The name Laplace appears in contemporary textbooks on stochastics only in passing, if at all. When his 1812 textbook on probability appeared, it immediately outshone the numerous books and articles on the subject by many important predecessors such as Huygens, Halley, Jacob Bernoulli, and de Moivre, and within a few years, it had been reprinted numerous times, with some of the editions being significantly expanded, and it remained a classic in its field for almost one hundred years. Today, Laplace is associated primarily with the most purely combinatorial aspect of probability, which rests on the assumption of finitely many equally probable elementary events and culminates in the easily understood formula for the probability “number of favourable outcomes : number of possible cases.”
But even this relatively trivial case had been treated long before Laplace. The hard mathematical core of Laplace’s contribution consists, in fact, in the introduction of so-called generating functions, a powerful aid in mastering the very long sums of arithmetic expressions that appear in calculations of probabilities and the formulas thereby obtained. This hard core is embedded in an abundance of applications and suggestions for applications, of which perhaps the most important concern error estimation (in particular for geodesy and astronomy), while others seem to us today to be mere curiosities, and in fact, their promotion led by the middle of the nineteenth century to a temporary rejection of probability theory by the majority of mathematicians. These concerned the believability of eye-witness testimony in court, the probability of just verdicts, and the accuracy of elections and voting procedures in the light of probability, and the integration of psychological points of view. Altogether, for Laplace and his followers, considerations of probability are motivated by nothing other than incomplete knowledge of something that has to be judged or decided.
Laplace’s Théorie analytique was preceded by numerous publications on separate topics, beginning in 1774 with the Memoir on recurrent series and their application to the theory of games of chance and the Memoir on the probability of causes of events. Finally, in 1814, to the second edition of the Théorie analytique was added a preface consisting of an edited version of a popular (that is, without formulas) lecture, “a philosophical essay on probabilities” that he had given in 1795 at the École normale.
We quote from the final sentences: “One sees in this essay that the theory of probabilities is basically only common sense reduced to a calculus. It makes one estimate accurately what right-minded people feel by a sort of instinct, often without being able to give a reason for it. It leaves nothing arbitrary in the choice of opinions and of making up one’s mind, every time one is able, by this means, to determine the most advantageous choice… if one observes also that even in matters which cannot be handled by the calculus it gives the best rough estimates to guide us in our judgements, and that it teaches us to guard ourselves from the illusions which often mislead us, one will see that there is no science at all more worthy of our consideration, and that it would be a most useful part of the system of public education” (translation by Andrew I. Dale, 1994).
In fact, with these sentences, Laplace staked out the limits of what probability theory could achieve before the introduction into stochastics of measure theory and before the discovery of indeterminate processes in nature. Yet with his oft cited “Laplace’s demon,” who at a given point in time knows everything and for whom, therefore, the entire future is foreseeable and predictable, he was practically the very symbol of a now defunct scientific worldview.
French postage stamp, 1955.
Pierre-Simon Laplace, born in 1749 in Beaumont-en-Auge in Normandy into a middle-class family, grew rapidly into one of France’s leading scientists. At the young age of 24, he was elected to membership in the French Academy of Sciences. The numerous offices that he later held, as well as his great achievements in analysis, celestial mechanics, and the organisation of science in France, may have contributed to the success of his probability theory. In the standard Dictionary of Scientific Biography, he is allotted 131 pages, jointly written by three authors. Euler and Gauss have only 17 pages each! When Laplace died in 1827, he bore the title of a marquis, awarded by King Louis XVIII to Laplace despite the latter’s political activity in previous regimes.
(By Peter Schreiber, Greifswald, Germany, August 2012; translated by David Kramer)