## 1. Introduction



Things change. This is a platitude but still a crucial feature of the
world, and one which causes many philosophical perplexities -
see for instance the entry on
 Zeno's paradoxes.
 For Aristotle, motion (he would have called it
'locomotion') was just one kind of change, like
generation, growth, decay, fabrication and so on. The atomists held on
the contrary that all change was in reality the motion of atoms into
new configurations, an idea that was not to begin to realize its full
potential until the Seventeenth Century, particularly in the work of
Descartes. (Of course, modern physics seems to show that the physical
state of a system goes well beyond the geometrical configuration of
bodies. Fields, while perhaps determined by the states of bodies, are
not themselves configurations of bodies if interpreted literally, and
in quantum mechanics bodies have 'internal states' such as
particle spin.)



Not all changes seem to be merely the (loco)motions of bodies in
physical space. Yet since antiquity, in the western tradition, this
kind of motion has been absolutely central to the understanding of
change. And since motion is a crucial concept in physical theories,
one is forced to address the question of what exactly it is. The
question might seem trivial, for surely what is usually meant by
saying that something is moving is that it is moving *relative*
to 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 speeding
train). Typically the relative reference body is either the
surroundings of the speakers, or the Earth, but this is not always the
case. For instance, it seems to make sense to ask whether the Earth
rotates about its axis West-East diurnally or whether it is instead
the heavens that rotate East-West; but if all motions are to be
reckoned relative to the Earth, then its rotation seems
impossible.



But if the Earth does not offer a unique frame of reference for the
description of motion, then we may wonder whether any arbitrary object
can be used for the definition of motions: are all such motions on a
par, none privileged over any other? It is unclear whether anyone has
really consistently espoused this view: Aristotle, perhaps, in the
*Metaphysics*. Descartes and Leibniz are often thought to have
done so; however, as we'll see, those claims are suspect.
Possibly Huygens, though the most recent and thorough reconstruction
of his position (Stan 2016) indicates not. Mach at some moments
perhaps. If this view were correct, then the question of whether the
Earth or heavens rotate would be ill-formed, those alternatives being
merely different but equivalent expressions of the facts.



However, suppose that, like Aristotle, you take ordinary language
accurately to reflect the structure of the world. Then you could
recognize systematic everyday uses of 'up' and
'down' that require some privileged standards - uses
that 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 usage
in terms of the fact that we and our language evolved in a very
noticeable gravitational field directed towards the center of the
Earth; but for Aristotle, as we shall see, this usage helped identify
an important structural feature of the universe, which itself was
required 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 point
privileged? One might expect that Aristotle simply identified it with
the center of the Earth, and so relative to that particular body;
however, we shall soon see that he did not adopt that tacit convention
as fundamental. So the question arises of whether the preferred point
is somewhere picked out in some other way by the bodies in the
universe - the center of the heavens perhaps? Or is it picked
out quite independently of the arrangements of matter?



The issues that arise in this simple theory help to frame the debates
between later physicists and philosophers concerning the nature of
motion; in this article, we will focus on the theories of Descartes,
Leibniz, and Newton. In the companion entry on
 absolute and relational space and motion: post-Newtonian theories,
 we study the approaches followed by Mach, Einstein, and certain
contemporary researchers. We will see that similar concerns pervade
all these works: is there any kind of privileged sense of motion: a
sense in which things can be said to move or not, not just relative to
this or that reference body, but 'truly'? If so, can this
true motion be analyzed in terms of motions relative to other bodies
- to some special body, or to the entire universe perhaps? (And
in relativity, in which distances, times and measures of relative
motion are frame-dependent, what relations are relevant?) If not, then
how is the privileged kind of motion to be understood, as relative to
space itself - something physical but non-material -
perhaps? Or can some kinds of motion be best understood as not being
spatial changes - changes of relative location or of place
- at all?


## 2. Aristotle



To see that the problem of the interpretation of spatiotemporal
quantities as absolute or relative is endemic to almost any kind of
mechanics one can imagine, we can look to one of the simplest theories
- Aristotle's account of natural motion (e.g., *On the
Heavens* I.2). According to this theory it is because of their
natures, and not because of 'unnatural' forces, that that
heavy bodies move down, and 'light' things (air and fire)
move up; it is their natures, or 'forms', that constitute
the gravity or weight of the former and the levity of the latter. This
account 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 relative
to the center of the universe, a point coincident with the center of
the Earth. That is, the theory holds that heavy bodies naturally move
towards the center, while light bodies naturally move away.



Does this theory involve absolute or merely relative quantities? It
depends on the nature of the center. If the center were identified
with the center of the Earth, then the theory could be taken to eschew
absolute quantities: it would simply hold that the natural motions of
any body depend on its position relative to another, namely the Earth.
But Aristotle is explicit that the center of the universe is not
identical with, but merely coincident with the center of the Earth
(e.g., *On the Heavens* II.14): since the Earth itself is
heavy, if it were not at the center it would move there! So the center
is not identified with any body, and so perhaps direction-to-center is
an absolute quantity in the theory, not understood fundamentally as
direction to some body (merely contingently as such if some body
happens to occupy the center). But this conclusion is not clear
either. In *On the Heavens* II.13, admittedly in response to a
different 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 after
all - namely up or down with respect to the mathematical center
of the stars.



It would be to push Aristotle's writings too hard to suggest
that he was consciously wrestling with the issue of whether mechanics
required absolute or relative quantities of motion, but what is clear
is that these questions arise in his physics and his remarks impinge
on them. His theory also gives a simple model of how they arise: a
physical theory of motion will say that 'under such-and-such
circumstances, motion of so-and-so a kind will occur' -
and the question of whether that kind of motion makes sense in terms
of the relations between bodies alone arises automatically. Aristotle
may not have recognized the question explicitly, but we see it as one
issue in the background of his discussion of the center of the
universe.


