David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Foundations of Science 17 (4):301-320 (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.
|Keywords||History of mathematics Mathematical analysis Complex analysis Power series expansion The logarithm function Ambiguities|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
Karin U. Katz & Mikhail G. Katz (2011). Cauchy's Continuum. Perspectives on Science 19 (4):426-452.
Citations of this work BETA
No citations found.
Similar books and articles
Mikhail G. Katz & David Tall (2013). A Cauchy-Dirac Delta Function. Foundations of Science 18 (1):107-123.
Alexandre Borovik & Mikhail G. Katz (2012). Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus. Foundations of Science 17 (3):245-276.
Tyler Marghetis & Rafael Núñez (2013). The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics. Topics in Cognitive Science 5 (2):299-316.
Henrik Kragh Sã¸Rensen (2009). Representations as Means and Ends: Representability and Habituation in Mathematical Analysis During the First Part of the Nineteenth Century. In Bart Van Kerkhove (ed.), New Perspectives on Mathematical Practices: Essays in Philosophy and History of Mathematics. World Scientific. 114--137.
Rp Bellamy (1993). Liberalism and Modern Society. Philosophical Quarterly 43 (172):383.
K. Vela Velupillai (2008). Sraffa's Mathematical Economics: Aconstructive1 Interpretation. Journal of Economic Methodology 15 (4):325-342.
Andrei Rodin (2010). How Mathematical Concepts Get Their Bodies. Topoi 29 (1):53-60.
Jeanne Peijnenburg & David Atkinson (2008). Achilles, the Tortoise, and Colliding Balls. History of Philosophy Quarterly 25 (3):187 - 201.
Pavel Materna (2012). Mathematical and Empirical Concepts. In James Maclaurin (ed.), Rationis Defensor.
Robert E. Abrams (2004). Landscape and Ideology in American Renaissance Literature: Topographies of Skepticism. Cambridge University Press.
Katharina A. Breckner (2003). F. Björling (Ed.), On the Verge. Russian Thought Between the Nineteenth and the Twentieth Centuries. Studies in East European Thought 55 (3):257-261.
Added to index2012-01-08
Total downloads8 ( #172,583 of 1,102,762 )
Recent downloads (6 months)2 ( #182,775 of 1,102,762 )
How can I increase my downloads?