SPACE
It is the aim of the _Aesthetic_ to deal with the _a priori_ knowledge
which relates to the sensibility. This knowledge, according to Kant,
is concerned with space and time. Hence he has to show _firstly_ that
our apprehension of space and time is _a priori_, i. e. that it is not
derived from experience but originates in our apprehending nature; and
_secondly_ that within our apprehending nature this apprehension
belongs to the sensibility and not to the understanding, or, in his
language, that space and time are forms of perception or sensibility.
Further, if his treatment is to be exhaustive, he should also show
_thirdly_ that space and time are the only forms of perception. This,
however, he makes no attempt to do except in one passage,[1] where the
argument fails. The first two points established, Kant is able to
develop his main thesis, viz. that it is a condition of the validity
of the _a priori_ judgements which relate to space and time that these
are characteristics of phenomena, and not of things in themselves.
[1] B. 58, M. 35.
It will be convenient to consider his treatment of space and time
separately, and to begin with his treatment of space. It is necessary,
however, first of all to refer to the term 'form of perception'. As
Kant conceives a form of perception, it involves three antitheses.
(1) As a _form_ of perception it is opposed, as a way or mode of
perceiving, to particular perceptions.
(2) As a form or mode of _perception_ it is opposed to a form or mode
of _conception_.
(3) As a form of _perception_ it is also opposed, as a way in which we
apprehend things, to a way in which things are.
While we may defer consideration of the second and third antitheses,
we should at once give attention to the nature of the first, because
Kant confuses it with two other antitheses. There is no doubt that in
general a _form_ of perception means for Kant a general capacity of
perceiving which, as such, is opposed to the actual perceptions in
which it is manifested. For according to him our spatial perceptions
are not foreign to us, but manifestations of our general perceiving
nature; and this view finds expression in the assertion that space is
a form of perception or of sensibility.[2]
[2] Cf. B. 43 init., M. 26 med.
Unfortunately, however, Kant frequently speaks of this form of
perception as if it were the same thing as the actual perception of
empty space.[3] In other words, he implies that such a perception is
possible, and confuses it with a potentiality, i. e. the power of
perceiving that which is spatial. The confusion is possible because it
can be said with some plausibility that a perception of empty
space--if its possibility be allowed--does not inform us about actual
things, but only informs us what must be true of things, if there
prove to be any; such a perception, therefore, can be thought of as a
possibility of knowledge rather than as actual knowledge.
[3] e. g. B. 34, 35, M. 22; B. 41, M. 25; _Prol._ Secs. 9-11.
The commonest expression of the confusion is to be found in
the repeated assertion that space is a pure perception.
The second confusion is closely related to the first, and arises from
the fact that Kant speaks of space not only as a form of _perception_,
but also as the form of _phenomena_ in opposition to sensation as
their matter. "That which in the phenomenon corresponds to[4] the
sensation I term its matter; but that which effects that the manifold
of the phenomenon can be arranged under certain relations I call the
form of the phenomenon. Now that in which alone our sensations can be
arranged and placed in a certain form cannot itself be sensation.
Hence while the matter of all phenomena is only given to us _a
posteriori_, their form [i. e. space] must lie ready for them all
together _a priori_ in the mind."[5] Here Kant is clearly under the
influence of his theory of perception.[6] He is thinking that, given
the origination of sensations in us by the thing in itself, it is the
business of the mind to arrange these sensations spatially in order to
attain knowledge of the spatial world.[7] Space being, as it were,
a kind of empty vessel in which sensations are arranged, is said
to be the form of phenomena.[8] Moreover, if we bear in mind that
ultimately bodies in space are for Kant only spatial arrangements
of sensations,[9] we see that the assertion that space is the form
of phenomena is only Kant's way of saying that all bodies are
spatial.[10] Now Kant, in thus asserting that space is the form of
phenomena, is clearly confusing this assertion with the assertion that
space is a form of perception, and he does so in consequence of the
first confusion, viz. that between a capacity of perceiving and an
actual perception of empty space. For in the passage last quoted he
continues thus: "I call all representations[11] _pure_ (in the
transcendental sense) in which nothing is found which belongs to
sensation. Accordingly there will be found _a priori_ in the mind the
pure form of sensuous perceptions in general, wherein all the manifold
of phenomena is perceived in certain relations. This pure form of
sensibility will also itself be called _pure perception_. Thus,
if I abstract from the representation of a body that which the
understanding thinks respecting it, such as substance, force,
divisibility, &c., and also that which belongs to sensation, such as
impenetrability, hardness, colour, &c., something is still left over
for me from this empirical perception, viz. extension and shape. These
belong to pure perception, which exists in the mind _a priori_, even
without an actual object of the senses or a sensation, as a mere form
of sensibility." Here Kant has passed, without any consciousness of a
transition, from treating space as that in which the manifold of
sensation is arranged to treating it as a capacity of perceiving.
Moreover, since Kant in this passage speaks of space as a perception,
and thereby identifies space with the perception of it,[12] the
confusion may be explained thus. The form of phenomena is said to be
the space in which all sensations are arranged, or in which all bodies
are; space, apart from all sensations or bodies, i. e. empty, being
the object of a pure perception, is treated as identical with a pure
perception, viz. the perception of empty space; and the perception of
empty space is treated as identical with a capacity of perceiving that
which is spatial.[13]
[4] 'Corresponds to' must mean 'is'.
[5] B. 34, M. 21.
[6] Cf. pp. 30-2.
[7] It is impossible, of course, to see how such a process
can give us knowledge of the spatial world, for, whatever
bodies in space are, they are not arrangements of sensations.
Nevertheless, Kant's theory of perception really precludes
him from holding that bodies are anything else than
arrangements of sensations, and he seems at times to accept
this view explicitly, e. g. B. 38, M. 23 (quoted p. 41),
where he speaks of our representing sensations as external
to and next to each other, and, therefore, as in different
places.
[8] It may be noted that it would have been more natural to
describe the particular shape of the phenomenon (i. e. the
particular spatial arrangement of the sensations) rather than
space as the form of the phenomenon; for the matter to which
the form is opposed is said to be sensation, and that of
which it is the matter is said to be the phenomenon, i. e.
a body in space.
[9] Cf. note 4, p. 38.
[10] Cf. _Prol._ Sec. 11 and p. 137.
[11] Cf. p. 41, note 1.
[12] Cf. p. 51, note 1.
