Mathematics and Revelation

Laplace experienced a second major change in his life that affected his career and provided an opportunity to develop his view on the calculating spirit. He suffered through the trauma of the French Revolution. Until the eve of the Revolution, his life had been singularly focused on his career. His social circle was limited to professional colleagues who served to establish or reinforce his academic stature. But marriage at the ripe age of thirty-nine and becoming the father of two children imposed a reorientation of his habits. He took time to be more accessible, mixing with the educated public from both the aristocracy and the urban middle class. Political events eventually forced him to find a refuge for his family outside of the capital. With the surge of populist power came the inevitable criticism of elitist science and its practitioners. Old patterns at the Academy were disrupted and ultimately destroyed; many friends went their separate ways to exile, the provinces, prison, or the guillotine; and his sources of income were seriously threatened. While remaining true to the professional ideals he had espoused earlier, Laplace was led by the new circumstances to add a public dimension to his life.

He cleansed himself of political suspicion by serving on several governmental committees connected with science. He became friendly with men of influence, but not so openly that he was

compromised when they happened to fall out of political favor. He wisely adopted the role of the technocrat, an expert on science and its applications, loyal to the nation rather than to any particular political party. This strategy paid off handsomely. As France emerged from the Terror, he was recognized as a leading spokesman of science and consulted on all the important phases of the reorganization of the cultural life of France. Good political instincts led him into Bonaparte's circle, and he wound up temporarily as Minister of Interior after the 1799 coup that brought the General to power. Kicked upstairs to the Senate, he became one of the favorite courtiers of Napoleon. When the Emperor fell, Laplace was so highly respected that the new Bourbon regime had to hold him in esteem despite his earlier close association with Napoleon. By then, he was a fixture in public circles and an integral part of France's cultural elite.

In late 1794, Laplace was selected with the aging Lagrange to lecture on mathematics at the Ecole normale before 1,200 auditors, the flower of French educators, sent to the capital to absorb the quintessence of learning from the foremost scientists of the day.^[20] The lectures were recorded by stenographers and immediately published in the media, which represented the event as evidence that the French could continue to absorb themselves in cultural activities despite their new-won reputation as blood-thirsty barbarians. Everything, even the childing remarks about Leibniz' silly belief in a deity and Newton's aberrant interest in the Apocalypse, was reported in the press and spread abroad. It was the first time Laplace had faced the world so openly.

He and Lagrange did a remarkable job, considering that neither had tried such a thing before. Setting aside the grand manner of philosophizing reminiscent of Condillac, d'Alembert, and the Académie française, Laplace chose a direct style of exposition on a fairly sophisticated level with little moralizing. He employed a minimum of mathematical symbols, mixing common-sense platitudes with profound conclusions about the nature of numbers, lengths, and their

manipulation. The so-called elementary lectures were, in fact, a superior condensation and synthesis of the best thought of the age about mathematics and its uses in everyday life and in the scientific enterprise. He included a lecture on the metric system and announced the composition of a new work meant for the same audience to cover astronomy, the Exposition du système du monde .

It is in this Exposition , published first in 1796, rather than in the ten lessons on mathematics, that one sees how Laplace's new role as a popularizer expanded his thoughts on mathematization. Not only was it presented as the central means to effect the progress of science, but that progress itself was offered as an example of the noblest aspects of humanity. In the midst of the Revolution, Laplace became a grand spokesman for his profession, justifying the activity he had chosen to pursue as the most glorious of all secular pursuits.

Challenged to expound his life's work to an educated audience presumed to be mathematically untutored, he eschewed all algebraic equations in the body of the work. That did not make it, however, "a handbook of cosmology."^[21] It was in fact a nonmathematical version of his planned Celestial mechanics , written for literate readers following the format of standard popularizations. It presents astronomy in a simulated inductive fashion, dealing first with direct observations of apparent motions in the heavens, followed by a description of the real (Copernican) movements of the planets and their satellites. These sections lead to a purely verbal discussion of the laws of motion and the theory of universal gravitation. The final section is a review and summary of the entire work offered in the form of an elementary history of astronomical discoveries. It is this last section that provides the key to its author's central purpose. Laplace wants to present the evolution of astronomy as a model for the finest features of modern science and the current state of celestial science as a reflection of the most elevated characteristics of liberated humanity.

