1. Introduction

In this article, I outline Emilie Du Châtelet’s philosophical justification of mathematical methods in philosophy and natural science. Like Christian Wolff, she regards space, time, and mathematical objects as ideal entities—as products, in part, of the imagination. While Wolff concludes that mathematics is not a good guide to how the world really is, Du Châtelet defends the indispensability of mathematics for an account of the physical world. In the final section, I briefly note some ways in which Du Châtelet’s approach differs from common contemporary accounts of the indispensability of mathematics; I suggest that her views are nevertheless defensible and worthy of further study.

2. Wolff’s Ambivalence about Mathematics in Physics

We now tend to assume both that physics is an importantly distinct discipline from mathematics, and physics nonetheless can and should be mathematical. In the seventeenth and eighteenth centuries, by contrast, the demarcation between mathematics and physics in this period was far from clear, and the mathematization of physics was vigorously resisted in some quarters. As Gingras (Reference Gingras2001, 385) puts it, historians tend to focus “on the ‘winners,’ those who … accepted the mathematical conception of natural philosophy,” rather than on various “resistances to mathematization.”

Furetière’s Dictionnaire (Reference [Furetière1690, s.v. “physique”) is typical in defining physics (physique) as “the science of natural causes.” The dictionary offers various definitions of mechanics (méchanique) under a separate heading. Most of these definitions designate mechanics as a branch of mathematics. Mechanics is not described as directly treating actual natural causes (s.v. “méchanique”; Shank Reference Shank2018, 46–47). The institutional structure of the early French Academy reflected these distinctions, scheduling mathematics sessions on days different from those on physique (George Reference George1938).

The separation of causal physics and mathematical mechanics was still common in the eighteenth century. Gingras (Reference Gingras2001) notes, for example, that Diderot and Buffon criticized the mathematization of physics throughout the 1740s and 1750s. Christian Wolff, arguably the most influential German philosopher of the early eighteenth century, was also ambivalent about mathematical physics. To be sure, Wolff famously defends what he calls the mathematical method, taking genuine explanations to involve deduction from axioms and definitions, broadly on the model of Greek geometry. Furthermore, Wolff considers definitions in mathematics to be models of clarity and distinctness.Footnote ¹

Praise for mathematical method and the precision of mathematical concepts, however, has historically been quite compatible with doubts about causal applications of mathematics (Mancosu Reference Mancosu1992; Dear Reference Dear1995, 161–68). Wolff fits this pattern. He is pessimistic about the role of mathematics in causal explanation, stressing that explanations ought to track how things are in themselves (“was in den Sachen selbst liege”), at the level of fundamental simple substances (Wolff Reference Wolff and École2001, 336). These substances are nonextended. We do not observe them directly, but their causal features can be partly deduced through metaphysical argument. Insofar as mathematical concepts depend on our imagination, however, they do not track properties of fundamental simple substances or mind-independent causal relations generally.Footnote ² Wolff thinks it also follows from these considerations that mathematical objects are at best merely possible entities. As such, mathematical propositions are hypotheses that do not prove that their objects are actually instantiated.

On the basis of arguments like these, Dunlop (Reference Dunlop2013) has shown, Wolff was willing to downgrade central claims in Newton’s Principia, such as those concerning force, to at best hypothetical status (and, at worst, to being mere figments of the imagination wrongly taken to be causes). Wolff interprets these claims as describing regularities in phenomena, not causes; they may be useful for discovery and prediction but cannot genuinely explain without metaphysical supplementation. Summing up his differences with Newton, Wolff wrote to Manteuffel on January 27, 1741, that Newton was no “metaphysician” and “only expressed imaginary notions [notions imaginaires]” rather than “real concepts” (Wolff Reference Wolff and Stolzenberg2019, 1:209). Dunlop even suggests that Wolff regards mathematical physics as merely “advantageous for metaphysics” but not as “necessary, or even especially important” (Dunlop Reference Dunlop2013, 467). That is, even in the order of discovery, Wolff may not regard mathematical methods as indispensable for doing rational cosmology.

Wolff’s alternative is, in brief, to do metaphysics—although in a way that he takes to be responsive to empirical constraints. Wolff contends that (mathematical) physics must be reconstructed as rational cosmology, a branch of axiomatic metaphysics that he thinks can fully account for real causes (Wolff Reference Wolff1737b, sec. 76; Reference Wolff and Arndt1983, sec. 143; Reference Wolff, Gawlick and Kreimendahl1996, sec. 59, 94–97, 139). It does so by tracking the causal powers and states “of each simple thing” (Wolff Reference Wolff, Gawlick and Kreimendahl1996, sec. 593; Dyck Reference Dyck2020, 123).