[13] The same confusion (and due to the same cause) is
implied _Prol_. Sec. 11, and B. 42 (b), M. 26 (b) first
paragraph. Cf. B. 49 (b), M. 30 (b).
The existence of the confusion, however, is most easily realized by
asking, 'How did Kant come to think of space and time as the _only_
forms of perception?' It would seem obvious that the perception of
_anything_ implies a form of perception in the sense of a mode or
capacity of perceiving. To perceive colours implies a capacity for
seeing; to hear noises implies a capacity for hearing. And these
capacities may fairly be called forms of perception. As soon as this
is realized, the conclusion is inevitable that Kant was led to think
of space and time as the only forms of perception, because in this
connexion he was thinking of each as a form of phenomena, i. e. as
something in which all bodies or their states are, or, from the point
of view of our knowledge, as that in which sensuous material is to be
arranged; for there is nothing except space and time in which such
arrangement could plausibly be said to be carried out.
As has been pointed out, Kant's argument falls into two main parts,
one of which prepares the way for the other. The aim of the former is
to show _firstly_ that our apprehension of space is _a priori_, and
_secondly_ that it belongs to perception and not to conception. The
aim of the latter is to conclude from these characteristics of our
apprehension of space that space is a property not of things in
themselves but only of phenomena. These arguments may be considered in
turn.
The really valid argument adduced by Kant for the _a priori_ character
of our apprehension of space is based on the nature of geometrical
judgements. The universality of our judgements in geometry is not
based upon experience, i. e. upon the observation of individual things
in space. The necessity of geometrical relations is apprehended
directly in virtue of the mind's own apprehending nature.
Unfortunately in the present context Kant ignores this argument and
substitutes two others, both of which are invalid.
1. "Space is no empirical conception[14] which has been derived from
external[15] experiences. For in order that certain sensations may be
related to something external to me (that is, to something in a
different part of space from that in which I am), in like manner, in
order that I may represent them as external to and next to each other,
and consequently as not merely different but as in different places,
the representation of space must already exist as a foundation.
Consequently, the representation of space cannot be borrowed from the
relations of external phenomena through experience; but, on the
contrary, this external experience is itself first possible only
through the said representation."[16] Here Kant is thinking that in
order to apprehend, for example, that A is to the right of B we must
first apprehend empty space. He concludes that our apprehension of
space is _a priori_, because we apprehend empty space _before_ we
become aware of the spatial relations of individual objects in it.
[14] _Begriff_ (conception) here is to be understood loosely
not as something opposed to _Anschauung_ (perception), but as
equivalent to the genus of which _Anschauung_ and _Begriff_
are species, i. e. _Vorstellung_, which maybe rendered by
'representation' or 'idea', in the general sense in which
these words are sometimes used to include 'thought' and
'perception'.
[15] The next sentence shows that 'external' means, not
'produced by something external to the mind', but simply
'spatial'.
[16] B. 38, M. 23-4.
To this the following reply may be made. (_a_) The term _a priori_
applied to an apprehension should mean, not that it arises prior to
experience, but that its validity is independent of experience. (_b_)
That to which the term _a priori_ should be applied is not the
apprehension of empty space, which is individual, but the apprehension
of the nature of space in general, which is universal. (_c_) We do not
apprehend empty space before we apprehend individual spatial relations
of individual bodies or, indeed, at any time. (_d_) Though we come to
apprehend _a priori_ the nature of space in general, the apprehension
is not prior but posterior in time to the apprehension of individual
spatial relations. (_e_) It does not follow from the temporal priority
of our apprehension of individual spatial relations that our
apprehension of the nature of space in general is 'borrowed from
experience', and is therefore not _a priori_.
2. "We can never represent to ourselves that there is no space, though
we can quite well think that no objects are found in it. It must,
therefore, be considered as the condition of the possibility of
phenomena, and not as a determination dependent upon them, and it is
an _a priori_ representation, which necessarily underlies external
phenomena."[17]
[17] B. 38, M. 24.
Here the premise is simply false. If 'represent' or 'think' means
'believe', we can no more represent or think that there are no
objects in space than that there is no space. If, on the other hand,
'represent' or 'think' means 'make a mental picture of', the assertion
is equally false. Kant is thinking of empty space as a kind of
receptacle for objects, and the _a priori_ character of our
apprehension of space lies, as before, in the supposed fact that
in order to apprehend objects in space we must begin with the
apprehension of empty space.
The examination of Kant's arguments for the _perceptive_ character
of our apprehension of space is a more complicated matter. By way of
preliminary it should be noticed that they presuppose the possibility
in general of distinguishing features of objects which belong to the
perception of them from others which belong to the conception of them.
In particular, Kant holds that our apprehension of a body as a
substance, as exercising force and as divisible, is due to our
understanding as conceiving it, while our apprehension of it as
extended and as having a shape is due to our sensibility as perceiving
it.[18] The distinction, however, will be found untenable in
principle; and if this be granted, Kant's attempt to distinguish in
this way the extension and shape of an object from its other features
can be ruled out on general grounds. In any case, it must be conceded
that the arguments fail by which he seeks to show that space in
particular belongs to perception.
[18] B. 35, M. 22 (quoted p. 39). It is noteworthy (1) that
the passage contains no _argument_ to show that extension and
shape are not, equally with divisibility, _thought_ to belong
to an object, (2) that impenetrability, which is here said
to belong to sensation, obviously cannot do so, and (3) that
(as has been pointed out, p. 39) the last sentence of the
paragraph in question presupposes that we have a perception
of empty space, and that this is a _form_ of perception.
There appears to be no way of distinguishing perception and conception
as the apprehension of different realities[19] except as the
apprehension of the individual and of the universal respectively.
Distinguished in this way, the faculty of perception is that in virtue
of which we apprehend the individual, and the faculty of conception is
that power of reflection in virtue of which a universal is made the
explicit object of thought.[20] If this be granted, the only test for
what is perceived is that it is individual, and the only test for what
is conceived is that it is universal. These are in fact the tests
which Kant uses. But if this be so, it follows that the various
characteristics of objects cannot be divided into those which are
perceived and those which are conceived. For the distinction between
universal and individual is quite general, and applies to all
characteristics of objects alike. Thus, in the case of colour, we can
distinguish colour in general and the individual colours of individual
objects; or, to take a less ambiguous instance, we can distinguish a
particular shade of redness and its individual instances. Further, it
may be said that perception is of the individual shade of red of the
individual object, and that the faculty by which we become explicitly
aware of the particular shade of red in general is that of conception.
