Gödel, Escher, Bach: An Eternal Golden Braid (144 page)

hemiolia, 257, 519 hemispheres, 257, 340-41 Henkin, Leon, 541

Henkin sentences, 541-53, 709; explicit and im

plicit versions, 542-43

Henkin's Theorem, 488

hereditary arguments, 36, 47-48, 217, 261 heterarchies, 134, 359, 534, 651-54, 662, 691

heterological adjectives,20-21, 22 heuristics, 587, 588, 590, 603, 629 Hewitt, Carl, 662

hiccups, 116, 254, 255, 673. 725-26, 736 hierarchy of variability, 643-45, 669

high-fidelity vs. low-fidelity, 77, 85, 101, 406-7,

470. 697-98

high-level explanatory power, 707-9 Hilbert, David, 20, 23-24, 230, 459-60 Hilbert's program, 23-24, 229-30

Hilbert's tenth problem, 459-60 H(n), 137

Hofstadter, D. R., 75, 310, 724, 728, 742 Hofstadter s law, 152

Hogen, 248

holes in formalized systems, 24, 26, 449, 451, 465, 468, 470-71

Contents

XVII

holism: defined, 254, 312; vs. reductionism, 284, 311-36, 389-90, 708-9; Zen and, 254

Hubel, David, 341, 343

Human Condition I, The (Magritte), 705-6 Hyakujo, 254

hydrogen bonds, 516, 522, 525

hyphen-strings, 47, 64-65, 66

hypothetical worlds, 95-100, 338, 360-62,

634-40, 641-44; groundedness in reality. 362,

378-79

hypotheticals, 44-45, 634-40

i, 454

I, 454

"I", referent of, 608

"I Can Be Played (Proven, etc.)...", 488, 541

"I Cannot Be Played (Proven, etc.)

, 76-77, 85,

406-7, 448, 465-67, 536, 541, 608 I-counts, 260-61

1-level, see inviolate level

I-mode, see Intelligent mode

iceberg, 495-96, 497

ideal numbers, 56-58

identification with artifacts, 609, 713-14 idiots savants, 566-67

IF-statements (BlooP), 411-12

images: blurry, 686-87; of thought, 623 "Imaginary Landscape no. 4" (Cage), 163-64, 699 imitation game, see Turing test Immunity Theorem, 536

implicit characterization, 41, 67, 72-73, 93 improvisation vs. introspection, 739

inaccessibility of lower levels to higher levels, 686-92, 706-10; in Aunt Hillary, 330-31, 630: in brains/minds, 302, 328-29, 362-65, 619, 677, 686-92, 697, 706-10, 739; in programs, 296, 300-1, 588, 630, 679; see also software and hardware, introspection, level-conflicts

incompleteness: Bach and, 86; of brains, 585; defined, 86; Escher and, 716-17; of extensions of TNT, 465-71; of formal arithmetics, 18, 86, 101-2, 407, 618-19; of list of mathematicians, 422; of list of reals, 421-24; of Lucas, 477; of phonographs, see record players, intrinsic vulnerability of; of Principia Mathematica, 18, 24, 618-19; of self-knowledge, 696-98; of TNT, 271-72, 430, 450-51; see also essential incompleteness, wincompleteness, etc.

inconsistency: defined, 94; with external world, 87-88, 95; internal, 87, 88, 94-96; of people, 197, 697-98; of Tortoise, 177-80; see also consistency, contradictions, su-inconsistency, Zen

increasing and decreasing rules, 73, 74, 260-61, 264, 269, 401-2, 407-x, 441; see also lengthening and shortening rules, chaos in number theory

index numbers for programs, 418-20, 427-28 index triplets for supernaturals, 455 India, 549, 551, 557, 562-66

