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Visualizations in mathematics
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. (Matematikens historia och didaktik)
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2008 (English)In: Erkenntnis, ISSN 0165-0106, E-ISSN 1572-8420, Vol. 68, no 3, 345-358 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Springer , 2008. Vol. 68, no 3, 345-358 p.
National Category
Other Mathematics Didactics
Research subject
Mathematics with specialization in Mathematics Education and the History of Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-96553DOI: 10.1007/s10670-008-9104-3OAI: oai:DiVA.org:uu-96553DiVA: diva2:171161
Available from: 2007-12-21 Created: 2007-12-21 Last updated: 2013-05-17Bibliographically approved
In thesis
1. On Axioms and Images in the History of Mathematics
Open this publication in new window or tab >>On Axioms and Images in the History of Mathematics
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This dissertation deals with aspects of axiomatization, intuition and visualization in the history of mathematics. Particular focus is put on the end of the 19th century, before David Hilbert's (1862–1943) work on the axiomatization of Euclidean geometry. The thesis consists of three papers. In the first paper the Swedish mathematician Torsten Brodén (1857–1931) and his work on the foundations of Euclidean geometry from 1890 and 1912, is studied. A thorough analysis of his foundational work is made as well as an investigation into his general view on science and mathematics. Furthermore, his thoughts on geometry and its nature and what consequences his view has for how he proceeds in developing the axiomatic system, is studied. In the second paper different aspects of visualizations in mathematics are investigated. In particular, it is argued that the meaning of a visualization is not revealed by the visualization and that a visualization can be problematic to a person if this person, due to a limited knowledge or limited experience, has a simplified view of what the picture represents. A historical study considers the discussion on the role of intuition in mathematics which followed in the wake of Karl Weierstrass' (1815–1897) construction of a nowhere differentiable function in 1872. In the third paper certain aspects of the thinking of the two scientists Felix Klein (1849–1925) and Heinrich Hertz (1857–1894) are studied. It is investigated how Klein and Hertz related to the idea of naïve images and visual thinking shortly before the development of modern axiomatics. Klein in several of his writings emphasized his belief that intuition plays an important part in mathematics. Hertz argued that we form images in our mind when we experience the world, but these images may contain elements that do not exist in nature.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2007. 16 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 53
Keyword
Mathematics, History of mathematics, axiomatization, intuition, visualization, images, Euclidean geometry
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-8345 (URN)978-91-506-1975-1 (ISBN)
Public defence
2008-01-17, Häggsalen, Ångström Laboratory, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2007-12-21 Created: 2007-12-21 Last updated: 2009-11-29Bibliographically approved
2. Studies in the Conceptual Development of Mathematical Analysis
Open this publication in new window or tab >>Studies in the Conceptual Development of Mathematical Analysis
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. 31 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 64
Keyword
History of mathematics, conceptual development, mathematical analysis
National Category
Other Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-101349 (URN)978-91-506-2080-1 (ISBN)
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

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