Article ID: CBB001252710

Ambiguities of Fundamental Concepts in Mathematical Analysis during the Mid-Nineteenth Century (2012)


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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Authors & Contributors
Katz, Mikhail G.
Grabiner, Judith V.
Lakatos, Imre
Cauchy, Augustin Louis
Smithies, F.
Belhoste, Bruno
Archive for History of Exact Sciences
Mathematical Intelligencer
Sciences et Techniques en Perspective
Mathesis: Filosofía e Historia de las Matemáticas
Bollettino di Storia delle Scienze Matematiche
Bulletin de la Société des Amis de la Bibliothèque de l'École Polytechnique
Oxford University Press
Université de Montréal (Canada)
Carocci Editore
Mathematical analysis
Philosophy of mathematics
Sequences and series (mathematics)
Functions (mathematics)
Cauchy, Augustin Louis
Bolzano, Bernard
Lewis, W. David
Ohm, Martin
Gauss, Carl Friedrich
Dirksen, Eeno Heeren
Time Periods
19th century
20th century, early
18th century
École Polytechnique, Paris

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