Article ID: CBB001221693

The Growth of Mathematical Knowledge---Introduction of Convex Bodies (2012)


The article addresses the topic of the growth of mathematical knowledge with a special focus on the question: How are mathematical objects introduced to mathematical practice? It takes as starting point a proposal made in a previous paper which is based on a case study on the introduction of Riemann surfaces. The claim is that (i) a new object first refers to previously accepted objects, and that (ii) reasoning is possible via a correspondence to the objects with reference to which it is introduced. In addition Riemann surfaces are geometrical objects, i.e., they are placed in a geometrical context, which makes new definitions possible. This proposal is tested on a case study on Minkowski's introduction of convex bodies. The conclusion is that the proposal holds also for this example. In both cases we notice that in a first stage is a close connection between the new object and the objects it is introduced with reference to, and that in a later stage, the new object is given an independent definition. Even though the two cases display similarity in these respects, we also point to certain differences between the cases in the process of the first stage. Overall we notice the fruitfulness of representing problems in different contexts.


Description A case study on Minkowski's introduction of convex bodies.

Citation URI

Similar Citations

Article Kjeldsen, Tinne Hoff; (2008)
From Measuring Tool to Geometrical Object: Minkowski's Development of the Concept of Convex Bodies (/isis/citation/CBB000773888/)

Book Gray, Jeremy; Parshall, Karen Hunger; (2007)
Episodes in the History of Modern Algebra (1800--1950) (/isis/citation/CBB000774194/)

Book Gray, Jeremy J.; (2004)
János Bolyai, Non-Euclidean Geometry, and the Nature of Space (/isis/citation/CBB000750880/)

Article Abardia, Judit; Reventós, Agustí; Rodríguez, Carlos J.; (2012)
What Did Gauss Read in the Appendix? (/isis/citation/CBB001251202/)

Thesis Braver, Seth Philip; (2007)
Lobachevski Illuminated: Content, Methods, and Context of the “Theory ofParallels” (/isis/citation/CBB001560945/)

Article Yan, Chen-guang; Deng, Ming-li; (2009)
Riemann's Idea of Geometry and its Impact on the Theory of Relativity (/isis/citation/CBB000952285/)

Article Christiansen, Andreas; (2009)
Bernt Michael Holmboe (1795--1850) and His Mathematics Textbooks (/isis/citation/CBB000931934/)

Book Voelke, Jean-Daniel; (2005)
Renaissance de la géométrie non euclidienne entre 1860 et 1900 (/isis/citation/CBB000830712/)

Chapter Giulia Giannini; (2009)
Poincaré, la nozione di gruppo e il Programma di Erlangen di F. Klein (/isis/citation/CBB227666735/)

Book de Saint-Gervais, Henri Paul; Poincaré, Henri; Klein, Christian Felix; (2010)
Uniformisation des surfaces de Riemann retour sur un théorème centenaire (/isis/citation/CBB001551391/)

Thesis Keranen, Jukka Petri Mikael; (2005)
Cognitive Control in Mathematics (/isis/citation/CBB001560623/)

Chapter Gray, Jeremy; (2006)
Modern Mathematics as a Cultural Phenomenon (/isis/citation/CBB000800129/)

Article Nastasi, Tina; (2008)
La storia del pensiero scientifico e il suo significato nell'opera di Federigo Enriques (/isis/citation/CBB001024085/)

Article Lambert, Kevin; (2013)
A Natural History of Mathematics: George Peacock and the Making of English Algebra (/isis/citation/CBB001320192/)

Authors & Contributors
Gray, Jeremy
Kieldsen, Tinne Hoff
Shoji, Kota
Parshall, Karen V. Hunger
Voelke, Jean-Daniel
Christiansen, Andreas
Archive for History of Exact Sciences
Isis: International Review Devoted to the History of Science and Its Cultural Influences
Historia Scientiarum: International Journal of the History of Science Society of Japan
Science in Context
British Society for the History of Mathematics Bulletin
Kexue Jishu Zhexue Yanjiu (Studies in Philosophy of Science and Technology)
The MIT Press
American Mathematical Society
University of Pittsburgh
University of Montana
Non-euclidean geometry
Philosophy of mathematics
Minkowski, Hermann
Bolyai, Janos (Johann) von
Lobachevskii, Nikolai Ivanovich
Riemann, Georg Friedrich Bernhard
Gauss, Carl Friedrich
Poincaré, Jules Henri
Time Periods
19th century
20th century, early
20th century
Great Britain
United States
University of Chicago

Be the first to comment!

{{ comment.created_by.username }} on {{ comment.created_on | date:'medium' }}

Log in or register to comment