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Torsten Broden's work on the foundations of Euclidean geometry
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2007 (English)In: Historia Mathematica, ISSN 0315-0860, E-ISSN 1090-249X, Vol. 34, no 4, 402-427 p.Article in journal (Refereed) Published
Abstract [en]

The Swedish mathematician Torsten Broden (1857-1931) wrote two articles on the foundations of Euclidean geometry. The first was published in 1890, almost a decade before Hilbert's first attempt, and the second was published in 1912. Broden's philosophical view of the nature of geometry is discussed and his thoughts on axiomatic systems are described. His axiomatic system for Euclidean geometry from 1890 is considered in detail and compared with his later work on the foundations of geometry. The two continuity axioms given are compared to and proved to imply Hilbert's two continuity axioms of 1903.

Place, publisher, year, edition, pages
2007. Vol. 34, no 4, 402-427 p.
Keyword [en]
Torsten Brodén, Euclidean geometry, Foundations of geometry, Axiomatic system, Continuity axioms
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-96552DOI: 10.1016/j.hm.2007.02.001ISI: 000252035600003OAI: oai:DiVA.org:uu-96552DiVA: diva2:171160
Available from: 2007-12-21 Created: 2007-12-21 Last updated: 2017-12-14Bibliographically 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

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