1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

nn2ESn = ∑ Eξi= na, var Sn = ∑ varξi = nσ , var Sn= σ n.i= 1 i=1For a random variable obeying the law N(0, 1) typical are absolutevalues of the order 1. For example, the probability of its absolute valueexceeding 3 is about 0.003 (hence the three sigma rule): we see thatthe inequality| Sn− E Sn|≤ 3, so that | Sn− na | ≤ 3σ nvar Snis practically certain.Let a ≠ 0. Then na is the typical value of S n and its randomdeviations do not exceed 3σ√n, a magnitude that increases with nessentially slower than na. Given a large n, the order of thedeterminate component na exceeds that of the random deviations.Such is the purely scientific result known (at least in some particularcases) to Laplace. Let us see now what philosophical and emotionalsuperstructure did he build above it. Here is one more quotation fromhis Essai (pp. 37 – 38):Every time that a great power, intoxicated by the love of conquest,aspires to world domination, the love of independence produces,among the threatened nations, a coalition to which that power almostalways becomes a victim. [...] It is important then, for both the stabilityand the prosperity of the states, that they not be extended beyond thoseboundaries to which they are continually restored by the action ofthese causes.This conclusion is reasonable, excellent and indeed typical for thepost-Napoleon France. But then Laplace adds: This is another result ofthe probability calculus. He bears in mind that, just as the determinatecomponent prevails over randomness, see above, so also in politics,what is destined actually happens. But was it necessary to justify thatstatement by the CLT? For the modern reader it is quite obvious thatwe can only see here a remote analogy, peculiar not for science butexactly for metaphysics, and a singular and very facile metaphysics atthat.A bit later Laplace (p. 38) states, again citing the theory ofprobability: When a vast sea or a great distance separates a colonyfrom the centre of the empire, the colony will sooner or later free itselfbecause it invariably attempts to get free. And elsewhere he (p. 123)says:The sequence of historic events shows us the constant action of thegreat moral principles amidst the passions and the various intereststhat disturb societies in every way.90