Surfaces and Essences

by Douglas Hofstadter

  KATY:Good grief, Anna — are you implying that categorization and analogy-making are exactly the same thing, and that there is no difference at all between them?

  ANNA:I’m glad to see that you’re starting to catch my drift, Katy. Hopefully this will make you more receptive to my ideas. Do you recall the category of faces that looked alike, back in Chapter 3? Seeing Mark Twain’s face gave rise to a first mental structure, and subsequently, seeing Edvard Grieg’s face enriched this initial structure, making it more complex. Each time a new face is seen that resembles the previous ones, the evolving mental structure gets further enriched. There is thus no box here, but simply a mental structure growing in complexity, and as the number of faces involved in this process increases, the mental structure starts to feel more abstract. If we step back and look at this type of process, we see that there is no box representing a category, with members neatly placed inside it, and that there is no upwards leap of abstraction involved in an act of categorization that makes it differ from the process of making an analogy.

  KATY:I’m not convinced. After all, human faces are very concrete, and mentally superimposing several faces has very little to do with the way categories are normally built up. More typical categories are much more abstract than that.

  ANNA:All right; your healthy resistance pushes me to seek a more abstract example to get my point across. Here’s an attempt. When I was little, our family went to an amusement park where we boarded a boat that sailed down a river through a jungle filled with scary beasts. I was thrilled by this great adventure, but when we got off the boat, my father pointed out that our vessel wasn’t really floating down the river but was actually rolling on wheels on some underwater tracks, which forced it down a predetermined pathway. I realized that our boat was in truth more a trolleycar than a boat, which made me sad. The disillusionment struck me, but I didn’t dwell on it for long. Several years later, when I was in high school, some friends convinced me to take part in a talent show in which they were doing a tango number. They were all good tango dancers but I knew nothing, and I had to work like the devil in order to get the routine down. Finally I learned it, and on the big day our number came off pretty well. Everyone congratulated me for my tango-dancing talent. But I knew I didn’t have any such talent! When I explained that the only bit of tango I knew was this tiny two-minute routine that I could perform only in the most inflexible way, I suddenly recalled that boat in the park that looked as if it was freely sailing down the river but was actually rigidly constrained by tracks. My apparent tango skill was just another boat held rigidly on its course by invisible tracks and capable of taking in naïve viewers!

  KATY:That’s an amusing analogy, but what’s the point?

  ANNA:Well, you’ve called it an analogy, which is fine with me, but the way I just described it, my deceptive tango skill was a new member of my childhood category fake-boat-on-tracks.

  KATY:Oh, now I see your point! My interpretation was that you made an analogy between two deceptive events in your life, whereas yours was that your deceptive tango skill seemed to you to belong to an old category that you knew — that of boats that look free but that are actually constrained by hidden tracks.

  ANNA:To be more precise, my interpretation is that the making of the analogy was tantamount to a categorization. The two acts are one and the same thing.

  KATY:Aha — one can look at things either way. I frankly have to admit, that’s most provocative… I can even see how this is parallel to the example of the faces. The boat rolling on tracks established a new category during your childhood; that event in your life is parallel to seeing a photo of Mark Twain. Then your tango-dancing fakery a few years later was the second member of the category, which is parallel to seeing a photo of Edvard Grieg, which makes the original category broader and richer. Now that you’ve transmitted to me the constrained-boats category, I can even think of a pretty good member of this very category in my own life!

  ANNA:I’d like to hear it.

  KATY:Well, one time I was on a plane sitting next to an elderly gentleman from Chile who hardly spoke a word of English. Just for the fun of it, I — who had never taken even a week of Spanish — recited a couple of lines from a poem by Pablo Neruda that a high-school friend of mine who loved South American literature had once taught me by rote because I was so taken with those lines’ beautiful sonority when she recited them. The Chilean gentleman jumped to the conclusion that I was a fluent speaker of his native language, and I think he was quite disappointed when he realized that my Spanish was limited to following that very short stretch of hidden tracks. Don’t you think that my Spanish on tracks is a nice example of your fake-boat (and also fake-tango) category?

