Lagrange's Theory of Analytical Functions and His Ideal of Purity of Method
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Lagrange’s Theory of Analytical Functions and his Ideal of Purity of Method Giovanni Ferraro, Marco Panza To cite this version: Giovanni Ferraro, Marco Panza. Lagrange’s Theory of Analytical Functions and his Ideal of Purity of Method. Archive for History of Exact Sciences, Springer Verlag, 2012, 66, pp.95-197. hal-00614606 HAL Id: hal-00614606 https://hal.archives-ouvertes.fr/hal-00614606 Submitted on 12 Aug 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LAGRANGE’S THEORY OF ANALYTICAL FUNCTIONS AND HIS IDEAL OF PURITY OF METHOD GIOVANNI FERRARO AND MARCO PANZA ABSTRACT. 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 interpre- tation of the algoritihms of the calculus. This was often referred to as the question of the “metaphysics of the calculus”1 (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 Theorie´ des fonctions analytiques (Lagrange 1797, 1813; henceforth the Theorie´ ); and the Lec¸ons sur le calcul des fonctions (Lagrange 1797, 1813; henceforth the Lec¸ons). His aim was to provide a new and non-infinitesimalist interpretation of these algorithms based on a general theory of power series.2 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 + x) in a power series of the indeterminate increment x.3 Lagrange called such coefficients ‘derivative functions [fonctions deriv´ ees´ ]’ (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 Theorie´ : Theorie´ des fonctions analytiques, contenant les principes du calcul differentiel,´ degag´ es´ de toute consideration´ d’infiniment petits, d’evanouissans, de limites et de fluxions, et reduite´ a` l’analyse algebrique´ des quantites´ 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:4 the reduction of all mathematics to an algebraic, purely formal theory5 centered of the manipulation of (finite or infinite) polynomials through the method of indeterminate coefficients.6 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).7 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: Zim- mermann 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 Theorie´ (Lagrange 1797, art. 7; 1813, Introduction, p. 5). 3Some clearly indicated exceptions apart, we shall use the term ‘power-series expansion of f (x + x)’ for referring to the expansion of f (x + x) in a power series of x. Lagrange uses ‘x’ to denote the increment of x in his earlier paper mentioned in footnote 2. In the Theorie´ and the Lec¸on, hep uses the Latin letter ‘i’ (as ‘increment’)—but´ we prefer using ‘x’ 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 mathe- matics 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. 1 2 GIOVANNI FERRARO AND MARCO PANZA The interpretation of the calculus and its algorithms presented a crucial difficulty for the complete accomplishment of such a program. The difficulty was not only concerned with the conceptual inaccuracy of the current infinitesimalist or pseudo-infinitesimalist accounts. Above all, it was their reliance on suppositions—like the negligibility of infinitesimals— that could hardly be explained in purely formal terms. The following passage, from a short report by Lagrange himself, clearly illustrates the point: [. ] I do not deny that one could rigorously prove the principles of the differential calculus through the consideration of limits envisaged in a particular way, as Maclaurin, d’Alembert and several others after them did. But the kind of metaphysics that has to be applied for this purpose is, if not contrary, at least foreign to the spirit of analysis, which should have no metaphysics but that which consists in the first principles and in the fundamental operations of calculation. [. ] je ne disconviens pas qu’on ne puisse, par le considerations´ des limites envisagees´ d’une maniere` partic- uliere,` demontrer´ rigouresement les principes du calcul differentiel,´ comme Maclaurin, d’Alembert et plusieurs autres auteurs apres` eux l’ont fait. Mais l’espece` de metaphysique´ qu’on est oblige´ d’y employer, est sinon con- traire, du moins etrang´ ere` a` l’esprit de l’analyse, qui ne doit avoir d’autre metaphysique´ que celle qui consiste dans les premiers principes et dans les operations´ fondamentales du calcul. (Lagrange 1799, p. 233.) Hence, Lagrange wanted not only to provide a conceptually more convenient basis for the calculus. His principal ambition was to incorporate it within a unitary conception of mathematics based on the “spirit of analysis”: a notion we shall try to shed light on.8 To do this, Lagrange should have shown that derivative functions could effectively replace differential quotients. Essen- tially, he should have shown that, for any function y = f (x), there exists an infinity of other functions f (k)(x) (k = 1;2;:::) that provide, apart from numerical factors, the coefficients of the power-series expansion of f (x + x), and formally coin- dky cide with the differential quotients dxk . We shall come to this in section 3, bearing in mind Lagrange’s notion of function, which we shall deal with in section 2. For the time being it is only important to remark that both the Theorie´ and the Lec¸ons begin with an argument that, if correct, would have convinced anybody already familiar with the differential cal- culus that this condition is met (even if it cannot really be stated, as such, within Lagrange’s theory, since it involves the notion of differential quotient, which has no place in this theory). For brevity let us call this argument ‘Lagrange’s fundamental proof’.9 Mathematicians and historians of mathematics have often discussed this argument (the relevant bibliographic refer- ences will be given below). Still, in our view, the current accounts, and, more generally, the usual interpretations of Lagrange’s theory, do not insist enough on the connections between his technical achievements and his understanding of the crucial notions of quantity, function, and power-series expansion, on which his foundational program is based.