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

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

by William Byers
     
 

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ISBN-10: 0691127387

ISBN-13: 9780691127385

Pub. Date: 05/07/2007

Publisher: Princeton University Press

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

Overview

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.

Product Details

ISBN-13:
9780691127385
Publisher:
Princeton University Press
Publication date:
05/07/2007
Edition description:
New Edition
Pages:
424
Product dimensions:
9.20(w) x 6.40(h) x 1.50(d)

Table of Contents


Acknowledgments     vii
Introduction: Turning on the Light     1
The Light of Ambiguity     21
Ambiguity in Mathematics     25
The Contradictory in Mathematics     80
Paradoxes and Mathematics: Infinity and the Real Numbers     110
More Paradoxes of Infinity: Geometry, Cardinality, and Beyond     146
The Light as Idea     189
The Idea as an Organizing Principle     193
Ideas, Logic, and Paradox     253
Great Ideas     284
The Light and the Eye of the Beholder     323
The Truth of Mathematics     327
Conclusion: Is Mathematics Algorithmic or Creative?     368
Notes     389
Bibliography     399
Index     407

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