  ANNA:Indeed! Let me be the first to welcome this new member into my childhood category, Katy. I think you now can see that any category, abstract or concrete, is launched by a first experience and then builds up gradually as, over one’s life, one runs into various analogous entities. And it’s crucial to see that there is no critical moment when the first memory suddenly switches status and turns into a category. It’s more like the process whereby a hamlet turns into a village, which may grow into a town, and possibly into a city. There isn’t a sharp baptismal moment at which one must henceforth say “town” or “city”, because the metamorphosis is gradual. And likewise, there is no hierarchical difference, no sudden jump in abstraction, between an initial memory and a category. The initial memory founds the category, just as a hamlet founds a potential city. We may be tempted to think that there is a qualitative difference between a category and its members, but that is an illusion coming from the naïve analogy “categories are boxes”.

  Categorization is objective; analogy is subjective

  KATY:I see I’m going to have to give up on this dichotomy, because your argument is persuasive. But I’m not done with you yet, Anna! I’m holding a pen in my hand. If I tell you it’s a ballpoint pen, that’s an objective fact. No one could say the opposite. Much the same holds for that sheet of paper and the paper clip that I see sitting over there on my desk. Ballpoint pen, sheet of paper, and paper clip are all categories that have the quality of total objectivity. An elephant is an elephant, an apple is an apple, and Paris will always be Paris. There are no two ways about it. When we categorize, we do something that is objective.

  ANNA:I don’t agree with you, but I think you might have Plato on your side, because I believe he argued, in Phædrus, that every human being is given the opportunity to look at situations ranging from the most general to the most specific, and to carve up the world in an objective fashion while making very fine distinctions. Am I not more or less correct in my memory of Plato?

  KATY:You’re absolutely right, and I can even cite the exact passage. It’s where Socrates tells Phædrus that humans have “the ability to separate things according to their natural divisions, without breaking any of the parts the way a clumsy butcher does.” This is how biologists proceed when they make taxonomies, for example, and it’s also how cultures and civilizations evolve, gradually moving towards the capability of sorting all the situations in the world into the categories to which they objectively belong.

  ANNA:Well, I’m sorry, but I have to contradict you (and needle your friend Plato as well): categorization is not objective, as you would have it, but is profoundly subjective. For instance, if I assert “Donald Trump is a troublemaker”, is it not the case that in thus categorizing Donald Trump I am making a highly subjective judgment?

  KATY:I understand your example, but troublemaker is an extremely blurry category. You deliberately chose a category that is as blurry as possible! I even think you did it just to be a troublemaker!

  ANNA:Me, a troublemaker? Never! And the truth is that my example is hardly unusual. Blurriness is par for the course with categories. With the greatest of ease we can find categorical blurriness all about us. Edgar Allan Poe’s poem “The Raven” might be seen by one literary critic as a masterpiece and by another as a piece of
junk; I might be judged reliable by one person and flaky by another; chicken liver might be considered mouth-watering by one person and disgusting by another. Doesn’t this show that categorization is enormously subjective? And there are so many other cases. Is our friend Virginia good-looking? Is she cordial ? Is our friend Stanley an artist ? Is he fluent in German? Was George’s retort to Virginia’s jab appropriate? Are Virginia’s clothes matching? Is George’s brother a sleazeball? Is it sprinkling outside, or is it raining? Is that a hill or a mountain over there? Was what Jane just said an insult or a joke? Is Jim impulsive? Is he straightforward ? Is he a patriot or is he a hypocrite? Is he ambitious, pushy, or driven? And think about bright people vehemently arguing over the nature of progress or about whether something is a piece of kitsch or not. Don’t you see how deeply blurriness and subjectivity pervade categorization?

