Light on the Subject
How mathematics can explain reflection and refraction.
Oct 6, 2008, Vol. 14, No. 04 • By DAVID GUASPARI
The Best of All Possible Worlds
The dust jacket of The Best of All Possible Worlds calls it "a journey through scientific attempts to envision the best of all possible worlds." Its more modest Introduction promises an account of optimization from its origins in mathematical physics to applications in biology and the social sciences. And that subject is introduced with a lucid and leisurely account of important ideas in classical mechanics: the physics, founded by Galileo and Newton, that describes spinning tops and bouncing billiard balls, projectiles, and planets.
Exposition then stops, and the carefully laid-out mathematics serves only as a peg on which the author can hang his views on a variety of large subjects--war, evolution, the common good, global warming. Those views are conventional, and it's hard to see the concluding op-ed style ruminations as anything but missed opportunities.
Ekeland centers his mathematical story on attempts to explain Snell's Law, which characterizes the way in which light refracts (bends) as it passes from air to water--or, more generally, from one medium to another. Refraction causes a stick dipped into a pond to appear bent at the water's surface. Snell's Law was established empirically, but Descartes showed that it could be derived from two assumptions: That one of its terms, called the refractive index, denotes the relative speeds of light in air and water, and that light--like sound--travels faster in water than in air.
The great mathematician Fermat also approached the problem from first principles, in a very bold and very different way, by formulating a single principle to explain both refraction and reflection. The law of reflection had been known since ancient times: The angle at which light strikes a mirror equals that at which it is reflected. That law is mathematically equivalent to saying that, in traveling from the original to the mirror to the beholder, a light ray uses the shortest possible path. And when one of Fermat's correspondents proposed that this expressed nature's economy, Fermat made something of the suggestion. He generalized the "least path" principle by proposing that a light ray passing through both water and air would follow the path that took the least possible time. The law of reflection follows immediately, and Fermat showed that Snell's Law would also follow if one of Descartes's assumptions were reversed, by supposing that light travels faster in air than in water.
It took two centuries to determine the facts about the speed of light. (Fermat's assumption was the correct one.) Meanwhile, the success of his "least time" principle presented a philosophical and scientific puzzle. It seemed to attribute purposive, predictive, reasoning abilities to inanimate nature, as if a light ray sniffed out all possible paths and then chose the one by which it would reach its destination most quickly.
Fermat argued that the philosophical question could be sidestepped: Since the principle yielded correct predictions, further speculation was unnecessary. Pierre Moreau de Maupertuis--scholar, courtier, and scientist--embraced the suggestion of purpose, which he attributed not to light rays but to the God who created both the light and its governing laws. He further generalized Fermat's principle by defining a quantity called "action," and proposing that all natural phenomena could be deduced from the single principle that nature acts so as to minimize its expenditure of action--an elegant behavior testifying to the care and wisdom of nature's Creator.
Many leading scientists dismissed the suggestion of teleology and purpose behind this "principle of least action" but fruitfully pursued its underlying mathematics. They developed methods to pose and attack a large class of optimization problems. An optimization problem asks us to determine, among all possible solutions to a set of constraints, one that maximizes or minimizes a chosen quantity: time, distance, profit, and so on. Here is an example: Find the shortest path that connects two given points and lies on some given surface. On a flat surface it will be a straight line, on the surface of a sphere some segment of a great circle, and so on.
Ekeland follows this thread through key developments of classical mechanics. The principle of least action eventually took its place not as a fundamental law but as a useful consequence of laws that are more basic. It lost its metaphysical appeal when its proper formulation was discovered: Nature does not always minimize the expenditure of action, but acts to satisfy a more technical and less demanding criterion less suggestive of divine perfection.