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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) 

1 Spinoza’s Ethics proven “mathematically, that is infallibly”1

Baruch Spinoza’s (1632–1677) Ethica ordine geometrico demonstrata, which was published posthumously in 1677, is renowned for its geometrical order, i.e. for its “intimidating array of definitions, axioms proposition, demonstrations and corollaries”, as Steven Nadler puts it (Nadler 2006, p. 35).2 If the concept “mathematization” can be understood broadly as “the application of concepts, procedures and methods developed in mathematics to the objects of other disciplines or at least to other fields of knowledge” (Roux 2010, p. 324), then we might consider Spinoza’s Ethics as an attempt to mathematize philosophy. By modelling his Ethics on Euclid’s Elements it seems that Spinoza was trying to impart the rigour and demonstrative certainty of mathematics to this own philosophical system. In the Ethics Spinoza oftentimes underscored the demonstrative certainty of his metaphysical conclusions. For instance, in the scholium to Proposition 17 in Part I of the Ethics, he emphasized that he has demonstrated “from God’s supreme power or infinite nature an infinity of things in infinite ways—that is, everything—has necessarily flowed or is always following from that same necessity [eâdem necessitate sequi; eodem modo], just as from the nature of a triangle it follows from eternity to eternity that its three angles are equal to two rights angles” and in the preface to Part III he pointed out that “I shall, then, treat of the nature and strength of the emotions, and the mind’s power over them, by the same method [eâdem Methodo] as I have used in treating of God and the mind, and I shall consider human actions and appetites just as if it were an investigation into lines, planes and bodies [ac si Quaestio de lineis, planis, aut de corporibus esset]” (Spinoza 2002, pp. 228, 278; Spinoza 1677, pp. 18, 94).

Spinoza’s claims baffled his compatriots for two reason. First of all, his Ethics obliterated the carefully installed separation between philosophy and theology that had emerged in the wake of the Voetian-Cocceian conflict by showing that philosophical reason could establish theological truths (e.g. Verbeek 1988; Ruler 1995; van Bunge 2001, pp. 50–54; Douglas 2015, ch. 2, p. 113). In addition, its mathematical format gave the impression that the contents of Spinoza’s Ethics, which were deemed heretical by many of his contemporaries (Israel 1996, 2001, ch. 16), were proven mathematically (Vermij 2003, pp. 190–191). A year after the publication of Spinoza’s Opera postuma, which contained the Ethics, the apostolic vicar of the Dutch Mission, Jan van Neercassel (1625–1686), reported to Rome that Dutch non-Catholics were increasingly being seduced by Spinoza’s “philosophy and the hollow fallacy according to the method of geometrical people [per Philosophiam et inanem fallaciam secundum traditionem hominum Geometrarum]” (quoted from Klever 1988, pp. 338–339). In the lemma on Spinoza in his Dictionaire historique et critique (1697), the Rotterdam based philosopher Pierre Bayle (1647–1706) characterized Spinoza as “the first to have reduced atheism to a system and to have made a doctrine out of it that is connected and bound together in the ways of the geometers [le premier qui ait reduit en systême l’Atheïsme, & qui en ait fait un corps de doctrine lié & tissu selon les manieres des Geometres]” (Bayle 1697, vol. 1, p. 1083). To the untrained eye the Ethics looks like a mathematical work and as a result of this many of its readers, especially the unskilled, will be inclined to consent to its contents, as the Dordrecht based grain broker Willem van Blijenbergh (1632–1696) warned in his Wederlegging van de Ethica of Zede-Kunst (‘Refutation of Ethics or Morality’) (van Blijenbergh 1682, p. 3 [in the chapter ‘Wederleggingh over het eerste deel der Zedekunst van God. Door B.D.S.’]). The geometrical order of the Ethics seemed to entail that Spinoza’s heterodox ideas—his equivocation “deus sive natura”, his rejection of final causes, and his necessitarian credo according to which “[t]hings could not have been produced by God in any other way or in any other order than is the case” (Spinoza 2002, p. 235)—were “based on absolute mathematical certainty”, which outraged his compatriots (Jorink and Zuidervaart 2012, p. 18; cf. Jorink 2009, p. 29). Because of the mathematical order of Spinoza’s Ethics, mathematics came to be associated with views that were considered atheistic by his contemporaries (Vermij 1989). It is telling that in this context Bernard Nieuwentijt, who we will discuss in the next section, recounted the story of a stubborn atheist he once knew who, because he was convinced that only Euclid’s work contained “the perfect truth in everything [de volmaakte waarheit in alles]”, came to doubt the veracity of the Bible (Nieuwentijt 1725, pp. 597–598). One of the main characters in the Spinozist roman à clef, Vervolg van ‘t leven van Philopater (‘Sequel to the life of Philopater’), maintained that reason when combined with mathematics enables one “to reason infallibly concerning all natural things [om infallibel, van alle natuurlijke dingen, [...] te kunnen redeneren]” ([Duijkerius] 1697, p. 51 [italics added]).

