Gödel, Escher, Bach: An Eternal Golden Braid (121 page)

The sameness-detectors in the Bongard problem-solver (Sams) could be implemented as enzyme-like subprograms. Like an enzyme, a Sam would meander about somewhat at random, bumping into small data structures here and there. Upon filling its two "active sites" with identical data structures, the Sam would emit a message to other parts (actors) of the program. As long as programs are serial, it would not make much sense to have several copies of a Sam, but in a truly parallel computer, regulating the number of copies of a subprogram would be a way of regulating the expected waiting-time before an operation gets done, just as regulating the number of copies of an enzyme in a cell regulates how fast that function gets performed. And if new Sams could be synthesized, that would be comparable to the seepage of pattern detection into lower levels of our minds.

Fission and Fusion

Two interesting and complementary ideas concerning the interaction of symbols are

"fission" and "fusion". Fission is the gradual divergence of a new symbol from its parent symbol (that is, from the symbol which served as a template off of which it was copied).

Fusion is what happens when two (or more) originally unrelated symbols participate in a

"joint activation", passing messages so tightly back and forth that they get bound together and the combination can thereafter be addressed as if it were a single symbol. Fission is a more or less inevitable process, since once a new symbol has been "rubbed off" of an old one, it becomes autonomous, and its interactions with the outside world get reflected in its private internal structure; so what started out as a perfect copy will soon become imperfect, and then slowly will become less and less like the symbol off of which it was

"rubbed". Fusion is a subtler thing. When do two concepts really become 'one? Is there some precise instant when a fusion takes place?

This notion of joint activations opens up a Pandora's box of questions. For instance, how much coo we hear "dough" and "nut" when we say "doughnut"? Does a German who thinks of gloves ("Handschuhe") hear "hand-shoes" or not? How about Chinese people, whose word "dong-xi" ("East-West") means "thing"? It is a matter of some political concern, too, since some people claim that words like "chairman" are heavily charged with undertones of the male gender. The degree to which the parts resonate inside the whole probably varies from person to person and according to circumstances.

The real problem with this notion of "fusion" of symbols is that it is very hard to imagine general algorithms which will create meaningful new symbols from colliding symbols. It is like two strands of
DNA
which come together. How do you take parts from each and recombine them into a meaningful and viable new strand of
DNA
which codes for an individual of the same species? Or a new kind of species? The chance is infinitesimal that a random combination of pieces of
DNA
will code for anything that will survive-something like the chance that a random combination of words from two books will make another book. The chance that recombinant
DNA
will make sense on any level but the lowest is tiny, precisely because there are so many levels of meaning in
DNA
. And the same goes for "recombinant symbols".

Epigenesis of the Crab Canon

I think of my Dialogue Crab Canon as a prototype example where two ideas collided in my mind, connected in a new way, and suddenly a new kind of verbal structure came alive in my mind. Of course I can still think about musical crab canons and verbal dialogues separately-they can still be activated independently of each other; but the fused symbol for crab canonical dialogues has its own characteristic modes of activation, too.

To illustrate this notion of fusion or "symbolic recombination" in some detail, then, I would like to use the development of my Crab Canon as a case study, because, of course, it is very familiar to me, and also because it is interesting, yet typical of how far a single idea can be pushed. I will recount it in stages named after those of meiosis, which is the name for cell division in which "crossing-over", or genetic recombination, takes place-the source of diversity in evolution.

PROPHASE: I began with a rather simple idea-that a piece of music, say a canon, could be imitated verbally. This came from the observation that, through a shared abstract form, a piece of text and a piece of music may be connected. The next step involved trying to realize some of the potential of this vague hunch; here, I hit upon the idea that

"voices" in canons can be mapped onto "characters" in dialogues-still a rather obvious idea.

Then I focused down onto specific kinds of canons, and remembered that there was a crab canon in the Musical Offering. At that time, I had just

begun writing Dialogues, and there were only two characters: Achilles and the Tortoise.

Since the Bach crab canon has two voices, this mapped perfectly: Achilles should be one voice, the Tortoise the other, with the one doing forwards what the other does backwards.

But here I was faced with a problem: on what level should the reversal take place? The letter level? The word level? The sentence level? After some thought, I concluded that the "dramatic line" level would be most appropriate.

Now that the "skeleton" of the Bach crab canon had been transplanted, at least in plan, into a verbal form, there was just one problem. When the two voices crossed in the middle, there would be a short period of extreme repetition: an ugly blemish. What to do about it? Here, a strange thing happened, a kind of level-crossing typical of creative acts: the word "crab" in "crab canon" flashed into my mind, undoubtedly because of some abstract shared quality with the notion of "tortoise"-and immediately I realized that at the dead center, I could block the repetitive effect, by inserting one special line, said by a new character: a Crab! This is how, in the "prophase" of the Crab Canon, the Crab was conceived: at the crossing over of Achilles and the Tortoise. (See Fig. 131.) FIGURE 131.
A schematic diagram of the Dialogue Crab Canon
.

