Gödel, Escher, Bach: An Eternal Golden Braid (147 page)

self-programmed objects, 685-86, 691-92 self-proving sentences, 542-43 self-quoting sentence, 426, 496-97

self-reference: Bach and, 86; banning, 21-23; as cause of essential incompleteness, 465, 470-71;

focusing of, 438, 443, 445-48; Gddelian, 17-18, 271, 447-49, 497, 502, 533, 667, 738; indirect, 21, 85, 204, 436-37, 502, 667, 738-39; many leveled. 742; near miss, 437; Quine method,

431-37, 445-46, 449, 497-99, 531; by translation, 502

self-reference and self-replication, compared, 530, 533-34, 541-43

self-referential sentences, 435-37, 477, 495-99, 501

self-rep: by augmentation, 503; canons and, 501, 503; differentiating, 546; epigenesis and, 160; by error message, 503; inexact, 500-503, 546; by retrograde motion, 500-501; by translation, 501; trivial, 499; typogenetical, 512-13 self-snuffing, 701-2

self-swallowing sets, 20

self-symbol, 385, 387-88, 709; free will and, 710-14; inevitability of, 388

self-transcendence, 477-78, 479 self-unawareness, irony of, 328, 330, 331, 630 semantic classes, 621, 630

semantic networks, 370-72; see also concept network

semi-interpretations, 189, 196

semiformal systems, 216; see also geometry, Euclidean

senseless loops, 679 sentences in TNT, 208-9 Sentences P and Q, 436-37

sequences of integers, 73, 135-39, 173-74, 408 set theory, 20-23

sets F and G, 73

1729, 204-5, 210-11, 345, 393, 551, 564-855 Shadows, The (Magritte), 480 Shakespeare, Wm., 96, 595, 598, 608, 736 Shandy Double-Dandy, 611 shared code, 387

Shepard, Roger, 717-19 Shepard tones, 717-19

shielding of lower levels, see inaccessibility SHRDLU, 586-93, 599, 627-32, 674 Shuzan, 251

Sierpifiski, W., 404

signals, crisscrossing, 322-23 signature, visual, 347-48 Silberescher, Lowen, 394

Silbermann, Gottfried, 3, 4 silver, 173

Simon, Herbert A., 303, 305

simple, complex, hypercomplex cells, see neurons Simpficio, see Salviati Contents

XXXIII

simplicity, 172, 560, 615

simulation: of entire brain, 572-73; of neural networks, 571-72

Six-Part Ricercar (Bach), 4-7, 719, 739-42 skater metaphor, 412-13

skeletons (recursion), 140-41; see also bottom 4

skimming off top levels, 309, 325, 326, 358-59, 568-79

slinky, 337

slippage, conceptual, 633-40. 641-44, 654-56,

672; see also conceptual mapping Sloth, 633-40, 643. 681-83, 722 Sloth Canon (Bach), 9, 666, 683 Sloth Canon (Dialogue), 738

SLOTHs, 684-719

slots, 645, 650-53, 656-57, 668 Smalltalk, 662

smart-stupids, 721-42

Smoke Signal 67, 702

SMUT, 6, 80, 155-57, 202, 682, 718, 740 soap cake, 497

soft-louds, see pianos

software and hardware: in brain, 346, 356-57, 686, 709; defined, 301

soldier ants, 318

Sonata for Unaccompanied Achilles, 502-3

Sonata for violin and clavier in F minor (Bach), 162

Sonatas and partitas for unaccompanied violin (Bach), 62, 63, 70-71, 257, 502 song, selfreproducing, 500

