Monday, April 23, 2007


Skorokhod on determinism and chaos

A. Skorokhod is a big name in probability. The following is excerpted from one of his books:

1.1.1 Determinism and Chaos

In a deterministic world, randomness must be absent -- it is absolutely subject to laws that specify its state uniquely at each moment of time. This idea of the world (setting aside philosophical and theological considerations) existed among mathematicians and physicists in the 18th and 19th centuries (Newton, Laplace, etc.). However, such a world was all the same unpredictable because of its complex arrangement. In order to determine a future state, it is necessary to know its present state absolutely precisely and that is impossible. It is more promising to apply determinism to individual phenomena or aggregates of them. There is a determinate relationship between occurrences if one entails the other necessarily. The heating of water to 100°C under standard atmospheric pressure, let us say, implies that the water will boil. Thus, in a determinate situation, there is complete order in a system of phenomena or the objects to which these phenomena pertain. People have observed that kind of order in the motion of the planets (and also the Moon and Sun) and this order has made it possible to predict celestial occurrences like lunar and solar eclipses. Such order can be observed in the disposition of molecules in a crystal (it is easy to give other examples of complete order). The most precise idea of complete order is expressed by a collection of absolutely indistinguishable objects.

In contrast to a deterministic world would be a chaotic world in which no relationships are present. The ancient Greeks had some notion of such a chaotic world. According to their conception, the existing world arose out of a primary chaos. Again, if we confine ourselves just to some group of objects, then we may regard this system to be completely chaotic if the things are entirely distinct. We are excluding the possibility of comparing the objects and ascertaining relationships among them (including even causal relationships). Both of these cases are similar: the selection of one (or several objects) from the collection yields no information. In the first case, we know right away that all of the objects are identical and in the second, the heterogeneity of the objects makes it impossible to draw any conclusions about the remaining ones. Observe that this is not the only way in which these two contrasting situations resemble one another. As might be expected, according to Hegel's laws of logic, these totally contrasting situations describe the exact same situation. If the objects in a chaotic system are impossible to compare, then one cannot distinguish between them so that instead of complete disorder, we have complete order.

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