In order to refute Spinoza’s Ethics, some of his critics, most notable Bernard Nieuwentijt, argued that Spinoza had abused the (pure) mathematical method by making knowledge claims about things that exist in rerum natura. By its very nature, Nieuwentijt explained, pure mathematics can only establish conclusions about ideas that exist only in our imagination. Conclusions about the natural world can only be established by a mixed-mathematical approach. Mixed mathematics or empirico-mathematics, i.e. the combination of observation cum experimentation and mathematics, is conducive to a religious understanding of nature because according to Nieuwentijt a better empirico-mathematical understanding of nature results in a better understanding of God’s careful design of the natural order.3 In Sect. 2, I will explore Nieuwentijt’s methodological criticism of Spinoza’s mathematical approach. In Sect. 3, I will argue that Nieuwentijt’s distinction between pure and mixed mathematics directly or indirectly influenced the methodological views of one of the Dutch Republic’s greatest eighteenth-century scientific minds, Pieter van Musschenbroek, who as I have shown elsewhere (Ducheyne 2014c), shared Nieuwentijt’s anti-Spinozist stance. In order to steer clear of epistemological and religious aberrations, our imagination is to be constrained by experience.

2 “The vain and fabulous discourses of your master”4: Bernard Nieuwentijt on Spinoza’s abuse of mathematics5

Bernard Nieuwentijt (1654–1718) was a Dutch physician and local politician, who developed himself into an amateur mathematician and experimenter.6 He was based in Purmerend, a small and wealthy town situated at roughly 20 km from Amsterdam. His father Emanuel, who was a Reformed preacher, wanted him to follow his career path. Despite his father’s wish, in 1675 Bernard inscribed at the university of Leiden to study medicine. He was soon expelled from the university because of his wanton conduct. A year later he received his doctorate in medicine from the university of Utrecht. After returning to his place of birth, he finally settled in Purmerend as a physician to the poor. During his lifetime Nieuwentijt made numerous experiments—for the most part demonstration experiments (Vermij 1987, 1991, pp. 36–42). In his obituary it is claimed that Nieuwentijt was drawn to experimental philosophy as it was practiced by Robert Boyle because he became disillusioned by Descartes’ speculative philosophical project (anon. 1718, p. 745).7 In Purmerend Nieuwentijt founded a circle of science aficionados who regularly met during 1695 and 1696. Most of his experiments date back to this period during which he also worked on mathematical problems. Nieuwentijt published several works on the calculus (Nieuwentijt 1694, 1695, 1696; Vermij 1991, pp. 16–36; Mancosu 1996, pp. 158–164; Nagel 2008 for discussion). In 1715 Nieuwentijt’s physico-theological apology Het regt gebruik der wereltbeschouwingen (‘The right use of contemplating the world’), appeared, which went through several editions and was translated in English in 1718, in French in 1725 and in German in 1731. This work made Nieuwentijt famous across Europe and it made him earn the epithet “the Dutch Ray or Derham” (Nieuwentijt 1718, vol. 1, iii). In this work Nieuwentijt sought to convince atheists of the existence of God and disbelievers, i.e. people who acknowledge God’s existence but not the authority of the Bible, of the divinity of the Scriptures (Nieuwentijt 1725, [iii]; for discussion see Freudenthal 1955; Petry 1979; Vermij 1988; for broader contextualization see Bots 1972, ch. 1; Jorink 2010). Moreover, he sought to make it clear to his readers that the workings of the universe are not governed by “blind necessity [blinde nootzakelykheit]” (Nieuwentijt 1725, pp. 811–812). In view of this Het regt gebruik is to be considered a physico-theological attack on Baruch Spinoza’s philosophical system (Nieuwentijt 1725, pp. 1–2, 6–7).

Nieuwentijt completed the manuscript of his second magnum opus, Gronden van zekerheid (‘Foundations of certitude’) just before his death. As the book’s subtitle indicates, it was meant to “refute Spinoza’s imaginary system”. Unfortunately, it was never translated in English. This time Nieuwentijt offered a methodological criticism of Spinoza’s Ethics. Pivotal to Nieuwentijt’s methodological criticism of Spinoza’s Ethics is his distinction between pure mathematics (“suyvere wiskunde”) and mixed mathematics (“gemengde wiskunde”) (Nieuwentijt 1720, pp. 2–3; Ducheyne 2017 for further discussion). The objects of pure mathematics are “entia rationis”, i.e. self-consistent ideas that exist only in our imagination and that therefore do not correspond to entities in the external world. It is exactly for this reason that pure mathematics is also called “imaginary mathematics [denkbeeldige wiskunde]”. Pure mathematics ex hypothesi deduces conclusions from entia rationis by analysing the relations between them (Nieuwentijt 1720, pp. 9, 11). Because pure mathematics deals with ideas that exist only in our imagination, it cannot teach us anything about actual things, i.e. about things that exist in the external world (Nieuwentijt 1720, p. 24). Note that God was an entity that exists in the external world, according to Nieuwentijt. By contrast, mixed mathematics deals with ideas that correspond to actually existing things (“wesentlyke zaken”) or to their properties, as it establishes ideas concerning things that exist in the empirical world through direct observation or experimentation or indirect observation or experimentation, i.e. observations and experimentations reported to us by witnesses (Nieuwentijt 1720, pp. 30, 33–36, 40). To rephrase this in Cartesian fashion, pure mathematics deals things that possess objective reality; mixed mathematics deals with things that possess formal reality. Mixed mathematicians will readily embrace “a learned ignorance [een geleert niet weten (eruditum nescire)]” (Nieuwentijt 1725, pp. 14–15), when the required experimental or observational data that lends support to the thesis they seek to confirm is lacking. Nieuwentijt furthermore differentiated between adequate ideas (“ideae adaequatae”) and abstracted ideas (“ideae abstractae”) (Nieuwentijt 1720, p. 45). Whereas an adequate idea contains all properties of the thing which it represents, an abstracted idea contains only some of the properties of the thing which it represents (Nieuwentijt 1720, pp. 44–45). Mixed mathematics can only provide abstracted ideas (Nieuwentijt 1720, pp. 44–48, 58–59, 62). More generally, Nieuwentijt endorsed the idea that we can only have abstracted ideas concerning things that exist in rerum natura—including God. Spinoza, by contrast, was convinced that philosophical reason alone can attain adequate knowledge of things that exist outside of our minds, as can be seen for instance from his assertion in Proposition XLVII of Part II of the Ethics that “[t]he human mind has an adequate knowledge of the eternal and infinite essence of God” (Spinoza 2002, p. 271).

