How mathematicians think using ambiguity, contradiction, and paradox to create mathematics

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...

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Bibliographic Details
Main Author:	Byers, William
Format:	eBook Book
Language:	English
Published:	Princeton, N.J. ; Woodstock Princeton University Press 2010
Edition:	1
Subjects:	Actual infinity Addition Algorithm Ambiguity Analogy Approximation Axiom Calculation Cardinal number Cardinality Complex number Complexity Conjecture Consciousness Continuous function Contradiction Counterexample David Hilbert Derivative Dimension Emergence Equation Euclidean geometry Existential quantification Geometry Gödel's incompleteness theorems History & Philosophy Inference Infinitesimal Informal mathematics Integer Irrational number Line segment Logic Logical reasoning Mathematical induction Mathematical practice Mathematical proof Mathematical theory Mathematician Mathematicians Mathematicians > Psychology MATHEMATICS Mathematics > Philosophy Mathematics > Psychological aspects MATHEMATICS / History & Philosophy Natural number Notation Paradox Parallel postulate Parity (mathematics) PDZ Philosophy Philosophy of mathematics Philosophy of science Principle Proof by contradiction Psychological aspects Psychology Pure mathematics Quantity Quantum mechanics Randomness Rational number Rationality Real number Reason Result Science Scientific theory Scientist Series (mathematics) Sign (mathematics) Subset Summation The Unreasonable Effectiveness of Mathematics in the Natural Sciences Theorem Theory Thought Variable (mathematics) Scientific theory Subset Series (mathematics) Rational number Counterexample Logical reasoning Result Complexity Gödel's incompleteness theorems Mathematical theory Mathematical induction Calculation Axiom Contradiction Paradox Addition Real number Approximation Cardinality Consciousness Rationality Conjecture Scientist Inference Continuous function Principle Algorithm Thought Existential quantification Cardinal number Irrational number Informal mathematics Derivative Philosophy of science Summation Ambiguity Natural number Sign (mathematics) Philosophy of mathematics Line segment The Unreasonable Effectiveness of Mathematics in the Natural Sciences Theory Science Mathematics Notation Reason Actual infinity Mathematical proof Pure mathematics Emergence Theorem Analogy Logic Proof by contradiction Euclidean geometry Equation Quantity Dimension Infinitesimal Mathematician Complex number Variable (mathematics) Geometry Integer David Hilbert Parity (mathematics) Quantum mechanics Randomness Mathematical practice Parallel postulate
ISBN:	0691127387, 9780691145990, 9780691127385, 0691145997, 1400833957, 9781400833955
Online Access:	Get full text
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Table of Contents:

How mathematicians think: using ambiguity, contradiction, and paradox to create mathematics -- Contents -- Acknowledgments -- Introduction: Turning on the Light -- Section I: The Light of Ambiguity -- 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.
Front Matter Table of Contents Acknowledgments INTRODUCTION CHAPTER 1: Ambiguity in Mathematics CHAPTER 2: The Contradictory in Mathematics CHAPTER 3: Paradoxes and Mathematics: CHAPTER 4: More Paradoxes of Infinity: CHAPTER 5: The Idea as an Organizing Principle CHAPTER 6: Ideas, Logic, and Paradox CHAPTER 7: Great Ideas CHAPTER 8: The Truth of Mathematics CHAPTER 9: Conclusion: Notes Bibliography Index
Cover -- Title -- Copyright -- Contents -- Acknowledgments -- INTRODUCTION: Turning on the Light -- SECTION I: THE LIGHT OF AMBIGUITY -- 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 -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z
Section I. The Light of Ambiguity --
INTRODUCTION. Turning on the Light
Chapter 5. The Idea as an Organizing Principle
Chapter 3. Paradoxes and Mathematics: Infinity and the Real Numbers
Chapter 1. Ambiguity in Mathematics
Chapter 6. Ideas, Logic, and Paradox
Index
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Chapter 2. The Contradictory in Mathematics
Chapter 7. Great Ideas
Chapter 8. The Truth of Mathematics
/
Section III. The Light and the Eye of the Beholder --
Chapter 9. Conclusion: Is Mathematics Algorithmic or Creative?
Section II. The Light as Idea --
Contents
Chapter 4. More Paradoxes of Infinity: Geometry, Cardinality, and Beyond
Acknowledgments
Introduction
Frontmatter --
Notes
Bibliography