Surfaces and Essences

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Surfaces and Essences Page 62

by Douglas Hofstadter


  Once we’ve chosen to map abc onto xyz with their physical directions reversed, then forwards motion in the alphabet is implicitly being mapped onto backwards motion in the alphabet, and thus implicitly, as part and parcel of this mapping, the concept of successor is being mapped onto the concept of predecessor. Another way to put this is that when abc is read c-wards, starting at the a, then its fabric is one of successorship, whereas when xyz is read x-wards, starting at the z, then its fabric is one of predecessorship. And so, from the simple act of seeing the a and the z as each other’s counterparts, a tight little cascade of conceptual slippages has flowed in a natural and (almost) irresistible manner. At the outset we saw first slipping to last, and then we saw left slipping to right. And now we’ve hit the final slippage in the cascade — the slippage from successor (the fabric of the abc world) to predecessor (that of the xyz world).

  This final conceptual slippage means that instead of wanting to take the successor of the x in xyz (an overly rigid thing to do), we’ll want to take its predecessor. All of this will give us, in the end, the string wyz. This is a surprisingly pretty answer to the question “Suppose abc changed to abd; can you make xyz change in the same way?” Several coordinated slippages have led us to a fluid analogy. And all this insight came from “stubbing our toe” on the letter z — that is, from the snag due to z’s lack of successor.

  This symmetry-based answer is, in a certain sense, an even stronger and more convincing analogy than when pqrs recalled turning into pqrt. That is, xyz ⇒ wyz is arguably more like abc ⇒ abd than pqrs ⇒ pqrt is, strong though that resemblance is. Why is this? Because both abc and xyz are strings wedged at their respective ends of the alphabet. There’s a perfect symmetry here, and the two changes are like perfect mirror images of each other. No flaw is perceivable anywhere in this analogy, and this is what makes it so appealing. Although pqrs has quite a bit in common with abc, it’s not as similar to abc as xyz is (what can rival a mirror for making symmetry?), even if pqrs is clearly much more similar to abc than tky is.

  In Chapter 5, we cited the striking conceptual-proximity slippage made by a grandfather who, as he and his son drove by a cemetery, observed, “This is where all four of your grandkids were born.” There we pointed out the very human drive towards internal consistency, pushing the grandfather towards making two slippages between opposite concepts, rather than just one. We even suggested that either of the slippages could have brought the other one along “on its coattails”. Such a “coattails” characterization would also describe the way in which slippages coordinatedly cascade in this Copycat analogy between abc ⇒ abd and xyz ⇒ wyz, although we are obviously talking about esthetic coattails rather than electoral ones.

  On Dizziness

  We wish now to venture momentarily into yet murkier waters, offering the event xyz ⇒ dyz as a potential analogue to our standard old tale abc ⇒ abd. What’s going on here? To come up with this analogy, a person would have to exhibit a highly implausible kind of confusion, which we will call “dizziness”. First they would discover the subtle idea of mapping of the a onto the z, and then they would notice the spatial reversal that this implies, involving the conceptual slippage right ⇒ left, which leads to the subtle idea of modifying the x instead of the z — but then, for some reason, they would drop the ball. Totally forgetting all the subtleties they had just noticed, they would blithely interpret the event abc ⇒ abd as nothing more than a replacement of a certain letter by the random letter d, without any deeper rhyme or reason, and would mindlessly apply this recipe to the x. This is a bizarre twist, because genuine initial insights are followed up by an act that totally betrays the spirit of finesse of what was done only moments before. If one can use the phrase “in bad taste” in such an austere domain, this would certainly seem an appropriate time to do so.

  In a word, this analogy is dizzy, in the sense that the hypothetical person who made it seems truly confused, exhibiting an incoherent mixture of smartness and stupidity, of insight and myopia, of brilliance and dullness. First there is a moment of genuine insight (mapping a onto z) and then, further into the process, the same high level of insight continues (mapping c onto x), and yet, at the very last moment (“What kind of insight is called for now, to modify the x?”), comes an act of myopic, local literality — seeing simply the raw letters c, d, and x, instead of the roles they play inside their worlds.