The same distinction can be drawn with respect to hardness, or shape,
or any other characteristic of objects. The distinction, then,
between perception and conception can be drawn with respect to any
characteristic of objects, and does not serve to distinguish one from
another.
[19] And _not_ as mutually involved in the apprehension
of any individual reality.
[20] This distinction is of course different to that
previously drawn _within_ perception in the full sense
between perception in a narrow sense and conception
(pp. 28-9).
Kant's arguments to show that our apprehension of space belongs to
perception are two in number, and both are directed to show not, as
they should, that space is a _form_ of perception, but that it is a
_perception_.[21] The first runs thus: "Space is no discursive, or, as
we say, general conception of relations of things in general, but a
pure perception. For, in the first place, we can represent to
ourselves only one space, and if we speak of many spaces we mean
thereby only parts of one and the same unique space. Again, these
parts cannot precede the one all-embracing space as the component
parts, as it were, out of which it can be composed, but can be thought
only in it. Space is essentially one; the manifold in it, and
consequently the general conception of spaces in general, rests solely
upon limitations."[22]
[21] Kant uses the phrase 'pure perception'; but 'pure' can
only mean 'not containing sensation', and consequently adds
nothing relevant.
[22] B. 39, M. 24. The concluding sentences of the paragraph
need not be considered.
Here Kant is clearly taking the proper test of perception. Its object,
as being an individual, is unique; there is only one of it, whereas
any conception has a plurality of instances. But he reaches his
conclusion by supposing that we first perceive empty space and then
become aware of its parts by dividing it. Parts of space are
essentially limitations of the one space; therefore to apprehend them
we must first apprehend space. And since space is _one_, it must be
object of perception; in other words, space, in the sense of the one
all-embracing space, i. e. the totality of individual spaces, is
something perceived.
The argument appears open to two objections. In the _first_ place, we
do _not_ perceive space as a whole, and then, by dividing it, come to
apprehend individual spaces. We perceive individual spaces, or,
rather, individual bodies occupying individual spaces.[23] We then
apprehend that these spaces, as spaces, involve an infinity of other
spaces. In other words, it is reflection on the general nature of
space, the apprehension of which is involved in our apprehension of
individual spaces or rather of bodies in space, which gives rise to
the apprehension of the totality[24] of spaces, the apprehension being
an act, not of perception, but of thought or conception. It is
necessary, then, to distinguish (_a_) individual spaces, which we
perceive; (_b_) the nature of space in general, of which we become
aware by reflecting upon the character of perceived individual spaces,
and which we conceive; (_c_) the totality of individual spaces, the
thought of which we reach by considering the nature of space in
general.
[23] This contention is not refuted by the objection that our
distinct apprehension of an individual space is always bound
up with an indistinct apprehension of the spaces immediately
surrounding it. For our indistinct apprehension cannot be
supposed to be of the whole of the surrounding space.
[24] It is here assumed that a whole or a totality can be
infinite. Cf. p. 102.
In the _second_ place, the distinctions just drawn afford no ground
for distinguishing space as something perceived from any other
characteristic of objects as something conceived; for any other
characteristic admits of corresponding distinctions. Thus, with
respect to colour it is possible to distinguish (_a_) individual
colours which we perceive; (_b_) colouredness in general, which we
conceive by reflecting on the common character exhibited by individual
colours and which involves various kinds or species of colouredness;
(_c_) the totality of individual colours, the thought of which is
reached by considering the nature of colouredness in general.[25]
[25] For a possible objection and the answer thereto, see
note, p. 70.
Both in the case of colour and in that of space there is to be found
the distinction between universal and individual, and therefore also
that between conception and perception. It may be objected that after
all, as Kant points out, there is only one space, whereas there are
many individual colours. But the assertion that there is only one
space simply means that all individual bodies in space are related
spatially. This will be admitted, if the attempt be made to think of
two bodies as in different spaces and therefore as not related
spatially. Moreover, there is a parallel in the case of colour, since
individual coloured bodies are related by way of colour, e. g. as
brighter and duller; and though such a relation is different from a
relation of bodies in respect of space, the difference is due to the
special nature of the universals conceived, and does not imply a
difference between space and colour in respect of perception and
conception. In any case, space as a whole is not object of perception,
which it must be if Kant is to show that space, as being one, is
perceived; for space in this context must mean the totality of
individual spaces.
Kant's second argument is stated as follows: "Space is represented as
an infinite _given_ magnitude. Now every conception must indeed be
considered as a representation which is contained in an infinite
number of different possible representations (as their common mark),
and which therefore contains these _under itself_, but no conception
can, as such, be thought of as though it contained _in itself_ an
infinite number of representations. Nevertheless, space is so
conceived, for all parts of space _ad infinitum_ exist simultaneously.
Consequently the original representation of space is an _a priori
perception_ and not a _conception_." In other words, while a
conception implies an infinity of individuals which come under it, the
elements which constitute the conception itself (e. g. that of
triangularity or redness) are not infinite; but the elements which go
to constitute space are infinite, and therefore space is not a
conception but a perception.
Though, however, space in the sense of the infinity of spaces may be
said to contain an infinite number of spaces if it be meant that it
_is_ these infinite spaces, it does not follow, nor is it true, that
space in this sense is object of perception.
The aim of the arguments just considered, and stated in Sec. 2 of
the _Aesthetic_, is to establish the two characteristics of our
apprehension of space,[26] from which it is to follow that space is a
property of things only as they appear to us and not as they are in
themselves. This conclusion is drawn in Sec. 4. Secs. 2 and 4 therefore
complete the argument. Sec. 3, a passage added in the second edition
of the _Critique_, interrupts the thought, for ignoring Sec. 2, it once
more establishes the _a priori_ and perceptive character of our
apprehension of space, and independently draws the conclusion drawn in
Sec. 4. Since, however, Kant draws the final conclusion in the same way
in Sec. 3 and in Sec. 4, and since a passage in the _Prolegomena_,[27]
of which Sec. 3 is only a summary, gives a more detailed account of
Kant's thought, attention should be concentrated on Sec. 3, together
with the passage in the _Prolegomena_.
[26] viz. that it is _a priori_ and a pure perception.
[27] Secs. 6-11.
It might seem at the outset that since the arguments upon which Kant
bases the premises for his final argument have turned out invalid, the
final argument itself need not be considered. The argument, however,
of Sec. 3 ignores the preceding arguments for the _a priori_ and
perceptive character of our apprehension of space. It returns to the
_a priori_ synthetic character of geometrical judgements, upon which
stress is laid in the Introduction, and appeals to this as the
justification of the _a priori_ and perceptive character of our
apprehension of space.