One of the causes of atheism which Nieuwentijt highlighted in Het regt gebruik is the human inclination to consider what we have derived from our own reason or ideas concerning God, his properties and the universe as truthful. This tendency often stems from a misguided use of the pure mathematical method as a result of which “one thinks to know everything” (Nieuwentijt 1725, pp. 8–9, 15). Although Spinoza’s Ethics is based on entia rationis only, he has made claims about entia realia based on a pure mathematical approach (Nieuwentijt 1725, pp. 10, 37, 138, 244–247, 389-390). Spinoza’s mos geometricus, Nieuwentijt asseverated, gives the impression that what is claimed in his Ethics “follows from legitimate mathematical reasoning [uit regte wiskundige redeneringen volgt]” (Nieuwentijt 1725, p. 9). However, the pure mathematical method pursued by Spinoza cannot make valid conclusions about the natural world. Close scrutiny of the Ethics reveals that it contains no more truth about the external world than Aesop’s fables (Nieuwentijt 1720, pp. 245, 250).

Criticisms on the Ethics’ mathematical pretence had been launched earlier by Adriaen Verwer (1655–1717) and Frans Kuyper (1629–1691). The Mennonite merchant Verwer distinguished between knowledge based on entia rationis and knowledge based on entia realia in the context of his criticism of Spinoza’s Ethics ([Verwer] 1683, pp. 2–5; furthermore Klijnsmit 1991).8 He argued that Spinoza misled the world by pretending that his ideas are adequately proven and by presenting them in a mathematical order ([Verwer] 1683, **3v\({^{\circ }})\). The Socinian Kuyper anonymously published a refutation of the Ethics, in which he called the veracity of Spinoza’s key definitions and axiom into question.9 His refutation, was appended to the Dutch translation of the Cambridge Platonist Henry More’s (1614–1687) criticism of Spinoza’s concept of substance ([Kuyper] 1687, pp. 75–114; for more background see Petry 1981). More questioned the demonstrative power of Spinoza’s mos geometricus in Ethics. For instance, he denied that Spinoza has demonstrated Proposition 7 “by some mathematical method or with certainty [Mathematicâ quâdam methodo ac certitudine demonstravit]” because its demonstration is based on false premises (More 1679, vol. 1, p. 615). Kuyper vehemently rejected that Spinoza had proven “his atheistic theses mathematically, that is infallibly [zijn Atheistische stellingen, wiskunstiglijk, dat is onfeijlbaarlijk]” ([Kuyper] 1687, p. 87).

The frontispiece of Het regt gebruik (see Fig. 1), which was made by Jan Goeree (1670–1731) in 1715, depicts Nieuwentijt’s view on how legitimate knowledge about things external to us is to be acquired (Vandevelde 1926; Vermij 2011, pp. 212–214). The lower left side of the engraving shows two philosophers who are engaged in pure mathematics. One of them turns away from physics (“Natuur-kunde”), who as “the teacher of things [rerum magistra]” stands on the altar that is shined upon by God’s light in the centre of the engraving. He is completely absorbed by his own imagination (“Verbeelding”), the “blind fold of his eyes [den blind-doek van syn oogen]”. Physics has removed the blindfold of the figure who is kneeling in front of her. She reveals truth to him, which is represented by the naked female figure on the upper right side, via mixed mathematics, which is depicted by the scientific instruments shown on the lower right side. The message could not be clearer: only a mixed mathematical approach will enable us the establish truths concerning the external world. Goeree’s engraving thus visually summarizes Nieuwentijt’s anti-Spinozist campaign.
Fig. 1

Engraving by Jan Goeree (1670–1731) as in the 1718 edition of Nieuwentijt’s Het regt gebruik der wereltbeschouwingen—private collection of the author