Indra's Net, 258, 359

inducers, 545

infinite bundle of facts, 397-98 infinite coincidence, 398, 421

Contents

XVIII

infinite regress, 111-13, 142, 146, 152, 231, 388-89, 426, 497, 738; in Carroll paradox, 43-45, 170, 192-93, 684-86. 693-94; halted, 127, 133-35, 170, 605, 684-86; of objectivity, 479; Zeno and, 31-32, 610; see also Carroll paradox, bottoming out, recursive acronyms, repeatability, etc.

infinite sentence, 497 infinite sky, 401

infinitesimals and nonstandard analysis, 455 infinity: Bach and, 10, 719; Escher and, 15; handled finitely, 59-60, 221-25, 461-64, 468; illustrated, 135-36, 138-43; names of, 475-76; supernaturals and, 454; types of, 421; see also nontermination, infinite regress, recursion, etc.

informal systems, see formal vs. informal systems information: accessibility of, see inaccessibility; creation of, 513; depth from surface, 234-35, 409, 427, 549-58, 606-7, 612-13, 628, 673, see also decoding; discardable, 649, 653, 657-59. 669-72: flow of, 513, 533, 545, 547; irrelevant, 560

information-bearers, 158, 166, 167

information -reveale rs, 158, 267

inhibition, cellular, 544

inner messages, 166-71, 174-76, 501, 524

input-output devices, 288

input parameters (BlooP), 411

insight, 613, 660-61, 665-76

instant replays, straight and subjunctive, 634-40, 641, 672

instructions: in machine language, 289-95; vs. templates, 497-99, 531, see also programs vs. data

INT(x), 138-41, 146, 661

intelligence: accidental inexplicability of, 707; essential abilities for, 26; extraterrestrial, see extraterrestrial intelligence; liftability of, see skimming off; limits of, 475-76, 679-80; necessary underpinning of, 324; simplicity of, 172-73; subtle features of, 566; tangled recursion and, 152; typical abilities of, 559; universality of, and intrinsic meaning, 158, 162-64, 170-76, 501, 661-62; see also brains, minds, A1, etc.

Intelligent mode, 38-39, 65, 193-94, 613-14

intensionality and extensionality, 337-39, 350, 361-62

intentions of machines, 684-85

interestingness, programmed, 615

interpretation-conventions, 687-88

interpretations: adjusted to avoid inconsistency, 87-88, 453, 456, see also undefined terms; multiple, 94-102, 153-57, 266-67, 271, 447-48; of pq-system, 49-53, 87-88, 101-2, 158; of Propositional Calculus, 186-87, 189, 191-92; of strands, 509-10; of TNT, 205-9, 266-67, 453, 533; of tq-system, C-system, P-system, 64-65, 73-74

interpreters: mechanisms in brain, 582-84; people, 293, 297, 524, 671; programs, 293, 504, 547, 616, 632, 662, 692

intrinsically high-level properties, 707-9

introspection, see self-monitoring, self-awareness, self-knowledge, inaccessibility, TNT.

introspection of

intuition, 560, 564, 613, 680, 713; programming of, 605, 609

inversion, 8-9, 81, 146, 681-83, 737-38; see also copies, complementary to original inviolate level, 686-92

Contents

XIX

irrationality vs, rationality in brain/mind, 575-78

itregularites, meta-irregularities, etc. 475-76

Iran, 254

ism, 254-55, 625, 704-6

isomorphisms: between Bongard problems, 660, 669; between brain-structures and reality, 82, 337-39, 350, 502, 569-71; between brains, 369-82; coarse-grained, 147-48, 503; in Contracrostipunctus, 83-85; between Crab's DNA and Crab Canon, 203, 667-68; defined, 9, 49-50; between earthworms. 342-43, 345; of emotions, 163; exotic, prosaic, 159-60; fluid, 338, 350, 362; between form and content in Dialogues, 84-85, 128-30, 204, 667-68; between formal systems and number theory, 408, 625; Godel-numbering and, see Godel isomor phism; between mathematicians, 566; between mathematics and reality, 53-60; between mental processes and programs, 568-73; between MIU-system and 310-system, 261-65; between models of natural numbers, 217; partial, 146-47, 371-82; as revelations, 159-61, 263; as roots of meaning, 49-53, 87-8, 94, 267, 337, 350; self-reps and, 501-3; between something and part of itself, 138-43, 146-47; between spiderwebs, 371-72; transparent, 82, 158, 267; on various levels between same objects, 369; between visual apparatuses, 345-46; in visual processing, 344; see also meaning, translation, copies, decoding, etc.

