THE SCHEMATISM OF THE CATEGORIES
As has already been pointed out,[1] the _Analytic_ is divided into
two parts, the _Analytic of Conceptions_, of which the aim is to
discover and vindicate the validity of the categories, and the
_Analytic of Principles_, of which the aim is to determine the use
of the categories in judgement. The latter part, which has now to be
considered, is subdivided into two. It has, according to Kant, firstly
to determine the sensuous conditions under which the categories are
used, and secondly to discover the _a priori_ principles involved in
the categories, as exercised under these sensuous conditions, such,
for instance, as the law that all changes take place according to
the law of cause and effect. The first problem is dealt with in
the chapter on the 'schematism of the pure conceptions of the
understanding', the second in the chapter on the 'system of all
principles of the pure understanding'.
[1] p. 141.
We naturally feel a preliminary difficulty with respect to the
existence of this second part of the _Analytic_ at all. It seems clear
that if the first part is successful, the second must be unnecessary.
For if Kant is in a position to lay down that the categories must
apply to objects, no special conditions of their application need be
subsequently determined. If, for instance, it can be laid down that
the category of quantity must apply to objects, it is implied either
that there are no special conditions of its application, or that they
have already been discovered and shown to exist. Again, to assert the
applicability of the categories is really to assert the existence of
principles, and in fact of just those principles which it is the aim
of the _System of Principles_ to prove. Thus to assert the
applicability of the categories of quantity and of cause and effect is
to assert respectively the principles that all objects of perception
are extensive quantities, and that all changes take place according
to the law of cause and effect. The _Deduction of the Categories_
therefore, if successful, must have already proved the principles
now to be vindicated; and it is a matter for legitimate surprise
that we find Kant in the _System of Principles_ giving proofs of
these principles which make no appeal to the _Deduction of the
Categories_.[2] On the other hand, for the existence of the account of
the schematism of the categories Kant has a better show of reason. For
the conceptions derived in the _Metaphysical Deduction_ from the
nature of formal judgement are in themselves too abstract to be the
conceptions which are to be shown applicable to the sensible world,
since all the latter involve the thought of time. Thus, the conception
of cause and effect derived from the nature of the hypothetical
judgement includes no thought of time, while the conception of which
he wishes to show the validity is that of necessary succession in
time. Hence the conceptions discovered by analysis of formal judgement
have in some way to be rendered more concrete in respect of time. The
account of the schematism, therefore, is an attempt to get out of the
false position reached by appealing to Formal Logic for the list of
categories. Nevertheless, the mention of a sensuous condition under
which alone the categories can be employed[3] should have suggested to
Kant that the transcendental deduction was defective, and, in fact, in
the second version of the transcendental deduction two paragraphs[4]
are inserted which take account of this sensuous condition.
[2] The cause of Kant's procedure is, of course, to be found
in the unreal way in which he isolates conception from
judgement.
[3] B. 175, M. 106.
[4] B. Secs. 24 and 26, M. Secs. 20 and 22.
The beginning of Kant's account of schematism may be summarized thus:
'Whenever we subsume an individual object of a certain kind, e. g.
a plate, under a conception, e. g. a circle, the object and the
conception must be homogeneous, that is to say, the individual must
possess the characteristic which constitutes the conception, or, in
other words, must be an instance of it. Pure conceptions, however, and
empirical perceptions, i. e. objects of empirical perception, are
quite heterogeneous. We do not, for instance, perceive cases of cause
and effect. Hence the problem arises, 'How is it possible to subsume
objects of empirical perception under pure conceptions?' The
possibility of this subsumption presupposes a _tertium quid_, which is
homogeneous both with the object of empirical perception and with the
conception, and so makes the subsumption mediately possible. This
_tertium quid_ must be, on the one side, intellectual and, on the
other side, sensuous. It is to be found in a 'transcendental
determination of time', i. e. a conception involving time and involved
in experience. For in the first place this is on the one side
intellectual and on the other sensuous, and in the second place it
is so far homogeneous with the category which constitutes its unity
that it is universal and rests on an _a priori_ rule, and so far
homogeneous with the phenomenon that all phenomena are in time[5].
Such transcendental determinations of time are the schemata of the
pure conceptions of the understanding.' Kant continues as follows:
[5] It may be noted that the argument here really fails. For
though phenomena as involving temporal relations, might
possibly be said to be instances of a transcendental
determination of time, the fact that the latter agrees with
the corresponding category by being universal and _a priori_
does not constitute it homogeneous with the category, in the
sense required for subsumption, viz. that it is an instance
of or a species of the category.
"The schema is in itself always a mere product of the imagination.
But since the synthesis of the imagination has for its aim no single
perception, but merely unity in the determination of the sensibility,
the schema should be distinguished from the image. Thus, if I place
five points one after another, . . . . . this is an image of the
number five. On the other hand, if I only just think a number in
general--no matter what it may be, five or a hundred--this thinking
is rather the representation of a method of representing in an image
a group (e. g. a thousand), in conformity with a certain conception,
than the image itself, an image which, in the instance given, I should
find difficulty in surveying and comparing with the conception. Now
this representation of a general procedure of the imagination to
supply its image to a conception, I call the schema of this
conception."
