Surfaces and Essences

by Douglas Hofstadter

  A source of problems, however, is the fact that the words used to express the membership criteria are not any more precise than the concept that one is trying to pin down — in this case, bird. What, for instance, is a foot? And what does “to possess” mean? What does “covered with” mean? And of course, everyone knows that there are all sorts of birds that don’t have two feet (perhaps because of an injury or a genetic defect) or that are not covered with feathers (ducklings and chicks, for example). And turning things around, we human beings have two feet, but if we hold a spray of feathers in our hand, this “possession” does not suffice to turn us into birds. And the famous plume de ma tante — my ancient aunt’s quill pen, which she loved to use to make beautiful calligraphy — would that count as a feather? And if so, would possession thereof make my bipedal old aunt a bird?

  At times one gets the impression that the actual goal of ancient philosophers was not to classify specific entities from the material world, such as individual birds, whose variety is bewildering, but rather to characterize the relationships that hold between generic, immaterial abstractions, such as the categories bee, bat, egg, chick, ostrich, pigeon, dragonfly, swallow, flying fish, and so forth. If this is one’s goal, then the crucial question would be “Which of these classes of entities are birds?” It’s clear that one has moved far from the specific and concrete, and has replaced it by an intellectual activity where everything is generic and abstract. This rarefied universe of Platonic concepts, since it lacks annoying exceptions like the plucked or the injured bird, not to mention the old aunt who keeps a quill in her drawer, might appear to be as pure, immutable, and objective as the universes of Euclidean geometry or chess, and this could suggest that in this universe there are a vast number of eternal verities lying in wait to be discovered, much like theorems in geometry. But appearances are deceptive. Even if one considers only abstract categories and pays no attention to their annoyingly problematic instances, one still faces enormous obstacles.

  Would a chick’s lack of plumage make it lose its membership in the category bird? That seems unlikely. Or is there a specific instant, for each chick, when it passes over from the category chick to that of bird? Would that switchover in status take place at the instant when its skin becomes “covered” with feathers? How many feathers does it take for a chick to be “covered” with them? Or what percentage of the skin’s area must be covered for it to count as “covered”? And how does one measure the surface area of a chick, if that is needed in order to decide if we are dealing with a bird or not?

  The closer one looks, the more such questions one will find, and the more they are going to seem absurd. And we have only scratched the surface of the issues. Consider the generic idea of a bird that has just died. Is it still a bird? And if so, for how long will this entity remain a member of the category bird? Will there be a sharp transitional moment at which the category membership no longer obtains? And let’s go backwards in time by a few million years. Where is the boundary line between birds and their predecessors (certain flying dinosaurs)? And to push matters in yet another direction, what about questions such as, “Is a plucked chicken still a bird?” The moment one has created the expression “plucked chicken”, the question we posed becomes a legitimate question in the hypothetical formal algebra that governs abstract categories. And with this, we have opened a Pandora’s box of questions: “Is a robin whose feet have been cut off still a bird?” (since the first noun phrase is the valid name of a category of entities), or “Is a snake onto which one has grafted some feathers and two eagle’s feet a bird?”, and so on, without any end in sight.

  Even without imagining such radical transformations, one can ask whether sandals are shoes, whether olives are fruits, whether Big Ben is a clock, whether a stereo set is a piece of furniture, whether a calendar hanging on one’s wall is a book, whether a wig is an article of clothing, and so forth. People turn out to have highly divided opinions on such questions. In an experiment conducted by the psychologist James Hampton, sinks turned out to be just barely included in the category kitchen utensils, while sponges were just barely excluded. Since these close calls are the result of averaging over many subjects in a large experiment, one might imagine that if one were to ask individuals instead, one would find clear-cut and fixed boundaries for each person (even if they would vary from individual to individual). However, even that idea, which runs considerably against the idea of Platonic concepts (which are supposed to be objective, not subjective), turns out to be quite wrong. Many people change their mind if they are asked whether pillows and night-table lamps are articles of furniture and then are asked the same question a few days later. Are these individuals suffering from a pathological state of permanent vacillation, never able to make up their mind about anything? It seems more likely that they are quite ordinary individuals whose categories simply grow blurry toward their edges; if these people were asked about more typical cases, such as whether dogs are animals, they would be extremely stable in their judgments about category membership.

  Anyone who has taken an interest in the letters of the alphabet will have savored the dazzling richness of a “simple” category like the letter “A”, whether capital or small. What geometric shapes belong to the category “A”, and what shapes do not? All that one needs to do is take a look at a few handwritten postcards or a collection of typefaces employed in advertising, or for that matter, the figure in the Prologue, in order to see why the boundaries of the twenty-six categories a, b, c, d, and so on are impossible to specify exactly. And, to be sure, what holds for the letters of the alphabet holds just as much for other familiar categories, such as bird, bill, boss, box, and brag.

