Leibniz on bodies and infinities: rerum natura and mathematical fictions

TL;DR

This paper interprets Leibniz's views on the nature of infinitesimals and infinite divisibility, arguing they are mathematical fictions consistent with Leibniz's philosophy, rather than logical contradictions or impossible entities.

Contribution

It offers a nuanced interpretation of Leibniz's letter, clarifying the status of infinitesimals as well-founded fictions aligned with his philosophical principles.

Findings

Infinitesimals are well-founded fictions, not logical contradictions.

The letter supports viewing infinitesimals as mental fictions, not absolute impossibilities.

Distinction between infinite divisibility in nature and mathematics is crucial.

Abstract

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part-whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals - as well as imaginary roots and other well-founded fictions - may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality.