From Fermat to Lenny Bernstein and Beyond

June 12, 2007

Book Review
Philip J. Davis

The Best of All Possible Worlds: Mathematics and Destiny. By Ivar Ekeland, University of Chicago Press, Chicago, 2006, 208 pages, $25.00.

Optimist: This is the best of all possible worlds.
Pessimist: I'm sorry that I have to agree with you.
---Old Joke

The book under review has two themes: The first relates to the rise and fall of the principle of least action in mechanics/dynamics, and its subsequent morphing into generalized optimization. Brought in to develop this first theme are such great names as Kepler, Galileo, Fermat, Newton, Leibniz, Maupertuis, Euler, Lagrange, Hamilton, Jacobi, Poincaré, Einstein, von Neumann. The second theme concerns morals, ethics, intent, purpose, metaphysics, and theological assertions. Here, in addition to some of the persons just mentioned, the author brings in Machiavelli, Guicciardini, Pascal, Voltaire, Ernst Mach, Robert Musil, Karl Popper, John Rawls, not to forget the playwright Bertolt Brecht.

These two themes constitute what the late evolutionist Stephen Jay Gould referred to as "two magisteria," suggesting that the two had little business overlapping and messing with each other; they ought, rather, to live peacefully, side by side, in a non-aggression pact. Alas, or fortunately---depending on one's point of view---these two themes do and always will overlap.

Ivar Ekeland, a professor of mathematics and economics at the University of British Columbia and a prolific writer of both technical and popular material, shows in this latest book how, over the years, the two themes have become interwoven. Ekeland is a great storyteller, and his book is a popularization--meaning that it has no equations. As he has no intention of leaving the reader with the abstract bones of the matter, there is in the book science, mathematics, history, personality, jealousy, conflicts of fact and opinion, horrors (Poincaré went into print with an erroneous result and proof for the three-body problem), and, at the very end, the suggestion or hope that this so-called best of all possible worlds, even in the sorry situation in which it finds itself, can be straightened out and put on the right path by a new infusion of rationalism.

Galileo (1564–1642) had his pendulum, his inclined plane, and his telescope, and heated up the conflict in the mix between what is found by observation and what is revealed in the Bible. Kepler (1571–1630) gave us his three planetary laws. Fermat (1601–1665), studying the refraction of light, showed that light reaches its destination in the speediest manner possible. At once, a certain Claude Clerselier (1614–1684) raised an objection to Fermat's demonstration:

"The principle on which you build your proof, namely that nature always acts by the shortest and simplest ways, is but a moral principle, not a physical one. . . ."

On which Ekeland comments:

"There is no awareness in nature says Clerselier. Attributing to nature any sense of purpose, suggesting for instance that it is trying to minimize some transition time, is not a scientific explanation. . . . Nature acts ‘without forethought, without choice.' It does not look ahead and it is never faced with choices."

Fermat rebutted with what has become a standard rhetorical ploy--say in quantum theory: Forget deeper meaning; what we've put forward agrees with experiment to seven decimal places. Newton (1643–1727) gave us his laws of motion and showed how they implied what Kepler had found.

Mathematical science is strongly motivated by the desire to condense, to find simple, universal principles that govern everything, and asserts, as an article of faith, that such a reduction can be found. Pierre-Louis Moreau de Maupertuis (1698–1759), an under-rated or at least under-reported polymath, stepped onto the stage and derived from Newton's equations the principle of "least action"---"action" meaning not steps taken by governments or individuals, but rather the line integral over a trajectory of the momentum (i.e., the product of mass and velocity). Maupertuis then went on to assert as a moral principle that, in running the universe, God saves on "action" to the extent possible. This leads without too much mind bending to the assertion that "this is the best of all possible worlds," an assertion of "intelligent design." Maupertuis' statement is so beautifully expressed, so sweeping in its generality, that it is worth noting here:

"The laws of movement thus deduced [from the principle of least action], being found to be precisely the same as those observed in nature, we can admire the application of it to all phenomena, in the movement of animals, in the vegetation of plants, in the revolution of the heavenly bodies: and the spectacle of the universe becomes so much the grander, so much the more beautiful, so much more worthy of its Author. . . . These laws, so beautiful and so simple, are perhaps the only ones which the Creator and Organizer of things has established in matter in order to effect all the phenomena of the visible world."

