Laplace, Pierre Simon de (1749–1827)

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LAPLACE, PIERRE SIMON DE
(1749–1827)

Pierre Simon de Laplace, the French astronomer and mathematician famous for his celestial mechanics and theory of probability, was born in Normandy. Upon coming to Paris, he attracted the attention of Jean Le Rond d'Alembert, who found him employment in the École Militaire. Here he taught mathematics to trainee artillery officers, among whom was Napoleon Bonaparte. When the revolutionary government established the École Polytechnique, Laplace was one of its founding professors. He served with distinction on many of the great committees of the French Academy of Sciences and of the government. He helped devise the meter, standardized weights and measures, and worked out an ingenious system of sampling to provide an economical and efficient census. The elegance of his mathematical work has yet to be rivaled, and his power of analysis is matched only by that of Isaac Newton and Joseph-Louis Lagrange. His philosophical opinions, especially those in his Exposition du système du monde (The System of the World ) and Essai philosophique sur les probabilités (A Philosophical Essay on Probabilities ), have a bluntness and clarity of expression that ensured their popularity.

Laplace's adult life was passed in conditions of civil strife and sometimes of chaos, but despite his revolutionary affiliations, the restoration of the Bourbons brought him neither poverty nor disgrace; he died honored by all, a newly created marquis. Against this background of political confusion, he came to believe that the theory of probability, properly and widely applied, would reduce most of the problems of society (like the attainment of justice) to something manageable; with the help of probability theory, he believed, a man of delicate intuition and wide experience could find practical solutions to most social difficulties.

Laplace's scientific work had a strong element of tidiness about it. It consisted largely of the final polishing of the Newtonian enterprise, knitting up its loose ends. Using the improved calculus devised by his colleagues, particularly Lagrange, he removed all known errors from, and explained all known anomalies in, the Newtonian cosmology and physics. It seemed to Laplace that there was no phenomenon that the improved and polished Newtonian physics was incapable of handling. He came to regard the enormous explanatory power of the system as practically a demonstration of its truth. New observations would only confirm it further, he thought, and their consequences were as certain as if they had already been observed.

What had produced this remarkable confidence was a series of complete successes. Newton had never been convinced of the stability of the solar system, which he suggested might need divine correction from time to time. Laplace showed, in effect, that every known secular variation, such as the changing speeds of Saturn and Jupiter, was cyclic and that the system was indeed entirely stable and required no divine maintenance. (It was this triumph that occasioned his celebrated reply to Napoleon's query about the absence of God from the theory; Laplace said that he had no need of that hypothesis.) He also completed the theory of the tides and solved another of Newton's famous problems, the deduction from first principles of the velocity of sound in air. Laplace added a very accurately estimated correction for the heating effect produced by rapidity of the oscillation, which was too short to allow the heat of compression to be dissipated.

Determinism and Probability

Not only was Laplace confident of the Newtonian theory, but he was also greatly struck by its determinist nature. Where one could gather accurate information about initial conditions, later states of a mechanical system could be deduced with both precision and certainty. The only obstacle to complete knowledge of the world was ignorance of initial conditions. Laplace's confidence in Newtonian theory is exemplified in the introduction to his Philosophical Essay on Probabilities, in which he envisaged a superhuman intelligence capable of grasping both the position at any time of every particle in the universe and all the forces acting upon it. For such an intelligence "nothing would be uncertain and the future, as the past, would be present to its eyes. The human mind offers, in the perfection which it has been able to give to astronomy, a feeble idea of this intelligence" (Philosophical Essay, p. 4).

But this ideal is difficult to attain, since we are frequently ignorant of initial conditions. The way to cope with the actual world, Laplace thought, is to use the theory of probability. The superhuman intelligence would have no need of a theory of probability. Laplace would have regarded as ridiculous the idea that there could be systems that would react to stimuli in only more or less probable ways. He said, "The curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits; the only difference between them is that which comes from our ignorance" (Philosophical Essay, p. 6). He then defined a measure of probability as follows:

The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible … and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible. (Philosophical Essay, p. 6)

This is the definition of probability known today as the proportion of alternatives. Then as now, it involves the very tricky notion of equipossible cases. Laplace deals with this notion by glossing equipossible cases as those that "we may be equally undecided about in regard to their existence" (Philosophical Essay, p. 6).

This account does have its difficulties. Equal indecision is not at all easy to determine and may, in the end, hinge upon states of mind quite irrelevant to a sound estimate of probabilities. Throughout his study of probability Laplace refers to such subjective factors as honesty, good judgment, and absence of prejudice, which are required in using probability theory. However, he does give a much sounder criterion for its practice; it encourages one to reckon as equally possible those kinds of events instances of which we have no special reason to believe will occur. Equality of ignorance then becomes his criterion for equality of possibility. Laplace is quite happy about this, since he believed—perhaps rightly—that the proper occasion for the recourse to probability is ignorance of the initial conditions, the relevant theory, or both. Actual estimates of probability are made statistically. In his practical examples he appears to depend on a further distinction, which also seems correct. It is the distinction between the meaning of the statement of probability for a certain kind of event (that is, ratio of number of favorable to equipossible kinds of events) and the usual estimate of this probability, which is the relative frequency of actual events of the kind under consideration among all appropriate cases.

