buy it for that alone. You will read along with the discussion, then get off and examine a footnote, and then pick the dialogue back up not having lost a step. On the contrary, Mr. Lakatos deepens your context with on-point explanations and math history.

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.

Very nice book if you are in high school or in college and would like to see how mathematics evolves. It makes a very pleasant reading although the mathematical ideas behind are not trivial.

It discusses polyhedra in 3 (or more) dimensions and Euler's formula that describes their numbers of vertices, edges, faces, e.t.c. The challenge is to determine what specific kinds of polyhedra satisfy the formula and conversely, how one could generalize the formula so as to describe more (if not all) polyhedra. Lots of historical references illustrate the fact that the discussion is not naive and that reflects the actual history of the subject.

One can realize through this book that math people are not Gods and do not produce theories out of nowhere, but they experiment with their objects like any other scientist, and then try to summarize in an elegant/rigorous way.

One should start right off admitting that this is a book about different approaches to what a mathematical proof is and should be. However, for some weird reason, it is hard to put down - more fun and fascinating than seems credible. It is dramatized, but still - in rather donnish style and all the characters (a teacher and a bunch of students) are named after letters of the Greek alphabet. In short, an anomoly - but one you must read.

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).

Probably the best philosophical book I have read. The book is both deep in its ideas, yet easy to read and comprehend