Practices and Principles in Cauchy’s Work

The age of Cauchy was a historical period in which scientific practices asserted an increasing degree of autonomy vis à visphilosophical reflections. If it is true, on the one hand; that the philosophers of the first half of the 19th century were less and less interested in the development of the sciences, with Auguste Comte as a notable exception, then it is also true that scholars—particularly mathematicians—increasingly insisted that they and they alone had the right to determine the content of their discipline and its standards of scientificality. The divorce between a particular scientific practice and general philosophical reflection, in a discipline whose foundations were quite uncertain at the time, gave rise to a blossoming of what might be termed local and spontaneous philosophies among the scholars and scientists who had dedicated themselves to this discipline. While these scholars’ philosophies were incomplete, approximate, and often anomalous, they nevertheless worked to the extent that they could justify current practices and thus mitigate existing theoretical shortcomings. Thus, the principle of rigor in mathematics seems to be an excellent example of what we earlier referred to as local and spontaneous philosophies.

Although Cauchy was not the only mathematician at the beginning of the 19th century to call for rigor in mathematics, he is justly regarded as one of the main initiators and supports of the movement to attain that goal. When Cauchy embarked on his career, France’s scientific institutions, which had been developed, for the most part, during the era of the Great Revolution and First Empire, were beginning to assume a definitive form, a shape that they would retain almost unchanged throughout the century. In particular, the professionalization of mathematical activity, together with the gradual appearance of a standard curriculum within this framework, could not remain without consequences: little by little both the state and the content of the mathematical sciences changed.

Certainly, mathematics was a great deal more than a means of making a living for Cauchy: it was a reason for living, as well as a vocation. Working, he resembles a monk cloistered away in his study:

His tastes and ways were simple and peaceable in the extreme. His study was small and modestly furnished. Most often, he did his writing at a small desk—a table, really, without pigeon holes—which, in the evenings and at night, was lighted by two simple wax candles with shades. Everything, whether in this small room or in his library, was in perfect order. He never went out without raising the wood in the fireplace and extinguishing the fire. He never wanted anything for himself, being oblivious to himself as it were, and having no desires to satisfy (1).

From time to time, in frustration, he would utter a complaint:

There is no let up! No end to it! Accursed problems! Innumerable calculations. Endless fighting. Signs. Formulas. Theorems besetting me from dawn to dusk (2)!

Far from living from his work as a mathematician, Cauchy used a considerable amount of money—especially during the 16 years that he was kept on the fringes of the scientific community—to guarantee the publication of his works. Institutional practices, both pedagogical and theoretical, had been rapidly evolving since the end of the 18th century, and it would be wrong to overlook the influence that these practices had on a mathematician such as Cauchy. The advancement and expansion of technological education, a development that civil and military engineers pushed to new heights, as well as the development of mathematical physics to a stage that was infinitely more ambitious than the rational mechanics of previous ages, revived the old theoretical and practical problem of the relationship between mathematics and the applications of mathematics. The creation and early success of the École Polytechnique, an institution at which Cauchy was first a student and later a professor, reveal the will and desire on the part of the political authorities and the scientific community to solve this problem. Within the span of a few years, the École was a center producing the best engineers for the public services, as well as exceptional scholars and scientists who quickly blazed a path of progress in mathematical physics.

But, even at the École, the problem of defining the relationship between mathematics and the applications of mathematics was sharply debated. From the very beginning, the École Polytechnique had accorded a special place to analysis in its educational program, and during the opening years of the 19th century, this special position for analysis was strengthened, as is witnessed by the fact that between 1801 and 1806 the time allotted to analysis in the instructional program, in the first and second years, grew from 16 to 29% and from 11 to 18% of the total instructional time (3). In spite of a slight decrease in these percentages during the following years, the leading place of analysis in the École’s educational program was never seriously questioned. Thus, it would appear that analysis was the most general and most widely taught subject at the École, required of all the engineering students, regardless of later areas of specialization. Placed at the very head of the official instructional program, analysis was the heart of the curriculum. Several courses in the area of applications, such as analysis applied to the geometry of three dimensions and mechanics depended on its very closely, and other areas, such as machines and physics, did so to a lesser degree, at least after Petit joined the faculty.

