Surfaces and Essences

by Douglas Hofstadter

  Why do so many people, perhaps even you, have an inner voice that so strongly resists the thesis that the building of analogies between things is just the same activity as the assigning of things to categories? How come your inner voice hasn’t gradually calmed down and grown silent over the course of reading this book? How come it hasn’t listened to all the reasons that we had hoped would convince you of our thesis?

  The answer — a rather ironic one — is that our thesis itself explains why so many people have so much trouble accepting it: namely, the belief that analogy-making and categorization are separate processes springs from none other than a certain naïve analogy about the nature of categories. This naïve analogy, which has the dubious honor of having seriously held back progress in the field of analogy and categorization for a long time, has already been cited in this book. Here it is, once again:

  Categories are boxes, and to categorize is to put items into boxes.

  This is the everyday, down-home view of categorization. Let’s think about it a little bit.

  If categories really were boxes and if there really were a reliable, precise mechanism for assigning things to their boxes, then it would make eminent sense to distinguish between two types of mental process. First would be categorization — a rigorous, exact algorithm reliably placing mental items in their proper boxes; second would be analogy-making — a subjective and fallible technique for dreaming up fanciful, unreliable bridges between mental items that do not enjoy a contents-to-container relationship.

  However, as research in psychology has shown, and as we have stated throughout this book, the vision categorization = placing things in their natural boxes is highly misleading, for categorization is every bit as subjective, blurry, and uncertain as is analogy-making. A categorization can be outright wrong, can be partially correct, can be profoundly influenced by the knowledge, prior experiences, prejudices, or goals, conscious or unconscious, of the person who makes it, and can depend on the local context or the global culture in which it is made. In addition, categorizations can be just as abstract as analogies, can be nonverbal as well as explicitly verbal, can be shaded, and so forth.

  Recent work, in fact, makes this point so obvious that the old view of categories as boxes is now usually called “the classical approach to categories”, because in cognitive science there is scarcely anyone around any longer who still puts stock in it; today the classical approach tends to be looked upon as simply a quaint historical stage in the development of a far more sophisticated theory of categories and categorization.

  And yet, the “categories are boxes” naïve analogy is still seductive, and leads us all to fall victim to the nearly irrepressible belief that all objects and situations we encounter have a privileged category to which they belong, and which constitutes their intrinsic identity. Let’s recall the object Mr. Martin purchased; though it subsequently bounced merrily back and forth among such motley categories as fragile object, dust-gatherer, spider carrier, and home for tadpoles, it always seemed to remain in truth just one thing — namely, a glass — and this label constituted its genuine, intrinsic identity. The naïve unspoken analogy “categories are boxes” implies that each item in the world, just like Mr. Martin’s glass, has a proper box to which it belongs, and that this connection between a thing and its natural box is universally shared in all people’s heads, and finally that this is simply the nature of the world, having nothing to do with thinking or psychology. In this view, not only would category identity exist, but it would be precise and objective.

  Over the years, the insidious motto “categories are boxes” has underwritten much scientific research reinforcing the belief that there is a clear-cut distinction between analogy-making and categorization. Indeed, the view of categorization that reigned in cognitive psychology for decades, though expressed in far more sophisticated terms, was essentially indistinguishable from this motto. That view was a restatement of the motto in technical terms borrowed from mathematical logic. It portrayed each mental category as possessing a set of “necessary and sufficient conditions” for membership. An entity belonged to the category if and only if it had all those properties. Thus each category-box was thought of as being precisely defined and having rigid, impermeable walls. In this theory of categories, there was no room for degrees of membership, nor for contextual effects on membership in categories. Only in the mid-1970s, when psychologist Eleanor Rosch published her seminal series of articles on categorization, was this erroneous but nearly universally accepted theory at last discredited.

  And so, after finally being liberated from the motto, how do today’s researchers view analogy-making and categorization? Has a clear consensus emerged on what, if anything, makes the two different? Well, let’s listen to some of the top authorities. On the one hand, Thomas Spalding and Gregory Murphy write, “Categories let people treat new things as if they were familiar”; on the other, Mary Gick and Keith Holyoak state, “Analogy is what allows us to see the novel as familiar”. If there is a distinction between these two characterizations, it eludes us! Or, to cite Catherine Clement and Dedre Gentner, “In an analogy, a familiar domain is used in order to understand a new domain, especially to predict new aspects of this new domain”, whereas for John Anderson, “If one establishes that a given object belongs to a certain category, then one can predict a great deal about the object.” Once more, these descriptions of supposedly different processes make them sound as alike as Tweedledum and Tweedledee.

  The great overlap of these experts’ definitions confirms (if confirmation was needed) the tight relationship of analogy-making and categorization, and provides grist for the mill we are defending — namely, that the idea that analogy-making and categorization are separate processes is illusory. Still, someone who wished to play devil’s advocate might argue that by overhauling the definitions of analogy-making and categorization, one might be able to show that they are indeed two processes and not just one. In fact, after Chapter 8 there is a dialogue that carefully explores this possibility, and we hope that that dialogue, in addition to closing our book, will also close the book on this issue.

