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

at point x = 1. However, preceding the criticism of Leibniz’ procedure,Laplace describes the following case, perhaps too far-fetched to betrue, but characteristic of his attitude to Leibniz. When considering thebinary number system, Leibniz thought that the unit represented God,and zero, Nothing. The Supreme Being pulled all the other creaturesout of Nothing just like in binary arithmetic zero is zero but all thenumbers are expressed by units and zeros. This idea so pleasedLeibniz, that he told the Jesuit Grimaldi, president of the mathematicalcouncil of China, about it in the hope that this symbolic representationof creation would convert the emperor of that time (who had aparticular predilection for the sciences) to Christianity 8 .Laplace goes on: Leibniz, always directed by a singular and veryfacile metaphysics, reasoned thus: Since at x = 1 the particular sums ofthe series (1.1) alternatively become 0 and 1, we will take theexpectation, i. e., 1/2, as its sum. We know now that such a method ofsumming is far from being stupid and may be sometimes applied, butLaplace hastens to defeat Leibniz, already compromised by thepreceding story.It is indeed remarkable that now, a century and a half later, we mayrightfully say the same about Laplace: directed by a singular and veryfacile metaphysics. This does not at all touch his concrete scientificwork but fully concerns his general ideas connected with concretescientific foundation. His Essai begins thus (p. 1):Here, I shall present, without using Analysis, the principles andgeneral results of the Théorie, applying them to the most importantquestions of life, which are indeed, for the most part, only problems inprobability.So, which most important questions of life did Laplace think about,and how had he connected them with the aims of the theory ofprobability? That theory includes the central limit theorem (CLT)which establishes that under definite conditions the sumS n = ξ 1 + ... + ξ nof a large number of random terms ξ i approximately follows thenormal law. When measuring the deviation of the random variable S nfrom its expectation ES n in terms of var Sn,we therefore obtainvalues of a random variable obeying the standard normal law. Brieflyit is written in the formSn− ESvar Snn→ N(0,1).Here, N(0, 1) is the standard normal distribution (with zeroexpectation and unit variance). Consider now the case of n → ∞. If theexpectations of all the ξ i are the same and equal a, the variance also thesame and equal σ 2 , and the random variables ξ i themselvesindependent. Following generally known rules, we get89