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Title: Proofs and Refutations : The Logic of Mathematical Discovery by Imre Lakatos, E. Zahar, J. Worrall ISBN: 0-521-29038-4 Publisher: Cambridge University Press Pub. Date: 01 January, 1976 Format: Paperback Volumes: 1 List Price(USD): $33.00 |
Average Customer Rating: 4.6 (10 reviews)
Rating: 5
Summary: nice reading for the general public
Comment: 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 Lots of historical references illustrate the fact that the discussion is not naive and that reflects the actual history of the subject.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.
Rating: 5
Summary: a study in mathematical thought
Comment: 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.
Rating: 5
Summary: Excellent Critical Reasoning Framework
Comment: 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.
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Title: For and Against Method: Including Lakatos's Lectures on Scientific Method and the Lakatos-Feyerabend Correspondence by Imre Lakatos, Paul K. Feyerabend, Matteo Motterlini ISBN: 0226467759 Publisher: University of Chicago Press Pub. Date: December, 2000 List Price(USD): $20.00 |
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Title: How to Solve It by G. Polya ISBN: 0691023565 Publisher: Princeton Univ Pr Pub. Date: 01 November, 1971 List Price(USD): $18.95 |
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Title: The Methodology of Scientific Research Programmes: Volume 1 : Philosophical Papers by Imre Lakatos, John Worrall, Gregory Currie ISBN: 0521280311 Publisher: Cambridge University Press Pub. Date: 16 October, 1980 List Price(USD): $42.00 |
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Title: What Is Mathematics?: An Elementary Approach to Ideas and Methods by Richard Courant, Herbert Robbins, Ian Stewart ISBN: 0195105192 Publisher: Oxford University Press Pub. Date: May, 1996 List Price(USD): $21.50 |
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Title: The Mathematical Experience by Phillip J. Davis, Reuben Hersh ISBN: 0395929687 Publisher: Houghton Mifflin Co Pub. Date: 14 January, 1999 List Price(USD): $19.00 |
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