The Southern Journal of Philosophy
Volume 50, Issue 3
September 2012

NEWTON AND KANT:
QUANTITY OF MATTER IN THE
METAPHYSICAL FOUNDATIONS OF NATURAL SCIENCE
Michael Friedman
abstract: Immanuel Kant’s Metaphysical Foundations of Natural Science (1786) provides
metaphysical foundations for the application of mathematics to empirically given
nature. The application that Kant primarily has in mind is that achieved in Isaac
Newton’s Principia (1687). Thus, Kant’s first chapter, the Phoronomy, concerns the
mathematization of speed or velocity, and his fourth chapter, the Phenomenology,
concerns the empirical application of the Newtonian notions of true or absolute space,
time, and motion. This paper concentrates on Kant’s second and third chapters—the
Dynamics and the Mechanics, respectively—and argues that they are best read as
providing a transcendental explanation of the conditions for the possibility of applying
the (mathematical) concept of quantity of matter to experience. Kant again has in
mind the empirical measures of this quantity that Newton fashions in the Principia, and
he aims to make clear, in particular, how Newton achieves a universal measure for all
bodies whatsoever by projecting the static quantity of terrestrial weight into the
heavens by means of the theory of universal gravitation. Kant is not attempting to
prove a priori what Newton has established empirically but, rather, to clarify the
character of Newton’s mathematization by building Newton’s empirical measures
into the very concept of matter that is articulated in the Metaphysical Foundations.

1. INTRODUCTION
Kant’s Metaphysical Foundations of Natural Science, which appeared in 1786
between the first (1781) and second (1787) editions of the Critique of Pure Reason,
takes Newton’s Principia (1687) as paradigmatic of the natural science for which
Kant provides a metaphysical foundation. The final two paragraphs of Kant’s
Michael Friedman is the Frederick P. Rehmus Family Professor of Humanities at Stanford
University, where he directs t; he Patrick Suppes Center for the History and Philosophy of
Science. He is the author of Foundations of Space-Time Theories: Relativistic Physics and Philosophy of
Science (Princeton University Press, 1983) and Kant and the Exact Sciences (Harvard University Press,
1992), and he is the editor of Kant: Metaphysical Foundations of Natural Science (Cambridge University
Press, 2004). His most recent book, Kant’s Construction of Nature: A Reading of the Metaphysical
Foundations of Natural Science, is forthcoming from Cambridge University Press.
The Southern Journal of Philosophy, Volume 50, Issue 3 (2012), 482–503.
ISSN 0038-4283, online ISSN 2041-6962. DOI: 10.1111/j.2041-6962.2012.00126.x

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Preface make the centrality of the Principia in his project especially clear. The
penultimate paragraph explains why Kant has “imitated the mathematical
method” in his treatise (4: 478), which is organized into a pattern of “explications,” “propositions,” “proofs,” “remarks,” and so on, in the following four
chapters.1 He has done so, Kant suggests, so that “mathematical natural
scientists should find it not unimportant to treat the metaphysical part, which
they cannot leave out in any case, as a special fundamental part of their general
physics, and to bring it into union with the mathematical doctrine of motion”
(478). The final paragraph is then explicit that Newton, in the Principia, is
representative of the “mathematical natural scientists” in question. Kant
quotes the famous sentence from the Preface to the Principia concerning how
much geometry can accomplish on such a meager basis of postulates and adds
that metaphysics, by contrast, “can nevertheless accomplish so little” (479).2
But, he continues, “this small amount is still something that even mathematics
unavoidably requires in its application to natural science” (479). There appears
to be very little doubt, therefore, that the metaphysical part of pure natural
science that Kant himself is articulating is intended primarily to ground the
mathematical part that has already been articulated in the Principia.
Toward the beginning of his Preface, Kant states that “[a]ll proper natural
science therefore requires a pure part, on which the apodictic certainty that
reason seeks therein can be based” (469). He explains that this pure part, in
turn, consists of both a metaphysical and a mathematical part.3 The reason
for the latter is that, “since in any doctrine of nature there is only as much
proper science as there is a priori knowledge therein, a doctrine of nature will
contain only as much proper science as there is mathematics capable of
application there” (470). The point of the metaphysical part, by contrast, is to
explain the possibility of this application of mathematics:
But in order to make possible the application of mathematics to the doctrine of body,
which only through this can become a [proper] natural science, principles for the
construction of the concepts that belong to the possibility of matter in general must be
1
I cite Kant’s works, except for the Critique of Pure Reason, by volume and page numbers of
Kant (1900–). Subsequent references to the Metaphysical Foundations omit the volume number.
The Critique of Pure Reason is cited by the standard A and B pagination of the first and second
editions, respectively. All translations from the German are my own.
2
Kant paraphrases Newton’s remarks in the Preface to Principia that geometry requires only
two mechanical operations as postulates (the description of straight lines and circles) and then
quotes (using the original Latin) the sentence in question: “[G]eometry can boast that with so few
principles taken from other fields, it can still do so much” (1999, 382).
3
The two parts are distinguished at the end of this paragraph: “Pure rational cognition from
mere concepts is called pure philosophy or metaphysics; by contrast, that which grounds its
cognition only on the construction of concepts, by means of the presentation of the object in an a
priori intuition, is called mathematics” (469). Compare the parallel distinction in the Doctrine of
Method of the first Critique (A712–27/B740–55).

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introduced first. Therefore, a complete analysis of the concept of a matter in general
will have to be taken as the basis, and this is a task for pure philosophy—which, for
this purpose, makes use of no particular experiences, but only that which it finds in
the isolated (although intrinsically empirical) concept itself, in relation to the pure
intuitions of space and time, and in accordance with laws that already essentially
attach to the concept of nature in general, and is therefore a genuine metaphysics of
corporeal nature. (472)

And what application of mathematics to corporeal nature does Kant have
most centrally in mind? The next paragraph of the Preface strongly suggests
that it is precisely that which is accomplished in Newton’s physics.
Kant begins by stating that “all natural philosophers who have wished to
proceed mathematically in their occupation have always, and must have
always, made use of metaphysical principles (albeit unconsciously), even if
they themselves solemnly guard against all claims of metaphysics upon their
science” (472). He concludes that “true metaphysics” (the transcendental
metaphysics of experience articulated in the first Critique) is unavoidable:
All true metaphysics is drawn from the essence of the faculty of thinking itself, and
it is in no way feigned [erdichtet] on account of not being borrowed from experience;
rather, [it] contains the pure actions of thought, and thus a priori concepts and
principles, which first bring the manifold of empirical representations into the lawgoverned connection through which it can become empirical cognition, that is,
experience. Thus these mathematical physicists could in no way avoid metaphysical
principles, and, among them, also not those that make the concept of their proper
object, namely, matter, a priori suitable for application to outer experience, such as
the concept of motion, the filling of space, inertia, and so on. (472)

So it seems clear that the “mathematical physicists” in question here (which
are surely the same as the “natural philosophers who have wished to proceed
mathematically” at the beginning of the paragraph) are the same as the
“mathematical natural scientists” in question in the two final paragraphs of
the Preface—and that the application of mathematics in Newton’s Principia is
paradigmatic of their enterprise.4
4
Kant’s notion of “feigning” (Erdichtung) appears to be an allusion to Newton’s well-known
protestations against “feigning” hypotheses in the General Scholium added to the second (1713)
edition of the Principia: “I have not as yet been able to deduce from phenomena the reason for
these properties of gravity, and I do not feign [fingo] hypotheses. For whatever is not deduced
from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or
physical, or based on occult qualities, or mechanical, have no place in experimental philosophy”
(1999, 943). Kant’s polemic against what he calls the “mechanical natural philosophy” (532) in the
general remark to his third chapter, the Dynamics, is entirely consistent with Newton’s protestations: “In the doctrine of nature, the absolutely empty and the absolutely dense are approximately what blind accident and blind fate are in metaphysical science, namely, an obstacle to
the governance of reason, whereby it is either supplanted by feigning [Erdichtung] or lulled to rest
on the pillow of occult qualities” (532).

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Kant here provides a list of partial concepts [Teilbegriffe]—“the concept[s]
of motion, the filling of space, inertia, and so on” (472)—for the analysis of
what he calls the empirical concept of a matter in general in the previous
paragraph.5 This list of concepts, moreover, corresponds to the four chapters
of Kant’s treatise, where “[a]ll determinations of the general concept of a
matter in general [are] brought under the four classes of [categories], those of
quantity, of quality, of relation, and finally of modality” (475–76). Thus, the
Phoronomy considers the construction of motion (of a mere moving point) as
a magnitude in accordance with the Axioms of Intuition in the first Critique;
the Dynamics considers the filling of space by repulsive force in accordance
with the Anticipations of Perception; the Mechanics considers the interactions between different such (space-filling) moving matters in accordance with
the Analogies of Experience; and the Phenomenology considers how their
motion or rest thereby becomes an object of experience in accordance with
the Postulates of Empirical Thought.6 It is in this way, for Kant, that the “a
priori concepts and principles, which first bring the manifold of empirical
representations into the law-governed connection through which it can become
empirical cognition, that is, experience” (472) come into play. And it is in this
same way, therefore, that the analysis of the concept of a matter in general
demanded in the previous paragraph is carried out “in relation to the pure
intuitions of space and time, and [carried out] in accordance with laws [i.e.,
the transcendental principles] that already essentially attach to the concept of
nature in general” (472).
But what does all this have to do with Newton’s Principia? The case of
Kant’s first determination of matter, that is, motion (of a mere moving point),
is relatively straightforward. The sole proposition of the Phoronomy is concerned with explaining how motion can be considered as a mathematical
magnitude in terms of both speed and direction, and it does so by providing
what Kant calls a mathematical construction in pure intuition exhibiting how
two speeds (in whatever direction) may be added or composed with one
another. It is only in this way, for Kant, that we can secure the application of
the mathematical theory of magnitude (the theory of proportion) to the
5
In sec. 15 of the Prolegomena, the relevant list of (partial) concepts reads: “the concept of
motion, of impenetrability (on which the empirical concept of matter rests), of inertia, and others”
(295).
6
The Phenomenology does not add a further conceptual determination of matter to the
preceding three but considers matter’s motion (or rest) merely in relation to the mode of
representation. The Postulates of Empirical Thought make a parallel point: “The categories of
modality have the following peculiarity: that, as determination of the object, they do not in the
least augment the concept to which they are ascribed as a predicate, but they only express the
relation [of this concept] to the faculty of cognition” (A219/B266). It is for this reason that
Kant’s lists of partial concepts have motion, the filling of space (or impenetrability), inertia, and
then trail off (see note 4 above, together with the sentence to which it is appended).

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concept of motion. According to the general concept of quantity or magnitude considered in the Axioms of Intuition, we need to show how two
magnitudes falling under a common magnitude kind (two lengths, areas, or
volumes, for example) may be composed or added together so that a new
magnitude having the properties of the mathematical sum of the two then
results.7 Our problem in the present case is therefore to show how any two
speeds or velocities may be summed or added together.8 Kant’s main contention in the proposition is that the addition or composition [Zusammensetzung] in question cannot be carried out (in pure intuition) in a single relative
space or reference frame, but we must rather consider two different frames
such that the moving point in question has the first velocity in one frame,
which, in turn, moves with the second velocity relative to the other frame.9
This, in particular, is why phoronomy, in Kant’s sense, is nothing but “the
pure doctrine of magnitude (Mathesis) of motions” (489), which aims simply to
explain how the concepts of speed and velocity are possible as mathematical
magnitudes.
Thus, whereas Newton assumes that the concept of velocity already has the
structure of a mathematical magnitude in his definitions initiating the Principia, Kant takes it to be a task of the Phoronomy to provide an a priori
explanation (here an a priori construction) that exhibits the relevant operation
of addition or composition underlying this structure. Similarly, but less
straightforwardly, whereas Newton assumes that the notions of true or absolute space, time, and motion are already well defined (prior to the Laws of
Motion) in his famous Scholium to the Definitions, Kant takes it to be the task
of the Phenomenology to explain how it is possible for us to apply the
distinction between true and merely apparent motion to our perceptual
experience of moving bodies. Here, as Kant makes clear, he is following the
example of the two balls connected by a cord rotating around a common
7
For Kant’s concept of magnitude in the Axioms of Intuition and its relation to the
traditional theory of ratios or proportion see Sutherland 2004 and 2006.
8
I develop what follows in much more detail in Friedman, forthcoming. For a bit more
detail, see my introduction to Kant 2004 and Friedman 2010, 602–03.
9
The relationship between the two frames is described by what we now call a Galilean
transformation, and what Kant is doing, in these terms, is exhibiting what we now call the
(classical) velocity addition law as a consequence of these transformations. Kant’s construction
is self-consciously similar to the parallelogram construction of the composition of motions that
Newton derives in the first corollary to his Axioms, or Laws of Motion (1999, 416–17). The
crucial point, however, is that Newton is presenting a parallelogram of forces and is therefore
depending on his Laws of Motion (here the Second Law). But Kant insists that we must be able
to exhibit motion as a mathematical magnitude prior to such “mechanical” laws—and, in
particular, in pure rather than empirical intuition. Indeed, it is only in his third chapter that
Kant establishes what he calls the three “Laws of Mechanics”: conservation of the total quantity
of matter, inertia, and the equality of action and reaction.

