Surfaces and Essences

by Douglas Hofstadter

  KATY:I concede that there are analogies that just bubble up and cannot be squelched, but analogy is at least sometimes under our control. This fact seals my argument, because categorization, in contrast, can never be controlled. Voilà! QED!

  ANNA:But it certainly can, dear Katy! There are categorizations that are totally under our control. Only moments ago we were talking about the categories innocent and guilty, and also about the category planet. Such categories don’t automatically jump to your mind all the time, and you can easily decide to ask, or to stop asking, whether a given entity belongs to a given category; this shows that the process can be controlled. For example, if you see a person walking down the street, you don’t instantaneously try to figure out their blood type or their political affiliation. Depending on the context, you can decide either to do so or not to do so. And thus, Katy, once more, despite all your valiant efforts, you haven’t shown me any clean line of demarcation separating analogy-making from categorization.

  Categorization favors similarities; analogy favors dissimilarities

  KATY:Enough said, Anna; your point is clear. But let me turn now to a criterion that seems to me to be absolutely central and undeniable — namely, degree of resemblance. The closer an item is to the core of a category, the easier categorizing it as such will be. That is, categorization is quickest when the resemblance is maximal, wouldn’t you agree? Take my dining-room table. I effortlessly call it “table” because it has so many commonalities with my mental image of tables. If you show me typical members of various categories, such as a typical glass, chair, dog, or eagle, I’ll have no trouble categorizing them. In a word, resemblance is crucial for categorization. But analogy-making is very different. In fact, it’s the exact opposite! What makes the quality of an analogy is how different the two analogous situations are. To put it another way, what lends great strength to an analogy is if someone spots an essence shared by two situations that to most people appear wildly different. An analogy’s power comes from the fact that it sees beyond the surface differences of two situations to reveal something deep and hidden they have in common. This is what gives an analogy power and interest. Do you see my point? At long last, I’ve put my finger on a rule of thumb fundamentally separating analogy-making from categorization!

  ANNA:Eureka! Congratulations, Katy! And so, while we’re speaking of fingers and thumbs, would you say that your right thumb and your left thumb are analogous?

  KATY:Of course not! My thumbs look just alike. That doesn’t make for an analogy!

  ANNA:Really? Think back to Chapter 6’s finger-wiggling puzzle. There it was self-evident that wiggling your left hand’s thumb is a great analogue to wiggling your right hand’s thumb. Doesn’t that suggest that two thumbs are indeed analogous?

  KATY:Surely you’re joking! Any analogy worth its salt is a mental bridge between two situations that look very different from each other. Take the atom and the solar system, for instance, or the heart and a pump, or a brain and a computer, or the countries of South America and a big family. In each case, the two items don’t look at all the same, and that’s what makes them such great analogues. By contrast, a lack of similarity will weaken or even totally block a categorization. So in sum, my point is that the more differences there are between two situations, the stronger the analogy linking them will be, whereas in categorization it’s the exact opposite: the fewer differences there are, the more apt the categorization will be.

  ANNA:Whoa there, Katy! You seem to have forgotten that any situation has both superficial and deep aspects, and what matters most, whether we’re talking about categorization or analogy, is that deep aspects be the same. In categorization no less than in analogy-making, there can be all sorts of superficial aspects that differ. My point is that in addition to obvious categorizations featuring resemblances at all levels, there are subtler categorizations where there are differences galore.

  KATY:How could the existence of many salient differences ever be consistent with strong category membership?

  ANNA:Well, tell me: what do a lion, a giraffe, a snake, a whale, an octopus, a bullfrog, a spider, a butterfly, and a millipede have in common?

  KATY:They’re all animals, obviously. But what are you driving at?