3 The anti-Spinozist underpinnings of van Musschenbroek’s mixed-mathematical programme10

After his studies in medicine at the university of Leiden, Pieter van Musschenbroek (1692–1761) made a trip to London in 1717 during which he attended John Theophilus Desaguliers’ (1683–1744) lectures and met Isaac Newton (1642–1727). After having practiced medicine between 1716 and 1719, from 1719 to 1723 he became professor of mathematics and philosophy at the University of Duisburg. Between 1723 and 1739 van Musschenbroek served as professor of mathematics and philosophy at the university of Utrecht—he would also serve there as professor of astronomy from 1732. In 1740 he returned to Leiden as professor of mathematics and philosophy—a position which he kept until the end of his career. Today, van Musschenbroek is mostly remembered for his experimental research on magnetism and for his invention of one of the first capacitators, the famous Leiden jar.11

In his inaugural oration at the university of Utrecht, Oratio de certa methodo philosophiae experimentalis (‘Oration on the certain method of experimental philosophy’), which was delivered on 13 September 1723, van Musschenbroek publicly explicated and defended what he saw as the correct method of philosophizing by means of which physics (‘physica’ in Latin; ‘natuurkunde’ in van Musschenbroek’s own vernacular) could be given a solid foundation.12 According to the method he promoted, observation and experiment, on the one hand, and mathematical demonstrations, on the other, are to be combined. If a certain physics is to be established, van Musschenbroek asserted, “it will either be based on observations of the senses and next on mathematical reasoning [in sensuum observationibus & Mathematico inde ratiocinio fundabitur] or it will never be founded at all” (van Musschenbroek 1723, p. 23).13 While mentioning other natural philosophers who had combined empirical research with mathematics, he singled out Isaac Newton, to whom in February 1726 he sent a letter in which he wrote that “I thought it no error to follow in your footsteps (though far behind), in embracing and propagating the Newtonian philosophy” (Hall 1982, p. 32), as an exemplary practitioner of physico-mathematics (van Musschenbroek 1723, pp. 22, 31, 33, 45, 48–49). In other words, van Musschenbroek was signalling that he intended to bring a new physico-mathematical research programme to the university of Utrecht that was to some extent inspired by Newton and “other British men who follow the same method of philosophizing [eandem Philosophandi methodum sequuntur]” (van Musschenbroek 1723, p. 45).14 Prior to van Musschenbroek’s arrival at the university of Utrecht the Aristotelian Johannes Luyts (1655–1721), to whom van Musschenbroek referred in his oration (van Musschenbroek 1723, p. 51), and the Cartesian Josephus Serrurier (1663–1742) were in charge of teaching mathematics and physics between 1677 to 1721 and 1706 to 1716, respectively (Sassen 1959, p. 147; Kernkamp 1936, vol. 1, pp. 214–216, 295–298). In his Epitome elementorum physico-mathematicorum (‘Summary of physico-mathematical elements’) (1726) van Musschenbroek pointed out that Newton was the first to systematically “reject all hypotheses” and to teach “by a more virtuous method of reasoning [magis casta ratiocinatus methodo]”15 how to lay a solid foundation for physics “by assuming nothing what has not be most clearly demonstrated by experiments and mathematical demonstrations” (van Musschenbroek 1726, *4). According to van Musschenbroek, Newton has revealed that the solar system could not have been created “mechanically” nor could have originated “from the accidental concourse of atoms [ex fortuito concurso atomorum]”, but that instead it was created “from the council and dominion of an intelligent and powerful being [ex consilio & dominio Entis intelligentis & potentis]”. In the Dutch Republic, van Musschenbroek added, Nieuwentijt has followed “the method of philosophizing and piety of Newton [Newtoni philosophandi methodum & pietatem]” (van Musschenbroek 1726, *4bis). Van Musschenbroek, it seems, wanted to follow not only Newton but also Nieuwentijt, who appropriated the famous Lucasian Professor’s programme in the context of his battle against Spinoza and his followers (Ducheyne 2017). In the following paragraphs, I will show that many of the anti-Spinozist distinctions that were introduced by Nieuwentijt in his Gronden der zekerheid resurfaced in van Musschenbroek’s oration.16 The point I want to argue here is that Nieuwentijt’s methodological criticism of the Ethics, which states that Spinoza mistakenly relied on a pure-mathematical method in order to make knowledge claims about actually existing things, resurfaced in van Musschenbroek’s work and that for this particular line of criticism he was indebted to Nieuwentijt’s Foundations of certitude. By implication, in the paper at hand I will not engage in an analysis of the relevant sources for van Musschenbroek’s anti-Spinozism in general. His physico-theological criticism of the Ethics, for instance, was indebted to Nieuwentijt’s The right use, to Newton’s General Scholium and to Roger Cotes’ (1682–1716) editorial introduction which were both introduced in the second edition of the Principia (1713), and to many other local and non-local physico-theological literature available at the time. As I have shown elsewhere (Ducheyne 2014c), van Musschenbroek’s criticism of the Ethics’ account of matter and motion was highly indebted to Samuel Clarke’s (1675–1729) A demonstration of the existence and attributes of God (1705), which contained a fierce criticism of Spinoza’s views on the matter (Schliesser 2012).

