This dissertation deals with aspects of axiomatization, intuition and visualization in thehistory of mathematics. Particular focus is put on the end of the 19th century, before DavidHilbert's (1862–1943) work on the axiomatization of Euclidean geometry. The thesis consistsof 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. Athorough analysis of his foundational work is made as well as an investigation into his generalview on science and mathematics. Furthermore, his thoughts on geometry and its nature andwhat consequences his view has for how he proceeds in developing the axiomatic system, isstudied. In the second paper different aspects of visualizations in mathematics areinvestigated. In particular, it is argued that the meaning of a visualization is not revealed bythe visualization and that a visualization can be problematic to a person if this person, due to alimited 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 whichfollowed in the wake of Karl Weierstrass' (1815–1897) construction of a nowheredifferentiable function in 1872. In the third paper certain aspects of the thinking of the twoscientists Felix Klein (1849–1925) and Heinrich Hertz (1857–1894) are studied. It isinvestigated how Klein and Hertz related to the idea of naïve images and visual thinkingshortly before the development of modern axiomatics. Klein in several of his writingsemphasized his belief that intuition plays an important part in mathematics. Hertz argued thatwe form images in our mind when we experience the world, but these images may containelements that do not exist in nature.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:hb-3481 |
| Date | January 2007 |
| Creators | Pejlare, Johanna |
| Publisher | Högskolan i Borås, Institutionen för Pedagogik, Department of Mathematics, Uppsala University |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, monograph, info:eu-repo/semantics/doctoralThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | Uppsala Dissertations in Mathematics, 1401-2049 ; 53 |
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