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FIGURE 3. The Royal Theme. When Bach sent a copy of his Musical Offering to the King, he included a dedicatory letter, which is of interest for its prose style if nothing else rather submissive and flattersome. From a modern perspective it seems comical. Also, it probably gives something of the flavor of Bach's apology for his appearance.2 MOST GRACIOUS KING! In deepest humility I dedicate herewith to Your Majesty a musical offering, the noblest part of which derives from Your Majesty's own august hand. With awesome pleasure I still remember the very special Royal grace when, some time ago, during my visit in Potsdam, Your Majesty's Self deigned to play to me a theme for a fugue upon the clavier, and at the same time charged me most graciously to carry it out in Your Majesty's most august presence. To obey Your Majesty's command was my most humble dim. I noticed very soon, however, that, for lack of necessary preparation, the execution of the task did not fare as well as such an excellent theme demanded. I resoled therefore and promptly pledged myself to work out this right Royal theme more fully, and then make it known to the world. This resolve has now been carried out as well as possible, and it has none other than this irreproachable intent, to glorify, if only in a small point, the fame of a monarch whose greatness and power, as in all the sciences of war and peace, so especially in music, everyone must admire and revere. I make bold to add this most humble request: may Your Majesty deign to dignify the present modest labor with a gracious acceptance, and continue to grant Your Majesty's most august Royal grace to Leipzig, July 7 1747 Your Majesty's most humble and obedient servant, THE AUTHOR Some twenty-seven years later, when Bach had been dead for twentyfour years, a Baron named Gottfried van Swieten-to whom, incidentally, Forkel dedicated his biography of Bach, and Beethoven dedicated his First Symphony-had a conversation with King Frederick, which he reported as follows: He [Frederick] spoke to me, among other things, of music, and of a great organist named Bach, who has been for a while in Berlin. This artist [Wilhelm Friedemann Bach] is endowed with a talent superior, in depth of harmonic knowledge and power of execution, to any 1 have heard or can imagine, while those who knew his father claim that he, in turn, was even greater. The King Introduction: A Musico-Logical Offering 14
is of this opinion, and to prove it to me he sang aloud a chromatic fugue subject which he had given this old Bach, who on the spot had made of it a fugue in four parts, then in five parts, and finally in eight parts.' Of course there is no way of knowing whether it was King Frederick or Baron van Swieten who magnified the story into larger-than-life proportions. But it shows how powerful Bach's legend had become by that time. To give an idea of how extraordinary a six-part fugue is, in the entire Well-Tempered Clavier by Bach, containing forty-eight Preludes and Fugues, only two have as many as five parts, and nowhere is there a six-part fugue! One could probably liken the task of improvising a six-part fugue to the playing of sixty simultaneous blindfold games of chess, and winning them all. To improvise an eight-part fugue is really beyond human capability. In the copy which Bach sent to King Frederick, on the page preceding the first sheet of music, was the following inscription: FIG URE 4. ("At the King's Command, the Song and the Remainder Resolved with Canonic Art.") Here Bach is punning on the word "canonic", since it means not only "with canons" but also "in the best possible way". The initials of this inscription are R I C E R C A R -an Italian word, meaning "to seek". And certainly there is a great deal to seek in the Musical Offering. It consists of one three-part fugue, one six-part fugue, ten canons, and a trio sonata. Musical scholars have concluded that the three-part fugue must be, in essence, identical with the one which Bach improvised for King Frederick. The six-part fugue is one of Bach's most complex creations, and its theme is, of course, the Royal Theme. That theme, shown in Figure 3, is a very complex one, rhythmically irregular and highly chromatic (that is, filled with tones which do not belong to the key it is in). To write a decent fugue of even two voices based on it would not be easy for the average musician! Both of the fugues are inscribed "Ricercar", rather than "Fuga". This is another meaning of the word; "ricercar" was, in fact, the original name for the musical form now known as "fugue". By Bach's time, the word "fugue" (or fuga, in Latin and Italian) had become standard, but the term "ricercar" had survived, and now designated an erudite kind of fugue, perhaps too austerely intellectual for the common ear. A similar usage survives in English today: the word "recherche" means, literally, "sought out", but carries the same kind of implication, namely of esoteric or highbrow cleverness. The trio sonata forms a delightful relief from the austerity of the fugues and canons, because it is very melodious and sweet, almost dance- Introduction: A Musico-Logical Offering 15
Page 2 and 3: Contents Overview List of Illustrat
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 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
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Achilles: Wasn't it terrifying to s
Page 72 and 73:
CHAPTER III Figure and Ground Prime
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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
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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
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and a global resolution. In fact, a
Page 140 and 141:
FIGURE 27. Recursive Transition Net
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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
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kinds of "jukeboxes"-intelligences-
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the phenotype. On the other hand, i
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sively more refined attempts along
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Tortoise: I certainly am not. Torto
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to entrap a poor, innocent, bumblin
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well formed strings. They will be d
Page 192 and 193:
square brackets `[' and ']', respec
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RULE OF DETACHMENT: If x and < x⊃
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substituted, and the resulting stri
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(9) . carry-over of line 4 (10) ~P
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Imprudence: Well, yes -- provided y
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and ~ are interchangeable. If this
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general method for synthesizing art
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FIGURE 42. “Crab Canon”, by M.
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Achilles: I don't know. But one thi
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Crab Canon 210
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CHAPTER VIII Typographical Number T
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zero: 0 one: SO two: SSO three: SSS
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One way of changing an open formula
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Ie:SSSSSSO=(SSO • e) Now these th
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(b+SSO), but there is a shorter way
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VARIABLES. a is a variable. If we'r
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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
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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
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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
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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
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arithmetized MIU-system, only there
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methods of reasoning, and therefore
Page 282 and 283:
FIGURE 54. Mobius Strip II, by M. C
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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
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PRINT the word pointed to in the in
Page 298 and 299:
Higher-Level Languages, Compilers,
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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,
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FIGURE 60. [Drawing by the author.)
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there are indeed three "HOLISM"'s,
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are sometimes quite involved, and o
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amount to anything coherent-especia
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practically nothing towards, the up
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if there is an inconsequential amou
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see "meaning" and "purpose", in the
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Crab: I can see that the signals ar
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Achilles: What about a description
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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'
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Crab: Well, what have we here? Oh,
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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!
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outer ring. If an on-center pattern
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each of them, the visual cortex bre
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place at some point after the recep
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Now what does a symbol do, when awa
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The Prototype Principle The list ab
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symbols for other people who are le
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the brain of every human who ever h
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Liftability of Intelligence Thus we
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The fact that a symbol cannot be aw
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"rubbing off" instances from classe
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that tone relationships could not b
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By Lewis Carroll .. . English Frenc
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'Twas brillig, and the slithy toves
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FIGURE 70. A tiny portion of the au
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nearsighted observer-for example an
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months. To make things a little eas
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ing paths in the same way. These co
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the cities represent not only the e
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"Stoliarny Lane" (or "Place"). This
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High-Level Comparisons between Brai
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whom the work is new. Presumably, a
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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
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play one or another of these thirty
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Achilles: But please don't let me d
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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
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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
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Loops and Upper Bounds If we try to
Page 418 and 419:
DEFINE PROCEDURE "MINUS" [M,N]: BLO
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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
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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
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all of life, and there are many mys
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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
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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
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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
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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
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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
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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
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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
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