  However, if you, even after having read this far in our book, still feel reluctant to accept our thesis, rest assured that you are in excellent company, for even experts in the fields concerned fall victim all too often to this same naïve vision. Indeed, we — the authors of this book, the most fervent proponents of our thesis — find ourselves from time to time falling into the very trap we’ve tried so hard to warn our readers about! Yes, we too fall occasionally for the tempting illusion that categories are boxes. The following way of putting it would probably have warmed the cockles of Aristotle’s heart:

  People make naïve analogies.

  Analogy experts are people.

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  Therefore, analogy experts make naïve analogies.

  Specialists of any ilk, be they experts in psychology, mathematics, physics, or any other field, do not belong to a different species from ordinary people; they make naïve analogies not only in their daily lives, but also in their professional lives, even ones that involve the concepts with which they are the most proficient.

  In short, we sympathize with those readers who still have some doubts about the identity of categorization and analogy-making. It is, after all, a counterintuitive view, and a little voice inside, prompted by a beguiling naïve analogy, continually whispers, “It’s wrong! It’s wrong!” Nonetheless, we harbor fond hopes that in the remaining pages of our book, we might still get some doubters to swing around to our view.

  In any case, speaking of the remaining pages, it is high time that we moved from the often troubling world of naïve analogies to the ever-admirable role of analogies in scientific discovery. And yet in so doing, will we really leave naïve analogies far behind?

  CHAPTER 8

  Analogies that Shook the World

  The Royal Role of Analogies in the Realm of Rigor

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nbsp; In Chapter 7, we saw that students who encounter new mathematical ideas for the first time lean heavily on analogies at every step, doing their best to keep from taking tumbles in the abstract world into which they are tremulously treading. We also saw that naïve analogies have a way of sticking in people’s minds for a long time. Indeed, naïve mathematical analogies often last a lifetime in the minds of non-mathematicians, tending to lead their makers into dead ends, confusions, and errors. To avoid this fate, one has to gradually refine one’s category system as one is exposed to mathematical notions having ever higher levels of sophistication and abstraction. But what about professional mathematicians? Do they, too, rely on naïve analogies in order to keep from stumbling left and right, or would such a vision of their professional lives itself be a result of making too naïve an analogy between beginners and experts?

  To some people, it might seem far-fetched to imagine that any role at all might be played by analogical thinking in the professional activity of a mathematician. After all, of all intellectual domains, mathematics is generally thought of as the one where rigor and logic reach their apogee. A mathematical paper can seem like an invincible fortress with ramparts built from sheer logic, and if it gives that impression it is no accident, because that is how most mathematicians wish to present themselves. The standard idea is that in mathematics, there is less place for intuitions, presentiments, vague resemblances, and imprecise instincts than in any other discipline. And yet this is just a prejudice, no more valid for mathematics than for any other human activity.

  Anecdotes as Antidotes

  The great French mathematician Henri Poincaré devoted much thought to the nature of scientific creativity. In a commentary on mathematicians, he wrote:

  Could anyone think… that they have always marched forward, one step after another, without having any clear idea of the goal they were trying to reach? It was necessary for them to guess at the proper route to get them there, and to do so they needed a guide. This guide is primarily analogy.

  This observation about mathematical thinking strikes your authors as being spot-on — but we don’t expect readers to take Poincaré’s word for it. In support of his thesis, and as antidotes to any possible skepticism about the role of analogies in mathematics, we now offer a bouquet of anecdotes.

  For starters, let’s go back to the early part of the sixteenth century, to a time when European mathematicians were struggling with the challenge of solving third-degree polynomial equations (of the form “ax3 + bx2 + cx + d = 0”), otherwise known as “cubics”. For more than fifteen centuries, the formula that gave the general solution of second-degree polynomial equations (that is, quadratic equations, which have the form “ax2 + bx + c = 0”) had been known, and at the heart of this now world-famous formula there was a square root — that of a certain quantity (specifically, b2 – 4ac) calculable from the three coefficients (a, b, and c) of the specific quadratic in question. For the few people who cared about such arcane matters, it was therefore natural to wonder if something of the same general sort might not also be the case for cubic equations.

  But why would a mathematician entertain such a blurry, imprecise thought? The answer is simple: any serious mathematician would suspect that these two equations, so similar in form, must be linked by a hidden connection — an analogy. More specifically, one would expect that there would be a general formula for the cubic and that it would contain, at its core, the cube root of some quantity analogous to b2 – 4ac, and that this quantity would involve the four coefficients (a, b, c, and d) of the given cubic equation.

