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Reviews elsewhere on the web:
Gregory Chaitin
Math. Assoc. of America
American Math. Soc.
Doron Zeilberger

William Byers

How mathematicians think

Mathematics seems to be the epitome of rational thinking. You follow straightforward procedures and end up with an unassailable result. William Byers thinks otherwise. In How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics he argues that in fact the development of mathematics is full of doubt and controversy.

The book gives plenty of examples of results that didn't seem to make sense. There's the irrationality of √2, the invention of non-Euclidean geometry and the paradoxes of self-reference. Even the introduction of zero to the number system took some getting used to. And there are any number of weird ideas that come from thinking about infinity. Byers argues that these caused consternation when they were first seen, and can still cause confusion when encountered by students today. These problems weren't aren't much solved, rather the decision is made to accept a workable resolution, and this leads to the opening up of new areas of mathematics.

Towards the end of the book Byers moves away from actual examples and towards a more philosophical discussion. I felt that the book lost its way a bit at this stage. In particular, in the last chapter Byers argues that computers will never be able to do mathematics - but he starts off assuming this, so the chapter seems a bit pointless. However, you shouldn't let the later parts put you off. I would recommend the book both to those interested in mathematics, who want a peek at some of the stranger parts of the subject, and to those teaching mathematics (including those teaching themselves), who will see that getting over some of the stumbling blocks of the subject involved more than an appeal to logic.

Amazon.com info
Hardcover 424 pages  
ISBN: 0691127387
Salesrank: 1584570
Weight:1.55 lbs
Published: 2007 Princeton University Press
Amazon price $45.00
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Amazon.co.uk info
Hardcover 424 pages  
ISBN: 0691127387
Salesrank: 2227862
Weight:1.55 lbs
Published: 2007 Princeton University Press
Amazon price £30.95
Marketplace:New from £25.41:Used from £3.70
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Amazon.ca info
Hardcover 424 pages  
ISBN: 0691127387
Salesrank: 2003573
Weight:1.55 lbs
Published: 2007 Princeton University Press
Marketplace:New from CDN$ 12.58:Used from CDN$ 10.55
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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.