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all of life, and there are many mysteries connected with it. But in essence it is easy to describe. Let us first give a picturesque image, and then render it more precise. Imagine the mRNA to be like a long piece of magnetic recording tape, and the ribosome to be like a tape recorder. As the tape passes through the playing head of the recorder, it is "read" and converted into music, or other sounds. Thus magnetic markings are "translated" into notes. Similarly, when a "tape" of mRNA passes through the "playing head" of a ribosome, the "notes" which are produced are amine acids, and the "pieces of music" which they make up are proteins. This is what translation is all about: it is shown in Figure 96. The Genetic Code But how can a ribosome produce a chain of amino acids when it is reading a chain of nucleotides This mystery was solved in the early 1960's by the efforts of a large number of people, and at the core of the answer lies the Genetic Code-a mapping from triplets of nucleotides into amino acids (see Fig. 94). This is in spirit extremely similar to the Typogenetic Code, except that here, three consecutive bases (or nucleotides) form a codon, whereas there, CUA GAU only two were needed. Thus , C u A g A u there must be 4x4x4 (equals 64) different entries in the A typical segment of mRNA table, instead of sixteen. A read first as two triplets ribosome clicks down a strand (above), and second as three of RNA three nucleotides at duplets (below): an example a time-which is to say, one. of hemiolia in biochemistry codon at a time -and each time it does so, it appends a single new amino acid to the protein it is presently manufacturing. Thus, a protein comes out of the ribosome amino acid by amino acid. Tertiary Structure However, as a protein emerges from a ribosome, it is not only getting longer and longer, but it is also continually folding itself up into an extraordinary three-dimensional shape, very much in the way that those funny little Fourth-of-July fireworks called "snakes" simultaneously grow longer and curl up, when they are lit. This fancy shape is called the protein's tertiary structure (Fig. 95), while the amino acid sequence per se is called the primary structure of the protein. The tertiary structure is implicit in the primary structure, just as in Typogenetics. However, the recipe for deriving the tertiary structure, if you know only the primary structure, is by far more complex than that given in Typogenetics. In fact, it is one of the outstanding problems of contemporary molecular biology to figure out some rules by which the tertiary structure of a protein can be predicted if only its primary structure is known. Self-Rep and Self-Rep 514
The Genetic Code. U C A G phe ser tyr cys U U phe leu leu ser ser ser tyr punt. punc. cys punt. trp C A G leu pro his arg U C leu leu Ieu pro pro pro his gin gln arg arg arg C A G A ile ile ile thr thr thr asn asn lys ser ser arg U C A met thr lys arg G G val ala asp gly U val ala asp gly C val ala glu gly A val ala glu gly G FIGURE 94. The Genetic Code, by which each triplet in a strand of messenger RNA codes for one of twenty amino acids (or a punctuation mark). Reductionistic Explanation of Protein Function Another discrepancy between Typogenetics and true genetics-and this is probably the most serious one of all-is this: whereas in Typogenetics, each component amino acid of an enzyme is responsible for some specific "piece of the action", in real enzymes, individual amino acids cannot be assigned such clear roles. It is the tertiary structure as a whole which determines the mode in which an enzyme will function; there is no way one can say, "This Self-Rep and Self-Rep 515
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Contents Overview List of Illustrat
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Overview Part I: GEB Introduction:
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of whether there exist rules for th
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Birthday Cantatatata ... In which A
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FIGURE 1. Johann Sebastian Bach, in
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pleased he would be to have the eld
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FIGURE 3. The Royal Theme. When Bac
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able. Nevertheless, it too is based
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which many ideas and forms have bee
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FIGURE 6. Ascending and Descending,
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Introduction: A Musico-Logical Offe
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FIGURE 9. Kurt Godel. Introduction:
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We shall examine the Godel construc
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ent kinds of "points" and "lines" i
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interest went no further than set t
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lenge: to demonstrate rigorously-pe
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ounds. This theory was tightly link
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present new concepts twice: almost
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Figure 10. Mobius strip by M.C.Esch
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ACHILLES: Hah! Zeno: And now the To
