Mind tools: The five levels of mathematical reality

MIND TOOLS ......Page 3
Dedication ......Page 5
Contents ......Page 7
00| INTRODUCTION -- The Five Modes of Thought ......Page 9
Mathematics as Information ......Page 11
Number and Space ......Page 13
Logic and Infinity ......Page 18
Psychological Roots of Mathematical Concepts ......Page 22
Information and Communication ......Page 33
A History of Ideas ......Page 39
01| NUMBER ......Page 45
Zero and One ......Page 47
Numbers and Logs ......Page 48
Number Patterns ......Page 57
Numerology, Numberskulls, and Crowds ......Page 80
The Mind Reckoner ......Page 85
Words as Numbers ......Page 91
The Limits of Knowledge ......Page 98
02| SPACE ......Page 105
Math Space and Real Space ......Page 107
Tiles, Cells, Pixels, and Grids ......Page 111
Algebraic Curves ......Page 127
Wiggles and Whorls ......Page 149
Endless Complexity ......Page 164
Fractals ......Page 169
Life Is a Fractal in Hilbert Space ......Page 186
Hilbert Space ......Page 191
03| LOGIC ......Page 203
The Laws of Thought ......Page 205
The Syllogism ......Page 208
Symbolic Logic ......Page 215
Exploring Logical Space ......Page 221
Godel's Theorem ......Page 217
Turing Machines ......Page 234
Unsolvable Problems ......Page 244
The Ocean of Truth ......Page 251
04| INFINITY AND INFORMATION ......Page 257
How Big Is Infinity? ......Page 259
Information and the Continuum Problem ......Page 269
Infinitesimals in Perspective ......Page 275
Algorithmic Complexity ......Page 287
Inconceivability ......Page 298
Runtime ......Page 309
Everything is Information ......Page 314
INDEX ......Page 323