The Emergence of Mathematical Probability from the Perspective of the Leibniz-Jacob Bernoulli Correspondence
This paper moves out from a defined body of evidence, namely the correspondence between Gottfried Wilhelm Leibniz and Jacob Bernoulli, to see what light it casts on the emergence of mathematical probability in the late seventeenth and early eighteenth centuries. At many times during his life Leibniz repeated his recommendation that serious attention should be paid to developing a theory of probabilistic reasoning. In his Ars Conjectandi (The Art of Conjecturing), Jacob Bernoulli tried to do just what Leibniz at many times recommended, but he did not finish the work before his death in 1705, partly because he lacked information about real-life situations. Although, when they discussed probability and what Bernoulli had accomplished and hoped still to do in letters written between April 1703 and April 1705, Leibniz and Bernoulli did not understand each other perfectly, nevertheless, when one looks more broadly in their writings, one sees that they were in fundamental agreement about their purposes: the new probability logic or the art of conjecturing was to play a role in a practical moral discipline and its problems were to be more like those of law than those of empirical science. Within this practical discipline, the role of mathematics was to supply a rigorous method of reasoning and, in particular, a way of insuring comprehensiveness and consistency when the complexity of the situation might otherwise overwhelm normal reason.
The Correspondence Between Leibniz and Jacob Bernoulli
There are twenty-one extant letters between Leibniz and Jacob Bernoulli (one of which may never have been sent) and evidence that at least five more once existed (see Appendix below for list of letters). The skimpiness of the correspondence was no doubt largely Jacob Bernoulli’s doing—in all there are only approximately 150 extant letters to and from Jacob Bernoulli, [End Page 41] whereas there are approximately 2,000 to and from his brother Johann Bernoulli and over 15,000 surviving letters from Leibniz to other people (Nagel in Jacob Bernoulli 1993a, p. 27, n. 2; Hess 1989, pp. 155–56; Ross 1984, p. 9). The correspondence between Jacob Bernoulli and Leibniz got off to a slow start because Leibniz was not in Hannover when Bernoulli first wrote him in 1687. Once Leibniz replied in 1690, Bernoulli allowed more than five years to pass before writing again in October 1695. Meanwhile, however, between 1693 and Jacob’s October 1695 letter to Leibniz, Johann Bernoulli, then living in Basel like his brother, exchanged 17 letters with Leibniz, often sending and receiving greetings between Leibniz and Jacob, but also reporting a growing conflict between the brothers. In April of 1695, Johann reported to Leibniz that he had shown Leibniz’s letters to his brother in the hopes they would convince Jacob of his culpability in his conflict with Johann. He said that Jacob’s failure to write to Leibniz himself was due not only to Jacob’s character, but also because Jacob had been sick (GM III, p. 173). Johann was sure that Jacob would write to Leibniz in the near future, but he was afraid that he would have nothing good to say about Johann. When Jacob then wrote to Leibniz on 9/19 October 1695, it was soon after Johann had left Basel to take up his new post in Groningen—Johann’s first letter to Leibniz from Holland was written from Amsterdam on 8/18 October 1695, just the day before his brother’s letter.
In the period after Johann left Basel for Groningen, there followed a more regular if not copious exchange of letters between Jacob Bernoulli and Leibniz. Leibniz replied to Jacob in December 1695. Each man wrote to the other at least once in 1696 and at least once in 1697. Then Jacob let another five years pass before writing to Leibniz again on 15 November 1702, his letter then being occasioned by receipt of documents making him a member of the Berlin Society of Sciences, which had been established at Leibniz’s instigation. Such a happy occasion was hardly one for complaining, but Jacob went on to explain that he had sent no letters for the last five years because he had been sick more than once and because he had been weighed down by unpleasant and unexpected mathematical quarrels with his brother Johann—Leibniz, too, he said, might have been choked in the weeds of the quarrel, because of his handling of mathematical papers Johann had sent him. At the beginning, Jacob said, he had considered sending Leibniz a short history of the work of the two brothers in mathematics, showing that it was Jacob who first understood Leibniz’s new calculus, and showing that both of the brothers had at first given more credit to Isaac Barrow than they should have. He had changed his mind about sending the history, because he saw no profit in it. Now he was resolved to let bygones be bygones (Jacob Bernoulli 1993a, p. 101). [End Page 42]
In that same period, while the correspondence between Leibniz and Jacob languished from late October 1695 to late November 1702, Leibniz and Johann Bernoulli in Groningen exchanged 138 letters, as compared to less than 10 letters between Leibniz and Jacob. Although Leibniz tried his best to maintain his good relations with both brothers, he could not help but be more aware of Johann’s point of view. While Johann and Jacob rarely or never wrote to each other, they published mathematical challenges to each other in the journals of the day. Often Leibniz was pressed to act as intermediary or umpire in the process. In May of 1697, Jacob published a mathematical challenge directed to his brother in the Acta Eruditorum of Leipzig. Johann sent his solution to Leibniz in June 1698, which Leibniz transferred in a sealed packet to the Académie des Sciences in Paris in February 1701, there to await Jacob’s own solution of the problem before being opened. Only after Jacob’s death in 1705, in fact, was the packet opened 17 April 1706 (Jacob and Johann Bernoulli 1991, p. 515). In interchanges like these, sometimes the correspondents told each other what problems they had solved or theorems they had proved without telling their methods, or they told each other their methods in private, but held them back from publication (see, e.g., GM III, p. 534). Jacob and Johann were often in the position of trying to guess how the other brother had solved some problem he had claimed to solve (e.g., GM III, pp. 442, 484, 533). More than once, Leibniz and Johann Bernoulli wrote to each other that they had “dissimulated” to a third person what they knew or did not know or about the source of their knowledge (e.g., GM III, p. 496).
When Leibniz replied to Jacob in April of 1703, then, he added in a postscript that he had heard that Jacob had cultivated fairly extensively a theory of the estimation of probabilities, a theory which Leibniz thought was both pleasant and useful and not at all unworthy of the most serious of mathematicians. Leibniz had wished that someone would treat mathematically the various kinds of games, which would be an attractive example of such a theory. He had seen, he said, only a few of Bernoulli’s available theses or dissertations on the subject, but he would like to have them all (Jacob Bernoulli 1993a, p. 109).
After receiving this letter, Jacob again let half a year pass before replying, and, when he did on 3 October 1703, he wanted to know who had told Leibniz about his work in probability. His suspicion was that it was his brother Johann, and he seemed to be further suspicious that Johann might not have been giving him the credit he deserved:
I would very much like to know, dear sir, from whom you have it that the theory of estimating probabilities has been cultivated by me. It is true that several years ago I took great pleasure in this sort [End Page 43] of speculations, so that I could hardly have thought any more about them. I had the desire to write a treatise on this matter. But I often put it aside for whole years because my natural laziness compounded by my illnesses made me most reluctant to get to writing. I often wished for a secretary who could easily understand my ideas and put them down on paper. Nevertheless, I have completed most of the book, but there is lacking the most important part, in which I teach how the principles of the art of conjecturing are applied to civil, moral, and economic matters. [This I would do after] having solved finally a very singular problem, of very great difficulty and utility. I sent this solution already twelve years ago to my brother, even if he, having been asked about the same subject by the Marquis de l’Hôpital, may have hid the truth, playing down my work in his own interest.(Jacob Bernoulli 1993a, p. 116)
Now Leibniz may have intended Bernoulli to believe that he knew about his work on probability only from his publications. Jacob had, for instance, published in the Acta Eruditorum of May 1690, “Some questions on usury, with the solution of a problem on the chance of players proposed in the Journal des Sçavans, 1685, art. 25” (Jacob Bernoulli  1967, pp. 427–31; Jacob Bernoulli 1975, pp. 91–93; Jacob Bernoulli 1993b, pp. 160–63). Although the word “probability” was never used in this article, nor was the idea of probabilistic reasoning as such mentioned, nevertheless both Bernoulli and Leibniz agreed that financial problems and games of chance were situations in which a mathematical theory of probability or a doctrine of chances could be profitably employed.
Aside from the Acta Eruditorum, which Leibniz could certainly have seen, Jacob had published in Basel a number of pamphlets, which Leibniz was less likely to have had access to. For instance, in 1685 Jacob published a pamphlet on The parallel of logical and algebraic reasoning, together with miscellaneous theses, 1 in which the nineteenth miscellaneous thesis stated:
How much that science stands out above other sciences, which, while the other sciences deal only with probability concerning the most certain and constant things, itself can deal apodictically and by most certain reasoning about things most fortuitous and subject to chance.(Jacob Bernoulli  1967, vol. 1, p. 221)
Bernoulli then went on in the next theses to propose problems of games [End Page 44] and marriage contracts, presumably exemplifying apodictic reasoning concerning things subject to chance. The next year Jacob published Logical Theses on the conversion and opposition of enunciations together with miscellaneous theses to be debated 12 February 1686. Among the miscellaneous theses were several related to the art of conjecturing, as well as more systematic theses about the certainty of mathematics versus the sciences (Jacob Bernoulli  1967, pp. 231–38). Thus the thirty-first thesis concerned the relation of life expectancies to the distribution of a dowry, and the thirty-second thesis concerned the game of tennis. On another occasion Jacob had published at Basel in 1687, The solution of a triplet of problems, arithmetical, geometrical, and astronomical, together with corollaries from general mathematics (Jacob Bernoulli  1967, p. 313). In this pamphlet his eleventh corollary was from the art of conjecturing (ex Arte Conjecturandi), and concerned the slowness with which the expectations in a lottery increase as more and more non-winning tickets are drawn out of the urn.