  Such a dizzy hodgepodge, where high insightfulness is found happily coexisting with mind-boggling literal-mindedness, is implausible, to put it mildly. We have never run into anyone who in all seriousness proposed the answer dyz. And yet anyone who carefully explores the subtle angles of this particular analogy problem in the Copycat domain will sooner or later bump into this answer, whatever they may think of it in the end; and if this answer is explained to people, they invariably understand its “logic”, even if they find it strange or uncomfortable. We may not like this answer, but we can still understand its strange logic, and when we do so, we can even find it humorous.

  In this sense, the dyz answer to the xyz problem is analogous to a joke that circulated in the 1980’s about Nancy Reagan, then the First Lady. It was stated that, upon hearing that there was a glut of butter, she perkily declared, “Oh, before it goes bad, we should distribute all the extra butter to the poor people of our country, so that they have something to dip their lobster tails in.” The First Lady in the world of the joke understood poverty in a ridiculously implausible manner: while realizing perfectly clearly that poor people have no butter, she totally failed to realize that they also don’t have lobster tails a-plenty. This joke is reminiscent of the infamous remark by Marie Antoinette, also about poor people, but in the streets of Paris: “Oh, have they no bread? Poor dears! Then let them eat cake!” This is what we mean by “dizziness”.

  During the heyday of the First Lady joke, a similar joke about the United States and the Soviet Union was making the rounds. An American, bragging to a Russian about our land’s great freedom of speech, proudly proclaimed, “We in America are so free that we can march up and down in front of the White House and shout, ‘Down with Reagan!’ ” Unimpressed, the Russian replied, “That’s no big deal. We in Russia can do exactly the same — we too can march up and down in front of the Kremlin and shout, ‘Down with Reagan!’ ” The analogy is almost complete, but the ball is fumbled on the one-yard line (much as in xyz ⇒ dyz).

  Frame Blends

  We are approaching the rich and important topic of frame blends. Rather than defining that term in an abstract fashion, we will do so through a series of very concrete examples in the Copycat microworld, since the phenomenon emerges with great clarity there. Once again we trot out our same old story of abc changing into abd, and this time (exactly as before) we ask, “Of what event in its past was iijjkk reminded?” Here is a set of possible answers:

  iijjkk ⇒ abd

  (“Replace the whole string by abd ”)

  iijjkk ⇒ iijjkk

  (“Replace the letter c by a d”)

  iijjkk ⇒ iidjkk

  (“Replace the third letter by a d ”)

  iijjkk ⇒ iikjkk

  (“Replace the third letter by its successor”)

  iijjkk ⇒ iijjkd

  (“Replace the rightmost letter by a d”)

  iijjkk ⇒ iijjd

  (“Replace the rightmost group by a d”)

  iijjkk ⇒ iijjdd

  (“Replace the rightmost group by an equal-sized group of d’s”)

  iijjkk ⇒ iijjkl

  (“Replace the rightmost letter by its successor”)

  iijjkk ⇒ iijjll

  (“Replace the rightmost group by its successor”)

  We’ve listed these answers in an order that echoes the amount of frame-blending involved, starting with a lot and finishing with none. Take the third answer, for instance, where the third letter — j — was replaced by a d. This answer seems very maladroit, for the concept d feels like an intruder in the iijjkk world; that
is, it belongs to the abc world, near the beginning of the alphabet. The presence of a d in the midst of the letters i, j, and k (which constitute their own little world, far away in the alphabet from the abc world) seems muddle-headed. But that’s not all; the concept third letter also seems like an intruder. Borrowed literally from the abc world, it has been bluntly thrust into the iijjkk world with no regard at all for the fact that iijjkk is twice as long as abc is, so that the richly pregnant meaning of third letter in the abc world is utterly lost when the concept is exported in such a literal fashion to the other world. Thus the third answer is the result of a great deal of blending, and esthetically speaking, it suffers for it.

