Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (53 page)

In the early twentieth century,
radio waves
(really just long-wavelength light waves) were used as a host medium for carrying
sound waves.
In other words, sound waves hitch a ride on the much faster medium of electromagnetic waves. Amplitude modulation (AM) is a kind of
transverse
way of letting the hitchhiking sound waves locally distort the medium through which they are traveling, whereas frequency modulation (FM) is essentially a
longitudinal
way of letting the hitchhiking sound waves hop aboard the very fast host waves. FM, in short, is a very abstract sort of compression wave. We can’t enter into the details here, but the brilliant though tricky idea of waves riding on waves gradually grew into an ever more common
leitmotiv
in physics.

Later during the twentieth century,
temperature waves
were discovered, in which the value of the temperature of a substance is what oscillates “up and down”. In other words, what is moving “up and down” is the number of degrees (Fahrenheit or Celsius) at each point in the substance, like millions of imaginary columns of mercury moving physically up and down in so many imaginary thermometers all through space (and of course the thermometers need not be in phase with each other).

Also discovered were
spin waves
, where the direction of spin of electrons (think of a room filled with millions of tiny spinning tops, some pointing up and some pointing down) can “ripple” across the medium, with spins periodically flipping from up to
down and then back to up. And then there are
gravitational waves
, where the amount of gravitational pull at some point in space oscillates periodically in time, as if an invisible ripple were silently shimmering through space and, by its local size, telling thousands of little pebbles floating in space how strongly, and in what direction, they are being pulled by a rapidly shifting, totally invisible force.

Last but not least, among the very most important kinds of waves in all of physics are
quantum-mechanical waves
, sometimes called
matter waves
or
probability waves.
Roughly speaking, at every point in space such a wave has a value that changes over time, and when that value is squared, it tells how likely one is to find a particle in the given spot at the given instant.

We could list dozens of other types of abstract waves, but this modest sampler will suffice. As we have seen, the notion of
wave
in physics has reached an enormous degree of abstraction and sophistication today, and yet all of the latest and most abstract forms of waves are tied by analogy and by heritage to the earliest kinds of extremely concrete, tangible, palpable waves in bodies of water and in fields of amber grain — waves that we can see with our eyes and feel with our bodies.

Oh, the poor fourth Earl of Sandwich! How often is his name abused these days! As is well known (or at least rumored), it was the august Earl who concocted the clever idea, around 1750, of putting some meat (or perhaps some cheese) between two slices of bread. Sandwich’s original “bread–meat–bread” pattern spurred an enormous number of copycat variations in the gastronomic world, which we don’t need to spell out here.

It might be amusing, however, to mention that one of the authors of this book has been known, to his great gustatory delight, to savor the act of devouring a handful of macadamia nuts placed on a square of American cheese that is then folded back on itself, thus becoming both an upper and a lower layer at once. The analogy is tremendously obvious, and yet there is pleasure in spelling it out: the two layers of surrounding cheese “are” the two layers of bread, and the macadamia nuts “are” the meat. The only question left, then, is whether we also need quotation marks around the word “is” if we assert that this
is
a sandwich. Of course it is
like
a sandwich, but what would the Earl himself say? Or is there any reason to presume that the fourth Earl of Sandwich would be the ultimate authority concerning membership in the category bearing his title’s name? Could there in fact be any ultimate authority on the topic? Would it not be a fine thing to create an elite Sandwich Memorial Board that would officially make all such rulings?

Let’s move on to sandwiches lying beyond the realm of the edible. Our first variation on the theme — our first inedible extension of this category — is not too far removed from that realm, however. For at least a couple of hundred years, restaurants in large cities have had the practice of hiring hungry souls to serve as walking advertisements, having them stroll the sidewalks wearing wooden or cardboard posters on front and back. Such people are usually called “sandwich men” (although of course
a sandwich man need not be a
man
₁
). Needless to say, there is many a discrepancy between a slice of bread and a piece of wood with slogans painted on it, and likewise there is many a discrepancy between a slab of meat and a living human being; these two facts already cast some doubt on the
sandwichhood
of the described item. But in addition, this kind of “sandwich” is not meant to be
consumed
, except in a very abstract sense — namely, by a visual system. And perhaps more subtly, a human sandwich of this sort challenges the great U.S.A. (that is, the Unspoken Sandwich Axiom), which is the tacit idea that a sandwich must always be
horizontal.
In sum, there are numerous reasons to wonder about the membership of this kind of entity in the category
sandwich.
The question arises as to the limits of the concept, and whether there are any limits at all to it. How far out does sandwichhood stretch, and in what directions? This is a most provocative question.

We are not the only thinkers, we hasten to add, to have posed such far-reaching questions. A bold web site called “The Sandwich Manifesto” addresses head-on the fundamental question “What is a sandwich?” One issue raised there is whether anything that has the form “A–B–A” is a sandwich. For example, do the books on a shelf, sandwiched between two identical bookends roughly a yard apart, form a sandwich? Or is the name “Einstein” a sandwich, given that it is spelled “Ein-st-ein”? Is America a sandwich, with the Atlantic and Pacific Oceans being the slices of bread? The non-identicality of those two great bodies of water, not to mention the non-identicality of “Ein” and “ein” (after all, the first boasts a capital “E” while the second does not), brings up the more general issue of the identicality, or lack thereof, of the two “slices of bread”.

