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Jeffrey K. McDonough Department of Philosophy

Harvard University jkmcdon@fas.harvard.edu

Leibniz’s Philosophy of Physics

Although better known today for his bold metaphysics and optimistic theodicy, Leibniz’s

intellectual contributions extended well beyond what is now generally thought of as

philosophy or theology. Remarkably in an era that knew the likes of Galileo, Descartes,

Huygens, Hooke and Newton, Leibniz stood out as one of the most important figures in

the development of the Scientific Revolution. This entry will attempt to provide a broad

overview of the central themes of Leibniz’s philosophy of physics, as well as an

introduction to some of the principal arguments and argumentative strategies he used to

defend his positions. The merits of Leibniz’s criticisms, contributions, and their relations

to his larger philosophical system remain fascinating areas for historical and

philosophical investigation.

1. The Historical Development of Leibniz’s Physics

2. Leibniz on Matter

2.1. The critique of atomism 2.2. The critique of Cartesian corpuscularianism 2.3. The passive powers of bodies

3. Leibniz’s Dynamics

3.1. A brief demonstration 3.2. The active powers of bodies 3.3. Forces and metaphysics

4. Leibniz on the Laws of Motion

4.1. Refuting the Cartesian laws of motion 4.2. A system of conservation principles 4.3. Absolute or relative motion?

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5. Leibniz on Space and Time

5.1. Against Absolute Space and Time 5.2. Space and Time as Ideal Systems of Relations

Bibliography Other Internet Resources Related Entries 1. The Historical Development of Leibniz’s Physics

In his earliest days, Leibniz read a wide range of traditional works drawn from his

father’s considerable library. Later he was formally educated at the University of Leipzig

(1661-1666), briefly at the University of Jena (1663), and finally at the University of

Altdorf (1666-1667). From these sources, Leibniz gained an early acquaintance with the

Aristotelian-Scholastic tradition, as well as a taste of neo-platonic themes common in

Renaissance humanism. By his own account, he quickly “penetrated far into the territory

of the Scholastics,” and derived “some satisfaction” from “Plato too, and Plotinus” (G III

606/L 655). Although he consciously broke with the Scholastic tradition while still quite

young, its doctrines clearly made a lasting impression on him, and served him both as a

font of ideas and a readily available target of criticism. Indeed, one can justifiably see

many of Leibniz’s mature doctrines as reactions – either positive or negative – to the

Scholastic views that he first became acquainted with while still a student in Germany.

According to his own recollection, it appears that Leibniz threw himself into the

mechanical philosophy sometime around the year 1661.1 In a well-known letter to

Nicolas Remond, Leibniz – then in the twilight of his years – recounted his early

conversion:

After having finished the trivial schools2, I fell upon the moderns, and I recall

walking in a grove on the outskirts of Leipzig called the Rosental, at the age of

fifteen, and deliberating whether to preserve substantial forms or not. Mechanism

finally prevailed and led me to apply myself to mathematics. (G III 606/L 655)

Although we have no records of Leibniz’s work from the years immediately following

his youthful adoption of mechanism, there is abundant evidence that by the late 1660’s,

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he had studied the writings of a wide range of mechanistic philosophers, 3 committed

himself to the “hypothesis of the moderns, which conceives no incorporeal entities within

bodies but assumes nothing beyond magnitude, figure, and motion,”4 and had begun to

search for ways in which to improve upon the mechanistic philosophy of his

predecessors. 5

The early influence of natural philosophers such as Gassendi and Hobbes -whom Leibniz

would later characterize as working in the Epicurean tradition – is particularly apparent in

his first pair of systematic works written on motion. In the Theoria motus abstracti

(TMA), dedicated to the French Academy in 1671, Leibniz introduces a set of abstract

laws of motion (A VI.ii.261-276/L 139-142). At the heart of those laws is the Hobbesian

notion of conatus, which Leibniz describes as “the beginning and end of motion,” and

which he seems to conceive of as a tendency to motion in a particular direction. Leibniz

argues that the motion of a body in isolation is determined entirely by its own conatus,

and that the motions of colliding bodies are determined solely by the combination of their

respective conatuses. In this way, Leibniz’s abstract theory of motion assigns no role

whatsoever, with respect to the fundamental laws of motion, to the sizes or masses of

material bodies. According to the TMA, a tiny pebble with a certain velocity striking a

large boulder at rest would, under idealized conditions, continue to move with the

boulder with the same velocity that it had initially.

Leibniz, of course, recognized that the laws of motion sketched in the TMA are radically

at odds with the testimony of everyday experience. After all, it would seem to be the

case that if the world were governed by the laws of the TMA, it should be no more

difficult to move a planet than a pebble.6 Leibniz sought to close this gap between the

fundamental laws and experience with the publication of his Hypothesis physica nova

(HPN) also known as the Theoria motus concreti, which he also dedicated in 1671, this

time to the Royal Society of London (A.VI.ii.221-257). In the HPN, Leibniz argues that

the laws of the TMA yield the Huygens-Wren laws of impact when taken together with

the contingent structure of the actual world. Central to Leibniz’s strategy is the idea that

all bodies in the actual world are elastic due to a pervasive aether, and are composed of

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discrete particles. Although mass continues to play no foundational role in his applied

system of physics, Leibniz is cleverly able to argue that larger bodies will generally be

more resistant to motion due to the larger number of particles from which they are

constituted. A pebble striking a planet will thus fail to have any significant effect

because the pebble’s velocity is propagated to the planet not at a single blow but particle

by particle; its motion is consequently diluted – and even reversed – as it is repeatedly

summed with the velocity of each subsequent particle.

In 1672, Leibniz was sent to Paris as part of a diplomatic mission where he stayed – short

trips aside – for the next four years. His time spent in the intellectually vibrant French

capital proved crucial to the development of his mature views in physics. While in Paris,

Leibniz gained an expert’s knowledge of the mathematics of his time, embarked on an

intensive study of Cartesian physics, and made contact with many of the leading natural

philosophers of his day.

In such fertile circumstances – which included tutoring by Huygens himself and direct

access to Descartes’s unpublished notebooks – Leibniz quickly fashioned his own

penetrating critique of Cartesian physics. As early as 1676,7 he had found what he

considered to be a fatal flaw in Descartes’s cornerstone conservation law, namely, that it

violates the principle of the equality of cause and effect. When Leibniz published

essentially the same objection in his Brief Demonstration of a Notable Error of

Descartes’s and Others Concerning a Natural Law in 1686, it sparked a now famous

dispute among natural philosophers that has become known as the vis viva controversy

(GM VI 117-119/L 296-298).8 For Leibniz the argument of the Brief Demonstration

marked not only an important event in his understanding of the Cartesian system, but also

a turning point in the development of his own distinctive physics.

On the one hand, the argument of the Brief Demonstration forced Leibniz to rethink the

foundations of his early theory of motion. For the charge of violating the principle of the

equality of cause and effect that he leveled against Cartesians spoke at least equally well

against the laws of the TMA and the HPN. The paradoxical result of the ideal laws of the

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TMA – that a tiny body moving with a given speed might impart the same speed to a

relatively enormous body – appeared, at least intuitively, as an example of a potentially

even greater increase in local force than anything allowed by Descartes’s rules of impact.

Conversely, according to the phenomenal laws of the HPN, since the combination by

impact of two conatuses can fall short, but never exceed, the strict sum of the conatuses

of the two individual bodies, the total conatus present in the world must naturally decline

over time – another violation of the equality principle. In short, in the argument of the

Brief Demonstration, Leibniz found a devastating criticism not only of Cartesian physics,

but also of Early Leibnizian physics.

On the other hand, Leibniz’s embrace of the principle of the equality of cause and effect

helped inspire his development of a series of ambitious positions that would collectively

serve as the moorings of his mature physics. The failure of Descartes’s conservation law

encouraged Leibniz to attach new significance to an alternative conservation principle

that he had learned from Huygens by 1669 at the latest.9 Leibniz came to see that if force

is taken to be equivalent to the quantity of vis viva – rather than the quantity of motion as

Descartes had implied – then, in cases such as those highlighted by the Brief

Demonstration, no violation of the principle of equality of cause and effect need be

tolerated. In subsequent works, including his Dynamica de Potentia et Legibus Naturae

corporea (1689), Essay of Dynamics on the Laws of Motion . . . (1690), and Specimen

Dynamicum (1695), Leibniz attempts to build on this discovery by suggesting further

implications not only for the conservation of force, but also for the laws of motion and

even the fundamental nature of physical bodies. Thus one can reasonably see in his

devastating critique of the Cartesian conservation law the seeds of much of what is

distinctive in his own mature physics.

Although Leibniz continued to refine, develop, and extend his views on the laws of

motion and impact, his work in the philosophy of physics was most prominently capped

by his famous correspondence with Samuel Clarke – Newton’s parish priest, intellectual

disciple, and possible mouthpiece.10 The controversy began when the Princess of Wales

passed along to Clark in 1715 a letter written by Leibniz decrying the decline of religion

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in England inspired by the rise of Newton’s natural philosophy. Taking place against the

backdrop of a bitter dispute over the priority of the calculus, Clark responded in

Newton’s defense and a series of five letters and replies were exchanged. Among the

many topics covered in the correspondence, the letters are best known for the opposing

views of space and time which they offer: Leibniz defending roughly the view that space

is an ideal system of relations holding between bodies, and Clarke defending the view

that space is something more like a container in which bodies are located and move. The

increasingly detailed and pointed exchange ended only with Leibniz’s death in 1716, with

Clarke, in the historical sense at least, having the last word.

2. Leibniz on matter

Leibniz’s views on the nature of matter are subtle and layered. From the time of his

youthful conversion in the Rosental, he remained broadly sympathetic to the explanatory

project of the new mechanical philosophy and its ambition to explain natural phenomena

in terms of matter and motion rather than in terms of a wide range of irreducible formal

natures. In spite of this general sympathy, however, he was also one of the most

penetrating critics of the dominant conceptions of matter in the mechanistic tradition.

This section looks first at Leibniz’s critiques of the two most important of those

conceptions before turning to his own positive account of the passive powers of bodies.

2.1. The critique of atomism

Although there was much disagreement over details, early modern atomists generally

affirmed that complex bodies are to be understood as being composed of naturally

indivisible material atoms moving about in an independently existing space. Such atoms

were commonly held to lack all but a few basic properties such as size, shape, and

possibly weight. While Leibniz himself was attracted to such a conception of body in his

early years, he eventually came to see atomism as deeply antithetical to his general

understanding of the natural world. Not surprisingly then, many of Leibniz’s arguments

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against atomism attempt to show how common atomist commitments conflict with

central principles of his mature natural philosophy.11

One such argument highlights a tension between Leibniz’s principle of continuity –

according to which “no change happens through a leap” – and the common assumption

that material atoms must be perfectly hard and inflexible. Taking for granted this

supposed perfect hardness and inflexibility, Leibniz argues that the collision of any two

atoms would lead to a discontinuous change in nature:

[I]f we were to imagine that there are atoms, that is, bodies of maximal hardness

and therefore inflexible, it would follow that there would be a change through a

leap, that is, an instantaneous change. For at the very moment of collision the

direction to the motion reverses itself, unless we assume that the bodies come to

rest immediately after the collision, that is, lose their force; beyond the fact that it

would be absurd in other ways, this contains, again, a change through a leap, and

instantaneous change from motion to rest, without passing through the

intermediate steps. (SD II.3/AG 132)

Leibniz’s suggestion here is that collisions between perfectly hard, inflexible atoms

would violate the principle of continuity since – being unable to flex or give – their

directions and speeds would have to change instantaneously upon contact. Tacitly

rejecting the possibility that continuity could be preserved by action at a distance, Leibniz

insists that continuity presupposes elasticity, and that elasticity in turn presupposes

having parts that can move relative to one another. Given that the original argument can

be invoked no matter how small the colliding bodies, Leibniz draws the inevitable

conclusion that all bodies must be elastic, and therefore must have parts, contrary to the

central thesis of atomism. He thus concludes that “no body is so small that it is without

elasticity, and furthermore ... there are no elements of bodies ... nor are there little solid

globes ... both determinate and hard. Rather, the analysis proceeds to infinity”

(SD.II.45/AG 132-133).

