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# Jakob Bernoulli

Bernoulli stemmed from a family of merchants. His grandfather Jakob Bernoulli became a citizen of Basel in 1622 by marriage. His father, Nikolaus Bernoulli, took over the drug business from his father and became a member of the town council. After finishing the master of arts in 1671 Jakob Bernoulli studied theology until 1676 when he received the licentiate in theology; at the same time he studied mathematics and astronomy secretly against the will of his father. After 1676 he left Basel to work as a tutor for four years which he spent in Geneva and France. He also travelled to France, the Netherlands, England and Germany in 1681, 1682. His first publication,in 1681, dealt with his observations of the comet of 1680 and his prediction of its reappearance in 1719. Back in Basel he began to give private lectures especially on the mechanics of solid and liquid bodies and he became interested in analysis. Two years after its foundation in 1682 Leibniz had published in the Acta Eruditorum a method to determine integrals of algebraic functions, a short presentation of the differential calculus in algorithmic form and some remarks concerning the fundamental ideas of the integral calculus. These papers occupied the interest of Jakob and his brother Johann Bernoulli. Jakob tried to get further information from Leibniz in 1687 but he answered Jakob's questions only three years later because of a journey undertaken in diplomatic mission. At that time Jakob and Johann had not only mastered the Leibnizian calculus but also had added so considerably to it that Leibniz wrote in a letter of 1694 that the infinitesimal calculus owes as much to the Bernoulli brothers as to himself. Jakob also cultivated the theory of series which were published in five dissertations between 1689 and 1704. He considered series as the universal means to integrate arbitrary functions, to square and rectify curves. In 1690 Jakob had introduced the term ``integral" in his solution of the problem to determine the curve of constant descent. In the 90's the relationship between Jakob and Johann deteriorated mainly because of the hot and excitable temper of the very ambitious younger Johann. The bitter quarrels between them became as notorious as the priority dispute between Leibniz and Newton, which began at about the same time, concerning the creation of the infinitesimal calculus. In 1687 Jakob became professor of mathematics at the university of Basel, in which position he remained until his death in 1705. He was honoured by the memberships of the Académie Royale des Science (1699) and of the Academy of Sciences in Berlin (1701). He had a daughter and a son from Judith Stupan whom he had married in 1684. His most famous monograph is the The The idea to give his book the title Jakob's interest in stochastics began in the 1680's. His first publication on the calculus of chances dates from 1685. The development of his ideas can at least in part be traced in the Meditationes, his so-called scientific diary, which he had begun in 1677. It shows the significance of jurisprudence for the transition from a calculus of chance to a calculus of probability. This is accompanied by Leibniz independent and lifelong interest in a doctrine or a logic of degrees of probabilities triggered by the hope to quantify conditional rights but also in the dissertation of Niklaus Bernoulli, Jakob's nephew, who sought to apply the findings of his uncle to a series of concrete problems in law. Different from Jakob Bernoulli Leibniz never worked out his ideas in this field. He left only a series of partly redundant drafts and manuscripts but no publication. Jakob Bernoulli learned only late in his life of Leibniz' interest in a doctrine of probabilities. This is testified by the correspondence between the two men from April 1703 until Jakob's death in 1705. Jakob Bernoulli took Leibniz' objections, especially against the significance of his law of large numbers, as representative of a critical reader and tried to refute them in part 4 of the The content of the ``Generally in civic and moral affaires things are to be understood, in which we of course know that the one thing is more probable, better or more advisable than another; but by what degree of probability or goodness they exceed others we determine only according to probability, not exactly. The surest way of estimating probabilities in these cases is not a priori, that is by cause, but a posteriori, that is, from the frequently observed event in similar examples." Bernoulli carried out the determination of probabilities a posteriori by adopting relative frequencies determined through observation as estimates of probabilities which could not be given a priori. He felt justified to proceed in this way by his fundamental theorem which at the same time served as the essential foundation of his program to extend the realm of application of numerically determinable probabilities. Bernoulli's understanding of chance excluded events as occurring indeterminately. He was convinced that through a more precise knowledge of the parameters affecting the motion of a die, for instance, it would be possible to specify the result of the throw in advance. In similar fashion he viewed changes in weather as a determinate process, just as the occurrences of astronomical events are. Chance, in his view and later in the view of Laplace, was reduced to a subjective lack of information. Thus, depending on the state of their information, an event may be described by one person as chance, but by another as necessary. With this anticipation of Laplacian determinism Bernoulli appears to solve the problem of the connection between chance and divine Providence. The entire realm of events which are described in daily life as uncertain or contingent in their outcome is such, he claims, merely because of incomplete information: nevertheless, these too fall within the field of the concept of ``probabilitas". Bernoulli's program to mathematize as much of this realm as possible with the aid of the classical measure of probability occupied researchers like de Moivre (q.v.) and Laplace (q.v.) throughout the 18th century and into the second half of the 19th.
A good account of Jakob Bernoulli's contribution to probability theory together with an appreciation of the mathematical methods used is given in: Hald, Anders, (1990). For Jakob Bernoulli's understanding of probability and chance see: Hacking, Ian, (1975). Shafer, Glenn, (1978). Non-additive probabilities in the work of Bernoulli and Lambert. The role of Jakob Bernoulli as the founder of a mathematical probability theory and his relationship to Leibniz is treated in: Schneider, Ivo, (1981). Why do we find the origin of a calculus of probabilities in the seventeenth century? In: Schneider, Ivo, (1981). Leibniz on the Probable. In: Ivo Schneider, (1984). The role of Leibniz and of Jakob Bernoulli for the development of probability theory. Very useful information concerning the background of Bernoulli's work in stochastics is contained in the commentaries by K. Kohli and B. L. van der Waerden in: Ivo Schneider |