Abstract
In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One example is how Björling interprets Cauchy’s definition of the logarithm function with respect to complex variables, which is investigated in the paper. Furthermore, in view of an article written by Björling (Kongl Vetens Akad Förh Stockholm 166–228, 1852) we consider Cauchy’s theorem on power series expansions of complex valued functions. We investigate Björling’s, Cauchy’s and the Belgian mathematician Lamarle’s different conditions for expanding a complex function of a complex variable in a power series. We argue that one reason why Cauchy’s theorem was controversial could be the ambiguities of fundamental concepts in analysis that existed during the mid-nineteenth century. This problem is demonstrated with examples from Björling, Cauchy and Lamarle.
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References
Belhoste B. (1991) Augustin-Louis Cauchy. A biography. Springer, New York
Björling E. G. (1845) Om betydelsen af tecknen x y, log b (x), sinx, cosx, arcsinx, arccosx i Analytisk mathematik. Kongliga Vetenskaps-Akademiens Förhandlingar 2: 75–156
Björling E. G. (1852) Om det Cauchyska kriteriet på de fall, då functioner af en variabel låta utveckla sig i serie, fortgående efter de stigande digniteterna af variabeln. Kongliga Vetenskaps-Akademiens Förhandlingar, 9: 166–228
Briot C. A., Bouqet J. C. (1856) Étude des fonctions d’une variable imaginaire. Journal l’école impériale polytechnique T. XXI: 85–131
Bråting K. (2007) A new look at E.G. Björling and the Cauchy sum theorem. Archive for History of Exact Sciences 61(5): 519–535
Bråting, K. (2009). Studies in the conceptual development of mathematical analysis. Dissertation, Uppsala University.
Cauchy A. L. (1823) Résumé des lecons données a l’ecole royale polytechnique, sur le calcul infinitesimal. Oeuvres Complètes T. IV: 9–261
Cauchy A. L. (1837) Extrait d’une lettre de M. Cauchy à M. Coriolis. Comptes Rendes T. IV: 216–218
Cauchy A. L. (1840) Note sur l’intégration des équations différentielles des mouvements planétaires. Exercises d’analyse et de Physique mathematique T. I: 1–153
Cauchy A. L. (1841) Note sur l’intégration des équations différentielles des mouvements planétaires. Exercises d’analyse et de Physique mathematique T. II: 41–97
Cauchy A. L. (1844) Mémoire sur quelques propositions fondamentales du calcul des résidus, et sur la théorie des intégrales singulières. Comptes Rendes T. XIX: 1343–1377
Cauchy A. L. (1846) Note sur le développement des fonctions en séries ordonnées suivant les puissances ascendantes des variables. Liouville’s Journal T. XI: 313–330
Cauchy A. L. (1851) Sur les fonctions de variables imaginaires. Comptes Rendes T. XXXII: 160–164
Fries, J. F. (1822). Die Matematische Naturphilosophie nach philosopher Methode bearbeitet: Ein Versuch. Heidelberg: C.F Winter.
Giusti E. (1984) Gli “errori” di Cauchy e i fondamenti dell’analisi. Bollettino di Storia delle Scienze Matematiche 4(2): 24–54
Grattan-Guinness I. (1986) The Cauchy–Stokes–Seidel story on uniform convergence: was there a fourth man?. Bulletin de la Société Mathématique de Belgique 38(series A): 225–235
Jahnke H.N. (1993) Algebraic analysis in Germany, 1780–1840: Some mathematical and philosophical issues. Historia Mathematica 20(3): 265–284
Katz K., Katz M. (2011) Cauchy’s continuum. Perspectives on science 19(4): 426–452
Kline M. (1972) Mathematical thought from ancient to modern times. Oxford University Press, Oxford
Lagrange J. L. (1847) Théorie des fonctions analytiques (2nd ed). Courcier, Paris
Lamarle A. H. E. (1846) Note sur le théorème de M. Cauchy rélatif au développ. des fonctions en séries. Liouville’s Journal T XI: 129–141
Lamarle A. H. E. (1847) Note sur la continuité considérée dans ses rapports avec la convergence des séries de Taylor et de Maclaurin. Liouville’s Journal T XII: 305–342
Laugwitz D. (1999) Bernhard Riemann—turning points in the conceptions of mathematics. Birkhäuser, Boston
Schlömilch O. (1850) Ueber das Theorem von MacLaurin in Mathematische Abhandlungen. Verlag von Moritz Katz, Dessau, pp 5–28
Sørensen H. K. (2005) Exceptions and counterexamples: Understanding Abel’s comment on Cauchy’s Theorem. Historia Mathematica 32(4): 453–480
Spalt D. (2002) Cauchy’s continuum—eine historiografische Annäherung via Cauchys Summensatz. Archive for History of Exact Sciences 56(4): 285–338
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Bråting, K. Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-nineteenth Century. Found Sci 17, 301–320 (2012). https://doi.org/10.1007/s10699-011-9274-3
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DOI: https://doi.org/10.1007/s10699-011-9274-3