Norman Kemp Smith.

A commentary to Kant's 'Critique of pure reason,' online

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limited space presupposes the representation of space as a
whole. Both partial and infinite space are of mental origin ;
sensation, as such, is non- spatial, purely subjective. And
lastly, the fact that Kant means to assert that space is not
only logically presupposed but is subjectively generated, is
sufficiently borne out by hr frequent employment elsewhere
in the Aesthetic of such phrases as " the subjective condition
of sensibility," " lying ready in our minds," and "necessarily
preceding [as the form of the subject's receptivity] all intuitions
of objects."

Second Argument. Having proved by the first "argument
that the representation of space is not of empirical origin,
Kant in the second argument proceeds to establish the posi-
tive conclusion that it is a priori}- The proof, when all its
assumptions are rendered explicit, runs as follows. Thesis :
Space is a necessary representation, and consequently is
a priori. Proof : It is impossible to imagine the absence of
space, though it is possible to imagine it as existing without
objects to fill it. A representation which it is impossible for
the mind to be without is a necessary representation. But
necessity is one of the two criteria of the a priori. The proof
of the necessary character of space is therefore also a proof
of its being a priori.

The argument, more freely stated, is that what is em-
pirically given from without can be thought away, and that
since space cannot be thus eliminated, it must be grounded
in our subjective organisation, i.e. must be psychologically a
priori. The argument, as stated by Kant, emphasises the
apriority, not the subjectivity, of space, but none the less
the asserted apriority is psychological, not logical in character. ,
For the criterion employed is not the impossibility of thinking
otherwise, but our incapacity to represent this specific element
as absent. The ground upon which the whole argument is
made to rest is the merely brute fact (asserted by Kant) of
our incapacity to think except in terms of space.

The argument is, however, complicated by the drawing
of a further consequence, which follows as a corollary from

1 This second argument is not in the Dissertation.


the main conclusion. From the subjective necessity of space
follows its objective necessity. Space being necessary a
priori, objects can only be apprehended in and through it.
Consequently it is not dependent upon the objects appre-
hended, but itself underlies outer appearances as the condition
of their possibility. This corollary is closely akin to the first
argument on space, and differs from it only in orientation.
The first argument has a psychological purpose. It maintains
that the representation of space precedes external experience,
causally conditioning it. The corollary has a more objective
aim. It concludes that space is a necessary constituent of
the external experience thus generated. The one proves
that space is a necessary subjective antecedent ; the other that
it is a necessary objective ingredient!

To consider the proof in detail. The exact words which
Kant employs in stating the nervus probandi of the argument
are that we can never represent (erne Vorstellung davon macheri)
space as non-existent, though we can very well think (denken)
it as being empty of objects. The terms Vorstellung and
denken are vague and misleading. Kant himself recognises
that it is possible to conceive that there are beings who intuit
objects in some other manner than in space. He cannot
therefore mean that we are unable to think or conceive space
as non-existent. He must mean that we cannot in imagina-
tion intuit it as absent. It is the necessary form of all our
intuitions, and therefore also of imagination, which is intuitive
in character. Our consciousness is dependent upon given
intuitions for its whole content, and to that extent space is
a form with which the mind can never by any possibility
dispense. Pure thought enables it to realise this de facto
limitation, but not to break free from it. Even in admitting
the possibility of other beings who are not thus constituted,
the mind still recognises its own ineluctable limitations.

Kant offers no proof of his assertion that space can be
intuited in image as empty of all sensible content ; and as a
matter of fact the assertion is false. Doubtless the use of
the vague term Vorstellung is in great part responsible for
Kant's mistaken position. So long as imagination and
thought are not clearly distinguished, the assertion is corre-
spondingly indefinite. Pure space may possibly be conceived,
but it can also be conceived as altogether non-existent. If,
on the other hand, our imaginative power is alone in question,

1 Cf. Vaihinger, ii. pp. 196-7. The corresponding argument on time, in the
form in which it is given in the second edition, is, as we shall find, seriously mis-
leading. It has caused Herbart and others to misinterpret the connection in which
this corollary stands to the main thesis. Herbart's interpretation is considered
below, p. 124.


the asserted fact must be categorically denied. With the
elimination of all sensible content space itself ceases to be a
possible image. Kant's proof thus rests upon a misstatement
of fact.

In a second respect Kant's proof is open to criticism. He
takes the impossibility of imagining space as absent as proof
that it originates from within. The argument is valid only
if no other psychological explanation can be given of this
necessity, as for instance through indissoluble association or
through its being an invariable element in the given sensations.
Kant's ignoring of these possibilities is due to his unquestion-
ing belief that sensations are non-spatial, purely qualitative.
That is a presupposition whose truth is necessary to the
cogency of the argument.

Third Argument. This argument, which was omitted in
the second edition, will be considered in its connection with
the transcendental exposition into which it was then merged.

Fourth (in second edition, Third) Argument. The next two
arguments seek to show that space is not a discursive or
general concept but an intuition. The first proof falls into
two parts, (a) We can represent only a single space. For
though we speak of many spaces, we mean only parts of one
and the same single space. Space must therefore be an
intuition. For only intuition is thus directly related to a
single individual. A concept always refers indirectly, per
notas communes, to a plurality of individuals, (b) The parts
of space cannot precede the one all -comprehensive space.
They can be thought only in and through it. They arise
through limitation of it. Now the parts (i.e. the attributes)
which compose a concept precede it in thought. Through
combination of them the concept is formed. Space cannot,
therefore, be a concept. Consequently it must, as the only
remaining alternative, be an intuition. Only in an intuition
does the whole precede the parts. In a concept the parts
always precede the whole. Intuition stands for multiplicity
in unity, conception for unity in multiplicity.

The first part of the argument refers to the extension, the
second part to the intension of the space representation. In
both aspects it appears as intuitional. 1

Kant, in repeating his thesis as a conclusion from the
above grounds, confuses the reader by an addition which is
not strictly relevant to the argument, viz. by the state-
ment that this intuition must be non-empirical and a priori.
This is simply a recapitulation of what has been established
in the preceding proofs. It is not, as might at first sight

1 Cf. Vaihinger, ii. p. 220.


appear, part of the conclusion established by the argument
under consideration. The reader is the more apt to be misled
owing to the fact that very obviously arguments for the non-
empirical and for the a priori character of space can be
derived from proof (b). That space is non-empirical would
follow from the fact that representation of space as a whole
is necessary for the apprehension of any part of it. Empirical
intuition can only yield the apprehension of a limited space.
The apprehension of the comprehensive space within which
it falls must therefore be non-empirical.

"As we intuitively apprehend (anschauend erkenneji) not only the
space of the object which affects our senses, but the whole space,
space cannot arise out of the actual affection of the senses, but must
precede it in time (vor ihr vorhergehen)" x

But in spite of its forcibleness this argument is nowhere
presented in the Critique.

Similarly, in so far as particular spaces can be conceived
only in and through space as a whole, and in so far as the
former are limitations of the one antecedent space, the in-
tuition which underlies all external perception must be a
priori. This is in essentials a stronger and more cogent mode
of formulating the second argument on space. But again, and
very strangely, it is nowhere employed by Kant in this form.

The concluding sentence, ambiguously introduced by the
words so werden auch, is tacked on to the preceding argument.
Interpreted in the light of 15 C of the Dissertation? and of
the corresponding fourth 3 argument 4 on time, it may be
taken as offering further proof that space is an intuition.
The concepts of line and triangle, however attentively con-
templated, will never reveal the proposition that in every
triangle two sides taken together are greater than the third.
An a priori intuition will alone account for such apodictic
knowledge. This concluding sentence thus really belongs to
the transcendental exposition ; and as such ought, like the
third argument, to have been omitted in the second edition.

