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 infinitesimalsnamely 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, 1 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.

When Does a Hyperbola Meet Its Asymptote? Bounded Infinities, Fictions, and Contradictions in Leibniz

“…34 In connection with Leibniz's work on perspective, Rodríguez Hurtado et al note that The characterisation of the point of view as the meeting point in the infinity of the parallels comes from Arguesian perspective. 56 [33, p. 16] 35 The reference is to Desargues' 1639 Brouillon Project [12]. Based on Leibniz's 1679 letter to Huygens, Rodríguez Hurtado et al point out a significant difference between the Cartesian model of magnitudes and Leibniz's search for a qualitative model for studying perspective:…”

“…Parallel lines can be considered converging or diverging 34 Field and Gray [16, p. 47]. 35 Their note 56 reads: " 56 It is worth mentioning that Leibniz transcribes the definition of point of view that Desargues makes at the end of the Brouillon Project (included in: A VII 7, 111)." The page number given is incorrect.…”