In summary, mathematics involves the making of analogies at all points along the spectrum, ranging from the blandest of memorized symbol-shunting recipes all the way up to the most vertiginous generalizations of ideas that are already highly abstract. In the sections preceding this one, we concentrated on quite sophisticated analogies in mathematics because this chapter is principally about what lies behind scientific creativity and “discoverativity”. It’s understandable that people may not at first see lowly symbol-manipulation recipes as requiring analogical thought, but this is only because they are letting themselves be subliminally guided by a naïve stereotype of analogy-making itself. This ironic fact is par for the course, though; after all, people seldom recognize at first how much analogy-making is involved in the mundane-seeming acts that result in the evocation of words and phrases. Once they get used to the idea, however, it becomes self-evident. And the same idea applies to routine techniques in mathematics.
Physics and Logical Thinking
We now shift our attention from mathematics to physics. In cartoons one often sees a wild-haired physicist standing before a blackboard overflowing with equations and esoteric symbols. Such images suggest that the prototypical thought processes of any physicist consist in the manipulation of symbols according to the laws of mathematics. High-school and college physics courses encourage this vision, as they abound in precise formulas for all sorts of sophisticated and elusive notions such as magnetic field, kinetic energy, and angular momentum. To be sure, all these formulas are perfectly correct, but they are nonetheless misleading, because they give the impression that physics is an axiomatic discipline in which, starting with a few basic principles, one can use the most rigorous logic and from them deduce all the key results of mechanics, thermodynamics, electromagnetism, the theory of waves, and so forth. This is the standard image of research in physics for the public at large, for intellectuals, and even for a good number of physicists — but it is, in the end, a myth.
Although many textbooks written by physicists paint physics as a purely deductive science, a good number of others effectively counter that image. And some of the greatest physicists have even tried to elucidate the pathways of thought that once led them to their most important discoveries. They invariably tell stories of analogies that came to them — analogies establishing links between just-discovered, fresh, unexplored physical phenomena and older, well-explored, and clearly understood physical phenomena.
And so, would the difference between a great and a mediocre physicist have to do with the degree of wildness and craziness of the analogies that each is willing to entertain? Would a genius in physics be someone who deliberately sets out to explore bizarre and far-fetched analogies between extremely distant concepts, someone who purposefully tries to make implausibly wide leaps of imagination connecting concepts that nobody would have dreamt had the slightest relation, all based on small pieces of evidence that other physicists would simply laugh at? And contrariwise, would the mediocre physicist be a timid individual who explores just a small neighborhood of ideas right under their nose, and who is never tempted to make bold mental leaps?
Such an image of genius versus mediocrity, though in some ways appealing, is not our view. Geniuses do not deliberately set off with the goal of concocting a wild-sounding analogy between some brand-new phenomenon, shimmering and mysterious, and some older phenomenon, conceptually distant and seemingly unrelated; rather, they concentrate intensely on some puzzling situation that they think merits deep attention, carefully circling around it, looking at it from all sorts of angles, and finally, if they are lucky, finding a viewpoint that reminds them of some previously known phenomenon that the mysterious new one resembles in a subtle but suggestive manner. Through such a process of convergence, a genius comes to see a surprising new essence of the phenomenon. This is high-level perception; this is discovery by analogy.
Albert Einstein, Analogizer Extraordinaire
Some years ago, this book’s senior author, who many years earlier had gotten his doctorate in physics and who had always been fascinated by the pathways of discovery in that discipline, decided to develop a lecture on the role of analogies in physics. Long before starting to prepare the talk, he impulsively chose as his title “The Ubiquity of Analogies in Physics”, and so, when he actually tackled the topic in earnest, it was with a sense of great relief that he discovered that each branch of physics is indeed as densely riddled with analogies as he could ever have hoped. Better yet, at the heart of nearly all the great discoveries of the last three centuries, he found that there lay a crucial, decisive analogy. Here is a small sampling from the list:
•gravitational potential: analogous to a hill — Lagrange and Laplace, 1770
•electric potential: analogous to gravitational potential — Poisson, 1811
•magnetic potential: analogous to electric potential — Maxwell, 1855
•four-dimensional space–time: analogous to three-dimensional space — Minkowski, 1907
•electrons as waves: analogous to photons as particles — de Broglie, 1924
•quantum-mechanical matrices: analogous to Fourier series — Heisenberg, 1925
•quantum-mechanical waves: analogous to classical waves — Schrödinger, 1926
•quantum commutators: analogous to classical Poisson brackets — Dirac, 1927
•“isospin” of related particles: analogous to quantum spin states — Heisenberg, 1936
•weak nuclear force: analogous to electromagnetic force — Fermi, 1931
•strong nuclear force: analogous to electromagnetic force — Yukawa, 1934
•vector bosons: analogous to photons — Yang and Mills, 1954
It thus appeared that just about every breakthrough by the greatest physicists, such as Newton, Maxwell, Dirac, Heisenberg, Fermi, and many others, had been the fruit of one or a number of analogies intuitively “sniffed” by its discoverer. Indeed, it seemed as if the lecturer’s early-chosen title had been confirmed far more powerfully than he had ever thought was possible. Curiously, however, he had deliberately shied away from looking at the work of Albert Einstein, as he was nagged by a fear that the genius of Einstein might have been so profound that he was able to do completely without those irrational leaps of the sort made by “ordinary geniuses”, and that Einstein might instead have relied exclusively on the purest of logical reasoning to arrive at his revolutionary ideas. Was it possible that Einstein’s exceptionally deep thinking style might undermine the thesis of the talk that he was going to give?