Kant's proof rests on the assumption that there are only -
two kinds of representation, intuitions and concepts, and also
in equal degree upon the further assumption that all concepts

1 Rejlexioncn, ii. 403.

2 " That in space there are no more than three dimensions, that between two
points there can be but one straight line, that in a plane surface from a given
point with a given straight line a circle is describable, cannot be inferred from
any universal notion of space, but can only be discerned in space as in the
concrete." Cf. also Prolegomena, g 12.

3 In the second edition, the third.

4 J or a different view cf. Vaihinger, ii. p. 233.


are of one and the same type. 1 Intuition is, for Kant, the
apprehension of an individual. Conception is always the
representation of a class or genus. Intuition is immediately re-
lated to the individual. Conception is reflective or discursive ;
it apprehends a plurality of objects indirectly through the
representation of those marks which are common to them all. 2
Intuition and conception having been defined in this manner,
the proof that space is single or individual, and that in it
the whole precedes the parts, is proof conclusive that it is an
intuition, not a conception. Owing, however, to the narrow-
ness of the field assigned to conception, the realm occupied
by intuition is proportionately wide, and the conclusion is not
as definite and as important as might at first sight appear.
By itself, it amounts merely to the statement, which no one
need challenge, that space is not a generic class concept.
Incidentally certain unique characteristics of space are, indeed,
forcibly illustrated ; but the implied conclusion that space on
account of these characteristics must belong to receptivity, not
to understanding, does not by any means follow. It has not,
for instance, been proved that space and time are radically
distinct from the categories, i.e. from the relational forms of

In 1770, while Kant still held to the metaphysical validity
of the pure forms of thought, the many difficulties which result
from the ascription of independent reality to space and time
were, doubtless, a sufficient reason for regarding the latter as
subjective and sensuous. But upon adoption of the Critical
standpoint such argument is no longer valid. If all our forms
of thought may be subjective, the existence of antinomies has
no real bearing upon the question whether space and time do
or do not have a different constitution and a different mental
origin from the categories. The antinomies, that is to say,
may perhaps suffice to prove that space and time are subjec-
tive ; they certainly do not establish their sensuous character.

But though persistence of the older, un-Critical opposi-
tion between the intellectual and the sensuous was partly
responsible for Kant's readiness to regard as radical the very
obvious differences between a category such as that of sub-
stance and attribute and the visual or tactual extendedness
with which objects are endowed, it can hardly be viewed as
the really decisive influence. That would rather seem to be
traceable to Kant's conviction that mathematical knowledge is
unique both in fruitfulness and in certainty, and to his further
belief that it owes this distinction to the content character of

1 Cf. above, pp. 99-100; below, pp. 126, 180-1, 184, 338-9.

- Cf. below, p. 180.


the a priori forms upon which it rests. For though the cate-
gories of the physical sciences are likewise a priori, they are
exclusively relational^ and serve only to organise a material
that is empirically given. To account for the superiority of
mathematical knowledge Kant accordingly felt constrained to
regard space and time as not merely forms in terms of which
we interpret the matter of sense, but as also themselves
intuited objects, and as therefore possessing a character
altogether different from anything which can be ascribed to
the pure understanding. The opposition between forms of
sense and categories of the understanding, in the strict
Kantian mode of envisaging that opposition, is thus insepar-
ably bound up with Kant's doctrine of space and time as
being not only forms of intuition, but as also in their purity
and independence themselves intuitions. Even the sensuous
subject matter of pure mathematics so Kant would seem to
contend is a priori in nature. If this latter view be questioned
and to the modern reader it is indeed a stone of stumbling
much of the teaching of the Aesthetic will have to be
modified or at least restated.

