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'Twas brillig, and the slithy toves Did gyre and gimble in the wabe: All mimsy were the borogoves, And the mome raths outgrabe. Il brilgue: les toves lubricilleux Se gyrent en vrillant dans le guave. Enmimes sont les gougebosqueux Et le momerade horsgrave. Es brillig war. Die schlichten Toven Wirrten and wimmelten in Waben: Und aller-mumsige Burggoven Die mohmen Rath' ausgraben. English French German Suite 374
CHAPTER XII Minds and Thoughts Can Minds Be Mapped onto Each Other? Now THAT WE have hypothesized the existence of very high-level active subsystems of the brain (symbols), we may return to the matter of a possible isomorphism, or partial isomorphism, between two brains. Instead of asking about an isomorphism on the neural level (which surely does not exist), or on the macroscopic suborgan level (which surely does exist but does not tell us very much), we ask about the possibility of an isomorphism between brains on the symbol level: a correspondence which not only maps symbols in one brain onto symbols in another brain, but also maps triggering patterns onto triggering patterns. This means that corresponding symbols in the two brains are linked in corresponding ways. This would be a true functional isomorphism-the same type of isomorphism as we spoke of when trying to characterize what it is that is invariant about all butterflies, It is clear from the outset that such an isomorphism does not exist between any pair of human beings. If it did, they would be completely indistinguishable in their thoughts; but in order for that to be true, they would have to have completely indistinguishable memories, which would mean they would have to have led one and the same life. Even identical twins do not approach, in the remotest degree, this ideal. How about a single individual% When you look back over things which you yourself wrote a few years ago, you think "How awful!" and smile with amusement at the person you once were. What is worse is when you do the same thing with something you wrote or said five minutes ago. When this happens, it shows that you do not fully understand the person you were moments ago. The isomorphism from your brain now to your brain then is imperfect. What, then, of the isomorphisms to other people, other species ... The opposite side of the coin is shown by the power of the communication that arises between the unlikeliest partners. Think of the barriers spanned when you read lines of poetry penned in jail by Francois Villon, the French poet of the 1400's. Another human being, in another era, captive in jail, speaking another language ... How can you ever hope to have a sense of the connotations behind the facade of his words, translated into English% Yet a wealth of meaning comes through. Thus, on the one hand, we can drop all hopes of finding exactly isomorphic software in humans, but on the other, it is clear that some people think more alike than others do. It would seem an obvious conclu Minds and Thoughts 375
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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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Page 328 and 329: see "meaning" and "purpose", in the
Page 330 and 331: Crab: I can see that the signals ar
Page 332 and 333: Achilles: What about a description
Page 334 and 335: full system. For example, Aunt Hill
Page 336 and 337: voices in a polyphonic piece (often
Page 338 and 339: Achilles: Ah ... "J. S. BACH". Oh!
Page 340 and 341: FIGURE 63. During emigrations arm'
Page 342 and 343: Crab: Well, what have we here? Oh,
Page 344 and 345: e mixed right in with the symbols t
Page 346 and 347: The type of decision which a neuron
Page 348 and 349: Earthworms have isomorphic brains!
Page 350 and 351: outer ring. If an on-center pattern
Page 352 and 353: each of them, the visual cortex bre
Page 354 and 355: place at some point after the recep
Page 356 and 357: Now what does a symbol do, when awa
Page 358 and 359: The Prototype Principle The list ab
Page 360 and 361: symbols for other people who are le
Page 362 and 363: the brain of every human who ever h
Page 364 and 365: Liftability of Intelligence Thus we
Page 366 and 367: The fact that a symbol cannot be aw
Page 368 and 369: "rubbing off" instances from classe
Page 370 and 371: that tone relationships could not b
Page 372 and 373: By Lewis Carroll .. . English Frenc
Page 376 and 377: FIGURE 70. A tiny portion of the au
Page 378 and 379: nearsighted observer-for example an
Page 380 and 381: months. To make things a little eas
Page 382 and 383: ing paths in the same way. These co
Page 384 and 385: the cities represent not only the e
Page 386 and 387: "Stoliarny Lane" (or "Place"). This
Page 388 and 389: High-Level Comparisons between Brai
Page 390 and 391: whom the work is new. Presumably, a
Page 392 and 393: Subsystems and Shared Code Typical
Page 394 and 395: Of course, this does not elevate co
Page 396 and 397: machine for the purposes of our dis
Page 398 and 399: play one or another of these thirty
Page 400 and 401: Achilles: But please don't let me d
Page 402 and 403: Achilles: I think you should call i
Page 404 and 405: Tortoise: You can put it that way i
Page 406 and 407: Tortoise: The first type of search-
Page 408 and 409: a single entity a property which it
Page 410 and 411: your number-theoretical entertainme
Page 412 and 413: CHAPTER XI11 BlooP and FlooP and Gl
Page 414 and 415: possibly lead to never-ending searc
Page 416 and 417: Loops and Upper Bounds If we try to
Page 418 and 419: DEFINE PROCEDURE "MINUS" [M,N]: BLO
Page 420 and 421: example of a full B1ooP program wou
Page 422 and 423: FIBO [N] = the Nth Fibonacci number
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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
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pleteness here is part and parcel o
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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
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A charming philosophy, is it not? A
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FIGURE 79. Tobacco Mosaic Virus. Fr
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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
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The sentence "The sentence "The sen
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TEMPLATE-is never interpreted as a
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the blooP-like language above, usin
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left, of course, for the reader to
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Molecular Biology, enunciated by Fr
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You can perhaps remember this molec
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and see how the enzyme acts on the
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primary structure is meant its amin
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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
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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
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called its active site, and any mol
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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
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FIGURE 101. The T4 bacterial virus
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Self-Rep and Self-Rep 534
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mechanisms for examining whether DN
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themselves. Somtiems enzymes may be
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prevented from being transcribed, w
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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
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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
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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
Page 648 and 649:
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
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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
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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
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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
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Forced matches occur every day in t
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measured on an imaginary "keyboard
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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
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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
Page 704 and 705:
The upshot is that the total pictur
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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
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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
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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
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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
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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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