Pierre-Simon de Laplace Research Paper

1. Programme of Studies

As Professor of Mathematics at the Ecole Royale Militaire, Laplace was literally to inundate the Academie des Sciences with mathematical papers, some fifteen in four years, certain ones of which ran to well over one hundred pages in quarto. On 24 April 1773, he was elected adjoint mecanicien at the Academie des Sciences in Paris. The subjects which he tackled were those which he would continue to explore for the whole of his scientific life, a life which was to prove exceptionally long, especially as he would continue publishing original and fundamental papers right up to his death in Paris on 5 March 1827.

Becoming very quickly aware of his intellectual superiority, Laplace set about tackling all the problems of his day in the three principal academies where exact sciences were fostered—Paris, Berlin, and St Petersburg. Indeed, one need only read the titles of his first papers to get a good idea of the state of mathematical sciences in the last quarter of the eighteenth century. First and foremost came differential and integral calculus, and in particular the study of differential equations and partial differential equations which governed the mechanics of solids as well as fluids and the stars. In a paper presented in 1773 Laplace 1912, Vol. 9, pp. 5–68), it was Laplace who was the first to demonstrate that in very general cases one could determine the whole integral of linear partial differential equations, and that in exceptional cases it was still possible to obtain from them particular solutions in the form of defined integrals that is, integral transforms which fitted with the initial data. This method would allow him, for example, in 1809, to solve the equation of heat in more general cases than those envisaged by Fourier.

The second subject tackled by Laplace equally concerns the solutions of differential systems, but in those cases where whole integrability is beyond reach, such as in the theory of the Moon or the large planets. These theories had not only eluded Newton, who had not used the infinitesimal algorithm in its full extension, but had also eluded Euler, Clairaut, d’Alembert, and Lagrange who were ingenious rivals in their use of it. Laplace, right from the outset, was to demonstrate his extraordinary faculty—often spoken of in terms of intuition—for reducing by approximation the irreducible difficulties present in equations dealing with the problems of multiple bodies. In March 1773, Laplace presented a paper (Laplace 1912, Vol. 8, pp. 201–75) on the ‘principle of universal gravitation’ in which he showed that Euler and Lagrange had been wrong in their planetary theory and that he, Laplace, at the age of twenty-four, through Newton’s single principle that ‘the planets gravitate against each other,’ could show the absence of secular inequalities in planetary mean motions, so that the planets could not accelerate their motion around the sun irreversibly by themselves. Furthermore, they only deviated from their perfectly elliptical trajectories through periodic and temporary inequalities, thus assuring a stability in the ‘system of the world’ which Laplace would encounter again in all areas of his work. The mechanics of the planets is quasi-stable, and the God of Newton played no useful role in maintaining them within rational limits. The foundations for Laplace’s astronomical work were established, a work which he never once doubted and which he would pursue from then on, right up to his death.

The third subject, the ‘theory of chance,’ is more surprising coming from a protege of d’Alembert, but it was of great topical interest in 1770. It was in fact in this year that the great academies suddenly resumed this subject, which had to some extent been abandoned for fifty years, and which was now brought back into fashion by works in political arithmetic—notably those of Sussmilch—and at the same time by the requirements of astronomy. Lambert, Euler, Daniel Bernoulli, Lagrange, and Condorcet simultaneously published important papers on probability calculus. It was therefore inevitable that the young Laplace would turn his efforts to it, and his first contributions were magnificent, notably the great paper ‘on the probability of causes’ published in 1774 (Laplace 1912 Vol. 8 pp. 27–65), which marked the beginning of Laplacian mathematical statistics. In this paper, one of the classic works of the theory, Laplace developed the analytical results of the proportionality principle from the probability of causes to that of effects, and, for the first time, proposed, as a criterion for estimation, a quantity known only from observations subject to unavoidable errors, this criterion serving to minimize the sum of potential errors through their respective probability. This method, which Laplace called the method of situation and which appeared to him to be the result of the nature of things, should, he believed, be applied to research in astronomical fields. In fact, it led to formidable mathematical difficulties which Laplace would, in the next fifty years, work on breaking down into a series of fundamental works which would culminate in 1812 in the publication of the Analytical Theory of Probability (Laplace 1820).

