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International Studies in the Philosophy of ScienceVol. 21, No. 3, October 2007, pp. 271–293

ISSN 0269–8595 (print)/ISSN 1469–9281 (online) © 2007 Inter-University FoundationDOI: 10.1080/02698590701589551

Newton’s Conceptual Argument for Absolute SpaceOri BelkindTaylor and FrancisCISP_A_258807.sgm10.1080/02698590701589551International Studies in the Philosophy of Science0269-8595 (print)/1469-9281 (online)Original Article2007Taylor & Francis213000000October 2007Professor OriBelkindobelkind@richmond.edu

While many take Newton’s argument for absolute space to be an inference to the bestexplanation, some argue that Newton is primarily concerned with the proper definition oftrue motion, rather than with independent existence of spatial points. To an extent thelatter interpretation is correct. However, all prior interpretations are mistaken in thinkingthat ‘absolute motion’ is defined as motion with respect to absolute space. Newton is alsousing this notion to refer to the quantity of motion (momentum). This reading reveals amisunderstood argument for absolute space, according to which absolute space is necessaryfor a workable definition of momentum.

1. Introduction

Newton argues for absolute space in the scholium to the definitions at the beginning ofthe Principia. Some features of the argument, especially Newton’s use of the bucketexperiment to demolish Descartes’s definition of true motion, are well understood.However, the overall strategy of the argument has been misconstrued primarilybecause of a fundamental misreading of what Newton means by ‘absolute motion’.

There are two common interpretations of Newton’s argument for absolute space(section 2). The first interpretation takes the argument to be an inference from inertialeffects to the best explanation thereof. The second interprets the argument as providingan adequate definition of true motion, rather than an explanation for inertial effects. Ishall attempt to show in this paper that, while the second interpretation matches moreclosely Newton’s strategy in the scholium, it still harbours a serious misunderstandingof Newton’s argument.

Newton’s arguments are easy to miscomprehend, since commentators do not readclosely enough Newton’s doctrine of place; consequently, they misunderstand hisargument for absolute space. An alternative reading of ‘place’ and ‘absolute motion’

Ori Belkind is at the Department of Philosophy, University of Richmond.Correspondence to: Department of Philosophy, University of Richmond, 28 Westhampton Way, Richmond, VA23173, USA. E-mail: obelkind@richmond.edu.

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reveals a misunderstood argument for absolute space. For Newton the notion ‘place’represents the particular volume a body occupies within a larger volume that containsit (section 3). Thus, by ‘absolute motion’, Newton means the body’s position relativeto an immovable container, not relative to a background space. Relative motion, on theother hand, is the change of position relative to a movable container. What commen-tators are missing is the significance of the quantity of motion for Newton’s argument.Newton argues that using a moveable container to define the motion of a body doesnot cohere with a property of the quantity of motion. Part of our task is to show a closeconnection in Newton’s thinking between the measure of ‘place’ and ‘quantity ofmotion’. For that purpose we show that Newton thinks of the ‘mass’ component in thequantity of motion of a body as the amount of impenetrable place a body occupies(section 4). The conflict between a relational definition of true motion and the quantityof motion threatens to undermine Newton’s dynamic theory (section 5). Thus, the setof ‘immovable places’, or absolute space, is indispensable for a working definition ofthe quantity of motion. The implication is that absolute space is not only required forthe explanation of a particular set of phenomena (i.e., inertial effects). Without theindependent existence of absolute space, a science of motion of the type that Newtonis pursuing is impossible.

2. Previous Readings of the Scholium

Before a close reading can be given of Newton’s scholium, previous readings of thescholium must be considered.

The more traditional line of interpreting Newton’s scholium takes Newton’s argu-ment to be an inference to the best explanation. This interpretation focuses on thefamous bucket experiment. Newton describes a bucket filled with water. Before thebucket rotates, the water stays level. The bucket is then given a big rotational push, afterwhich it starts to rotate. The water remains levelled since it is not moving yet. Due tofriction with the bucket, the water begins to rotate until it rotates at the same rate as thebucket. At that point the water receives a concavity along the surface as a result of thetendency of the water to recede from the axis of rotation. But this inertial effect cannotbe correlated with the motion of the water relative to the bucket. At first the relativemotion is zero and the inertial effect is not present. After the water rotates with thebucket, the relative motion is zero, and the inertial effect is present. Thus, it is incon-gruous to define the water’s true or proper motion as its motion relative to bucket. It isalso implausible that any remote body, such as the earth or the fixed stars, could beused to define the water’s true motion.1

Sklar summarizes Newton’s argument as follows:

So accelerations give rise to observable forces. And these accelerations are not accel-erations relative to ordinary material objects. But all accelerations are relative to some-thing, and so must these be. The something is clearly space itself. (Sklar 1974, 184)

According to the traditional reading, the first step of Newton’s argument is that relativeaccelerations cannot be part of any explanation of inertial effects. The second step is

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that we require some object relative to which we can define the acceleration of the body.Since no material body provides the reference for these accelerations, we have to positthe existence of absolute space.

According to Sklar, there is no difference between absolute space and otherunobservable entities we include in our theories:

[Space] cannot be immediately apprehended by the senses, any more than can amagnetic field; but like the field it can be indirectly inferred from observablephenomena by a legitimate scientific inference. (Sklar 1974, 162)

Thus for Sklar, Newton’s argument is a scientific inference from an observablephenomenon to the unobservable entity which best explains it.2

Since Newton’s argument for absolute space is an inference to the best explanation,we can also correct a flaw in his argument. Absolute rest is not really essential to histheory, since various states of rectilinear motions are indistinguishable in his physics.We can posit what is known as Galilean space-time, which consists of a four-dimensionaldifferentiable manifold with a flat affine connection. Galilean space-time includes auniversal temporal metric and a Euclidean metric on planes of simultaneity. Such aspace-time does not distinguish between absolute motion and rest, but it does distin-guish between bodies moving uniformly in a straight line and accelerating bodies. Thus,since today we have a more powerful mathematical apparatus at our disposal, we canprovide Newton with the minimal structure that is still empirically adequate for hisphysics (Friedman 1983, ch. 2; Stein 1970).

The inference-to-the-best-explanation account of Newton’s argument is seriouslyflawed, since Newton’s argument is not really an explanation. Absolute space doesn’tseem to function like other unobservable entities. Inferences to the best explanationordinarily employ a causal story that connects the unobservable entity to the phenom-enon being explained. The kinetic theory of gases, for example, describes a causalprocess that connects the molecules with phenomenological measures such as pressureand temperature. But inferences from inertial effects to absolute accelerations, andfrom absolute accelerations to absolute space, do not consider or rely on any causalconnections.