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center of gravity in the final paragraph of Newton’s Scholium (1999, 414–
15).10
Newton begins by acknowledging the difficulty in determining true
motions: “It is certainly very difficult to find out the true motions of individual
bodies and actually to differentiate them from apparent motions, because the
parts of the immovable space in which the bodies truly move make no
impression on the senses. Nevertheless, the case is not utterly hopeless”
(Newton 1999, 414). He then presents the example of the two balls and
considers it twice: first as a rotation in empty space, or an “immense
vacuum,” second as a rotation relative to “some distant bodes [that] were set
in that space, and maintained given positions with respect to one another, as
the fixed stars do in the regions of the heavens” (1999, 414). This second
consideration points toward Newton’s later procedure in Book 3, which
begins by recording the initial Keplerian “Phenomena” as motions relative to
the fixed stars centered on the relevant primary bodies (the sun, Jupiter,
Saturn, and the earth) and concludes by establishing a privileged relative
space fixed at the center of gravity of the solar system for describing all true
motions therein—in modern terms, an (approximately) inertial (nonrotating)
frame of reference. This procedure begins from perceptually observable relative or apparent motions, and, by applying the Laws of Motion and the law
of universal gravitation (which is established at the same time and by the same
argument), it finally succeeds in determining, from the phenomena, motions
that are by no means merely relative or apparent.11
10
There are two references to this paragraph in the Phenomenology. The first is in the
remark to the second proposition: “Newton’s Scholium to the definitions he has prefixed to his
Principia may be consulted on this subject, towards the end, where it becomes clear that the
circular motion of two bodies around a central point (and thus also the axial rotation of the
earth) can still be known by experience even in empty space, and thus without any empirically
possible comparison with an external space; so that a motion, therefore, which is a change of
external relation in space, can be empirically given, even though this space is not itself empirically given, and is no object of experience” (557–58). The second is in the general remark to the
Phenomenology, where Kant appends a footnote that begins by quoting Newton’s first two
sentences and continues: “He then lets two spheres connected by a cord revolve around their
common center of gravity in empty space, and shows how the actuality of their motion, together
with its direction, can nonetheless be discovered by means of experience. I have attempted to
show this also in the case of the earth moved around its axis, in somewhat altered circumstances” (562).
11
This famous argument begins with the Phenomena initiating Book 3 and concludes with
Propositions 11–13. In the first corollary to Proposition 14, Newton is then able to conclude
(rather than presuppose) that the fixed stars are at rest. He appears to be anticipating just this
argument in the concluding paragraph of the Scholium to the Definitions: “But in what follows,
a fuller explanation will be given of how to determine true motions from their causes, effects,
and apparent differences, and, conversely, of how to determine from motions, whether true or
apparent, their causes and effects. For this was the purpose for which I composed the following
treatise” (1999, 415).

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It is precisely this Newtonian procedure, on my reading, that Kant has in
mind in his Phenomenology chapter—which describes a process of reducing
all motion and rest to “absolute space” that is intended to generate a determinate distinction between true and merely apparent motion despite the
acknowledged relativity of all motion as such to some given empirically
specified relative space or reference frame. Kant begins by considering our
position on the earth, indicates how the earth’s state of true (axial) rotation
can nonetheless be empirically determined, and concludes by considering the
cosmos as a whole—together with the “common center of gravity of all
matter” (563)—as the ultimate relative space (or reference frame) for correctly
determining all true motion and rest. The upshot, from a modern point of
view, is that Kant takes his three Laws of Mechanics (implicitly) to define a
privileged inertial frame of reference in which they are satisfied. These laws
are thereby revealed to be (synthetic) a priori principles governing the (true or
actual) motion of matter, and Kant has thus explained their “a priori sources”
in the sense of the paragraph on the relationship between metaphysics and
mathematical natural philosophers from the Preface (472). While taking the
argument of Book 3 as his model, Kant again diverges from Newton in
viewing the crucial notions of absolute space, time, and motion to be constituted or “constructed” within our perceptual experience rather than being
already well defined independently of this experience.12
Yet the “construction” that Kant envisions here, unlike the construction of
the addition or composition of motions that he describes in the sole proposition of the Phoronomy, cannot be carried out in pure intuition. On the
contrary, the aim of the Phenomenology is to explain how the true motions
can be determined from the appearances or phenomena—and thus in what
Kant calls empirical intuition.13 In particular, we cannot, as in the Phoronomy,
consider only the motions of mere mathematical points; rather, we must
consider full-fledged moving bodies in space, which fill the space they occupy
with some empirical content. For Kant, therefore, we must also presuppose
the further conceptual determinations of matter introduced in the Dynamics
and Mechanics: the property of filling a space (through impenetrability)
12
I develop this reading in most detail in Friedman, forthcoming. See Friedman 2012 for
my most recent discussion already in print. But one can also find detailed discussion in my
introduction to Kant 2004 and, e.g., in Friedman 1992a and 1992b. Observe, however, that
Kant’s three Laws of Mechanics—as derived from the three Analogies of Experience—diverge
from Newton’s three Laws of Motion (see note 8 above). I shall return to this point below.
13
Explaining how we can transform mere appearances of (relative) motions into determinate
experience of (true) motions is precisely how the motion of matter becomes an object of experience
in the Phenomenology: “[A]ll motion and rest must be reduced to absolute space, if the
appearance thereof is to be transformed into a determinate concept of experience (which unites
all appearances)” (560).

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introduced in the former and the property of inertia introduced in the latter.14
We are thereby involved with both the traditional concept of quantity of
matter that Newton introduces in the first definition of the Principia and
the new concept of what we now call inertial mass that Newton introduces
in the third. Kant’s problem, once again, is to explain how these notions,
too, acquire the structure of a measureable mathematical magnitude, and
to do this, in a certain sense, a priori. He is guided by the intricate system
of empirical connections that Newton establishes between quantity of
matter, weight, and inertial mass—and, most importantly, between these
notions and what we now call gravitational mass. Nevertheless, Kant again
gives this Newtonian argument his own characteristically transcendental
interpretation.
2. QUANTITY OF MATTER IN THE DYNAMICS
Kant characterizes quantity of matter in the second explication of the Mechanics
as “the aggregate of the movable in a determinate space” (538), and he
characterizes both mass and quantity of motion in the remainder of this explication. I shall return to these last two concepts in what follows, but I first want
to observe that the concept of quantity of matter (but not yet its explication)
is introduced much earlier, in the previous Dynamics chapter. It is initially
introduced in the first remark to the fourth proposition of the Dynamics—
which states that “[m]atter is divisible to infinity, and, in fact, into parts such that
each is matter in turn” (503). In this remark, Kant considers the expansion
and compression of such infinitely divisible (and therefore continuously distributed) matter in terms of the representation of contact at an infinitely small
distance, “which must also necessarily be so in those cases where a greater or
smaller space is to be represented as completely filled by one and the same
quantity of matter, that is, one and the same quantum of repulsive forces” (505;
emphasis added).15 The next occurrence, however, is in the fifth proposition,
14
See notes 4 and 5 above, together with the paragraph to which they are appended.
Observe from the former note that the Prolegomena characterizes the first property (here impenetrability) as that “on which the empirical concept of matter rests” (4: 295; emphasis added), and
recall from the latter note that no further conceptual determination is introduced in the
Phenomenology.
15
The bulk of the first remark leading up to this statement consists of a refutation of Kant’s
earlier conception of the filling of space in his precritical Physical Monadology (1756), according to
which a “physical monad” can fill the space that it occupies by the “sphere of activity” of
repulsive force exerted only by the central point of the (spherical) space in question. The goal
of the Physical Monadology, in fact, was to explain how a material substance or physical monad
could fill a space that is geometrically infinitely divisible without being divisible itself. In Kant’s
critical conception articulated in the Dynamics, by contrast, this is impossible: “[T]he possible
physical division of the substance that fills space extends as far as the mathematical divisibility

