Abstract
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 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 (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).
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Notes
Whether Spinoza succeeded in proceeding in a mos geometricus in the Ethics and whether he himself thought that he had succeeded in doing so are issues from which I will abstract here. For discussion see e.g. Curley (1986), Steenbakkers (1994, pp. 141–151, 2009), Garrett (2003, pp. 99–103), Nadler (2006, ch. 2), van Bunge (2012, pp. 40–44), and Schliesser forthcoming.
On Boyle’s influence on Nieuwentijt, see Vermij (1987).
For the above biographical details I have relied on de Pater (1979, chapter 2).
In order to contextualize the claims that van Musschenbroek develops in his 1723 oration, I will rely on other parts of his oeuvre. Since the core ideas of van Musschenbroek’s methodological programme on which I will focus here remained quite fixed during his career, as will be shown, his later work can, in my view, be legitimately used to shed light in his earlier work.
Van Musschenbroek did not spell out in much detail how empirical data and mathematics are to be combined. See Ducheyne (2015, pp. 288–294) for further discussion.
In Ducheyne (2015) I have argued that Newton’s methodology influenced van Musschenbroek’s only in a fairly general way.
‘Castus’ can both mean ‘virtuous’ and ‘pious’. Van Musschenbroek may have deliberately chosen to exploit this polysemy.
The book sale catalogue of van Musschenbroek’s library contains a copy of Nieuwentijt’s Regt gebruik der wereldbeschouwingen en Gronden der zekerheid (see anon. 1762, p. 29, item no. 328).
Willem Jacob ’s Gravesande, on whom more is to follow, also underscored that mathematical physics essentially explores relations between quantities. For discussion see Ducheyne (2014a, p. 101).
This obviously put him in opposition with Spinoza, but also with Descartes. In the annotations in one of his copies of Institutiones physicae (1748) van Musschenbroek pointed out that teleology was banished from philosophy by “Descartes and his disciples [Cartesius cum suis Sectatoribus]” (BPL 240, no. 54, interleaved folio facing p. 2).
Translation of: “Mathesis dividitur in mathesin puram et impuram: sive in abstractam et mixtam, quod idem significat. Pura considerat magnitudinis ac numeri generalem naturam, ac proprius affectiones, et consistit in Geometria et Arithmetica. Mixta vel concreta considerat magnitudines ut certis corporibus et subjectis specialibus applicatas [...]”.
Unfortunately, van Musschenbroek did not provide further explication of what correspondence or homology amounts to.
I am indebted to Jip Van Besouw for these two references, which he uses in Van Besouw ms., and for discussion of ’s Gravesande’s strategy of debunking Spinoza’s Ethics.
Translation of: “Post varias Propositiones, per multas ambagas, tandem incidit in conclusionem, quam ex solis definitionibus potuisset deducere, si clare hae propositae fuissent. Quare autem ita agat, dicam. Definitiones captiosae sunt, & hac de causâ non statim patet, in his non servari verborum significationem vulgarem. Ineptum fuisset, conclusionem ex solis arbitrariis definitionibus deductam, ad res veras, vulgari sermone ipsis illis vocibus designatas, transferre: [...].”
See Van Besouw ms. for an interesting analysis of the a priori demonstrations which ’s Gravesande relied on to make claims about God.
Descartes’ view on the divisibility of matter was aired by his early eighteenth-century followers. In his Oratio pro philosophia (‘Oration for philosophy’) (1706), which he delivered upon becoming professor in mathematics and philosophy in Utrecht, Serrurier claimed that it can be shown mathematically that matter is infinitely divisible (Serrurier 1706, pp. 39–40).
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Acknowledgements
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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Ducheyne, S. Constraining (mathematical) imagination by experience: Nieuwentijt and van Musschenbroek on the abuses of mathematics. Synthese 196, 3595–3613 (2019). https://doi.org/10.1007/s11229-017-1392-1
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DOI: https://doi.org/10.1007/s11229-017-1392-1