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| Rucker R. — Mind Tools. The Five Levels of Mathematical Reality |
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Numeration systems, zeros-and-ones 280 312
Numerology 74
Numerology and labeling 31
Octagon, as number pattern 54 64 69
Octahedron as number pattern 53 56 57 64 66 70
Octahedron as regular 63
Omega  bit 264—267
One 39—40
Opposites See also "Dyads"
Opposites, table of 16
parabola 12 124—127 134 141
Paraboloid 158
Paradox of the Liar 221—222
Paradox, Chaitin's theorem based on 287
Parameter space 125—127 129 131 137
Parameters 122 137
Pascal, Blaise 133
Patterns 3 See spatial" "Mathematical "Shapes"
Patterns as inconceivable 290 291
Patterns, chemical elements composed of 73—74
Patterns, complexity of 304
Patterns, information in 89
Patterns, numbers as 49—72 91—92 293
Patterns, Wittgenstein's world as 215
Peano, Giuseppe 159—160 208 209
Pearls of Sluze 132 137
Penrose, Roger 108—111
Pentad 21
Pentagon as number pattern 53—55 58 60—65 67 68
Pentagram 54 56
Pentahedroid (pentahedron) 21 23 24
Perspective 272—277
Phase space 187
Phenomenology 185
Physical name for numbers 93—95
Physics and halting problem 241
Physics and higher-dimensional spaces 187—188
Physics and information processing 35
Physics and space 102
Physics, master equations for 313
Piero della Francesca 273
Pixel 112 116—117
Pixel and perspective 277 278
Plane 158
Plane and Euclid's axioms 12
Plato 19 20 156 200—201 290
Platonic solids 63
Point as degenerate conic 128 130
Polar coordinates 150—151
Politics and numbers 75—76
POLYGONS 103—104
Polygons and simplicity 161
Polygons, tessellations 104—111 221
Polyhedron, regular 63
Polynomial equation, endless 144—145
Population growth as exponential 146
Post, Emil 226
Postmodern era 34—35
Power, and number 99
Power-series curves 158—159
Precision and infinite bit 264
Precision golf 256—258
Precision, bits of 118 120 188—194 277
Predicate calculus 209 210 216
Presentation information 282 303
Prime numbers 56
Prime numbers, computability of 244
Prime numbers, Goldbach's conjecture on 241—242
Primordial soup for Life game 112
Principia Mathematica (Russell and Whitehead) 216
Principles of thought See "Laws or principles of thought"
Procrustean bed 16—17
Propositional calculus 201 209—212 216
Provability and Church's theorem 237—239
Provability and formal logic 236—237
Provability and Godel's theorem 218 220—225
Pyramid as number pattern 55 59 63—65 71
Pyramid as tetrahedron 20
Pythagoras 50
Pythagoreans 32 50—51
Pythagoreans and dyads 16
Pythagoreans and even/odd numbers 50
Pythagoreans and number five 51 55
Pythagoreans and number six 56
Pythagoreans and number ten 60
Pythagoreans on numbers as reality 73—74
Pythagoreans, quaternity of 20
Quadric surfaces 158
Quadrivium 19—21
quantum mechanics 32 34
Quantum mechanics and Hilbert space 102 187—188
Quartic curves 129 130 132 136
Quaternity 17—21
Questions See also "Explanations"
Questions in information measurement 4 27—29
Questions, "Why is there something instead of nothing?" 40
Questions, endless number of 264
Quincunx 21 23
Quintad as archetype 50
R-complexity 282
Radio waves, AM and FM 143—144
Randomization, in fractal curve construction 174 175
Randomness and simple explanations 289 314
Randomness as hidden predictability 198
Randomness in inconceivable and conceivable bit strings 300
Rating system, numerical 84
Rationality and Godel's theorem 223
Ratios, reciprocal 270—271
Real number line 13 261 268 See
reality See also "World"
Reality as information pattern 31
Reality, complementary aspects of 6—9
Reality, mathematics and number based on 14—16 50 73
Reality, Wittgenstein's view of 215
Reciprocals 269—272
Reductio ad absurdum 211 212
Reflection principle 253 258 261
Relations, logical treatment of 207
Relativity theory 32 34
Renaissance 25 32 33 253
Rep-numbers 57
Reproduction See also "DNA"
Reproduction by cellular automata 115—116
Richardson, Lewis Fry 166 168
Robinson, Abraham 268
Robots, self-reproduction by 115—116 See
Roman number system 40
Rosser, J. Barkley 225
runtime 219 301—306 314 See
Runtime for production of natural phenomena 314
Russell, Bertrand 208 209 216
Sand Reckoner, The (Archimedes) 77—78
Sand, infinitesimal grains of 278
science See also "Astronomy" "Chemistry" "Physics"
Science and mathematics 14
Secret of Life or the universe 214 289 306 See
Segment as simplex 23
Self-knowledge, endless regress in 255—256
Self-reference and Godel's theorem 222 224 240
Self-reference and Turing's theorem 240
Self-reproduction by cellular automata 115—116
Self-similarity in logarithmic spiral 151—153
Self-similarity in snail shells 153
Self-similarity of fractal curve 167 175 177
Self-similarity of trees 162 163
Sensation 18 19
Set theory and infinity 4 32 251 259 278
Shannon, Claude 27 28 47
Shapes See also "Geometric patterns" "Surfaces"