Analysis, celestial mechanics, and the theory of probability were three overlapping subjects which Laplace did not invent and which he already found in the academic papers of his time, but which he grappled with and never abandoned, to such an extent that he was to change the view scholars had formed of the world which surrounded them, a physical world, of course, but also a moral world from the moment when it could be converted into numbers.

2. Celestial Mechanics

The years which were to follow were a long succession of scientific triumphs. One after another, Laplace solved the most difficult problems in celestial mechanics posed since Newton, problems which had foiled the attempts of the most eminent scholars of the eighteenth century to such an extent that it was sometimes doubted whether the single principle of attraction could really account for the inequalities of motion—such as the secular inequality of the Moon, the inequalities of Jupiter and Saturn, the theory of the satellites of Jupiter, and so on—which were being brought to light by the increasingly precise observations of modern astronomy. Laplace’s works as a whole—which were spread over twenty years—were brought together in a vast treatise, Celestial Mechanics (Laplace 1825), the publication of which began in 1799, and which Laplace would continue to expand and amend up to the end of his life. Concurrently, in 1796 he published Exposition of the System of the World (Laplace 1984), which was aimed at a wider readership and presented, in layman’s terms, the principal results which demonstrated with the utmost clarity that Newton’s law of universal gravity ‘represents all celestial phenomena, right down to their smallest details’ and that ‘through this, empiricism has been entirely removed from Astronomy, which is now a major problem of mechanics.’

However, Laplace was well aware of the fact that whole areas of the system of the world existed where empiricism seemed difficult to exclude entirely, such as the theory of oceanic or atmospheric tides; in other words, the areas where ‘irregular causes’ occurred in an anarchic fashion which could, if one did not bear them in mind, conceal the true causes of phenomena and their fixed order. Laplace encountered this problem from the beginning of the 1770s to the moment when the theory of errors began to take shape. Observations of all parts of nature, even the most precise, were affected by ‘unavoidable errors’ which could prove themselves predominant if left to accumulate by chance, and remove all scientific value from the apparent objectivity. According to Laplace, it was therefore necessary to multiply the observations, adjust the value of the uncertain elements progressively, and reveal gradually the phenomena which were ‘until this point shrouded in the errors of observations.’ This progressive revealing could not be done without a method, and this is precisely the purpose of Laplace’s probability calculus.

3. The Analytical Theory Of Probability

For Laplace, probability theory was first conceived as a means of critically analysing data from observation, a critical analysis which gradually became more refined and revealed the true system of the world. Two periods in Laplace’s development are usually distinguished. The first, which runs from 1774 to 1785, saw the development of Laplace’s first method. In the spirit of Bayes, but essentially asymptotic in that it was independent of the special nature of a priori laws, it did not allow Laplace to deal with the major problem of the theory of errors, that is, to determine in the best possible way the elements known from a large number of indirect observations. It did, however, allow him to solve the population theory problems posed by Daniel Bernoulli and Condorcet by giving political arithmetic the scientific rigor it lacked, left open as it was to the most trivial empirical digressions; for example, in deciding whether in truth more boys were born in London than in Paris for the same number of births, or even whether the population of France was increasing or decreasing.

The second Laplacian asymptotic theory, this time non-Bayesian, began in 1810 after its author had solved one of the major problems of probability theory, the central limit problem as it is called today, where under very general conditions the total sum of errors is asymptotically normal, which allows the calculation of probability even if the particular law of errors is unknown. This remarkable theorem and its applications in the method of least squares, published the following year, makes up Chapter IV of Book II of the Analytical Theory of Probability, published by Laplace in 1812 (Laplace 1820), and bringing together his work on probability as a whole. This second great treatise from Laplace would be expanded with various different chapters until 1825. From 1816, Laplace gradually extended the applications of probability calculus into increasingly wider fields, anticipating a large number of the applications which the twentieth century would develop. The theory of probability, therefore, became for Laplace ‘the most felicitous addition to the ignorance and weakness of the human spirit’ as he asserts in conclusion to his Philosophical Essay on Probability (Laplace 1986), the first edition of which dates back to 1814 and the last to 1825.