"Song Without Time or Season, A", 483 songs, recombinant, 607-8

sonnets, 596, 608, 736

soulism, 385, 472, 574, 597, 686, 708

spaces: abstract, 457; of behavior, 306, 307, 363, 621; of integers, 502

spare tire, 670

Sperry, Roger, 710

Sphex wasp, 360-61, 613-14

spiderwebs, 371-72, 617

splitting-off, 353-54, 664; see also classes vs. instances

stacks, pushdown, 127-35, 136, 610-11

starting positions in Art of Zen strings, 239-40, 241

State of Grace (Magritte), 481

statistical mechanics, see gases

Sterner, Geo., 166-67, 642-43, 670 Stent, Gunther, 514

stomach processes, 572

strands: of DNA and RNA, 514-18; in Typogenetics, 508-13, 514

Strange Loops: abolishment of, 21-23; with Babbage and Turing, 737; in Bach, 10, 719; in Central Dogmap, 534; consciousness and, 709-10; contrasted with feedback, 545, 691; defined, 10; in Escher, 10-15, 714-17; in Godel, 15-17, 24; in government, 692-93; in language, 22, 691; in the mind, 27, 691-92; in molecular biology, 231, 532-34, 545, 546-48; in Principia Mathematica, 24; surprise element necessary in, 691; in TNT, see TNT, introspection of; see also Tangled Hierarchies, level-mixing, levelconflicts, level-confusion, etc.

street nuisances, 25, 726-27, 729

stretto, 314, 738

Contents

XXXIV

String Manipulation Rules, 240-42

strings: defined, 33-34; folded, 234-44, 427 structure vs. function, 522, 670-71; see also use vs.

mention, syntax vs. semantics style, 148, 371

SUB (TNT-formula), 444-45

subbrains, see subsystems

subdescriptions, etc. 650-51; see also recursive structure of ideas subframes, etc., 644-45; see abo recursive structure of ideas

subject vs. object, 698-99; see also dualism, use vs. mention, symbol vs. object subjunc-TV, 635110

subjunctive instant replays, see instant replays subjunctives, see counterfactuals suborganisms, see subsystems subroutines, 150, 292, 471, 677 substitution notation in TNT, defined, 224 substitution relationship (ingredient in Godel's proof), 443-45, 497

substrate: of Epimenides sentence, 581, 584-85; mental, necessity of, for analogical thought, 569-71; mental, noninterpretability of, 570-71; mental, simulation of, 571-72; of proteins, 529

subsystems of brain, 385-88, 725 subtraction, 52-53, 411-12

sufficiently powerful systems, 86, 101, 406-7, 430, 460, 530

Suites for unaccompanied cello (Bach), 70-71 summarizing string, 221-22, 450-51

superconductivity, 304-5 superinielligence, 679

supernatural numbers, 223, 453-56, 458-59, 467 supernatural proofs, 454-55

supertangling, 688

surrealism, 700

Sussman, Gerald, 664

Swieten, Baron Gottfried van, 6-7

Switcheroo, Q. q., 187

symbol-level description of brain, 349-51

symbol vs. object, 699-706; see also subject vs. object, use vs. mention, etc.

symbols: active vs. passive, 324-25, 327-28, 337-38; Al realizations of, 662-63, 665; in ant colonies, 324-28, 330; borderlines between, 354-57, 359-60; as brush strokes, 351; compared with neurons, 350, 371; compared with ripples, 356-37; conceptual scope of, 350-51; dormant, 327, 349, 355-56, 384; form of, 348, 356-57, 361; free will and, 712-14; funneling and, 347; of insects, 360-61; invariant core of, 349; joint activation of, 351, 354-56, 359, 361, 365, 584-85, 664-65, 675; message-exchange and, 350, 371, 662-63; modes of activation of, 349-57, 361; need of, for originality, 609; neural substrate of, 356-57, 570; no access to substrate by, see inaccessibility; overcrowding of, 358; overlap of, 348-49, 356-57; potential, 355-56, 382-84; universal, 375-76; vs. neurons, 348, 356-57, 361; vs. signals, 325-27, 349-50

synapses, 339

syntax vs. semantics, 626-27, 630-32, 676; see also form, syntactic vs. semantic system, boundaries of, 37-38; see also jumping out of the system