## 3. Descartes



The issues are, however, far more explicit in the entry on
 Descartes' physics;
 and since the form of his theory is different, the 'kinds of
motion' in question are quite different - as they change
with all the different theories that we discuss. For Descartes argued
in his 1644 *Principles of Philosophy* (see Book II) that the
essence of matter is extension (i.e., size and shape), because any
other attribute of bodies could be imagined away without imagining
away matter itself. But he also held that extension constitutes the
nature of space, and hence he concluded that space and matter are one
and the same thing. An immediate consequence of the identification is
the impossibility of the vacuum; if every region of space is a region
of matter, then there can be no space without matter. Thus
Descartes' 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 different
sizes, which has been stirred around. Since fundamentally the pieces
of matter are nothing but extension, the universe is in fact nothing
but 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: if
the space that a body occupies literally is the matter of the body,
then when the body - i.e., the matter - moves, so does the
space that it occupies. Thus it doesn't change place, which is
to say that it doesn't move after all! Descartes resolved this
difficulty by taking all motion to be the motion of bodies relative to
one another, not a literal change of space.



Now, a body has as many relative motions as there are bodies, but it
does not follow that all are equally significant. Indeed, Descartes
uses several different concepts of relational motion. First there is
'change of place', which is nothing but motion relative to
this or that arbitrary reference body (II.13). In this sense no motion
of a body is privileged, since the speed, direction, and even curve of
a 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, but
strictly it differs because according to the rules of ordinary speech
one correctly attributes motion only to bodies whose motion is caused
by some action, *not* to arbitrary relative motion. (For
instance, a person sitting on a speeding boat is ordinarily said to be
at rest, since 'he feels no action in himself'.) This
distinction is important in some passages, but arguably not in those
that we discuss. Finally, he defined motion 'properly
speaking' (II.25) to be 'the transference of one part of
matter or of one body, from the vicinity of those bodies immediately
contiguous 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, Descartes
argued (questionably) that this standard of motion was unique.



What we see here is that Descartes, despite holding motion to be the
motion of bodies relative to one another, also held there to be a
privileged sense of motion; in a terminology sometimes employed by
writers of the period (see Rynasiewicz 2019, SS3), he held there
to be a sense of 'true motion', over and above the merely
relative motions. In logical terms we can make the point this way:
while *moves-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: let
*contiguous-surroundings* be a function from bodies to their
contiguous surroundings, then *x*
*moves-properly-speaking* is defined as *x*
*moves-relative-to-contiguous-surroundings*(*x*).)



This example illustrates why it is crucial to keep two questions
distinct: on the one hand, is motion to be understood in terms of
relations between bodies or by invoking something additional,
something absolute; on the other hand, are all relative motions
equally significant, or is there some 'true', privileged
notion of motion? Descartes' views show that eschewing absolute
motion is logically compatible with accepting true motion; which is of
course not to say that his definitions of motion are themselves
tenable.


### 3.2 Motion and Dynamics



There is an interpretational tradition which holds that Descartes only
took the first, 'ordinary' sense of motion seriously, and
introduced the second notion to avoid conflict with the Catholic
Church. Such conflict was a real concern, since the censure of
Galileo's Copernicanism took place only 11 years before
publication of the *Principles*, and had in fact dissuaded
Descartes from publishing an earlier work, *The World*. Indeed,
in the *Principles* (III.28) he is at pains to explain how
'properly speaking' the Earth does not move, because it is
swept around the Sun in a giant vortex of matter - the Earth
does not move relative to its surroundings in the vortex.



The difficulty with the reading, aside from the imputation of
cowardice to the old soldier, is that it makes nonsense of
Descartes' mechanics, a theory of collisions. For instance,
according to his laws of collision if two equal bodies strike each
other at equal and opposite velocities then they will bounce off at
equal and opposite velocities (Rule I). On the other hand, if the very
same bodies approach each other with the very same *relative*
speed, but at different speeds then they will move off together in the
direction of the faster one (Rule III). But if the operative meaning
of motion in the Rules is the ordinary sense, then these two
situations are just the same situation, differing only in the choice
of reference frame, and so could not have different outcomes -
bouncing apart *versus* moving off together. It seems
inconceivable that Descartes could have been confused in such a
trivial
 way.[3]



Thus Garber (1992, Chapter 6-8) proposes that Descartes actually
took the unequivocal notion of motion properly speaking to be the
correct sense of motion in mechanics. Then Rule I covers the case in
which the two bodies have equal and opposite motions *relative to
their contiguous surroundings*, while Rule VI covers the case in
which the bodies have different motions *relative to those
surroundings* - one is perhaps at rest in its surroundings.
That is, exactly what is needed to make the rules consistent is the
kind of privileged, true, sense of motion provided by Descartes'
second definition. Insurmountable problems with the rules remain, but
rejecting the traditional interpretation and taking motion properly
speaking seriously in Descartes' philosophy clearly gives a more
charitable reading.


## 4. Newton



Newton articulated a clearer, more coherent, and more physically
plausible account of motion that any that had come before. Still, as
we will see, there have been a number of widely held misunderstandings
of Newton's views, and it is not completely clear how best to
understand 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 a *Scholium* to the definitions given in
his 1687 *Mathematical Principles of Natural Philosophy*,
Newton attacked both of Descartes' notions of motion as
candidates for the operative notion in mechanics. (Newton's
critique is studied in more detail in the entry
 Newton's views on space, time, and motion.)[4]



The most famous argument invokes the so-called 'Newton's
bucket' experiment. Stripped to its basic elements one
compares:


1. a bucket of water hanging from a cord as the bucket is set
spinning about the cord's axis, with
2. the same bucket and water when they are rotating at the same rate
about the cord's axis.