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which argues that repulsive force alone cannot sufficiently explain matter’s
filling of space: “[M]atter, by its repulsive force alone (containing the ground
of impenetrability), would [through itself] alone and if no other moving force
counteracted it, be confined within no limit of extension; that is, it would
disperse itself to infinity, and no specified quantity of matter would be found in
any specified space” (508; emphasis added).
Kant concludes that the relevant counteracting force must be a fundamental or original force, so that, in addition to the fundamental force of repulsion,
“an original attraction is attributed to all matter, as a fundamental force
belonging to its essence” (509). This is the famous “balancing argument.”
Matter has an original tendency to expand (and thereby resist compression),
but if this were the only fundamental force belonging to the essence of matter,
it would “disperse itself to infinity” and thus would not fill any space after at
all. Therefore, we need a second fundamental force of attraction that can
resist this expansive tendency and thereby ensure that matter can in fact fill a
space to a determinate degree. The determinate degree in question—that is,
a “specified quantity of matter” in a “specified space”—results precisely by a
balancing or state of equilibrium between the two fundamental forces.16
I shall not pause over the details of this argument here.17 I instead want to
emphasize that Kant proceeds to link his fundamental force of attraction to
Newtonian universal gravitation. The eighth (and final) proposition of the
Dynamics attributes the property of universality to this force: “The original
attractive force, on which the very possibility of matter as such rests, extends
immediately to infinity throughout the universe, from every part of matter to
every other part” (516). The following second note then makes the connection
with universal gravitation explicit:
The original attraction is proportional to the quantity of matter and extends to
infinity. Therefore, the determinate filling, in accordance with its measure, of a

of the space filled by matter. But the mathematical divisibility extends to infinity, and thus so
does the physical [divisibility] as well. That is, matter is divisible to infinity, and, in fact, into
parts such that each is itself material substance in turn” (504).
16
Kant runs a converse version of the balancing argument in the sixth proposition: “No
matter is possible through mere attractive force without repulsion” (510). The argument reads:
“[W]ithout repulsive forces, through mere convergence, all parts of matter would approach one
another unhindered, and would diminish the space that they occupy. But since, in the case
assumed, there is no distance of the parts at which a greater approach due to attraction would
be made impossible by a repulsive force, they would move toward one another so far, until no
distance at all would be found between them; that is, they would coalesce into a mathematical
point, and space would be empty, and thus without matter” (511).
17
Again, I provide a detailed discussion in Friedman, forthcoming; see also Friedman 2010,
601–10.

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space by matter, can in the end be effected only by the attraction of matter extending
to infinity, and imparted to each matter in accordance with the measure of its
repulsive force.
The action of the universal attraction immediately exerted by each matter on all
matters, and at all distances, is called gravitation; the tendency to move in the
direction of greater gravitation is weight. (518)

The fundamental force of attraction is universal (exerted by each part of
matter on every other part), proportional to mass, and the cause of weight;
and for precisely this reason it can be called gravitation—that is, universal
gravitation.18
The preceding seventh proposition, however, is even more striking.19 In his
second remark to this proposition, Kant not only explicitly refers to the
crucial Proposition 6 of Book 3 of the Principia, he also ventures a rare
criticism of Newton for leaving it open (in the Queries to the Opticks and
elsewhere) that the action of universal gravitation may be due to the pressure
exerted by an external aether. Kant’s seventh proposition, on the contrary,
insists that the action of the universal attraction is a genuine action at a
distance: “The attraction essential to all matter is an immediate action of matter on
other matter through empty space” (512). Moreover, the argument for this
proposition, as emphasized in the second remark, crucially involves the
concept of quantity of matter and the property of universal attraction of being
proportional to this quantity: “[H]ow could [Newton] ground the proposition
that the universal attraction of bodies, which they exert at equal distances
around them, is proportional to the quantity of their matter, if he did not
assume that all matter, merely as matter, and through its essential property,
exerts this moving force?” (514). This, it appears, is precisely why Kant then
refers to Proposition 6 of Book 3, which generalizes the (Galilean) equality of
the acceleration of fall for all bodies near the surface of the earth to the
gravitational fields of any planet: “All bodies gravitate toward each of the planets, and
at any given distance from the center of any one planet the weight of any body whatever
toward that planet is proportional to the quantity of matter which the body contains” (1999,
806).

18
Compare the end of the general remark to the Dynamics: “[N]o law of either attractive
or repulsive force may be risked on a priori conjectures. Rather, everything, even universal
attraction as the cause of weight, must be inferred, together with its laws, from data of
experience” (534). This passage occurs toward the end of Kant’s polemic against the mechanical
philosophy mentioned in note 3 above, and it thus links his conception of universal attraction
with the methodology of Newtonian “experimental philosophy.”
19
There are four propositions in the Dynamics governing the fundamental force of attraction: the fifth, sixth, seventh, and eighth. The fifth, discussed above, initiates the balancing
argument. For the sixth, see note 15 above.

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Proposition 6 amounts to the claim that the specifically gravitational force
on an attracted body (i.e., its “weight”) is always proportional (at a given
distance) to this body’s mass. This claim, as Kant emphasizes, does not hold
for other forces such as magnetism, and, accordingly, it is quite distinct from
what he calls the mechanical law of the equality of action and reaction
(whereby the accelerations mutually produced in two interacting bodies by
any force are in inverse proportion to their masses). What is at issue in
Proposition 6 is the singular property of gravitational force that it produces
the same acceleration in any body whatsoever at equal distances—entirely
independently, in particular, of the attracted body’s inertial mass. And so it
essentially depends, in particular, on what we now call the equality of inertial
and passive gravitational mass. Kant’s claim in the above quotation from the
second remark to his seventh proposition (514), however, depends on what we
now call the equality of inertial and active gravitational mass: the property that
the acceleration produced in the attracted body is always proportional (at a
given distance) to the attracting body’s mass.20
Newton establishes this property in Proposition 7 of Book 3: “Gravity exists
in all bodies universally and is proportional to the quantity of matter in each” (1999, 810).
The proof depends on Proposition 69 of Book 1, which appeals to the
Galilean property (the equality of inertial and passive gravitational mass), the
universality of the attraction in question (in this case gravitation), and, crucially, the equality of action and reaction expressed in the Third Law of
Motion. Consider the system of bodies consisting of Jupiter, Saturn, and their
satellites. By the Galilean property, there are what Howard Stein (1967) has
helpfully called “acceleration fields” around the two planets, whose actions
are independent of the mass of any body “falling” in such a field. By universality, these fields extend (in the inverse-square proportion to distance) arbitrarily far into space and, in particular, to the two planets themselves.
Applying the equality of action and reaction to this interaction, however,
implies that the acceleration produced by Jupiter on Saturn, for example, is
proportional (at the given distance) to the inertial mass of Jupiter (and conversely for Saturn)—and so, by measuring the acceleration of one of Jupiter’s
satellites in the same acceleration field, we are thereby measuring the inertial
20
In the finished theory of universal gravitation, the force between any two bodies A and B
is directly proportional to the product of their masses and inversely proportional to the square
of the distance between them. In modern notation, if FAB is the (gravitational) force that B exerts
on A, then (by the Second Law of Motion) FAB = mAaA, where FAB = GmAmB/r2. That the same
mA occurs in both equations expresses the equality of inertial and passive gravitational mass, and
this implies that aA (at a given distance) is independent of mA. It then follows from the finished
theory of universal gravitation that aA is directly proportional (at a given distance) to mB, now
understood as the active gravitational mass of B (i.e., the source of the gravitational force on A).
But it is precisely this last proportionality that has yet to be established in Newton’s argument.