  Summing up, then, the ancient hope of making the categories describing physical objects in the world into precise and rigorous theoretical entities is a vain hope. Such categories are as fleeting and elusive, as blurry and as vague, as clouds. Where are the boundaries of a cloud? How many clouds are there in the sky today? Sometimes, when looking at the sky, one has the impression that such questions have clear and exact answers, and perhaps that’s the case on some particular day; however, the next day, the sky will have a radically more complex appearance, and the idea of applying such notions to it as how many and boundary will simply be a source of smiles.

  Concepts Seen in a More Contemporary Fashion

  Since the classical view of categories is now generally perceived as a dead end, some contemporary psychologists have tackled the challenge of making the very blurriness and vagueness of categories into a precise science. That is, their goal is to explore those mental nebulas that are our concepts. This has led them to formulating theories of categorization that reject the role of precise membership criteria and instead invoke either the notion of a prototype (a generic mental entity found in long-term memory, which summarizes all one’s life’s experiences with the given category) or else the notion of the complete set of exemplars of the given category that one has encountered over one’s lifetime. Another influential view involves stored “mental simulators” of experiences one has undergone, which, in response to a fresh stimulus, reactivate certain regions of the brain that were once stimulated by the closest experiences to the current stimulus.

  Behind all these efforts lies the appealing idea of non-homogeneous categories — that is, categories having stronger and weaker members — which amounts to distinguishing between more central and less central members. For example, if one times the responses of experimental subjects when they are asked questions of the form “Is an X a Y?”, or if one asks them to write down a list of members of a certain category, or if one gives them a list and asks them to indicate, for each item, its degree of typicality as a member of a specific category, one finds that some very striking trends emerge, and these trends turn out to be stable across all these different ways of testing. Certain members of the category turn out to belong more to the given category than others do (recalling how some animals in Orwell’s Animal Farm were “mor
e equal” than others). For instance, ostriches and penguins turn out to lie close to the outer fringes of the category bird, whereas sparrows and pigeons are near its core.

  This phenomenon can affect the difficulty one has in understanding a sentence inside a passage that one has been asked to read. Thus, it turns out that the time taken to read and understand a sentence such as “The bird was now just a few yards away” depends on whether, earlier in the passage, there was a reference to an ostrich (an atypical bird) or to a pigeon (a typical bird), in preparatory sentences such as “The ostrich was approaching” or “The pigeon was approaching”. The link in memory between ostrich and bird turns out to be less strong than that between pigeon and bird, and this tends to impair the understanding of the passage in the first case.

  It’s important to point out that categorization goes well beyond the intellectual realm of connections among words, which is to say, the names of various categories (such as “sparrow”, “ostrich”, and “bird”). If, for example, someone were to ask Eleanor “Is a spider an insect?”, she might well reply, on the basis of her knowledge from books, “No”, and yet if she were to espy a dark blob hanging from the ceiling of her bedroom, it is likely that she would cry out, “Yikes! Get it out of here! I hate insects in my room — they’re scary!” If someone were to object to her word choice, Eleanor would say that she knows very well that the “insect” was in fact not an insect but a spider.

  Generally speaking, context has a great influence on categorization. The spider in this anecdote was seen as an insect in the bedroom, but it would not have been seen as such in the context of a biology test, for instance. And much the same holds in general: a single item in the world belongs to thousands of categories, which can be extremely different from each other, and a good fraction of our mental life consists in placing entities in one category and then in reassigning them to another category. During a basketball game, everyone is aware of the fact that basketballs roll, but it has been experimentally shown that only situations that involve water (such as the loading of a bunch of basketballs on board a ship) evoke the notion that basketballs float.

  Context thus changes categorization and can modify how we perceive even the most familiar of items. For example, an object can slip in the blink of an eye from the category chair to that of stool when a light bulb has just burned out and one needs something to stand on in order to change it. Usually one is unaware of these category shifts because one is mentally immersed in a specific context and such shifts are carried out in a totally unconscious manner. In a given context, just one categorization seems possible to most people. Their lack of awareness of the contextual blinders that they are wearing reinforces the widespread belief in a world in which every object belongs to one and only one Platonic category — its “true” category.

  On the other hand, one cannot help but recognize how complex category membership is if one considers the fact that a single entity can easily belong to many diverse categories, such as, for instance:

  60-kilogram mass, mirror-symmetric object, living entity, biped, mammal, primate, mosquito attractor, arachnophobe, human being, forty-something, book-lover, nature-lover, non-compromiser, non-speaker of Portuguese, romantic, Iowan, blood-type A+, possessor of excellent long-distance vision, insomniac, idealist, vegetarian, member of the bar, mother, mother hen, beloved daughter, sister, big sister, little sister, best friend, sworn enemy, blonde, woman, pedestrian, car driver, cyclist, feminist, wife, twice-married woman, divorcée, neighbor, Dalmatian owner, intermediate-level salsa dancer, breast-cancer survivor, parent of a third-grader, parents’ representative…

  To be sure, this is but a small excerpt from a much longer list one could draw up, a list having essentially no end, and whose entries would all be terms that anyone and everyone would, without any trouble, recognize as designating various categories.

