*Andrew Janiak. Journal of the History of Philosophy. Volume 45, Issue 1. January 2007.*

Although natural philosophers in the late seventeenth century engaged in various disputes regarding the proper interpretation of forces, the debate became especially acute with the appearance of Newton’s theory of gravity in Principia Mathematica in 1687. Newton and his critics tended to agree that if some force were “real”—such as the force one material body impresses on another by colliding with it—this would mean that the force causes a change in some body’s state of motion. A contrast class would be a so-called Coriolis force, a fictitious force that does not alter the state of any body. This type of force would be a mere calculating device, where we would treat the body’s motion as if it arose from the force in question. Against this background of agreement, Newton’s interlocutors then argued that his treatment of gravity in the Principia saddles him with a substantial dilemma. if Newton contends that gravity is a real force, he must ultimately rely on action at a distance because of his well-known failure to characterize the mechanism underlying gravity. However, if he seeks to avoid distant action, he must admit that gravity is not a real force, and that he has therefore failed to discover the cause of the phenomena his theory associates with gravity, such as the planetary orbits and the tides. This is precisely the dilemma that Leibniz famously attempted to foist upon Newton in response to his apparent deviation from the norms established in the mechanical philosophy. Since Newton himself insists that he presents a merely “mathematical” treatment of force in the Principia, it may seem natural to conclude that he aims to avoid invoking action at a distance by denying that gravity is a real force, construing it as a mere calculating device. Many prominent eighteenth-century Newtonians read the Principia in just this way.

This reading is bolstered by Newton’s famous proclamation to Richard Bentley that action at a distance is unintelligible, and therefore to be rejected by natural philosophers. He wrote to Bentley in 1693, six years after the Principia first appeared:

It is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact … That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that i believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.

In forcefully rejecting the very idea of action at a distance, Newton at least appears to accept one horn of the dilemma. And it might seem reasonable to infer that this rejection entails that Newton does not conceive of gravity as a real force.

Yet matters do not rest there, for elsewhere Newton appears to avoid accepting this horn of the dilemma. As is well known, he acknowledges in the General Scholium that he has failed to discover the cause of, or the mechanism underlying, gravity.6 He then adds a rather striking contention, which bears quoting at length because it is nestled in between other famous pronouncements:

Thus far i have explained the phenomena of the heavens and of our sea by the force of gravity [Hactenus phæ nomena cæ lorum & maris nostri per vim gravitatis exposui], but i have not yet assigned a cause to gravity. indeed, this force arises from some cause that penetrates as far as the centers of the sun and the planets without any diminution of its power to act, and that acts not in proportion to the quantity of the surfaces of the particles on which it acts (as mechanical causes are wont to do) but in proportion to the quantity of solid matter, and whose action is extended everywhere to immense distances, always decreasing as the squares of the distances … I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and i do not feign hypotheses [hypotheses non fingo] … And it is enough that gravity really exists [Et satis est quod gravitas revera existat], acts according to the laws that we have set forth, and suffices for all the motions of the heavenly bodies and of our sea [& ad corporum caelestium & maris nostri motus omnes sufficiat].

If Newton intended to treat gravity as purely mathematical, as a mere calculating device, we would not expect him to contend that it “really exists.” And he does not dodge the implication that, as a real force, gravity bears causal relations, for he concludes the Scholium to Proposition 5 of Book III with the dramatic pronouncement: “And therefore that force by which the moon is kept in its orbit is the very one that we generally call gravity.” So although Newton is agnostic in the Principia on the underlying cause of gravity-by which he means its underlying physical basis, say in some medium between bodies, or in some type of mediating particle—his agnosticism does not hinder him from claiming that gravity prevents the moon from following the inertial trajectory along the tangent to its orbit. He therefore does not appear to take the Principia to be neutral regarding the causes of certain motions, including the motions that constitute the lunar orbit. As some have argued, discovering such causes might be one of the text’s primary goals.

The interpretive difficulty we face, therefore, is rather stark. Newton appears to claim both that gravity is a real force—which means that it causes various natural phenomena—and that action at a distance must be rejected within natural philosophy. But given his agnosticism concerning the physical basis of gravity, how can he contend that it is causally efficacious without invoking action at distance? How can it be specified as the cause of anything without characterizing its underlying physical basis, or what Newton sometimes calls its “physical seat?” if Newton is committed to finding the “physical seat” of gravity, how can he proclaim in the General Scholium that gravity “really exists”? Given what Newton knew, even at the time of the third edition of the Principia in 1726, an ether might eventually have been discovered as the medium in question, in which case one would think that the proper conclusion is that the ether-but nothing above and beyond it called “gravity”—is what “really exists.” One would presume that if the “physical seat” has yet to be discovered, referring to the force of gravity at all would have to be merely provisional-proper only for the purposes of calculation. And yet Newton’s statements do not seem to be provisional.

This paper attempts to answer this set of questions. I argue that the solution lies in reinterpreting Newton’s distinction in the Principia between what he calls the “mathematical” and the “physical” treatment of force. As we will see, it may be natural to assume that the merely “mathematical” treatment of force brackets questions concerning causation, leaving them for the future “physical” treatment. One reason that interpretation seems compelling is that there is a relevant historical precedent in astronomy. According to a prominent tradition, Copernican astronomy is a complex mathematical theory that eschews any causal account of the heavenly orbits. Hence it was said to “save”—but not to explicate the cause of—the phenomena. However, I will suggest that Newton’s mathematical treatment is intended to identify a real force—a genuine cause of motion-and not merely to employ a calculating device. What the mathematical treatment of force leaves for future research is not the discovery of causes, in my view, but rather the discovery of the physical characterization of the real force it has identified. This may enable Newton to avoid the dilemma with which Leibniz confronts him, but it will take the rest of the paper to spell this out.