  ANNA:What I’m driving at is that they all belong to a single category — that of animal — and yet they look very different. Superficially, they have differences galore. So you see, as soon as we start considering categories that are a bit general and abstract, then for their members to be highly variegated is par for the course! And this is not a rare phenomenon. Take the categories of vegetable, mammal, clothing, furniture, and musical instrument. Wouldn’t you agree that each of these is a very familiar category whose members differ enormously from each other? None of them is a category whose members are like so many peas in a pod. And I’ve just scratched the surface of such cases. Think, for instance, of categories named by phrases, such as “slippery slope”, or of categories named by proverbs, such as “beggars can’t be choosers”. In such cases, there are countless excellent members of the given category that differ from one another in myriads of superficial ways. Do you recall the long list in Chapter 2 of members of the abstract category Once bitten twice shy? There’s no doubt that the proverb names a genuine category, but the list showed that there can be enormous differences between situations that bring the saying “Once bitten twice shy” to our lips.

  KATY (momentarily shaken but still confident): I have to say, Anna, that once again your points are well taken. What can I do but admit that some categories have members that are enormously different? And yet I am going to push onward because I am convinced that for an analogy to be strong, the existence of differences is necessary, whereas for a strong categorization, the existence of differences is merely optional.

  ANNA:Come now, Katy! Do you seriously believe that your two thumbs are not analogous, whereas your thumb and big toe are fairly analogous, and your thumb and your little toe are even more analogous? That sounds backwards to me! Let me be more explicit. Some analogies are very simple and down-to-earth, in that they link objects or situations that look very much alike, and by making such analogies all the time we survive very well in our everyday world. Once again, think back to the novel staircase, novel doorbell, novel doorknob, and so on, which we spoke about not long ago. As we move about in the world, we make analogies left and right between novel situations we encounter and situations that we encountered in the past, and we don’t place higher value on connections that are distant; indeed, the more closely the precedents that we retrieve from memory resemble the novel situations, the more confident we feel in our analogies. The rule of thumb “The fewer the differences, the better”, which you proposed for categorization, holds just as much for these everyday, down-to-earth analogies. Unconscious, irrepressible, run-of-the-mill analogies are all around us, and they don’t feature any major differences at all. Every time you say “me too”, you’ve made one!

  KATY:But Anna, who would ever be proud of having made analogies of that sort? They are flat and utterly devoid of interest.

  ANNA:Agreed, but there’s no law that says every useful analogy has to be a source of great pride. To be sure, these down-to-earth analogies are not ones that will make humanity swell with pride for having made them; they’re not in the same league as highly subtle and abstract analogies that evoke feelings of astonishment. But even so, they still belong, beyond a shadow of a doubt, to the category analogy! Analogies linking Fido and Spot, linking my dining-room table and yours, linking my elbow to yours — these are all perfectly fine analogies, even if they fall short of being deep new insights. And while we’re at it, let’s not forget Chapter 3’s analogy between my eyelid and yours, thanks to which I can tell you my technique for getting an annoying eyelash out of my eye (and thus out of yours, too). Analogous situations can have many superficial resemblances, but they can also have many superficial differences. Such considerations will determine how subtle or deep the ana
logy is. Just as huge differences are par for the course in abstract categorizations but are rarer in very concrete ones, so huge differences are standard in deep analogies but are rarer in rudimentary, everyday ones. The simplest bread-and-butter analogies, just like the simplest bread-and-butter acts of categorization, involve easily-seen resemblances and don’t need dissimilarities to be thrown into the bargain.

  KATY:Well, all right, then. I can see that categorizations, like analogies, can exist perfectly well despite all kinds of dissimilarities. So be it. But we come now to a key difference between analogy-making and categorization, and because of this key difference, no one should ever confuse the two.

  ANNA:I’m eager to hear your new proposed distinction.

  Categorization applies to entities; analogy to relations

  KATY:I’ll explain it with pleasure. The fact is that when one makes a categorization, one takes some entity from one’s surroundings and associates it with a mental category, whereas when one makes an analogy, one takes the relationships in two situations and one finds that it’s these relations that are shared. For instance, Anna, if I categorize the object I’m holding as a hammer, it’s because I’m taking an element of my environment and linking it to a particular mental structure in my brain — the category hammer. So this object is matched with that category by a perceptual process that involves recognizing that it possesses many typical hammer traits. However, no relationships between parts of the hammer are involved.

