that space be conceived as the a priori form of external sense.
Only in reference to applied geometry does the Critical
problem arise : viz. how we can form synthetic judgments
a priori which yet are valid of objects ; or, in other words,
how judgments based upon a subjective form can be objectively
valid. But any attempt, at this point, to define the nature
and possibility of applied geometry must anticipate a result
which is first established in Conclusion b. 1 Though, therefore,
the substitution of this transcendental exposition for the third
space argument is a decided improvement, Kant, in extending
it so as to cover applied as well as pure mathematics, over-
looks the real sequence of his argument in the first edition.
The employment of the analytic method, breaking in, as it
does, upon the synthetic development of Kant's original
argument, is a further irregularity. 2
It may be noted that in the third paragraph Kant
takes the fact that geometry can be applied to objects as
proof of the subjectivity of space. 3 He refuses to recognise
the possibility that space may be subjective as a form of re-
ceptivity, and yet also be a mode in which things in them-
selves exist. This, as regards its conclusion, though not as
regards its argument, is therefore an anticipation of Conclusion
a. In the last paragraph Kant is probably referring to the
views both of Leibniz and of Berkeley.
CONCLUSIONS FROM THE ABOVE CONCEPTS 4
Conclusion a. Thesis : Space is not a property of things
in themselves, 5 nor a relation of them to one another. Proof :
The properties of things in themselves can never be intuited
prior to their existence, i.e. a priori. Space, as already proved,
is intuited in this manner. In other words, the apriority of
space is by itself sufficient proof of its subjectivity.
1 Cf. below, p. Ii4ff.
2 Cf. below, pp. 115-16.
3 Cf. Lose Blatter, i. p. 18 : " This is a proof (Beweis] that space is a sub-
jective condition. For its propositions are synthetic and through them objects
can be known a priori. This would be impossible if space were not a subjective
condition of the representation of these objects." Cf. Reflexionen, ii. p. 396, in
which this direct proof of the ideality of space is distinguished from the indirect
proof by means of the antinomies.
4 By " concepts" Kant seems to mean the five arguments, though as a matter
of fact other conclusions and presuppositions are taken into account, and quite
new points are raised.
5 This, according to Vaihinger (ii. p. 287), is the first occurrence of the phrase
Dinge an sich in Kant's writings.
SUBJECTIVITY OF SPACE 113
This argument has been the subject of a prolonged
controversy between Trendelenburg and Kuno Fischer. 1
Trendelenburg was able to prove his main point, namely, that
the above argument is quite inconclusive. Kant recognises
only two alternatives, either space as objective is known
a posteriori, or being an a priori representation it is subjective
in origin. There exists a third alternative, namely, that though
our intuition of space is subjective in origin, space is itself
an inherent property of things in themselves. The central
thesis of the rationalist philosophy of the Enlightenment
was, indeed, that the independently real can be known by a
priori thinking. Even granting the validity of Kant's later
conclusion, first drawn in the next paragraph, that space is
the subjective form of all external intuition, that would only
prove that it does not belong to appearances, prior to our
apprehension of them ; nothing is thereby proved in regard
to the character of things in themselves. We anticipate by a
priori reasoning only the nature of appearances, never the
constitution of things in themselves. Therefore space, even
though a priori, may belong to the independently real. The
above argument cannot prove the given thesis.
Vaihinger contends 2 that the reason why Kant does not
even attempt to argue in support of the principle, that the
a priori must be purely subjective, is that he accepts it
as self-evident. This explanation does not, however, seem
satisfactory. But Vaihinger supplies the data for modifica-
tion of his own assertion. It was, it would seem, the exist-
ence of the antinomies which first and chiefly led Kant to
assert the subjectivity of space and time. 3 For as he then
believed that a satisfactory solution of the antinomies is
possible only on the assumption of the subjectivity of space
and time, he regarded their subjectivity as being con-
clusively established, and accordingly failed to examine
with sufficient care the validity of his additional proof from
their apriority. This would seem to be confirmed by the
fact that when later, 4 in reply to criticisms of the argu-
ments of the first edition, he so far modified his position as
to offer reasons in support of the above general principle, even
then he nowhere discussed the principle in reference to the
forms of sense. All his discussions concern only the possible
independent reality of the forms of thought. 5 To the very
last Kant would seem to have regarded the above argument
1 Cf. Vaihinger's analysis of this discussion, ii. pp. 290-313.
2 ii. pp. 289-90. 3 Cf. below, pp. 415 ff., 515 ff., 558 ff. 4 In B 166 ff.
