Young Lagrange meets the "metaphysics" of mathematics: In the context of French infinitesimal controversy

Abstract

Joseph-Louis Lagrange, one of the eminent mathematicians in the eighteenth century, had given a lecture on calculus as early as in the latter half of the 1750s. One will find this lecture interesting not only because he introduces the differential calculus with special emphasis on the concept of limit, which seemed novel for that time, but also that Lagrange refers to several textbooks most of which were published in France. Given these points, the present article attempts to consider Lagrange's early thought on the foundation of calculus in the context of French controversy. In France, after the reception of Leibnizian calculus through l'Hôpital's textbook (1696), Fontenelle had constructed a mathematical system based on infinite quantities (1727). In 1740s, however, the supposition of infinitesimals were criticized by supporters of fluxions, among others Maclaurin (1742, translated in 1749 into French), and then in 1750s d'Alembert proposed the concept of limit as the "basis of the true metaphysics of differential calculus." Although Lagrange does not mention d'Alembert's name, perhaps he has been influenced by the latter. It is in this context of French infinitesimal controversy that one can get a better view of young Lagrange's concern with the foundation problem