Prologue: The Rolling of the Golden Apple
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In 1776, a year in which political rebels in Philadelphia were proclaiming their independence and freedom, a physicist in Europe was proclaiming total dependence and determinism. According to Pierre-Simon Laplace, if you knew the initial conditions of any situation, you could determine the future far in advance: "The present state of the system of nature is evidently a consequence of what it was in the preceding moment, and if we conceive of an intelligence which at a given instant comprehends all the relations of the entities of this universe, it could state the respective positions, motions, and general effects of all these entities at any time in the past or future."
The Laplacian universe is just a giant pool table. If you know where the balls were, and you hit and bank them correctly, the right ball will always go into the intended pocket.
Laplace's hubris in his ability (or that of his "intelligence") to forecast the future was completely consistent with the equations and point of view of classical mechanics. Laplace had not encountered nonequilibrium thermodynamics, quantum physics, or chaos. Today some people are frightened by the very notion of chaos. (I have explored this at length in an essay devoted to chaos from a philosophical perspective. But the same is also true with respect to the somewhat related mathematical notion of chaos.) Today there is no justification for a Laplacian point of view.
At the beginning of this century, the mathematician Henri Poincaré, who was studying planetary motion, began to get an inkling of the basic problem:
"It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible" (1903).
In other words, he began to realize "deterministic" isn’t what it’s often cracked up to be, even leaving aside the possibility of other, nondeterministic systems. An engineer might say to himself: "I know where a system is now. I know the location of this (planet, spaceship, automobile, fulcrum, molecule) almost precisely. Therefore I can predict its position X days in the future with a margin of error precisely related to the error in my initial observations."
Yeah. Well, that’s not saying much. The prediction error may explode off to infinity at an exponential rate (read the discussion of Lyapunov exponents later). Even God couldn’t deal with the margin of error, if the system is chaotic. (There is no omniscience. Sorry.) And it gets even worse, if the system is nondeterministic.
The distant future? You’ll know it when you see it, and that’s the first time you’ll have a clue. (This statement will be slightly modified when we discuss a system’s global properties.) (
continued)
