Contents vii
3 The Wing of the Hummingbird . . . . . . . . . . . . . . . . . . . . . 43
3.1 Parsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Number sense and grammar . . . . . . . . . . . . . . . . . . . . 46
3.3 What about music? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Palindromes and mirrors . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Parsing, continued: do brackets matter? . . . . . . . . . . 52
3.6 The mathematics of bracketing and Catalan
numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.7 The mystery of Hipparchus . . . . . . . . . . . . . . . . . . . . . 57
4 Simple Things . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 Parables and fables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Cryptomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Israel Gelfand on languages and
translation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 Isadore Singer on the compression of
language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.3 Cognitive nature of cryptomorphism . . . . . 69
4.3 Some mathlets: order, numerals, symmetry. . . . . . . 70
4.3.1 Order and numerals . . . . . . . . . . . . . . . . . . . . 70
4.3.2 Ordered/unordered pairs . . . . . . . . . . . . . . . . 72
4.3.3 Processes, sequences, time . . . . . . . . . . . . . . 74
4.3.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 The line of sight and convexity . . . . . . . . . . . . . . . . . . 75
4.5 Convexity and the sensorimotor intuition . . . . . . . . 78
4.6 Mental arithmetic and the method of Radzivilovsky 81
4.7 Not-so-simple arithmetic: “named” numbers . . . . . . 82
5 Infinity and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1 Some visual images of infinity. . . . . . . . . . . . . . . . . . . 89
5.2 From here to infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 The Sand Reckoner and potential infinity . . . . . . . . 97
5.4 Achilles and the Tortoise . . . . . . . . . . . . . . . . . . . . . . . 100
5.5 The vanishing point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.6 How humans manage to lose to insects in mind
games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.7 The nightmare of infinitely many (or just many)
dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 Encapsulation of Actual Infinity . . . . . . . . . . . . . . . . . . . 117
6.1 Reification and encapsulation . . . . . . . . . . . . . . . . . . . 117
6.2 From potential to actual infinity. . . . . . . . . . . . . . . . . 119
6.2.1 Balls, bins, and the Axiom of
Extensionality . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2.2 Following Cantor’s footsteps . . . . . . . . . . . . . 123
6.2.3 The art of encapsulation . . . . . . . . . . . . . . . . 123
6.2.4 Can one live without actual infinity? . . . . . 124
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