This leading role had at least two important implications for the content of the analysis courses. On the one hand, it largely determined the structure and organization of the courses to the extent that the basic idea was always to proceed from the general to the particular, from algebraic analysis to analysis applied to the geometry of three dimensions. On the other hand, it had a direct influence on methodology. Two notions come together at this juncture: some who were associated with the École—and indeed they were the majority—thought that analysis should be presented in the most simple and straightforward was possible, always keeping in mind the possible applications in the engineering sciences. This approach encouraged systematic appeals to intuition and to geometric representations. Others—and they were supported by Laplace and Lagrange—held the opposite view. They felt that it was necessary to present analysis in a way that was sufficiently abstract so as to permit its methods to be easily used in quite diverse situations. Cauchy, who, of course, shared this latter view, went so far as to propose in 1816 that instruction in mechanics, a subject for which he was responsible, be relegated to the second-year program in order that the entire first year could be devoted to infinitesimal calculus (4).

In his research, Cauchy, despite his training as an engineer, rarely bothered with applications per se. Being little inclined to the physicists’ view, his efforts were really focused, except, for his fundamental contributions to the theory of elasticity, on the development of mathematical tools that would be applicable to physics and mathematical astronomy, as well as on the development of the calculus of residues, the calculus of limits, and on characteristics, i.e., on differential operators, etc.

These works illustrate the taste and flair for formalism that is characteristic of Cauchy’s creative works. This is particularly so of his research in mathematical physics. Cauchy tried to develop various formal calculi with universal claims, not only in analysis but in algebra also, which his work on the calculus of permutations, his work in geometric calculus, and his theory of algebraic keys. On the basis, then, he was an 18th-century mathematician, a worthy successor of Euler and Lagrange.

Nevertheless, as far as history of mathematics is concerned, Cauchy remains firmly connected with the development of rigor. This trend, as we have seen, legitimized the dominant position that analysis enjoyed at the École Polytechnique, as well as that generally enjoyed by mathematics at scientific institutions relative to the applied, empirical sciences. Cauchy went further in this respect than most. He justified the requirement of rigor in mathematics by a philosophical and religious theory of knowledge that did not change significantly during the 46 years of his scientific life.

To understand this, we will examine several texts of a philosophical and epistemological nature that he wrote at one time or another—introductions to books and papers, as well as nonscientific discussions, lectures, and publications. The fact that Cauchy’s epistemological concepts did not really change will facilitate our work.

The philosophical and religious ideas to which Cauchy remained faithful all his life were well in line with traditional Catholicism and extreme conservatism as espoused by the Jesuits, and during the Restoration Era, by the Congrégation, to which he then belonged.

Rightfully anointed kings have regained the throne [wrote Haller in 1816]. We will similarly enthrone a rightful, lawful science, a science which will serve the Sovereign Lord and whose truth is confirmed by the entire Universe (5).

Here we have it. Truth: truth was an essential term to Cauchy:

Truth is a treasure beyond value, and no remorse or anguish in the soul ever stems from its acquisition. The mere contemplation of these heavenly wonders, of their divine beauty, suffices to compensate us for all that we may sacrifice in discovering it, and the joy of heaven itself is no more than the full and complete possession of immortal truth (6).

Cauchy, of course, did not define truth:

Here, on earth, truth will never be complete, it will never be wholly revealed (7).

That could only be encompassed by God. Nevertheless, he did distinguish two orders of truth.

The first order consisted of philosophical moral truths. These were revealed truths, verities ‘too sublime for our thoughts and understanding ever to attain them’ (8). In 1811, he stated:

The most submissive [obedient] persons are also the wisest; and by force of various sophistries, an individual might very well come to doubt the truths that have been taught to him, but he will not learn any new ones (9).

Accordingly, it would be far better to leave the teaching of these truths to those whom the creator of the universe has entrusted with that mission and responsibility. This mission, according to Cauchy, was entrusted to the priests, a group whom, Jullien wrote, Cauchy revered (10). All the doctrines other than the revealed truths as contained in the Holy Scriptures, and as interpreted and taught by the Church, were false and dangerous. For this reason, then, religious education was necessary. The Université’s monopoly was culpable not because it was a monopoly, but because it planted ‘chaos and anarchy’ in beliefs by not accepting religious precepts as the foundation of education (11). This kind of reasoning, which was quite widespread among the supporters of the free school movement, provoked mockery and raillery from their opponents who accused the Catholics, and specially the Jesuits, of wanting to use the cover of religious education to reestablish the Church’s control over the Université.

