Metaphysical Foundations of Natural Science

Immanuel Kant

1786

Copyright © Jonathan Bennett 2017. All rights reserved

[Brackets] enclose editorial explanations. Small ·dots· enclose material that has been added, but can be read as
though it were part of the original text. Occasional •bullets, and also indenting of passages that are not quotations,
are meant as aids to grasping the structure of a sentence or a thought. Every four-point ellipsis indicates the
omission of a brief passage that seems to present more difficulty than it is worth. Longer omissions are reported
on, between [brackets], in normal-sized type. Numerals in the margins refer to the pages in the Akademie edition
of the work; these numbers are also supplied in both the existing English translations, which can thus easily be
correlated with the present version.
First launched: June 2009

Contents
Preface 1

Chapter 1: Metaphysical Foundations of Phoronomy 7

Definition 1................................................................................................................................................................................................... 7
Definition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Definition 3................................................................................................................................................................................................ 11
Definition 4................................................................................................................................................................................................ 12
Definition 5................................................................................................................................................................................................ 14
Metaphysical Foundations of Natural Science Immanuel Kant

Chapter 2: Metaphysical Foundations of Dynamics 19

Definition 1................................................................................................................................................................................................ 19
Definition 2................................................................................................................................................................................................ 20
Definition 3................................................................................................................................................................................................ 22
Definition 4................................................................................................................................................................................................ 22
Definition 5................................................................................................................................................................................................ 23
Definition 6................................................................................................................................................................................................ 30
Definition 7................................................................................................................................................................................................ 34
GENERAL REMARK ON DYNAMICS............................................................................................................................................ 39

Chapter 3: Metaphysical Foundations of Mechanics 48

Definition 1................................................................................................................................................................................................ 48
Definition 2................................................................................................................................................................................................ 48
GENERAL REMARK ON MECHANICS.......................................................................................................................................... 58

Chapter 4: Metaphysical Foundations of Phenomenology 59

Definition.................................................................................................................................................................................................... 59
GENERAL REMARK ON PHENOMENOLOGY..............................................................................................................................62
 Metaphysical Foundations of Natural Immanuel Prefa

467 connect items of knowledge so that they constitute a whole:

Preface
•
empirical principles and •rational principles. This could

[Kant launches this work by distinguishing two senses of

the word ‘nature’. There is (n) its use in phrases of the
form ‘the nature of. . . ’, where a nature is a quality, and
(N) its use as a proper name, ‘Nature’, which signifies not a
quality but the sum-total of everything that can be an
object of our senses and thus also an object of experience.
Kant also lingers on (n) in order to distinguish ‘the nature
of. . . ’ from ‘the essence of. . . , but this won’t concern us in the
present work, whose central concern is with (N) Nature—
the whole world that we can know about through our
senses. [Because he was writing in German, Kant had to use a capital
‘N’ for Natur in each of those senses. In English we have a choice; and
this version will use ‘nature’ for the ‘nature of . . . ’ concept and ‘Nature’ for
the name of a single entity.]]
Kant continues:] Nature taken in
this sense of the word has two main parts, corresponding to
the main line through our senses:
One part contains the objects of the •external senses.
Any theory about that will be a doctrine of •body,
dealing with extended Nature.
The other part contains the object of the •internal
sense. Any theory about it will be a doctrine of •soul,
dealing with thinking Nature.
[Kant’s Lehre has to be translated as ‘doctrine’, but really that is mislead-
ing. He will label as a Lehre any disciplined body of fact-and-theory
about a given subject; any respectable university department will be
dedicated to some Lehre; but a Lehre can be much too small to support a
department—e.g. Kant will speak about ‘the doctrine of the properties of a
468 straight line’!] If a doctrine is a system—i.e. a knowledge-total
ordered according to principles—then it’s what we call a
science. Now, there are two sorts of principles that can

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 Metaphysical Foundations of Natural Immanuel Prefa
l natural science’ from ‘rational natural science’; but it •aren’t apodeictically certain. So that entire structure
p turns out that this is a bad way of stating things. doesn’t strictly count as a ‘science’, and would be better
r [Kant’s explanation of why is confusing: he announces referred to as a systematic art. [This uses ‘art’, as Kant uses the
o it as focussing on the meaning of ‘Nature’ but states it corresponding word Kunst, to mean something like ‘disciplined
m in terms of the meaning of ‘science’. [We’ll see that both assemblage of skills’.]
p items are involved.] The core of the explanation is that
t any natural science, properly so-called, must include
princi- ples that rationally hold items of knowledge
u together. Kant continues:] So the doctrine of Nature—
s whether extended or thinking—might better be
divided into
t (a) the historical doctrine of Nature, which
o contains nothing but the systematically
ordered facts about natural things—
d presenting •Nature as a system of classes of
i natural things ordered according to simi-
s larities, and the •history of Nature as a
t systematic account of natural things in
i different times and in different places; and (b)
n natural science.
g And natural science properly so-called would treat its
u subject- matter wholly according to a priori principles,
i while natural science improperly so-called would
s treat its subject-matter according to laws of
h experience.
Nothing counts as science proper unless it is
‘ •apodeictically certain, ·i.e. certain because it is
h absolutely necessary·; any cognitive structure that
i makes use of merely •empirical certainty is only
s improperly called ‘science’......................................An
t example
o of the latter is chemistry, the basic premises of which
r are merely empirical; the laws from which the given
i facts are logically deduced in chemistry are merely
c laws of experience, which •don’t bring with them any
a consciousness of their necessity and therefore

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 Metaphysical Foundations of Natural Immanuel Prefa

So a rational doctrine of Nature deserves the label ‘nat- kept out of sight while we expound the pure part on its own,
ural science’ only when the laws of Nature that underlie it as completely as we possibly can, so as to discover exactly
are (1) known a priori and aren’t mere (2) laws of experi- •
what reason can accomplish unaided, and •where it starts
ence. Knowledge of Nature of kind (1) is called pure rational to need help from principles of experience. . . . ·And now I
knowledge; knowledge of kind (2) is called applied rational need to introduce another distinction·:
knowledge. Since the word ‘Nature’ already carries with it •
Pure philosophy (= metaphysics) is pure rational
the concept of laws, and since that carries with it the concept knowledge from mere concepts; •Mathematics is pure
469 of. . . .necessity, it’s easy to see •why something can count rational knowledge that is based entirely on the con-
as natural science only because of the pure part of it, i.e. struction of concepts by means of the presentation of
the part containing the a priori principles of all the other the object in a priori intuition.
explanations of Nature, and •why it’s only because of this [That account of mathematics comes from a theory of Kant’s which is
easiest to grasp in application to geometry. Take the proposition that the
pure part that it is a science. Thus, every discipline dealing total length of any two sides of a triangle is greater than the length of the
with Nature must, according to reason’s demands, eventually third side; how do you know that this is true? Not empirically, by (1)
come to be natural science, because the very concept of measuring the sides of triangular things, or by (2) reading it off from the
Nature has the necessity of laws inseparably attached to it concept triangle.
By method (1) we could only get truths known a posteriori, i.e.
and required for Nature to be thoroughly understood. [•This from experience.
removes the confusion mentioned in an earlier note. Kant holds that both By method (2) we could only derive analytic truths—ones know-
the concepts of Nature and those of science conceptually involve neces- able through conceptual analysis.
sary law; so any disciplined treatment of Nature must bring in such laws, What is remarkable about geometrical truths is that they are known a
thereby helping to qualify itself as a science. •Why ‘reason’s demands’? priori and yet are synthetic—i.e. known without appeal to experience
Because of Kant’s doctrine—expounded in his Critique of Pure Reason but but not by being derived purely from concepts. Well, then, how are
not here—that reason constantly urges us to interconnect our various they known? Kant’s answer is this: If you know that proposition about
items of knowledge, always restlessly trying to get it all into a single triangles (and haven’t merely taken it on trust from someone else), you
That is why the most complete
rigidly interconnected system.] must have constructed a triangle in your mind’s eye and seen from this
explanation of certain phenomena by chemical principles that the proposition is true. In our present text Kant writes here and
always leaves us dissatisfied, because it has involved only below of ‘constructing concepts’, but that is misleading. He doesn’t think
contingent laws learned by mere experience, with no input that in this process you construct any concept. Rather, you construct,
from anything a priori. Natural science
under the guidance of a concept, a mental triangle.]
Thus all genuine natural science requires a pure part properly so-called presupposes the metaphysics of Nature,
which could be the basis for the apodeictic certainty that ·i.e. pure rational knowledge from mere concepts·. Why? Be-
reason looks for in such science. And since the principles cause a science properly so-called has to include necessary
at work in the pure part make it completely different from propositions, and in this science they must be necessary
the part whose principles are only empirical, there is a lot truths having to do with the existence of things; so they can’t
to be gained from a procedure in which the empirical part is be based on a priori intuition, because no such intuition

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 Metaphysical Foundations of Natural Immanuel Prefa

can present anything concerning existence. The necessary mere possibility. But the possibility of specific natural things
propositions involved in natural science, therefore, have to be ·such as bodies and minds· can’t be discovered from their
the concept-based ones that define ‘metaphysics of Nature’. mere concepts. ·For example·: from the concept of body
There are two possibilities for what they might be: we can discover the possibility of having a self-consistent
(1) The metaphysics of Nature might deal with the laws thought about a body, but we can’t discover the possibility
that make possible the concept of a thing’s nature, without of a body as a natural thing that could exist outside of
bringing in any specific object of experience, and therefore the thought of it. So if we are to have knowledge of the
not saying anything specific about any particular kinds of possibility of specific kinds of natural things, and hence to
empirical object. The part of the metaphysics of Nature know ·truths about· them a priori, we’ll need to be given
470 that does this is its transcendental part. [For Kant a a priori an intuition corresponding to the concept, i.e. we
‘transcendental’ need the concept to be constructed. And rational
principle is one that has to do with the conditions that make possible knowledge through the construction of concepts is
some kind of knowledge.] mathematical. It may be possible to dispense with
(2) The metaphysics of Nature might instead deal with mathematics in developing a
the special nature of this or that kind of thing, of which it •
pure philosophy of Nature in general, i.e. one whose only
has an empirical concept; doing this in such a way it doesn’t topic is what constitutes the concept of a nature in general;
look to experience for anything except this concept. (·If it but a pure doctrine of Nature concerning specific natural
looked to experience for information, it wouldn’t count as things (a doctrine of body or a doctrine of soul) is possible
‘metaphysics’·.) For example, it takes as its foundation the only through mathematics. . . .
empirical concept of a material thing or the empirical concept [That’s why chemistry can’t be a science, Kant says. For
of a thinking thing, and searches for anything that reason it to be a science it would have to derive chemical laws about 471
can teach us a priori regarding these things. This science how different sorts of matter react with one another from
would still count as a ‘metaphysic’ of Nature—specifically, of an a priori intuition—something constructed in our minds.
corporeal or of thinking Nature—but it wouldn’t be a •general And there is no chance of that. And so, Kant continues,]
metaphysic but rather •a special metaphysical natural sci- chemistry can’t be anything more than a systematic art or
ence (physics and psychology), in which the transcendental experimental doctrine, never a science proper, because the
principles mentioned in (1) are applied to the two sorts of principles of chemistry are merely empirical and can’t be
sense-objects. presented a priori in intuition. . . .
In any special doctrine of Nature there is only as much But the empirical study of the soul must always be even
genuine science as there is mathematics. As I have explained, further from qualifying as a natural science than chemistry
a science (properly so-called) of Nature must have a pure is. Why? Because mathematics can’t be applied to the
part that is the foundation for the empirical part and is based phenomena of inner sense and their laws. (‘But the flow of
upon a priori knowledge of natural things. ·Let us now look inner sense’s internal changes is continuous, and
very carefully into this notion of a priori knowledge of natural continuity can be treated mathematically.’ Yes, but •what
things·. To know something a priori is to know it from its that could add to the content of the doctrine of the soul is

4
 Metaphysical Foundations of Natural Immanuel Prefa
v
a
s
t
l
y

l
e
s
s

5
 Metaphysical Foundations of Natural Immanuel Prefa

than •what mathematics can add to the content of the So all natural philosophers who have wanted to proceed
doctrine of body; in about the way that •the doctrine of mathematically in their work have availed themselves (with-
the properties of a straight line are less than •the whole out realizing it) of metaphysical principles; they had to do
of geometry! In each case, the tiny doctrine concerns only so, despite their solemn declarations that metaphysics has
a single dimension—in the case of the soul it’s the single no claims on their science. No doubt they took ‘metaphysics’
dimension of time.) Anyway, if we ·keep mathematics out of to be a light-minded activity of •inventing possibilities at will
the picture and· think of the doctrine of the soul merely as as and •playing with concepts which might be incapable of being
a •systematic art of analysis or as an •experimental doctrine, presented in intuition and have as their only claim to objec-
it still falls wells short of chemistry, ·in three ways·. (i) Given tive reality the mere fact that they aren’t self-contradictory!
any two elements in the complex of events observed through All true metaphysics comes from the essential nature of our
inner sense, I can think of them them separately, but I can’t thinking faculty, so it’s not something we invent. The content
separate them and then bring them together as I choose. (ii) of metaphysics doesn’t come from experience; ·it’s nearer
I can’t investigate the mental events in someone else’s mind. the truth to say that experience comes from metaphysics! ·.
(iii) With mental events, ·unlike chemical ones·, an observed Metaphysics consists in the pure operations of thought—a
event can be altered and distorted by the mere fact of being priori concepts and principles whose basic role is to bring the
observed. So the doctrine of the soul can’t be anything more elements of the tangle of empirical representations into law-
than a •natural description of the soul, not a •science of ful connection with one another, thereby turning the tangle
it, and not even a •psychological experimental doctrine. That into experience. That’s why those mathematical physicists
is why in the title of this work—which really contains only couldn’t do without some metaphysical principles, includ-
the principles of the doctrine of body—I have followed ing the ones that make the concept of their own special object
standard usage in employing the general name ‘natural —matter—available a priori for application to external
science’; for strictly speaking it’s only the doctrine of body experience, as with the concepts of motion, of the filling of
that is entitled to be called ‘science’. space, of inertia, and so on. But they rightly held that the
472 But it can’t be natural science unless mathematics is apodeictic certainty they wanted their natural laws to have
brought into it, and that can’t happen until. a complete couldn’t be had by any merely empirical principles; so they
analysis of the absolutely general concept of matter has been preferred to postulate such laws without investigating their
provided. Providing that is the business of pure a priori sources.
philosophy. That general concept is an empirical one, but In the pure part of natural science as ordinarily con-
pure philoso- phy ·in dealing with it· doesn’t make use of ducted, metaphysical and mathematical constructions criss-
any particular experiences; it employs only what it finds in cross with one another; ·and that is very unsatisfactory·. It
the concept of matter that relates to pure space and time. is enormously beneficial for the sciences to keep principles of
(Such relations come from laws that depend essentially on different kinds at a distance from each other, putting •each 473
the concept of Nature.) Such a doctrine of body is, kind into a separate system which constitutes a science
therefore, an actual metaphysics of corporeal Nature. of •that kind. If this isn’t done, people can confuse them

6
Metaphysical Foundations of Natural Immanuel Prefa

with one another, failing to see which kind is relevant to The schema for the completeness of a metaphysical sys-
a particular problem. . . . That is why I have thought it tem, whether of Nature in general or of corporeal Nature in
necessary to segregate •metaphysical principles from the particular, is the table of the categories.1 That is because this 474
pure part of natural science that has usually been the
stamping-ground of metaphysical as well as mathematical 1
[In an enormous footnote Kant reports that something published in a
constructions, putting •them into a system of their own—a recent issue of one of the learned journals expresses doubts
relating to his use of his ‘table of the pure concepts of the
system that will also contain the principles of the construc- understanding’. He continues:] The doubts aren’t aimed at the table
tion of those ·mathematical· concepts, and therefore the itself, but at the con- clusions I have drawn from it regarding the
principles of the possibility of a mathematical doctrine of limitations of the whole faculty of pure reason and therefore of all
Nature itself. ·But this system won’t contain any metaphysics. These
doubts are supposed to touch the main foundation of my system,
mathemat- ics·. . . . as set out in the Critique of Pure Reason This main foundation
Here is a second advantage of this procedure. In anything is said ·by my critic· to be my deduction of the pure concepts of
that is called ‘metaphysics’ we can hope for absolute com- the understanding, expounded partly in the Critique and partly in
pleteness, which can’t be expected in any other branch of the Prolegomena. That part of the Critique (·says my critic·) should
have been the clearest but is actually the most obscure or indeed
knowledge; and we can confidently expect such completeness argues in a circle, and so on. The chief point in these objections is
not only for the metaphysics of Nature in general but also the claim that without a completely clear and adequate deduction of
for our present topic of the metaphysics of corporeal Nature. the categories, the system of the Critique of Pure Reason, far from
Why can we expect this? Because in metaphysics the object— being apodeictically certain, would totter on its foundation; and
that is what I shall answer here. [Kant’s answer is long, dense,
·the item you are studying ·—is considered merely as it has difficult, and not needed for present purposes. The gist of it involves
to be represented in accordance with the universal necessary his taking his critic to agree •that the categories are forms of
laws of thought; this confines the possible results to a thought that we have to use in intellectually dealing with whatever
definite number of items of knowledge, and it’s possible to we have to think about, and •that all we can ever have to think
about are appearances. These concessions, Kant says, give him his
come to have all of these. In contrast with this, in any
core thesis in theCritique, namely that the categories represent the
other science we consider the object as it has to be limits to what thoughts we can have, what propositions we can
represented in accordance with data of intuition; there is a entertain, and so on; and he represents his critic as accepting that
limitless web of intuitions, and therefore of objects of the categories do this while complaining that Kant hasn’t explained
thought, so that the science can never achieve absolute how they can do it. He replies that his system doesn’t need thehow,
which is mere icing on the cake [not his formulation!]. He says that
completeness, but can be endlessly extended, as in pure if his account of how were a failure, he would still be in good
mathematics and the empirical doctrine of Nature. [In that company:] Newton’s system of universal gravitation is well
sentence, Kant twice specifies that the intuitions he is talking about established, despite our continuing difficulty about explaining how
I think that ·in the
include pure as well as empirical ones.] attraction at a distance is possible. Difficulties are not doubts. [And
then Kant re-states all this at much greater length, ending up with a
present work· I completely exhaust the metaphysical slap at his critic, saying that when certain things are made clearer in
doctrine of body, extend it as far as you like; but I don’t the second edition of the Critique,] that will spare my critic from
regard that as much of an achievement. having to resort to a pre-established harmony because of the
surprising agreement of appearances with the laws of the
understanding. This ‘remedy’ is much worse than the

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 Metaphysical Foundations of Natural Immanuel Prefa

table contains all the pure concepts of the understanding Science can therefore be divided into four chapters.
that have something to do with the nature of things. It 1. Phoronomy: In this, motion is considered as pure
475 must be possible to bring under the four kinds of cate- gory— quantum—portions of which can be combined in various
quantity, quality, relation, and modality—all detailed special ways—with no attention being paid to any quality of the
476 cases of the universal concept of matter, and therefore matter that moves. [See note on page 18]
everything that 2. Dynamics: This treats motion as belonging to the quality
can be •thought a priori concerning matter, •presented of the matter under the label ‘basic moving force’. [See note on
in mathematical constructions, or •given in experience page 39]
as a determinate object of experience. 3. Mechanics: This deals with how the movements of
There’s nothing more to be discovered or added; but there portions of matter bring them into ·causal· relations with
may be room for improvements in clearness or thoroughness. one another. [See page 58]
Accordingly, the present work contains four chapters, 4. Phenomenology: In this chapter, matter’s motion or
each dealing with matter brought under one of the four rest is handled purely in terms of how it is represented—its
kinds of concepts of the understanding. Something that modality—and thus in terms of its status as an appearance
is present in all the chapters is motion. The senses can’t of the external senses. [See note on page 62]
be affected by matter unless something moves; so motion I have shown the necessity of distinguishing •the meta-
is the basic fact about anything that is to be an object of physical foundations of the doctrine of body not only from
the external senses; and the understanding leads all other •
physics (which employs empirical principles) but even from
477 predicates that express the nature of matter back to motion; •
physics’s rational premises, which concern the employment
so natural science is, throughout, either a pure or an of mathematics in physics. The reasons for that were internal
applied doctrine of motion. The Metaphysical Foundations of to metaphysics; but there’s also an external reason to deal
Natural thoroughly with the doctrine of body as a separate unit, not
mixing it up with the general system of metaphysics. This
evil it is meant to cure. Such a pre-established harmony can’t
generate the objective necessity that characterizes the principles in external reason is only accidental—·it depends on a sheer
which pure concepts of the understanding are applied to appear- fact about how certain people behave·—but it is important.
ances. For example, it provides no basis for cause-effect We can mark the boundaries of a science not merely in terms
connections to be objectively necessary (though it allows
subjective necessity
of •its subject-matter and of •the specific kind of knowledge
(·when we experience C we can’t help expecting E to follow·). The only of that subject-matter, but also in terms of •what those
possible basis for this objective necessity is the a priori principles who pursue the science have in mind as a use for it. Well,
that lie at the foundation of the possibility of thought itself, these what do all the people who have busied their heads with
being needed if we are to have knowledge of objects whose appear-
metaphysics—and will continue to do so —had in mind as a
ance is given us, i.e. if we are to be able to have experience. And
even if there could be no adequate explanation of how experience is use for it? They have planned for it to
possible in the first place, it would still be indisputably certain that
•
extend natural knowledge
experience is possible only through those concepts and, conversely,
that the only meaningful use for those concepts is in relation to
objects of experience.