Applications of Probability

Laplace made several practical applications of probability theory. In science he applied it to the problem of sampling for the census and to the theory of errors; to both of these studies he made valuable contributions. He also believed that probability theory would have great utility in the moral sciences. He studied the optimum size for a jury to give the least doubtful verdict and the voting procedures of assemblies both on candidates for office and on propositions. He discussed the advantages and disadvantages of voting by ranking in order of merit and of voting by the knockout majority system. In this study and in his reflections on what it is reasonable to risk and in what kind of game, one gets the occasional glimpse of Laplace's basic moral principle, "Only bet on a reasonably sure thing."

Philosophy of Science

In his philosophy of science and in his views on the nature of scientific method, Laplace expressed himself somewhat along the same lines as Newton, but more liberally. He saw quite clearly that science is not the accumulation of isolated and particular items of information. "It is by comparing phenomena together, and by endeavouring to trace their connection with each other, that he [man] has succeeded in discovering these laws, the existence of which may be perceived even in the most complicated of their effects" (System, Vol. I, p. 205). In searching for connections we do not need to shun hypotheses. Laplace said of hypotheses what Newton should have said, considering the use he made of them: that if we refuse to attribute them to reality and regard them merely as the means of connecting phenomena in order to discover the laws (which we correct according to further observations), they can lead us to the real causes or at least enable us to infer from observed phenomena those which given conditions ought to produce.

In fact, it is by excluding on the basis of decisive experiments all those hypotheses that are false that "we should arrive … at the true one." Ideally, Laplace sees scientific method as the formulation of generalizations of connection between phenomena, proceeding inductively from phenomena to laws (which are the ratios connecting particular phenomena), and from these to forces. When these forces reveal some general principle, that principle is verified by direct experience, if possible, or by examination of its agreement or disagreement with known phenomena.

Testing consists both of trying to formulate a deductive system based upon the highest hypotheses and designed to explain the phenomena, "even in their smallest details," and of seeing whether the theory agrees with as varied and as numerous phenomena as are relevant to it. If a theory passes these tests, it "acquires the highest degree of certainty and of perfection that it is able to obtain."

Laplace saw that our confidence in predictions had to be based upon confidence in some principle of the uniformity of nature. The sources of his confidence in some principle of uniformity were twofold. First, there is the condition of the absence of interference. If there is no reason why a change should occur, a change will not occur—a principle deeply embedded in Newtonian science. As Laplace put it, "Being assured that nothing will interfere between these causes and their effects, we venture to extend our views into futurity, and contemplate the series of events which time alone can develop" (System, Vol. I, p. 206). Second, simplicity was to be regarded as a mark of future reliability. The principle of induction, said Laplace, is that "the simplest ratios are the most common." He said, too, "We judge by induction that if various events, movements for example, appear constantly and have been long connected by a simple ratio, they will continue to be subjected to it" (Philosophical Essay, p. 178). The theory of probability supplies a connection between the two sources of confidence, for, said Laplace, we conclude from the fact that a simple ratio is found among quantities in nature "that the ratio is due, not to hazard, but to a regular cause." Thus, if no other causes intervene, we may expect a likeness of effects, in fact, a uniformity of nature.

Summing up scientific method, Laplace said, "Induction, analogy, hypotheses founded upon facts and rectified continually by new observations, a happy tact given by nature and strengthened by numerous comparisons of its indications with experience, such are the principal means for arriving at truth" (Philosophical Essay, p. 176).

Bibliography

works by laplace

Oeuvres complètes. 14 vols. Paris: Gauthier-Villars, 1878–1914.

Exposition du système du monde. Paris, 1798. Translated by J. Pond as The System of the World. London: R. Phillips, 1809. Contains two important conjectures: that the planets might have been formed by the condensation of a large, diffuse solar atmosphere as it contracted, and the hypothesis, since confirmed, that the nebulae are clouds of stars and that the Milky Way is our view of that nebula of which our sun is a star.

Traité de la mécanique céleste, 5 vols. Paris: Chez J.B.M. Duprat, 1799–1825. Translated by N. Bowditch as Mécanique Céleste, 4 vols. Boston: Hillard, Gray, Little, and Wilkins, 1829–1839.

Théorie analytique des probabilités. Paris: Courcier, 1812.

Essai philosophique sur les probabilités. Paris: Courcier, 1814. Translated by F. W. Truscott and F. L. Emory as A Philosophical Essay on Probabilities. New York: Wiley, 1902; New York: Dover, 1951. A semipopular introduction to the Théorie analytique.