Sklar claims that absolute accelerations ‘give rise’ to inertial effects. The notion ofone property ‘giving rise’ to another phenomenon intimates the existence of somecausal process, but this turn of phrase is ambiguous. Either this means that the propertyof absolute acceleration itself is the cause of the effect; or we take absolute space to be thecausal agent. According to the first reading, the function of the space container ismerely to provide an idle reference for the definition of absolute acceleration, whilespace itself is causally inert. Newton himself argued that inertial effects are produced bythe inherent force of matter, which is ‘the power of resisting by which every body, sofar as it is able, perseveres in its state either of resting or of moving uniformly straightforward’ (Newton 1999, 404). For him inertial effects originate from the inherent forceof a body, a force which is ‘proportional to the body and does not differ in any way formthe inertia of the mass’ (Newton 1999, ibid.). However, Newton’s ‘explanation’ is seri-ously problematic. The sole function of inherent forces is to account for inertial effects

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by asserting that bodies have an inherent ‘tendency’ to continue moving in a straightline. But this tendency has no theoretical significance other than to provide us withsome causal story about the sources of inertial effects.3

Another way of interpreting the claim that absolute accelerations ‘give rise’ to iner-tial effects is to assert that absolute space itself is the cause of inertial effects. Accordingto this reading, a body’s tendency to move in a straight line results from the space ‘guid-ing’ the moving body. In the wake of Mach’s criticism many commentators in the 20thcentury insisted that Newton’s argument is an inference to the best causal explanation,even though Newton did not take absolute space to be explaining inertial effects. Giventhat inherent forces of inertia seem metaphysically superfluous, it is tempting toattribute to space the power to influence material bodies, which renders spacesubstance-like (hence the misnomer ‘substantivalism’).4 However, this reading ignoresobvious disanalogies between space and other unobservable entities and overlooksNewton’s assertion in the De Gravitatione that ‘Space’ is not exactly a substance becauseit has no power to act on bodies.5

Newton’s argument for absolute space is not at all an attempt to explain inertialeffects. In recent decades, attempts have been made to deflate substantivalist readingsof Newton’s argument and refrain from thinking of it as a metaphysical claim about thenature of absolute space. Stein (1970) initiated a line of interpretation emphasizingNewton’s argument against relative accelerations. According to this reading, Newton’sargument is not in itself an inference from inertial effects to an unobservable entity.The traditional reading of Newton’s argument ignores the historical context of thescholium, especially the intimate and argumentative relation it bears to Descartes’sphysics (Stein 1970, 268). Other notable examples of commentators objecting to thetraditional reading are Laymon (1978), Rynasiewicz (1995, 2004) and DiSalle (1994,2002).

According to the alternative line of interpretation, the arguments in the scholiumfocus on the appropriate definition of true motion, or on the procedure for singling outtrue motions from apparent ones (Rynasiewicz 1995, 134). Newton’s contemporaries,including Descartes, Huygens, and Leibniz, agreed that true motions are conceptuallydistinct from apparent ones. The true motion of a body is the one motion we can pred-icate of the body, which stands out against the background of innumerable apparentmotions that are too irregular or random to reveal the causal structure of the world.Thus crucial to the debate was whether one can specify a definition of true motion orsome procedure that would enable one to isolate the one true motion from all theapparent ones. One can then proceed with the claim that true motion is relative,namely, that out of the many possible relations a body holds with other bodies, there isone relative motion that could be predicated of the body. Or, if one believes that noparticular relation can be used to isolate true motion, one could conclude that truemotion is absolute.

Stein (1970) argued that Newton is mainly concerned with making precise his use ofvarious theoretical notions. The precise meaning of ‘absolute motion’ is implicit inNewton’s dynamic theory, since it allows us to pick the one true rotational motion outof the many relational ones:

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there is singled out within the theory one and only one unique rotational state, andwhile we are free to call this [unique state of rest] the state of absolute rotational restor refrain from using that term, if we do choose to use it at all, and to base our useupon the theory, there is really no other way. (Stein 1970, 281)

Whether we choose to call the unique motion singled out by Newton’s dynamics ‘abso-lute motion’ or not, Newton is justified in claiming that he has provided a valid empir-ical distinction in meaning between absolute and relative motion.6

DiSalle developed Stein’s suggested reading and argued similarly that Newton’stheory provides the empirical content for the definition of a term:

Newton has not tried to justify the causal link between rotation and centrifugaleffects, but simply to identify it as definitive of true rotation. Thus he has defined atheoretical quantity, absolute rotation, by exhibiting how it is detected and measuredby centrifugal effects. (DiSalle 2002, 44)

According to DiSalle, true absolute accelerations can be singled out by measuring iner-tial effects, and so we have an empirical procedure for separating true rotations fromapparent ones. The introduction of a theoretical entity is not an explanation of thephenomenon, nor does it consider the causal process by which the phenomenon arises:

Newton’s argument, in sum, was never an argument from physical phenomenon tometaphysical conclusions about the ‘absoluteness’ of rotation. Instead, it was anargument of the sort that is fundamental to every empirical science: an argumentthat a novel theoretical concept has a well-defined empirical content. Like the defi-nition of absolute time, and unlike the definition of absolute translation, the defini-tion of absolute rotation does indeed have a basis in Newton’s Laws; if the universein fact obeys those laws, we can always measure the true rotation of any body.(DiSalle 2002, 44)

Thus, DiSalle emphasizes the empirical procedure for singling out true motions overthe metaphysical significance of Newton’s argument. Whether or not a proposed set oflaws obtains in this world is an empirical question, but if these laws do obtain, so do theunique motions implicitly defined in them. DiSalle’s reading receives support fromNewton’s last comment in the scholium:

But in what follows, a fuller explanation will be given of how to determine truemotions from their causes, effects, and apparent differences, and conversely, of howto determine from motions, whether true or apparent, their causes and effects. Forthis was the purpose for which I composed the following treatise. (Newton 1999, 415)

Thus, according to DiSalle’s reading, it is the treatise as a whole which underwrites theprocedure for singling out the true motions from all the apparent ones.

The main problem with emphasizing the empirical adequacy of Newton’s Laws asunderwriting the definition of true motion is that it fails to make sense of Newton’sconceptual arguments for absolute space. These philosophical arguments are presentin the arguments from the properties and causes of motion. Given the conceptualnature of these arguments, philosophers who are worried that Newton has outsteppedhis empiricist principles have often taken these to be inconsequential or meaningless.But Newton’s arguments in these parts of the scholium are important, as they paintNewton’s overall argument in a different light from what is usually assumed.

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Rynasiewicz (1995, 2004) is one of the few commentators who take these argumentsto be significant. Rynasiewicz believes that Newton is primarily concerned with show-ing that relational definitions of true motion are inadequate, and therefore isconcerned with the procedure for singling out the true motion from all the apparentones. But he disagrees with Stein and DiSalle in their claim that Newton is providingempirical means for defining ‘absolute motion’, since from the outset Newton presup-poses that there must be some procedure for singling out true motions from apparentones:

the arguments take as their point of departure the assumption, common to Cartesianand Aristotelian philosophy, that each body has a unique state of true motion (orrest). Throughout the arguments, the terms ‘true motion’ and ‘absolute motion’ aretreated synonymously. At issue is whether true motion (and rest) can be reduced tosome special instance of relative motion (or rest) with respect to other bodies.(Rynasiewicz 2004, sec. 5.3)

Rynasiewicz’s reading has the benefit of providing room for arguments that are notfocused solely on explanations of inertial effects or the empirical adequacy of Newton’slaws. It is quite clear that Newton is concerned to show, on general philosophicalgrounds, that relational definitions of true motion are inadequate. The problem withRynasiewicz’s reading is that his account does not make clear the conceptual groundsfor thinking that relational definitions of true motion are inadequate. If we want tomake sense of Newton’s philosophical arguments for absolute space, we have to correcta common misreading of the scholium. Newton argues that there is a particular prop-erty of motion which conflicts with relational definitions of true motion. But to makesense of this conflict, we first have to correct a misreading of Newton’s doctrine ofplace.

