The Magazine

Light on the Subject

How mathematics can explain reflection and refraction.

Oct 6, 2008, Vol. 14, No. 04 • By DAVID GUASPARI
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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.
For that matter, a review may not tell the author he should have chosen to write a different book. It may, however, suggest that he has misunderstood his strengths and, as a result, written a book that is--well, suboptimal.

David Guaspari is a writer in Ithaca.