On one memorable occasion in Paris, the métro station Odéon was described by a fairly bored métro rider as being “sandwiched” in between two métro stations bearing the names of saints (Saint-Michel and Saint-Germain-des-Prés). This trio of stations might thus, with good reason, be called a “subway sandwich”. In a (somewhat) similar fashion, car-borne criminals are sometimes said to be “caught in a sandwich” if two police cars manage to maneuver into a position such that one of them is behind the criminals and the other is ahead of them. Another flesh/flesh/flesh configuration is the sexual position known as “The Sandwich”. But leaving the body behind and moving on to the mind, we have rhyme sandwiches: consider a rhyming quatrain whose rhyme scheme is “ABBA”, where the “A” rhymes play a
bread
role to the “B” rhymes’
meat.
And surely we would not want to overlook the crucial role of inedible sandwiches in the realm of solid-state physics. Specifically, there are important types of semiconductors called “P” (for “positive”) and “N” (“negative”), and from them are formed three-layer structures of the form “PNP” and also of the complementary form “NPN”. These are both standardly called “sandwiches” in academic papers by physicists — and such sandwiches are incidentally also members of the category
transistor
(hardly a concept to be sneezed at).

The bold interchange of the
meat
and
bread
roles in solid-state physics raises another fundamental question of sandwichology — namely, whether there are certain categories of things that are more eligible to play the
bread
role in a sandwich, with
other categories being more eligible to play the
meat
role. For example, we all know that
peanut butter and jelly
constitutes a fine member of the
meat
category (at least in the context of sandwich-making), but would it ever be able to serve in the role of
bread
? Let us pose this question in a more point-blank fashion. One can envision, without the least difficulty, an “NPN” sandwich in which the “N” stands for a delicious Indian
naan
and the “P” stands for peanut butter and jelly — but what about an inside-out “PNP” sandwich? Or is this simply going too far? Have the ultimate limits of the
sandwich
category been transcended, or could it be that our era is simply not yet ready for such bold new visions?

Arguably the most burning conundrum in sandwichology is under what conditions an entity that has the form “A–B–C” should be counted as a
sandwich.
For instance, if one’s Bostonian bosom buddy Bradley (“B”, for short) happened to be fast asleep on his comfortable Chesterfield couch (“C”, for short), and if Bradley’s Abyssinian feline friend Adele (“A”, for short) were suddenly to leap atop dormant Bradley, might the resulting A–B–C configuration count as a
sandwich
? And if A had been an
armchair
rather than an
Abyssinian
? Armchairs being presumably a bit more couch-like than Abyssinians, would one not be ever so slightly closer to the canonical A–B–A form?

The last few paragraphs have been rather fanciful, but it is worth noting that the word “sandwich” is routinely used in colloquial speech and even in formal contexts, sometimes as a noun and sometimes as a verb, to denote abstract and definitely non-edible patterns. How many times, for instance, have you casually remarked, “My meeting with the dean this morning was sandwiched right between my dermatologist’s appointment and my dentist appointment”? You probably can’t even remember! Yes, in an era when using the word “sandwich” to describe all sorts of inedible things has become a routine worldwide phenomenon, we are no longer talking about wild, self-indulgent flights of fancy. We are talking popular culture.

As you can no doubt sense by now, the questions of shadowology, wavology, and sandwichology open up vast conceptual horizons without an abstraction ceiling lurking anywhere in sight.

As we have just seen, the extension of concepts by analogy seems limitless. It allows us to see one’s homeland as one’s
mother
, to see a snow-free area under a tree as a
shadow
, to see a pattern of sequential brakings by drivers along a stretch of freeway as a kind of
wave
, a sequence of three appointments as a
sandwich
, and also to see the story of a person who finds reasons for satisfaction in their failure to purchase tickets for a concert they had dearly hoped to attend as just a differently dressed retelling of the
sour grapes
fable. Sometimes leading to extending the boundaries of a category, such as
bird
or
book, moon
or
marriage, eat
or
undress, much
or but, or to the construction of a new and more abstract category, such as
lump
or
desk
, or even a whole range of new concepts, as in the case of
shadow
and
wave
, the process is simply part and parcel of the human condition, and as such is unstoppable. Indeed, almost as if to show off its irresistibility,
the process of category extension survives even in extremely austere environments where one would suspect it could not — namely, in the world of proper nouns, a world where everything comes in ones, and thus a world that, by its very definition, would seem to prohibit the extension of categories.

In contrast to common nouns, which are obviously the names of categories, proper nouns, which are singled out by the capital letters with which they start, might seem to be of a completely different nature. Although they are certainly definable through language, proper nouns aren’t supposed to need definitions because the set of entities that they refer to seems to be unambiguously defined. Often, they designate one and only one entity: a planet, a continent, a country, a city, a monument, a human being, a work, and so forth — and when they designate more than one entity, as does a first name, a last name, a commercial brand, a nationality, and so forth, the set of entities to which they apply seems nonetheless so sharp and clearly defined that one might have a hard time imagining that there is anything about them that resembles the “halos” that surround the cores of all typical categories, as we have been describing them all through our book so far.

Indeed, one might well go so far as to question the use of the word “category” when there is just one member. What kind of sense does it make to speak of categories such as
Paris, Galileo, Earth
, and
Moon
, when each of those exists (or at least once existed) in but a single case? But upon analysis, this kind of argument is quickly seen to hold no water. Our irrepressible human tendency to extend categories by the making of analogies applies in the case of proper nouns no less than it does for all other nouns (and other words, for that matter).

Chapter 1
recounted the story of Galileo, the Moon, and the many moons that subsequently came along. That story may have seemed like a very unusual case, but leaps such as that from “Moon” to “moons” (or from “Sun” to “suns”) take place all the time around us, far removed from the specialized world of scientific discoveries. In particular, they take place whenever categories are extended by an act of marking, in which a proper noun loosens its belt a bit and in so doing becomes the label of a more general category.