In his inaugural oration upon taking up the position of Dean of Philosophy in 1692, Jacob had spoken “On the Combinatorial Art,” starting and ending with the observations that the theory of combinations lay at the foundation of the wisdom of the philosopher, the exactness of the historian, the dexterity of the physician, and the prudence of the politician. It could be used for deciphering codes, compounding medicines, forming legal cases, computing chances in games, estimating conjectures and the probabilities of future events, and thousands of other situations (Jacob Bernoulli 1975, pp. 99, 106). This inaugural oration would certainly have been of interest to Leibniz, but it is unlikely he would have known about it, because it seems not to have been published at the time (Jacob Bernoulli 1975, pp. 98–106).
In any case, when Leibniz replied to Jacob on 26 November 1703, he said that he did not remember first learning of Jacob’s work on probability from his brother, but from somewhere else. As might be guessed from Leibniz’s “weasel words,” “first” and “I remember,” this was clearly a dissimulation to some degree (Jacob Bernoulli 1993a, p. 123). Though Leibniz may have read what Bernoulli published on usury in the Acta Eruditorum of May 1690, we also know that Johann Bernoulli wrote to Leibniz about his brother’s work on the art of conjecturing on 16 February 1697 (GM III, p. 367). As Jacob wrote to Leibniz in April 1703 (as quoted above), he had told his brother about twelve years before about his work in probability—this would put it around 1691. Though the letter is no longer extant, the editors of the Bernoulli edition conclude from the evidence that about September 1692 Jacob wrote to his brother Johann, then temporarily in Paris, boasting of the mathematical discoveries he had made, including his proof of the law of large numbers (Johann Bernoulli [End Page 45] 1955, p. 119). Johann, it seems, quickly told the Marquis de l’Hôpital about the contents of Jacob’s letter. In a letter of 8 December 1692, l’Hôpital wrote to Johann asking that he give his compliments to his brother and saying that he was anxious to know more about Jacob’s mathematical work, including that on infinite series and about the unprecedented (inoüis) problems Jacob had spoken of in his last letter to Johann. And l’Hôpital went on, “[A]sk him also what is the proposition in his book on the art of conjecturing which he rates as highly as the discovery of the quadrature of the circle” (this was, of course, his proof of the weak law of large numbers; Johann Bernoulli 1955, p. 160).
From roughly this time until Jacob’s death in 1705, the animus between the brothers continued unabated. In the many letters that Johann wrote to Leibniz, there is hardly one that does not have something critical to say about Jacob. These problems, if not first caused, were at least exacerbated by the fact that when he was in Paris Johann had entered an agreement with l’Hôpital according to which, for remuneration, he would in effect be his confidential mathematical consultant, feeding him mathematical problem solutions and the like, without divulging the fact (Johann Bernoulli 1955, pp. 131ff.). As Johann wrote to Leibniz on 8 February 1698, he had, in this way told, dictated, or written to l’Hôpital nearly all of what was to become l’Hôpital’s 1696 textbook on the calculus, Analyse des infiniment petites (GM III, p. 480). Whatever Johann’s motivations may have been for entering into this contract, and whatever benefits he may have derived to compensate him for the costs to his personal reputation, it is understandable that Jacob would not have been pleased if he suspected that his own original mathematical results were also being funneled confidentially to l’Hôpital in this way.
The mention of Jacob’s work on probability in the correspondence of Johann Bernoulli and Leibniz was occasioned when Leibniz wrote to Johann Bernoulli on 29 January 1697, saying, “I remember having read once in the Journal des Sçavans that he [Sauveur] wrote something mathematical about the game of Bassette, which, however, I didn’t see” (GM III, p. 363). Johann replied on 16 February:
Until now I didn’t know that he wrote something mathematical on the game of Bassette. I would like to see it, because my brother already a long time ago, endeavored to write a book entitled Ars Conjectandi, in which would be provided not only a mathematical treatment of all kind of games, but also means of reducing to calculations other probabilities in every area of life. I don’t know, however, whether the book is still unfinished or not, only that while he once judged that to consult anything of mine was not to [End Page 46] be spurned, now, agitated by his usual rivalry, he scarcely ever fails to attack. There is also what Huygens wrote about games of chance. In mathematical works in folio (Ouvrages de Mathematique), published a few years ago in Paris, there was something to be seen by Freniclio on combinations, where he also investigated something about the chances of competitors in connection with the Genoese Senate elections.(GM III, p. 367)
In the letters that followed between Leibniz and Johann, Jacob’s work on probability was mentioned again more than once. On 5 March 1697, Leibniz wrote:
It comes to mind again what you wrote about the Ars Conjecturandi of your brother. It will, without doubt, be something not to be disregarded. I too once thought about such things, especially in connection with jurisprudence and politics. I call it the doctrine of degrees of probability. Does your brother also consider the so-called art of deciphering? It also would merit treatment by a mathematician. What exists up to now is of little value. I have been wishing someone would appear who would treat mathematically every kind of game.(GM III, p. 377)
On the basis of Johann’s report and of his correspondence with Jacob himself, Leibniz believed that there was indeed something worthwhile in the Ars Conjectandi. Thus on 27 June 1708 (by which time Leibniz could also have known of Jacob Hermann’s report of the contents of the book manuscript used by those who composed éloges for Jacob), Leibniz wrote Johann:
I don’t remember hearing anything about Montmort who is preparing a book on games of fortune. I was wishing that this argument would be treated well. But I also wished that the considerations of your brother of pious memory on such things not perish. I believe that there is something in them not to be despised. And above all I would hope that selections from this work would sometime be published. For the public utility, I wish that a way might be agreed to by everyone.(GM III, p. 836)
All in all, one must be sympathetic to Leibniz’s difficulties trying to mediate between Jacob and Johann. Leibniz was certainly pleased to have both brothers as part of his intellectual circle in the development of the calculus. He did not want to alienate either one, yet it was almost impossible to please them both at the same time. If, in order to correspond with [End Page 47] Jacob about probability he had to dissimulate where he had learned about Jacob’s work on the subject, that was a small price to pay.
Leibniz’s View of the History of Mathematical Probability
At almost the same time as Leibniz and Johann Bernoulli were mentioning probability and Jacob’s Ars Conjectandi in their correspondence, Leibniz wrote to Thomas Burnett in London, giving his view of the history of the development of mathematical probability and urging that the logic of probability be developed further. Whereas Leibniz wrote to Johann Bernoulli about Saveur’s mathematical work on games on 29 January 1697, he wrote to Thomas Burnett on 1/11 February of the same year a several page letter describing the need for establishing on firm foundations an art of measuring degrees of proofs (l’art d’estimer les degrés des probations; GP III, p. 193).
By the time that Leibniz wrote to Jacob Bernoulli in April 1703 saying that he would like to know everything he had done in the area of estimating probabilities, he was soon to begin composing his New Essays on Human Understanding, something he worked on between August 1703 and 25 April 1704 (Müller and Krönert 1969). In the New Essays, Leibniz/Theophilus recommends the study of degrees of probability:
Perhaps opinion based on likelihood also deserves the name of knowledge; otherwise nearly all historical knowledge will collapse, and a good deal more. But, without arguing about names, I maintain that the study of the degrees of probability would be very valuable and is still lacking, and that this is a serious shortcoming in our treatises on logic. For when one cannot absolutely settle a question, one could still establish the degree of likelihood ex datis, and so one can judge rationally which side is the most plausible. And when our moralists—I mean the wisest of them, such as the present General of the Jesuits—bring in the question of what is safest as well as of what is most probable, and even put safety ahead of probability, they do not really abandon the most probable. For the question of safety is the question of the improbability of a feared evil. Moralists who are lax about this have gone wrong largely because they have had an inadequate and over-narrow notion of probability, which they have confused with Aristotle’s endoxon. . . . But probability or likelihood is broader: it must be drawn from the nature of things; and the opinion of weighty authorities is one of the things which can contribute to the likelihood of an opinion, but it does not produce the entire likelihood by itself.(NE, Bk. IV., Ch. 2, pp. 372–73; I have made some changes in the translation.) [End Page 48]
As Leibniz said repeatedly during his life, it was the jurists who had done the most in the past to quantify levels of proof. In the New Essays he went on:
When jurists discuss proofs, presumptions, conjectures, and evidence, they have a great many good things to say on the subject and go into considerable detail. They begin with common knowledge, where there is no need for proof. They deal next with complete proofs, or what pass for them: judgments are delivered on the strength of these, at least in civil actions. . . . in criminal actions . . . there is nothing wrong with insisting on more-than-full proofs, and above all for the so-called corpus delicti if it is that sort of case. . . . Then there are presumptions, which are accepted provisionally as complete proofs—that is as long as the contrary is not proved. There are proofs which are, strictly speaking, more than half full; a person who founds his case on such a proof is allowed to take an oath to make up its deficiency (juramentum suppletorium). And there are others that are less than half full; with these, on the contrary, the oath is administered to the one who denies the charge, to clear him (juramentum purgationis). Apart from these, there are many degrees of conjecture and of evidence. And in criminal proceedings, in particular, there is evidence (ad torturam) for applying torture. . . . The entire form of judicial procedures is, in fact, nothing but a kind of logic, applied to legal questions.(NE, pp. 464–65)
Outside of the area of law, much less had been done to develop a systematic logic of probabilities, but mathematicians had recently made a start:
Mathematicians have begun, in our own day, to calculate the chances (hazards) in games. It was the Chevalier de Méré . . . who prompted them by raising questions about the division of the stakes, wanting to know how much [a given player’s part in] a game would be worth if the game were interrupted at such and such a point. Accordingly he enlisted his friend M. Pascal to take a brief look at the problem. The question caused a stir and prompted M. Huygens to write his treatise on chance. Other learned men joined in. Certain principles were established, and were also employed by Pensionary de Witt in a little discourse, published in Dutch, on annuities. The foundations they built on involved prosthaphaeresis, i.e. arriving at an arithmetic mean between several equally admissible hypotheses (suppositions également recevables). Our peasants have used this method for a long time, guided by their natural mathematics. For instance, when some inheritance or piece [End Page 49] of land is to be sold, they appoint three teams of assessors—these teams are called Schurzen in Low Saxon—and each team assesses the commodity in question. . . . This is the axiom aequalibus aequalia—equal hypotheses (suppositions égales) must receive equal consideration. But when the hypotheses are unlike (les suppositions sont inégales), we compare them with one another. Suppose, for instance, that with two dice one player will win if he throws a 7 and the other if he throws a 9. We want to know the ratio between their expectations for winning (apparences de gagner). I say that the expectation (apparence is only two thirds the expectation (apparence) for the first player, since there are three ways in which the first can throw a 7 with two dice . . . whereas there are only two ways in which the second can throw a 9. . . . And all these are equally possible. So that the expectations (apparences), which are in the ratio of (sont comme) the numbers of equal possibilities, will be as 3 to 2, or as 1 to 2/3. I have said more than once that we need a new kind of logic, concerned with degrees of probability. . . . Anyone wanting to deal with this question would do well to pursue the investigation of games of chance. In general, I wish that some able mathematician were interested in producing a detailed study of all kinds of games, carefully reasoned and with full particulars. This would be of great value in improving the art of invention, since the human mind appears to better advantage in games than in the most serious pursuits.(NE, pp. 465–66, translation slightly modified)
