C:\Documents and Settings\dave\Desktop\Godel, Escher, Bach ...

More documents

Recommendations

Info

Contents Overview List of Illustrations Words of Thanks viii xiv xix Part I: GEB Introduction: A Musico-Logical Offering 3 Three-Part Invention 29 Chapter I: The MU-puzzle 33 Two-Part Invention 43 Chapter II: Meaning and Form in Mathematics 46 Sonata for Unaccompanied Achilles 61 Chapter III: Figure and Ground 64 Contracrostipunctus 75 Chapter IV: Consistency, Completeness, and Geometry 82 Little Harmonic Labyrinth 103 Chapter V: Recursive Structures and Processes 127 Canon by Intervallic Augmentation 153 Chapter VI: The Location of Meaning 158 Chromatic Fantasy, And Feud 177 Chapter VII: The Propositional Calculus 181 Crab Canon 199 Chapter VIII: Typographical Number Theory 204 A Mu Offering 231 Chapter IX: Mumon and Gödel 246 Contents II
Part II EGB Prelude ... 275 Chapter X: Levels of Description, and Computer Systems 285 Ant Fugue 311 Chapter XI: Brains and Thoughts 337 English French German Suit 366 Chapter XII: Minds and Thoughts 369 Aria with Diverse Variations 391 Chapter XIII: BlooP and FlooP and GlooP 406 Air on G's String 431 Chapter XIV: On Formally Undecidable Propositions of TNT and Related Systems 438 Birthday Cantatatata ... 461 Chapter XV: Jumping out of the System 465 Edifying Thoughts of a Tobacco Smoker 480 Chapter XVI: Self-Ref and Self-Rep 495 The Magn fierab, Indeed 549 Chapter XVII: Church, Turing, Tarski, and Others 559 SHRDLU, Toy of Man's Designing 586 Chapter XVIII: Artificial Intelligence: Retrospects 594 Contrafactus 633 Chapter XIX: Artificial Intelligence: Prospects 641 Sloth Canon 681 Chapter XX: Strange Loops, Or Tangled Hierarchies 684 Six-Part Ricercar 720 Notes 743 Bibliography 746 Credits 757 Index 759 Contents III
Page 4 and 5: Overview Part I: GEB Introduction:
Page 6 and 7: of whether there exist rules for th
Page 8 and 9: Birthday Cantatatata ... In which A
Page 10 and 11: FIGURE 1. Johann Sebastian Bach, in
Page 12 and 13: pleased he would be to have the eld
Page 14 and 15: FIGURE 3. The Royal Theme. When Bac
Page 16 and 17: able. Nevertheless, it too is based
Page 18 and 19: which many ideas and forms have bee
Page 20 and 21: FIGURE 6. Ascending and Descending,
Page 22 and 23: Introduction: A Musico-Logical Offe
Page 24 and 25: FIGURE 9. Kurt Godel. Introduction:
Page 26 and 27: We shall examine the Godel construc
Page 28 and 29: ent kinds of "points" and "lines" i
Page 30 and 31: interest went no further than set t
Page 32 and 33: lenge: to demonstrate rigorously-pe
Page 34 and 35: ounds. This theory was tightly link
Page 36 and 37: present new concepts twice: almost
Page 38 and 39: Figure 10. Mobius strip by M.C.Esch
Page 40 and 41: ACHILLES: Hah! Zeno: And now the To
Page 42 and 43: But although all of these are legit
Page 44 and 45: (3) MIII from (2) by rule II (4) MI
Page 46 and 47: oring ritual of checkmating. Althou
Page 48 and 49: Step 3: Apply every applicable rule
Page 50 and 51: FIGURE 12. Sky Castle, by M. C.: Es
Page 52 and 53:
Schools are invented, which will no
Page 54 and 55:
CHAPTER 11 Meaning and Form in Math
Page 56 and 57:
hyphen-group. for instance, --p--q
Page 58 and 59:
It is cause for joy when a mathemat
Page 60 and 61:
in a language, when we have learned
Page 62 and 63:
system. Its symbols do not move aro
Page 64 and 65:
many cubes assembled to form a larg
Page 66 and 67:
decided to try to capsulize all of
Page 68 and 69:
it by using phrases like "whatever
Page 70 and 71:
Achilles: Wasn't it terrifying to s
Page 72 and 73:
CHAPTER III Figure and Ground Prime
Page 74 and 75:
Illegally Characterizing Primes It
Page 76 and 77:
FIGURE 16. Tiling of the plane usin
Page 78 and 79:
you will see "FIGURE" everywhere, b
Page 80 and 81:
prime number of hyphens. So far, th
Page 82 and 83:
RULE: If x D N Dy is a theorem, the
Page 84 and 85:
Achilles: Naturally, I suppose you
Page 86 and 87:
Achilles: Compassion for the Crab o
Page 88 and 89:
FIGURE 19. The last page of Bach's
Page 90 and 91:
Chapter IV Consistency, Completenes
Page 92 and 93:
FIGURE 20. Visual rendition of the
Page 94 and 95:
The Art of the Fugue A few words on
Page 96 and 97:
Suppose, for instance, that we rein
Page 98 and 99:
definition, it is a foregone conclu
Page 100 and 101:
Adrien-Marie Legendre came up with
Page 102 and 103:
The Possibility of Multiple Interpr
Page 104 and 105:
an extremely vague, unsatisfactory
Page 106 and 107:
FIGURE 22. Relativity, by M. C. Esc
Page 108 and 109:
Is Number Theory the Same In All Co
Page 110 and 111:
How an Interpretation May Make or B
Page 112 and 113:
Voce: Allow me to introduce myself.
Page 114 and 115:
Tortoise: I don’t precisely know,
Page 116 and 117:
Enters the sinister “Tunnel of Lo
Page 118 and 119:
I Genie: Hello, my friends - and th
Page 120 and 121:
Meta-Meta-Genie: I am the MetaMeta-
Page 122 and 123:
Achilles: I see. You mean GOD sits
Page 124 and 125:
granted, it had to be denied - yet
Page 126 and 127:
Tortoise (muttering): Eh? This stor
Page 128 and 129:
(To emphasize his point, he sticks
Page 130 and 131:
manipulate your emotions, and to bu
Page 132 and 133:
Achilles-are you all right? Achille
Page 134 and 135:
forward one whit to the long walk b
Page 136 and 137:
Pushing, Popping, and Stacks In the
Page 138 and 139:
and a global resolution. In fact, a
Page 140 and 141:
FIGURE 27. Recursive Transition Net
Page 142 and 143:
whenever it wants a relative clause
Page 144 and 145:
FIGURE 30. Diagram G, further expan
Page 146 and 147:
It is reminiscent of the Fibonacci
Page 148 and 149:
What corresponds to the bottom in t
Page 150 and 151:
Gplot shows that distribution. The
Page 152 and 153:
Let us be a little more concrete, n
Page 154 and 155:
the physicist has to be able to tak
Page 156 and 157:
FIGURE 37. Butterflies, by M. C. Es
Page 158 and 159:
epeated several times in larger loo
Page 160 and 161:
program) was world champion. But af
Page 162 and 163:
less as cookies than as message bea
Page 164 and 165:
Achilles: I don't understand that a
Page 166 and 167:
CHAPTER VI The Location of Meaning
Page 168 and 169:
The isomorphism between DNA structu
Page 170 and 171:
logic to its structure that its mes
Page 172 and 173:
Now imagine that this is the piece
Page 174 and 175:
Three Layers of Any Message In thes
Page 176 and 177:
' The Location of Meaning 176
Page 178 and 179:
explicit language. To find an expli
Page 180 and 181:
kinds of "jukeboxes"-intelligences-
Page 182 and 183:
the phenotype. On the other hand, i
Page 184 and 185:
sively more refined attempts along
Page 186 and 187:
Tortoise: I certainly am not. Torto
Page 188 and 189:
to entrap a poor, innocent, bumblin
Page 190 and 191:
well formed strings. They will be d
Page 192 and 193:
square brackets `[' and ']', respec
Page 194 and 195:
RULE OF DETACHMENT: If x and < x⊃
Page 196 and 197:
substituted, and the resulting stri
Page 198 and 199:
(9) . carry-over of line 4 (10) ~P
Page 200 and 201:
Imprudence: Well, yes -- provided y
Page 202 and 203:
and ~ are interchangeable. If this
Page 204 and 205:
general method for synthesizing art
Page 206 and 207:
FIGURE 42. “Crab Canon”, by M.
