Metamagical Themas, page 76
It seems to me that people do not generally recognize how deeply mplicated the analogical capacity is in decisions that affect the course of heir lives. On a global level, it is evident, once pointed out. Is the embroilment of the United States in Lebanon "another Viet Nam"? How about in El Salvador? How does the American invasion of Grenada map onto the Soviet invasion of Afghanistan, or the British invasion of the Falklands? Is the Soviet Union more like an irrational paranoid person, or someone rational who has been badly bullied recently? Does the current arms race have valid precedents in history to which it can be compared?
On a more local scale, our system of law very obviously sanctifies analogy as the ultimate justification for making a reasonable and even a wise decision. The term "precedent" is just a legalistic way of saying "well-founded analogue". Two cases that at a surface level have nothing to do with each other (a bank robbery, say, and a kidnapping) may be mapped onto each other in exquisite detail at a more abstract level, with the napped kid being the loot, for instance. Lawyers attempt to sway the jury by bringing in new ways of looking at the situation that discredit their opponents' analogies, as well as by making and buttressing their own rival analogies. (Peter Suber has written a nice article connecting Copycat and Seek-Whence analogies with legal reasoning. It is called "Analogy Exercises for Teaching Legal Reasoning" and can be gotten from him at the Philosophy Department of Earlham College in Richmond, Indiana 47374.)
In our private lives, most of our important judgments are made by conscious or unconscious analogy. Should I fight this bureaucracy or accept some annoying inconvenience? Should I buy this computer or wait for a better one to come along at the same price? Should we have children now or wait a few years? Should I retire or continue working beyond retirement age? Questions concerning what to buy, what to think of someone, whom to marry, whether to move to a new city, how to talk to someone who has suffered a calamity, and on and on-all of them are influenced in a myriad ways by prior experiences of the same general sort. And remember that even in cases where there is not any obvious analogy guiding the judgment, all the categorization of the situation is being made by a mind exposed to many thousands of words, and the purpose of words is to label situation types and thus implicitly to make use of stored analogical mappings.
As was discussed in the column, the boundary line between making creative analogies and recognizing pre-existent categories is very blurry. It is signaled when we feel a desire to pluralize a proper noun ("your Einsteins and your Mozarts") or to prefix a proper noun by a definite article ("the Podunk of Albania"). Most common words hide an enormous degree of analogical abstraction. For instance, the abstractions "female" and "male" are not nearly as simple as most people think, especially when you consider how they extend to plants. What makes Middle Eastern pita, Indian purl, French baguettes, and American Wonder all be examples of the concept "bread"? When you migrate from nouns toward verbs and prepositions, the difficulties escalate. What do all "x-is-on-y" situations have in common?
All of this points out how analogies determine the course of our lives in the present. But I would go much further than that. In pre-civilized days, when people (or proto-people) lived in caves and hunted bison, analogy played no less important a role. Samenesses that we have absorbed into our perception as being obvious were great insights back then. For instance, the idea that one could chart out a plan for trapping a wild beast by drawing a map on the ground must have been a fabulous advance. All that is involved, in some sense, is a change in scale-one of the most obvious of analogical transforms, yet when it was first invented, it must have been revolutionary. On the other hand, proto-humans who tried burying meat underground in an attempt to imitate squirrels' underground hoarding of acorns might thereby seriously damage their chances for survival. Some analogies help, others hinder.
Our current mechanisms for analogy-making must certainly have emerged as a consequence of natural selection. Good mechanisms were selected for, bad ones were selected against, way back when, in the old times when you and I were but monkeys and rodents scampering about in tree branches ('member?). The point, then, is that far more than being just a matter of taste, variations in analogy-making skill can spell the difference between life and death. That's why "right answer" means something even for analogies; it's why analogies are only to some degree a matter of taste.
