Gödel, Escher, Bach: An Eternal Golden Braid (122 page)

a structure having two parts which do the same thing,

only moving in opposite directions.

This is a concrete geometrical image which can be manipulated by the mind almost as a Bongard pattern. In fact, when I think of the Crab Canon today, I visualize it as two strands which cross in the middle, where they are joined by a "knot" (the Crab's speech). This is such a vividly pictorial image that it instantaneously maps, in my mind, onto a picture of two homologous chromosomes joined by a centromere in their middle, which is an image drawn directly from meiosis, as shown in Figure 132.

FIGURE 132.

In fact, this very image is what inspired me to cast the description of the
Crab Canon's
evolution in terms of meiosis-which is itself, of course, vet another example of conceptual mapping.

Recombinant Ideas

There are a variety of techniques of fusion of two symbols. One involves lining the two ideas up next to each other (as if ideas were linear!), then judiciously choosing pieces from each one, and recombining them in a new symbol. This strongly recalls genetic recombination. Well, what do chromosomes exchange, and how do they do it? They exchange genes. What in a symbol is comparable to a gene? If symbols have frame-like slots, then slots, perhaps. But which slots to exchange, and why? Here is where the crabcanonical fusion may offer some ideas. Mapping the notion of "musical crab canon"

onto that of "dialogue" involved several auxiliary mappings; in fact it
induced
them. That is, once it had been decided that these two notions ,ere to be fused, it became a matter of looking at them on a level where analogous parts emerged into view, then going ahead and
mapping the parts
onto each other, and so on, recursively, to any level that was found desirable. Here, for instance, "voice" and

"character" emerged as corresponding slots when "crab canon" and "dialogue" were viewed abstractly. Where did these abstract views come from, though? This is at the crux of the mapping-problem-where do abstract views come from? How do you make abstract views of specific notions?

Abstractions, Skeletons, Analogies

A view which has been abstracted from a concept along some dimension is what I call a
conceptual skeleton
. In effect, we have dealt with conceptual skeletons all along, without often using that name. For instance, many of the ideas concerning Bongard problems could be rephrased using this terminology. It is always of interest, and possibly of importance, when two or more ideas are discovered to share a conceptual skeleton. An example is the bizarre set of concepts mentioned at the beginning of the Contrafactus: a Bicyclops, a tandem unicycle, a teeter-teeter, the game of ping-ping, a one-way tie, a two-sided Mobius strip, the "Bach twins", a piano concerto for two left hands, a one-voice fugue, the act of clapping with one hand, a two-channel monaural phonograph, a pair of eighth-backs. All of these ideas are "isomorphic" because they share this conceptual skeleton:

a plural thing made singular and re-pluralized wrongly.

Two other ideas in this book which share that conceptual skeleton are (1) the Tortoise's solution to Achilles' puzzle, asking for a word beginning and ending in "HE" (the Tortoise's solution being the pronoun "HE", which collapses two occurrences into one), and (2) the Pappus-Gelernter proof of the Pons As' norum Theorem, in which one triangle is reperceived as two. Incidentally, these droll concoctions might be dubbed "demidoublets".

A conceptual skeleton is like a set of constant features (as distinguished from parameters or variables)-features which should not be slipped in a subjunctive instant replay or mapping-operation. Having no parameters or variables of its own to vary, it can be the invariant core of several different ideas. Each
instance
of it, such as "tandem unicycle", does have layers of variability and so can be "slipped" in various ways.

Although the name "conceptual skeleton" sounds absolute and rigid, actually there is a lot of play in it. There can be conceptual skeletons on several different levels of abstraction. For instance, the "isomorphism" between Bongard problems 70 and 71, already pointed out, involves a higher-level conceptual skeleton than that needed to solve either problem in isolation.

Multiple Representations

Not only must conceptual skeletons exist on different levels of abstraction; also, they must exist along different conceptual dimensions. Let us take the following sentence as an example:

"The Vice President is the spare tire on the automobile of government."

How do we understand what it means (leaving aside its humor, which is of course a vital aspect)? If you were told, "See our government as an automobile" without any prior motivation, you might come up with any number of correspondences: steering wheel =

president, etc.. What are checks and balances? What are seat belts? Because the two things being mapped are so different, it is almost inevitable that the mapping will involve functional aspects. Therefore, you retrieve from your store of conceptual skeletons representing parts of automobiles, only those having to do with function, rather than, say, shape. Furthermore, it makes sense to work at a pretty high level of abstraction, where

"function" isn't taken in too narrow a context. Thus, of the two following definitions of the function of a spare tire: (1) "replacement for a flat tire", and (2) "replacement for a certain disabled part of a car", certainly the latter would be preferable, in this case. This comes simply from the fact that an auto and a government are so different that they have to be mapped at a high level of abstraction.

Now when the particular sentence is examined, the mapping gets forced in one respect-but it is not an awkward way, by any means. In fact, you already have a conceptual skeleton for the Vice President, among many others, which says,

"replacement for a certain disabled part of government". Therefore the forced mapping works comfortably. But suppose, for the sake of contrast, that you had retrieved another conceptual skeleton for "spare tire"-say, one describing its physical aspects. Among other things, it might say that a spare tire is "round and inflated". Clearly, this is not the right way to go. (Or is it? As a friend of mine pointed out, some Vice Presidents are rather portly, and most are quite inflated!)

Ports of Access

One of the major characteristics of each idiosyncratic style of thought is how new experiences get classified and stuffed into memory, for that defines the "handles" by which they will later be retrievable. And for events, objects, ideas, and so on-for everything that can be thought about-there is a wide variety of "handles". I am struck by this each time I reach down to turn on my car radio, and find, to my dismay, that it is already on! What has happened is that two independent representations are being used for the radio. One is "music producer", the other is "boredom reliever". I am aware that the music is on, but I am bored anyway, and before the two realizations have a chance to interact, my reflex to reach

down has been triggered. The same reaching-down reflex one day occurred just after I'd left the radio at a repair shop and was driving away, wanting to hear some music. Odd.

Many other representations for the same object exist, such as

shiny silver-knob haver

overheating-problems haver

lying-on-my-back-over-hump-to-fix thing

buzz-maker

slipping-dials object

multidimensional representation example

All of them can act as ports of access. Though they all are attached to my symbol for my car radio, accessing that symbol through one does not open up all the others. Thus it is unlikely that I will be inspired to remember lying on my back to fix the radio when I reach down and turn it on. And conversely, when I'm lying on my back, unscrewing screws, I probably won't think about the time I heard the
Art of the Fugue
on it. There are

"partitions" between these aspects of one symbol, partitions that prevent my thoughts from spilling over sloppily, in the manner of free associations. My mental partitions are important because they contain and channel the flow of my thoughts.

One place where these partitions are quite rigid is in sealing off words for the same thing in different languages. If the partitions were not strong, a bilingual person would constantly slip back and forth between languages, which would be very uncomfortable. Of course, adults learning two new languages at once often confuse words in them. The partitions between these languages are flimsier, and can break down.

Interpreters are particularly interesting, since they can speak any of their languages as if their partitions were inviolable and yet, on command, they can negate those partitions to allow access to one language from the other, so they can translate. Steiner, who grew up trilingual, devotes several pages in
After Babel
to the intermingling of French, English, and German in the layers of his mind, and how his different languages afford different ports of access onto concepts.

Forced Matching

When two ideas are seen to share conceptual skeletons on some level of abstraction, different things can happen. Usually the first stage is that you zoom in on both ideas, and, using the higher-level match as a guide, you try to identify corresponding subideas.

Sometimes the match can be extended recursively downwards several levels, revealing a profound isomorphism. Sometimes it stops earlier, revealing an analogy or similarity.

And then there are times when the high-level similarity is so compelling that, even if there is no apparent lower-level continuation of the map, you just go ahead and make one: this is the
forced match
.

Forced matches occur every day in the political cartoons of newspapers: a political figure is portrayed as an airplane, a boat, a fish, the Mona Lisa; a government is a human, a bird, an oil rig; a treaty is a briefcase, a sword, a can of worms; on and on and on. What is fascinating is how easily we can perform the suggested mapping, and to the exact depth intended. We don't carry the mapping out too deeply or too shallowly.

Another example of forcing one thing into the mold of another occurred when I chose to describe the development of my Crab Canon in terms of meiosis. This happened in stages. First, I noticed the common conceptual skeleton shared by the Crab Canon and the image of chromosomes joined by a centromere; this provided the inspiration for the forced match. Then I saw a high-level resemblance involving "growth", "stages", and

"recombination". Then I simply pushed the analogy as hard as I could. Tentativity-as in the Bongard problem-solver-played a large role: I went forwards and backwards before finding a match which I found appealing.

A third example of conceptual mapping is provided by the Central Dogmap. I initially noticed a high-level similarity between the discoveries of mathematical logicians and those of molecular biologists, then pursued it on lower levels until I found a strong analogy. To strengthen it further, I chose a Godel-numbering which imitated the Genetic Code. This was the lone element of forced matching in the Central Dogmap.

Forced matches, analogies, and metaphors cannot easily be separated out.

Sportscasters often use vivid imagery which is hard to pigeonhole. For instance, in a metaphor such as "The Rams [football team are spinning their wheels", it is hard to say just what image you are supposed to conjure up. Do you attach wheels to the team as a whole% Or to each player? Probably neither one. More likely, the image of wheels spinning in mud or snow simply flashes before you for a brief instant, and then in some mysterious way, just the relevant parts get lifted out and transferred to the team's performance. How deeply are the football team and the car mapped onto each other in the split second that you do this?

Recap

Let me try to tie things together a little. I have presented a number of related ideas connected with the creation, manipulation, and comparison of symbols. Most of them have to do with slippage in some fashion, the idea being that concepts are composed of some tight and some loose elements, coming from different levels of nested contexts (frames). The loose ones can be dislodged and replaced rather easily, which, depending on the circumstances, can create a "subjunctive instant replay", a forced match, or an analogy. A fusion of two symbols may result from a process in which parts of each symbol are dislodged and other parts remain.

Creativity and Randomness

It is obvious that we are talking about mechanization of creativity. But the this not a contradiction in terms? Almost, but not really. Creativity s essence of that which is not mechanical. Yet every creative act is mechanical-it has its explanation no less than a case of the hiccups does. The mechanical substrate of creativity may be hidden from view, but it exists. Conversely, there is something unmechanical in flexible programs, even today.

It may not constitute creativity, but when programs cease to be transparent to their creators, then the approach to creativity has begun.