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Introduction: A Musico-Logical Offering 22
haziness in level-counting, not only in Escher pictures, but in hierarchical, many-level systems. We will sharpen our understanding of this haziness later on. But let us not get too distracted now' As we tighten our loop, we come to the remarkable Drawing Hands (Fig. 135), in which each of two hands draws the other: a two-step Strange Loop. And finally, the tightest of all Strange Loops is realized in Print Gallery (Fig. 142): a picture of a picture which contains itself. Or is it a picture of a gallery which contains itself? Or of a town which contains itself? Or a young man who contains himself'? (Incidentally, the illusion underlying Ascending and Descending and Waterfall was not invented by Escher, but by Roger Penrose, a British mathematician, in 1958. However, the theme of the Strange Loop was already present in Escher's work in 1948, the year he drew Drawing Hands. Print Gallery dates from 1956.) Implicit in the concept of Strange Loops is the concept of infinity, since what else is a loop but a way of representing an endless process in a finite way? And infinity plays a large role n many of Escher's drawings. Copies of one single theme often fit into each' other, forming visual analogues to the canons of Bach. Several such patterns can be seen in Escher's famous print Metamorphosis (Fig. 8). It is a little like the "Endlessly Rising Canon": wandering further and further from its starting point, it suddenly is back. In the tiled planes of Metamorphosis and other pictures, there are already suggestions of infinity. But wilder visions of infinity appear in other drawings by Escher. In some of his drawings, one single theme can appear on different levels of reality. For instance, one level in a drawing might clearly be recognizable as representing fantasy or imagination; another level would be recognizable as reality. These two levels might be the only explicitly portrayed levels. But the mere presence of these two levels invites the viewer to look upon himself as part of yet another level; and by taking that step, the viewer cannot help getting caught up in Escher's implied chain of levels, in which, for any one level, there is always another level above it of greater "reality", and likewise, there is always a level below, "more imaginary" than it is. This can be mind-boggling in itself. However, what happens if the chain of levels is not linear, but forms a loop? What is real, then, and what is fantasy? The genius of Escher was that he could not only concoct, but actually portray, dozens of half-real, half-mythical worlds, worlds filled with Strange Loops, which he seems to be inviting his viewers to enter. Gödel In the examples we have seen of Strange Loops by Bach and Escher, there is a conflict between the finite and the infinite, and hence a strong sense of paradox. Intuition senses that there is something mathematical involved here. And indeed in our own century a mathematical counterpart was discovered, with the most enormous repercussions. And, just as the Bach and Escher loops appeal to very simple and ancient intuitions-a musical scale, a staircase-so this discovery, by K. Gödel, of a Strange Loop in Introduction: A Musico-Logical Offering 23
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 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 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
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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
Page 242 and 243:
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
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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
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Tortoise: You can put it that way i
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Tortoise: The first type of search-
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a single entity a property which it
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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
Page 428 and 429:
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
Page 470 and 471:
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
Page 476 and 477:
pleteness here is part and parcel o
Page 478 and 479:
Among other things, it has to be ab
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FIGURE 76. Dragon, by M. C. Escher
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There Is No Recursive Rule for Nami
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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
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Edifying Thoughts of a Tobacco Smok
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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
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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
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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
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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
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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
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Crab: Well, it was this way. This f
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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
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It was proven in 1936 by the Americ
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FIGURE 105. Srinivasa Ramanujan and
Page 572 and 573:
mediately jumped to fourth powers.
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"Idiots 'Savants" There is another
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Representation of Knowledge about t
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contrast to processes which are sup
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FIGURE 108. Crucial to the endeavor
Page 582 and 583:
This view is prevalent among certai
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level is determined by whether or-
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power equal to that of FlooP-that i
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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
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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
Page 602 and 603:
telligence when it first came under
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line of thought, usually turning up
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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
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as accurately as possible. The effe
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following the program does not serv
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human intellect, and the computer h
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the infiniteness of the goal stack,
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than not come as sudden flashes of
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is a collection of programs develop
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the same information in several dif
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subsidiary goals, subsubgoals, etc.
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haiku-for example. the final sample
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My program produced the rest. Numbe
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as-perhaps emptier than-the p and q
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able in any way at all (which I bel
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One of the basic viewpoints underly
Page 638 and 639:
in the box" as a single phrase indi
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This gives you a glimpse behind the
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next-to-last (and only) movement wa
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Crab: So that you can tune it to th
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Announcer: -but it seems to be a sp
Page 648 and 649:
Achilles: What about you Mr. Sloth?
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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
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FIGURE 123. A small portion of a co
Page 662 and 663:
The difference between a triangle a
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etween Class I and Class II in BP 8
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FIGURE 126. Bongard problem 55. [Fr
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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
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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.
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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
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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
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unlikely threesome, at first though
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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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