  KATY:Well, perhaps I misspoke myself, because you’ve pointed out very effectively that some categories are subjective. But that doesn’t affect the crux of my point, which is that analogies are always subjective. For example, wouldn’t you agree that when one event reminds you of another, it’s an analogy, and that such remindings are totally subjective because they depend completely on the idiosyncratic memories that you’ve built up over the course of your life?

  ANNA:I’ll certainly grant you that some analogies are subjective.

  KATY:Excellent! I see that we’re on the same wavelength. So let me continue. We can come up with analogies between anything and anything else, depending on what’s recently been passing through our minds. And if one is focused on one thing in particular, then analogies by the bucketful will come to mind. Back in Chapter 5, we saw how an obsession of any sort — golf, dogs, physics, a video game, or who knows what — can trigger a raft of analogies with just about anything that one encounters. One can unwittingly wind up in a kind of analogy-mania! And it doesn’t even take an obsession for this kind of thing to happen. Just yesterday, in fact, something of the sort took place. A friend told me about a science-fiction story he’d read and enjoyed. The gist of it was that some guy was listening to the news on the radio and he heard that a woman in a nearby town had been killed when she’d been hit by a car driven by a wild driver. For some reason he heard this piece of news at 7:30 PM but it said that the accident had taken place at 8 PM. So maybe you can anticipate what happened in the story?

  ANNA:Mmm, not really…

  KATY:Well, the guy dashes to his garage, jumps in his car, and makes a beeline for the nearby town in order to warn the victim to get far from all roads. But as he’s approaching the main square, he loses control of his car and careens right into the woman, killing her at exactly 7:59 PM.

  ANNA:That’s a striking and original story. But where is it leading us?

  KATY:Well, if you think about it, your first thought is probably going to be that it’s paradoxical, and if you think a little further about what lies at its essence, you will come to the conclusion that it’s the idea of an unfortunate incident brought about by the very act of trying to avoid it. Such a thing might seem to be totally unique — a clever fantasy dreamt up by an inventive author. And yet, if you start to look around you with this idea in mind, you’ll find a rich harvest of analogous events.

  ANNA:Oh, so you came up with a bunch? That sounds interesting.

  KATY:Yes, and it wasn’t even very hard. They just spontaneously flooded my mind, and I couldn’t rightly say why. For example, the first memory that came to my mind was from way back when I was a little girl. One day I saw a tall vase sitting on a table that I knew was rickety, and so, trying to be helpful, I reached out to try to grab hold of the vase to make it stabler. What do you know, I accidentally banged the table with my hand, and immediately the vase toppled and shattered!

  ANNA:That anecdote about little Katy clearly has something central in common with the science-fiction story — they share a conceptual skeleton.

  KATY:To be sure. That’s exactly why it came to mind. But that wasn’t the only memory that came to me. I next remembered another time when I offered a friend a gift to make up for an argument we’d had, but for some reason she found my gift offensive, and this eventually led to a total break between us. And then I suddenly remembered that time a while back that you, Anna, baked a delicious cake for a party you were throwing, and how at the last minute you decided to brown it to make it even tastier, but you got distracted by a few guests arriving early and you overcooked it, thus ruining your cake. You’ll never forget that incident, will you?

  ANNA:How could I? It was such a disappointment to me! And I agree with you that it is analogous to the other events that you were reminded of. I think anyone would. So doesn’t that show the objectivity of this analogy?

  KATY:No, no! All these analogies are totally subjective. They came to my mind only because I was thinking about the story of the woman killed by the driver who was doing his utmost to prevent her death, and if no one had told me that story, none of these memories would have been triggered in my mind. None of them would have bubbled up, although any of them could have bubbled up in some completely different context. So you see how subjective analogy-making is!

