Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
List of Photographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv
Part I
Simple Things:
How Structures of Human Cognition Reveal Themselves in
Mathematics
1 A Taste of Things to Come. . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Simplest possible example . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Switches and flows: some questions for cognitive
psychologists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Choiceless computation. . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Polynomial time complexity . . . . . . . . . . . . . 7
1.3.2 Choiceless algorithms. . . . . . . . . . . . . . . . . . . 9
1.4 Analytic functions and the inevitability of choice . . 10
1.5 You name it—we have it . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6
pleasurable than others? . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 What lies ahead?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 What You See Is What You Get . . . . . . . . . . . . . . . . . . . . . 23
2.1 The starting point: mirrors and reflections . . . . . . . 23
2.2 Image processing in humans . . . . . . . . . . . . . . . . . . . . 25
2.3 A small triumph of visualization: Coxeter’s proof
of Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Mathematics: interiorization and reproduction . . . 30
2.5 How to draw an icosahedron on a blackboard . . . . . 33
2.6 Self-explanatory diagrams . . . . . . . . . . . . . . . . . . . . . . 38
vi
Why are certain repetitive activities more
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