System crash, 116

systems, reliable vs. unreliable, 307

T (Tarski's formula), 580-81

Contents

XXXV

't'-concept, 'h'-concept, 'c -concept, 326 T-even phages, 540, 542 T-level, see tangled level table of nontheorems, 66 Tagore, Rabindranath, 169

Tangled Hierarchies: in art, 704; defined, 10; of

genetics, 532-34, 546-48, 688; of meta

mathematics, 458, 532-34; of mind, 691-92, 709-10, 719; near miss, 691; in Propositional Calculus, 194; of self-modifying chess, 688; of

Tortoises reasoning, 177-80; of Typogenetics,

513, 688

tangled level, 688 Tanguy, Yves, 700

tape recorders, 485, 519, 523, 525-28 Tarski, Alfred, 579-81 Tarski-Church-Turing Theorem. 561, 581 Tarski's Theorem, 579, 580-81, 584-85, 697 Taube, Mortimer, 574

Taurinus, F..4., 92

IrC_battles, 75-78, 406-7, 424, 467-70, 478, 483-88, 536-41, 721

tea. 153-54, 231, 275, 322-23, 333, 549, 557, 558, 561

teleological vs. evolutionary viewpoint, 320-22 telephone calls, 61-63, 127-28. 161; obscene, 431, 437

telephone systems, 296, 663 telephones, 302

television, 285, 348, 478, 484, 487-93, 634-40 telophase, 667-68

templates: for Bongard problems, 650-53,

656-57; vs. instructions, see instructions vs. tem

plates, programs vs. data

tension and resolution, 121-23, 129-30, 227 tentativity, 646, 651, 654-56, 672 termination tester, 425-29 terminators (FlooP), 425-29 terms (TNT), 206-7, 213, 214

tertiary structure: of proteins, 519-22, 525-27; of

tRNA, 524; of typoenzymes, 510-11, 512, 519 Tesler, Lawrence G., 601

Tesler's Theorem, 601, 623; see also paradox of Al, jumping out of the system, essential in completeness

tesselations, 68, 69, 198, 667

tests vs. functions in BlooP and FlooP, 413, 418 text-handling by computers, 301 T4

phage, 537-41

"the" 586, 629-30

theorem-numbers, 264-67, 269-70, 440-43, 451 theorem-proving, mechanical, 602, 609, 617-19 theorems: defined, 35; systematic enumeration of,

39-40, 48, 471-73, 578, 615, 617-18; vs. non

theorems, 39-41, 66-67, 70, 71-73, 190-91,

416-17, 560, 579-80; vs. rules, 43-45, 193-94,

509-10; vs. Theorems, 35, 193; vs. truths,

49-54, 70-71, 86-102, 190-97, 213, 221-23,

228-30, see also consistency, completeness,

Godel's Theorem, consequences of

Theseus and Ariadne, 130 thinking, speed of, 679

30, as possible MIU-number, 265-67 "this sentence", 436, 495-98, 499 thought, substrate of, 559

3n+1 problem, 400-2; see also wondrous and un wondrous numbers

Three Spheres If (Escher), 258 310-system, 261-67

Three Worlds (Escher), 247, 256 thymine, see nucleotides tilde, 183, 191-92, 554

Contents

XXXVI

timesharing, 296, 354-55, 387, 730

TNT, 204-30; absolute geometry and, 451-52; austere, 211, 214, 216, 268, 442, 534; axioms of,

215-17, 222-23; as code, 265-67; consistency of, 229-30, 449-50; extended, axioms of, 451-52, 466-68; extensions of, 451-59, 465-68; FIGURE-FIGURE figure and, 70; as general metalanguage, 265-67; goals of, 60; incompleteness of, see incompleteness; introspection of, 17, 194, 267-72, 406, 438, 443, 449-50, 698, 707-8; as its own metalanguage, 267-72, 441-46, 514; plight of, pigorial version, 71; relaxed, 228; rules of formation, table of, 213-14; rules of inference, 215, 217-15; sixth axiom of, 222-23, 451-55, 459, 465-68; well-formedness in, 205-15