  ANNA:I can’t help stating that this claim strikes me as extremely dubious.

  KATY:Well, first let me finish, please, because by the time I’m done I bet you’ll have changed your mind. No relationships are needed to categorize this object as a hammer. On the other hand, if I am told about the analogy between a heart and a pump, what goes on is that I see some relationships among the parts of a heart and also some relationships among the parts of a pump, and then I map those relationships onto each other. The relationships on either side of this specific analogy involve such concepts as output, pressure, expansion, compression, and so forth. The analogy is convincing because the relationships on one side match up with the relationships on the other side. In the case of the atom and the solar system, the relationships among the parts on either side include such notions as massive entity exerting an attractive force on lighter entities and rotation around a central entity, among others, and once again, these relationships match up very nicely across the two sides. So you see, categorization pays no attention to relationships among parts of a situation, whereas analogy-making depends vitally upon them. This is surely a crucial distinction, wouldn’t you agree?

  ANNA:What I hear you saying is that if a process associates some entity to a mental structure, then it’s a case of categorization, while if a process maps relationships in situation #1 to relationships in situation #2, then it’s a case of analogy-making. Well, I’ll simply say that this proposed distinction may seem appealing at first, but under scrutiny it falls apart, just as all the others have. In fact, your own examples will serve well to undermine your thesis! Let’s take the heart–pump analogy. Is it really all that obvious that when one matches up the ideas of heart and pump, one is necessarily making an analogy and not a categorization? To be sure, a heart is analogous to a pump, but why not also assign a heart to the category pump?

  KATY:Oh, I suppose you could do that, but you would just be making a metaphor.

  ANNA:Really? Are you suggesting that hearts don’t deserve membership in the category pump? I’m sorry, but that’s quite an untenable viewpoint; all you need to do is open any random biology or anatomy textbook and there you’ll see it printed in black and white that the heart pumps blood, that it’s the pump of the circulatory system, and so on. The fact is, hearts are genuine pumps, pumps as canonical and exemplary as one could ever want. They just happen to be made out of muscle and nerve instead of metal and plastic — but then again, artificial hearts, which are made out of plastic and metal, are also pumps. That’s quite obvious.

  KATY:Well, all right, but I don’t see what you’re driving at.

  ANNA:If a heart is a pump, the reason it is one is precisely because the relationships among parts of the heart and the blood passing through it are the same as those among parts of a metal pump and the liquid passing through it. So we see that categorization does indeed involve taking into account the relationships among the components of things, and that takes the wind out of your prior claim.

  KATY:I see your point for the category pump, Anna, because a pump is a complex device having numerous moving parts, but I don’t buy it for more basic categories, such as hammer or plate. Surely you aren’t going to argue that there, too, category membership depends on relationships?

  ANNA:Oh, yes, I surely am! For something to be a hammer, we expect it to have a handle that is grasped by a human hand, and a metal head that is raised (by the human hand) above the handle and then brought down onto a nail, allowing the latter to penetrate into various materials, such as wood. So here we are talking about a handle, a head (the hammer’s), a hand (a person’s), a nail, and a piece of wood, at the very least, and in order to describe the situation, one needs to refer to many relationships connecting these items. How does this differ from the case of the category pump (which, as we just saw, has internal parts and relationships among them)? It’s quite the same.

  KATY:Maybe that’s true of hammer, but plate is different! All that counts for a thing to be a plate are its size, shape, and strength, and these are just traits, not relationships among parts. In fact, a plate has no parts to speak of. So my point is proven: at least some categories don’t involve relationships, whereas all analogies do.