It happened that a few years later, the same speaker was invited to take part in a conference celebrating the centennial of the year 1905, which is known as Einstein’s annus mirabilis — his “miraculous year” — since that was the year that he revolutionized physics with a series of five unparalleled articles. For this special occasion, the speaker decided to take the bull by the horns and to look carefully into the pathways of Einstein’s thought in the hopes of finding at least a handful of analogies in them. And mirabile dictu, to his great relief, he found that analogies abounded in the thought processes of Einstein, every bit as much as in those of any other physicist.
There was just one difference — namely, that often what at first seemed to be a shaky analogical leap between two phenomena would, years later, turn out to be the unification of two phenomena that up till then had been thought of as totally separate. Over and over again, two things were revealed to be really just one thing. Could it be that Einstein always knew from the very start that he had discovered an identity as opposed to a mere analogy? In fact, no. Even the genius of Einstein could not foresee that his newborn intuitive hunch that two physical phenomena were related — his new analogy — would someday wind up totally unifying them. From his perspective, he was merely offering a promising analogy whose consequences would unfold with time. But in Einstein’s case, unification happened so frequently that one is forced to say that this is one of the most salient traits of Einstein’s thinki
ng style. By contrast, most other physicists find analogies that, though revealing a fruitful network of similarities between phenomena, do not lead to a deep unification.
To make this idea a bit more concrete (via a caricature analogy), suppose that a chemist of a few centuries ago, after observing a number of similar characteristics of coal and diamonds, had made the wild suggestion that these two totally different-seeming substances were linked by some kind of hidden analogy. What a thrill it would have been for this chemist to learn one day that diamonds and coal are just two superficially different forms of one single chemical element — namely, carbon!
Low-level and High-level Einsteinian Analogies
Before we step aboard the train of grand Einsteinian analogies, we wish to make it clear that it was not solely in the highest and most exalted flights of his scientific imagination that Albert Einstein resorted to making analogies. Just like every human being, Einstein perceived the world around him by making analogies on many levels, all the way down to the tiniest mental connections.
A rather obvious example of an Einsteinian analogy comes from a reminiscence he wrote, late in life, about why he choose, as a teen-ager, not to become a mathematician: “I saw that mathematics was split up into numerous specialties, each of which could absorb the short lifetime granted to us. Consequently I saw myself in the position of Buridan’s ass, which was unable to decide upon any specific bundle of hay.” The image is amusing, especially when one notes that Albert Einstein is comparing himself to a ravenous but confused donkey that is utterly immobilized because it is surrounded by a number of tempting things to eat, and which, in the end, perishes because it can’t bring itself to make a choice. The analogy isn’t bad — indeed, it has charm — but it certainly didn’t take an Einstein to make it!
And there are smaller and humbler analogies aplenty lurking unnoticed in this same pair of sentences. To take just one, Einstein describes each specialized area of mathematics as potentially “absorbing” one’s entire short lifetime. Any reader will effortlessly conjure up an image of a professional life utterly “sucked up”, “eaten up”, “swallowed up”, or “devoured” by the need to master a vast field of knowledge. Almost surely this choice of a verb by Einstein was not the result of a prolonged linguistic rumination but was made casually and instinctively, as are most word choices when one writes or says anything. One can also point to many other word choices in Einstein’s writings that have a similar flavor — for instance, his choice of the verb “to fall in love” in the sentence “I really began to learn [to play the violin] only when I was about 13 years old, mainly after I had fallen in love with Mozart’s sonatas.”
The point is that all of Einstein’s word choices, whether he was writing, speaking, or just thinking, were likewise the fruit born from myriads of tiny, evanescent analogies, as is the case for all humans. Whenever Albert Einstein saw a shoe and the word “Schuh” came to his mind (or the English word “shoe”, once he had moved to America), or whenever he saw someone ambling down the street and the word “walk” came to his mind — in any such case, he was building an analogical bridge between a current percept and a stored structure that had been built up in his memory, no less than in yours or ours, by untold numbers of previous analogies made all through his lifetime.
To be sure, such tiny, mundane Einsteinian analogies are far from being “analogies that shook the world”, and if a chapter advertised as being largely about analogies made by the great Albert Einstein did nothing but discuss occasions on which he made such forgettable pronouncements as, “Where are my old shoes?” or “Oh, look, there’s my old friend Gödel walking home”, readers would be forgiven for feeling that they had been sold a bill of goods, so to speak. But we felt it important to bring out the fact that even the great Einstein’s thought was utterly pervaded by unspectacular, throwaway, workaday analogies that allowed him to live. But from here on out, when we speak of “Einstein’s analogies”, we will mean his few dozen great analogies, not his millions of mundane ones. With this out of the way, we now head straight for the big game.
