- 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 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
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system. Its symbols do not move aro
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many cubes assembled to form a larg
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decided to try to capsulize all of
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it by using phrases like "whatever
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Achilles: Wasn't it terrifying to s
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CHAPTER III Figure and Ground Prime
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Illegally Characterizing Primes It
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FIGURE 16. Tiling of the plane usin
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you will see "FIGURE" everywhere, b
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prime number of hyphens. So far, th
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RULE: If x D N Dy is a theorem, the
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Achilles: Naturally, I suppose you
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Achilles: Compassion for the Crab o
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FIGURE 19. The last page of Bach's
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Chapter IV Consistency, Completenes
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FIGURE 20. Visual rendition of the
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The Art of the Fugue A few words on
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Suppose, for instance, that we rein
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definition, it is a foregone conclu
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Adrien-Marie Legendre came up with
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The Possibility of Multiple Interpr
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an extremely vague, unsatisfactory
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FIGURE 22. Relativity, by M. C. Esc
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Is Number Theory the Same In All Co
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How an Interpretation May Make or B
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Voce: Allow me to introduce myself.
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Tortoise: I don’t precisely know,
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Enters the sinister “Tunnel of Lo
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I Genie: Hello, my friends - and th
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Meta-Meta-Genie: I am the MetaMeta-
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Achilles: I see. You mean GOD sits
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granted, it had to be denied - yet
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Tortoise (muttering): Eh? This stor
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(To emphasize his point, he sticks
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manipulate your emotions, and to bu
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Achilles-are you all right? Achille
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forward one whit to the long walk b
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Pushing, Popping, and Stacks In the
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and a global resolution. In fact, a
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FIGURE 27. Recursive Transition Net
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whenever it wants a relative clause
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FIGURE 30. Diagram G, further expan
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It is reminiscent of the Fibonacci
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What corresponds to the bottom in t
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Gplot shows that distribution. The
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Let us be a little more concrete, n
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the physicist has to be able to tak
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FIGURE 37. Butterflies, by M. C. Es
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epeated several times in larger loo
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program) was world champion. But af
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less as cookies than as message bea
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Achilles: I don't understand that a
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CHAPTER VI The Location of Meaning
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The isomorphism between DNA structu
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logic to its structure that its mes
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Now imagine that this is the piece
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Three Layers of Any Message In thes
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' The Location of Meaning 176
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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
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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
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ack the universal quantifier on the
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Illegal Shortcuts Now here is an in
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there was no way to express general
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pattern is a theorem in itself. Tha
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(26) (d+SO)=(Sd+o) transitivity (li
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Formal Reasoning vs. Informal Reaso
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needed in proving that such a typog
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whole Oriental mysticism trip, with
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Achilles: Well, there is one proble
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look like (picks up a nearby napkin
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that if you took all such stories e
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Tortoise: What are the other four l
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(He writes down the phonetic transc
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Tortoise: Quite a yarn. It's hard t
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CHAPTER IX Mumon and Gödel What Is
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Koan: Hogen of Seiryo monastery was
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FIGURE 48. Another World, by M. C.
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Mumon's Commentary: FIGURE 49. Day
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This curious statement seems to abo
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FIGURE 51. Puddle, by M. C. Escher
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FIGURE 53. Three Spheres II, by M.
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We made two crucial observations in
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It is easy. Clearly this mapping be
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I (1) 31 given (2) 311 rule 2 (m=1,
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The Dual Nature of MUMON In order t
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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
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FIGURE 54. Mobius Strip II, by M. C
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FIGURE 55. Pierre de Fermat. Anteat
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Crab: Don't tell me it's a recordin
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FIGURE 56. Cube with Magic Ribbons,
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Crab: I have never seen that illust
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trick to making a machine play well
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takes on a completely different fee
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PRINT the word pointed to in the in
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Higher-Level Languages, Compilers,
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omplished first step up from machin
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machine language program and whatev
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means, then it is quite conceivable
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FIGURE 59. To create intelligent pr
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them out. I like to think of softwa
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neutrons". But the forces which pul
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the chemical bond, the structure of
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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
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voices in a polyphonic piece (often
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Achilles: Ah ... "J. S. BACH". Oh!
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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
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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
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Of course, this does not elevate co
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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-
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a single entity a property which it
- Page 410 and 411:
your number-theoretical entertainme
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CHAPTER XI11 BlooP and FlooP and Gl
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possibly lead to never-ending searc
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Loops and Upper Bounds If we try to
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DEFINE PROCEDURE "MINUS" [M,N]: BLO
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example of a full B1ooP program wou
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FIBO [N] = the Nth Fibonacci number
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computer within a predictable lengt
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The Diagonal Method Very well-now w
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example, that we subtract 1 from th
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the whole trick, lock, stock, and b
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a very long Gödel number. For inst
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A complete pool of all call-less Fl
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vided into two realms: (1) those wh
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FIGURE 74. Above and Below, by M.C.
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"preceded" to the idea of precedenc
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Tortoise: What do you mean? Sentenc
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CHAPTER XIV On Formally Undecidable
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the `611' codon comes in. Its purpo
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Since it is a property of two numbe
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Formula a=a We now replace all free
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of the substitution operation we de
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-or if you prefer, "I am not a theo
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the day: how to prove a system cons
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This suggestion seems rather innocu
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long as those properties are given
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oth instances, one is powerfully dr
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Bifurcations in Number Theory, and
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Now for the simplification of G. It
- Page 468 and 469:
And it is only for that reason that
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are yet further Answer Schemas, suc
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too. The trick will be to find a st
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Naturally, after a while, the whole
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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
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There Is No Recursive Rule for Nami
- Page 484 and 485:
However, it is important to see the
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sufficed: Simplicio, the educated s
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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
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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
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CHAPTER XVI Self-Ref and Self-Rep I
- Page 504 and 505:
The sentence "The sentence "The sen
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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
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FIGURE 91. The four constituent bas
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Messenger KNA and Ribosomes As was
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all of life, and there are many mys
- Page 528 and 529:
amino acid's presence means that su
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FIGURE 96. A section of mRNA passin
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This is accomplished by having the
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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
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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
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and further attempts to improve it
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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
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preter in this context, I mean not
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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
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Dr. Tony Earrwig: There is no recor
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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
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Calculus-and thus took the first st
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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
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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
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subsidiary goals, subsubgoals, etc.
- Page 628 and 629:
haiku-for example. the final sample
- Page 630 and 631:
My program produced the rest. Numbe
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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
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mean.is that any of them can vary (
- Page 654 and 655:
which I can give no answers. There
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FIGURE 120. Bongard problem 47. [Fr
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Templates and Sameness-Detectors On
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FIGURE 123. A small portion of a co
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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
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e triggered. In cells, all the comp
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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
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FIGURE 133. "Sloth Canon",from the
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CHAPTER XX Strange Loops, Or Tangle
- Page 694 and 695:
Unless a person designed himself an
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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
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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
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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
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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