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But although all of these are legit
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(3) MIII from (2) by rule II (4) MI
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oring ritual of checkmating. Althou
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Step 3: Apply every applicable rule
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FIGURE 12. Sky Castle, by M. C.: Es
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Schools are invented, which will no
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CHAPTER 11 Meaning and Form in Math
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hyphen-group. for instance, --p--q
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It is cause for joy when a mathemat
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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
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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
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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
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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
Page 476 and 477: pleteness here is part and parcel o
Page 478 and 479: Among other things, it has to be ab
Page 480 and 481: FIGURE 76. Dragon, by M. C. Escher
Page 482 and 483: There Is No Recursive Rule for Nami
Page 484 and 485: However, it is important to see the
Page 486 and 487: sufficed: Simplicio, the educated s
Page 488 and 489: Crab: You mean a normal pipe with a
Page 490 and 491: A charming philosophy, is it not? A
Page 492 and 493: FIGURE 79. Tobacco Mosaic Virus. Fr
Page 494 and 495: to tighten up my defenses against t
Page 496 and 497: FIGURE 80. The Fair Captive, by Ren
Page 498 and 499: Edifying Thoughts of a Tobacco Smok
Page 500 and 501: astonishment crosses his face.) Goo
Page 502 and 503: CHAPTER XVI Self-Ref and Self-Rep I
Page 504 and 505: The sentence "The sentence "The sen
Page 506 and 507: TEMPLATE-is never interpreted as a
Page 508 and 509: the blooP-like language above, usin
Page 510 and 511: left, of course, for the reader to
Page 512 and 513: Molecular Biology, enunciated by Fr
Page 514 and 515: You can perhaps remember this molec
Page 516 and 517: and see how the enzyme acts on the
Page 518 and 519: primary structure is meant its amin
Page 520 and 521: cycle goes on and on. This can go o
Page 522 and 523: FIGURE 91. The four constituent bas
Page 524 and 525: Messenger KNA and Ribosomes As was
Page 528 and 529: amino acid's presence means that su
Page 530 and 531: FIGURE 96. A section of mRNA passin
Page 532 and 533: This is accomplished by having the
Page 534 and 535: the Prelude, Ant Fugue. The global
Page 536 and 537: called its active site, and any mol
Page 538 and 539: Comparison of DNA's Self-Rep Method
Page 540 and 541: Self-Rep and Self-Rep 528
Page 542 and 543: FIGURE 100. The Godel Code. Under t
Page 544 and 545: FIGURE 101. The T4 bacterial virus
Page 546 and 547: Self-Rep and Self-Rep 534
Page 548 and 549: mechanisms for examining whether DN
Page 550 and 551: themselves. Somtiems enzymes may be
Page 552 and 553: prevented from being transcribed, w
Page 554 and 555: novel features. We have seen, in th
Page 556 and 557: The Magnificrab, Indeed, It is spri
Page 558 and 559: Crab: Well, it was this way. This f
Page 560 and 561: Crab: I don't understand. What stat
Page 562 and 563: and further attempts to improve it
Page 564 and 565: numbers-such as the square root of
Page 566 and 567: CHAPTER XVII Church, Turing, Tarski
Page 568 and 569: It was proven in 1936 by the Americ
Page 570 and 571: FIGURE 105. Srinivasa Ramanujan and
Page 572 and 573: mediately jumped to fourth powers.
Page 574 and 575: "Idiots 'Savants" There is another
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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
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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
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FIGURE 112. "Will you please stack
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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
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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
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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
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Achilles: What about you Mr. Sloth?
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The New The New Yorker commented: A
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mean.is that any of them can vary (
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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
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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.
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problems lies very close to the cor
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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
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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.
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the picture, this is unlikely-but f
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evidence: C. And for the validity o
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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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