'Jabberwocky" (Carroll), 366-68, 372-73 Jacquard loom, 25

Jaki, Stanley, 574

"Jammerwoch, Der", (Carroll-Scott), 366-68 "Jaseroque, Le" (Carroll-Warrin), 366-68

Jauch, J. M., 408, 409, 478-79 Jefferson, G., 598

Joan of Arc, 20 Johns, jasper, 703

J6shti, 233, 237, 238, 240, 253, 259, 272 JOSHU (TNT-string), 443

jukeboxes, 154-57, 160-61, 164, 170-71, 174-76,

500

jumping out of a subsystem, 477

jumping out of the system: in advertisement, 478;

by answer-schemers, 462-64; Godel's Theorem

and, see Godefization, essential incompleteness;

illusion of, 478-79, 698; as method to resolve

contradictions, 196-97; in political systems, 692;

by programs, 36-38, 476-78, 678; from 2-D to

3-D, see 2-D vs. 3-D; Zen arid, 255, 479; see also Godelization, Todelization, Escherization, TC battles, repeatability, nonprogrammability, etc.

Kaiserling, Count, 391-92

Kay, Alan, 662 Kennedy, John F., 641

keys, musical, 10, 299, 466, 501; see also modulation

Kim, Scott, 68-69, 503, 523, 719 Kimberger, Johann Philipp, 9, 726 kitchen sink, the, 315 Kleene, Stephen C., 476 Klein bottle, 691

Kliigel, G. S., 91 knitting, 149-50

knots, 341-44, 272, 628

knowledge: accessible vs. inaccessible, 362, 365,

616, 619; encoded in ant colonies, 319-28, 359;

Contents

XX

explicit vs. implicit, 617-18; modularity of,

615-18, 628; procedural v4. declarative. 363-65, 615-17, 630, 654; knowledge transplantation, surgical, 618

koans, 30, 189-91, 233-45, 246-59, 625-26; generated by computer, 625-26; genuine vs.

phony, 234-35, 239, 242, 244, 427, 625-26 Kronecker, Leopold, 216 Kuhn, Thomas, 660

Kupfergiidel, Roman, 394 Kvogen,244-45

La Mettrie, Julien Offroy de, 3, 27, 729

labeling technique, 487-88, 540-41

Lambert, J. H., 91, 92, 99

lamp, meta-lamp, etc., 108-13, 216

language(s): acquisition of, 170, 294, 302; active meanings in, 51-52; Arabic, 623-24; of bees, 360; of the brain, 570; Chinese, 164, 665, 676; collage of, see scripts; computers and, 130-34, 300-1, 363, 586-93, 599-600, 601-3, 619-32, 674-75, 721; effect on thought, 376-77; English, 169, 372-73, 377, 379-80, 619-32, 674-75; flexibility of, 649, 674-75; French, 297, 366-68, 372-73, 377, 501, 618; German, 366-68, 372-73, 380, 665; Hebrew, xviii, 377; hierarchy of, 22; imprecise, 674-5; invisible isomorphisms and, 82; Japanese, 169; as medium for proofs, 88-90, 195; necessary underpinning of, 324; partitions between, 671; procedural grammars for, 131-34, 619-32; reading meaning into computer-produced, 599-600, 625; on Rosetta stone, 165; Russian, 297, 379-80, 642; self-refs in, 431-37, 495-98, 501; see also meaning, translation, etc.

