How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics (Hardcover)

by William Byers (Author) "This chapter begins the process of developing a new way of describing what mathematics is and what mathematicians do..." (more)
Key Phrases: creative certainty, objective subjectivity, rational number system, David Hilbert, North Pole, Roger Penrose (more...)

Editorial Reviews

Review
"Byers isn't content merely to rescue maths. His goals are even higher: he dares to oppose the entire zeitgeist." -- Gregory Chaitin, New Scientist

"Byers writes with verve and clarity about deep and difficult mathematical and philosophical issues." -- Andrew Robinson, Physics World

"In his view, thinking about math requires creativity and the use of non-logical forms of thought." -- Siobhan Roberts, The Globe and Mail

Review
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 )

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Product Details

• Hardcover: 425 pages
• Publisher: Princeton University Press; 1 edition (May 7, 2007)
• Language: English
• ISBN-10: 0691127387
• ISBN-13: 978-0691127385
• Product Dimensions: 9.2 x 6.3 x 1.5 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars See all reviews (9 customer reviews)
• Amazon.com Sales Rank: #555,895 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

7 of 8 people found the following review helpful:

5.0 out of 5 stars Clear, Accessible Book on Philosophy of Mathematics, October 27, 2007

By	J. Herman (Keene, NY) - See all my reviews (REAL NAME)

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.

8 of 10 people found the following review helpful:

5.0 out of 5 stars Ubiquity of Ambiguity, September 23, 2007

By	Roger E. Frye (Santa Fe, NM USA) - See all my reviews (REAL NAME)

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!

2 of 2 people found the following review helpful:

4.0 out of 5 stars Ambiguity and paradox as inspirations for mathematicians???, March 6, 2008

By	Raczek's Roughnecks (Milwaukee, WI USA) - See all my reviews

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.