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Studies in the Conceptual Development of Mathematical Analysis
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This dissertation deals with the development of mathematical concepts from a historical and didactical perspective. In particular, the development of concepts in mathematical analysis during the 19th century is considered. The thesis consists of a summary and three papers. In the first paper we investigate the Swedish mathematician E.G. Björling's contribution to uniform convergence in connection with Cauchy's sum theorem from 1821. In connection to Björling's convergence theory we discuss some modern interpretations of Cauchy's expression x=1/n. We also consider Björling's convergence conditions in view of Grattan-Guinness distinction between history and heritage. In the second paper we study visualizations in mathematics from historical and didactical perspectives. We consider some historical debates regarding the role of intuition and visual thinking in mathematics. We also consider the problem of what a visualization in mathematics can achieve in learning situations. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. In the third paper we consider Cauchy's theorem on power series expansions of complex valued functions on the basis of a paper written by E.G. Björling in 1852. We discuss Björling's, Lamarle's and Cauchy's different conditions for expanding a complex valued function in a power seris. In the third paper we also discuss the problem of the ambiguites of fundamental concpets that existed during the mid-19th century. We argue that Cauchy's and Lamarle's proofs of Cauchy's theorem on power series expansions of complex valued functions are correct on the basis of their own definitions of the fundamental concepts involved.

 

Place, publisher, year, edition, pages
Uppsala: Matematiska Institutionen , 2009. , p. 31
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 64
Keywords [en]
History of mathematics, conceptual development, mathematical analysis
National Category
Other Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-101349ISBN: 978-91-506-2080-1 (print)OAI: oai:DiVA.org:uu-101349DiVA, id: diva2:212686
Public defence
2009-06-05, Häggsalen, Ångström Laboratory, Lägerhyddsvägen 1, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2009-05-15 Created: 2009-04-23 Last updated: 2011-11-07Bibliographically approved
List of papers
1. Visualizations in mathematics
Open this publication in new window or tab >>Visualizations in mathematics
2008 (English)In: Erkenntnis, ISSN 0165-0106, E-ISSN 1572-8420, Vol. 68, no 3, p. 345-358Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Springer, 2008
National Category
Other Mathematics Didactics
Research subject
Mathematics with specialization in Mathematics Education and the History of Mathematics
Identifiers
urn:nbn:se:uu:diva-96553 (URN)10.1007/s10670-008-9104-3 (DOI)
Available from: 2007-12-21 Created: 2007-12-21 Last updated: 2017-12-14Bibliographically approved
2. E.G. Björling's view of power series expansions of complex valued functions
Open this publication in new window or tab >>E.G. Björling's view of power series expansions of complex valued functions
(English)Manuscript (Other academic)
Keywords
History of mathematics
Identifiers
urn:nbn:se:uu:diva-101347 (URN)
Available from: 2009-04-23 Created: 2009-04-23 Last updated: 2010-01-14
3. E.G Björling's version of the Cauchy sum theorem
Open this publication in new window or tab >>E.G Björling's version of the Cauchy sum theorem
2004 (English)In: HPM 2004 & ESU 4, Uppsala / [ed] F. Furinghetti, S. Kaijser & T. Tzanakis, Uppsala: Uppsala universitet, 2004, p. 48-52Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Uppsala: Uppsala universitet, 2004
National Category
Didactics
Identifiers
urn:nbn:se:uu:diva-23470 (URN)
Conference
HPM 2004
Available from: 2007-01-29 Created: 2007-01-29 Last updated: 2017-12-12Bibliographically approved

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