METAPHASE: This was the skeleton of my
Crab Canon
. I then entered the second stage-the "metaphase"-in which I had to fill in the flesh, which was of course an arduous task. I made a lot of stabs at it, getting used to the way in which pairs of successive lines had to make sense when read from either direction, and experimenting around to see what kinds of dual meanings would help me in writing such a form (e.g.,

"Not at all"). There were two early versions both of which were interesting, but weak. I abandoned work on the book for over a year, and when I returned to the
Crab Canon
, I had a few new ideas. One of them was to mention a Bach canon inside it. At first my plan was to mention the "Canon per augmentationem, contrario motu", from the
Musical
Offering (Sloth Canon
, as I call it). But that started to seem a little silly, so reluctantly I decided that inside my Crab Canon, I could talk about Bach's own
Crab Canon
instead.

Actually, this was a crucial turning point, but I didn't know it then.

Now if one character was going to mention a Bach piece, wouldn't it be awkward for the other to say exactly the same thing in the corresponding place? Well, Escher was playing a similar role to Bach in my thoughts and my book, so wasn't there some way of just slightly modifying the line so that it would refer to Escher? After all, in the strict art of canons, note-perfect imitation is occasionally foregone for the sake of elegance or beauty. And

no sooner did that idea occur to me than the picture
Day and Night
(Fig. 49) popped into my mind. "Of course!" I thought, "It is a sort of pictorial crab canon, with essentially two complementary voices carrying the same theme both leftwards and rightwards, and harmonizing with each other!" Here again was the notion of a single "conceptual skeleton" being instantiated in two different media-in this case, music and art. So I let the Tortoise talk about Bach, and Achilles talk about Escher, in parallel language; certainly this slight departure from strict imitation retained the spirit of crab cano.is.

At this point, I began realizing that something marvelous was happening namely, the Dialogue was becoming self-referential, without my even having intended it! What's more, it was an indirect self-reference, in that the characters did not talk directly about the Dialogue they were in, but rather about structures which were isomorphic to it (on a certain plane of abstraction). To put it in the terms I have been using, my Dialogue now shared a "conceptual skeleton" with Gödel’s G, and could therefore be mapped onto G in somewhat the way that the Central Dogma was, to create in this case a "Central Crabmap". This was most exciting to me, since out of nowhere had come an esthetically pleasing unity of Gödel, Escher, and Bach.

ANAPHASE: The next step was quite startling. I had had Caroline MacGillavry's monograph on Escher's tessellations for years, but one day, as I flipped through it, my eye was riveted to Plate 23 (Fig. 42), for I saw it in a way I had never seen it before: here was a genuine crab canon-crab-like in both form and content! Escher himself had given the picture no title, and since he had drawn similar tessellations using many other animal forms, it is probable that this coincidence of form and content was just something which I had noticed. But fortuitous or not, this untitled plate was a miniature version of one main idea of my book: to unite form and content. So with delight I christened it
Crab Canon
, substituted it for
Day and Night
, and modified Achilles' and the Tortoise's remarks accordingly.

Yet this was not all. Having become infatuated with molecular biology, one day I was perusing Watson's book in the bookstore, and in the index saw the word

"palindrome". When I looked it up, I found a magical thing: crab-canonical structures in DNA. Soon the Crab's comments had been suitably modified to include a short remark to the effect that he owed his predilection for confusing retrograde and forward motion to his genes.

TELOPHASE: The last step came months later, when, as I was talking about the picture of the crab-canonical section of
DNA
(Fig. 43), 1 saw that the
'A', 'T', 'C'
of Adenine, Thymine, Cytosine coincided-
mirabile dictu
-with the
'A', 'T', 'C'
of Achilles, Tortoise, Crab; moreover, just as Adenine and Thymine are paired in
DNA
, so are Achilles and the Tortoise paired in the Dialogue. I thought for a moment and, in another of those level-crossings, saw that
'G
', the letter paired with '
C
' in
DNA
, could stand for

"Gene". Once again, I jumped back to the Dialogue, did a little surgery on the Crab's speech to reflect this new discovery, and now I had a mapping between the
DNA
's structure, and the Dialogue's structure. In that sense, the
DNA
could be said to be a genotype coding for a phenotype: the

Structure of the Dialogue. This final touch dramatically heightened the self-reference, and gave the Dialogue a density of meaning which I had never anticipated.

Conceptual Skeletons and Conceptual Mapping

That more or less summarizes the epigenesis of the Crab Canon. The whole process can be seen as a succession of mappings of ideas onto each other, at varying levels of abstraction. This is what I call conceptual mapping, and the abstract structures which connect up two different ideas are conceptual skeletons. Thus, one conceptual skeleton is that of the abstract notion of a crab canon: