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.
William Byers is professor of mathematics at Concordia University in Montreal. He has published widely in mathematics journals.
Table of Contents
Acknowledgments vii INTRODUCTION: Turning on the Light 1 SECTION I: THE LIGHT OF AMBIGUITY 21 CHAPTER 1: Ambiguity in Mathematics 25 CHAPTER 2: The Contradictory in Mathematics 80 CHAPTER 3: Paradoxes and Mathematics: Infinity and the Real Numbers 110 CHAPTER 4: More Paradoxes of Infinity: Geometry, Cardinality, and Beyond 146 SECTION II: THE LIGHT AS IDEA 189 CHAPTER 5: The Idea as an Organizing Principle 193 CHAPTER 6: Ideas, Logic, and Paradox 253 CHAPTER 7: Great Ideas 284 SECTION III: THE LIGHT AND THE EYE OF THE BEHOLDER 323 CHAPTER 8: The Truth of Mathematics 327 CHAPTER 9: Conclusion: Is Mathematics Algorithmic or Creative? 368 Notes 389 Bibliography 399 Index 407
What People are Saying About This
David Ruelle
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"
David Tall
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"
From the Publisher
"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
Reuben Hersh
An amazing tour de force. Utterly new, utterly truthful. — Reuben Hersh, author of "What Is Mathematics, Really?"
Joseph Auslander
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