Another line of argument offered by Leibniz against material atomism highlights a

tension with what might be called his “principle of plentitude.” That principle, grounded

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in Leibniz’s broader theological and metaphysical views, maintains that existence itself is

good, and as a consequence God creates as much being as is consistent with the laws of

logic and his own moral goodness. Naturally, Leibniz sees the principle of plentitude as

being inconsistent with the existence of a barren void or interspersed vacua:

[T]o admit the void in nature is ascribing to God a very imperfect work ... I lay it

down as a principle that every perfection which God could impart to things,

without derogating from their other perfections, has actually been imparted to

them. Now let us fancy a space wholly empty. God could have placed some

matter in it without derogating, in any respect, from all other things; therefore, he

has actually placed some matter in that space; therefore, there is no space wholly

empty; therefore, all is full. (G VII.378/AG 332)

Interestingly, Leibniz uses the principle of plentitude not only to argue against the

atomists’ postulation of empty space, but also against the possibility of simple indivisible

atoms themselves. For, Leibniz argues, no matter how small one imagines atoms to be,

as long as they are reckoned internally simple and homogenous, the world could still

contain more variety, richness, and being if they were more finely divided. He thus

draws the characteristic conclusion that “The least corpuscle is actually subdivided in

infinitum and contains a world of other creatures which would be wanting in the universe

if that corpuscle were an atom, that is, a body of one entire piece without subdivision” (G

VII.377-378/AG 332).

A third, somewhat obscure, but especially intriguing line of argument takes off from a

position defended by Descartes in his Principles of Philosophy and touches on Leibniz’s

work on the so-called “Labyrinth of the Continuum.” In the Principles, Descartes had

argued that (i) the world is a plenum, (ii) all motion in a plenum must be circular, and (iii)

such circular motion presupposes the “infinite, or indefinite, division of the various

particles of matter” as they adjust and shift to accommodate a continuous flow around

bends and through narrows (AT VIIIA.59/CSM 1:239). Concerning these propositions,

Leibniz approvingly remarks, “What Descartes says here is most beautiful and worthy of

his genius, namely, that every motion in filled space involves circulation and that matter

must somewhere be actually divided into parts smaller than any given quantity” (L 393).

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But whereas Descartes had insisted only on the “indefinite” division of “merely some

part of matter” (AT VIIIA.60/CSM 239) Leibniz pushes for a stronger conclusion.

Dismissing Descartes’s cautious “indefinite division” as “being not in the thing, but in the

thinker,” he takes the argument to show that every part of matter is actually infinitely

divided (A VI.ii.264). Similarly, rather than apparently restricting the infinite division of

matter to some select moving parts, Leibniz maintains that every part of matter is

everywhere moving, and so the sort of accommodation envisioned by Descartes must

occur everywhere (A. 6.3.565f; 6.3.58f). Thus, for Leibniz, not only are some parts of

matter actually infinitely divided, but every part of matter is divided to infinity (Levey

1998, fn6)!

The actual infinite division of all matter is, of course, sufficient to rule out any standard

picture of material atomism since any body that might lay claim to being an indivisible

atom would itself be actually subdivided into smaller sub-bodies. But Leibniz is not

done. Even more radically, he argues that Descartes’s initial considerations lead not

“merely” to the conclusion that there are no smallest bodies, but furthermore show that

strictly speaking no determinate shapes can be ascribed to bodies at all.

[W]ith respect to shape, I uphold another paradox, namely, that there is no shape

exact and real, and that neither sphere, nor parabola, nor other perfect shape will

ever be found in body. . . . One will always find there inequalities to infinity.

That comes about because matter is actually subdivided to infinity. (translated,

Sleigh 1990, 112)12

Although how to best interpret Leibniz’s arguments for the ideality of exact shape – and

indeed how to best interpret his admittedly paradoxical conclusion – remains open for

debate, the general idea here seems clear enough. While mechanistic physics appeals to

exact shapes like cubes, spheres and hooks, the actual infinite division of matter reveals

that these can be at best approximations of the shapes of real bodies. For the shapes of

real bodies must – again at best – be infinitely complex since they are everywhere

divided into bodies actually distinct in virtue of their differing accommodating motions.

Although Leibniz’s conclusion is especially bold and enigmatic, it is nonetheless not

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untypical for him in suggesting that mechanism carefully thought through points towards

a radically different underlying metaphysical reality.

2.2. The critique of Cartesian corpuscularianism

Leibniz allied himself with Descartes and most later Cartesians in opposing material

atomism. Nonetheless he was equally concerned to rebut the Cartesian account of matter

according to which the whole essence of matter is extension – that is, the thesis that

matter is something like geometrical extension made concrete. Although Leibniz

characteristically offers a wide range of arguments against the Cartesian identification of

matter with extension, it will perhaps best serve our purposes to focus on three especially

important arguments all rooted in the guiding idea that the notion of extension is simply

too impoverished to provide an intelligible foundation for physics.

The first of these arguments presses on the relationship between the nature of matter and

the laws of motion. Leibniz, in effect, argues that the thesis that the whole essence of

matter is extension saddles Cartesians with a dilemma: they must either hold that the

laws of motion are grounded in the nature of extension, or that God acts directly to bring

about the lawful regularities that are observed in the world. Leibniz maintains that the

first horn is inconsistent with the true laws of motion, while the second horn leads to the

untenable postulation of perpetual miracles. In order to better appreciate Leibniz’s line of

reasoning here, it might be worth unpacking a little his thinking with respect to each horn

of the dilemma.

Behind Leibniz’s rejection of the first horn lies the assumption that if bodies were

nothing but matter, and matter nothing but extension, then bodies would be wholly and

essentially passive. But if bodies were wholly and essentially passive, Leibniz reasons,

they would necessarily be entirely indifferent to motion, and consequently they would

obey radically different laws of motion than the ones we actually observe:

If there were nothing in bodies but extended mass and nothing in motion but

change of place and if everything should and could be deduced solely from these

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definitions by geometrical necessity, it would follow ... that upon contact the

smallest body would impart its own speed to the largest body without losing any

of this speed; and we would have to accept a number of such rules which are

completely contrary to the formation of a system. (DM 21/AG 53-54)

There is, of course, some irony in Leibniz’s argument here. For, as we noted above, in

his early theory of motion Leibniz himself held that bodies offer no resistance to being

moved, and thus that it would take no more effort to budge a boulder than a bobby pin.

In his later works, however, Leibniz rejects the ideal laws of motion that he had espoused

in the TMA, while nonetheless continuing to hold that if matter were indifferent to

motion, then the laws of his abstract theory of motion would hold. Leibniz thus

concludes that since his early laws of motion do not govern actual bodies, bodies must

not be indifferent to motion, and thus must not be – as Descartes had maintained – simply

bits of geometrical extension made real.

All of this, of course, might seem to simply put greater pressure on the second horn of

Leibniz’s dilemma. For it might seem that it is open to Descartes’s defenders to maintain

that the laws of motion are the result of God’s direct decree, and that consequently they

do not – as Leibniz seems to assume – follow from the nature of body as such. (Indeed,

Descartes himself implies that the laws of motion follow inexorably not from the nature

of matter, but rather from the nature of God (AT VIIIA 61-62/CSM 1:240).) For Leibniz,

however, such a view is equivalent to conceding that the laws of nature hold by dint of

divine miracle since “properly speaking, God performs a miracle when he does

something that surpasses the forces he has given to creatures and conserves in them

(Letter to Arnauld, 30 April 1687, G II 93/AG 83). Thus, for Leibniz a miracle occurs

when a creature performs an action that does not flow from its own natural powers, and

so bodies could obey laws of motion not grounded in their own natural powers only

through the advent of a perpetual miracle. If Leibniz’s understanding of miracles is

granted, his second horn is thus much harder to resist than one might have otherwise

supposed.

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Leibniz’s second argument focuses on the ability of Cartesian physics to account for

qualitative variety in the world. The thesis that the whole essence of matter is extension

blocks Descartes from holding that matter is itself intrinsically diverse as well as from

supposing that matter is differentially distributed in empty space. Clearly recognizing the

constraints imposed by his reductive account of body, Descartes elegantly maintains that

all qualitative variety is to be grounded in the motion of bodies, that is, that “All the

variety in matter, all the diversity of its forms, depends on motion (Principles 2:23/CSM

1:232). Qualitative variety amongst bodies for Descartes thus appears to be grounded not

in the intrinsic natures of bodies or their parts, but rather in the way in which the parts of

bodies move relative to one another. To take an example friendly to Descartes, it might

be supposed that the qualitative differences between a block of ice, a puddle of water, and

a cloud are to be explained not by appeal to intrinsic differences in the elements that

constitute them, nor by appeal to the density of their elements in space, but rather in

terms of the relative speeds holding between those elements. One might thus imagine –

incorrectly of course – that the hardness of ice depends only on water molecules being at

rest with respect to one another, the fluidity of water only on water molecules moving

with respect to one another at a moderate speed, and the etherealness of a cloud only on

water molecules moving with an extreme relative speed.

Leibniz’s ingenious attack on this Cartesian model of qualitative variety proceeds in two

steps. The first step charges that motion alone is unable to account for qualitative variety

at an instant: since all qualitative variety in the Cartesian system depends on motion, and

there is no motion in an instant, it follows that in a Cartesian world there could be no

qualitative variety at an instant.13 The second step of Leibniz’s argument charges that if

the world is qualitatively homogenous at every instant, then it must be qualitatively

homogenous over time as well. For if the world is qualitatively undifferentiated at each

instant, then every instant will be qualitatively identical, and so the world as a whole will

not undergo any qualitative change as it passes from one instant to the next. To use an

anachronistic analogy, the two steps taken together imply that a Cartesian world would be

like a filmstrip whose every frame was blank, and thus whose projection would not only

be homogenous at each instant, but through time as well.

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A third argument, especially important in connection with Leibniz’s more general

metaphysics, takes issue with the implication that Cartesian bodies – or at least Cartesian

matter taken as a whole – might be on a metaphysical par with created minds. Again, it

might be helpful to see Leibniz’s thinking here as proceeding in two steps. First, drawing

on a principle integral to his metaphysics, Leibniz insists that independence and unity are

marks of true substances. The independence criterion suggests that we can usefully

distinguish between created substances that depend upon only God for their existence,

and derivative creatures that depend not only upon God but upon the existence of created

substances as well. The unity criterion implies a distinction between creatures that are

true, indivisible unities, and creatures that are mere accidental unities, or “aggregates.”

Leibniz maintains that only the former have a rightful claim to being genuine created

substances, since, as he famously puts it in a letter to Arnauld, “what is not truly one

being is not truly one being either” (G II.97/AG 86).

With his two criteria in hand, Leibniz argues that the bodies of Cartesian physics fail to

meet the standards for created substances twice over. For on the one hand, every

Cartesian body must be extended and divisible, and so must be composed of parts. But

this, Leibniz maintains, is sufficient by itself to show that Cartesian bodies are not

fundamental created entities: for if every Cartesian body is composed of parts, then by

the independence criterion, the parts will be more fundamentally real than the wholes

which they compose. No body whose essence is simply extension could therefore be a

genuine created substance according to Leibniz. On the other hand, according to

Descartes’s physics the only principle available to unite bodies – at least as they are

studied by physicists – is motion. But it is clear that Leibniz thinks that the sort of unity

that might be provided by mere common motion is insufficient for genuine, substantial

unity. Even if an army were to always march in perfect step, it would still not be a unity

per se, and thus would not be “truly one being either” (G II 76/AG 79). Thus measured

either by the standards of the independence criterion or the unity criterion, Cartesian

bodies can be shown to be not fully real in the deepest sense available to created beings.

In this thought, Leibniz sees not only an important contrast with human minds – which as

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partless simples are plausibly independent and unified – but also another indication that

mechanistic physics must itself rest on a deeper metaphysical reality quite different from

the “manifest image” with which we are familiar from everyday experience.