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This finally gets us back to the rivalry among answers in the pqrs case. The domain does admittedly appear abstract and of course it is totally decoupled from the cruel world. People who prefer dddd are not suddenly going to get swallowed by a tiger or topple off a cliff. But people who genuinely believe in dddd's superiority over pgrt will still have a rough time in life, because they lack the means to size up a situation and catch its essence in their mind's mesh, letting the trivial pass through. Something about their analogy making mechanisms is defective.
To be sure, there is room for argument about answers in this mini-world, just as there is in a courtroom. But just as a lawyer who suggested that killing a human being is analogous to breaking a window because both are nasty or because both can be done with a brick would lose the case in a snap, so .anyone who prefers dddd or abt to pqrt can be safely assumed to be totally off base. There are absurd answers, there are good answers, and there are in-between ones, just as there are degrees of edibility of food. Some foods lead to no survival, some to bare survival, and others to comfortable survival; the same is true of analogies.
One can liken the various levels of quality to the concentric circles surrounding a bull's-eye on a target (see Figure 24-5). In the middle are the totally edible foods (or insightful analogies); further out are semi-edible substances; such as grass, hay, or ants (weak analogies) and worse, leather or wood (dubious analogies); and then way out are the completely inedible hings, such as nails, shards of glass, or Anglican cathedrals (these correspond to analogies that lead to disaster, such as forming a higher-level category that lumps tigers together with zebras simply because both have stripes). In the very center, to be sure, one can argue about taste and it is indubitably good for the human race that there are people who see analogies differently in that region, but you cannot get too far-fetched.
FIGURE 24-5. Targets representing the survival values of various actions taken by (a) a mind seeking answers to the analogy "If abc changes to abd, what does pqrs change to?", and (b) a mind choosing things for its body to ingest. To be sure, there is room for debate in the bull's-eye region (hard to say whether having a fried egg sandwich or a plate of spaghetti is better for you), but as you edge further out, it gets less and less debatable. Fried dust just doesn't match up to pea soup! Similarly for analogies. Among the various answers to any analogy problem, some will be decidedly weaker than others, even if you find that no one answer emerges as the clear victor.
There is a radius beyond which analogies will be very likely to bring bad consequences to their proposers, at least if they are acted upon.
It is for this kind of reason that I unbudgeably believe that there are better and worse answers to analogies, whether in life or in the Copycat domain. Elegance is more than just a frill in life; it is one of the driving criteria behind survival. Elegance is just another way of talking about getting at the essence of situations. If you don't trust the word "elegance" in this context, then you may substitute "compactness", "efficiency", or "generality"-in short, survivability. Insight into the mechanisms behind this sense of elegance is the goal of the Copycat project. And personally, I would not shy away from equating elegance with wit. I would venture that one cannot be a successful Copycat without a sense of humor.
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Having seen numerous wild and woolly (and sometimes witty) answers to the question "What happens to pqrs ?", you might now wish to see some alternate targets. They are extremely important, because they bring out a variety of new ways of perceiving the original "abc goes to abd" change. Here are a few fascinating challenges that I urge you, once again, to actually devote some time to considering. All of them are based upon our old stock event, abc goes to abd.
1. What does cab go' to?
2. What does cba go to?
3. What does pct go to?
4. What does pxgxrx go to?
5. What does aabbcc go to?
6. What does aaabbbcck go to?
7. What does srgp go to?
8. What does spsqsrss go to?
9. What does abcdeabcdabc go to?
10. What does bcdacdabd go to?
11. What does ace go to?
12. What does xyz go to?
Each one of these questions shakes some fundamental assumptions about how you should perceive the original change. Does it necessarily affect just one letter? Need the object affected be at the righthand extremity? Does the changed piece always get replaced by its successor? In short, what should be taken literally, and what slipperily ? Although I would love to do so, I am not going to discuss all twelve examples here, for that would take a good long time. Each one merits at least a page on its own. (A set of "answers" is given at the end of this P.S.) I will discuss just one of them, number 12, in some depth. It has real beauty and raises all the central issues, so I hope you will give it some thought before reading on.
