Since antiquity, natural philosophers have struggled to comprehend thenature of three tightly interconnected concepts: space, time, andmotion. A proper understanding of motion, in particular, has been seento be crucial for deciding questions about the natures of space andtime, and their interconnections. Since the time of Newton andLeibniz, philosophers’ struggles to comprehend these conceptshave often appeared to take the form of a dispute betweenabsolute conceptions of space, time and motion, andrelational conceptions. This article guides the readerthrough some of the history of these philosophical struggles. Ratherthan taking sides in the (alleged) ongoing debates, or reproducing thestandard dialectic recounted in most introductory texts, we havechosen to scrutinize carefully the history of the thinking of thecanonical participants in these debates – principally Descartes,Newton, Leibniz, Mach and Einstein. Readers interested in following upeither the historical questions or current debates about the naturesof space, time and motion will find ample links and referencesscattered through the discussion and in theOther Internet Resources section below.

• 1. Introduction
• 2. Aristotle
• 3. Descartes
  • 3.1 The Nature of Motion
  • 3.2 Motion and Dynamics
• 4. Newton
  • 4.1 Newton Against the Cartesian Account of Motion – The Bucket
  • 4.2 Absolute Space and Motion
• 5. Newtonian Absolute Space in the Twentieth Century
  • 5.1 The Spacetime Approach
  • 5.2 Substantivalism
• 6. Leibniz
  • 6.1 The Ideality of Space
  • 6.2 Force and the Nature of Motion
  • 6.3 Motion and Dynamics
  • 6.4 Where Did the Folk Go Wrong?
  • 6.5 Leibniz’s Response to Newton’sScholium
• 7. ‘Not-Newton’ versus ‘Be-Leibniz’
  • 7.1Non Sequiturs Mistakenly Attributed to Newton
  • 7.2 The Best Explanation Argument Mistakenly Attributed to Newton
  • 7.3 Substantivalism and The Best Explanation Argument
• 8. Beyond Newton
• Bibliography
  • Works cited in text
  • Notable Philosophical Discussions of the Absolute-Relative Debates
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

Things change. This is a platitude but still a crucial feature of theworld, and one which causes many philosophical perplexities –see for instance the entry onZeno’s paradoxes. For Aristotle, motion (he would have called it‘locomotion’) was just one kind of change, likegeneration, growth, decay, fabrication and so on. The atomists held onthe contrary that all change was in reality the motion of atoms intonew configurations, an idea that was not to begin to realize its fullpotential until the Seventeenth Century, particularly in the work ofDescartes. (Of course, modern physics seems to show that the physicalstate of a system goes well beyond the geometrical configuration ofbodies. Fields, while perhaps determined by the states of bodies, arenot themselves configurations of bodies if interpreted literally, andin quantum mechanics bodies have ‘internal states’ such asparticle spin.)

Not all changes seem to be merely the (loco)motions of bodies inphysical space. Yet since antiquity, in the western tradition, thiskind of motion has been absolutely central to the understanding ofchange. And since motion is a crucial concept in physical theories,one is forced to address the question of what exactly it is. Thequestion might seem trivial, for surely what is usually meant bysaying that something is moving is that it is movingrelativeto something, often tacitly understood between speakers. For instance:the car is moving at 60mph (relative to the road and things along it),the plane is flying (relative) to London, the rocket is lifting off(the ground), or the passenger is moving (to the front of the speedingtrain). Typically the relative reference body is either thesurroundings of the speakers, or the Earth, but this is not always thecase. For instance, it seems to make sense to ask whether the Earthrotates about its axis West-East diurnally or whether it is insteadthe heavens that rotate East-West; but if all motions are to bereckoned relative to the Earth, then its rotation seemsimpossible.

But if the Earth does not offer a unique frame of reference for thedescription of motion, then we may wonder whether any arbitrary objectcan be used for the definition of motions: are all such motions on apar, none privileged over any other? It is unclear whether anyone hasreally consistently espoused this view: Aristotle, perhaps, in theMetaphysics. Descartes and Leibniz are often thought to havedone so; however, as we’ll see, those claims are suspect.Possibly Huygens, though the most recent and thorough reconstructionof his position (Stan 2016) indicates not. Mach at some momentsperhaps. If this view were correct, then the question of whether theEarth or heavens rotate would be ill-formed, those alternatives beingmerely different but equivalent expressions of the facts.

However, suppose that, like Aristotle, you take ordinary languageaccurately to reflect the structure of the world. Then you couldrecognize systematic everyday uses of ‘up’ and‘down’ that require some privileged standards – usesthat treat things closer to a point at the center of the Earth as more‘down’ and motions towards that point as‘downwards’. Of course we would likely explain this usagein terms of the fact that we and our language evolved in a verynoticeable gravitational field directed towards the center of theEarth; but for Aristotle, as we shall see, this usage helped identifyan important structural feature of the universe, which itself wasrequired for the explanation of weight. Now a further question arises:how should a structure, such as a preferred point in the universe,which privileges certain motions, be understood? What makes that pointprivileged? One might expect that Aristotle simply identified it withthe center of the Earth, and so relative to that particular body;however, we shall soon see that he did not adopt that tacit conventionas fundamental. So the question arises of whether the preferred pointis somewhere picked out in some other way by the bodies in theuniverse – the center of the heavens perhaps? Or is it pickedout quite independently of the arrangements of matter?

The issues that arise in this simple theory help to frame the debatesbetween later physicists and philosophers concerning the nature ofmotion; in this article, we will focus on the theories of Descartes,Leibniz, and Newton. In the companion entry onabsolute and relational space and motion: post-Newtonian theories, we study the approaches followed by Mach, Einstein, and certaincontemporary researchers. We will see that similar concerns pervadeall these works: is there any kind of privileged sense of motion: asense in which things can be said to move or not, not just relative tothis or that reference body, but ‘truly’? If so, can thistrue motion be analyzed in terms of motions relative to other bodies– to some special body, or to the entire universe perhaps? (Andin relativity, in which distances, times and measures of relativemotion are frame-dependent, what relations are relevant?) If not, thenhow is the privileged kind of motion to be understood, as relative tospace itself – something physical but non-material –perhaps? Or can some kinds of motion be best understood as not beingspatial changes – changes of relative location or of place– at all?

2. Aristotle

To see that the problem of the interpretation of spatiotemporalquantities as absolute or relative is endemic to almost any kind ofmechanics one can imagine, we can look to one of the simplest theories– Aristotle’s account of natural motion (e.g.,On theHeavens I.2). According to this theory it is because of theirnatures, and not because of ‘unnatural’ forces, that thatheavy bodies move down, and ‘light’ things (air and fire)move up; it is their natures, or ‘forms’, that constitutethe gravity or weight of the former and the levity of the latter. Thisaccount only makes sense if ‘up’ and ‘down’can be unequivocally determined for each body. According to Aristotle,up and down are fixed by the position of the body in question relativeto the center of the universe, a point coincident with the center ofthe Earth. That is, the theory holds that heavy bodies naturally movetowards the center, while light bodies naturally move away.

Does this theory involve absolute or merely relative quantities? Itdepends on the nature of the center. If the center were identifiedwith the center of the Earth, then the theory could be taken to eschewabsolute quantities: it would simply hold that the natural motions ofany body depend on its position relative to another, namely the Earth.But Aristotle is explicit that the center of the universe is notidentical with, but merely coincident with the center of the Earth(e.g.,On the Heavens II.14): since the Earth itself isheavy, if it were not at the center it would move there! So the centeris not identified with any body, and so perhaps direction-to-center isan absolute quantity in the theory, not understood fundamentally asdirection to some body (merely contingently as such if some bodyhappens to occupy the center). But this conclusion is not cleareither. InOn the Heavens II.13, admittedly in response to adifferent issue, Aristotle suggests that the center itself is‘determined’ by the outer spherical shell of the universe(the aetherial region of the fixed stars). If this is what he intends,then the natural law prescribes motion relative to another body afterall – namely up or down with respect to the mathematical centerof the stars.

It would be to push Aristotle’s writings too hard to suggestthat he was consciously wrestling with the issue of whether mechanicsrequired absolute or relative quantities of motion, but what is clearis that these questions arise in his physics and his remarks impingeon them. His theory also gives a simple model of how they arise: aphysical theory of motion will say that ‘under such-and-suchcircumstances, motion of so-and-so a kind will occur’ –and the question of whether that kind of motion makes sense in termsof the relations between bodies alone arises automatically. Aristotlemay not have recognized the question explicitly, but we see it as oneissue in the background of his discussion of the center of theuniverse.

3. Descartes

The issues are, however, far more explicit in the entry onDescartes’ physics; and since the form of his theory is different, the ‘kinds ofmotion’ in question are quite different – as they changewith all the different theories that we discuss. For Descartes arguedin his 1644Principles of Philosophy (see Book II) that theessence of matter is extension (i.e., size and shape), because anyother attribute of bodies could be imagined away without imaginingaway matter itself. But he also held that extension constitutes thenature of space, and hence he concluded that space and matter are oneand the same thing. An immediate consequence of the identification isthe impossibility of the vacuum; if every region of space is a regionof matter, then there can be no space without matter. ThusDescartes’ universe is ‘hydrodynamical’ –completely full of mobile matter of different sized pieces in motion,rather like a bucket full of water and lumps of ice of differentsizes, which has been stirred around. Since fundamentally the piecesof matter are nothing but extension, the universe is in fact nothingbut a system of geometric bodies in motion without any gaps.^[1]

3.1 The Nature of Motion

The identification of space and matter poses a puzzle about motion: ifthe space that a body occupies literally is the matter of the body,then when the body – i.e., the matter – moves, so does thespace that it occupies. Thus it doesn’t change place, which isto say that it doesn’t move after all! Descartes resolved thisdifficulty by taking all motion to be the motion of bodies relative toone another, not a literal change of space.

Now, a body has as many relative motions as there are bodies, but itdoes not follow that all are equally significant. Indeed, Descartesuses several different concepts of relational motion. First there is‘change of place’, which is nothing but motion relative tothis or that arbitrary reference body (II.13). In this sense no motionof a body is privileged, since the speed, direction, and even curve ofa trajectory depends on the reference body, and none is singled out.Next, he discusses motion in ‘the ordinary sense’ (II.24).This is often conflated with mere change of arbitrary place, butstrictly it differs because according to the rules of ordinary speechone correctly attributes motion only to bodies whose motion is causedby some action,not to arbitrary relative motion. (Forinstance, a person sitting on a speeding boat is ordinarily said to beat rest, since ‘he feels no action in himself’.) Thisdistinction is important in some passages, but arguably not in thosethat we discuss. Finally, he defined motion ‘properlyspeaking’ (II.25) to be ‘the transference of one part ofmatter or of one body, from the vicinity of those bodies immediatelycontiguous to it and considered as at rest, into the vicinity of[some] others.’^[2] Since a body can only be touching one set of surroundings, Descartesargued (questionably) that this standard of motion was unique.

What we see here is that Descartes, despite holding motion to be themotion of bodies relative to one another, also held there to be aprivileged sense of motion; in a terminology sometimes employed bywriters of the period (see Rynasiewicz 2019, §3), he held thereto be a sense of ‘true motion’, over and above the merelyrelative motions. In logical terms we can make the point this way:whilemoves-relative-to is a two-place predicate,moves-properly-speaking is a one-place predicate. (And this,even though it is defined in terms of relative motion: letcontiguous-surroundings be a function from bodies to theircontiguous surroundings, thenxmoves-properly-speaking is defined asxmoves-relative-to-contiguous-surroundings(x).)

This example illustrates why it is crucial to keep two questionsdistinct: on the one hand, is motion to be understood in terms ofrelations between bodies or by invoking something additional,something absolute; on the other hand, are all relative motionsequally significant, or is there some ‘true’, privilegednotion of motion? Descartes’ views show that eschewing absolutemotion is logically compatible with accepting true motion; which is ofcourse not to say that his definitions of motion are themselvestenable.

3.2 Motion and Dynamics

There is an interpretational tradition which holds that Descartes onlytook the first, ‘ordinary’ sense of motion seriously, andintroduced the second notion to avoid conflict with the CatholicChurch. Such conflict was a real concern, since the censure ofGalileo’s Copernicanism took place only 11 years beforepublication of thePrinciples, and had in fact dissuadedDescartes from publishing an earlier work,The World. Indeed,in thePrinciples (III.28) he is at pains to explain how‘properly speaking’ the Earth does not move, because it isswept around the Sun in a giant vortex of matter – the Earthdoes not move relative to its surroundings in the vortex.

The difficulty with the reading, aside from the imputation ofcowardice to the old soldier, is that it makes nonsense ofDescartes’ mechanics, a theory of collisions. For instance,according to his laws of collision if two equal bodies strike eachother at equal and opposite velocities then they will bounce off atequal and opposite velocities (Rule I). On the other hand, if the verysame bodies approach each other with the very samerelativespeed, but at different speeds then they will move off together in thedirection of the faster one (Rule III). But if the operative meaningof motion in the Rules is the ordinary sense, then these twosituations are just the same situation, differing only in the choiceof reference frame, and so could not have different outcomes –bouncing apartversus moving off together. It seemsinconceivable that Descartes could have been confused in such atrivial way.^[3]

Thus Garber (1992, Chapter 6–8) proposes that Descartes actuallytook the unequivocal notion of motion properly speaking to be thecorrect sense of motion in mechanics. Then Rule I covers the case inwhich the two bodies have equal and opposite motionsrelative totheir contiguous surroundings, while Rule VI covers the case inwhich the bodies have different motionsrelative to thosesurroundings – one is perhaps at rest in its surroundings.That is, exactly what is needed to make the rules consistent is thekind of privileged, true, sense of motion provided by Descartes’second definition. Insurmountable problems with the rules remain, butrejecting the traditional interpretation and taking motion properlyspeaking seriously in Descartes’ philosophy clearly gives a morecharitable reading.

4. Newton

Newton articulated a clearer, more coherent, and more physicallyplausible account of motion that any that had come before. Still, aswe will see, there have been a number of widely held misunderstandingsof Newton’s views, and it is not completely clear how best tounderstand the absolute space that he postulated.

4.1 Newton Against the Cartesian Account of Motion – The Bucket

In an unpublished essay –De Gravitatione (Newton,2004) – and in aScholium to the definitions given inhis 1687Mathematical Principles of Natural Philosophy,Newton attacked both of Descartes’ notions of motion ascandidates for the operative notion in mechanics. (Newton’scritique is studied in more detail in the entryNewton’s views on space, time, and motion.)^[4]

The most famous argument invokes the so-called ‘Newton’sbucket’ experiment. Stripped to its basic elements onecompares:

1. a bucket of water hanging from a cord as the bucket is setspinning about the cord’s axis, with
2. the same bucket and water when they are rotating at the same rateabout the cord’s axis.

As is familiar from any rotating system, there will be a tendency forthe water to recede from the axis of rotation in the latter case: in(i) the surface of the water will be flat (because of theEarth’s gravitational field) while in (ii) it will be concave.The analysis of such ‘inertial effects’ due to rotationwas a major topic of enquiry of ‘natural philosophers’ ofthe time, including Descartes and his followers, and they wouldcertainly have agreed with Newton that the concave surface of thewater in the second case demonstrated that the water was moving in amechanically significant sense. There is thus an immediate problem forthe claim that proper motion is the correct mechanical sense ofmotion: in (i) and (ii) proper motion isanti-correlated withthe mechanically significant motion revealed by the surface of thewater. That is, the water is flat in (i) when it is in motion relativeto its immediate surroundings – the inner sides of the bucket– but curved in (ii) when it is at rest relative to itsimmediate surroundings. Thus the mechanically relevant meaning ofrotation is not that of proper motion. (You may have noticed a smalllacuna in Newton’s argument: in (i) the water is at rest and in(ii) in motion relative to that part of its surroundings constitutedby the air above it. It’s not hard to imagine smallmodifications to the example to fill this gap.)

