We all have an intuitive sense of what "simplicity" means. In science, the word is often used as a term of praise. We expect that simple explanations are more natural, sounder, and more reliable than complicated ones. We abhor epicycles, or long lists of exceptions and special cases.
But can we take a crucial step further, to refine our intuitions about simplicity into precise, scientific concepts? Is there a simple core to "simplicity"? Is simplicity something we can quantify, and measure?
When I think about big philosophical questions, which I probably do more than is good for me, one of my favorite techniques is to try to frame the question in terms that could make sense to a computer. Usually it's a method of destruction: It forces you to be clear, and once you dissipate the fog you discover that very little of your big philosophical question remains. Here, however, in coming to grips with the nature of simplicity, the technique proved creative, for it led me straight toward a (simple) profound idea in the mathematical theory of information, the idea of description length. (The idea goes by several different names in the scientific literature, including algorithmic entropy and Kolmogorov-Smirnov-Chaitin complexity. Naturally, I chose the simplest one.)
Description length is actually a measure of complexity, but for our purposes that's just as good, since we can define simplicity as the opposite—or, numerically, the negative—of complexity. To ask a computer how complex something is, we have to present that "something" in a form the computer can deal with, that is as a data file, i.e. a string of 0s and 1s. That's hardly a crippling constraint: We know that data files can represent movies, for example, so we can ask how about the simplicity of anything we can present in a movie; since our movie might be a movie recording scientific observations or experiments, we can ask about the simplicity of a scientific explanation.
Interesting data files might be very big, of course. But big files need not be genuinely complex; for example, a file containing trillions of 0s and nothing else isn't genuinely complex. The idea of description length is, simply, that a file is only as complicated as its simplest description. Or, to put it in terms a computer could relate to: A file is as complicated as the shortest program that can produce it from scratch. This defines a precise, widely applicable, numerical measure of simplicity.
An impressive virtue of this notion of simplicity is that illumines and connects up other attractive, successful ideas. Consider, for example, the method of theoretical physics. In theoretical physics, we try to summarize the results of a vast number of observations and experiments in terms of a few powerful laws. We strive, in other words, to produce the shortest possible program that outputs the world. In that precise sense, it's a quest for simplicity.
It's appropriate to add that symmetry, a central feature of the physicist's laws, is a powerful simplicity enabler. For example, if we work with laws that have symmetry under space and time translation—in other words, laws that apply uniformly, everywhere and everywhen—then we don't need to spell out new laws for distant parts of the universe or for different historical epochs, and we can keep our world-program short.
Simplicity leads to depth: For a short program to unfold into rich consequences, it must support long chains of logic and calculation, which are the essence of depth.
Simplicity leads to elegance: The shortest programs will contain nothing gratuitous. Every bit will play a role, for otherwise we could expunge it, and make the program shorter. And the different parts will have to function together smoothly, in order to make a lot from a little. Few processes are more elegant, I think, than the construction, following the program of DNA, of a baby from a fertilized egg.
Simplicity leads to beauty: For it leads, as we've seen, to symmetry, which is an aspect of beauty. As, for that matter, are depth and elegance.
Thus simplicity, properly understood, explains what it is that makes a good explanation deep, elegant, and beautiful.