So at the time that Leibniz was showing an interest in Jacob Bernoulli’s work in probability theory, he was writing in the New Essays in the person of Theophilus an exhortation in favor of the development of that theory. When he came to write the same history again in a letter to Louis Bourguet on 22 March 1714, after the publication of Ars Conjectandi, he inserted mention of Jacob’s work into the history as he had previously written it:
The art of conjecture is based on what is more or less easy (facile) or, better, more or less doable (faisable), for the Latin facile derived from faciendo literally means doable; for example, with two dice, it is as doable to throw a twelve as to throw an eleven for each can only be done in one way;2 but it is three times more doable to throw seven, for that can be done by throwing six and one, five and [End Page 50] two, and four and three, and one of these combinations is as doable as the other. Chevalier de la Méré (author of Livre des Agrémens) was the first to give occasion to these thoughts, which Pascal, Fermat, and Huygens pursued. The Pensioner de Witt and Hudde have also worked on these questions since. The late M. Bernoulli cultivated the matter on my exhortations. One also evaluates plausibilities (vraisemblances) a posteriori, by experience; one should have recourse to this in the absence of a priori ratios (raisons); for example, it is equally plausible (vraisemblable) that a baby about to be born will be a boy or a girl, because the numbers of boys and girls are found to be nearly equal in this world. We may say that what is done more or done less is also more or less doable in the present state of things, putting all the considerations together which must concur in the production of what is done (d’un fait).(GP III, pp. 569–70)
Here Leibniz adds Jacob Bernoulli to his history of the evaluation of degrees of probability and he furthermore accepts the determination of ratios of cases a posteriori. By adding Hudde to de Witt he also puts more emphasis on empirical information about birth and death rates, values of annuities, and the like. In light of Leibniz’s 1714 version of the history of mathematical probability, we can now understand more about the 1703–5 correspondence on probability between Leibniz and Jacob Bernoulli.
The A Posteriori or Empirical Basis for Probability Theory
After Bernoulli in his letter of 3 October 1703 had asked Leibniz how he had come to know about his probability work, he nevertheless then went on to give Leibniz an idea of what he had tried to prove—although not of his method of proof—using phrasing often very similar to what would later appear in the Ars Conjectandi:
I will reveal to you briefly what the problem is. It is known that the probability of any event depends on the number of cases in which it may happen or not happen, so that we know, for example, how much more probable it is that with two dice 7 rather than 8 points will fall. We do not know, on the other hand, how much more likely it is that a young person of 20 years will survive an old person of 60 than the reverse. This is so because we know the numbers of cases in which 7 or 8 points may occur with dice, but we do not know the number of cases in which a young person meets death before an old one or vice versa. From whence I began to think whether or not what was hidden to us a priori might not at least be known to us a posteriori from the outcome in similar examples many times observed. For example, if I perceive, having made the experiment [End Page 51] in very many pairs of young and old, that it happens 1000 times that the young person outlives the paired old person and it happens only 500 times that the reverse happens, then I may safely enough conclude that it is twice as probable that a young person will outlive an old one than the reverse.
People very commonly reason from past experience, Bernoulli said, and they even seem to realize that the more experience they have, the less danger there is of using it as a guide to the future. No one, however, before him has demonstrated accurately and mathematically that, as the number of observations increases, the danger of erring decreases. This is what he has been able to do. Moreover, he has shown there is no limit to the degree of probability that can be achieved in this way:
It is to be sought besides whether, as the number of observations increases the probability also continually increases, so that finally, given any probability, it becomes more probable to me that I have found the true ratio between the numbers of cases than [that I have found] some other [ratio] different from the true one, or whether I will eventually come to some degree of probability beyond which it cannot be made more probable to me that I have discovered the true ratio. Now if the latter is so, it is all over with our effort to explore the numbers of cases by experiment. If, however, the former is true, then we can find out the ratio of cases just as certainly a posteriori as if it were known to us a priori. And this is how I have found things to be, and I can now determine how many observations must be made in order that it be a hundred, a thousand, ten thousand, etc. times more likely (or finally that it be morally certain), that the ratio between the numbers of cases that I have obtained in this way is the legitimate and real one.
This was what Jacob regarded as the greatest mathematical achievement of the Ars Conjectandi—this was the result, as referred to by l’Hôpital, which he considered more important than if he had squared the circle (Jacob Bernoulli 1975, p. 88). 3 Since, however, he gave no hint of the mathematics he used for the proof, it is not surprising that Leibniz’s attention did not focus on the proof. Instead, he concentrated on more philosophical issues. When he replied on 26 November 1703, then, after saying he did [End Page 52] not remember hearing about Jacob’s probability work first from his brother, he went on:
When we estimate probabilities empirically by results in succession, you ask whether in that way finally a perfect estimate could be obtained, and you write that this has been found by you. The difficulty in it seems to me to be that contingent things or things that depend on infinitely many circumstances cannot be determined by finitely many results, for nature has its habits, following from the return of causes, but only for the most part. Who is to say that the following result will not diverge from the law of all the preceding ones—because of the mutabilities of things? New diseases attack humankind. Therefore even if you have observed the results for any number of deaths, you have not therefore set limits on the nature of things so that they could not vary in the future. When we determine the path (lineam) of a comet from some number of observations, we assume that it is one of the conics or one of the simpler kinds. Given any number of points, infinitely many curves (lineae) may be found that traverse them. . . . Since a point may be varied infinitely many ways, infinitely many other curves (lineae) are also possible. These points, however, may be compared to the cases observed, and the curves (lineae) drawn through them may be compared to the rules or estimates deduced from them.(Jacob Bernoulli 1993a, p. 123)
Thus, not seeing how Jacob’s proof worked, Leibniz assumed it was something like fitting a curve to a number of data points, and he argued that it was not possible in such a way to achieve certainty—there are always infinitely many other lines that could be drawn through all of the same points. Bernoulli’s response to this in his letter of 20 April 1704 was to say:
Your difficulty concerning my empirical way of determining the ratio between the numbers of cases is not more telling for the examples in which it is not possible to establish these numbers by another means than for those in which it is also possible to know them a priori. I have said that I can demonstrate it for the examples you mention [in istis] and my brother saw and approved of the demonstration more than twelve years ago. So that you may understand what I want more clearly, I will give you an example. I posit that some number of black and white tokens is hidden in an urn and that the number of white tokens is double the number of black ones, but that you don’t know the ratio of the two types but want [End Page 53] to determine it by experiments. You take one token after another out of the urn (replacing each one after you take it so that the number of tokens in the urn is not reduced), and you observe whether each one you select is light or dark. Now I say that, taking two ratios as near as you like to a double ratio, one larger and one smaller, say 201:100 and 199:100, I will scientifically determine the number of observations necessary for it to be ten or a hundred or a thousand times more probable to you that the ratio of the number of times you select a white token to the number of times you select a black token will fall within rather than outside the limits 201:100 and 199:100 of a double ratio—until finally you can be morally certain that the ratio to be found by experiment will approach the true double ratio as closely as you like. If you now replace the urn with an old or young human body, which contains in itself the germs of diseases like the urn contains tokens, then you can determine in the same way by observations how much closer to death the former is than the latter. Nor does it make any difference to say that the number of diseases to which either one is exposed is infinite. Let this be so. It is known that even in the infinite there are degrees and that a ratio of one infinite to another infinite can be expressed by finite numbers either exactly or closely enough to suffice in practice. If the diseases multiply by the passage of time, it would be necessary to make new observations. It is certain that anyone who wanted to judge the life spans of those who lived before the flood by observations made today in London or Paris or elsewhere would stray widely from the truth.(Jacob Bernoulli 1993a, pp. 128–29)
Here Jacob gave Leibniz more mathematical detail about his theorem, but still gave no hint of the basis of his proof. Moreover, he made the surprising claim that Leibniz’s difficulty concerning his empirical way of determining the ratios between numbers of cases was “not more telling for the examples in which it is not possible to establish the numbers by another means than for those in which it is also possible to know them a priori.” Then he says, “I have said that I can demonstrate it for the examples you mention.” The correct interpretation of this statement is not entirely clear. What is meant by “in istis posse demonstrare”? In the preceding sentence, “istis” occurs only in the phrase “these numbers”, “illa exempla, in quibus de numeris istis aliunde constare nequit” referring to those examples in which it is not possible to establish these numbers otherwise. And he contrasts the examples in which “these numbers,” i.e., numbers of cases, can not be known in another way and those in which they can also be known a priori (“illa, in quibus priori cognosci possunt”). In fact, as has been noted by several [End Page 54] commentators (Keynes 1921, p. 369; Gini 1946, p. 403; Hacking 1975, pp. 154ff.), Bernoulli’s proof of the law of large numbers assumes that unknown ratios of cases exist at the same time as it is expected to undergird the possibility of determining these ratios within narrow limits with probability as high as one wishes a posteriori. Not having seen Bernoulli’s proof, Leibniz could hardly have been expected to understand what Bernoulli intended. What Bernoulli had done, however, was to demonstrate the theorem for a case in which, although it was unknown, the ratio of cases was in fact determined, that is, it was taken to be the ratio of tokens hidden in an urn. Then his claim was that considering the effects of germs on the human body was analogous to considering the drawing out of tokens from an urn.
