home email

Gottlieb Wilhelm Leibniz

b. 1 July 1646
d. 14 November 1716

Introduction

Leibniz was a philosopher, mathematician, logician and political advisor. For our purposes we select from among his voluminous writings those parts most directly related to probability theory. This includes mention of his analyses of games, combinatorics, annuities, and correspondence with others. 

Works

We list in approximate chronological order the writings of Leibniz related to probability. In the Opuscules et Fragments Inédits de Leibniz of Louis Couturat published in 1903 we have an inventory of the manuscripts residing at the Royal Library at Hanover together with short extracts from them. More recently, Marc Parmentier has collected and published in French translation 21 selected writings related to games and annuities. See L'estime des apparences: 21 manuscrits de Leibniz sur les probabilités, la théorie des jeux, l'espérance de vie, Vrin, 1995. Few of these writings were published in his lifetime. Parmentier's inventory is as follows with notes taken from Couturat.

Combinations

In 1666 was published the Dissertatio de ArteCombinatoria of Leibniz. This is the earliest work of him connected to mathematics, but it is of little interest. More properly it should be classified among his philosophical works.

Correspondence on the Ars Coniectandi

In 1697, Johann Bernoulli, brother of Jacob, exchanged letters with Leibniz in which Johann informed Leibniz that Jacob had been writing a treatise on probability called the Ars Conjecturandi and which concerned applications to life beyond the study of games. Leibniz replied that he himself had once considered such things and hoped that mathematicians would pursue this line of inquity.

Six years later Leibniz returns to the subject in a letter to Jacob Bernoulli. Within the two year period of April 1703 and April 1705 Jacob Bernoulli and Leibniz exchanged a number of letters regarding topics relevant to the Ars Conjectandi.

Nouveaux Essais sur l'Entendement Humain

The English philosopher John Locke wrote An Essay Concerning Human Understanding Vol. I and Vol. II which appeared in 1690. As early as 1696, Leibniz engaged in commentary on this work, but it was with the subsequent publication of the French translation of it by Pierre Coste in 1700 that Leibniz gained full access to Locke's work. In response to it, Leibniz ultimately wrote his Nouveaux Essais sur l'Entendement Humain as a dialogue between himself and Locke. The actors Philalethes and Theophilus represent Locke and Leibniz respectively. This work was likely completed in its present form around 1709 but it was not published until 1765. It may be found printed in Gerhardt's Leibnizen Philosophische Schriften Vol. V. An English translation as New Essays Concerning Human Understanding by Alfred Langley published in 1896 is available as well as others more recent. From Langley, we select two passages out of  Book IV Of Knowledge. These are Chapter XV to illustrate his thoughts on probability and a portion of Chapter XVI.

OF PROBABILITY
§ 1. Philalethes. If demonstration shows the connection of ideas, probability is nothing else than the appearance of this connection based upon proofs in which immutable connection is not seen.  § 2. There are several degrees of assent from assurance down to conjecture, doubt, distrust. § 3. When there is certainty, there is intuition in all parts of the reasoning which show its connection; but what makes me believe is something extraneous. § 4. Now probability is grounded in its conformity with what we know, or in the testimony of those who know.

 Theophilus. I prefer to maintain that it is always grounded in likelihood (vraisemblance) or in conformity with the truth; and the testimony of another is also a thing which the truth has been wont to have for itself as regards the facts that are within reach. It may be said then that the similarity of the probable and the truth is taken either from the thing itself, or from some extraneous thing. The rhetoricians emply two kinds of arguments: the artificial, drawn from things by reasoning, and the non-artificial, based only upon the express testimony either of man or perhaps also of the thing itself. But there are mixed arguments also, for testimony may itself furnish a fact which serves to from an artificial argument.

§ 5. Ph. It is for lack of similarity to truth that we do not readily believe that which has nothing like that which we know. Thus when an ambassador told the king of Siam that with us the water was so hardened in winter that an elephant might walk thereon without breaking through, the king said to him: Hitherto I have believed you as a man of good faith; now I see that you lie. § 6. But if the testimony of others can render a fact probable, the opinion of others should not pass of itself as a true ground of probability. For there is more error than knowledge among men, and if the belief of those whom we know and esteem is a legitimate ground of assent, men have reason to be Heathen in Japan, Mahometans in Turkey, Papists in Spain, Calvinists in Holland, and Lutherans in Sweden.

Th. The testimony of men is no doubt of more weight than their opinion, and in reason it is also the result of more reflection. But you know that the judge sometimes makes them take the oath de credulitate, as it is called; that in the examinations, we often ask witnesses not only what they have seen but also what they think, demanding of them at the same time the reasons of their judgment, and whether they have reflected thereupon to such an extent as behooves them. Judges also defer much to the views and opinions of experts in each profession; private individuals, in proportion as it is inconvenient for them to present themselves at the appropriate examination, are not less compelled to do this. Thus a child, or other human being whose condition is but little better in this respect, is obliged, whenever he finds himself in a certain situation, to follow the religion of the country, so long as he sees nothing bad therein, and so long as he is not in a condition to find out whether there is a better. A tutor of pages, whatever his sect, will compel them each to go to the church where those who profess the same belief as this young man go. The discussions between Nicole and others on the argument from the great number in a matter of  faith may be consulted, in which sometimes one defers to it too much and another does not consider it enough. There are other similar prejudgments by which men would very easily exempt themselves from discussion. These are what Tertullian, in a special treatise, calls Prescriptiones, [De Praescriptione Haereticorum] availing himself of a term which the ancient jurisconsults (whose language was not known to him) intended for many kinds of exceptions or foreign and predisposing allegations, but which now means merely the temporal prescription when it is intended to repel the demand of another because not made within the time fixed by law. Thus there was reason for making known the legitimate prejudgments both on the side of the Roman Church and on that of the Protestants.  It has been found that there are means of opposing novelty, for example, on the part of both in certain respects; as, for example, when the Protestants for the most part abandoned the ancient form of ordination of clergymen, and the Romanists changed the ancient canon of the Old Testament books of Holy Scripture, as I have clearly enough shown in a discussion I had in writing, and from time to time, with the bishop of Meaux, whom we have just lost, according to the news which came some days since. although it presents suspicion of error in these matters, is not a certain proof thereof.
ON THE DEGREE OF ASSENT
Th. ... The mathematicians of our times have begun to calculate chances upon the occations of games. Chevalier de Méré, whose "Agrémens"and other works have been printed, a man of penetrating mind who was both a player and a philosopher, gave them an opportunity by forming questions regarding the profits in order to know how much the game would be worth, if interrupted at such or such a stage. In this way he induced Pascal, his friend, to examine these things a little. The question made a stir and gave Huygens the opportunity to produce his treatise "de Alea." Other learned men entered into the subject. Some principles were established of which the Pensioner De Witt also availed himself in a brief discourse printed in Dutch on annuities. The foundation on which they have built goes back to the prosthaphaeresis, i. e. the taking of an arithmetical mean between several equally receivable suppositions. Our peasants also have made use of it for a long time according to their natural mathematics. For example, when some inheritance or land is to be sold, they form three bodies of appraisers; these bodies are called Schurzen in Low Saxon, and each body makdes an estimate of the property in question. Suppose, then, that the first estimates its value to be 1000 crowns, the second 1400, the third 1500; the sum of these three estimates is taken, viz. 3900, and because there were three bodies, the third, i.e. 1300, is taken as the mean value asked for; or rather, they take the sum of the third part of each estiamte which is the same thing. This is the axiom: aequalibus aequalia, equal suppositions must have equal consideration. But when the suppositions are unequal they compare them with each other. Suppose, for example, that with two dice, the one ought to win if it makes 7 points, the other if it makes 9, the question is asked what proportion obtains between their probabilities of winning? I reply that the probability of the last is worth only two-thirds of the probability of the first, for the first can make 7 in three ways with two dice, viz.: by 1 and 6, or 2 and 5, or 3 and 4; and the other can make 9 in two ways only, by throwing 3 and 6, or 4 and 5; and all these methods are equally possible. Then the probabilities, which are as the numbers of equal possiblilities, will be as 3 to 2, or as 1 to 2/3. I have more than once said that a new kind of logic would be required which would treat of the degrees of probability, since Aristotle in his "Topics" has done nothing less than this, and has contented himself with putting in a certain order certain popular rules distributed according to the topics, which may be of use on some occasion where the question concerns the amplification of the discourse and the giving to it probability without putting it to the trouble of furnishing us a necessary balance for weighing probabilities and forming thereupon a solid judgment. It would be well for him who should treat of this matter to pursue the examination of games of chance;  and in general I wish that some skillful mathematician would produce an ample work with full details and thoroughly reasoned upon all sorts of games, which would be very useful in perfecting the art of invention, the human mind appearing to better advantage in games than in the most serious matters.

Mathematische Schriften

Leibniz' Mathematische Schriften was edited by Gerhardt, published at Berlin between 1849 and 1863. These volumes are