Newton also points out that the height that the water climbs up theinside of the bucket provides a measure of the rate of rotation ofbucket and water: the higher the water rises up the sides, the greaterthe tendency to recede must be, and so the faster the water must berotating in the mechanically significant sense. But suppose, veryplausibly, that the measure is unique, that any particular heightindicates a particular rate of rotation. Then the unique height thatthe water reaches at any moment implies a unique rate of rotation in amechanically significant sense. And thus motion in the sense of motionrelative to an arbitrary reference body is not the mechanical sense,since that kind of rotation is not unique at all, but depends on themotion of the reference body. And so Descartes’ change of place(and for similar reasons, motion in the ordinary sense) is not themechanically significant sense of motion.

4.2 Absolute Space and Motion

In our discussion of Descartes we called the sense of motion operativein the science of mechanics ‘true motion’, and the phraseis used in this way by Newton in theScholium. ThusNewton’s bucket shows that true (rotational) motion isanti-correlated with, and so not identical with, proper motion (asDescartes proposed according to the Garber reading); and Newtonfurther argues that the rate of true (rotational) motion is unique,and so not identical with change of place, which is multiple. Newtonproposed instead that true motion is motion relative to a temporallyenduring, rigid, 3-dimensional Euclidean space, which he dubbed‘absolute space’. Of course, Descartes also defined motionas relative to an enduring 3-dimensional Euclidean space; thedifference is that Descartes’ space was divided into parts (hisspace was identical with a plenum of corpuscles) in motion, not arigid structure in which (mobile) material bodies areembedded. So according to Newton, the rate of true rotation of thebucket (and water) is the rate at which it rotates relative toabsolute space. Or put another way, Newton effectively defines the1-place predicatex moves-absolutely asxmoves-relative-to absolute space; both Newton and Descartes offercompeting 1-place predicates as analyses ofxmoves-truly.

4.2.1 Absolute Space vs. Galilean Relativity

Newton’s proposal for understanding motion solves the problemsthat he posed for Descartes, and provides an interpretation of theconcepts of constant motion and acceleration that appear in his lawsof motion. However, it suffers from two notable interpretationalproblems, both of which were pressed forcefully by Leibniz (in theLeibniz-Clarke Correspondence, 1715–1716) – which is notto say that Leibniz himself offered a superior account of motion (see below).^[5] First, according to this account, absolute velocity is a well-definedquantity: more simply, the absolute speed of a body is the rate ofchange of its position relative to an arbitrary point of absolutespace. But the Galilean relativity of Newton’s laws (see theentry onspace and time: inertial frames) means that the evolution of a closed system would have been identicalif it had been moving at a different (constant) overall velocity: asGalileo noted in his (see the entry onGalileo Galilei), an experimenter cannot determine from observations inside his cabinwhether his ship is at rest in harbor or sailing smoothly. Put anotherway, according to Newtonian mechanics, in principle Newton’sabsolute velocity cannot be experimentally determined.

So in this regard absolute velocity is quite unlike acceleration(including rotation). Newtonian acceleration is understood in absolutespace as the rate of change of absolute velocity, and is, according toNewtonian mechanics, generally measurable; for instance by measuringthe height that the water ascends the sides of the bucket.^[6] Leibniz argued (rather inconsistently, as we shall see) that sincedifferences in absolute velocity are unobservable, they are not begenuine differences at all; and hence that Newton’s absolutespace, whose existence would entail the reality of such differences,must also be a fiction. Few philosophers today would immediatelyreject a quantity as unreal simply because it was not experimentallydeterminable, but this fact does justify genuine doubts about thereality of absolute velocity, and hence of absolute space.

4.2.2 The Ontology of Absolute Space

The second problem concerns the nature of absolute space. Newton quiteclearly distinguished his account from Descartes’ – inparticular with regards to absolute space’s rigidity versusDescartes’ ‘hydrodynamical’ space, and thepossibility of the vacuum in absolute space. Thus absolute space isdefinitely not material. On the other hand, presumably it is supposedto be part of the physical, not mental, realm. InDeGravitatione, Newton rejected both the traditional philosophicalcategories of substance and attribute as suitable characterizations.Absolute space is not a substance for it lacks causal powers and doesnot have a fully independent existence, and yet it is not an attributesince it would exist even in a vacuum, which by definition is a placewhere there are no bodies in which it might inhere. Newton proposesthat space is what we might call a ‘pseudo-substance’,more like a substance than property, yet not quite a substance.^[7] In fact, Newton accepted the principle that everything that exists,exists somewhere – i.e., in absolute space. Thus he viewedabsolute space as a necessary consequence of the existence ofanything, and of God’s existence in particular – hencespace’s ontological dependence. Leibniz was presumably unawareof the unpublishedDe Gravitatione in which these particularideas were developed, but as we shall see, his later works arecharacterized by a robust rejection of any notion of space as a realthing rather than an ideal, purely mental entity. This is a view thatattracts even fewer contemporary adherents, but there is somethingdeeply peculiar about a non-material but physical entity, a worry thathas influenced many philosophical opponents of absolute space.^[8]

5. Newtonian Absolute Space in the Twentieth Century

This article is largely a historical survey of prominent views.However, it is hard to fully understand those debates without knowingsomething about scientific and mathematical developments have changedrecent understanding of the issues. In particular, a spacetimeapproach can clarify the situation with which the interlocutors werewrestling, and help clarify their arguments. This is a point widelyrecognized in the secondary literature, and indeed colors much of whatis said there (for good and bad, as we shall touch on later). So ashort digression into these matters is important for engaging theliterature responsibly; that said, while §7 presupposes thissection, the reader only interested in §6 could skip thissection.

5.1 The Spacetime Approach

After the development of relativity theory (a topic that we address inthe companion article), and its interpretation as aspacetimetheory, it was realized that the notion of spacetime had applicabilityto a range of theories of mechanics, classical as well asrelativistic. In particular, there is a spacetime geometry –‘Galilean’ or ‘neo-Newtonian’ spacetime (theterms are interchangeable) – for Newtonian mechanics that solvesthe problem of absolute velocity; an idea exploited by a number ofphilosophers from the late 1960s onwards (e.g., Stein 1968, Earman1970, Sklar 1974, and Friedman 1983). For details the reader isreferred to the entry onspacetime: inertial frames, but the general idea is that although a spatial distance iswell-defined between any two simultaneous points of this spacetime,only the temporal interval is well-defined between non-simultaneouspoints. Thus things are rather unlike Newton’s absolute space,whose points persist through time and maintain their distances: inabsolute space the distance betweenp-now andq-then(wherep andq are points) is just the distancebetweenp-now andq-now. However, Galilean spacetimehas an ‘affine connection’ which effectively specifies forevery point of every continuous curve, the rate at which the curve ischanging from straightness at that point; for instance, the straightlines are picked out as those curves whose rate of change fromstraightness is zero at every point.^[9]

Since the trajectories of bodies are curves in spacetime, the affineconnection determines the rate of change from straightness at everypoint of every possible trajectory. The straight trajectories thusdefined can be interpreted as the trajectories of bodies movinginertially (i.e., without forces), and the rate of change fromstraightness of any trajectory can be interpreted as the accelerationof a body following that trajectory. That is, Newton’s First Lawcan be given a geometric formulation as ‘bodies on which no netforces act follow straight lines in spacetime’; similarly, theSecond Law can be formulated as ‘the rate of change fromstraightness of a body’s trajectory is equal to the forcesacting on the body divided by its mass’. The significance ofthis geometry is that while acceleration is well-defined, velocity isnot – in accordance with the empirical determinability ofacceleration (generally though not universally) but not of velocity,according to Newtonian mechanics. Thus Galilean spacetime gives a verynice interpretation of the choice that nature makes when it decidesthat the laws of mechanics should be formulated in terms ofaccelerations not velocities. (In fact, there are complications here:in light of Newton’s Corollary VI mentioned above, one mightwonder whether even Galilean spacetime is the appropriate spacetimestructure for Newtonian mechanics. Saunders (2013), for example,argues that in fact only a yet more impoverished spacetime structure– ‘Newton-Huygens spacetime’ – is needed.)

5.2 Substantivalism

Put another way, one can define the predicatex acceleratesastrajectory(x)has-non-zero-rate-of-change-from-straightness, wheretrajectory maps bodies onto their trajectories in Galileanspacetime. And this predicate, defined this way, applies to the waterin Newton’s bucket if and only if it is rotating, according toNewtonian mechanics formulated in terms of the geometry of Galileanspacetime; it is the mechanically relevant sense ofaccelerate in this theory. But this theoretical formulationand definition have been given in terms of the geometry of spacetime,not in terms of the relations between bodies; acceleration is‘absolute’ in the sense that there is a preferred (true)sense of acceleration in mechanics and which is not defined in termsof the motions of bodies relative to one another. Note that this senseof ‘absolute’ is broader than that of motion relative toabsolute space, which we defined earlier. In the remainder of thisarticle we will use it in this new broader sense. The reader should beaware that the term is used in many ways in the literature, and suchequivocation often leads to significant misunderstanding.

If any of this analysis of motion is taken literally then one arrivesat a position regarding the ontology of spacetime rather like that ofNewton’s regarding space: it is some kind of‘substantial’ (or maybe pseudo-substantial)thingwith the geometry of Galilean spacetime, just as absolute spacepossessed Euclidean geometry. This view regarding the ontology ofspacetime is usually called ‘substantivalism’ (Sklar,1974). The Galilean substantivalist usually sees themselves asadopting a more sophisticated geometry than Newton but sharing hissubstantivalism (though there is plenty of room for debate onNewton’s exact ontological views; see DiSalle, 2002, and Slowik2016). The advantage of the more sophisticated geometry is thatalthough it allows the absolute sense of acceleration apparentlyrequired by Newtonian mechanics to be defined, it does not allow oneto define a similar absolute speed or velocity –xaccelerates can be defined as a 1-place predicate in terms ofthe geometry of Galilean spacetime, but notx moves ingeneral – and so the first of Leibniz’s problems isresolved. Of course we see that the solution depends on a crucialshift from speed and velocity to acceleration as the relevant sensesof ‘motion’: from the rate of change of position to therate of rate of change.

While this proposal solves the first kind of problem posed by Leibniz,it seems just as vulnerable to the second. While it is true that itinvolves the rejection of absolute space as Newton conceived it, andwith it the need to explicate the nature of an enduring space, thepostulation of Galilean spacetime poses the parallel question of thenature of spacetime. Again, it is a physical but non-materialsomething, the points of which may be coincident with material bodies.What kind of thing is it? Could we do without it? As we shall seebelow, some contemporary philosophers believe so.

6. Leibniz

There is a ‘folk-reading’ of Leibniz that one often findseither explicitly or implicitly in the philosophy of physicsliterature which takes account of only some of his remarks on spaceand motion. For instance, the quantities captured by Earman’s(1999) ‘Leibnizian spacetime’ do not do justice toLeibniz’s view of motion (as Earman acknowledges). But it isperhaps most obvious in introductory texts (e.g., Huggett 2000).According to this view, the only quantities of motion are relativequantities, relative velocity, acceleration and so on, and allrelative motions are equal, so there is no true sense of motion.However, Leibniz is explicit that other quantities are also‘real’, and his mechanics implicitly – but obviously– depends on yet others. The length of this section is ameasure, not so much of the importance of Leibniz’s actualviews, but the importance of showing what the prevalent folk viewleaves out regarding Leibniz’s views on the metaphysics ofmotion and interpretation of mechanics. (For further elaboration ofthe following points the reader is referred to the entry onLeibniz’s philosophy of physics.)

That said, we shall also see that no one has yet discovered a fullysatisfactory way of reconciling the numerous conflicting things thatLeibniz says about motion. Some of these tensions can be put downsimply to his changing his mind (see Cover and Hartz 1988 or Arthur2013 for explications of how Leibniz’s views on spacedeveloped). However, we will concentrate on the fairly short period inthe mid 1680–90s during which Leibniz developed his theory ofmechanics, and was most concerned with its interpretation. We willsupplement this discussion with the important remarks that he made inhisCorrespondence withSamuel Clarke around 30 years later (1715–1716); this discussion is broadlyin line with the earlier period, and the intervening period is one inwhich he turned to other matters, rather than one in which his viewson space were evolving dramatically.

6.1 The Ideality of Space

Arguably, Leibniz’s views concerning space and motion do nothave a completely linear logic, starting from some logicallysufficient basic premises, but instead form a collection of mutuallysupporting doctrines. If one starts questioning why Leibniz heldcertain views – concerning the ideality of space, for instance– one is apt to be led in a circle. Still, exposition requiresstarting somewhere, and Leibniz’s argument for the ideality ofspace in theCorrespondence with Clarke is a good place tobegin. But bear in mind the caveats made here – this argumentwas made later than a number of other relevant writings, and itslogical relation to Leibniz’s views on motion is complex.

Leibniz (LV.47 – this notation means Leibniz’s Fifthletter, section 47, and so on) says that (i) a body comes to have the‘same place’ as another once did, when it comes to standin the same relations to bodies we ‘suppose’ to beunchanged (more on this later); (ii) that we can define ‘aplace’ to be that which any such two bodies have in common (herehe claims an analogy with the Euclidean/Eudoxan definition of arational number in terms of an identity relation between ratios); andfinally that (iii) space is all such places taken together. However,he also holds that properties areparticular, incapable ofbeing instantiated by more than one individual,even at differenttimes; hence it is impossible for the two bodies to be inliterally the same relations to the unchanged bodies. Thus the thingthat we take to be the same for the two bodies – the place– is something added by our minds to the situation, and onlyideal. As a result, space, which is constructed from these idealplaces, is itself ideal: ‘a certain order, wherein the mindconceives the application of relations’.

Contrast this view of space with those of Descartes and of Newton.Both Descartes and Newton claim that space is a real, mind-independententity; for Descartes it is matter, and for Newton a‘pseudo-substance’, distinct from matter. And of coursefor both, these views are intimately tied up with their accounts ofmotion. Leibniz simply denies the mind-independent reality of space,and this too is bound up with his views concerning motion.^[10]

6.2 Force and the Nature of Motion

So far (apart from that remark about ‘unchanged’ bodies)we have not seen Leibniz introduce anything more than relations ofdistance between bodies, which is certainly consistent with the folkview of his philosophy. However, Leibniz sought to provide afoundation for the Cartesian/mechanical philosophy in terms of theAristotelian/scholastic metaphysics of substantial forms (here wediscuss the views laid out in Sections 17–22 of the 1686Discourse on Metaphysics and the 1695Specimen ofDynamics, both in Garber and Ariew 1989). In particular, heidentifies primary matter with what he calls its ‘primitivepassive force’ of resistance to changes in motion and topenetration, and the substantial form of a body with its‘primitive active force’. It is important to realize thatthese forces are not mere properties of matter, but actuallyconstitute it in some sense, and further that they are not themselvesquantifiable. However, because of the collisions of bodies with oneanother, these forces ‘suffer limitation’, and‘derivative’ passive and active forces result.^[11] Derivative passive force shows up in the different degrees ofresistance to change of different kinds of matter (of ‘secondarymatter’ in scholastic terms), and apparently is measurable.Derivative active force, however, is considerably more problematic forLeibniz. On the one hand, it is fundamental to his account of motionand theory of mechanics – motion fundamentallyispossession of force. But on the other hand, Leibniz endorses themechanical philosophy, which precisely sought to abolish Aristoteliansubstantial form, which active force represents. Leibniz’s goalwas to reconcile the two philosophies, by providing an Aristotelianmetaphysical foundation for modern mechanical science; as we shallsee, it is ultimately an open question exactly how Leibniz intended todeal with the inherent tensions in such a view.

