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How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics Paperback – May 2, 2010

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Editorial Reviews


Winner of the 2007 Best Sci-Tech Books in Mathematics, Library Journal

One of Choice's Outstanding Academic Titles for 2007

"Ambitious, accessible and provocative...[In] How Mathematicians Think, William Byers argues that the core ingredients of mathematics are not numbers, structure, patterns or proofs, but ideas...Byers' view springs from the various facets of his career as a researcher and administrator (and, he says, his interest in Zen Buddhism). But it is his experience as a teacher that gives the book some of its extraordinary salience and authority...Good mathematics teaching should not banish ambiguity, but enable students to master it...Everyone should read Byers...His lively and important book establishes a framework and vocabulary to discuss doing, learning, and teaching mathematics, and why it matters."--Donal O'Shea, Nature

"From Byers's book, if you work at it, you will learn some mathematics and, more important, you may begin to see how mathematicians think."--Peter Cameron, Times Higher Education Supplement

"As William Byers points out in this courageous book, mathematics today is obsessed with rigor, and this actually suppresses creativity.... Perfectly formalized ideas are dead, while ambiguous, paradoxical ideas are pregnant with possibilities and lead us in new directions: they guide us to new viewpoints, new truths.... Bravo, Professor Byers, and my compliments to Princeton University Press for publishing this book."--Gregory Chaitin, New Scientist

"Many people assume that mathematicians' thinking processes are strictly methodical and algorithmic. Integrating his experience as a mathematician and as a Buddhist, Byers examines the validity of this assumption. Much of mathematical thought is based on intuition and is in fact outside the realm of black-and-white logic, he asserts. Byers introduces and defines terms such as mathematical ambiguity, contradiction, and paradox and demonstrates how creative ideas emerge out of them. He gives as examples some of the seminal ideas that arose in this manner, such as the resolution of the most famous mathematical problem of all time, the Fermat conjecture. Next, he takes a philosophical look at mathematics, pondering the ambiguity that he believes lies at its heart. Finally, he asks whether the computer accurately models how math is performed. The author provides a concept-laden look at the human face of mathematics."--Science News

"This book is a radically new account of mathematical discourse and mathematical thinking...What Byers's book reveals is that ambiguity is always present...You can't quite say that nobody has said this before. But nobody has said it before in this all-encompassing, coherent way, and in this readable, crystal clear style...This book strikes me as profound, unpretentious, and courageous."--Reuben Hersh, Notices of the AMS

"This is a truly exceptional work. In an almost gripping tour de force, Byers examines the creative impulse of mathematics, which to him is the notion of ambiguity, understood to 'involve a single idea that is perceived in two self-consistent but mutually incompatible frames of reference'...[I]t is a sorely needed complement to often-formulaic textbooks.... An incredible book."--J. Mayer, Choice

"William Byers...has written a passionate defense of the uniquely human aspect of mathematics...Byers [demonstrates] that the insights of mathematicians come about through a discipline that...has something in common with Zen practice. First, there is a positive use of difficulty: 'the paradox has the enormous value of highlighting a fertile area of thought.' Then the breakthrough: 'An idea emerges in response to the tension that results from the conflict inherent in ambiguity.' These sentences from Byers's book apply equally to scientific and spiritual work."--Eliot Fintushel, Tricycle

"After a lifetime of research and teaching, [Byers argues] that mathematical breakthroughs do not come from simply manipulating symbols according to strict rules. Byers writes with verve and clarity about deep and difficult mathematical and philosophical issues such as the relationship between great mathematical ideas and cultural crises. Byers discusses in depth some examples of great ideas and crises...and explains why he is dead against seeing the mind as a computer."--Andrew Robinson, Physics World

"It is a pleasure to read [Byers'] well written, carefully referenced, and clearly illustrated arguments. Byers describes what 'doing math is: a process characterized by the complementary poles of proof and idea, of ambiguity and logic.' Byers' book has given me a greater appreciation for mathematics. I recommend it to anyone interested in, and open-minded about, the attempt to define mathematics."--Lee Kennard, Math Horizons

"Byers subverts the widely held notion that mathematicians are a form of computer, or robotic followers of unbending rules. In his view, thinking about math requires creativity and the use of non-logical forms of thought. Thus the ambiguity, paradox and contradiction of the subtitle."--The Globe and Mail

"Well-organized and carefully written the present book is very useful to all who are interested in How Mathematicians Think!"--Ioan A. Rus, Mathematica

"[A] brilliant and easily accessible book on the creative foundations of math and psychology."--Ernest Rossi, Psychological Perspectives

"What does one like to learn when one reads a book? Because the reading of a book is a union between its text and the reader's consciousness, one answer is the wedding custom of 'something old, something new, something borrowed, something blue'. All are there in this book. . . . It is a useful book for the apprentice mathematician by clarifying the importance of boldness in making mistakes and declaring that one does not fully understand some technical details which at first sight appear to be more complex than they really are."--Bob Anderssen, Australian Mathematical Society Gazette

"Excellent discussions are presented."--EMS Newsletter

"[Byers'] book helps us not to eliminate the myths surrounding mathematics and mathematicians, but to master them."--David Cohen, European Legacy

"The author is a mathematician, and he plainly knows what he is talking about. In my opinion he has done a good job of getting it across. . . . The book has a lot of worthwhile material to recommend."--Robert Thomas, Philosophia Mathematica

"Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself."--World Book Industry

From the Inside Flap

"An amazing tour de force. Utterly new, utterly truthful."--Reuben Hersh, author of What Is Mathematics, Really?

