Gödel, Escher, Bach: An Eternal Golden Braid (65 page)

But for the moment, suppose that issue had been solved. Suppose we now agree that there are certain drawings of nodes, connected by links (let us say they come in various colors, so that various types of conceptual nearness can be distinguished from each other), which capture precisely the way in which symbols trigger other symbols.

Then under what conditions would we feel that two such drawings were isomorphic, or nearly isomorphic? Since we are dealing with a visual representation of the network of symbols, let us consider an analogous visual problem. How would you try to determine whether two spiderwebs had been spun by spiders belonging to the same species? Would you try to identify individual vertices which correspond exactly, thereby setting up an exact map of one web onto the other, vertex by vertex, fiber by fiber, perhaps even angle by angle? This would be a futile effort. Two webs are never exactly the same: yet there is still some sort of "style", "form", what-have-you, that infallibly brands a given species'

web.

In any network-like structure, such as a spiderweb, one can look at local properties and global properties. Local properties require only a very nearsighted observer-for example an observer who can only see one vertex at a time; and global properties require only a sweeping vision, without attention to detail. Thus, the overall shape of a spiderweb is a global property, whereas the average number of lines meeting at a vertex is a local property. Suppose we agree that the most reasonable criterion for calling two spiderwebs "isomorphic" is that they should have been spun by spiders of the same species. Then it is interesting to ask which kind of observation-local or global-tends to be a more reliable guide in determining whether two spiderwebs are isomorphic. Without answering the question for spiderwebs, let us now return to the question of the closeness-or isomorphicness, if you will-of two symbol networks.

Translations of "Jabberwocky"

Imagine native speakers of English, French, and German, all of whom have excellent command of their respective native languages, and all of whom enjoy wordplay in their own language. Would their symbol networks be similar on a local level, or on a global level? Or is it meaningful to ask such a question? The question becomes concrete when you look at the preceding translations of Lewis Carroll's famous "Jabberwocky".

I chose this example because it demonstrates, perhaps better than an example in ordinary prose, the problem of trying to find "the same node" in two different networks which are, on some level of analysis, extremely nonisomorphic. In ordinary language, the task of translation is more straightforward, since to each word or phrase in the original language, there can usually be found a corresponding word or phrase in the new language. By contrast, in a poem of this type, many "words" do not carry ordinary meaning, but act purely as exciters of nearby symbols. However, what is nearby in one language may be remote in another.

Thus, in the brain of a native speaker of English, "slithy" probably activates such symbols as "slimy", "slither", "slippery", "lithe", and "sly", to varying extents. Does

"lubricilleux" do the corresponding thing in the brain of a Frenchman? What indeed would be "the corresponding thing"? Would it be to activate symbols which are the ordinary translations of those words? What if there is no word, real or fabricated, which will accomplish that? Or what if a word does exist, but is very intellectual-sounding and Latinate ("lubricilleux"), rather than earthy and Anglo-Saxon ("slithy")? Perhaps

"huilasse" would be better than "lubricilleux"? Or does the Latin origin of the word

"lubricilleux" not make itself felt to a speaker of French in the way that it would if it were an English word ("lubricilious", perhaps)?

An interesting feature of the translation into French is the transposition into the present tense. To keep it in the past would make some unnatural turns of phrase necessary, and the present tense has a much fresher flavor in French than the past. The translator sensed that this would be "more appropriate"-in some ill-defined yet compelling senseand made the switch. Who can say whether remaining faithful to the English tense would have been better?

In the German version, the droll phrase "er an-zu-denken-fing" occurs; it does not correspond to any English original. It is a playful reversal of words, whose flavor vaguely resembles that of the English phrase "he out-to-ponder set", if I may hazard a reverse translation. Most likely this funny turnabout of words was inspired by the similar playful reversal in the English of one line earlier: "So rested he by the Tumtum tree". It corresponds, yet doesn't correspond.

Incidentally, why did the Tumturn tree get changed into an "arbre T6-t6" in French? Figure it out for yourself.

The word "manxome" in the original, whose "x" imbues it with many rich overtones, is weakly rendered in German by "manchsam", which hack-translates into English as "maniful". The French "manscant" also lacks the manifold overtones of

"manxome". There is no end to the interest of this kind of translation task.

When confronted with such an example, one realizes that it is utterly impossible to make an exact translation. Yet even in this pathologically difficult case of translation, there seems to be some rough equivalence obtainable. Why is this so, if there really is no isomorphism between the brains of people who will read the different versions? The answer is that there is a kind of rough isomorphism, partly global, partly local, between the brains of all the readers of these three poems.

ASU's

An amusing geographical fantasy will give some intuition for this kind of quasi-isomorphism. (Incidentally, this fantasy is somewhat similar to a geographical analogy devised by M. Minsky in his article on "frames", which can be found in P. H. Winston's book The Psychology of Computer Vision.) Imagine that you are given a strange atlas of the USA, with all natural geological features premarked-such as rivers, mountains, lakes, and so on-but with nary a printed word. Rivers are shown as blue lines, mountains b color, and so on. Now you are told to convert it into a road atlas for a trip which you will soon make. You must neatly fill in the names of all states, their boundaries, time zones, then all counties, cities, towns, all freeways and highways and toll routes, all county roads, all state and national parks, campgrounds, scenic areas, dams, airports, and so on

... All of this must be carried out down to the level that would appear in a detailed road atlas. And it must be manufactured out of your own head. You are not allowed access to any information which would help you for the duration of your task.

