The Stochastic Man

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Causality is a decent, honorable principle, but it doesn’t have all the answers. If we want to make sense of things, we have to move on beyond it. We have to recognize that many important phenomena refuse to be packed into neat casual packages but can be interpreted only by stochastic methods.

A system in which events occur according to a law of probability but aren’t individually determined in accordance with the principle of causality is a stochastic system. The daily rising of the sun isn’t a stochastic event; it’s inflexibly and invariably determined by the relative positions of the earth and the sun in the heavens, and once we understand the causal mechanism there’s no risk in predicting that the sun will rise tomorrow and the next day and the next. We can even predict the exact time of sunrise, and we don’t guess it, we know it in advance. The tendency of water to flow downhill isn’t a stochastic event either; it’s a function of gravitational attraction, which we hold to be a constant. But there are many areas where causality fails us and stochasticity must come to our rescue.

For instance we’re unable to predict the movements of any one molecule in a liter of oxygen, but with some understanding of kinetic theory we can confidently anticipate the behavior of the whole liter. We have no way of foretelling when a particular uranium atom will undergo radioactive decay, but we can calculate quite accurately how many atoms in a block of U-235 will disintegrate in the next ten thousand years. We don’t know what the next spin of the roulette wheel will bring, but the house has a good idea of what its take is likely to be over the course of a long evening. All sorts of processes, however unpredictable they may seem on a minute-to-minute or case-by-case basis, are predictable by stochastic techniques.

Stochastic.According to the Oxford English Dictionary this word was coined in 1662 and is now rare or obs. Don’t believe it. It’s the OED that’s obs., not stochastic, which gets less obs. every day. The word is from the Greek, originally meaning “target” or “point of aim,” from which the Greeks derived a word meaning “to aim at a mark,” and, by metaphorical extension, “to reflect, to think.” It came into English first as a fancy way of saying “pertaining to guesswork,” as in Whitefoot’s remark about Sir Thomas Browne in 1712: “Tho’ he were no prophet … yet in that faculty which comes nearest it, he excelled, i.e., the stochastick, wherein he was seldom mistaken, as to future events.”

In the immortal words of Ralph Cudworth (1617-1688), “There is need and use of this stochastical judging and opinion concerning truth and falsehood in human life.” Those whose way of life is truly governed by the stochastic philosophy are prudent and judicious, and tend never to generalize from a skimpy sample. As Jacques Bernoulli demonstrated early in the eighteenth century, an isolated event is no harbinger of anything, but the greater your sampling the more likely you are to guess the true distribution of phenomena within your sample.

So much for probability theory. I pass swiftly and uneasily over Poisson distributions, the Central Limit Theorem, the Kolmogorov axioms, Ehrenhaft games, Markov chains, the Pascal triangle, and all the rest. I mean to spare you such mathematical convolutions. ("Let p be the probability of the happening of an event in a single trial, and let s be the number of times the event is observed to happen in n trials …") My point is only that the pure stochastician teaches himself to observe what we at the Center for Stochastic Processes have come to call the Bernoulli Interval, a pause during which we ask ourselves, Do I really have enough data to draw a valid conclusion?

I’m executive secretary of the Center, which was incorporated four months ago, in August, 2000. Carvajal’s money pays our expenses. For now we occupy a five-room house in a rural section of northern New Jersey, and I don’t care to be more specific about the location. Our aim is to find ways of reducing the Bernoulli Interval to zero: that is, to make guesses of ever-increasing accuracy on the basis of an ever-decreasing statistical sample, or, to put it another way, to move from probabilistic to absolute prediction, or, rephrasing it yet again, to replace guesswork with clairvoyance.

So we work toward post-stochastic abilities. What Carvajal taught me is that stochasticity isn’t the end of the line: it’s merely a phase, soon to pass, in our striving toward full revelation of the future, in our struggle to free ourselves from the tyranny of randomness. In the absolute universe all events can be regarded as absolutely deterministic, and if we can’t perceive the greater structures, it’s because our vision is faulty. If we had a real grasp of causality down to the molecular level, we wouldn’t need to rely on mathematical approximations, on statistics and probabilities, in making predictions. If our perceptions of cause and effect were only good enough, we’d be able to attain absolute knowledge of what is to come. We would make ourselves all-seeing. So Carvajal said. I believe he was right. You probably don’t. You tend to be skeptical about such things, don’t you? That’s all right. You’ll change your mind. I know you will.