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mass of Jupiter as well (and similarly for Saturn and its satellites). In other
words, the equality of inertial and passive gravitational mass, in the context of
universality and the Third Law of Motion, implies the equality of inertial and
active gravitational mass as well.21
At the end of the second remark to his seventh proposition, Kant appeals
to precisely this example to clinch the argument that Newton’s professed
agnosticism concerning action at a distance “sets him at variance with
himself” (515).22 For Newton appeals to such measurements in determining
the quantities of matter of the primary bodies in the solar system (the sun,
Jupiter, Saturn, and the earth) in the corollaries to Proposition 8. Moreover,
this same argument—the heart of the argument for universal gravitation—
centrally involves the “moon test” carried out in Proposition 4, whereby the
acceleration of the moon toward the earth is “brought down” to the earth’s
surface (using the inverse-square proportion) and there found to be equal to
the Galilean acceleration of terrestrial gravity.23 Terrestrial weight, in this
way, is projected into the heavens and thereby extended to a universal
measure of quantity of matter for all bodies whatsoever, regardless of their
relation to the surface of the earth. In the first remark to his seventh proposition, Kant alludes to the moon test as well, and so this crucial proposition
of the Dynamics—that the “attraction essential to all matter is an immediate
action of matter on other matter through empty space” (512)—ultimately
21
Let the acceleration field on Saturn’s moons be given by a1 = kS/r12 and the acceleration
field on Jupiter’s moons by a2 = kJ/r22, where kS and kJ are constants characterizing the absolute
strengths of the fields independently of distance. Since the two acceleration fields extend far
beyond the regions of their satellites, we also have accelerations aJ = kS/r2 of Jupiter and
aS = -kJr2 of Saturn, where r is the distance between them. But, according to the Third Law,
mJaJ = -mSaS. Therefore (using the above two equations for aJ and aS), mS/mJ = -aJ/aS = kS/kJ:
the absolute strengths of the two fields (i.e., the active gravitational masses of their respective
sources) are proportional to the (respective) inertial masses. For further discussion in the context
of the Metaphysical Foundations see Friedman 1992b, 152–59.
22
Kant writes: “[I]t is clear that the offense taken by his contemporaries, and perhaps even
by Newton himself, at the concept of an original attraction sets him at variance with himself; for
he could absolutely not say that the attractive forces of two planets, e.g., of Jupiter and Saturn,
manifested at equal distances of their satellites (whose mass is unknown), are proportional to the
quantity of matter of these heavenly bodies, if he did not assume that they attracted other matter
merely as matter, and thus according to a universal property of matter” (515).
23
The moon test figures centrally in the argument for Proposition 6. After testing the
Galilean property of gravity near the surface of the earth by means of experiments with
pendulums, Newton continues: “Now, there is no doubt that the nature of gravity toward the
planets is the same as toward the earth. For imagine our terrestrial bodies to be raised as far as
the orbit of the moon and, together with the moon, deprived of all motion, to be released so as
to fall to the earth simultaneously; and by what has already been shown, it is certain that in
equal times these falling terrestrial bodies will describe the same spaces as the moon, and
therefore that they are to the quantity of matter in the moon as their own weights are to its
weight” (1999, 807). Newton proceeds to derive the Galilean property for Jupiter and Saturn
from the inverse-square proportion (which entails the acceleration-field property).

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MICHAEL FRIEDMAN

comprehends all of the most important properties of Newtonian universal
gravitation.24
3. QUANTITY OF MATTER IN THE MECHANICS
Let us now return to the second explication of the Mechanics, which, as
observed, characterizes not only quantity of matter but also mass and quantity of
motion:
The quantity of matter is the aggregate of the movable in a determinate space. This
same [quantity of matter], insofar as all its parts in their motion are considered as
acting (moving) simultaneously [zugleich], is called mass [Masse], and one says that a
matter acts in mass, when all its parts, moved in the same direction, simultaneously
[zugleich] exert their moving force externally. A mass of determinate figure is called
a body (in the mechanical meaning). The quantity of motion (estimated mechanically) is
that which is estimated by the quantity of the moved matter and its velocity together
[zugleich]; phoronomically it consists merely in the degree of velocity. (537)

The first concept (quantity of matter) corresponds to the category of substance,
and it thereby corresponds to Kant’s First Law of Mechanics (541–43).25 The
second concept (mass) is explicitly causal (the aggregate of parts in question is
considered as “acting” on another such aggregate by “exerting their moving
force externally”), and it thereby corresponds to Kant’s Second Law (543–
44). The third concept (quantity of motion) is then crucial for considering
what Kant calls the “communication of motion” between the two, via a
“construction” of their interaction [Wechselwirkung] in accordance with the
equality of action and reaction [Wirkung und Gegenwirkung], and it thereby
corresponds to his Third Law (544–51). I shall concentrate in what follows on
24
In the first remark to his seventh proposition, Kant is replying to the objection that action
at a distance is impossible because a body cannot act at a place where it is not. He responds with
the example of the earth and the moon, first considered as interacting over a distance of “many
thousands of miles” and then “in contact with one another,” such that their “places” are separated
only “by the sum of their radii” (513). In the remark to his fifth proposition, Kant alludes to
Proposition 8 of Book 3, according to which a sufficiently uniform spherical distribution of matter
will attract a body external to it precisely as if all of its matter were concentrated at its central
point—so that, as Newton says, the “[inverse-square] proportion which holds exactly enough at
very great distances” cannot “be markedly in error near the surface of the planet” (1999, 811). In
the remark to his fifth proposition, Kant illustrates with “the attractive force of all parts of the
earth,” which can act “in no other way, than as if it were wholly united in the earth’s center”
(509)—and he thereby alludes to both Proposition 8 (and its corollaries) and the moon test.
25
Kant’s three Laws of Mechanics—articulated in the second, third, and fourth
propositions—are the conservation of the total quantity of matter, inertia, and the equality of
action and reaction (note 8 above). I observed that these laws, as derived from the three
Analogies of Experience, diverge from Newton’s three Laws of Motion (note 12 above). The
most important points of divergence are that Newton does not state the conservation of quantity
of matter, while Kant, for his part, does not state the Second Law of Motion.

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the category of substance and the first part of Kant’s explication, touching on
the remaining parts only in passing.
The second remark to Kant’s first proposition of the Mechanics states that
“the quantity of matter is the quantity of substance in the movable” (540) and
that the quantum of substance “can have no other magnitude than that of the
aggregate of homogeneous [elements] external to one another” (541). The
remark to the second proposition explains that the spatially extended aspect
of quantity of matter is “essential” in Kant’s argument for conservation: “For,
since all quantity of an object possible merely in space must consist of parts
external to one another, these [parts], if they are real (something movable), must
therefore necessarily be substances” (542). This links the second proposition
of the Mechanics to the earlier fourth proposition of the Dynamics, whose
conclusion states that every part of a material substance is also substance:
“[M]atter is divisible to infinity, and, in fact, into parts such that each is itself
material substance in turn” (504; see note 15 above).
The point of this earlier proposition is to establish the physical (and not
merely mathematical) infinite divisibility of material substance, which is therefore continuously distributed in the space that it fills. Kant’s characterization of
quantity of matter in the Mechanics thereby emphasizes that matter is precisely
such a space-filling continuum. There can be no such thing, for Kant, as an
isolated point-mass, and so, more generally, he now rejects all versions of an
atomism of discrete force-centers—whether that of Boscovich, for example, or
his own precritical Physical Monadology.26 By the same token, he also rejects all
forms of hard-body atomism, according to which the ultimate structure of
matter consists of absolutely hard, dense, and impenetrable corpuscles distributed in empty space—so that, in particular, the quantity of matter in a given
spatial volume is given simply by the volume of the absolutely hard matter
contained therein. Matter is compressible, on this view, only in virtue of
containing empty interstices within the volume in question; it then becomes
increasingly dense as these interstices are “squeezed out” until only the
absolutely hard matter is left.27
26
See again note 15 above. It is because there is no such thing, for Kant, as an isolated
point-mass that, if matter “coalesce[d] into a mathematical point,” “space would be empty, and
thus without [quantity of] matter” (511; see note 16 above).
27
Although he does not, of course, accept empty space, this conception of density and its
relationship to compression was paradigmatically exemplified in the early modern period by
Descartes, who, in this respect, is in fundamental agreement with the more classical version of
corpuscularianism postulating atoms and the void. It was also paradigmatically exemplified, for
Kant, by the views of his friend and correspondent J. H. Lambert, whom Kant explicitly names
in the remark to his first proposition of the Dynamics (497–98) as representative of that view of
impenetrability or solidity he is now concerned to reject. It was also paradigmatically exemplified,
finally, by the “solid, massy, hard, impenetrable, movable Particles” of Query 31 to Newton’s
Opticks (1952, 400). I discuss Kant’s relationship to all of these views in Friedman, forthcoming.

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MICHAEL FRIEDMAN

Kant rejects this last view in his first remark to the fourth explication of the
Dynamics, where he takes it to follow from “the merely mathematical concept
of impenetrability” according to which “matter is not capable of compression
except insofar as it contains empty spaces within itself” (502). He rejects it
again, under the rubric of the “mechanical natural philosophy,” in the general
remark to the Dynamics (532–33; see note 3 above).28 But what is most
important, in the present context, is that Kant’s rejection of precisely this view
underlies his proof of the pivotal first proposition of the Mechanics, according
to which “[t]he quantity of matter, in comparison with every other matter, can
be estimated only by the quantity of motion at a given velocity” (537):
Matter is infinitely divisible; so its quantity cannot be immediately determined by an
aggregate of its parts. For even if this occurs in comparing the given matter with
another of the same kind, in which case the quantity of matter is proportional to the
size of the volume, it is still contrary to the requirement of the proposition, that it is
to be estimated in comparison with every other (including the specifically different).
Hence matter cannot be validly estimated, either immediately or mediately, in
comparison with every other, so long as we abstract from its inherent motion. Therefore, no other generally valid measure [of matter] remains except the quantity of its
motion. But here the difference of motion, resting on the differing quantity of
matters, can be given only when the velocity of the compared matters is assumed to
be the same; hence, etc. (537–38)

On the mechanical concept of filling a space that Kant opposes, one could in
principle transform any species of matter into any other by compression or
expansion, whereby the proportion of absolutely dense matter is increased or
diminished. On Kant’s own dynamical conception of filling a space, by
contrast, this kind of comparison is impossible between matters of specifically
different kinds (between water and mercury, for example), and so, he concludes, the only way to compare quantities of matter in general is through
their “quantit[ies] of motion at a given velocity.”29
What exactly does Kant have in mind by the estimation of quantity of
matter by its “quantity of motion at a given velocity” (537), and why, since
quantity of motion is “estimated by the quantity of moved matter and its velocity
together” (537), is the first estimation not viciously circular? Kant addresses
28
Kant also rejects this view in the Anticipations of Perception of the first Critique, where he
describes it as an (unwarranted) presupposition adopted by most “investigators of nature”:
“Almost all investigators of nature, because they observe a large difference in the quantity of
matter of different kinds at the same volume (partly by means of the moment of gravity, or
weight, partly by means of the moment of resistance to other moved matters), unanimously
conclude from this that the volume in question (extensive magnitude of the appearance) must in
all matters be empty in different amounts” (A173/B215).
29
Compare Kant’s earlier discussion of density in the first numbered subsection of his
general remark to the Dynamics (525–26).

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these questions in the following remark: “The quantity of the movable in
space is the quantity of matter; but this quantity of matter (the aggregate of
the movable) manifests itself in experience only by the quantity of motion at the
same velocity (for example, by equilibrium [Gleichgewicht])” (540). There is
thus an important difference, Kant says, between “the explication of a
concept, on the one hand, and that of its application to experience, on the other”
(540; emphasis added). Martin Carrier (2001) has convincingly argued that
Kant here has primarily in mind the procedure of terrestrial weighing in an
equal arm balance, for example, where, in accordance with Galileo’s law, the
downward accelerations (or infinitesimal velocities) of the two weights are
necessarily equal, so that the pressures (gravitational forces) exerted on the
two arms are proportional to the masses.
It appears, then, that Kant’s parenthetical example of how quantity of
matter manifests itself in experience involves a determination in static equilibrium by means of the balance.30 We already know, however, that extending
this kind of static determination for bodies near the surface of the earth (under
the influence of Galilean or terrestrial gravity) into the heavens so as thereby
to obtain a universal measure of quantity of matter for all bodies whatsoever
is a central achievement of the theory of universal gravitation for both
Newton and Kant. It is no wonder, therefore, that Kant explicitly considers
this extension in the same remark.31 After stating that “the quantity of substance in a matter must only be estimated mechanically, i.e., by the quantity
of its inherent motion, and not dynamically, by the magnitude of the originally moving forces” (541), Kant qualifies this assertion in the case of universal attraction:
Nevertheless, original attraction, as the cause of universal gravitation, can still yield a
measure of the quantity of matter, and of its substance (as actually happens in the
comparison of matters by weighing), even though a dynamical measure—namely,
attractive force—seems here to be the basis, rather than the attracting matter’s own
inherent motion. But since, in the case of this force, the action of a matter with all
its parts is exerted immediately on all parts of another, and hence (at equal distances)
is obviously proportional to the aggregate of the parts, the attracting body also
thereby imparts to itself a velocity of its own inherent motion (by the resistance of the
30
Compare a parallel passage in Kant’s precritical essay on Negative Magnitudes (1763),
where, after discussing the conflict of two moving forces in a state of rest, he continues: “In
precisely the same way, the weights on the two arms of the balance are at rest, if they are placed
on the lever in accordance with the laws of equilibrium [Gleichgewicht]” (2: 199).
31
See note 23 above, together with the paragraph to which it is appended. Here is where I
differ from Carrier, who argues (2001 §6) that the discussion of estimating quantity of matter in
the Mechanics is limited to the terrestrial procedure of weighing and does not involve celestial
procedures for determining this quantity via universal gravitation. For a discussion of how
Newton employed the balance as a model for universal gravitation see Machamer, McGuire,
Kochiras (this volume).

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MICHAEL FRIEDMAN

attracted body), which, in like external circumstances, is exactly proportional to the
aggregate of its parts; so the estimation here is still in fact mechanical, although only
indirectly so. (541)

Kant here returns to the procedure discussed in the second remark to the
seventh proposition of the Dynamics for determining the quantity of matter
of a celestial body (such as Jupiter or Saturn) by means of the attractive force
it exerts on other such bodies (their satellites).
Consider, for example, determining the quantity of matter of Jupiter from
the acceleration produced on one of its moons. Inferring the mass of Jupiter
from the moon’s acceleration is a dynamical procedure in Kant’s sense,
because no motion of Jupiter is yet being considered.32 Nevertheless, Jupiter’s
force of attraction, by the resistance of its moon and the equality of action and
reaction, produces a corresponding acceleration—and therefore change of
momentum—in Jupiter itself. And this change of momentum, in the given circumstances, is also proportional to Jupiter’s mass. Just as the moon falls toward Jupiter,
Jupiter falls toward its moon—and Jupiter’s “weight” toward this moon (like
all gravitational forces) is, at a given distance, directly proportional to Jupiter’s
mass.33 So determining the quantity of matter of an attracting body by the
acceleration it produces in an attracted body is mechanical in Kant’s sense,
because it ultimately rests, like all mechanical comparisons, on an exchange
of momentum or quantity of motion between the two interacting bodies (note
32 above). It is only “indirectly” mechanical, however, because the change of
momentum of the attracting body itself, despite the fact that it is indeed
proportional to this same body’s quantity of matter, is not what is actually
measured.
4. KANT AND NEWTON ON QUANTITY OF MATTER
In the second explication and the first proposition of the Mechanics, Kant
begins with his characterization of quantity of matter as the aggregate of the
movable in a given space. This points back to the fourth proposition of the
32
Kant explains at the beginning of the Mechanics that, while a fundamental dynamical
force (attraction or repulsion) is “an originally moving force for imparting [erteilen] motion[; i]n
mechanics, by contrast the force of a matter set in motion is considered as communicating [mitteilen]
this motion [i.e., momentum] to another” (536). From the point of view of mechanics, therefore,
both bodies must be viewed as moving in accordance with the equality of action and reaction.
33
The gravitational force acting on Jupiter is given by mJaJ = GmJmM/r2, where mM is the
mass of the moon in question. Holding mM and r constant, then, it follows that mJaJ is
proportional to mJ. This is precisely analogous, in the context of universal gravitation, to the way
in which the gravitational force or weight of a falling body near the earth’s surface is proportional to its mass. Both depend on the Galilean property of gravitational force (equal accelerations produced at equal distances) that figures centrally in Propositions 6 and 7 of Book 3 of the
Principia: see notes 18–21 above, together with the paragraphs to which they are appended.

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Dynamics and thereby emphasizes that material substance is necessarily
infinitely divisible and thus continuously fills the space that it occupies. We
are therefore led to the balancing argument of the fifth and sixth propositions,
according to which both an original repulsion and an original attraction
belong to the essence of matter—for, only so, Kant argues, can a “specified
quantity of matter” be found in a “specified space” (508).34 This argument, as
we have seen, leads to the seventh and eighth propositions, which, by means
of the Galilean property of original attraction, support the identification of
quantity of matter with both weight and inertial mass as well as with both
passive and active gravitational mass. It is precisely this sequence of identifications, for Kant, that explains how it is possible to turn the traditional
concept of quantity of matter into a measurable mathematical magnitude, by
first connecting it with the traditional static concept of weight and then
extending this concept, in turn, into a universally applicable measure of mass
valid for all bodies in the universe independently of their relation to the
surface of the earth. Kant incorporates these key properties of Newtonian
universal gravitation into his metaphysical foundation for natural science by
taking them to be constitutive of the empirical concept of matter that he is in
the process of articulating—“in relation to the pure intuitions of space and
time, and in accordance with laws that already essentially attach to the
concept of nature in general” (472).35
Newton characterizes quantity of matter in the first definition of the Principia:
“Quantity of matter is a measure of matter that arises from its density and volume jointly”
(1999, 403). His comments on this definition show the same kind of concern
with the relationship between density and the possibility of compression that
are found in Kant’s discussion and other traditional discussions of the time:
If the density of air is doubled in a space that is also doubled, there is four times as
much air, and there is six times as much if the space is tripled. The case is the same
for snow and powders condensed by compression or liquefaction, and also for all
bodies that are condensed in various ways by any causes whatsoever. For the
34
See notes 15 and 16 above, together with the paragraphs to which they are appended.
The argument for physical divisibility in the fourth proposition explicitly depends on repulsive
force: “[I]n a space filled with matter, every part of it contains repulsive force, so as to
counteract all the rest in all directions, and thus to repel them and be repelled by them, that is,
to be moved a distance from them. Hence every part of a space filled with matter is in itself
movable, and thus separable from the rest as material substance by physical division” (503–04).
It thereby sets the stage for the fifth proposition.
35
See notes 2–5 above, together with the paragraphs to which they are appended. The
quotation from the Prolegomena in note 4 above suggests that the specifically empirical content of
Kant’s concept of matter enters in with his (dynamical) concept of impenetrability—which, as
it later becomes clear, is developed in explicit contrast with the “merely mathematical concept
of [absolute] impenetrability” (502) that he attributes to most “investigators of nature” (A173/
B215): see notes 27 and 28 above, together with the paragraphs to which they are appended.

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MICHAEL FRIEDMAN

present, I am not taking into account any medium, if there should be any, freely
pervading the interstices between the parts of bodies. Furthermore, I mean this
quantity whenever I use the term ‘body’ or ‘mass’ in the following pages. It can
always be known from a body’s weight, for—by making very accurate experiments
with pendulums—I have found it to be proportional to the weight, as will be shown
below. (1999, 403–04)

These comments, however, exhibit several important points of difference
from the corresponding discussion in Kant’s second explication. First, while
Kant rejects the empty interstices view of density and compression (see note
27 above, together with the paragraph to which it is appended), Newton is
careful, “for the present,” to leave it open. Second, while Kant also characterizes mass explicitly, Newton simply equates it with quantity of matter.
Third, while Kant does not mention weight in his second explication but only
introduces it implicitly in his first proposition, Newton explicitly introduces
weight as the proper measure of quantity of matter and alludes to experimental evidence on behalf of this claim.
The experiments in question play a central role in Proposition 6 of Book 3
concerning the Galilean property of gravitational attraction and thus, as we
would now put it, the equality of inertial and (passive) gravitational mass.
What is most important, in the present context, is that Newton carefully
constructs pendulum bobs of equal weight and volume filled with a variety of
specifically different materials (e.g., gold, silver, glass, wood, and water), each
of which has a different specific density. He observes (1999, 806–07) that such
pendulums undergo synchronous oscillations over very long periods of time,
and it follows, by the corollaries to Proposition 24 of Book 2, that their
quantities of matter—that is, their (inertial) masses—are proportional to their
weights. So Newton is here not only verifying the equality of the acceleration
of fall (at a given distance) independently of the differing (inertial) masses of
the bodies in question, he is also establishing such independence with respect
to their differing specific densities as well.36 He thereby links his notions of
density and quantity of matter to questions concerning the ultimate structure
of matter, and he puts himself in a position, in particular, to develop one more
argument against the Cartesian commitment to a subtle aetherial medium
“freely pervading the interstices between the parts of bodies” (1999, 403).37
36
Proposition 24 of Book 2 belongs to Newton’s discussion of hydrostatics and concerns the
motions of pendulums in a buoyant (and possibly resisting) medium (such as water or air).
Newton thus characterizes the densities of his bobs in terms of their buoyant or specific weights
using the well-known method of Archimedes, and he extends the already-known results for
motion in a vacuum to motion in air. I discuss this in more detail in Friedman, forthcoming, and
I am here (and in the next note) especially indebted to George E. Smith.
37
This argument occurs at the end of the General Scholium to Section 6 of Book 2, which
is initiated by Proposition 24. Newton argues (1999, 722–23) against the Cartesian opinion that

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Although he does not discuss Newton’s first definition or pendulum experiments, Kant, on my reading, has reason to have qualms concerning Newton’s
insertion of the empty interstices conception of matter into his comments on
the definition of its quantity. Such qualms, in fact, are analogous to those that
Kant does express concerning Newton’s agnosticism about action at a
distance—which, according to Kant, “sets him at variance with himself” (515;
see note 22 above). For, if matter did consist of absolutely hard (absolutely
dense) matter separated by empty interstices, there would be a purely geometrical measure of its quantity (i.e., the volume of such absolutely dense
matter) that is already well defined independently of the empirical measures
that Newton himself provides. One would eventually have to establish a
connection between these empirical measures (such as weight) and the underlying (so far completely unobservable) geometrical measure, and this Newton
never does. Rather, he introduces a new dynamical quantity of (inertial) mass
along with his Three Laws of Motion and shows, by his pendulum experiments, that this quantity (near the surface of the earth) is always proportional
to weight.38 Propositions 6 and 7 of Book 3 then extend this quantity into the
heavens, where it becomes equated with both active and passive gravitational
mass. So it is only the theory of universal gravitation, in the end, that extends
the traditional static quantity of weight into a universal empirical measure
that is valid for all matter whatsoever (regardless of its relation to the earth).
Kant, by contrast, is defining quantity of matter in such a way that, on the
one side, the empty interstices view of this quantity is excluded, and, on the
other, the very empirical measures that Newton has developed are thereby
incorporated into the concept of matter (which, for Kant, is explicitly empirical). Thus, in particular, Kant links quantity of matter to both (inertial) mass
and weight in the first proposition of the Mechanics. Precisely because this
quantity cannot, in general, be estimated purely geometrically (in terms of
volume and empty space), the only remaining option, Kant argues, is “the
quantity of motion at a given speed” (537)—that is, the momentum of the
moved body in question. A purely geometrical estimation must therefore be
replaced by what Kant calls a “mechanical” one, and he appeals to the
Galilean property to connect quantity of matter in his sense with weight
(terrestrial gravity). Moreover, since Kant has also built a universal and
original attraction into his (dynamical) concept of matter, the Newtonian
extrapolation of (terrestrial) weight into the heavens then follows as well.
“a certain aetherial medium” exerts resistance to the motions of bodies by penetrating into them
through their pores: he finds that any such resistance would be extremely small in comparison
with the resistance (of the air) exerted on the external surfaces of his pendulum bobs.
38
The proof of Proposition 24 of Book 2 depends entirely on the Second Law of Motion and
thus on equating mass in the sense of the Second Law with quantity of matter in the sense of the
first definition.

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Kant is not, on my reading, attempting to prove a priori what Newton has
established empirically. Just as in his reinterpretation of absolute space, time,
and motion, Kant is using precisely the empirical measures that Newton has
developed to define or “construct” a possibly problematic unobservable entity
(absolute space, quantity of matter) from materials that are themselves
unproblematically observable (the observed relative motions, the now available Newtonian measurements of this quantity). Nor is the conception of
matter that Kant articulates in the Dynamics a speculative theory of its
ultimate internal structure in competition with the then extant forms of
atomism (force-center and hard-body).39 It is instead best understood as a
conceptual exploration of precisely those features of matter that make it
suitable for the empirically measurable mathematization of its quantity that
Newton has achieved in the Principia.40
REFERENCES
Carrier, Martin. 2001. Kant’s mechanical determination of matter in the Metaphysical
foundations of natural science. In Kant and the sciences, ed. Eric Watkins, 117–35. Oxford:
Oxford University Press.
Friedman, Michael. 1992a. Causal laws and the foundations of natural science. In The
Cambridge companion to Kant, ed. Paul Guyer, 161–99. Cambridge: Cambridge University Press.
———. 1992b. Kant and the exact sciences. Cambridge, MA: Harvard University Press.
———. 2010. Synthetic history reconsidered. In Discourse on a new method: Reinvigorating the
marriage of history and philosophy of science, ed. Mary Domski and Michael Dickson,
571–813. Chicago: Open Court.
———. 2012. The Prolegomena and natural science. In Kant: Prolegomena. Ein gemeinschaftlicher Kommentar, ed. Holger Lyre and Oliver Schliemann, 231–66. Frankfurt am Main:
Klostermann.
———. Forthcoming. Kant’s construction of nature: A reading of the Metaphysical Foundations
of Natural Science. Cambridge: Cambridge University Press.
Kant, Immanuel. 1900–. Kant’s gesammelte Schriften. Berlin: Georg Reimer, later Walter de
Gruyter.

39
The hard-body atomism that Newton proposes in the Opticks (note 27 above) is explicitly
hypothetical or conjectural, and it is by no means presupposed in his treatment of quantity of
matter in the Principia. The one qualm that Kant could have concerning Newton’s comments to
his first definition is that they raise a (presently unanswerable) speculative question, which, in the
context of the empirical mathematization that Newton in fact achieves, is potentially quite
misleading.
40
In my understanding of Newton (in relation to Kant), I am especially indebted to two of
the leading philosophical Newton scholars of our time: Howard Stein and George E. Smith.
The influence of both is flagged at crucial points in this paper. I am also indebted to Mary
Domski for helpful comments that significantly enhanced its clarity.

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———. 2004. Metaphysical foundations of natural science. Trans. and ed. Michael Friedman.
Cambridge: Cambridge University Press.
Newton, Isaac. 1952. Opticks, or A treatise of the reflections, refractions, inflections & colours of light.
New York: Dover.
———. 1999. The Principia: Mathematical principles of natural philosophy. Trans. and ed. I.
Bernard Cohen and Anne Whitman, assisted by Julia Budenz. Berkeley: University of
California Press.
Stein, Howard. 1967. Newtonian space-time. Texas Quarterly 10: 174–200.
Sutherland, Daniel. 2004. The role of magnitude in Kant’s critical philosophy. Canadian
Journal of Philosophy 34: 411–42.
———. 2006. Kant on arithmetic, algebra, and the theory of proportion. Journal of the
History of Philosophy 44: 533–58.