3. Newton’s Doctrine of Place

Our philosophical heritage distorts our reading of Newton’s argument for absolutespace. A misreading of Newton’s doctrine of place obscures Newton’s argument againstrelational definitions of true motion.

To understand Newton’s argument for absolute space, we have to juxtapose his defi-nition of place over those of Descartes and the Aristotelian tradition that preceded it.According to Descartes,

We always take a space to be an extension in length, breadth and depth. But withregard to place, we sometimes consider it as internal to the thing which is in the placein question, and sometimes as external to it. Now internal place is exactly the same asspace; but external place may be taken as being the surface immediately surroundingwhat is in the place. (Descartes 1985, 229)

Descartes introduces his doctrine of place in order to give a coherent account ofmotion. For Descartes there is no distinction between extension and bodies, and sothere is no distinction between the (internal) place a body occupies and the body itself.To explain how a body can move from one place to another, Descartes must distinguishbetween the internal and external place of a body. The internal place of a body is the

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particular part of space a body takes up; this part of space is the body’s essence. Externalplace is the relation a body has to other bodies. By external place Descartes meanssomething like the inner surface of the ambient bodies which enclose the body in itsplace. The contours of this surface provide a reference for the position of a body. Thus,the place of a body is also its position in relation to the surrounding bodies.

Descartes’s definition of external place is very much in line with the Aristoteliantradition. Aristotle defined ‘place’ as the boundaries of the surrounding bodies whichcontain the body in its place. Like his Aristotelian teachers, Descartes believed in aplenum, i.e., for him there are no regions of the world which are empty of materialbodies. When a body moves from one place to another it simply changes its relation tothe surrounding bodies. Descartes differed from the Aristotelian tradition in that hereduced all change in material bodies to change of place and eliminated all essentialforms other than thought and extension. His definition of place differs from Aristotle’sin that he includes the notion of internal place, which describes the volume a bodytakes up independently of the bodies that surround it.

In the beginning of the scholium to the definitions in the Principia, Newton definestime, space, place, and motion. According to him, ‘Absolute, true, and mathematicaltime, in and of itself and of its own nature, without reference to anything external,flows uniformly and by another name is called duration’ (Newton 1999, 408). Thisnotion of absolute time is followed with his definition of absolute space, which ‘of itsown nature without reference to anything external, always remains homogeneous andimmovable’ (Newton 1999, 408). Then Newton defines the notion of place:

Place is the part of space that a body occupies, and it is, depending on the space, eitherabsolute or relative. I say the part of space, not the position of a body or its outersurface. For the places of equal solids are always equal, while their surfaces are for themost part unequal because of the dissimilarity of shapes; and positions, properlyspeaking, do not have quantity and are not so much places as attributes of places.(Newton 1999, 409)

In this passage Newton emphasizes that by place he means ‘part of space’, not the posi-tion of the body or its outer surface. As we shall see from his discussion of the shipexample further in the scholium, it is natural to assume that ‘part of space’ means thevolume a body occupies within a larger volume that contains it.7 It is plausible, I think,to assume that while writing the scholium, Newton has Descartes’s notion of ‘externalplace’ in mind as a target, since Descartes defines external place exactly as the positionof a body relative to the inner surface of the surrounding bodies, which for Descartes isidentical in shape to the outer surface of the body. Given the close attention Newtonpaid to Descartes’s physics in the De Gravitatione, it is plausible to assume that he istrying to distance himself from Descartes.

Newton provides an explicit argument for rejecting Descartes’s notion of externalplace. The place of a body has a measure, a size we can attribute to the body’s volume.This measure only describes the internal place of a body, not its external place. We cancompare the volumes of different bodies, i.e., the amount of space they take up, even ifthey have different positions relative to their surroundings or different shapes. Newtonargues that the position of a body, whether relative to distant bodies or relative to the

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immediate surrounding bodies, cannot provide the definition of place, since positiondoes not carry the measure of place. Similarly, the outer shape of the body does notcarry this measure. Since the properties of place in Descartes’s definition of externalplace do not carry the essential attribute of place, Descartes’s notion of external placeshould be rejected. Descartes’s definition of external place is therefore rejected and hisnotion of internal place is kept as the sole definition of place. This is Newton’s first stepin his argument for absolute space, since his definition of place has to allow for thedistinction between bodies and empty space.

For Newton, the distinction between relative and absolute place depends on thecharacter of the space of which the place is part. That is, to understand the nature of theplace, we have to consider the larger space that contains it. If the larger space ismoveable, then the place is relative. If the larger space is immovable, then the place isabsolute. Changes in relative place are defined as relative motion and changes inabsolute place are defined as absolute motion.

In the next step of the argument, Newton claims that the measure of place is concep-tually derived from the measure of motion:

The motion of a whole is the same as the sum of the motions of the parts; that is, thechange in position of a whole from its place is the same as the sum of the changes inposition of its parts from their places, and thus the place of a whole is the same as thesum of the places of the parts and therefore is internal and in the whole body.(Newton 1999, 409)

Here Newton makes a connection between ‘the motion of the whole which is the sameas the sum of the motions of the parts’ and ‘the place of the whole which is the same asthe sum of the places of the parts.’ It seems as if the first claim about the compositionalityof motion enables us to infer the second claim about the quantities that measure place.

The only way to explain Newton’s remark about the measure of place is to considerthe notion of the quantity of motion, which Newton defines as the product of mass andvelocity. This is how Newton explicates the quantity of motion after he defines it in thePrincipia:

The motion of the whole is the sum of the motions of the individual parts, and thusif a body is twice as large as another and has equal velocity there is twice as muchmotion, and if it has twice the velocity there is four times as much motion. (Newton1999, 404)

Newton here is moving freely between the quantity of matter and the size of the body.He is saying that a body ‘twice as large’ as another body with the same velocity carriestwice as much motion. This description of the quantity of motion seems to presupposeDescartes’s definition of the quantity of motion, which equates it with the product ofvolume and speed. Assume Newton is thinking of the equation , where is themomentum of the object; s is its volume; and its velocity. Assume also that ,and are the momentum, size, and velocity of the composite object, and that ,and are the same properties of each individual part i. In the case of solid body, thevelocities equal the velocity . From the part-whole relation governing quantitiesof motion, or from the equation

r rP s v= ⋅

rP

rv

rP sΠ Π,

rvΠ

rP si i=

rvi r

virvΠ

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we can derive the measure of volume which governs places:

The notion of place, which is measured with the quantity of volume, is therefore inti-mately connected with our ability to amass the (quantity of) motion of a whole fromthe (quantities of) motions of the parts.

While the connection between the quantity of motion and the measure of place is nowclear, we still have an interpretive problem. The definition of the quantity of motion werelied on was that of Descartes. Newton replaced the size in Descartes’s quantity ofmotion with mass and distinguished between bodies that have mass and empty space.How is it that Newton was thinking of Descartes’s definition of the quantity of motionin his argument for absolute space while advancing a different definition in his explicitdefinition? We will address this problem in the next section and will examine Newton’sremark about place in relation to his account of bodies in the De Gravitatione.

We have seen that Newton’s doctrine of place is conceptually inseparable from thequantity of motion. The essential attribute of place, i.e., the volume we attribute to it,is logically derived from the quantity of motion we attribute to bodies that occupy theplace. As we shall see in section 5, this conceptual relation underwrites his argumentfor absolute space.

Our interpretation of Newton’s doctrine of ‘place’ also changes our understandingof the distinction between absolute and relative motion. According to Newton,

Absolute motion is the change of position of a body from one absolute place toanother; relative motion is change of position from one relative place to another.(Newton 1999, 409)

The standard way of reading this passage takes Newton to be distinguishing betweenthe change in the position of a body relative to the backdrop of absolute space, on theone hand, (absolute motion) and the change in the position of a body relative to otherbodies, on the other hand (relative motion). In both cases, motion is conceived as achange in the relation between two mereologically distinct entities that share no parts.However, the change in relative or absolute place is not merely a change in distancerelations. Newton defines relative place as the particular volume a body takes up inrelation to the moveable space which contains it. Newton’s ship example demonstratesthis account of relative places:

Thus, in a ship under sail, the relative place of a body is that region of the ship inwhich the body happens to be or that part of the whole interior of the ship which thebody fills and which accordingly moves along with the ship, and relative rest is thecontinuance of the body in that same region of the ship or same part of its interior.(Newton 1999, 409)

The ship example makes clear that relative place is the particular region the bodyoccupies that is part of the composite object which includes both the ship and the

r r r rP s v P s vi

ii i

iΠ Π Π= = =∑ ∑ ,

s sii

Π =∑

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object. The change of relative place depends on the additional entities which areincluded in the composite body. The position of relative place is the position of theregion a body occupies relative to the composite whole. Relative rest therefore is the‘continuance’ of the body within the same region of the ship.

Absolute rest is defined as taking up the same part of immovable space:

true rest is the continuance of a body in the same part of that unmoving space inwhich the ship itself, along with its interior and all its contents, is moving. (Newton1999, 409)

Thus, Newton does not define absolute rest as a body changing its position relative toa space ‘backdrop’. Rather, absolute rest is the continuance of the body in the same partof the unmoving space in which the composite body, the ship together with its interioris moving. That is, he imagines absolute space as providing a large enough ‘immovable’region that contains the moving part.

It is easy to overlook the significance of treating places as the particular volumeswithin places of a larger size rather than just relative positions. When discussingNewtonian physics, we usually think, unconsciously, about dimensionless pointparticles. This latter perspective, however, makes it difficult to understand Newton’sargument against relative definitions of true motion.

4. Impenetrable Place and Mass

Interpreting Newton’s remarks about the measure of place as intimately connected tothe quantity of motion creates an interpretive problem. Such an account seems to‘forget’ that Descartes’s definition of the quantity of motion is incorrect. Newton’sdefinition of momentum involves the mass of a body, not its size. However, thistension can be alleviated by comparing Newton’s remarks about place with his writingin the De Gravitatione. In the De Gravitatione Newton tells us a story about the way inwhich God created material bodies. The purpose of the story is not to describe the actof creation, since that would be beyond the capacity of finite minds. Rather, its purposeis to give us a possible way in which material bodies could have been created by God:

Thus we may suppose that there are empty spaces scattered through the world, oneof which, defined by certain limits, happens by divine power to be impervious tobodies, and by hypothesis it is manifest that this would resist the motions of bodiesand perhaps reflect them, and assume all the properties of a corporeal particle, exceptthat it will be regarded as motionless. If we should suppose that that impenetrabilityis not always maintained in the same part of space but can be transferred here andthere according to certain laws, yet so that the quantity and shape of that impenetra-ble space are not changed, there will be no property of body which it does not possess.(Newton 2004, 28)

Thus it is enough for God to designate parts of space to be impervious to others and toallow these impenetrable regions to move hither and thither, to create bodies that areindistinguishable from the ones we experience. Bodies can therefore be defined as ‘deter-mined quantities of extension which omnipresent God endows with certain conditions’(Newton 2004, 28; emphasis in original). These conditions are (1) that bodies are

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mobile; (2) that two bodies do not coincide (i.e., that they are impenetrable); and (3)that they excite various perceptions of the senses and the imagination in created minds.8

In the De Gravitatione therefore Newton thinks of bodies as impenetrable regionsthat can move from one place to another. The notion of mass is not mentioned in theDe Gravitatione, but we can see the thought process that leads Newton from the notionof impenetrable place to the notion of mass. In definitions 5–13 in the De GravitationeNewton defines the notion of force, its extension and intension. Definition 5 assertsthat ‘Force is the causal principle of motion and rest’ (Newton 2004, 36). Newtonproceeds to define the intension of the force, which is ‘the degree of its quality’(Newton 2004, 36), and the extension of the force, which is ‘the quantity of space andtime in which it operates’ (Newton 2004, 36). These definitions are not very clear, butthey become clearer when Newton discusses the absolute quantities of the force, whichare the product of its extension and intension. Newton explains,

And thus motion is either more intense or more remiss, as the space traversed in thesame time is greater or less, for which reason a body is usually said to move moreswiftly or more slowly. Again, motion is more or less extended as the body moved isgreater or less, or as it is diffused through a larger or smaller body. And the absolutequantity of motion is composed of both the velocity and the magnitude of themoving body. So force, conatus, impetus, or inertia are more intense as they aregreater in the same or an equivalent body: they are more extensive when the body islarger, and their absolute quantity arises from both. (Newton 2004, 37)

In this passage Newton is thinking of the absolute force of conatus, which is also knownas impetus or inertia. Newton defines this absolute force as the product of ‘the magni-tude of the moving body’ and its velocity; i.e., the force is the product mv, where m isthe extension of the impetus force; and v is the intension of the force. Notice that theaccount here bears similarities to the one given in the Principia for the quantity ofmotion. Newton here is not referring explicitly to the notion of mass; rather, it seemshe has Descartes’s definition of the quantity of motion in mind, sv, where s is the sizeof the body; and v its speed. Here in the De Gravitatione it is not yet clear whether by vNewton, like Descartes, means speed, or he has the notion of velocity in mind, whichis speed together with the inclination to go in a particular direction. However, it seemsclear from the context that an important difference between Descartes’s and Newton’sdefinition is that Newton takes bodies to consist of impenetrable places, rather thanjust bounded regions in space. Thus, ‘the magnitude of the moving body’ is the amountof impenetrable place the body occupies and is the extension of the force of inertia mv.In the Principia this magnitude will be replaced with mass.

We can even see how Newton conceived of mass already in the De Gravitatione, eventhough mass is not mentioned explicitly. Definition 15 of the De Gravitatione describesthe notion of density (Newton 2004, 37). The increase and decrease in inertia iscorrelated with the increase and decrease in the body’s density. To clarify the notion ofdensity Newton discusses a body which is shaped like sponge with pores. The body hasregions that are impenetrable and pores which do not contain matter. The inertia of abody increases or decreases in proportion to the density as the pores diminish orincrease in overall size. Thus, density is defined as the amount of impenetrable volume

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a body occupies relative to its overall volume, pores included. This is where Newton’sdefinition of the quantity of motion departs from that of Descartes, since for Descartesthe notion of density does not make sense. There can be no difference for Descartesbetween impenetrable and non-impenetrable places—no body has regions that areempty of matter.

Furthermore, Newton conducts a thought experiment where the impenetrable partsof the body and the empty pores are thought to be equally distributed throughout auniform body:

But in order that you may conceive of this composite body as a uniform one, supposeits parts to be infinitely divided and dispersed everywhere throughout the pores, sothat the whole composite body there is not the least particle of extension without anabsolutely perfect mixture of infinitely divided parts and pores. Certainly suchreasoning is suitable for contemplation by mathematicians. (Newton 2004, 38)

Thus, the De Gravitatione shows us the process in which Newton conceived of thenotion of mass. Bodies are distinguished from empty space in that they possess theproperty of being impenetrable. In addition, Newton revises Descartes’s definition ofthe quantity of motion and defines it as the product of impenetrable volume andvelocity. Finally, Newton considers various contractions and expansions in which thesame amount of impenetrable place is distributed over different volumes, in perfectmixtures of infinitely divided parts and pores. Although the technical notion of ‘mass’was not used until the Principia, it is evident that Newton was conceptually ready forthe notion in the De Gravitatione. Mass can be said to be identical to the quantity ofimpenetrable place which is spread over different volumes in different proportions ofimpenetrable places and empty space.

The concept of impenetrability may also give us an idea of how Newton conceivedof the causal roles of mass. An impenetrable place resists another body’s attempt to‘enter’ the impenetrable region, so that impenetrability gives rise to the power of a bodyto resist impressed contact forces. But the notion of impenetrable place does notexplain why a body resists the action of forces that act at a distance (since no externalbody is really attempting to ‘enter’ the impenetrable region of the body), nor does itexplain why the resisting force is proportional to the acceleration of a body. However,some evidence that the notion of impenetrability still guides Newton’s thinking in thePrincipia is the fact that he lists impenetrability as one of the qualities that should beassigned to all bodies universally (‘Rule 3 for the Study of Natural Philosophy’, Newton1999, 795).

Newton’s explication of the measure of place, which follows from the measure of thequantity of motion, now becomes clear. When we amass the quantity of motion of acomposite body from its parts, we also have to make sure that the quantity of theirimpenetrable place is amassed in the same proportion from the parts. Without acorrelation between the quantity of motion and quantities of impenetrable place, thedefinition of the quantity of motion as the product of mass and velocity does not makesense. Moreover, as we shall see in the next section, Newton’s inference from thedefinition of the quantity of motion to the measure of places (which is the volume ofimpenetrable places) parallels his inference from the definition of the quantity of

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motion to the existence of absolute places. Absolute space will be shown to be necessaryfor a workable definition of the quantity of motion.

5. Distinguishing by Properties

Newton’s distinction between relative and absolute motion is the distinction betweenthe motion of the part relative to a moveable or relative to an immovable container.Newton’s argument against relational definitions of true motion is essentially that thesedefinitions do not cohere with a property of the quantity of motion. If the containerrelative to which the body’s motion is defined is itself moveable, then the containercarries the quantity of motion. Thus, the quantity of motion we attribute to thecontainer as a whole must cohere with the true motion we attribute to each of the parts.But to understand this argument it is important to consider the ways in which Newtonappropriates and criticizes the Cartesian theory of motion.9

According to Descartes no distinction is to be made between empty space and mate-rial bodies. Accordingly, motion has to be relative. However, one has to distinguishbetween the ordinary, or vulgar, sense of motion and what is meant by motion in the‘strict’ sense of the term:

if we consider what we should understand by motion, not in common usage but inaccordance with the truth of the matter, and if our aim is to assign a determinate natureto it, we may say that motion is the transfer of one piece of matter, or one body, from thevicinity of the other bodies which are in immediate contact with it, and which are regardedas being at rest, to the vicinity of other bodies. (Descartes 1985, 233; emphasis in original)

For Descartes, out of the many apparent relative motions there is one true motion thathas determinate value. This determinate value is the motion of a body relative to thebodies in its immediate vicinity. True motion for Descartes is therefore change inexternal place. To represent a body’s true motion, we have ‘to regard’ the bodies in theimmediate vicinity as being at rest and then measure the motion of the body relative tothe ambient bodies. Notice that for Descartes, motion is both relative and objective.There is no single reference frame appropriate for defining the motions of all bodies.The motion of each body is defined in relation to its surroundings. Thus, if we walkwithin the interior of a ship, our motion is defined in relation to the ship. The motionof the ship is defined in relation to the water in the river, and the motion of the wateris defined in relation to the river bank. However, each of these motions is uniquelydefined. There is a single, objective reference for the motion of each body. ForDescartes, therefore, there is an objective distinction between motion and rest, since wecan determine for each body whether it moves relative to its surrounding or whether itis at rest.

After the above definition of true motion, Descartes formulates his laws of nature.According to Descartes, since God’s nature is immutable, he created the universe byimparting to it a quantity of motion that does not increase or decrease over time:

Admittedly motion is simply a mode of the matter which is moved. But neverthelessit has a certain determinate quantity; and this, we easily understand, may be constant

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in the universe as a whole while varying in any given part. Thus if one part of mattermoves twice as fast as another which is twice as large, we must consider that there isthe same quantity of motion in each part; and if one part slows down, we mustsuppose that some other part of equal size speeds up by the same amount. (Descartes1985, 240)

According to Descartes, the immutable nature of God implies that the total quantity ofmotion we find in the universe is conserved. The various motions in the individualparts may change, but the quantity of motion amassed from these parts must beconserved. From this Descartes concludes that the quantity of motion of a particularbody must be proportional to the speed of the body and its size. The speed of the bodyrepresents the ‘intensity’ of motion. The size of the body is proportional to the numberof parts the body has. The overall quantity of motion is conserved, when in collisionsthe quantity of motion is transferred from one body to another, and is never created ordestroyed (Garber 1992, ch. 8).

From the conservation of the quantity of motion, Descartes ‘deduces’ his three lawsof motion.10 The first law states that ‘each thing, in so far as it simple and undivided,always remains in the same state, as far as it can, and never changes except by externalcauses’ (Descartes 1985, 240). The second law states that ‘every piece of matter,considered in itself, always tends to continue moving, not in any oblique path but onlyin a straight line’ (Descartes 1985, 241). The two laws taken together are similar toNewton’s first law of motion.

Descartes’s third law of motion asserts that the ‘force for proceeding in a straightline’ is conserved in collisions:

when a moving body collides with another, if its power of continuing in a straightline is less than the resistance of the other body, it is deflected so that, while thequantity of motion is retained, the direction is altered; but if its power of continu-ing is greater than the resistance of the other body, it carries that body along withit, and loses a quantity of motion equal to that which it imparts to the other body.(Descartes 1985, 242)

The rules of collision Descartes derived from his definition of the quantity of motionare terribly inadequate. The correct rules were discovered by Wren, Wallis, andHuygens. Nevertheless, his definitions of true motion and the quantity of motionprovide the context for Newton’s scholium. Newton attempts to show, among otherthings, the absurdity of the Cartesian doctrine of motion. The conservation of thequantity of motion presupposes that the quantity of motion of the whole is the sum ofthe quantities of motions of the parts, and it is this property that is inconsistent withDescartes’s definition of true motion as the motion of a body relative to the immediatesurrounding bodies.

Newton proposes three types of arguments for distinguishing between absolute andrelative motion, ‘which are distinguished from each other by their properties, causes,and effects’. This section will focus on the argument from ‘properties’, where we canfind Newton’s conceptual argument for absolute space. The first argument in thearguments from ‘properties’ attempts to distinguish between absolute and relative restby showing that we are ignorant about whether a body is in fact at rest, especially when

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it is located in regions inaccessible to us. Whatever the merits of this argument, it doesnot involve the quantity of motion, so I shall not pursue it further.

The following argument from the properties of motion is more relevant to thisdiscussion:

It is a property of motion that parts which keep given positions in relation to wholesparticipate in the motions of such wholes. (Newton 1999, 409)

Newton’s doctrine of place and Descartes’s theory of motion should make Newton’sremark clear. Newton discusses the parts ‘which keep given positions in relations towholes’. Thus, it is the position of the place a body occupies within a larger volume thatcontains it which concerns Newton. The property of motion, whether this motion isrelative or absolute, is such that volume-parts that are to be found within a body alsoparticipate in the motion of the whole body. The notion of ‘participation’ refers tocontribution of the part to the quantity of motion of the whole. That Newton has thequantity of motion in mind is made clear by the next remark:

For all the parts of bodies revolving in orbit endeavour to recede from the axis ofmotion, and the impetus of bodies moving forward arises from the joint impetus ofthe individual parts. Therefore, when bodies containing others move, whatever isrelatively at rest within them also moves. (Newton 1999, 411)

Newton reminds us here that the impetus of bodies moving in a straight line arisesfrom the joint impetus of the individual parts. Thus it is clear that Newton is thinkingof the quantity of motion. First, the notion of impetus is used interchangeably with thenotion of the quantity of motion in the De Gravitatione. Second, the language in thispassage parallels the language used in the definition of the quantity of motion alreadyquoted. Newton is concerned here with the ‘property’ of absolute motion which relatesthe quantity of motion of the parts to the quantity of motion of the whole. In fact, theequation P = mv reflects this property. The quantity of motion of the whole arises fromthe quantity of motions of the parts. Thus, the quantity motion is proportional to boththe mass, which describes the number of parts a body has (or the size of its impenetra-ble place, before being contracted or expanded), and motion, which for a solid body isidentical to all the parts.

The above property of the quantity of motion is inconsistent with Descartes’sdefinition of true motion. When a body contains another and moves together with it,the contained body is at rest in relation to the whole body, but its quantity of motionis said to contribute to the quantity of motion of the whole:

And thus true and absolute motion cannot be determined by means of change ofposition from the vicinity of bodies that are regarded as being at rest. For the exteriorbodies ought to be regarded not only as being at rest but also as being truly at rest.Otherwise all contained bodies, besides being subject to change of position from thevicinity of the containing bodies, will participate in the true motions of the contain-ing bodies and, if there is no such change of position, will not be truly at rest but onlybe regarded as being at rest. For containing bodies are to those inside them as theouter part of the whole to the inner part or as the shell to the kernel. And when theshell moves, the kernel also, without being changed in position from the vicinity ofthe shell, moves as a part of the whole. (Newton 1999, 411)

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Newton is here reducing Descartes’s doctrine of motion to an absurdity. A movingbody supposedly carries a quantity of motion from all its parts. However, if we takemotion in the ‘strict’ sense to be defined relative to the bodies in the immediate vicinity,the interior parts of the moving bodies should be at rest, since they do not change theirplaces relative to their surroundings. But then, if the interior parts are at rest, theycannot contribute any quantity of motion to the whole. If this is the case, any solid bodytreated as a whole must be at rest, since all of its interior parts are at rest according toDescartes’s definition of true motion. The result is simply absurd. If solid bodies aresaid to carry the quantity of motion, Descartes must face the dilemma of denying eitherhis doctrine of true motion or his definition of the quantity of motion. Either the inte-rior parts of moving bodies are not truly at rest, if we take them to be contributing tothe motion of the whole; or the interior parts of moving bodies are truly at rest, whichimplies that any moving solid body does not carry a quantity of motion. The firstoption would lead to the rejection of Descartes’s doctrine of true motion, the secondoption would render the notion of the ‘quantity of motion’ meaningless, since it wouldimply that no solid body can carry a quantity of motion.

Previous commentators, such as Stein and Rynasiewicz, pointed out the incoherencybetween Descartes’s definition of true motion and his theory of conatus, which assertsthat rotating bodies have a tendency to recede from the axis of rotation (Newton 2004,15). But Newton’s argument is much more general here, and emphasizes the incoher-ency between Descartes’s definition of true motion and the property of true motion,namely the part-whole relation which is exhibited by it. While the theory of conatusdepends on this property, it is the very definition of the quantity of motion whichNewton is concerned with, not just Descartes’s dynamic theory as it applies to celestialbodies.

Descartes therefore needs to reject either his definition of the quantity of motion, orhis definition of true motion. This dilemma stems from a conceptual incoherence.Newton is committed to keeping the definition of the quantity of motion and rejectingDescartes’s definition of true motion. It is the quantity of motion that is really crucialto his science, as this notion provides the basis for his whole work. Also, Newtonprovides in his Rule 3 a reason for taking the quantity of motion to be a universalproperty. According to this rule,

Those qualities of bodies that cannot be intended and remitted [i.e., qualities thatcannot be increased and diminished] and that belong to all bodies on whichexperiments can be made should be taken as qualities of all bodies universally.(Newton 1999, 795)

According to Newton, then, we have a criterion for taking a quality to be universallyapplicable to all bodies. If the quality cannot be ‘intended’ or ‘remitted’, it is a universalproperty. It is not entirely clear what Newton means by this criterion, but his followingremarks clarify his intention:

The extension, hardness, impenetrability, mobility and force of inertia of the wholearise from the extension, hardness, impenetrability, mobility and force of inertia ofeach of the parts; and thus we conclude that every one of the least parts of all bodies

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is extended, hard, impenetrable, movable, and endowed with a force of inertia. Andthis is the foundation of all natural philosophy. (Newton 1999, 795)

Newton takes some properties to be universally valid based on their reducibility tomicrostructure. The extension, hardness, impenetrability and force of inertia of thewhole arise from the same properties that apply to the parts. Since we observe theseproperties in all physical bodies, and since these properties are always reducible toproperties of microstructure, we can safely assume that all bodies retain them.

Newton’s argument for absolute space includes the premise that the quantity ofmotion and the part–whole relation exhibited by it is a universally valid property. Thispremise is partially rooted in experience. Newton conducted thorough experiments toverify that the quantity of motion is conserved (for example, he showed in his pendu-lum experiments that action equals reaction). However, the universal validity ofconservation of the quantity of motion is partially based on his Rule 3 for the Study ofNatural Philosophy. Once experience shows that the quantity of motion of isconserved, we can conclude that the quantity of motion of the whole arises from thequantity of motion of the parts. Rule 3 then allows us to take the additive nature of thequantity of motion to be a universally valid rule. If the conservation of the quantity ofmotion conflicts with relational definitions of motion, we have to find an alternativeway of singling out true motions. Thus, the premise of the argument is partiallysupported by experiment and partially by a methodological rule. The argument itself isa conceptual one. Newton relies on conceptual analysis to show that the property of thequantity of motion conflicts with relational definitions of motion.

Newton’s argument effectively destroys Descartes’s definition of true motion. Butperhaps there are other relational definitions of true motion consistent with the part-whole relation exhibited by the quantity of motion. Newton’s next argument seems toprovide a universal argument against relational definitions of true motion:

A property akin to the preceding one is that when a place moves, whatever is placedin it moves along with it, and therefore a body moving away from a place that movesparticipates also in the motion of its place. Therefore, all motions away from placesthat move are only parts of whole and absolute motions, and every whole motion iscompounded of the motion of a body away from its initial place, and the motion ofthis place away from its place, and so on, until an unmoving place is reached, as inthe above-mentioned example of the sailor. (Newton 1999, 412)

The crux of the argument seems to be that if no distinction is made between movableand non-movable (i.e., ‘absolute’) places, one runs the risk of generating an infiniteregress. Exactly why there is this threat of a regress is difficult to ascertain. Nevertheless,I offer the following analysis as a possible reading.

Newton discusses the property of moving bodies, which ‘carry’ the motions of what-ever is placed within them. If there are only moveable volumes, the motion of body A1has to be defined relative to another movable place A2, which takes up the former placein which the body was located before moving. Keep in mind that by relative motion,Newton is referring to the change in the position of relative place, which is the volumeof A1within a larger body B1 that contains A1. Newton’s comments make it clear thatin this case, the relative motion is defined as the change of position of A1 within a

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volume B1 which contains both A1 and A2. But A1 should participate in the quantity ofmotion to be found in B1 (this is what he means by ‘all motions … are only parts ofwhole and absolute motions’). And here is where the regress begins. If A2 is itselfmovable, then we must treat A1’s quantity of motion as contributing only part of thequantity of motion in B1, since B1 carries the quantities of motion of both A1 and A2.However, to define the quantity of motion of B1 we have to treat this composite bodyas moveable, which implies that we have to consider the motion of B1 relative toanother movable place A3, which is the place from which B1 is moving away. But thisimplies that the quantity of motion of B1 contributes only part of the quantity ofmotion of body B2, which contains bodies A1, A2, and A3, since B2 carries the quantityof motion in A3. However, B2 is moving away from another body A4, and so on and soon. If we do not admit a distinction between movable and immovable volumes, orvolumes that participate in and volumes that do not participate in the quantity ofmotion of a body, we run the risk of having to include an infinite number of bodies inthe definition of a system’s quantity of motion.

Newton’s argument suggests that relational definitions of true motion will preventus from treating a physical system as isolated from external causal influences. Therelational definition requires that we include more and more bodies in the analysis adinfinitum. The only way to stop the regress is to assume the existence of immovablecontaining places that cannot interact causally with material bodies. These would beimmovable places which are excluded in principle from any physical system and areunable to carry momentum.

We can get a sense of the role of absolute space from what Newton says in remarksthat follow:

Thus, whole and absolute motions can be determined only by means of unmovingplaces, and therefore in what has preceded I have referred such motions to unmovingplaces and relative motions to movable places. Moreover, the only places that areunmoving are all those that keep given positions in relation to one another frominfinity to infinity and therefore always remain immovable and constitute the spacethat I call immovable. (Newton 1999, 412)

Newton arrives at the claim that immovable places are necessary in his system if theyare to support the concept of momentum. Without introducing absolute space, thequantity of motion of a whole system cannot be determined, since relational defini-tions of true motions require that we include an infinite regress of moveable places.

The argument from ‘properties’ is the heart of Newton’s argument for absolutespace. However, this argument has often been treated as inconsequential. Burtt, forexample, argues that this argument from properties is weak: ‘But immovable space isquite inaccessible to observation or experiment: our difficulty persists, how can we tellwhether a given body is at rest or moving in it?’ (Burtt 1954, 250). Obviously Burtt isnot considering the conceptual problem Newton is addressing. Whether one canascertain the motion of a body relative to absolute space has nothing to do with theconceptual coherence of defining the quantity of motion. Earman takes these argu-ments to be either ‘claims about the meaning of words’ or an a priori analysis ofconcepts (Earman 1989, 42). The first option seems to him to be an inconsequential

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quibble over the meaning of words, and the second argument a type of philosophizingthat we should no longer do. If Newton’s argument is something like a conceptual anal-ysis of motion, ‘the claims would be interesting if we were inclined to think thatconceptual analysis can reveal synthetic a priori truths. But today we are not soinclined’ (Earman 1989, 42). But Earman is obviously unaware of the conceptual inco-herence Newton is pointing out in the argument from ‘properties’. If various propertiesof motion seem essential to the definition of true motion, then one has to provide anapplicable and coherent conceptual framework.

Commentators who are careful to situate Newton’s scholium in relation to Descartesare aware that Newton is arguing against Descartes’s relational definition of truemotion. Stein certainly does not miss Newton’s implicit reference to Descartes:

Newton’s criticism of Descartes is that Descartes’s several formulations of the funda-mental meaning of the word ‘motion’ are inconsistent, and that none of them cansatisfy what for Newton is the crucial test of a philosophical conception of motion:namely that it make possible an adequate expression of the principles of dynamics.(Stein 1970, 274)

But the inconsistency referred to by Stein is not the one we were discussing but the onebetween relational definitions of true motion and Newton’s entire theoretical appara-tus. Stein’s reconstruction of the scholium is substantiated by Newton’s explicit analy-sis in the De Gravitatione of the inconsistency between Descartes’s definition of truemotion and his assertion that the planets have a tendency to recede from their axis ofrotation. The reconstruction offered here does not necessarily conflict with Stein’s; itonly adds a conceptual dimension to it. Newton is not just providing empirical contentto the meaning of ‘absolute motion’ by subsuming this notion under Newton’s laws ofmotion and his account of the system of the world. He also shows the conceptualincoherence between the quantity of motion and relational definitions of true motion(compare Stein 1970, 281).

Laymon (1978) and DiSalle (1994, 2002) have a view similar to that of Stein. Accord-ing to Laymon, ‘Newton is not claiming that absolute time and space exist becauseabsolute time and motion can be calculated on the basis of some observations. Thesecalculations are applications of theories already assumed correct. The acceptance ofthese theories—astronomy or Newton’s dynamics—depends on their empiricalsuccess’ (Laymon 1978, 402). DiSalle similarly asserts that ‘the definition of absoluterotation does indeed have a basis in Newton’s Laws; if the universe in fact obeys thoselaws, we can always measure the true rotation of any body’ (DiSalle 2002, 44). To besure, the non-relational definition of true motion gets further empirical justification inNewton’s bucket experiment and the empirical adequacy of his whole system.However, as I’ve tried to show here, the inconsistency Newton points out in theargument from properties is a theoretical inconsistency between the quantity of motionand relational definitions of true motion.

Rynasiewicz, despite his careful analysis of the distinction between true and apparentmotions, overlooks the implicit reference to the quantity of motion in Newton’sargument from properties. This oversight prevents him from making fully explicitNewton’s reasons for rejecting relational definitions of true motion. According to him,

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‘the remainder of the elaboration [of the argument from “properties”] speaks of howit is that a body together with the surrounding bodies fit the model of part and whole’(Rynasiewicz 1995, 145). But Rynasiewicz does not explain the significance of the‘model of part and whole’, and why exactly this model does not cohere with a relationaldefinition of true motion.

6. Conclusions

Newton’s arguments reduce Descartes’s definition of true motion to an absurdity byshowing that they conflict with the property of absolute motion, which takes themotion of the whole to be amassed from the motions of the parts. Thus, Newton’sargument is primarily an internal critique of an accepted scientific theory. His ownapproach should be seen as providing a reasonable alternative to a defunct, incoherenttheory. However, the third argument in the arguments from properties, I believe,makes a more general point. An attempt to define the quantity of motion using rela-tional definitions of motion runs the risk of an infinite regress. Newton’s argumentpresupposes that the quantity of motion and its mereological nature is indispensablefor formulating a scientific theory and that any such theory requires a definition of truemotion which stands apart from all the apparent motions. Thus, any attempt to expli-cate Newton’s argument for absolute space has to begin with these two assumptions.

Space then, according to Newton, is not a substance-like entity with causal powersthat influence bodies. The essential property of absolute motion which Newton reliesin his argument is that which takes the motion of the part to participate in the motionof the whole. If there were only moveable places, namely, if there were only materialbodies which carry momentum, then there could never be a way of demarcating onesystem of interacting bodies from another. Thus we need immovable places, oneswhich can serve as containers of absolute motion, but which could not be made to carryany quantity of motion. The fear is that the removal of immovable places from theworld would undermine the ability to isolate any physical system.

We can now see why the traditional reading of Newton’s scholium, which takes it tobe an inference to the best explanation, is wrong. The argument is not that byremoving absolute containing places merely inertial effects are left without a causalprecedence. The consequence of allowing all containing places to be moveable is thedestruction of causal explanations in general. Without an immovable containing spaceone would not be able to provide a stable definition of physical systems and thequantities of motions they carry, and without such definitions you have no method ofrevealing causal relations. Absolute space, according to this reading, supportsNewton’s entire structure.

However, we also see that Newton is not just looking for a definition of true motionwhich is empirically adequate. He does exclude relational definitions of motion on thegrounds that they do not cohere with his theoretical apparatus. His argument showsthat relational definitions are inadequate from an empirical standpoint, but their inad-equacy also stems from a tension between these definitions and a proper definition ofthe quantity of motion. Newton’s belief in absolute space is not an idle metaphysical

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speculation, but an attempt to provide a solid conceptual foundation to a scientifictheory.

Acknowledgements

I would like to thank Arthur Fine, Marc Lange, Robert DiSalle, Sona Gosh, and twoanonymous referees for providing comments on previous and current versions of thepaper. I would also like to thank Walt Stevenson for his help in reading the scholiumin Latin and clarifying the various uses of the word ‘situs’.

Notes

[1] Newton himself did not consider the possibility that remote bodies can provide a reference forthe definition of motion. The reason could be that none of these other definitions of truemotion were ever proposed by Newton’s contemporaries. I’m here of course ignoring Mach’slater objection that the bucket experiment doesn’t preclude the possibility that inertial effectscould be correlated with the motion of the water relative to the universe as a whole.

[2] See Earman (1989, 64) for an explicit discussion of the two steps of the argument, whichaccording to him amount to an inference to the best explanation.

[3] According to Mach, for example, Newton’s definition of inherent forces is ‘superfluous’, sincethe tendency to move in a straight line is included as an external case in the second law ofmotion (Mach 1893, 300). Mach devotes the majority of his powers to deny Newton his firstinference, i.e., that non-relative motion could not be reasonably correlated with inertialeffects. But he also criticizes Newton for including unobservable entities such as absolutespace and inherent forces which contribute nothing to our understanding of inertial effects. IfNewton wanted to use a legitimate scientific inference, he could have discussed a medium thatpervades the entire universe, which causally influences the behaviour of bodies (Mach 1893,282). Nevertheless, such a full-blown material medium is not Newton’s account of absolutespace. Mach, therefore, takes Newton to be positing an absolute space which has no casualeffect on any particular phenomenon. And it is this construction which he finds ‘metaphysical’and ‘meaningless’. For a similar argument, see Einstein (1967 [1923], 13–17).

[4] For the history of this misreading of the argument, see Reichenbach (1927, 210–218), Burtt(1954, 244–255), Jammer (1994, 106), Lacey (1970), and Westfall (1971, 443).

[5] Newton explains why space is not a substance in the following passage: ‘Perhaps it may beexpected that I should define extension as substance, or accident, or else nothing at all. But byno means, for it has its own manner of existing which is proper to it and which fits neithersubstances nor accidents. It is not a substance: on the one hand, because it is not absolute initself, but as it were an emanative effect of God and an affection of every kind of being; on theother hand because it is not among the proper affections that denote substance, namelyaction, such as thoughts in minds and motions in body.’ (Newton 2004, 21)

[6] Laymon introduces a similar reading, although he takes the implicit definition of absoluterotation to provide some inductive support to the existence of absolute space: ‘the function ofthe [bucket] experiment clearly is to be an example of a case where it is possible to determinethe state of absolute rotation, assuming the existence of absolute and relative space, and thetruth of Newtonian mechanics. The function of the bucket experiment is not per se to showthat absolute space exists. It shows this existence only indirectly in the sense that to show thata concept has application is to give inductive support to the claim that the entities presup-posed by the concept do exist.’ (Laymon 1978, 403)

[7] This understanding of ‘place’ is quite in line with the commonplace use of ‘situs’ in Latin textsbefore the advent of modern science. The Oxford Latin Dictionary (OLD) (Glare 1996 [1983])

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292 O. Belkind

defines ‘situs’ as ‘the position (of a thing) in relation to its surroundings, situation.’ However,the quotations the OLD mentions in support of the definition actually use ‘situs’ with theconnotation of a volume a body occupies within a larger volume that contains it. For example,in his discussion of the physical organs of speech as the ‘voice’ of the brain, Cicero tells us, ‘inore sita lingua est finita dentibus; ea vocem immoderate profusam fingit et terminat atquesonos vocis distinctos et pressos efficit cum et dentes et alias partes pellit oris’ (‘the tongue islocated within the mouth and confined by the teeth; it modulates and contains the inarticu-late flow of the voice and renders its sounds distinct and clear by striking the teeth and otherparts of the mouth’) (Cicero, 266, modified translation). Here the word ‘situs’ is used to indi-cate that the tongue has a designated place within the larger organ that contains it, the mouth.Another citation by the OLD also gives support to the meaning we attribute to ‘situs’: ‘supe-rius autem umeri caput rotundius quam cetera ossa, de quibus adhuc dixi, parvo excessuvertici lati scapularum ossis inseritur, ac maiore parte extra situm nervis deligatur’ (’the upperhead of the humerus is more rounded than any other bone hitherto described and is insertedby a small excrescence into the top of the wide bone of the shoulder-blades, and the greaterpart of it is held fast by sinews outside its socket’) (Celsus 1935, 8.1.19, modified translation).As in Cicero, the word ‘situs’ is used to indicate the place of an organ created by the surround-ing organs. The humerus joint is said to be held in its place by the ligaments within the shoul-der-blade bone.

[8] Newton’s account suggests that the distinction between space and bodies is that betweenpenetrable and impenetrable places. However, there may also be another distinction at workin Newton’s thinking. Newton seems to follow More in distinguishing between mathematicaland physical divisibility. While space is infinitely divisible, there seems to be a limit to divisi-bility in physical bodies and material bodies must comprise of indivisible atoms. See Janiak(2000) for an account of Newton’s views on mathematical and physical divisibility.

[9] In what follows I will mostly follow Garber (1992).[10] Descartes’s initial claim is that that the quantity of motion of the whole universe is

conserved. From this claim Descartes deduces that an isolated individual body continues tomove in a straight line and that quantity of motion is conserved during collisions. However,the particular form in which the quantity of motion is conserved does not follow from thephilosophical claim that quantity of motion is conserved throughout the universe as awhole.

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