The argument of Sec. 3 runs as follows: "Geometry is a science which
determines the properties of space synthetically and yet _a priori_.
What, then, must be the representation of space, in order that such a
knowledge of it may be possible? It must be originally perception, for
from a mere conception no propositions can be deduced which go beyond
the conception, and yet this happens in geometry. But this perception
must be _a priori_, i. e. it must occur in us before all
sense-perception of an object, and therefore must be pure, not
empirical perception. For geometrical propositions are always
apodeictic, i. e. bound up with the consciousness of their necessity
(e. g. space has only three dimensions), and such propositions cannot
be empirical judgements nor conclusions from them."
"Now how can there exist in the mind an external perception[28] which
precedes[29] the objects themselves, and in which the conception of
them can be determined _a priori_? Obviously not otherwise than in so
far as it has its seat in the subject only, as the formal nature of
the subject to be affected by objects and thereby to obtain _immediate
representation_, i. e. _perception_ of them, and consequently only as
the form of the external sense in general."[30]
[28] 'External perception' can only mean perception of what
is spatial.
[29] _Vorhergeht._
[30] 'Formal nature _to be affected by objects_' is not
relevant to the context.
Here three steps are taken. From the _synthetic_ character of
geometrical judgements it is concluded that space is not something
which we _conceive_, but something which we _perceive_. From their _a
priori_ character, i. e. from the consciousness of necessity involved,
it is concluded that the perception of space must be _a priori_ in a
new sense, that of taking place _before_ the perception of objects in
it.[31] From the fact that we perceive space before we perceive
objects in it, and thereby are able to anticipate the spatial
relations which condition these objects, it is concluded that space is
only a characteristic of our perceiving nature, and consequently that
space is a property not of things in themselves, but only of things as
perceived by us.[32]
[31] Cf. B. 42, M. 26 (a) fin., (b) second sentence.
[32] Cf. B. 43, M. 26-7.
Two points in this argument are, even on the face of it, paradoxical.
Firstly, the term _a priori_, as applied not to geometrical judgements
but to the perception of space, is given a temporal sense; it means
not something whose validity is independent of experience and which is
the manifestation of the nature of the mind, but something which takes
place before experience. Secondly, the conclusion is not that the
perception of space _is the manifestation of_ the mind's perceiving
nature, but that it _is_ the mind's perceiving nature. For the
conclusion is that space[33] is the formal nature of the subject to be
affected by objects, and therefore the form of the external sense in
general. Plainly, then, Kant here confuses an actual perception and a
form or way of perceiving. These points, however, are more explicit in
the corresponding passage in the _Prolegomena_.[34]
[33] Kant draws no distinction between space and the
perception of space, or, rather, habitually speaks of space
as a perception. No doubt he considers that his view that
space is only a characteristic of phenomena justifies
the identification of space and the perception of it.
Occasionally, however, he distinguishes them. Thus he
sometimes speaks of the representation of space (e. g.
B. 38-40, M. 23-4); in _Prol._, Sec. 11, he speaks of a pure
perception of space and time; and in B. 40, M. 25, he says
that our representation of space must be perception. But this
language is due to the pressure of the facts, and not to his
general theory; cf. pp. 135-6.
[34] Secs. 6-11.
It begins thus: "Mathematics carries with it thoroughly apodeictic
certainty, that is, absolute necessity, and, therefore, rests on no
empirical grounds, and consequently is a pure product of reason, and,
besides, is thoroughly synthetical. How, then, is it possible for
human reason to accomplish such knowledge entirely _a priori_?... But
we find that all mathematical knowledge has this peculiarity, that it
must represent its conception previously in _perception_, and indeed
_a priori_, consequently in a perception which is not empirical but
pure, and that otherwise it cannot take a single step. Hence its
judgements are always _intuitive_.... This observation on the nature
of mathematics at once gives us a clue to the first and highest
condition of its possibility, viz. that there must underlie it _a pure
perception_ in which it can exhibit or, as we say, _construct_ all its
conceptions in the concrete and yet _a priori_. If we can discover
this pure perception and its possibility, we may thence easily explain
how _a priori_ synthetical propositions in pure mathematics are
possible, and consequently also how the science itself is possible.
For just as empirical perception enables us without difficulty to
enlarge synthetically in experience the conception which we frame of
an object of perception through new predicates which perception itself
offers us, so pure perception also will do the same, only with the
difference that in this case the synthetical judgement will be _a
priori_ certain and apodeictic, while in the former case it will be
only _a posteriori_ and empirically certain; for the latter [i. e. the
empirical perception on which the _a posteriori_ synthetic judgement
is based] contains only that which is to be found in contingent
empirical perception, while the former [i. e. the pure perception on
which the _a priori_ synthetic judgement is based] contains that which
is bound to be found in pure perception, since, as _a priori_
perception, it is inseparably connected with the conception _before
all experience_ or individual sense-perception."
This passage is evidently based upon the account which Kant gives in
the _Doctrine of Method_ of the method of geometry.[35] According to
this account, in order to apprehend, for instance, that a three-sided
figure must have three angles, we must draw in imagination or on paper
an individual figure corresponding to the conception of a three-sided
figure. We then see that the very nature of the act of construction
involves that the figure constructed must possess three angles as well
as three sides. Hence, perception being that by which we apprehend the
individual, a perception is involved in the act by which we form a
geometrical judgement, and the perception can be called _a priori_, in
that it is guided by our _a priori_ apprehension of the necessary
nature of the act of construction, and therefore of the figure
constructed.
[35] B. 740 ff., M. 434 ff. Compare especially the following:
"_Philosophical_ knowledge is _knowledge of reason_ by means
of _conceptions_; mathematical knowledge is knowledge by
means of the _construction_ of conceptions. But the
_construction_ of a conception means the _a priori_
presentation of a perception corresponding to it. The
construction of a conception therefore demands a
_non-empirical_ perception, which, therefore, as a
perception, is an _individual_ object, but which none the
less, as the construction of a conception (a universal
representation), must express in the representation universal
validity for all possible perceptions which come under that
conception. Thus I construct a triangle by presenting the
object corresponding to the conception, either by mere
imagination in pure perception, or also, in accordance with
pure perception, on paper in empirical perception, but in
both cases completely _a priori_, without having borrowed
the pattern of it from any experience. The individual drawn
figure is empirical, but nevertheless serves to indicate the
conception without prejudice to its universality, because in
this empirical perception we always attend only to the act of
construction of the conception, to which many determinations,
e. g. the magnitude of the sides and of the angles, are
wholly indifferent, and accordingly abstract from these
differences, which do not change the conception of the
triangle."
