Gödel, Escher, Bach: An Eternal Golden Braid (146 page)

Phonographs, see record players

phonons, 304

photocopy machines, 499 photons, 142-46, 258

Contents

XXVII

physics laws of: as basis for choosing between rival mathematical theories, 100, 456-57; as blocking infinite regress, 170, 685; as formal system, 53-54; inconsistency and, 95-96, 99,

584-85; intuitive, 362-63, 711; levels and,

303-5, 307-8, 693; no escape from, 477, 575;

Reductionist s Dilemma and, 522, 709; underlying consciousness, 575, 685, 710

IT, 277, 306, 408, 415, 421, 546, 568, 605, 673,

728

piano postulates, 552-53

pianos, 3-4, 302, 305, 633-34, 700, 726; inverted, 681-83

Pickruick papers, The (Dickens), 24. 326, 595, 598 pinball machine, 307

pipe Dream, 703

pipes, 480-82, 486, 488, 493-94, 521, 638, 701-3 planets and satellites, 353-54

PLANNER, 629-32

plurals, 354

pocket calculators, 568-70, 616, 678, 710 pointers in computers, 289-90, 619

points (geometrical), 19-20, 90, 92-93, 100, 207,

222, 452, 456

Polanyi, Michael, 574 polarons, 304-5 polypeptides, 523, 525, 528 polyribosomes, 526-28

Pons Asinorum proof, 606-7, 669 Pool B, 418

Pool F, 427

popcorn, 104, 124-25 popping, 127-35, 184-85 popping-tonic, 105-6, 116-17, 125

porridge, 431

ports of access, 670-71 Post, Emd, 33

post-ending endings, 392, 403 postal system metaphor, 663

postulates of geometry, 90-91, 92-93, 407 potentially endless searches, 396, 400-402, 425,

400-401, 444, 582-83

pq-system: completeness and consistency of, 101;

decision procedure for, 47-49; expressive weak

ness of, 101, 221-22, 407, 417; horse-apple

happy interpretation of, 51, 88, 215; isomor

phisms and, 49-53, 158, 625; modified, 87-88,

92-93, 102; surprise interpretation of, 52-53,

94

predicate calMus, 609

predicates, number-theoretical, 208-9

predictable termination, 400, 407, 409-18, 420,

441, 582; see also unpredictable but guaranteed

termination, terminators

Prelude, 337, 383, 460, 686, 737, 739

preludes and fugues, 280-84, 335; see also

fugues, Well-Tempered Clavier

premise, 184

Preprocessing, 647, 650, 659 President v. Supreme Court, 692

Contents

XXVIII

primary structure: of proteins, 519-22; of ty

poenzymes, 511, 512

prime numbers, 58-59, 64-67, 72-74, 149,

211-12, 413, 551-58, 615; differences of, 393, 395-98, 400, 416; sums of, 393-96, 400, 414

primitive recursive truths, 407

primitive recursivity, 406, 407, 414-20, 422, 424, 429-30, 440, 441, 444, 451, 466, 472

Principus Mathematiaa, 18-19,• 21, 23-24, 228, 618-19

Print Gallery (Escher), 15, 714-17

printer of computer, 301, 307

problem reduction, 609-13; self-applied, 613

problem spaces, representation of, 611-13

procedures, 132-34, 150-51, 292, 410-15, 418-20, 424-28; chains of, 413-14, 415, 418

processors (computers), 504, 513, 547; see also. central processing unit producible numbers, 264-65, 269-70

program space, 299

programs: in Analytical Engine, 25; in BlooP and FlooP, 410-15, 424-26; Blue, Green, Red, see Blue, Green, Red programs; chess-playing, see chess programs; constructed by programs, 589, 629-32, 664; as data, 293, 692; for determining enzyme function, 521-22; for determining phenotype, 532; for determining tertiary structure, 521-22; families of, 503, 546; for generating theorems, 471-73, 578, 615, 617-18; high-level comparison of, 380-81; for naming ordinals, 476; recursive structure of, 149-50; secondorder, third-order, etc., 476; self-modifying, 152, 692; self-reproducing, 498-504, 547; for translating programs, 291-94; vs. data, 499, 513, 531, 546-48, 616-17, 630, see also use vs. mention; vs. programmers, 306, 734-37; see also computers, computer languages, Al programs, etc.

Prokofiev, Sergei, 150

pronoun reference, 587, 591, 592

proof-pairs, 416, 438-43, 446-47, 450-51, 452-54, 466, 468, 469

proofs: nature of, 18-24, 58-60, 88-93, 192-97, 227-28, 458-59, 578, 707-8; as never absolute, 191-94; of proofs, 192-93; vs. derivations, 35, 193, 194-95

prophase, 665-666

Propositional Calculus, 181-97; embedded in TNT, 195, 197, 207, 215-17; as an epiphenomenon, 578; interpretations of symbols of, 186, 189, 191-92; rules of inference, justified, 188-89; rules of inference, presented, 181-87; rules of inference, table of, 187; streamlined, 193-94; variants of, 195; ways to improve, 193-94; 196-97; weaknesses of, 195-97, 578; well-formedness in, 181-83

