NOTES Titles of books, papers and journals abbreviated here are given more fully in the bibliography. The following abbreviations for Laplace's works are used throughout these notes: Essai - Essai Philosophique sur les Probabilites. T.A.P. I. (or II.) - Theone Analytique des Probabilites, Book I (or II), as reprinted in Volume 7 of the (Euvres Completes de Laplace. When "Supple­ ment" is added to "T.A.P." , the reference is to the appropriately numbered Supplement appearing in the third edition of the T .A.P. as printed in the (Euvres Completes de Laplace. o. C. - (Euvres Completes de Laplace. Le~on - Sur les probabilites, a lecture given in the tenth session of the Le~ons de Mathematiques. The 1932 German translation of the Essai, edited by R. von Mises, will be referenced throughout as ''von Mises." A reference "see Note n" will always indicate Note n to the article cur­ rently under comment. My indebtedness to the writer (Bernard Bru) of the notes in the Bru/Thom edition [1986] of the Essai will be evident to anyone who has studied that work. The notes by Hilda Pollaczek-Geiringer in von Mises [1932] have also been useful; these notes should be referred to for details of the "collective" approach to probability. 125 126 Notes: Foreword Foreword 1. This lecture, originally entitled "Sur les probabilites," was given in the tenth session of the Le~ons de Mathimatiquesj it was published in 1812. An early version of the opening part of the Essai was published in 1810j it is reprinted in full in Gillispie [1979]. 2. Writing of Laplace's work on the apparent anomalies of the mean motions of Jupiter and Saturn, Wilson explains that, in Laplace's terminology, Analyse refers primarily to the set of operational symbols and sym­ bolic procedures developed by Leibniz, the Bernoullis, Eu­ ler, and others, for the formulation and solution of differen­ tial equations. Although the Analyse employed by Laplace consisted of algorithmic processes, their successful use was far from being merely a matter of calculation. The user needed to assess and choose among procedures with a clear idea of the goal at which he aimed, the kind of solution to be expected or hoped for. [1985, pp. 24-25] However, not too much violence will be done to this notion if we think of Analyse as "mathematics." 3. The word "chance," in French as in English, is patient of a number of interpretations, the most common being "probability" and "for­ tuitous event." It is with the latter meaning that Laplace uses the word in the Essai. The use of "chances" in the sense of ''things which may befall" was not uncommon in the 18th and 19th centuries. For example, Emerson writes Chance is an event, or something that happens without the design or direction of any agentj and is directed or brought about by nothing but the laws of nature. [1776, p. 2] This is perhaps why Bayes felt it necessary to state in his Essay of 1763 "By chance I mean the same as probability" [po 376]. On probability 1. A similar sentiment had been expressed earlier by David Hume, who, in Section VIII of his Inquiry Concerning Human Understanding, wrote "It is universally allowed, that nothing exists without a cause of its existence" [Hume, 1894, p. 366]. Laplace is using "cause" in a fairly usual sense here. Keynes [1973] notes the "metaphysical diffi­ culties which surround the true meaning of cause" [po 306], and goes on to say Notes: On probability 127 I have followed a practice not uncommon amongst writers on probability, who constantly use the term cause, where hypothesis might seem more appropriate. [loco cit.] De Morgan is even more forthright: he says By a cause, is to be understood simply a state of things antecedent to the happening of an event, without the in­ troduction of any notion of agency, physical or moral. [1838, p. 53] Quetelet in fact distinguishes three principal classes of causes, viz. Constant causes are those which act in a continuous man­ ner, with the same intensity, and in the same direction. Variable causes act in a continuous manner, with ener­ gies and tendencies which change either according to de­ termined laws or without any apparent law ... Accidental causes only manifest themselves fortuitously, and act indifferently in any direction. [1849, p. 107] One is tempted to say of cause what Mr. Pickwick said of politics, viz. "The word ... comprises, in itself, a difficult study of no incon­ siderable magnitude." (See Chap. XV of Dickens [1837].) 2. What appears in earlier editions as un effet ("an effect") appears mistakenly in the Bru/Thom edition as en effet ("indeed"). 