5 This is likewise true of the references in the letter to Herz, 2ist Feb. 1772.
Cf. below, pp. 219-20.
ii4 THE TRANSCENDENTAL AESTHETIC
as an independent, and by itself a sufficient, proof of the
subjectivity of space.
The refutation of Trendelenburg's argument which is
offered by Caird 1 is inconclusive. Caird assumes the chief
point at issue, first by ignoring the possibility that space may
be known a priori in reference to appearances and yet at the
same time be transcendently real ; and secondly by ignoring
the fact that to deny spatial properties to things in them-
selves is as great a violation of Critical principles as to
assert them. One point, however, in Caird's reply to
Trendelenburg calls for special consideration, viz. Caird's
contention that Kant did actually take account of the third
alternative, rejecting it as involving the " absurd " hypothesis
of a pre-established harmony. 2 Undoubtedly Kant did so.
But the contention has no relevancy to the point before us.
The doctrine of pre-established harmony is a metaphysical
theory which presupposes the possibility of gaining knowledge
of things in themselves. For that reason alone Kant was
bound to reject it. A metaphysical proof of the validity of
metaphysical judgments is, from the Critical point of view, a
contradiction in terms. As the validity of all speculations
is in doubt, a proof which is speculative cannot meet our
difficulties. And also, as Kant himself further points out, the
pre-established harmony, even if granted, can afford no
solution of the Critical problem how a priori judgments can
be passed upon the independently real. The judgments, thus
guaranteed, could only possess de facto validity ; we could
never be assured of their necessity. 3 It is chiefly in these
two inabilities that Kant locates the "absurdity" of a theory of
pre-established harmony. The refutation of that theory does
not, therefore, amount to a disproof of the possibility which
we are here considering.
Conclusion b. The next paragraph maintains two theses :
(a) that space is the form of all outer intuition ; (#) that this
fact explains what is otherwise entirely inexplicable and para-
doxical, namely, that we can make a priori judgments which
yet apply to the objects experienced. The first thesis, that
the pure intuition of space is only conceivable as the form
of appearances of outer sense, is propounded in the opening
sentence without argument and even without citation of
grounds. The statement thus suddenly made is not anticipated
1 The Critical Philosophy of Kant, i. pp. 306-9.
2 Cf. letter to Herz, W. x. p. 126. It is, Kant there says, the most absurd
explanation which can be offered of the origin and validity of our knowledge,
involving an illegitimate circulus in probando, and also throwing open the door to
the wildest speculations. Cf. above, p. 28; below, pp. 141-2, 290, 590.
3 Cf. B 167-8.
SPACE THE FORM OF OUTER INTUITION 115
save by the opening sentences of the section on space. 1 It is
an essentially new doctrine. Hitherto Kant has spoken of
space only as an a priori intuition. The further assertion that
as such it must necessarily be conceived as the form of outer
sense (i.e. not only as a formal intuition but also as a form of
intuition), calls for the most definite and explicit proof. None,
however, is given. It is really a conclusion from points all too
briefly cited by Kant in the general Introduction, namely, from
his distinction between the matter and the form of sense.
The assertions there made, in a somewhat casual manner,
are here, without notification to the reader, employed as
premisses to ground the above assertion. His thesis is not,
therefore, as by its face value it would seem to profess to be,
an inference from the points established in the preceding
expositions. It interprets these conclusions in the light of
points considered in the Introduction ; and thereby arrives at
a new and all-important interpretation of the nature of the
a priori intuition of space.
The second thesis employs the first to explain how prior
to all experience we can determine the relations of objects.
Since (a) space is merely the form of outer sense, and (b)
accordingly exists in the mind prior to all empirical intuition,
all appearances must exist in space, and we can predetermine
them from the pure intuition of space that is given to us a priori.
