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

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.