The Classificatory Function of Diagrams: Two Examples from Mathematics
In a recent paper, De Toffoli and Giardino analyzed the practice of knot theory, by focusing in particular on the use of diagrams to represent and study knots . To this aim, they distinguished between illustrations and diagrams. An illustration is static; by contrast, a diagram is dynamic, that is, it is closely related to some specific inferential procedures. In the case of knot diagrams, a diagram is also a well-defined mathematical object in itself, which can be used to classify knots. The objective of the present paper is to reply to the following questions: Can the classificatory function characterizing knot diagrams be generalized to other fields of mathematics? Our hypothesis is that dynamic diagrams that are mathematical objects in themselves are used to solve classification problems. To argue in favor of our hypothesis, we will present and compare two examples of classifications involving them: (i) the classification of compact connected surfaces (orientable or not, with or without boundary) in combinatorial topology; (ii) the classification of complex semisimple Lie algebras.
KeywordsDiagrammatic reasoning Knot diagrams Mathematical classifications Combinatorial topology Compact connected surfaces Lie algebras
This collaboration is one of the results of the project “Les mathématiques en action” (2016–2017), funded by the Archives Henri-Poincaré and the Pôle Scientifique CLCS of the Université de Lorraine. We also acknowledge support from the ANR-DFG project FFIUM for which we thank Gerhard Heinzmann.
- 3.Klein, F.: Über Riemanns Theorie der algebraischen Functionen und ihrer Integrale. Teubner, Leipzig (1882)Google Scholar
- 7.Poincaré, H.: Papers on Topology: Analysis Situs and Its Five Supplements. Translated by John Stillwell. American Mathematical Society, London Mathematical Society (2010)Google Scholar
- 8.Alexander, J.W.: Normal forms for one- and two-sided surfaces. Ann. Math. 16(1/4), 158–161 (1914–1915)Google Scholar
- 10.Hilbert, D., Cohn-Vossen, S.: Geometry and the imagination; transl. by P. Nemenyi. Chelsea publishing company, New York (1990)Google Scholar
- 11.Eckes, C.: (with the collaboration of Amaury Thuillier): Les groupes de Lie dans l’œuvre de Hermann Weyl. Presses universitaires de Nancy, Éditions universitaires de Lorraine (2014)Google Scholar
- 12.Cartan, E.: Sur la structure des groupes finis et continus. Doctoral thesis, Paris (1894)Google Scholar
- 14.Artin, E.: Die Aufzählung aller einfachen kontinuierlichen Gruppen. Lecture at Göttingen University, 13–15 July. The written version of this lecture is due to H. Heesch and E. Witt (1933)Google Scholar
- 18.Dynkin, E.: The structure of semi-simple Lie algebras. Uspekhi Mat. Nauk 2(4(20)), 59–127 (1947)Google Scholar