With regard to Leibniz’s argument concerning the fitting of a curve to data points in the motion of a comet, Bernoulli—perhaps to Leibniz’s surprise—agreed with Leibniz, but said that the example was irrelevant since he did not use any such procedure to prove his theorem. If one did want to make an analogy between finding the path of a comet and what he was concerned with, you might say that if you had observed five points in the path of a comet and they all fit on a parabola, you would have a greater suspicion that the rest of the path was a parabola than if you had examined only four points. Furthermore even if there are infinitely many different curves that pass through these five points, there are infinitely many others that go through four points and not the fifth. In any case, Bernoulli said, he thought all conjectures based on observations in this way were rather slight and hazardous unless it was taken as given that the curve sought would belong to one of the simpler kinds of curves—which, indeed, he thought plausible, since we see that nature everywhere follows the simplest paths (Jacob Bernoulli 1993a, p. 129).
In previous discussions of Bernoulli’s law of large numbers and of Leibniz’s correspondence with him about it, it has not been sufficiently noticed that with regard to empirical curve fitting Leibniz and Bernoulli are in agreement. What Bernoulli tells Leibniz here is that his law of large numbers is not about empirical curve fitting, but about probability in the sense of epistemology or how we are justified in increasing the confidence with which we make given assertions (how our “suspicions” increase with more evidence). In Bernoulli’s terms as in our own, this is subjective, not objective probability. For him there is no “objective probability,” because for him “probable” is a moral term, applying to human statements and behavior, not to the physical world. We should understand, however, that for Bernoulli subjective probability is supposed to be rational and not personal, as some modern connotations of “subjective” might seem to imply.
Leibniz’s next two or three letters are lost, so we do not know what he [End Page 55] might have said in them, or how he may have responded to Bernoulli’s claim that the analogy to curve fitting was irrelevant. When Bernoulli wrote again on 2 August 1704, however, he said:
The ratio between the numbers of deaths, even if the numbers are infinite, we can determine by a finite number of observations not exactly but closely enough for practice, approaching it more and more closely until the error becomes undetectable. It is well known in geometry that the ratio of the diameter to the circumference, even if it cannot be determined accurately except by the infinite series of numbers of Ludolph, nevertheless was defined closely enough for practice by Archimedes as between 7:22 and 71:223.
Leibniz replied on 28 November 1704:
In some things that are not sufficiently connected (that is for our comprehension), it is not certain that when the number of givens has been increased or the years of observations continued we will come closer to some mean truth in general (veritatem mediam in universum), even if prudence bids us to take things in this way. But in the series of Ludophine numbers, when one continues one always approaches closer to the mean (Jacob Bernoulli 1993a, p. 136).
Given that Bernoulli did not tell Leibniz the nature of his proof, it was perhaps inevitable that they would not come to a complete meeting of minds, but here, by inserting the explanation “that is for our comprehension,” Leibniz seems to show that he understood Bernoulli to be talking about epistemological probability. Now Leibniz’s argument seems to be that even though prudence may dictate that we should regulate our opinions by the evidence, in reality, given our irrationality, hopes, and desires, etc., we might not do so. In this also, however, Bernoulli would have agreed—his theory of probability was not meant as a description of how our opinions actually vary and why, but as a normative prescription for how we should regulate our confidence based on the evidence. Once this is understood, it is less surprising that in his letter to Louis Bourguet in 1714, Leibniz accepted the determination of ratios of cases a posteriori. The reason for this is that the a posteriori ratios were to be used in a practical moral art of probabilistic reasoning, not in an empirical science, as I will explain further below. Before that, however, I want to mention the [End Page 56] other connected topic that was repeatedly mentioned in the Leibniz-Jacob Bernoulli correspondence, and that was Jacob’s effort to borrow from Leibniz his copy of Jan de Witt’s pamphlet on annuities and any other relevant materials Leibniz might have available.
The Problem of Obtaining Real Life Data
In his first 3 October 1703 reply to Leibniz’s request to know everything he had done on mathematical probability, Jacob wrote:
I do not know, dear sir, whether there may seem to you to be anything of solidity in this sort of speculation. If so, would you be so kind to submit to me some legal questions, which, in your judgment, I might usefully investigate? Just lately I saw in Menstruis Excerptis Hanoverae (Hannoverian Monthly) the citation of a Treatise unknown to me of a Pensionarius de Witte von Subtiler Ausrechnung des valoris der Leib-Renten (On Subtly Calculating Values of Life Annuities). Perhaps this has something for doing this. If so, I would be very happy to receive a copy (Jacob Bernoulli 1993a, p. 117).
To this request, Leibniz replied on 26 November 1703 that he had once had the work of de Witt and that in it he used the familiar method of estimating equal results from the possibility of equal cases. This was sufficient to show equitable pricing of annuities for the Dutch. Because he wanted his work to be commonly known, Leibniz said, de Witt wrote in Dutch (Jacob Bernoulli 1993a, p. 124). Having said this, Leibniz did not send Bernoulli the work.
In the subsequent correspondence between Leibniz and Bernoulli, the question of de Witt’s work was a repeated theme. Bernoulli thought it might have information that would be useful for his purposes. He had been right in thinking that Leibniz might have a copy of de Witt’s pamphlet, because he had seen a notice of it in the Monatliche Auszug, Leibniz’s own journal under the editorship of his assistant Johann Georg von Eckhart (Ross 1984, p. 8; Biermann and Faak 1959, p. 170). Bernoulli asked again that Leibniz send him de Witt’s work in letters of 20 April 1704, 2 August 1704, 15 October 1704, and 28 February 1705, and Leibniz replied in letters of 28 November 1704, and April 1705 that he had the work but could not find it in his papers or that, anyway, there was nothing in it that would be very new to Bernoulli. Bernoulli’s failure to obtain de Witt’s work or other sources of data on life expectancies or other frequencies may explain why he never finished or published the Ars Conjectandi before his death (Jacob Bernoulli 1975, e.g., p. 18). That Leibniz had mislaid the pamphlet in the mess of his papers can easily be believed: such things happened to him regularly. So in July of 1698, for instance, he [End Page 57] received a letter from Johann while he was on his way to Herrenhausen. Later he wrote to Johann that he had lost it and was afraid he left it or dropped it somewhere. In September, he finally found it in a pile of papers where he had previously searched in vain (GM III, pp. 514–15, 541).
To a partisan of Jacob Bernoulli, Leibniz’s claim in his 1714 letter to Louis Bourguet that Jacob Bernoulli cultivated the matter of probabilities under his exhortations may seem particularly galling, since Jacob had finished nearly all of the Ars Conjectandi before the first mention of probability in a letter from Leibniz to Jacob, and since Leibniz helped to stall Jacob’s further work by failing to find and send him de Witt’s pamphlet. At the very least, it could be argued that Leibniz should have made more effort to find and send Bernoulli the copy of de Witt’s pamphlet he had in his possession. Apparently, few copies of the pamphlet had been printed, and it was difficult to come by. Biermann and Faak, however, believe that Leibniz honestly tried to find the pamphlet. They point out that Leibniz, having failed to find his own copy among his papers, asked Johann Bernoulli to try to find another copy of the pamphlet for Jacob in Holland—albeit not until July 1705, a month before Jacob’s death (Biermann and Faak 1959, pp. 168–73; GM III, p. 767). 5
Leibniz may have made less effort to get a copy of the pamphlet to Jacob than he might otherwise have done because he assumed, as he wrote to Jacob, that there was not much Jacob could learn from it. He seems to have thought Jacob wanted to see the mathematical methods it used, whereas in fact Jacob was looking for institutional information or statistics. In fact, Leibniz himself continued to see a connection between the pamphlet and the development of mathematical probability. In his own earlier papers he had made use of de Witt’s work more than once in discussing such topics as annuities and how their prices or the amount paid per year should be determined using life expectancies (A IV.iii, pp. 440, 457). When Johann Bernoulli sent Leibniz a copy of Nicolaus Bernoulli’s Dissertatio Inauguralis Mathematico-Juridica De Usu Artis Conjectandi in Jure on 2 July 1709, Leibniz wrote back on 6 September 1709 mentioning de Witt’s pamphlet, the book that Jacob Bernoulli had been writing before his death, and his own long-standing interest that the part of logic dealing with degrees of probability should be developed. Degrees [End Page 58] of plausibility should be estimated from degrees of possibility, Leibniz thought, or from the number of equal possibilities (GM III, pp. 842, 844–45). It was shown in some sort of political booklet, he said, that sometimes estimates are made by addition and sometimes by multiplication. 6 On 1/11 February 1697 he had written to Thomas Burnett that this anonymous booklet was in fact his own work about the election of a king of Poland (See Cohen 1994, pp. 112–13).
