Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (16 page)

Although the repertoire of human concepts is in a sense hierarchical, in that some concepts are prerequisites to other ones, thus implying a rough temporal order in which various concepts generally are acquired, it is nonetheless extremely different in nature from the precise and rigid way that concepts are built up systematically and strictly hierarchically in mathematics or computer science. In the latter contexts, formal definitions are introduced that make each new concept depend explicitly and in an ironclad fashion on a well-defined set of prior concepts. Ordinary concepts have none of this rigidity or precise dependence. True, a person probably needs some familiarity with such concepts as
wheel, spoke, takeoff, landing, leg of a trip, jetway, concourse
, and
transit area
, for instance, before they can acquire the concept of hub, but it’s by no means clear what precise role such concepts play in any specific person’s notion of what a hub is, nor how deeply such concepts have to have been internalized by someone who feels perfectly comfortable with the sentence “Denver is a hub for Frontier Airlines.”

Over the course of our lives, we humans build up concept after concept after concept. This process continues incessantly until we die. This is not the case for many animals, whose conceptual repertoires seem fixed from an early age, and in some cases very limited (think of the conceptual inventory of a frog or a cockroach). And each new concept depends on a number (often very large, as we’ve just seen in the case of hub) of previously existent concepts. But each of those old concepts depended, in its turn, on previous and more primitive concepts. The regress all the way back to babyhood is an extremely long one, indeed. And as we stated earlier, this buildup of concepts over time does not in any way establish a strict and rigid hierarchy. The dependencies are blurry and shaded rather than precise, and there is no strict sense of “higher” or “lower” in the hierarchy, since, as we’ve shown, dependencies can be reciprocal. New concepts transform the concepts that existed prior to them and that enabled them to come into being; in this way, newer concepts are incorporated inside their “parents”, as well as the reverse. Moreover, this continual process of conceptual chunking goes hand in hand with a continual process of conceptual refinement.

Until quite recently, philosophers believed that the physical world was divided into natural categories — that is, that each and every thing, by its very nature, belonged eternally to an objective category. These philosophers focused primarily on categories such as
bird, table, planet
, and so on, whose members were visible entities. In part as a result of these conjectures from long ago, there remains a tendency, even among most
contemporary thinkers, to link the notion of
category
with the idea of classifying physical objects, especially objects that we can perceive visually. The idea that situations of someone being
nursed
back to health, for example, or situations of
hoping
for an outcome or of
changing one’s mind
, might constitute categories with just as much legitimacy as
table
or
bird
was far from such philosophers’ beliefs, let alone the even further-out idea that words such as “and”, “but”, “so”, “nevertheless”, “probably” (and so forth) are the names of important categories. If you find it difficult to imagine that a word like “but”, which seems so general and perhaps even bland, denotes a category, don’t worry; we will come soon enough to this matter, but for the time being we would like to make some observations on the more classical types of categories, since over the millennia certain ideas have become so entrenched in our culture that it is very difficult to overcome them and to start afresh down new pathways. It will thus be helpful for us to make some elementary observations that will paint a picture of concepts that is markedly different from the classical one.

We might begin by asking what a bird is. According to classical philosophers, whose view went essentially unchallenged in philosophy for centuries, until the studies of philosopher Ludwig Wittgenstein, published in the 1950’s, and which also reigned supreme in psychology until the pioneering research of Eleanor Rosch two decades later, the category
bird
should have a precise definition consisting of necessary and sufficient conditions for an entity’s membership in the category, such as “possesses two feet”, “has skin covered with feathers”, “has a beak”, “lays eggs”. (Obviously one could add further or more refined membership criteria for the category
bird
; these few simply constitute a gesture towards the idea.) The set of membership criteria (the defining properties) is said to be the
intension
of the category, while the set of actual entities that meet the criteria (the
members)
is said to be the
extension
of the category. The notions of intension and extension, borrowed from mathematical logic, are thought of as being just as precise and rigorous as that discipline itself, and the use of these terms reveals the ardent desire to render crystal-clear that which at first seems utterly elusive — namely, the abstract essence of all the highly variegated objects that surround us.

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.

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 “more 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.