Surfaces and Essences

by Douglas Hofstadter

  Chapter 8 then looks at the extreme other end of the spectrum — namely, how great discoveries are made by insightful scientists. We show how the history of mathematics and physics consists of a series of snowballing analogies. By examining from up close certain great moments in the history of these disciplines, we reveal the crucial role played over and over again by analogies — sometimes very obvious ones, sometimes very hidden ones. In particular, the deep analogies of Albert Einstein play a starring role in this chapter, including a little-known analogy that led to his hypothesis in 1905 that light consists of particles, an idea that was mightily resisted by the entire physics community for nearly two decades. The most carefully examined historical episode is that of Einstein’s own slow and gradual process of coming to understand the various levels of meaning of his celebrated equation “E = mc2”.

  The epilogue to our book is a dialogue — thus it is entitled “Epidialogue” — in which categorization and analogy-making are compared and contrasted along many dimensions, and although at first the two processes may seem very different, at the end of this careful comparison, the spirited debaters conclude that there is no difference between them, and they realize that in fact they are one and the same.

  CHAPTER 1

  The Evocation of Words

  How do Words Pop to Mind?

  At every moment we are faced with a new situation. Actually, the truth is much more complicated than that. The truth is that, at every moment, we are simultaneously faced with an indefinite number of overlapping and intermingling situations.

  In the airport, we are surrounded by strangers whom we casually observe. Some seem interesting to us, others less so. We see ads everywhere. We think vaguely about the cities whose names come blaring out through loudspeakers, yet at the same time we are absorbed in our private thoughts. We wonder if there’s time enough to go get a frozen yogurt, we worry about the health problems of an old friend, we are troubled by the headline we read in someone’s newspaper about a terrorist attack in the Middle East, we smile to ourselves at a clever piece of wordplay in an ad on a television screen, we are puzzled as to how the little birds flying around and scavenging food survive in such a weird environment… In short, far from being faced with one situation, we are faced with a seething multitude of ill-defined situations, none of which comes with a sharp frame delineating it, either spatially or temporally. Our poor besieged brain is constantly grappling with this unpredictable chaos, always trying to make sense of what surrounds it and swarms into it willy-nilly.

  And what does “to make sense of” mean? It means the automatic triggering, or unconscious evocation, of certain familiar categories, which, once retrieved from dormancy, help us to find some order in this chaos. To a large extent, this means the spontaneous coming to mind of all sorts of words. Without any effort, one finds oneself thinking, “cute little girl”, “funny-looking coot”, “same dumb ad as at the airport I was at yesterday”, “an Amish family”, “sandals”, “what’s she reading?”, “who’s whistling?”, “where is their nest?”, “when are we going to board?”, “what an annoying ring tone”, “how could I have left my cell-phone charger at home?”, “and I did it last time, too”, “the air-conditioning is on too high in here”, and so on.

  All these words! No experience is more familiar to us than this ceaseless barrage of words popping up in our mind extremely efficiently and without ever being invited. But where do these words come from, and what kind of invisible mechanism makes them bubble up? What is going on when one merely thinks silently to oneself, “a mother and her daughter”?

  It All Starts with Single-member Categories

  To be able to attach the label “mother” to some entity without thinking about it, one has to be intimately familiar with the concept mother, which is denoted by the word. For most of us, this intimacy with the concept goes all the way back to our earliest childhood, to our first encounters with the notion. For one-year-old Tim, the core of the concept is clearly his own mother — a person who is much bigger than he is, who feeds him, comforts him when he cries, sings him lullabies, picks him up, plays with him in the park, and so forth. Once this first mental category bearing the name “Mommy” has a toehold, Tim will be able to see that in the world around him there are similar phenomena, or as we prefer to put it, analogous phenomena.

  We take a momentary break here to explain a typographical convention of our book. When speaking about a word, we will put it in quotation marks (“table”), whereas when speaking about a concept, we will use italics (table). This is an important distinction, because whereas a word is a sequence of sounds, a set of printed letters, or a chunk of silent inner language, a concept is an abstract pattern in the brain that stands for some regular, recurrent aspect of the world, and to which any number of different words — for instance, its names in English, French, and so forth, or sometimes no word at all — can be attached. Words and concepts are different things. Although the distinction between them is crucial and often very clear, there will unavoidably be cases in our text where it will be ambiguous and blurry, and in such cases, we’ll make a choice between italics and quotation marks that might seem a bit arbitrary. Another source of ambiguity is the fact that here and there we’ll use italics for emphasis, just as we’ll use quotation marks to suggest a sense of doubt or approximation (which could sometimes be conveyed equally well by the word “so-called”), and of course we will use quotation marks when we are making a quotation. Alas, the world is simply filled with traps, but we hope that the ambiguities are more theoretical than actual. And with that said, we return to our main story.

