The Best of All Possible Worlds: Mathematics and Destiny

Does nature operate according to an optimization principle, and — if so — can we discover it? The Best of All Possible Worlds takes up those questions and offers an informal history of developments since the time of Galileo. For Galileo and Leibnitz, God was the designer of the world and the laws of nature were simply the rules that God set down when creating the world. The role of science for them was to uncover those rules by observation. Maupertuis took a bold and decisive step when he asserted the principle of least action: “the quantity of action necessary to cause any change in Nature, always is the smallest possible.” Although he initially applied this only to mechanics, Maupertuis soon expanded its scope significantly:

“Once it becomes known that the laws of motion are founded on the principle of the better, no one will doubt that they are due to an all-powerful and all-wise Being, who may have given bodies the power to act upon each other, and who may have used some other way which is even less known to us.”

Leibnitz takes a far more subtle approach, but he arrives at a similar position. Ours is the best of all possible worlds because a perfect being could do no less in his creation. (Leibnitz does not, however, argue that the best of all possible worlds is the best from a human perspective.) Throughout the book, the author chips away at the notion of an optimal world, at least as envisioned by Maupertuis and Leibnitz. He starts with the principle of least action, which Jacobi and Hamilton discovered to be wrong as stated: action is not always minimized, but the action functional is always stationary. Light does not always follow the shortest path when, for example, it is reflected a spherical mirror (and hence does not always minimize action).

The author proceeds to dispose of optimality even in mechanical systems. He uses examples of chaotic motion to argue that disruption of “chains of causality” leads to essentially random motion and disposes of optimality in all but integrable mechanical systems. He points to the introduction of uncertainty in quantum mechanics as yet another indication that there is no operative optimum principle in world of physics. Likewise, neither biology nor human societies show any evidence of optimized behavior.

While the writing is elegant and the author’s erudition is evident, I often had trouble tracking the argument from chapter to chapter. In the end, I think, the question the author asks is ill-posed. What is optimality — really — and by what criteria to we identify it? Why are randomness and uncertainty sufficient arguments against the existence of an optimal world?

All in all, this is an intriguing and stimulating contribution to a centuries-old discussion, but it is hardly the last word.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.