3. See Leibniz [1710]. The following extracts are pertinent: (a) All we have just said likewise agrees perfectly with the maxims of philosophers, who teach that a cause could not operate, without having a disposition to the action; and it is this disposition that contains a predetermina­ tion, be it what the agent has received from without, or be it what he has had by virtue of his own previous constitution. [§46] (b) Therefore I acknowledge indifference only in one sense, which it gives notice of in the same way as contingency, or non-necessity. But as I have carefully explained more than once, I do not acknowledge an indifference of equi­ librium, and I do not believe that one ever chooses, when one is completely indifferent. Such a choice would be a kind of pure chance, without conclusive reasoning, and as much visible as hidden. But such a chance, such an absolute and real fortuitousness, is a chimera that is never found in nature. Wise men all agree that chance is only a specious thing, like fortune; it is ignorance 128 Notes: On probability of causes that leads to it. But if there had been such a vague indifference, or else if one had chosen with­ out there having been anything to lead one to choose, the chances would be something real, similar to what is found in that small change of direction in particles, occurring for no rhyme or reason, to the feeling of Epi­ curus, who introduced it to eschew necessity, of which Cicero has justifiably made so much fun. [§303] 4. Laplace had expressed this sentiment before: in his [1773] he wrote The present state of the system of nature is clearly a sequel to what it was a moment before, and, if we conceive of an intelligence that, at a given instant, encompasses all the relationships of the beings of this universe, it would be able to determine at any time whatsoever, in the past or in the future, the respective position, the movements and generally the attachments of all these beings. [O.C. VIII, p.I44] Laplace's position was by no means unreservedly accepted. Poincare in fact wrote A very slight cause, which escapes us, determines a con­ siderable effect which we can not help seeing, and then we say this effect is due to chance. If we could know exactly the laws of nature and the situation of the universe at the initial instant, we should be able to predict exactly the sit­ uation of this same universe at a subsequent instant. But even when the natural laws should have no further secret for us, we could know the initial situation only approxi­ mately. If that permits us to foresee the subsequent situa.­ tion with the same degree of approximation, this is all we require, we say the phenomenon has been predicted, that it is ruled by laws. But this is not always the case; it may hap­ pen that slight differences in the initial conditions produce very great differences in the final phenomena; a slight error in the former would make an enormous error in the latter. Prediction becomes impossible and we have the fortuitous phenomenon. [1946, pp. 397-398] 5. The word ''world'' may be interpreted throughout this translation in the broad sense of ''universe.'' 6. An anonymous tract of 1716 was devoted to a discussion of the evils to be expected from the appearance of a particularly noteworthy aurora borealis. De Morgan comments on the work as follows: Notes: On probability 129 The prodigy, as described, was what we should call a very decided and unusual aurora borealis. The inference was, that men's sins were bringing on the end of the world. [1915, vol. I, p. 134) 7. The evil effects foreshadowed by the appearance of comets had long been recognized. In his essay entitled An Excellent Discourse of the Names, Genus, Species, Efficient and Final Causes of all Comets, &c., Wharton wrote It has been a received Opinion in all Ages, that Comets are certain FUnebrious Appearances, secret Fires and Torches of Death rather than of Life, and were ever look'd upon as the threatening Eyes of Divine Vengeance, and the Tongue of an Ireful Deity, portending the Death of Princes, Plague of the People, Famine, and Earthquakes, with horrible and terrible Tempests. [1683, p. 159) As holders, or at least reporters, of these opinions Wharton cited Aristotle, Cicero, Pliny, the Holy Fathers (Tertullian, St Augustine), meteorologers (Fromandus) and astronomers (Tycho Brahe, Kepler). He also noted the role played by the fixed stars near to which the comet appears - for example, The Dominion of Mercury portends great Calamity unto all those that Live by their own Industry, and such as love and favour the Muses, with the Death of some great Personage, Wars, Famine, and Pestilence; of Diseases, the Phrenzy, Lethargy, Epilepsie, and griefs of the Head.