6.2.1Vis Viva and True Motion

The texts are sufficiently ambiguous to permit dissent, but arguablyLeibniz intends that one manifestation of derivative active force iswhat he callsvis viva – ‘living force’.Leibniz had a famous argument with the Cartesians over the correctdefinition of this quantity. Descartes defined it assizetimesspeed – effectively as the magnitude of themomentum of a body. Leibniz gave a brilliant argument (repeated in anumber of places, for instance Section 17 of theDiscourse onMetaphysics) that it wassize timesspeed² – so (proportional to) kineticenergy. If the proposed identification is correct then kinetic energyquantifies derivative active force according to Leibniz; or looked atthe other way, the quantity ofvirtus (another term used byLeibniz for active force) associated with a body determines itskinetic energy and hence its speed. As far as the authors know,Leibniz never explicitly says anything conclusive about the relativityofvirtus, but it is certainly consistent to read him (asRoberts 2003 does) to claim that there is a unique quantity ofvirtus and hence ‘true’ (as we have been usingthe term) speed associated with each body. At the very least, Leibnizdoes say that there is a real difference between possession andnon-possession ofvis viva (e.g., in Section 18 of theDiscourse) and it is a small step from there to true, privilegedspeed. Indeed, for Leibniz, mere change of relative position is not‘entirely real’ (as we saw for instance in theCorrespondence) and only when it hasvis viva as itsimmediate cause is there some reality to it.^[12] An alternative interpretation to the one suggested here might saythat Leibniz intends that while there is a difference betweenmotion/virtus and no motion/virtus, there is somehowno difference between any strictly positive values of thosequantities.

It is important to emphasize two points about the preceding account ofmotion in Leibniz’s philosophy. First, motion in the everydaysense – motionrelative to something else – isnot real. Fundamentally motion is possession ofvirtus,something that is ultimately non-spatial (modulo its interpretation asprimitive force limited by collision). If this reading is right– and something along these lines seems necessary if wearen’t simply to ignore important statements by Leibniz onmotion – then Leibniz is offering an interpretation of motionthat is radically different from the obvious understanding. One mighteven say that for Leibniz motion is not movement at all! (We willleave to one side the question of whether his account is ultimatelycoherent.) The second point is that however we should understandLeibniz, the folk reading simply does not and cannot take account ofhis clearly and repeatedly stated view that what is real in motion isforcenot relative motion, for the folk reading allowsLeibnizonly relative motion (and of course additionally,motion in the sense of force is a variety of true motion, againcontrary to the folk reading).

6.3 Motion and Dynamics

However, from what has been said so far it is still possible that thefolk reading is accurate when it comes to Leibniz’s views on thephenomena of motion, the subject of his theory of mechanics. The casefor the folk reading is in fact supported by Leibniz’sresolution of the tension that we mentioned earlier, between thefundamental role of force/virtus (which we will now take tomeanmass timesspeed²) and itsassociation with Aristotelian form. Leibniz’s way out (e.g.,Specimen of Dynamics) is to require that while considerationsof force must somehow determine the form of the laws of motion, thelaws themselves should be such as not to allow one to determine thevalue of the force (and hence true speed). One might conclude that inthis case Leibniz held that the only quantities which can bedetermined are those of relative position and motion, as the folkreading says. But even in this circumscribed context, it is at bestquestionable whether the interpretation is correct.

6.3.1 Leibniz’s Mechanics

Consider first Leibniz’s mechanics. Since his laws are what isnow (ironically) often called ‘Newtonian’ elasticcollision theory, it seems that they satisfy both of his requirements.The laws include conservation of kinetic energy (which we identifywithvirtus), but they hold in all inertial frames, so thekinetic energy of any arbitrary body can be set to any initial value.But they do not permit the kinetic energy of a body to take on anyvalues throughout a process. The laws are only Galilean relativistic,and so are not true in every frame. Furthermore, according to the lawsof collision, in an inertial frame, if a body does not collide thenits Leibnizian force is conserved while if (except in special cases)it does collide then its force changes. According to Leibniz’slaws one cannot determine initial kinetic energies, but one certainlycan tell when they change. At the very least, there are quantities ofmotion implicit in Leibniz’s mechanics – change in forceand true speed – that are not merely relative; the folk readingis committed to Leibniz simply missing this obvious fact.

6.3.2 The Equivalence of Hypotheses

That said, when Leibniz discusses the relativity of motion –which he calls the ‘equivalence of hypotheses’ about thestates of motion of bodies – some of his statements do suggestthat he was confused in this way. For another way of stating theproblem for the folk reading is that the claim that relative motionsalone suffice for mechanics and that all relative motions are on a paris a principle of general relativity, and could Leibniz – amathematical genius – really have failed to notice that his lawshold only in special frames? Well, just maybe. On the one hand, whenhe explicitly articulates the principle of the equivalence ofhypotheses (for instance inSpecimen of Dynamics) he tends tosay only that one cannot assigninitial velocities on thebasis of the outcome of a collision, which requires only Galileanrelativity. However, he confusingly also claimed (On Copernicanismand the Relativity of Motion, also in Garber and Ariew 1989) thatthe Tychonic and Copernican hypotheses were equivalent. But if theEarth orbits the Sun in an inertial frame (Copernicus), then there isno inertial frame according to which the Sun orbits the Earth (TychoBrahe), and vice versa: these hypotheses are simply not Galileanequivalent (something else Leibniz could hardly have failed torealize). So there is some textual support for Leibniz endorsinggeneral relativity for the phenomena, as the folk readingmaintains.

A number of commentators have suggested solutions to the puzzle of theconflicting pronouncements that Leibniz makes on the subject: Stein1977 argues for general relativity, thereby imputing amisunderstanding of his own laws to Leibniz; Roberts 2003 argues forGalilean relativity, thereby discounting Leibniz’s apparentstatements to the contrary. Jauernig 2004 and 2008 points out that intheSpecimen, Leibniz claims that all motions are composed ofuniform rectilinear motions: an apparently curvilinear motion isactually a series of uniform motions, punctuated by discontinuouscollisions. This observation allows one to restrict the scope ofclaims of the kind ‘no motions can be attributed on the basis ofphenomena’ to inertial motions, and so helps read Leibniz asmore consistently advocating Galilean relativity, the reading Jauernigfavors (see also Huggett’s 2006 ‘Can Spacetime Help SettleAny Issues in Modern Philosophy?’, in the Other InternetResources, which was inspired by Jauernig’s work). Note thateven in a pure collision dynamics the phenomena distinguish a body inuniform rectilinear motion over time, from one that undergoescollisions changing its uniform rectilinear motion over time: the lawswill hold in the frame of the former, but not in the frame of thelatter. That is, apparently contrary to what Jauernig says,Leibniz’s account of curvilinear motion does not collapseGalilean relativity into general relativity. In that case,Leibniz’s specific claims of the phenomenal equivalence ofCopernican and Tychonic hypotheses still need to be accommodated.

6.4 Where Did the Folk Go Wrong?

So the folk reading simply ignores Leibniz’s metaphysics ofmotion, it commits Leibniz to a mathematical howler regarding hislaws, and it is arguable whether it is the best rendering of hispronouncements concerning relativity; it certainly cannot be acceptedunquestioningly. However, it is not hard to understand the temptationof the folk reading. In hisCorrespondence with Clarke,Leibniz says that he believes space to be “something merelyrelative, as time is, … an order of coexistences, as time is anorder of successions” (LIII.4), which is naturally taken to meanthat space is at base nothing but the distance and temporal relationsbetween bodies. (Though even this passage has its subtleties, becauseof the ideality of space discussed above, and because inLeibniz’s conception space determines what sets of relations arepossible.) And if relative distances and times exhaust thespatiotemporal in this way, then shouldn’tallquantities of motion be defined in terms of those relations?

We have seen two ways in which this would be the wrong conclusion todraw.Force seems to involve a notion of speed that is notidentified with any relative speed. And (unless the equivalence ofhypotheses is after all a principle of general relativity), the lawspick out a standard of constant motion that need not be any constantrelative motion. Of course, it is hard to reconcile these quantitieswith the view of space and time that Leibniz proposes – what isspeed insize times speed² orconstant speed if not speed relative to some body or toabsolute space? Given Leibniz’s view that space is literallyideal (and indeed that even relative motion is not ‘entirelyreal’) perhaps the best answer is that he tookforceand hencemotion in its real sense not to be determined bymotion in a relative sense at all, but to be primitive monadicquantities. That is, he tookx moves to be a 1-placepredicate, but he believed that it could be fully analyzed in terms ofstrictly monadic predicates:x moves iffxpossesses-non-zero-derivative-active-force. And this readingexplains just what Leibniz took us to be supposing when we‘supposed certain bodies to be unchanged’ in theconstruction of the idea of space: that they had no force, nothingcausing, or making real any motion.

6.5 Leibniz’s Response to Newton’sScholium

It’s again helpful to compare Leibniz with Descartes and Newton,this time regarding motion. Commentators often express frustration atLeibniz’s response to Newton’s arguments for absolutespace: “I find nothing … in theScholium thatproves or can prove the reality of space in itself. However, I grantthat there is a difference between an absolute true motion of a bodyand a mere relative change …” (LV.53). Not only doesLeibniz apparently fail to take the argument seriously, he then goeson to concede the step in the argument that seems to require absolutespace! But with our understanding of Newton and Leibniz, we can seethat what he says makes perfect sense (or at least that it is not asdisingenuous as it is often taken to be).

Newton argues in theScholium that true motion cannot beidentified with the kinds of motion that Descartes considers; but bothof these are purely relative motions, and Leibniz is in completeagreement that merely relative motions are not true (i.e.,‘entirely real’). Leibniz’s ‘concession’merely registers his agreement with Newton against Descartes on thedifference between true and relative motion; he surely understood whoand what Newton was refuting, and it was a position that he hadhimself, in different terms, publicly argued against at length. But aswe have seen, Leibniz had a very different analysis of the differenceto Newton’s; true motion was not, for him, a matter of motionrelative to absolute space, but the possession of quantity of force,ontologically prior to any spatiotemporal quantities at all. There isindeed nothing in theScholium explicitly directed againstthat view, and since it does potentially offer an alternative way ofunderstanding true motion, it is not unreasonable for Leibniz to claimthat there is no deductive inference from true motion to absolutespace.

7. ‘Not-Newton’ versus ‘Be-Leibniz’

7.1Non Sequiturs Mistakenly Attributed to Newton

The folk reading which belies Leibniz has it that he sought a theoryof mechanics formulated in terms only of the relations between bodies.As we’ll see in the companion article, in the NineteenthCentury, Ernst Mach indeed proposed such an approach, but Leibnizclearly did not; though certain similarities between Leibniz and Mach– especially the rejection of absolutespace –surely helps explain the confusion between the two. But not only isLeibniz often misunderstood, there are influential misreadings ofNewton’s arguments in theScholium, influenced by theidea that he is addressing Leibniz in some way. Of course thePrincipia was written 30 years before theCorrespondence, and the arguments of theScholiumwere not written with Leibniz in mind, but Clarke himself suggests(CIV.13) that those arguments – specifically those concerningthe bucket – are telling against Leibniz. That argument isindeed devastating to the parity of all relative motions, but we haveseen that it is highly questionable whether Leibniz’sequivalence of hypotheses amounts to such a view. That said, hisstatements in the first four letters of theCorrespondencecould understandably mislead Clarke on this point – it is inreply to Clarke’s challenge that Leibniz explicitly denies theparity of relative motions. But, interestingly, Clarke does notpresent a true version of Newton’s argument – despite someinvolvement of Newton in writing the replies. Instead of the argumentfrom the uniqueness of the rate of rotation, he argues that systemswith different velocities must be different because the effectsobservedif they were brought to rest would be different.This argument is of course utterly question begging against a viewthat holds that there is no privileged standard of rest (the viewClarke mistakenly attributes to Leibniz)!

As we discuss further in the companion article, Mach attributed toNewton the fallacious argument that because the surface of the watercurved even when it was not in motion relative to the bucket, it mustbe rotating relative to absolute space. Our discussion of Newtonshowed how misleading such a reading is. In the first place he alsoargues that there must be some privileged sense of rotation, and hencenot all relative motions are equal. Second, the argument isadhominem against Descartes, in which context a disjunctivesyllogism – motion is either proper or ordinary or relative toabsolute space – is argumentatively legitimate. On the otherhand, Mach is quite correct that Newton’s argument in theScholium leaves open the logical possibility that theprivileged, true sense of rotation (and acceleration more generally)is some species of relative motion; if not motion properly speaking,then relative to the fixed stars perhaps. (In fact Newton rejects thispossibility inDe Gravitatione (1962) on the grounds that itwould involve an odious action at a distance; an ironic position givenhis theory of universal gravity.)

7.2 The Best Explanation Argument Mistakenly Attributed to Newton

The kind of folk-reading of Newton that underlies much of thecontemporary literature replaces Mach’s interpretation with amore charitable one: for instance, Dasgupta 2015, is a recentinfluential presentation of the following dialectic, and its relationtosymmetry arguments. According to this reading, Newton’s point is thathis mechanics – unlike Descartes’ [specialcharacter:mdash] couldexplain why the surface of therotating water is curved, that his explanation involves a privilegedsense of rotation, and that absent an alternative hypothesis about itsrelative nature, we should accept absolute space. But our discussionof Newton’s argument showed that it simply does not have an‘abductive[special character:rsquo], ‘bestexplanation’ form, but shows deductively, from Cartesianpremises, that rotation is neither proper nor ordinary motion.

That is not to say that Newton had no understanding of how sucheffects would be explained in his mechanics. For instance, inCorollaries V and VI to the Definitions of thePrinciples hestates in general terms the conditions under which different states ofmotion are not – and so by implicationare –discernible according to his laws of mechanics. Nor is it to say thatNewton’s contemporaries weren’t seriously concerned withexplaining inertial effects. Leibniz, for instance, analyzed arotating body (in theSpecimen). In short, parts of arotating system collide with the surrounding matter and arecontinuously deflected, into a series of linear motions that form acurved path. (Though the system as Leibniz envisions it –comprised of a plenum of elastic particles of matter – is fartoo complex for him to offer any quantitative model based on thisqualitative picture. So he had no serious alternative explanation ofinertial effects.)

7.3 Substantivalism and The Best Explanation Argument

7.3.1 The Rotating Spheres

Although the argument is then not Newton’s, it is still animportant response to the kind of relationism proposed by thefolk-Leibniz, especially when it is extended by bringing in a furtherexample from Newton’sScholium. Newton considered apair of identical spheres, connected by a cord, too far from anybodies to observe any relative motions; he pointed out that their rateand direction of rotation could still be experimentally determined bymeasuring the tension in the cord, and by pushing on opposite faces ofthe two globes to see whether the tension increased or decreased. Heintended this simple example to demonstrate that the project heintended in thePrincipia, of determining the absoluteaccelerations and hence gravitational forces on the planets from theirrelative motions, was possible. However, if we further specify thatthe spheres and cord arerigid and that they are the onlythings in their universe, then the example can be used to point outthat there are infinitely many different rates of rotation all ofwhich agree on the relations between bodies. Since there are nodifferences in the relations between bodies in the differentsituations, it follows that theobservable differencesbetween the states of rotation cannot be explained in terms of therelations between bodies. Therefore, a theory of the kind attributedto the folk’s Leibniz cannot explain all the phenomena ofNewtonian mechanics, and again we can argue abductively for absolute space.^[13]

This argument is not Newton’s, neither the premises norconclusion, and must not be taken as a historically accurate reading,However, that is not to say that the argument is fallacious, andindeed many have found it attractive, particularly as a defense not ofNewton’s absolute space, but of Galilean spacetime. That is,Newtonian mechanics with Galilean spacetime can explain the phenomenaassociated with rotation, while theories of the kind proposed by Machcannot explain the differences between situations allowed by Newtonianmechanics; but these explanations rely on the geometric structure ofGalilean spacetime – particularly its affine connection, tointerpret acceleration. And thus – the argument goes –those explanations commit us to the reality of spacetime: a manifoldof points whose properties include the appropriate geometric ones.This final doctrine, of the reality of spacetime with its componentpoints or regions, distinct from matter, with geometric properties, iswhat we earlier identified as ‘substantivalism’.

7.3.2 Relationist Responses

There are two points to make about this line of argument. First, therelationist could reply that they need not explain all situationswhich are possible according to Newtonian mechanics, because thattheory is to be rejected in favor of one which invokes only distanceand time relations between bodies, but which approximates toNewton’s if matter is distributed suitably. Such a relationistwould be following Mach’s proposal, which we will discuss in thesequel article. Such a position would be satisfactory only to theextent that a suitable concrete replacement theory to Newton’stheory is developed; Mach never offered such a theory, but recentlymore progress has been made (again, see the companion article fordiscussion).

Second, one must be careful in understanding just how the argumentworks, for it is tempting to gloss it by saying that in Newtonianmechanics the affine connection is a crucial part of the explanationof the surface of the water in the bucket, and if the spacetime whichcarries the connection is denied, then the explanation fails too. Butthis gloss tacitly assumes that Newtonian mechanics can only beunderstood in a substantial Galilean spacetime; if an interpretationof Newtonian mechanics that does not assume substantivalism can beconstructed, then all Newtonian explanations can be given withoutpostulating a connection in an ontologically significant sense. BothSklar (1974) and van Fraassen (1970) have made proposals along theselines.

Sklar proposes interpreting ‘true’ acceleration as aprimitive quantity not defined in terms of motion relative toanything, be it absolute space, a connection or other bodies. (Ray1991 points out the family resemblance between this proposal andLeibniz’s suggestion thatvis viva addressesNewton’sScholium arguments.) Van Fraassen proposesformulating mechanics as ‘Newton’s Laws hold insome frame’, so that the form of the laws and thecontingent relative motions of bodies – not absolute space or aconnection, or even any instantaneous relations – pick out astandard of true motion, namely with respect to such an‘inertial frame’. These proposals aim to keep the fullexplanatory resources of Newtonian mechanics, and hence admit‘true acceleration’, but deny any relations between bodiesand spacetime itself. Like the actual Leibniz, they allow absolutequantities of motion, but claim that space and time themselves arenothing but the relations between bodies.

Some may question how the laws can be such as to privilege frameswithout prior spacetime geometry. In reply, Huggett 2006 proposes thatthe laws be understood as a Humean ‘best system’ (see theentry onlaws of nature) for a world of bodies and their relations; the laws don’treflect prior geometric structure, but systematic regularities inpatterns of relative motions. For obvious reasons, this proposal iscalled ‘regularity relationism’. (Several authors havedeveloped a similar approach to a variety of physical theories: forinstance, Vassallo & Esfeld 2016.) This approach is committed tothe idea that in some sense Newton’s laws are capable ofexplaining all the phenomena without recourse to spacetime geometry;that the connection and the metrical properties are explanatorilyredundant. This idea is also at the core of the ‘DynamicalApproach’, discussed in the companion article.

Another approach is to consider fully spatiotemporal relations. Forinstance, Maudlin 1993 discusses the possibility of a ‘Newtonianrelationism’ which addscross-temporal distancerelations, i.e., distances between bodies at distinct moments of time.With such distances, relationists can capture (almost) the fullstructure of Newtonian space, and time, including the affine structurerequired for Newton’s first and second laws.

8. Beyond Newton

In sum: we have seen how historical authors, from Aristotle through toNewtonian and Leibniz, tackled the puzzles of motion and change inphysical theorising. In a sequel entry onabsolute and relational space and motion: post-Newtonian theories, we will see how post-Newtonian authors, from Mach through to Einsteinand other contemporary physicists and philosophers, have brought newconceptual and technical resources to bear on (arguably) the selfsameissues. The sequel also includes a longer conclusion, reflecting onthe themes running through both articles.

For now we will just note that we have focussed on authors who madecontributions to the science of mechanics, and so a significantphilosophical lacuna is a discussion of Kant’s views on spaceand motion. For recent treatments, see Friedman 2013 and Stan2015.

Bibliography

Works cited in text

• Aristotle, 1984,The Complete Works of Aristotle: The RevisedOxford Translation, J. Barnes (ed.), Princeton: PrincetonUniversity Press.
• Arthur, R.T., 2013, “Leibniz’s theory of space,”Foundations of Science, 18(3), pp.499–528.
• Biener, Z., 2017, “De Gravitatione Reconsidered: TheChanging Significance of Experimental Evidence for Newton’sMetaphysics of Space,”Journal of the History ofPhilosophy, 55(4), pp.583–608.
• Brill, D. R. and Cohen, J., 1966, “Rotating Masses and theireffects on inertial frames,”Physical Review 143:1011–1015.
• Dasgupta, S., 2015, “Substantivalism vs relationalism aboutspace in classical physics,”Philosophy Compass, 10(9),pp.601–624.
• Descartes, R., 1983,Principles of Philosophy, R. P.Miller and V. R. Miller (trans.), Dordrecht: D. Reidel.
• Earman, J., 1970, “Who’s Afraid of AbsoluteSpace?,”Australasian Journal of Philosophy, 48:287–319.
• Friedman, M., 2013,Space in Kantian idealism. Space: AHistory (Oxford Philosophical Concepts), A. Janiak (ed.), NewYork: Oxford University Press.
• –––, 1983,Foundations of Space-TimeTheories: Relativistic Physics and Philosophy of Science,Princeton: Princeton University Press.
• Garber, D., 1992,Descartes’ Metaphysical Physics,Chicago: University of Chicago Press.
• Garber, D. and J. B. Rauzy, 2004, “Leibniz on Body, Matterand Extension,”The Aristotelian Society (SupplementaryVolume), 78: 23–40.
• Hartz, G. A. and J. A. Cover, 1988, “Space and Time in theLeibnizian Metaphysic,”Noûs, 22:493–519.
• Huggett, N., 2012, “What Did Newton Mean by ‘AbsoluteMotion’,” inInterpreting Newton: CriticalEssays, A. Janiak and E. Schliesser (eds.), Cambridge: CambridgeUniv Press, 196–218.
• –––, 2006, “The Regularity Account ofRelational Spacetime,”Mind, 115: 41–74.
• –––, 2000, “Space from Zeno to Einstein:Classic Readings with a Contemporary Commentary,”International Studies in the Philosophy of Science, 14:327–329.
• Janiak, A., 2015,Space and motion in nature and Scripture:Galileo, Descartes, Newton. Studies in History and Philosophy ofScience Part A, 51, pp.89–99.
• Jauernig, A., 2008, “Leibniz on Motion and the Equivalenceof Hypotheses,”The Leibniz Review, 18:1–40.
• –––, 2004,Leibniz Freed of Every Flaw: AKantian Reads Leibnizian Metaphysics, Ph.D. Dissertation,Princeton University.
• Leibniz, G. W., 1989,Philosophical Essays, R. Ariew andD. Garber (trans.), Indianapolis: Hackett Pub. Co.
• Leibniz, G. W., and Samuel Clarke, 1715–1716,“Correspondence”, inThe Leibniz-ClarkeCorrespondence, Together with Extracts from Newton’s“Principia” and “Opticks”, H. G.Alexander (ed.), Manchester: Manchester University Press, 1956.
• Maudlin, T., 1993, “Buckets of Water and Waves of Space: WhySpace-Time is Probably a Substance,”Philosophy ofScience, 60: 183–203.
• Newton, I. and I. B. Cohen, 1999,The Principia: MathematicalPrinciples of Natural Philosophy, I. B. Cohen and A. M. Whitman(trans.), Berkeley: University of California Press.
• Pooley, O., 2002,The Reality of Spacetime, D. Phil.thesis, Oxford: Oxford University.
• Ray, C., 1991,Time, Space and Philosophy, New York:Routledge.
• Roberts, J. T., 2003, “Leibniz on Force and AbsoluteMotion,”Philosophy of Science, 70: 553–573.
• Rynasiewicz, R., 2019, “Newton’s Scholium on Time,Space, Place and Motion,” inThe Oxford Handbook ofNewton, E. Schliesser and C. Smeenk (eds.), published online 8January 2019. doi:10.1093/oxfordhb/9780199930418.013.28 [Preprint available online]
• –––, 1995, “By their Properties, Causes,and Effects: Newton’s Scholium on Time, Space, Place, and Motion– I. The Text,”Studies in History and Philosophy ofScience, 26: 133–153.
• Sklar, L., 1974,Space, Time and Spacetime, Berkeley:University of California Press.
• Slowik, E., 2016,Deep Metaphysics of Space, Cham,Switzerland: Springer.
• Stan, M., 2015, “Absolute Space and the Riddle of Rotation:Kant’s Response to Newton”. Oxford Studies in EarlyModern Philosophy, vol. 7, Daniel Garber and Donald Rutherford(eds.), pp. 257–308. Oxford: Oxford University Press.
• Stan, M., 2016, “Huygens on Inertial Structure andRelativity,”Philosophy of Science, 83(2),pp.277-298.
• Stein, H., 1977, “Some Philosophical Prehistory of GeneralRelativity,” inMinnesota Studies in the Philosophy ofScience 8: Foundations of Space-Time Theories, J. Earman, C.Glymour and J. Stachel (eds.), Minneapolis: University of MinnesotaPress.
• –––, 1967, “Newtonian Space-Time,”Texas Quarterly, 10: 174–200.
• Van Fraassen, B. C., 1970,An introduction to the philosophyof time and space, New York: Columbia University Press.
• Vassallo, A. and M. Esfeld, 2016, “Leibnizian relationalismfor general relativistic physics,”Studies in History andPhilosophy of Science Part B: Studies in History and Philosophy ofModern Physics, 55. pp. 101–107.

Notable Philosophical Discussions of the Absolute-Relative Debates

• Barbour, J. B., 1982, “Relational Concepts of Space andTime,”British Journal for the Philosophy of Science,33: 251–274.
• Belot, G., 2000, “Geometry and Motion,”BritishJournal for the Philosophy of Science, 51: 561–595.
• Butterfield, J., 1984, “Relationism and PossibleWorlds,”British Journal for the Philosophy of Science,35: 101–112.
• Callender, C., 2002, “Philosophy of Space-TimePhysics,” inThe Blackwell Guide to the Philosophy ofScience, P. Machamer (ed.), Cambridge: Blackwell, pp.173–198.
• Carrier, M., 1992, “Kant’s Relational Theory ofAbsolute Space,”Kant Studien, 83: 399–416.
• Dasgupta, S., 2015, “Substantivalism vs Relationalism AboutSpace in Classical Physics”,Philosophy Compass 10, pp.601–624.
• Dieks, D., 2001, “Space-Time Relationism in Newtonian andRelativistic Physics,”International Studies in thePhilosophy of Science, 15: 5–17.
• DiSalle, R., 2006,Understanding Space-Time, Cambridge:Cambridge University Press.
• Disalle, R., 1995, “Spacetime Theory as PhysicalGeometry,”Erkenntnis, 42: 317–337.
• Earman, J., 1986, “Why Space is Not a Substance (at LeastNot to First Degree),”Pacific Philosophical Quarterly,67: 225–244.
• –––, 1970, “Who’s Afraid of AbsoluteSpace?,”Australasian Journal of Philosophy, 48:287–319.
• Earman, J. and J. Norton, 1987, “What Price SpacetimeSubstantivalism: The Hole Story,”British Journal for thePhilosophy of Science, 38: 515–525.
• Hoefer, C., 2000, “Kant’s Hands and Earman’sPions: Chirality Arguments for Substantival Space,”International Studies in the Philosophy of Science, 14:237–256.
• –––, 1998, “Absolute Versus RelationalSpacetime: For Better Or Worse, the Debate Goes on,”BritishJournal for the Philosophy of Science, 49: 451–467.
• –––, 1996, “The Metaphysics of Space-TimeSubstantialism,”Journal of Philosophy, 93:5–27.
• Huggett, N., 2000, “Reflections on ParityNonconservation,”Philosophy of Science, 67:219–241.
• Le Poidevin, R., 2004, “Space, Supervenience andSubstantivalism,”Analysis, 64: 191–198.
• Malament, D., 1985, “Discussion: A Modest Remark aboutReichenbach, Rotation, and General Relativity,”Philosophyof Science, 52: 615–620.
• Maudlin, T., 1993, “Buckets of Water and Waves of Space: WhySpace-Time is Probably a Substance,”Philosophy ofScience, 60: 183–203.
• –––, 1990, “Substances and Space-Time:What Aristotle would have Said to Einstein,”Studies inHistory and Philosophy of Science, 21(4): 531–561.
• Maudlin, T., 2012,Philosophy of Physics: Space and Time,Princeton, NJ: Princeton University Press.
• Mundy, B., 1992, “Space-Time and Isomorphism,”Proceedings of the Biennial Meetings of the Philosophy of ScienceAssociation, 1: 515–527.
• –––, 1983, “Relational Theories ofEuclidean Space and Minkowski Space-Time,”Philosophy ofScience, 50: 205–226.
• Nerlich, G., 2003, “Space-Time Substantivalism,” inThe Oxford Handbook of Metaphysics, M. J. Loux (ed.), Oxford:Oxford Univ Press, 281–314.
• –––, 1996, “What SpacetimeExplains,”Philosophical Quarterly, 46:127–131.
• –––, 1994,What Spacetime Explains:Metaphysical Essays on Space and Time, New York: CambridgeUniversity Press.
• –––, 1973, “Hands, Knees, and AbsoluteSpace,”Journal of Philosophy, 70: 337–351.
• Pooley, O., 2013 “Substantivalism and Relationalism AboutSpace and Time”, in R. Batterman (ed.),The Oxford Handbookof Philosophy of Physics, OUP.
• Rynasiewicz, R., 2000, “On the Distinction between Absoluteand Relative Motion,”Philosophy of Science, 67:70–93.
• –––, 1996, “Absolute Versus RelationalSpace-Time: An Outmoded Debate?,”Journal ofPhilosophy, 93: 279–306.
• Teller, P., 1991, “Substance, Relations, and Arguments aboutthe Nature of Space-Time,”Philosophical Review,363–397.
• Torretti, R., 2000, “Spacetime Models for the World,”Studies in History and Philosophy of Modern Physics (Part B),31(2): 171–186.

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Acknowledgments

Carl Hoefer’s research for this entry was supported by hisemployer, ICREA (Pg. Lluís Companys 23, 08010 Barcelona,Spain), and by Spanish MICINN grant FFI2016-76799-P.