2.3. The passive powers of bodies

Much about Leibniz’s own positive conception of matter as it is studied by physics can

be gleaned from his criticisms of his predecessors. So, for example, his critique of

atomism already suggests that Leibnizian bodies must always be flexible or “soft” to

some degree,14 fill every region of the natural world, and be infinitely divisible, indeed,

be infinitely divided. Likewise, his critique of the Cartesian conception of matter implies

that the properties of bodies are not limited to their passive powers, that bodies must

always admit of intrinsic variety, and that ordinary physical objects such as desks and

chairs must be ontologically dependent upon a deeper level of metaphysical reality.

Nonetheless, we should be able to get an even better grip on Leibniz’s positive

conception of matter by looking more explicitly at four passive powers, or forces, which

he attributes to bodies as they are studied by the physicist.

(1) As we noted above, the physics of the TMA assigns to bodies no resistance to motion,

and thus predicts that under idealized conditions it should be no harder to move

something massive than something miniscule. In distancing himself from this early view,

Leibniz comes to argue that, in fact, matter “resists being moved through a certain

natural inertia it has . . . so that it is not indifferent to motion and rest, as is commonly

believed, but requires more active force for motion in proportion to its size” (G

IV.510/AG 161). In short, Leibniz came to maintain that bodies have an intrinsic power

resistant to motion, which he calls, following Kepler, “natural inertia.” Significantly,

Leibnizian natural inertia is a force which is opposed to motion itself, and not merely, as

Newton held, to changes in velocity. Thus, while Newton maintains that no active force

is required to keep a body moving with a constant velocity under idealized conditions,

Leibniz maintains that a body in motion in the absence of any countervailing active force

will naturally come to rest.

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(2) In addition to offering resistance to motion, Leibniz thought it essential to bodies that

they resist mutual collocation – they must be at least to some degree “solid” or

impenetrable. In attributing solidity to bodies, Leibniz rejects the Cartesian suggestion

that impenetrability simply follows from the property of a body’s being extended. For,

Leibniz argues, it could be the case that a body’s solidity is “due to a body's having a

certain reluctance – but not an unconquerable one – to share a place with another body,”

and thus he implies that it is, for example, at least conceivable that two bodies might be

forced to overlap in much the same way that we generally imagine bodies and regions of

space to mundanely coincide (NE, Book II, Ch. iv). Having distinguished solidity from

extension, Leibniz goes on to further distinguish solidity from hardness. He suggests that

whereas solidity concerns the ability of a body to resist being collocated with another

body, hardness concerns the ability of a body to resist changing its shape or structure.

Thus a body, according to Leibniz, might be perfectly solid (i.e. impenetrable) and yet

not hard, a point he illustrates by noting that “if two bodies were simultaneously inserted

into the two open ends of a tube, into which each of them fitted tightly, the matter which

was already in the tube, however, fluid [or “soft”] it might be, [the matter] would resist

just because of its sheer impenetrability [i.e. “solidity”]” (NE, Book II, Ch. iv).

(3) In addition to the (relatively) basic powers of natural inertia and solidity, Leibniz

recognizes two further derived passive powers of bodies. The first, which we might call

“firmness” or “cohesiveness,” concerns the ability of a body to resist being scattered or

torn into pieces. (An egg and a baseball differ not only with respect to their “natural

inertia” and impenetrability, but also with respect to their ability to hold themselves

together, as it were, under impact.) In the New Essays, Leibniz writes:

But now a new element enters the picture, namely firmness or the bonding of one

body to another. This bonding often results in one’s being unable to push one

body without at the same time pushing another which is bonded to it, so that there

is a kind of traction of the second body. Because of this bonding, there would be

resistance even if there were no inertia or manifest impetus. For . . . if space were

full of small cubes, a hard body would encounter resistance to its being moved

15

among them. This is because the little cubes – just because they were hard, i.e.

because their parts were bonded together – would be difficult to split up finely

enough to permit circular movement in which the position being evacuated by the

moving body would at once be refilled by something else. (NE Book II, Ch. iv)

Although Leibniz recognizes “firmness” as an important passive power attributable to

bodies, like many other mechanists, he was reluctant to posit the existence of primitive

attractive forces in nature (SD II.52/AG 136; see also Leibniz’s Fifth Letter to Clark,

paragraph 35/Alexander 66). He thus insists that firmness is a derived passive power of

bodies since “we should not explain firmness except through the surrounding bodies

pushing a body together” (SD II.52/AG 136).

(4) A final passive property of bodies might go without mention if not for the special

twist that Leibniz puts on it. Even while arguing against the Cartesian thesis that the

whole essence of matter is extension, Leibniz could hardly deny that bodies as they are

studied by physicists are extended, or if one prefers, that they have the property of

extension. Nonetheless, Leibniz intriguingly suggests that extension – far from

constituting the whole essence of matter – is not even a basic property of bodies. In an

informative piece dated to 1702, Leibniz writes:

... I believe that the nature of body does not consist in extension alone ... since

extension is a continuous and simultaneous repetition ... it follows that whenever

the same nature is diffused through many things at the same time, as, for example,

malleability or specific gravity or yellowness is in gold, whiteness is in milk, and

resistance or impenetrability is generally in body, extension is said to have place .

. . From this, it is obvious that extension is not an absolute predicate, but is

relative to that which is extended or diffused, and therefore it cannot be separated

from the nature of that which is diffused ... (G IV.393f/AG 251).

Leibniz thus grants that while extension is to be ascribed to the bodies of physics, it is not

to be treated as a basic or fundamental property of matter. Rather, extension itself is to

be understood as a repetition, or distribution, of more basic forces. Leibniz’s radical

suggestion would, in effect, turn the Cartesian understanding of mater on its head:

whereas, for example, Descartes attempted to explain solidity in terms of extension,

16

Leibniz proposes to explain extension in terms of solidity – a body isn’t solid because it

is extended, it is extended because it has the ability to exclude other bodies!

3. Leibniz’s Dynamics

One of the cornerstones of Leibniz’s mature physics is to be found in his thesis that the

bodies studied by physicists must be viewed not only in terms of passive powers and

motion, but also active forces. This section sketches Leibniz’s chief physical argument

for the postulation of active forces and then looks respectively at the roles that active

forces play in his mature physics and how they are related to his deeper metaphysics.

3.1. A Brief Demonstration

In his Principles of Philosophy, Descartes had argued that the quantity of motion in the

world remains constant, where the quantity of motion of a body is determined by its

speed times its size. So, for example, at Principles 2:36, Descartes argues that “if one

part of matter moves twice as fast as another which is twice as large, we must consider

that there is the same quantity of motion in each part; and if one part slows down, we

must suppose that some other part of equal size speeds up by the same amount” (AT

VIIIA 61/CSM 240). This general conservation principle, which Descartes takes to be

grounded in God’s immutable nature, in turn serves as the foundation for his three laws

of motion as well as his rules of impact.

In the short piece fully and tellingly entitled A Brief Demonstration of a Notable Error of

Descartes and Others Concerning a Natural Law, According to which God is Said

Always to Conserve the Same Quantity of Motion; a Law which They also Misuse in

Mechanics, Leibniz publicly attacked Descartes’s conservation principle and thereby the

foundations of Cartesian physics (GM VI.117-119/L 296-302). The central argument of

the Brief Demonstration is ingenious and elegant, but only deceptively straightforward,

and Leibniz continued to develop and expand upon its central theme for many years after

its initial public presentation in 1686.15

17

At the heart of Leibniz’s main line of argument lie four premises: (1) The amount of

force a body acquires in virtue of falling from a certain altitude is equal to the amount of

force that would be required to raise that same body to the same altitude. (2) The amount

of force a one pound body acquires by falling from a height of four meters is equal to the

amount of force a four pound body acquires by falling from a height of one meter. (3)

Galileo’s Law: the distance traveled by a falling body is directly proportional to the

square of the time it falls (i.e. d = at2 where a is a constant); so, to illustrate, a falling

body traveling a distance of one meter in one second will travel four meters in two

seconds, nine meters in three seconds, and sixteen meters in four seconds, etc.16 (4) The

total amount of force in the world is conserved both locally and globally with the result

that there is always as much force in a cause as in its effect.

Using these four premises, Leibniz attempts to establish three principal conclusions.

First, he argues that Descartes’s quantity of motion is not an adequate measure of force.

To see this, suppose that a four pound brick falls from a height of one meter in one

second. Its quantity of motion = 1 m/s X 4lbs = 4 units. By Galileo’s Law it follows that

a one pound brick dropped under similar circumstances will travel four meters in two

seconds (since the distance traversed is proportional to the square of time). The quantity

of motion of a one pound brick falling through a distance of four meters therefore = 2 m/s

X 1lbs = 2 units. Since by the second premise the force of the two bodies is equal, but

their quantities of motion are unequal, Leibniz concludes that quantity of motion is an

inadequate measure of force.

Second, Leibniz is similarly able to argue that the quantity of vis viva (mv2) is an

adequate measure of force. For, the quantity of vis viva of the four pound brick falling

from a height of one meter = (1m/s)2 X 4 lbs. = 1 m/s X 4lbs = 4 units, and the quantity

of vis viva of the one pound brick falling from a height of four meters = (2 m/s)2 X 1 lbs

= 4 m/s X 1 lbs. = 4 units. Thus if force is measured by the quantity of vis viva rather

than the quantity of motion the equality of the force acquired by the one pound body

18

during its four meter fall will be equal to the force acquired by the four pound body

during its one meter fall.

Third, Leibniz is further able to argue that the quantity of vis viva rather than the quantity

of motion is conserved. To see this, suppose that a one pound brick, having been raised

to a height of four meters, falls on the end of a teeter-totter imparting a motion to a four

pound brick on the other end. Now if the amount of active force is conserved – if there is

as much force in the effect as in the cause – then (by the first premise) the brick should be

raised to a height of one meter. That being the case, however, it is clear from the

preceding demonstrations that the quantity of vis viva but not the quantity of motion will

be conserved (since the quantity of motion has decreased from 4 units to 2 units). 17

It is important to note that the fourth premise plays a crucial role not only in Leibniz’s

proof, but also in the larger picture of what he takes his proof to show. Early in his

career, Leibniz took the principle of the equality of cause and effect to be a necessary

truth. Later, however, he came to hold that it is only hypothetically or morally necessary

– a contingent feature that must nonetheless be found in the best of all possible worlds.

From the status of the principle of equality of cause and effect, Leibniz in turn drew two

important conclusions, one metaphysical, one methodological. With respect to

metaphysics, he took the contingent status of the principle of equality of cause and effect

to be evidence that the laws of motion and force are themselves contingent, and thus he

argued that the elegance of the laws of nature provides evidence of God’s benevolent

design of the universe. With respect to methodology, he took the derivation of the

conservation of vis viva as confirmation of the utility – indeed the practical necessity – of

considerations of divine teleology in making scientific discoveries even in the domain of

physics.

3.2. Physics and active forces

In his important summary of his dynamics – generally known by its Latin title “Specimen

Dynamicum” – Leibniz distinguishes between two kinds of active force. The first, and

19

most important, is of course vis viva itself. In the context of his physics, vis viva, or

“living force,” represents for Leibniz a measure of a body’s ability to bring about effects

in virtue of its motion. It is an active force which allows a moving body to, say, raise

itself up to a given height or impart a motion to a slower body. (Intuitively, we might

think of it as the “force” that a bowling ball, for example, has in virtue of its falling with

a given speed, or the power a baseball has once released from a pitcher’s hand.) As note

above, Leibniz maintains that vis viva is conserved both locally in particular cases of

impact, and globally in the created world taken as a whole. The living force a body

expends in raising itself to a given height must therefore be equal to the living force it

gains by falling from that height; the vis viva a body is able to transfer to another body

through impact must be equal to the measurement of vis viva it loses during that impact.18

The fact that collisions between actual bodies are never perfectly elastic represents a

prima facie objection to the conservation of vis viva. Although the law looks tolerably

accurate if we consider the collision of, say, two billiard balls, or two steel spheres, it

seems woefully off if we consider, say, the collision of two lumps of clay, or two scoops

of ice cream. Indeed, in non-elastic collisions it appears that much – even all – of the

active force of the colliding bodies as measured by mv2 may be lost. In addressing this

fairly obvious worry, Leibniz returns to the idea that any body – no matter how small –

must be composed of infinitely many smaller bodies. He then argues that the apparent

loss of vis viva in cases of inelastic collision is to be attributed to the fact that living force

has been transferred to the smaller parts of the gross bodies in a way that does not

contribute fully to the motion of the whole. It is thus lost in the sense that it does not

contribute to the motion of the larger bodies, but it is nonetheless conserved in the deeper

sense that it is still present in the motion of the smaller bodies from which the larger

bodies are constituted. Leibniz therefore insists, “when the parts of the bodies absorb the

force of the impact, as a whole, as when two pieces of rich earth or clay come into

collision . . . when, I say, some force is absorbed by the parts, it is as good as lost . . . But

this loss . . . does not detract from the inviolable truth of the law of the conservation of

the same force in the world. For that which is absorbed by the minute parts is not

20

absolutely lost for the universe, although it is lost for the total force of the concurrent

bodies” (GM VI.230/Langley 670).

In his Specimen, Leibniz terms his second postulated active force “vis mortua” or “dead

force,” although it also appears in connection with the titles, “solicitation,” “conatus,”

and “impetus.” In spite of their correlative labels, the notion of dead force appears to be

less carefully worked out by Leibniz than the notion of living force. In the Specimen he

tells us:

One force is elementary which I also call dead force, since motion does not yet

exist in it, but only a solicitation to motion, as with . . . a stone in a sling while it

is still being held by a rope. . . . An example of dead force is centrifugal force

itself, and also the force of heaviness or centripetal force, and the force by which

a stretched elastic body begins to restore itself. But when we are dealing with

impact, which arises from a heavy body which has already been falling for some

time, or from a bow that has already been restoring its shape for some time, or

from a similar cause, the force in question is living force, which arises from an

infinity of continual impressions of dead force. (GM VI.238f/AG 121f)

From this passage and surrounding texts, we can glean three central ideas Leibniz

associates with the notion of dead force. First, just as we might think of vis viva as a

measure of the force a moving body has in virtue of its being in motion, so we might

think of vis mortua as a measure of the force a body has to bring about motion even while

at rest, or at an instant. Second, living force is related to dead force by an infinite

summation; Vis viva is, as it were, an infinite accumulation of individual instances of vis

mortua.19 Third, dead force is thus more immediately related to the study of statics than

to the study of dynamics, and Leibniz repeatedly suggests that Cartesians have indeed

been misled into maintaining the conservation of quantity of motion, by confusing the

laws of statics with the laws of dynamics.

On some persisifications of Leibniz’s texts, the measure of dead force approaches very

closely the measure of force more clearly at work in Newton’s Principia.20 It is therefore

worth noting that although he could hardly deny the technical achievement of Newton’s

21

masterpiece, Leibniz was nonetheless eager to distance his own thinking about force from

that of his great rival’s. In a polemical piece whose title has been translated as “Against

Barbaric Physics,” Leibniz writes:

It is, unfortunately, our destiny that, because of a certain aversion toward light,

people love to be returned to darkness. . . . That physics which explains

everything in the nature of body through number, measure, weight, or size, shape

and motion, and so teaches that, in physics, everything happens mechanically, that

is, intelligibly, this physics seems excessively clear and easy. . . . It is permissible

to recognize magnetic, elastic, and other sorts of forces, but only insofar as we

understand that they are not primitive or incapable of being explained, but arise

from motions and shapes. However, the new patrons of such things don’t want

this. And it has been observed that in our own times there was a real suggestion

of this view among certain of our predecessors who established that the planets

gravitate and tend toward one another. It pleased them to make the immediate

inference that all matter essentially has a God-given and inherent attractive power

and, as it were, mutual love, as if matter had senses, or as if a certain intelligence

were given to each part of matter by whose means each part could perceive and

desire even the most remote thing. (G VII.337-339/AG 312=313; see also

Leibniz’s fourth paper to Clark, paragraph 45/Alexander 43)

The grounds for Leibniz’s negative reaction to Newton’s conception of force, and

specifically his apparent postulation of a universal force of gravitation, are various and

complex. One especially important theme, however, indicated in the passage above,

concerns what conception of force should be allowed to operate in the study of physics.

On the negative side, Leibniz thought that by postulating what he understood to be an

irreducible force holding between bodies and acting at a distance, Newton had abandoned

the intelligible explanations of the mechanical philosophy, and returned to uninformative

scholastic accounts that rested content with the postulation of primitive powers. On the

positive side, Leibniz thought that by treating forces as inherent powers of bodies tied

inextricably to their ability to move and be moved, his own conception of force could

preserve the intelligibility that was the great hallmark of mechanism, while nonetheless

improving upon the work of those such as Huygens, Descartes and Galileo. For, by

22

Leibniz’s lights, the active and passive forces he postulates not only render physics itself

more accurate and consonant with reason but at the same time set the stage for its

intelligible grounding in his own deeper metaphysics. In this way, Leibniz’s

understanding of dynamics becomes inextricably bound up with his more thoroughly

metaphysical views to which we must now briefly turn.

3.3. Forces and Metaphysics

If the most important distinction of the Specimen Dynamicum with respect to physics is

between active and passive forces, its most important distinction with respect to

metaphysics is between what Leibniz calls “primitive” and “derivative” forces. He holds

that while derivative forces are of primary interest to the working physicist, they are

metaphysically secondary to primitive forces, and he speaks of the former as being

“modifications” or “limitations” of the latter (GM VI.236/AG 119). Although the topic

of the relationship between derivative and primitive forces quickly takes us away from

Leibniz’s treatment of physics and into the heart of his metaphysics, it should be

worthwhile to at least call attention to this interface where what we are inclined to think

of as Leibniz’s physics so clearly bumps up against – indeed overlaps with – what we are

inclined to think of as his metaphysics.

It is widely accepted that Leibniz’s primitive forces are supposed to serve as the

intelligible metaphysical grounds for the forces that are of concern in physics, and more

specifically that active derivative forces are to be grounded in active primitive forces

while passive derivative forces are to be grounded in passive primitive forces.

Furthermore, there is no denying that Leibniz sees his distinction between active and

passive primitive forces as being in some way analogous to the distinction between

Aristotelian form and matter. Thus in the Specimen, he tells us that “primitive [active]

force (which is nothing but the first entelechy) corresponds to the soul or substantial form

[of the scholastics]” while “the primitive force of being acted upon or of resisting

constitutes that which is called primary matter in the schools” (GM VI.236-237/AG 119-

23

120). What has remained less certain is the question, “Where, as it were, does Leibniz

think active and passive primitive forces are located?”

The answer to this question is complicated by Leibniz’s embrace of the seemingly

fantastic thesis that the gross bodies studied in physics are to be understood as being

composed of infinitely many organisms:

I am very far removed from the belief that animate bodies are only a small part of

the others. For I believe rather that everything is full of animate bodies . . . and

that since matter is endlessly divisible, one cannot fix on a part so small that there

are no animate bodies within, or at least bodies endowed with a basic entelechy or

. . . with a vital principle, that is to say corporeal substances, about which it may

be said in general of them all that they are living. (G II.118/Garber 1998, 294)

Thus, according to Leibniz, the block of wood that is pushed up an inclined plane, or the

ball dropped from a leaning tower must be composed of infinitely many organisms, with

each of those organisms being composed of further organisms, etc. These organisms

therefore must in some way be counted as more ontologically basic than the gross bodies

which they compose.

Drawing on his commitment to panorganicism, his talk of corporeal substances, and his

explicit invocation of the Aristotelian notions of form and matter, it has been suggested

that in his middle years Leibniz held to an essentially Aristotelian ontology according to

which the fundamental level of reality is occupied by organisms composed of substantial

forms and matter. As Daniel Garber – the reading’s most influential defender - puts the

view, it is “a world whose principal inhabitants are corporeal substances understood on

an Aristotelian model as unities of form and matter, organisms of a rudimentary sort, big

bugs which contain smaller bugs, which contain smaller bugs still, all the way down

(1985, 29). According to this picture, the derivative forces studied by physicists would

be grounded in the active and passive natures of the organisms from which they are

composed. Organisms strive and are acted upon at the most basic level of ontology, and

those strivings and passions serve as the ground for the active and passive derivative

forces examined, for example, in collisions between elastic spheres. Although Leibniz’s

24

panorganicism remains striking, this picture promises a rather elegant account of the

foundations of Leibniz’s physics, and squares tolerably well with many of his texts,

especially from his middle years.

The “bugs all the way down” model is not, however, the only model of fundamental

metaphysics that one can find in Leibniz’s writings. In his most mature works, Leibniz

suggests that at the deepest level of ontology, we find only truly simple, mind-like

substances, or “monads.” Although strictly indivisible, Leibniz insists that monads can

nonetheless be thought of as unities of form, insofar as they are active, and matter, insofar

as they are passive. According to this metaphysics, the gross bodies studied by physicists

are at least twice removed from monadic reality. Gross bodies are grounded in

organisms, or “corporeal substances,” while corporeal substances are themselves

grounded in simple substances, or “monads.” This same chain of ontological

dependence, however, likewise holds for forces: the active forces attributed to gross

bodies are “well-founded” in the active forces attributable to organisms, which are “well-

founded” on the active forces of monads, just as the passive forces attributable to gross

bodies are “well-founded” in the passive forces attributable to organisms, which in turn

are “well-founded” on the passive forces of monads. A full understanding of these

grounding relations would of course require a more detailed explication of the relations

between monads, corporeal substances, and gross bodies, as well as a fuller account of

the “grounding” relations involved. Nonetheless, even this rough sketch should suffice to

indicate the broad outlines of Leibniz’s commitment to founding the active and passive

forces he postulates as part of his physics in the most fundamental level of reality which

he postulates as part of his most mature metaphysics.

4. Leibniz on the Laws of Motion

The laws of motion held a privileged place in the mechanical philosophy of the early

modern period. Together with bodies they served as the chief explanatory postulates of

the new physics. It is thus not surprising that Leibniz held strong views concerning the

justification of the laws of motion, the content of those laws, and their implications for

25

the epistemology and metaphysics of motion. In thinking about Leibniz’s positive views

on the laws of motion, it might prove once again helpful to first look at his reasons for

being dissatisfied with the account offered by Descartes.

4.1. Refuting Cartesian Laws of Motion

In addition to maintaining that the quantity of motion in the world is conserved,

Descartes held that material bodies are governed by three laws of motion and seven rules

of impact (AT VIIIA 62-71/CSM 1:240-245).21 The first two laws in the Principles

concern the movements of bodies in isolation, and taken together constitute Descartes’s

formulation of the principle of inertia. The third law in the Principles regulates the

behavior of bodies colliding under idealized conditions:

When a moving body comes upon another, if it has less force for proceeding in a

straight line than the other has to resist it, then it is deflected in another direction,

and retaining its motion, changes only its determination. But if it has more, then

it moves the other body with it, and gives the other as much of its motion as it

itself loses. (AT VIIA.65/CSM 1:242)

Significantly, the third law thus distinguishes between two kinds of cases that Descartes

believes to be importantly different. The first kind of case occurs when a moving body

with a given power for proceeding collides with another body that has a greater power for

resisting. Descartes maintains that in such situations the moving body’s direction or

determination is altered, but that nonetheless the quantity of motion for each of the

colliding bodies remains the same. The second kind of case occurs when a moving body

with a given power for proceeding collides with another body that has a lesser power for

resisting. Descartes maintains that in such situations, the moving body’s determination

remains the same, and that the moving body carries the resisting body along with it in

such a way that their shared total quantity of motion remains the same. The seven rules

further spell out the implications Descartes takes his laws of motion to have for more

specific cases of impact.

26

Although Descartes’s laws of motion go beyond his conservation law in placing further

constraints on the directions of bodies, they nonetheless presuppose that the total quantity

of motion of bodies remains the same. Leibniz is therefore able to argue that Descartes’s

laws of motion are untenable because they would lead to violations of the conservation of

force as measured by mv2. Thus, in a letter to Bayle of 1687, Leibniz introduces, as an

example, a case where a ball B moving with 100 degrees of speed collides head on with a

ball C moving with 1 degree of speed. Before the collision the two balls collectively

have 101 units of quantity of motion as measure by ms, and 10,001 units of vis viva as

measured by mv2. According to Descartes’s Third Rule, after the collision the two balls

should move together in the direction of B with a speed of 50 ½ units. Their quantity of

motion will be 101 units as dictated by Descartes’s conservation law, but their quantity of

vis viva will be 2(50 ½)2 or 5100 ½ units. Leibniz thus objects that Descartes’s laws of

motion would violate the conservation of vis viva and with it the principle that the whole

cause must be equal to the entire effect (G III.46).22

In addition to attacking Descartes’s laws of motion on the basis of the principle of the

equality of cause and effect, Leibniz also maintains that they would violate another

“metaphysical” principle, namely, the principle of continuity according to which

continuous changes in inputs should lead to continuous changes in outputs.23 Thus, for

example, in a letter to Malebranche of July 1687, Leibniz writes:

I shall not repeat here what I have said before about the other source of

[Descartes’s] errors, in taking the quantity of motion for the force. But his first

and second rules, for example, do not agree with each other. The second says that

if two bodies B and C collide in a straight line and with equal velocities, but B is

but the least amount greater than C, C will be reflected with its former velocity,

but B will continue its motion. But according to his first rule, if B and C are equal

and collide in a straight line, both will be reflected and return at a velocity equal

to that of their approach. This difference in the outcome in these two cases is

unreasonable, however, for the inequality of the two bodies can be made as small

as you wish, and the difference between the assumptions in the two cases, that is,

the difference between such inequality and a perfect equality, becomes less than

27

any difference; therefore according to our principle, the difference between the

effects or consequences ought also to become less than any given difference. (G

III.53/L 352)

Leibniz’s argument here is that as the sizes of the bodies B and C change continuously

from inequality to equality the effects of that change should be continuous as well.

Descartes’s rules of impact, however, would have an infinitesimal change in input – a

change from B’s being infinitesimally bigger than C, to C’s being equal in size to B –

result in a leap of output – from C rebounding while B remains stationary, to both B and

C rebounding. Leibniz maintains that Descartes’s rules of impact must therefore be false

since they violate the principle of continuity.

It is worth noting that, for Leibniz, the significance of the principle of continuity runs

deeper than providing yet another reason for thinking that the Cartesian laws of motion

are flawed.24 For him the principle of continuity is a contingent principle of order

grounded not in brute necessity, but in divine benevolence – a discontinuous world would

not be impossible, but merely sub-optimal. As such, Leibniz takes it to yield further

support for the metaphysical and methodological points noted above in connection with

his “proof” of the conservation of vis viva. Metaphysically, Leibniz takes the principle of

continuity to support the claim that the true laws of motion are contingent since they

follow not from God’s immutable nature or eternal truths, but rather from God’s wisdom

and benevolence. Methodologically, he takes it to support once again his view that the

most promising route to the discovery of nature’s secrets is neither blind empiricism, nor

deductive rationalism, but a combination of observation, pure reason, and reflection on

the constraints imposed by considerations of the best.

4.2. A system of conservation principles

From the perspective of kinematics, there is little new in Leibniz’s positive account of the

laws of motion. As we have seen, Leibniz’s earliest systematic physics sought to

accommodate the laws of impact as developed by Huygens and Wren by showing how

those results might be derived from more fundamental laws of motion and the structure of

28

the actual world. In his later work, this strategy is replaced by attempting to show how

essentially those same (“concrete”) laws of motion may be derived from a set of three

conservation laws. Although Leibniz’s contribution to the kinematics of motion and

impact was thus not revolutionary, he nonetheless had the good sense to champion the

best accounts going, and made them his own through his elegant derivations and by

relating them to the broader themes of his dynamics and metaphysics.

One of the conservation laws that Leibniz takes to govern the behavior of material bodies

is what he calls the conservation of relative velocity (GM VI.227/Langley 667).

According to this principle, two perfectly elastic bodies will maintain the same relative

velocity with respect to their common center of gravity before and after collision.25 That

is, letting A and B represent two elastic balls involved in a head on collision:

Velocity A before – Velocity B before = Velocity B after – Velocity A after

or more simply:

VA before – VB before = VB after – VA after

Leibniz suggests that the conservation of relative velocity is rooted in the conservation of

the ability of the two colliding bodies to perform work on one another. The guiding idea

here seems to be that the ability of, say, the balls A and B to act on one another in virtue

of their relative motion, should not be lost at all (at least in idealized cases), so that after a

collision they should continue to have the same ability to act on one another as they had

before they collided, and thus that they should have the same relative velocity after the

collision as they had before.26

A second conservation law Leibniz calls the conservation of quantity of progress.

According to it, two bodies will maintain the same relative progress – where progress is

measured by the quantity of “mass” times velocity – before and after collision (GM

29

VI.227/Langley 667). Thus, letting A and B once again represent two balls involved in a

head on collision:

(Mass A x VA before) – (Mass B x VB before) = (Mass B x VB after) – (Mass A x VA

after)

or more simply:

MAVA before – MB x VB before = MB x VB after – MA x VA after

Leibniz’s conservation of progress is closely related to the Cartesian law of conservation

of quantity of motion (and the more familiar law of the conservation of momentum). As

Leibniz is at pains to emphasize, however, his law differs from the Cartesian law at least

in that it traffics in “signed” velocities rather than scalar speeds. Leibniz is thus able to

maintain that although “it will be found that the total progress is conserved, or that there

is as much progress in the same direction before or after the impact” nonetheless the

quantity of motion, as measured simply by speed times “mass” is not conserved (GM

VI.217/Langley 658).

Leibniz’s third law applies the conservation of vis viva to cases of impact. It therefore

maintains that the “motive” or “living” force, as measured by “mass” times velocity

squared for a pair of bodies is the same before and after collision (GM VI.227/Langley

667-668):

MA (VA)2 before – MB x (VB)2

before = MB x (VB)2 after – MA x (VA)2 after

Unlike the quantity of progress, but like relative velocity, the quantity of vis viva appears

to be conserved only in elastic collisions. At the macro-level, when a lump of soft clay

strikes another lump of soft clay, their momentum is conserved but kinetic energy is lost.

As noted above, rather than abandon the universality of his third law, Leibniz suggests

instead that energy is conserved but redistributed to the minute parts of which the clay is

30

composed. In this way, he tells us “that which is absorbed by the minute parts is not

absolutely lost for the universe, although it is lost for the total force of the concurrent

bodies” (GM VI.231/Langley 670).

In his Essay on Dynamics, Leibniz shows how from any two of his conservation laws the

third law may be derived. This might suggest that Leibniz sees all three as being on a par

with one another. In fact, however, he insists that the conservation of vis viva is more

fundamental than the conservation of relative velocity or common progress. His reasons

for privileging mv2 in this way are far from clear, but he must have recognized them to be

necessarily metaphysical. Although the measurement of mv2 is relative to a choice of

reference frame, Leibniz probably thought that it had a better claim to tracking an

intrinsic property of bodies in light of the considerations he raises in his Brief

Demonstration. For Leibniz, vis viva is a force attributable to particular bodies in virtue

of which they are able to perform work on other bodies – or as in the case of a pendulum

- on themselves; he thus held the conservation of vis viva to be the most fundamental,

overarching conservation law for the physical world.

4.3. Absolute or relative motion?

As far as kinematics is concerned, Leibniz, like most of his contemporaries, accepted the

observational under-determination of constant linear motion. That is to say, he granted

that – if we bracket considerations of force – there’s no saying which of two bodies

moving relative to each other with a constant velocity is really moving. So, for example,

Leibniz would have conceded that we can’t tell just by looking whether Train A or Train

B is really moving, even if they are moving with a constant velocity relative to one

another. This observational under-determination or “invariance” – often called Galilean

invariance – is still accepted today, although the assumption that it makes sense to speak

of any body as “really” moving with a constant velocity independently of an arbitrarily

chosen frame of reference is not.

31

Continuing to restrict ourselves to kinematics, Leibniz appears to embrace something

even stronger than Galilean invariance. He suggests that not only is constant linear

motion observationally underdetermined, but furthermore that “If we consider change in

position alone, or that which is merely mathematical in motion,” then all motion is

observationally underdetermined (A VI.iv.2017/Lodge 2003, 278). That is to say, he

seems to accept that not even accelerations – changes in direction or speed – can be

detected by empirical observation. Thus, for example, he writes:

The law of nature concerning the equivalence of hypotheses that we established

earlier, namely, that a hypothesis which once corresponds to the present

phenomena will always correspond to the subsequent phenomena in that way, is

not only true for rectilinear motion but more generally, however the bodies act on

one another, just as long as the system of bodies is isolated from others, or no

external agent comes along. (GM VI.507/Lodge 2003, 280)

Even if we grant that we cannot tell whether it is our train that is gliding along with a

constant velocity or the train that we see through the window, we might nonetheless

maintain that we can tell – by, say, the sudden jerk we feel – if our train has just

accelerated by increasing its speed, or – because we feel ourselves pushed against the

wall – that it is rounding a sharp corner. Leibniz, however, seems to deny this,27 insisting

instead that “no eye, wherever in matter it might be placed, has a sure criterion for telling

from the phenomena where there is motion, how much motion there is and of what sort it

is, or even whether God moves everything around it, or whether he moves that very eye

itself” (AG 91).28

From the observational under-determination of all motion considered kinematically,

Leibniz infers that if there were nothing more to motion than change of position relative

to other bodies, then there would be no real or genuine motion in the world at all.29 He is

thus committed to maintaining that if there were nothing more to motion than relative

change of position, then, since motion could be ascribed with equal right to, say, Train A

or Train B, then there would be no fact of the matter as to whether Train A or Train B is

moving, and thus it would make no sense to say that either Train A or Train B is moving.

The suggestion that motion be treated as irreducibly relational – so that motion could be

32

ascribed to Train A relative to one reference frame, and to Train B relative to another

reference frame – would have had no attraction for Leibniz, who consistently denied that

there are any genuine external relations (i.e. relations that do not wholly supervene on

intrinsic properties).

Accepting the observational under-determination of motion understood as mere change of

relative position, and finding absurd the consequence that there is no genuine motion in

the world, Leibniz denies the premise that there is nothing more to motion than relative

change of position. He thus maintains that rather than grant that there is no real motion,

“in order to say that something is moving, we will require not only that it change its

position with respect to other things but also that there be within itself a cause of change,

a force, an action” (G IV.396/L 393). Intuitively, Leibniz’s suggestion is that genuine

motion requires, in addition to relative change of place, a cause of that relative change.

To return to our earlier example, if Train A moves relative to Train B, we should say,

according to Leibniz, that Train A really moves if and only if, it is the active cause of

their relative motion. Since, for Leibniz, being an active cause, or locus of force, is a

non-relativistic property attributable to individual bodies, it is in principle capable of

breaking the “equivalence of hypotheses” and thus grounding true or genuine motion.

While force, however, thus provides the necessary metaphysical grounds for the

existence of genuine motions, it does not, at least as far as the physicist is concerned,

solve the empirical issue of which bodies can be ascribed genuine motion (Garber 1995,

307; but see also Lodge 2003). For, as we have seen, the ascription of force – of vis viva

or mv2 – is itself empirically relative to a frame of reference. The postulation of force

makes genuine motion possible; it does not tell us which bodies are genuinely in motion

and which are moved merely relatively. In practice Leibniz thus counsels that – as in

astronomy – “one can hold the simplest hypothesis (everything considered) as the true

one” (GM II 184/AG 308). Although the metaphysician can rest assured that true motion

must be absolute, the physicist must therefore be content to work with relative motions

and simplifying assumptions.

33

5. Leibniz on Space and Time30

In his Principia, Newton had suggested that the absolute motion of bodies is to be

defined relative to absolute space and time, and to be discovered by its properties, causes

and effects.31 Leibniz, as we have just seen, opposed such a view, holding instead that

true motions are to be defined with respect to the active forces that he took to be inherent

in truly moving bodies. This disagreement over the nature of true motion surfaced more

explicitly in their disagreement over the nature of space and time in the Leibniz-Clark

correspondence. In the five letters he managed to write before his death, Leibniz

succeeded in articulating not only his reasons for opposing what he took to be Newton’s

conception of absolute space and time, but also sketching an alternative picture according

to which they are to be understood as abstract systems of relations.

5.1. Against Absolute Space and Time

In his correspondence with Leibniz, Clarke defends what has become known as an

absolute theory of space and time. The version championed by Clarke on Newton’s

behalf might briefly be characterized for our purposes as having four central theses.

First, space and time are logically and metaphysically prior to physical bodies and events.

That is to say, although space and time could exist even if there were no physical bodies

or events, the existence of things like planets and flashes could not exist without space

and time. Second, physical bodies and events exist within space and time – the beach ball

is collocated with a region of space equal to its volume; the explosion endures through a

determinate measure of absolute time. Third, although we may distinguish regions, or

“parts,” of space and time, neither space nor time strictly speaking are divisible since no

region of space or time could be separated, or “pulled apart,” from any other region.

Fourth, ontologically speaking, space and time may be identified with attributes of God:

infinite space just is the attribute of God’s Immensity, while infinite time is the attribute

of God’s Eternity.

34

Leibniz introduces three main lines of attack against the Clarke-Newton conception of

absolute space and time. The first line focuses on the suggestion that space and time

might be identified with the divine attributes, and on Newton’s claim – made in his

Optics32 – that space is, as it were, the sensorium of God. Leibniz, at root, argues that

such claims are deeply misleading at best, heretical at worst. Thus, for example, against

the suggestion that space might be identified with God’s Immensity, he writes to Clarke,

“if space is a property of God . . . space belongs to the essence of God. But space has

parts: therefore there would be parts in the essence of God” (Fifth Paper, paragraph 43;

G VII.399/Alexander 68). Likewise, he argues that if time were identified with God’s

Immensity, then we would have to say that since things are in time, they are in God’s

Immensity, and thus in God’s essence, “Strange expressions; which plainly show, that the

author [i.e. Clarke] makes a wrong use of terms” (Fifth Paper, paragraph 44; G

VII.399/Alexander 68).33 Finally, from his first letter on, Leibniz seizes on what he takes

to be the impious implication of Newton’s suggestion that space might in some sense be

considered the seat of divine perception or cognition, writing, “Sir Isaac Newton says,

that space is an organ, which God makes use of to perceive things by. But if God stands

in need of any organ to perceive things by, it will follow, that they do not depend

altogether upon him, nor were produced by him” (First Paper, paragraph 3; G

VII.352/Alexander 11).

A second, more philosophical, line of attack pivots on Leibniz’s commitment to the

Principle of Sufficient Reason (PSR). In the present context we can understand PSR as

demanding that there be some reason for God’s creating the world in one way rather than

another since “A mere will without any motive, is a fiction, not only contrary to God’s

perfection, but also chimerical and contradictory” (Fourth Paper, paragraph 2; G VII.371-

2/Alexander 36). Leibniz argues that if the Principle of Sufficient Reason is granted, the

apparent possibility of absolute space and time can be undermined. For on the

supposition that God creates the world in an infinite, homogenous, absolute space,

Leibniz argues, there could be no reason for his creating the world oriented in one way

with respect to that space rather than another way – that is, there could be no reason to

prefer the world situated in one way rather than, say, rotated in space by ninety degrees.

35

Since the supposition of absolute space thus leads to a violation of PSR, the supposition

itself must be rejected as chimerical or confused according to Leibniz. Similarly, on the

supposition that God creates the world in an infinite, homogenous, absolute time, there

could be no reason for God’s creating the world at one time rather than at another time.

Again, since the supposition leads to a violation of PSR, Leibniz maintains that the

supposition itself must be rejected as chimerical or confused.

A third line of attack offered by Leibniz against the Newtonian conception of space and

time draws on another principle familiar from Leibniz’s metaphysics, namely, the

Principle of the Identity of Indiscernibles (PII). In the present context we may

understand the PII as ruling out the possibility of two things being distinct, but not

distinct in virtue of some discernible property. It thus suggests that where we cannot

identify a recognizable difference between two things or possibilities, those two are in

fact only one – that is, as Leibniz puts it, that “To suppose two things indiscernible, is to

suppose the same thing under two names” (Fourth Paper, paragraph 6; G

VII.372/Alexander 37). Armed with the PII, Leibniz argues once again that the apparent

possibility of absolute space and time can be undermined. For on the supposition of

absolute space, the world oriented in one way with respect to space would have to be a

distinct possibility from the world oriented in another way with respect to absolute space.

But, according to Leibniz, two such purported possibilities would be indiscernible since

no being – not even God or an angel – could recognize any difference between them.

Leibniz thus concludes that since the supposition of absolute space leads to a violation of

the PII, the supposition itself must be rejected. By essentially the same reasoning,

Leibniz argues similarly that the apparent possibility of absolute time is also inconsistent

with the PII and so too must be rejected as chimerical or confused.

5.2. Leibnizian Space and Time

Leibniz’s positive account of space and time might be thought of as resting on two

primary pillars and being filled out by a number of ancillary theses. The first pillar

consists in an alternative model, or conception, of space and time offered in conscious

36

opposition to the Newtonian conception absolute space and time. According to Leibniz,

space and time are not so much things in which bodies are located and move as systems

of relations holding between things. He thus famously tells Clark in his Third Paper:

As for my own opinion, I have said more than once, that I hold space to be

something merely relative, as time is, that I hold it to be an order of coexistences,

as time is an order of successions. (Third Paper, paragraph 4; G

VII.363/Alexander 25-26)

The main idea is illustrated more clearly by a helpful example that Leibniz introduces in

his Fifth Paper. There he suggests that space and time are analogous to a family tree.

Unlike the relationship between, say, a mighty oak and its leaves, a genealogical tree is

not something which exists independently of, and prior to, its members, but is itself rather

a reification of the relations holding between brothers, sisters, parents, children, aunts,

uncles, etc. Analogously for Leibniz, space and time are not to be thought of as

containers in which bodies are located and through which they move, but rather as a

construction out of the relations that hold between the actual (and even possible) events

and bodies.

The second pillar of Leibniz’s positive account of space and time is rooted in his view

that – even understood as systems of relations – space and time must supervene on, or be

reducible to, more ontologically basic entities. In claiming that space and time are

“merely beings of reason,” Leibniz seems to have thought that they must be at least two

steps removed from the monads of his mature metaphysics. (i) Although bodies may be

held to stand in spatial relations to one another, Leibniz claims, space itself must be

considered an abstraction or idealization from those relations. For while relations

between bodies and events are necessarily variable and changing, the relations

constituting space and time must be viewed as determinate, fixed, and ideal. In this

sense, space and time are one level removed from whatever reality might be attributed to

physical bodies and events. (ii) As we have briefly noted, however, according to

Leibniz’s most mature metaphysics, physical bodies and events are themselves merely

well-founded phenomena. Space and time must thus not only be abstractions, but

abstractions from merely well-founded phenomena. As such they can be at best two

37

steps removed from fundamental reality. However one counts the ontological “levels”

involved, Leibniz maintains that space must ultimately be grounded in the various

perceptions by which monads represent the world, each from its own “point of view,” and

that time must find its ultimate root in the various appetites in accordance with which

monad sequentially unfold in synchronized harmony. Although the details of this

grounding story remain less than perspicuous, there can be no doubt that Leibniz saw his

relationalism about space and time as dovetailing with the foundations of his monadic

metaphysics.

Beyond arguing that space and time are ideal systems of relations, Leibniz also defends a

number of less central theses that help to flesh out his positive conceptions of space and

time. For our purposes it might be worth quickly calling attention to three of the most

significant. First, although he takes space and time to be ideal systems of relations,

Leibniz nonetheless insists that they are conceived of as infinite. In doing so he takes a

characteristically intermediate position, partially siding with most early moderns in

rejecting the notion of an imaginary space surrounding a finite cosmos, but also partially

siding with earlier medieval thinkers in affirming that “Since space is itself an ideal thing

. . . space out of the world must needs be imaginary [but] . . . The case is the same with

empty space within the world; which I also take to be imaginary” (Fifth Paper, paragraph

33; G VII.396/Alexander 64).34 Second, Leibniz maintains that although the existence of

empty space is possible – logically speaking, two bodies could exist at a spatial distance

with nothing between them – nonetheless it is morally certain that the actual world is a

plenum. For, as we have already noted in connection with his critique of atomism,

Leibniz maintains the existence of empty space would be inconsistent with God’s

decision to create the best of all possible worlds. Third, in spite of his critique of

Newton, Leibniz affirms that space and time are continuous, homogenous, and infinitely

divisible (although not actually divided to infinity). He holds that the ascription of such

properties to space and time is only possible once they are recognized to be ideal, or

“imaginary,” and in this way Leibnizian space and time are further distinguished not only

from Newtonian absolute space and time, but also from extended Leibnizian bodies.

38

Bibliography

Selected Primary Texts in Original Languages with Abbreviations A: German Academy of Sciences, ed., Gottfried Wilhelm Leibniz: Sämtliche Schriften und Briefe, Darmstadt and Berlin: Akademie Verlag, 1926-. Reference is to series, volume and page. AT: Oeuvres de Descartes, eds., Charles Adam and Paul Tannery, Paris: J. Vrin, 1964-74. Reference is to volume and page. G: C. I. Gerhardt, ed., Die philosophischen Schriften von Gottfried Wilhelm Leibniz, Berlin: Weidmann, 1875-90; reprinted Hildesheim: Georg Olms, 1965. Reference is to volume and page. GM: C. I. Gerhardt, ed., Mathematische Schriften von Gottfried Wilhelm Leibniz, Berlin: A. Asher; Halle: H.W. Schmidt, 1849-63. Reference is to volume and page. Selected Primary Texts in English Translation with Abbreviations Alexander: H. G. Alexander, ed., The Leibniz-Clarke Correspondence, Manchester: Manchester University Press. AG: R. Ariew and D. Garber, eds., G. W. Leibniz: Philosophical Essays, Indianapolis: Hackett, 1989. CSM: The Philosophical Writings of Descartes, ed. and trans., J. Cottingham, R. Stoothoff, and D. Murdoch, 2 vols. Cambridge: Cambridge University Press, 1985. Reference is to volume and page. FW: R. Franks and R. Woolhouse, G. W. Leibniz: Philosophical Texts, New York, Oxford University Press, 1998. L: L. E. Loemker, ed. and trans., G. W. Leibniz: Philosophical Papers and Letters, 2nd ed., Dordrecht: Reidel, 1969. Langely: Langely, Alfred G. 1949. New Essays Concerning Human Understanding together with An Appendix of Some of His Shorter Pieces, La Salle, Illinois: The Open Court Publishing Company). NE: P. Remnant and J. Bennett, eds. and trans., G. W. Leibniz: New Essays on Human Understanding, Cambridge: Cambridge University Press, 1981. The pagination of Remnant and Bennett is identical with that of the Academy edition (A VI. vi); references to both editions are therefore the same.

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Selected General Studies and Collections Adams, Robert. 1994. Leibniz: Determinist, Theist, Idealist, New York: Oxford University Press. Broad, C. D. 1975. Leibniz: An Introduction, Cambridge: Cambridge University Press. Costabel, P. 1973. Leibniz and Dynamics, Ithaca, New York: Cornell University Press. Cover, J. A. and John O’Leary-Hawthorne. 1999. Substance and Individuation in Leibniz, Cambridge: Cambridge University Press. Garber, Daniel. 1998. “Leibniz: Physics and Philosophy” in Nicholas Jolley, ed., The Cambridge Companion to Leibniz, New York: Cambridge University Press. Gueroult, M. 1967. Leibniz: Dynamique et Métaphysique, Paris: Aubier-Montaigne. Hannequin, A. 1908. La première philosophie de Leibnitz, in Hannequin, Etudes d’histoire des sciences et d’histoire de la philosophie, Paris: Alcan. Hooker, M., ed. 1982. Leibniz: Critical and Interpretive Essays, Minneapolis: University of Minnesota Press. Jolley, N. 1995. Cambridge Companion to Leibniz, Cambridge: Cambridge University Press. Mercer, C. 2001. Leibniz’s Metaphysics: Its Origins and Development, Cambridge: Cambridge University Press. Nelson, Alan. 2005. The Blackwell Companion to Rationalism, Malden, MA: Blackwell Publishing, Ltd. Okruhlik, Kathleen and J. R. Brown. 1995. The Natural Philosophy of Leibniz, Dordrecht: Reidel. Russell, B. 1937. A Critical Exposition of the Philosophy of Leibniz, Allen and Unwin, 2nd Edition. Rutherford, Donald. 1998. Leibniz and the Rational Order of Nature, Cambridge: Cambridge University Press. Sleigh, R. C., Jr. 1990. Leibniz and Arnauld: A Commentary on Their Correspondence, New Haven, Connecticut: Yale University Press.

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Lodge, Paul. 2001. “Leibniz’s Notion of an Aggregate,” British Journal of Philosophy (9:3) 467-486. Lodge, Paul. 1998. “Leibniz’s Heterogeneity Argument Against the Cartesian Conception of Body,” Studia Leibnitiana (30:1) 83-102. Lodge, Paul. 1997. “Force and the Nature of Body in Discourse on Metaphysics Paragraph 17-18,” Leibniz Society Review (7) 116-124. Miller, Richard. 1988. “Leibniz on the Interaction of Bodies,” History of Philosophy Quarterly (5) 245-255. Leibniz’s Dynamics Bernstein, Howard. 1981. “Passivity and Inertia in Leibniz’s ‘Dynamics’,” Studia Leibnitiana (13) 97-113. Bertoloni Meli, Domenico. 1993. Equivalence and Priority: Newton versus Leibniz, Oxford: Oxford University Press. Costabel, P. 1973. Leibniz and Dynamics, Ithaca, New York: Cornell University Press. Cover, J. A. and Glenn Hartz. 1994. “Are Leibnizian Monads Spatial?” History of Philosophy Quarterly (11:3) 295-316. Gabby, Alan. 1971. “Force and Inertia in Seventeenth-Century Dynamics,” Studies in History and Philosophy of Science (2) 1-68. Gale, George. 1988. “The Concept of ‘Force’ and Its Role in the Genesis of Leibniz’s Dynamical Viewpoint,” Journal of the History of Philosophy (26) 45-67. Gale, George. 1973. “Leibniz’s Dynamical Metaphysics and the Origins of the Vis Viva Controversy,” Systematics (11) 184-207. Garber, Daniel. 1985. “Leibniz and the Foundations of Physics: The Middle Years,” in Kathleen Okruhlik, ed., The Natural Philosophy of Leibniz, Dordrecht: Reidel, 27-130. Garber, Daniel. 2005. “Leibniz and Idealism,” in Donald Rutherford and J. A. Cover, eds., Leibniz: Nature and Freedom, New York: Oxford University Press. Iltis, C. 1974. “Leibniz’ Concept of Force: Physics and Metaphysics,” Studia Leibnitiana Supplementa, Volume XIII, Band II, Weisbaden: Franz Steiner Verlag. Iltis, C. 1979. “Leibniz and the Vis Viva Controversy,” Isis (62) 21-35.

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Jolley, Nicholas. 1986. “Leibniz and Phenomenalism,” Studia Leibnitiana (18) 38-51. Laudan, L. 1968. “A Postmortem on the Vis Viva Controversy,” Isis (59) 296-300. Rutherford, Donald. 1998. Leibniz and the Rational Order of Nature, Cambridge: Cambridge University Press. Rutherford, Donald. 1992. “Leibniz’s Principle of Intelligibility,” History of Philosophy Quarterly (9:1) 35-49. Rutherford, Donald. 1990. “Leibniz’s ‘Analysis of Multitude and Phenomena into Unities and Reality’,” Journal of the History of Philosophy (28) 525-552. Rutherford, Donald. 1990. “Phenomenalism and the Reality of Body in Leibniz’s Later Philosophy,” Studia Leibnitiana (22) 11-28. Westfall, Richard. 1971. Force in Newton’s Physics: The Science of Dynamics in the Seventeenth Century, New York: Elsevier. Leibniz on The Laws of Motion Bernstein, Howard. 1984. “Leibniz and Huygens on the ‘Relativity’ of Motion.” Studia Leibnitiana (13) 97-113. Brown, Gregory. 1992. “Is There a Pre-Established Harmony of Aggregates in the Leibnizian Dynamics, or Do Non-Substantial Bodies Interact?” Journal of the History of Philosophy (30:1) 53-75. Crockett, Timothy. 1999. “Continuity in Leibniz’s Mature Metaphysics,” Philosophical Studies (94: 1-2) 119-138. Earman, John. 1989. “Remarks on Relational Theories of Motion,” Canadian Journal of Philosophy (19) 83-87. Gabbey, Alan. 2003. “New Doctrines of Motion,” in Michael Ayers and Daniel Garber, eds., The Cambridge History of Seventeenth-Century Philosophy, Two Volumes, Cambridge: Cambridge University Press. Garber, Daniel. 1983. “Mind, Body, and the Laws of Nature in Descartes and Leibniz,” Midwest Studies in Philosophy (8) 105-134. Hacking, Ian. 1985. “Why Motion is Only a Well-founded Phenomenon,” in Kathleen Okruhlik, ed., The Natural Philosophy of Leibniz, Dordrecht: Reidel.

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Lariviere, Barbara. 1989. “Leibnizian Relationalism and the Problem of Inertia,” Canadian Journal of Philosophy (17) 437-448. Lariviere, Barbara. 1989. “Reply to Earman’s ‘Remarks on Relational Theories of Motion,” Canadian Journal of Philosophy (19) 89-90. Levey, Samuel. 2003. “The Interval of Motion in Leibniz’s Pacidius Philalethi,” Nous (37:3) 371-416. Lodge, Paul. 2003. “Leibniz on Relativity and the Motion of Bodies,” Philosophical Topics (31:1&2). Lodge, Paul. 2001. “The Debate over Extended Substance in Leibniz’s Correspondence with De Volder,” International Studies in the Philosophy of Science (15:2) 155-165. Lodge, Paul. 1998. “The Failure of Leibniz’s Correspondence with De Volder,” Leibniz Society Review (8) 47-67. Roberts, John. 2003. “Leibniz on Force and Absolute Motion,” Philosophy of Science (70:3) 553-573. Rutherford, Donald. 1993. “Natures, Laws and Miracles: The Roots of Leibniz’s Critique of Occasionalism,” in Steven Nadler, ed., Causation in Early Modern Philosophy, University Park: Penn State University Press. Stein, H. 1977. “Some Philosophical Prehistory of General Relativity,” in J. Earman, C. Glymour, and J. Stachel, eds., Foundations of Space-Time Theories, Minnesota Studies in the Philosophy of Science, Volume VIII, Minneapolis: University of Minnesota Press, 3-49. Woolhouse, R. S. 2000. “Leibniz’s Collision Rules for Inertialess Bodies Indifferent to Motion,” History of Philosophy Quarterly (17:2) 143-157. Leibniz on Space & Time Arthur, Richard. 1994. “Space and Relativity in Newton and Leibniz,” British Journal for the Philosophy of Science (45:1) 219-240. Arthur, Richard. 1985. “Leibniz’s Theory of Time,” in Okruhlik and Brown, eds., The Natural Philosophy of Leibniz, Dordrecht: Reidel, 263-313. Barbour, Julian. 1982. “Relational Concepts of Space and Time,” British Journal for the Philosophy of Science (33) 251-274.

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Broad, C. D. 1981. “Leibniz’s Last Controversy with the Newtonians,” in R.S. Woolhouse, ed., Leibniz: Metaphysics and Philosophy of Science, Oxford: Oxford University Press, 157-174. Carriero, John. 1990. “Newton on Space and Time: Comments on J. E. McGuire,” in Philosophical Perspectives on Newtonian Science, Cambridge: MIT Press. Cover, J. A. 1997. “Non-Basic Time and Reductive Strategies: Leibniz’s Theory of Time,” Studies in the History and Philosophy of Science (28:2) 289-318. Earman, John. 1989. World Enough and Space-Time: Absolute Versus Relational Theories of Space and Time, Cambridge, Mass. MIT Bradford. Friedman, Michael. 1983. Foundations of Spacetime Theories, Princeton, NJ: Princeton University Press. Fox, Michael. 1970. “Leibniz’s Metaphysics of Space and Time,” Studia Leibnitiana (2) 29-55. Futch, Michael. 2002. “Leibniz’s Non-Tensed Theory of Time,” International Studies in the Philosophy of Science. Futch, Michael. 2002. “Supervenience and (Non-Modal) Reductionism in Leibniz’s Philosophy of Time,” Studies in the History and Philosophy of Science. Garber, Daniel, et. al. 1998. “New Doctrines of Body and Its Powers, Place, and Space,” in Michael Ayers and Daniel Garber, eds., The Cambridge History of Seventeenth-Century Philosophy, Two Volumes, New York: Cambridge University Press, 553-623. Grant, Edward. 1981. Much Ado About Nothing: Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution, Cambridge: Cambridge University Press. Hartz, Glenn and J. A. Cover. 1988. “Space and Time in the Leibnizian Metaphysic,” Nous (22) 493-519. Khamara, Edward. 1993. “Leibniz’s Theory of Space: A Reconstruction,” Philosophical Quarterly (43) 472-488. Khamara, Edward. 1988. “Indiscernibles and the Absolute Theory of Space and Time,” Studia Leibnitiana (20) 140-159. McRae, Robert. 1979. “Time and the Monad,” Nature and System (1) 103-109. Mundy, Brent. 1992. “Space-Time and Isomorphism,” in Proceedings of the Biennial Meetings of the Philosophy of Science Association (1) 515-527.

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Nerlich, Graham. 1991. “How Euclidean Geometry Has Misled Metaphysics,” Journal of Philosophy (88:4) 169-189. Nerlich, Graham. 1994. The Shape of Space, 2nd Edition, Cambridge: Cambridge University Press. Pooley, Oliver and Harvey Brown. 2002. “Relationalism Rehabilitated? I: Classical Mechanics,” British Journal for the Philosophy of Science (53:2) 183-204. Rynasiewicz, Robert. 1995a. “By Their Properties, Causes and Effects: Newton’s Scholium on Time, Space, Place and Motion – I. The Text,” Studies in the History and Philosophy of Science (26:1) 133-153. Rynasiewicz, Robert. 1995b. “By Their Properties, Causes and Effects: Newton’s Scholium on Time, Space, Place and Motion – II. The Context,” Studies in the History and Philosophy of Science (26:2) 295-321. Sayre-McCord, Geoffrey. 1984. “Leibniz, Materialism, and the Relational Account of Space and Time,” Studia Leibnitiana (16) 204-211. Sklar, Lawrence. 1974. Space, Time and Spacetime, Berkeley: University of California Press. Vailati, Ezio. 1993. “Clarke’s Extended Soul,” Journal of the History of Philosophy (31:3) 387-403. Vailati, Ezio. 1997. Leibniz and Clarke: A Study of Their Correspondence, New York: Oxford University Press. Winterbourne, A. T. 1982. “On the Metaphysics of Leibnizian Space and Time,” Studies in History and Philosophy of Science (13) 201-214. Other Internet Resources Leibniz Society of North America [http://philosophy2.ucsd.edu/~rutherford/Leibniz/leibsoc.htm] Gregory Brown’s “Leibnitiana” contains many helpful links and resources [http://www.gwleibniz.com/] Leibniz: Texts and Translations: A site providing texts and translations maintained by Don Rutherford [http://philosophy2.ucsd.edu/%7Erutherford/Leibniz/]

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The Newton Project: Hosted by the Centre for History of Science, Technology and Medicine at Imperial College London [http://www.newtonproject.sussex.ac.uk/prism.php?id=1] Related Entries Atomism Descartes, René • physics Galileo, Galilei Gassendi, Pierre Hobbes, Thomas Kant and Leibniz Leibniz, Gottfried Wilhelm • modal metaphysics • on causation Malebranche, Nicholas Newton 1 There has been considerable debate over the exact date and extent of Leibniz’s

conversion to mechanism. See, for starters, Kabitz (1909, 51-53), Brown (1984, chapter

3), and Mercer (2001, 24-48). For further references see especially Mercer (2001, 26, fn.

11). 2 I.e. lower level liberal arts studies, traditionally consisting of grammar, rhetoric, and

logic. Loemker reports that “The curriculum of the Nicolai School in Leipzig, while not

conforming completely to the medieval trivium, still consisted of Latin and Greek,

rhetoric, and logic, together with Scholastic theology” (L 660, fn. 2). 3 See, for example, his letters to his former instructor Jacob Thomasius of 26 September/6

October 1668 and 20/30 April 1669 (A II:i, no. 9 and 11). An English translation of the

letter of 20/30 April 1669 can be found in L 93-103. 4 A.II.i 16/L 95 5 It should be noted that, for Leibniz the adoption of mechanical philosophy was not

tantamount to a wholesale repudiation of Aristotelian natural philosophy. Thus, for

example, in a letter of 1669 to his former mentor Jacob Thomasius, Leibniz argues not

only that “the reformed philosophy can be reconciled with Aristotle’s and does not

conflict with it” but, even more aggressively, that “the one … must be explained through

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the other … that the very views which the moderns are putting forth so pompously are

derived from Aristotelian principles” (A.II.i 16/L 95; italics added). For more on

Leibniz’s conciliatory attitude towards Aristotelianism, see especially, Mercer (2001). 6 Recalling the theory of the TMA later in life, Leibniz writes:

. . . I showed that it ought to follow that the conatus of a body entering into a

collision, however small it might be, would be impressed on the whole receiving

body, hwoever larger it might be, and thus, that the largest body at rest would be

carried off by a colliding body however small it might be, without regarding it at

all, since such a notion of matter contains not resistance to motion, but

indifference. From this it follows that it would be no more difficult to put a large

body into motion than a small one . . . (SD 19/AG 124) 7 This dating is based on a quotation attributed to Leibniz in FC, page LXIV, which

reads, “Spinoza did not see the mistakes in Descartes’s rules of motion; he was surprised

when I began to show him that they violate the equality of cause and effect.” I first

learned of the quotation from Garber (1998, 278). Garber reports that he has been unable

to confirm the date in a more reliable source, and notes “there is some uncertainty that

attaches to the dating” (Garber 1998, 339, fn. 25). 8 For discussion of the controversy, see especially Iltis (1979), Costabel (1973), and

Laudan (1968). 9 See A VI.ii, 158. 10 For more on the Leibniz-Clarke correspondence, and further references, see especially,

Broad (1981) and Vailati (1997). 11 Indeed, Leibniz sometimes argues against atomism directly from the Principle of

Sufficient Reason. In his Fourth Letter to Clark, for example, he first argues that there

can be no sufficient reason for any ratio of void to matter other than 0:1, and then argues

that “the case is the same with atoms: what reason can anyone assign for confining

nature in the progression of subdivision?” (G VII.378/AG 332). 12 The passage can be found in its original language in Eduar Bodemann, Der

Briefwechsel des Gottfried Willhelm Leibniz in der Königlichen öffentlichen Bibliothek zu

Hannover, Hannover, 1895; reprinted Hildesheim: Georg Olms, 1966.

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13 It is perhaps worth noting that since Leibniz maintains that true motion is grounded in

forces, the same conclusion will not follow on his own account; even at an idealized

instant the physical world for Leibniz could still enjoy a qualitative variety in virtue of a

differential distribution of derivative forces. 14 Cf. “Nothing is really solid or fluid, absolutely speaking, and everything has a certain

degree of solidity or fluidity; which term we apply to a thing derives from the

predominant appearance it presents to our senses” (SD 51/AG 135). 15 See, for example, “Preliminary Specimen: On the Law of Nature Relating to the

Power of Bodies” BM VI 287-92/AG 105-111 and “A Specimen of Dynamics” GM VI

234-254/AG 117-138. 16 It should be noted that Leibniz also attempts to prove the conservation of vis viva

without the help of this obviously empirical principle. See, for example, “Preliminary

Specimen: On the Law of Nature Relating to the Power of Bodies” BM VI 291-92/AG

110-111; “A Specimen of Dynamics, Part I, par. 15” GM VI 243-44/AG 127. For

discussion see Garber (1998, 313f). 17 Given the controversy that erupted in the wake of Leibniz’s argument, it may be

worthwhile to make two brief remarks in connection with the conservation principles of

Newtonian mechanics. First, the Cartesian quantity of motion is not a vector quantity – it

doesn’t take account of the direction of the moving body – and therefore it must be

distinguished from the Newtonian notion of momentum (mv). In fact, Leibniz accepts

the conservation of momentum and thus must be understood only to be arguing against

the non-vectorial quantity of motion. Second, although kinetic energy (1/2 mv2) is

conserved only in elastic collisions, Leibniz maintains that, at root, all fundamental

collisions are elastic, and that inelastic collisions must therefore be analyzed as collisions

of composite, and ultimately elastic, bodies. 18 It should be noted that for metaphysical reasons, Leibniz denies that strictly speaking

forces are ever transferred from one created entity to another. Thus, for example, he

writes,

It should be known, however, that forces do not cross from body into body, since

any body whatever already has in itself the force that it exerts, even if it does not

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show it or convert it into motion of the whole prior to a new modification. For

example, when a ball that is at rest is struck by another, it is moved by an

implanted force, namely by elastic force, without which there would be no

collision. Moreover, the Elastic force in the body arises from an internal motion

invisible to us. And the Entelechy itself is modified corresponding to these

mechanical or derivative [forces]. Therefore it can be said that force is already

present in every body, and it is determined only by modification. (LW

131/Adams (1994, 383) 19 Leibniz clearly takes his infinitesimal calculus to relate vis viva and vis mortua. It is

not, however, so clear whether he thinks the two quantities are related by a single or a

double integration. The passage quoted just above in the main text suggests that dead

force is related to living force by a single integration so that if dead force were as x,

living force would be as ∫xdx. As Westfall (1971) notes, however, Leibniz sometime

suggests that the move from dead force to living force requires us to integrate twice.

Thus, for example, in a letter to De Volder, Leibniz writes “Hence according to the

analogy of geometry or of analysis, solicitation [i.e. dead forces] are as dx, velocities as x

and [living] forces as xx or ∫xdx” (G II.154-156). For further discussion, see especially

Westfall (1971, 298f). 20 That is, if we take dead force to be equal to be the derivative of the body’s velocity

with respect to time multiplied by its mass, i.e. dead force = mass (dv/dt). 21 For discussion of Descartes’s treatment of the laws of motion, see especially Garber

1992, Chapters 7 and 9. 22 Mais pour faire mieux voir comment il s'en faut servir, et pourquoy M. des Cartes et

d'autres s'en sont éloignés, considerons sa troisieme regle du mouvement, pour servir

d'exemple, et supposons que deux corps B et C, chacun d'une livre, aillent l'un contre

l'autre, B avec une vistesse de 100 degrés, et C avec une vistesse d'un degré. Toute leur

quantité de mouvement sera 101. Mais si C avec sa vistesse peut monter à un pouce de

hauteur, B pourra monter avec la sienne à 10000 pouces; ainsi la force de tous les deux

sera d'elever une livre à 10001 pouces. Or, suivant cette troisieme Regle Cartesienne,

apres le choc ils iront ensemble de compagnie avec une vistesse comme 50 et demy, à fin

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qu'en la multipliant par 2 (nombre des livres qui vont ensemble apres le choc) il revienne

la premiere quantité de mouvement, 101. Mais ainsi ces 2 livres ne se pourront elever

ensemble qu'à une hauteur de 2550 pouces et un quart (qui est le quarré de 50 et demy)

ce qui vaut autant que s'ils avoient la force d'elever une livre à 5100 et demy, au lieu

qu'avant le choc il y avoit la force d'elever une livre à 10001 pouces. Ainsi presque la

moitié de la force sera perdue en vertu de cette regle sans aucune raison, et sans estre

employée à rien. Ce qui est aussi peu possible, que ce que nous avons monstré

auparavant dans un autre cas, où en vertu du même Principe Cartesien general, on

pourroit gagner le triple de la force sans aucune raison. 23 See, for example, letter to Malebranche July 1687:

When the difference between two instances in a given series or that which is presupposed

can be diminished until it becomes smaller than any given quantity whatever, the

corresponding difference in what is sought or in their results must of necessity also be

diminished or become less than any given quantity whatever. Or to put it more

commonly, when two instances or data approach each other continuously, so that one at

last passes over into the other, it is necessary for the consequences or results (or the

unknown) to do so also” (G. III, 51/L. 351). 24 By the time Leibniz introduced the principle of continuity, the defects of Descartes’s

collision rules were already widely acknowledge, even by his staunchest defenders. 25 Leibniz may have followed Huygens in measuring the velocity of the bodies relative to

their common center of gravity, but any inertial reference frame will do. 26 For more on this argument, see Dynamica GM VI.495; for a brief discussion see

Garber 1998, 316-317, 350-51 fn. 124. 27 It is rather difficult to say whether or not Leibniz actually is committed to this denial.

Certainly, he would have denied that if we consider the motion of the train as merely a

change of relative position, then there is no saying whether or not the train is accelerating

independently of an arbitrarily chosen frame of reference. But, as soon as we add that we

feel the jerk of the train as it accelerates, or feel ourselves pressed up against its walls,

Leibniz might insist that we have moved from kinematics to dynamics. Cf. A VI.iv.2019.

For helpful discussion of this point, see especially Lodge 2003.

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28 The text in the original language can be found in Louis Couturat, Opuscules et

fragments inédits de Leibniz, Paris 1903, 590. 29 In his critical notes on Descartes’s Principles, Leibniz writes:

If motion is nothing but the change of contact or of immediate vicinity, it follows

that we can never define which thing is moved. For just as the same phenomena

may be interpreted by different hypotheses in astronomy, so it will always be

possible to attribute the real motion to either one or the other of the two bodies

which change their mutual vicinity or position. Hence, since one of them is

arbitrarily chosen to be at rest or moving at a given rate in a given line, we may

define geometrically what motion or rest is to be ascribed to the other, so as to

produce the given phenomena. Hence if there is nothing more in motion than this

reciprocal change, it follows that there is no reason in nature to ascribe motion to

one thing rather than to others. The consequence of this will be that there is no

real motion. (GP IV.369/L 393)

30 This section is especially indebted to the elegant discussion in Broad 1981. 31 For a helpful discussion of Newton’s view and argument, see especially Rynasiewicz

1995a and 1995b. 32 Isaac Newton, 1952 [1730] Opticks or A Treatise on the Reflections, Refractions,

Inflections and Colours of Light, (New York: Dover) Query 28, 31. 33 Similarly, Leibniz writes:

I shall give another instance of this. God’s immensity makes him actually present

in all spaces. But now if God is in space, how can it be said that space is in God,

or that it is a property of God? We have often heard that a property is in its

subject; but we never heard that a subject is in its property. In like manner, God

exists in all time. How then can time be in God; and how can it be a property of

God? These are perpetual alloglossies [i.e. verbal oddities]. (Fifth Paper,

paragraph 45; G VII.399/Alexander 68) 34 On the doctrine of imaginary space, see Grant (1981), especially Chapter 6.

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