Evolution of Leibniz's thought in the matter of fictions and infinitesimals

Abstract

In this paper we offer a reconstruction of the evolution of Leibniz's thought concerning the problem of the infinite divisibility of bodies, the tension between actuality, unassignability and syncategorematicity, and the closely related question of the possibility of infinitesimal quantities, both in physics and in mathematics. Some scholars have argued that syncategorematicity is a mature acquisition, to which Leibniz resorts to solve the question of his infinitesimals namely the idea that infinitesimals are just signs for Archimedean exhaustions, and their unassignability is a nominalist maneuver. On the contrary, we show that sycategorematicity, as a traditional idea of classical scholasticism, is a feature of young Leibniz's thinking, from which he moves away in order to solve the same problem, as he gains mathematical knowledge. We have divided Leibniz's path toward his mature view of infinitesimals into five phases, which are especially significant for reconstructing the entire evolution. In our reconstruction, an important role is played by Leibniz's text De Quadratura Arithmetica. Based on this and other texts we dispute the thesis that fictionality coincides with syncategorematicity, and that unassignability can be bypassed. On the contrary, we maintain that unassignability, as incompatible with the principle of harmony, is the ultimate reason for the fictionality of infinitesimals.