Lagrange’s Theory of Analytical Functions and his Ideal of Purity of Method

We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline. 1. PRELIMINARIES AND PROPOSALS Foundation of mathematics was a crucial topic for 18th-century mathematicians. A pivotal aspect of it was the interpretation of the algoritihms of the calculus. This was often referred to as the question of the “metaphysics of the calculus” (see Carnot 1797, as an example). Around 1800 Lagrange devoted two large treatises to the matter, both of which went through two editions in Lagrange’s lifetime: the Théorie des fonctions analytiques (Lagrange 1797, 1813; henceforth the Théorie); and the Leçons sur le calcul des fonctions (Lagrange 1797, 1813; henceforth the Leçons). His aim was to provide a new and non-infinitesimalist interpretation of these algorithms based on a general theory of power series. He viewed the direct algorithm as a rule for transforming functions, which—applied reiteratively to any function y = f (x)—gives, apart from numerical factors, the coefficients of the expansion of f (x+ ξ ) in a power series of the indeterminate increment ξ .3 Lagrange called such coefficients ‘derivative functions [fonctions dérivées]’ (Lagrange 1797, art. 17; 1801, p. 5; 1806, p. 5; 1813, Introduction, p. 2): a term whose meaning has since changed. In what follows, we shall use this term in Lagrange’s sense. Throughout his theory, Lagrange certainly pursues an ideal of conceptual clarity involving the elimination of any sort of infinitesimalist insight. This has been often noticed, and is emphasized by Lagrange himself from the very complete title of the Théorie: Théorie des fonctions analytiques, contenant les principes du calcul différentiel, dégagés de toute considération d’infiniment petits, d’evanouissans, de limites et de fluxions, et réduite à l’analyse algébrique des quantités finies. We shall not dwell on this point, then. We shall rather argue that this ideal was part of a more general ideal of purity of method: the reduction of all mathematics to an algebraic, purely formal theory centered of the manipulation of (finite or infinite) polynomials through the method of indeterminate coefficients. This was a sweeping project rooted in a mathematical program going back to the early mathematical work of Newton (see Panza 2005), and whose manifesto was the first volume of Euler’s Introductio in analysin infinitorum (Euler 1748). Its main purpose was the development of a fairly general and formal theory of abstract quantities: quantities merely conceived as elements of a net of relations, expressed by formulas belonging to an appropriate language and subject to appropriate transformation rules. 1We use double inverted commas for quotation and simple ones for mention. We never use inverted commas for other purposes, namely for emphasizing a term or phrase. 2A few years earlier, Arbogast had proposed a similar interpretation in an unpublished treatise (Arbogast ESSAI; for a commentary, see: Zimmermann 1934; Panza 1985; Grabiner 1990, pp. 47-59). According to Grabiner (1981b, p. 316), this treatise was inspired by an even earlier paper by Lagrange himself (Lagrange 1772; see footnote 3.2, below). Lagrange mentions Arbogast’s treatise at the beginning of the Théorie (Lagrange 1797, art. 7; 1813, Introduction, p. 5). 3Some clearly indicated exceptions apart, we shall use the term ‘power-series expansion of f (x+ξ )’ for referring to the expansion of f (x+ξ ) in a power series of ξ . Lagrange uses ‘ξ ’ to denote the increment of x in his earlier paper mentioned in footnote 2. In the Théorie and the Leçon, he uses the Latin letter ‘i’ (as ‘incrément’)—but we prefer using ‘ξ ’ in order to avoid any possible confusion with the symbol that is now used to denote √ −1. Our quotations from Lagrange’s treatises are altered accordingly. 4On the notion of purity of method, see Arana (2008), Detlefsen (2008)—which explicitly mentions Lagrange’s “purification program”, in footnote 6, p. 182—, and Hallett (2008). In this last paper (ibid., p. 199), M. Hallet describes as follows the concern for purity of method , by referring to Hilbert’s mention of such a concern in Hilbert (1899, p. 199): “one can enquire of a given proof or of a given mathematical development whether or not the means it uses are ‘appropriate’ to the subject matter, whether one way of doing things is ‘right’, whereas another, equivalent way is ‘improper’.” 5The clarification of the exact sense in which the adjectives ‘algebraic’ and ‘formal’ have to be understood here is one of the main purposes of our paper. 6The centrality of the method of indeterminate coefficients is the common denominator of Lagrange’s foundational programs both in pure mathematics and in mechanics. For the case of mechanics, see Panza (1991-1992) and (2003). 7An attempt to reconstruct the sources and evolution of this program up to Lagrange is made in Panza (1992). The present paper is partly based on chapter III.6.