## 1.

Near the opening of the Principia, Newton contrasts what he calls the “mathematical” and the “physical” treatment of force. In the definitions—after defining various sorts of motion and of force, and in particular after defining what he takes to be the various quantities of centripetal force—Newton writes of the concept of force as he employs it in general: “This concept is purely mathematical, for i am not now considering the physical causes and seats of these forces.” Similarly, he describes his use of the term ‘impulse’ by noting that he considers “not the species of forces and their physical qualities but their quantities and mathematical proportions, as I have explained in the definitions.” So whereas a physical treatment of force describes, among other things, its “causes and qualities,” a mathematical treatment eschews such a description, providing instead a characterization of its “quantities.” But what precisely does this mean?

As a first step toward clarification, I propose that we think of this Newtonian distinction as a technical one. By this I mean that the distinction cannot be understood antecedently to its articulation in the text, despite the fact that it appears to be a perfectly familiar distinction. There are at least two potential contrasts that Newton might have in mind, and it is important to determine which one is salient. On the one hand, we might distinguish things into those that are mathematical—such as numbers, sets, and points—and those that are physical—such as rocks, comets, and planets. if we were to apply this to forces, we would distinguish (e.g.) between some impressed force and a Coriolis force, between a genuine cause of motion and a useful calculating device that in fact causes nothing. On the other hand, we might distinguish our treatment of things into a “mathematical” type and a “physical” type; in this case, ‘mathematical’ and ‘physical’ modify the treatment of the thing in question, not the thing itself. Here, then, is a preliminary suggestion concerning Newton’s view: a treatment of force can be mathematical in his technical sense even if it characterizes perfectly ordinary physical entities, such as planets, perfectly ordinary physical processes, such as the progress of the tides, and perfectly ordinary quantities, such as the distance between two material bodies. As we will see, there is even a sense in which it can specify the cause of ordinary phenomena such as the tides and the orbit of the moon (it can identify a real force). At this point, of course, this is simply one step toward clarification.

Unfortunately, Newton’s own discussions of his “mathematical point of view” can sometimes obfuscate his intentions. Consider, for instance, his caveat at the very end of the Principia’s definitions, just before he begins to discuss space and time in the Scholium:

Further, it is in this same sense that I call attractions and impulses accelerative and motive. Moreover, I use interchangeably and indiscriminately words signifying attraction, impulse, or any sort of propensity toward a center, considering these forces not from a physical but only from a mathematical point of view. Therefore, let the reader beware of thinking that by words of this kind I am anywhere defining a species or mode of action or a physical cause or reason, or that I am attributing forces in a true and physical sense to centers (which are mathematical points) if I happen to say that centers attract or that centers have forces. (Newton, Principia, 408)

Although material bodies are perfectly real, their “centers” (as Newton considers them in his mathematical account) are merely mathematical points. Gravity is very nearly as the inverse square of the distance between the centers of material bodies, where the center is taken as a merely mathematical object. As Newton says, he should not be read as attributing any force to mathematical points, for the latter obviously do not generate any alterations in the states of motion of material bodies. This might be taken to mean that one way for an account of force to be “mathematical” is for it to treat forces as if they were borne by mathematical entities, such as points. In that case, we would not be dealing with a fictitious force, but we would be attributing a force to a point only for the purposes of calculation. The question, then, is whether gravity should be understood on the model of a material body, i.e. as a real force, or on the model of a “center” of a material body, as a merely mathematical entity, which can be treated as if it bore causal relations.

We can move one step closer to clarifying the mathematical treatment of force by considering Newton’s description of a force’s “physical” treatment, which serves as the relevant contrast class. As indicated in definition Viii of the Principia, the physical treatment involves at least the following two elements, which highlight the technical character of Newton’s distinction:

(1) A characterization of what is called the force’s “physical mode of action.” For instance, does it involve some sort of medium, or the exchange of particles? What are the relevant properties of the constituents of the entities in question?

(2) A characterization of the force’s relation to other phenomena. For instance, is gravity due to some medium or process that also accounts for magnetism or electricity?

Consider Newton’s familiar assertion that gravity is as the masses of the bodies and inversely proportional to the square of the distance between them: that is a physical claim in an ordinary sense, for it highlights two perfectly ordinary physical quantities-masses and distances-as salient. It does not appear to be a claim about mathematical entities. But the claim does not constitute a physical treatment of gravity in Newton’s technical sense because it does not take a position on issues (1) and (2) above. As I discuss in more detail below, Newton is committed to the view that gravity causes the planetary orbits. Note that in presenting that claim, he intends to be agnostic on whether this is due to an ether or to some other medium; he brackets the question of what particles or entities might constitute such a medium; and, he brackets the question (e.g.) of how the medium’s properties may render the inverse square of the distance between any two given masses a salient quantity. In that sense, the claim is an appropriate element of a mathematical treatment of force.

But of course, this only dispels one potential misinterpretation. The fact that one is dealing with ordinary physical processes and objects, rather than mathematical ones, does not entail that one is dealing with a real force, rather than a fictitious one. After all, one could treat the perfectly real motion of some perfectly real material body as if it arose from a fictitious force. And thus far, it remains entirely unclear what it means to contend, as Newton frequently does, that gravity causes certain phenomena, such as the lunar orbit. If his assertion of a causal relation is intended to be “mathematical” because it eschews the question of (say) whether an ether, or some other medium, is gravity’s “physical seat,” what meaning can the contention that gravity causes the phenomena possibly retain? In what sense can we say that gravity is a real force if we are ignorant of its physical basis?

Before attempting to clarify this claim in what follows, i want to indicate what is at stake in the proper interpretation of Newton’s “mathematical point of view” by considering its place in the work of his most prominent critic and most important defender. Both Leibniz and Clarke invoke Newton’s distinction between the mathematical and the physical treatment of force in the course of pressing their cases against one another, and the question of the place of causation within the mathematical treatment of force is central for each of them. In the next section, i first consider Clarke’s attempt to employ this distinction to defend Newton against the charge that he invokes action at a distance—a criticism leveled famously and forcefully by Leibniz. I then argue that Leibniz developed a surprisingly compelling interpretation of the Principia, one on which Clarke’s defense ultimately fails because it overlooks the depth of Newton’s causal commitment.

## 2.

Among Clarke’s myriad defenses of Newtonian natural philosophy in the face of Leibnizian criticisms, we find the argument that the Principia’s treatment of force can be construed as involving a kind of causal agnosticism, one that allows Newton to avoid the charge that he relies on action at a distance. Before this attribution of agnosticism can be characterized, a point of clarification regarding the concept of causation is in order. Newton’s own view of force-along with Leibniz’s criticism of it and Clarke’s rebuttal of that criticism—does not require a definition or technical understanding of causation. Leaving aside the distinction between mathematical and physical treatments of force, which requires its own discussion, Newton deals with technical terms and concepts in the Principia in at least two ways. In the definitions, he introduces technical terms such as ‘centripetal force,’ the ‘quantity of matter’ (mass), and so on, that would have been unfamiliar to his readers (Newton, Principia, 403-08). in the Scholium, which follows the definitions, Newton explicitly notes that space, time, place and motion are understood by everyone, and therefore require no definition-but he adds that the “common” (vulgare) conception of these notions must be rigorously distinguished from the “mathematical” (or “absolute”) conception of them (ibid., 408-09). Since Newton never defines ’cause,’ or various related terms such as ‘action,’ and does not distinguish an ordinary from a more precise conception of causation, he apparently thinks that our ordinary conception is adequate for interpreting the Principia. Hence when Newton characterizes forces as causes of changes in states of motion, he typically does not elaborate. Significantly, in interpreting Newton’s mathematical treatment of force, Leibniz and Clarke each appear to be satisfied with employing this ordinary conception in order to present their differing characterizations of Newton’s causal commitments. That is to say, each deploys this ordinary notion when determining whether Newton is committed to thinking of gravity as a cause of certain phenomena. So their debate is not merely semantic.

Having read Leibniz’s constant criticisms of Newton’s theory of gravity, in his last letter to Leibniz, Clarke insists:

It is very unreasonable to call attraction a miracle, and an unphilosophical term; after it has been so often distinctly declared, that by that term we do not mean to express the cause of bodies tending toward each other, but barely the effect, or the phenomenon it self, and the laws or proportions of that tendency discovered by experience; whatever be or be not the cause of it.

And then in the same letter he writes:

The phenomenon itself, the attraction, gravitation, or tendency of bodies towards each other (or whatever other name you please to call it by) and the laws, or proportions, of that tendency, are now sufficiently known by observations and experiments. if this or any other learned author can by the laws of mechanism explain these phenomena, he will not only not be contradicted, but will moreover have the abundant thanks of the learned world.

So for Clarke, the Principia’s mathematical treatment of gravity precisely describes certain tendencies to motion, but remains neutral as to the cause of those tendencies. He construes gravity as a “matter of fact” concerning various motions, rather than as a cause. Although Clarke does not use this language here, he does not take gravity to be a real force, a maneuver that seems intended to eschew action at a distance.

Clarke’s attempted defense of Newton has an intriguing and influential echo in the contemporary literature, especially in two recent papers by Ernan McMullin in which he presents an illuminating interpretation of Newton’s mathematical treatment of force. McMullin notes that there is a tension between Newton’s use of terms like ‘attraction’ and his caveats about those terms, claiming that ‘attraction’ can be given two distinct meanings in the Principia. The meanings are as follows:

(1) In contending, for instance, that the sun “attracts” the earth, the sun is conceived of as an “agent;” i.e., the sun is construed as the cause of the earth’s motion.

(2) The contention that the sun “attracts” the earth is construed to mean only that the earth has a “disposition” to move in various ways.

It is natural, McMullin thinks, to interpret Newton’s eschewing of the “physical” treatment of gravity as a rejection of (1) in favor of (2), which can be construed as a “merely mathematical” treatment. McMullin is, of course, quite correct, as we have seen, that Newton employs (e.g.) the terms ‘attract’ and ‘attraction’ at various stages of his argument; McMullin’s view is that to provide a merely mathematical understanding of the use of such terms is to construe them as referring to dispositions to motion.

