King Heron asked Archimedes to determine whether his new crown was made of pure gold or whether, as he suspected, the goldsmith to whom he had given the gold was a crook and had adulterated it by mixing silver into it. It was simple for the King to check that the crown weighed exactly the same as a standard gold ingot. But how to know if the metalsmith had replaced some of the gold by exactly the same mass of silver? If only he could determine the crown’s volume! In case of adulteration, the crown’s volume would have to be greater than the ingot’s, since silver is less dense than gold. Unfortunately, though, the crown was far too irregular to allow Archimedes to determine its volume by using geometry. What then to do? According to legend, Archimedes hit upon the solution while taking a bath. He noticed that as his body slid into the water, it made the water level rise exactly in proportion to the volume that was submerged; it occurred to him that the same would hold for the crown and a gold ingot. And so, by submerging them and observing the changes in water level, he discovered that the crown’s volume was greater than the ingot’s, and so the metalsmith was indeed a crook.
Archimedes made an analogy between the crown and his body, noting that either one, when fully submerged, would displace a volume of water equal to its volume. Now if one names various categories to which a crown belongs, one comes up with such things as symbol of royalty, made of precious metal, worn on head, and so on. These have little if anything in common with categories that come to mind in thinking of a human body. Only when one rises to a highly rarefied level of abstraction — all the way to the category object endowed with volume — does the analogical connection appear explicitly. One has to ignore many facts, such as that a human body is alive, in order to map it onto a crown, which is not alive. Here, then, is a case where an analogy depends on a categorization far more abstract than anything that springs to mind spontaneously. The creative act in this story thus resides in the ability to relate entities that are extremely different in every way except one, which allows the discovery to be made.
The preceding discussion might give the impression that we are trying to portray creativity as the natural outcome of meticulous searching for commonalities among situations that, on first glance, would seem to have little or nothing in common. Unfortunately, though, even if we are exploring the mechanisms underlying creativity, we cannot pass along a recipe for genius, because a deliberate process of search for features linking two situations plays little or no role in the matter. The process of abstraction here described does not emerge from systematic, conscious search. Indeed, it often surprises the discoverer as much as it would surprise an observer.
When leaps are made at a very high level of abstraction, small though they may be, they can give rise to highly important discoveries, as the Archimedes story shows. This is certainly part of the explanation of the mystery of creative discovery, but it would be misleading to see it as a recipe, for in general, it is only after the fact that two seemingly distant situations strike an average person as being analogous, and the common traits that give rise to an abstract analogy linking them may have been hidden by unconscious presuppositions, tending to make the sought-after generalization very hard to find.
Of Mice and Men
There is a well-known recent invention that came out of one simple abstraction and resulted in a major revolution. In the early 1980s, communicating with computers was not easy. Aside from a tiny coterie of people who had worked at or visited a handful of advanced and little-known research centers in the preceding decade, no one had ever seen or used a mouse. Everything was done by typing symbols on the keyboard. One had to explain to the machine, using arcane strings of symbols expressing commands, what one wanted it to do, and it would then execute those instructions. All human–machine interaction was conducted via such a “command language”. A human would order the machine to do something, and the computer would obediently carry out the order, as long as the command was syntactically well-formed.
Over the last few decades, things have changed enormously. Today, just about the only use for keyboards is to write text, as in an email or a book; users almost never write out commands to be obeyed. When we want some action to be taken, we use a mouse (or the successor to a mouse on laptops — a touchpad). People who have grown up with this natural-seeming technology have a hard time imagining how things could ever have been otherwise. That this forward step was dependent on technological advances is self-evident, but it also depended on a small-seeming but nontrivial intellectual leap that we will now discuss.
A mouse is a kind of electronic counterpart to a biological limb; it is a prosthesis with magical powers. While our biological limbs allow us to act on the three-dimensional physical world all around us, the mouse constitutes our interface with the two-dimensional virtual world on the screen. Like Alice, who followed the white rabbit into its hole and thus discovered a world of wonders, we gain access, via the mouse, to an immaterial world. The mouse is our rabbit.
Indeed, choosing a file to work on, opening it, closing it, shifting it from one folder to another, copying it, printing it, deleting it, and so forth — all these are ways of interacting with immaterial objects. To carry out such actions in the material world, our biological limbs serve us well. To do such things in the virtual world, we rely on a mouse. We sidestep the writing of commands; instead, all we do is act and see what happens, exactly as we do in the material world.
To say that the situation has changed since the command-language era would be a vast understatement. It’s as if we had once had to do everything without using our limbs at all, always being forced to describe what we wanted in some technical jargon, or as if we had always been separated from the computational world by a thick sheet of glass, our sole channel of communication being slips of paper. After writing down our desires, we would slide our slip of paper through a narrow slot in the glass made for this purpose alone (corresponding to the “Enter” key), and the computer would then execute our written command. Such a modus operandi would obviously be extremely inefficient.