Laplace did not limit the possibilities for the application of probability calculus to traditional fields such as games, the theory of errors, and annuities. From the moment when a shred of calculation seemed to him possible, whatever the field and however tenuous it may be, he got to grips with it and generally exceeded previous results by a great deal. For example, inspired by the works of Daniel Bernoulli, Laplace moved from population theory to a general reflection on the chance series of draws from a ballot box which, at first, only display disorder and chaos and gradually adjust and give way to the simplest and most admirable order, the irregular causes cancelling each other out to display the eternal action of constant causes. These reflections on order in chaos anticipate the work on statistical physics at the beginning of the twentieth century. What is more, Laplace, inspired by Condorcet, proposed a theory of decisions in majority voting where the basic concepts of modern test theory can be seen. Further still, inspired once again by Daniel Bernoulli, Laplace established a general result for the concave utility functions, and applied it to questions of maritime insurance and so on. One can also see many more things in the Analytical Theory (Laplace 1820), which make this work a monument of science for all time.

4. Career

After the flamboyant beginnings we have seen, Laplace was very quickly recognised by his peers, and if his arrogance had caused him some trouble at the beginning, he had an exceptional career. Elected in 1783 to the position of Associate Academician, he became a member of the Academie des Sciences in Paris in 1785. During the French Revolution, Laplace retired to a property near Paris to draft out Celestial Mechanics and kept himself at a distance from events. But with the fall of Robespierre, he returned to Paris and took an active part in the creation of three great scholarly institutions emerging out of the Revolution—the Institut de France, the Bureau de Longitudes, and the Ecole Polytechnique. Bonaparte, who was concerned about his image and the scientific glory of France, was to make Laplace, albeit short-lived, his First Minister of the Interior and soon one of the highest dignitaries of the Empire, the Chancellor of the Senate and Count of the Empire.

In 1814 Laplace voted with the majority of the Senate for the forfeiture of the Emperor. He was immediately made a peer of France, then marquis, by King Louis XVIII. In the numerous positions he held, Laplace made himself the herald and the defender of exact sciences. He involved himself little in politics, and when he did, only with questions he could reduce to numbers.

In his lifetime, Laplace enjoyed an immense scientific prestige. As founder of the Societe d’Arcueil in 1807, he addressed all the scientific debates of his time, and created a brilliant school which would make his mark on part of his century, up to the point at which Laplacian physics of corpuscular fluids was superseded by positive mathematical physics and experimental physics (on this subject, see Gillispie 1997).

In 1788, Laplace married Charlotte de Courty (1769–1862) with whom he had two children, Charles (1789–1874), general and peer of France who died without descendants, and Sophie (1792–1813) who died in childbirth. Angelique (1813–1889), the only daughter of Sophie, was brought up by her grandparents and married the marquis de Colbert Chabanais and had three children, through whom the family line was established.

5. Influence

Laplace was one of the most influential men of science of all time, as much through his scientific work as through his two expository works which met with considerable success. There was no public or private library which did not possess them, and Laplace remains among the most cited scholars in present-day literature. His contribution to the formal assertion of mechanical determinism is sometimes summarized thus: ‘All events, even those which by their insignificance do not seem to be due to the great laws of nature, are a consequence of them which is as necessary as the revolutions of the sun.’ This introductory sentence from the Philosophical Essay (Laplace 1986) of 1814 which can already be found in one of Laplace’s first philosophical texts written in 1773, is without doubt the most frequently occurring scholarly quotations in the whole of world literature. In fact, as is well known, the assertion of determinism is a recurrent theme in the history of human thought, and countless authors in the eighteenth century in particular could be credited with it. Laplace’s originality lies in producing a brilliant proof; that the tiniest disturbances in the system of the world—for example the slow acceleration of the Moon—are a necessary consequence of the single best established law of nature, that of universal gravity. It was because such a deduction was possible, as Laplace demonstrated, that determinism could be asserted and, even more, that it was then a question of one of the highest scholarly ambitions of humanity. Science in the nineteenth century became a race for precision made possible by the local determinism of phenomena in the whole of nature.

