Surfaces and Essences

by Douglas Hofstadter

  If we examine the pathway that Einstein took to reach the extended principle of equivalence (which applies to all phenomena of physics, not just to those of mechanics), we see that he re-exploited, in a new context, an analogy that he had already exploited once. More specifically, he carried out the same conceptual extension in two different contexts, each time starting with the concept of mechanical experiment and ending up with the more abstract concept of physical experiment. Each of these extensions was of course due to an analogy — namely, the vertical analogical leap that amounts to the thought: “Physics is analogous to mechanics; it’s just that it includes more.” (Recall the similar vertical analogical leap of Doctor Gelenk, who thought, “Joints are analogous to knees, only they are a more general biological notion.”) And Einstein used this analogical leap in two situations that were themselves analogous (first in extending the principle of Galilean relativity to yield special relativity, and later in extending the restricted principle of equivalence to yield the founding ideas of general relativity). Thus, the most advanced breakthrough of Einstein’s life came out of an analogical leap that was analogous to another analogical leap — thus an analogy between analogies, or, if you will, a meta-analogy.

  This recalls a remark once made by the Polish mathematician Stefan Banach, as recounted by his friend Stanislaw Ulam: “Good mathematicians see analogies between theorems or theories, but the very best ones see analogies between analogies.” And along the same lines, Einstein’s Scottish predecessor James Clerk Maxwell once observed that rather than being attracted by parallels between different principles of physics, he was attracted by parallels between parallels, which would certainly seem to be the quintessence of abstraction.

  To sum things up, what Einstein’s creative life illustrates so clearly is that the perception of profound and abstract analogies in the vast tree of science has the effect of shaking not just a twig or a branch, but the trunk itself. If ever anything made the earth tremble, it was the analogies discovered by Albert Einstein.

  EPIDIALOGUE

  Katy and Anna Debate the Core of Cognition

  Categorization versus Analogy-making

  Our book ends with a dialogue in which two friends, Katy and Anna, argue about what lies at the core of cognition. Katy sees categorization as playing that role, and she is persuaded that it differs in many ways from analogy-making. Anna sees analogy-making as lying at cognition’s core, though she agrees with Katy about categorization’s importance; indeed, she does her best to show that analogy-making and categorization are one and the same thing by finding weaknesses, point by point, in Katy’s arguments.

  The following table summarizes the nine dimensions along which Katy claims that categorization and analogy-making differ significantly.

  Categorization

  Analogy-making

  Frequency

  Nonstop

  Occasional

  Originality

  Routine

  Creative

  Level of awareness

  Unconscious

  Conscious

  Controllability

  Automatic

  Voluntary

  Degree of similarity

  Disparities are undesirable

  Disparities are desirable

  Focus of the activity

  Entities

  Relations

  Levels of abstraction

  A jump between two levels

  A bridge on just one level

  Degree of objectivity

  Objective

  Subjective

  Trustworthiness

  Reliable

  Suspect

  A careful examination of these potential distinctions leads, as will shortly be seen, into territory lying well beyond the question as to whether analogy-making and categorization constitute just one thing or different things. It raises issues concerning how we perceive the world, how we form concepts, how we understand, and how we communicate. In short, it opens up the entire question of the nature of thought.

  But our two protagonists, in their spirited exchange, will surely make all this far clearer than we could, and so, without further ado, we’ll give them the floor.

  The telephone rings.

  KATY:Hello! Who is it?

  ANNA:Hello, Katy, it’s Anna. I hope I’m not waking you, at this early hour.

  KATY:Oh, no — not at all. As a matter of fact, just a few minutes ago I woke up from a strange dream in which you and I were having a lively telephone call. Actually, it wasn’t an ordinary phone call — it was a heated argument! And it was taking place in Chinese, of all things. How that was possible, I don’t have the foggiest idea (or the cloudiest, for that matter), since I don’t speak a single word of Chinese!

  ANNA:Are you serious? Exactly the same thing just happened to me!

  KATY:How odd! But exactly the same?

  ANNA:Well, sort of — you know what I mean. Something very similar — almost exactly the same. To be specific, I too just woke up from a strange dream, and in my dream, just as in yours, you and I were having a heated argument on the telephone! But in mine, our debate was all taking place in Russian — and as to how that could have happened, well, I don’t have any concept (not even the tiniest one), since, as you well know, I couldn’t utter a word in Russian to save my life!

  KATY:What a weird coincidence! One dream, dreamt simultanously by two different people! It sounds like a fairy tale! What was the nature of our heated argument in your dream?

  ANNA:To tell the truth, I don’t recall at all. I guess that makes sense, since I literally didn’t know what I was talking about, speaking as I was in Russian! And in your dream, what was our argument about, Katy?

  KATY:Well, I have to admit that I don’t remember a single word of it either, since everything I said was in Chinese, of which, as you well know, I am totally ignorant. Is truth not stranger than fiction?

  ANNA:Well, I wouldn’t know, but dreaming is certainly a strange phenomenon. The mind leaps about in such fantastic ways.

