## 1. The Historical Development of Leibniz's Physics




In his earliest days, Leibniz read a wide range of 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 
>  schools[2],
>  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 1660s, 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 to improve 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 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 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 as well
against the laws of the TMA and the HPN. The paradoxical result of the
ideal laws of the 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. One can 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 famously capped by his correspondence with
Samuel Clarke [10]
 The controversy between Leibniz and Clarke began when the 
Princess of Wales passed along to 
Clarke in 1715 a letter written by Leibniz decrying the decline of 
religion 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, Clarke 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 
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 
concludes that all bodies must be elastic, and must
therefore 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 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). 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) 




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 and spheres, 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 
untypical for him in so far as it suggests 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 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 is 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 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. 




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.[12]
 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 is blank, and thus whose projection would not only be 
homogenous at each instant, but through time as well.




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 principles 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 only upon God for their existence, and 
derivative creatures that depend not only upon God but also 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 next 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 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,[13]
 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 minuscule. 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. 




(2) In addition to offering resistance to motion, Leibniz thought it
essential to bodies that they resist mutual collocation - to
some degree, they must be "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 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 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 
is reluctant to posit the existence of primitive attractive forces 
in nature (SD II.52/AG 136; see also Leibniz's Fifth Letter to 
Clarke, 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 physical extension. Nonetheless, Leibniz intriguingly 
suggests that physical 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; see also GP II 269/AG 179, and GP II 183/L 519).
> 




Leibniz thus grants that while physical 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, physical 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 matter on its head: whereas, for example, 
Descartes attempted to explain solidity in terms of physical 
 extension,[14]
 Leibniz proposes to explain physical 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 a 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]





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* (*mv*2) 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 
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.[18]
 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 (see, for example, G IV 
281/L272). 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 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 from some height, or the power a baseball 
has once released from a pitcher's hand.) As noted 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.[19]





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 
*m**v*2 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 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*.[20]
 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 ways of reading of Leibniz's texts, the measure of dead force
approaches very closely the measure of force more clearly at work in 
Newton's 
 *Principia*.[21]
 It is therefore worth noting that although he could hardly deny the 
technical achievement of Newton's 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 Clarke, paragraph 45/Alexander 43) 




The grounds for Leibniz's negative reaction to Newton's conception of
force, and specifically Newton's 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 
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
119-120). What has remained less certain is the answer to the
question, "Where, as it were, does Leibniz think active and
passive primitive forces are located?" 




Answering 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 are 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 panorganicism remains striking, 
this picture promises a relatively straightforward 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 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" in 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" in 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 what"grounding" amounts to in this
context. 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 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).[22]
 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. 




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 *mv*2. 
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 1/2 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 1/2)2 or 
5100 1/2 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).




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 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 the
actual world. In his later work, that strategy is replaced by the
attempt 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: 



\[\begin{align}
\Velocity\ A\_{\before} &- \Velocity\ B\_{\before} = \\
 & \Velocity\ B\_{\after} - \Velocity\ A\_{\after}
\end{align}\]



or more simply:



\[
V\_{A\ \before} - V\_{B\ \before} = V\_{B\ \after} - V\_{A\ \after}
\]



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 (*molem*) times velocity - before and 
after collision (GM VI.227/Langley 667). Thus, letting *A* and
*B* once again represent two balls involved in a head on 
collision:



\[\begin{align}
(\Mass\ A\times V\_{A\ \before}) &+ (\Mass\ B\times V\_{B\ \before}) = \\
 &(\Mass\ B\times V\_{B\ \after}) + (\Mass\ A\times V\_{A\ \after}) 
\end{align}\]



or more simply:



\[\begin{align}
M\_A \times V\_{A\ \before} &+ M\_B \times V\_{B\ \before} = \\
 &M\_B \times V\_{B\ \after} + M\_A \times V\_{A\ \after}
\end{align}\]



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): 



\[\begin{align}
 M\_A \times (V\_{A\ \before})^2 &+ M\_B \times (V\_{B\ \before})^2 = \\
 &M\_B \times (V\_{B\ \after})^2 + M\_A \times (V\_{A\ \after})^2
\end{align}\]



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 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 
*mv*2 in this way are far from clear, but he must 
have recognized them to be necessarily metaphysical. Although the 
measurement of *mv*2 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 natural 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. 




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 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 *mv*2 
- 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. 


## 5. Leibniz on Space and Time




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.[30]
 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
between Newton and Leibniz over the nature of true motion surfaced
more explicitly in their disagreement over the nature of space and
time in the Leibniz-Clarke 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 just is 
the attribute of God's *Eternity*. 




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
 *Optics[31]*
 - 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).[32]
 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 the 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 PSR 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, 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. Since the supposition of absolute space thus
leads to a violation of the 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 the 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 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 Clarke 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 perhaps illustrated more intuitively 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 as a thing 
independently of, and prior to, its members, but is itself rather 
something like an abstract system of 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 literally located and through which they move, but 
rather as an abstract structure of relations in which actual (and 
even possible) bodies might be embedded. 




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 as "beings of reason" 
are in a sense at least two steps removed from the monads of his 
mature metaphysics. (i) Although bodies may be held to stand in 
spatial and temporal relations to one another, Leibniz claims, space 
and time themselves must be considered abstractions or idealizations 
with respect to 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. (ii) As we have briefly noted, however, according to Leibniz's
most mature metaphysics, physical bodies and events are themselves to
be understood as merely well-founded phenomena. Relations of relative
distance and duration holding between bodies must therefore 
themselves be a step removed from monadic reality, and thus space and
time must be, as it were, a second step removed from the most basic 
non-relational entities of Leibniz's most mature metaphysics. In 
spite of this double relation holding between ideal space and time on
the one hand and monadic reality on the other, Leibniz nonetheless 
insists that ideal space must be intelligibly related to the various 
perceptions by which monads represent the world, each from its own 
"point of view," just as ideal time must be intelligibly 
related to the various appetites in accordance with which monads 
sequentially unfold in synchronized harmony. Although the details of 
this intelligible 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).[33]
 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.