Providing comic relief in a subplot, a certain Johann put forward the claim that Leibniz (1646–1716) was the first to arrive with an assertion of minimal action. The claim was backed by a letter purportedly written by Leibniz, but charged to be a forgery. In any case, mathematics apart, Leibniz was known for his optimistic view, arrived at on moral grounds, that in some sense our world is the best that God could have made. This led to the satirization of both Leibniz and Maupertuis by Voltaire (1694–1778) in his famous Candide and in more recent times (1956) to Leonard Bernstein's smash Broadway musical of the same name.

Euler (1707–1783) and Lagrange (1736–1813) introduced generalized coordinates and rewrote least action in those terms. Euler, who was a devout Calvinist, agreed more or less with the theology of Maupertuis. D'Alembert (1717–1783) disagreed, and Lagrange, busy working on his Analytical Mechanics, a theory that needed no geometry whatsoever, stayed away from the issue. Après Lagrange, mechanics could be taught without a single mention of the Deity or, indeed, with the exception of the names of a few authors, without a single mention of humans and their role in the universe. A famous anecdote had Laplace assuring an incredulous Napoleon on this point.

But humankind, some of our most creative mathematicians and physicists, philosophers and moralists, have been loathe to let it go at that. They have derived from these equations various consequences about philosophical, metaphysical, eschatological questions: the problems of free will and of why evil exists.

Hamilton (1805–1865) and Jacobi (1804–1851) came on stage pointing out that Euler's mathematics wasn't exactly correct. They reformulated the principle of least action more correctly, as a principle of stationarity rather than as a minimum principle. Stationarity implying a morality much less punchy than minimality, theology was halfway out the window. At this point, in the middle of the 19th century, the positivism implied by the French Enlightenment was taking hold. The rhetoric about God was tamped down. The physicist/philosopher Ernst Mach (1838–1916) summed up the principle this way (in Eke-land's reformulation):

"The least action principle tells us nothing; except that the universe is deterministic, that is, the motion is uniquely determined by its initial conditions."

Yet the spirit of minimization is so attractive that it is not easily exorcised. Here again is Mach, recalling "Ockham's Razor":

"Science itself can be considered as a minimum problem, consisting in accounting for facts, as perfectly as possible, at the smallest intellectual expense."

Henri Poincaré (1854–1912), struggling with the limits of computation, stressed "soft qualitative" topological methods as opposed to the "hard quantitative." He employed the stationary action principle to show the existence of closed trajectories of non-integrable dynamical systems. Today, the principle is simply one possible representation of the equations of motion, to be used or not at your convenience. Thus, Stephen Weinberg, in his Quantum Theory of Fields (1995), notes that the dynamical Lagrangian formalism becomes the principle of stationary action and is the natural framework for the implementation of symmetry principles.

But enough along historic lines; readers are invited to look into Ekeland's splendid book and see how optimality or stationarity diffuses into game theory, biology, evolution, chance . . . .

The principle of least action has been totally defrocked. "The grandiose views of Maupertuis have been laid to rest," Ekeland writes. What remains is the strong underlying desire---even the strong necessity---to moralize. The final three chapters of the book constitute a lamentation on the present sorry state of the world and thoughts as to what can be done about it.

"Nature is indifferent," Ekeland writes. The laws of physics do not tell us how to behave, and "we are at the mercy of our own malice." We should strive, presumably, for the elusive notion known as "the common good," but how is it to be defined and implemented? We should optimize some "quantity" or thing according to some criterion. And there's the rub, for Dick's criterion may contradict Charlie's. Ayn Rand (as I interpret her) says: Maximize selfishness (the philosophy of "objectivism"). Eastern philosophies say: Minimize your desires.

Ekeland's last chapter is a plea for a critical rationalism, the strength of which he asserts has not yet been exhausted. But what is rationalism, and what are its difficulties? Struggling with this question, he quotes, among others, Pareto, Lévi-Strauss, Karl Popper, Chomsky, and Robert Musil, and arrives at a view that might have been expressed by the logical positivists in one of their more tender moments. Ekeland hopes for a surge in rationality at a time when rationality finds itself crushed within the clash of criteria.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at [email protected].

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