The account in the _Prolegomena_, however, differs from that of the
_Doctrine of Method_ in one important respect. It asserts that the
perception involved in a mathematical judgement not only may, but
must, be pure, i. e. must be a perception in which no spatial object
is present, and it implies that the perception must take place
_before_ all experience of actual objects.[36] Hence _a priori_,
applied to perception, has here primarily, if not exclusively, the
temporal meaning that the perception takes place _antecedently to all
experience_.[37]
[36] This becomes more explicit in Sec. 8 and ff.
[37] This is also, and more obviously, implied in Secs. 8-11.
The thought of the passage quoted from the _Prolegomena_ can be stated
thus: 'A mathematical judgement implies the perception of an
individual figure antecedently to all experience. This may be said to
be the first condition of the possibility of mathematical judgements
which is revealed by reflection. There is, however, a prior or higher
condition. The perception of an individual figure involves as its
basis another pure perception. For we can only construct and therefore
perceive an individual figure in empty space. Space is that _in which_
it must be constructed and perceived. A perception[38] of empty space
is, therefore, necessary. If, then, we can discover how this
perception is possible, we shall be able to explain the possibility of
_a priori_ synthetical judgements of mathematics.'
[38] _Pure_ perception only means that the space perceived is
empty.
Kant continues as follows: "But with this step the difficulty seems to
increase rather than to lessen. For henceforward the question is '_How
is it possible to perceive anything a priori?_' A perception is such a
representation as would immediately depend upon the presence of the
object. Hence it seems impossible _originally_ to perceive _a priori_,
because perception would in that case have to take place without an
object to which it might refer, present either formerly or at the
moment, and accordingly could not be perception.... How can
_perception_ of the object precede the object itself?"[39] Kant here
finds himself face to face with the difficulty created by the
preceding section. Perception, as such, involves the actual presence
of an object; yet the pure perception of space involved by
geometry--which, as pure, is the perception of empty space, and which,
as the perception of empty space, is _a priori_ in the sense of
temporally prior to the perception of actual objects--presupposes that
an object is not actually present.
[39] _Prol._ Sec. 8.
The solution is given in the next section. "Were our perception
necessarily of such a kind as to represent things _as they are in
themselves_, no perception would take place _a priori_, but would
always be empirical. For I can only know what is contained in the
object in itself, if it is present and given to me. No doubt it is
even then unintelligible how the perception of a present thing should
make me know it as it is in itself, since its qualities cannot migrate
over into my faculty of representation; but, even granting this
possibility, such a perception would not occur _a priori_, i. e.
before the object was presented to me; for without this presentation,
no basis of the relation between my representation and the object can
be imagined; the relation would then have to rest upon inspiration. It
is therefore possible only in one way for my perception to precede the
actuality of the object and to take place as _a priori_ knowledge,
viz. _if it contains nothing but the form of the sensibility, which
precedes in me, the subject, all actual impressions through which I am
affected by objects_. For I can know _a priori_ that objects of the
senses can only be perceived in accordance with this form of the
sensibility. Hence it follows that propositions which concern merely
this form of sensuous perception will be possible and valid for
objects of the senses, and in the same way, conversely, that
perceptions which are possible _a priori_ can never concern any things
other than objects of our senses."
This section clearly constitutes the turning-point in Kant's argument,
and primarily expresses, in an expanded form, the central doctrine of
Sec. 3 of the _Aesthetic_, that an external perception anterior to
objects themselves, and in which our conceptions of objects can be
determined _a priori_, is possible, if, and only if, it has its seat
in the subject as its formal nature of being affected by objects, and
consequently as the form of the external sense in general. It argues
that, since this is true, and since geometrical judgements involve
such a perception anterior to objects, space must be only the[40] form
of sensibility.
[40] _The_ and not _a_, because, for the moment, time is
ignored.
Now why does Kant think that this conclusion follows? Before we can
answer this question we must remove an initial difficulty. In this
passage Kant unquestionably identifies a form of perception with an
actual perception. It is at once an actual perception and a capacity
of perceiving. This is evident from the words, "It is possible only in
one way for my perception to precede the actuality of the object ...
viz. _if it contains nothing but the form of the sensibility_."[41]
The identification becomes more explicit a little later. "A pure
perception (of space and time) can underlie the empirical perception
of objects, because it is nothing but the mere form of the
sensibility, which precedes the actual appearance of the objects, in
that it in fact first makes them possible. Yet this faculty of
perceiving _a priori_ affects not the matter of the phenomenon, i. e.
that in it which is sensation, for this constitutes that which is
empirical, but only its form, viz. space and time."[42] His argument,
however, can be successfully stated without this identification. It is
only necessary to re-write his cardinal assertion in the form 'the
perception of space must be nothing but the _manifestation_ of the
form of the sensibility'. Given this modification, the question
becomes, 'Why does Kant think that the perception of empty space,
involved by geometrical judgements, can be only a manifestation of our
perceiving nature, and not in any way the apprehension of a real
quality of objects?' The answer must be that it is because he thinks
that, while in empirical perception a real object is present, in the
perception of empty space a real object is not present. He regards
this as proving that the latter perception is only of something
subjective or mental. "Space and time, by being pure _a priori_
perceptions, prove that they are mere forms of our sensibility which
must precede all empirical perception, i. e. sense-perception of
actual objects."[43] His main conclusion now follows easily enough. If
in perceiving empty space we are only apprehending a manifestation of
our perceiving nature, what we apprehend in a geometrical judgement is
really a law of our perceiving nature, and therefore, while it _must_
apply to our perceptions of objects or to objects as perceived, it
_cannot_ apply to objects apart from our perception, or, at least,
there is no ground for holding that it does so.
[41] _Prol._, Sec. 9.
[42] _Prol._, Sec. 11.
[43] _Prol._, Sec. 10.
If, however, this fairly represents Kant's thought, it must be allowed
that the conclusion which he should have drawn is different, and even
that the conclusion which he does draw is in reality incompatible with
his starting-point.
