Bernoulli, Jakob

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Jakob BERNOULLI

b. 6 January 1655 - d. 16 August 1705

Summary. Jacob Bernoulli, together with his brother Johann one of the pioneers of the Leibnizian form of the calculus, transformed Huygens' calculus of expectations to make probability its main concept. He formulated and proved the weak law of large numbers, the cornerstone of modern probability and statistics.

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 "Ars conjectandi" (or Art of Conjecturing) which was only published after his death in 1713. This represents the transition from the calculus of chances as developed by Christiaan Huygens in 1657 to a new mathematical theory centered around the concept of probability the classical measure of which was also introduced by Jakob Bernoulli.

The Ars conjectandi, generally considered as Bernoulli's most original work, remained incomplete. It consists of four parts. The first is a reprint of Huygens' De ratiociniis in aleae ludo, which was published in 1657, complemented by extensive annotations which contained important modifications and generalizations preparing something new, the theory of probability theory. In the second part Bernoulli deals with the theory of combinations. In the third part Bernoulli gives twenty-four examples concerning the determination of the modified Huygenian concept of expectation in various games. The fourth part is the most interesting and original. At the same time it is the part which Bernoulli was not able to complete. In the first three of its five chapters it deals with the new central concept, probability, its relation to certainty, necessity and chance, and ways to estimate and measure it. In chapter 4 Bernoulli distinguishes two ways to determine exactly or approximately the classical measure of probability. The first presupposes equipossibility of the outcomes of certain elementary events like drawing either one of $n$ balls numbered from 1 to $n$ out of an urn. So the probability to draw a ball of a certain colour out of an urn filled with balls of different colours is determined a priori by the ratio of the number of balls of this special colour to the number of all balls in the urn. For the determination of the probability of an event like dying of a certain person within the next ten years a reduction to numbers of equipossible cases which are favourable or unfavourable for the event is impossible. But according to Bernoulli we can inductively, by experiments, or a posteriori in his sense, get as close as we desire to the true measure of such a probability. The possibility to estimate the unknown probability of such an event by the relative frequency of the outcome of this event in a series of supposedly independent trials is secured according to Jakob Bernoulli by his theorema aureum, which was called later, by Poisson, Bernoulli's law of large numbers. The proof of this theorem is contained in chapter 5. In an appendix Jakob Bernoulli treats the jeu de paume, a ball game usually considered as a predecessor of tennis, as a game of chance.

The idea to give his book the title Ars conjectandi was stimulated by the Ars Cogitandi, better known as the Logic of Port Royal, in the very last chapter of which the chances for future contingent events are equated with the ratios of the associated degrees of probability. One can see how Bernoulli, beginning from this notion, developed the classical concept of probability, and how he established as its measure a generalization of Huygens' expectation, namely, the ratio of favorable to all possible cases. At the same time he became the first to set down the prerequisites for consciously formulating a program for the mathematization of all the fields of application subject to "probabilis". Bernoulli himself sought to execute this program, but his premature death prevented him.

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 "Ars conjectandi. Leibniz emphasized that in the area of jurisprudence and politics, which was so important for Bernoulli's program, no extended calculations were usually required, since an enumeration of the relevant conditions would suffice. Against the possibility of attaining a better approximation to a sought after probability with an increasing number of observations, Leibniz suggested that contingent events, here identified with dependence on infinitely many conditions, could not be determined by a finite number of experiments. In addition, e. g., the appearance of new diseases could change the probability of survival of a person with a certain age. However, Bernoulli saw no problem with applying an urn model to human mortality, with the stones corresponding to diseases with which a person can be taken ill. So Bernoulli's research program stood firm after his discovery of the law of large numbers in 1689.

The content of the Ars conjectandi in its most important sections went back to preliminary studies he had done in the 1680's as testified by the Meditationes. A key passage for the transformation of Huygens' expectation to probability involves the treatment of a problem of law, on which Bernoulli worked in 1685/86. It has to do with a marriage contract, which, assuming that the couple have children and that the wife (Caja) dies before the husband (Titius), will govern the division of their common property between the father and the children. The portion of the groom will be larger if he has already entered into his inheritance, smaller if not, unless both fathers have died. The bride's father objects to his initial proposal; this induces Titius to make a second proposal, according to which he will receive the same portion of the common property regardless of what happens to the fathers. On this basis Bernoulli poses the question: which suggestion would be more favorable for the children? To this end he has to make assumptions about the possible order of death of the three people involved, the two fathers and Caja. He first assumes that all six possible orders have equal weight. But this assumption does not satisfy him, since the youth of the bride has not been taken into account. Thus, he assumes that for every two instances e.g., diseases, symptoms or events which might bring about the death of either father, there is only one which threatens Caja with death. There are thus five cases in all, each equally likely to take its victim first. Since Caja is affected by only one of these, while the two fathers are affected by four, her situation is evaluated as one-fifth of certainty of her being first to die, that is, "one probability, five of which make the entire certainty". Here Bernoulli uses the plural ``probabilities", where these are equated with the no more precisely distinguished individual cases; this usage does not permit the conception of ``probability" as ``degree of certainty" which is observed in the next stage. Aided by Huygens' formula for determining expectation, Bernoulli then derives a certainty of 4/15, written 4/15 c, where ``c" stands for ``certitudo", or 4 probabilities our of 15, that Caja will die second, and finally 8/15 c that she will die third. At the end of his treatment of the marriage contract he speaks of the degree of probability:

"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 and Laplace throughout the 18th century and into the second half of the 19th.

References

[1]	Hald, Anders, (1990). A History of Probability and Statistics and their Applications Before 1750, Wiley, Chapters 15 and 16. For Jakob Bernoulli's understanding of probability and chance see:
[2]	Hacking, Ian, (1975). The Emergence of Probability, Cambridge UP, Chapters 16 and 17 and
[3]	Shafer, Glenn, (1978). Non-additive probabilities in the work of Bernoulli and Lambert. Archive for History of Exact Sciences, 19, 309-370, especially pp. 323-341. The role of Jakob Bernoulli as the founder of a mathematical probability theory and his relationship to Leibniz is treated in:
[4]	Schneider, Ivo, (1981). Why do we find the origin of a calculus of probabilities in the seventeenth century? In: Probabilistic Thinking, Thermodynamics, and the Interaction of the History and Philosophy of Science (Eds. J. Hintikka, D. Gruender and E. Agazzi) Vol. II, Dordrecht: Reidel (= Synthese Library, Bd. 146), pp. 3-24.
[5]	Schneider, Ivo, (1981). Leibniz on the Probable. In: Mathematical Perspectives, Essays on Mathematics and its Historical Development (Ed. Joseph W. Dauben), New York: Academic Press, pp. 201-219.
[6]	Ivo Schneider, (1984). The role of Leibniz and of Jakob Bernoulli for the development of probability theory. LLULL, Boletin de la Sociedad Espanola de Historia de las Ciencias}, 7, 68-89. 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: Die Werke von Jakob Bernoulli, vol. III (ed. by the Naturforschende Gesellschaft in Basel), Birkhäuser Verlag Basel 1975.

Reprinted with permission from Christopher Charles Heyde and Eugene William Seneta (Editors), Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.