  Each of the other answers shown, except for the last one, involves some type of frame-blending, in the sense that some concept belonging to the abc world has been borrowed and thrust into the iijjkk world, contaminating it. Even the second answer, iijjkk, which on its surface exhibits no contamination from the abc world, is in fact infected, because the logic behind it (“replace the letter c by a d”) is based on the concepts c and d, which are both alien to the iijjkk world.

  On the other hand, the last answer, iijjll, doesn’t borrow any concept from the abc world that doesn’t belong just as much to the iijjkk world. In that sense, the final line constitutes a “purer” form of analogy than all the lines above it do. All the other lines feature some degree of frame-blending, sometimes very blatant and sometimes less so. And there are plenty of other possible frame-blended answers, such as iijjkk ⇒ aabbdd or even iijjkk ⇒ aabbll.

  Finally, it’s amusing to observe that all the answers in our list, except for the first one, are covered by the single frame-blending recipe “Replace the c of iijjkk by its successor”. Of course this formula is highly ambiguous and subjective, since inside iijjkk there is no more an instance of c than there is a Golden Gate Bridge in Indiana, and yet that wouldn’t in the least keep one from speaking of “the Golden Gate of Indiana”, which is actually a much less wacky idea than our old friends from Chapter 4, “the Meryl Streep of spitting” and “the Mussolini of mulligatawny”.

  Strengths and Weaknesses of Frame-blending

  We don’t wish to portray frame-blending as being intrinsically feeble-minded or confused. Some frame blends, in fact, are carefully designed to exhibit creativity and humor. The xyz ⇒ dyz story is a fine example of this. In it, the letter d was quite intentionally stuffed into a world where it has no business at all, like a bull that has been led into a china shop. This move, deliberately gauche, induces a feeling of complete nonsensicality, and thus of humorousness. One needs quite a sense of fantasy to come up with as off-the-wall an answer as this.

  But a frame blend can also be made inadvertently, simply by overlooking something, in which case it is more like a makeshift solution to a real-world dilemma: acceptable but not optimal. Indeed, many people come up with the fairly flat answer xyd rather than wyz for exactly this reason — they don’t examine the strings deeply enough to let the subtle a–z connection, followed by its rapid chain reaction of conceptual reversals, jump out at them.

  A frame blend can also be made when one simply doesn’t possess the knowledge required to see certain aspects of a situation, and so one necessarily leaves them out. If, for example, someone knew the alphabet but didn’t know how to count, then the answer mrrjjj ⇒ mrrjjjj would simply be unavailable to them. On the other hand, with the set of concepts they possess, they could easily come up with mrrjjj ⇒ mrrkkk. To some observers, this answer may seem filled with insight and elegance, while to others it may look like the most glaring of frame blends (what is the concept of alphabetic successor doing in a world whose essence is not alphabetic but numerical?).

  Or suppose that someone with a strangely defective kind of vision were looking at the string iijjkk, and could just barely make out that it contained six letters but had no idea that any of them were repeated. They might say, “Change the rightmost letter to its successor!”, thus defining the event iijjkk ⇒ iijjkl. To us this looks awkward, but if we take into account the lack of information that gave rise to it, we can understand the oversight. They performed as well as possible, given that their information was partial and incomplete. This is an echo of the earlier idea that people make the deepest analogies that they can, as constrained by their conceptual repertoires.

  Although these examples may seem artificial, scientific analogies are often made in just this way — as brave leaps way out into the thickest of fogs. Decades or centuries later, some of those leaps may look silly, because, having been based on precedents well known at their time, they incorporated various naïve assumptions that in the end turned out to be utterly inappropriate to the new situation — but such ex post facto judgments of naïveté are a luxury. They are the wisdom that one acquires when one is blessed with, as the phrase goes, “20–20 hindsight”.