  One day in the park, Tim, aged eighteen months, sees a tot playing in the sandbox and then notices a grown-up near her who is taking care of her. In a flash, Tim makes a little mental leap and thinks to himself more or less the following (although it’s far from being fully verbalized): “That person is taking care of her just like Mommy takes care of me.” That key moment marks the birth of the concept mommy with a small “m”. The lowercase letter is because there are two members of this new category now (and of course using uppercase and lowercase letters is just our way of hinting at what’s going on in Tim’s head, not his way). From this point on, it won’t take Tim long to notice yet other instances of this concept.

  At the outset, Tim’s concept of mommy still floats between singular and plural, and the analogies in his head will be quite concrete, a comparison always being made to the first mommy, which is to say, with Mommy (the one with the capital “M”), but as new instances of the concept mommy are superimposed and start to blur in his memory, the mental mapping that Tim will automatically carry out, each time he spots a new grown-up in the park, will start to be made not onto Mommy, but onto the nascent and growing concept of mommy — that is, onto a generalized, stereotyped, and even slightly abstract situation, centered on a generic grown-up (i.e., stripped of specific details) and involving a generic child who is near the grown-up and whom the grown-up talks to, smiles at, picks up, comforts, watches out for, and so on.

  It’s not our goal here to lay out a definitive theory of the growth of the specific concept mommy, as our purpose is more general than that. What we are proposing is that the birth of any concept takes place more or less as described above. At the outset, there is a concrete situation with concrete components, and thus it is perceived as something unique and cleanly separable from the rest of the world. After a while, though — perhaps a day later, perhaps a year — one runs into another situation that one finds to be similar, and a link is made. From that moment onward, the mental representations of the two situations begin to be connected up, to be blurred together, thus giving rise to a new mental structure that, although it is less specific than either of its two sources (i.e., less detailed), is not fundamentally different from them.

  And so the primordial concept Mommy and the slightly more sophisticated concept mommy act in very similar ways. In particular, both of them are easily mapped onto newly
encountered situations “out there”, which leads both of them to extending themselves outwards — a snowball effect that will continue all throughout life. It’s this idea of concepts extending themselves forever through a long series of spontaneous analogies that we wish to spell out more carefully in the next few sections.

  Passing from Mommy to mommy and then to mother

  One day, Tim, who sadly has never met his father, is playing in the park, and he runs into a little girl accompanied by a grown-up who is encouraging the girl to play with the other children. He thinks to himself that this grown-up is the mommy of the little girl. That is, Tim’s mind makes a link between what he’s observing and his new concept of mommy. This is an act of categorization. Perhaps the new person is not actually the child’s mother but the child’s father, or perhaps it’s her grandmother, or even her older brother or sister, but even so, that doesn’t make Tim’s mapping of this new person onto the category mommy irrational, because his notion of Mommy/mommy is wider than ours is (not richer, of course, but less discriminating, due to his lack of experience). This simple analogy Tim has made is flawless; it’s just that he hasn’t taken into account certain details that an adult would have used. If Sue, his mother, explains to him that this person isn’t the little girl’s mommy but her daddy, then Tim may well modify his concept of mommy, thereby coming into closer alignment with the people around him.

  Gradually, as Tim uses the word “mommy” more and more, his initial image — that of his own mother — will start to recede from view, like a root being grown over ever more as time passes. He will overlay his earliest image with traits of other people whom he assigns to this mental category, and the vivid and unique features of his own Mommy will become harder and harder to find in it. Nonetheless, even when Tim is himself a grown-up, there will remain in his concept of mommy some residual traces of his primordial concept Mommy.

  One day, a friendly woman who’s come all the way from her home in Canada turns up and treats Tim very sweetly. He hears the word “mommy” used several times to refer to this grown-up, and so for a while he concludes that maybe he has more than one mommy. For Tim this is conceivable, since he has not yet built up a set of expectations that would rule this possibility out. Sometimes his “second mommy” takes him to the park and she, too, chats with the other mommies. But after a week or so, Tim’s second mommy vanishes, which quite understandably saddens him. The next day, one of the mommies in the park asks Tim, “Did your grandmother go back home?” Tim doesn’t answer, because he doesn’t yet know the concept of grandmother. So she reformulates her question: “Where’s your mommy’s mom today, Tim?” But this question makes even less sense to Tim. He knows perfectly well that he’s the one who has the mommy (he even had two of them in the past few days!), and so his mommy (that is, the remaining one) can’t have a mommy. After all, it’s children who have mommies (and sometimes also daddies) whose purpose is to be sweet to them, to watch over them, and to help them, and Tim knows that his mommy isn’t a child, and so she doesn’t have a mommy. That’s obvious! The woman doesn’t push her strange question, and Tim goes back to his playing.

