Procedures of Leibnizian infinitesimal calculus: An account in three modern frameworks

TL;DR

This paper critically examines various modern frameworks for Leibnizian calculus, highlighting misunderstandings in recent scholarship and proposing Robinson's infinitesimal analysis as the most faithful formalization.

Contribution

It clarifies foundational issues in Leibnizian calculus and formalizes it within Robinson's SPOT framework, emphasizing key distinctions like boundedness and the law of continuity.

Findings

Robinson's framework best captures Leibnizian procedures

Identifies errors in Arthur's comparison of frameworks

Formalizes Leibnizian concepts using SPOT

Abstract

Recent Leibniz scholarship has sought to gauge which foundational framework provides the most successful account of the procedures of the Leibnizian calculus (LC). While many scholars (e.g., Ishiguro, Levey) opt for a default Weierstrassian framework, Arthur compares LC to a non-Archimedean framework SIA (Smooth Infinitesimal Analysis) of Lawvere-Kock-Bell. We analyze Arthur's comparison and find it rife with equivocations and misunderstandings on issues including the non-punctiform nature of the continuum, infinite-sided polygons, and the fictionality of infinitesimals. Rabouin and Arthur claim that Leibniz considers infinities as contradictory, and that Leibniz' definition of incomparables should be understood as nominal rather than as semantic. However, such claims hinge upon a conflation of Leibnizian notions of bounded infinity and unbounded infinity, a distinction emphasized by early Knobloch. The most faithful account of LC is arguably provided by Robinson's framework. We exploit an axiomatic framework for infinitesimal analysis called SPOT (conservative over ZF) to provide a formalisation of LC, including the bounded/unbounded dichotomy, the assignable/inassignable dichotomy, the generalized relation of equality up to negligible terms, and the law of continuity.