  To be concrete, let’s recall the way that, in the seventeenth century, light waves were hypothesized to be very similar to sound waves: they were presumed to be longitudinal compression waves carried by some all-pervasive elastic substance very much like air but even more pervasive and even more tenuous than air. (A sufficiently bold spirit might even dare to describe that elusive substance as “ethereal”!) This conceptual leap was an audacious one and contained far more than a grain of truth, as was revealed centuries later, but in those early days, people’s perception of the nature of light was contaminated by incorrect analogical importations from the world of sound — in other words, by a frame blend in which concepts that were to some degree inappropriate were imported from the sound side of the bridge.

  To be specific, both the idea that light had to be a longitudinal compression wave and the idea that it would need an elastic medium to propagate it were practically perfect carbon copies of what was known to be the case for sound — but in the case of light, both presumptions were eventually discovered to be wrong. Nonetheless, these guesses about light, even if they were later realized to have been slightly off, constituted a great and bold leap in the dark, affording humanity its first inklings of light’s true nature. It would be unjustified to criticize such boldness simply because some of it eventually turned out to be wrong. Why would anyone have postulated a transverse wave propagating through vacuum when such a thing was unheard of, and indeed, virtually inconceivable to people at that time? To criticize the natural even if somewhat cautious leap (as seen from today’s viewpoint) that was made would be nonsensical. After all, even the greatest thinkers of the 1600s couldn’t possibly have leapt so far ahead as to anticipate Maxwell’s equations and Einstein’s theory of special relativity, which are indispensable ingredients in today’s understanding of light as a wave.

  The greatest scientists of any era venture into the wild unknown by making bold but often naïve analogies about phenomena that they do not fully understand, and almost inevitably, in making these frame-blending analogies, they inadvertently bring along irrelevant baggage that comes from their own limited knowledge. They borrow a neat little “package” of various facts about phenomena that they do know, taking (or mistaking) those facts for universal truths, and indeed, some of the facts in the packet will turn out later to be inapplicable to the new phenomena. And yet the best of such analogies contain so many grains of truth that they open up whole new perspectives, despite various incorrect assumptions that have been unwittingly imported from other domains. This kind of partly-insightful, partly-defective analogy-making is a hallmark of all humans, and will be discussed extensively in the next two chapters.

  We might add that frame-blending is at times deliberately exploited in literature and other artistic domains for its lively and stimulating qualities. And the activity of reading a book or watching a film, play, or opera takes for granted the act of projection by each reader or viewer into the scenes, identifying with one and then another of the characters. Such mental blends are done rapidly and often without any realization on the part of the viewer, but they are what give to any dramat
ic work its emotional meaning, since without them the work would merely be a cold, third-person recitation of events.

  Fauconnier and Turner’s Conceptual Blends

  We have brought up the concept of frame-blending because it is an enormously rich source of insight into many phenomena in human cognition, and under the name “conceptual integration” it has been beautifully and richly explored and described by cognitive scientists Gilles Fauconnier and Mark Turner and their colleagues and students. They have shown time and again that frame-blending is found throughout human thought, sometimes using marvelous examples that seem exotic, just as often using examples that are as down-to-earth as can be, but in any case demonstrating the fundamental importance of the phenomenon.

  One of the examples analyzed by Fauconnier and Turner in their book The Way We Think is a billboard that was put out by the state of California in a campaign to combat smoking. As they describe it, the billboard featured a large photo of a macho cowboy (much like a classic Marlboro man) riding a horse and conspicuously smoking a cigarette, and at the bottom it stated, in big bold letters, “WARNING: SMOKING CAUSES IMPOTENCE.” The crucial frame-blending touch was that the cigarette was clearly bent and drooping downwards. A complex mapping was of course intended, in which the drooping cigarette would be seen as analogous to a non-erect penis, and in which the act of smoking would be seen as somehow causing both droops at once. It’s obvious that the drooping cigarette is a mental contamination of the scene, and that the drooping has simply been borrowed from an analogous situation (a cigarette being in some ways clearly analogous to a penis, Sigmund Freud’s famous disclaimer “Sometimes a cigar is just a cigar” notwithstanding). Whether you consider it cheap or clever, there’s no doubt that this advertising maneuver constituted a frame blend par excellence, and that its clever ploy induced some long-time male smokers to reconsider their habit.

 

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