How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematicsby William Byers
Pub. Date: 04/22/2010
Publisher: Princeton University Press
To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodicallyeven algorithmicallyfrom 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
To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodicallyeven algorithmicallyfrom 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.
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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
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