Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (111 page)

This explanation, although clear, does not allude to the rather salient
resemblance
between elbows and knees, which played a critical role in Doctor Ellenbogen’s discovery. This is a regrettable oversight, since one might well wonder if she thought that Doctor von Knie’s treatment for knee ailments might not also work equally well in combating diseases affecting the eyes, ears, stomach, kidneys, and so forth. But the truth is that Doctor Ellenbogen never made any such mental leaps, which suggests that she did not see enough of a resemblance between knees and stomachs (for example), or between knees and eyes, to lead her to guess that she might extend the virtues of Doctor von Knie’s miraculous cure to those organs.

Once one has seen that the concept of
knee
can be analogically extended outwards, yielding the more general concept of
knee-plus-elbow
(which is the extension found by Doctor Ellenbogen), then this wider category exists in its own right in one’s mind. This kind of addition to a conceptual repertoire typifies the process of conceptual broadening by analogy. We will call this kind of broadening
horizontal
.

By contrast, in a
vertical
category leap, one would make an upwards move from the concept of
knee
to the more abstract concept of
joint
, which, in the minds of most adults, would subsume the concepts
knee
and
elbow
(and in addition, the concepts
ankle, shoulder, knuckle
, and so on). So let us now imagine that the brilliant Doctor Gregorius Gelenk starts with Doctor von Knie’s treatment for knee diseases and makes a mental leap allowing him to announce a uniform treatment for a varied group of illnesses that afflict all the different joints, including ankles, knuckles, and so forth. In this case, we would be dealing with a
vertical
jump: the result of following a preexistent link between
knee
and
joint
in Doctor Gelenk’s personal repertoire of concepts.

Such horizontal categorical broadenings and vertical category leaps are natural and unsurprising — the bread and butter of human thinking. But on occasion, such broadenings and leaps can also be insightful and admirable. Suppose Doctor Zygmund Zeigefinger discovers a cure for an ailment of the
index
finger, and his colleague Doctor Renate Ringfinger modifies this cure so that it works also for the
fourth
finger. It would certainly be shocking, would it not, if the Nobel Prize in Medicine were awarded to Dr. Ringfinger for her “remarkable breakthrough”? The analogy on which such a “breakthrough” would be based is far too obvious to deserve such a great honor. People would protest, “Come on! Two fingers are as alike as two peas in a pod!”

On the other hand, suppose Doctor Zora von Zehe adapted Doctor Zeigefinger’s cure so that it also worked for
toe
diseases. It goes without saying that we would applaud her contribution more than that of Doctor Ringfinger, but we would still be bewildered were she to receive a Nobel Prize for her work.

Finally, suppose that Doctor Hartmut Herz, in a bold moment of inspiration, had the sudden insight that there might be a connection between (of all things) the index finger and the human heart, and that then, by slightly tweaking certain aspects of
Doctor Zeigefinger’s cure, he stumbled upon a miraculous cure for certain
coronary
diseases. In such a case, we would have no problem understanding why the Nobel Committee had seen fit to award a Nobel Prize in Medicine to Doctor Herz.

Language can play a catalytic role in such situations; for example, the French terms “doigt” (“finger”) and “doigt de pied” (“toe”, but literally “foot finger”) make it clear
a priori
that we are dealing with essentially the same thing in both cases. The finger–toe analogy is thus extremely simple and natural for French speakers, whereas for English speakers there is no silver platter on which the analogy is delivered to them. However, for easily understood
visual
reasons, it is still quite obvious. In neither language is just one word used to denote both elbows and knees, and since the resemblance of elbows to knees is perhaps slightly subtler than that of fingers to toes, it would be less probable for someone to connect these concepts. On the other hand, if in German the word for “elbow” were “Armknie” (regrettably, it is not), we would expect there to be a somewhat larger amount of unconscious crosstalk between the two concepts in the minds of German speakers than in the minds of English or French speakers.

And lastly, if in German the human heart, by some odd quirk, were called “der Brustzeigefinger” (that is, “the chest-index-finger”), and if Doctor Herz were a native speaker of German, then his mother tongue could have furnished him with an intuitive hunch that a cure for a disease affecting index fingers might be adaptable to the heart (that is, to the chest’s index finger — at least from his German-speaking viewpoint). The idea of such a compound word in German is not totally far-fetched, by the way, since, as readers may recall from
Chapter 2
, the German for “glove” is “Handschuh”, or even “Fingerhandschuh”, as contrasted with “Fausthandschuh”, a compound that, when taken apart, means “fist-hand-shoe” or “fist-glove” — which is to say, “mitten”.

When pathways between concepts are handed to one in advance — for example, by extremely salient physical resemblances (as between fingers and toes), or by related linguistic expressions (“hair” is a clear example in English, since the hair on one’s head and the hair on one’s body are denoted by one and the same word, whereas in French they are denoted by two extremely different words — namely, “cheveux” and “poils” — which break a single anglophone category into a pair of francophone ones) — then the corresponding horizontal category extensions via analogy will be obvious and inviting. Much the same holds for vertical category leaps. If in one’s mind there already exists a connection between a specific concept and a more general concept, then the shift in perspective involved in making the leap from one to the other, such as seeing knees or elbows as
joints
, is a simple and natural act.