  ANNA:Well, not so fast. I greatly appreciate the variety of the episodes in your life that this story reminded you of, but as we both just agreed, they all share a conceptual skeleton, which you formulated as an unfortunate incident brought about by the very act of trying to avoid it. These episodes are all seen by everyone as analogous because it’s clear that they all share this central essence. This is why I insist that analogy is not always subjective — no more than categorization is.

  KATY:All right, all right… I see that the common core is clear here, but maybe this analogy, or this family of analogies, is a special case. Maybe this wasn’t the best possible example.

  ANNA:No, that’s not the problem. The same would hold for analogies all across the board. For instance, think of the analogy in Chapter 6 between a thin wooden stick offered for stirring coffee and some javelins offered to row a boat in a lake. Who could ever claim that these situations have nothing in common? No one, since their shared essence is clear as a bell! Or take the analogy between sound waves in air and ripples in water — where’s the subjectivity in that? Or the analogy between a point (x, y) in a plane and a complex number x + yi, or the analogy between lungs and gills, or that between a bullet and an arrow, or between a table in your house and a table in my house. What could be more objective than these analogies?

  KATY:But I still insist that there is something highly subjective about analogy-making, because the memories that flashed to my mind when I heard that science-fiction story were completely idiosyncratic, and depended entirely on the chance events of my life and on how they happened to be stored in my memory. Yes, they all share the same conceptual skeleton with the science-fiction story, but they wouldn’t have occurred to anyone but me! They are products of my brain, and my brain alone! And so this definitively proves the subjectivity of analogy-making!

  ANNA:No one could deny, Katy, that what the science-fiction story brought to your mind — that specific set of curious and paradoxical episodes — is completely unique to you. As you say, this is a set of memories that is yours and yours alone, and it’s a result of the random vicissitudes of your life, the events that chanced to happen to you or to your friends, or perhaps events that you had read about or seen in films. And whenever you hear any story, analogous situations will come floating up to your consciousness, and they will certainly be a function of your personal experiences. Your unique experiences and how you encode them will determine when they will come to mind on later occasions. What this clearly shows is that categorization (or analogy-making, whatever you want to call it) comes from the perspective one adopts on a situation. And for this reason, categorization is profoundly subjective.

  KATY:Your words are confusing me, Anna. I carefully chose an example to show you that analogy is subjective, and yet you’re turning my own example against me, to arg
ue that categorization is subjective! What kind of sleight of hand is this, anyway?

  ANNA:There’s nothing underhanded or tricky going on here, Katy. It’s just an inevitable outcome of clear thinking. Your example was all about a set of stories that we agreed are all analogous, stories that all share a conceptual skeleton — and it was also all about a set of stories that all belong to a single category, a category whose core and whose fringes were very nicely fleshed out by your highly diverse set of examples. The verbal label “unfortunate incidents brought about by the very act of trying to avoid them” attempts to pinpoint the subtle shared essence that makes them all analogous.

  KATY:I can easily envision a little category centered on the story of the woman whose death was caused by the man whose goal was to save her.

  ANNA:Why call it “little”? It wouldn’t be hard to add members galore to this category. For instance, one might think of a person strangled by their seatbelt, or an attempt to stave off a war between two hostile countries that goes wrong and winds up triggering the feared war. One could also think of a face deformed by plastic surgery that came out unexpectedly, of a singer who loses her voice on the very morning of her recital as a result of too much practicing, of the big stain in a tablecloth left by a stain remover, of a résumé that is so jam-packed with lists of achievements and honors that all potential employers are immediately put on their guard, and who knows what else. This shows how diverse are the potential members of such a category, and all the situations that you and I described could be looked at in plenty of other ways and thus could be seen as members of plenty of other categories. For instance, the singer whose voice went hoarse as a result of too much practicing could be assigned to the category too much of a good thing, and the grotesque face caused by plastic surgery could be seen as a member of the category should have left well enough alone. This kind of shift in point of view gives rise to a shift in categorization. In short, I believe that I’ve just demonstrated the profoundly subjective nature of categorization.