Page 208 and 209:
Achilles: I don't know. But one thi
Page 210 and 211:
Crab Canon 210
Page 212 and 213:
CHAPTER VIII Typographical Number T
Page 214 and 215:
zero: 0 one: SO two: SSO three: SSS
Page 216 and 217:
One way of changing an open formula
Page 218 and 219:
Ie:SSSSSSO=(SSO • e) Now these th
Page 220 and 221:
(b+SSO), but there is a shorter way
Page 222 and 223:
VARIABLES. a is a variable. If we'r
Page 224 and 225:
which can be derived in the same wa
Page 226 and 227:
ack the universal quantifier on the
Page 228 and 229:
Illegal Shortcuts Now here is an in
Page 230 and 231:
there was no way to express general
Page 232 and 233:
pattern is a theorem in itself. Tha
Page 234 and 235:
(26) (d+SO)=(Sd+o) transitivity (li
Page 236 and 237:
Formal Reasoning vs. Informal Reaso
Page 238 and 239:
needed in proving that such a typog
Page 240 and 241:
whole Oriental mysticism trip, with
Page 242 and 243:
Achilles: Well, there is one proble
Page 244 and 245:
look like (picks up a nearby napkin
Page 246 and 247:
that if you took all such stories e
Page 248 and 249:
Tortoise: What are the other four l
Page 250 and 251:
(He writes down the phonetic transc
Page 252 and 253:
Tortoise: Quite a yarn. It's hard t
Page 254 and 255:
CHAPTER IX Mumon and Gödel What Is
Page 256 and 257:
Koan: Hogen of Seiryo monastery was
Page 258 and 259:
FIGURE 48. Another World, by M. C.
Page 260 and 261:
Mumon's Commentary: FIGURE 49. Day
Page 262 and 263:
This curious statement seems to abo
Page 264 and 265:
FIGURE 51. Puddle, by M. C. Escher
Page 266 and 267:
FIGURE 53. Three Spheres II, by M.
Page 268 and 269:
We made two crucial observations in
Page 270 and 271:
It is easy. Clearly this mapping be
Page 272 and 273:
I (1) 31 given (2) 311 rule 2 (m=1,
Page 274 and 275:
The Dual Nature of MUMON In order t
Page 276 and 277:
will have to Gödel-number TNT itse
Page 278 and 279:
arithmetized MIU-system, only there
Page 280 and 281:
methods of reasoning, and therefore
Page 282 and 283:
FIGURE 54. Mobius Strip II, by M. C
Page 284 and 285:
FIGURE 55. Pierre de Fermat. Anteat
Page 286 and 287:
Crab: Don't tell me it's a recordin
Page 288 and 289:
FIGURE 56. Cube with Magic Ribbons,
Page 290 and 291:
Crab: I have never seen that illust
Page 292 and 293:
trick to making a machine play well
Page 294 and 295:
takes on a completely different fee
Page 296 and 297:
PRINT the word pointed to in the in
Page 298 and 299:
Higher-Level Languages, Compilers,
Page 300 and 301:
omplished first step up from machin
Page 302 and 303:
machine language program and whatev
Page 304 and 305:
means, then it is quite conceivable
Page 306 and 307:
FIGURE 59. To create intelligent pr
Page 308 and 309:
them out. I like to think of softwa
Page 310 and 311:
neutrons". But the forces which pul
Page 312 and 313:
the chemical bond, the structure of
Page 314 and 315:
Because of many canceling effects,
Page 316 and 317:
FIGURE 60. [Drawing by the author.)