May I go on now? All right, Many people are inclined to say that xyz should go to xya. But who said the alphabet was circular? To make that leap, you almost need to have had prior experience with circularity in some form, which we all have. For instance: The hours of a clock form a closed cycle, as do the days of the week, the months of the year, the cards in a suit, the digits 0-9, and so on. But not all linear orders are cyclic. The bottom rung on a ladder is not above the top rung! The Empire State Building's top floor is not the same as its basement! It is a premise of the Copycat world that z has no successor. Sure, a machine could posit that a is the successor to z, but to do so would be an act of far greater creativity than it would be for you, because you have all these prior experiences with wuWaround structures. The Copycat program does not. Therefore, let us consider xya as admirable but simply too daring, and look for something more humble yet no less apt. What else remains? Again, I urge you to think about this before reading on. This is the crucial point where there simply is no substitute for your own experimentation.
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Okay. You've thought it over. You've got an answer, perhaps even a few, ranked more or less according to the pleasure they give you. Great! Some people suggest xy, pure and simple. Since z has no successor, they just let the third term drop away, as if it had fallen off the edge of the world. Some think, "Why not just leave z alone, producing xyz ?" Some say, "Since the rightmost letter has no successor, why not slip over to the next-to-rightmost one and take its successor, thus producing xzz ?"
Those are all right, but far more insightful answers are possible. To find them, one can let the unexpected "snag" (namely, the problem of trying to take the successor of a successor-less object) trigger a search for something crucial possibly overlooked earlier. What people tend to see at this stage is the potential correspondence of a and z-the two extreme letters of the alphabet, our two twin "monarchs". If z is the a of xyz, then what is the c ? Quite obviously, it is x. Now the question arises: "What to do to x ?" Should we take its successor, thereby producing yyz ? To my mind, there is something almost repulsively rigid about this suggestion. After all, the very fabric out of which abc was constructed has now been reversed in our new way of looking at xyz. Leftward motion has seized the role of rightward motion, and with it, predecessorship has taken on the role that successorship played in abc. Therefore elegance, in the form of a drive toward abstract symmetry, very strongly pushes for the answer wyz. Now that's a beautiful answer, in my estimation.
There is one other answer that I have encountered fairly often, and that I admire and decry at one and the same time. That is wxz. To be sure, it has the same inner structure as does abd: a jump of size one followed by a jump of size two. That much is good, but there is something peculiar about wxz nonetheless. To illustrate my ambivalence toward this answer, I will relate a micro-allegory.
Arphabelle Snerxis built a lovely house, ultra-modern in every respect. Then one day she capriciously removed her snazzy, sleek, new doorknob and replaced it by a most conspicuous creaky, rusty, old doorknob. Now Zulips Twankler, a great admirer of Arphabelle Snerxis, happened to have built a lovely house, old-fashioned in every respect; and when he saw Arphabelle's action, he determined to do "the same thing" to his house. And how did he do that? You might guess that Zulips Twankler removed his creaky, rusty, old doorknob and replaced it by a most conspicuous snazzy, sleek, new one! But no, Zulips left his creaky, rusty, old doorknob intact and instead he tore down the rest of his lovely house, old-fashioned in every respect. Then Zulips built another quite different but also lovely house, ultra-modern in every respect-except for the creaky, rusty, old doorknob. And that's how Zulips Twankler did "the same thing" to his house as Arphabelle Snerxis did to her house (except that when he'd finished, it wasn't his house any more-it was a different house altogether).