Space, when thus viewed as the a priori form of outer sense,
renders comprehensible the validity of applied mathematics.
As we have already noted, 2 Kant in the second edition
obscures the sequence of his argument by offering in
the new transcendental exposition a justification of applied
as well as of pure geometry. In so doing he anticipates the
conclusion which is first drawn in this later paragraph. This
would have been avoided had Kant given two separate tran-
scendental expositions. First, an exposition of pure mathe-
matics, placed immediately after the metaphysical exposition ;
for pure mathematics is exclusively based upon the results of
the metaphysical exposition. And secondly, an exposition
:>f applied mathematics, introduced after Conclusion b. The
explanation of applied geometry is really the more essential
md central of the two, as it alone involves the truly Critical
problem, how judgments formed a priori can yet apply to
)bjects. Conclusion b constitutes, as Vaihinger rightly insists, 3
he very heart of the Aesthetic. The arrangement of Kant's
irgument diverts the reader's attention from where it ought
)roperly to centre.
1 That is, in the first edition. Cf. above, p. 85 ff. ; and below, p. 116.
2 Above, pp. 1 1 1- 12. 3 ii. p. 335.
ii6 THE TRANSCENDENTAL AESTHETIC
The use which Kant makes of the Prolegomena in his
statement of the new transcendental exposition is one cause
of the confusion. The exposition is a brief summary of the
corresponding Prolegomena^ sections. In introducing this
summary into the Critique Kant overlooked the fact that in
referring to applied mathematics he is anticipating a point
first established in Conclusion b. The real cause, however, of
the trouble is common to both editions, namely Kant's failure
clearly to appreciate the fundamental distinction between the
view that space is an a priori intuition and the view that it is
the a priori form of all external intuition, i.e. of outer sense.
He does not seem to have fully realised how very different
are those two views. In consequence of this he fails to dis-
tinguish between the transcendental expositions of pure and
applied geometry. 2
Third paragraph. Kant proceeds to develop the subjectivist
conclusions which follow from a and b.
" We may say that space contains all things which can appear
to us externally, but not all things in themselves, whether intuited
or not, nor again all things intuited by any and every subject." 3
This sentence makes two assertions : (a) space does not
belong to things in and by themselves ; (b} space is not a
necessary form of intuition for all subjects whatsoever.
The grounds for the former assertion are not here con-
sidered, and that is doubtless the reason why the oder nicht
is excised in Kant's private copy of the Critique. As we
have seen, Kant does not anywhere in the Aesthetic even
attempt to >ffer argument in support of this assertion. In
defence of j(^j\ Kant propounds for the first time the view of
sensibility as a limitation. Space is a limiting condition to
which human intuition is subject. Whether the intuitions of
other thinking beings are subject to the same limitation, we
have no means of deciding. But for all human beings, Kant
implies, the same conditions must hold universally. 4
In the phrase "transcendental ideality of space " 5 Kant,
it may be noted, takes the term ideality as signifying subjec-
tivity, and the term transcendental as equivalent to trans-
cendent. He is stating that judged from a transcendent point
of view, i.e. from the point of view of the thing in itself, space
has a merely subjective or " empirical " reality. This is an
2 This identification of the two is especially clear in A 39 = B 56.
8 A 27 = 643.
4 Cf. above, p. xxxv ; below, pp. 117-20, 142, 185-6, 241-2, 257, 290-1.
6 A 28 = 644, cf. A35 = B 52.
KANT AND MODERN GEOMETRY 117
instance of Kant's careless use of the term transcendental.
Space is empirically real, but taken trans cendently, is merely
KANT'S ATTITUDE TO THE PROBLEMS OF
This is an appropriate point at which to consider the
consistency of Kant's teaching with modern developments
in geometry. Kant's attitude has very frequently been
misrepresented. As he here states, he is willing to recognise
that the forms of intuition possessed by other races of finite
beings may not coincide with those of the human species.
But in so doing he does not mean to assert the possibility
of other spatial forms, i.e. of spaces that are non-Euclidean.
In his pre-Critical period Kant had indeed attempted to
deduce the three-dimensional character of space as a conse-
quence of the law of gravitation ; and recognising that that
law is in itself arbitrary, he concluded that God might, by
establishing different relations of gravitation, have given rise
to spaces of different properties and dimensions.
" A science of all these possible kinds of space would un-
doubtedly be the highest enterprise which a finite understanding
could undertake in the field of geometry." 2
But from the time of Kant's adoption, in 1770, of the
Critical view of space as being the universal form of our outer
sense, he seems to have definitely rejected all such possibilities.
Space, to be space at all, must be Euclidean ; the uniformity
of space is a presupposition of the a priori certainty of geo-
metrical science. 3 One of the criticisms which in the Dis-
sertation 4 he passes upon the empirical view of mathematical
1 Cf. Vaihinger, i. pp. 351-4; and above, p. 76; below, p. 302. Cf. Caird,
The Critical Philosophy, i. pp. 298-9, 301 ; and Watson, Kant Explained, p. 91.
2 Gedanken von der ivahren Schatzung der lebendigen Kraft e (1747), 10.
3 This important and far-reaching assertion we cannot at this point discuss.
Kant's reasoning is really circular in the bad sense. Kant may legitimately argue
from the a priori character of space to the apodictic character of ptire mathematical
science ; but when he proceeds similarly to infer the apodictic character of
applied mathematics, he is constrained to make the further assumption that space
is a fixed and absolutely uniform mode in which alone members of the human
species can intuit objects. That, as we point out below (p. 120), is an assumption
which Kant does not really succeed in proving. In any case the requirements of
the strict synthetic method preclude him from arguing, as he does both in the
Dissertation ( 15) and in the third space argument of the first edition, that the
a priori certitude of applied mathematics affords proof of the necessary uniformity
of all space.
4 IS D.
u8 THE TRANSCENDENTAL AESTHETIC
science is that it would leave open the possibility that "a
space may some time be discovered endowed with other
fundamental properties, or even perhaps that we may happen
upon a two-sided rectilinear figure." This is the argument
which reappears in the third argument on space in the first
edition of the Critique! The same examples are employed
with a somewhat different wording.
" It would not even be necessary that there should be only one
straight line between two points, though experience invariably shows
this to be so. What is derived from experience has only comparative
universality, namely, that which is obtained through induction. We
should therefore only be able to say that, so far as hitherto observed,
no space has been found which has more than three dimensions."
But that Kant should have failed to recognise the possibility
of other spaces does not by itself point to any serious defect in
his position. There is no essential difficulty in reconciling the
recognition of such spaces with his fundamental teaching. He
admits that other races of finite beings may perhaps intuit
through non-spatial forms of sensibility ; he might quite well
have recognised that those other forms of intuition, though not
Euclidean, are still spatial. It is in another and more vital
respect that Kant's teaching lies open to criticism. Kant is
convinced that space is given to us in intuition as being
definitely and irrevocably Euclidean in character. Both our
intuition and our thinking, when we reflect upon space, are,
he implies, bound down to, and limited by, the conditions of
Euclidean space. And it is in this positive assumption, and
not merely in his ignoring of the possibility of other spaces,
that he comes into conflict with the teaching of modern
geometry. For in making the above assumption Kant is
asserting that we definitely know physical space to be three-
dimensional, and that by no elaboration of concepts can we
so remodel it in thought that the axiom of parallels will cease
to hold. Euclidean space, Kant implies, is given to us as an
unyielding form that rigidly resists all attempts at conceptual
reconstruction. Being quite independent of thought and
being given as complete, it has no inchoate plasticity of
which thought might take advantage. The modern geometer
is not, however, prepared to admit that intuitional space
has any definiteness or preciseness of nature apart from
the concepts through which it is apprehended ; and he
therefore allows, as at least possible, that upon clarification
of our concepts space may be discovered to be radically
different from what it at first sight appears to be. In any
1 Cf. above, p. in.
KANT AND MODERN GEOMETRY 119
case, the perfecting of the concepts must have some effect
upon their object. But even as the modern geometer further
maintains should our space be definitely proved, upon
analytic and empirical investigation, to be Euclidean in
character, other possibilities will still remain open for specula-
tive thought. For though the nature of our intuitional data
may constrain us to interpret them through one set of concepts
rather than through another, the competing sets of alternative
concepts will represent genuine possibilities beyond what the
actual is found to embody.
Thus the defect of Kant's teaching, in regard to space,
as judged in the light of the later teaching of geometrical
science, is closely bound up with his untenable isolation of
the a priori of sensibility from the a priori of understanding. 1
Space, being thus viewed as independent of thought, has to
be regarded as limiting and restricting thought by the un-
alterable nature of its initial presentation. And unfortunately
this is a position which Kant continued to hold, despite his
increasing recognition of the part which concepts must play
in the various mathematical sciences. In the deduction of the
first edition we find him stating that synthesis of apprehen-
sion is necessary to all representation of space and time. 2 He
further recognises that all arithmetical processes are syntheses
according to concepts? And in the Prolegomena 4 there occurs
the following significant passage.
" Do these laws of nature lie in space, and does the understanding
learn them by merely endeavouring to find out the fruitful meaning
that lies in space ; or do they inhere in the understanding and in
the way in which it determines space according to the conditions of
the synthetical unity towards which its concepts are all directed?
Space is something so uniform and as to all particular properties so
indeterminate, that we should certainly not seek a store of laws of
nature in it. That which determines space to the form of a circle or to
the figures of a cone or a sphere, is, on the contrary, the understanding,
so far as it contains the ground of the unity of these constructions.
The mere universal form of intuition, called space, must therefore be
the substratum of all intuitions determinable to particular objects,
and in it, of course, the condition of the possibility and of the variety
of these intuitions lies. But the unity of the objects is solely
determined by the understanding, and indeed in accordance with
conditions which are proper to the nature of the understanding . . ."
Obviously Kant is being driven by the spontaneous de-
velopment of his own thinking towards a position much more
1 Cf. above, pp. 40-2, 93-4; below, pp. 131-3, 338-9, 418 ff. 2 A 99-100.
3 A 78 = B 104. Cf. A 159 = B 198, B 147.
4 38, Eng. trans, p. 81.
120 THE TRANSCENDENTAL AESTHETIC
consistent with present-day teaching, and completely at
variance with the hard and fast severance between sensibility
and understanding which he had formulated in the Dissert-
ation and has retained in the Aesthetic. In the above
Prolegomena passage a plasticity is being allowed to space,
sufficient to permit of essential modification in the conceptual
processes through which it is articulated. But, as I have just
stated, that did not lead Kant to disavow the conclusions
which he had drawn from his previous teaching.
This defect in Kant's doctrine of space, as expounded in
the Aesthetic, indicates a further imperfection in his argument.
He asserts that the form of space cannot vary from one
human being to another, and that for this reason the judg-
ments which express it are universally valid. Now, in so far
as Kant's initial datum is consciousness of time, 1 he is entirely
justified in assuming that everything which can be shown to
be a necessary condition of such consciousness must be upi-
form for all human minds. But as his argument is not that
consciousness of Euclidean space is necessary to consciousness
of time, but only that consciousness of the permanent in space
is a required condition, he has not succeeded in showing the
necessary uniformity of the human mind as regards the specific
mode in which it intuits space. The permanent might still
be apprehended as permanent, and therefore as yielding a
possible basis for consciousness of sequence, even if it were
apprehended in some four-dimensional form.
Fourth Paragraph. The next paragraph raises one of the
central problems of the Critique, namely, the question as
to the kind of reality possessed by appearances. Are
they subjective, like taste or colour? Or have they a
reality at least relatively independent of the individual per-
cipient ? In other words, is Kant's position subjectivism or
phenomenalism ? Kant here alternates between these positions.
This fourth paragraph is coloured by his phenomenalism,
whereas in the immediately following fifth paragraph his
subjectivism gains the upper hand. The taste of wine, he
there states, is purely subjective, because dependent upon the
particular constitution of the gustatory organ on which the
wine acts. Similarly, colours are not properties of the objects
which cause them.
"They are only modifications of the sense of sight which is
affected in a certain manner by the light. . . . They are connected
1 Cf. p. 241 ff.
SPACE AND SENSE QUALITIES 121
with the appearances only as effects accidentally added by the
particular constitution of the sense organs." l
Space, on the other hand, is a necessary constituent of the