With Cauchy, as with all the counterrevolutionaries, religion was necessary for the maintenance of the established order, because it served to ‘hold man’s passions in check and make him practice virtues (12). ‘The vain and pernicious philosophy’ of the last century, ‘after having overrun the higher classes in society, then descended into the huts of the poor and there turned the lower classes into its toys, making these classes the authors of its misery and the instruments of its crimes’ (13). From this stemmed all the evils that beset the 19th century. Cauchy was thus being quite specific in 1844 when he declared:

Unless it be accompanied by a good education, instruction can become more troublesome than useful . . . Of what use is it for the child of a poor man to learn how to write, if he only takes up the pen to snare the innocent, to deceive and undermine the good faith of others…(14).

The second order of truth was composed of scientific truths; these were ‘conquered’ verities as opposed to ‘revealed’ truths, which constituted the first order. ‘The pursuit of truth should be the sole aim of any science’ (15). In 1811, Cauchy paid homage to the efforts that generations had made in increasing the scope of human knowledge (16), and he regarded the times he lived in as ‘an extraordinary era in which a renascent, ceaseless activity devours all thought’ (17). In spite of remarks such as these, Cauchy did not set forth any distinctive defining criteria for scientific truths. Indeed, if he regarded exactness as an ‘essential and necessary feature of any true science’, then he also saw exactness as a crucial feature of ‘the most beautiful creations of the human mind, even in literature, even in poetry.’ (18).

(1) F.N. Moigno, preface to A.L. Cauchy, Sept Leçons de Physique Générale, Paris, 1868

(2) A.L. Cauchy, ‘La Chandeleur’, Bulletine de L’Institut Catholique, February 1, 1843

(3) A. Fourcy, Histoire de l’École Polytechnique, Paris, 1828, Summary Table, pp. 255-257.

(4) See Chapter 5, pp. 61-63.

(5) Cited by G. Bertier de Sauvigny, La Restauration, Paris, Flammarion, 1955, p. 346.

(6) A.L. Cauchy, ‘Sur la recherche de la verité’, Bulletin de L’Institut Catholique, April 14, 1842, p. 21.

(7) A.L. Cauchy, Sept Leçons de Physique Générale, o.c., 2, 15, p. 413

(8) Ibid., p. 413.

(9) A. L. Cauchy, ‘Sur les limites des connaissances humaines’, o.c.,2, 15, p. 7

(10) M. Jullien, ‘Quelque souvenirs d’un étudiant jésuite à la Sorbonne et au Collège de France, 1852-1856’, Les Études, 127, 1911, pp. 329-348, especially, p. 336

(11) A. L. Cauchy, Quelques Réflexions sur la Liberté de l’Enseignement, Paris, 1844, especially Chapter 3, ‘De l’enseignement scientifique et de l’ensiegnement religieux’, pp. 14-16

(12) A.L. Cauchy, Considérations sur les Moyens de Prévenir les Crimes et de Réformer les Criminels, Paris, 1844

(13) A. L. Cauchy, ‘Sur la recherche de la vérité’, Bulletin de l’Institut Catholique, April 14, 1842, p. 21

(14) A. L. Cauchy, Considérations sur les Moyens de Prévenir les Crimes et de Réformer les Criminels, Paris, 1844

(15) Sept Leçons de Physique Générale, o.c., 2, 15, p. 413.

(16) Practically in the same terms in Sur les limites . . . of 1811, o.c., 2, 15, pp. 5-6; Sept Leçons de Physique Générale of 1833, ibid., pp. 412-413; and ‘Sur la recherche de la vérité’, Bulletin de l’Institut Catholique, April 14, 1842, p. 20

(17) Sept Leçons de Physique Générale, o.c., 2, 15, p. 412.

(18) A.L. Cauchy, ‘Sur quelques préjugés contre les physiciens et les géomètres’, Bulletin de l’Institut Catholique, March 3, 1842, p. 43.