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.
"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
Table of Contents:
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
Another Princeton book authored or coauthored by William Byers: