# Antimathematicism in Early Modern Philosophy and Science

**DOI:**https://doi.org/10.1007/978-3-319-20791-9_447-1

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Mathematics Mathematical physics Explanation Descartes Spinoza Hume## Introduction

The seventeenth century saw many striking advances in pure mathematics and in the application of mathematics to physics. As a result, the status of mathematics rose markedly, and the expectation was established that mathematical techniques were the key to tracking fundamental truths about the natural world. Less familiar is the battery of considerations that various writers brought out to advocate limits, sometimes strict limits, on the epistemic status of mathematics and its applicability to the natural world. This more cautious perspective on the growing influence of mathematics can be called *antimathematicism.*

## Antimathematicism

*Dialogue Concerning the Two Chief World Systems*, he argued that mathematics was the key to the intelligibility of natural phenomena. And in

*The Assayer*, one finds what is perhaps the most often quoted philosophical idea in all his writings.

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these one wanders about in a dark labyrinth. (in Drake 1957, pp. 237–238)

The idea is that important truths about the natural world are themselves mathematical, so knowledge of them requires the application of mathematical techniques.

Along with Galileo, Descartes has been almost unanimously received as a central figure in the growth of the importance of mathematics. His identifying matter with geometrical extension seemed to make his mathematical innovations in analytic geometry into a kind of theoretical physics. Physics is geometry applied to extension regarded as actually created in motion by God. Nevertheless, Descartes’ writings provide the basis for almost all of the crucial antimathematicist arguments.

Although Descartes’s admiration for the certainty of sound mathematics is well known, he held that mathematical knowledge is not uniquely distinguished by its certainty or intelligibility. Mathematics can meet the high standard of being “clearly and distinctly perceivable” with its guarantee of truth, but the knowledge of the human mind and of God equally meet the standard. Moreover, these non-mathematical subject matters are both prior in knowledge to mathematics. Knowledge of God is prior because creation is ontologically dependent on God and because knowledge of God is methodologically required for discharging Descartes’ famous methodological skepticism. Knowledge of the mind is prior to knowledge of mathematics because knowledge itself is dependent on the knowing mind (Nelson 2018). Descartes does acknowledge that practicing mathematics is very important in the development of intellectual virtues that are necessary for pursuits more important than mathematics itself. These more important pursuits include morality and the development of technologies to improve human well-being. The principal value of mathematics, then, is instrumental; it provides training in the mental virtues that promote human ends (Jones 2006; Lachterman 1989).

Many readers of Descartes’s *Meditations* assume that he regarded mathematical objects as having “true and immutable natures” that conferred upon them the exalted status of being true whether or not they actually exist (Kenny 1968). Recent scholarship overturns this assumption by showing how Descartes’ invocation of true and immutable natures in the Fifth Meditation is an artifact of the methodological procedure of that work. At that point in the *Meditations*, the meditator clearly and distinctly perceives mathematical truths. She can, however, still raise a methodological doubt about what is the subject matter of these truths. It might be extension, the essence of matter. But the skeptical hypothesis that the mind itself or God is the object of these truths cannot be discharged until the existence of extension is proved beyond skeptical doubt in the Sixth Meditation (Cunning 2010). This circumstance also reinforces Descartes’ position that knowledge of God and mind are prior to mathematical knowledge. It also coheres with his deflationary stance regarding universals and other abstracta, including putative mathematical universals (Nolan 1998).

Another antimathematicist doctrine of Descartes’ that is often overlooked is the dependence of mathematical cognition on the faculty of imagination. God and the mind can be cognized by the pure intellect, without any assistance from the partly corporeal imagination. But mathematical thinking (possibly excepting a general, undifferentiated idea of extension) requires the cooperation of the imagination. Extension taken generally must be delimited by imagined planes, lines, and points. This detracts from its epistemic status because of the special status accorded pure intellection by Descartes and those strongly influenced by him – Malebranche – for example (Nolan 2005; Sepper 1996). Algebraic thinking, which makes use of symbols, also requires directly imagining the symbols even apart from what Descartes say as the overall dependence of algebra on geometry (Liu 2017).

Apart from these strictures regarding pure mathematics, antimathematicism is also found in Descartes’ understanding of the application of mathematics to the natural world. The qualitative, non-mathematical character of his mature scientific writings is unmistakeable. This feature has drawn much scorn and many attempts to account for it. It is often suggested that Descartes tried but failed to emulate Galileo’s deployment of mathematics (e.g., in Burtt 1932; Heilbron 1982). Or perhaps he regarded his qualitative explanations as hypotheses that he intended to convert to mathematical treatments at some later date (e.g., Clarke 1992). Another suggestion is that Descartes’ intended his *Principles of Philosophy* as a textbook, so he needed to uphold the traditional division between the physics of natural phenomena and the mechanics of artifacts (Gabbey 1993). On this view, a quantitative, mathematical mechanics would have to wait for a separate textbook. The reason is that Descartes’ fundamental physical laws, because they are derived from metaphysical considerations, are too general to be expressed mathematically.

Descartes’ preference for scientific explanations that depend on contrived stories about the behavior of corpuscles and vortexes is, however, hard to explain apart from his antimathematicism. Other interpretations of Descartes’ failure to produce quantitative science belie the fact that he repeatedly displayed these skills in his correspondence (Schuster 2013; Gaukroger 1995). Since his early encounter with Isaac Beeckman, Descartes showed considerable interest in the project that his Dutch friend (and later enemy) called physico-mathematics. Garber (2000) suggests that Descartes preferred not to publish this work under his own name and gives one convincing reason why. Already in his early, unpublished *Rules*, Descartes was convinced that explanations of natural phenomena had to terminate in purely extensive properties. The Galilean paradigm was more promiscuous allowing appeal to inherent qualities of bodies such as “heaviness” or “magnetic virtue” that were not analyzed into size, shape, and motion. On the assumption that bodies have such occult qualities, it is possible to give quantitative descriptions of phenomena, but for Descartes these assumptions are certainly false. So while Descartes was, for a time, interested in problems of this kind, he did not regard their solutions as good science (Garber 2000, 2002). If Descartes wandered from his supposed program of quantitative science, as Burtt asserted, it was not due to a lack of ability. His prioritizing his metaphysics of matter could be one part of the explanation of his neglect of the Galilean style in his published works.

Descartes’ metaphysical doctrine that extension is the essence of matter provides even deeper grounds for his antimathematicism. Because there is no absolute space in which to situate Cartesian bodies, their boundaries depend on adjacent bodies. And because there is nothing to distinguish adjacent bodies from each other except for their relative motion, bodies are everywhere engaged in “mutual transfer” with adjacent bodies (Garber 1992; Sowaal 2004, 2005). One important ramification of this feature of extension that Descartes notices is that no parcel of matter is strictly determined by geometrical shape for even a moment. Exact geometrical shapes in matter must be idealized to points of time owing to matter’s being everywhere actually divided by motion (Crockett 2009). Descartes allows that the human body retains its identity through continual change only because it is united with a soul. The same cannot be said about the identity of cannonballs, corpuscles, or (notoriously) nonhuman animals. Descartes’ position here means, against Galileo, that material things cannot, even in principle, exactly correspond to mathematical figures. Furthermore, since the volume of a body is simply its extension, there is no variation in “quantity of matter” within a volume. This leaves no room for Newtonian mass or for a pointlike “center of mass” to trace out mathematically expressible trajectories through space.

An even more far-reaching metaphysical argument is based on Descartes’ requirement that motion in a plenum involves circuits owing to the rejection of both the vacuum and the occult quality of vacuum abhorrence. Because the regular periodic celestial vortices are not perfectly concentric, when the channel through which the vortical material flows becomes narrower, that material must divide into smaller parts to increase the rate of flow so that the period is maintained. This process takes place, however, at a scale smaller than any specified scale. From this it follows that the number of particles into which matter is divided is in fact infinite (although Descartes prefers to say “indefinite”). Once one sees how this example works, it is easy to generalize it. Recall that bodies are, without exception, in mutual transference with respect to adjacent bodies. This means that everybody, without exception, is involved in a circuit in which it is being divided without limit. The same goes for each of its parts and so on without limit. Although Descartes does not explicitly draw this conclusion, it follows that there is no microscopic level at which matter is not blooming into particles each spawning indefinitely many vortices, sub-particle, sub-vortices, and so on without limit (Nelson 1995). The Cartesian metaphysics of body does not allow for the application of mathematical idealization to any truth about actual bodies (Nelson 2017).

This explanation of the qualitative character of Descartes’ preferred scientific explanations raises another interpretive puzzle. Despite the reasons reviewed above for Descartes to prefer qualitative explanations, he repeatedly makes the claim that the science presented in the *Principles of Philosophy* and elsewhere *is mathematical*. This seems incompatible with his anti-mathematical doctrines and raises the possibility that he revised his views about the applicability of mathematics before finishing the composition of the *Principles*. In the Preface to the work, he claims to *deduce* from his first principles of mind, extension, and God “knowledge of all the other things to be found in the world.” And at the end of Part Two which is about fundamental physics, he states that the principles of geometry and pure mathematics suffice for physics and enable us to provide “certain demonstrations” of natural phenomena. Later in the same passage, he again claims to have provided “mathematical demonstrations.”

Statements like these certainly invite the negative appraisals that Descartes’ scientific theorizing has received. He seems to purport to produce mathematically informed explanations, but he actually trades almost entirely in qualitative descriptions of imaginative mechanisms. Magnetism, for example, is explained as the effect of grooved corpuscles twisting through grooved passages in iron. Descartes’s characterization of his scientific explanations as mathematical can, however, be consistent with his antimathematicism. In the theory of deduction that Descartes worked out in his *Rules for the Direction of the Native Intelligence*, he replaces traditional formal logic with a system of analysis and synthesis grounded in clear and distinct perceptions (Nelson 2017). This means that the constraints on what counts as a deduction or demonstration are lax compared to formalist approaches. If a desired conclusion can be analyzed into components including simple, clearly, and distinctly perceivable items, then the analyzed components can be rearranged into a synthetic deduction. All that is required is that each step in the deduction be understood to contain the simplest residues of analysis as components (Rogers and Nelson 2015; Smith 2010). This general theory of deduction applies equally in mathematical and non-mathematical contexts, so Descartes would have thought himself justified in thinking that his scientific explanations had all the certainty and rigor of quantitative mathematical explanations of the sort produced by Galileo.

Versions of some of these Cartesian grounds for antimathematicism are found in later seventeenth-century thinkers despite their deep disagreements with Descartes and with each other on some fundamental issues. Spinoza, for example, shared Descartes strictures on mathematics deriving from its dependence on imagination (Schliesser 2014). Like Spinoza, Margaret Cavendish rejected Descartes substance dualism and included self-motion and thinking in the essence of matter (Cunning 2016). The material universe is, moreover, a plenum. In agreement with Descartes, Cavendish’s considered view seems to be that this entails the infinite divisibility of matter and rules out atomism (Detlefsen 2006). Again agreeing with Descartes, she sees this as leading to a complexity of phenomena that is unfathomable by the application of mathematics. There is also a version of the neo-Platonic idea that all souls are sympathetically related that she adapts to her materialism. For Cavendish, every parcel of (thinking) matter has some knowledge of every other parcel. And because of the infinity of the universe, this knowledge entails an infinite complexity that cannot be tracked by applied mathematics. Moreover, details of her theory of perception also preclude the possibility of mathematics tracking metaphysically real phenomena (Peterman 2017). And finally, Cavendish is in accord with Descartes’ and Spinoza’s nominalism and anti-abstrationism regarding the objects of mathematics (Cunning 2016; Peterman 2017).

By the end of the seventeenth century, anti-mathematical arguments about the applicability of mathematics lost much of their force in light of the development of sophisticated techniques for calculating and approximating based on the calculus. These were applied with great skill by Newton and, in short order, by many others as well. Anti-mathematical arguments about the ontological status of mathematical objects were not affected by these developments. Philosophers such as Berkeley and Hume went farther than Descartes and Spinoza in stressing the role of imagination in mathematical cognition. For them, all reasoning deals in images and geometrical precision cannot exceed what can be mentally discriminated in those images. Others conceded the exact applicability of mathematics to the traditional disciplines of mixed mathematics, i.e., optics and astronomy, but denied it to more complicated terrestrial phenomena. Such a “containment strategy” can be found in Buffon, in Mandeville, and, arguably, in Adam Smith (Schliesser 2017). This anti-mathematical strategy is, like the others, ultimately traceable to Cartesian considerations about the complexity of phenomena.

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