Page 318 and 319:
there are indeed three "HOLISM"'s,
Page 320 and 321:
are sometimes quite involved, and o
Page 322 and 323:
amount to anything coherent-especia
Page 324 and 325:
practically nothing towards, the up
Page 326 and 327:
if there is an inconsequential amou
Page 328 and 329:
see "meaning" and "purpose", in the
Page 330 and 331:
Crab: I can see that the signals ar
Page 332 and 333:
Achilles: What about a description
Page 334 and 335:
full system. For example, Aunt Hill
Page 336 and 337:
voices in a polyphonic piece (often
Page 338 and 339:
Achilles: Ah ... "J. S. BACH". Oh!
Page 340 and 341:
FIGURE 63. During emigrations arm'
Page 342 and 343:
Crab: Well, what have we here? Oh,
Page 344 and 345:
e mixed right in with the symbols t
Page 346 and 347:
The type of decision which a neuron
Page 348 and 349:
Earthworms have isomorphic brains!
Page 350 and 351:
outer ring. If an on-center pattern
Page 352 and 353:
each of them, the visual cortex bre
Page 354 and 355:
place at some point after the recep
Page 356 and 357:
Now what does a symbol do, when awa
Page 358 and 359:
The Prototype Principle The list ab
Page 360 and 361:
symbols for other people who are le
Page 362 and 363:
the brain of every human who ever h
Page 364 and 365:
Liftability of Intelligence Thus we
Page 366 and 367:
The fact that a symbol cannot be aw
Page 368 and 369:
"rubbing off" instances from classe
Page 370 and 371:
that tone relationships could not b
Page 372 and 373:
By Lewis Carroll .. . English Frenc
Page 374 and 375:
'Twas brillig, and the slithy toves
Page 376 and 377:
FIGURE 70. A tiny portion of the au
Page 378 and 379:
nearsighted observer-for example an
Page 380 and 381:
months. To make things a little eas
Page 382 and 383:
ing paths in the same way. These co
Page 384 and 385:
the cities represent not only the e
Page 386 and 387:
"Stoliarny Lane" (or "Place"). This
Page 388 and 389:
High-Level Comparisons between Brai
Page 390 and 391:
whom the work is new. Presumably, a
Page 392 and 393:
Subsystems and Shared Code Typical
Page 394 and 395:
Of course, this does not elevate co
Page 396 and 397:
machine for the purposes of our dis
Page 398 and 399:
play one or another of these thirty
Page 400 and 401:
Achilles: But please don't let me d
Page 402 and 403:
Achilles: I think you should call i
Page 404 and 405:
Tortoise: You can put it that way i
Page 406 and 407:
Tortoise: The first type of search-
Page 408 and 409:
a single entity a property which it
Page 410 and 411:
your number-theoretical entertainme
Page 412 and 413:
CHAPTER XI11 BlooP and FlooP and Gl
Page 414 and 415:
possibly lead to never-ending searc
Page 416 and 417:
Loops and Upper Bounds If we try to
Page 418 and 419:
DEFINE PROCEDURE "MINUS" [M,N]: BLO
Page 420 and 421:
example of a full B1ooP program wou
Page 422 and 423:
FIBO [N] = the Nth Fibonacci number
Page 424 and 425:
computer within a predictable lengt
Page 426 and 427:
The Diagonal Method Very well-now w
Page 428 and 429:
example, that we subtract 1 from th
Page 430 and 431:
the whole trick, lock, stock, and b
Page 432 and 433:
a very long Gödel number. For inst
Page 434 and 435:
A complete pool of all call-less Fl
Page 436 and 437:
vided into two realms: (1) those wh
Page 438 and 439:
FIGURE 74. Above and Below, by M.C.