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of

(which they could do much more easily and certainly by as this is

observation, experiment, and the application of mathematics
to external phenomena), and also to
•
give them knowledge of what lies entirely beyond all
the boundaries of experience, namely God, freedom,
and immortality.
These things being so, there is a lot to be gained by
handling the metaphysics of the doctrine of body in
isolation from the rest of metaphysics ·rather than
letting it get caught up in that jumble of concerns ·. It does
in fact grow from general metaphysics, and that shouldn’t
be forgotten; but it will grow better if we treat it as
having been planted in its own ground. This won’t affect
the completeness of the system of •general metaphysics. It
will indeed make it easier for •this science to progress
478 smoothly towards its goal if, whenever it needs to bring in
the general doctrine of body, it can call upon the separate
system of such a doctrine without having had to include it
in its baggage all along. And there’s another significant fact
(which I can’t go into in detail here), namely that general
metaphysics, whenever it needs to provide examples
(intuitions) to give meaning to its pure concepts of the
understanding, always has to take them from the general
doctrine of body, i.e. from the form and principles of
external intuition. And when such examples are not ready
at hand, general metaphysics gropes, shaking with
uncertainty, among mere meaningless concepts. So
a separate metaphysics of corporeal Nature does excellent
and indispensable service to general metaphysics..........In
the present work I have modelled my procedure on the
mathematical method —not making my work strictly
math- ematical (I hadn’t time for that), but treating
mathematics as something to imitate. This isn’t meant as a
display of profundity that might earn the work a better
reception. Rather, it reflects my belief that a system such

9
 Metaphysical Foundations of Natural Immanuel 1: Foundations of
q ent, and that it may some day be completed by the space in which all motion occurs—space that is
u someone cleverer than I am. That could happen when therefore absolutely immovable—is called ‘pure’ space or
i mathematical investigators of Nature, stimulated by ‘absolute’ space.
t this sketch of mine, think it worthwhile to extend
e their studies to the metaphysical portion ·of the
doctrine of body· and to bring it into unison with
c the
a mathematical doctrine of motion.
p In the preface of his Principia, Newton follows up
a his remark that geometry needs to postulate only two
b mechanical actions, the ones that trace a straight line
l and a circle, by saying: ‘Geometry is prioud of being
e able to produce so much, with so little taken from
elsewhere.’ In contrast with that, one might say of
o
metaphysics: It stands astonished that with
f
so much offered to it by pure mathematics, it can achieve so
479
a
little! Nevertheless, this ‘little’ is something that
mathematics absolutely has to have in its application
m
to natural science; and since mathematics must here
a
necessarily borrow from metaphysics, it shouldn’t be
t
ashamed to be seen in the company of the latter.
h
e [From here on, displayed occurrences of ‘Def- inition’ translate
m Kant’s Erklärung, which usually means ‘explanation’. Kant hinself
a licenses this somewhat loose use of ‘definition’ in his Critique of
t Pure Reason B 75.]
i
c
a
Chapter 1
l Metaphysical Foundations of Phoronomy

t Definition 1
r
I call something ‘material’ if and only if it is movable in
e
480
a
space. Any space that is movable is what we call
t
‘mate- rial’ or ‘relative’ space. What we think of as
m
1
 Metaphysical Foundations of Natural Immanuel 1: Foundations of

Remark 1 of our senses in application to the external world; it stands in contrast

The whole topic of phoronomy is motion; so the only property with a priori intuition—see the long note on page 2.] In contrast to
that is here attributed to the subject of motion, i.e. matter, this form, matter would be what our outer senses give us
is its movability. So we are free to take ·any portion of· sensations of ; so it would be the properly empirical part of
matter as a point. In phoronomy we set aside all the internal external sensory intuition, because matter cannot be given
characteristics of matter, thereby setting aside anything at all a priori. In all experience •something must be sensed,
involving the quantitative notion of how much matter we and •this is the real component in sensible intuition. So
are dealing with; all we are concerned with is the motion of the space in which we are to set up experience concerning
matter, and the only quantitative notion that we need is not motions must also be perceptible, i.e. must be indicated by
how much matter but only how fast and in what direction the what is perceptible; and this space—
matter moves. [Why does Kant imply that direction-of-movement is the sum-total of all objects of experience, and itself an
quantitative? Because he is thinking of 180-degree changes of direction object of experience
of straight-line movements; an N movement in one direction can be —is called ‘empirical space’. Now, if such a space is material,
If I
thought of as a minus-N movement in the opposite direction.] it is itself movable. But a movable space, if its motion is to
sometimes use the expression ‘body’— ·meaning a body, not be perceptible, presupposes a larger material space for it to
merely undifferentiated matter·—that will be because I am move in, this enlarged space presupposes one larger still,
deliberately getting ahead of myself, making my discourse and so on to infinity.
less abstract and more comprehensible by bringing into Thus, all motion that is an object of experience is merely
phoronomy some of the more determinate concepts of matter relative. We have
that we shall come to later. an object x which we perceive to move;
a space S1 relative to which we perceive x to move;
Remark 2
a larger space S2 relative to which S1 may move.
481 If I explain the concept of matter not by a predicate that It might happen that S1 does move relative to S2, and indeed
applies to it as object—
moves in the opposite direction to x ·and at the same speed ·;
·i.e. not by saying anything of the form ‘any item is in which case we can describe x as ’moving’ in relation to S 1
matter if it has property P·’ and at the same time ‘motionless’ with respect to S 2. These
—but only by how it relates to the knowledge-faculty through
varying accounts of whether and how x moves continue in-
which it is basically represented to me—
finitely as we bring in larger and larger relative spaces. [Kant
·i.e. by saying ‘matter is whatever is represented to now has a long sentence that is hideously unclear, apparently
me by outer sense’· because it is too compressed. The gist of it seems to be as
—then ‘matter’ is being explained as applying to every object follows. An absolute space —i.e. a space that isn’t material 482
of the external senses; and this would be the mere metaphys- because it isn’t movable—is something we assume because it
ical definition of it. But space would be simply the form of all is required for the possibility of experience. But in doing this
external sensible intuition. [That last phrase refers to the use we are assuming something that can’t be perceived •in itself

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of

or •in its consequences. (Perceiving it in its consequences feel a bit of matter being cut into
would be perceiving something that we knew was a case of
some object moving relative to absolute space, and there’s
no way we can perceive that.) Furthermore, although we
need this assumption for the possibility of experience, we
never have any experience in which absolute space plays
a part. The whole story of what we perceive can’t give any
role to absolute space. Kant continues:] So absolute space
is in itself nothing; it’s not any kind of object. All it signifies
is this: Whenever I am thinking about some object that is
moving relative to some space S— ·e.g. a leaf blowing through
the window and falling onto the carpet in my study ·—my
thought of ‘absolute space’ AS is just my thought of every
other relative space that I can think of as containing S,
the series of such ever-larger spaces running to infinity.
This is just a thought that I have; I’m not confronted by
anything—any matter—that indicates this space AS; so my
thought represents AS as pure, nonempirical, and absolute.
I can compare any empirical space S with AS, representing
S as movable in AS, which is therefore always taken to be
immovable. If you regard AS as an actual thing, you have
mistaken
•
the logical universality that consists in our ability to
regard any empirical space as being included in it,
for
•
a physical universality that consists in its actually
containing every empirical space. . . .
[Kant speaks of this mistake as a case of ‘misunderstanding reason in
its idea’, using ‘idea’ (German Idee) as a technical term that he
introduced in the Critique of Pure Reason and employs just seven times
in the present work. (This version will use ‘idea’ only in translating
Idee.) For a grasp of how it works, you need to start with the
understanding and the concepts that are its tools. We can have a
concept of x only if we could be ‘given’ an example of x in experience; so
we have a concept of division of a bit of matter because we can see or

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of
two or four or. . . What about •the thought of division of a bit of if we are to define the concept of matter in a comprehensive
matter carried the whole way? Unless you think that there are way that covers moving bodies,
‘atoms’, smallest bits of matter that can’t be further divided, •this
thought goes with the thought of an infinitely small bit of matter;
that is something we couldn’t conceivably encounter in
experience; so we have no concept of it; but we do have the idea,
this being a thought that takes some concept and subjects it to the
thought of going the whole way or (in terminology that Kant uses
a lot in the Critique but not in the present work) the thought of a
certain kind of totality. It is the role of reason, he holds, to engage
in this totalising sort of thought, which is why he links ideas with
reason, as he links concepts with understanding. In the use of
‘idea’ that we have just encountered, Kant speaks of the totalising
activity as involving a ‘logical universality’, and he is referring to
the totalising that is involved in the thought of the whole of
space.]One last remark: An object’s movability in
space can’t be known a priori, i.e. without instruction
from experience; which is why in the Critique of
Pure Reason I couldn’t count such movability as one
of the pure concepts of the understanding. The
concept of movability, just because it is empirical, can
find a place in a natural science only as a bit of
applied metaphysics, which is where concepts given
through experience are dealt with, though according
to a priori principles.

Definition 2
The motion of a thing is the change of its external
relations to a given space.
Remark 1
I have based the concept of •matter on the concept of
•motion. That’s because I wanted to fix the concept of
matter without bringing in the concept of •extension, so
that I could consider matter as a point, helping myself to
the common definition of motion as change of place. But

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of

that ‘change-of- place’ definition won’t do. The place of any phoronomy merely because the word ‘speed’ is generally
body is a point. The distance of the moon from the earth is used in one sense for movements in class (1) and a different
given by the shortest line between their places, i.e. betwen sense for movements in class (2), as you will see in a
their central points. (That’s the only way to get a determinate moment.
single distance between them; any other approach will have
Remark 3
us measuring from some arbitrarily chosen pair of points—
In any motion we have just two factors to think about—speed
·say the distance from the lowest point in the Dead Sea
and direction—once we have set aside all the other
to the highest point in the Mare Frigoris ·.) Now, taking a
properties of the moving thing. I am here taking for
body’s place to be its central point, a body can move
granted the usual definitions of both of these, but various
without changing its place, as the earth does by turning on
limitations have to be built into the definition of direction. .
its axis. But although the rotating earth doesn’t change its
..
place, it does change its relation to external space, because
Consider two snails that are exactly alike in shape and
at different times it turns different sides toward the moon,
even size, except that one winds to the right and the other
and these differences produce all kinds of different effects
to the left. What does this difference rest on? Or the
on the earth. The equation of ‘motion’ with ‘change of
difference between the winding of beans around their pole
place’ holds only for movable points, i.e. physical points.
(like a corkscrew—‘against the sun’, as sailors would say)
[The next bit is awkwardly written, but its content can be
and the winding of hops, which go around their pole with 484
made clear. Its point is just that the change-of-place
the sun? We have here an internal difference between
definition omits more things than just rotation; it omits, for
the two snails, or between the pole-climbing plants— ·it’s
483 example, the movements that go on when beer is ‘internal’ in the sense that we can’t make it disapear by re-
fermenting in a cask. What the definition applies to is
movement of the cask-and-contents as a unit—movement arranging other things in certain ways·. Now, the concept of
of the cask, not movement in the cask.] . . . . this internal difference can be constructed, but it can’t be
expressed in general terms. It can happen that two things
Remark 2 differ only in this way, i.e. without this difference bringing
Motions can be divided into two classes. (1) Progressive others in its train. Take the rare case of a human being
movements, which enlarge their space; straight-line move- who is found through an autopsy to have all his organs
ments and curved-line movements that don’t return in on inter-related according to the physiological rules that hold for
themselves. (2) Rotatory movements, which don’t enlarge other human beings except that they are left/right
their space, but keep returning in on themselves, staying reversed. This can’t possibly have made any difference to
with the same limited space. And these can be divded in the internal workings of that person’s body. And yet there
turn, into (2a) circular movements like those of the planets is a real mathematical and indeed internal difference
around the sun and (2b) oscillatory movements like that of a between two motions that differ only in that way, e.g. two
pendulum. I mention these different kinds of motion in circular motions differing in direction but exactly alike in all
other respects. [Kant adds his claim that this left/right
matter confirms his view that ‘space in general doesn’t
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 Metaphysical Foundations of Natural Immanuel 1: Foundations of
b
e
l
o
n
g

t
o

t
h
e

p
r
o
p
e
r
t
i
e
s

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of

or relations of things in themselves’ but ‘belongs merely to This can’t happen unless the body doesn’t spend any time—
the subjective form of our sensible intuition’. He remarks not the smallest portion—at B. Why? Because it is present
that he has dealt with this elsewhere [in Prolegomena section at B only once in Oscillate; allow its presence there to occupy
13]. He continues:] But this is a digression from our a tiny period of time and you’ll have the problem of which of
present business, in which we have to treat space as a the two journeys—AB or BA—to assign it to. Either way, the
property of the things we are considering, namely bodies, times for the two sub-journeys won’t be equal. Now change
because bodies themselves are only phenomena of the the example to a movement (·I’ll call it Straight·) in which
external senses and need to be explained here only as such. a body moves exactly as in Oscillate except that in-
—So much for direction. As for speed: the meaning of this stead of switching back at B it continues straight on
expression also varies in different contexts. We say that the to a further point C.
earth rotates on its axis ‘faster’ than the sun because it In Straight the body is moving at B, not at rest. (Why?
completes a rotation in a shorter time, although the Because B is just one point in a continuously moving journey,
motion of the earth in this rotation is slower than that of with nothing special about it except that we have chosen to
the sun. [Kant gives other examples, without suggesting talk about it. If the body weren’t moving at B it wouldn’t be
that this point matters much for his present work. He moving at any point along the A–C line, which means Straight
concludes:] In phoronomy we use the word ‘speed’ with a didn’t occur.) But Straight is supposed to be exactly like
merely spatial meaning—the measure of how far a thing Oscillate except for the directional difference; so if the body
travels in a given period of time. is moving at B in Straight then it is moving at B in Oscillate
too—but we have just shown that it can’t be! Now consider
Definition 3 a third example, of a movement ·that I’ll call Updown·, in
485 which
Rest is time-taking presence in the same place; for some- a body rises from A up to B which is directly above A,
thing to be time-taking is for it to exist throughout a time. and then—having lost its motion by means of gravity
[The translation makes this look trivial, but it doesn’t in the German. ] when it reaches B—it falls back again from B to A.
Remark In this case is the body moving at B or at rest there? The
A moving body is momentarily at each point of the line that it most plausible answer is this:
traverses. Is it at rest at each point or is it moving? No doubt In Updown the body is at rest at point B; because
you’ll want to say that it is moving, because it is precisely when it is there it has been deprived by gravity of all
by moving that it came to be at this point. But let’s consider its upward motion, and the downward motion that
what is going on in a movement ·I’ll call Oscillate·, in which gravity will also give to it hasn’t yet begun. And
a body tracks the line AB, from A to B and then back something that doesn’t have any motion is at rest.
to A again, doing this with a uniform speed so that But if that is all right for Updown, why isn’t it also all right
the total time is exactly one second—half a second for Oscillate; for in the latter also the return journey from 486
from A to B and half a second for the return journey. B to A can’t start until the forward journey from A to B has

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Metaphysical Foundations of Natural Immanuel 1: Foundations of

ended; so that in Oscillate also we seem to have to

Definition 4
conclude that the body is not, after all, moving at B. But we
can’t draw that conclusion, because something that is To construct the concept of a composite motion means to
moving with a uniform speed can’t be at rest anywhere present a priori in intuition a motion as the result of two or
along its journey. What, then, makes the crucial difference more given motions united in one movable thing.
between Oscillate and Updown? It is that in Updown the
Remark
body’s motion isn’t uniform—it is uniformly decelerated
In constructing a concept one mustn’t make use of any
and then uniformly accelerated, in such a way that its
input from experience, e.g. presupposing some force that
speed at B is reduced not to nothing but only to a speed S
one knows about only from experience. Putting the point in 487
that is smaller than any assignable speed. Keep gravity out
its most general form: in constructing a concept one musn’t
of this for a moment, and suppose that the body with speed
use any concept that can’t be given a priori in intuition—such
S in Updown doesn’t start to fall at B but keeps moving
as the concepts of cause and effect, and of action and
upwards. ·How far would it get, in how much time, if it
resistance, etc. Don’t lose sight of the fact that
stayed at speed S? The answer is this·:
phoronomy’s only concern is with the construction of
Take any distance you like, however small, along the motions in general as amounts, so that it takes matter
line up from B, the body wouldn’t cover that distance, merely as something mov- able, ignoring any facts about
however long it kept moving with speed S. how much matter is moving in any given case. So
This implies that (for any possible experience) the body phoronomy has from the outset to characterize these
would remain at B for all eternity. Consequently, it is put motions solely as amounts determined by their speed,
into a state of •time-taking presence in the same place, i.e. a their direction and their composition. That much has to be
state of settled entirely a priori and indeed through intuition,
•
rest, although owing to the continuous influence of gravity, setting things up for applied mathematics. For the rules
i.e. the change of this state, the rest is immediately abolished. governing how motions are inter-connected through
To •be in a time-taking state is conceptually different from physical causes—i.e. forces—can’t be properly explained
•
spending time in that state. . . . Thus rest can’t be defined until there’s a mathematically constructed basis containing
as lack of motion, because that is negative and so can’t the principles of their composition in general.
be constructed. It must instead be defined as time-taking
presence in the same place. This can be constructed, by Principle
representing a motion with infinitely small speed through a
period of time that is not infinitely short; and because it Every motion that could be an object of experience can be
can be constructed it can be used in applying mathematics viewed either as •the motion of a body in a space that is at
to natural science. rest or as •the rest of a body in a space that is moving in the
opposite direction with equal speed. It’s a free choice.
Remark

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of

We can experience the motion of a body only if both •the them with other phenomena in our theories.
body and •the space in which it moves are objects of external Also, our experience can’t enable us to pick out a fixed
experience—hence, only if they are both material. [Remember point by reference to which we could give sense to a
that Kant has said that he calls a space ‘material’ if it can move relative distinc- tion between •absolute motion and •absolute rest.
So an absolute motion—i.e. a motion related
to a larger space.] Why not? Because everything we confront in experience is
to an immaterial space—can’t possibly be experienced and •material, and therefore •movable, and therefore •perhaps
is hence nothing at all for us (even if we allow that absolute actually moving without our being able to perceive this
space is something in itself ). But in all relative motion the motion. . . .
space itself, because it is assumed to be material, can be When a body moves in empirical space, I can think of any
represented as at rest or as moving. I represent the space proportion of the given speed—from none to all—as belong-
as at rest when it isn’t included in some larger space in ing to the body, and the remainder—from all to none—as
relation to which I could see it as moving. And I represent belonging to the space moving in the opposite direction.
the space as moving when it is included in some such larger There can’t be any empirical evidence that would favour any
space; an example would be seeing a ball roll along a table in particular distribution. In saying this I am assuming that
the cabin of a ship, where there is a larger space (including we are dealing only with motion in a straight line. When
the shore) beyond the space of the cabin, in relation to other motions are concerned, there isn’t the same freedom
which •the cabin’s space is moving and—it may happen— of choice about what to attribute to the body and what to the
488 • the ball is at rest. ·But then the shore’s space may be space. For example, as between
enclosed in a •still larger space relative to which the •shore’s
•
the earth rotates daily on its axis, while the surround-
space is moving and the •cabin’s space is at rest and the ing space (the starry heavens) stay at rest
•
ball is moving after all·! With respect to any empirically and
given space, we can’t rule out its being enclosed in a still
•
the earth remains still while the starry heavens re-
larger space in relation to which it may be moving or not volve around it,
moving. Thus, for all experience and for every inference from there are empirically detectable differences. I shall discuss
experience, it can’t make any difference whether I choose this later on [starting on page 61]. In phoronomy, then, where
to •consider a body as moving or rather to •consider the I consider the motion of a body only in relation to space
body as at rest and the space it is in as moving in the (upon whose motion or rest the body has no influence at
opposite direction with the same speed. The two ways of all), it is an arbitrary matter how much (if any) of the speed
looking at it are strictly equivalent. You might think that of a given motion I attribute to the body in question and
in relation to absolute space one of the accounts is right how much (if any) I attribute to the space that contains it.
and the other wrong, but absolute space can’t possibly enter Later on, ·in mechanics·, where we’ll consider how a moving
into any experience of ours, so we can set it aside. The only body interacts causally with other bodies in the space of
difference between body-moving-in-motionless-space and its motion, it will make a discoverable difference how we
space-moving-around-motionless-body is in how we connect distribute the speed between the moving body and the space
containing it. I’ll show this in the proper place [starting at

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of

page 53]. Note that this concerns two or motions that constitute one
motion; ·all there is to the one motion is those two or more
Definition 5 put together·; we are not concerned here with two or more
motions that cause some single motion to occur. In order
489 The composition of motion is the representation of the to find the motion arising from the composition of several
motion of a point as identical with two or more motions of motions—as many as you want—you have to proceed piece-
the point combined. meal (as we do with the production of all quantities): start by
Remark working out the motion that comes from compounding two of
Since in phoronomy I don’t have thoughts of any quality the motions, then compound this with a third. . . and so on.
of matter other than its movability, I can consider matter So the doctrine of the composition of all motions comes down
itself only as a mere point, and can consider any motion as to the composition of two. [Kant goes on to say that there are
a track through a space. But that doesn’t mean that I am three different ways in which two motions—whether of equal
attending only to the space that geometry deals with; or unequal speeds—can be happening in a single point at
because I also bring in the •time involved and hence the the same time: They may be going (1) in a straight line in the
•speed of the point’s movement through space. So same direction, (2) in a straight line in opposite directions, or 490
phoronomy is the pure doctrine of the amounts of motions. (3) along different lines that are at an angle to one another.]

what comes next, conservatively translated: The determinate Proposition

concept of a an amount is the concept of the production of
the representation of an object through the composition of The only way to think of two motions as composing the
the homogeneous. motion of a single point is by representing •one of the two as
occurring in absolute space, and •the other as consisting in
what Kant seems to have meant: Any how-much thought is
the movement of a relative space in the opposite direction.
the thought of a process of mentally assembling something
out of parts that are all of the same kind as it. In thinking Proof
about (say) a gallon of water one is somehow thinking of First case: A single point undergoes two motions in the
mentally building up a gallon drop by drop. same direction along the same line at one time.

Now, nothing is homogeneous with motion except motion, so

phoronomy is a doctrine of
the putting together of different motions of a single
point according to their direction and speed,
which is the same as
the representation of a single motion as comprising
within itself two or more motions occurring at the [Kant’s presentation of this part of his proof is very hard to
same time. follow. It starts with this line of thought: In phoronomy we

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of

can equate a speed with a distance/time pair, as we do when point at the same time, we’ll have to think of it as involving a
we name a speed in terms of ‘miles per hour’. Now, suppose point x that moves in a certain direction relative to absolute
that a point is subject to two movements at once, both in a space while the relative space that also contains x is moving
straight line and in the same direction, and think about how in the same direction at the same speed. The upshot of this,
we can represent their speeds. If they are equal, then their of course, that relative to the relative space x doesn’t move
speeds can be represented by the AB and ab lines in Figure at all. [When in contexts like this Kant speaks of ‘the relative space’
1. But. . . The preparer of this version of the text is defeated that is involved, we can take him to mean something like ‘the smallest
by what comes next. We are threatened with some kind of intuitively convenient space’ that is involved, out of the possibly infinite
incoherence or contradiction in representing the speeds of series of ever-larger relative spaces that x is contained in.]
491 the two movements on the assumption that they are both Third case: Two motions of a single point go in different 492
movements of a single point x relative to a single space. directions—not opposite directions but different ones that
[The difficulty is solved, Kant tells us, if we take one of enclose an angle.
the movements to be a left-to-right movement by x from A to
B and take the other to be a right-to-left movement of some
relative space that also contains x.
[That is straightforward enough, but the difficulty it is
supposed to remedy defeats understanding. The passage
in question is presented, closely following the two currently
available translations of this work (which differ very little in
their handling of this passage) on page 66.]
Second case: Two motions in exactly opposite directions are
to be combined at one and the same point.

·To start with, ignore the dotted lines and attend to the
square.· Let the two motions we are concerned with be AB
and AC. (The angle BAC could be any non-acute angle; it
Let AB be one of these motions and AC the other in the doesn’t have to be a right angle as it is here.) Now, if these
opposite direction (and again let’s take the speeds to be two motions occur at the same time in the same space, they
equal). In this case the very thought of representing two will go in the directions AB and AC but they won’t follow the
such motions of a single point x in relation to single space at lines AB and AC, but only lines parallel to these. The moving
the same time is plainly impossible. If we are to make sense point will go through m; and this will be as though the AB
of the notion of two equal and opposite motions of a single movement had pulled the AC movement over to the line Mm,

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of

and the AC movement had pulled the AB movement down to to be congruous. All geometrical construction of complete
the line Em. [Of course all of that should be said first about •a point identity rests on congruity. This congruity of two combined
between A and m, and before that about a point between A and •that motions with a third (as what is composed by the two) can
point, and. . . and so on. To draw this properly we would need infinitely never take place when the two are represented in a single
many smaller squares within the big one! But jumping across and down space, e.g. in a single relative space. Hence each attempt to
If the directions are to
to m is sufficient for Kant to make his case. ] •
disprove the Proposition on page 14 has failed because it is
remain the same, therefore, one of the two motions must come up with merely mechanical solutions—saying how two
be altering the other. [Actually, each motion must alter the other.] motions m1 and m2 combine with one another to produce
But the Proposition we are proving is about what the two m3 a third motion. Such attempts didn’t prove that m1 and
motions compose; and the meaning of that (see Definition m2 were identical with m3 and that because of this identity
5 on page 14) is that the two will jointly be the motion in they could be presented in pure intuition a priori. [Kant wrote
question, not that by changing one another they’ll produce ‘Each attempt to •prove the Proposition’, but that must have been a slip.]
it.
On the hand, that our moving point x undergoes motion
Remark 2
AC in absolute space while—instead of x’s undergoing motion
When a speed AC is termed ‘double’, this can only mean
AB—some relative space that x is in undergoes motion BA.
that it consists of two simple and equal speeds AB and
Then while x moves AE in absolute space, the relative space
BC (see the diagram on page 14). But if a ‘double speed’
moves Ee, that is, moves to the left, so that x’s position in
is explained as ‘a motion whereby a doubly great space
the relative space is m. And the same story holds for x’s
is traversed in the same time’, then something is being
493 absolute move AF while the relative space moves Ff; and for
assumed that shouldn’t be taken for granted, namely that
x’s entire absolute move AC while the relative space moves
two equal speeds can be combined in the same way as two
Cc. From the standpoint of the relative space, therefore, x
equal spaces. It isn’t obvious that a given speed consists of
moves smoothly down the diagonal, through m and n ·and of
smaller speeds—that a speed is made up of slownesses!—in
course all the intermediate positions · to D, which is exactly
the way that a space consists of smaller spaces. The parts
the same result as if it had undergone movements AB and
of the speed aren’t external to one another, as the parts
AC. So we get the result we want without having to postulate
of the space are; and if a speed is to be considered as an
two motions that affect one another.
amount, then the concept of its amount (·‘How fast?’·) can’t
Remark 1 be constructed in the same way as the concept of the size of
•
Geometrical construction requires that two amounts a space (·’How big?’·), because the former is intensive and
when put together are a third amount, not that they the latter extensive. [Except for a passing mention (not included
produce the third in a causal way—for that would be in this version), this is the first time Kant has used ‘intensive’ in this
•mechanical con- struction. For two items to be completely work. Examples: ‘How severe was the pain?’ and ‘How hot is the water?’
similar and equal in every way that can be known about in ‘How fast did the train go?’ ask about intensive magnitude, ‘How long
intuition is for them