3. Du Châtelet: Mathematical Physics and Mathematical Idealism

Du Châtelet repeatedly points out the influence of Christian Wolff on her work, and there is a growing body of recent scholarship on the topic.Footnote ³ Here I focus on affinities between Wolff’s views and Du Châtelet’s idealism about space, time, and mathematics. She holds that the fundamental level of created reality consists of real, simple substances, which are not in space and time. The bodies we can observe are partly grounded in our cognitive faculties.

As Carson (Reference Carson2004, 170) put it, mathematical concepts are for Du Châtelet “doubly removed from the real.” Geometrical extension, for example, is a product of a second step of abstraction from bodies, which are already nonfundamental (Du Châtelet Reference Du Châtelet1740, 179–80). As such, Du Châtelet presents mathematical objects as “fictions” or “imaginary beings” (99–100, 105, 118–19). She also dismisses some scientific hypotheses, such as the Cartesian vortex theory of gravitation, as merely imaginary (Du Châtelet Reference Du Châtelet1738, 534; Reference Du Châtelet1759, 32–33, 111–13).Footnote ⁴

Given these commitments, we might expect Du Châtelet to follow Wolff and deny that Newtonian physics can track the real causal structure of the world without supplementation. Instead, she is optimistic about mathematical physics. The preface of the Institutions stresses the crucial explanatory role of geometry in physics: “With no resort to it, one would hope in vain to make great progress in the study of nature. It is the key to all discoveries; and if there are still several inexplicable things in physics, that is because geometry has been insufficiently used to explain them, and one has perhaps not yet gone far enough in this science” (Du Châtelet Reference Du Châtelet1740, 3). While this passage specifically concerns geometry, it is noteworthy that when Du Châtelet recasts a number of Newton’s key proofs in the language of Leibnizian calculus (in her commentary on Book III of Newton’s Principia),Footnote ⁵ she does not explicitly suggest that she regards calculus as raising distinctive logical or ontological problems.Footnote ⁶

Her account of the argumentative structure of Newton’s Principia also expresses optimism about freestanding mathematical physics. In Principia Book III, she writes, Newton “applies … the propositions of the first [i.e., Book I] to the explication of celestial phenomena” (Du Châtelet Reference Du Châtelet1759, 9). This is so even though much of Book I presents abstract mathematical propositions, or what she calls the foundations of “the geometry of the infinite” or infinitesimal (9). Such propositions abstract from manifest kinds or “species” of matter and from the kind of medium in which bodies move (9). Nevertheless, for Du Châtelet Book I “provides” Newton’s “entire theory” of gravitation (9).

Propositions referring to imaginary mathematical objects can thus yield a true theory (if not the full details of its application to actual phenomena).Footnote ⁷ The general propositions of Book I are premises in sound physical demonstrations, despite their mathematical status. And, conversely, Du Châtelet suggests that the specific applications of mathematics laid out in Book III can be used to address puzzles in “mere geometry [la seule Géométrie],” that is, in pure mathematics (Reference Du Châtelet1759, 37). So, unlike some early French readers of Newton, Du Châtelet understands the Principia as an integrated work of mathematical and physical science—not as a mere treatise in mathematics.Footnote ⁸

4. Vindicating the Use of Mathematics in Physics

In this section, I present three ways in which Du Châtelet attempts to justify the role of mathematics in physics. Fully evaluating these justifications is a project for another time.

First, Du Châtelet generalizes from particular cases of the legitimate scientific use of idealized mathematical objects, in order to defuse worries about the fictional or imaginary character of mathematical objects. Here we can make use of work distinguishing abstractions (which merely leave out some properties of the target phenomena) from idealizations (which represent target phenomena falsely; see, e.g., Koyré Reference Koyré1968, 45–46; McMullin Reference McMullin1985). As we will see, some of Du Châtelet’s remarks show that she regards mathematical objects as products of abstraction. Yet she also implies that mathematics involves strictly false idealizations. For example, she holds that the ideas of mathematical space and time are images “produced by confused ideas” (Du Châtelet Reference Du Châtelet1740, 113).

Du Châtelet frequently discusses fairly uncontroversial cases of idealized mathematical representation—cases where even those taking a more realist view of mathematical objects might accept some role for idealization. To take one example, Newton’s calculations of the shape of the earth began with a series of mathematical idealizations. Newton had treated the earth as a fluid body of homogeneous density: first as a sphere and then as an elliptical spheroid. Like Newton himself, Du Châtelet regards these assumptions as probably false. She notes that Newton voiced doubts about homogeneity and suggested an alternative hypothesis: the increasing density of the earth toward its core (Du Châtelet Reference Du Châtelet1759, 56–57, 194).

Nevertheless, she contends that Newton’s idealized treatment of the earth as homogeneous was fruitful: it was testable and could be refined through an evolving research program. She is in a position to note that Newton’s theory had been substantially refined by subsequent observations.Footnote ⁹ Moreover, this is a rare case in which the universality of Newtonian gravitation (i.e., the attraction of “all the particles composing the earth” taken distributively and not just collectively) plays a direct role in Newton’s reasoning (Du Châtelet Reference Du Châtelet1759, 57–59).Footnote ¹⁰

This example shows that, for Du Châtelet, abstract representations that do not truly represent empirical reality can still play a role in correct reasoning. What matters is whether the reasoning in question leads to empirically accurate conclusions and not whether each of its premises is strictly speaking true. Some problems in astronomy, for example, can be solved using simple but false Ptolemaic assumptions “because we can in these instances substitute one hypothesis for the other [i.e., a Copernican hypothesis] without damaging the truth [sans faire tort à la vérité]” (Du Châtelet Reference Du Châtelet1740, 106). A physicist may make such assumptions so long as doing so does not result in any “error” in “explications and … experiments” (Du Châtelet Reference Du Châtelet1742, 206). Strictly false or distorting representations should be distinguished from erroneous judgments and inferences that may be made on their basis.Footnote ¹¹

Du Châtelet considers many other cases that illustrate this point. One can treat the moon as a point-mass, comets as subject solely to gravitational forces, outer space as if it were a void, and so on. All these cases involve successive mathematical approximation, which begins with the simplest ideal cases and continues through gradually more complex cases as more relevant evidence is obtained, refining mathematical parameters as needed. These passages strongly imply that Du Châtelet accepts Newton’s method of approximation—discussed by Cohen (Reference Cohen1980) and Smith (Reference Smith and Cohen2002), among others—as a legitimate use of mathematical idealization. For example, she argues that an advantage of Newtonian gravitational theory is that, as “knowledge of phenomena” becomes “more exact … the easier it is to apply the attractive principle to their explication” (Du Châtelet Reference Du Châtelet1759, 184). She is suggesting that, as more precise observations and experiments are obtained, the principle of attraction, and in particular what we would now call the gravitational constant, may itself be subject to revision.Footnote ¹²

More broadly, Du Châtelet defends what she calls general or physical explanations; this provides a more general framework for making sense of idealizations in explanation. Du Châtelet distinguishes between the “general” or physical explanation of phenomena and engagement “in detail with phenomena and their mechanical causes” (Reference Du Châtelet1740, 321; Reference Du Châtelet1742, 203). General explanations are at best incomplete and may rely on strictly false assumptions. But they do pick out genuine features of a “force,” such as the force of attraction, even if “the cause of this tendency” is not thereby explained in mechanistic terms (Du Châtelet Reference Du Châtelet1759, 10).Footnote ¹³ So as Brading (Reference Brading2019, 89–91) has emphasized, although Du Châtelet takes mechanistic explanation as an ultimate goal of inquiry, her account does not thereby undercut all nonmechanistic characterizations of force.

Her position faces a serious challenge, however: mathematical representations appear to be different in kind from bodies, which themselves differ from the nonextended simples Du Châtelet takes to be most fundamental in created reality. It is not clear that quantitative approximation can usefully be applied across these distinctions in kind.Footnote ¹⁴

Mathematical representations and bodies, however, have an important feature in common: they are essentially spatiotemporal. Mathematical objects are derived from acts of abstraction from bodies, which are themselves nonfundamental yet real substances. What this means is that mathematical objects are partly grounded in bodies and may approximate the properties of bodies.Footnote ¹⁵ Du Châtelet holds that our representations can have degrees of clarity and distinctness. She could make use of this assumption to further work out the relevant notion of approximation, via the suggestion (discussed further below) that mathematical representations are in some sense indeterminate.

There is a difference in kind, on Du Châtelet’s view, between bodies and simple substances. But the latter are essentially nonspatiotemporal, and Du Châtelet affirms that it is “impossible” for us to determinately “represent” their causal properties (Reference Du Châtelet1740, 169–70). So it is natural for her to deny that approximation can yield determinate knowledge of simple substances.