In his inaugural oration van Musschenbroek stated that the objects of experimental physics (physica experimentalis), which ultimately leads us to “the demonstration of a first, infinitely powerful and most wise cause [ad demonstrationem causae primae, Infinite potentis, sapientissimae]”, i.e. a “most free creator [Liberalissimus Creator]” (van Musschenbroek 1723, pp. 9, 18), are “all bodies in the universe” (van Musschenbroek 1723, pp. 9–10). The attributes of these bodies cannot be known a priori and need to be determined observationally or experimentally, for we can only acquire proper ideas about the attributes of bodies by means of the senses. Since the “ideas of the attributes of bodies can be represented by quantities” they can be treated mathematically and it can be investigated which mathematical relation obtains between those quantities (van Musschenbroek 1723, p. 21).17 While the conclusions concerning “the works of nature” derived from a priori reasoning will be “uncertain, defective and loose”, physics founded on observation and experiment will be “true, certain and stable” (van Musschenbroek 1723, pp. 8, 10). The senses, van Musschenbroek contended, will never deceive us when they are “enclosed within their limits”. Those who maintain that our senses deceive us oppose to “the sanctity of the most perfect Entity, who is the fountain of all truths [sanctitati Entis perfectissimi, omnium veritatum fontis]” (van Musschenbroek 1723, pp. 19–20). Van Musschenbroek frequently underscored the theological significance of physics. For instance, in the second edition of Elementa physicae (‘Elements of physics’) (1741) he urged that physics “places the divine wonders in the clearest light [haec scientia, miracula Divina in clarissima luce ponit]” and “leads us directly to know and prove the existence of God and his providence, and to a right understanding of many of his attributes as his power wisdom, goodness,&c. [nos ducit ad existentiam Dei, ejusque providentiam cognoscendam, probandam; tum ad ejus attributa plurima, praecipue potentiam, sapientiam, bonitatem&c. optime intelligenda]” (van Musschenbroek 1741, p. 6, 1744, vol. 1, p. 9). As I have shown elsewhere (Ducheyne 2017), just as in Nieuwentijt’s thought (Nieuwentijt 1725, pp. 16–17), teleological and voluntarist considerations played an important role in van Musschenbroek’s experimental philosophy.18

Experimental physics will only reveal the “superficies” of bodies; the internal substance of bodies will forever remain unknown (van Musschenbroek 1723, pp. 13–14). “Our science”, i.e. physics, van Musschenbroek wrote, is truly most far removed “from an adequate cognition of bodies [ab adaequata cognitione corporum]”, “because a further knowledge is often denied to mortals” as, he emphasized, has been shown by Nieuwentijt (van Musschenbroek 1723, pp. 29–30). Only God perfectly knows all attributes of bodies “without any ignorance [absque ignorantione ulla]” (van Musschenbroek 1748, p. 20). Therefore many things will forever remain hidden to us. Instead of hypothesizing about those things it is better “to candidly confess our ignorance [candide ignorantiam confiteri]” (van Musschenbroek 1723, p. 36), a claim that resonates well with the ethos of Nieuwentijt’s ‘learned ignorance’. The concept of ‘learned ignorance’ has a wider history and is, it should be noted, not exclusive to Nieuwentijt’s work. In the Dutch Republic it was also used by Gisbertus Voetius (1589–1676) in the context of his campaign against Descartes (Verbeek 1993; Schuurman 2010). However, as I have shown elsewhere (Ducheyne 2017), while Voetius promoted learned ignorance as a tool to subdue natural-philosophical findings to claims made in the Bible about natural things, for Nieuwentijt’s learned ignorance consisted in admitting our ignorance about natural phenomena when the required observations and experiments are lacking. The latter position is more closer to van Musschenbroek’s than the former. In view of his refusal to attribute a cause to gravity, Nieuwentijt portrayed Newton as an exemplary mixed-mathematician who endorsed learned ignorance (Nieuwentijt 1720, pp. 32–33, 38–39). Learned ignorance also featured in Boyle’s work (Corneanu 2011, p. 179). In The Christian Virtuoso (1690) Boyle, who Nieuwentijt is known to have read, characterized “learned ignorance” as the state “wherein a man, after having taking pains to be instructed, is, by the utmost knowledge he has attained, made sensible, that that knowledge is but imperfect, and very disproportionate to the admired object” (Boyle 1772, vol. 6, p. 761). In other words, there are different sources in which ‘learned ignorance’ features from which van Musschenbroek could have drawn on. Yet the expression “to confess one’s ignorance” is exclusive to Nieuwentijt’s work (Niewentijt 1720, p. 37).

