Abstract
To say “scientific developments have social and epistemological causes” is to state a generality to prove it requires an empirical base of specific examples. I wish to focus on one example from the history of mathematics: the foundations of the calculus between the late eighteenth and early nineteenth centuries. We will see, in some detail, the ways external factors can influence the choice of problems in mathematical research. And, because of the particular subject chosen - the foundations of the calculus - we will see the mechanisms by which external factors worked to increase the rigor of a branch of mathematics.
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References
Poincaré, H.: in: Compte rendu du 2me Congrès internationale des mathématiciens. 1900, Paris 1902, pp. 120–122.
Grabiner, J.V.: The Origins of Cauchy’s Rigorous Calculus, forthcoming; compare Grabiner, J.V.: Cauchy and Bolzano. Tradition and Transformation in the History of Mathematics, in Mendelsohn, E. (ed.): Transformation and Tradition in the Sciences, forthcoming.
E. g. Newton, Euler, Maclaurin, l’Hospital, Lacroix, etc. See the bibliographies in: Boyer, C.: History of the Calculus, New York 1959, and in: Cajori, F.: History of the Conceptions of Limits and Fluxions in Great Britain from Newton to Woodhouse, Chicago and London 1919.
Especially the articles ‘Limité’ and ‘Différentiel’.
See Cajori (ref. 3).
Berkeley, G.: The Analyst, or a discourse addressed to an infidel mathematician (1734).
Limité’ and ‘Différentiel’; though he did not cite Berkeley, both the similarities in language and the coincidence of problems support this conclusion.
Carnot, L.N.M.: Réflêxions sur la métaphysique du calcul infinitesimal, Paris 1797.
Lagrange, J.-L.: Théorie des fonctions analytiques, Paris, 1797, preface, repr. in:-Oeuvres de Lagrange, vol. IX, pp. 16–20. Compare Lagrange, J.-L.: Discours sur l’object de la théorie des fonctions analytiques, in: J. Ecole poly. (1799); repr. in: Oeuvres de Lagrange, vol. VII, p. 325.
Struik, D.J.: Concise History of Mathematics, New York 1967, p. 137. Compare the assessment by J. Dieudonne that mathematics in 1797 had just entered “a period of stagnation”: Abrégé d’histoire des mathématiques, Paris 1978, p. 337.
Diderot, D.: De l’interprétation de la nature, Section IV, in: Verniere, P. (ed.): Oeuvres philosophiques de Didérot, Paris 1961, pp. 180–181.
Lagrange, letter to d’Alembert, 24 February 1772; in: Oeuvres de Lagrange, vol. XIII, p. 229.
Sur une nouvelle éspece du calcul rélatif à la differentiation et à l’intégration des quantités variables, in: Nouv. Mem. Beri. (1772), pp. 185–221; Oeuvres de Lagrange, vol. Ill, pp. 329–476.
Lacroix, S.-F.: Traité du calcul différentiel et intégral, Paris 1797, Ch. 1.
See, e. g., Struik, History (réf. 10); Ovâert, J.-L. et al.: Philosophie et calcul de l’infini, Paris 1976; and Mendelsohn, E.: The Emergence of Science as a Profession in: Hill, K. (ed.): The Management of Scientists, Boston 1964, pp. 3–48.
Lagrange, J.-L.: Theorie des fonctions analytiques (1797) (2d ed. 1813 ); Levons sur le calcul des fonctions (1799–1801); Cauchy, A.-L.: Cours d’analyse, Paris 1821; Résumé des leçons… sur le calcul infinitesimal, Paris 1823.
Arbogast, L.F.A.: Essai sur de nouveaux principes de calcul différentiel et intégral, indépendants de la théorie des infiniment–petits et celle des limites, Biblioteca Medicea-Laurenziana, Florence, MS Codex Ashburnham Appendix Sign. 1840. I thank the Laurentian Library for providing me a copy of this manuscript. Grattan-Guinness, I.: Development of the Foundations of Mathematical Analysis from Euler to Riemann (Cambridge, Mass., 1970), cites another manuscript version, p. 155 of his bibliography, which I have not seen: MS 2089, Ecole des ponts et chaussées, Paris. Arbogast’s manuscript “Essai” will be discussed below.
Carnot, op. cit.; l’Huilier, S.: Exposition élémentaire des principes des calculs supérieurs, Berlin 1787.
See Dickstein, S.: Zur Geschichte der Prinzipien der Infinitesimal-Rechnung. Die Kritiker der ‘théorie des fonctions analytiques’ von Lagrange, in: Abh. z. Geschichte der Math. 9 (1899) (= Z. fuer Math. u. Phys., Supplement to 44(1899)), pp. 65–79.
Grabiner, Cauchy and Bolzano, (réf. 2).
Dickstein, op. cit.
Grabiner, J. V.: The Origins of Cauchy’s Theory of the Derivative, in: Hist. Math. (1978), pp. 379–409. Compare Dugac, P. in: Histoire du theorème des accroissements finis, Paris 1979, which includes extensive citations from the primary sources with a connecting narrative.
The letter is reprinted in: Oeuvres de Lagrange, vol. XIV, p. 173.
