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
Previous Page Next Page