Let us now focus on what van Musschenbroek had to say on mathematics in his oration. In his oration van Musschenbroek distinguished between pure and mixed mathematics (van Musschenbroek 1723, p. 26). In his lecture notes ‘Praefatio ad Collegium Mathematicorum’, which are undated, he provided the following characterization of pure versus mixed mathematics:

Mathematics is divided into pure and impure mathematics or in abstract and mixed mathematics, which is the same. Pure mathematics considers the general nature and particular affections of magnitude and quantity and consists of geometry and arithmetic. Mixed or concrete mathematics considers the magnitudes as they are applied to certain bodies or to special subjects [...]. (BPL 240, no. 1, f. 21\(^{\mathrm{r}}\), cf. BPL 240, no. 35, f. 28\(^{\mathrm{v}})\)19

Branches of mixed mathematics include astronomy, music, optics, mechanics, hydrostatics, gnomonics, geography, aerometry, the art of warfare, etc. Whereas (pure) mathematicians are not concerned whether their concepts correspond to things that exist in nature, the mathematical physicist attempts to establish concepts that correspond to nature. In his Institutiones logicae (‘Logical instructions’) (1748), van Musschenbroek contrasted homologous or inadequate ideas, i.e. ideas that contain only some of the properties on the thing they represent, to adequate ideas, i.e. ideas that perfectly represent that what they represent. Adequate ideas are at the same time clear and distinct and perfect, i.e. they are “complete” in the sense that they “exhibit all the properties of the things they represent” (van Musschenbroek 1748, p. 19). Moreover, he distinguished between ideas whose objects are in the mind itself (for instance, mathematical figures) and ideas whose objects of outside the mind, i.e. “all created things that exist in this universe, and next God himself [omnia, quae in hoc universe existunt creata, tum & ipse Deus]” (van Musschenbroek 1748, p. 5). According to van Musschenbroek, a mathematician “begins his reasoning in a pure science and reasons from ideas of things [...] that do not exists unless be the power of imagination [non nisi vi imaginationis existunt]” (van Musschenbroek 1723, pp. 23–24). Only the essences of ideal things, i.e. things that exist only in the mind, are known to us (van Musschenbroek 1748, pp. 6–7; cf. BPL 240, no. 25, f. 55\(^{\mathrm{r}})\). Pure mathematics, which “builds on clear and distinct ideas” (BPL 240, no. 1, f. 14\(^{\mathrm{v}})\), leads to certainty, if the mathematical inference are correctly deduced. Mixed mathematics, by contrast, leads to certainty if the conclusions from certain ideas are correctly deduced and if those ideas correspond, i.e. are homologous,20 to nature. Van Musschenbroek’s distinction between pure and mixed mathematics, on the one hand, and between homologous and adequate ideas, on the other, very likely derive from Nieuwentijt’s Gronden der zekerheid.

What entitles me to conclude that van Musschenbroek’s ideas were ‘very likely’ influenced by the distinctions which Nieuwentijt introduced in his Gronden der zekerheid? As one of the reviewers of this paper suggested, could Nieuwentijt’s ideas not have reached van Musschenbroek indirectly, more particularly through Willem Jacob ’s Gravesande (1688–1742), who corresponded with Nieuwentijt and was familiar with his work? In order to answer this question let us consider ’s Gravesande objections against Spinoza’s Ethics.

First of all, like Nieuwentijt ’s Gravesande distinguished between pure and mixed mathematics as can be seen from his preface to the first volume of the first edition of Physices elementa mathematica, experimentis confirmata (1720) (‘Mathematical elements of physics confirmed by experiments’) and in his Oratio de evidentia (1724) (‘Oration on evidence’) (’s Gravesande 1720–1721, vol. 1, p. [2], 1721, vol. 1, xliii).21 In his ‘Oratio inauguralis de matheseos, in omnibus scientiis, praecipue in physicis, usu, nec non de astronomiae perfectione ex physica haurienda’ (1717) (‘Inaugural oration on the utility of mathematics in all sciences, especially in physics and also on the perfection of astronomy that is to be borrowed from physics’), which he delivered when he became professor of mathematics and astronomy, ’s Gravesande assured those who attended his oration—the curators, professors and students of the university of Leiden as well as the local burgomasters—that his mathematical physics programme would not lead to religious aberrations. Certain mathematicians, ’s Gravesande wrote, who “desire demonstrations everywhere”, always “deduce their reasoning through an excessive series of consequences, what in religion often precipitates errors [per nimiam consequentiarum seriem ratiocinia sua deducunt, quod in Religione saepe in errores praecipitat]” (’s Gravesande 1717, p. 10). They venture to submit everything to reason, although there being many things in the Scriptures, he warned, that surpass human reason (’s Gravesande 1717, p. 10). What these mathematicians fail to realize is that apart from things that have mathematically evidence there are also things that have moral evidence and that for human beings both forms of evidence are legitimate (for further discussion of this distinction, see de Pater 1995; Ducheyne 2014b). Things that have mathematically evidence are things whose negation is absolutely impossible. Mathematical evidence do not pertain to mathematics exclusively, for it also features in logic and in metaphysics. Things that have moral evidence, by contrast, are things whose negation is possible. Morally evidence features in history, in physics and in theology as far as it is concerned with knowledge that is revealed to us in the Scriptures. It is mistaken to identify moral evidence knowledge— as ’s Gravesande understood it—with probable evidence, for he claimed in Oratio de evidentia that, although mathematical and moral evidence have different foundations, “yet a different persuasion does not result therefrom [non tamen diversa inde sequitur persuasio]” (’s Gravesande 1721, vol. 1, lv). In other words, according to ’s Gravesande, things that have moral evidence things and things that have mathematical evidence are intuited by human beings as being equally certain. Although ’s Gravesande did not mention Spinoza’s name, it seems very likely that his criticism was directed at him and his followers. What is important for our present purpose is that in order to counter Spinoza’s Ethics ’s Gravesande did not invoke the distinction between pure and mixed mathematics, as Nieuwentijt and van Musschenbroek did.

