Leibniz’s Laws of Continuity and Homogeneity

Abstract

We explore Leibniz's understanding of the differential calculus, and argue that his methods were more coherent than is generally recognized. The foundations of the historical infinitesimal calculus of Newton and Leibniz have been a target of numerous criticisms. Some of the critics believed to have found logical fallacies in its foundations. We present a detailed textual analysis of Leibniz's seminal text Cum Prodiisset, and argue that Leibniz's system for differential calculus was free of contradictions.

Leibniz�s well-founded fictions and their interpetations

“…A re-evaluation of Leibniz's contribution to analysis was developed in 2012 (Katz-Sherry [58]), in 2013 (Katz-Sherry [59]), in 2014 (Bascelli et al [15], Sherry-Katz [112]), in 2016 (Bair et al [16]), in 2017 (Bair et al [12], B laszczyk et al ( [24]), and elsewhere. In an apparent reaction to this work, Arthur wrote in 2018:…”

The Mathematical Intelligencer Flunks the Olympics

“…The revolutionary idea that there does exist a system, sometimes called hyperreal numbers, satisfying such a transfer principle is due to the combined effort of [Hewitt 1948], [ Loś 1955], and [Robinson 1961], and has roots in Leibniz's Law of continuity and his distinction between assignable and inassignable numbers; see [Katz & Sherry 2012], , [Bair et al 2016], [Bascelli et al 2016], as well as [B laszczyk et al 2016a]. We will provide an explanation of the extension R ⊆ * R in Section 5.…”

A cognitive analysis of Cauchy’s conceptions of function, continuity, limit and infinitesimal, with implications for teaching the calculus

“…(see Sherry (2012 and2013) and Sherry and Katz (2013) for a fuller discussion). Based on this general heuristic, Leibniz was able to assume that infinitesimals enjoy the same properties as ordinary numbers, and to operate on them accordingly.…”