A second line of argument emphasizes the epistemic advantages of abstracting from the details of particulars and insists that there is no genuine alternative to mathematical abstraction in particular. Du Châtelet contends that all the sciences “must” employ abstract mathematical notions, both in the context of discovering laws or principles—this is the art d’inventer or ars inveniendi—and to address problems that cannot be solved by the unaided understanding (Reference Du Châtelet1740, 106). Here she breaks with a widely held Aristotelian doctrine that is still to be found in Locke. What is “by us … measurable” in numerical terms, Locke writes, is “principally … Expansion and Duration,” that is, pure spatial and temporal quantities rather than qualities (Reference Locke and Nidditch1975, sec. II.XVI.8).Footnote ¹⁶

For Du Châtelet, by contrast, it is unavoidable that the sciences represent qualities mathematically. The need for abstraction is grounded in limitations of our intellect. Given the finitude of our mental faculties, in most cases we cannot reason about a multiplicity of particulars without abstracting from many “internal determinations,” in order to represent their relational, quantitative features (Du Châtelet Reference Du Châtelet1740, 107).

This point applies not just in scientific contexts but in everyday reasoning. To take Du Châtelet’s example, an iron bar has numerous internal determinations, some of which explain its manifest solidity and cohesion. But we can characterize the length of the bar numerically; the parts of this quantity exist “outside of one another” and contingently “are one by their union” in a way that may remain constant across variations in internal determinations (Du Châtelet Reference Du Châtelet1740, 98).Footnote ¹⁷ Quantitative descriptions have a generality that is lacking in accounts that appeal to the manifest secondary qualities of bodies. This is no accident: mathematical objects are not completely determinate, according to Du Châtelet. For example, they are not subject to some versions of the principle of the identity of indiscernibles.Footnote ¹⁸ Indeterminacy has epistemic advantages, however. Mathematical representation is a privileged way of knowing general properties of matter—in a way that is not possible through singular perceptions of bodies—even if the entities it posits are metaphysically ideal.

A third argument reinforces the indispensability of mathematical representation by undercutting Wolff’s alternative, namely, rational (or broadly metaphysical) physics and cosmology. Du Châtelet takes metaphysics to have a specific, limited role to play. From Wolff and Leibniz, she borrows fundamental principles of reasoning and some minimal ontological structure, such as Wolff’s version of the substance/attribute/mode distinction (Stan Reference Stan2018). But empirical physics, rather than Wolffian rational cosmology, supplies determinate truths about the actual world, including its most general descriptive laws and principles.

Breaking with Wolff, Du Châtelet argues that the objects of metaphysics can be conceived only by the understanding and not the imagination. But it is precisely the imagination, and the spatiotemporal properties it deals with, that is needed for making determinate claims about actual bodies.Footnote ¹⁹ Du Châtelet does appear sympathetic to some aspects of Wolff’s mathematical method but repeatedly suggests that metaphysics has not yet learned enough from mathematical practice to attain anything like mathematical rigor (Du Châtelet Reference Du Châtelet1740, 131–33, 150–51).

5. Conclusion

I have argued that despite her broadly Wolffian metaphysics, Du Châtelet is far more optimistic than Wolff about the explanatory and epistemic status of mathematics and that this position is linked to core features of her metaphysics and epistemology. I would like to conclude by stepping away from the historical narrative provided so far and drawing connections to recent discussions of the scientific indispensability of mathematics. Despite thematic similarities between Du Châtelet’s work and contemporary debates, some of her assumptions are strikingly different from those commonly made today. Connecting these contemporary and historical discussions can help highlight the distinctiveness of Du Châtelet’s position. She arguably stakes out a viable position that has relatively few contemporary adherents.

First, we have seen that Du Châtelet asserts that geometry explains physical things, rather than merely representing them. She presents this as a natural claim, but in the contemporary philosophy of mathematics literature, it is fairly controversial (Baker Reference Baker2009; Saatsi Reference Saatsi2011).

Second, Du Châtelet does not rest her account of the ontology of mathematics solely on what it can do for physics. It is unclear whether she takes any ontological conclusions to follow merely from the indispensability of mathematics. By contrast, indispensability arguments today are not merely based on the fact that mathematical objects play a crucial role in science. They make an additional epistemological assumption that it is rational to “believe in the existence of any entity that plays an indispensable explanatory role in our best scientific theories” (Baker Reference Baker2009, 613). One plausible worry about this approach is that it determines the ontological status of mathematics merely in terms of their relation to scientific theories, rather than providing a freestanding account of mathematics. This is a reason to take a second look at Du Châtelet’s use of metaphysics rather than epistemology in grounding the status of mathematics.

A final, elementary point is that Du Châtelet does not deny the existence of mathematical objects outright but also does not endorse Platonism or some other form of heavyweight realism.Footnote ²⁰ This type of intermediate position—not metaphysically realist about mathematical objects but committed to mathematical indispensability—seems underrepresented in the literature today and deserves further attention.