Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (57 page)

In a closely related type of situation (but still in the realm of the first kind of metaphorical understanding), the abstract category may be lost in the fog of etymology and may no longer exist in the collective mind of native speakers. In such cases, the abstract category belongs to the history of concepts but not to their psychology. These cases include familiar expressions that are perfectly understood by native speakers but whose origin is mysterious and for which one feels at a loss to connect the two different meanings of the same term (such as the two senses of the verb “to bore”, which, although they probably sprang from a common origin, are not heard by English speakers as being in any way related). For instance, consider the sentences “Walter is a wolf”, “Tim is a turkey”, “Denise is a dodo”, and “Belle is a bitch”. In each case, one can get involved in endless guessing games as to why this particular kind of animal is the source of this particular metaphor in English, because there is no obvious superordinate category in one’s memory that includes the two in a natural way. But the key to understanding such sentences is that there already exists in the listener’s mind a non-animal category that shares the same lexical label as the animal category.

In the second type of metaphorical understanding, there is likewise no need to construct a new abstract category on the spot because it is already in one’s mental lexicon, but that category has to be made use of in order for the sentence to be understood. Thus, suppose someone were to exclaim, “This damn word processor is a pig!” A suitable scenario to evoke such an outcry would be if one’s word processor frequently made wrong hyphenations, ugly interword spacings, and random insertions of characters. In this case, one would have to utilize the most general unmarked category of
pig
₃
in order to understand the utterance. This would be a completely natural act of categorization, not requiring the construction of a new category, because one can simply take advantage of the abstract category
pig
₃
.
To understand “This
damn word processor is a pig”, the unmarked sense of the word “pig” is activated, just as unmarked abstract senses of words are activated in all sorts of everyday situations — as, for instance, when somebody refers to Liz’s pickup truck as “Liz’s car”, or refers to an atheist friend as “the pope of popsicles”, or announces, “I need to buy some kleenex”, even if the Kleenex brand isn’t sold in the store.

We come now to the third type of metaphorical understanding. Here the abstract category does not exist
a priori
, so the listener is forced to construct it in order to understand the statement. This would be the case for utterances such as “Bill is a bridge”, “Steve is a stone”, “Florence is a firefly”, “Patsy is a prawn”, and so forth. In such cases, one has to distill on the fly a new essence from an already-existing category in order to see Bill, Steve, Florence, and Patsy as members of new ad-hoc categories. The challenge is similar to that for a French speaker who is told by an English speaker, “My new car is a lemon.” Since this metaphor doesn’t exist in French, the French speaker would have to concoct a new ad-hoc category in order to understand the statement, and there is no guarantee of success, even if to English speakers it’s obvious that the key property of lemons in this case is their sourness, which theoretically would allow anyone to make a new abstract category that includes both citrus fruits and cars.

And how would we English speakers understand a French person who told us, “The movie last night was a turnip”? French speakers understand immediately that the movie was mediocre, because “être un navet” (“to be a turnip”) is a stock phrase applying to films, and so to understand the sentence they simply exploit this familiar preexisting category. They do not need to jump to a yet higher level of abstraction, that of
turnip
₃
; the level of
turnip
₂
suffices. By contrast, for people who are not familiar with this expression, it will be necessary to concoct a new abstract ad-hoc category based on some new-found essence of the concept
turnip
, and of which certain films will be members. Thus some people might imagine that what’s crucial for this kind of abstract
turnip-ness
is
being purple
, or
growing underground
, or
being used in salads.
Then again, to someone more in tune with the speaker’s tone, it might occur that
insipidity
or
blandness
is the key quality.

In all of these cases, whether the category is abstract or concrete, preexisting or invented on the fly, the understanding of metaphorical statements depends on applying a category to a situation. Our take-home lesson is thus:

If there’s any domain that people think of as having precise and unambiguous concepts, it would have to be mathematics. Here, where contradictions and blurriness should play absolutely no role, one would naturally suppose that the subjective, context-dependent phenomenon of marking, which by definition conflates two categories by assigning them the same label, would surely be nonexistent. And yet this is not the case. Even in mathematics, our human style of fluently jumping between categories, relying on context to make things clear, trumps the desire for pure logicality, as we’ll now see.

Tom is in seventh grade. He’s just finished a geometry class in which his teacher gave him some homework problems on the topic of quadrilaterals. The first exercise said, “Write ‘S’ in each square, ‘R’ in each rectangle, ‘Rh’ in each rhombus, and ‘P’ in each parallelogram.” Tom diligently carried out his assignment, and the figure below shows what he did.

Tom was very careful not to fall for any trick questions. For example, he wasn’t fooled into thinking that the square balanced on one of its corners was a rhombus. And he also correctly identified rectangles that were tilted, and even rhombi that were tilted at strange angles.

The assignment that came next was called “definitions”, and it featured the following questions:

Tom had learned all of this material very well, and he replied as follows:

1. A square has four right angles and four equal sides, parallel in pairs.

2. A rectangle has four right angles and four sides, parallel in pairs.

3. A rhombus has four equal sides, parallel in pairs.

4. A parallelogram has four sides, parallel in pairs.

Tom’s teacher graded the homework and when Tom got home, he proudly told his parents that he’d gotten a perfect grade on each of the exercises. Is there anything worrisome in this situation? Well, yes, something is wrong; in fact, there are interesting contradictions here. Indeed, what would have happened if Tom’s acts of writing letters inside shapes had been guided by the written answers that he gave?

Why, then, did Tom’s teacher give him a perfect grade, when in fact his answers to her two assignments have just been shown to be inconsistent?

Indeed: are squares rectangles? The answer comes down to a classic case of marking. From a mathematician’s point of view, a square is certainly a rectangle, because it satisfies the criteria that define rectangles. In that sense, the question is unambiguous, and the answer is simply “yes”.

If one looks at the definitions of the various types of quadrilaterals, one can easily draw a diagram showing their relationships:

The different categories shown in the figure illustrate the different points of view that one can adopt for any of the quadrilaterals drawn. According to the diagram above, a square belongs to five different categories:
square, rectangle, rhombus, parallelogram
, and
quadrilateral.
If it’s seen as a quadrilateral, a square has two diagonals, but there’s nothing special to say about them. If it’s seen as a parallelogram, a square’s two diagonals slice through each other exactly in their midpoints. If it’s seen as a rhombus, its diagonals are perpendicular; and if it’s seen as a rectangle, they have the same length as each other.

And yet this chart of categories in the world of quadrilaterals isn’t sophisticated enough for us to anticipate how Tom’s geometry teacher would react if one of her pupils were to draw nothing but a single square as the answer to the following exercise: “Draw a square, a rectangle, a rhombus, and a parallelogram.” Strictly speaking, she should be very respectful of the precise understanding of the definitions implicit in such an answer, and she should give the young showoff the highest possible grade. So let’s suppose that she was a good sport, and that she did so. Even so, she would have to be taken by surprise by this playful answer, and depending on her personality, she might be either charmed or exasperated by the highly unusual interpretation of her question. But why would she be so surprised to see such an answer, given that it’s perfectly correct and elegantly economical, to boot?