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Title: How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics
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Manufacturer: Princeton University Press
List Price: $35.00
Our Price: $21.99
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| How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics by Princeton University Press Extremely simpleminded | This is yet another naive rehash of the same old pop-math clichés. Since Byers knows nothing about mathematics beyond the meat-and-potatoes undergraduate curriculum, he has to use dishonest tricks to make these tired topics seem interesting. One time-tested trick is to claim that every single theorem is "surprising". For example, "we usually forget how surprising" it is that logic can be applied to geometry (p. 219). Only eight pages earlier we were surprised that the natural numbers have the same cardinality as the even numbers. And only four pages later we are surprised again, this time that there are infinitely many primes (p. 223). And so it goes. What a thrilling ride! Another underhand trick is to make completely unsubstantiated claims as to some mysterious metaphysical importance of every mathematical concept, e.g., "the notion of countable infinity captures something quite fundamental about the human nature and limits of what human beings can know" (p. 165).
Stylewise, Byers' amateur prose is made all the more unbearable by his obnoxious habit of putting at least four or five words per page in quotation marks for no apparent reason. We learn, for example, of a conjecture which "seems" true (sic, p. 281); elsewhere we study "infinity", "zero", and the notion of continuity, which means that f(x) gets "close" to f(a) when x gets "close" to a (sic, p. 239).
Byers' pathetic use of footnotes is a parody of scholarship. For example, the claim that "Poincaré call[ed] Cantorism a disease" (p. 286) is backed up by a footnote saying "Gardner (2001)" with no page reference. This is Gardner's "Colossal Book of Mathematics", obviously a deeply unscholarly source, and a colossal one at that. In this case, of course, Byers could not have provided the original reference because there is none. Poincaré never made this statement in print; and even if he did make it informally he was most likely referring to the paradoxes of set theory and not Cantor's transfinite numbers, as Byers would have known if he had not done his research at a high school library. | | How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics by Princeton University Press Bits of interesting mathematics mixed with unremarkable philosophy | I suspect that Prof. Byers is an excellent mathematics teacher and I very much enjoyed the snippets of mathematics in this book. However, most of the book was devoted to philosophy, which I found to be at least one of the following: (1) repetitive, (2) unoriginal, or (3) wrong. Repetitive for sure. Unoriginal in that he repeats many of the points made more eloquently and clearly by folks like Lakatos (though Byers does do a good job of giving credit where it is due). Wrong in the philosophy of mind, as in the section toward the end of the book where he tries to argue (a la Searle?) that machines can't think, and that computers might be able to write proofs but they can't do the inherently creative aspects of mathematics. It's very strange to me to run into a mathematician who holds these kinds of mystical views about minds, that they are not machines!
I feel like this would make a truly excellent 50 page book, with just a few of the key philosophical points clearly explained and illustrated with some of the excellent mathematical examples in this book. It could even be expanded -- but with more of Byers' mathematical illustrations, not his philosophical ramblings.
If I focused just on my favorite 50 pages of this book, it would get at least four stars; but the other 300 pages average it down to two. | | How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics by Princeton University Press Ambiguity and paradox as inspirations for mathematicians??? | Those of us who spent painful hours learning how to do "proofs" in geometry, or tried to keep in mind all the rules and procedures for solving polynomial expressions will probably exclaim "I Knew IT!" about a third of the way through the introduction. The author makes clear that he does not share that "Middle School" view of mathematics. In fact, it seems apparent that he considers that teaching approach responsible for the sorry state of mathematical knowledge in this society. Most of the book is an earnest attempt to "rescue" mathematics from the prevailing opinion that it is made up of well-defined processes and fully developed principles, with a list of known "problems" yet to be solved. The author makes clear that "doing math" is less like following blueprints and more like wandering in a garden, picking the prettiest flowers. As he makes his point, the non-mathematical reader will find insights into concepts and theories that were confusing, difficult, or just plain unknown.
Readers who found T.S.Kuhn's "The Structure of Scientific Revolutions interesting and thought-provoking will enjoy this book. Those who are more comfortable with a view of mathematics and mathematicians as ruled by logic and devoid of emotion, will be challenged and disconcerted. All readers will come away with a much better understanding of the current "state of the art" of mathematics. | | How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics by Princeton University Press interesting book on the process of doing mathematics | | The central thesis of How Mathematicians Think is that mathematics is more creative than algorithmic. The author describes mathematicians as dealing with ambiguity most of the time rather than simply adhering to a formula and plugging in numbers to get somewhere. The book is very interesting. The author covers some aspects of mathematics such as the concept of infinity and Cantor's work on the same. Although this book is about the creative process in mathematics don't expect to end up with a set of rules by which to create new mathematics. As there is no algorithmic approach to creativity in math there is no set of rules you can use to produce new math. I think this is as it should be, as no field as complex and profound as mathematics should devolve into a simple set of rules. I highly recommend this book to budding mathematicians and to lay people, like myself, who want a peek into the stuff of mathematical creativity. | | How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics by Princeton University Press Clear, Accessible Book on Philosophy of Mathematics | | I've been looking for a book like this for years. It presents major issues in the philosophy of mathematics (e.g., what is mathematical truth?) in a clear manner and takes an unconventional view towards many of the big questions (e.g., is proof the essence of math?). You do need to be comfortable with basic algebra and geometry to follow most of the arguments, but it never delves into anything more complicated than basic ideas on complex numbers or simple calculus. The ideas make you think about more basic questions of epistemology. It's not light reading but it's not dry or too technical either. | | How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics by Princeton University Press Product Description | To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself |
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