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Constraining (mathematical) imagination by experience: Nieuwentijt and van Musschenbroek on the abuses of mathematics

S.I.: Use & Abuse of Maths


Like many of their contemporaries Bernard Nieuwentijt (1654–1718) and Pieter van Musschenbroek (1692–1761) were baffled by the heterodox conclusions which Baruch Spinoza (1632–1677) drew in the Ethics. As the full title of the EthicsEthica 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 (type 1 abuse). In addition, by insisting that mixed mathematics should be painstakingly based on mathematical ideas that correspond to nature, van Musschenbroek argued that René Descartes’ (1596–1650) natural-philosophical project (and that of others who followed his approach) abused mathematics by introducing hypotheses, i.e. (mathematical) ideas, that do not correspond to nature (type 2 abuse).


René Descartes (1596–1650) Early eighteenth-century Dutch Republic Bernard Nieuwentijt (1654–1718) Pure versus mixed mathematics Baruch Spinoza (1632–1677) Pieter van Musschenbroek (1692–1761) 



Research for this paper was funded by the Vrije Universiteit Brussel under the form of a Research Professorship. Parts of this essay were delivered at the international workshop ‘The Uses and Abuses of Mathematics in Early Modern Philosophy’ which took place in Budapest on 10 March 2015. I am grateful to its audience for feedback. I am also indebted to the Special Collections Department at Leiden University Library for permission to quote from material in their care, to Ronald Desmet, Koen Lefever, and Jip Van Besouw for comments on an earlier version of this essay, to the editors of this special issue for their encouragement and hard work, and to the two anonymous referees of this journal for valuable feedback.


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Authors and Affiliations

  1. 1.Department of Philosophy and Moral Sciences, Centre for Logic and Philosophy of ScienceVrije Universiteit BrusselBrusselsBelgium

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