Page 440 and 441:
"preceded" to the idea of precedenc
Page 442 and 443:
Tortoise: What do you mean? Sentenc
Page 444 and 445:
CHAPTER XIV On Formally Undecidable
Page 446 and 447:
the `611' codon comes in. Its purpo
Page 448 and 449:
Since it is a property of two numbe
Page 450 and 451:
Formula a=a We now replace all free
Page 452 and 453:
of the substitution operation we de
Page 454 and 455:
-or if you prefer, "I am not a theo
Page 456 and 457:
the day: how to prove a system cons
Page 458 and 459:
This suggestion seems rather innocu
Page 460 and 461:
long as those properties are given
Page 462 and 463:
oth instances, one is powerfully dr
Page 464 and 465:
Bifurcations in Number Theory, and
Page 466 and 467:
Now for the simplification of G. It
Page 468 and 469:
And it is only for that reason that
Page 470 and 471:
are yet further Answer Schemas, suc
Page 472 and 473:
too. The trick will be to find a st
Page 474 and 475:
Naturally, after a while, the whole
Page 476 and 477:
pleteness here is part and parcel o
Page 478 and 479:
Among other things, it has to be ab
Page 480 and 481:
FIGURE 76. Dragon, by M. C. Escher
Page 482 and 483:
There Is No Recursive Rule for Nami
Page 484 and 485:
However, it is important to see the
Page 486 and 487:
sufficed: Simplicio, the educated s
Page 488 and 489:
Crab: You mean a normal pipe with a
Page 490 and 491:
A charming philosophy, is it not? A
Page 492 and 493:
FIGURE 79. Tobacco Mosaic Virus. Fr
Page 494 and 495:
to tighten up my defenses against t
Page 496 and 497:
FIGURE 80. The Fair Captive, by Ren
Page 498 and 499:
Edifying Thoughts of a Tobacco Smok
Page 500 and 501:
astonishment crosses his face.) Goo
Page 502 and 503:
CHAPTER XVI Self-Ref and Self-Rep I
Page 504 and 505:
The sentence "The sentence "The sen
Page 506 and 507:
TEMPLATE-is never interpreted as a
Page 508 and 509:
the blooP-like language above, usin
Page 510 and 511:
left, of course, for the reader to
Page 512 and 513:
Molecular Biology, enunciated by Fr
Page 514 and 515:
You can perhaps remember this molec
Page 516 and 517:
and see how the enzyme acts on the
Page 518 and 519:
primary structure is meant its amin
Page 520 and 521:
cycle goes on and on. This can go o
Page 522 and 523:
FIGURE 91. The four constituent bas
Page 524 and 525:
Messenger KNA and Ribosomes As was
Page 526 and 527:
all of life, and there are many mys
Page 528 and 529:
amino acid's presence means that su
Page 530 and 531:
FIGURE 96. A section of mRNA passin
Page 532 and 533:
This is accomplished by having the
Page 534 and 535:
the Prelude, Ant Fugue. The global
Page 536 and 537:
called its active site, and any mol
Page 538 and 539:
Comparison of DNA's Self-Rep Method
Page 540 and 541:
Self-Rep and Self-Rep 528
Page 542 and 543:
FIGURE 100. The Godel Code. Under t
Page 544 and 545:
FIGURE 101. The T4 bacterial virus
Page 546 and 547:
Self-Rep and Self-Rep 534
Page 548 and 549:
mechanisms for examining whether DN
Page 550 and 551:
themselves. Somtiems enzymes may be
Page 552 and 553:
prevented from being transcribed, w
Page 554 and 555:
novel features. We have seen, in th
Page 556 and 557:
The Magnificrab, Indeed, It is spri
Page 558 and 559:
Crab: Well, it was this way. This f
Page 560 and 561:
Crab: I don't understand. What stat
Page 562 and 563:
and further attempts to improve it
Page 564 and 565:
numbers-such as the square root of
Page 566 and 567:
CHAPTER XVII Church, Turing, Tarski
Page 568 and 569:
It was proven in 1936 by the Americ
Page 570 and 571:
FIGURE 105. Srinivasa Ramanujan and
Page 572 and 573:
mediately jumped to fourth powers.
Page 574 and 575:
"Idiots 'Savants" There is another
Page 576 and 577:
Representation of Knowledge about t
Page 578 and 579:
contrast to processes which are sup
Page 580 and 581:
FIGURE 108. Crucial to the endeavor
Page 582 and 583:
This view is prevalent among certai
Page 584 and 585:
level is determined by whether or-
Page 586 and 587:
power equal to that of FlooP-that i
Page 588 and 589:
When interpreted, it says: "The ari
Page 590 and 591:
preter in this context, I mean not
Page 592 and 593:
far, then presumably the rest of th
Page 594 and 595:
FIGURE 111. "Find a block which is
Page 596 and 597:
FIGURE 112. "Will you please stack
Page 598 and 599:
Dr. Tony Earrwig: SHRDLU remembers
Page 600 and 601:
Dr. Tony Earrwig: There is no recor
Page 602 and 603:
telligence when it first came under
Page 604 and 605:
line of thought, usually turning up
Page 606 and 607:
does the card in my right hand belo
Page 608 and 609:
Calculus-and thus took the first st
Page 610 and 611:
creating original thoughts or works
Page 612 and 613:
as accurately as possible. The effe
Page 614 and 615:
following the program does not serv
Page 616 and 617:
human intellect, and the computer h
Page 618 and 619:
the infiniteness of the goal stack,
Page 620 and 621:
than not come as sudden flashes of
Page 622 and 623:
is a collection of programs develop
Page 624 and 625:
the same information in several dif
Page 626 and 627:
subsidiary goals, subsubgoals, etc.
Page 628 and 629:
haiku-for example. the final sample
Page 630 and 631:
My program produced the rest. Numbe
Page 632 and 633:
as-perhaps emptier than-the p and q
Page 634 and 635:
able in any way at all (which I bel
Page 636 and 637:
One of the basic viewpoints underly
Page 638 and 639:
in the box" as a single phrase indi
Page 640 and 641:
This gives you a glimpse behind the
Page 642 and 643:
next-to-last (and only) movement wa
Page 644 and 645:
Crab: So that you can tune it to th
Page 646 and 647:
Announcer: -but it seems to be a sp
Page 648 and 649:
Achilles: What about you Mr. Sloth?
Page 650 and 651:
The New The New Yorker commented: A
Page 652 and 653:
mean.is that any of them can vary (
Page 654 and 655:
which I can give no answers. There
Page 656 and 657:
FIGURE 120. Bongard problem 47. [Fr
Page 658 and 659:
Templates and Sameness-Detectors On
Page 660 and 661:
FIGURE 123. A small portion of a co
Page 662 and 663:
The difference between a triangle a
Page 664 and 665:
etween Class I and Class II in BP 8
Page 666 and 667:
FIGURE 126. Bongard problem 55. [Fr
Page 668 and 669:
FIGURE 130. Bongard problems 70-71.
Page 670 and 671:
problems lies very close to the cor
Page 672 and 673:
e triggered. In cells, all the comp
Page 674 and 675:
egun writing Dialogues, and there w
Page 676 and 677:
Structure of the Dialogue. This fin
Page 678 and 679:
Multiple Representations Not only m
Page 680 and 681:
Forced matches occur every day in t
Page 682 and 683:
measured on an imaginary "keyboard
Page 684 and 685:
seriously. taut in some ways, those
Page 686 and 687:
similar reasons. It will represent
Page 688 and 689:
Speculation: Probably the differenc
Page 690 and 691:
FIGURE 133. "Sloth Canon",from the
Page 692 and 693:
CHAPTER XX Strange Loops, Or Tangle
Page 694 and 695:
Unless a person designed himself an
Page 696 and 697:
Now it is possible to go considerab
Page 698 and 699:
FIGURE 135. Drawing Hands, by M. C.
