Unité et pluralité de l’espace mathématique chez Leibniz

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.

Analysis Situs, the Foundations of Mathematics, and a Geometry of Space

This chapter focuses on Gottfried Wilhelm Leibniz’s writings on analysis situs—a collection of mathematical and philosophical investigations into the foundations, development, and formalization of geometry, examining how they contributed to the development of modern geometry. It first explains the concept of situation (situs) as the first source of Leibniz’s theory of a relational space and its use in metaphysics, then considers Leibniz’s development of a new formalism for geometry (a characteristica geometrica propria) and his use of (Euclidean) metric relations between figures or points as the main subject of his analysis situs. It also discusses Leibniz’s definition of space as an order of situations, his grand ambition to establish the principles of Euclidean geometry based on the abstract concept of a metric space, and how his writings on analysis situs became intertwined with metaphysics and the theory of knowledge. Finally, the chapter reviews Leibniz’s definition of space in relation to monads and bodies.