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.
Ekeland's discussion culminates with an excellent account of "chaos" that will take a layman well beyond the famous sound bite that, because weather is chaotic, a butterfly's flutter in China could cause a tornado in Kansas. Ekeland makes clear the difference between a system that is deterministic--its state at any moment uniquely determines its entire future--and one that is predictable, a distinction that would have surprised the founders of our physics. Classical mechanics provides equations of motion that determine how a mechanical system will evolve, but that evolution will not be predictable unless those equations can, in a precise technical sense, be "solved."
The technical term for "solvable" is "integrable." If a system's equations of motion are integrable we can, from a description of its state at any moment, estimate its state a short time later--and can predict more distant futures by repeating that procedure. Although the initial measurements may contain errors, and errors may be introduced each time we take another step into the future, those errors are nonetheless manageable and the procedure can yield accurate predictions.
But even idealized descriptions of real-world systems are rarely integrable: We cannot manage the accumulation of errors, and that can drastically limit how far ahead we are able to see. The ubiquity of non‑integrable systems forms part of Ekeland's argument that we are unlikely to find a single principle or single set of laws that accounts for all physical phenomena--a truth he clearly regards as applicable well beyond the bounds of physics.
The physics is worth the price of admission, but Best of All Possible Worlds has two weaknesses deserving note. One seems endemic to popular books on science, if not to human nature: Using other people's ideas as foils to (or anticipations of) one's own, but not taking them seriously enough to get them right. Ekeland cites, for example, the Meno, a Platonic dialogue about teaching and learning. It portrays, as usual, a character called Socrates disposed to arguing and telling stories; but to identify particular views of the dramatic character Socrates with those of Plato, or to take his stories as straightforward assertions of belief, requires a leap--one that is often questionable.
In the Meno Socrates poses questions that guide a slave boy to solve a geometrical problem, and then asks how the boy could have learned the answer, which he hadn't known and Socrates hadn't told him. Faced with this commonplace, but near-miraculous event, Socrates offers a story: We have all had a previous existence in which we saw truth directly, and when we seem to be acquiring knowledge we are, in fact, recalling it. Ekeland takes that myth at face value and asserts that, according to "the Platonic tradition" (as if there were just one), all learning is recollection. This suits his expository purposes, but does not provide a reliable guide to Plato.
Then come the op-eds. "The Common Good" uses the preceding mathematics lesson as a segue: If you wish to regulate society scientifically by solving an optimization problem you would need to define the quantity to be optimized, something like "the common good." So we're off on a package tour--two paragraphs on utilitarianism, one on John Rawls, a bit of Rousseau, etc.--wending to the unimpeachable but unsurprising insight that an agreed-upon definition of the common good would be awfully hard to come by. The gist of "May the Best One Win" has been well said in many an essay by the late Stephen Jay Gould, whose influence Ekeland acknowledges: Evolution cannot be modeled as an optimization problem, either, since there is no universal measure of "fitness" to be optimized.
Ekeland sprinkles these conventional views with faculty lounge sneers, such as "the basic claim to superiority of Western civilization" is that "we have an overwhelming military advantage, which enables us to take the land and resources away from others and put it to our own use." How's that for coming to grips with opinions other than one's own? Of course, the point of such one-liners is not to assert an intelligible proposition but to advertise that one is the right sort of chap.
David Guaspari is a writer in Ithaca.