The Best of All Possible Worlds: Mathematics and Destiny by Ivar Ekeland, University of Chicago Press, $25
History has not been kind to Pierre-Louis Maupertuis. Voltaire ruined his reputation among his 18th-century contemporaries by skewering him twice, first in the figure of the inane optimist Dr. Pangloss in Candide and then as an incompetent plagiarist of Gottfried Wilhelm Leibniz in The Story of Doctor Akakia and the Native of Saint-Malo. Now, Ivar Ekeland, an accomplished mathematician and historian of math, calls forth Maupertuis’s spirit again, only to banish it the more thoroughly in the context of a broad meditation on the idea that the world we live in might in some sense be optimal.
Maupertuis’s major contribution to mathematics is the least-action principle, which he arrived at erroneously, named badly, and failed to apply helpfully. The basic idea is that a body in motion will follow the path that minimizes what Maupertuis called the “quantity of action.” The exact nature of this quantity, which relates to the mass and velocity of the object and the distance it travels, is difficult to grasp intuitively, but the metaphysical implications he drew from it are easy to follow. If physical objects act to conserve a quantity, natural laws seem designed to be thrifty. Maupertuis boldly argued that this was direct evidence of the hand of God in the design of the universe and that the same principle writ large suggests that He has also designed the world to be optimal in a moral sense. In short, we must be living in the best of all possible worlds.
By reading metaphysical assumptions into science, Maupertuis fell victim to a philosophical fallacy common among Renaissance mathematicians. The intellectual milieu of the time was not so far removed from the medieval worldview in which Nature was held to be the book of God’s work and inquiry into its perfection was a form of worship. The classical model of the pursuit of science, though inspirational in many respects, was scarcely less metaphorical. Plato conceived science to be the recognition of truths first encountered before birth in the ideal world of the forms (the Greek theoreia literally means “things which have been seen”), a semisecular version of the idea that natural laws reflect perfection of design.
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In this context, and into the modern era, greater thinkers than Maupertuis have gazed deep into nature and found purpose in it. Descartes believed he could deduce physical properties from the fact of God’s perfection; Einstein rejected quantum uncertainty on the theory that He does not play dice. They all turned out to be wrong: subatomic particles do behave randomly; light actually moves faster in air than in water, contrary to Descartes’ deduction; and nature does not in fact act to minimize the quantity of action—although it does sometimes act to hold it constant, and this modified version of Maupertuis’s principle was used to good effect when applied more empirically by Henri Poincaré in the 20th century.
As Ekeland moves forward in time, the scientific method gains ascendancy, and his discussion moves away from found optimality (or the lack thereof) and toward the question of active optimization. Here Maupertuis is, in a small way, redeemed, for he appears as the inventor of optimization problems that seek to use equations to maximize or minimize a certain quantity. The world turns out not to be such a problem, but the form has been employed to great effect on a smaller scale. For example, Newton designed an ideal bullet by solving for minimal air resistance, and the field of operations research is founded on optimization theory. The obvious question is whether, having stripped optimization of its metaphysical weight, we can usefully apply it to the ancient philosophical problem of how best to arrange society.
In a brisk, wide-ranging final chapter, which ultimately takes on too much, Ekeland considers and rejects a variety of possible quantities that we might attempt to maximize in order to perfect the human world. Economic efficiency is frighteningly silent on the question of distributive justice; utilitarian social value runs into the problems of asymmetry of information and corruption; and individual preference is limited by the Arrow impossibility theorem, which states that such preferences cannot be aggregated in a perfectly coherent way that is also immune to manipulation.
In the end, Ekeland determines that, although some kind of equilibrium is desirable for consistency’s sake, optimization theory cannot help us choose among equilibriums because there is at present no way to define the common good as a maximizable quantity. He nevertheless concludes with a surprising optimism grounded in an almost mystical idea of rationality. By embracing the scientific method deeply, actively, and patiently, Ekeland asserts, we will eventually arrive at entirely new moral conceptions. He closes the books with this quote from Robert Musil:
The truth is that science has developed the idea of a raw and sober intellectual power which makes mankind’s old metaphysical and moral representations simply unbearable, even though it can put in its place no more than a hope: that some day will come, in a long time, when a race of intellectual conquerors will settle in the valleys of spiritual abundance.
How far this vision really is from Maupertuis’s is open to debate.
Ekeland’s broad range of reference includes Galileo’s establishment of time and motion as measurable, scientific concepts; the philosophy of history set out by Thucydides and Francesco Guicciardini; and the fragile, non-optimal equilibriums described by game theory. It is impossible for Ekeland to do justice to all this in the span of his brief book. (This volume is less focused and ultimately less compelling than his earlier books, Mathematics and the Unexpected and The Broken Dice.) But if the book’s discursive style is sometimes repetitive and occasionally leads into too-familiar territory (the idea that evolution could somehow lead to better species is by now a straw man), The Best of All Possible Worlds shows throughout the hand of an intelligent, erudite, and witty power. Maupertuis would think he had found God.
