Manipulative imagination: how to move things around in mathematics

Résumé En

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 [1]. 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.