Our fearful symmetry
Yesterday, I spent an hour fixing my garage door, which got stuck halfway up and refused to budge. I went about it in the way I usually approach such household tasks: I took a flashlight and a pair of vise-grips and stared at it for a while. In this case, for once, it worked, even if I’m only postponing the inevitable service call. But it wouldn’t have occurred to me to tackle it myself in the first place—when I probably wouldn’t have tried to fix, say, my own car by trial and error—if it hadn’t been for two factors. The first is that the workings were all pretty visible. On each side, there’s basically just a torsion spring, a steel cable, and two pulleys, all of it exposed to plain sight. The second point, which was even more crucial, is that a garage door is symmetrical, and only one side was giving me trouble. Whenever I wasn’t sure how the result should look, I just had to look at the other half and mentally reflect it to its mirror image. It reminded me of how useful symmetry can be in addressing many problems, as George Pólya notes in How to Solve It:
Symmetry, in a general sense, is important for our subject. If a problem is symmetric in some ways we may derive some profit from noticing its interchangeable parts and it often pays to treat those parts which play the same role in the same fashion…Symmetry may also be useful in checking results.
And if I hadn’t been able to check my work along the way using the other side of the door, I doubt I would have attempted to fix it at all.
But it also points at a subtle bias in the way we pick our problems. There’s no question that symmetry plays an important role in the world around us, and it provides a solid foundation for the notion that we can use beauty or elegance as an investigative tool. “It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has a really sound insight, one is on a sure line of progress,” Paul Dirac famously said, and Murray Gell-Mann gave the best explanation I’ve ever found of why this might be true:
There’s a quotation from Newton, I don’t remember the exact words but lots of other physicists have made the same remark since—that nature seems to have a remarkable property of self-similarity. The laws—the fundamental laws—at different levels seem to resemble one another. And that’s probably what accounts for the possibility of using elegance as a criterion [in science]. We develop a mathematical formula, say, for describing something at a particular level, and then we go to a deeper level and find that in terms of mathematics, the equations at the deeper level are beautifully equivalent. Which means that we’ve found an appropriate formula.
Gell-Mann concludes: “And that takes the human being, human judgment, out of it a little. You might object that after all we are the ones who say what elegance is. But I don’t think that’s the point.”
He’s right, of course, and there are plenty of fields in which symmetry and self-similarity are valuable criteria. Yet there’s also a sense in which we’re drawn to problems in which such structures appear, while neglecting those that aren’t as amenable to symmetrical thinking. Just as I was willing to take apart my garage door when I wouldn’t have done the same with my car—which, after all, has a perfectly logical design—it’s natural for us to prefer problems that are obviously symmetrical or that hold out the promise of elegance, much as we’re attracted to the same qualities in the human face. But there are plenty of important questions that aren’t elegant at all. I’m reminded of what Max Perutz, who described the structure of hemoglobin, said about the work of his more famous colleague James Watson:
I sometimes envied Jim. My own problem took thousands of hours of hard work, measurements, calculations. I often thought that there must be some way to cut through it—that there must be, if only I could see it, an elegant solution. There wasn’t any. For Jim’s there was an elegant solution, which is what I admired. He found it partly because he never made the mistake of confusing hard work with hard thinking; he always refused to substitute one for the other.
In The Eighth Day of Creation, Horace Freehand Judson calls this “the most exact yet generous compliment I have ever heard from one scientist to another.” But there’s also a wistful acknowledgement of the luck of the draw. Both Perutz and Watson were working on problems of enormous importance, but only one of them had an elegant solution, and there was no way of knowing in advance which one it would be.
Given the choice, I suspect that most of us would prefer to work on problems that exhibit some degree of symmetry: they’re elegant, intuitive, and satisfying. In the absence of that kind of order, we’re left with what Perutz calls “thousands of hours of hard work, measurements, calculations,” and it isn’t pretty. (As Donald Knuth says in a somewhat different context: “Without any underlying symmetry properties, the job of proving interesting results becomes extremely unpleasant.”) When we extrapolate this preference to the culture as a whole, it leads to two troubling tendencies. One is to prioritize problems that lend themselves to this sort of attack, while overlooking whole fields of messier, asymmetrical phenomena that resist elegant analysis—to the point where we might even deny that they’re worth studying at all. The other is to invent a symmetry that isn’t there. You can see both impulses at work in the social sciences, which tend to deal with problems that can’t be reduced to a series of equations, and they’re particularly insidious in economics, which is uniquely vulnerable to elegant models that confirm what existing interest groups want to hear. From there, it’s only a small step to more frightening forms of fake symmetry, as Borges writes in “Tlön, Uqbar, Orbis Tertius”: “Any symmetry with a resemblance of order—dialectical materialism, anti-Semitism, Nazism—was sufficient to entrance the minds of men.” And the first habit has a way of leading to the second. The more we seek out problems with symmetry while passing over those that lack it, the more likely we become to attribute false symmetries to the world around us. Symmetry, by definition, is a beautiful thing. But it can also turn us into suckers for a pretty face.