However, Laplace’s influence extends well beyond that, particularly in the social and moral sciences of the nineteenth century, most notably in the work of Quetelet. Indeed, ‘Laplacian determinism’ was not limited to determinist phenomena. In the system of the world, there are numerous phenomena which are subject to irregular accidental causes which constantly alter the course of these phenomena. Necessity, order, and determinism are nevertheless not absent from them. For example—and it is an example dear to Pastor Sussmilch which Daniel Bernoulli took up again—the sex at birth does not seem to be subject to any determinism, yet one observes that always or almost always slightly more boys are born than girls as soon as one considers quite a large number of births. Laplace concludes that the invariability involved here is ‘a result of regular causes which override the anomalies due to the chance’ of irregular causes Laplace 1912, Vol. 7, p. xlviii. Chance is thus furnished with a special meaning; it is the combination of irregular causes which take effect in all directions, without favouring any one in particular. Regular causes, on the other hand, compel nature along enforced paths which become apparent in the long term. This is Laplace’s theory of constant causes which one can clearly see in the works of other authors, for example Buffon, Moivre, and the Bernoullis, but also in the works of the ancient Stoics. It is Laplace, however, who, just as in the case of determinism in minor astronomical perturbations, provides this theory with a mathematical provision which justifies its implementation. This is the full import of his theorem of 1810; the variations due to irregular causes are enclosed in narrow limits determined by the Laplace formula, which gradually move closer to genuine elements in such a way that the constant causes determine them. The analysis of causes which Adolphe Quetelet undertook of the ‘phenomena which organised beings present us with’ is no different, except in that it integrates into the Laplacian constant causes–accidental causes schema a third type of causes, variable causes which, for example, produce seasonal effects, causes which Poisson, Cournot, and Bienayme had envisaged in order to account for irregularly periodic statistical data. Quetelet’s ‘average man,’ and the ‘tendencies to crime,’ can only be understood in the perspective of the constant causes in Laplace’s Essay. In a still more obvious way, Poisson’s ‘law of large numbers’ is a triumphant metaphor for Laplacian theory, or at least for certain aspects of it—showing as it does the order of constant causes within the apparent chaos of accidental causes.

The decision-making aspect of Laplace’s work—the theory of probability seen as ‘common sense reduced to calculation’—experienced a less glorious fate. After the Laplace school—Poisson, Cournot, and Bienayme—had attempted to give it a more acceptable form in the first half of the nineteenth century, it was for a long time rejected, and with this rejection went an increasingly significant part of Laplace’s work on probability. Causes were multiple and, moreover, ‘variable’ since this aspect a century later was reinstated at the highest level in the way that it is best known today—the theory of games, the theory of decisions, the theory of tests, and judgments based on samples. For Laplace, chance was also the name for our ignorance, an ignorance which was often incurable owing to those multiple accidental causes which distort the natural order of things and of our reason, so that the eminent role of the theory of chance, of probability calculus, is to ‘criticise’ the statistical data from all parts of nature; and even in the absence of data, by analogy, image, model, and calculation when possible, the theory still allowed one to find one’s way more confidently through the maze of social and economic life. In France one would have to wait for Borel and his theory of games before one considered afresh along with Laplace that the theory of probability ‘leaves nothing arbitrary in the choice of opinions and in the choice of decisions to make, every time one can determine, in one’s way, the most advantageous choice.’ Laplace’s legacy—to perfect everything through the powerful means that mathematical analysis places at the scholar’s disposal—is in no danger of being forgotten.

References:

1. Gillispie C C 1997 Pierre-Simon de Laplace 1749–1827. A Life in Exact Science. Princeton University Press, Princeton, NJ
2. Hald A 1998 A History of Mathematical Statistics from 1750 to 1930. Wiley, New York
3. Laplace P-S de 1820 Theorie Analytique des Prob (Analytical Theory of Probability). Ve Courcier, Paris
4. Laplace P-S de 1825 Traite de Mecanique Celeste (Celestial Mechanics). Gauthier-Villars, Paris
5. Laplace P-S de 1912 Oeuvres Completes de Laplace (Complete Works), 14 vol, (1878–1912). Ve Courcier, Paris
6. Laplace P-S de 1984 Exposition du Systeme du monde (The System of the World ). Artheme Fayard, Paris
7. Laplace P-S de 1986 Essai philosophique sur les probabilities (Philosophical Essay on Probability). Bourgois, Paris
8. Stigler S M 1986 The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press, Cambridge, MA