Contents Preface xi Credits xiii Acknowledgments xvii Overture 1 The Ideal Mathematician (with Philip J. Davis) 5 Manifesto 13 Self-introduction 17 Chronology 21 Mathematics Has a Front and a Back 35 Part 1. “Mostly for the right hand” 41 Introduction 43 True Facts About Imaginary Objects 47 Mathematical Intuition (Poincar´ e, Polya, Dewey) 51 Summary 51 Mathematical Intuition 59 Polya 61 Mental Models 63 Mental Models Subject to Social Control 66 Dewey and Pragmatism 68 Acknowledgments 72 To Establish New Mathematics, We Use Our Mental Models And Build On Established Mathematics 73 Introduction 73 Established mathematics 75 Mathematicians’ proof vs. axiomatic proof 77 Mathematicians’ proof is semantic, not syntactic 78 Established mathematics is fallible 80 Published vs. private, rigorous vs. plausible 82 Established mathematics is not controversial 83 Acknowledgments 85 vii
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