  ANNA:Sorry, but you’ve jumped a little too fast, Katy. Relationships don’t have to hold among the pieces of an object; they can connect an object to its surroundings. Thus for an object to be a plate, it has to be able to sit on a table without tipping over, and it has to be able to hold such things as gravy without the gravy leaking off. Moreover, it has to be washable, not poisonous, not prone to shedding small flakes, not damageable by the process of cutting, not prone to dissolve in water, and so forth. These key qualities that make something be a plate bring in many other notions, such as tables, gravity, foods that flow, water, soap, human health, knives, cutting, dissolving, and so forth. An object is a plate by virtue of its relationships to all these other things in the world. And so we see that any category, no matter how simple, is a mental structure that involves relationships among various entities in the world, and the act of categorization depends on taking these relationships into account. And thus, dear Katy, your hoped-for distinction goes down the drain.

  KATY:Oh, my… You’re quite right, Anna, that on more careful analysis, hammers and even plates are no less “relational” than are hearts or pumps, and so I have no choice but to withdraw my proposed dichotomy. This is getting frustrating and confusing, but luckily I have a new dichotomy up my sleeve, and with this one I’m pretty sure I’m going to put you into something of a pickle. Are you ready?

  ANNA:Be my guest!

  Categorization involves two levels of abstraction; analogy involves just one

  KATY:In the examples you’ve just given, whenever an analogy is involved, such as between an atom and the solar system, or between a heart and a pump, the two entities involved are both situated at the same level of abstraction. But when you categorize something, that’s not the case, because by definition any category is more abstract than all of its members. In other words, in contrast to what happens in analogies, there is a difference of abstraction levels between things that are being categorized and the categories into which they are getting placed.

  ANNA:Shall we try to focus on a concrete example? That will help a lot, I think.

  KATY:Sure! I assume you’re sitting on a chair, right? That tangible object is a special case of the general category chair, which in turn is a special case of the more general category furniture, and on upwards it goes. And s
o there’s a big difference between comparing two things of equal generality, such as a heart and a pump, and comparing two things that have different levels of generality, such as a specific chair and the abstract category chair, or the abstract category chair and the even more abstract category furniture. And by the way, there was a good reason that when we broached this topic, you carefully steered clear of the classic old analogy between the atom and the solar system: since both items are concrete things, thus at the same level of abstraction, there is no way that we could be dealing with a case of categorization here. To be specific, an atom is not a solar system, nor is the solar system an atom; they’re at the same level. And so to summarize, in an analogy, source and target reside at the same level of abstraction, while in a categorization, the category is more abstract than the thing being categorized. All right; the ball’s in your court now, Anna, and it’s a foregone conclusion that I’m going to win this point!

  ANNA:Don’t be so sure, Katy. I believe I can handle your volley without too much trouble. Your claim that there is necessarily a level-distinction whenever one categorizes something is, I’m sorry to say, just a simplistic stereotype. In fact, this way of looking at things comes straight from the naïve analogy “categories are boxes”, which suggests that one should look at categories as containers and entities in the environment as things to stick in them. Your image reflects the idea that categories in the mind are like Russian dolls, one nested inside another inside another, going many levels up or down. For example: Ollie is a dog; dogs are canines; canines are mammals; mammals are animals; animals are living things; and so forth. And indeed, if categories really were boxes containing their members, then your proposed distinction might be justified. But your idea collapses as soon as it is examined carefully. I suggest you go back and take a look at the end of Chapter 3, where the mental process of categorization or analogy-making — call it whichever of these two labels you prefer — is portrayed as the construction of a mental bridge between two structures that themselves are also mental entities. This happens not only when we make a link between a heart and a pump (or more precisely, between mental structures representing them) but also when we assign an object on a shelf to the category plate. In either case, we take a fresh new mental representation of something and we link it to some preexisting mental structure. And as I just stressed, this process of building a link between two mental structures can be called either “analogy-making” or “categorization” — whichever label one prefers.