Why do we claim that the invention of the mouse was more than a technological advance — indeed, that it constituted a marvelous conceptual breakthrough?
Is it possible to see the invisible? No, since the invisible is precisely that which one cannot see.
Is it possible to touch the intangible? No, since the intangible is precisely that which one cannot touch.
Is it possible to move an object without touching it, to open or close a folder without touching it, to throw an object into the wastebasket without touching it? Moving, opening, closing, and getting rid of intangible objects — these are things that we do continually with our mouse.
Acting on the immaterial world is no longer a self-contradiction. The mouse plays the role of a biological limb, but acts on intangible entities. Thus with the help of a mouse, one can flip through a virtual book, adjust the volume of a loudspeaker, place items into a “shopping cart” in a virtual store, and so forth. Immaterial entities can be acted upon no less than physical objects.
The mouse is the fruit of a substantial mental leap. From time immemorial there has been an unbridgeable gap between the categories of material and immaterial. Thanks to this gap there has always been a clear-cut distinction between the world we touch and perceive and the world inside our minds, between what is real and what is imagined, between the concrete and the abstract, between matter and pattern, between tangible and intangible. If the idea that a physical action can have an effect on a distant entity is strange, the idea that it can have an effect on an immaterial entity is even stranger; indeed, it verges on the paradoxical. Thus the creation of the mouse involved generalizing a property that till then had been limited to the world of material objects — that is, the property of being actable-upon. A new category — that of actable-upon objects, far broader than the category of material objects — came into being when the mouse was conceived of. This new category toppled our prior assumptions about how we relate to the worl
d around us, dramatically altering our ontological categories — that is, the set of basic categories with whose aid we carve up the world.
What Makes Homo Sapiens Sapiens Sapiens?
An auto mechanic in the small town of Martinsville, Indiana had the curious idea of hoisting an old VW bug onto a sturdy stump in front of his garage, in order to attract the attention of people driving by. After all, it’s quite striking to see a car perched on a tree stump, several feet above the ground. And come Christmas season, the ex-bug was often duly decked out with Christmas lights of various colors, and as a result, no passing driver could fail to see it as a member of the category Christmas tree.
And when one checks into any hotel these days, chances are good that one will be handed a plastic card and pleasantly told that it is the key to one’s room. Well, if this card belongs to the category key, then of course the slot near the door handle in which one “swipes” it will belong to the category lock. And the traveling couple who emerge from their motel room right next door to the VW repair place in Martinsville with their keys in their pockets and who spot the nearby Christmas tree aren’t exactly in the prototypical situation that the words “key” and “Christmas tree” would tend to evoke in one’s mind…
This small example, just one among myriads that we could have chosen, is the result of the special capacity that distinguishes humans from (other) animals. Every normal person, from the most primitive to the most sophisticated, has this capacity, and it comes in various degrees, much as does any human quality, and it drives our mental development from the earliest age. Of course the capacity we are referring to is that of endlessly extending categories through analogy. We humans construct categories, extend them, multiply them, link them together, and glide gracefully among them. If we think in a subtle fashion, it is due to the richness of our repertoire of concepts and to our flexibility in seeing freshly-encountered situations in terms of our concepts. We categorize in ways that go well beyond the most common meaning of the term “categorize”, for not only do we categorize objects, actions, and situations in preexisting categories, but at the drop of a hat we transform any already-perceived entity into a category, a process exemplified by proper nouns that are run through the categorization mill and emerge as full-fledged plurals — ranging from one’s sister to the corner druggist, from Lincolns to Leonardos, from Picassos to Puccinis, from Mozarts to Mataharis. To put it in a nutshell, we humans are compulsive category-extenders, insatiable tourists who cannot get enough of the world of categories, old hands at semantic slippages, similarity-spotters par excellence, and last but not least, inveterate abstractors.
The phenomenon of marking illustrates perfectly our irrepressible tendency to extend categories. Even as we keep a lexical label fixed, we relentlessly widen the set of objects or situations to which it applies as our knowledge gradually expands, and meanwhile, behind the scenes, there is always a strong and stable set of analogical links — a conceptual skeleton — uniting its many members. Think, for instance, of the lumps that, for a child, are inevitably the result of bumping into something, or of the bottles that the child imagines as made for holding liquids; for an adult, these are far richer categories all of whose members share a deep essence. Semantic enrichment via analogy is what justifies the use of a single word or phrase as the label of more than one category, allowing us to glide smoothly up and down a spectrum of abstraction, choosing the appropriate point along the way as a function of the context in which we find ourselves. Thus Megan, who would never make the egregious error of calling even a single truck in her toy chest a “car”, will instinctively and naturally avoid trucks (and motorcycles, bikes, scooters, etc.) when her father tells her to “Watch out for cars!” just before she heads off for school. And thus Roy, who would never dream of ordering a coffee, can sip his tea while “having coffee” with some friends.