  When Ann had to be hospitalized on an emergency basis and a transfusion was needed, her membership in the category blood-type A+ dominated all her other category memberships, but in a restaurant she is above all a vegetarian, while at work she is a lawyer, at home a mother, in a PTA meeting a parents’ representative, and so forth. It may seem useless to point out such obvious facts, but such simple observations carry one well outside the realm of classical categories.

  When I Imitate Tweety, Am I a Bird?

  Let’s come back to the one-word category bird, which still has some lessons to teach us. Consider the following candidates for membership in the category:

  •a bat;

  •an airplane;

  •a bronze seagull;

  •an eagle in a photograph;

  •the shadow of a vulture in the sky;

  •Tweety the (cartoon-inhabiting) canary;

  •an entire avian species, such as eagle or robin;

  •a chick inside an egg two hours before it hatches;

  •a flying dinosaur (or rather, a dinosaur that once flew);

  •a pigeon on the screen in a showing of Hitchcock’s film The Birds;

  •the song of a nightingale recorded and played back fifty years after it died;

  •a rubber-band-powered wing-flapping plastic object that swoops about in the air.

  If you are like the vast majority of humans, you probably felt a keen desire to say “yes” or “no” to each of the candidates in the list above, as if you were taking an exam in school and had to demonstrate the precision of your knowledge, and as if, in each of these cases, there really were a correct answer to the question. A sparrow — is it a bird? Yes! When you spot a black spot moving unpredictably through the air against a light cloudy background, are you seeing a bird? Of course! And when one sees the shadow of a vulture on the ground, is one seeing a bird? Of course not! When one hears a loud hooting during the night, is one hearing a bird? Yes! And if one hears a recorded hooting (perhaps without being aware that it is recorded)? And what about the case where some person imitates hooting extremely well? And if one dreams about an owl, is there a bird involved? And if one reads a comic book featuring Tweety?

  No one ever taught us the boundaries of categories. Our spontaneous sense for their boundaries is an outcome of what we often call “common sense”, and no one teaches that in any school. There are no courses on category membership, and even if there were, there would be endless arguments among the students as well as between teachers and students, not to mention the passionate debates that would take place among the teachers themselves. Indeed, expertise doesn’t help at all. Here we borrow an anecdote from the psychologist Gregory Murphy, who quoted from a keynote speech once delivered by a world-renowned metallurgist at a conference of world experts in that field: “I’ll tell you something. You really don’t know what a metal is. And there is a big group of people that don’t know what a metal is. Do you know what we call them? Metallurgists!”

  The recent vehement debates among astronomers over whether Pluto should or should not be deemed a planet (which, as of this writing, it no longer officially is) were due to the blurriness of the concept of planet, even in the minds of this planet’s greatest specialists, which made the question extremely thorny. For similar reasons, although there is considerable agreement among experts today that it is not correct to refer to our “five senses”, since proprioception, thermoception, and nociception (among others) would be left out of such a roll call, there remains a major blur about what our senses really are. Since the experts can’t even agree on how many senses we have, let alone on what they all are, they often talk about “our five main senses”. And in a similar vein, a standard definition of life is still missing, even if biologists, hoping to pin it down for once and for all, are constantly juggling the details of taxonomies that laypersons would have presumed had long ago been cast in concrete. The classification of living organisms has come a long way since Linnæus, and today, many classic terms that he employed in his classification, such as “reptile”, “fish”, and “algæ”, remain present in school texts, but no longer
appear in modern phylogenetic classifications. All this goes to show that the blur of categories is not due to some kind of lack of expertise, but is part and parcel of the act of categorization.

  How Many Languages do You Speak?

  Although psychologists have done a good job in making it clear that no category has precise boundaries, our everyday language and thought are still permeated with residual traces of the classic vision in which category boundaries are as sharp as those of nations (which, to be sure, are often not all that clear, but we’ll leave that matter aside). Our intense human desire to avoid ambiguity, to pinpoint the true and to discard the false, to separate the wheat from the chaff, tends to make us seek and believe in very sharp answers to questions that have none.

  For instance, people who enjoy studying foreign languages are frequently asked the question, “How many languages do you speak?” Despite how perfectly natural this question might seem, it is based on the tacit idea that the languages of the world fall into two precise bins: languages that person X does speak, and languages that X does not speak, as if this were a black-and-white matter. But in fact, for each language one has studied, one speaks it to a different degree, depending on many factors, such as when one first studied it, the context in which one studied it, how long it has been since one spoke it, and so forth. When pressed, the questioner may retreat, saying, “All I meant was, ‘How many languages can you have an everyday conversation in?’ ”

  But once again, even if this new question sounds reasonable at first, it’s just as blurry. For example, it presumes that the category everyday conversation is sharp and well-defined. But it might mean a conversation of two minutes about the cost of postage stamps with someone standing next to one in a line in the post office. Or it might mean a half-hour conversation about one’s children and family, or about the World Series, or about the sad state of the world economy, with a stranger sitting next to one in an airplane. Then again, it might mean a three-hour conversation ranging over twenty different random topics with seven other people, all native speakers, seated around the table at a lively dinner party. Most people say they speak a language when they have surpassed a far lower threshold than that, but in any case, the threshold for “speaking a language” is not well-defined.