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 (
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.