TNT-derivations compared with machine language, 291

TNT-level, explanations on, 708

TNT-numbers, 269-70; see also theorem-numbers

TNT + G, 465-67, 471

TNT + G + G', etc., 467-71

TNT + -G, 467

tobacco mosaic virus, 484-85, 542, 543

Todelization, repeatability of, 76-78, 424, 467-70, 483-86; see also Godelization, essential incompleteness, answer schemas, etc.

Todel's Theorem, 486, 536

Tokusan, 189-90

tonic, musical, 121-23, 129-30

Tortoise: answer-schemas and, 475; Carroll paradox and, 46, 193, 684-85, 693; Crab Canon and, 204, 666-67; degenerate solution by, 669; Diophantine equations and, 4590; as harpsichord, 502; initial letter of, 231, 507, 667; mentioned, 102, 170, 267; origin of, 28-29; picture of, 42; recursion and, 128-30; 149; use of words by, 181; vs. Crab, 84-86, 271, 406, 424, 467-69, 540, 543; Zen string made by, 272; in ZET cycle, 94-95

Tortoise-pairs, 416, 441, 448; compared with proof-pairs, 441, 448

Tortoise property, 395-97, 415-18, 425, 426, 441, 448

Tortoise's love song, 435-36

Tortoise's method, see Todelization

T6zan, 190, 255, 257, 479

tq-system, 64-67

transcendentalism, 704

transcription: DNA to mRNA, 517, 524, 527-28, 530, 533, 536, 538, 540-41, 544-45; DNA to tRNA, 425; koans to messengers, 235-36, 238, 239, 242; letters to notes, 83; prevention of, 544-45

transfer RNA, see tRNA

translation: between computer languages, 192-95, 297-98, 306, 380-81, 547, 632; between levels of a brain, 349, 381-84, 709; between natural languages, difficulties of, 372-73, 379-80; between TNT and meta-TNT, 267-72, 441-46, 709; of Crime and Punishment, 379-80.; English to TNT, 209-13, 215, 417; of "Jabberwocky", 372-73, 379; levels of fidelity in, 379-80; mechanical, 380, 601, 603; of messengers into strings, 234-36; mRNA to proteins, 485, 518-19, 522-25, 527-28, 531-36, 538, 545, 546-48; from N

to Meta-TNT, 533; from score to sounds, 83; in Typogenetics, 509-10, 512-13

transparency to user, 629, 632

Contents

XXXVII

trees: look-ahead, see look-ahead trees; recursive diagrams, 40„ 71, 135-37; of theorems, 40, 71

triggering patterns of symbols: dependence on meaningless lower levels, 569; isomorphism between minds and, 369, 376; isomorphism between physical law and, 362; as key to meaning, 325, 327, 350, 360, 385, 609; mediated by mes-triggering patterns of symbols (continued)

sages, 350, 371; for melodies, 364; nouns vs. verbs, 361; randomness in, 673

triggers: DNA as, 160-61; dormant symbols and, 281, 383, 384; frame messages and, 162; jukeboxes and, 160-61, 170-71, 174, 500; koans as, 246; music and, 162-63, 281, 383, 583; outer messages and, 166, 169, 170-71, 174, 501

Trio Sonata from the Musical Offering (Bach), 7-8, 720, 724, 726

Tripitaka, 257

trip-lets, cover, xiv, 1, 28, 273 tRNA, 522-24, 547, 548

Trojan Horse, 538

truth: capturability by symbol manipulation, 53-60; elusiveness of, 694-95; inexpressible in TNT, 580-81; not fully mirrorable in brain, 584-85; vs. beauty, 554-58, 584; vs.

commercials, 478; vs. falsity, 70, 71, 213, 228-29, 417, 561, 579-81