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How Mathematicians Think : Using Ambiguity, Contradiction, and Paradox to Create Mathematics: Byers, William: BOOKS KINOKUNIYA
詳細
How Mathematicians Think : Using Ambiguity, Contradiction, and Paradox to Create Mathematics
How Mathematicians Think : Using Ambiguity, Contradiction, and Paradox to Create Mathematics
著者名 Byers, William
出版社 : Princeton Univ Pr
出版年月 : 2010/04
Binding : Paperback
ISBN : 9780691145990

BookWeb価格 : A$ 37.95
会員価格 : A$ 34.16

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言語 : English
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内容情報
Source: ENG
Subject Development: History
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    110
  and the Real Numbers
  CHAPTER 4 More Paradoxes of Infinity:            146
  Geometry, Cardinality, and Beyond
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           323
BEHOLDER
  CHAPTER 8 The Truth of Mathematics               327
  CHAPTER 9 Conclusion: Is Mathematics             368
  Algorithmic or Creative?
Notes                                              389
Bibliography                                       399
Index                                              407
 

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
Acknowledgments vii INTRODUCTIONTHE 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