Page 700 and 701:
the picture, this is unlikely-but f
Page 702 and 703:
evidence: C. And for the validity o
Page 704 and 705:
The upshot is that the total pictur
Page 706 and 707:
undermines itself in a Gödelian wa
Page 708 and 709:
e doing Cage justice, but to me it
Page 710 and 711:
FIGURE 139. Smoke Signal. [Drawing
Page 712 and 713:
wooden crate on a museum floor is j
Page 714 and 715:
without ideas-which, however you in
Page 716 and 717:
this string a theorem of TNT?" Now
Page 718 and 719:
At the crux, then, of our understan
Page 720 and 721:
If you play a game against certain
Page 722 and 723:
FIGURE 142. Print Gallery, by M. C.
Page 724 and 725:
We have eliminated the "town" level
Page 726 and 727:
FIGURE 148: Two complete cycles of
Page 728 and 729:
Six-Part Ricercar Achilles has brou
Page 730 and 731:
what would the smart-stupid itself
Page 732 and 733:
Achilles: Big Deal! It was just som
Page 734 and 735:
Author: But I am also comparing you
Page 736 and 737:
welcome the challenge of trying out
Page 738 and 739:
Case, due to my own Insufficiencies
Page 740 and 741:
FIGURE 149. Verbum, by M. C. Escher
Page 742 and 743:
Grab: I understand fully your demur
Page 744 and 745:
Turing: My test. Please, consider i
Page 746 and 747:
Author: to be sure. My Crab Canon e
Page 748 and 749:
Crab: Quite Gödelian, Tell me -doe
Page 750 and 751:
unlikely threesome, at first though
Page 752 and 753:
1 Lewis Carroll. The Annotated Alic
Page 754 and 755:
Bibliography The presence of two as
Page 756 and 757:
set theory-is here explained to non
Page 758 and 759:
Goodman's famous problem-words "ble
Page 760 and 761:
• Jeffrey, Richard. Formal Logic:
Page 762 and 763:
Meyer, can. ''Essai d'application d
Page 764 and 765:
* Sagan, Carl, ed. Communication wi
Page 766 and 767:
Tietze, Heinrich. Famous Problems o
Page 768 and 769:
Credits Figures: Fig. 1, Johann Seb
Page 770 and 771:
INDEX AABB form, 130, 227 Abel, Nie
Page 772 and 773:
axiom schemata, 47, 48, 65, 87, 468
Page 774 and 775:
centrality, 374-75 centromere, 668
Page 776 and 777:
573-74, 579-81; receives presents a
Page 778 and 779:
emulation, 295 Endlessly Rising Can
Page 780 and 781:
Friedrich, 92, 100 Gebstadter, Egbe
Page 782 and 783:
679; see also software and hardware
Page 784 and 785:
616, 619; encoded in ant colonies,
Page 786 and 787:
meaning: built on triggering-patter
Page 788 and 789:
tioning of, 575-77; firing of, 83,
Page 790 and 791:
points (geometrical), 19-20, 90, 92
Page 792 and 793:
ecord players: alien-rejecting, 487
Page 794 and 795:
490, 492; total, 493 self-knowledge
Page 796 and 797:
compared with ripples, 356-37; conc
Page 798 and 799:
triggering patterns of symbols: dep
Page 800:
Well-Tested Conjecture (Fourmi), 33
show all

C:\Documents and Settings\dave\Desktop\Godel, Escher, Bach ...

Create successful ePaper yourself

Delete template?

Save as template?