In the end, perhaps both Leibniz and Jacob Bernoulli had more interest in developing the mathematics of probability than in obtaining the data necessary for its application. Perhaps the errors in Leibniz’s examples of a priori and a posteriori ratios of cases in his 22 March 1714 letter to Louis Bourguet—saying that it is equally easy to throw 11 and 12 with two dice, when in fact there are two ways to throw 11, but only one way to throw 12, and saying that approximately the same number of boys and girls are born, when it was commonly said at the time that the ratio of boys to girls was about 18 to 17—show that he was not primarily interested in the numbers per se. Nevertheless, Bernoulli needed statistical data on frequencies for use in his mathematical formulas. As in Part III of the [End Page 59] Ars Conjectandi Bernoulli had calculated expectations for various games, so in Part IV he should have calculated, for instance, what would be a fair price for an annuity or how much the expectation of an inheritance would be worth if it was contingent on one person dying before another. De Witt himself had shown in his pamphlet such things as the comparative values of annuities for one or two lifetimes (Rowen 1978, pp. 418–19). This kind of evaluation depended on a knowledge of life expectancies, which Bernoulli hoped to find in de Witt’s work.
But it was not only data that Bernoulli needed, but also knowledge of law. On 2 August 1704 he wrote to Leibniz:
expecting again from you at these markets the writing of Pensioner de Witt, to which I wish you could add what you once wrote about conditions. I wish that you would also give me an example of conditional legacies and also that you explain by an example what you understand about annuities on several lives, since I have never applied myself professionally to the study of law.(Jacob Bernoulli 1993a, p. 132)
In examples of the ways in which jurists had measured degrees of proofs, Leibniz frequently mentioned cases where the degrees of proof were legally, rather than scientifically or otherwise determined. If Roman law required two eyewitnesses to a capital crime, then the testimony of two eyewitnesses would count as a full proof. If the law also required there be a corpse before putting an alleged murderer to death, then the corpus delicti, leading to more than a full proof, would be needed, as Leibniz mentioned in the New Essays, as quoted above. In similar ways, laws about dowries and bequests would affect the expectations of inheritance. To complete Part IV of the Ars Conjectandi, Bernoulli needed this kind of institutional or legal information as well as statistics about life expectancies and the like.
Bernoulli’s Proof of the Weak Law of Large Numbers
If Leibniz had seen Bernoulli’s proof of the law of large numbers, then he would have known that it did not involve something like empirical curve fitting as his responses in his letters seemed to assume. Bernoulli never doubted that Leibniz would admire his demonstration once he saw it. So he wrote on 28 February 1705 that he was convinced he was right and that he was sure that when Leibniz saw his proof he would be convinced:
With regard to verisimilitudes and their increase as the number of observations increases, the matters are entirely as I have written, [End Page 60] and I am sure you will be pleased with the demonstration when I shall have published it.(Jacob Bernoulli 1993a, p. 141)
Completed in the 1680s (by 1689 at the latest and likely started by 1687), Bernoulli’s proof made no use of the calculus, then in its first stages of development. Instead, it used properties of binomial expansions. The key to the proof was a lemma showing that when a binomial (r + s) is taken to a very high power nt, where t = r + s, then more and more of the total quantity of the resulting expression is contained in terms within an interval around the largest term. But such a binomial expansion, as Bernoulli argued, can be used to represent the possible results of nt trials of some situation in which the cases for a positive result are r and of a negative result s. Then the largest term of the binomial expansion to the nt power will be that involving rnrsns, and this term will represent the number of cases in which, of nt trials, nr come out positive and ns come out negative, or in other words that the experimental outcome will manifest the a priori ratios of cases. The terms close on either side of this maximum term will represent the number of cases in which the ratio of outcomes will be close to the a priori ratio of cases. Jacob’s proof shows that by taking n large enough, the ratio between the sum of all the terms in a defined interval around the maximum term and the sum of all the other terms can be made as large as you choose.
Knowing, as we do, what Jacob’s proof involved, it is interesting to note that Johann on 10 June 1702 sent to Leibniz a list of problems he had proposed to Varignon, including as the fourth:
For a given polynomial raised to any power, for instance a trinomial raised to the twentieth power (a+b+c)20, to find the largest term.(GM III, p. 702)
To this Leibniz replied on 24 June:
I have the first, second, and third problem in my power, but I don’t sufficiently understand the fourth. What is it that you call the maximum term?(GM III, p. 703)
When he explained his meaning on 24 August, Johann went on to say that he was sure that, once he understood the meaning, Leibniz could easily solve the problem, since he already knew how to determine the coefficients of the given terms (GM III, p. 709).
How Bernoulli’s theorem might be expected to be used can be seen in an application that his nephew Nicolaus Bernoulli made of it even before the Ars Conjectandi had been published. In this period data about birth and death rates was first becoming available. The data on births for the city of [End Page 61] London showed that more boys had been born than girls in every year for eighty-two years. It was argued by John Arbuthnot in the Philosophical Transactions of the Royal Society published in 1712 that this shows Divine Providence, because more men are killed in war and other masculine activities than women die in childbirth or other feminine activities, so that since it is desirable to have equal numbers of men and women as adults, more boys than girls should be born. Nicolaus Bernoulli, however, argued that this argument for Divine Providence, or, as he called it, for miracles, was invalid. Suppose, he said in effect, that the mechanism for determining the sex of a child is like a die that has 18 sides leading to the birth of a boy and 17 sides leading to the birth of a girl. The effect of throwing very many such dice could explain the variations in numbers of boys and girls born in London over those eighty-two years. Given the numbers of births, the observed small variation in the ratio of boys to girls born was not at all surprising. Nicolaus proved this result using a variant of Jacob Bernoulli’s law of large numbers in a letter sent to Pierre Remond Montmort that was published in the second edition of his Essay d’Analyse sur les Jeux de Hazard, which appeared in 1713 (actually he proved that, if the process were something like throwing dice, there should have been less variation than actually observed—but he seems not to have noticed this inconsistency).
It may be noted that in arguing for this result, Nicolaus Bernoulli made no suggestion that what determines the sex of a child is something like a thirty-five sided die. Rather his point was to describe how much variation might be expected as the result of a law with some randomizing factor, like the throwing of a die. Similarly, when Jacob Bernoulli sought to use a posteriori ratios, he did not try to find fundamental underlying ratios. In the one case that Bernoulli discusses in most detail, the problem is to predict who will win at tennis. We have no way of knowing or measuring all the physical and psychological factors that combine to determine who will win at tennis. What we should do, however, is to count the relative numbers of times that one player or another wins a point and then use these a posteriori ratios as we might have used the a priori ratios of cases in dice games. These a posteriori ratios are not immediate ratios of physical or psychological factors, but ratios of points won.
This status of the law of large numbers fits with Bernoulli’s concept of concrete mathematical disciplines, as stated in Bernoulli’s publication of theses for oral disputation, the Logical Theses on the conversion and opposition of enunciations together with miscellaneous theses to be debated February 12, 1686. Bernoulli wrote:
XI. Concrete mathematical disciplines, such as physics, medicine, astronomy, optics, statics, ballistics, (and if you wish astrology) add [End Page 62] only some principles or foundations to abstract mathematics. Some of these are proven elsewhere, some are taken only from experiment. On these principles one can reason further with no less geometrical rigor than in abstract mathematics based on axioms or common notions. Thus physics assumes the laws of motion, medicine assumes the fabric of the human body, astronomy assumes the fabric or the system of the world, astrology the influx of the stars on the sublunar world and that the fates of men, cities, regions, depend on the configuration of the heavens. . . .
XII. It is therefore clear that the certainty of these sciences depends only on the certainty of these principles, not on the way of reaching conclusions, which all should be deduced from the principles by the most evident reasoning. This is the reason why abstract mathematics is of invincible certainty and why astrology is vain and futile. The other sciences have a middling certitude between these, because such are the principles on which they are erected.(Jacob Bernoulli  1967, vol. I, p. 234)
Thus Jacob’s proof of the law of large numbers proceeded with complete certainty. Its application was open to doubt, however, if one doubted that there are unvarying ratios of cases in the world. For Bernoulli himself, God’s creation of the world implied that it was determined (cf. Jacob Bernoulli  1968, pp. 210–12).