  Such an analogy is irresistible here, just as were the me-too analogies described in Chapter 3. Mathematicians, even the most exceptional ones, are also ordinary human beings, and without any conscious thought, they automatically anticipate that there will be an analogy between, on the one hand, the two solutions of any quadratic equation, which one can calculate by extracting the square root of a special number determined by the equation’s three coefficients, and, on the other hand, the three solutions of any cubic equation, which one can calculate by extracting the cube root of some special number determined by the equation’s four coefficients. This little guess, sliding a couple of times from two-ness to three-ness, and also once from three-ness to four-ness (which in itself comes from a mini-analogy: “4 is to 3 as 3 is to 2”) seems like an utter triviality, but without very simple-seeming conceptual slippages of this sort, which crop up absolutely everywhere in mathematics, it would be impossible to make any kind of progress at all.

  Let’s return to the story of the solution of “the” cubic equation (the reason for the quote marks will emerge shortly). It all took place in Italy — first in Bologna (Scipione del Ferro), and a bit later in Brescia (Niccolò Tartaglia) and Milan (Gerolamo Cardano). Del Ferro found a partial solution first but didn’t publish it; some twenty years later, Tartaglia found essentially the same partial solution; finally, Cardano generalized their findings and published them in a famous book called Ars Magna (“The Great Art”). The odd thing is that, as things were coming into focus, in order to list all the “different” solutions of the cubic equation, Cardano had to use thirteen chapters! Nowadays, by contrast, the whole solution is covered by just one formula that can be written out in a single line, and which could easily be taught in high schools. What lay behind such diversity, of which we no longer see any trace today?

  The problem was that no one in those days accepted the existence of negative numbers. For us today, it’s self-evident that the coefficient in the third term of the equation x3 + 3x2 – 7x = 6 is the negative number –7. It jumps right out at us, since we are completely used to the idea that a subtraction is equivalent to the addition of a negative quantity. We could rewrite the equation as follows: x3 + 3x2 + (–7)x = 6. For us who live five centuries after Cardano, these two equations are trivially interchangeable. The conceptual slippage on which their equivalence is based is so minute that we don’t even perceive it at all. But for the author of the vast tome on the third-degree equation, the concept of negative seven simply didn’t exist. For him, the only legitimate way to get rid of a subtraction in a polynomial (that is, a term with a negative coefficient) was to move the misfit term to the other side of the equation, thus yielding a different but related equation — namely, x3 + 3x2 = 7x + 6 — all of whose coefficients are positive.

  The upshot of all this is that before he could handle all the different cases of the general problem of cubic equations, Cardano had to move terms around so that there were no more subtractions anywhere, thus obtaining new equations that had only plus signs (and therefore positive coefficients). As it turns out, this procedure gave rise to thirteen different types of cubic equation, each one being — in the eye of specialists of the time — essentially different from the twelve others. And thus, in order to publish the general solution to “the” cubic equation (now the raison d’être for the quote marks should be apparent!), Cardano had to write thirteen chapters, each of which contained a complicated recipe covering one of the thirteen types of cubic. All in all, Cardano’s book on cubic equations, Ars Magna, was a long, heavy, and formidable tome, but its reception was, nonetheless, positive, shall we say.

  From a contemporary viewpoint, what Cardano did is comparable to someone who invents thirteen kinds of can-openers, each one working for just one type of can. It was a great feat, but what was lacking was an umbrella formulation, laying bare the hidden unity lying behind all this apparent diversity. That is, what was missing for “the” cubic equation was its universal can-opener. But this goal was unthinkable until someone recognized that all these different equations were, in fact, just one equation.

  Indeed, although all thirteen of Cardano’s recipes were somewhat different, there were nonetheless striking similarities between them — analogies, that is, that inspired his successors to try to combine them all into just one formula. However, in order for such a unification to come about, some concept was going to have to stretch, expand
, or bend. In this case, the concept in question was the most basic of mathematical concepts — that of number. The unification of Cardano’s recipes for solving cubic equations depended on a conceptual extension, and a quite significant one, which would allow thirteen different types of algebraic recipes, as seen by as highly skilled a mathematician as Cardano, to melt down into just a single one.

  Obviously, what was needed was a new conceptual leap, this time extending the category number to include negative numbers. This was by no means an easy step to take. Ever since the ancient Greeks, it had been known that there were very simple equations that lacked solutions, such as 2x + 6 = 0. The idea of giving such equations solutions had been considered but was always rejected (at least in Europe). Cardano himself understood that “fictitious” numbers (as he referred to them) could satisfy such an equation, but he rejected the idea with disdain. To him, the concept of negative three, being in no way visualizable, was an absurdity, somewhat like the concept of an object that violated the laws of physics. Such an idea might be stimulating to the mind, but it had to be recognized as absurd, because there was no way of actually realizing it in the world. Since Cardano was unable to associate negative numbers with any kind of entity in the real world, he labeled them “fictitious” and discarded them.