Evolution of Leibniz’s Thought in the Matter of Fictions and Infinitesimals

Monica Ugaglia, Mikhail Katz

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

In this chapter, 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 syncategorematicity, 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. (In this chapter, we employ “syncategorematic” as a shorthand for “eventually identifiable with a procedure of exhaustion” and, as a consequence, “involving only assignable quantities.” We also identify syncategorematicity with potentiality, as suggested by some part of the scholastics, and in Sect. 2.2, we show that the two characterizations are equivalent, provided that “potentiality” is intended in the correct way: namely as in the unending iterative procedures of Greek mathematics.) On the contrary, we maintain that unassignability, as incompatible with the principle of harmony, is the ultimate reason for the fictionality of infinitesimals.

Original languageEnglish
Title of host publicationHandbook of the History and Philosophy of Mathematical Practice
Subtitle of host publicationVolume 1-4
PublisherSpringer International Publishing
Pages341-384
Number of pages44
Volume1
ISBN (Electronic)9783031408465
ISBN (Print)9783031408458
DOIs
StatePublished - 1 Jan 2024

Bibliographical note

Publisher Copyright:
© Springer Nature Switzerland AG 2024.

Keywords

  • Fictionality
  • Infinita terminate
  • Infinite divisibility
  • Infinitesimals
  • Leibnizian metaphysics
  • Potential infinity
  • Syncategoremata
  • Unassignability

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