You are told that it will pay off, in ways that will become clear at a later date, to make your map as true as you can. Of course, you will begin by filling in large cities and major roads, etc., which you know. And when you have exhausted your factual knowledge of an area, it will be to your advantage to use your imagination to help you reproduce at least the flavor of that area, if not its true geography, by making up fake town names, fake populations, fake roads, fake parks, and so on. This arduous task will take

months. To make things a little easier, you have a cartographer on hand to print everything in neatly. The end product will be your personal map of the "Alternative Structure of the Union"-your own personal "ASU".

Your personal ASU will be very much like the USA in the area where you grew up. Furthermore, wherever your travels have chanced to lead you, or wherever you have perused maps with interest, your ASU will have spots of striking agreement with the USA: a few small towns in North Dakota or Montana, perhaps, or the whole of metropolitan New York, might be quite faithfully reproduced in your ASU.

A Surprise Reversal

When your ASU is done, a surprise takes place. Magically, the country you have designed comes into being, and you are transported there. A friendly committee presents you with your favorite kind of-automobile, and explains that, "As a reward for your designing efforts, you may now enjoy an all-expense-paid trip, at a leisurely pace, around the good old A. S. of U. You may go wherever you want, do whatever you wish to do, taking as long as you wish-compliments of the Geographical Society of the ASU. And-to guide you around-here is a road atlas." To your surprise, you are given not the atlas which you designed, but a regular road atlas of the USA.

When you embark on your trip, all sorts of curious incidents will take place. A road atlas is being used to guide you through a country which it only partially fits. As long as you stick to major freeways, you will probably be able to cross the country without gross confusions. But the moment you wander off into the byways of New Mexico or rural Arkansas, there will be adventure in store for you. The locals will not recognize any of the towns you're looking for, nor will they know the roads you're asking about. They will only know the large cities you name, and even then the routes to those cities will not be the same as are indicated on your map. It will happen occasionally that some of the cities which are considered huge by the locals are nonexistent on your map of the USA; or perhaps they exist, but their population according to the atlas is wrong by an order of magnitude.

Centrality and Universality

What makes an ASU and the USA, which are so different in some ways, nevertheless so similar? It is that their most important cities and routes of communication can be mapped onto each other. The differences between them are found in the less frequently traveled routes, the cities of smaller size, and so on. Notice that this cannot be characterized either as a local or a global isomorphism. Some correspondences do extend down to the very local level-for instance, in both New Yorks, the main street may be Fifth Avenue, and there may be a Times Square in both as well-yet there may not be a single town that is found in both Montanas. So the local-global

distinction is not relevant here. What is relevant is the centrality of the city, in terms of economics, communication, transportation, etc. The more vital the city is, in one of these ways, the more certain it will be to occur in both the ASU and the USA.

In this geographic analogy, one aspect is very crucial: that there are certain definite, absolute points of reference which will occur in nearly all ASU's: New York, San Francisco, Chicago, and so on. From these it is then possible to orient oneself. In other words, if we begin comparing my ASU with yours, I can use the known agreement on big cities to establish points of reference with which I can communicate the location of smaller cities in my ASU. And if I hypothesize a voyage from Kankakee to Fruto and you don't know where those towns are, I can refer to something we have in common, and thereby guide you. And if I talk about a voyage from Atlanta to Milwaukee, it may go along different freeways or smaller roads, but the voyage itself can still be carried out in both countries. And if you start describing a trip from Horsemilk to Janzo, I can plot out what seems to me to be an analogous trip in my ASU, despite not having towns by those names, as long as you constantly keep me oriented by describing your position with respect to nearby larger towns which are found in my ASU as well as in yours.

My roads will not be exactly the same as yours, but, with our separate maps, we can each get from a particular part of the country to another. We can do this, thanks to the external, predetermined geological facts mountain chains, streams, etc.-facts which were available to us both as we worked on our maps. Without those external features, we would have no possibility of reference points in common. For instance, if you had been given only a map of France, and I had been given a map of Germany, and then we had both filled them in in great detail, there would he no way to try to find "the same place"

in our fictitious lands. It is necessary to begin with identical external conditions-otherwise nothing will match.

Now that we have carried our geographical analogy quite far, we return to the question of isomorphisms between brains. You might well wonder why this whole question of brain isomorphisms has been stressed so much. What does it matter if two brains are isomorphic, or quasi-isomorphic, or not isomorphic at all? The answer is that we have an intuitive sense that, although other people differ from us in important ways, they are still "the same" as we are in some deep and important ways. It would be instructive to be able to pinpoint what this invariant core of human intelligence is, and then to be able to describe the kinds of "embellishments" which can be added to it, making each one of us a unique embodiment of this abstract and mysterious quality called "intelligence".

In our geographic analogy, cities and towns were the analogues of symbols, while roads and highways were analogous to potential triggering paths. The fact that all ASU's have some things in common, such as the East Coast, the West Coast, the Mississippi River, the Great Lakes, the Rockies, and many major cities and roads is analogous to the fact that we are all forced, by external realities, to construct certain class symbols and trigger