Let us take a look at an instance where ’s Gravesande mentioned Spinoza’s name. In his oration ‘De vera, & nunquam vituperata philosophia’ (‘On the true and never reprimanded philosophy’) which he delivered in 1734 when he became professor of philosophy, he openly criticised Spinoza’s Ethics. There he stated that Spinoza has abused mathematics in his Ethics:

After various propositions and through many detours, he [Spinoza, whose name is mentioned a couple of lines later] finally reaches a conclusion which he would have been able to deduce from his definitions alone, if these had been proposed clearly. I will however say why he does so. His definitions are deceitful and because of this it is not immediately clear that in these definitions the ordinary meaning of words is not preserved. It would have been improper to apply the conclusion deduced only from arbitrary definitions to real things that are designated by the same words in their ordinary meaning. (’s Gravesande 1734, pp. 24–25)22

As a result of this, Spinoza has “distorted reasoning” and he has deduced “many incautious and the most absurd and worst errors” (’s Gravesande 1734, p. 25). The crux of ’s Gravesande’s criticism is that in the definitions of the Ethics Spinoza introduced terms which he gave a different meaning from their ordinary meaning. Again, he did not debunk the Ethics by arguing that it pursues a pure mathematical method. Instead, he criticized it because of the deceitful definitions its contains—thereby leaving open the possibility that truths about God could be attained a priori.23 Given that he uses different strategies than Nieuwentijt to criticize the Ethics and given that van Musschenbroek’s strategies to do so are closer to Nieuwentijt’s, it seems more likely that van Musschenbroek was indebted to Nieuwentijt on this point.

However, did these distinctions have the same anti-Spinozist underpinning for van Musschenbroek as they clearly had for Nieuwentijt? This seems to be confirmed by a series of lectures that van Musschenbroek delivered in 1725 at Utrecht. The notes for these lectures have been preserved (BPL 240, no. 12, ff. 1\(^{\mathrm{r}}\)–118\(^{\mathrm{v}}\), esp. ff. 49\(^{\mathrm{r}}\)–53\(^{\mathrm{v}}\); for an analysis see Ducheyne 2014c). In these notes van Musschenbroek characterized Spinoza as someone “who cultivates reason and believes by the force of his arguments that there probably is no God [qui rationem excolit, et credit ex viribus suorum argumentorum probabiliter \(\downarrow \)fluere\(\downarrow \) non dari Deum]” (BPL 240, no. 12, f. 2\(^{\mathrm{r}})\). In these lecture notes van Musschenbroek vehemently attacked Spinoza’s idea of God and the metaphysical necessity implied in his work by criticizing several propositions of Ethica. Although van Musschenbroek did not explicitly repeat anti-Spinozist criticism later in his career, it seems rather unlikely that he ever changed his opinion of Spinoza’s system in view of the voluntarist and teleological views which he continued to air until late in his career—as has been documented in Ducheyne (2016). Moreover, as Wiep van Bunge has recently argued, during the first half of the eighteenth century scholars gradually felt less pressed to attack Spinoza because of the increasing alleviation of theological tensions and because of the expanding popularity of Newton’s natural philosophy and physico-theology (van Bunge 2017). This could explain why later in his career van Musschenbroek did no longer mentioned Spinoza.

Van Musschenbroek identified another kind of abuse of mathematics. The aim of physics is to establish ideas that correspond to nature. Truthful ideas, van Musschenbroek commented, represent that “which exists truly in the things themselves [quod vere in rebus ipsis existit]” (van Musschenbroek 1723, p. 12). The application of mathematical ideas to nature is, however, “not always without error [non semper sine errore]” (van Musschenbroek 1723, p. 23). When concepts are not homologous to nature, the reasoning (‘ratiocinium’) based upon such ideas will be “badly deduced [male deductum]”—in the sense that the inferred conclusion will not correspond to nature (van Musschenbroek 1723, p. 26). In trying to uncover the properties of bodies which are composed of several different properties that are not fully known to us, apply mathematical ideas to things that exist in nature errors cannot be excluded. When we apply mathematical ideas to things in nature that are not, as van Musschenbroek put it, ‘homologous’ to those things, we “abuse mathematics [Mathesi abutimur]” and provide explanations of things “from fictitious hypotheses [ex hypothesibus fictis]” (van Musschenbroek 1723, pp. 27, 32), he underscored. Such “laxity [laxitas]” amounts to “an aberration from truth [aberratio à veritate]” (van Musschenbroek 1723, p. 27). For instance, he pointed out that the mathematical idea of divisibility ad infinitum is not to be applied to matter which in fact consists of indivisible atoms. Although van Musschenbroek did not drop any name, many of his contemporaries would have easily grasped that he was targeting René Descartes’ doctrine according to which matter is infinitely divisible (Adam and Tannery 1897–1909–1909, vol. 6, pp. 238–239 and vol. 8, pp. 51–52).24 Hereby van Musschenbroek diagnosed Descartes as someone who abused mathematics because he feigned mathematical ideas, instead of performing the painstaking observational and experimental work required in order to arrive at ideas that are homologous to nature. In his Oratio de methodo instituendi experimenta physica (‘Oration on the method of setting up physical experiments’), which was delivered on 27 March 1730, he later also mocked the hypothesis of “that ludicrous subtle matter of Descartes [illa ludicra Cartesii materia subtilis]”, by which he had attempted in vain to explain the motions of the celestial bodies (van Musschenbroek 1731, p. xxxiv). In the preface to the second edition of Beginselen der natuurkunde (1736), Beginsels der natuurkunde (‘Principles of physics’) (1739), van Musschenbroek emphasized that true physics is to be distinguished from the “vain conjectures by which philosophy has been impregnated by Descartes and his followers [ydele gissingen, waarmee de Wijsbegeerte door DESCARTES en zyne navolgers bezwangerd is geworden]” (van Musschenbroek 1739, p. [ii]). There he also pointed out that “most followers of Descartes first presuppose causes that pop-up in their minds and simply reason on the basis of that hypothesis [de meeste navolgers van Descartes onderstellen eerst de oorzaken, welke hun in de gedachten schieten, en gaan maar uit de onderstelling redeneeren]” (van Musschenbroek 1739, p. 105).

4 Conclusion

In this essay, I have shown that the posthumous publication of Spinoza’s Ethics (1677) led to a vigorous debate in the Dutch Republic on the epistemic authority of mathematics as a tool for making knowledge claims about the natural world. More precisely, I have highlighted that the heterodox conclusions established by Spinoza’s mos geometricus were vehemently opposed by Nieuwentijt and van Musschenbroek. Spinoza’s approach led Nieuwentijt, first, to distinguish between pure and mixed mathematics—as we have seen, this distinction was taken over by van Musschenbroek without loss of its anti-Spinozist underpinnings—and, second, to insist that only by carefully following to the latter approach one is entitled to make valid knowledge claims about the empirical world.

By insisting on the necessity of the connubium, i.e. marriage, between mathematics and painstaking empirical investigation in the establishment of legitimate knowledge about the world Nieuwentijt and van Musschenbroek immunized the atheistic dangers of unbridled mathematical imagination by constraining it by experience, on the one hand, and they avoided the errors that originated from Descartes’ programme of fictitious hypotheses, on the other. By rejecting pure mathematics as an adequate tool for establishing knowledge about the world, Nieuwentijt and van Musschenbroek had, however, to give up knowledge based on ‘clear and distinct’ or ‘adequate’ ideas. Instead, they had to settle with ‘abstracted’ ideas that correspond approximately to certain aspects of the empirical world, but this kind of modest knowledge actually fitted hand in glove with their religious ethos according to which man can only arrive at a very limited understanding of God’s creation.


  1. 1.

    The source of this quotation is [Kuyper] 1687, p. 87. Unless otherwise indicated, all translations are mine.

  2. 2.

    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.

  3. 3.

    For more background on the religious context in the Dutch Republic, see van der Wall (2003) and Jorink (2010).

  4. 4.

    Translation of: “de ydele en Fabuleuse Betogingen van uwen Meester” (Nieuwentijt 1720, p. 255).

  5. 5.

    In the first three paragraphs of this section I draw on material from Ducheyne (2017).

  6. 6.

    For bibliographical details I have relied on anon. (1718) and Vermij (1991, ch. 1).

  7. 7.

    On Boyle’s influence on Nieuwentijt, see Vermij (1987).

  8. 8.

    On Verwer, see van Bunge et al. (2003, vol. 2, pp. 1026–1028). The selling catalogue of Nieuwentijt’s library, which appeared 2 years after his death, contains no copy of [Verwer] 1683. It does contain, however, a copy of Verwer (1698) (anon. 1720, p. 35, item n\({^{\circ }}\) 54).

  9. 9.

    On Kuyper, see van Bunge et al. (2003, vol. 2, pp. 578–580). I have found no evidence that Nieuwentijt was familiar with [Kuyper] 1687.

  10. 10.

    In this section I occasionally draw on material from Ducheyne (2014c).

  11. 11.

    For the above biographical details I have relied on de Pater (1979, chapter 2).

  12. 12.

    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.

  13. 13.

    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.

  14. 14.

    In Ducheyne (2015) I have argued that Newton’s methodology influenced van Musschenbroek’s only in a fairly general way.

  15. 15.

    ‘Castus’ can both mean ‘virtuous’ and ‘pious’. Van Musschenbroek may have deliberately chosen to exploit this polysemy.

  16. 16.

    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).

  17. 17.

    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).

  18. 18.

    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).

  19. 19.

    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 [...]”.

  20. 20.

    Unfortunately, van Musschenbroek did not provide further explication of what correspondence or homology amounts to.

  21. 21.

    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.

  22. 22.

    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: [...].”

  23. 23.

    See Van Besouw ms. for an interesting analysis of the a priori demonstrations which ’s Gravesande relied on to make claims about God.

  24. 24.

    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).



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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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

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

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