Interpreting the dispositional construal of the mathematical treatment of force requires care, because one does not want it to represent a way of weighing in on precisely the sort of issue that Newton takes to be “physical” in the technical sense. if we construe the claim that the sun “attracts” the earth to mean that the earth has a disposition to follow a particular elliptical orbit with the sun lying at one focus of the ellipse, and if we think of this as a way of avoiding the ascription of action at a distance to any of the bodies in question, then we can avoid the “physical” characterization of gravity in the technical sense, provided that we do not take this account to rule out various possibilities, such as the possibility that the earth is continually pushed by an ether along its orbit. Perhaps the earth would be pushed in virtue of its having mass, so its disposition to motion would be a relational property dependent on its mass and on the characteristics of the ether.

The point is that by Newton’s lights, the use of the terms ‘attraction’ and ‘attract’ is intended to be compatible with any physical account of gravity in the sense that it is not intended to rule out any physical medium that “pushes” or “pulls” bodies, as long as that medium’s properties and modes of action are consistent with the law of universal gravitation. (i discuss the issue of consistency below.) Of course, that is perfectly compatible with the claim that our ignorance concerning the ether’s operation-assuming for a moment that we have empirical evidence indicating the existence of an ether, but no understanding of how it operates on bodies-might be expressed by contending that it is the ether that has a disposition to push, or pull, the planetary bodies in various directions. That is certainly a way of expressing agnosticism about the physical operation of the ether, but it is not (it seems to me) a way of expressing the type of agnosticism signaled in Newton’s contention that his treatment of force is “merely mathematical.”

McMullin’s own construal of the dispositional account of the mathematical treatment of force may press us in a slightly different direction than the one i have just outlined. After noting that Newton was committed to rejecting action at a distance per se, he discusses an implication of this rejection:

The significance of this point for my theme is that it means that the “mathematical” or “dispositional” interpretation of the mechanics of the Principia was, in Newton’s own eyes, incomplete; in the long run it would have to be supplemented by a properly “physical” account of just how the active disposition of one body could affect the state of motion of another causally at a distance from it. The only reading of the mathematical-dispositional account of gravity that would transform it directly into a physical account, i.e. that would allow it to pose as complete, would necessarily limit it to a postulation of unmediated action at a distance. And this Newton did not want. (McMullin, “Field Concept,” 24)

As will become clear below, I concur with McMullin’s (perhaps surprising) contention that the mathematical account will eventually have to be supplemented with, but not replaced by, a physical account of gravity. But his way of characterizing the dispositional account in the passage above indicates that it already rules out any medium between bodies to account for gravitational phenomena like the planetary orbits, for the claim that the earth and the sun are causally distant would appear to express the fact that we know there to be no causally efficacious medium between them. And according to the construal above, to assert that the earth is “disposed” to orbit the sun is ipso facto to assert that the two are causally distant. But from Newton’s point of view, that is precisely the sort of proposition that we do not know.

Of course, this concerns only McMullin’s gloss on his own dispositional construal of the mathematical treatment of force. if we do not endorse that gloss, we are left with something akin to Clarke’s view: the mathematical account’s use of terms such as ‘attraction’ describes only dispositions or tendencies to motion and is, in fact, causally neutral; it is neutral in the sense that it is silent on what leads to the various motions of the heavenly bodies. It is precisely this interpretation of Newton’s commitments that Leibniz rejects.

One of Leibniz’s principal tactics is to praise the Principia’s mathematical treatment of force, but then to contend that Newton ultimately strays from its safe harbor into philosophically problematic territory. To his credit, in the Tentamen of 1689, which represents one of Leibniz’s most extensive responses to the Principia, and which outlines a vortex theory of planetary motion, Leibniz acknowledges the significance of Newton’s mathematical achievement in the Principia. He even notes that the planets are “attracted” by the sun as 1/r2, something that was apparently “also known” to the “renowned Newton!” Several years later, when writing to Newton directly in 1693, Leibniz makes a related acknowledgment, while expressing his commitment to his own vortex theory of gravity:

You have made the astonishing discovery that kepler’s ellipses result simply from the conception of attraction or gravitation [attractio sive gravitatio] and passage in a planet. And yet I would incline to believe that all these are caused or regulated by the motion of a fluid medium, on the analogy of gravity and magnetism as we know it here. Yet this solution would not at all detract from the value and truth of your discovery.

The obvious question here is this: can Leibniz genuinely acknowledge that the planetary orbits are due to “attraction?” Could this be anything more than a disingenuous remark intended to keep the peace? I will return to that issue shortly.

Unlike Clarke, Leibniz did not allow Newton’s account of gravity in the Principia to be interpreted only as a causally neutral mathematical theory of a force-that is, as a mathematically precise description of certain motions, or dispositions to motion—that is agnostic on their cause. He consistently interpreted Newton as committed to the view that the world actually contains an “attractive” force, i.e., one that is causally efficacious without contiguity because it acts independently of any medium, whether vortex-like or otherwise. Leibniz often bemoaned this deviation from the contemporaneous philosophical consensus. As he insists in his last letter to Clarke:

For it is a strange fiction [étrange fiction] to make all matter gravitate, and that toward all other matter, as if all bodies equally attract all other bodies according to their masses and distances, and this by an attraction properly so called [une attraction proprement dite], which is not derived from an occult impulse of bodies, whereas the gravity of sensible bodies toward the center of the earth ought to be produced by the motion of some fluid. And it must be the same with other gravities [d’autres pesanteurs], such as is that of the planets toward the sun, or toward each other. A body is never moved naturally except by another body that touches it and pushes it; and after that it continues until it is prevented by another body that touches it. Any other operation on bodies is either miraculous or imaginary. (Leibniz, Philosophischen Schriften, vol. 7, 397-98)

So Leibniz reads Newton as presenting a twofold description of gravity. On the one hand, when writing strictly from a mathematical point of view, Newton construes “attraction” agnostically to express the fact that it is as if the sun attracts the earth, or the earth the moon, etc. This construal does not express what actually causes the motions of these bodies toward one another. This view, according to Leibniz, is perfectly defensible because it does not take ‘attraction’ to name any real force. in that sense, Leibniz’s praise for Newton in his 1693 letter was genuine.

On the other hand, Newton insists on articulating a causal claim concerning the phenomena associated with gravity (e.g. in his response to Leibniz’s 1693 letter, as we will see below), and in so doing, he transcends the limits of a merely mathematical treatment of gravity. He wrongly attributes the motions associated with gravity to attraction properly so-called; that is, he takes ‘attraction’ to name a real force and thereby invokes action at a distance. In his 1693 letter to Newton, Leibniz insists that a mathematical construal of “attraction” should not press us into invoking a genuine attraction. Rather, for the relevant causal chain, we must look to the motion of a vortex, thereby indicating that the apparent attraction is reducible without remainder to contact action of a familiar and acceptable variety. But of course Newton consistently dismisses any vortex theory of gravity à la Leibniz.

The question we now face, therefore, is whether Clarke’s attribution of causal agnosticism to Newton is defensible in the light of Leibniz’s contention that Newton spoils his mathematical account of attraction by rejecting vortices and by consistently invoking causal claims that invoke “attraction properly so called.” As I will argue in the next section, Leibniz is quite right to contend that Newton is committed to the view that, in some sense, “gravity” causes various natural phenomena-a view that Clarke takes Newton to eschew. But, as I will argue in section 3, Leibniz ultimately mischaracterizes the implications of Newton’s causal commitment.

## 3.

The claim that Newton is genuinely committed to the view that gravity itself is the cause of the planetary orbits is buttressed by the fact that Newton viewed his theory as a competitor to a type of causal theory that was prevalent at the time the Principia first appeared: the theory that gravity is propagated through some kind of vortex. Leibniz himself, of course, consistently defended such a theory, as Newton well knew; so we can gauge the depth of Newton’s causal commitment by considering his response to the type of theory Leibniz defends in the Tentamen. I will first provide a brief sketch of Leibniz’s theory.

In the Tentamen, Leibniz emphasizes that, before any empirical research is completed within physics, we already know both the nature of motion and the nature of bodies; these not only constrain any empirical research, they help us to understand (e.g.) basic astronomical data as providing evidence for certain conclusions rather than others. In particular, the “nature of motion” is expressed by the principle of inertia; so, for instance, moving bodies tend to recede along the tangent to any curve. It is the “nature of bodies” to be such that the state of motion of any given body can be altered only by something “contiguous” to that body. Hence it is part of the nature of bodies that there can be no action between distant bodies. If we begin from the assumption that the planets follow curvilinear paths around the sun, it follows from the nature of motion that something must intervene to prevent them from following the tangents to those paths; and it follows from the nature of bodies that whatever alters their motion in this respect must be “contiguous” to them.

Leibniz’s argument then proceeds as follows. We introduce, ex hypothesi, the claim that a fluid surrounds, and is contiguous to, the various planetary bodies, and we then make the following argument to prove that this imagined fluid must be in motion:

To tackle the matter itself, then, it can first of all be demonstrated that according to the laws of nature all bodies which describe a curved line in a fluid are driven by the motion of the fluid. For all bodies describing a curve endeavor to recede from it along the tangent (from the nature of motion), and it is therefore necessary that something should constrain them. There is, however, nothing contiguous except for the fluid (by hypothesis), and no conatus is constrained except by something contiguous in motion (from the nature of the body), therefore it is necessary that the fluid itself be in motion.

Venturing beyond a merely mathematical treatment by proposing a causal analysis of the planetary orbits must involve an attribution of contact action between the planets themselves and some physically characterized entity or medium, such as a vortex, that is contiguous to the planets. This is a bedrock assumption for Leibniz.

Newton was perfectly familiar with this Leibnizian account of gravity. How would he respond to it? He would obviously accept aspects of it, such as the principle of inertia, although his understanding of inertia differs from Leibniz’s: Newton tended to think that it expresses, not the nature of motion, but the nature of matter, though little seems to hinge on that difference here. And Leibniz speaks of “motion” rather than the “state of motion,” which may indicate his failure to appreciate an implication of the principle of inertia. More importantly, Newton would obviously object to Leibniz’s ex hypothesi premise, which introduces the claim that the bodies in question-the planets-sit in a fluid, presumably on the grounds that this is, as Leibniz would obviously agree, a mere hypothesis. By that Newton would mean (perhaps among other things) that there is no independent empirical evidence for the existence of such a fluid. The disagreement, therefore, would partially involve Leibniz’s willingness to proceed in this avowedly “hypothetical” manner.

But the disagreement would not end there, and this is particularly salient for our purposes. Just four years after the Tentamen was written in 1689, Newton responded to Leibniz’s 1693 letter with a rather startling claim-one which Leibniz clearly had in mind when contending that Clarke failed to acknowledge Newton’s commitment to causal claims regarding the phenomena associated with gravity. in his response to Leibniz, Newton contends that the Principia indicates the following:

For since celestial motions are more regular than if they arose from vortices and observe other laws, so much so that vortices contribute not to the regulation but to the disturbance of the motions of planets and comets; and since all phenomena of the heavens and of the sea follow precisely, so far as I am aware, from nothing but gravity acting in accordance with the laws described by me [cumque omnia caelorum et maris phaenomena ex gravitate sola secundum leges a me descriptas]; and since nature is very simple, i have myself concluded that all other causes are to be rejected and that the heavens are to be stripped as far as may be of all matter, lest the motions of planets and comets be hindered or rendered irregular.

The claim is that vortices cannot be the cause of the planetary orbits—for complex but reasonably well-known reasons that i will not delve into here—and that gravity alone should be singled out as their cause. Newton’s causal view, therefore, is an integral component of an ongoing program of gravitational research—a program that he takes to have refuted the vortex-centered program favored by Leibniz.

This perspective appears to represent Newton’s considered view. In May of 1712, Leibniz wrote to Nicholas Hartsoeker to present several familiar criticisms of the Newtonians; his letter was later published in english in the Memoirs of Literature. After learning of the letter from Roger Cotes (the editor of the second edition of the Principia and a subscriber to the Memoirs), Newton wrote an only-posthumously published rebuttal in which he rejects the Leibnizian criticism that his theory renders gravitation a “perpetual miracle” because it fails to specify what Leibniz would countenance as a relevant physical mechanism. Newton first paraphrases a Leibnizian criticism, one repeated in the Clarke correspondence:

But he [Leibniz] goes on and tells us that God could not create planets that should move round of themselves without any cause that should prevent their removing through the tangent. For a miracle at least must keep the planet in.

Newton’s rebuttal is illuminating:

But certainly God could create planets that should move round of themselves without any other cause than gravity that should prevent their removing through the tangent. For gravity without a miracle may keep the planets in.

So Newton repeats his assertion from his 1693 letter to Leibniz that gravity itself causes the planets to follow their orbital paths instead of their inertial trajectories along the tangents to those orbits.

Newton therefore consistently asserts in various pieces of correspondence between 1693 and 1712 that gravity—as he understands it—is to be interpreted as a genuine cause, as a real force. Even more importantly, this view is reflected in the Principia itself, and not just in the assertion in the General Scholium that gravity “really exists.” There are at least two independent ways in which Newton endorses this view in the Principia. Since Leibniz was a reasonably subtle interpreter of Newton’s text on this score, he may have been aware of these passages, and perhaps even thought of them as buttressing his case against Clarke.

In the definitions that open the Principia, Newton characterizes the two forces that he will later identify with one another via Rule II (see below) in a particularly salient way. In definition V, he defines “centripetal force” as the force by which bodies tend toward a point as a center. Two examples of this force are terrestrial gravity, and whatever force compels the planets to retain their solar orbits. In definition IV, we have already learned that centripetal forces are sources of impressed force, and the latter is an action that alters the state of motion of any body. As he writes later, in the Scholium: “The causes which distinguish true motions from relative motions are the forces impressed upon bodies to generate motion.” So Newton conceives of centripetal forces as causes in an ordinary sense: they alter the states of motion of material bodies. Thus, when he identifies these two examples of centripetal force together in Book III, Newton clearly conceives of the force of gravity itself as altering the states of motion of material bodies.

Moreover, Newton explicitly employs the so-called Rules of Reasoning in later editions of the Principia to derive, in Book III, the universality of gravity. For instance, consider Rule II, which says: “Therefore, the causes assigned to natural effects of the same kind must be, so far as possible, the same.” This rule is employed to identify the cause of the weight of objects near the surface of the earth—which Newton calls “gravity” throughout the Principia, before he has derived universal gravity—and the force that maintains the planets in their solar orbits, and the moon in its terrestrial orbit. He contends that these are the same effects, and thus, by Rule II, they ought to be assigned the same cause. That is to say, Newton’s use of his own methodological principles commits him explicitly to thinking of gravity as a genuine cause, as a real force.

We are therefore left with the following question: if Clarke’s defense of Newton is not viable, and if Leibniz is correct in claiming that Newton is committed to the view that gravity is a real force, then how can Newton hope to avoid an entanglement with action at a distance? I briefly suggested above that Leibniz ultimately mischaracterizes Newton’s theory of gravity as involving a commitment to action at a distance; but given what we have seen thus far, it is far from clear how Newton can avoid invoking a type of action that he himself takes to be “inconceivable.” In attempting to answer this question in the next section, i will return to my interpretation of Newton’s contrast between the mathematical and the physical treatment of force sketched above.

## 4.

Given the specter of action at a distance, and given Leibniz’s powerful criticism of Newton’s text, what possible meaning can Newton’s claim that gravity causes various phenomena still retain? it seems to me that Newton’s causal claim retains at least the following three meanings. First, it means that a wide-range of previously disparate phenomena-including the free fall of bodies on earth, the tides, and the planetary and satellite orbits-have the same cause. it is precisely to highlight this startling result, of course, that we call this cause ‘gravity’; indeed, before reaching this conclusion in Book III of the Principia, Newton employs such phrases as “whatever force maintains the planetary bodies in their orbits,” as we witnessed at the end of the last section. The culmination of Newton’s argument is the identification of this force as “gravity.” So although the use of the term ‘gravity’ is intended to be neutral with respect to the physical characterization of the cause of these phenomena in Newton’s technical sense, it is not intended to be neutral on the question of whether the free fall of some rock toward the surface of the earth, and the orbital path of some extremely distant comet, have the same cause. They do.

The second meaning is this: the various phenomena caused by gravity are such that mass and distance are the only salient variables in the causal chain that involves them. we express this precisely through the law of universal gravitation, asserting that gravity is as the masses of the objects in question and is inversely proportional to the square of the distance between them. This raises several crucial issues, one of which leads us to the third meaning of Newton’s contention.

Third, and perhaps most surprisingly, given that mass is one of the salient variables in the causal chain involving the previously disparate phenomena taken by Newton to be caused by gravity, we already know-as Newton himself makes perfectly clear-that gravity is not a mechanical cause. At least, we attain such knowledge if we adopt Newton’s own understanding of ‘mechanical’ in the Principia. As we have seen above, Newton writes in the General Scholium:

Indeed, this force arises from some cause that penetrates as far as the centers of the sun and the planets without any diminution of its power to act, and that acts not in proportion to the quantity of the surfaces of the particles on which it acts (as mechanical causes are wont to do) but in proportion to the quantity of solid matter, and whose action is extended everywhere to immense distances, always decreasing as the squares of the distances … I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and i do not feign hypotheses [hypotheses non fingo].

Newton’s important contention that gravity is not a mechanical cause—or, more specifically, that it does not act “as mechanical causes are wont to do”—can easily be interpreted as undermining his claim to have avoided action at a distance, as Leibniz would surely insist. So the plot thickens.

To understand this third aspect of Newton’s contention, we should consider his insistence that some defenders of the mechanical philosophy, including Leibniz, were guilty of conflating local with surface action (or impact). According to the overarching view that Newton would attribute to Leibniz, a cause must involve some mechanism-it must be “mechanical”—in the following two senses: (1) the cause cannot alter the state of motion of any material body at a spatial distance from it; and (2) it cannot alter the state of motion of any material body without impacting on one or more surfaces of that body (recall Leibniz’s argument in his Tentamen). One might think that the only way to avoid violating condition (1) is to cite causes that meet condition (2). From Newton’s point of view, however, this is the conflation of which Leibniz is guilty: (1) and (2) should be seen as distinct. Whereas “mechanical causes” operate on surfaces, we must reserve room for local action that involves the penetration of a material body by some other body, such as a material particle, or perhaps by another phenomenon, such as a ray of light (leaving aside whether light consists of particles or of waves).

The import of this point is not difficult to find. Newton’s imagined physical theory of gravity-based on the ether in Query 21 of the Opticks-involves an acceptance of (1) but a rejection of (2), for the ether would act on bodies by “penetrating” them. Newton goes so far as to contend, in Query 28, that the mechanical philosophers unjustly introduce “hypotheses” into physics precisely by presupposing that physical accounts must meet condition (2); for him, it is an empirical question whether any cause meets that condition. So although all action between material bodies must be local-on pain of there being an “inconceivable” distant action-we cannot presuppose that two macroscopic material bodies must interact with one another through impact. it may turn out that the constituent microscopic particles of any two bodies do interact by impacting upon one another’s surfaces, but we cannot establish this in advance, and it does not follow from this fact that the macroscopic bodies interact with one another via impact. From Newton’s point of view, the development of empirical science has indicated the inadequacy of the position that all causation must be mechanical in his sense. This represents an unjust insertion of a metaphysical requirement into physical theory.

To unite these three elements of Newton’s view, we might say that ‘gravity’ names a real force-a genuine cause-in the following sense. ‘Gravity’ refers to whatever it is that non-mechanically causes various motions of bodies near the surface of the earth, of our oceans, and of the heavenly bodies, in such a way that distance and mass are the salient variables in their changes in states of motion.

This interpretation should help to illuminate how Newton can justifiably assert that gravity causes various motions while nonetheless avoiding the invocation of action at a distance. The contention that gravity causes the planetary orbits-which means, as we have seen, that the planetary orbits and the free fall of bodies on earth have the same cause-does not amount to, or entail, the contention that there is no causally efficacious medium between material bodies that serves as the basis of their gravitational interactions. On the contrary, since Newton’s theory is neutral with respect to the “seat” of gravity, it is perfectly compatible with the discovery that some type of medium does, in fact, exist. Hence, it does not amount to, nor does it entail, the claim that bodies act on one another at a distance through a vacuum; it is in fact entirely compatible with a state of affairs in which all action is local action. This is as it should be, for Newton considered any non-local action to be simply “inconceivable.”