As is familiar from any rotating system, there will be a tendency for
the water to recede from the axis of rotation in the latter case: in
(i) the surface of the water will be flat (because of the
Earth's gravitational field) while in (ii) it will be concave.
The analysis of such 'inertial effects' due to rotation
was a major topic of enquiry of 'natural philosophers' of
the time, including Descartes and his followers, and they would
certainly have agreed with Newton that the concave surface of the
water in the second case demonstrated that the water was moving in a
mechanically significant sense. There is thus an immediate problem for
the claim that proper motion is the correct mechanical sense of
motion: in (i) and (ii) proper motion is *anti*-correlated with
the mechanically significant motion revealed by the surface of the
water. That is, the water is flat in (i) when it is in motion relative
to its immediate surroundings - the inner sides of the bucket
- but curved in (ii) when it is at rest relative to its
immediate surroundings. Thus the mechanically relevant meaning of
rotation is not that of proper motion. (You may have noticed a small
lacuna in Newton's argument: in (i) the water is at rest and in
(ii) in motion relative to that part of its surroundings constituted
by the air above it. It's not hard to imagine small
modifications to the example to fill this gap.)



Newton also points out that the height that the water climbs up the
inside of the bucket provides a measure of the rate of rotation of
bucket and water: the higher the water rises up the sides, the greater
the tendency to recede must be, and so the faster the water must be
rotating in the mechanically significant sense. But suppose, very
plausibly, that the measure is unique, that any particular height
indicates a particular rate of rotation. Then the unique height that
the water reaches at any moment implies a unique rate of rotation in a
mechanically significant sense. And thus motion in the sense of motion
relative to an arbitrary reference body is not the mechanical sense,
since that kind of rotation is not unique at all, but depends on the
motion of the reference body. And so Descartes' change of place
(and for similar reasons, motion in the ordinary sense) is not the
mechanically significant sense of motion.


### 4.2 Absolute Space and Motion



In our discussion of Descartes we called the sense of motion operative
in the science of mechanics 'true motion', and the phrase
is used in this way by Newton in the *Scholium*. Thus
Newton's bucket shows that true (rotational) motion is
anti-correlated with, and so not identical with, proper motion (as
Descartes proposed according to the Garber reading); and Newton
further argues that the rate of true (rotational) motion is unique,
and so not identical with change of place, which is multiple. Newton
proposed instead that true motion is motion relative to a temporally
enduring, rigid, 3-dimensional Euclidean space, which he dubbed
'absolute space'. Of course, Descartes also defined motion
as relative to an enduring 3-dimensional Euclidean space; the
difference is that Descartes' space was divided into parts (his
space was identical with a plenum of corpuscles) in motion, not a
*rigid* structure in which (mobile) material bodies are
embedded. So according to Newton, the rate of true rotation of the
bucket (and water) is the rate at which it rotates relative to
absolute space. Or put another way, Newton effectively defines the
1-place predicate *x moves-absolutely* as *x
moves-relative-to absolute space*; both Newton and Descartes offer
competing 1-place predicates as analyses of *x
moves-truly*.


#### 4.2.1 Absolute Space vs. Galilean Relativity



Newton's proposal for understanding motion solves the problems
that he posed for Descartes, and provides an interpretation of the
concepts of constant motion and acceleration that appear in his laws
of motion. However, it suffers from two notable interpretational
problems, both of which were pressed forcefully by Leibniz (in the
Leibniz-Clarke Correspondence, 1715-1716) - which is not
to say that Leibniz himself offered a superior account of motion (see
 below).[5]
 First, according to this account, absolute velocity is a well-defined
quantity: more simply, the absolute speed of a body is the rate of
change of its position relative to an arbitrary point of absolute
space. But the Galilean relativity of Newton's laws (see the
entry on
 space and time: inertial frames)
 means that the evolution of a closed system would have been identical
if it had been moving at a different (constant) overall velocity: as
Galileo noted in his (see the entry on
 Galileo Galilei),
 an experimenter cannot determine from observations inside his cabin
whether his ship is at rest in harbor or sailing smoothly. Put another
way, according to Newtonian mechanics, in principle Newton's
absolute velocity cannot be experimentally determined.



So in this regard absolute velocity is quite unlike acceleration
(including rotation). Newtonian acceleration is understood in absolute
space as the rate of change of absolute velocity, and is, according to
Newtonian mechanics, generally measurable; for instance by measuring
the height that the water ascends the sides of the
 bucket.[6]
 Leibniz argued (rather inconsistently, as we shall see) that since
differences in absolute velocity are unobservable, they are not be
genuine differences at all; and hence that Newton's absolute
space, whose existence would entail the reality of such differences,
must also be a fiction. Few philosophers today would immediately
reject a quantity as unreal simply because it was not experimentally
determinable, but this fact does justify genuine doubts about the
reality 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 quite
clearly distinguished his account from Descartes' - in
particular with regards to absolute space's rigidity versus
Descartes' 'hydrodynamical' space, and the
possibility of the vacuum in absolute space. Thus absolute space is
definitely not material. On the other hand, presumably it is supposed
to be part of the physical, not mental, realm. In *De
Gravitatione*, Newton rejected both the traditional philosophical
categories of substance and attribute as suitable characterizations.
Absolute space is not a substance for it lacks causal powers and does
not have a fully independent existence, and yet it is not an attribute
since it would exist even in a vacuum, which by definition is a place
where there are no bodies in which it might inhere. Newton proposes
that 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 viewed
absolute space as a necessary consequence of the existence of
anything, and of God's existence in particular - hence
space's ontological dependence. Leibniz was presumably unaware
of the unpublished *De Gravitatione* in which these particular
ideas were developed, but as we shall see, his later works are
characterized by a robust rejection of any notion of space as a real
thing rather than an ideal, purely mental entity. This is a view that
attracts even fewer contemporary adherents, but there is something
deeply peculiar about a non-material but physical entity, a worry that
has 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 knowing
something about scientific and mathematical developments have changed
recent understanding of the issues. In particular, a spacetime
approach can clarify the situation with which the interlocutors were
wrestling, and help clarify their arguments. This is a point widely
recognized in the secondary literature, and indeed colors much of what
is said there (for good and bad, as we shall touch on later). So a
short digression into these matters is important for engaging the
literature responsibly; that said, while SS7 presupposes this
section, the reader only interested in SS6 could skip this
section.


### 5.1 The Spacetime Approach



After the development of relativity theory (a topic that we address in
the companion article), and its interpretation as a *spacetime*
theory, it was realized that the notion of spacetime had applicability
to a range of theories of mechanics, classical as well as
relativistic. In particular, there is a spacetime geometry -
'Galilean' or 'neo-Newtonian' spacetime (the
terms are interchangeable) - for Newtonian mechanics that solves
the problem of absolute velocity; an idea exploited by a number of
philosophers from the late 1960s onwards (e.g., Stein 1968, Earman
1970, Sklar 1974, and Friedman 1983). For details the reader is
referred to the entry on
 spacetime: inertial frames,
 but the general idea is that although a spatial distance is
well-defined between any two simultaneous points of this spacetime,
only the temporal interval is well-defined between non-simultaneous
points. Thus things are rather unlike Newton's absolute space,
whose points persist through time and maintain their distances: in
absolute space the distance between *p*-now and *q*-then
(where *p* and *q* are points) is just the distance
between *p*-now and *q*-now. However, Galilean spacetime
has an 'affine connection' which effectively specifies for
every point of every continuous curve, the rate at which the curve is
changing from straightness at that point; for instance, the straight
lines are picked out as those curves whose rate of change from
straightness is zero at every
 point.[9]



Since the trajectories of bodies are curves in spacetime, the affine
connection determines the rate of change from straightness at every
point of every possible trajectory. The straight trajectories thus
defined can be interpreted as the trajectories of bodies moving
inertially (i.e., without forces), and the rate of change from
straightness of any trajectory can be interpreted as the acceleration
of a body following that trajectory. That is, Newton's First Law
can be given a geometric formulation as 'bodies on which no net
forces act follow straight lines in spacetime'; similarly, the
Second Law can be formulated as 'the rate of change from
straightness of a body's trajectory is equal to the forces
acting on the body divided by its mass'. The significance of
this geometry is that while acceleration is well-defined, velocity is
not - in accordance with the empirical determinability of
acceleration (generally though not universally) but not of velocity,
according to Newtonian mechanics. Thus Galilean spacetime gives a very
nice interpretation of the choice that nature makes when it decides
that the laws of mechanics should be formulated in terms of
accelerations not velocities. (In fact, there are complications here:
in light of Newton's Corollary VI mentioned above, one might
wonder whether even Galilean spacetime is the appropriate spacetime
structure 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 predicate *x accelerates*
as *trajectory*(*x*)
*has-non-zero-rate-of-change-from-straightness*, where
*trajectory* maps bodies onto their trajectories in Galilean
spacetime. And this predicate, defined this way, applies to the water
in Newton's bucket if and only if it is rotating, according to
Newtonian mechanics formulated in terms of the geometry of Galilean
spacetime; it is the mechanically relevant sense of
*accelerate* in this theory. But this theoretical formulation
and 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 terms
of the motions of bodies relative to one another. Note that this sense
of 'absolute' is broader than that of motion relative to
absolute space, which we defined earlier. In the remainder of this
article we will use it in this new broader sense. The reader should be
aware that the term is used in many ways in the literature, and such
equivocation often leads to significant misunderstanding.



If any of this analysis of motion is taken literally then one arrives
at a position regarding the ontology of spacetime rather like that of
Newton's regarding space: it is some kind of
'substantial' (or maybe pseudo-substantial) *thing*
with the geometry of Galilean spacetime, just as absolute space
possessed Euclidean geometry. This view regarding the ontology of
spacetime is usually called 'substantivalism' (Sklar,
1974). The Galilean substantivalist usually sees themselves as
adopting a more sophisticated geometry than Newton but sharing his
substantivalism (though there is plenty of room for debate on
Newton's exact ontological views; see DiSalle, 2002, and Slowik
2016). The advantage of the more sophisticated geometry is that
although it allows the absolute sense of acceleration apparently
required by Newtonian mechanics to be defined, it does not allow one
to define a similar absolute speed or velocity - *x*
*accelerates* can be defined as a 1-place predicate in terms of
the geometry of Galilean spacetime, but not *x moves* in
general - and so the first of Leibniz's problems is
resolved. Of course we see that the solution depends on a crucial
shift from speed and velocity to acceleration as the relevant senses
of 'motion': from the rate of change of position to the
rate 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 it
involves the rejection of absolute space as Newton conceived it, and
with it the need to explicate the nature of an enduring space, the
postulation of Galilean spacetime poses the parallel question of the
nature of spacetime. Again, it is a physical but non-material
something, the points of which may be coincident with material bodies.
What kind of thing is it? Could we do without it? As we shall see
below, some contemporary philosophers believe so.


## 6. Leibniz



There is a 'folk-reading' of Leibniz that one often finds
either explicitly or implicitly in the philosophy of physics
literature which takes account of only some of his remarks on space
and motion. For instance, the quantities captured by Earman's
(1999) 'Leibnizian spacetime' do not do justice to
Leibniz's view of motion (as Earman acknowledges). But it is
perhaps most obvious in introductory texts (e.g., Huggett 2000).
According to this view, the only quantities of motion are relative
quantities, relative velocity, acceleration and so on, and all
relative 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 a
measure, not so much of the importance of Leibniz's actual
views, but the importance of showing what the prevalent folk view
leaves out regarding Leibniz's views on the metaphysics of
motion and interpretation of mechanics. (For further elaboration of
the following points the reader is referred to the entry on
 Leibniz's philosophy of physics.)



That said, we shall also see that no one has yet discovered a fully
satisfactory way of reconciling the numerous conflicting things that
Leibniz says about motion. Some of these tensions can be put down
simply to his changing his mind (see Cover and Hartz 1988 or Arthur
2013 for explications of how Leibniz's views on space
developed). However, we will concentrate on the fairly short period in
the mid 1680-90s during which Leibniz developed his theory of
mechanics, and was most concerned with its interpretation. We will
supplement this discussion with the important remarks that he made in
his *Correspondence* with
 Samuel Clarke
 around 30 years later (1715-1716); this discussion is broadly
in line with the earlier period, and the intervening period is one in
which he turned to other matters, rather than one in which his views
on space were evolving dramatically.


### 6.1 The Ideality of Space



Arguably, Leibniz's views concerning space and motion do not
have a completely linear logic, starting from some logically
sufficient basic premises, but instead form a collection of mutually
supporting doctrines. If one starts questioning why Leibniz held
certain views - concerning the ideality of space, for instance
- one is apt to be led in a circle. Still, exposition requires
starting somewhere, and Leibniz's argument for the ideality of
space in the *Correspondence* with Clarke is a good place to
begin. But bear in mind the caveats made here - this argument
was made later than a number of other relevant writings, and its
logical relation to Leibniz's views on motion is complex.



Leibniz (LV.47 - this notation means Leibniz's Fifth
letter, section 47, and so on) says that (i) a body comes to have the
'same place' as another once did, when it comes to stand
in the same relations to bodies we 'suppose' to be
unchanged (more on this later); (ii) that we can define 'a
place' to be that which any such two bodies have in common (here
he claims an analogy with the Euclidean/Eudoxan definition of a
rational number in terms of an identity relation between ratios); and
finally that (iii) space is all such places taken together. However,
he also holds that properties are *particular*, incapable of
being instantiated by more than one individual, *even at different
times*; hence it is impossible for the two bodies to be in
literally the same relations to the unchanged bodies. Thus the thing
that we take to be the same for the two bodies - the place
- is something added by our minds to the situation, and only
ideal. As a result, space, which is constructed from these ideal
places, is itself ideal: 'a certain order, wherein the mind
conceives 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-independent
entity; for Descartes it is matter, and for Newton a
'pseudo-substance', distinct from matter. And of course
for both, these views are intimately tied up with their accounts of
motion. 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 of
distance between bodies, which is certainly consistent with the folk
view of his philosophy. However, Leibniz sought to provide a
foundation for the Cartesian/mechanical philosophy in terms of the
Aristotelian/scholastic metaphysics of substantial forms (here we
discuss the views laid out in Sections 17-22 of the 1686
*Discourse on Metaphysics* and the 1695 *Specimen of
Dynamics*, both in Garber and Ariew 1989). In particular, he
identifies primary matter with what he calls its 'primitive
passive force' of resistance to changes in motion and to
penetration, and the substantial form of a body with its
'primitive active force'. It is important to realize that
these forces are not mere properties of matter, but actually
constitute it in some sense, and further that they are not themselves
quantifiable. However, because of the collisions of bodies with one
another, these forces 'suffer limitation', and
'derivative' passive and active forces
 result.[11]
 Derivative passive force shows up in the different degrees of
resistance to change of different kinds of matter (of 'secondary
matter' in scholastic terms), and apparently is measurable.
Derivative active force, however, is considerably more problematic for
Leibniz. On the one hand, it is fundamental to his account of motion
and theory of mechanics - motion fundamentally *is*
possession of force. But on the other hand, Leibniz endorses the
mechanical philosophy, which precisely sought to abolish Aristotelian
substantial form, which active force represents. Leibniz's goal
was to reconcile the two philosophies, by providing an Aristotelian
metaphysical foundation for modern mechanical science; as we shall
see, it is ultimately an open question exactly how Leibniz intended to
deal with the inherent tensions in such a view.


#### 6.2.1 *Vis Viva* and True Motion



The texts are sufficiently ambiguous to permit dissent, but arguably
Leibniz intends that one manifestation of derivative active force is
what he calls *vis viva* - 'living force'.
Leibniz had a famous argument with the Cartesians over the correct
definition of this quantity. Descartes defined it as *size*
times *speed* - effectively as the magnitude of the
momentum of a body. Leibniz gave a brilliant argument (repeated in a
number of places, for instance Section 17 of the *Discourse on
Metaphysics*) that it was *size* times
*speed2* - so (proportional to) kinetic
energy. If the proposed identification is correct then kinetic energy
quantifies derivative active force according to Leibniz; or looked at
the other way, the quantity of *virtus* (another term used by
Leibniz for active force) associated with a body determines its
kinetic energy and hence its speed. As far as the authors know,
Leibniz never explicitly says anything conclusive about the relativity
of *virtus*, but it is certainly consistent to read him (as
Roberts 2003 does) to claim that there is a unique quantity of
*virtus* and hence 'true' (as we have been using
the term) speed associated with each body. At the very least, Leibniz
does say that there is a real difference between possession and
non-possession of *vis viva* (e.g., in Section 18 of the
Discourse) and it is a small step from there to true, privileged
speed. Indeed, for Leibniz, mere change of relative position is not
'entirely real' (as we saw for instance in the
*Correspondence*) and only when it has *vis viva* as its
immediate cause is there some reality to
 it.[12]
 An alternative interpretation to the one suggested here might say
that Leibniz intends that while there is a difference between
motion/*virtus* and no motion/*virtus*, there is somehow
no difference between any strictly positive values of those
quantities.



It is important to emphasize two points about the preceding account of
motion in Leibniz's philosophy. First, motion in the everyday
sense - motion *relative* to something else - is
not real. Fundamentally motion is possession of *virtus*,
something that is ultimately non-spatial (modulo its interpretation as
primitive force limited by collision). If this reading is right
- and something along these lines seems necessary if we
aren't simply to ignore important statements by Leibniz on
motion - then Leibniz is offering an interpretation of motion
that is radically different from the obvious understanding. One might
even say that for Leibniz motion is not movement at all! (We will
leave to one side the question of whether his account is ultimately
coherent.) The second point is that however we should understand
Leibniz, the folk reading simply does not and cannot take account of
his clearly and repeatedly stated view that what is real in motion is
force *not* relative motion, for the folk reading allows
Leibniz *only* relative motion (and of course additionally,
motion in the sense of force is a variety of true motion, again
contrary to the folk reading).


### 6.3 Motion and Dynamics



However, from what has been said so far it is still possible that the
folk reading is accurate when it comes to Leibniz's views on the
phenomena of motion, the subject of his theory of mechanics. The case
for the folk reading is in fact supported by Leibniz's
resolution of the tension that we mentioned earlier, between the
fundamental role of force/*virtus* (which we will now take to
mean *mass* times *speed2*) and its
association with Aristotelian form. Leibniz's way out (e.g.,
*Specimen of Dynamics*) is to require that while considerations
of force must somehow determine the form of the laws of motion, the
laws themselves should be such as not to allow one to determine the
value of the force (and hence true speed). One might conclude that in
this case Leibniz held that the only quantities which can be
determined are those of relative position and motion, as the folk
reading says. But even in this circumscribed context, it is at best
questionable whether the interpretation is correct.


#### 6.3.1 Leibniz's Mechanics



Consider first Leibniz's mechanics. Since his laws are what is
now (ironically) often called 'Newtonian' elastic
collision theory, it seems that they satisfy both of his requirements.
The laws include conservation of kinetic energy (which we identify
with *virtus*), but they hold in all inertial frames, so the
kinetic 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 any
values throughout a process. The laws are only Galilean relativistic,
and so are not true in every frame. Furthermore, according to the laws
of collision, in an inertial frame, if a body does not collide then
its Leibnizian force is conserved while if (except in special cases)
it does collide then its force changes. According to Leibniz's
laws one cannot determine initial kinetic energies, but one certainly
can tell when they change. At the very least, there are quantities of
motion implicit in Leibniz's mechanics - change in force
and true speed - that are not merely relative; the folk reading
is 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 the
states of motion of bodies - some of his statements do suggest
that he was confused in this way. For another way of stating the
problem for the folk reading is that the claim that relative motions
alone suffice for mechanics and that all relative motions are on a par
is a principle of general relativity, and could Leibniz - a
mathematical genius - really have failed to notice that his laws
hold only in special frames? Well, just maybe. On the one hand, when
he explicitly articulates the principle of the equivalence of
hypotheses (for instance in *Specimen of Dynamics*) he tends to
say only that one cannot assign *initial* velocities on the
basis of the outcome of a collision, which requires only Galilean
relativity. However, he confusingly also claimed (*On Copernicanism
and the Relativity of Motion*, also in Garber and Ariew 1989) that
the Tychonic and Copernican hypotheses were equivalent. But if the
Earth orbits the Sun in an inertial frame (Copernicus), then there is
no inertial frame according to which the Sun orbits the Earth (Tycho
Brahe), and vice versa: these hypotheses are simply not Galilean
equivalent (something else Leibniz could hardly have failed to
realize). So there is some textual support for Leibniz endorsing
general relativity for the phenomena, as the folk reading
maintains.



A number of commentators have suggested solutions to the puzzle of the
conflicting pronouncements that Leibniz makes on the subject: Stein
1977 argues for general relativity, thereby imputing a
misunderstanding of his own laws to Leibniz; Roberts 2003 argues for
Galilean relativity, thereby discounting Leibniz's apparent
statements to the contrary. Jauernig 2004 and 2008 points out that in
the *Specimen*, Leibniz claims that all motions are composed of
uniform rectilinear motions: an apparently curvilinear motion is
actually a series of uniform motions, punctuated by discontinuous
collisions. This observation allows one to restrict the scope of
claims of the kind 'no motions can be attributed on the basis of
phenomena' to inertial motions, and so helps read Leibniz as
more consistently advocating Galilean relativity, the reading Jauernig
favors (see also Huggett's 2006 'Can Spacetime Help Settle
Any Issues in Modern Philosophy?', in the Other Internet
Resources, which was inspired by Jauernig's work). Note that
even in a pure collision dynamics the phenomena distinguish a body in
uniform rectilinear motion over time, from one that undergoes
collisions changing its uniform rectilinear motion over time: the laws
will hold in the frame of the former, but not in the frame of the
latter. That is, apparently contrary to what Jauernig says,
Leibniz's account of curvilinear motion does not collapse
Galilean relativity into general relativity. In that case,
Leibniz's specific claims of the phenomenal equivalence of
Copernican 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 of
motion, it commits Leibniz to a mathematical howler regarding his
laws, and it is arguable whether it is the best rendering of his
pronouncements concerning relativity; it certainly cannot be accepted
unquestioningly. However, it is not hard to understand the temptation
of the folk reading. In his *Correspondence* with Clarke,
Leibniz says that he believes space to be "something merely
relative, as time is, ... an order of coexistences, as time is an
order of successions" (LIII.4), which is naturally taken to mean
that space is at base nothing but the distance and temporal relations
between bodies. (Though even this passage has its subtleties, because
of the ideality of space discussed above, and because in
Leibniz's conception space determines what sets of relations are
*possible*.) And if relative distances and times exhaust the
spatiotemporal in this way, then shouldn't *all*
quantities of motion be defined in terms of those relations?



We have seen two ways in which this would be the wrong conclusion to
draw. *Force* seems to involve a notion of speed that is not
identified with any relative speed. And (unless the equivalence of
hypotheses is after all a principle of general relativity), the laws
pick out a standard of constant motion that need not be any constant
relative motion. Of course, it is hard to reconcile these quantities
with the view of space and time that Leibniz proposes - what is
*speed* in *size times speed2* or
*constant speed* if not speed relative to some body or to
absolute space? Given Leibniz's view that space is literally
ideal (and indeed that even relative motion is not 'entirely
real') perhaps the best answer is that he took *force*
and hence *motion* in its real sense not to be determined by
motion in a relative sense at all, but to be primitive monadic
quantities. That is, he took *x moves* to be a 1-place
predicate, but he believed that it could be fully analyzed in terms of
strictly monadic predicates: *x moves* iff *x
possesses-non-zero-derivative-active-force*. And this reading
explains just what Leibniz took us to be supposing when we
'supposed certain bodies to be unchanged' in the
construction of the idea of space: that they had no force, nothing
causing, or making real any motion.


### 6.5 Leibniz's Response to Newton's *Scholium*



It's again helpful to compare Leibniz with Descartes and Newton,
this time regarding motion. Commentators often express frustration at
Leibniz's response to Newton's arguments for absolute
space: "I find nothing ... in the *Scholium* that
proves or can prove the reality of space in itself. However, I grant
that there is a difference between an absolute true motion of a body
and a mere relative change ..." (LV.53). Not only does
Leibniz apparently fail to take the argument seriously, he then goes
on to concede the step in the argument that seems to require absolute
space! But with our understanding of Newton and Leibniz, we can see
that what he says makes perfect sense (or at least that it is not as
disingenuous as it is often taken to be).



Newton argues in the *Scholium* that true motion cannot be
identified with the kinds of motion that Descartes considers; but both
of these are purely relative motions, and Leibniz is in complete
agreement that merely relative motions are not true (i.e.,
'entirely real'). Leibniz's 'concession'
merely registers his agreement with Newton against Descartes on the
difference between true and relative motion; he surely understood who
and what Newton was refuting, and it was a position that he had
himself, in different terms, publicly argued against at length. But as
we have seen, Leibniz had a very different analysis of the difference
to Newton's; true motion was not, for him, a matter of motion
relative to absolute space, but the possession of quantity of force,
ontologically prior to any spatiotemporal quantities at all. There is
indeed nothing in the *Scholium* explicitly directed against
that view, and since it does potentially offer an alternative way of
understanding true motion, it is not unreasonable for Leibniz to claim
that there is no deductive inference from true motion to absolute
space.


## 7. 'Not-Newton' versus 'Be-Leibniz'


### 7.1 *Non Sequiturs* Mistakenly Attributed to Newton



The folk reading which belies Leibniz has it that he sought a theory
of mechanics formulated in terms only of the relations between bodies.
As we'll see in the companion article, in the Nineteenth
Century, Ernst Mach indeed proposed such an approach, but Leibniz
clearly did not; though certain similarities between Leibniz and Mach
- especially the rejection of absolute *space* -
surely helps explain the confusion between the two. But not only is
Leibniz often misunderstood, there are influential misreadings of
Newton's arguments in the *Scholium*, influenced by the
idea that he is addressing Leibniz in some way. Of course the
*Principia* was written 30 years before the
*Correspondence*, and the arguments of the *Scholium*
were not written with Leibniz in mind, but Clarke himself suggests
(CIV.13) that those arguments - specifically those concerning
the bucket - are telling against Leibniz. That argument is
indeed devastating to the parity of all relative motions, but we have
seen that it is highly questionable whether Leibniz's
equivalence of hypotheses amounts to such a view. That said, his
statements in the first four letters of the *Correspondence*
could understandably mislead Clarke on this point - it is in
reply to Clarke's challenge that Leibniz explicitly denies the
parity of relative motions. But, interestingly, Clarke does not
present a true version of Newton's argument - despite some
involvement of Newton in writing the replies. Instead of the argument
from the uniqueness of the rate of rotation, he argues that systems
with different velocities must be different because the effects
observed *if they were brought to rest* would be different.
This argument is of course utterly question begging against a view
that holds that there is no privileged standard of rest (the view
Clarke mistakenly attributes to Leibniz)!



As we discuss further in the companion article, Mach attributed to
Newton the fallacious argument that because the surface of the water
curved even when it was not in motion relative to the bucket, it must
be rotating relative to absolute space. Our discussion of Newton
showed how misleading such a reading is. In the first place he also
argues that there must be some privileged sense of rotation, and hence
not all relative motions are equal. Second, the argument is *ad
hominem* against Descartes, in which context a disjunctive
syllogism - motion is either proper or ordinary or relative to
absolute space - is argumentatively legitimate. On the other
hand, Mach is quite correct that Newton's argument in the
*Scholium* leaves open the logical possibility that the
privileged, 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 this
possibility in *De Gravitatione* (1962) on the grounds that it
would involve an odious action at a distance; an ironic position given
his 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 the
contemporary literature replaces Mach's interpretation with a
more charitable one: for instance, Dasgupta 2015, is a recent
influential presentation of the following dialectic, and its relation
to
 symmetry
 arguments. According to this reading, Newton's point is that
his mechanics - unlike Descartes' [special
character:mdash] could *explain* why the surface of the
rotating water is curved, that his explanation involves a privileged
sense of rotation, and that absent an alternative hypothesis about its
relative nature, we should accept absolute space. But our discussion
of Newton's argument showed that it simply does not have an
'abductive[special character:rsquo], 'best
explanation' form, but shows deductively, from Cartesian
premises, that rotation is neither proper nor ordinary motion.



That is not to say that Newton had no understanding of how such
effects would be explained in his mechanics. For instance, in
Corollaries V and VI to the Definitions of the *Principles* he
states in general terms the conditions under which different states of
motion are not - and so by implication *are* -
discernible according to his laws of mechanics. Nor is it to say that
Newton's contemporaries weren't seriously concerned with
explaining inertial effects. Leibniz, for instance, analyzed a
rotating body (in the *Specimen*). In short, parts of a
rotating system collide with the surrounding matter and are
continuously deflected, into a series of linear motions that form a
curved path. (Though the system as Leibniz envisions it -
comprised of a plenum of elastic particles of matter - is far
too complex for him to offer any quantitative model based on this
qualitative picture. So he had no serious alternative explanation of
inertial 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 an
important response to the kind of relationism proposed by the
folk-Leibniz, especially when it is extended by bringing in a further
example from Newton's *Scholium*. Newton considered a
pair of identical spheres, connected by a cord, too far from any
bodies to observe any relative motions; he pointed out that their rate
and direction of rotation could still be experimentally determined by
measuring the tension in the cord, and by pushing on opposite faces of
the two globes to see whether the tension increased or decreased. He
intended this simple example to demonstrate that the project he
intended in the *Principia*, of determining the absolute
accelerations and hence gravitational forces on the planets from their
relative motions, was possible. However, if we further specify that
the spheres and cord are *rigid* and that they are the only
things in their universe, then the example can be used to point out
that there are infinitely many different rates of rotation all of
which agree on the relations between bodies. Since there are no
differences in the relations between bodies in the different
situations, it follows that the *observable* differences
between the states of rotation cannot be explained in terms of the
relations between bodies. Therefore, a theory of the kind attributed
to the folk's Leibniz cannot explain all the phenomena of
Newtonian mechanics, and again we can argue abductively for absolute
 space.[13]



This argument is not Newton's, neither the premises nor
conclusion, and must not be taken as a historically accurate reading,
However, that is not to say that the argument is fallacious, and
indeed many have found it attractive, particularly as a defense not of
Newton's absolute space, but of Galilean spacetime. That is,
Newtonian mechanics with Galilean spacetime can explain the phenomena
associated with rotation, while theories of the kind proposed by Mach
cannot explain the differences between situations allowed by Newtonian
mechanics; but these explanations rely on the geometric structure of
Galilean spacetime - particularly its affine connection, to
interpret acceleration. And thus - the argument goes -
those explanations commit us to the reality of spacetime: a manifold
of points whose properties include the appropriate geometric ones.
This final doctrine, of the reality of spacetime with its component
points or regions, distinct from matter, with geometric properties, is
what we earlier identified as 'substantivalism'.


#### 7.3.2 Relationist Responses



There are two points to make about this line of argument. First, the
relationist could reply that they need not explain all situations
which are possible according to Newtonian mechanics, because that
theory is to be rejected in favor of one which invokes only distance
and time relations between bodies, but which approximates to
Newton's if matter is distributed suitably. Such a relationist
would be following Mach's proposal, which we will discuss in the
sequel article. Such a position would be satisfactory only to the
extent that a suitable concrete replacement theory to Newton's
theory is developed; Mach never offered such a theory, but recently
more progress has been made (again, see the companion article for
discussion).



Second, one must be careful in understanding just how the argument
works, for it is tempting to gloss it by saying that in Newtonian
mechanics the affine connection is a crucial part of the explanation
of the surface of the water in the bucket, and if the spacetime which
carries the connection is denied, then the explanation fails too. But
this gloss tacitly assumes that Newtonian mechanics can only be
understood in a substantial Galilean spacetime; if an interpretation
of Newtonian mechanics that does not assume substantivalism can be
constructed, then all Newtonian explanations can be given without
postulating a connection in an ontologically significant sense. Both
Sklar (1974) and van Fraassen (1970) have made proposals along these
lines.



Sklar proposes interpreting 'true' acceleration as a
primitive quantity not defined in terms of motion relative to
anything, be it absolute space, a connection or other bodies. (Ray
1991 points out the family resemblance between this proposal and
Leibniz's suggestion that *vis viva* addresses
Newton's *Scholium* arguments.) Van Fraassen proposes
formulating mechanics as 'Newton's Laws hold in
*some* frame', so that the form of the laws and the
contingent relative motions of bodies - not absolute space or a
connection, or even any instantaneous relations - pick out a
standard of true motion, namely with respect to such an
'inertial frame'. These proposals aim to keep the full
explanatory resources of Newtonian mechanics, and hence admit
'true acceleration', but deny any relations between bodies
and spacetime itself. Like the actual Leibniz, they allow absolute
quantities of motion, but claim that space and time themselves are
nothing but the relations between bodies.



Some may question how the laws can be such as to privilege frames
without prior spacetime geometry. In reply, Huggett 2006 proposes that
the laws be understood as a Humean 'best system' (see the
entry on
 laws of nature)
 for a world of bodies and their relations; the laws don't
reflect prior geometric structure, but systematic regularities in
patterns of relative motions. For obvious reasons, this proposal is
called 'regularity relationism'. (Several authors have
developed a similar approach to a variety of physical theories: for
instance, Vassallo & Esfeld 2016.) This approach is committed to
the idea that in some sense Newton's laws are capable of
explaining all the phenomena without recourse to spacetime geometry;
that the connection and the metrical properties are explanatorily
redundant. This idea is also at the core of the 'Dynamical
Approach', discussed in the companion article.



Another approach is to consider fully spatiotemporal relations. For
instance, Maudlin 1993 discusses the possibility of a 'Newtonian
relationism' which adds *cross-temporal* distance
relations, i.e., distances between bodies at distinct moments of time.
With such distances, relationists can capture (almost) the full
structure of Newtonian space, and time, including the affine structure
required for Newton's first and second laws.


## 8. Beyond Newton



In sum: we have seen how historical authors, from Aristotle through to
Newtonian and Leibniz, tackled the puzzles of motion and change in
physical theorising. In a sequel entry on
 absolute and relational space and motion: post-Newtonian theories,
 we will see how post-Newtonian authors, from Mach through to Einstein
and other contemporary physicists and philosophers, have brought new
conceptual and technical resources to bear on (arguably) the selfsame
issues. The sequel also includes a longer conclusion, reflecting on
the themes running through both articles.



For now we will just note that we have focussed on authors who made
contributions to the science of mechanics, and so a significant
philosophical lacuna is a discussion of Kant's views on space
and motion. For recent treatments, see Friedman 2013 and Stan
2015.