As claimed by Jourdain, P.E.B.: The idea of the ‘Fonctions analytiques’ in Lagrange’s early work, in: Proc. Int. Cong. Math. II (1912), pp. 540–1; Jourdan’s reasons are: (1) Lagrange had introduced the prime-notation for the coefficients in the Taylor series in 1761, and (2) in 1772 Lagrange did define the derivative as the coefficient of h in the Taylor-series expansion for f(x-fh). However, (1) does not support Jourdain7s conclusion, since the “reform” in the 1761 paper is merely one of notation; for our explanation of (2), see our text.
Gerdil, H.S.: De l’infini absolu, in: Misc. Taur. (1760–61), pp. 1–45.
Lagrange, J.-L.: Note sur la métaphysique du calcul infinitésimal, in: Misc. Taur. (1760–61), pp. 17–18; repr. in: Oeuvres de Lagrange, vol. VII, pp. 597–9.
As Boyer seems to imply, op. cit., pp. 257–8, and as Carnot apparently believed, op. cit.
Lagrange, op. cit., p. 598.
FA, preface, p. 4, in: Oeuvres de Lagrange IX, p. 17.
Grabiner, J.V.: Is Mathematical Truth Time-Dependent?, in: Am. Math. Monthly 1974, pp. 354–365; see pp. 356–7.
Lagrange certainly was acquainted with some of Berkeley’s ideas, since he uses the concept and the term “compensation of errors.” There is no direct evidence about when, if ever, Lagrange read the Analyst itself; Lagrange did read English, as his citations of Landen and Maclaurin show. And Lagrange does use Berkeleyan arguments in his critique of older foundations; see FA, preface, in: Oeuvres de Lagrange, vol. IX, pp. 16–20.
In the preface to his FA; see: Oeuvres de Lagrange, vol. IX, p. 18.
Sur une nouvelle espece du calcul, in: Nouv. Mem. Berl.(1772), pp. 185–221; repr. in: Oeuvres de Lagrange, vol. III, pp. 439–476, p. 443. The notation here is confusing, since it is not the prime-notation for derivatives which he introduces later in the same paper; but it is his notation nevertheless.
Loc. cit.
The prize competition was set by the “Classe de mathématiques11 of the Academy, which in 1784 included Lagrange, Johann (II) Bernoulli, and Johann Karl Gottlieb Schulze. Lagrange was the leading light, and had long been concerned with this problem. That it was his idea is supported by Hofmann, J.E.: Geschichte der Mathematik, vol. I”, Berlin 1957, p. 68, and by Youschkevitch, A.P.: Lazare Carnot and the Competition of the Berlin Academy in 1787 on the Mathematical Theory of the Infinite, in which he drew on the manuscript collection of the Berlin Academy; the article is to be found in: Gillispie, C.C.: Lazare Carnot Savant, Princeton 1971, pp. 149–168; see p. 155.
Since the whole idea of “infinite magnitude” seemed to be inherently contradictory, the Academicians asked “that it can be explained how so many true theorems have been deduced” from such a problematic concept. See Hist. Acad. Royale, Berlin 1784, pp. 12–13.
Hist. Acad. Royale, Berlin 1786, p. 8.
Vivanti, G.: Infinitesimalrechnung, in: Cantor, M.: Vorlesungen ueber Geschichte der Mathematik, vol. IV, Leipzig 1908, p. 645, has claimed this; however, the time lag is simply too great.
As he said explicitly in the preface to the second edition of that work, the “Mechanique analytique” uses infinitesimals, but these can be given a rigorous basis by using the theory of analytic functions; Oeuvres de Lagrange, vol. XI, p. xiv.
FA (1797), preface, p. 5; Oeuvres de Lagrange, vol. IX, p. 19.
Loc. cit.
Quoted from the MS by Zimmermann, K.: Arbogast als Mathematiker und Historiker der Mathematik (Heidelberg, 1934), p. 45.
Arbogast: Calcul des derivations, Paris 1800, p. xiii, in his own account of this manuscript.
Loc. cit. Cp. Zimmermann, Arbogast (ref. 42 ), pp. 47–48.
Grabiner, Cauchy’s Theory of the Derivative, (ref. 22), p. 385.
See ref. 29.
Grabiner, Cauchy’s Theory of the Derivative (ref. 22), p. 384; see CF, 2d ed., in: Oeuvres de Lagrange, vol. X, p. 87; the notation is his, exept that I have substituted the more usual h for his i.
See ref. 2.
For Weierstrass, see Klein, F.: Entwicklung der Mathematik im 19ten Jahrhundert, vol. I, Berlin 1926, p. 283ff.; compare Dieudonne, Abrege d’histoire (ref. 10), pp. 370–3. For Dedekind, see his own introduction to: Continuity and Irrational Numbers [1872], New York 1963, p. 1. For Bolzano, see. e. g., his: Paradoxes of the Infinite; on Cantor, see first Dauben, J. W.: Georg Cantor, Cambridge (Mass.) 1979.
I owe this observation to the remarks at the Bielefeld conference of L. J. Daston and E. Mendelsohn. And, as I. Toth pointed out at the conference, within mathematics itself the development of non-Euclidean geometry provides an instructive example of the definition of a new - and autonomous - subdiscipline, and of the intercation between mathematics and philosophy.
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Grabiner, J.V. (1981). Changing Attitudes Toward Mathematical Rigor: Lagrange and Analysis in the Eighteenth and Nineteenth Centuries. In: Jahnke, H.N., Otte, M. (eds) Epistemological and Social Problems of the Sciences in the Early Nineteenth Century. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8414-1_18
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