Unité et pluralité de l’espace mathématique chez Leibniz

Abstract

Abstract In this paper, we study the possibility of accepting a plurality of spaces in Leibniz – a question which was already tackled by Yvon Belaval and Nicolas Rescher, both of whom answered it positively. Starting from a famous passage in the Principes de la Nature et de la Grâce where God is said to have chosen a world “with terrain, place, and time arranged in the best way possible”, we ask ourselves whether this implies that God chooses amongst a variety of spatial settings and, if so, whether this variety would imply a plurality of spaces. We recall a first existing solution to this issue, which consists in distinguishing between an “abstract” and a “concrete” space. After a quick survey of the many variants of this interpretation, we demonstrate that this disctinction is neutral, in the best case scenario, with regard to our initial problem. We then confront the question of the necessity of geometrical truths – which are, for Leibniz, eternal truths inscribed in divine understanding. At first glance, this seems to prevent the possibility of a plurality of geometries (i. e. of spatial structures endowed with incompatible properties). We show that this is not the case and that one must pay attention to the “conditional” nature of geometrical truths, which Leibniz insists upon in some places and which appears compatible with their being absolutly necessary. Finally, we confront the results of the two preceding sections with the current knowledge on the project of an analysis situs. We put particular emphasis on the way in which metaphysical principles were put forward by Leibniz in the elaboration of his geometry.