proteins, 517-18, 544-45; as procedural knowledge, 616-17; as programs, data, interpreters, processors, 547; see also enzymes

prototype principle, 352

provability, 18, 101

Prudence and Imprudence, 191-92, 229

pruning, explicit vs. implicit, 286

pseudo-epigenesis, 531-32

psychic powers, 693; see also ESP

Ptolemy V Epiphanes, 165

Contents

XXIX

Puddle (Escher), 256

pulling-out (mechanisms), see decoding (mechanisms)

punctuation, 33, 268, 440, 510, 512, 520, 524-25

purities, 506-7, 514, 516, 534; see also nudeotides, bases, base-pairing purposeful vs. purposeless behavior, 320-22

pushcorn, 124-25

pushing, 127-34, 184-85

pushing-potion, 105-6

Pushkin, Alexander, 124

puzzles. 8, 33-35, 62-63, 67, 73, 137, 182, 212,

215, 220, 401, 415-17, 425, 442-43, 444,

512-13, 564-565, 609, 621-23, 646-60, 689 pyramidal family of theorems, 221-25, 450-53 pyrimidines, 506-7, 514, 516, 534: see also nu

cleotides, bases, base-pairing Pythagoras, 418, 556-57

Q(n), 137-38, 152, 265, 409

quantifiers, 207-9, 210, 211-12. 214, 217-19 quantum mechanics, 19, 54. 140-46, 350, 455,

457, 699; see also partides

Quantz, Joachim, 4 quarks, 304, 305, 350

quasi-isomorphisms, see isomorphisms, fluid quatemarv structure, 525 Questions and Speculations, 676-80

Quine, Willard Van Orman, 435, 446, 449, 699 quining, 431-37, 445, 446, 449, 497-99, 531 QUIT (BlooP), 412

quotation, 426, 431, 433-37, 496-97, 702, 738 quotation marks, 33, 434, 498, 499, 702

Rachmaninoff, Sergei, 150

RACRECIR, 738

radio broadcasts, 128, 163, 169, 353, 478, 545, 720 Ramanujan, Srinivasa, 562-66, 614

randomness, 408-9, 620, 673, 712 rational and irrational numbers, 140-42, 418, 452, 556-57

rational vs. irrational, in human head, 575-77 rats in mazes, 342

Rauschenberg, Robert, 703

r.e. sets, see recursively enumerable sets reading frame shift, 154, 525 reality, nature of, 409

rearrangement of parts, 78, 333-35, 484; see also self-assembly

reasoning about formal systems, 38-39, 66,

260-72, 438-52, 465-71, 579-81

reasoning by programs, 569-70, 577-78, 586-93,

606-7, 609-11, 614-15, 618-19, 628-32 recognition: molecular, 540-41; visual, 346-48, 646-62; vs. production, 648-49 recognizable forms, 68

recombination, 657, 665-69

record players: alien-rejecting, 487-88; EpsilonZero, 486; family of, in Crab's jukebox, 154-57; Grand Self-assembling, see record player Epsilon-Zero; as information-revealers, 158-61, 164; intrinsic vulnerability of, 75-78, 102, 424, 470, 483-86, 536, 543, 584, 721, see also Tbdelization, TC-battles, etc.; likened to formal systems, 84, 85; low-fidelity, 77, Contents

XXX

85, 101, 406-7, 470; Omega, 78, 468, 483-84; Numbers l, 2 ... etc., 76-77; Tortoise-chomping, 483, 487-88; two-channel monaural, 634, 669; see also jukeboxes records: defective, 102, as information-bearers, 158, 160-61, 164; as labyrinths, 120-24; with multiple melodies, 154-57; as phonographbreakers, 75-78, 83-85, 271, 406-7, 424, 469, 484, 486, 536, 543, 584; smashed, information in, 161; in space, 162-64, 172, 174-75; of WellTempered Clavier given to Crab, 275, 278-80

records and record players, likened to cellular constituents and cells, 83, 158-64, 167, 175, 536

recursion: avoidance of infinite regress in, 127, 134-35; avoidance of paradox in, 127; defined, 127-29, 131-35; elementary particles and, 142-46; fantasy rule and, 184-85; in game-playing programs, 150-51, 604-5; indirect, 134, 137; in language. 130-34, 588, 591, 592; in music, 121-23, 129-30; and unpredictability, 152; see also nesting, levels, distinct vs. similar, level confusion, etc.

