Ars Combinatoria

 

This section contains as yet unpublished manuscripts by Leibniz that concern the Ars Combinatoria and more particularly his tabular methods. The German philosopher's combinatorial practice is based on two tools: the table, which Leibniz links to inventories, and thus to the Ars Inveniendi, and the form, which he links to the characteristic, and thus to the Ars Characteristica. The table and the form are in fact two sides of the same coin: one develops and the other envelops. Indeed, the table is a tool that allows Leibniz to arrange different terms in bidimensional space. The relations between these terms then establish a synoptic panorama of the different possible combinations and the way they are generated. The form, on the other hand, envelops all the combinations within it. The combinatorial structure is expressed by means of the identical. Indeed, the equality between two forms expresses the invariants that define the combinatorial subclasses of a structure.

It is through the tabular practice that combinatorics is first integrated within Leibnizian mathematics, at the beginning of the Parisian stay. These works, published in the third volume of the Akademie's series VII, deal with questions of series and quadratures. From 1674 onwards, Leibniz worked intensively on algebraic questions and the tables gradually took on a purely heuristic role in his work. 

The texts we propose here are texts from after the Paris period (1677-1680), in which Leibniz uses tables to analyse problems related to series and quadratures, despite his considerable interest in formal aspects and symbolic problems at that time. These texts constitute a double counterweight in the image that one tends to construct of Leibniz's mathematical practice. On the one hand, Leibniz uses diagrams as a tool for calculation and not only as a support for the imagination, which should be distrusted. On the other hand, he is an experimental mathematician who gives great importance to induction and does not resign himself to reducing all reasoning to a mechanical and deductive derivation.