gregory_benford's picture
Emeritus Professor of Physics and Astronomy, UC-Irvine; Novelist, The Berlin Project
The Intrinsic Beauty and Elegance Of Mathematics Allows It to Describe Nature.

Many believe this seeming axiom, that beauty leads to descriptive power. Our experience seems to show this, mostly from the successes of physics. There is some truth to it, but also some illusion.

There is a ready explanation of how a distant primate came into the beginnings of a mathematical appreciation of nature. Hunting, that primate found it easier to fling rocks or spears at fleeing prey than chase them down. Some of his fellows found the curve of a flung stone difficult to achieve, but he did not. He found the parabola beautiful and simpler to achieve, because that pleasurable sensation provided evolutionary feedback. Over eons this led to an animal that invented complex geometries, calculus and beyond.

This is a huge leap, of course, an evolutionary overshoot. We seem to be smarter than needed simply to survive in the natural world—earlier hominids did, even spreading over most of the planet. We did go through some population bottlenecks in our past, perhaps as recent as about 130,000 years ago. Perhaps those recent eras of intense selection explain why we have such vastly disproportionate mental abilities.

Still there remain, beyond evolutionary arguments, two mysteries in math: whence its amazing ability to describe nature, and why its intrinsic beauty and elegance?

Parabolas are elegant, true. They describe how hard bodies fly through the air under gravity. But the motion of a falling leaf, on the other hand, demands several differential equations taking into account wind velocity, gravity, geometry of the leaf, fluid flow and much else. A cruising airplane is even harder to describe. Neither case is elegant or simple.

So the utility of math stands separately from its intrinsic beauty. Mathematics is most elegant when we simplify the system considered. So with a baseball we account for the initial acceleration and angle, the air and gravity, and out comes a parabola as a good approximation. Not so the leaf.
And that parabola? We see its simple beauty far too slowly to be of any use in real time. Our appreciation comes afterward. To actually make a parabola work for us in baseball, we learn how to throw. Such learning builds on hard-wired neuronal networks in the brain, selected for over evolutionary times, since knowing how to throw a missile is adaptive. A human pitcher can more subtly affect the trajectory by throwing curves, knuckle balls etc. Those are certainly more complex trajectories and probably less elegant, but still well within the capability of our nervous systems. But for well-learned actions, all that processing goes on at unconscious levels. In fact, too much conscious attention to the details of action can interfere. Athletes know this—it's the art of staying in the zone. Probably that zone is where the mind runs on its sense of rightness, beauty, economy of effort.

Further, elegance is hard to define, as are most aesthetic judgments. Richard Feynman once noted that it is simple to make known laws more elegant, say by starting with Newton's force law, F=ma, then defining R=F – ma. The equation R=0 is visually more elegant, but contains no more information. The Lagrangian method in dynamics is elegant—just write the expression for kinetic energy minus the potential energy—but one must know a fundamental theory to do so; the elegance of the Lagrangian comes later, as a mathematical aid.

More recently, it is hard to devise an elegant cosmological theory that yields directly the small cosmological constant we observe. Some solve this problem by invoking the Anthropic Principle, and thus multiverses of some sort. But this ventures near a violation of another form of the elegance standard, Occam's Razor. Imagining a vast sea of multiverses, with us arising in one where conditions produce intelligent beings, seems to many excessive. It invokes a plentitude we can never see. The scientific test of multiverse cosmology is whether it leads to predictable consequences.

Can multiverses converse with each other? That would be a way of verifying the basis of such theories. Most multiverse models seem to say there is no possible communication between the infinitude of multiverses. Brane theory, though, comes from models where no force law operates between branes, except gravitation. Perhaps someday an instrument like LIGO, the Laser Interferometer Gravitational-Wave Observatory, can detect such waves from branes. But is it elegant to shift confirmation onto some far future technology? Sweeping dust under a rug seems inelegant to me.

Evolution doesn't care about beauty and elegance, just utility. Beauty does play a secondary role, though. The male who best throws the spear to bring down prey is appreciated and may have a choice of many mates. It just so happens that the effective and now beautiful act of spear throwing is describable with fairly simple math.
We make the short step to say the underlying math is also beautiful.

Math's utility implies that for a suitably simple model of the universe there should be a fairly simple mathematical theory of everything, something like general relativity, describable by a one-line equation. Searching for it on that intuitive basis may lead us to such a theory. I suspect a model that captures the full complexity of the universe, though, would take up a lot more than one line.

When we say a math model is elegant and beautiful, we express the limits of our own minds. It is not a deep description of the world. In the end, simple models are much easier to comprehend than complex ones. We cannot expect that the path of elegance will always guarantee we are on the right track.