recursive acronyms, 113, 133, 134-35, 738, 742 recursive diagrams, 135-37 recursive figures, 67-70. 72, 73 recursive formula, of thinking, 560

recursive functions, 136-40, 152, 430, 455; see also

genera] recursivity, primitive recursivity. BlooP,

HooP

recursive graphs. 138-43 recursive sequences, 135-38, 139 recursive sets, 72-74, 152, 191

recursive structure of ideas, 386-87, 560, 621,

644-45, 650, 656-57, 669. 671-72

Recursive Transition Networks, 131-34, 136, 145,

150, 620-21

recursively enumerable sets, 72-74, 152, 191, 265,

269

recursively related notation-systems, 475 Red Programs, 427-28 Reddiag [NJ, 428, 429

redness, subjective and objective, 710 reductionism: defined, 312; proteins and, 520-22; see also holism vs- reductionism, sealing-off Reductionist's Dilemma, 522, 709 reentrant code, 387

refrigerators, see record players, low fidelity registers, in computers, 289 relativity, 19, 96, 100, 680 Relativity (Escher), 97-98 relevant implication, 197 renormalization, 142-46, 258, 304-5, 309 repeatability, see Godelization, Todelization, diag onal method, Escherization, TC-batteries, answer-schemers

representability, 407, 417-18, 430, 441, 443, 444,

451, 466, 468, 579-80

representation of knowledge: in All, 569, 615-21, 626-32, 641-59, 664-65, 668-72; in brains, see symbols, localization

repressors, 544-45 Reptiles (Escher), 116-17

Requirement of Formality, 33, 52, 65 retrogression, 8-9, 81, 146, 200, 208, 500-501, 549, 666-68, 723-25, 737-38

return addresses, 128, 133 revelation, 160-61, 175 ribo, some, 236

ribonucleic acid, see mRNA, rRNA, tRNA ribosomal RNA, see rRNA

ribosomes: as models for Al, 662, 663; molecular

canons and, 527-28; need for in DNA's self

rep, 530; origin of, 528, 548; as self-assembling

Contents

XXXI

objects, 485-86, 542; structure of, 528; as translators of Genetic Code, 485, 518-19, 522-25, 547; in Typogenetics, 512 ricercar, defined, 7 RICERCAR (E), 7, 727-42 Ripplad Surface (Escher), 256-57 RNA, see mRNA, rRNA, tRNA RNA polymerise, 527, 530, 544 robot in T-maze, 711-13 Rogers, Hartley ropes, thin and thick, 229-30

Rose, Steven, 342

Rosetta stone, 165, 166 Rosaak, Theodore, 574 Rousseau, Henri, 680

Royal Theme, 4-10, 96, 719. 739-40 rRNA, 528

RTN's, see Recursive Transition Networks rule-less systems, 598, 685; see also formal vs. in formal systems

rules: arithmetical vs. typographical, 262-64. 269;

flattened into strings, see theorems vs. rules; intelligence and, 26-27, 559, see also brains and formal systems

rules of inference: of C-system, 65; compared with enzymes, 509-10, 513, 531; defined, 34-35;

derived, 193-94; of MIU-system, 34, 260; of P system. 74; of pq-system, 47; proposed, 66, 221;

of Propositional Calculus, 187; recursive enumerability and, 152; run backwards, 48-49, 182; of 310-system, 263; of TNT, 215, 217-20, 223-25; of tq-system, 65; of Typogenetics,

509-10

rules of production, see rules of inference run-of-the-mill sets, 20-21 Russell, Bertrand, 18-24 Russell's paradox, 20-21, 685

Saccheri, Girolamo, 91-93, 99, 452, 456

Sagredo, see Salviati, et al

Salviati, Simplicio, Sagredo, 408-9, 478-79, 673, 694

sameness: of ASU's, 375; of BACH and CAGE,

153-57; in Bongard world, 650-53, 657, 660,

664; of butterflies, 147, 369; of demi-doublets,

669; elusiveness of, 14619; of Escher drawings,

147; of human and machine intelligence, 337,

379, 679-80; of human minds, 341-42, 369-72,

375-77, 382; intensionality and, 338; mech

anisms underlying perception of abstract,

646-62, 665-69, 671-72; overlooked, 614, 674; of programs, 380-82; in self-refs and selfreps,

500-4; of semantic networks, 371; of translations between languages, 372, 379-80; universality of intelligence and, 158, 501; vs.differentness, 153-57; visual, 344-48, 662; see also copies, isomorphisms, conceptual mapping sameness-detectors, see Sams Sams, 650-53, 657, 664

Samuel, Arthur, 604-5, 684-86

Samuel's argument, pro and con, 684-86 San Francisco Chronicle example, 351 sand castles, 725-26

Contents

XXXII

sanity vs. insanity, 192, 696 satellite-symbols, see splitting-off satoti, see enlightenment scale, cyclic, see Shepard tones Schmidt, Johann Michael, 27 Schnirelmann, Lev G., 394

Schdnberg, Arnold, 125 Schrodinger, Erwin, 167 Schweikart, F. K., 92

science: and Bongard problems, 659-61; self-applied, 699

Scott, Robert, 366

scripts, collage of, 168-69 sealingoff, 305, 309, 350, 534 secondary structure, 521, 525

self, nature of, 316-17, 327-28, 384-85, 387-88, 695-96, 709-14

self-assembly, spontaneous, 485-86, 542-43

self-awareness, 406, 479, 573

self-descriptive adjectives, see autological adjectives self-engulfing, 489-94; failed; 490, 492; total, 493 self-knowledge, possibility of, 696-98, 706 self-modifying games, 687-88

self-monitoring, 328, 385,-387-88, 697, 713 self-perception, 695-98; vs. self-transcendence, 478