The process of marking plays an important role in clarifying what the essence of a concept is. From our first concrete examples of any concept, we proceed to more abstract ones (thus from tangible paper documents found on our wooden desks to intangible electronic documents found on our screens, or from postal addresses to email addresses). Once there are two levels of abstraction (as in these examples), then the earlier, lower level is the marked category, while the later, higher level is the unmarked one. The two levels share an abstract essence that was not visible to those who knew the marked category alone. Thus only when files and folders became familiar to people in their electronic guises did the heavy solidity of one’s traditional desk come to be seen as a dispensable attribute of deskness. The same holds for the traditional desk’s properties of occupying considerable space in a room and possessing drawers. Thanks to technological advances, the essence of the concept desk was revealed as being the fact that desks are workplaces for documents, a description that applies equally well to old-fashioned wooden desks and new-fangled electronic ones.
Such a process can carry one far afield from the first members of a category, for the pathways that slowly reveal the hidden essence are many, and finding them takes time. They may involve several levels of abstraction, as we saw with the notion of waves. Such extensions of categories are very important for our thinking, since they help us see what is central and what is peripheral in a given concept, and thus they show us how to generalize appropriately.
We are all able to overlook certain superficial qualities and to focus on more important ones in order to see what two situations have deeply in common, and this ability to abstract, which we depend on constantly, allows us to use our previous knowledge efficiently as, over the course of our lives, we confront one new situation after another. Abstraction is thus the key tool that allows human beings to move gracefully from one category to others, and to perceive the world efficiently and to interact with it profitably.
It is in their way of doing this in a given domain that novices and experts differ. Novices depend on a very coarse-grained categorization of what is out there — a single genus and a number of species beneath it. This is a perfectly good way of starting out in any domain, since it serves the basic purpose of categorization: to make distinctions based on recognizable features. For example, a square (one species) is a quadrilateral (the genus) that has four equal sides that come in parallel pairs, and it also has four right angles. As one’s familiarity with a domain grows, class inclusions become increasingly common, through the addition of categories at intermediate levels (sub-genera). This often takes place by using a concrete term as the name of a more general category — that is, through marking. Thus, for a zoologist, who is by definition an expert, a dromedary is a kind of camel, while for most people the two categories are mutually exclusive; in like manner, a square is a kind of rectangle for mathematics-savvy people, while for most people the two categories are mutually exclusive.
The proliferation of levels of abstraction grants us the freedom to shift viewpoint whenever one of the categories that we are currently focusing on seems to be leading us into a box canyon. But the recognition of which act of categorization is responsible for the lack of progress is extremely difficult, and much of the subtlety of creativity lies here. For example, who would have been able to say, a priori, that it was an overly hasty categorization of the box of thumbtacks that caused the impass in Duncker’s classic candle problem? Recognizing this fact — that is, being able to put one’s finger on the categorization that was at fault — is very difficult, but that skill can open up new horizons, because a more abstract categorization, by bringing into focus certain features that had been unseen till that moment, allows one to circumvent the dead end. From the picnic in the woods to the principle of Archimedes and the invention of the mouse, the key move is making the appropriate upward leap in the space of categories.
We humans all think thanks to categories, and as we collectively expand our shared repertoire of categories, we all continually expand own own personal repertoire. We have very abstract categories on which we unconsciously ba
se our understanding of the world around us, such as the category of what is actable-upon and the category of objects with volume, and we also have very concrete categories, which allow us to distinguish between a Yorkshire terrier and a Staffordshire terrier, to hold forth on the qualities of a Gevrey-Chambertin Premier Cru Les Cazetiers Dominique Laurent 1996, or to select between anchellini and primaverine as an ingredient for a minestrone. It would seem that the flexibility of our category systems is the key to what distinguishes Homo sapiens sapiens from a Yorkshire terrier.
CHAPTER 5
How Analogies Manipulate Us
At the Mercy of Uninvited Guests
Is it possible that analogies have the power to manipulate us, to twist us around their little fingers? Certainly; in fact, they do so in two senses of the term “manipulate”. First, analogies often arise in our minds without our even being aware of them: they invade us surreptitiously and seize center stage. Second, analogies coerce us: they force our thoughts to flow along certain channels. And thus, far from being just appealing and colorful pedagogical or rhetorical devices, analogies are wily creatures with a will of their own. They shape our interpretations of situations and determine the conclusions of arguments. Put otherwise, an analogy will not be content with merely crashing the party; having shown up, it then dictates the rest of the evening.
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