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 Why are certain repetitive activities more
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
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
6.2.5 Finite differences and asymptotic at zero ...... 125
6.3 Proofs by handwaving .................................. 126
Part II Mathematical Reasoning
7 What Is It That Makes a Mathematician? ..................... 135
7.1 Flies and elephants ................................... 135
7.2 The inner dog ......................................... 138
7.3 Reification on purpose ................................ 140
7.4 Plato vs. Sfard ....................................... 143
7.5 Multiple representation and de-encapsulation .......... 143
7.5.1 Rearrangement of brackets ...................... 147
7.6 The Economy Principle ................................. 148
7.7 Hidden symmetries ..................................... 151
7.8 The game without rules ................................ 153
7.9 Winning ways .......................................... 155
7.10 A dozen problems ...................................... 160
7.10.1 Caveats ........................................ 160
7.10.2 Problems ....................................... 161
7.10.3 Comments ....................................... 163
8 "Kolmogorov's Logic" and Heuristic Reasoning ............... 169
8.1 Hedy Lamarr: a legend from the golden era of moving
pictures .............................................. 169
8.2 Mathematics of frequency hopping ...................... 171
8.3 "Kolmogorov's Logic" and heuristic reasoning .......... 173
8.4 The triumph of the heuristic approach: Kolmogorov's
"5/3" law ............................................. 178
8.5 Morals drawn from the three stories ................... 181
8.6 Women in mathematics .................................. 181
9 Recovery vs. Discovery ..................................... 187
9.1 Memorize or rederive? ................................. 187
9.2 Heron's formula ....................................... 189
9.3 Limitations of recovery procedures .................... 190
9.4 Metatheory ............................................ 192
10 The Line of Sight .......................................... 197
10.1 The Post Office Conjecture ............................ 197
10.2 Solutions ............................................. 202
10.3 Some philosophy ....................................... 205
10.4 But is the Post Office Conjecture true? ............... 207
10.5 Keystones, arches, and cupolas ........................ 209
10.6 Military applications ................................. 212
Part III History and Philosophy
11 The Ultimate Replicating Machines .......................... 217
11.1 Mathematics: reproduction, transmission, error
correction ............................................ 219
11.2 The Babel of mathematics .............................. 220
11.3 The nature and role of mathematical memes ............. 222
11.4 Mathematics and Origami ............................... 228
11.5 Copying by squares .................................... 231
11.6 Some stumbling blocks ................................. 235
11.6.1 Natural language and music ..................... 235
11.6.2 Mathematics and the natural sciences ........... 235
11.6.3 Genotype and phenotype ......................... 236
11.6.4 Algorithms of the brain ........................ 236
11.6.5 Evolution of mathematics ....................... 237
11.7 Mathematics as a proselytizing cult ................... 238
11.8 Fancy being Euclid? ................................... 240
12 The Vivisection of the Cheshire Cat ........................ 247
12.1 A few words on philosophy ............................. 247
12.2 The little green men from Mars ........................ 251
12.3 Better Than Life ...................................... 252
12.4 The vivisection of the Cheshire Cat ................... 253
12.5 A million dollar question ............................. 256
12.6 The boring, boring theory of snooks ................... 260
12.6.1 Why are some mathematical objects more
important than others? ......................... 260
12.6.2 Are there many finite snooks around? ........... 262
12.6.3 Snooks, snowflakes, Kepler, and Pálfy .......... 264
12.6.4 Hopf algebras .................................. 267
12.6.5 Back to ontological commitment ................. 270
12.7 Zilber's Field ........................................ 271
12.8 Explication of (in)explicitness ....................... 273
12.9 Testing times ......................................... 276
References .................................................... 281
Index ......................................................... 307
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