2
 Metaphysical Foundations of Natural Immanuel 1: Foundations of

did the pain last?’ and ‘How much water is there?’ and ‘How far did the Nature or art through certain tools and forces. . . . 495
494 train go?’ ask about extensive magnitude.] But the only way this
Remark 3
construction can be done is by putting together two equal
So there we have phoronomy—a pure doctrine not of •motion
motions, the motion of the body in one direction and the
but of the •quantity of motion, in which matter is thought
equal motion of the relative space in the opposite direction.
of wholly in terms of its mere movability. All it contains
Two equal speeds can’t be combined in one body except
is this single proposition—·the one on page 14·—about the
through external moving causes—e.g. a ship carries the body
composition of motion, applied to the three kinds of cases I
with one of these speeds while another moving force
have discussed. And it only concerns straight-line motions,
within the ship gives the body a second speed equal to the
not motions along curves; because curved-line motion is
first. . . .
continuously changing in direction, and there has to be a
So much for the addition of speeds to one another. But when
cause for this change, a cause that can’t be merely space.
it’s a matter of subtracting one speed from another, it is
People usually take the phrase ‘composite motion’ to refer
easy enough to •think of such subtraction once we have the
only to the case where the directions of the motion enclose
notion of a speed as an amount by addition; but it’s not so
an angle; this. hasn’t done any harm to physics, because
easy to •construct the concept of this subtraction. To do this
in physics all three kinds of combination can be adequately
one must combine two opposite motions in one body—and
treated as versions of the third case, ·the enclosed-angle·
how is that to happen? It can’t happen if we work with only
one. If the angle enclosing the two given motions is
one space that doesn’t move. ‘Isn’t the concept of opposite
thought of as infinitely small [i.e. as approaching 0 degrees ], it
and equal motions of a single body in a single space simply
contains the first case; and if the angle is represented as only
the concept of rest?’ No, it is not! What we get out of this
infinitely little different from a single straight line [i.e. as
is not the concept of rest but merely the fact that what we
approaching 180 degrees], it contains the second case. So all
are trying to do is impossible. As I have already shown, the
three of the cases I have listed can indeed be covered by
composition that is assumed in the proposition ·on page 14·
the single familiar enclosing-an-angle formula. But a
to be possible has to be done by combining the motion of the
proper a priori grasp of the quantitative doctrine of
body with the motion of the ·relative· space ·that contains motion isn’t provided by that
it·. Finally, the composition of two motions whose directions formula, and such a grasp is useful for many purposes. [Per-
enclose an angle: this also can’t be thought of in the body haps that last remark goes with Kant’s saying that confining ‘composite
by reference to a single space. We can make sense of there motion’ to the enclosed-angle kind of case is harmful to ‘the principle of
being a body which is acted on by a northward-pushing force the classification of a pure philosophical science in general’.]
and a westward-pushing one, which between them produce [Kant ends this chapter with a needlessly difficult paragraph
a movement of the body in the north-westerly direction. But connecting the •three kinds of composition of motion with the
that is the •mechanical account of the concept of this kind •
three categories—i.e. pure concepts of the understanding—
of composition, not the •mathematical construction of it. A that he lists under heading ‘Quantity’ in the Critique of Pure
mathematical construction has only to make intuitive what Reason. In that work the division is into •unity, •plurality,
the combined movement is, not how it can be produced by

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 Metaphysical Foundations of Natural Immanuel 1: Foundations of

•
totality (corresponding to propositions of the form •‘Henry
is a tyrant’, •‘Some husbands are tyrants’, •‘All weak
husbands are tyrants’). Kant hopes to link that with
phoronomy by speaking of the latter in terms of •unity of line
and direction, the •plurality of directions in one and the same
line, and finally the •totality of directions as well as of lines.]
[In that paragraph, the Critique’s Quantity trio are labelled first by Größe
and then by Quantität. But in the Critique the only label is Quantität,
whereas Größe is regularly used there for ‘size’ or ‘magnitude’. Quite
apart from questions of consistency, Größe just does mean ‘size’ or ‘amount’
or something like it, and has nothing to do with that one/some/all trio of
categories; the only two places where Kant writes as though it were the
right label for that trio is the paragraph reported above and in the list
of category-trios on page 6. —Setting aside issues about the terminology
of the Critique (which won’t concern us much), the present version will
mainly translate
Größe by ‘size’ or ‘magnitude’ or ‘amount’ or by phrases using
‘how much’ or ‘how strong’ etc., and Quantität by ‘quantity’.
The standard meanings of the German words are confirmed by Kant’s
uses of them: Größe stands for a universal—bigness, how-much-ness,
something that a thing has; whereas Quantität stands for a particular
portion—e.g. the portion of coffee that I drank a moment ago—this
being something that a thing is. Both Quantität and ‘quantity’ can also be
used to name a universal, but they have this other option, which is the
one Kant sometimes employs. Quite often he uses Quantität to stand for
a universal—i.e. as equivalent to Größe—and in those cases the relevant
English word will have a subscript q, as in ‘the amount q of matter in it’
on page 47. Don’t think or worry about this; it is put there just for the
record. Just twice he uses Größe to mean ‘quantity’ in the non-universal
sense.]

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

itself a space;
Chapter 2
Metaphysical Foundations of Dynamics
Definition 1
496 Matter is whatever is movable and fills a space. To fill a
space means to resist every ·other· movable thing that tries
to move into that space. A space that is not filled is an empty
space.
Remark
This is, now, the dynamical definition of the concept of
matter. This definition presupposes the phoronomic one [page
7] but adds to it a causal property, namely the capacity to
resist a motion within a certain space. This property couldn’t
have any role in phoronomy, even when we were dealing with
the motions of a single point in opposite directions. This
filling of space keeps a certain space free from the intrusion of
any other movable thing, whatever direction it is coming
from. Now we must investigate what matter’s all-around
resistance is based on and what it is. Definition 1 makes it
clear that we aren’t talking about matter’s resistance to
•being pushed from one place to another (that’s a
497 mechanical phenomenon,
·to be dealt with in chapter 3·), but only its resistance to
•
being squeezed into a smaller amount of space. The
phrase ‘occupying a space’, i.e. being immediately present
at every point in the space, is used merely to indicate the
extension of a thing in space; and this concept of a thing’s
spatial extension or presence-in-space implies nothing
about what if anything the thing does to resist other things
that try to force their way into that space. It doesn’t
even rule out the possibility that something present in a
given space acts causally to attract other movable things into
that space. The concept might also apply to something that,
rather than being an instance of matter in a space, is

2
 Metaphysical Foundations of Natural Immanuel 2: Foundations of
bec ce·. . . . Bestrebung (and not for anything else), but remember that it isn’t a
aus Because it leaves all these possibilities open, the psychological term.]
e concept of occupying a space is broader and less
eve determinate than the concept of filling a space.
ry
spa Proposi
ce tion 1
is
Matter fills a space not by its mere existence but by a special
an
ass moving force.
em Proof
bla Penetration into a space is motion. The cause of
ge motion’s be- coming less, or even changing into
of immobility, is resistance to it. Now, the only thing that
sm can be combined with a motion in such a way as to
alle lessen or destroy it is another motion, in the opposite
r direction, of the same movable thing. [Kant adds
spa ‘(phoronomic proposition)’; but what he has just said doesn’t
ces, come from the Proposition on page 14. Perhaps it comes from
·an the various proofs and comments relating to that Proposition. ]
d Consequently, when a portion of matter x fills a space
one and thus resists all intrusion into that space by
of another portion of matter y, the resistance that it puts
the up against y’s coming into the space is a cause of y’s
m moving in the opposite direction. But our label for
cou any cause of motion is ‘moving force’. Consequently,
ld matter fills its space not by merely being there but by
be ·exerting· moving force. [At the start of this paragraph,
sai Kant says that the very first instant of a thing’s movement is
d called Bestrebung, which can mean ‘attempt’ or ‘endeavour’ or the
to like. Like other early modern philosophers he used that term (or
be its equivalent in other languages) to stand for an active tendency
in that a body may have to move in a certain way. To say that thing
the has a Bestrebung to enter a given space is not to say •that it is
lar consciously trying to move in, but it is to say more than merely
ger •that it is in a state such that it will move in unless something
spa stops it. From now on in this version, ‘endeavour’ will be used for

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

Remark from it (or, equivalently, through which it resists another

Lambert and others used the rather ambiguous word portion’s moving towards it).
‘solidity’ to name the property of a portion of matter by [In English we have the verb ‘move’ both as transitive (as in ‘She moved
which it fills a space; and they maintained that solidity the jar to the end of the shelf’) and intransitive as in ‘You spoiled the
must be possessed by every thing that exists (every picture: just as I clicked, you moved’. In the phrase translated as ‘moving
substance), or at least by every thing in the external sensible force’ Kant is referring not to a force that moves-intransitive but rather
498 world. According to their way of thinking, a real thing x in a to a force that moves-transitive; not a force that roams, but one that
region of space must by its very concept carry with it this shoves. In fact, German doesn’t have a verb that exactly matches the
resistance: the principle of contradiction rules out there English intransitive ‘move’. In the present version of this work, Kant is
being anything else in the space containing x. But a portion often translated as saying of some item that it ‘moves’; but he does this
of matter that is moving towards penetrating a space that with a German expression which would be mechnically translated as ‘is
already contains another portion of matter isn’t pushed moved’.]
back by the principle of contradiction! Note
The only way I can make sense of the suggestion that These are the only two moving forces that can be thought of,
a contradiction is involved in a space’s containing one ·as I shall now prove·. In the context of questions about one
thing x and being penetrated by another y portion of matter impressing some motion on another, the
is by attributing to x a force through which it pushes back two portions must be regarded as points; so any transaction
an external movable thing that approaches it. Here the of that kind must be regarded as happening between two
mathematician (·Lambert·) has assumed, as an initial datum points on a single straight line. Now, there are only two ways
in constructing the concept of matter, something that doesn’t for two points to move ·relative to one another· on a single
admit of being further constructed. Well, he can indeed straight line: either
begin his construction with any datum he pleases, treating •
they approach one another, caused to do so by an
the datum as unanalysed; but he isn’t entitled to block attractive force; or
the route back to the first principles of natural science by •
they recede from one another, caused to do so by a
analysing this datum as something wholly incapable of any repelling force. 499
mathematical construction. Consequently, these two kinds of forces are the only ones we
can make sense of; and all the forces of motion in material
Definition 2 Nature must come down to them.
Attractive force is the moving force through which a portion Proposition 2
of matter can be the cause of another portion’s moving
towards it (or, equivalently, through which it resists another (a) Matter fills its space by the repelling forces of all its parts,
portion’s moving away from it). i.e. by its own force of extension, and (b) this ·repelling force·
Repelling force is the moving force through which a portion has a definite degree that can be thought of as smaller or
of matter can be the cause of another portion’s moving away

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

greater to infinity. [This use of ‘degree’ translates what is almost the ‘greater force’·.
first occurrence of Grad in the original. From here on, Grad/degree will
occur often; in Kant’s usage it is firmly linked to the notion of intensive
magnitude [see note on page 16]. We’ll later see him writing about the
degree to which a given portion of space is filled; this doesn’t mean
(extensive) how much of the space is filled but (intensive) how strongly
the space is filled.]
Proof
(a) Matter fills a space only through moving force
(Proposition 1), specifically by a moving force that resists
the penetration,
i.e. the approach, of other matter; and this is a repelling force
(Definition 2). So matter fills its space only through repelling
forces, and indeed through the repelling forces of all its parts.
(Why ‘all its parts’? Well, try to suppose that some part x of
a portion of matter doesn’t exert repelling force. That means
that the portion of space assigned to x is not filled, which
means that that x isn’t a portion of matter after all, but only a
region of space contained within a portion of matter.) And the
force of something that is extended by virtue of the repulsion
of all its parts is a force of extension. [Kant adds in brackets that
this is ‘expansive’ force—the first time this word has occurred in the work.
We’ll see a lot of it from now on. ] Therefore, matter fills its space
only by its own force of extension. (b) Given any particular
force, it is conceivable that there should be a greater one.
If for a given force F it was inconceivable that there should
be a greater force, that would mean that F was the greatest
conceivable force, which could make something travel an
infinite distance in a finite length of time; which is impossible.
·Why ‘an infinite distance’? Well, suppose that the best F
can do is to make something travel N miles in a year, where
N is a finite number; then it is conceivable that some force
F+ should make a thing travel N+1 miles in a year, so that
F+ would be greater than F. Where there’s room for the
thought ‘greater distance’ there’s room for the thought

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of
Also, given any particular force, it is conceivable that
there should be a lesser one. If that weren’t so, there
could be a force F such that a weaker force was
inconceivable, which implies that the distance F could
make a thing travel in a year was zero; meaning that it
couldn’t make anything move at all; meaning that F
isn’t a force of movement after all. (·The explanation
of zero in this half of the proof of (b) can easily be
worked out from the explanation of infinity in the first
half·.) Putting (a) and (b) together: The force of
extension through which every portion of matter
fills its space has a degree that is never the greatest
or smallest, but beyond which greater as well as
smaller degrees can always be found. [Kant presumably
means ‘can be found in the realm of possibilities’ = ‘can be
conceived’, not ‘can be found in the material world’. His later uses
of ‘can be found’ will be translated without comment.]
Note 1
The expansive force of matter is also called elasticity. This
500 force is the basis for the filling of space as an
essential property of all matter, so it is basic, not a
consequence
of any other property of matter. So all matter is
basically elastic.
Note 2
Given any extensive force there can be found a greater
moving force that can work against it and diminish the
space that the extensive force is trying to expand. In
this case the latter force is called a ‘compressive’ one.
Thus, for any given portion of matter a compressive
force can be found that can squeeze this matter into a
smaller space than the one it is currently occupying.

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

Definition 3 be greater when enclosed in a smaller space, and must be

infinite when compressed into an infinitely small space. (a)
A portion of matter x in its motion penetrates another
Now, for any given extensive force that a portion of matter
portion y when by compression it completely abolishes the has, there can be found a greater compressive force that
space of y’s extension. [Kant’s verb aufhebt apparently has to mean squeezes this matter into a smaller space, and so on to
‘abolishes’ in this context. But we’ll see in a moment that what he means
infinity. But (b) penetrating the matter would require its
is that x takes over the space through which y was extended, depriving y
compression into an infinitely small space, and thus would
of it.]
require an infinitely strong compressive force; but such a
Remark force is impossible. Consequently, a portion of matter cannot
When an air-pump’s piston is pushed ever closer to the be penetrated by the compression of any other portion of
bottom of the cylinder, the air-matter is compressed. If this matter.
compression could be carried so far that the piston came flat Remark
against the bottom with no air escaping, then the air-matter I have assumed at the start of this proof that the more an
would be penetrated ·in the sense laid down in Definition extensive force is constricted the more strongly it must resist.
3·. For it is between two portions of matter that leave no This might not hold for a •derivative elastic force, but it can
space for it, so that it’s to be met with between the bottom of
be postulated of ·any •basic elastic force, i.e. · any elastic
the cylinder and the piston without occupying a space. This
force that a portion of matter has essentially, just because
penetrability of matter by external compressive forces would it is matter filling a space. Expansive force exercised from
be called ‘mechanical’, if there were such a thing—or indeed if all points toward all sides constitutes the very concept of
such a thing were conceivable. I distinguish this impossible elasticity. And the smaller the space in which a given amount
penetrability of matter from another kind of penetrability of expanding force has to exercise itself, the more strongly
which is perhaps equally impossible. I may need to say a the force must exercise itself at every point in the space.
little about this second kind of penetrability later on. [We’ll see
that in this second kind of penetrability, which Kant will call ‘chemical’,
Definition 4
x penetrates y by coming to share all y’s space with y (see page 44).
This is a much more natural meaning for ‘penetrate’ than the present The impenetrability of matter that comes from its resis-
‘mechanical’ one.] tance ·to being squeezed·—impenetrability that increases
proportionally to the degree of compression—I call ‘relative’.
Proposition 3
The impenetrability that comes from the assumption that 502
501 (a) Matter can be •compressed to infinity, but (b) it can never matter as such can’t be compressed at all is called ‘absolute’
be •penetrated by other matter, however great the latter’s impenetrability. The filling of space with absolute impenetra-
pressing force may be. bility can be called ‘mathematical’; that with merely relative
Proof A basic force through which a portion of matter impenetrability can be called ‘dynamical’.
tries to extend itself all through the space that it occupies
must
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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

Remark 1 force; but it doesn’t (ii) lack helpful complexity, because it

According to the merely mathematical concept of impene- involves the concept of an •active cause and of •the laws
trability (which doesn’t assume that any moving force is of this cause in accordance with which the strength of the
basically inherent in matter), no matter can be compressed force can be measured by how strongly the space in question
except to the extent that it contains empty spaces within resists penetration.
itself. So matter, just as matter, resists all penetration
unconditionally and with absolute necessity. According to
my discussion of it, however, impenetrability has a phys- Definition 5
ical basis; for the extensive force makes matter itself, as Material substance is whatever it is in space that is movable
something extended filling its space, first of all possible. on its own, i.e. separated from everything else existing out-
But this force has a degree that can be overcome, so the side it in space. The motion of a portion of matter whereby 503
space occupied by a portion of matter can be diminished, it ceases to be a part ·of some larger portion of matter· is
i.e. its space can be somewhat penetrated by a given separation. The separation of the parts of a portion of matter
compressive force; but complete penetration is impossible, is physical division.
because it would require an infinite compressive force.
Because of all this, the filling of space must be regarded Remark
only as relative impenetrability. The concept of substance signifies the ultimate subject of
existence, i.e. everything that doesn’t exist merely as a predi-
Remark 2 cate [here = ‘property’] of some other existing thing, ·in the way
In fact absolute impenetrability is nothing more or less a blush exists merely as a property of a face, or a storm exists
than a qualitas occulta. [Kant here refers (in Latin) to the ‘occult (= merely as a property of some wind and water ·. Now, matter is
hidden) qualities’ that were postulated by various mediaeval philosophers
the subject of everything existent in space; for besides matter
to ‘explain’ certain phenomena; by Kant’s time, everyone agreed that
no other spatial subject can be thought of except space itself;
these explanations were no good. There were two basic complaints about and the concept of space hasn’t any content relating to
them: (i) They weren’t derived from anything deeper or more general; existence, and merely contains the necessary conditions for
they were always treated as basic, fundamental. (ii) Their ‘explanations’ things we can perceive through the external senses to have
were always slam-bang one-sentence affairs, with no complexity that external relations to one another. So •matter—as what is
might enable them to connect fruitfully with other explanations of other movable in space—is •substance in space. Similarly every
phenomena.] We ask ‘Why can’t portions of matter penetrate part of a portion of matter will also be a substance, because
one another in their motion?’ and are given the answer it too is itself a subject and not merely a predicate of other
‘Because they are impenetrable’! The appeal to repelling force portions of matter; so every part of any portion of matter is
is not open to this complaint. It is true that (i) this force also itself a portion of matter. . . .
can’t be shown to be possible through our giving a further
analysis of it, so that we have to accept it as a fundamental

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

Proposition 4 repelling force. The region containing such a physical

point is divided, but the substance acting in it—the
Matter is divisible to infinity, and indeed into parts each of physical point—is not divided.’
which is again matter. Thus, this monadist can have matter made up of physically
Proof indivisible parts while still allowing it to occupy space in
Matter is impenetrable because of its basic force of extension a dynamical way, ·i.e. to occupy space by exerting force
(Proposition 3 [page 22]), but this force of extension is only throughout it·.
the consequence of the repelling forces of each point in a But the proof I have given completely undermines this
space filled with matter. Now, the space that matter fills monadist dodge. My proof makes it clear that every point
is mathematically divisible to infinity, i.e. its parts can be in a filled space must push back against whatever pushes
differentiated to infinity; although they can’t be moved and in upon it. This •can be the case if the point contains a
so can’t be pulled apart. Now, in a space filled with matter reacting subject that is separately movable and distinct from
every part of the space contains repelling force to hold at every other repelling point; and it’s clear that it •can’t be
bay on all sides all the parts surrounding it, and hence to the case if all you have is a mere driving force exerting itself
repel them and be repelled by them, i.e. to be moved to through a region of space. To get an intuitive grasp of this
504 a distance away from them. Hence every part of a space (and, therefore, of the proof I have given for Proposition 4),
filled by matter is movable and is therefore separable by consider this diagram:
physical division from any of the other parts that are material
substances. Consequently, every mathematical division of a
region of space has corresponding to it a possible physical
division—a pulling apart—of the substance that fills the
region of space; and such mathematical divisions can be
A is stipulated to be a monad whose sphere of repulsive
continued to infinity, so all matter is physically divisible to
force has the line aAb as a diameter. Then penetration of
infinity—divisible indeed into parts each of which is itself
A’s sphere of influence is resisted at the point a. But now
also a material substance.
consider a point c that is within the sphere, between a and
Remark 1 A (there must be such a point, because space is infinitely
Proving the infinite divisibility of space is far from proving divisible); and ask yourself what the state of affairs is at c.
the infinite divisibility of matter unless one first shows that in The answer is that there must be at c something that holds A
every part of space there is material substance, i.e. separately apart from a:
movable parts. ·To see the need for this further premise, A force emitted from A can’t make itself felt at a unless
consider this position, which· a monadist might adopt: ·the contents of· those two points are kept apart;
‘Matter consists of physical points, each of which— without that, they would penetrate one another ·so
just because it is a point—has no separately movable that the entire sphere would condense into a point·. 505
parts, but nevertheless fills a region of space by mere

3
Metaphysical Foundations of Natural Immanuel 2: Foundations of

So something at c resists penetration by a and by A; so it Remark 2

repells the monad A at the same time as being repelled by When mathematicians are just doing mathematics, they can
it. And repelling is a kind of motion. So we get the result ignore the tricks played by mistaken metaphysics. They
that c is something movable, ich means that it is matter. can be sure of the obvious mathematical truth that space is
This shows that the filling of that sphere can’t consist infinitely divisible, without caring about objections that may
merely in a repelling force’s being exerted throughout it by be brought against this by foolish nit-pickers. But when they
a one-point monad in its centre. On the contrary, the are ·not merely doing mathematics but · taking mathematical
sphere must be filled with matter. (We are assuming, of
propositions that are valid for space and applying them to
course, that the argument about the point c could be
substance filling space, they have to submit what they are
repeated for any point within the sphere.)
saying to purely conceptual tests, which means that they
Mathematicians represent the repelling forces of the parts
have to attend to metaphysics. Proposition 4 [page 24] is
of elastic portions of matter as increasing or decreasing in
already a proof of this. For although matter is infinitely
proportion to their distances from one another. The smallest
divisible mathematically, it doesn’t follow that matter is
parts of air, for instance, repel each other in inverse
physically divisible to infinity. Granted that every part of
propor- tion to the distance between them, because their
space is also a space, so that every part of space includes
elasticity is inversely proportional to the spaces that they
within itself parts that are external to one another, it doesn’t
are squeezed into. Don’t misunderstand the thought and
follow that in every possible part of this filled space there
mistake the language of the mathematicians by taking
is substance, which is separated from everything else and
•something that necessarily belongs to the process of
is independently movable. [Notice that Kant says ‘filled space’—a
constructing the concept to be •something that applies to
phrase that he uses quite often to mean ‘space filled with matter’. So the
the object of the concept.
mathematicians’ account of space as infinitely divisible stands firm even
·Here’s why they are different·. In the construction process,
if the space in question is thought of as ‘full of matter’, provided (Kant
two things’ being in contact can be represented as their
warns) that this is left unexplained and (in particular) is not understood
being an infinitely small distance apart; and indeed the
construction has to handle contact in that way in cases as meaning that every part of space contains a material substance. To
where a single quantity [Quantität] of matter, i.e. a single the proposition that he is allowing the mathematicians to assert he might
quantum of repelling forces, is represented as completely give the label ‘the mathematical proposition of the infinite divisiblity of
filling spaces of different sizes ·at different times·. For us matter’, setting this off against (a phrase that he does use) ‘the physical
to •have an intuitive sense of the expansion of a portion of So there has always
proposition of the infinite divisibility of matter’. ]
matter to fill a larger space— ·•that being what constructions been something missing from mathematical proof ·of the
are for·—we have to make use of the idea of an infinitely infinite divisibility of matter·, and there has been no guar-
small distance. [See the note on ‘idea’ on page 9.] But if matter is antee that that proof could be securely applied in natural
infinitely divisible, there can’t be any actual distance between science. This gap has now been filled—by ·my proof of·
Proposition 4 above. Now we have the physical proposition 506
any two ·nearest· parts; however much a portion of matter
expands, it is still a continuum. of the infinite divisibility of matter; and when it comes to

3
Metaphysical Foundations of Natural Immanuel 2: Foundations of

metaphysical attacks on that, the mathematician must ·back of things in themselves, is denying (1). So the philosopher
off and· leave them entirely to the philosopher. When the sees himself as forced to depart from the assertion (2) that
philosopher tries to deal with these attacks, he ventures into matter is a thing in itself and space a property of things
a labyrinth that is hard enough to get through when he just in themselves—maintaining instead that space is only the
approaches it philosophically; he can do without interference form of our external sensible intuition [see note on page 8], so
from mathematician! ·Here’s a sketch of the labyrinthine that matter and space are not things in themselves but
problem· (stated for portions of matter, though it applies only subjective modes of representation of objects that are
equally to regions of space): in themselves unknown to us. Proposition (2) is common
(a) A whole must already contain within itself all and commonsensical; the philosopher denies it only on the
the parts into which it can be divided. Therefore understanding that this will get him out of the difficulty
(b) if matter is infinitely divisible, then it consists of about matter’s being infinitely divisible yet not consisting
infinitely many parts. But (c) a portion of matter can’t of infinitely many parts. That matter consists of infinitely
possibly have infinitely many parts, because (d) the many parts can indeed be thought by reason, though this
concept of infiniteness is the concept of something that thought can’t be constructed and made intuitable [see note
can’t ever be wholly complete, from which it follows on page 2]. If something x is •actual only by •being given in
that ‘There are infinitely many of them, and they are a representation, all you are given ·when you think of it·
all there, complete, settled’ is self-contradictory. is what’s met with in the representation, i.e. as far as the
That is the difficulty as it presents itself to the dogmatic sequence of representations reaches. If something is an
metaphysician, who is thinking of wholes as things in appearance that can be divided to infinity, what can we
them- selves, the crucial point being that proposition (a) is say about how many parts it has? Only that it has as many
true only of wholes considered as things in themselves. So parts as we give it, i.e. as many as result from whatever
we have to choose between two options: division of
•
Defy the geometer by denying (1) that space is divisi- it we choose to make. That’s because the parts of something 507
ble to infinity. that is merely an appearance exist only in thought, i.e. only
•
Annoy the metaphysician by denying (2) that •matter in ·the thought of· the division itself. The division does
is a thing in itself and •space a property of a thing in indeed go on to infinity, but it is never given as infinite; so
itself, saying instead that matter is a mere appearance we can’t infer that the divisible item contains within itself
of our external senses and that space is just the infinitely many parts ·that are things· in themselves existing
essential form of matter, ·i.e. of that appearance·. independently of our representation of them. Why can’t we?
The philosopher is now squeezed between the horns of a Because the division that can be infinitely continued is the
dangerous dilemma. It’s no use denying (1) that space is division not •of the thing but only •of its representation. A
divisible to infinity; that’s a mathematical result, and you great man who perhaps contributes more than anyone else to
can’t get rid of it by tricky argument! But regarding matter the reputation of mathematics in Germany has several times
as a thing in itself, and thus regarding space as a property rejected the impudent metaphysical claim to overturn what
geometry teaches concerning the infinite divisibility of space.