How, then, does the proof of the law of large numbers support the prudent use of relative numbers of points won in previous games in predicting who will win future games of tennis? The point here is not that by studying past tennis games one has found the true empirical law or the causes why one or another person wins. The point is rather that the law of large numbers shows that, assuming there are underlying ratios of cases, then you can calculate how much data you need to collect in order to be as safe as you want to be. The art of conjecturing or the logic of probabilities is not an empirical physical science, but rather a practical or moral discipline supporting prudent decision-making in situations in which complete certainty is impossible. Here there is a similarity to the status of the Leibnizian calculus, as stated in Leibniz’s 26 November 1703 letter to Jacob:
It is shown that in infinite series and in our calculus of sums and differences the calculations are not to be extended beyond the cases in which the matter can be reduced to a rigorous demonstration in the traditional way. Our method is only a contraction of the traditional one, suited for invention. I take large or small quantities of any size in place of infinitely large or infinitely small ones, and if in [End Page 63] this way the error can be made smaller than any given error, then the method is safe.(Jacob Bernoulli 1993a, p. 121)
Probability Mathematics and Practice
I claim, then, that because when he wrote his letters to Jacob Bernoulli, Leibniz had no idea how Bernoulli’s proof of the law of large numbers was structured, he at first mistakenly assumed that Bernoulli was arguing for something like empirical curve fitting. In fact, both Leibniz and Bernoulli in the end conceived of mathematical probability as a practical or moral and not a theoretical discipline. All along in his letters and elsewhere, Leibniz had said that even though he doubted Bernoulli’s claim that there was no limit to the degree of certainty one could obtain about an a posteriori ratio if one made enough trials, nevertheless the limited certainty one could obtain would be sufficient in practice. When Bernoulli first wrote Leibniz about his work on probability on 3 October 1703, immediately after he had stated what he had proved in the law of large numbers, he said:
This suffices in the practice of civil life for directing our conjectures no less scientifically in contingent matters than in games of chance. In my opinion, all the prudence of the statesman consists only in this.
Note that Bernoulli’s comparison here is of conjectures in civil life to conjectures in games of chance. Leibniz wrote back on 26 November 1703, saying first of all that the estimation of probabilities was extremely useful, before he went on to say that in juridical and political examples the problem was often not subtle calculations, but accurate enumeration of all the circumstances (Jacob Bernoulli 1993a, p. 123). Coming to the end of his discussion of the reliability of a posteriori ratios, Leibniz concluded:
Even if a perfect estimate cannot be obtained empirically, nonetheless an empirical estimate would be useful and sufficient in practice.(Jacob Bernoulli 1993a, p. 124)
In his reply on 20 April 1704, Jacob concurred with the practical slant on his work, although claiming to have shown that “subtle calculation” was useful:
various questions on insurance, annuities, dowry contracts, presumptions, and so forth, teach me that the theory of estimating probabilities in legal matters requires not only an enumeration of the circumstances, but also the same sort of reasoning and calculation that we are accustomed to use in calculating in games of [End Page 64] chance. I will show this clearly when the time comes.(Jacob Bernoulli 1993a, p. 128)
In his next extant letter, on 28 November 1704, Leibniz again wrote:
In games of pure reason or games partly of chance . . . even if it is not easy to calculate how much closer one person is to winning than another, nevertheless we can very often rationally determine that one person is closer to winning from the data. We see clever players determining more or less what is better to do, as is done in military matters or in medicine, using reasoning that is more broad than deep, which is also part of the art.(Jacob Bernoulli 1993a, pp. 136–37)
In comparing what Leibniz first wrote to Bernoulli to what he said later in his letter to Louis Bourguet, Ian Hacking has suggested that Leibniz changed his mind because he was persuaded by Jacob Bernoulli (Hacking 1975, p. 128), while Robert Adams disagrees (Adams 1994, p. 202). If we understand that all along Leibniz conceived of probability logic as a practical art something like legal reasoning, then we can see that Leibniz’s position was consistent on this matter rather than changing.
Indeed, Robert Adams’s discussion of the legal notion of “presumption” in Leibniz suggests that Jacob Bernoulli may have intended to propose the use of well established a posteriori ratios in calculating probabilities as something like a presumption (Adams 1994, pp. 192–213; de Olaso 1975; Menochio 1587–90). The definition of a legal presumption is that it is to be taken as true until the opposite is proved, as we say that in a legal trial the defendant is to be presumed innocent until proven guilty. This is a matter of law and regulates where the burden of proof lies in a trial: are you guilty unless you are proven innocent or are you innocent unless you are proven guilty? Leibniz makes a clear distinction between a presumption and a supposition. So he wrote in 1702:
For every being ought to be judged possible until the contrary is proved, until it is shown that it is not possible at all. This is what is called presumption, which is incomparably more than a simple supposition, since most suppositions ought not to be admitted unless they are proved, but everything that has presumption for it ought to pass for true until it is refuted. . . . So this Argument has the force to shift the burden of proof to the opponent (onus probandi in adversarium), or to make the opponent responsible for the proof.(Adams 1994, p. 192; French in GP III, p. 444)
Another familiar legal case of presumption is that possession is nine-tenths [End Page 65] of the law, or “a possessor is presumed to be an owner” (Adams 1994, p. 194; A VI, iii, p. 608).
Leibniz wanted to establish practical disciplines on firm foundations. In his letter to Thomas Burnett on 1/11 February 1697, which, as I have already mentioned, advocates the development of probability theory, Leibniz said:
I have noted several times, both in philosophy and in theology and even in medicine, jurisprudence and history, that we have an infinity of good books and good thoughts dispersed here and there, but that we have almost no Establishments (Establissemens).(GP III, pp. 191–92)
Here Leibniz means by an “Establishment” something that would be like the Constitution in American legal theory. He goes on to explain:
I call it an “Establishment” when one determines and achieves at least certain points and puts certain theses out of contention to gain ground and to have foundations on which one can build. . . . Here is how one should proceed. I distinguish the propositions of which one should make the Establishments into two kinds. One kind can be demonstrated absolutely by metaphysical necessity and in a way incontestible. The other kind can be demonstrated morally, that is to say, in a way which gives what is called moral certainty, in the way that we know that there is a China or a Peru, although we have never seen them and have no absolute demonstration that they exist.(GP III, pp. 192–93)
Given this distinction, theology has two parts. There is a part that is metaphysically certain, and there is a part that depends on history, facts, and the interpretation of texts. To make use of texts, moreover, we must establish their antiquity and sanctity, and to understand the texts it is necessary to have a true philosophy and a natural jurisprudence:
For Philosophy has two parts, the theoretical and the practical. Theoretical Philosophy is founded on the true analysis, of which the Mathematicians give samples, but which ought also to be applied to Metaphysics and to natural theology, in giving good definitions and solid axioms. But practical Philosophy is founded on the true Topics or Dialectics—that is to say, on the art of estimating the degrees of proofs, which is not yet found among the authors who are Logicians, but of which only the Jurists have given samples that are not to be despised and that can serve as a beginning for forming the science of proofs proper for verifying historical facts [End Page 66] and for giving the meaning of texts. For it is the Jurists who are occupied ordinarily with the one and the other in [legal] processes. Thus before Theology can be treated by the method of Establishments, as I call it, a Metaphysics, or demonstrative natural Theology is needed, and so is a moral Dialectic, and a natural Jurisprudence, by which the way to estimate the degrees of proofs may be learned demonstratively. For several probable arguments joined together sometimes make a moral certainty, and sometimes don’t. There is therefore need of a sure method to be able to determine it. It is often said, with justice, that reasons should not be counted, but weighed; however no one has yet given us that balance that should serve to weigh the force of reasons. This is one of the greatest defects of our Logic; we feel the effects of it even in the most important and most serious matters of life, which concern justice, the peace and well-being of the State, human health, and even religion. It is almost thirty years since I made these remarks publicly, and since that time I have done a quantity of research, to lay the foundations of such works; but a thousand distractions have prevented me from giving final form to those Philosophical, Juridical, and Theological Elements that I had projected. If God still gives me life and health, I will make it my principal business. I still would not prove all that can be proved, but I would prove at least a very important part, in order to begin the method of Establishments, and to give others occasion to go further.(Adams 1994, pp. 198–99; translating GP III, pp. 193–94; I have made a few minor changes in the translation.)
Thus Leibniz envisioned establishing practical philosophy and providing it with a sure methodology for probabilistic reasoning. This is consistent with what he had written thirty years before in the “Nova Methodos Discendae Docendaeque Jurisprudentiae,” of 1667:
But there is missing so far the most useful and practical part of logic concerning the degrees of probability or the scale for weighing arguments when opposing opinions vie with each other with verisimilitudes. This is the “Logocritica” of contingents, which Aristotle only gave for what is necessary.(A VI, i, p. 281)7
To establish practical philosophy, then, empirical induction alone was [End Page 67] insufficient. As Leibniz wrote to Bernoulli several times and as he wrote in many other places, by induction certainty is impossible. Analysis alone is also insufficient. In a letter to Louis Bourguet 5 August 1715 Leibniz wrote:
There is a difference between the analysis of what is necessary and the analysis of contingents. The analysis of what is necessary, which is that of essences, moving from what is posterior in nature to what is prior in nature, ends in primitive notions, in the same way as the numbers are resolved into units. But in contingents or existing things, this analysis from what is posterior in nature to what is prior in nature goes to infinity without one being able to reduce it to primitive elements.(GP III, p. 592)
In his preface to Nizolius written in 1670 (at age 24), Leibniz said:
It is evident that induction by itself produces nothing, not even a moral certainty, without the support of propositions that depend not on induction but on universal reason.(Adams 1994, p. 201; A VI, ii, p. 432; L, pp. 129–30)
Hacking and Adams both seem to imply that the propositions referred to here as depending on universal reason will be conditional probabilities, conditional on given data (Adams 1994, p. 201). Perhaps this is the case, but, if so, both Hacking and Adams may be misunderstanding how the conditional probabilities are to fit into rational philosophy. Consider Bernoulli’s proposal to use a posteriori ratios of cases to make practical judgments about expected outcomes in future tennis games. Probability mathematics (“propositions that depend . . . on universal reason”) will allow him to reason from the premise that Peter has 3 cases for winning to every 2 cases for Paul (if, in the past Peter has won 300 points for every 200 Paul has won) to the conclusion that Peter’s chances for winning will be such-and-such in a given future situation. This is not a conditional probability from observational data to an empirical generalization, but from an assumed ratio of cases to a calculation about the future. What Bernoulli’s law of large numbers was supposed to provide in this circumstance, then, was information about the degree of confidence (probability) you should associate with your conclusion. If your 3/2 ratio of cases was based on knowledge of 500 past points, then you would have some degree of confidence in your prediction. If the 3/2 ratio was based on 5000 past points, you would have greater confidence, and so forth.