  KATY:The human mind is profoundly mysterious, I agree. Thinking is the most elusive phenomenon under the sun, even though we do it all the time. Shouldn’t its nature be crystal-clear to us, since it’s the medium in which we swim? Or perhaps it is murky and miraculous precisely because it’s so ubiquitous.

  ANNA:I fully agree with you, Katy, and actually, this brings me right around to the reason that I phoned you so early in the morning. Lately, you see, I have been thinking a great deal about thinking, and I’ve come to the surprising conclusion that there is one special type of mental process that lies at the very core of it all. I wonder if you can guess what it is.

  KATY:What a coincidence! I, too, have been thinking about thinking, Anna, and I, too, have identified a special mental process that I believe lies at thought’s very core. Would it not be astonishing and wonderful if both of us had independently stumbled on the very same idea?

  ANNA:Oh, yes — that would be a delightful surprise. So let me tell you straight off what my candidate mental process is. For me, the key mechanism underlying all of thinking is the making of analogies — the spotting of a link between something one is just experiencing now and something one has experienced before. An analogy can summon up any aspect of our past, and thus, when we face a new situation, we can bring to bear the closest experiences we have ever had. In a word, analogy-making is the mechanism at the basis of all thought.

  KATY:Analogy-making, eh? Now there’s an off-the-wall candidate! It’s certainly not what I would have proposed. I see we have rather divergent views on the topic. So let me come straight out as well, and put my cards on the table: categorization, not analogy-making, is where I think the secret of our minds lies. Categories grow out of our experiences and they organize our mental libraries; categories are the building blocks of thought, and categorization is the magical key to all of thinking.

  ANNA:Oh, really? So you would say that categorization is more important than analogy-making in the
life of the mind? Tell me how you see things, Katy.

  Categorization is a constant necessity; analogy is a rare luxury

  KATY:Gladly! In the process of thinking, nothing is more pervasive or essential than the assignment of things and situations to known categories. To simplify the world, we constantly have to carve it up into standard, familiar pieces; otherwise, we would find ourselves overwhelmed by a tidal wave of constant novelty. After all, each moment and each situation that we face is different from all other ones we have already experienced, even if only slightly so. Each time you blink, shift your gaze, breathe in or breathe out, move your lips or nostrils or eyebrows, your face is a little different from how it was a hundredth of a second earlier. And each of these micro-movements you make changes, even if only infinitesimally, the perspective from which you see the table in front of us. And as our surroundings change from moment to moment, the only way for us to orient ourselves is to sort the incoming stimuli into familiar, reliable categories, such as table, even if we have never seen that specific table before. What is around us is always changing, and if we didn’t continually categorize it all, thus simplifying it into stable regularities, our environment would seem to us like utter chaos. Everything would be novel and unknown, and our heads would forever be spinning. My ability to recognize a novel thing as being, say, a chair, a table, a breeze, a look of surprise, or a thinly veiled threat, is due to the fact that I have already developed these categories, and I have a huge repertoire of them. Without that storehouse, I would be unable to recognize any of the recurrent features of the world around me. So it’s categorization that allows me to survive in this unpredictable world. That’s my view, my dear Anna.

  ANNA:You’ve made valid and insightful points, Katy, and you’ve clearly shown why conception and perception are such closely related phenomena. As a matter of fact, they are interchangeable notions, even if common usage tends to distinguish between them, with perception supposedly guided by the senses and conception supposedly guided by the mind.

  KATY:All of this is well and good, but in my view, categorization must be ranked far above the making of analogies. I understand, Anna, that you hold analogies in very high esteem, and surely they merit your esteem, but nonetheless, just think for a moment: analogies are rare mental events, far rarer than the sorting of things into their proper categories. Analogies take place only when we are being inventive, when we are inspired to add sparkle and pizzazz to our mental life. When we make an analogy, we rock the boat, connecting two things in a way that we’d never dreamt of before. That lovely frisson when we spot a novel connection between ideas or objects or situations is a very gratifying feeling, but alas, it does not arise often. If we never made analogies, our mental lives would be a tad less zesty, no doubt, but our existence would not otherwise suffer in any way. Let me put it this way, Anna: if categorization is the meat and potatoes of cognition, then analogy-making is an exotic spice that one can easily do without. Coming up with an innovative analogy brings delight, that’s for sure, but doing so is a cognitive luxury; one could easily live one’s whole life without ever making a single analogy!

  ANNA:I subscribe enthusiastically to how you have portrayed categorization, Katy. Categories are the heart and the breath, the motor and the fuel, the roads and the highways of thought, and it would take someone without an ounce of imagination to fail to recognize the utter pervasiveness and indispensability of categorization in the activity of cognition. Nonetheless, I think you delude yourself concerning analogy-making, for it is in fact ubiquitous, every bit as much as categorization.

  KATY:That makes no sense to me. I make an analogy only once in a blue moon!

  ANNA:You underestimate yourself, my friend. Do you recall how in Chapter 7 of this very book several quotations were presented, all taken from writings by various specialists, some of which characterized categorization and others of which characterized analogy-making, and yet the statements were virtually identical? Specifically, there it was stated that both of these mental processes cast new and strange things in a familiar light (“categories allow us to treat new things as if they were familiar” and analogy is what allows us “to make the novel seem familiar”) — and being able to do that is an absolute necessity for survival, is it not? So you see, you make an analogy whenever you recognize the familiar in something unfamiliar.