We now return to the genesis of
E = mc
²
.
Any given entity belongs simultaneously to an unlimited number of categories. Nonetheless, in daily life one often has the illusion of dealing with an entity belonging to just one category. In general, our surroundings often strike us as being clear and unambiguous, as if there were just one correct, objective way to perceive them; indeed, it’s this illusion that allows us to live. If
we had to take into account, at every moment, the boundless number of categories that the situations we run across might belong to, we would constantly be spinning our wheels in utter mental confusion and we would be incapable of taking any action.

Labels and stereotypes found and stuck on in a flash are indispensable, but the flip side is that they also limit us tremendously. How does one find one’s way efficiently in the space of all possible categorizations without taking forever to do so? Where is the happy medium in daily life, or in scientific thinking, between finding a quick-and-dirty categorization and putting one’s finger on the perfect one in a given context? In a new situation, how can one know, when one is trying to pinpoint its essence, whether one should settle for the fastest and easiest categorization that comes to mind? How can one recognize situations in which expanding horizontally outwards, thus constructing a
broader category
, or else jumping vertically, thus reaching a
higher level of abstraction
, would be a wise move (or an unwise one)?

When one reads works by Einstein himself, as well as his more scientifically oriented biographies, it is clear that the great physicist frequently ran into just such dilemmas. Indeed, this is one of the most salient traits of his intellectual style. For example, in 1905, in coming up with special relativity, Einstein made a very fecund category extension based on what might seem to be the most innocuous of analogies.

Taking a fresh look at an old and fundamental principle known to us today as “Galilean relativity”, he extended it outwards in the most innocent-seeming fashion, suggesting that it held not only within the domain of
mechanics
, but also within the larger domain consisting of
mechanics together with electromagnetism.
(Mechanics is the earliest branch of physics, and it deals solely with the movement of tangible bodies in space — thus with speed, acceleration, rotation, gravity, friction, orbits, collisions, springs, pendulums, vibration, tops, gyroscopes, and so on — but it does not include optics, electricity, or magnetism, let alone nuclear forces; none of these branches of physics were known in Galileo’s day.)

More precisely, the principle of Galilean relativity said: “Given two frames of reference moving at a constant relative velocity, there is no mechanical experiment whatsoever that will distinguish one from the other.” To make this more concrete, if one is inside an airplane that is flying in a straight line at a fixed altitude at a speed of 500 miles an hour, the principle of Galilean relativity states that no mechanical experiment carried out inside the plane will be able to reveal that it is not standing stock still in a hangar — or conversely, if the plane is sitting on the ground, no mechanical experiment performed inside it will be able to reveal that it isn’t streaking along at 500 miles an hour. We all know that when we are flying in the sky at a great speed, as long as the speed is constant, we can pour ourselves a glass of water without taking into account the fact that we are moving. It will feel exactly as if we are perfectly still. And the same holds, of course, when we are in a train moving along a straight stretch of track at a fixed speed, whatever that speed may be.

We can easily imagine experiments of various sorts that one could carry out inside a plane or train to see if their outcomes do or do not depend on the vehicle’s state of motion. Mechanical experiments might involve pouring water into a glass, spinning a
gyroscope on a table, swinging a pendulum from the train car’s ceiling, making weights bob up and down on springs from which they dangle, sliding and colliding hockey pucks on a frictionless surface, rolling a ball down an inclined plane, floating a helium balloon above our heads, and so on. And as it turns out, such phenomena look and feel exactly the same inside a parked plane or train and a smoothly moving one, thus fully confirming Galilean relativity.

But what can be said about optical and electromagnetic phenomena in trains and planes? We have all, while traveling, turned lights on and off, looked at ourselves in mirrors, checked what time it is by consulting our digital watches, used laptops and video games, and so forth. All these devices seem to us to work “just like normal” — every bit as normally as does water poured from a pitcher into a glass. That is, in a plane or train they look just as they look when we are sitting in a chair in our living room. The devices that are involved — glasses, watches, telephones, computers — are combinations of simpler things such as lenses, mirrors, prisms, batteries, bulbs, coils, magnets, currents, and so forth, and the physical laws governing this class of things are those of optics and electromagnetism.

With this prelude, let’s return to Albert Einstein in his
annus mirabilis
in Bern. In that year, as he pondered the principle of relativity formulated by Galileo, Einstein asked himself why this profound-seeming principle should be limited to just
mechanical
experiments. If observations of a pendulum, a spring, or a gyroscope did not allow one to figure out whether one was in motion or not, then why should observations of a candle, a magnet, a mirror, or an electrical circuit have a better chance at doing so? Einstein saw no reason that they should. He was pushing Galileo’s principle outwards in his mind, generalizing it by analogy, but only in the gentlest of fashions.

It’s not clear, incidentally, whether it’s better to call this a horizontal category-broadening act or a vertical category leap, because one can see it either way. On the one hand, all that Einstein did was to replace the phrase “any kind of
mechanical
experiment” by the more general phrase “any kind of
mechanical or electromagnetic
experiment”; in this sense it seems like a horizontal extension justified by a simple analogy between one area of physics (mechanics) and another (electromagnetism). This is similar to a student saying, “I’m not going to take the electricity-and-magnetism course next semester, because this semester’s mechanics course was so hard for me.”

On the other hand, one might also say that Einstein replaced the idea “any kind of
mechanical
experiment” by the more abstract idea “any kind of
physical
experiment” — that is, he made a leap from a narrow concept to a wider, more general one that encompasses it. This would be like a discouraged student saying, “I’ll never take another physics course again, because this semester mechanics was such a bear.”

And how did Einstein himself see his analogical move? Well, he once described his feelings at the time in the following manner: