Abstract
Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet in this debate, this chapter argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key premise in the argument is that mathematical representations and material nature correspond.
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Notes
- 1.
For example, in a 1669 letter to Thomasius, Leibniz endorses these two broadly Aristotelian criteria for science, and adds that the “scholastici”—though not Leibniz himself—deny they can be met by geometry (1875–90, IV.168–70).
- 2.
- 3.
I focus here on Du Châtelet’s Institutions de Physique, and still more narrowly on the foundational and methodological discussions early in the Institutions, as opposed to more specific discussions of physics later in the work. It is undisputed that mathematics plays a pivotal role in the Institutions. The puzzle is why and how Du Châtelet takes this to be justified.
- 4.
One recent statement: “there must be some kind of correspondence between the mathematics in which we formulate our theories and the nature of the physical world, a correspondence that helps explain, on the one hand, the effectiveness of the mathematics in describing the world, and on the other, the success of our inferences about the world on the basis of that mathematics” (North 2021, 5).
- 5.
- 6.
Aristotle’s Physics distinguishes between pure geometry and optics, which is “hupó” or subordinate to it (194a6–11). But optics is still a mathematical rather than natural science: the Meteorology expounds observations about reflection drawn “from the theory of optics” (1984, 372a29), then separately lays out causal natural-scientific theses concerning halos, rainbows, and so on (372b13–378b6). Early modern optical texts continued to classify optics as applied mathematics, citing Aristotle as an authority (Dear 1995, 41–58).
- 7.
See for example a letter to Hartsoeker of 8 February 1712 (Leibniz 1875–1890, III:534–5; cf. VII:452).
- 8.
- 9.
Instead, Voltaire praises the psychological account given in Berkeley’s New Theory of Vision. Voltaire presents Berkeley’s theory as both empirical and metaphysical: the intended contrast is with the merely mathematical and acausal style of classical optics. Du Châtelet seems sympathetic to this point (1738a, b, 537). But given the broad scope of the term ‘metaphysics’ here, we cannot assume that Voltaire influenced Du Châtelet’s shift towards the a priori metaphysical systems of Leibniz and Wolff in the Institutions, as Gessell (2019, 868–71) has argued.
- 10.
Kant, in 1790, still cites Keill as giving an authoritative “demonstration” of matter’s infinite divisibility (2004, 8:202). We will see below that Du Châtelet could endorse Keill’s argument, so long as it is not taken to show that matter is made up of an infinity of actual parts (Du Châtelet 1742, 191–92).
- 11.
Voltaire comments that this example shows how the mathematics of the infinitely large and small, despite its apparent absurdity (“déraison”), is “founded on simple ideas” (1961/1734, 71; 76). Yet he denies that we can directly draw any physical consequences from it.
- 12.
If the division of matter were actually completed, there would only be empty “pores,” not matter; thus the actual existence of matter contradicts its infinite divisibility (1738, 102). Clarke’s Fourth Reply to Leibniz presents a similar argument that also identifies pores with void (Leibniz and Clarke 2000, 35). But Aristotle already discusses the gist of this argument, attributing it to the Presocratic atomists (1984, 324b25–5a16). In the same passage, Aristotle discusses a theory of pores (poroi)—though as it appears in the Empedoclean theory of causation and change, rather than an atomist argument.
- 13.
Wolff famously claims to follow a mathematical method in philosophy, though Lambert and other critics soon objected that his method has little to do with mathematical practice (Basso 2008). Wolff presents the method as in the first instance logical: it could be grasped “even if mathematics did not exist” (Wolff 1965/1726, §139).
- 14.
The Wolffian Samuel Formey, secretary of the Berlin Academy, took this position to extremes. He argued in a reply to Euler that quantity, as an essentially imaginary concept, cannot be applied to “real and existing” objects at all (1754/1747, 281). For Formey, then, attempts at mathematical physics, including Newton’s, issue in “absurdity,” “occult qualities,” and contradictions (282; 285; 294). He rejects not only geometrical arguments for the indivisibility of body (284), but all quantitative conceptions of motion and force, taking these to be based on “notions imaginaires & confuses” (292).
- 15.
- 16.
The 1740 edition of the Institutions suggests that souls (but not all simple substances) are individuated by their unique representational states, and cites Leibniz (§128). However, Du Châtelet deletes this discussion from the 1742 edition.
- 17.
By an internal property, in turn, I take Du Châtelet to mean one that reliably appears intrinsic to a thing. Because of Du Châtelet’s idealist commitments, magnitudes are not absolutely intrinsic properties of material things. Material magnitudes are partly mind-dependent, and so can’t be instantiated in worlds without minds. But for practical purposes, one can abstract from the dependent status of material beings and draw a contrast between the intrinsic and the relational at the level of what Du Châtelet calls substantial phenomena.
- 18.
Wolff also emphasizes that qualitatively similar things can differ in quantity (1720, §20–22, 2001/1731, §196). He denies, however, that things can be individuated by difference in quantity (1720, §20; 1973/1710, 118). It is debatable whether this is a merely epistemic point, or also concerns metaphysical individuation.
- 19.
Compare Wolff: “If I am supposed to tell someone how large something is, I must tell him what relation it has to a certain measure that he is familiar with” (1720, §20; 1713, I, 38). Wolff concludes from this that magnitude cannot be grasped by the understanding, but must be sensibly “given” (gegeben) (§20; cf. Sutherland 2005, 147). Du Châtelet does not explicitly draw this epistemological consequence.
- 20.
This is the sense of ‘part’ (méros) at issue in Definition 1 of Book V of Euclid’s Elements: “A magnitude is a part of a magnitude, the less of the greater, when it measures the greater” (Euclid 1908, II.113). In turn, Definition 4, the so-called Axiom of Archimedes, stipulates that magnitudes in ratio to one another must be “capable, when multiplied, of exceeding one another” (II.114; White 1992, 148–154). Thus, infinitely small or large magnitudes are incommensurable with finite ones. In Commandino’s widely used edition of the Elements (1572) this was presented not just as a definition of magnitudes in ratio, but as a genuine axiom, ranging necessarily over all magnitudes whatsoever (De Risi 2016, 626). The point was contested even before the development of calculus, however. Cavalieri and Torricelli, among others, treated infinite collections of indivisibles as summing to finite magnitudes, thus violating the Axiom of Archimedes (608).
- 21.
For example, she defines the “measure” of dead force algebraically, as the product of mass and initial velocity, rather than in terms of parts of a homogeneous magnitude (1742, 438–39).
- 22.
- 23.
The Port-Royaliens define abstraction in terms of analysis into parts or aspects (Arnauld and Nicole 1996, 37). Du Châtelet does not define abstraction this way. Locke and Wolff introduce abstraction to give a nominalist account of how we acquire general ideas (Locke 1975, 2.11.9; Wolff 1713, I, 26). Du Châtelet does not explicitly share this aim, or their nominalism. Abstractionism about Lockean general ideas need not go together with abstractionism concerning mathematical objects. Kant and the Frege of the Grundlagen, for example, retain an abstractionist account of concepts such as <tree> or < cat>, but deny that it applies to mathematical objects. Conversely, one could endorse mathematical abstractionism without applying the theory to other general ideas.
- 24.
- 25.
Crousaz also objected to Leibnizianism, and initiated a correspondence on this point with Du Châtelet in 1741 (Du Châtelet 2018, II.43–47). Du Châtelet commented to Maupertuis two months later that “les Institutions m’ont encore attiré une drôle d’adversaire, c’est Crousaz” (II.54), and sent a cutting reply to Crousaz the next day (II.56).
- 26.
For the early Ockham, ficta exist in the mind, but lack the reality of the soul’s powers or dispositions: they have intentional existence. Ockham went on to eliminate ficta from his epistemology. Pasnau (1997, 82) argues that Ockham found a more parsimonious alternative to ficta, but never had “doubts about the concept of fictive existence” itself.
- 27.
Nonetheless, the manner in which mathematical objects exist is “special” and qualified (1984, 1077b16). Substances exist in an unqualified way. So Aristotle’s mathematical objects are not substances: they are in substances. Interpreters disagree on further details (Hasper 2021).
- 28.
A letter from Leibniz to Varignon—published in 1702 and perhaps known to Du Châtelet—sketches this position, though Leibniz is cautious about labelling infinitesimals as fictions (Leibniz 1849, IV:91–95; Reichenberger 2016, 152n54). Further details were not publicly available, and remain controversial (Jesseph 2015; Rabouin and Arthur 2020).
- 29.
Du Châtelet wrote, but never published, a Grammaire Raisonée. The known surviving chapters discuss substantive terms in detail, stating that the grammar of substantives is based on a generalization from the metaphysics of substance (Du Châtelet n.d., ff. 133v–135v; 138r; 145r).
- 30.
Similarly, when the Port-Royal Grammaire turns from linguistic signs themselves to their signification, it asserts that understanding the latter requires an account of “what occurs in our minds” (Arnauld and Lancelot 1975, 65–68). These mental operations include conceiving concepts and objects of perception, judging propositions, and reasoning through syllogisms, which are all discussed in the Port-Royal Logic. Humanly universal logical principles, then, underlie all languages qua “signifying…thoughts” (41). We find similar assumptions in Maupertuis’s (1740) essay on the origin of language, which treats mental capacities expressed in all language, rather than the features of any specific language. For broad discussion of early modern general grammar as irreducible to either formal logic or empirical linguistics, see Foucault (1966, 92–136).
- 31.
Descartes makes the still stronger claim that ideas “cannot strictly speaking be false” (1984, 26). Leibniz, for his part, stresses that an “abstraction n’est pas une erreur,” including in mathematics, so long as one “knows” that the material world is not really as the abstraction presents it (1875–1890, V.50; these New Essays were only published after Du Châtelet’s death).
- 32.
Du Châtelet’s later commentary on the Principia confirms this point. She holds for example that the mathematical Lemmas in Book I of the Principia not only lay out Newton’s “method of first and last ratios,” but also establish “general solutions” for Newton’s “entire theory” of gravitation (1759, 9; 32). These solutions then “explain astronomical phenomena” in Principia Book III (9).
- 33.
- 34.
Other examples Du Châtelet considers include treating the moon as a point-mass, comets as subject solely to gravitational forces, outer space as if it were a void, and so on. These ideas are further developed in her commentary on the Principia.
- 35.
In the early manuscript “Of Attomes,” for example, Newton contends that objections to the indefinite divisibility of body apply equally in mathematics. But they have absurd consequences in the mathematical case. Therefore, these objections are unpersuasive in the case of bodies as well (Janiak 2000, 213). Leibniz, throughout his career, regards matter as actually infinitely divided (1875–1890, II:77; II:268; Rutherford 1990, 544–49). Yet in early works, he presents infinite division as compatible with atomism (IV:15–26; IV:228–29).
- 36.
Aristotle’s Physics defines the continuous as that which can be endlessly divided into further divisibles (1985, 200b19; 232b25). While apparently allowing some completed actual infinities, Aristotle does not countenance actual infinite collections of parts of continua (White 1992, 111–12n54; 156–61; Coope 2005, 80–81). Broadly Aristotelian approaches to continuity were still authoritative in the eighteenth century. Outside of relatively isolated mathematical works, continuity lacked a clear, positive definition; it was usually conceived as a mere lack of division or determination (De Risi 2021). See for example Wolff’s definition of continuity (2001/1730, §554).
- 37.
Leibniz’s arguments for continuity sometimes use the premise that “nature…observes the same [rule]” as does geometry (1875–1890, IV:375–76; IV:568–69). Du Châtelet probably did not know these texts, however.
- 38.
- 39.
An influence here may be Jacques Rohault’s introductory work on physics, which Du Châtelet owned (Du Châtelet 2018, I.367). Near the beginning of this work, Rohault offers a proof that matter cannot have simple parts, but he also rules out an actual infinity of material points. Instead, “matter is indefinitely divisible” and has no definite number of actual parts (Rohault 1723, I.32).
- 40.
As for the continuity of change, McNulty (2019) argues that Kant considers it an “a priori…truth of metaphysics,” linked to a demand for intelligible explanation, though also grounded in the “geometric” continuity of space and time (1595; 1600; 1605–7). McNulty’s reading suggests similarities between Du Châtelet and Kant on the continuity of change.
- 41.
Many thanks to Wolfgang Lefèvre and Anat Schechtman for helpful written comments, as well as to William Marsolek for discussion. I presented drafts of some of this material at the Philosophy of Science Association meeting in Baltimore, a Du Châtelet discussion group at Paderborn University, and the British Society for the History of Philosophy conference in Edinburgh. I thank the participants on these occasions, particularly Clara Carus, Manuel Fasko, Ruth Hagengruber, William Harper, Jil Muller, Hanns-Peter Neumann, Areins Pelayo, and Edward Slowik. I also thank Katherine Dunlop, Andrea Reichenberger, and Maja Sidzińska for sharing relevant work in progress. Research on this paper was partly supported by the Deutsche Forschungsgemeinschaft (DFG), project number 435124693. Section 4.4.1 draws on my “Du Châtelet on the Need for Mathematics in Physics,” Philosophy of Science 88, 1137–48 (2021).
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Wells, A. (2023). “In Nature as in Geometry”: Du Châtelet and the Post-Newtonian Debate on the Physical Significance of Mathematical Objects. In: Lefèvre, W. (eds) Between Leibniz, Newton, and Kant. Boston Studies in the Philosophy and History of Science, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-031-34340-7_4
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