Analogies for Better or for Worse

Douglas Hofstadter wrote about the relationship between analogies and intelligence in the September 1981 installment of his Scientific American column series Metamagical Themas, entitled "Analogies and Roles in Human and Machine Thinking." His central point is that being able to see similarities between different situations and to capitalize on those similarities to make predictions is core to the nature of human intelligence (and by extension, to fruitful research on machine intelligence as well). "Being attuned to vague resemblances," he writes, "is the hallmark of intelligence, for better or for worse."

As if to highlight the "worse" side of the ledger, somewhere toward the middle of the column, he discusses the pitfalls of taking analogies too far. Ultimately, situations don't map perfectly onto each other, and the greater the demands placed on any given analogy, the more likely it will stretch so far it snaps.

Analogies are particularly useful for teaching purposes. Students seem often to learn something better when it is explained in terms of something they already know. We might learn about electrons orbiting an atomic nucleus by analogy with planets orbiting the Sun, for instance. To the extent that principles in one domain apply to the other, we can understand and explain behaviors in the new, unfamiliar domain in terms of the old, familiar one.

There are dangers to this path to learning, though. The famed Caltech physicist Richard Feynman—surely one of the great physics teachers of all time—was extremely conscientious when it came to teaching by analogy. He avoided analogies that he found misleading or circular. It might be natural to think of electromagnetism as being mediated by imaginary "rubber bands," he said, but in the first place, rubber bands draw things together more the further apart they get, whereas electromagnetism gets weaker with distance, and secondly, rubber bands themselves work through electromagnetism interactions at the molecular level, so any understanding students derived through this analogy must needs be circular.

Care must be taken, too, not to stretch the analogy beyond its limits. The fact is that electrons don't orbit the nucleus in neat circles (or even ellipses) like planets orbiting the Sun. If we study further, we find that although planets can apparently orbit the Sun at any distance whatsoever, electrons are constrained to orbit the nucleus only at specific distances, which we can characterize as those distances which allow an integral number of electron waves to circle the nucleus. If we study still further, we find that electrons don't travel in any kind of orbit at all, but instead can be found at any location around the nucleus according to a probability distribution (or, equivalently, are simultaneously at all different points according to that distribution—at least prior to observation).

The problem is that analogies are so darned appealing. The good ones yield correct answers to our questions so often that we lose track of where the limits of the analogies are, or even that there are any. We simply trust the analogies, often to our detriment. It's tempting to understand the budgetary situation of, say, the United States in relation to our personal budget; after all, there are many similar concepts and relationships: income, expenses, debt, balance, and so forth. It's tempting, but it's often misleading. But because we do understand many things correctly using that analogy, we become overconfident in areas where the analogy was never going to hold water.

My pet peeve in this regard is the rubber sheet analogy for general relativity. Given that general relativity was one of the major developments of 20th-century physics, you'd expect that there'd be significant time spent in explaining it to the lay public. I mean, even people who only vaguely have a notion of what physics is about have heard of Albert Einstein and "warped space."

Gravity is everywhere; we feel its effects all the time. And we've sort of internalized the Newtonian theory of gravity, which is that any two particles exert a gravitational force on each other, no matter how far apart they are; although the degree of force drops off quite rapidly with distance, it never quite shrinks down to zero. We've internalized it so well that we hardly ever wonder how that force is mediated. How does that force get exerted across all that distance? By the Newtonian theory, I wiggle my finger here, and my finger's gravitational influence on the most distant galaxy, however faint, oscillates with the same frequency as my wiggling finger. Newton himself felt this conundrum most keenly, never mind his insistence that he did not "feign hypotheses."

Einstein's general theory of relativity ostensibly resolves all of that. It posits space not simply as a theater in which gravitational interactions take place, but a physical, almost tangible thing that is affected by masses and in turn affects them. The usual term for this is curved space—a term that is justified in a mathematical sense but which is almost certain to mean nothing directly to anyone who isn't already a physicist. I imagine that the most common response is mute incomprehension.

So we explain what we mean by "curved space" by analogy. First of all, we should really be calling it "curved space-time," since in Einstein's theory time and space are interwoven almost irrevocably. With three dimensions of space and one of time—well, that's a lot of dimensions. People don't visualize four dimensions very well. So we abstract away two of them: one of the spatial dimensions, and the one time dimension, leaving two spatial dimensions. The one spatial dimension is OK, probably, but already there are problems. You've lost the one temporal dimension you have; it's possible that you might lose something essential there!

But we're pressing on. We lay down an infinite rubber sheet, typically marked with grid lines. We plop down a big heavy ball, like a bowling ball. This is the Sun, we are told. It bends or curves or warps space. Sure enough, the rubber sheet is seen to dimple significantly. Then, we roll a smaller ball around the bowling ball, and because of the warping caused by the bowling ball—err, Sun—the smaller ball (representing the Earth, say) sweeps around in a neat circular or elliptical orbit. Just like the real planets.

This is an enormously popular representation of general relativity; even Carl Sagan's Cosmos, my favorite science documentary series of all time, uses it. And yet, in my opinion, it's fatally flawed. In the first place, it's circular, just like Feynman's rubber bands. We're told that the effect of the Sun's gravity can be interpreted in terms of the Sun's warping of nearby space, by analogy with the warping of the rubber sheet caused by the bowling ball. But what is it that causes the bowling ball to warp the rubber sheet? Gravity itself! We can't rightly claim to understand gravity if gravity is involved in the explanation as well.

Even that would be excusable for pedagogical purposes if the analogy were actually accurate. But it's not. In all of the rubber-sheet depictions of general relativity I've seen, and I've seen quite a few, only one includes a disclaimer that demonstrates what's wrong with it—a little-known primer on relativity written by Lewis Carroll Epstein called, appropriately enough, Relativity Visualized. (I heartily recommend it.) He makes the following point: In space, there is no universally preferred direction up or down; those directions are only understood in reference to some gravitational field. So the rubber sheet analogy, if it's really right, should work just as well if you flip the rubber sheet upside down, so that the warp goes upward (like a volcano) rather than downward. After all, it's not supposed to be the bowling ball itself that makes the other ball go 'round, but the warp. But if you roll the smaller ball toward the volcano, what happens? As any miniature golfer knows, it certainly doesn't orbit the volcano; instead, it either goes into the volcano, or it veers away from it, never to return.

But even that's not the worst of it. The irony of this analogy is that even though it's not a very accurate depiction of general relativity, it's a dead-on match for Newtonian potential energy wells. That's right: This immensely popular analogy, which is supposed to highlight how general relativity differs from Newtonian gravity, is instead a much better illustration of the very theory general relativity was intended to supplant! I was so struck by this that I wrote up an exposition of general relativity for my astronomy Web site, which (on the off chance you've actually read this far) you can find here. In it, you'll find an analogy to general relativity which is hopefully understandable but hits much closer to the mark. (I even asked a physicist!)

But does anyone care? Nooooo, I'm sure we'll continue to see the rubber-sheet analogy trotted out at regular intervals on the Discovery Channel, with no disclaimer regarding its appropriateness.