Norman Kemp Smith.

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

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upon the nature of those changes (of quantity) which are possible
only in connection with a specific property of inner sense and its
form (time). The scie?ice of number, notwithstanding the succession
which every construction of quantity demands, is a pure intellectual
synthesis which we represent to ourselves in thought. But so far as
quanta are to be numerically determined, they must be given to us

1 Cf. Vaihinger, ii. pp. 388-9.
IVerke (Frauenstadt's ed., 1873), i- P- I 33-
3 P. 129.


in such a way that we can apprehend their intuition in successive
order, and such that their apprehension can be subject to time. . . ." l

No more definite statement could be desired of the fact
that though in arithmetical science as in other fields of study
our processes of apprehension are subject to time, the quanti-
tative relations determined by the science are independent of
time and are intellectually apprehended.

But if the above psychological interpretation of Kant's
teaching is untenable, how is his position to be defined ? We
must bear in mind the doctrine which Kant had already
developed in his pre-Critical period, that mathematical differs
from philosophical knowledge in that its concepts can have
concrete individual form. 2 In the Critique this difference is
expressed in the statement that the mathematical sciences
alone are able to construct their concepts. And as they are
pure mathematical sciences, this construction is supposed to
take place by means of the a priori manifold of space and of
time. Now though Kant had a fairly definite notion of what
he meant by the construction of geometrical figures in space,
his various utterances seem to show that in regard to the
nature of arithmetical and algebraic construction he had never
really attempted to arrive at any precision of view. To
judge by the passage already quoted 3 from the Dissertation^
Kant regarded space as no less necessary than time to
the construction or intuition of number. " [The intellectual
concept of number] demands for its concrete actualisation
the auxiliary notions of time and space (in the successive
addition and in the juxtaposition of a plurality')" A similar
view appears in the Critique in A 140 = 6 179 and in B 15.
In conformity, however, with the general requirements of his
doctrine of Schematism, Kant defines the schema of number
in exclusive reference to time ; and, as we have noted, it is to
this definition that Schulze appeals in support of his view of
arithmetic as the science of counting and therefore of time.
It at least shows that Kant perceived some form of connection
to exist between arithmetic and time. But in this matter
Kant's position was probably simply a corollary from his
general view of the nature of mathematical science, and in
particular of his view of geometry, the " exemplar" 4 of all
the others. Mathematical science, as such, is based on in-
tuition ; 5 therefore arithmetic, which is one of its departments,

1 W. x. p. 530. Italics not in Kant.

2 Untersuchung iiber die Deutlichkeit der Grundsatze : Erste Betrachtimg, 2,
3 ; dritte Betrachtung, I ; Dissertation, 12, 15 C.

3 P. 128. * Dissertation, 15 C.
5 Cf. above, pp. 40-2, 118-20 ; below, pp. 338-9.


must be so likewise. No attempt, however, is made to define
the nature of the intuitions in which it has its source.
Sympathetically interpreted, his statements may be taken as
suggesting that arithmetic is the study of series which find
concrete expression in the order of sequent times. The follow-
ing estimate, given by Cassirer, 1 does ample justice both to
the true and to the false elements in Kant's doctrine.

"[Even discounting Kant's insistence upon the conceptual
character of arithmetical science, and] allowing that he derives
arithmetical concepts and propositions from the pure intuition of 'time ;
this teaching, to whatever objections it may lie open, has certainly
not the merely psychological meaning which the majority of its critics
have ascribed to it. If it contained only the trivial thought, that the
empirical act of counting requires time, it would be completely refuted
by the familiar objection which B. Beneke has formulated: 'The
fact that time elapses in the process of counting can prove nothing ;
for what is there over which time does not flow ? ' It is easily seen
that Kant is only concerned with the ' transcendental ' determination
of the concept of time, according to which it appears as the type of
an ordered sequence. William [Rowan] Hamilton, who adopts
Kant's doctrine, has defined algebra as ' science of pure time or order
in progression? That the whole content of arithmetical concepts can
really be obtained from the fundamental concept of order in unbroken
development, is completely confirmed by Russell's exposition. As
against the Kantian theory it must, of course, be emphasised, that it
is not the concrete form of time intuition which constitutes the ground
of the concept of number, but that on the contrary the pure logical
concepts of sequence and of order are already implicitly contained
and embodied in that concrete form."

Much of the unsatisfactoriness of Kant's argument is
traceable to his mode of conceiving the "construction" 2 of
mathematical concepts. All concepts, he seems to hold, even
those of geometry and arithmetic, are abstract class concepts
the concept of triangle representing the properties common
to all triangles, and the concept of seven the properties
common to all groups that are seven. Mathematical con-
cepts differ, however, from other concepts in that they are
capable of a priori construction, that is, of having their
objects represented in pure intuition. Now this is an ex-
tremely unfortunate mode of statement. It implies that
mathematical concepts have a dual mode of existence, first
as abstracted, and secondly as constructed. Such a position
is not tenable. The concept of seven, in its primary form,
is not abstracted from a variety of particular groups of

1 Kant und die moderne Mathematik in Kant-Studien^ xii. (1907) p. 34 n.

2 Cf. A 713 ff. =B 741 ff. ; A4 = B 8; B 15-16; A 24; A 47-8 = 6 64-5.


seven ; it is already involved in the apprehension of each of
them as being seven. Nor is it a concept that is itself con-
structed. It may perhaps be described as being the repre-
sentation of something constructed ; but that something is
not itself. It represents the process or method generative of
the complex for which it stands. Thus Kant's distinction
between the intuitive nature of mathematical knowledge and
the merely discursive character of conceptual knowledge is at
once inspired by the very important distinction between the
product of construction and the product of abstraction, and
yet at the same time is also obscured by the quite inadequate
manner in which that latter distinction has been formulated.
Kant has again adhered to the older logic even in the very
act of revising its conclusions ; and in so doing he has sacri-
ficed the Critical doctrines of the Analytic to the pre- Critical
teaching of the Dissertation and Aesthetic. Mathematical
concepts are of the same general type as the categories ; their
primary function is not to clarify intuitions^ but to make them
possible. They are derivable from intuition only in so far
as they have contributed to its constitution. If intuition
contains factors additional to the concepts through which it
is interpreted, these factors must remain outside the realm
of mathematical science, until such time as conceptual analysis
has proved itself capable of further extension.

I may now summarise this general discussion. Though
Kant in the first edition of the Critique had spoken of the
mathematical sciences as based upon the intuition of space
and time, he had not, despite his constant tendency to conceive
space and time as parallel forms of existence, based any
separate mathematical discipline upon time. His definition
of number, in the chapter on Schematism, had recognised the
essentially conceptual character of arithmetic, and had con-
nected it with time only in a quite indirect manner. A
passage in the Prolegomena is the one place in all Kant's
writings in which he would seem to assert, though in brief
and quite indefinite terms, that arithmetic is related to time
as geometry is related to space. No such view of arithmetic
is to be found in the second edition of the Critique. In the
transcendental exposition of time, added in the second edition,
only pure mechanics is mentioned. This would seem to
indicate that Kant had made the above statement carelessly,
without due thought, and that on further reflection he found
himself unable to stand by it. The omission is the more
significant in that Kant refers to arithmetic in the passages
added in the second edition Introduction. The teaching of
these passages, apart from the asserted necessity of appealing


to fingers or points, 1 harmonises with the view so briefly out-
lined in the Analytic. Arithmetic is a conceptual science ;
though it finds in ordered sequence its intuitional material,
it cannot be adequately defined as being the science of time.


These Conclusions do not run parallel with the correspond-
ing Conclusions in regard to space. In the first paragraph
there are two differences, (a) Kant takes account of a view
not considered under space, viz. that time is a self-existing
substance. He rejects it on a ground which is difficult to
reconcile with his recognition of a manifold of intuition as
well as a manifold of sense, namely that it would then be
something real without being a real object. In A 39 =B 57
and B 70 Kant describes space and time, so conceived, as
unendliche Undinge. (b) Kant introduces into his first Con-
clusion the argument 3 that only by conceiving time as the
form of inner intuition can we justify a priori synthetic
judgments in regard to objects.

Second Paragraph (Conclusion b). This latter statement is
repeated at the opening of the second Conclusion. The
emphasis is no longer, however, upon the term " form " but
upon the term " inner " ; and Kant proceeds to make asser-
tions which by no means follow from the five arguments,
and which must be counted amongst the most difficult and
controversial tenets of the whole Critique, (a) Time is not a
determination of outer appearances. For it belongs neither
to their shape nor to their position and prudently at this
point the property of motion is smuggled out of view under
cover of an etc. Time does not determine the relation of
appearances to one another, but only the relation of representa-
tions in our inner stated It is the form only of the intuition
of ourselves and of our inner state. 5 Obviously these are
assertions which Kant cannot possibly hold to in this un-
qualified form. In the very next paragraph they are modified
and restated, (b) As this inner intuition supplies no shape
(Gestalt\ we seek to make good this deficiency by means of
analogies. We represent the time-sequence through a line

1 Cf. below, pp. 337-8.

2 Cf. above, pp. 112 ;;. 4.

3 The content of the second Conclusion in regard to space.

4 This expresses the matter a little more clearly than Kant himself does.
The term representation is ambiguous. In the first paragraph it is made to cover
the appearances as well as their representation.

6 Cf. Dissertation, 15 Coroll. : "Space properly concerns the intuition of the
object ; time the state, especially the representative state."


progressing to infinity in which the manifold constitutes a
series of only one dimension. From the properties of this
line, with the one exception that its parts are simultaneous
whereas those of time are always successive, we conclude to
all the properties of time.

The wording of the passage seems to imply that such
symbolisation of time through space is helpful but not in-
dispensably necessary for its apprehension. That it is indis-
pensably necessary is, however, the view to which Kant finally
settled down. 1 But he has not yet come to clearness on this
point. The passage has all the signs of having been written
prior to the Analytic. Though Kant seems to have held
consistently to the view that time has, in or by itself, only one
dimension, 2 the difficulties involved drove him to recognise that
this is true only of time as the order of our representations. It
is not true of the objective time apprehended in and through
our representations. When later Kant came to hold that
consciousness of time is conditioned by consciousness of space,
he apparently also adopted the view that, by reference to space,
time indirectly acquires simultaneity as an additional mode.
The objective spatial world is in time, but in a time which
shows simultaneity as well as succession. In the Dissertation 3
Kant had criticised Leibniz and his followers for neglecting
simultaneity, "the most important consequence of time."

" Though time has only one dimension, yet the ubiquity of time
(to employ Newton's term), through which all things sensuously
thinkable are at some time, adds another dimension to the quantity of
actual things, in so far as they hang, as it were, upon the same point
of time. For if we represent time by a straight line extended to
infinity, and simultaneous things at any point of time by lines
successively erected [perpendicular to the first line], the surface thus
generated will represent the phenomenal world both as to substance
and as to accidents."

Similarly in A 182 = B 226 of the Critique Kant states that
simultaneity is not a mode of time, 4 since none of the parts
of time can be simultaneous, and yet also teaches in A 177 =
B 219 that, as the order of appearances, time possesses in addition
to succession the two modes, duration and simultaneity. The
significance of this distinction between time as the order of our
inner states, and time as the order of objective appearances,
we shall consider immediately.

A connected question is as to whether or not Kant teaches
the possibility of simultaneous apprehension. In the Aesthetic

1 Cf. below, pp. 309 ff., 347-8, 359. 2 Cf. Reflexionem, ii. 365 ff.

3 14, 5 and note to 5. 4 The opposite is, however, asserted in B 67.


and Dialectic he certainly does so. Space is given as con-
taining coexisting parts, and 1 can be intuited as such without
successive synthesis of its parts. In the Analytic^ on the
other hand, the opposite would seem to be implied. 2 The
apprehension of a manifold can only be obtained through the
successive addition or generation of its parts.

(c) Lastly, Kant argues that the fact that all the relations
of time can be expressed in an outer intuition is proof that the
representation of time is itself intuition. But surely if, as
Kant later taught, time can be apprehended at all only in and
through space, that, taken alone, would rather be a reason for
denying it to be itself intuition. In any case it is difficult to
follow Kant in his contention that the intuition of time is
similar in general character to that of space. 3

Third Paragraph (Conclusion c). Kant now reopens the
question as to the relation in which time stands to outer
appearances. As already noted, he has argued in the begin-
ning of the previous paragraph that it cannot be a determina-
tion of outer appearances, but only of representations in
our inner state. External appearances, however, as Kant
recognises, can be known only in and through representations.
To that extent they belong to inner sense, and consequently
(such is Kant's argument) are themselves subject to time.
Time, as the immediate condition of our representations, is also
the mediate condition of appearances. Therefore, Kant con-
cludes, " all appearances , i.e. all objects of the senses, are in
time, and necessarily stand in time-relations."

Now quite obviously this argument is invalid if the dis-
tinction between representations and their objects is a real
and genuine one. For if so, it does not at all follow that
because our representations of objects are in time that the
objects themselves are in time. In other words, the argument
is valid only from the standpoint of extreme subjectivism,
according to which objects are, in Kant's own phraseology,
blosse Vorstellungen. But the argument is employed to
establish a realist conclusion, jthat outer objects, as objects,
stand in time-relations to one another. In contradiction of
the previous paragraph he is now maintaining that time is a
determination of outer appearances, and that it reveals itself in
the motion of bodies as well as in the flux of our inner states.

1 Cf. A 427-8 n. - B 456 .

2 A 99. Cf. A 162 = B 203: "I cannot represent to myself a line, however
small, without drawing it in thought, i.e. generating from a point all its parts one
after another." Cf. pp. 94, 347-8.

3 Cf. Lose Blatter, i. 54 : " Without space time itself would not be represented
as quantity (Grosse), and in general this conception would have no object." Cf.
Dissertation, 14. 5.


The distinction between representations and their objects
also makes it possible for Kant both to assert and to deny that
simultaneity is a mode of time. " No two years can be co-
existent. Time has only one dimension. But existence (das
Daseiri], measured through time, has two dimensions, succession
and simultaneity." There are, for Kant, two orders of time, sub-
jective and objective. Recognition of the latter (emphasised
and developed in the Analytic] * is, however, irreconcilable with
his contention that time is merely the form of inner sense.

We have here one of the many objections to which
Kant's doctrine of time lies open. It is the most vulnerable
tenet in his whole system. A mere list of the points
which Kant leaves unsettled suffices to show how greatly he
was troubled in his own mind by the problems to which it
gives rise, (i) The nature of the a priori knowledge which
time yields. Kant ascribes to this source sometimes only the
two axioms in regard to time, sometimes pure mechanics, and
sometimes also arithmetic. (2) Whether time only allows
of, or whether it demands, representation through space.
Sometimes Kant makes the one assertion, sometimes the
other. (3) Whether it is possible to apprehend the coexistent
without successive synthesis of its parts. This possibility is
asserted in the Aesthetic and Dialectic, denied in the Analytic.
(4) Whether simultaneity is a mode of time. (5) Whether,
and in what manner, appearances of outer sense are in time.
Kant's answer to 4 and to 5 varies according as he identifies
or distinguishes representations and empirical objects.

The manifold difficulties to which a theory of time thus
lies open are probably the reason why Kant, in the Critique,
reverses the order in which he had treated time and space in
the Dissertation? But the placing of space before time is none
the less unfortunate. It greatly tends to conceal from the
reader the central position which Kant has assigned to time
in the Analytic. Consciousness of time is the fundamental
fact, taken as bare fact, by reference to which Kant gains his
transcendental proof of the categories and principles of under-
standing. 3 In the Analytic space, by comparison, falls very
much into the background. A further reason for the reversal
may have been Kant's Newtonian view of geometry as the
mathematical science par excellence^ In view of his formu-
lation of the Critical problem as that of accounting for
synthetic a priori judgments, he would then naturally be led
to throw more emphasis on space.

1 Cf. below, p. 365 ff. 2 In the Dissertation time is treated before space.

3 Cf. above, pp. xxxiv, 120; below, pp. 241-2, 365, 367-70, 390-1.

4 Cf. Dissertation, 15 C.


To sum up our main conclusions. Kant's view of time as-
a form merely of inner sense, and as having only one dimension, '
connects with his subjectivism. His view of it as inhering in
objects, and as having duration and simultaneity as two of its
modes, is bound up with his phenomenalism. Further dis-
cussion of these difficulties must therefore be deferred until we
are in a position to raise the more fundamental problem as to
the nature of the distinction between a representation and its
object. 1 Motion is not an inner state. Yet it involves time
as directly as does the flow of our feelings and ideas. Kant's
assertion that " time can no more be intuited externally than
space can be intuited as something in us," 2 if taken quite
literally, would involve both the subjectivist assertion that
motion of bodies is non-existent, and also the phenomenalist
contention that an extended object is altogether distinct from
a representation.

The fourth and fifth paragraphs call for no detailed
analysis. 3 Time is empirically real, transcendentally ideal
these terms having exactly the same meaning and scope as in
reference to space. 4 The fourth sentence in the fifth paragraph
is curiously inaccurate. As it stands, it would imply that
time is given through the senses. In the concluding sentences
Kant briefly summarises and applies the points raised in these
fourth and fifth paragraphs.


First and Second Paragraphs. Kant here replies to a criti-
cism which, as he tells us in his letter of 1772 to Herz, was
first made by Pastor Schulze and by Lambert. 5 In that letter
the objection and Kant's reply are stated as follows.

" In accordance with the testimony of inner sense, changes are
something real. But they are only possible on the assumption of
time. Time is, therefore, something real which belongs to the deter-
minations of things in themselves. Why, said I to myself, do we
not argue in a parallel manner: 'Bodies are real, in accordance
with the outer senses. But bodies are possible only under the
condition of space. Space is, therefore, something objective and
real which inheres in the things themselves.' The cause [of this
differential treatment of space and of time] is the observation
that in respect to outer things we cannot infer from the reality
of representations the reality of their objects, whereas in inner sense
the thought or the existing of the thought and of myself are one

1 Cf. below, pp. 272 ff., 294-5, 308 ff., 365 ff. 2 A 23 = B 37.

3 They correspond to the third paragraph dealing with space. Cf. above,
p. 116. * Cf. above, pp. 116-17.

8 Cf. W. x. p. 102. Mendelssohn had also protested ; cf. op. cit. x. p. no.


and the same. Herein lies the key to the difficulty. Undoubtedly
I must think my own state under the form of time, and the form of
the inner sensibility consequently gives me the appearance of changes.
Now I do not deny that changes are something real any more than
I deny that bodies are something real, but I thereby mean only that
something real corresponds to the appearance. I may not even say
the inner appearance undergoes change (verdndere sich\ for how could
I observe this change unless it appeared to my inner sense ? To
the objection that this leads to the conclusion that all things in the world
objectively and in themselves are unchangeable, I would reply that they
are neither changeable nor unchangeable. As Baumgarten states in 18
of his Metaphysica^ the absolutely impossible is hypothetically neither
possible nor impossible, since it cannot be mentally entertained under
any condition whatsoever; so in similar manner the things of the
world are objectively or in themselves neither in one and the same state
nor in different states at different times, for thus understood [viz. as
things in themselves] they are not represented in time at all." *

Thus Kant's contention^, both in this letter and in the
passage before us, is thatyeven our inner states would not
reveal change if they could oe apprehended by us or by some
other being apart from the subjective form of our inner sense.J
We may not say that our inner states undergo change, or that
they succeed one another, but only that to us they necessarily
appear as so doing. 2 Time is no more than subjectively real. 3
As Korner writes to Schiller : " Without time man would
indeed exist but not appear. Not his reality but only his
appearance is dependent upon the condition of time." " Man
is not, but only appears, when he undergoes change." 4 The
objects of inner sense stand in exactly the same position as

Online LibraryNorman Kemp SmithA commentary to Kant's 'Critique of pure reason,' → online text (page 20 of 72)