In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis, (...) as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton’s Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse. “If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.”—(Bos 1974–1975, 82–83). (shrink)

Like many of their contemporaries Bernard Nieuwentijt and Pieter van Musschenbroek were baffled by the heterodox conclusions which Baruch Spinoza drew in the Ethics. As the full title of the Ethics—Ethica ordine geometrico demonstrata—indicates, these conclusions were purportedly demonstrated in a geometrical order, i.e. by means of pure mathematics. First, I highlight how Nieuwentijt tried to immunize Spinoza’s worrisome conclusions by insisting on the distinction between pure and mixed mathematics. Next, I argue that the anti-Spinozist underpinnings of Nieuwentijt’s distinction between (...) pure and mixed mathematics resurfaced in the work of van Musschenbroek. By insisting on the distinction between pure and mixed mathematics, Nieuwentijt and van Musschenbroek argued that Spinoza abused mathematics by making claims about things that exist in rerum natura by relying on a pure mathematical approach. In addition, by insisting that mixed mathematics should be painstakingly based on mathematical ideas that correspond to nature, van Musschenbroek argued that René Descartes’ natural-philosophical project abused mathematics by introducing hypotheses, i.e. ideas, that do not correspond to nature. (shrink)