Fifth (in second edition, Fourth) Argument. This argument
is quite differently stated in the two editions of the Critique,
though the purpose of the argument is again in both cases
to prove that space is an intuition, not a general concept.
In the first edition this is proved by reference to the fact
that space is given as an infinite magnitude. This character-
istic of our space representation cannot be accounted for so
long as it is regarded as a concept. A general conception
of space which would abstract out those properties and
relations which are common to all spaces, to a foot as
well as to an ell, could not possibly determine anything in
regard to magnitude. For since spaces differ in magnitude,
any one magnitude cannot be a common quality. Space is,
however, given us as determined in magnitude, namely, a<
being of infinite magnitude ; and if a general conception o:
space relations cannot determine magnitude, still less can ii
determine infinite magnitude. Such infinity must be derivec
from limitlessness in the progression of intuition. Our con
ceptual representations of infinite magnitude must be deriva
tive products, acquired from this intuitive source.

In the argument of the second edition the thesis is agaii
established by reference to the infinity of space. But in al
other respects the argument differs from that of the firs
edition. A general conception, which abstracts out commoi

1 Cf. above, p. xxxvi ; below, pp. 176 ff., 191, 195-6, 257, 290-1, 404 ff.


qualities from a plurality of particulars, contains an infinite
number of possible different representations under it ; but it
cannot be thought as containing an infinite number of repre-
sentations in it. Space must, however, be thought in this
latter manner, for it contains an infinite number of coexisting
parts. 1 Since, then, space cannot be a concept, it must be an

The definiteness of this conclusion is somewhat obscured
by the further characterisation of the intuition of space as a
priori, and by the statement that it is the original (ursprnng-
liche) representation which is of this intuitive nature. The
first addition must here, again, just as in the fourth argu-
ment, be regarded as merely a recapitulation of what has
already been established, not a conclusion from the present
argument. The introduction of the word ' original ' seems to
be part of Kant's reply to the objections which had already
been made to his admission in the first edition that there is
a conception as well as an intuition of space. It is the original
given intuition of space which renders such reflective concep-
tion possible.

The chief difficulty of these proofs arises out of the
assertion which they seem to involve that space is given as
actually infinite. There are apparently, on this point, two
views in Kant, which were retained up to the very last, and
which are closely connected with his two representations of
space, on the one hand as a formal intuition given in its purity
and in its completeness, and on the other hand as the form of
intuition, which exists only so far as it is constructed, and
which is dependent for its content upon given matter.

Third Argument, and Transcendental Exposition of Space.
The distinction between the metaphysical and the transcend-
ental expositions, introduced in the second edition of the
Critique? is one which Kant seems to have first made clear
to himself in the process of writing the Prolegomena? It is
a genuine improvement, marking an important distinction.
It separates out two comparatively independent lines of
irgument. The terms in which the distinction is stated are
lot, however, felicitous. Kant's reason for adopting the title
metaphysical is indicated in the Prolegomena : 4

"As concerns the sources of metaphysical cognition, its very
concept implies that they cannot be empirical. . . . For it must not

1 This statement occurs in a parenthesis ; it has already been dwelt upon in
he fourth (third) argument.

2 It has led Kant to substitute erortern for betrachten in A 23 = B 38.

3 Cf. Vaihinger, ii. p. 151.

4 i (Eng. trans, p. 13). Cf. above, p. 64.


be physical but metaphysical knowledge, i.e. knowledge lying beyond
experience. ... It is therefore a priori knowledge, coming from
pure understanding and pure Reason."

The metaphysical exposition, it would therefore seem, is
so entitled because it professes to prove that space is a priori,
not empirical, and to do so by analysis of its concept. 1 Now
by Kant's own definition of the term transcendental, as the
theory of the a priori, this exposition might equally well have
been named the transcendental exposition. In any case it is
an essential and chief part of the Transcendental Aesthetic.
Such division of the Transcendental Aesthetic into a meta-
physical and a transcendental part involves a twofold use,
wider and narrower, of one and the same term. Only as
descriptive of the whole Aesthetic is transcendental employed
in the sense defined.

Exposition (Erorterung, Lat. expositio) is Kant's substitute
for the more ordinary term definition. Definition is the term
which we should naturally have expected ; but as Kant
holds that no given concept, whether a priori or empirical,
can be defined in the strict sense, 2 he substitutes the term
exposition, using it to signify such definition of the nature
of space as is possible to us. To complete the parallelism
Kant speaks of the transcendental enquiry as also an ex-
position. It is, however, in no sense a definition. Kant's
terms here, as so often elsewhere, are employed in a more
or less arbitrary and extremely inexact manner.

The distinction between the two expositions is taken by
Kant as follows. The metaphysical exposition determines
the nature of the concept of space, and shows it to be a
given a priori intuition. The transcendental exposition
shows how space, when viewed in this manner, renders com-
prehensible the possibility of synthetic a priori knowledge.

The omission of the third argument on space from the
second edition, and its incorporation into the new transcend-
ental exposition, is certainly an improvement. In its location
in the first edition, it breaks in upon the continuity of Kant's
argument without in any way contributing to the further
definition of the concept of space. Also, in emphasising that

1 This is, no doubt, one reason why Kant employs, in reference to space, the
unfortunate and confusing term concept (Begriff} in place of the wider term repre-
sentation ( Vorstellung). Cf. B 37, and above, p. 64.

2 Cf. A 729 = 6 757 : "In place of the term definition I should prefer to
employ the term exposition. For that is a more guarded expression, the claims of
which the critic may allow as being in a certain degree valid even though he
entertain doubts as to the completeness of the analysis." Cf. Logic, 99 ff., 105.
Cf. also Untersuchung iiber die Deutlichkeit der Grundsatze, IV. ii. pp. 183-4 :
" Augustine has said, ' I know well what time is, but if any one asks me, I
cannot tell.'"


mathematical knowledge depends upon the construction of
concepts, 1 Kant presupposes that space is intuitional ; and
that has not yet been established.

The argument follows the strict, rigorous, synthetic method.
From the already demonstrated a priori character of space,
Kant deduces the apodictic certainty of all geometrical prin-
ciples. But though the paragraph thus expounds a conse-
quence that follows from the a priori character of space,
not an argument in support of it, something in the nature
of an argument is none the less implied. The fact that this
view of the representation of space alone renders mathematical
science possible can be taken as confirming this interpreta-
tion of its nature. Such an argument, though circular, is none
the less cogent. Consideration of Kant's further statements,
that were space known in a merely empirical manner we
could not be sure that in all cases only one straight line is
possible between two points, or that space will always be
found to have three dimensions, must meantime be deferred. 2

In the new transcendental exposition Kant adopts the
analytic method of the Prolegomena, and accordingly presents
his argument in independence of the results already established.
He starts from the assumption of the admitted validity of
geometry, as being a body of synthetic a priori knowledge.
Yet this, as we have already noted, does not invalidate the
argument ; in both the first and the last paragraphs it is
implied that the a priori and intuitive characteristics of space
have already been proved. From the synthetic character of
geometrical propositions Kant argues 3 that space must be an
intuition. Through pure concepts no synthetic knowledge is
possible. Then from the apodictic character of geometry he
infers that space exists in us as pure and a priori \^ no
experience can ever reveal necessity. But geometry also
exists as an applied science ; and to account for our power
of anticipating experience, we must view space as existing
only in the perceiving subject as the form of its sensibility.
If it precedes objects as the necessary subjective condition of
their apprehension, we can to that extent predetermine the
conditions of their existence.

In the concluding paragraph Kant says that this is the
only explanation which can be given of the possibility of
geometry. He does not distinguish between pure and applied

1 For explanation of the phrase "construction of concepts " cf. below, pp. nz-T

2 Cf. below, p. ii;ff.

3 Cf. conclusion of fourth argument on space.

4 A priori is here employed in its ambiguous double sense, as a priori in so
far as it precedes experience (as a representation), and in so far as it is valid
independently of experience (as a proposition}. Cf. Vaihinger, ii. p. 268.


geometry, though the proof which he has given of each differs
in a fundamental respect. Pure geometry presupposes only
that space is an a priori intuition ; applied geometry demands

Online LibraryNorman Kemp SmithA commentary to Kant's 'Critique of pure reason,' → online text (page 17 of 72)