In the brief chapters of Part IV of Ars Conjectandi that precede the proof of the law of large numbers, Jacob Bernoulli gives just such a pragmatic context for the application of probability theory to civil, moral, and economic [End Page 68] matters. He begins by setting down some general rules or axioms “which simple reason may dictate to any person of sound mind, and which the more prudent constantly observe in civil life.” Among these rules are:
(5). In matters that are uncertain and open to doubt, we should suspend our actions until we learn more. But if the occasion for action brooks no delay, then between two actions we should always choose the one that seems more appropriate, safer, more carefully considered, or more probable, even if neither action is such in a positive sense. . . .
(8). In our judgments we should be careful not to attribute more weight to things than they have. Nor should we consider something that is more probable than its alternatives to be absolutely certain, or force it on others. For it is necessary that the confidence we ascribe to any particular thing be proportioned to the degree of certainty the thing has and also that it be diminished in proportion as the probability of the thing is diminished. . . .
(9). Because, however, it is rarely possible to obtain certainty that is complete in every respect, necessity and use ordain that what is only morally certain be taken as absolutely certain.(Jacob Bernoulli  1968, pp. 216–17)
He follows this last rule with the comment:
It would be useful, accordingly, if definite limits for moral certainty were established by the authority of the magistracy. For instance, it might be determined whether 99/100 of certainty suffices or whether 999/1000 is required. Then a judge would not be able to favor one side, but would have a reference point to keep constantly in mind in pronouncing a sentence.(Jacob Bernoulli  1968, p. 217)
The logic of probability that both Leibniz and Bernoulli hoped to establish, was not, need I say it once again, a logic of frequencies or randomness, but a logic of decision making in practical affairs (cf. Daston 1992). Neither Bernoulli nor Leibniz believed that chance or genuine randomness played a role in the universe. The world was, on Bernoulli’s view, genuinely determined, at least with respect to secondary causes. Probability has to do with our knowledge of things (Jacob Bernoulli  1968, pp. 210–14). Moreover, necessity may be physical, hypothetical, or contractual/institutional. As far as games are concerned, it is contractual or institutional necessity that determines the players’ expectations, that is, the agreement with the other players on the rules of the game. Indeed, perhaps strangely to modern eyes, Christiaan Huygens’s had founded his De ratiociniis in ludo aleae (Calculations in Games of Chance) not on a consideration [End Page 69] of the propensities or frequencies of occurrences with various game pieces, but on his notion of the fair price for playing a game, defining expectation in this way:
I use the fundamental principle that the chance or expectation in a game of chance should be judged to be worth as such as the amount needed to obtain a like chance or expectation contending under fair conditions.(Jacob Bernoulli  1968, pp. 3–4)
To make equity the foundation of a theory of games rather than taking the propensities of game pieces to fall out in various ways may seem backward from the viewpoint of modern probability theory, but it seemed not at all remarkable to Jacob Bernoulli, who, in commenting on this definition in reproducing Huygens’s book as Part I of the Ars Conjectandi, said:
Here the author of this treatise is explaining the fundamental principle of the whole art. . . . Since it is very important that this principle be correctly understood, I will try to demonstrate it by reasoning that is more popular than the previous and more adapted to anyone’s comprehension. I posit only this as an axiom or definition: Everyone may expect, or should be said to expect, just as much as he will infallibly acquire.(Jacob Bernoulli  1968, p. 5; my translation)
Why may a player infallibly acquire what is said to be his expectation? Huygens had said at the end of his introduction that if you have as much money as your expectation, you can equitably buy your way back into the same or a similar game situation—that is the existence of other players willing to engage you in a fair game or business venture is assumed. Bernoulli may be thinking of this or of the possibility that everyone may decide not to play the game and to divide up the stakes fairly. Thus the idea that there could be an institutional, social, or legal basis for Leibniz’s “Establishments,” and hence for probability reasoning, was quite normal.
Leibniz, for his part, agreed with Bernoulli that in practical life one has to act on uncertain knowledge, treating what is not certain as if it is certain, while he also agreed that, in reality, nothing is accidental. In his letter to Bourguet on 22 March 1714, from which I have already quoted the history of mathematical probability, Leibniz wrote:
I understand that someone who had sensitive organs penetrating enough to sense the small parts of things would find that everything is organized. And if he could increase his penetration continually according to need, he would always see in the same mass new organs that were imperceptible by his preceding degree of penetration. . . . I understand that each creature is presently full of its future [End Page 70] state and that it follows naturally a certain path if nothing prevents it. . . . But nevertheless I don’t say that the future state of the creature follows from its present state without the concourse of God.(GP III, pp. 565–66)
Slightly later the same year, he wrote to Bourguet, “In relation to God, there is nothing accidental in the universe” (GP III, p. 575).
Although, given his claim to have had no training in law, it would be dangerous to read too much legal meaning into Bernoulli’s text, it is notable that, when in the Ars Conjectandi he came to his argument for the use of a posteriori ratios, he uses the root word presumere from which the legal notion of presumption was derived:
Another way is open to us by which we may obtain what is sought. What cannot be elicited a priori may at least be found out a posteriori from the results many times observed in similar situations, since it ought to be presumed (praesumi debet) that something may happen or not happen in the future in as many cases as it was observed to happen or not to happen in the past in a similar state of things. If, for example, there once existed three hundred people with the same age and body type as Titius now is, and you observed that two hundred of them died before the end of a decade, while the rest prolonged their lives further, you could safely enough conclude (satis tuto colligere poteris) that there are twice as many cases in which Titius also may die within a decade as there are cases in which he may live beyond a decade.(Jacob Bernoulli  1968, pp. 224–25)
By his language here, Bernoulli is not making a scientific claim about induction, but a moral or legal recommendation about what should be taken as a presumption and about what you should do to be safe. As he had said in his initial rules five and eight, we should not claim too much for uncertain conclusions, but when it is necessary to act, one should take what is only morally certain as absolutely certain. The art of conjecturing will allow you to measure the degrees of probability, which you should know before making such decisions.
The very same line of reasoning is found in Leibniz’s New Essays. Philalethes/Locke had said:
The conduct of our lives, and the management of our great concerns, will not bear delay: [and] the determination of our judgment [is absolutely necessary] in points, wherein we are not capable of certain . . . knowledge.(NE, p. 460)
Theophilus/Leibniz replied: [End Page 71]
What you have just said, sir, is thoroughly sound and good. . . . Let me add that although it is not usually permitted in the courts to rescind a judgment after it has been delivered, or to do a revision after having ‘cast up the account’ (otherwise we would have to be in perpetual disquiet, which would be all the more intolerable because we cannot always keep records of past events), nevertheless we are sometimes allowed to appeal to the courts on new evidence. . . . It is like that also in our personal affairs and especially in the most important matters, in cases where it is still open to us to plunge in or to draw back, and is not harmful to postpone action or to edge cautiously ahead: the pronouncements that our minds make on the grounds of probabilities should never be taken in rem judicatam, as the jurists say—i.e. to such an extent that we shall be unwilling to revise our reasoning in the light of substantial new reasons to the contrary. But when there is no time left for deliberation, we must abide by the judgment we have made as resolutely as if it were infallible, although not always as inflexibly.(NE, pp. 460–61)
The Value of the History of Scientific Thinking
In his 22 March 1714 letter to Bourguet from which I have quoted above more than once, before giving his revised history of mathematical probability including Jacob Bernoulli as someone who had studied probability on his exhortations, Leibniz wrote:
It is good to study others’ discoveries in a way that reveals to us the source of inventions and which makes them in a sense our own. I wish authors would give us the history of their discoveries and the routes by which they came to them. When they don’t do it, it is necessary to guess in order to profit more from their works. If journalists would do it in their reviews of books, they would render a great service to the public.(GP III, p. 568)
Leibniz, himself, would have liked to develop his own work in a way that would win followers, but, he went on to admit, located as he was, he lacked a community of people who might take up his viewpoint:
I have constructed a number of definitions, which I hope to put in order one day; but unfortunately, given where I am, I lack conversation and the company of people suited to enter into my opinions.(GP III, p. 569)
When he then went on to say that he had encouraged Jacob Bernoulli’s work in probability theory, this is what he would have liked to believe, even if it was not strictly speaking so. Both Jacob Bernoulli and Leibniz [End Page 72] had been interested in the development of a logic or mathematics of probability long before they began to correspond about it. If Leibniz had sent an exhortation like the one he sent to Thomas Burnett in February 1697 to Jacob Bernoulli before Bernoulli did the work for Ars Conjectandi, it would have been natural to think that it was worth mentioning Leibniz’s encouragement of Jacob Bernoulli’s work in probability, but the evidence from Bernoulli’s Meditations and related published works indicates that Bernoulli had already proved a version of the law of large numbers by sometime between 1687 and 1689 (Jacob Bernoulli 1975, vol. 3, p. 76). Leibniz knew Jacob had developed theories of probability before he first wrote him on the subject: in his first request to Jacob for information, Leibniz said that he had heard that Jacob had cultivated not a little the doctrine of estimating probabilities (Audio a Te doctrinam de aestimandis probabilitatibus [quam ego magni facio] non parum esse excultum). Nevertheless, if Leibniz came in his later years to believe that Jacob had cultivated mathematical probability under his exhortations, this would have fit in his mind with the general concept that the Bernoullis were part of his intellectual circle in this as in the development of the calculus. In his Éloge of Jacob Bernoulli published in the Journal des Sçavans in 1706, Saurin said:
We would be omitting an essential point of this Éloge if we did not mention the part that the Bernoullis played, both of them, with regard to the differential calculus of Leibniz. . . . Thus without having the right to pretend to the glory of having invented the calculus, which was entirely due to Leibniz, they have some merit with regard to it. . . . They did so much to aid in the perfection of the calculus, that this great man [Leibniz] ceded to them part of his glory and had the generosity to recognize that the new calculus was not more his than theirs and ought no more to carry his name than theirs.(Saurin 1706, p. 85)
If Leibniz shared with the Bernoullis the glory for the development of the calculus, it would be small-minded to deny him part of the glory for encouraging the development of mathematical probability in return.
When we first look at the letters that Leibniz and Jacob Bernoulli exchanged, including their discussions about probability in the years 1703–5, we may notice that Leibniz was skeptical about what Jacob regarded as his greatest accomplishment. We also notice how many times Bernoulli asked Leibniz to loan him de Witt’s pamphlet to no avail. They may seem to have disagreed. When we consider the matter in greater detail, however, we can see that many of Leibniz’s doubts about Bernoulli’s work stemmed [End Page 73] from the fact that Bernoulli gave him no idea of the structure of his proof of the law of large numbers. Both men agreed, in fact, that induction or empirical curve fitting can never lead to certainty. Observed frequencies may change—if new diseases develop, life expectancies will change. In the end, both Leibniz and Bernoulli conceived the art of conjecturing or the logic of probability not as an empirical science, but as a practical moral discipline, one that would be modelled on law, and that would tell Christian statesmen, honest businessmen, and prudent citizens how to order their lives and societies in regular and harmonious ways, taking the safest course, avoiding risk, and maximizing their expectations for good. In this practical moral discipline, probability mathematics would play an instrumental role, expanding the ability of citizens to reason soundly and to act safely in complex situations.
Edith Sylla is professor of history and interim head of the Department of History at North Carolina State University. She is co-chair of the program committee for the 1999 History of Science Society annual meeting. Her research interests include medieval scholasticism (the Oxford Calculators and Parisian Aristotelian commentators), the early history of mathematical probability theory, and alternative perspectives on the Scientific Revolution. She has published other articles on Jacob Bernoulli.
The correspondence between Jacob Bernoulli and Leibniz consists of the following letters (totaling slightly more than a hundred pages in both GM and Jacob Bernoulli 1993a, pp. 47–151). See also Jacob Bernoulli 1975, pp. 509–13; Hess 1989. Passages relevant to the emergence of mathematical probability occur on the indicated pages in Jacob Bernoulli 1993a.
1. JaB to GWL, Basel, 15/25 December 1687.
2. GWL to JaB, Hannover, 24 September 1690.
3. JaB to GWL, Basel, 9/19 October 1695.
4. GWL to JaB, Hannover, 2/12 December 1695.
5. JaB to GWL, Basel, 4/14 March 1696.
6. GWL to JaB, after 4/14 March 1696 and before June 1696.
7. GWL to JaB, Hannover, June 1696 (not sent?).
7. 1. GWL to JaB, Hannover, 13/23 September 1696 (no longer exists).
8. JaB to GWL, Basel, 27 January/6 February 1697.
9. GWL to JaB, Hannover, 15/25 March 1697.
9. 1. GWL to JaB, before 26 May 1699 (attested).
10. JaB to GWL, Basel, 15 November 1702
11. GWL to JaB, Berlin, April 1703 (p. 109).
12. JaB to GWL, Basel, 3 October 1703 (pp. 116–17).
13. GWL to JaB, Hannover, 26 November 1703 (pp. 123–24).
14. JaB to GWL, Basel, 20 April 1704 (pp. 128–29).
141. 14 2 GWL to JaB, before August 1704 (letters mentioned in 15).
15. JaB to GWL, Basel, 2 August 1704 (pp. 132–33).
151. GWL to JaB, before 15 October 1704 (mentioned in number 16).
16. JaB to GWL, Basel, 15 October 1704 (p. 134).
17. GWL to JaB, Berlin, 28 November 1704 (pp. 135–37).
18. JaB to GWL, Basel, 28 February 1705 (pp. 138, 141).
19. GWL to JaB, Hannover?, April 1705 (p. 143).
20. JaB to GWL, Basel, 25 April 1705.
21. JaB to GWL, Basel, 3 June 1705.
In addition to the letters between Jacob Bernoulli and Leibniz, the following letters are also relevant to understanding Leibniz’s point of view:
Leibniz to Thomas Burnett, 1/11 February 1697 (GP III, pp. 190–94).
Leibniz to Louis Bourguet, Vienna, 22 March 1714 (GP III, pp. 564–70).
1. The pamphlet corresponded to a live disputation taking place 9 September 1685 in which Jacob presided and Johann responded: Parallelismus ratiocinii logici et algebraici, quem, Una cum Thesibus Miscellaneis, Defendendum suscepit Par Fratrum Jacobus et Joannes Bernoulli, Ille Praesidis, Hic Respondentis vices agens. Ad diem 9 Septembris Anni M. DC. LXXV.
2. This, of course, is wrong by modern methods, since there are two ways to throw an 11 (the 5 can be on one die or the other) and only one way to throw 12.
3. From Bernoulli’s notebook Meditationes, p. 191: “NB. Hoc inventum pluris facio quam si ipsam circuli quadraturam dedissem, quod si maximè reperiretur, exigui usûs esset.”
4. The Ludophine numbers were involved in the decimal expansion of π, given by Ludoph van Ceulen (1540–1610), professor of mathematics at Leyden. They are referred to several times in the Leibniz-Jacob Bernoulli correspondence.
5. Leibniz to Johann Bernoulli, 14 July 1705: “Dominus Frater Tuus valde desiderat Schediasma olim a Pensionario De Witte typis editum circa reditus ad vitam. Habeo, sed in mole chartarum invenire non possum. Tibi ex Batavis nancisci eique adferre facile erit. Itaque rogo hanc mihi gratiam facias.” Johann, however, replied on 25 July 1705 that he was unable to get it, “Schediasma Pensionarii de Witt circa reditus ad vitam vix putem reperire posse. Agnatus aliquis meus, qui nuper ex Batavis venit, se illud ibi reperiri non potuisse mihi retulit. Inquiram tamen et ego, ubi ea transivero.”
6. Johann had written on 15 April 1709 (GM III, p. 842): “Quidam Remundus de Montmort scripsit nuper mihi se ad me missurum librum suum, cujus Titulus Essay d’Analyse sur les jeux de hazard; dubito autem an bene satis tractaverit hanc materiam. Fratris mei Filius ad prelum parat Dissertationem inauguralem Juridicam de simili materia, nempe De usu artis conjectandi in jure, ubi tractandus suscipit quaestiones varias in Jure agitare solitas, praecipue circa absentes pro mortuis habendos, reditus item vitales etc. adeo ut, quam ego olim ad Medicinam, ille nunc ad Jurisprudentiam non inutiliter applicare instituat Mathesin nostram, quod quidem apud Jurisconsultos (qui hunc tractandi modum insuper habent) aliquid novi et insoliti erit. Ubi prelum evaserit dissertatio, eam quoque ad Te mittam, modo mihi commodam mittendi viam indices. Spero Tibi illam non displicituram.” On 2 July 1709, he wrote: “Nunc Tibi mitto per eum, qui has ad Te deferet, repetens Patriam suam Daniam et ex Italia veniens, hanc, de qua nuper Tibi scripsi, Dissertationem De Usu artis conjectandi in Jure, quae si Tibi placuerit, erit de quo sibi gratulari queat Auctor; hunc ut commendatum habeas, enixe rogo” (GM III, p. 844). Leibniz replied on 6 September 1709: “In aestimandis reditibus ad vitam occupati olim fuere, cum Hugenio, Huddenius non tantum, sed et Pensionarius Johannes de Wit, cujus breve ea de re extat Schediasma Belgico Sermone, sed in usum popularium, ut intelligerent rei aequitatem. Quae de Amstelodamensi aestimatione refert Tuus ex Fratre nepos, credo ex Huddenianis computationibus fuisse profecta. Quidam Fergusonius, Belga ex Scotis oriundus, qui olim aliquo mensibus Hanoverae egit et librum quemdam Algebraicum Belgice edidit, multa mihi de Wittii et Hudenii in hac disquisitione meditationibus memorabat, etsi eorum methodos ratiocinationesque non teneret. Vellem talia ex schedis ipsorum eruerentur, neque enim dubito, quin plurima inde disceremus. . . . P.S. Mereri puto Domini Fratris Tui p.m. Librum, qui edatur, idque scribenti plus semel significavi. Saepe monui deesse nobis partem Logicae de gradibus verisimilitudinis; aestimandos autem censeo ex gradibus possibilitatibus, seu ex multitudine aequalium possibilitatum. Ostendi olim in schediasmate quodam politico Principis jussu edito, quasdam aestimationes fieri per additionem, quasdam per multiplicationem” (GM III, p. 844–45).
7. In his “Commentatiuncula de Judice Controversiarum” of 1669–71, Leibniz likewise wrote: “In other controversies which do not touch upon the foundations of faith, complete infallibility is not required, but only moral certitude, that is practical infallibility” (A VI, i, p. 554).