Shapes, "most natural" 21—23
| Shapes, curves 100 see
Shapes, fractal 103 160—177 see
Shapes, polygons 103—104 see
Shapes, Sphere 158
Sierpinski carpet 172 173
Similarity and logic 15
Simple or short explanations 289—290 306 313—314
Simplexes 23
Sine curve 142—145 158
Size and infinite sets 259—260
Slide rule 46—47
Sluze, Baron Rene Francaise de 132—133
Smooth magnitude 5 6 10 See
Society, helical development in 155
Socrates 200—201
Software form compared to 157
Solids See also "Cube" "Polyhedron"
Solids, fractal 175—177
Solids, sphere 158
Solomonoff, R.J. 284
Sophists 200—201
Sorites 206 213—214
Space 15—16
Space and feeling 20
Space and freedom 99
Space and Godel's theorem 220—222
Space and motion or change 99
Space and perspective 273—277
Space and Renaissance 25 33
Space and right brain 9
Space and simplexes 23—24
Space and vanishing point 274
Space as three-dimensional 100
Space in notation of line 122
Space patterns, numbers as 51—72
Space vs. number 5—6 9 10
Space, four-dimensional 63 186—187
Space, hand as 3—4
Space, Hilbert space 32 100 182 186—194 see
Space, logical space 213—217 220—221
Space, mathematical vs. real 100—103 273
Space, parameter space 125—127 129 131 137
Space, phase space 187
Spatial forms 100 156—157 177 See "Curves" "Fractals" "Polygons" "Sphere" "Surfaces"
Sphere 158
spirals 141 142 145 148 150—154
Spiritual development, helical pattern of 155
Spotty magnitudes 5 6 10 See
Square as number pattern 53—55 58 62—64 66 67 70
Square pentagon as number pattern 54 58 61—63 65 67
Star of David 54 56
Straight line as degenerate conic 128 130
String name 92 93
Surfaces, algebraic 157—158
Surfaces, fractal 170—175
Surfaces, quadric 158
Syllogism 202—207
Symbolic logic 32 207—212
Tally system 40 43
Tarski, Alfred 225—226
Tautologies 211—212
Taylor, Brook 159
Tessellations 104—111
Tessellations and Godel's theorem 221
Tetrad 17—20
Tetrad as archetype 21 50
Tetrahedral number 65 67—68 70
Tetrahedron(a) 20—22
Tetrahedron(a) as number pattern 53 54 57 61 67 69—70
Tetrahedron(a) as regular 63
Tetrahedron(a) as simplex 23
Theory and complexity 285—286
Theory and discovery of results 237
Theory, Chaitin's theorem on 286—290
Theory, ideal properties of 218—219
Thinking and Godel's theorem 226
Thinking and logic 20 see
Thinking, ingrained modes of 15—16 39
Thinking, Jung on 18
Thomas Aquinas, Saint 251
Thompson, D'Arcy 153 313
Thoughts, number of 81—83 88—89
Tiles, Penrose 108—110
Toffoli, Tommaso 308
Tone of voice as analog information 117 184
topology 32 160
Torsion of curve 154 155
Tractatus Logico-Philosophicus (Wittgenstein) 214—215
Transcendental curves, exponential curves as 145—147 158
Transcendental curves, logarithms as 147—148
Transcendental curves, sine curve as 144 158
Transfinite computer 258
Transfinite numbers 253—254 258
Transfinite numbers, countable 294—295
Transfinite numbers, nonstandard 267—269
Transfinite processes 254—258
Trees See also "Binary trees" "Branches"
Trees and fractals 177
Triads 16—18 20
Triads as archetype 21 50
Triads in tetrahedron 20
Triangle(s) as number patterns 51 53—55 57 59 61 64 65 67 69—71
Triangle(s) as simplex 23
Triangle(s), God as eye in 51
Triangular number 56 60 63
Truth and Godel's theorem 221—223 225—226
Truth in modern times 34
Truth table 210—211 244
Truth, complementarity of 7
Turing halting problem See "Halting problem"
Turing Machines 228—236 See
Turing machines and computability 226 243 244
Turing machines and listability 243
Turing machines and omega bits 264
Turing machines and runtime 302
Turing machines as CA 309—311
Turing machines as serial 307
Turing machines, integers/fractions from 271—272
Turing's theorem 238 See
Turing, Alan 226 228 238 247
Twenty Questions and information theory 27—29
Uccello, Paolo 273 275
Ulam, Stanislaw 11 115—116 283
Ultimate Desert and perspective 275—277
Uncertainty (unpredictability) and information theory 28—29 47
Universal Turing machine 234—236 283 See
Universal Turing machine and algorithmic complexity 282—285 296 300
Universal Turing machine and inconceivability 291—292 296 300
Universal Turing machine and runtime 302
Universal Turing machine, game of Life as 311
Universe See also "World"
Universe and reflection principle 253
Universe as "this" 223
Unsolvable problems 236—242
Volta, Alessandro 33
von Koch, Helge 167
von Neumann, John 11 115—116
Wang, Hao 221
Waves 141 142
Waves, sine curves of 142—145
Whitehead, Alfred North 208 216
William of Shyreswood 205
Wittgenstein, Ludwig 214—215 217
Wolfram, Stephen 309 311—313
Words See also "Information"
Words as numbers 73 84—89 92 222 243 280—282
World See also "Reality" "Universe"
World, atomistic vs. phenomenological view of 185
World, mathematics as approximation of 156 177
World, simple explanation of 214 289—290 306 313—314
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