  And time passes. A few months later, Tim starts to realize that grown-ups are sometimes accompanied by other grown-ups that they refer to as their “mother”. Suddenly everything starts to be clear… What children have are mommies, and what grown-ups have are mothers. That makes sense! And into the bargain, there’s even an analogical bond between mommy and mother. Of course Tim isn’t aware of having made an analogy — neither this concept nor the word for it will be known to him for another ten or more years! — but he has nonetheless made one. And as is often the case with analogies, this one helps clarify things for Tim but it also misleads him a little.

  We now will skip over the details, simply adding that the two concepts of mommy and mother gradually merge to create a more complex concept at whose core there is the primordial concept of Mommy. This doesn’t mean that the primordial image of Sue springs to Tim’s mind every time that he hears the word “mother” or even the word “mommy”, but merely that the invisible roots are structured in that manner.

  As any concept grows in generality, it also becomes more discriminating, which means that at some point it’s perfectly possible that some early members of the category might be demoted from membership while new members are being welcomed on board. Thus the dad at the park whom Tim had first taken for a mommy is stripped of the label, and although Tim’s grandmother stays on as a member of the category mother, she winds up in a less central zone than the mommy zone, which is reserved for the mothers of small children. And of course as time goes by, Tim will come to understand that his grandmother herself was once a member of the category mommy (just as his own Mommy was once a member of the category small child), but at present all of that is well beyond his grasp.

  The Cloud of Concepts of Mother

  One might think that the concept of mother is very precise — perhaps as precise as that of prime number. That would imply that to every question of the form “Is X a mother or not?”, there would always be a correct, objective, black-and-white answer. But let’s consider this for a moment. If a little girl is playing with two dolls, one bigger and one smaller, and she says that the big one is the small one’s mother, is this an example of motherhood? Does the large doll belong to the category mother? Or contrariwise, could one state without risk of contradiction that she does not belong to that category?

  And if we read a certain book in which a certain Sue is described as the mother of a certain Tim, then does this Sue, who is never anything but a made-up character in a book, truly belong to the category mother? Does it make any difference that Sue was modeled on a real person, and Tim on her son? Is Sue more of a mother than the doll is? What indeed is Sue? If in the book it states that she is 34 years old, that she has light brown hair, that she weighs 120 pounds, that she is five feet five inches tall, and that she’s the mom of a small boy, does that mean that Sue has a body and once gave birth? A doll, at least, is a physical object, but what is Sue, when you come down to it? An abstract thought triggered by some words on a page, by some black marks on a white background. Does this thought even deserve the pronoun “she”?

  When Tim gets to be six, if someone tells him that Lassie is Spot’s mother, he certainly won’t protest, but if he were told that the queen bee is the mother of all the bees in her hive, it’s less clear what he would say, and in any case some mental effort would be needed before he could absorb this idea. And if he were told that a drop of water that he just watched dividing into two drops is the mother of the two new drops, he would almost surely find this suggestion very surprising. Everyone knows phrases that use the word “mother” in ways that go far beyond the senses that apply to Lassie, the queen bee, or even the splitting drop of water — for instance, “my motherland”, “a mother cell”, “the mother lode”, “Mother Earth”, “Greece is the mother of democracy”, and “Necessity is the mother of invention”. Are these true instances of the concept of mother, genuine cases of maternity? What is the proper way to understand such usages of the word?

  Some readers may feel inclined to say that these are all “metaphorical mothers”, and indeed, such a viewpoint is not without merit, but we have to point out that there is no sharp boundary that separates “true” mothers from those that are metaphorical, for categories in general don’t have sharp boundaries; most of the time, metaphorical and literal meanings overlap so greatly that when one tries to draw a clear boundary, one discovers that things only get blurrier and blurrier.

  When he turns seven or eight, Tim will start to be able to handle phrases in which the word “mother” is used with greater fluidity than back in nursery school. He might run into the statement “Mary is the mother of the Lord Jesus” in a religious context. This is a mild extension of the usual meaning, since Mary is imagined as a woman whereas the Lord Jesus is imagined as a divine being, magical and omnipotent in some ways, even if also, in
some sense, as a baby like all others. At age seven, though, Tim probably won’t have much trouble envisioning Mary giving birth to the Lord Jesus.

  On the other hand, having given physical birth to a baby is not a prerequisite for attributing motherhood to an entity, since even if no one ever teaches us this explicitly, we all come to know that motherhood pulls together several different properties, such as that of female biological parent, that of female nurturer, and that of female protector, and these properties do not all need to be present simultaneously. For example, the familiar fact of adoption reminds us that giving birth is only one possible route for becoming a mother.

  If at age nine, Tim is reading a book on Egypt or on mythology and runs into the sentence “Isis is the mother of Nature”, he’ll have to extend his prior conceptions of motherhood at least slightly, because this time, Isis is not a human being but a deity who, in Tim’s mind, looks much like a woman but in some sense is not one, and who is capable of giving birth to some rather abstract things, such as Nature, yet without anything emerging from her body. And yet Tim will rather easily absorb this new instance of motherhood, because she looks enough like hundreds of other members of the category mother that are already installed in his memory.