  KATY:Are you suggesting that analogies are as widespread as categorizations are? I don’t see them everywhere — in fact, hardly anywhere. To be convinced, I would need some examples, because to me analogies seem as rare as hens’ teeth.

  ANNA:In that case, Katy, think of the most ordinary moments in our lives — if you look carefully, you will see that analogies abound in them, like flowers in a spring meadow. Given that analogies allow us to understand the new in terms of the old, they are every bit as common as the assignment of things to categories. Didn’t you just point out, a few moments ago, that any new situation differs in all sorts of ways from all prior situations? Well, then, in order to make sense of a new situation, we have no choice but to relate it to things we have experienced before. This means that we have to make analogies left and right. Moreover, researchers who have studied the role of analogy in human thinking have often pointed out its ubiquity. For instance, the mathematician George Polya wrote: “Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.” Along similar lines, the theoretical physicist Robert Oppenheimer declared: “When faced with something new, we cannot help but relate to it except by comparing it with what is familiar and known to us.” Psychologists who specialize in the study of analogy-making are also on the same page. Thus Dedre Gentner has written: “Analogies and metaphors are pervasive in language and thought.” And Keith Holyoak and Paul Thagard have stated: “Analogy is ubiquitous in human thinking.”

  KATY:So far you’ve merely given me a bunch of generalities. They are good food for thought, but I told you I want some concrete examples.

  ANNA:All right, let me try. On what basis do you walk up a staircase that you’ve never laid eyes on before? How do you know how to use a doorbell you’ve never encountered before? How do you know how to turn the knob of a door that you’ve never seen before? How do you know how to use a shower you’ve never used before? And what about sitting on a new chair for the first time, or picking up a magazine you’ve never touched before? How do you know how to flip its pages, effortlessly telling its ads from its articles? Life is filled to the brim with rudimentary analogies like these.

  KATY:All these examples are not analogies; they’re just actions, not thoughts.

  ANNA:I challenge you to find the borderline between acting and thinking. Mental activity lies behind every action, and the term “mental activity” is simply a fancy synonym for “thinking”. But if these examples don’t satisfy you, then take a look at Chapters 1 and 2 of this book. They show how the choice of every single word or phrase comes out of analogy-making. What could be more ubiquitous than choosing what word to say next? Then Chapter 3 shows how, through analogy, we are constantly reminded by one situation of prior situations, and how, when we come out with such common phrases as “me too”, “next time”, “in general”, “it won’t happen again”, “that’s what always happens”, and many others, we are relying on tacit analogies. For instance, if you say “me too”, don’t you mean “the analogous thing holds in my case”? And if you say “next time”, what could you mean other than “as soon as an analogous event takes place”? Doesn’t “like that” mean “analogous to that”, just in more everyday language? And so, in short, analogies have no reason to feel outnumbered by categorizations; they are everywhere under foot, being neither rare luxuries nor exquisite delicacies. Analogies are not merely the icing on the cake of cognition — they are the full cake, including the icing!

  Categorization is routine; analogy is creative

  KATY (still convinced that categorizat
ion and analogy-making are as different as day and night): I’ve followed your argument, Anna, and I’ll withdraw my claim about the rarity of analogies. I’ll concede that we make analogies all the time, even if most of them are trivialities. It would never have occurred to me before to call such mini-thoughts “analogies”, but you’ve convinced me. Nonetheless, analogies and categorizations are very different beasts. You yourself implicitly admitted this point only moments ago, with your words “the icing on the cake of cognition”. Let me explain. To categorize is to rely on preexisting mental categories. Let’s say that a noisy, furry, four-footed entity is trotting down the street by your side and I assign it to the category dog. This connection to a previously acquired notion gives me instant access to many facts, such as that I’m dealing with a living entity, that it has many internal organs, that it could possibly bite and transmit rabies, that it tends to eat meat, that it enjoys being taken for walks, that it might smell bad, and so on. The moment I decide that something belongs to this category, I effortlessly retrieve all sorts of facts that I’ve accumulated over the course of my life about dogs in general or about certain breeds of dogs. But my use of all this knowledge has nothing creative to it. To the contrary, it’s entirely humdrum.

  ANNA:I have no objection to this; calling a dog “dog” is certainly very mundane.

  KATY:Precisely. Whenever we categorize, we merely take advantage of facts acquired in our past. We rely on prior experiences, expecting them to repeat themselves, but we create nothing new. Although categorization pervades every moment of one’s mental life, it is neither inspired nor inspirational. By contrast, your “icing on the cake of cognition” involves originality, creativity, inspiration, and flashes of insight, all of which go way beyond bland uses of prior knowledge. In this sense, analogy deserves praise, because it’s what allows us to creatively connect things that one would tend to think are utterly unrelated. This is how analogy-making is so distinct from categorization. I admit that my observation saddens me a bit, because I would have liked categorization to play a more noble role in thought, but in any case this shows that categorizing and analogy-making are not cut from the same cloth. The former is routine, while the latter is creative.