In this "analogory", Zulips let the identity of his house slip in a manner determined by its doorknob's properties, while for most people it would seem more natural to have the slippabilities reversed. There is a parallel in the fight between answers wxz and wyz. The former sees preservation of the literal intervals (1, then 2) as necessary at all costs, and it allows the bulk of the original entity xyz to be shifted in order to achieve those aims. The latter sees only one small role (analogous to the doorknob) as singled out for modification, and keeps the bulk intact. The contrast is vividly portrayed in Figure 24-6. Another way to see the contrast is this: wyz is based on a better imitation of the change from abc to abd, while wxz is a better imitation of the end product, abd. Which do you find the more satisfying or elegant action? For me it is wyz, hands down. Still, I find a strange charm in the Zulipian answer. This is one of those cases where two different answers (three, if you count xya, that deeply creative answer that comes so easily to us old circularity hands) can lie inside the innermost "delicious and nutritious" circle, within which de gustibus non disputandum est.
FIGURE 24-6. A visual comparison of two good answers to the question "If abc changes to abd, what does xyz change to?" In (a), wyz is depicted. The total symmetry of this answer is its virtue, In (b), wxz is depicted. Its virtue is that its spacing imitates the spacing of abd literally, even slavishly. The question is, which of these answers creates a better overall analogy? Is it wise or is it a cop-out to intone "De gustibus non est disputandum" and leave it at that?
There are a host of other "possible" answers to this problem, but many of them lie in the risky outer circles and are dangerous to their proposers. (Or, to speak more precisely, the mental mechanisms that would allow them to be considered seriously are dangerous to their owners, since those same mechanisms could produce very untrustworthy suggestions for courses of action in cases that are not decoupled from consequences.) Some of these answers are so far-fetched that they are actually quite humorous. In fact, some of them are so outlandish that I would claim no conceivable survivor of evolution's harsh pruning would ever come up with them, unless deliberately for humorous purposes. Let us look.
First of all, not all that funny but very much in the Feynman "village idiot" spirit, is xyd. Just' plain old dullsville. What can you say to such an answer? Certainly, it has more merit than pqrd did earlier, for here, at least, there is an excuse for such rigidity: z is a trouble spot, whereas s was not. Another answer, definitely more pitiful, is dyz. This one just goes "Thud!" very loudly in my mind's ear. Anyone who would seriously suggest this answer `has seen the light but then dropped the ball, mixed-metaphorically speaking. That is, they go quite a long ways in mapping a onto z, c onto x, but then they seemingly totally forget this level of sophistication and revert to an infantile literality: "Replace whatever plays the c role by d. " It is so inept that it is funny. In fact, I would say that in the answer dyz to this analogy problem there is the germ of a rich theory of humor, based on the idea of level-mixing slips of this sort. Now if only I could say just exactly what I mean by "slips of this sort", I'd be in business ...
Equally scramble-brained is dba. Its hypothetical serious proposer has clearly seen that, in some abstract sense, xyz is a "mirror image" of abc. An attempt is therefore made to take a mirror image of abd, but somewhere in the shuffle, the type' of mirror image involved got forgotten, and for it was -substituted the crudest possible notion of "reversal". The result is another heavy-handed thud. By the way, nobody has ever seriously suggested to me either the clunky dyz or dba, or even yyz, interestingly enough. Oh well, I guess such people's ancestors must have all gotten gobbled up by dreaded human-eating zebras.
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These analogies, as must be abundantly clear by now, are themselves analogous to real-world analogies. Or better yet, they are allegories or fables for analogy-land children. They capture the essence of the dilemmas that analogy-makers face over and over again. Pressures for literality vie mightily with pressures for "literarity"-that is, for high-flown abstractions that disdain rigidity of reference. In an analogy where real insight is needed, often some initial forays are made that reveal some literal-minded ways of seeing the situation, but they ring false or overly crude, and so one does not stop there. Instead, one continues searching, guided by a number of small cues, and at some unpredictable moment, something simply snaps, and a host of things fall into place in a new conceptual schematization of what is going on. What was once important becomes suddenly trivial, and a new essence emerges, an organizing concept or set of concepts that seem far superior.