"Byers gives a compelling presentation of mathematical thinking where ambiguity, contradiction, and paradox, rather than being eliminated, play a central creative role."--David Ruelle, author of Chance and Chaos

"This is an important book, one that should cause an epoch-making change in the way we think about mathematics. While mathematics is often presented as an immutable, absolute science in which theorems can be proved for all time in a platonic sense, here we see the creative, human aspect of mathematics and its paradoxes and conflicts. This has all the hallmarks of a must-read book."--David Tall, coauthor of Algebraic Number Theory and Fermat's Last Theorem

"I strongly recommend this book. The discussions of mathematical ambiguity, contradiction, and paradox are excellent. In addition to mathematics, the book draws on other sciences, as well as philosophy, literature, and history. The historical discussions are particularly interesting and are woven into the mathematics."--Joseph Auslander, Professor Emeritus, University of Maryland

--This text refers to an out of print or unavailable edition of this title.

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Product Details

  • Paperback: 424 pages
  • Publisher: Princeton University Press; Sixth edition (May 2, 2010)
  • Language: English
  • ISBN-10: 0691145997
  • ISBN-13: 978-0691145990
  • Product Dimensions: 9.1 x 6 x 1.1 inches
  • Shipping Weight: 1.4 pounds (View shipping rates and policies)
  • Average Customer Review: 3.9 out of 5 stars  See all reviews (18 customer reviews)
  • Amazon Best Sellers Rank: #1,180,601 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

16 of 17 people found the following review helpful By J. Herman on October 27, 2007
Format: Hardcover
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.
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17 of 21 people found the following review helpful By RogerFrye on September 23, 2007
Format: Hardcover
Byers demonstrates the ubiquity of ambiguity, rather than of absolute certainty, in mathematics. It is easy to dismiss the contradiction in 0 (the nothing that is), because we have become so familiar with it. More people have trouble equating the infinite process indicated by 0.99999... with the integer captured in the symbol 1.

Who could be confused about 'x + 2 = 5'? Students will be confused until they have absorbed the strange idea that before you solve the equation, 'x' represents any number, but afterwards only 3. Where is the difficulty in proving that the angles of a triangle add up to 2 right angles? Once you get the ideas to focus on one vertex and extend a side and draw a parallel, it becomes straight-forward to match up the angles.

Byers structures his book around Andrew Wiles' metaphor of turning on the lights in unexplored rooms of a mansion for the long process of disproving Fermat's conjecture. In the introduction, he says "This book is written in the conviction that we need to talk about mathematics in a way that has a place for the darkness as well as the light and, especially, a place for the mysterious process whereby the light switch gets turned on." Exactly so, and well done!
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4 of 4 people found the following review helpful By Michael George on October 1, 2011
Format: Paperback Verified Purchase
There is a vastness about mathematics that is daunting: Many works are identified as mathematical; mathematics has a long and colorful history, extending back to the dawn of civilization (and with "primitive" concepts that extend even into primate behavior and the behavior of other animals); and there are many people today, all over the world, who identify themselves as mathematicians, including teachers and university professors who dedicate their lives to mathematics. Thus, the task of considering "how mathematicians think" is a huge one.

It is little wonder, that in faced with this task, Prof. Byers focuses on ambiguity, contradictions and paradox. However, since the common perception of mathematics (apart from the mathematicians) is as a field of formal purity and certainty, such a viewpoint seems to present us with an unusual view of mathematics.

The central feature of Prof. Byers account is "duality", i.e. that ambiguity, for example, which he takes to mean seeing mathematics and mathematical ideas from multiple perspectives, cannot be separated from the "certainty" of mathematics. He insists that the dualities are inherent in mathematics, and in particular that we cannot just make a clean separation between the objective and the subjective. He sees this view as not only having consequences with respect to the way in which mathematician think, qua mathematicians, but also with respect to appreciating the history of mathematics, and in how we teach and study mathematics.

Mathematicians were challenged to an extraordinary degree in the twentieth century. The work in metamathematics, especially by Godel, raised the issue of the limitations of mathematics.
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4 of 4 people found the following review helpful By Wu Bing on August 19, 2008
Format: Hardcover
I would classify this book as a Mathematical Philosophy Book. The author definitely places Philosophy more than the hard-core Mathematics, so don't be disappointed if the reader's main goal is on Math.
Overall this book is a great book, but definitely not for the weaker math students. It brings you to the higher platform to look down on Math issues in a pensive way - ie "Switch on the light" à la Andrew Wiles.

This book should be best read not in sequential manner, because of the writing style of the author which is quite verbose.

Some chapters are very well written:
Chap 8: (Pg 363) Obstacles to Learning Mathematics : Many great ideas and truths are hidden behind the math theoretical structures, unfortunately in the university math profs emphasize more on structures and leave the poor students to find out the 'beauty' of truth themselves - because 'Beauty' is not tested in Exams :(
Chapter 4: Paradoxes of Infinity. The "Cantor Set" Construction example is very refreshing.
Chapter 5: on "Quotient Space" (X/R) is excellent.
I also like the Isomorphism ideas (Pg 216-217): as Isometry (in Geometry), homeomorphism (in Topology), besides various isomorphism in Groups, Rings, Fields...

In summary, this book will elevate your math philosophical thinking like a Mathematician.
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5 of 6 people found the following review helpful By Raczek's Roughnecks on March 6, 2008
Format: Hardcover
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.
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