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

[Who? Leibniz is a good guess (see below), except that the tenses in the of the monadology of Leibniz, which they saw as trying to
foregoing sentence don’t seem right for someone who had been dead for explain natural appearances whereas really it is a platonic
70 years when Kant wrote this work. ]His basis for this rejection concept of the world. There’s nothing wrong with Leibniz’s
was the reminder that space belongs only to the appearance concept ·of the world as a system of sizeless monads ·, as
of external things; but his readers didn’t understand him. long as the world is being regarded not as •an object of
They took him to mean: the senses but as •a thing in itself, i.e. as merely an object
M: Space is a thing in itself or a relation amongst of the understanding, though it is the foundation of the
things in themselves; but it appears to us, and the appearances of the senses. [From here down to the next mention
mathematicians ·aren’t vulnerable to metaphysical of Leibniz, this version expands on Kant’s words in ways that the ·small
attack because they· are talking only about space as dots· convention can’t easily signify. ]
Now, any composite thing
it appears, ·not about actual space itself·. made up of things in themselves must certainly consist of
What they should have understood him to mean is this: simple things, because a composite thing in itself can’t exist
Space isn’t a property of anything outside of our except as an upshot of the existence of its parts, all its
senses; it is only the subjective form of our sensibility. parts, right down to the smallest ones that don’t have
Objects of our external senses appear to us under parts. But
this form, and we call this appearance matter. As for a composite thing that is an appearance doesn’t consist of 508
what these objects are like ·in themselves·—we know simple things, because its parts exist only as upshots of
nothing about that. a division of the thing; so that they, rather than existing
According to the misinterpretation M, space was always independently of the composite thing of which they are parts,
thought of as a quality that things have independently of exist only in that composite thing. For a thing in itself x:
our power of representation, and the mathematicians ·were x exists as an upshot of the putting together of its
being criticised because they· thought of this quality only parts;
through common concepts (i.e. thought of it confusedly, whereas for an appearance y:
for appearance is commonly thought of confusedly). This y’s parts exist as upshots of the division of y.
meant that according to M the geometricians had used So it seems to me that Leibniz didn’t intend to explain
a •confused representation of space as their basis for a space in terms of an order of simple entities side by side,
mathematical proposition—asserting the infinite divisibility but rather to claim that this order corresponds to space
of matter—which presupposes the highest •clarity in the while still belonging to a merely intelligible world that is
concept of space. Thus the door was left open for the M- unknown by us. And this is to assert just what I said
accepting metaphysicians to bring clarity into this concept of elsewhere [in the Critique of Pure Reason], namely that space
space (they thought!) by supposing that space is made up of along with matter
points and matter is made up of simple parts, ·i.e. parts . . . . doesn’t make up the world of things in themselves but
that did not in their turn have parts·. This error was based only the appearance of such a world, and that what space
on another misinterpretation—namely a misunderstanding itself is is only the form of our external sensible intuition.

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Metaphysical Foundations of Natural Immanuel 2: Foundations of

Proposition 5 expanded infinitely, there wouldn’t be as big a portion of matter as M in a

space as
The possibility of matter requires a force of attraction, as the
second essential basic force of matter.
Proof
Impenetrability, as the fundamental property of matter
through which it first reveals itself as something real in
the space of our external senses, is nothing but matter’s
power of extension (Proposition 2). Now, an essential moving
force by which parts of matter pull away from one another
cannot
(1) be limited by itself, because such a force works on
matter to drive it towards continuously expanding the
space that it occupies;
and it cannot
(2) be kept within limits by space itself. Why not?
Because the most that space can do is to bring it
about that when the volume of a portion of matter is
in- creasing the extensive force becomes
correspondingly weaker; such weakenings can go on
to infinity—·the strength of a force is continuous·—
but they can’t reach zero, which is to say that
space can’t bring it about that the extensive force
stops.
Therefore, if matter were driven only by its repelling force
(the source of its impenetrability), with no other moving
force counteracting this repelling one, there would be
nothing to limit matter’s extension; every portion of matter
would disperse itself to infinity, so that no assignable
quantity [Quantität] of matter would be found in any
assignable ·region of· space. Consequently, if there were only
repelling forces in matter, all regions of space would be
empty—so that strictly
speaking there wouldn’t be any matter! [The thought is this:
Let R be a region of space measuring a billion cubic kilometers, and
let M be a portion of matter weighing a billionth of a gram: if matter
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 Metaphysical Foundations of Natural Immanuel 2: Foundations of
small as R, because that amount of matter would have been spread
still more thinly through a still larger region of space. Repeat the
argument, making M ever smaller and R ever larger; you will always
have too much matter for that amount of space.]For matter to
exist, therefore,
it must have compressive forces opposed to the extensive
509 forces. ‘Might not the force that keeps material portion
x within limits be the ·expansive· force of a different portion y?’
No, that can’t be the basic account of the situation, because
this ‘different portion y’ can’t exist as matter unless some
compressive force is acting upon it. So we have to
assume that matter has a basic force acting in an
opposite direction to the repelling force; this force must
tend to bring things closer to one another, which is to say
that it must be an attractive force. Now, this attractive
force is needed for any matter to be possible, so it is more
basic than any differences between kinds of matter; and
therefore it must be attributed not merely to some one
species of matter but to all matter. Thus, a basic attraction
belongs to all matter as a basic force that is part of its
essence.
Remark
We need to look more closely into what happens in our
thinking when when we move from •one property ·that is
contained in· the concept of matter to •a radically different
property that equally belongs to the concept of matter
with- out being contained in it. If attractive force is
basically required for matter to be possible, why don’t we
use it, along with impenetrability, as the primary sign of
matter? Impenetrability is given immediately with the
concept of matter, while attraction isn’t thought in the
concept but only associated with it by inference—what’s
going on here? You might think: ‘Well, our senses don’t let
us perceive attraction as immediately as repulsion and
the resistance of impenetrability’—but that doesn’t
properly answer the

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

question. Suppose that we could perceive attraction as of touch this property tells us the size and shape of an
easily as repulsion: our understanding would still choose to extended thing, thus creating the concept of a determinate
differentiate space from matter—i.e. to designate substance object in space—a concept that underlies everything else
in space—in terms of the filling of space (otherwise known as that can be said about this thing. No doubt this is what
solidity). Attraction, however well we perceived it, ·couldn’t explains the fact that although there are very clear proofs
do the job. It· would never reveal to us any portion of matter that attraction must belong to the basic forces of matter just
with a definite volume and shape. All it could reveal to as much as repulsion does, there are people who strenuously
us would be our perceiving organ’s being tugged towards a reject attractive forces and won’t allow matter to have any
point outside us, namely the central point of the attracting forces except those of impact and pressure (both by means
body. [Translated more strictly, Kant speaks not of the organ’s being of impenetrability). ‘What space is filled by is substance’,
tugged but of its ‘endeavouring’ to reach that external point. Either way, they say; and this is correct enough, ·but its correctness
it is initially surprising, but it is not unreasonable. How do we perceive has led these people astray·. The substance that they talk
repelling forces? By feeling ourselves being pushed away from things. about reveals its existence to us through the sense by which
So how would we (if we could) perceive attractive forces? By feeling we perceive its impenetrability, namely the sense of touch;
ourselves being pulled towards things! This interpretation presupposes so it reveals its existence only through the contact of one
that the ‘perceiving organ’ is the perceiver’s body, the ‘organ’ of the sense portion of matter with another—a process that starts with
That experience wouldn’t reveal to us any material
of touch.] collision and continues with pressure. And because of this
things with definite sizes and shapes, because the only way it seems as though the only way for one material thing to
the attractive force of all parts of the earth could affect act immediately on another is by colliding with it or putting
us is exactly the same as if that force were concentrated pressure on it—these being the two influences that we can
entirely in the centre of the earth and this point alone were immediately perceive. Whereas it’s very hard for us to think
tugging us; similarly with the attraction of a mountain, or of of attraction as a basic force, because it doesn’t give us any
a stone, etc.—the pull would always be to the central point, sensation at all, or anyway no definite object of sensation.
and would give no sense of the relevant body’s shape or
Proposition 6
510 size of even its location. (·Why not its location? Because·
although we would be able to perceive the direction of the Matter isn’t made possible by mere attraction, without repul-
attraction, as it is perceived in our experience of weight, we sion.
wouldn’t know how far away it was in that direction.) The
Proof
attracting point would be unknown, and I don’t see how
Attractive force is the moving force of matter whereby one
it could even be discovered through inferences unless we
material thing gets another to approach it. If every part of
already had perceptions of matter as filling space, ·i.e. as
the material world exercises such a force, all those parts are
having repelling force·. This makes it clear that our first
led to cluster together, thus shrinking the region of space
application of our concepts of size to matter. . . .is based
that they jointly occupy. Now, the only thing that can block
only on matter’s space-filling property. Through our sense

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

the action of a moving force is a moving force opposed to 511

only repulsion and attraction?· Because they are the only
it; and the force that is opposite to attraction is the force
ones that are thinkable.
of repulsion. If that didn’t exist, there would be nothing to
stop the force of attraction from pulling portions of matter
together closer and closer, constantly shrinking the region of Definition 6
space containing matter. There would be no such thing as Contact in the physical sense is the immediate action and
two material things so close together that repelling reaction of impenetrability. The action of one portion of
forces block them from coming even closer, matter on another when there is no contact between them
so that the force of attraction would eventually pull material is action at a distance. When this action at a distance
things closer and closer together until they shrank into a occurs without the mediation of matter lying between the
mathematical point; and at that stage space would be empty, two portions of matter it is called unmediated action at a 512
i.e. wouldn’t contain any matter. So matter is impossible distance, or the action of portions of matter on one another
through mere attractive forces without repelling ones. [Notice through empty space. [Kant’s word unmittelbar is usually translated
the elegant shape of Kant’s arguments about the two kinds of force. as ‘immediate’; and that is not incorrect. But it’s natural for us to think
Allow only repulsion/expansion and matter is spread so widely and of x’s ‘immediate’ influence on y as ruling out not only (a) any mediating
thus thinly that it disappears; allow only attraction/contraction and thing between them but also (b) any distance between x and y as well.
matter is packed so densely that it is all contained in a single point and Therefore, in cases where Kant is ruling out (a) and emphatically not
disappears from all space except that point.] ruling out (b), ‘unmediated’ will be used instead.]
Note
Remark
Any property that is required for something to be
Contact in the mathematical sense of the word is the
intrinsically possible (·whether or not possible in relation to
shared boundary of two regions of space—so it isn’t in
other things·) is itself an essential element of that intrinsic either of them. So straight lines can’t be in contact (in this
possibility. So repelling force belongs to the essence of sense) with one another: when two straight lines have a
matter as much as attractive force does—the two can’t be point in common, that is because they intersect, and their
separated in the concept of matter. common point belongs to each of them. But a circle and a
Remark straight line can be in contact at a point, and so can a circle
I had first to consider the forces of repulsion and attraction and another circle; two planes can be in contact at a line, and
•
separately, in order to see what each on its own could two solids can be in contact at a plane. Mathematical
contribute to the presentation of matter. The upshot was contact lies at the basis of physical contact, but it doesn’t
an a priori proof that they are both present, •united, in the constitute it. To get from the concept of mathematical contact
general concept of matter. We found that space remains to that of physical contact you have to add the thought of a
empty, with no matter to be found in it, unless both these dynamical relation—not that of the attractive forces but
forces are at work in it. ·Why only these two forces—why the relation of the repelling ones, i.e. of impenetrability.
Physical contact

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

is the two-way interaction of repelling forces at the common give rise to it. Just because it is a basic force it can’t be
boundary of two portions of matter. derived from anything.
[This use of ‘conceivable’ may seem odd. It comes from the fact that Kant
Proposition 7 is running the proposition
The concept of attraction can’t be analysed into simpler or more
basic concepts
The attraction that is essential to all matter is an unmediated
in the same harness as the proposition
action through empty space of one portion of matter on The attractive force can’t be shown to be derived from and depen-
another. dent on some more basic forces.
On page 40 we shall find Kant inferring from propositions of the type
Proof
The. . . force can’t be shown to be derived from and dependent
The possibility of matter as the thing that fills a space in a on some more basic forces
determinate degree depends on the basic attractive force, the corresponding propositions of the form
and so the possibility of physical contact between portions It isn’t possible for us to comprehend the possibility of the. . . force.
of matter also depends on it. [Until now Kant hasn’t spoken He regards this as an inevitable drawback of any theory that postulates
explicitly of regions as being filled to a greater or lesser degree, more basic forces; but we’ll see that it’s a drawback he is willing to put up
or less intensively filled; but he has done so implicitly, by saying that But the basic
with because of the advantages of that kind of theory.]
the repelling force that constitutes space-filling is a matter of degree, i.e. attractive force isn’t even slightly more inconceivable than
can be more or less strong at a given point. This concept of the degree to the basic force of repulsion. The difference is merely that the
which a given region of space is filled will be crucially important in what basic attractive force doesn’t offer itself so immediately to
follows.]Thus, physical contact presupposes the attractive our senses as impenetrability—the repelling force—does in
force, so the force can’t depend on there being physical giving us concepts of determinate objects in space. Because
contact. Now, the action of a moving force that •doesn’t it’s not •felt but only •inferred, the attractive force gives
depend on any contact •doesn’t depend either on the filling the impression of being ·not a •basic force but· a •derived
of space between the moving thing and the thing moved, one, as though repulsion were the upshot of a hidden play
·because ‘the space between x and y is filled’ is equivalent of ·more basic· moving forces. But when we take a closer
to ‘from x to y there is a series of portions of matter, each look at attraction, we see that it can’t be derived from any
in contact with the next’·. This means that such action source, least of all from the moving force of portions of matter
must occur without the intervening space being filled, and through their impenetrability, because its action is exactly
so it’s action that operates through empty space. Therefore the opposite of impenetrability. The most common objection
the basic essential attraction of all matter is an unmediated to unmediated action at a distance is the claim that a portion
action of portions of matter upon one another through empty of matter can’t directly act at a place if it isn’t there. ·But·
space. when the earth directly influences the moon to come closer,
it is acting unmediatedly on a thing thousands of miles away;
Remark 1
and the space between the earth and the moon might as well
513 It is completely impossible to make any basic force conceiv- be regarded as entirely empty, because even if there is matter
able, i.e. to present one or more other forces that somehow

3
 Metaphysical Foundations of Natural Immanuel 2: Foundations of

there it has no effect on the attraction. So the earth acts a constant law of the form ‘If two portions of matter
directly in a place without itself being there. That may seem have relation R1 between them, they move towards
to be self-contradictory, but it isn’t. The truth of the matter one another’.
in fact is that whenever anything in space acts on anything And for x to repel y unmediatedly and without contact is for
else, it acts in a place without itself being in it! If something this to be the case:
were to act in the same place where it itself is present, then it (2) x and y move away from one another in accordance
wouldn’t be acting on anything outside it, but only on itself. with a constant law of the form ‘If two portions of
For a thing x to be ‘outside’ a thing y is for x to be in a place matter have relation R2 between them, they move
that doesn’t have y in it. If the earth and the moon touched away from one another’.
each other, the point of contact would be a place that has Now, there is not the slightest difficulty about supposing
neither the earth nor the moon in it. . . . It wouldn’t even that repelling force doesn’t come into R1 and that attractive
have anyI part of either the earth or the moon in it, because force doesn’t come into R2. These two moving forces are
this point lies at the boundary of the two filled regions, and wholly different in kind, and there’s not the slightest basis
this boundary isn’t a part of either of them. It follows from for claiming, of either of them, that it depends on the other
this that the ·widely accepted· proposition that and isn’t possible without the intervention of the other.
•
portions of matter cannot unmediatedly act on each
Remark 2
other at a distance
Attraction between two things that are in contact can’t result
amounts to the proposition that
in any motion. Why not? Because for two bodies to be in
•
portions of matter can’t unmediatedly [unmittelbar] act
contact is for the impenetrability of each to act against the
on each other without the intervention [Vermittelung] of
impenetrability of the other, and that impedes all motion. So
the forces of impenetrability.
there must be some unmediated attraction without contact,
This amounts to saying that repelling forces are the only ones
i.e. unmediated attraction at a distance. To see why, suppose
by which portions of matter can be active, or at least that
that it is not so, and see where you get. We have two bodies
they must be involved when portions of matter act on one
that are approaching one another, without unmediated at-
another; which implies that the force of attraction is either
traction being at work. In that case, the situation must be
•
impossible or •always dependent on the action of repelling that they are being pushed towards one another by forces
forces; and there is no basis for either of those assertions.
of pressure and impact. This is only apparent attraction,
The ·widespread· misunderstanding of this matter is a result as against true attraction in which repelling forces have no
514 of confusing •the mathematical contact of regions of space role at all. But even such an apparent attraction must, deep
with •their physical contact through repelling forces. [The rest down, involve true attraction, because the portions of matter
of this paragraph expands Kant’s words in ways that the ·small dots· whose pressure or impact is at work wouldn’t even be matter
convention can’t easily signal.] For x to attract y unmediatedly if they didn’t have attractive forces (Proposition 5 [page 28]).
and without contact is for this to be the case: So the attempt to ·get rid of true attraction and· explain
(1) x and y come closer together in accordance with

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

all phenomena of approach in terms of apparent attraction will impart more relative motion to x than y alone
moves in a circle. did on the first occasion, despite the fact that the
There is a view about Newton that is widely accepted, wood, which contributes to the quantity [Quantität] of
namely: the matter in y-plus-box, adds nothing at all to y’s
He didn’t see any need for his system to postulate attractive force and exerts no magnetic attraction.
unmediated attraction of portions of matter. Behaving Newton ·regarded attraction as something that all matter, of
strictly like a pure mathematician, he •kept right out whatever kind, must have. He· wrote:
of this issue, •left the physicists completely free to ‘If the ether or any other body had no weight, it would
explain the possibility of such attraction in differ from any other portion of matter only in its
whatever way they thought best, and •avoided form, so that it could be transformed little by little
mixing up his propositions with their play of through a gradual change of this form into a portion
hypotheses. of matter of the heaviest kind on earth; and conversely
But how could he establish the proposition that the universal the heaviest kind could become weightless through a
attraction of bodies—across a given distance—is proportional change of its form. This is contrary to experience’ and
to the quantity [Quantität] of matter in the bodies if he didn’t so on. [Newton’s Principia II.vi.cor.2]
assume that it’s an essential feature of matter as such, Thus he didn’t exclude even the ether (much less other kinds
·matter simply qua matter·, that it exercises this motive of matter) from the law of attraction. If Newton held that
force? For when one body pulls another, their approach the approach of bodies to one another was a case of mere
515 to one another (according to •the law of the equality of apparent attraction, created ·somehow· by impact, what kind
reciprocal action) must always occur in inverse proportion of matter would he be left with to provide the impact? So you
to ·the quantity of· the matter in those bodies—and it makes can’t claim this great founder of the theory of attraction as
no difference what kinds of matter are involved. Still, •this your predecessor, if you take the liberty of replacing the •true
law is not attraction that he did maintain by an •apparent attraction
a principle of •dynamics, i.e. a law about the distribu- that forces you to explain the appproach of bodies in terms
tion of attractive forces, of impact. ‘What causes the universal attraction of matter?’
but rather Newton declined to get into any hypotheses to answer this
a law only of •mechanics, i,e, a law about the question; and he was right to do so, because the question
motions that attractive forces cause. belongs to physics or metaphysics, not mathematics. It’s
And not just attractive forces; it is valid for moving forces true that in the preface of the second edition of his Optics
generally, of whatever kind. ·Here is an illustrative example·: he says: ‘And to show that I do not take gravity to be an
A magnet x is attracted by an exactly similar magnet essential property of bodies, I have added one question
y on two occasions: on one occasion there are just concerning its cause’ and so on [Kant quotes this in Newton’s
the two magnets, on the other occasion magnet y Latin]. Well, perhaps he shared his contemporaries’ shock at
is enclosed in a wooden box that weighs twice as the concept of basic attraction, and was led by this to be at
much as y does. On the second occasion, y-plus-box

4
 Metaphysical Foundations of Natural Immanuel 2: Foundations of

variance with himself. ·There can be no question of taking Proposition 8

that remark from the Optics as his most fundamental and
most considered view, because· he held that the attractive The basic attractive force, on which the very possibility of
forces that two planets. exercise on their satellites (mass matter depends, reaches out directly from every part of the
unknown), when at the same distance from those satellites, universe to every other part, to infinity.
are proportional to the quantity [Quantität] of the matter in the Proof
two planets; and he absolutely could not say this unless he Because the basic attractive force......is essential to matter,
assumed that just by being matter they had attractive force, every portion of matter has it. Now, suppose there were a
in which case all matter must have it. distance beyond which the force of attraction didn’t reach: 517
what could explain this limitation of the sphere of its efficacy?
Definition 7 It would have to be explained either (a) by the matter lying
516 within this sphere or (b) by the sheer size of the sphere. It
A superficial force is a moving force by which portions of couldn’t be (a), because this attraction is a penetrative force,
matter can directly act on one another only at the common which acts unmediatedly at a distance; it goes across every
surface of their contact; a penetrating force is a moving region of space as though the space were empty,
force by which one portion of matter can directly act on the unaffected by any intervening portions of matter. And (b)
parts of another that are not at the surface of contact. can’t be right either. Every case of attraction involves a
Note moving force that has a degree ·of strength·, given any
The repelling force through which matter fills a space is a such degree a smaller one is thinkable, and then one
mere superficial force. That is because the parts touching smaller than that. and so on
each other limit one another’s sphere of action; the repelling to infinity. Now, the great distance between two portions of
force can’t move any more distant part except by means of matter would reduce the strength of the attraction between
those lying between. On the other hand, no intervening them—reducing it in inverse proportion to the amount of the
matter limits an attractive force. That kind of force enables diffusion of the force—but it wouldn’t destroy the attractive
a portion of matter to •occupy a region of space without force between them completely. So there is nothing that
filling it [see Remark on page 20]; and to •act through empty could bring about a limit to the sphere of efficacy of the basic
space upon other distant portions of matter, without this attraction of any part of matter, so this attraction reaches
action’s being limited by any intervening matter. That is how throughout the universe to infinity.
we must think of the basic ·force of· attraction that makes Note 1
matter itself possible. So it’s a penetrative force, and for We have here a basic attractive force—a penetrating force—
that reason alone it is always proportional to the quantity which is exercised
[Quantität] of the matter. •
by every portion of matter (in proportion to its quan-
tity [Quantität] of matter), •upon all portions of matter,
•
across any possible distance.