A Crazy “Swimming Pool Table” Analogy Emits Quanta of Light
Toward the end of the nineteenth century, one of the great mysteries in physics was that of the famous “blackbody spectrum”. A spectrum is a graph that shows what proportion of the energy carried by a bunch of waves is associated with each different wavelength, while a black body can be thought of as a hollow cavity whose walls are held at a specified temperature, and which is filled with electromagnetic waves that are bouncing against the walls and crisscrossing the empty space — the vacuum — in the cavity, in the manner of ripples crisscrossing on the surface of a swimming pool. The most familiar type of electromagnetic wave is visible light, which includes all the colors of the rainbow, but the notion also includes ultraviolet, infrared, X-rays, gamma rays, microwaves, radio waves, and so forth. The essence of a black body thus resides in the co-presence of electromagnetic waves of many wavelengths, continually bouncing off the walls of the cavity just as ripples reflect off the walls of a swimming pool.
This analogy between a black body and a swimming pool will in fact be very useful to us. If you toss an object — a wedding ring, a bowling ball, a grand piano — into a swimming pool, the splash will create circular waves having various wavelengths, which will then be reflected off the pool’s walls. Roughly speaking, the lighter the object, the daintier will be the waves (both in their amplitude, or height, and in their wavelength). A physicist might well ask, “Given the mass of the object thrown into the water, what will be the dominant wavelength of the waves it causes?” Along the same lines but more sophisticated would be the question, “Given the mass of the thrown object and also some particular wavelength, how much of the total energy of the agitated water do the waves at that wavelength have?” One easily imagines that for each different object tossed in, there will be a specific wavelength that will be dominant in the pool, and that waves having much longer or much shorter wavelengths will hardly be triggered at all by the splash. A graph showing the distribution of energy among different wavelengths would constitute the spectrum of waves on the pool.
Much the same holds for a black body, except that there is no palpable medium that wiggles up and down or back and forth, like the water in the pool; the vacuum through which the electromagnetic waves course, crisscrossing each other all the time, contains no matter (whether solid, liquid, or gas), by the very definition of the word “vacuum”. The waves themselves are ghostlike entities, consisting of electric and magnetic fields oscillating in space in time, which have the ability to make charged particles move. An electromagnetic wave is thus much like a fluctuating gravitational field; it is something that pervades empty space and whose value at any point in space changes according to a “wave equation”, or possibly a set of intertwined equations — in this case, Maxwell’s equations, discovered in the early 1860’s.
The black body’s temperature corresponds, in our analogy, to the size of the object tossed into the pool, which gives rise to the ripples on the surface. The hotter the black body’s walls are kept, the more electromagnetic energy there will be in the cavity, and, perhaps surprisingly, the shorter will be the dominant wavelength of the waves. This may seem counterintuitive, since it contradicts what one would guess on the basis of the swimming-pool analogy. After all, a sofa heaved into a pool will make very long-wavelength ripples, while a pebble will make short-wavelength waves. But we won’t worry about the conflict with the water-wave situation; physicists at the time were well aware of these phenomena in blackbody radiation. They were very familiar with the bell-curve shape of the graph of the blackbody spectrum, as determined through many careful experiments. They simply had no coherent explanation for that shape.
Admittedly, in 1900, the German physicist Max Planck had discovered an elegant mathematical formula that, for any given temperature, exactly reproduced the experimentally found graph of the blackbody spectrum, but at the outset he
had no justification for his formula. It was as if he had pulled it out of his hat, like a rabbit. Then, after several months of obsessive toil, he showed how one could deduce his formula from a strange and arbitrary-seeming assumption about the atoms in the walls of the cavity — namely, that their vibrational energies were limited to taking on just certain special values, determined by a new constant of nature, to which Planck gave the name “h”, which later was dubbed “Planck’s constant”. An atom, instead of being able to vibrate with any arbitrary amount of energy (as all physicists would have expected), could only assume energies that were exact multiples of a certain very small chunk or “quantum” of energy, as Planck called it. The idea of energy coming in such chunks struck physicists, even Planck himself, as very weird; however, everyone realized that Planck’s formula, based on this idea, matched the experimental results perfectly.
For macroscopic creatures such as humans, the fact that the atoms in the walls couldn’t take on arbitrary amounts of vibrational energy would be unnoticeable, as the magnitude of h was so tiny. Indeed, if h were equal to zero, then all possible energies would be allowed and the quantum restriction would not hold. However, the graph that could be calculated from the assumption that h equaled 0 didn’t look at all like the true blackbody spectrum, as found in experiments; contrariwise, Planck’s theoretical graph, which he had calculated using his tiny but non-zero constant, matched the mysterious blackbody spectrum with total accuracy. These developments certainly had a promising feel to them, but for most physicists of the time, Planck’s theory of the “quanta of excitation” of the atoms in the cavity’s walls seemed too arbitrary, and for that reason almost no one took it seriously. Even Planck felt very uncomfortable. Although he had led himself to the quantum pond, he did not want to sip from it.
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