His starting-point is the view that the truth of geometrical
judgements presupposes a perception of empty space, in virtue of which
we can discover rules of spatial relation which must apply to all
spatial objects subsequently perceived. His problem is to discover the
presupposition of this presupposition. The proper answer must be, not
that space is a form of sensibility or a way in which objects appear
to us, but that space is the form of all objects, i. e. that all
objects are spatial.[44] For in that case they must be subject to the
laws of space, and therefore if we can discover these laws by a study
of empty space, the only condition to be satisfied, if the objects of
subsequent perception are to conform to the laws which we discover, is
that all objects should be spatial. Nothing is implied which enables
us to decide whether the objects are objects as they are in themselves
or objects as perceived; for in either case the required result
follows. If in empirical perception we apprehend things only as they
appear to us, and if space is the form of them as they appear to us,
it will no doubt be true that the laws of spatial relation which we
discover must apply to things as they appear to us. But on the other
hand, if in empirical perception we apprehend things as they are, and
if space is their form, i. e. if things are spatial, it will be
equally true that the laws discovered by geometry must apply to things
as they are.
[44] Kant expresses the assertion that space is the form of
all objects by saying that space is the form of _phenomena_.
This of course renders easy an unconscious transition from
the thesis that space is the form of objects to the quite
different thesis that space is the form of sensibility; cf.
p. 39.
Again, Kant's starting-point really commits him to the view that space
is a characteristic of things as they are. For--paradoxical though it
may be--his problem is to explain the possibility of _perceiving a
priori_, i. e. of _perceiving_ the characteristics of an object
anterior to the actual presence of the object in perception.[45] This
implies that _empirical_ perception, which involves the actual
presence of the object, involves no difficulty; in other words, it is
implied that empirical perception is of objects as they are. And we
find Kant admitting this to the extent of allowing _for the sake of
argument_ that the perception of a present thing can make us know the
thing as it is in itself.[46] But if empirical perception gives us
things as they are, and if, as is the case, and as Kant really
presupposes, the objects of empirical perception are spatial, then,
since space is their form, the judgements of geometry must relate to
things as they are. It is true that on this view Kant's first
presupposition of geometrical judgements has to be stated by saying
that we are able to perceive a real characteristic of things in space,
before we perceive the things; and, no doubt, Kant thinks this
impossible. According to him, when we perceive empty space no object
is present, and therefore what is before the mind must be merely
mental. But no greater difficulty is involved than that involved in
the corresponding supposition required by Kant's own view. It is
really just as difficult to hold that we can perceive a characteristic
of things as they appear to us _before_ they appear, as to hold that
we can perceive a characteristic of them as they are in themselves
_before_ we perceive them.
[45] Cf. _Prol._, Section 8.
[46] _Prol._, Sec. 9 (cf. p. 55).
The fact is that the real difficulty with which Kant is grappling in
the _Prolegomena_ arises, not from the supposition that spatial bodies
are things in themselves, but from the supposed presupposition of
geometry that we must be able to perceive empty space before we
perceive bodies in it. It is, of course, impossible to defend the
perception of empty space, but _if_ it be maintained, the space
perceived must be conceded to be not, as Kant thinks, something mental
or subjective, but a real characteristic of things. For, as has been
pointed out, the paradox of pure perception is reached solely through
the consideration that, while in empirical perception we perceive
objects, in pure perception we do not, and since the objects of
empirical perception are spatial, space must be a real characteristic
of them.
The general result of the preceding criticism is that Kant's
conclusion does not follow from the premises by which he supports it.
It should therefore be asked whether it is not possible to take
advantage of this hiatus by presenting the argument for the merely
phenomenal character of space without any appeal to the possibility of
perceiving empty space. For it is clear that what was primarily before
Kant, in writing the _Critique_, was the _a priori_ character of
geometrical judgements themselves, and not the existence of a
perception of empty space which they were held to presuppose.[47]
[47] The difficulty with which Kant is struggling in the
_Prolegomena_, Secs. 6-11, can be stated from a rather different
point of view by saying that the thought that geometrical
judgements imply a perception of empty space led him to apply
the term '_a priori_' to perception as well as to judgement.
The term, _a priori_, applied to judgements has a valid
meaning; it means, not that the judgement is made prior to
all experience, but that it is not based upon experience,
being originated by the mind in virtue of its own powers of
thinking. Applied to perception, however, '_a priori_' must
mean prior to all experience, and, since the object of
perception is essentially individual (cf. B. 741, M. 435),
this use of the term gives rise to the impossible task of
explaining how a perception can take place prior to the
actual experience of an individual in perception (cf.
_Prol._, Sec. 8).
If, then, the conclusion that space is only the form of sensibility
can be connected with the _a priori_ character of geometrical
judgements without presupposing the existence of a perception of empty
space, his position will be rendered more plausible.
This can be done as follows. The essential characteristic of a
geometrical judgement is not that it takes place prior to experience,
but that it is not based upon experience. Thus a judgement, arrived at
by an activity of the mind in which it remains within itself and does
not appeal to actual experience of the objects to which the judgement
relates, is implied to hold good of those objects. If the objects were
things as they are in themselves, the validity of the judgement could
not be justified, for it would involve the gratuitous assumption that
a necessity of thought is binding on things which _ex hypothesi_ are
independent of the nature of the mind. If, however, the objects in
question are things as perceived, they will be through and through
conditioned by the mind's perceiving nature; and, consequently, if a
geometrical rule, e. g. that a three-sided figure must have three
angles, is really a law of the mind's perceiving nature, all
individual perceptions, i. e. all objects as perceived by us, will
necessarily conform to the law. Therefore, in the latter case, and in
that only, will the universal validity of geometrical judgements be
justified. Since, then, geometrical judgements are universally valid,
space, which is that of which geometrical laws are the laws, must be
merely a form of perception or a characteristic of objects as
perceived by us.
This appears to be the best form in which the substance of Kant's
argument, stripped of unessentials, can be stated. It will be
necessary to consider both the argument and its conclusion.
The argument, so stated, is undeniably plausible. Nevertheless,
examination of it reveals two fatal defects. In the first place, its
starting-point is false. To Kant the paradox of geometrical judgements
lies in the fact that they are not based upon an appeal to experience
of the things to which they relate. It is implied, therefore, that
judgements which are based on experience involve no paradox, and for
the reason that in experience we apprehend things as they are.[48] In
contrast with this, it is implied that in geometrical judgements the
connexion which we apprehend is not real, i. e. does not relate to
things as they are. Otherwise, there would be no difficulty; if in
geometry we apprehended rules of connexion relating to things as they
are, we could allow without difficulty that the things must conform to
them. No such distinction, however, can be drawn between _a priori_
and empirical judgements. For the necessity of connexion, e. g.
between being a three-sided figure and being a three-angled figure, is
as much a characteristic of things as the empirically-observed shape
of an individual body, e. g. a table. Geometrical judgements,
therefore, cannot be distinguished from empirical judgements on the
ground that in the former the mind remains within itself, and does not
immediately apprehend fact or a real characteristic of reality.[49]
Moreover, since in a geometrical judgement we do in fact think that we
are apprehending a real connexion, i. e. a connexion which applies to
things and to things as they are in themselves, to question the
reality of the connexion is to question the validity of thinking
altogether, and to do this is implicitly to question the validity of
our thought about the nature of our own mind, as well as the validity
of our thought about things independent of the mind. Yet Kant's
argument, in the form in which it has just been stated, presupposes
that our thought is valid at any rate when it is concerned with our
perceptions of things, even if it is not valid when concerned with the
things as they are in themselves.
[48] Cf. p. 17.
[49] For the reasons which led Kant to draw this distinction
between empirical and _a priori_ judgements, cf. pp. 21-2.
This consideration leads to the second criticism. The supposition that
space is only a form of perception, even if it be true, _in no way
assists_ the explanation of the universal validity of geometrical
judgements. Kant's argument really confuses a _necessity_ of relation
with the _consciousness of a necessity_ of relation. No doubt, if it
be a law of our perceiving nature that, whenever we perceive an object
as a three-sided figure, the object as perceived contains three
angles, it follows that any object as perceived will conform to this
law; just as if it be a law of things as they are in themselves that
three-sided figures contain three angles, all three-sided figures will
in themselves have three angles. But what has to be explained is the
universal applicability, not of a law, but of a judgement about a law.
For Kant's real problem is to explain why _our judgement_ that a
three-sided figure must contain three angles must apply to all
three-sided figures. Of course, if it be granted that in the judgement
we apprehend the true law, the problem may be regarded as solved. But
how are we to know that what we judge _is_ the true law? The answer is
in no way facilitated by the supposition that the judgement relates to
our perceiving nature. It can just as well be urged that what we think
to be a necessity of our perceiving nature is not a necessity of it,
as that what we think to be a necessity of things as they are in
themselves is not a necessity of them. The best, or rather the only
possible, answer is simply that that of which we apprehend the
necessity must be true, or, in other words, that we _must_ accept the
validity of thought. Hence nothing is gained by the supposition that
space is a form of sensibility. If what we judge to be necessary is,
as such, valid, a judgement relating to things in themselves will be
as valid as a judgement relating to our perceiving nature.[50]
[50] The same criticism can be urged against Kant's appeal to
the necessity of _constructing_ geometrical figures. The
conclusion drawn from the necessity of construction is stated
thus: "If the object (the triangle) were something in itself
without relation to you the subject, how could you say that
that which lies necessarily in your subjective conditions of
constructing a triangle must also necessarily belong to the
triangle in itself?" (B. 65, M. 39). Kant's thought is that
the laws of the mind's constructing nature must apply to
objects, if, and only if, the objects are the mind's own
construction. Hence it is open to the above criticism if, in
the criticism, 'construct' be substituted for 'perceive'.
This difficulty is concealed from Kant by his insistence on the
_perception_ of space involved in geometrical judgements. This leads
him at times to identify the judgement and the perception, and,
therefore, to speak of the judgement as a perception. Thus we find him
saying that mathematical judgements are always _perceptive_,[51] and
that "It is only possible for my perception to precede the actuality
of the object and take place as _a priori_ knowledge, if &c."[52]
Hence, if, in addition, a geometrical judgement, as being a judgement
about a necessity, be identified with a necessity of judging, the
conformity of things to these universal judgements will become the
conformity of things to rules or necessities of our judging, i. e. of
our perceiving nature, and Kant's conclusion will at once follow.[53]
Unfortunately for Kant, a geometrical judgement, however closely
related to a perception, must itself, as the apprehension of what is
necessary and universal, be an act of thought rather than of
perception, and therefore the original problem of the conformity of
things to our mind can be forced upon him again, even after he thinks
that he has solved it, in the new form of that of the conformity
within the mind of perceiving to thinking.
[51] _Prol._, Sec. 7.
[52] _Prol._, Sec. 9.
[53] Cf. (_Introduction_, B. xvii, M. xxix): "But if the
object (as object of the senses) conforms to the nature of
our faculty of perception, I can quite well represent to
myself the possibility of _a priori_ knowledge of it [i. e.
mathematical knowledge]."
The fact is simply that the universal validity of geometrical
judgements can in no way be 'explained'. It is not in the least
explained or made easier to accept by the supposition that objects
are 'phenomena'. These judgements must be accepted as being what we
presuppose them to be in making them, viz. the direct apprehension of
necessities of relation between real characteristics of real things.
To explain them by reference to the phenomenal character of what is
known is really--though contrary to Kant's intention--to throw doubt
upon their validity; otherwise, they would not need explanation. As a
matter of fact, it is _impossible_ to question their validity. In the
act of judging, doubt is impossible. Doubt can arise only when
we subsequently reflect and temporarily lose our hold upon the
consciousness of necessity in judging.[54] The doubt, however, since
it is non-existent in our geometrical consciousness, is really
groundless,[55] and, therefore, the problem to which it gives rise is
unreal. Moreover if, _per impossibile_, doubt could be raised, it
could not be set at rest. No vindication of a judgement in which we
are conscious of a necessity could do more than take the problem a
stage further back, by basing it upon some other consciousness of a
necessity; and since this latter judgement could be questioned for
precisely the same reason, we should only be embarking upon an
infinite process.
[54] Cf. Descartes, _Princ. Phil._ i. Sec. 13, and _Medit._ v
sub fin.
[55] The view that kinds of space other than that with which
we are acquainted are possible, though usually held and
discussed by mathematicians, belongs to them _qua_
metaphysicians, and not _qua_ mathematicians.
We may now consider Kant's conclusion in abstraction from the
arguments by which he reaches it. It raises three main difficulties.
In the first place, it is not the conclusion to be expected from
Kant's own standpoint. The phenomenal character of space is inferred,
not from the fact that we make judgements at all, but from the fact
that we make judgements of a particular kind, viz. _a priori_
judgements. From this point of view empirical judgements present no
difficulty. It should, therefore, be expected that the qualities which
we attribute to things in empirical judgements are not phenomenal, but
belong to things as they are. Kant himself implies this in drawing his
conclusion concerning the nature of space. "Space does not represent
any quality of things in themselves or things in relation to one
another; that is, it does not represent any determination of things
which would attach to the objects themselves and would remain, even
though we abstracted from all subjective conditions of perception. For
neither absolute nor relative[56] determinations of objects can be
perceived prior to the existence of the things to which they belong,
and therefore not _a priori_."[57] It is, of course, implied that in
experience, where we do not discover determinations of objects prior
to the existence of the objects, we do apprehend determinations of
things as they are in themselves, and not as they are in relation to
us. Thus we should expect the conclusion to be, not that all that we
know is phenomenal--which is Kant's real position--but that spatial
(and temporal) relations alone are phenomenal, i. e. that they alone
are the result of a transmutation due to the nature of our perceiving
faculties.[58] This conclusion would, of course, be absurd, for what
Kant considers to be the empirically known qualities of objects
disappear, if the spatial character of objects is removed. Moreover,
Kant is prevented by his theory of perception from seeing that this is
the real solution of his problem, absurd though it may be. Since
perception is held to arise through the origination of sensations by
things in themselves, empirical knowledge is naturally thought of as
knowledge about sensations, and since sensations are palpably within
the mind, and are held to be due to things in themselves, knowledge
about sensations can be regarded as phenomenal.
[56] The first sentence shows that 'relative determinations'
means, not 'determinations of objects in relation to us', but
'determinations of objects in relation to one another.' Cf.
B. 37, M. 23; and B. 66 fin., 67 init., M. 40 (where these
meanings are confused).
[57] B. 42, M. 26.
[58] This conclusion is also to be expected because,
inconsistently with his real view, Kant is here (B. 41-2, M.
25-6) under the influence of the presupposition of our
ordinary consciousness that in perception we are confronted
by things in themselves, known to be spatial, and not by
appearances produced by unknown things in themselves. Cf. (B.
41, M. 25) "and thereby of obtaining immediate representation
of them [i. e. objects];" and (B. 42, M. 26) "the receptivity
of the subject to be affected by objects necessarily precedes
all perceptions of these objects." These sentences identify
things in themselves and bodies in space, and thereby imply
that in empirical perception we perceive things in themselves
and as they _are_.
On the other hand, if we consider Kant's conclusion from the point of
view, not of the problem which originates it, but of the distinction
in terms of which he states it, viz. that between things as they are
in themselves and things as perceived by us, we are led to expect the
contrary result. Since perception is the being affected by things, and
since the nature of the affection depends upon the nature of our
capacity of being affected, in _all_ perception the object will become
distorted or transformed, as it were, by our capacity of being
affected. The conclusion, therefore, should be that in all judgements,
empirical as well as _a priori_, we apprehend things only as
perceived. The reason why Kant does not draw this conclusion is
probably that given above, viz. that by the time Kant reaches the
solution of his problem empirical knowledge has come to relate to
sensation only; consequently, it has ceased to occur to him that
empirical judgements could possibly give us knowledge of things as
they are. Nevertheless, Kant should not have retained in his
formulation of the problem a distinction irreconcilable with his
solution of it; and if he had realized that he was doing so he might
have been compelled to modify his whole view.
The second difficulty is more serious. If the truth of geometrical
judgements presupposes that space is only a property of objects as
perceived by us, it is a paradox that geometricians should be
convinced, as they are, of the truth of their judgements. They
undoubtedly think that their judgements apply to things as they are in
themselves, and not merely as they appear to us. They certainly do not
think that the relations which they discover apply to objects only as
perceived. Not only, therefore, do they not think that bodies in space
are phenomena, but they do not even leave it an open question whether
bodies are phenomena or not. Hence, if Kant be right, they are really
in a state of illusion, for on his view the true geometrical judgement
should include in itself the phenomenal character of spatial
relations; it should be illustrated by expressing Euclid I. 5 in
the form that the equality of the angles at the base of an isosceles
triangle belongs to objects as perceived. Kant himself lays this down.
"The proposition 'all objects are beside one another in space'
is valid under[59] the limitation that these things are taken as
objects of our sensuous perception. If I join the condition to the
perception, and say 'all things, as external phenomena, are beside
one another in space', the rule is valid universally, and without
limitation."[60] Kant, then, is in effect allowing that it is possible
for geometricians to make judgements, of the necessity of which
they are convinced, and yet to be wrong; and that, therefore, the
apprehension of the necessity of a judgement is no ground of its
truth. It follows that the truth of geometrical judgements can no
longer be accepted as a starting-point of discussion, and, therefore,
as a ground for inferring the phenomenal character of space.
[59] A. reads 'only under'
[60] B. 43, M. 27.
There seems, indeed, one way of avoiding this consequence, viz. to
suppose that for Kant it was an absolute starting-point, which nothing
would have caused him to abandon, that only those judgements of which
we apprehend the necessity are true. It would, of course, follow that
geometricians would be unable to apprehend the necessity of
geometrical judgements, and therefore to make such judgements, until
they had discovered that things as spatial were only phenomena. It
would not be enough that they should think that the phenomenal or
non-phenomenal character of things as spatial must be left an open
question for the theory of knowledge to decide. In this way the
necessity of admitting the illusory character of geometry would be
avoided. The remedy, however, is at least as bad as the disease.
For it would imply that geometry must be preceded by a theory of
knowledge, which is palpably contrary to fact. Nor could Kant accept
it; for he avowedly bases his theory of knowledge, i. e. his view
that objects as spatial are phenomena, upon the truth of geometry;
this procedure would be circular if the making of true geometrical
judgements was allowed to require the prior adoption of his theory of
knowledge.
The third difficulty is the most fundamental. Kant's conclusion (and
also, of course, his argument) presupposes the validity of the
distinction between phenomena and things in themselves. If, then, this
distinction should prove untenable in principle, Kant's conclusion
with regard to space must fail on general grounds, and it will even
have been unnecessary to consider his arguments for it. The importance
of the issue, however, requires that it should be considered in a
separate chapter.
NOTE to page 47.
The argument is not affected by the contention that, while
the totality of spaces is infinite, the totality of colours
or, at any rate, the totality of instances of some other
characteristic of objects is finite; for this difference
will involve no difference in respect of perception and
conception. In both cases the apprehension that there is a
totality will be reached in the same way, i. e. through the
_conception_ of the characteristic in general, and the
apprehension in the one case that the totality is infinite
and in the other that it is finite will depend on the
apprehension of the special nature of the characteristic in
question.