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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

From this force, in combination with the opposing repelling can then be imparted to every portion of matter in
force, it must be possible to derive the limitation of the accordance with the degree of its repelling force. The action
repelling force and hence the possibility of the filling of a of universal attraction—exercised by all matter directly on
region of space to a determinate degree. And in this way all matter and at all distances—is called gravitation; the
the dynamical concept of matter as what is movable, and endeavour [see long note on page 19] to move in the dominant
fills a region of space to some determinate degree can be gravitational direction is weight. The action of the universal
constructed. This construction requires a law governing how repelling force of the parts of each portion of matter is
basic attraction and basic repulsion relate to one another at called its basic elasticity. Weight involves an external
various distances. Finding this relation is a purely mathe- relation, while elasticity is internal. These two are the only a
matical problem, because the relation rests solely on •the priori compre- hensible universal characteristics of matter;
opposite directions of these two forces (one drawing points ·they are a priori graspable because· they are the foundations
together, the other pushing them apart) and on •the size on which rests the very possibility of matter. When cohesion
of the space into which each force diffuses itself at various is explained as the reciprocal attraction of portions of matter
distances; metaphysics has nothing to do with this. If the that are in contact with one another, it doesn’t belong to the
attempt to construct matter in this way meets with failure, possibility of matter in general and therefore can’t be known
that won’t be the fault of metaphysics. Its only responsibility a priori to be bound up with matter. This property ·of
518 is for the correctness of the elements of the construction that cohesion through contact· would be physical, not
reason leads us to; it isn’t responsible for the insufficiency metaphysical, so it wouldn’t belong to our present
and limitedness of our reason in doing the construction. considerations.
Note 2 Remark 1
Each portion of matter succeeds in being a determinate mate- I can’t forbear adding a small preliminary remark for the
rial thing only by filling a region of space with a determinate sake of any attempt that may be made toward such a
degree of repelling force; and such a filling of a determinate possible construction.
region of space can happen only through a conflict between a (1) Let F be some force—any force—that acts unmedi-
basic attraction and the basic repulsion. Now, the attraction atedly at different distances, with the amount of moving
involved in this filling of a determinate region of space may force that it exerts at any given point being limited only 519
arise either ·internally· from •the attractions that the parts of by how far it had to travel to reach that point. However
the compressed matter exert on one another or ·externally· much or little space F is spread through, the total amount
from •the attraction exerted upon this compressed matter of it is the same; but the intensity of its action upon a
by all the matter of the world. The basic attraction is pro- given point x will always be inversely proportional to the
portional to the quantity [Quantität] of matter, and it reaches space F had to get through to reach x. Think of light being
to infinity. So the only way a determinate region of space propagated from a point P, surrounded by a series of spheres
can be filled by matter is through matter’s infinitely-reaching each with P as its centre. The total amount of light falling
attraction; such a determinate degree of the filling of space on any sphere is the same as the total amount falling on

4
Metaphysical Foundations of Natural Immanuel 2: Foundations of

any other; but the amount of light falling on (say) a square

are; the centre leaves this undetermined.2
inch of one sphere will be greater than the amount falling
(3) If the force is an unmediated repulsion by which a 520
on a square inch of a larger sphere. And that’s how it is
point. fills a space dynamically, and if the question is
with all other forces, and the laws according to which these
What law of infinitely small distances (here = contacts)
forces must diffuse themselves, either in two dimensions
governs how a basic repelling force acts at different
or in three, in order to act according to their nature upon
distances?. . . .
distant objects. If you want to do a drawing of the diffusion
then it is even further from being correct to represent this
of a moving force from one point, it is better not to do it
force by diverging rays of repulsion coming from the repelling
in the ordinary way (as in optics, for example), namely by
point, even though the direction of the motion has this point
means of straight-line rays diverging from a central point.
as its starting-point. That’s because the space in which
However many lines you put into such a diagram, they’ll
the force must be diffused in order to act at a distance is
get further apart the further they get from the central point;
a corporeal space that is to be thought of as filled. There’s
so they can never fill the space through which they pass
no way of mathematically representing how a point can
or (therefore) fill the surface that they reach. This makes
dy- namically fill a space; and the repelling force of a
them a source of troubles that can be avoided if we ·get rid
corporeally filled space can’t possibly be represented by
of straight-line rays, and· think of the situation merely in
diverging rays coming from a point. What we must do,
terms of the size of the whole spherical surface that is to
rather, is to assign a value to the repulsion at various
be uniformly illuminated by the same quantity [Quantität] of
infinitely small distances of
light, so that—quite naturally—the intensity of illumination
these mutually repelling points simply in inverse proportion 521
of any given area of a surface is inversely proportional to
to the ·volumes of the· corporeal spaces that each of these
the size of the whole surface; and similarly with every other
points dynamically fills, so that the value will be in inverse
diffusion of a force through spaces of different sizes.
(2) If the force is an unmediated attraction at a distance, 2
It’s impossible to represent surfaces at given distances as wholly
the lines of the direction of the attraction must be repre- filled with the action of lines spreading out from a point like rays,
whether the action is illumination or attraction. Draw the situation
sented as rays not •diverging from the attracting point but,
in that way and you make it look as though the inferior illumina-
rather •converging at the attracting point from all points tion of a distant spherical surface consists in its having relatively
of the surrounding spherical surface. Why? Because the large unilluminated and widely spaced illuminated ones! Euler’s
line of direction of the motion to this point—a point that hypothesis ·that light consists of waves, not streams of particles·
causes the motion and is its goal—assigns the points from avoids this inconvenience, but at the cost of making it harder to get
a conception of the rectilinear motion of light. [The footnote goes
which the lines must begin, namely from all points of the on at some length, recommending that we think of light as
surface. These lines get their direction from this surface to consisting not of waves or of straight-line streams of particles but
the attracting centre ·of the sphere·, and not vice versa. For rather an infinitely divisible fiuid. Kant seems to acknowledge that
only the size of the surface determines how many lines there there is no convenient way to draw this account of the matter; and
recommends that we resort to the device of straight-line rays but only
after getting firmly and clearly in mind what the truth is, so as not
to be misled by the lines.]

4
 Metaphysical Foundations of Natural Immanuel 2: Foundations of

proportion to the cube of the distances of these points regions are inter-related, but •isn’t in fact a region of
from one another. . . . space.
(4) So the basic attraction of matter would act in inverse [See note on Idee on page 9.] In the case cited of a supposed
proportion to the square of the distance—any distance— physical monadology, there were to be actual spaces that
while the basic repulsion would act in inverse proportion were filled by a point dynamically, i.e. through repulsion;
to the cube of the infinitely small distances. It’s that action for they existed as points before any possible production of
and reaction of the two fundamental forces that make ·a matter from these points, and through the proper sphere of
portion of· matter possible, by filling its space to a deter- their activity they fixed the part of the space to be filled that
minate degree. The point is that as parts move closer could belong to them. In this physical monadology, therefore,
together the •repulsion between them increases faster matter can’t be regarded as infinitely divisible and as a con-
than the •attraction does; and that sets a limit to the tinuous quantum, because the parts that unmediatedly repel
approach— the limit at which the available attractive force one another are at a determinate distance from one another
loses out to the available repelling force—and that limit (the sum of the radii of their spheres of repulsion); whereas
determines how intensely the space is filled. the thought of matter as a continuous quantity [Größe] doesn’t
Remark 2 allow for any distance between the unmediately repelling
I’m well aware of the difficulty about this way of explaining parts, or, therefore, for any increase or decrease of the
the possibility of a portion of matter ·considered as separate spheres of their unmediated activity. However, portions
from other portions of matter·. It consists in the fact that if of matter can expand or be compressed (like the air), and
a point can’t unmediatedly [see note on page 30] drive another ·within the framework of the physical monadology· this can
point by repelling force without at the same time filling the be represented in terms of increase and decrease of the
whole intervening corporeal space with its force, then it distance between their nearest parts. But ·in actual fact·
seems to follow that this ·intervening· space must contain the closest parts of a continuous portion of matter touch
several driving points. That conflicts with the hypothesis one another, even when it is being expanded or compressed;
·of the discussion, namely that we are talking here about so their distances from one another have to be thought of 522
action at a distance·, and it was ruled out above through as infinitely small, and this infinitely small space must be
the label ‘sphere of repulsion of the simple in space’. [Ruled understood to be filled in a greater or lesser degree [see note
out where? Kant cites Proposition 4, but that seems wrong. Definition 6 on page 21] by their force of repulsion. But two things’ •having
is better, though neither there nor anywhere else has he spoken of ‘the an infinitely small space between them is their •being in
But we should distinguish •the concept
repulsion of the simple’.] contact. Hence it is only the idea of space that enables us
of an actual region of space, which could exist, from •the to intuit [= ‘see in our mind’s eye’] the expansion of matter as
mere idea of a continuous quantity [Größe], although it can’t actually be
a region of space that •is entertained in thought only conceived in this way. Thus, when it is said that the repelling
for the purpose of determining how various given forces that two parts of matter unmediatedly exercise on one
another are

4
Metaphysical Foundations of Natural Immanuel 2: Foundations of

•
in inverse proportion to the cube of the distance
make it easy to conceive that these ·heated-air· vibrations
between them,
give to the air’s parts a force that •causes them to move
this means only that they are
away from one another and •stands in inverse proportion to
•
in inverse proportion to the corporeal spaces that
the distances between the parts. [The phrase ‘communication of
one thinks of between the parts,
motion’ is a common translation of the German Mitteilung der
though in fact the parts are immediately in contact (which
Bewegung. It would be closer to the German to put ‘sharing of motion’,
is why we have to call the distance between them
but we would have to remember to liken this to ‘thank you for sharing
‘infinitely small’ so as to distinguish it from every actual
that news with me’ rather than to ‘thank you for sharing your cake with
distance). We mustn’t raise any objection to a concept
me’. Or we might use ‘the passing on of motion’; but on page 56 Kant
itself because of difficulties in the construction of it or
writes about those who thought of the Mitteilung der Bewegung as a
rather in the misinter- pretation of the construction of it. . . .
literal passing over of some motion, from one body that loses it to
The universal law of dynamics would in both cases be
another that gains it. That is one theory about this phenomenon; so
this:
terminology that strongly suggests it can’t be used as a neutral name for
•
The action of the moving force that one point exerts
the phenomenon.] But let me explain: I do not want my
on each other point external to it is inversely propor-
exposition of the law of basic repulsion to be seen as
tional to •the space through which that moving force
essential to the aim of my
has had to spread in order to act unmediatedly upon
metaphysical treatment of matter. All I needed for that 523
the other point at the given distance.
treatment was to present the filling of space as a dynamic
From the law that the parts of matter basically repel one
property of matter; and I don’t want that to be mixed up
another in inverse cubic proportion to their infinitely small
with the disputes and doubts that might arise from ·further
distances, there must necessarily follow a law of the expan-
details of· my exposition.
sion and compression of these parts that is entirely different
from Mariotte’s law regarding the air. Mariotte’s law GENERAL NOTE ON DYNAMICS
proves that the forces causing the closest parts of the air to
move away from one another are in inverse proportion to Looking back over everything I have said about the meta-
the dis- tances between parts (Newton proves this in the physical treatment of matter, we find that the treatment has
scholium to Proposition 23 of Book II of the Principia). But dealt with
the expansive force of the parts of the air can’t be an (1) what is real in space (otherwise known as what is
example of the action of basic repelling forces. Why not? ‘solid’) in its filling of space through repelling force;
Because this expansive force comes from heat, which (2) what relates in a negative way to the real in space
compels the proper parts of the air (which, incidentally, are . . . ., namely, attractive force, ·which negates the real
at actual distances from each other) to move away from one in space in the sense that· if this attractive force were
another, doing this, apparently, by vibrations. . . . But the left to itself it would permeate the whole of space and
laws of the communication of motion through the completely abolish everything solid;
vibration of elastic portions of matter (3) the limitation of (1) by (2), yielding an empirically
accessible degree of the filling of space.
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 Metaphysical Foundations of Natural Immanuel 2: Foundations of

So we see that the quality of matter has been completely dealt every portion of matter exerts on every other and also 524
with under the headings of reality, negation, and limitation. the attraction that holds the portion together as a
When I say ‘completely dealt with’, I mean that the treatment unit.
contains everything needed for a metaphysical dynamics. Now, (1) doesn’t run in harness with (2); on the contrary, we
[The terms ‘reality’ etc. are Kant’s labels for the categories of Quality in can think of their relationship to one another as infinitely
his Critique of Pure Reason.] diverse. This is because (2) rests on the amount [Menge]
of matter in a given space, while (1) matter rests on the
GENERAL REMARK ON DYNAMICS degree to which the space is filled—and this degree can vary
enormously (as the same quantity [Quantität] of air in the
In what I am about to say, I use ‘real’ [German real, from Latin
same volume exhibits more or less elasticity according to its
res = ‘thing’] to apply only to things and not to mere states or
temperature). The underlying difference is this:
qualities; for example a thing’s location and size and shape
(2) in true attraction all particles of matter act directly on
are not real because they are not themselves things but are
all other particles of matter; whereas
spatial qualities of things. Now, the universal principle of
(1) by expansive force there is only action between the
the dynamics of material Nature is this:
particles at the surface of contact between the two
Everything that is real in the objects of our external
portions, and it makes no difference what the state
senses must be regarded as a moving force.
of affairs is—whether there is much or little of this
This principle banishes from natural science the empty
matter—behind this surface.
concept of the so-called solid, i.e. the concept of absolute
All this brings a great advantage to natural science, by
impenetrability, and replaces it by the concept of repelling
relieving it of the burden of imagining a world built up out
force. On the other hand, the true and unmediated attraction
of full ·parts of space· and empty ones, allow it instead to
is •defended against all the bad arguments of a
think of all regions of space as full, but filled in varying
metaphysics that misunderstands itself, and •is explained
measure [= ‘in different degrees’]. This at least deprives empty
as a funda- mental force that is necessary for the very
space of its status as necessary. It used to be thought of
possibility of the concept of matter. One consequence of
as required to explain differences in the weight or density
this is that we can if necessary think of space as filled
etc. of different portions of matter, but now the thesis that
throughout but in varying degrees, ·so that we can think of
there is absolutely empty space is reduced to the status
a portion of matter as light or soft or undense· without of an hypothesis. [From here to the end of this chapter Kant will
having to suppose that it has pockets of empty space repeatedly contrast two different accounts of the fundamental nature of
scattered through it. To understand this, consider these the physical world. To make it easier to keep the thread, the two will be
two: given numerical labels within curly brackets, which aren’t used for any
(1) The basic repelling forces of matter, which are the other purpose in this document.]
basis for matter’s first property, namely the filling of [Kant begins his next paragraph by speaking of the advan-
space; tage that {2} ‘a methodically employed metaphysics’ has over
(2) The basic attraction of matter—the attraction that

4
 Metaphysical Foundations of Natural Immanuel 2: Foundations of

{1} ‘principles that are also metaphysical but haven’t been

They can only be assumed; ·and it is all right to assume
subjected to the test of criticism’. That last word translates
them· if they inseparably belong to a concept that is provably
Kritik, which occurs in the German title of the Critique of Pure
basic and not further derivable from any other (such as the
Reason. Its appearance here is sudden and surprising; it
concept of the filling of space). These basic forces are the
hasn’t occurred earlier in this work except as part of that •
repelling forces in general and the counteracting •attractive
title; but Kant evidently expects us to gather that the
forces in general. We can quite well form a priori judgments
difference between
concerning their inter-relations and consequences; the in-
{1} the common atomist metaphysic that deals in basic
vestigator is free to think up any relations he likes among
solidity, absolute impenetrability, and empty space these forces, provided he doesn’t contradict himself. But he
and
mustn’t assume either of them as actual, because he is
{2} his metaphysic of basic forces and degrees of flatly not entitled to set up a hypothesis unless the
intensity of fullness of space possibility of what is assumed in it is entirely certain; and
is the difference between {1} a metaphysic that •hasn’t been the possibility of the basic forces can never be
subjected to the kind of criticism that is central to the
comprehended ·and so can never be entirely certain ·. And
Critique of Pure Reason and {2} a metaphysic that •has. He
this points to an advantage that {1} the mathematico-
says that the advantage of {2} over {1} is ‘apparently only
mechanical kind of definition has
negative’. (Perhaps his thought is that {2} seems at first sight
over {2} the metaphysico-dynamical kind, namely: Starting 525
to do nothing but stop {1} from saying some of the things it
with
is saying.) Anyway, {2} does in an indirect way enlarge the
(a) a single completely homogeneous basic kind of
scope of the investigator of Nature, Kant continues:] because
material
the conditions by which he previously limited his field, and
—·namely absolutely solid matter·—this {1} mathematico-
by which all basic moving forces were philosophized away,
mechanical mode can provide for a great variety of sorts
now lose their validity, ·so that he has at his disposal some of matter that differ in density and (if it adds forces from
good concepts that he had thought were illegitimate; and outside the basic material) different modes of action. To do
that advantage is not ‘only negative’ ·. But he—·this liberated this, it needs the help of
investigator of Nature·—must be careful not to go beyond (b) the different shapes that matter can have, and
what makes the •universal concept of matter in general (c) empty spaces between the portions of matter,
possible by trying to explain a priori any •specific facts about ·But the addition of those two doesn’t weaken the system
kinds of matter, let alone facts about •particular material in any way·, because the possibility of (b) the shapes and
things. The concept of matter is reduced to nothing but of (c) the empty intermediate spaces can be proved with
moving forces; that was to be expected, because in space mathematical evidentness. In contrast with this, if {2} the
the only activity, the only change, that is conceivable is basic •material is transformed into basic •forces, then we
motion. But who would claim to comprehend the possibility don’t have the means for constructing this concept or for
of fundamental forces? [See note on ‘conceivable’ on page 31.] presenting as possible in intuition what we thought
univer- sally. Why not? Because there’s no secure way of
4
 Metaphysical Foundations of Natural Immanuel 2: Foundations of
explaining

4
 Metaphysical Foundations of Natural Immanuel 2: Foundations of

different sorts of matter in terms of different patterns of dynamical system, which has only relative impenetrability,
the basic forces; indeed, we can’t even determine a priori there is no maximum or minimum of density. In that context
what the laws are that govern those forces. But {1} a merely any portion of matter can be called ‘fully dense’ if it has no
mathematical physics pays a high price for that advantage, empty spaces within its boundaries. i.e. if it is a continuum 526
because •it has to base itself on an empty concept (absolute and not an interruptum; and this implies nothing about
impenetrability), and because •it must forgo all matter’s how thin—·airy, light, etc.·—it is. And one portion of matter
own forces, ·and make do with forces from outside ·. ·And in counts as ‘less dense’, in {2} the dynamical sense, than
addition to those two defects, {1} also runs a risk ·: Employing another if it entirely fills its space but not to the same degree
its basic patterns of portions of solid matter interspersed as the other. But even in the dynamical system it’s not
with empty spaces, it is required to provide explanations satisfactory to make a ‘density’ comparison between two
·of the variety in sorts of matter·, and this requires it to portions of matter unless they are homogeneous with one
allow—and to insist on its right to—a greater freedom of another, ·i.e. of the same kind·, so that one can be produced
imagination than is prudent. from the other by mere compression. Now, it doesn’t appear
Starting with {2} basic forces I can’t adequately show to be essential to the nature of matter as such that any
the possibility of matter or explain the different sorts of portion of it could be made indistinguishable from any other
matter. But all that variety can be brought a priori under by compression, we shouldn’t make density comparisons
a few intermediate headings, and I do hope to present a between heterogeneous portions or kinds of matter, as
complete account of those. (Not that this will provide a way people customarily do when they say that water is less dense
of conceiving the possibility of matter.). . . . ·This material than mercury. [The {1}/{2} labelling will turn up again on page 46.]
will be presented in four groups·. (ii) Attraction when considered as acting between things
(i) A body in the physical sense of the word is a portion of that are in contact is called cohesion.
matter that has determinate boundaries and therefore has (It’s true that some very good experiments have shown
a shape. The size of the space within these boundaries is that the force that is called ‘cohesion’ when it operates
the body’s volume. The degree to which a space is filled is between things that are in contact with one another is
called density The {1} system of absolute impenetrability also active at a very small distance. But attraction across
provides for something to have absolute density, by having a small distances is hardly perceivable; so when we speak
portion of matter that has absolutely no empty spaces inside of ‘cohesion’ we are thinking of things that are in contact.
it. Using this concept of the filling of space, one portion of Cohesion is commonly taken to be a property that all matter
matter counts as less dense than another if it contains less has—not •derivable from the concept of matter but •shown
empty space than the other, the extreme case being that of by experience to be a feature of all matter. This universality
a portion of matter that is called perfectly dense because mustn’t be misunderstood as meaning (a) that every portion
there is no empty space within its boundaries. The phrase of matter is constantly exerting this kind of attraction on
‘perfectly dense’ has a use in the context of {1} the merely every other portion of matter —like gravitation—but rather
mathematical concept of matter, and only there. In {2} the as meaning (b) that every portion of matter acts in this way

5
 Metaphysical Foundations of Natural Immanuel 2: Foundations of

on any other portion of matter that comes into contact if its parts can’t be re-arranged without its breaking, so that
with it. [Kant describes these two versions of the force’s the way its parts cohere can’t be changed without its losing
universality as ‘collective’ (he could have said ‘conjuctive’) cohesion altogether. It is quite wrong to say (·as some do·)
and ‘disjunctive’ respectively: in (a) a portion x acts in the that •the difference between fluid and solid portions of
relevant way on y and z and w and. . . etc, while in (b) it matter comes from the •difference in the degree to which
acts on y or w or z or. . . and so on, depending on which their parts cohere. When we call a portion of matter ‘fluid’,
of these comes into contact with it.] For that reason, and we aren’t talking about •how resistant it is to being broken
also because this attraction is not a penetrating force but apart, but only about •how resistant it is to being re-
only a superficial one (there’s plenty of evidence for that), arranged. Its
its strength isn’t always proportional to the density ·of the •
cohesion can be as strong as you like, but its •resistance to
matter involved·. What is needed for two portions of matter re-arrangement equals zero. Consider a drop of water. If a
to cohere with full strength is for them to be first fluid and particle within the drop is drawn to one side by a very strong
then rigid; . . . . when a looking-glass has a crack across it, the attraction of the parts touching it on that side, then it will
portions of glass on the two sides of the crack are nowhere be drawn just as strongly to the opposite side; and since
near to being as strongly attracted as they were when they the attractions cancel out, the particle is as easily movable
became solid after being fluid. For all these reasons I regard as if it were in empty space. That’s because any force that
this attraction-in-contact as only a derivative force of nature, might move it has no cohesion to overcome; the only
not a fundamental one. But more of this later.) resistance to it would be the matter’s so-called inertia,
A portion of matter is fluid if any moving force, however and that has to be overcome in making any matter move,
small, is sufficient to re-arrange its parts. The parts of a even matter that doesn’t cohere at all. Therefore, a
portion of matter are re-arranged when they are made to microscopic bug will move as easily within this drop as if
527 switch places while remaining completely in contact with there were no cohesion to overcome. For in fact it doesn’t
one another. Portions of matter—including the parts of a have to lessen the water’s cohesion—to pull particles of
portion of matter—are separated if they lose all contact the water apart from one another—but only to re-arrange
with one another or the amount of contact is lessened. A them. [Kant goes on to explain that if the bug tries to
rigid body is one whose parts can’t be re-arranged by any escape from the drop, then it does now have to overcome the
force—so these parts must be resisting re-arrangement by water’s cohesion, but not in a way that lessens the strength
a certain degree of force ·of their own·. The resistance to of the water’s holding together as a cohering drop. He
the re-arrangement of portions of matter is friction. The continues:] So it is clear that an increase of the cohesion of
resistance to the separation of portions of matter that are the parts of a portion of matter hasn’t the slightest effect on
in contact is cohesion. So fluid portions of matter don’t its fluidity. Water coheres in
undergo friction when they divide; and where friction is its parts much more strongly than is commonly thought. 528
found the portions of matter are assumed to be more or less What is quite decisive with regard to our concept of fluidity
rigid, at least in their smaller parts. A rigid body is is this: fluid portions of matter can be defined as those in
brittle which the forces exerted by or acting upon each point are the

5
 Metaphysical Foundations of Natural Immanuel 2: Foundations of
s aw of hydrostatics is based
a
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e

i
n

e
v
e
r
y

d
i
r
e
c
t
i
o
n
.

T
h
e

f
i
r
s
t

5
 Metaphysical Foundations of Natural Immanuel 2: Foundations of

on the property of fluidity; and it can’t be a property of an parts. I have presented two definitions of fluidity:
aggregation of smooth solid particles The considerations (a) A portion of matter is fluid if any moving force,
we are in among here enable us to show that fluidity is a however small, is sufficient to re-arrange its parts.
basic property. ·If it were not basic but derivative, there (b) Fluid portions of matter can be defined as those in
would be portions of matter that were very but not which the forces exerted by or acting upon each point
perfectly fluid, and there aren’t any·. If in a fluid portion of are the same in every direction.
matter there was a tiny hindrance to re-arrangement We can derive (b) from the conjunction of (a) and the (c) prin-
and hence a tiny amount of friction, this friction would ciple of general dynamics saying that all matter is basically
grow with the strength of the pressure with which the elastic, as follows: A portion of matter that is (c) elastic will
portion’s parts are pressed against one another, and a resist, by stretching, any force of compression to which it is
strong enough pressure would have the effect that the subjected; and if it is (a)-fluid, its force of recovery will equal
parts of this portion of matter couldn’t be re-arranged by the force of compression (nothing will be lost to friction);
any small force, ·i.e. the effect that the portion would no which is to say that the forces at work in it will be the
longer be fluid·. ·Here is a concrete example·: same in every direction, i.e. that this portion of matter is (b)-
Take a U-shaped tube, of which one arm is very wide fluid. So friction, properly so-called, can be had only by rigid
and the other very narrow (but not as narrow as a portions of matter Some portions of matter that may
capillary, ·because that would have effects that would have no more force of cohesion than some fluids nevertheless
cloud our result·). Let both arms be a few hundred feet strongly resist the re-arrangement of their parts, so that they
high. According to the laws of hydrostatics, the fluid in can’t be pulled apart except by destroying the cohesion of
the narrow arm would reach exactly the height of all parts in a given surface, thus creating an illusion that
the fluid in the wide arm (they are arms of a single they do have more cohesion. ·An example would be a cake
tube). But now let us keep adding fluid to the tube, of chocolate: you can break it in two, but you can’t pull the
steadily increasing the pressure on the matter at the two halves of it apart·. Such portions are rigid bodies. But
bottom of the tubes. It there were the tiniest why this is so, i.e. how rigid bodies are possible, is still an
potentiality for friction there, then at some height-of- unsolved problem, though the ordinary doctrine of Nature
fluid the movement of matter between the arms [see note on page 1] thinks it has easily solved it.
would stop: adding a small quantity of water to the (iii) A portion of matter may be able, after it has been
narrower tube wouldn’t affect the height of the water deformed by an external force, to regain its original size and
529 in the wider tube; so that the narrow arm’s column shape when the deforming force is removed; that ability is
of fluid could be made to rise higher and higher elasticity. When something can return to its previous size
above the wider arm’s column. after being compressed, that is expansive elasticity; some-
And this is contrary to experience and even to the concept thing that returns to its previous size after being stretched
of fluidity. The same thing holds if, instead of unlimited has attractive elasticity. The elasticity that consists only in
pressure by weight, we postulate unlimited cohesion of the the recovery of the previous shape is always attractive—e.g.

5
 Metaphysical Foundations of Natural Immanuel 2: Foundations of

part, however small, consists

with a bent sword in which the parts on the convex surface
have been pulled away from one another and try to resume
their former closeness to one another (Attractive
elasticity
. . . . is obviously •derivative. An iron wire stretched by a
weight springs back into its original size when the weight
is removed. The attraction we have here is the cause of the
530 cohesion of the wire. Expansive elasticity, on the other
hand, may be either •basic or •derivative. Every portion of
matter must, just because it is matter, have basic elasticity,
but some also have derived elasticity. [Kant cites the example
of hot air, which he thinks has—in addition to its basic
elasticity—–a further elasticity from being hot. He thinks
that air’s being hot is its being mixed with a special fluid, and
that the elasticity of this—which may be basic—is passed on
to or shared by the air. Finally:] It isn’t always possible to
know for sure whether a given instance of elasticity is basic
or derived.
(iv) When moving bodies collide and alter one anothers’
motion, that is called mechanical action. When a body is
at rest [= ‘motionless’] as a whole though its parts are mov-
ing around within it and interacting, their action is called
chemical. When this chemical influence has the effect of
pulling apart the parts of a portion of matter ·and inserting
between them parts of another portion·, it is called
dissolv- ing. When the influence has the effect of
separating out two portions of matter that have been
dissolved in one another, it is called chemical analysis. . . .
Absolute dissolving is the dissolving-in-one-another of two
portions of matter of different kinds—·call them X and Y·
—in such a way that every part of the X portion is united
with a part of the Y portion in the same proportion as that
of the solution as a whole. ·For example, if 5 units of X
matter are merged into 10 units of Y matter, and this is
‘absolute’ dissolving, then in the resultant solution every

5
 Metaphysical Foundations of Natural Immanuel 2: Foundations of
o s could also be called chemical penetration. (Whether infinite division, because the dissolving takes place
f the dissolving forces that actually occur in Nature are contin-
capable of bringing about a complete ·or absolute·
o dissolving doesn’t have to be decided here. Here the
n question is only whether such a solution can be
e thought of.) Obviously if the parts of a dissolved
portion of matter are still particles, it is as possible for
t •them to be dissolved as it was for •the larger parts to
h be dissolved; and if the dissolving force continues, it’s
i not merely possible but inevitable that the dissolving
r will continue until every part of the solution is
d composed of X matter and Y matter in the same
proportion as they have in the solution as a whole.
X Because in this case every part of the solution contains
a part of the X matter, this matter must completely fill
a the volume in a continuous way. And the same holds
n for the Y matter. And when each of two portions of
d matter entirely fill a single space, they penetrate one
another. That is why a perfect ·or absolute· chemical
t dissolving would involve penetration of the portions of
w matter. This chemical penetration would be entirely
o different from mechanical penetration. In the latter, the
thought is that as ·two· portions of matter approach
t one another the repelling force of one could entirely
h outweigh that of the other, so that the extent of one or
i both of these portions of matter is reduced to
r nothing. In contrast with
d that, in chemical penetration each portion of matter retains
s 531 its extent, but the portions are not outside one another
but within one another. It’s hard to see any objection to the
Y thesis that such perfect dissolving is possible, and thus
. that chemical penetration is possible. It does involve a
· completed division to infinity, ·and that seems to clash
with the thesis that an infinite such-and-such is one
T that ‘can’t ever be wholly complete’· [see page 26], but:
h (a) There there is no contradiction in this case of
i
5
Metaphysical Foundations of Natural Immanuel 2: Foundations of

uously throughout a period of time, i.e. through an You might want to suggest that chemical
can’t easily indicate.]
infinite series of ever-shorter moments. dissolving will never be complete because the Y solvent will
(b) Moreover, as the division proceeds the sums of the always be somewhat viscous, i.e. a bit thick and sticky. But
surfaces of the not-yet-divided portions of matter that thought rests on the assumption that all dissolving
increase; and since the dissolving force acts contin- consists in some X matter’s coming apart and allowing some
uously, the whole dissolving can be completed in a more fluid Y matter to flow between the parts; and this view
specifiable time. of what dissolving is doesn’t fit with the great force that the
If you think you can’t conceive of such a chemical penetration more solvent fluids exert on dissolved portions of matter—
of two portions of matter, that will be because the divisibility e.g. the action of dilute acids on metals. They don’t merely
to infinity of every continuum in general really is inconceiv- touch the metallic bodies, which is what would happen if the
able. If you won’t accept this complete dissolving ·of one kind particles of metal merely swam in the acid; rather, the acids
of matter in another · then you’ll have to settle for an account exert great attractive force to pull these bodies apart and
that ends with certain small particles of the dissolved matter disperse them throughout the entire space of the containing
swimming around in the solvent at fixed distances from flask. And another point: Even if •our knowledge and skills
one another; these are still divisible portions of matter but didn’t put at our disposal any chemical forces of dissolving
·according to you· they aren’t also dissolved, and you won’t that could bring about a complete dissolving, •Nature might
be able to give the slightest explanation of why! It may exhibit such forces in the operations of plants and animals,
be true in Nature, as far as our experience goes, that the perhaps producing portions of matter that were products
solvent goes a certain distance and doesn’t act further, ·but of complete or absolute dissolving though we had no way
that is beside my present point ·. My question concerns the of separating the components out again. [Kant sketches 532
possibility of a dissolving force that acts on smaller and two possible examples of this, one involving heat and the
smaller particles until the dissolving is completed. The other magnestism. They are hard to follow, and rest on now-
volume of the resultant solution can be equal to the sum of exploded theories about those two phenomena. Then:] Our
the volumes of the two portions of matter before the mixture, present search, though, is not for •hypotheses to explain
or it can be smaller than this sum, or even larger than it, particular phenomena but for •the principle according to
depending on how the attractive forces relate to the repelling which such hypotheses are all to be judged. Everything
forces, These mutually dissolving portions of matter that frees us from the necessity of invoking empty spaces
consti- tute in solution, each of itself and both combined, is an actual gain for natural science. Why? Because empty
an elastic medium. This elasticity provides the only spaces leave the imagination far too free to invent fictions
sufficient reason why the dissolved X matter doesn’t by its to make up for the lack of real knowledge of Nature. In
weight pull itself out from the Y solvent; it’s because the the doctrine of Nature, absolute emptiness and absolute
solvent Y’s attraction, since it occurs equally strongly toward density play about the same role that blind chance and blind
all sides, destroys the resistance of the X dissolved matter fate play in metaphysics, namely that of a bar to reason’s
[The next sentence and
dominance—either replacing it with fictions or lulling it to
a half expands Kant’s words in ways that the convention of ·small dots·

5
 Metaphysical Foundations of Natural Immanuel 2: Foundations of

sleep on the pillow of occult qualities! •

the absolute homogeneity of this matter, with no
The chief problem in natural science is to explain how differences except in shape, and
there can be an infinite variety of kinds of matter. There •
the absolute unconquerability of the cohesion of the
are just two ways in which this can be attempted: {1} the matter in these basic particles.
mechanical way, by combinations of the absolutely full with Those were the materials for generating different kinds of
the absolutely empty; and in opposition to that there is {2} matter in a manner that ·has two seemingly attractive fea-
a dynamical way in which all the varieties of matter are tures; it· •avails itself of a single basic kind of matter, varied
explained merely through combinations of the basic forces only by the shapes of its portions; and •it explains Nature’s
of repulsion and attraction. {1} The raw materials of the various actions mechanically, as arising from the shape of
first are atoms and the void. An atom is a small portion these basic parts considered as machines that only needed
of matter that is physically indivisible. A portion of matter an externally impressed force. But the claim of this system to
is physically indivisible if its parts cohere with a force that be accepted depended, first and foremost, on the supposedly
can’t be overcome by any existing moving force in Nature. unavoidable necessity of explaining the different densities
An atom marked off from other atoms by its shape is called of kinds of matter in terms of empty spaces, which were
a primary particle. A body (or particle) whose moving force assumed to be distributed among the particles and within
depends on its shape is called a machine. The mechanical each particle. . . . {2} A dynamical mode of explanation is
natural philosophy is the process of explaining the variety far better suited to experimental philosophy [here = ‘science’],
of kinds of matter in terms of the nature and composition of because it leads directly to the discovery of •the moving
their smallest parts, considered as machines. {2} And we can forces that are inherent in portions of matter and •the laws
label as ‘the dynamical natural philosophy’ the explanation of those forces, but doesn’t freely allow the assumptions
of the variety of kinds of matter not in terms of of empty intermediate spaces and fundamental particles
•
particles considered as machines, i.e. as mere imple- with fixed shapes, neither of which can be discovered and
ments used by external moving forces, determined by any experiment. To go to work in {2} this way
but rather in terms of we don’t have to devise new hypotheses; all we need is to
•
the moving forces of attraction and repulsion that refute {1}’s postulate that it’s impossible to think of different
are inherent in these particles. kinds of matter in any way except through the intermixture
The {1} mechanical kind of explanation is very convenient for of portions of matter and empty spaces. And we can refute
533 mathematics, which is why it has—under the label ‘atom- it, simply by showing how the different densities of kinds of
ism’ or ‘the corpuscular philosophy’—always maintained its matter can be consistently thought of without bringing in
authority over and influence on the principles of natural empty spaces. . . . This move rests on the fact that matter
science, with little change from Democritus to Descartes does not fill its space by absolute impenetrability but by
and even to our own times. Its essentials consist in the repelling force; this force is a matter of degree, which can be
assumption of different in different portions of matter. The attractive force 534
•
the absolute impenetrability of the basic matter, of a portion of matter is proportional to the amountq of matter

5
Metaphysical Foundations of Natural Immanuel 2: Foundations of

in it, and is not correlated with its degree of repelling force; unconditioned basic givens such as a purely mathematical
so the proportions of repelling to attractive force in different treatment would postulate.
portions of matter can vary greatly. So there is no difficulty in I end this chapter with some remarks about the familiar
the thought of a portion of matter that entirely fills its space question of the admissibility of empty spaces in the world.
without any empty parts and yet with only a tiny amount q of The possibility of such spaces can’t be disputed. All forces
matter—so little that we can’t detect it experimentally. This of matter presuppose space; the laws governing the spread
is one way to think about the ether. . . . The only reason
of these forces have the form ‘If a region of space is. . . ,
for assuming an ether is to counter the claim that ‘rarefied’
then. . . ’, so space is necessarily presupposed before all
matter can’t be thought of except in terms of empty spaces.
matter. Thus, attractive force is attributed to matter because
·The outright assertion· that there is ether should not be
matter occupies a space around itself by attraction, yet 525
made a priori, nor should any ·supposed· law about attractive
without filling the space. So a region of space can be thought
or repelling forces. Everything must be concluded from data
of as empty even when matter is active in it, so long as
of experience—and that includes •the thesis that universal
the activeness doesn’t involve repelling forces, i.e. doesn’t
attraction is the cause of gravity, and •the laws of gravity.
involve the matter’s being in the space. But no •experience,
Still less will conclusions regarding chemical affinities be •
inference from experience, or •necessary hypothesis for
tested in any way except experimentally. Why? Because it
explaining empty spaces can justify us in assuming that they
lies right outside the scope of our reason to come at basic are actual. Experience presents us only with comparatively
forces a priori. What natural philosophy [here = ‘science’] empty spaces; and these can be perfectly explained in terms
does is to explain the variety of empirically encountered of the strength of the expansive force with which a portion
forces in terms of a smaller number of forces and powers;
of matter fills its space— ·the whole of its space·—a strength
but these explanations go only as far as the fundamental
that can be thought of as lesser and lesser to infinity, through
forces—our reason can’t get further down than that.
all possible degrees, without requiring ·absolutely· empty
Metaphysical investigation into the underpinnings of the
spaces.
empirical concept of matter is useful only for the purpose of
leading natural philosophy as far as possible in the
investigation of the dynamical grounds of explanation,
because they provide our only hope of finding determinate
laws and a system of explanations that hangs together in a
rational way.
That is all that metaphysics can ever do for •the construc-
tion of the concept of matter, and thus on behalf of •the
application of mathematics to the part of natural science
dealing with •the properties by which a portion of matter fills
a region of space in determinate measure. All metaphysics
can do is to regard •these properties as dynamical and not as

5
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

anything
Chapter 3
Metaphysical Foundations of Mechanics
Definition 1
536 Matter is what can be moved, considered as having—just
because it can be moved—a moving force.
Remark
This is the third definition of matter. The merely
dynamical concept ·is different from this because it·
applies also to matter that is motionless. The moving force
that was in question back there concerned merely the filling
of a certain region, and we weren’t permitted to regard the
matter that filled the space as itself moving. So repulsion
was a basic moving force for imparting motion, whereas in
mechanics a force is regarded as actually at work in one
portion of matter imparting motion to another portion.
·Very briefly and schematically: the movement of portions
of matter is considered as a •potential in dynamics, and as
•actual in mechanics·. Clearly, a portion of matter won’t
have
•
the power to make other things move by its own
motion
unless it has
•
basic moving forces through which it is active in
every place where it exists,
this being an activeness that comes before any proper motion.
·Breaking that point down into its two constituents ·: Clearly
a portion of matter moving in a straight line and
encountering another portion won’t make the other move
unless both of them have basic forces [Kant wrote Gesetze = ‘laws’,
presumably a slip] of repulsion; and a portion of matter couldn’t
in moving drag another portion after it unless they both had
537 attractive forces [Kräfte = ‘forces’]. So all mechanical laws
presuppose dynamical ones. You’ll notice that I shan’t say

5
 Metaphysical Foundations of Natural Immanuel 3: Foundations of
m on by •attraction (such as might happen if a comet is in another, if the two are of exactly the same kind, because
o with a stronger attractive power than the earth came in that case the amountsq are proportional to the volumes.
r close to the earth and dragged it out of its orbit). I’ll But
e be talking only about the agency of
•
repelling forces—i.e. agency by pressure (as by
a means of tensed springs) or by impact. I’ll do this
b because applying the laws of repulsion is exactly the
o same as applying the laws of attraction except for
u the difference in direction.
t
Definition 2
t
h The amountq of matter there is in a certain space is
e given by how many movable ·parts· there are there.
When this matter is thought of as having all its parts
c in motion at once, is called the mass; when all the parts
o of a portion of matter move in the same direction,
m exercising their moving force externally, the portion is
m said to act in mass. A mass with a determinate shape is
u called a body (in the mechanical sense). Mechanically
n estimated, •the amount of motion is ·a function of two
i variables, namely· •how muchq matter is moved and
c
•
how fast. . . .
a
t Proposi
i tion 1
o
The only way of comparing the amounts q of any two
n
portions of matter is by comparing their amountsq of
motion at a single speed.
o
f Proof
Matter is infinitely divisible, so the amountq of
matter in a given portion can’t be determined
m
directly by how many
o
parts it has. How muchq matter there is in one portion of
t
538
i
matter can be directly compared with how much q there

6
Metaphysical Foundations of Natural Immanuel 3: Foundations of

Proposition 1 concerns quantitative comparisons between its •moving force from •how it is moving, this turns into a me-
any two portions of matter, including ones of different kinds. chanical concept of the amountq of the motion. But
So there is no all-purpose method—direct or indirect— for ·actually this is a blind alley; we can’t get at a mechanical
comparing any two portions of matter with one another, if
concept of amount-of-motion in this way, because · in
we ignore their motions. If we bring motion into the story we
phoronomy we can’t represent a motion as composed of
do get a universally valid procedure for such comparisons—
many motions existing externally to one another. Why not?
and it is the only one we can have. It involves measuring
Because in phoronomy movable items are represented as
the amounts of matter in terms of the amounts q of motion. mere points, with no moving force, so that the only basis for
But this comparison gives us what we want only if the two distinguishing the amountsq of motion of two things is in
portions of matter are going at the same speed. Therefore terms of their
etc. ·differences of· speed. [In a spectacularly obscure passage,
Note Kant goes on from there to compare measuring amounts of 539
How muchq motion a body has is how much q matter it has •
motion with measuring amounts of •action, and to criticise
and how fastq it is moving. One body has twice the motion a wrong idea that some theorists have had about the latter.
of another body if His purpose in going into all this seems to be to present
•
they have the same speed, and one has twice as some thoughts about differentiating ‘dead forces’ from ‘living
muchq matter as the other, or forces’. We hear no more of that distinction in the present
•
they have the same mass and one has twice the speed work, and Kant invites us to bypass it when he ends by
of the other. saying:] . . . .if indeed the terminology of ‘dead force’ and
That is because the determinate concept of a size or amount ‘living force’ deserves to be retained at all.
is possible only through the construction of the quantum;
and such a construction involves putting together many Remark
items that are equivalent to one another [see ‘wwhat Kant aseems I have things to say in explanation of the preceding three
to have meant’ on page 14]. Thus, the construction of a motion’s statements—[i.e. Definition 2, Proposition 1, and the following Note ]—
amountq is the putting together of many equivalent motions. and in the interests of concentration I shall condense them
Now, in the context of phoronomy there is no difference into a single treatment.
between Definition 2 says that the quantity [Quantität] of a portion of
•
giving to a movable thing a speed S, and matter can only be thought of in terms of how many movable
•
giving to each of n equivalent movable things a speed parts (external to one another) it has. This is a remark-
of S/n. able and fundamental statement of universal mechanics,
The first thing we get from this is an apparently phoronomic ·because it supplies an answer to the important question
concept of the amount q of a motion, as composed of many ‘Can we have a concept of the intensive magnitude of an
motions that are external to one another but constitute a instance of moving force? The answer is that we cannot·.
single united whole. And if we think of each point as Such an intensive magnitude would have to be independent
getting of •the amount of matter and of •the speed, both of which

6
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

are extensive magnitudes; and Definition 2 tells us that muchq moved matter.
those are the only quantitative notions that are applicable This seems to revolve in a circle, offering no prospect of a
to a portion of matter. Intensive magnitude would have determinate concept of either of the terms. It really would
a place if matter consisted of monads. ·A monad has (by be circular if these were definitions of concepts in terms of
definition) no parts; so· any monad—however it was related one another, but that’s not what is going on. The Proposition
to anything else—could be more or less real in some way that does define a concept, but all the Note does is to explain how
didn’t depend on how many parts-external-to-one-another that concept applies to experience. . . .
540 it had, ·which means that its reality could be an intensive This should be noted: For any given portion of matter,
magnitude·. As for the concept of •mass in Definition 2: the question ‘How much [Quantität] matter is there in this?’
it is usually equated with the concept of •quantity, but is the question ‘How much [Quantität] substance is there in
this is wrong. Fluid portions of matter can by their own this?’ and not ‘How much [Größe] of quality Q is there in
motion act in mass [see Definition 2] but they can also act this?’, where Q is some special quality such as the powers
in fiow. In the so-called water-hammer— ·which causes a of repulsion or attraction that are cited in dynamics. And
knocking sound in the pipes when a flow of water is suddenly what is meant in this context by ‘the amount [Quantum] of
blocked·—the water in striking acts in mass, i.e. with all substance’ is merely ‘how many movable parts’ there are
its parts simultaneously; the same is true when a pot full in the given portion of matter.
of water is weighed on a scale. But when the water of a [Throughout this paragraph, ‘how many’ translates the German noun
Menge. Of the two other currently available English translations of this
millstream acts on the lower paddles of the wheel, it doesn’t work, one says
do so in mass, i.e. with all its parts together colliding with (1) ’the mere aggregate of the movable’
the wheel; rather, the parts act successively. So if in this while the other says
case we want to determine how much q matter is being moved (2) ‘the mere number of the movable parts’.
with a certain speed and exerting moving force, we must first Of these, the second is not quite right, but is nearer to right than the
of all look for the body of water, i.e. find out how much q other. As you might guess, Menge has two meanings. (1) It can be a
matter can produce the same action when it acts in mass (by concrete noun, meaning something like ‘multitude’ or ‘crowd’ or, if you
bringing its weight to bear) with a certain speed. That’s why like, ‘aggregate’. ‘I looked along the street and saw a Menge of angry
we usually understand by the word ‘mass’ the amountq of people coming towards me’. (2) It also has a sense in which it is an
a solid body (a fluid can be treated as solid on the strength abstract noun, meaning something like ‘how-many-ness’. Why say it in
of the vessel containing it). Finally, there’s something odd that clumsy way, rather than just using ‘number’ as the translator did?
about the Proposition and its appended Note. According to Because Kant sometimes—notably in the Critique of Pure Reason—uses
the Proposition, Menge as his more general how-many concept while reserving Zahl =
•
how muchq matter must be estimated by how much q ‘number’ to mean ‘Menge that is determinate’. He holds that when there
motion at a given speed, are infinitely many Fs, the Menge of Fs is not determinate, and so there
whereas according to the Note, is no such thing as the Zahl of Fs; the phrase ‘infinite number’ is, he
•
how muchq motion. . . .must be estimated by how holds, self-contradictory. In the present work, most occurrences of
Menge are

6
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

in the context of items of which there are infinitely or endlessly many, so here is an account of what substance in matter is.) In every
that Kant couldn’t have used Zahl (which in fact occurs only twice in the portion of matter the movable in space is the ultimate subject
whole work). The more important point, however, is that all through this of all qualities that matter has, and •how many movable parts
work Menge is being used as an abstract ‘how-many’ noun and not as a external to one another a portion of matter has is •how much 542
concrete noun meaning ‘crowd’ or the like.] substance there is to it. Hence the amount of any portion of
[What follows expands and re-arranges what Kant wrote, in ways that material substance is nothing but how many substances it
the usual conventions of ·small dots· can’t easily indicate. The thoughts consists of. So the only way the amountq of matter could be
expressed here are all present, explicitly or implicitly, in the paragraph increased or lessened would be for material substances to
This emphasis on how-many-parts is
that is being replaced.] go out of existence or for new ones to come into existence.
justified by a deep theoretical point about the line between But substances never come into or go out of existence in
substance and quality. The concept of substance is the changes of matter. So the over-all amount q of matter in the
concept of the rock-bottom subject, i.e. a subject that isn’t world is neither increased nor lessened in these changes, but
in its turn a quality of another subject. Now, we want to remains always the same.
get a grip on a notion of how much substance there is in a Remark
given portion of matter; and we can’t get at it through any The essential thing about substance as it figures in this
such notion as that of how-much-force the portion has or proof—only as existing in space and subject to the condi-
how-big the portion is, because force and size belong on tions of space, and hence as having to be an object of the
541 the quality side of the fundamental substance/quality line. outer senses—is that the amount of it can’t be increased
Well, then, what isn’t on that side of the line? The only or diminished unless ·some· substance comes into or goes
candidate is how-many-ness: ‘How many movable parts of out of existence. Why not? Because if x is something that
substance S are there?’ isn’t a question about any of S’s can exist only in space, the amount of x that there is has to
qualities, so the answer to it doesn’t slide across to the consist in ·facts about· the parts that x has external to one
wrong side of the line. [The remainder of the ‘Remark’ is omitted another, and if these are real (i.e. are movable) they must
because the preparer to this version has been defeated by it. On page 67 necessarily be substances. On the other hand, something
the passage is presented in each of the two currently available English regarded as an object of inner sense can as substance
translations of it.] have an amount or magnitude that doesn’t consist of parts
Proposition 2 external to one another, so that the parts that it does have
are not substances. When this item comes into or goes out
First law of mechanics: Through all changes of corporeal of existence, that doesn’t involve any substance’s doing so;
Nature, the over-all amount q of matter remains the same— so the magnitude of the item can increase or lessen
neither increased nor lessened. without detriment to the principle of the permanence of
Proof substance. [On this and the next two pages Kant uses the phrase ‘the
(Universal metaphysics contains the proposition that through permanence (Beharrlichkeit) of substance’ to mean ‘the fact that no
all changes of Nature no substance either comes into exis- substance comes
tence or goes out of existence; all that mechanics is adding
6
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

into, or goes out of, existence’. On pages 11–12 the cognate adjective, up by saying that the person who uses ‘I’ isn’t employing
beharrlich, being used for a quite different purpose, was translated as any concept of himself as a substance, and he is clearly
How can that happen? Well, I can be more
‘time-taking’.] implying that there is no such concept. Then:] In contrast 543
conscious or less conscious, so my mental representations with that, the concept of a portion of matter as substance
can be clearer or less clear, and this gives to my faculty of is the concept of something that is movable in space. So it’s
consciousness—I call it ‘Self-awareness’—a degree ·of not surprising that the permanence of substance can be
reality·, and we can even say that the substance of my soul proved of matter but not of the soul. This is because from
has such a degree; and none of this in any way requires that the concept of matter as what is movable in space it follows
any substance come into existence or go out of existence. This that the quantitative or how-much aspect of matter depends
faculty of Self-awareness can gradually diminish, to the point on there being many •real parts external to one another—and
where it finally goes right out of existence, so the substance thus many •substances. Thus, the going out of existence of
of the soul can gradually go out of existence. [In this sentence a portion of matter ·would involve the going out of existence
and the preceding one, Kant doesn’t say that the soul is a substance; he of many substances, and that· is impossible according to
speaks of the ‘substance of the soul’. He doesn’t explain the ‘substance the law of permanence. [Kant has Gesetz der Stetigkeit = ‘law of
of ’ locution (which occurs nowhere else in this work, and nowhere in continuity’ here, an obvious slip.] (The portion of matter could be
the Critique of Pure Reason). It does save him from having contradicted diminished by being taken apart, but that isn’t the same
If a thing
himself about whether substances can go out of existence.] as going out of existence.) The thought ‘I’, on the other
has parts external to one another, the only way it could hand, isn’t a concept at all but only an inner perception.
go out of existence gradually is by being slowly dismantled, And nothing follows from this thought (except that an object
pulled apart; but the soul can go out of existence gradually of inner sense is completely distinct from anything that
in a different way, through being gradually lessened and is thought of merely as an object of outer sense); so the
eventually extinguished. [Kant’s next sentence is hard to permanence of the soul as substance doesn’t follow from it.
follow. In it he sketches, in a condensed form, some doctrine
from the Critique of Pure Reason. He is facing the challenge Proposition 3
‘Don’t we know that the soul is a substance? Isn’t it obvious
that when I say “I see something red” I am attributing the Second law of mechanics: Every change in matter has an
predicate “sees something red” to the mental thing, the external cause. (Every motionless body remains at rest, and
substance, that I call “I”?’ Kant rejects this, and gestures every moving body continues to move in the same
towards the Critique’s account of how ‘I’ does work in all direction at the same speed, unless an external cause
its uses. Fortunately, we don’t really need that account compels it to change.)
for present purposes. All that matters here (and even it Proof
doesn’t here matter much) is his negative thesis that ‘I’ or the (Universal metaphysics contains the proposition that every
German Ich does not serve to pick out an individual thing, change has a cause. All we have to do here ·in mechanics· is
and therefore isn’t the name of a substance. Kant winds this to prove with regard to matter that every change in it must

6
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

have an external cause.) Because matter is a mere object with it. That’s because in gathering knowledge about Nature
of outer sense, the only facts about it are facts about how we must •first discover the laws of matter as such, not mixing
portions of matter relate to one another in space; and from them up with any other active causes, and •then connect
this it follows that the only way there can be any change these laws with any other causes there may be, in order to
in matter is through motion—i.e. through •changes from get a clear view of exactly what each law of matter brings
motion to rest or vice versa, or •changes in direction and about unaided. The possibility of a natural science proper
speed of motion. The principle of metaphysics says that each rests entirely on the law of inertia (along with the law of
such change must have a cause; and this cause can’t be the permanence of substance). Hylozoism [= ‘the thesis that
internal, because matter has no absolutely inner states or matter itself is alive’] is the opposite of this, and is therefore
inner causal resources. Hence all change of a portion of the death of all natural philosophy! Just from the concept
matter is based on an external cause. of inertia as •lifelessness we can infer that ‘inertia’ doesn’t
Remark signify a thing’s •positive effort to maintain its state. Only
The name ‘law of inertia’ should be given only to •this law living things can be called ‘inert’ in this positive sense; it
involves their having a thought of another state that they
544 of mechanics, and not to •the law that every action has an
equal and opposite reaction. The latter says what matter don’t want to be in and do their best to avoid.
does, but the former says only what it doesn’t do, and that
Proposition 4
is a better fit for the word ‘inertia’. To say that matter ‘has
inertia’ is just to say that matter in itself is lifeless. For a Third mechanical law: In all communication of motion, ac-
substance to have life is for it to be able to get itself, through tion and reaction are always equal to one another.
its own inner resources, to act—i.e. to change in some way
Proof
(for any finite substance) or start or stop moving (for any
[In a notably obscure explanation—omitted here—of why he had to deal
material substance). Now, the only inner resource we know
with this third law, ‘for the sake of completeness’, Kant refers to it as ‘the
of through which a substance might change its state is
law of two-way causal interaction of universal metaphysics’. His word
desire, along with its dependents— •feelings of pleasure and
for ‘two-way causal interaction’ is Gemeinschaft, which is standardly but 545
unpleasure, •appetite, and •will—and the only inner activity
unhelpfully translated as ‘community’.] Active relations of portions
that we know of is thought. But none of these causes and
of matter in space, and changes of these relations, have
activities have anything to do with the representations of
to be represented as reciprocal if they are to be ·thought
outer sense, and so they don’t belong to matter as matter.
of as· causes of certain effects. Now, any change of such
Therefore all matter as such is lifeless; and that is what
relations is motion; so we get the result that whenever one
Proposition 3, the one about inertia, says—and it’s all it
body causes a change in another body, the other must also
says. If we want to explain any change in a material thing in
be in motion (so that the interaction can go both ways); so
terms of life, we’ll have to look for this cause in some other
we can’t allow for any case in which one body A causes
substance that is different from matter although bound up
motion in another body B which until that moment was
•absolutely

6
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

at rest. What we can do is to represent B as being at rest two-way causal interaction of the bodies is constructed as
•
relative to the space to which it is referred; B must be follows. 546
represented as moving with its reference-space towards A,
moving at the same speed in absolute space as A is moving
towards B. For the change of relation (and hence the motion)
is completely reciprocal between both bodies; by as much as
A approaches every part of B, by that much B approaches
every part of A. What we are dealing with here is not the
empirical space surrounding the two bodies but only the line Let a body A be moving into a collision with the body B
stretching between them (because our whole topic is just the with a speed = AB with regard to the relative space in
effect that the movement of each has on the state of the other, relation to which the body B is at rest. Let the speed AB
and for that we can abstract from all relation to empirical be divided into two parts, Ac and Bc, in such a way that
space); and therefore we think about their motion only in their respective speeds are inversely proportional to their
terms of absolute space, in which they share equally in the respective masses. Represent A as moved with the speed Ac
motion attributed to A, the one in relative space, because in absolute space, and ·the larger body· B (together with
there’s no basis for attributing more motion to A than to B. the relative space) as moving with the ·smaller· speed Bc in
On this footing, the motion of a body A toward an immobile the opposite direction. Thus the two motions are opposite
body B is handled in terms of absolute space, i.e. the and equal to one another. [Kant is relying here on the thesis (page
49) that in mechanics the concept of how much motion is a function of
motion in question is treated as a relation of two causes
interacting with one another and not with anything else; speed and mass.]·Because they are equal and opposite,
and so the motion which appears to us as only A’s is neither is the winner, and· they destroy one another and
considered as something shared between A and B. This can both come to be, relatively to one another, i.e. in absolute
occur only in the following way. The speed which in the space, in a state of rest. [In a helpful footnote in his translation of
this work (Cambridge University Press 2004), Michael Friedman points
relative space is attributed only to A is divided between A
out that Kant is here discussing the collision of perfectly inelastic
and B in inverse proportion to their masses; A’s share is
only its speed in absolute space, whereas B (along with So we
bodies, i.e. ones that have no bounce-back from a collision. ]

the relative space in which it is at rest) is assigned its speed have B moving with its relative space in the BA direction, and
in the opposite direction; and in this way the same losing its motion when it collides with A; but the collision
appearance—·i.e. the appearance that A moves towards B, doesn’t automatically cancel the motion of B’s relative
which is motionless·—is perfectly retained. [We are about to space as well. So we have two equivalent ways of
see Kant representing speeds by lines, in accordance with his statement
describing the state of affairs after the collision:
•
The bodies A and B are now at rest in absolute space,
on page 11 that ‘In phoronomy we use the word “speed” with a merely
and relative to them the relative space moves in the
spatial meaning—the measure of how far a thing travels in a given
direction BA with the speed Bc.
period of time’—which has the result: the longer the line, the faster
the motion.] What happens in the

6
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

The bodies A and B move with equal speed Bd = Bc in

•
Note 1
the direction AB, i.e. the direction that the impacting From this we can infer a natural law that is of some im- 548
body A had. portance for universal mechanics, namely that every body,
Now, according to this, however great its mass may be, must be movable by the
the amountq of motion of B in the direction and with impact of any other body, however small its mass or speed
the speed Bc, may be. This is because the motion of body A in the direction
which is the same as AB must encounter an equal motion of B in the opposite
the amountq of motion of B in the direction Bd with direction BA [this refers to the diagram on page 54]. The two
the speed Bd, motions cancel one another in absolute space by impact.
is equal to But thereby both bodies receive a speed Bd = Bc in the
the amountq of motion of the body A with the speed direction of the impacting one; consequently, the body B is
and in the direction Ac.
direction’ or ‘the body stays still and the space moves in that direc-
Consequently, the effect ·of the collision·, i.e. the motion Bd tion’, i.e. the opposite direction; these two were indistinguishable
that B receives by impact in relative space, and hence also by any appearance. The quantity of motion of the space was
the action of the body A with the speed Ac, is always equal to merely its speed, so the quantity of motion of the body was nothing
the reaction Bc. That is a part of what was to be proved—the but its speed (which is why we could regard the body as a mere
part saying that whenever a moving body causes a change movable point). But in mechanics we deal with a body x that is
moving relative to another body y and, through that motion, is
in the motion of a stationary body, action and reaction are causally related to y. Whether by •moving towards y and exercising
equal. It’s a thesis in mathematical mechanics that this its force of impenetrability or by •moving away from y and exercising
same law ·about action and reaction· holds for the impact its force of attraction, x comes to be in a two-way causal
of one moving body on another moving body just as well interaction with y.
So here ·in mechanics· there is a difference between ‘x moves this
as it does for the impact of one body on another that is way’ and ‘x is stationary while the space containing it moves the
motiononless. Also, the communicating of motion by impact opposite way’. That is because we are now working with a different
differs from the communicating of motion by traction only concept of quantity of motion—it involves not merely a thing’s speed
in the direction in which the portions of matter oppose one but also the thing’s quantity of substance, which is relevant to its
role as a moving cause. And we now •have to assume that both
547 another in their motions. From all of this it follows that bodies are moved (in phoronomy we had our •choice about that),
in all communication of motion, action and reaction are and indeed that they are moved with the same quantity of motion
always equal to one another: an impact can communicate in opposite directions. When one body is not moving in relation to
the motion of one body to another only by means of an its space, we have to attribute the required motion to this body
equal counter-impact, a pressure only by means of an together with its space. For the only way the body x can act on body
y other through x’s s own motion is by repulsive force as it
equal counter-pressure, and a traction only by an equal
approaches y or by attractive force as it moves away. Now, given
counter- traction).3 that the two forces always act with equal strength and in opposite
3 directions, any action by one of those forces in one body requires a
In phoronomy a body’s motion was considered merely as the change
counter-action by the other; so no body can pass motion on to an
in its relation to space, so we could say that ‘the body moves in this
absolutely immobile body; the second body must be moved in the
opposite direction . . . .

6
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

movable by every force of impact, however small. Remark 1

Note 2 This then is the construction of the communication of motion.
This is, then, the mechanical law of the equality of action This construction necessarily carries with it the law of the
and reaction. It is based on the fact that motion is never equality of action and reaction. Newton didn’t venture to
communicated ·from one body to another · except in a two- prove this law a priori, but appealed to experience to prove it.
Others tried to secure this law by introducing into natural
way causal interaction between the two. And on the
science a special force of ‘inertia’ (Kepler’s name for it); so
following: It is trivially obvious that a body A can’t hit a
second body B that is motionless in relation to A; what basically they were also deriving it—·i.e. the law of action
happens is that A hits B which is motionless in relation to and reaction·—from experience. Yet others tried to get it
B’s space. So when A hits B, it must be that B together with from the mere concept of the communication of motion [see the
its space is moving towards A. How fast? Well, the initial note on page 38]. They thought of this as involving a gradual
speed that we would attribute to A if we thought of it as transfer of body A’s motion into another body B, so that A
moving in absolute space has to be divided between A and loses exactly as much as B gains; ·their view was that· the
B-with-its-space, ·so that we get the right account of how transfer stops when A and B are moving at the same speed
in the same direction as that of the latter; and that rules out
long it takes A to reach B from the given distance away·;
any reaction, i.e. any reacting force of B acting back against
and the division must assign to each body not the same
A that collides with it.4 ·That is bad enough, but there is
speed but the same amount of motion, so that the speed of
each is inversely proportional to its mass. [See the discussion of 4
·This theory didn’t have to say that the motion-transfer is gradual·.
amount-of-motion on page 49.]. . . . The equality of A’s action with B’s. is secured just as well if the
There is also a dynamical law of the equality of the action transfer of motion is supposed to be instantaneous, with body A com-
ing to rest immediately after the collision; and that’s the form these
and reaction of portions of matter. It doesn’t concern one
theorists’ account would have to take if they were thinking of the two
portion A’s sharing its motion with another portion B, but bodies not as elastic but as absolutely hard. But the law of motion
rather A’s giving its entirer motion to B and having motion that that left them with doesn’t square with experience and isn’t even
produced in itself through B’s resistance. This can be easily consistent, so their only way out would be to deny that there are any
absolutely hard bodies (thus making their law contingent, because
demonstrated in a similar way. For if A attracts B, then A
dependent on a special quality of colliding bodies). But I can’t see
compels B to approach A, i.e. resists the force with which how the transfer-of-motion theorists could explain what happens
B tries to pull away. But there’s no difference between in collisions if the colliding bodies are elastic. It is clear that when
•
B’s pulling away from A and •A’s pulling away from B; elastic body A collides with immobile elastic body B, it is not the case
that B merely receives motion that A loses; rather, B exercises actual
549 so traction and countertraction are equal to one another. force in the opposite direction against A, as though it were pushing
Similarly, if A repels B, then A resists the approach of B; but against a spring lying between them; and for this it requires just
B’s approaching A is just the same as A’s approaching B, so as much actual motion (but in the opposite direction) as A needs
it is just as correct to say that B equally resists the approach for its part in this transaction. In my version of this law, on the
of A; so pressure and counterpressure are always equal. other hand, ·no such difficulty arises, because· it doesn’t make the
slightest difference whether the colliding bodies are absolutely hard

6
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

550 mechanical law.]

worse·. They don’t show what their account means, i.e.
they haven’t explained ‘communication of motion’ in a way
that shows that such communication is possible. [In the
original, this ‘Remark’ down to here is a single sentence. ] The mere
words ‘transfer of motion from one body to another’ don’t
explain anything. If they are understood literally, implying
that motion is poured from one body into another like
pouring water from one glass into another, they conflict
with the principle that qualities don’t wander from one
substance to another [Kant gives this in Latin ]. If that literal
reading is rejected, then the theorists I am discussing must
face the problem of how to make this possibility
conceivable. The
only way to make sense of A’s motion’s being necessarily
connected with B’s motion is by attributing to both bodies
dynamic forces (e.g. the force of repulsion) that precede all
motion—·i.e. so that the forces explain the motions, not
vice versa·. And then we can prove that the motion of A
towards B is necessarily connected with the approach of B
toward A; and if B is regarded as immobile, then A’s
motion is connected with the motion of B-together-with-
its-space towards A, so that the bodies with their (basic)
moving forces are considered to be moving relatively to one
another. We can fully grasp this course of events a priori:
whether or not the body B is moving in relation to its
empirically knowable space, we have to regard it as
moving in relation to the body A, and indeed moving in the
opposite direction. Otherwise, A’s movement couldn’t bring
into action the repelling forces of itself and of B, in which
case portions of matter couldn’t act mechanically on one
another in any way, i.e. there couldn’t
or not. [Kant is referring to his version of the ‘third mechanical law’—
Proposition 4 on page 53. So his topic in this footnote, where he
writes about ‘the law’ that these transfer-of-motion theorists have to
accept, is really their version of, their presentation of, the third

6
 Metaphysical Foundations of Natural Immanuel 3: Foundations of
be of inertia’ must be dismissed from natural science
any (despite the fame of its inventor). This must be done
co •
because the phrase is inherently self-contradictory;
m and because because the ·so-called· ‘law of inertia’
mu (law of life- lessness!) could easily be confused with
nic the law of reaction; and above all •because this
ati confusion would support and encourage the wrong
on account given by those who don’t have a proper grasp
of of the mechanical laws. According to their account,
mo
the reactions of bodies—now described as ·exercises
tio
n of· ‘the force of inertia’—would lead to
thr (i) the lessening or annihilation of all the
oug motion in the world,
h and to
coll (ii) collisions in which no motion is
isio communicated. The reason for (i) is this: according
ns. to the account that I am attacking, the moving body A
would have to ‘spend’ some of its motion in
Re
overcoming the inertia of the immobile body B, and
ma
rk that ‘expense’ would be sheer loss. And the reason
2 for
T (ii) is this: If B were very massive ·and A much less so·, A
h wouldn’t have enough motion both to overcome B’s ’inertia’
e and then to make B move; so that this would be a collision
in which no motion was communicated. ·Summing up·:
n 551 A motion can’t be resisted by anything except an
a opposite motion; it can’t be resisted by a body’s
m immobility! So the ‘inertia’ of matter, i.e. its mere
e incapacity to get itself moving,
isn’t the cause of any resistance. It could be defined:
‘ ‘inertia’ = ‘a unique force to resist a body but
f not to move it’
o and that would make ‘inertia’, ·despite its definition·, a word
r without any meaning. We could put ‘inertia’ to a
c better use by designating the three laws of universal
e mechanics as:

7
 Metaphysical Foundations of Natural Immanuel 3: Foundations of

•
the law of matter’s subsistence [Proposition 2, page 51], which has to balance, or to equal, what there is on the
•
the law of matter’s inertia [Proposition 3, page 52], and weight’s side, namely
•
the law of the reaction of portions of matter [Proposition •
a body of finite mass with an infinitely small speed.
4, page 53].
Whereas expansive force operates only at the surface, and is
These laws, and hence all the propositions of mechanical therefore a force exercised by an infinitely small amount of
science, correspond exactly to the categories of •substance, matter, attraction is a penetrating force: a body’s attractive
•
causality, and •two-way interaction. There is no need for force penetrates the body itself, so that the body’s inner
me to discuss this here. parts contribute to the attractive force of the body as a whole.
If attraction were not a penetrating force, the equations
GENERAL REMARK ON MECHANICS implied by the mechanical proposition 4 wouldn’t come out
right. [That somewhat simplifies what Kant wrote.] Cohesion is 552
The communication of motion takes place only by means often thought of as a force operating only at surfaces; but
of moving forces—impenetrability and attraction—that a we now see that if cohesion is to be true attraction and not
portion of matter also has when it is not moving. The merely external compression—i.e. if it’s to be thought of in
action of a moving force on a body at an instant is the terms of the parts of a body pulling together rather than
solicitation of the body. [That’s the first appearance of being pushed together—it can’t be thought of in this way.
‘solicitation’ in this work. It is or was a technical term in mechanics. An absolutely hard body would be one whose parts at-
You can safely think of it as meaning ‘instantaneous tug or push’. ] The tracted one another so strongly that no weight could
speed of the body brought about by its solicitation— •separate or •re-arrange them. This means that the
understood in terms of how this speed can increase parts of such a body would have to pull on one another
uniformly through time—is the acceleration-at-a-moment infinitely more strongly than gravity pulls on them
value. (The latter must involve only an infinitely small (·because: however strong the gravitational pull, the part-on-
speed, because otherwise the body would attain through the part pull will defeat it·). But . . . [and then Kant proceeds
acceleration value an infinite speed in a given time, which is with a defeatingly technical reason why this fact, conjoined
impossible. . . .) As an example of the solicitation of matter with some others, implies that absolute hardness is
by expansive force, let us consider compressed air holding impossible. He follows this with what seems to be an
up a weight. In this situation, the air’s exercise of expansive entirely different and much more accessible reason,
force must have a finite speed. [In this paragraph, ‘finite’ always namely:] An absolutely hard body is impossible because
means ‘more than infinitely small’.] Why? Because expansive in a collision between body x and absolutely hard
force occurs only at the surface, which means that it is the body y, x would be moving with a finite speed and y
motion of an infinitely small amount of matter; and so we would react instantaneously with a resistance equal
have on the air’s side of the transaction to the whole of x’s force.
•
an infinitely small amount of matter with a finite And this is impossible. A portion of matter produces by its
speed, impenetrability or cohesion only an infinitely small instan-

7
 Metaphysical Foundations of Natural Immanuel 4: Foundations of

taneous resistance to the force of a body that collides with Chapter 4

it. A consequence of this is the mechanical law of continuity, Metaphysical Foundations of Phenomenol-
namely:
A collision can’t make a body move, or stop moving, ogy
or change speed or direction, instanteously. Any such
Definition
change occurs through a time-taking infinite series of
intermediate states whose difference from one Matter is whatever is movable and can be an object of 554
another is smaller than that between the first and last experience.
such states.
Remark
[That is what Kant wrote; but the reference to ‘the first and last’ states
doesn’t help to pin down the notion of continuity, and may be a mere slip
Like everything that is represented through the senses, mo-
on Kant’s part. What he needed, and perhaps what he meant, was this:
tion is given only as appearance. For the representation of
‘Given any two members of this series, there is an intermediate state that motion to become experience, there has to be—in addition ·to
(i) what is received through the senses·—also (ii) something
is more like each of them than they are like one another.’]
Thus, a moving body x that collides with a portion of matter thought by the understanding. As well as (i) being a state of
y is halted by y’s resistance not •instanteously but through the perceiving •subject, the representation ·of motion· must
a continuous slowing down; similarly for the other changes also determine an •object. So something movable becomes
that a collision can subject a body to—starting to move, an object of experience when a certain object (here, a material
changing speed, changing direction. When the direction of thing) is thought of as falling under the predicate ‘moves’.
a body’s motion changes, it goes through all possible Now, motion is change of relation in space. So there are
directions intermediate between its first and last ones, always two correlates here, ·namely matter and space, and
which means that it changes direction by moving in a we have some options·:
curved line. This law ·of continuity· also applies. . . .to (a) In appearance, we can handle things in terms of the
motion of matter or the motion of space; it doesn’t
553 changes in a body’s state by means of attraction. This
·mechanical· law of continuity is based on the law of the matter which we choose, because the two accounts
inertia of matter. On the other hand, the metaphysical law of are equivalent.
(b) In the experience of motion we must think of one of
continuity applies quite generally to all change (internal as
well as external), and its basis is provided by concepts: the two—matter or space—as moving and the other
the concept of change as a magnitude, and the concept of as staying still.
generation of a magnitude (which necessarily happens (c) Reason must necessarily represent both of these cor-
continuously through a period of time). So this relates as moving at the same time.
metaphysical law has no place here. All we get in the appearance of motion is the change in
the relation of matter to space; and that doesn’t pick out 555
any of those three options as the right one. But ·we can’t

7
 Metaphysical Foundations of Natural Immanuel 4: Foundations of

just leave it at that·: we have to settle the conditions under opposite way—around it
which a movable thing can be thought of as moving in this
or that specific way, because without that there can’t be
experience of a moving thing. (The difference ·that I am
invoking here· between •appearance and •experience is
not the same as the difference between •illusion and •truth—
·i.e. the difference between •how things seem and •how
they are·. That’s because •illusion ·or seeming· is nothing
like
•
appearance: something’s seeming to be the case always
involves judgments about what is objectively the case;
such judgments are always in danger of going wrong by
taking the subjective to be objective, but in appearance
there is no judgment of the understanding. This distinction is
significant not only here but all through philosophy,
because there is always confusion when what is said about
‘appearances’ is taken to be referring to illusion or
seeming.)

Proposition 1

(a) ‘That portion of matter is moving in a straight line in

relation to that empirical space’—as distinct from ‘The space
is moving in the opposite direction in relation to the portion
of matter’—is a merely possible predicate. (b) ‘That portion
of matter is moving in a straight line period’, i.e. its
movement is absolute, not thought of as a changing relation
to matter outside it, is impossible. [Kant is assuming here that if
you don’t relate a moving thing to any body outside it you can’t be relating
it to an empirical space.]
Proof
In the case of a body x moving in a relative space y, these
two:
(1) y is at rest and x is moving this way within it, and
(2) x is at rest and y is moving in that way—the

7
 Metaphysical Foundations of Natural Immanuel 4: Foundations of
tell the same story about what is objectively Remark
happening out there; they differ only in what they This proposition determines the modality of motion with
imply about the subject,
·the person whose experience is being reported on·. So there
is no difference between them at the level of experience,
only at the level of appearance. If the spectator puts
himself into the space y, then he says that the body
moves; if he puts himself (at least in thought) into
another space z that encloses y, with x being at rest in
relation to z, then he (the spectator) will say that
space y is moving. Therefore,
in experience. . . .there is no difference whatever between
556
(1) and (2). [In a very repetitious passage, Kant
belabours the point that (1) and (2) represent a pair of
choices that one might make, not rival accounts of what
is objectively the case. And yet they do apply
conflicting predicates to x—‘moving’ and ‘at rest’—
from which Kant concludes:] Something that is in
itself undetermined as regards two mutually opposed
predicates is to that extent merely possible. So the
straight- line motion of a portion of matter in
empirical space—as against the opposite motion of the
space—is in experience a merely possible predicate.
This was (a) the first thing to be proved.
Next: For any •relation to be an object of
experience, each of the related items must be an
object of experience; this holds also for any •change
of relation, including the special case of the relation-
change that is •motion. Now pure space (in contrast
to empirical space), i.e. absolute space (in contrast to
relative space) is not an object of experience;
basically it is nothing. So straight-line motion without
reference to anything empirical, i.e. absolute motion, is
utterly impossible. This was (b) the second thing to
be proved. . . .

7
 Metaphysical Foundations of Natural Immanuel 4: Foundations of

respect to phoronomy—namely possibility. with •the circular motion of the body, nevertheless in the
complex of all appearances, i.e. of possible experience, the
Proposition 2 •
former motion conflicts with the •latter; and hence the
former is nothing but mere illusion [Schein].
The circular motion of a portion of matter x, as against the
opposite motion of the space, is an actual predicate of x. Remark
557 On the other hand, the opposite motion of a relative space, This proposition determines the modality of motion with re-
taken as a substitute for the motion of x, is not an actual gard to dynamics—namely, actuality. For a motion that can’t
motion of x—at most it may seem to be an actual motion of take place without the influence of a continuously acting
that body, but this is a mere illusion. external moving force exhibits—directly or indirectly—basic
moving forces of matter, either of repulsion or of attraction.
Proof
In connection with this topic, see Newton’s scholium to
Circular motion is (like every curved-line motion) a continu-
the definitions at the start of his Mathematical Principles
ous change of •straight-line motion; and since •this motion
of Natural Philosophy. This makes it very clear that
is itself a continuous change of relation to external space; so
the circular motion of rotating around a common 558
circular motion is a change of the change of these external
centre,
spatial relations, and is therefore a continuous arising of
and therefore also
new motions. Now, according to the law of inertia, a motion
the rotation of the earth on its axis,
can’t start up without having an external cause. But the
can be known by experience even in empty space; which
circulating body x at every point of this circle is (also by
means that a motion that is a change of external relations in
the law of inertia) endeavouring to proceed in the straight
space can be empirically given, even though this space itself
line at a tangent to the circle, and this straight-line motion
is not empirically given and is not an object of experience.
acts against the external cause ·of x’s circular movement ·.
This paradox deserves to be solved.
[Re ‘endeavouring’, see the long note on page 19.] Hence every body
in circular motion manifests by its motion a moving force. Proposition 3
Now, the motion of the space, in contrast to the motion of
the body, is merely phoronomic and has no moving force. In every motion of a body whereby it is moving with
Consequently, the judgment that here either the body is regard to another body, an opposite and equal motion of
moved or else the space is moved in the opposite direction this other body is necessary.
is a disjunctive one, by which, if the one member, namely, Proof
the motion of the body, is posited, then the other member, According to the third law of mechanics (Proposition 4 on
namely, the motion of the space, is excluded. Therefore, the page 53), the communication of the motion of the bodies is
circular motion of a body, in contradistinction to the motion possible only through the two-way causal interaction of their
of the space, is an actual motion. Even though according to basic moving forces, and this two-way causal interaction is
the appearance [Erscheinung] •the motion of the space agrees possible only through mutually opposite and equal motion.

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 Metaphysical Foundations of Natural Immanuel 4: Foundations of

So the motion of both bodies is actual. But the actuality idea. [See the note on ‘idea’ on page 9.] ·Here is how it comes into
of this motion doesn’t come from. the influence of exter- play·. For there to be even an appearance of motion, there
nal forces, but follows immediately and inevitably from a has to be an empirical representation of the space with which
concept—the concept of how something that moves relates to the moving thing is changing its relation; but that space—the
each other thing that can be moved by it. So motion of the space that is perceived—must be material and therefore
‘other thing’ is necessary. itself movable. [Kant says that this last ‘therefore’ depends on ‘the
Remark concept of matter in general’. Perhaps he is referring to the equation of

This Proposition determines the modality of motion with ‘material space’ with ‘relative space’ in the phoronomic Definition of

regard to mechanics, namely, necessity. It is immediately Now, we can’t think of this space as moving
‘matter’ on page 7.]

obvious that these three propositions determine the motion except by thinking of it as contained in a more extensive
of matter with regard to its possibility, actuality, and ne- space that is at rest. But this latter space can be related in
cessity, and thus hence with regard to all three categories just the same way to a still larger space . . . and so on to
of modality. [This completes Kant’s attempt to tie his four chapters infinity, without ever arriving empirically at an immovable
severally to his four trios of categories.] (immaterial) space with regard to which any portion of
matter could be said to be outright moving or at rest.
Rather, we have to keep changing our concept of these
GENERAL REMARK ON PHENOMENOLOGY
relational set-ups depending on what we are thinking of as
So we have here three concepts that have to be employed in moving relative to what. ·I’ll say it again·:
universal natural science, and which therefore have to be The condition for regarding something as at rest, or
understood in precise detail—though the details are hard to as moving, is always its being placed in a relative
pin down and hard to understand. space—always, again and again ad infinitum, as
They are these: we enlarge our view.
(1) the concept of motion in relative (movable) space; From this we can draw two conclusions: (1) All motion or
(2) the concept of motion in absolute (immovable) space; rest must be merely relative; neither can be absolute. That is,
559 (3) the concept of the across-the-board distinction be- matter can be thought of as moving or at rest only in relation
tween relative motion and absolute motion. to •matter and never in relation to •mere space without
The concept of absolute space lies at the foundation of all of matter. It follows that absolute motion— ·i.e. motion that
these. How do we come by this unusual concept, and why doesn’t consist in one portion of matter changing its relation
do we have to use it? to another portion·—is simply impossible. (2) For this very
•
The concept of absolute space can’t be ·a concept of reason, there can’t be, out of all the ever-wider concepts of
the understanding, because absolute space can’t be· an motion or rest in relative space, one that is ·so wide as to
object of experience—space without matter isn’t an object be· valid for every appearance. ·To have such an all-purpose
of perception. But •it is a necessary concept of reason, and concept·, we have to make room in our minds for the thought
that gives it the status of an idea, but that is all it is—a mere of a space that isn’t nested within any larger space, i.e. an

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 Metaphysical Foundations of Natural Immanuel 4: Foundations of

560 absolute space in which all relative motions are nested. In might simultaneously have are grounded in the concept of
such a space everything empirical is movable,. . . . 5 but none experience that unites them all, namely the concept of merely
can be valid as absolute motion or rest. So absolute relative motion and rest.
space According to Proposition 2 [page 61], circular motion can
is necessary not as a concept of an actual object but as an be experienced as actual motion, even if no external
idea that is to regulate all our thoughts about relative motion. empirically given space comes into the story; so it seems to
If we want all the appearances of motion and rest to be held be absolute motion. ·I’ll say that again, explaining it a little
together by a determinate empirical concept, we must put as I go·. A motion such as (a) the earth’s rotation on its
them within the framework of the idea of absolute space. axis relative to the stars is an appearance that can be
[Actually, Kant writes that these appearances must all auf den absoluten matched by (b) the opposite motion of the space of the
Raum reducirt werden which literally = ‘be reduced to absolute space’; stars, and these two are ·empirically· fully equivalent.
but his meaning, in this sentence and the next, seems to be something But Proposition 2
about framing or handling-in-terms-of.] forbids us ever to postulate (b) intead of (a); so (a) is not to be 561
Thus the straight-line motion of a body in relative space represented as externally relative—which sounds as though
is handled in terms of [reducirt auf ] absolute space (i) when I it is being assumed to be absolute.
think of a body as being at rest and think of the relative space But ·that’s a mistake·. What we are dealing with here is
·that it is in· as moving in the opposite direction—moving in the ·humdrum everyday· diference between what seems to
non-empirical absolute space—·and (ii) when I think of the be the case and what is really the case, not the
body as moving and the relative space as being motionless in ·metaphysical· distinction between relative space and
absolute space·. The two ways of representing the situation absolute space. Em- ploying the former distinction, we can
are empirically exactly alike. By means of this representation and do have empirical evidence that the earth is really
all possible appearances of rectilinear motions which a body spinning and thus that the stars may be at rest, although
5
the space they move in can’t be perceived. The earth’s
[Kant offers here a longish footnote, to the following effect: Any
empirically knowable fact about something’s moving can be con-
circular motion doesn’t present us with any appearance of
strued either as a body moving in a relative space or a relative change of place, i.e. any
space moving around a body. In the context of phoronomy, these •
phoronomic change in the earth’s relation to the (empirical)
two are alternatives; we shouldn’t think of them as a disjunction—’P space surrounding it; but it exhibits a continuous •dynamic
or Q, one of which must be wrong’—because the difference between change in the relations amongst portions of matter within
them is a difference of viewpoint, a difference in how the knowing
subject relates to the state of affairs, not in what is objectively the the space that it occupies, and this change is provable by
case. In the context of dynamics, on the other hand, such a pair of experience. For example, the attraction ·that holds the earth
propositions are rivals, which can’t both be true. And in mechanics together· is constantly lessened by an endeavour to escape,
there is a different pattern again: When one body is rushing towards
another, we must attribute an equal proportion of the total motion
·i.e. by centrifugal force·; we know about this empirically,
to each body. Kant is here presenting a trio of ways of understand- and it’s a result of the earth’s rotation, which shows that
ing a certain thesis of the form ’P or Q’: disjunctively (dynamics), the rotation is real and not illusory. [Kant wrote this paragraph
distributively (mechanics), alternatively (phenomenology).] down to here as a single sentence. ] Thus, for instance, we can

7
 Metaphysical Foundations of Natural Immanuel 4: Foundations of
r s axis in infinite empty
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 Metaphysical Foundations of Natural Immanuel 4: Foundations of

space, and can produce empirical evidence for this motion of appearance.] ·Our evidence· that this rotation is true ·or
even though it doesn’t involve any phoronomic change (i.e.
change in the appearance) in how the earth’s parts relate •to
one another or •to the space surrounding the earth. . . . ·Here
is a description of one course of events that would provide
such evidence·. I let a stone fall down a deep hole running
to the centre of the earth; and I find that although gravity
keeps taking it downwards, its fall continuously diverges
from the vertical direction by tending towards the east;
from which I conclude that the earth is rotated on its axis
from evening to morning. This is good enough evidence
of the earth’s
actiually rotating in that way; and we don’t get such evidence
from •the change in the earth’s relation to external space (the
starry heavens). Why not? Because •that change is a mere
appearance, which could come from either of two opposed
causes—from the earth’s spinning on its axis or from the
stars revolving around the earth. But the earth’s
rotation,
even though it isn’t a change of relation to empirical space
(·I am now returning to the imagined case of a rotating world
in a space that is otherwise empty·), isn’t a case of
absolute motion. Rather, it is a continuous change in how
portions of matter •relate to to one another, so it really is
only a case of •relative motion, although we represent it to
ourselves as as happening in absolute space. And it’s just
because this movement of the earth is relative that it is
true ·or actual· motion.
[Kant is here recalling us to his point that the line between
(a1) illusory and (a2) real or actual
is not the same as the line between
(b1) how things appear and (b2) how they are in themselves,
or the line between
(c1) relative space and (c2) absolute space.
He is emphasizing the difference by saying that the status of the earth’s
rotation as something (a2) real depends on its belonging to (b1) the realm

7
 Metaphysical Foundations of Natural Immanuel 4: Foundations of
actual· rests upon our encounter with the fact that would be the straight-line motion of the universe,
parts of the earth outside its axis of rotation tend to i.e. of the system of all matter. ·It is easy to see why·: If
fly off, i.e. the fact that any two parts of the earth that outside of a portion of matter x there is any other matter,
are exact antipodes
of one another tend to move apart. . . . [Kant likens this
562 to a slightly different consideration that Newton used
—two bodies joined by a cord and rotating, pulling on the
cord—see
his remark about a ‘paradox’ on page 61.]
As for the third proposition: to show the truth ·or ac-
tuality· of the motions of two bodies moving relatively
to one another, showing this without reference to
empirical space, we don’t need to learn from experience
about an active dynamical influence (of gravity or of a
taut cord), though we needed this in the case of the
second proposition. Rather, we can get this result
from the mere dynamical possibility of such an
influence, as a property of matter (repulsion or
attraction). That possibility brings with it the result that
any motion by one of the two bodies is matched by an
equal and opposite motion of the other at the same time.
indeed such action and reaction stem from mere
concepts of a relative motion when this motion is
regarded as in absolute space,
i.e. according to truth. Therefore, this third
proposition is, like everything adequately provable
from mere concepts, a law of an absolutely necessary
countermotion.
So there is no absolute motion even if a body in
absolute space is thought of as moving in relation to
another body. The motions of the two bodies are here
not relative to the space surrounding them but only to
the space between them, which is the sole determinant
of their external relation to one another. . . . So these
motions are only relative. Thus, absolute motion would
have to be motion that a body has without a relation to
any other matter, and the only candidate for this role
8
 Metaphysical Foundations of Natural Immanuel 4: Foundations of

even if separated from x by empty space, then x’s motion the world or •outside of the world (if the world is
would certainly be relative. Thus, if you can show regarding represented as limited). An empty space within the world can
any law of motion L that be further subdivided into
denying L implies that there is a straight-line motion (a) scattered all through the world, so that a part of the
of the whole universe, volume of any body may be empty space; and
that proves that L is absolutely necessary, because such (b) occurring between bodies, e.g. as space between the
563 motion is utterly impossible. There is a law of that kind, stars.
namely the law of the reaction of portions of matter in all This distinction is not theoretically deep, because it doesn’t
two- way causal interactions that depend on motion [see mark off different kinds of empty space but only different
Proposition 4, page 53]. Any divergence from this law would •
places in the world where empty space might occur. Still,
·consist in a shove in one direction without an equal shove in the distinction is put to use, because the two sides of it are
the opposite direction; so it would· create a straight-line used for different explanatory purposes. (a) Space within
movement of the common centre of gravity of all matter, and bodies is used to explain differences in the density of bodies;
hence of the whole universe. No such result follows from the and (b) space between bodies is used to explain how motion
thesis that the entire universe rotates on its axis; so there is is possible. It isn’t necessary to (a) assume empty space for
never any obstacle to thinking of the universe in this way, the first purpose, as I have shown in the General Remark
though I can’t see any conceivable use for it. on Dynamics [see pages 39–41 and 46–47]; but there’s no way
Corresponding to the ·three· concepts of motion and of showing that empty space is impossible because its
moving forces, there are ·three· concepts of empty space. concept is self-contradictory. Still, even if it can’t be ruled
(1) What passes for ‘empty space’ (or ‘absolute space’) in out on merely •logical grounds, there might be general
the context of phoronomy really shouldn’t be called •physical grounds for banishing empty space from the
empty space. It is only the idea [see the note on ‘idea’ on page 9] doctrine of Nature. Suppose that the following turned
of a space from which I filter out all particular matter that out to be the
would make it an object of experience, in order to think of case (there are many reasons for thinking that it is the case):
it as the space within which every material or empirical What holds bodies together is not •true but only
space can move; this being something I want so as to think •
apparent attraction; what really holds a body to-
of every truth of the form ‘x moves’ not as predicating gether is pressure from the outside, pressure from 564
something of x alone but as relating x to something else. So matter (the ether) that is distributed everywhere in
this ·‘ideal’· space belongs not to the •existence of things the universe. What leads this matter to exert this
but merely to the •fixing of concepts; so no empty space pressure is gravitation, this being a basic attraction
·on this pattern· exists. (2) in the context of dynamics, that all matter exerts.
empty space is space that isn’t filled, i.e. space in which If this is how things stand, then empty space within portions
things move without being resisted by other things, i.e. a of matter would be impossible—not •logically but •dynamically
space in which no repelling force acts. Such a space might impossible, and therefore physically impossible. Why? Be-
be either •empty space within cause in this state of affairs every portion of matter would

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Metaphysical Foundations of Natural Immanuel 4: Foundations of

expand into the empty spaces assumed to be within it (be- same fate as every other attempt by •reason to get back to
cause there’s nothing here to resist such expansion), so that the principles of the first causes of things. •It fails in these
those spaces would always be kept filled up. As for (b) an attempts because it brings to them its own nature, which
empty space outside of the world (i.e. outside the totality is such that the only things •it can grasp are ones that are
of . . . .large heavenly bodies) would be impossible for the specified as satisfying certain conditions, and yet it can never 565
very same reasons. ·In the scenario we are exploring ·, these be satisfied with anything conditioned. When it is gripped by
large bodies are surrounded by ether which, driven by the a thirst for knowledge that invites it to reach for the absolute
attractive force, presses in on the stars and maintains them totality of all conditions, all it can do is to turn back from
in their density. The further any portion of this ether is objects to itself in order to investigate and determine the
from the star-totality that we are calling ‘the world’, the less ultimate boundary of its powers, instead of investigating and
dense it is; this lessening of density continues ad infinitum determining the ultimate boundary of things.
as the distance grows; but it never gets to the point where
the density is zero and that portion of space is therefore *****
empty. [Kant does not try to explain why the density of portions of
ether is proportional to their distance from ‘the world’.] Don’t be ·THE PASSAGE THAT CREATED A DEFEAT ON PAGE 15·
surprised that this elimination of empty space is in the First case: Two motions in one and the same line and direc-
meantime entirely hypothetical; the assertion that there is tion belong to one and the same point.
empty space doesn’t fare any better! Those who venture Two speeds AB and ab are to be represented as contained
to decide this controversial question dogmatically, whether in one speed of motion. Let these speeds be assumed to be
for empty space or against it, basically rely on nothing but equal for the moment, so that AB = ab; then I say that they
metaphysical suppositions, as you’ll have noticed in the can’t be represented at the same time in one and the same
dynamics; and I had at least to show here that the question space (whether absolute or relative) in one and the same
can’t be answered by metaphysics. (3) Concerning empty point. For, since the lines AB and ab designating the speeds
space in a mechanical sense—i.e. the ·supposed· emptiness are, properly speaking, the spaces they traverse in equal
accumulated in the universe to provide the heavenly bodies times, the composition of these spaces AB and ab = BC, and
with room to move—it is obvious that the possibility or hence the line AC as the sum of the spaces, would have to
impossibility of this doesn’t rest on metaphysical grounds express the sum of the two speeds. But neither the part AB
but on Nature’s secrets (so hard to unravel!) concerning how nor the part BC represents the speed = ab, for they are not
matter sets limits to its own force of extension. . . . traversed in the same time as ab. Therefore, the doubled
line AC, traversed in the same time as the line ab, does not
*****
represent the twofold speed of the latter, which, however,
This brings us to the end of the metaphysical doctrine was required. Therefore, the composition of two speeds in
of body, and we end with the empty and therefore with the one direction cannot be represented intuitively in the same
inconceivable! On this topic, the doctrine of body meets the space.

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Metaphysical Foundations of Natural Immanuel 4: Foundations of

***
·AND THEN IN FRIEDMAN’S:·
·THE PASSAGE THAT CREATED A DEFEAT ON PAGE 51, FIRST Now since the inherent motion of matter is a predicate that
IN E LLINGTON ’ S TRANSLATION : · determines its subject (the movable), and indicates in a mat-
ter, as an aggregate of movables, a plurality of the subjects
Now, the proper motion of matter is a predicate which
moved (at the same speed and in the same way), which is
determines such motion’s subject (the movable) and with
not the case for dynamical properties, whose magnitude can
regard to matter as a multitude of movable parts indicates
also be that of the action of a single subject (where an air
the plurality of the moved subjects (at equal velocity in
particle, for example, can have more or less elasticity); it
the same direction); this is not the case with dynamical
therefore becomes clear how the quantity of substance in
properties, whose quantity can also be the quantity of the
a matter has to be estimated mechanically only, that is, by
action of a single subject (e.g., a particle of air can have
the quantity of its own inherent motion, and not dynami-
more or less elasticity). Because of all of this it is clear
cally, by that of the original moving forces. Nevertheless,
that the quantity of substance in a matter must be estimated
original attraction, as the cause of universal gravitation,
mechanically, i.e., by the quantity of the proper motion of the
can still yield a measure of the quantity of matter, and of its
matter, and not dynamically, by the quantity of its original
substance (as actually happens in the comparison of matters
moving forces. Nevertheless, original attraction as the cause
by weighing), even though a dynamical measure—namely,
of universal gravitation can indeed provide a measure of the
attractive force—seems here to be the basis, rather than the
quantity of matter and its substance (as actually happens
attracting matter’s own inherent motion. But since, in the
in the comparison of matters by weighing), although there
case of this force, the action of a matter with all its parts is
seems to be laid at the foundation here not the proper motion
exerted immediately on all parts of another, and hence (at
of the attracting matter but a dynamical measure, namely,
equal distances) is obviously proportional to the aggregate
attractive force. But in the case of this force, the action of
of the parts, the attracting body also thereby imparts to
one matter occurs with all its parts directly on all parts of
itself a speed of its own inherent motion (by the resistance of
another matter; and hence the action is (at equal distances)
the attracted body), which, in like external circumstances,
obviously proportional to the number of the parts. Because
is exactly proportional to the aggregate of its parts; so the
of this fact the attracting body itself thereby also imparts
estimation here is still in fact mechanical, although only
the velocity of its proper motion (by means of the resistance
indirectly so.
of the attracted body). This velocity is directly proportional,
in equivalent external circumstances, to the number of the
attracting body’s parts; because of this the estimation takes
place here, as a matter of fact, mechanically, although only
indirectly so.