Here is his sermon-like peroration:^[22]

Taken as a whole, astronomy is the most beautiful monument of the human mind—the noblest voucher of its intelligence. Seduced by the illusion of the senses and of vanity, man considered himself for a long time as the center about which the celestial bodies revolved, and his pride was punished by the vain fears they inspired. The labor of many ages has at length withdrawn the screen that concealed the system of the world. And man now appears [to dwell] upon a small planet, almost imperceptible in the vast extent of the solar system, itself only an insensible point in the immensity of space. The sublime results to which this discovery has led may console him for the [inferior] rank assigned to him in the universe.

In a later edition, he added that thinking beings should especially take pride in their ability to have measured this universe, given the tiny base from which they were operating. "Let us carefully preserve, and even augment the number of these sublime discoveries, which constitute the delight of thinking beings."^[23] The loss of anthropomorphic centrality is compensated by the power of the human mind, which despite its translocation, is able to contemplate and provide a proper assessment of the heavens.

The originality of Laplace's treatment lies in his characterization of the historic path to progress by the systematic movement from observation to induced laws of nature, and from laws to their causes. Within this framework, he identifies improved observation and mathematics as the two propellants responsible for progress, while unsupported speculation and religious obsession are its most notorious obstacles. Time and again he proclaims "observation and calculation as the only [solid] grounds for human knowledge."^[24] His historical analysis does not rest alone, however, on this trite generalization. By "calculation" he means several things, each of which denotes the spirit of the géomètre philosophe .

Most obvious is the way increased precision has led to the discovery of celestial regularities (and hence to the possibility of establishing laws of nature). Laplace points to Hipparchus, Ulugh Beg, Tycho Brahe, Galileo, and Kepler. Improvement in precision, he

observes, may come about in a variety of ways: through the construction of better observatories and instruments; with the introduction of simplifying techniques of calculation, such as logarithms; by the more systematic tabular collection of data; or simply through the greater conscious attention to detail by observers.

A second dimension is the critical evaluation of data that comes from juxtaposing expected positions calculated from elements of planetary orbits alongside the results of observation. As the degree of conviction about the validity of the calculations increases over time (especially since Newton), observation is subjected to ever greater criticism. Laplace repeatedly reminds his readers that it is "this analytical connection of particular with general facts that constitutes theory," and that this theoretical outlook distinguishes modern Western astronomy from its predecessors.^[25]

A third and more complex notion advanced by Laplace refers to the new potentialities of algebra and the calculus for astronomy. It was an issue close to his heart because in the Traité de mécanique céleste , he replaced all of Newton's geometrical demonstrations with what was then called analyse . The same term was used to denote a particular epistemological method, and Laplace at times shifts from one usage to another without realizing he is dealing with distinct issues. Thus when praising Newton's methodological approach, he adopts the distinction between induction (analyse ) and deduction (synthèse ). In the Principia , Newton argued the truth of his system by means of synthesis, using a geometrical form. While admiring Newton's use of induction to discover his principles, however, Laplace criticizes him for having chosen a geometrical form of exposition to establish their truth. To be sure, he recognized that there were extenuating circumstances: "The state of imperfection in which the infinitesimal calculus was in the hands of its inventor did not permit him to resolve completely the difficulties pertaining to the theory of the system of the world; and he was often given to positing uncertain conjectures, until such time as they were to be verified by rigorous analysis."^[26] Nonetheless, the further advancement of

astronomy depended upon abandoning the geometrical approach, of which Newton was the last grand master. Laplace credits calculus not only with bettering the inductive or analytical processes that led from evidence to general principles, but also with advancing the deductive or synthetic phase that allows one to derive particulars from the new theory.

Whatever confusion exists in Laplace's use of terminology, he emerges as a strong advocate of the value of calculus for further progress:^[27]

Geometrical synthesis has the advantage of never allowing us to lose sight of its goal. . .; whereas algebraic analysis quickly allows us to forget the principal goal in the form of abstract combinations, and it is only at the end [of the operation] that it brings us back to it. But in isolating itself from this goal after having taken what is needed to arrive at the required result; and then by giving ourselves over to the operations of analysis. . . one is led by the power and generality of this method [and by the inestimable advantage of transfering reasoning into a mechanical process to arrive at] results often inaccessible to synthesis.

No other [mathematical] language lends itself so elegantly. . .to the long train of interconnected expressions, all flowing from one fundamental equation. Analysis also offers the advantage of always leading us to the simplest methods. One need only make a judicious selection of unknowns using the proper methods and give the results the form most easily reducible to. . .numerical calcualtion.

He offers as an illustration of the truth of his observation the solution of lunar inequalities, which "would be impossible to arrive at by synthesis."^[28]