Lashley, Karl, 342, 343, 348

"last step", 462-63, 468

lateral geniculate, 343-44

layers: of deception, 478; of messages, 166-71, 524, 703-4; of stability, 643-45

leakage, between levels of science, 305-6 Legendre, Adrien-Marie, 92 Lehninger, Albert, 504 Leibniz, Wilhelm Gottfried, 24-25, 600 lemmas, 227

Lenat, Douglas, 615

lengthening and shortening rules: decision procedures and, 48-49, 182, 407-8; MIU-system and, 39-40, 260-61, 264, 613; TNT and, 213, 266, 269; see also increasing and decreasing rules, problem reduction

Leonardo of Pisa, see Fibonacci Lermontov, Mikhail, 642

level-conflicts: in Aunt Hillary, 330, 630; in mind/brain, 575-78; in messages, 164, 170, 699-704; between object language and metalanguage,

194, 449-50; in SHRDLU, 630

level-con fusion: ants and, see ants vs. ant colonies;

in art, see 2-D vs. 3-D; authorship and, 3. 608, 720-26; in computer systems, 287, 291, 295, 300-2, 308; of Kimian self-rep, 503; minds/

brains and, 287, 575-77; in Propositional Calculus, 185, 194; subjunc-TV and, 608; self and, 709

level-crossing, in thought, 666, 668

level-mixing in genetics, 509-10, 513-14, 546-48 level-shifting, conceptual, see abstraction, levels of levels: of computer languages, 290-99; distinct vs.

similar, 285, 287; in Escher, 11-15, 6

levels (continued)

Contents

XXI

715-16; haziness of, 13-15, 54618, 715-16; intermediate, 302-3, 317, 324, 532, 632; of irreality, 243, 641; of MU-picture, 311-13, 328-29, 525-26; of partides, 305; in radio news, 128; of reality, 15, 103-25, 128-29, 184-85, 481, 493, 640, 725-26, 737, 739; in recursive processes, 128-29; of rules in thought, 26-27

levels of description: of ant colonies, 315-33; of brain, 349-50, 382. 559, 570-77, 584-85; of caste distribution, 319-29; of chess boards, 285-86; of errors, 294-95; of gases, 308; of human body, 285; of human psyche, 287; of mental processes, 568-73, 575-78, 584-85; of programs, 294-95, 380-81; of television screen, 285; see also holism vs. reductionism levels of meaning: in ant colonies, 319-27; in Contracrostipunctus, 82-85; of DNA, 160, 531-32, 665; in Epimenides paradox, 496, 581, 584-85; of groove-patterns, 83-84; of Mumon, 248; of MUMON, 266-67; of music, 162-63; of neural activity, 575-77; of TNT-strings, 266, 270-71

levels of structure: of enzymes, 510-11, 519, 521, 525-27, 532; of music, 525

liar paradox, see Epimenides paradox

Liberation (Escher), 57-58, 65

lightning calculators, see idiots savants limericks, 483, 736

limitative results, in general, 19, 74, 609, 697, 699 Lincoln, Abraham, 454

lines, geometrical, 19-20, 90-93, 100, 222, 452, 456

LISP, 293, 381, 626, 652, 692, 738-39

List of All Great Mathematicians, 404, 422

Little Harmonic Labyrinth (Bach), 121-23, 129, 130 Little Harmonic Labyrinth (Dialogue), 127, 128-30,

149, 216, 610-11

Little Harmonic Labyrinth (of Majotaur), 119-25 Littlewood, J. E., 564

lizards, 108-9, 110, 115-17, 125 Lobachevskiy, Nikolay, 91

local vs. global properties, 21, 160, 359, 363, 371-75, 543, 582-84, 678

localization of knowledge, in brains and programs, 342, 348, 365, 617-18 Lockwood, Anna, 700

logic, 19-24, 43-45, 99-100, 177-80, 181-97, 461-64, 618-19

Loocus the Thinker, 477