This interpretation may also help to illuminate Newton’s understanding of the relation between the mathematical and the physical treatment of force, for it allows us to recognize a surprising fact: Newton thinks that any future physical characterization of gravity must somehow cohere with, or account for, the facts established in the mathematical treatment. in his 1693 correspondence with Leibniz, he explicitly makes this point. After asserting that gravity, rather than some combination of vortices, causes the planetary orbits-in a passage quoted above-Newton goes on to write:

But if, meanwhile, someone explains gravity along with all its laws by the action of some subtle matter, and shows that the motion of planets and comets will not be disturbed by this matter, I shall be far from objecting.

Newton allows that there might someday be a physical account of gravity via some “subtle matter,” which is to say, not a vortex or some kind of fluid of the sort favored by Leibniz, but most likely some kind of ether. Yet such a future physical explanation would not contravene the conclusion that gravity itself is the cause of various phenomena-a claim presented just before the passage above. Why should that be? Because as we have seen, the claim that gravity causes the planetary orbits should be interpreted to mean that the planetary orbits and the free fall of bodies on earth have the same cause, and that is precisely the sort of surprising datum that a physical explanation of gravity must somehow elucidate. Thus the task of Newton’s imagined “subtle matter” theory is precisely to indicate how that matter interacts both with bodies near the surface of the earth and with bodies on the other side of the solar system in such a fashion that mass and distance are the salient quantities in their interactions. The “subtle matter” could not be mechanical in Newton’s sense, but would have to flow through material bodies, interacting somehow with their masses. Similarly, it would have to exhibit differential density, or some other feature that renders the distance between masses their salient relation. As Newton knew, no physical account of gravity in this period-including those of Huygens and Leibniz-was remotely capable of accounting for these aspects of gravity. In sum, gravity is real because regardless of what its “physical seat” turns out to be, Newton takes it to name a non-mechanical cause of a huge amalgam of previously disparate phenomena.

This view that the mathematical account of force can highlight certain physical quantities as salient, and thereby guide the future physical account of that force, is not confined to Newton’s 1693 letter to Leibniz. In broad terms, he expresses a similar sentiment elsewhere. For instance, in the preface to Book III of the Principia, he writes:

In the preceding books I have presented principles of philosophy that are not, however, philosophical but strictly mathematical-that is, those on which the study of philosophy can be based. These principles are the laws and conditions of motions and of forces, which especially relate to philosophy. (Newton, Principia, 793)

Determining the “laws and conditions of motions and forces” is itself the basis for future “physical” investigations of the forces in question. By unifying previously disparate phenomena under a single force, and by expressing the law governing that force, we guide its future physical characterization, and once we have sufficient empirical data to tackle this characterization, all action attributed to material bodies and their constituents must be local in character, as far as Newton is concerned. That sort of two-step maneuver-disallowed by Leibniz and by other prominent defenders of the mechanical philosophy-proved to be of considerable significance.

If it is correct to read Newton’s distinction between the two treatments of force as a technical one-in part because it helps to clarify the notion that gravity is a real force-we might conclude by noting that the correspondence between Leibniz and Clarke appears to be hampered by their joint acceptance of the interpretation that the distinction is non-technical. This results, in turn, in their failure to recognize the subtlety in Newton’s contention that gravity causes various phenomena. Failing to recognize the subtlety leads each of them to misstep, in my view.

As for Clarke’s interpretation, it seems that Newton’s agnosticism appears to be less strict than Clarke’s expression of it. For Clarke, as we have seen, the mathematical treatment of force ought to be understood as causally neutral, as a description of various motions that eschews any determination of their cause; the latter is left, apparently, until someone can present a physical treatment of gravity. This is, it seems to me, a perfectly reasonable interpretation if we take Newton’s distinction as a non-technical one. But as we have seen, by ‘gravity’ Newton does not intend to refer to certain tendencies to motion; rather, he means to signal the startling conclusion that the force that maintains the moon in its orbit is the very same one that we call ‘gravity’ on earth. A description of certain tendencies to motion à la Clarke presumably does not commit one to the claim that the motions in question have the same cause.

As I say, Leibniz also appears to construe the mathematical treatment of gravity in a non-technical fashion. In the course of providing an exact mathematical description of planetary motions, we are free to treat the motions as if they arose from an attraction between the heavenly bodies; this treatment must explicitly avoid any causal claims concerning the motions in question. Thus far, Leibniz and Clarke concur. Since the mathematical treatment of force is causally neutral on this view, any causal claim must be a component of a physical characterization of gravity, such as that provided by Leibniz’s vortex theory. It is therefore natural for Leibniz to understand Newton’s claim that gravity causes the planetary orbits as a component of a physical treatment of force in an ordinary or non-technical sense. But because Newton explicitly eschews any vortex-type theory, and because he explicitly admits that he does not know what the physical basis of gravity is, Leibniz thinks that he must retract the contention that gravity causes various phenomena. He then takes Newton’s stubborn resistance to that retraction to entail that he is actually committed to the view, not that we do not know what physical basis gravity has, but that gravity is efficacious without any physical basis or medium, i.e., that it is a property of material bodies in virtue of which they alter one another’s states of motion across empty space. In other words, to insist that gravity itself causes motions in the absence of any attribution of a mechanism is, from Leibniz’s point of view, to assert that there is action without a mechanism. And as we have seen, for Leibniz, that means distant action. Clarke concurs with Leibniz’s interpretation in the sense that he sees the denial of any causal commitment on Newton’s part as his only viable means of escape from this pitfall. For his part, Newton does not follow the lead of his most prominent defender in that respect.