How mathematicians think : using ambiguity, contradiction, and paradox to create mathematics / William Byers.

Format:

Book

Author/Creator:

Byers, William, 1943-

Language:

English

Subjects (All):

Mathematicians--Psychology.

Mathematicians.

Mathematics--Psychological aspects.

Mathematics.

Mathematics--Philosophy.

Physical Description:

1 online resource (vii, 415 pages) : illustration

Edition:

Course Book

Place of Publication:

Princeton, NJ : Princeton University Press, c2007.

Language Note:

English

Summary:

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.

Contents:

Frontmatter

Contents

Acknowledgments

INTRODUCTION. Turning on the Light

Section I. The Light of Ambiguity

Introduction

Chapter 1. Ambiguity in Mathematics

Chapter 2. The Contradictory in Mathematics

Chapter 3. Paradoxes and Mathematics: Infinity and the Real Numbers

Chapter 4. More Paradoxes of Infinity: Geometry, Cardinality, and Beyond

Section II. The Light as Idea

Chapter 5. The Idea as an Organizing Principle

Chapter 6. Ideas, Logic, and Paradox

Chapter 7. Great Ideas

Section III. The Light and the Eye of the Beholder

Chapter 8. The Truth of Mathematics

Chapter 9. Conclusion: Is Mathematics Algorithmic or Creative?

Notes

Bibliography

Index

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references (p. 399-405) and index.

ISBN:

9786612531453

OCLC:

979779657