"The fact is that it is not images of objects, but schemata, which lie
at the foundation of our pure sensuous conceptions. No image could
ever be adequate to our conception of a triangle in general. For it
would not attain the generality of the conception which makes it valid
for all triangles, whether right-angled, acute-angled, &c., but would
always be limited to one part only of this sphere. The schema of the
triangle can exist nowhere else than in thought, and signifies a rule
of the synthesis of the imagination in regard to pure figures in
space. An object of experience or an image of it always falls short of
the empirical conception to a far greater degree than does the schema;
the empirical conception always relates immediately to the schema of
the imagination as a rule for the determination of our perception in
conformity with a certain general conception. The conception of 'dog'
signifies a rule according to which my imagination can draw the
general outline of the figure of a four-footed animal, without being
limited to any particular single form which experience presents to me,
or indeed to any possible image that I can represent to myself _in
concreto_. This schematism of our understanding in regard to phenomena
and their mere form is an art hidden in the depths of the human soul,
whose true modes of action we are not likely ever to discover from
Nature and unveil. Thus much only can we say: the _image_ is a product
of the empirical faculty of the productive imagination, while the
_schema_ of sensuous conceptions (such as of figures in space) is a
product and, as it were, a monogram of the pure _a priori_
imagination, through which, and according to which, images first
become possible, though the images must be connected with the
conception only by means of the schema which they express, and are in
themselves not fully adequate to it. On the other hand, the schema of
a pure conception of the understanding is something which cannot be
brought to an image; on the contrary, it is only the pure synthesis in
accordance with a rule of unity according to conceptions in general, a
rule of unity which the category expresses, and it is a transcendental
product of the imagination which concerns the determination of the
inner sense in general according to conditions of its form (time)
with reference to all representations, so far as these are to be
connected _a priori_ in one conception according to the unity of
apperception."[6]
[6] B. 179-81, M. 109-10.
Now, in order to determine whether schemata can constitute the desired
link between the pure conceptions or categories and the manifold of
sense, it is necessary to follow closely this account of a schema.
Kant unquestionably in this passage treats as a mental image related
to a conception what really is, and what on his own theory ought to
have been, an individual object related to a conception, i. e. an
instance of it. In other words, he takes a mental image of an
individual for the individual itself.[7] On the one hand, he treats a
schema of a conception throughout as the thought of a procedure of the
imagination to present to the conception its _image_, and he opposes
schemata not to objects but to _images_; on the other hand, his
problem concerns subsumption under a conception, and what is subsumed
must be an instance of the conception, i. e. an individual object of
the kind in question.[8] Again, in asserting that if I place five
points one after another, . . . . . this is an image of the number
five, he is actually saying that an individual group of five points is
an image of a group of five in general.[9] Further, if the process of
schematizing is to enter--as it must--into knowledge of the phenomenal
world, what Kant here speaks of as the images related to a conception
must be taken to be individual instances of the conception, whatever
his language may be. For, in order to enter into knowledge, the
process referred to must be that by which _objects of experience_ are
constructed. Hence the passage should be interpreted as if throughout
there had been written for 'image' 'individual instance' or more
simply 'instance'. Again, the process of schematizing, although
_introduced_ simply as a process by which an individual is to be
subsumed indirectly under a conception, is assumed in the passage
quoted to be a process of _synthesis_. Hence we may say that the
process of schematizing is a process by which we combine the manifold
of perception into an individual whole in accordance with a
conception, and that the schema of a conception is the thought of the
rule of procedure on our part by which we combine the manifold in
accordance with the conception, and so bring the manifold under the
conception. Thus the schema of the conception of 100 is the thought of
a process of synthesis by which we combine say 10 groups of 10 units
into 100, and the schematizing of the conception of 100 is the process
by which we do so. Here it is essential to notice three points. In the
first place, the schema is a conception which relates not to the
reality apprehended but to us. It is the thought of a rule of
procedure on our part by which an instance of a conception is
constructed, and not the thought of a characteristic of the reality
constructed. For instance, the thought of a rule by which we can
combine points to make 100 is a thought which concerns us and not the
points; it is only the conception corresponding to this schema, viz.
the thought of 100, which concerns the points. In the second place,
although the thought of time is involved in the schema, the succession
in question lies not in the object, but in our act of construction
or apprehension. In the third place, the schema presupposes the
corresponding conception and the process of schematizing directly
brings the manifold of perception under the conception. Thus the
thought of combining 10 groups of 10 units to make 100 presupposes the
thought of 100, and the process of combination brings the units under
the conception of 100.
[7] Cf. pp. 240-1. The mistake is, of course, facilitated by
the fact that 'objects in nature', being for Kant only
'appearances', resemble mental images more closely than they
do as usually conceived.
[8] Cf. B. 176, M. 107. That individuals are really referred
to is also implied in the assertion that 'the synthesis of
imagination has for its aim no single _perception_, but
merely unity in the determination of sensibility'. (The
italics are mine.)
[9] Two sentences treat individual objects and images as
if they might be mentioned indifferently. "An object of
experience or an image of it always falls short of the
empirical conception to a far greater degree than does
the schema." "The conception of a 'dog' signifies a rule
according to which my imagination can draw the general
outline of the figure of a four-footed animal without being
limited to any single particular form which experience
presents to me, or indeed to any possible image that I can
represent to myself _in concreto_."
If, however, we go on to ask what is required of schemata and of the
process of schematizing, if they are to enable the manifold to be
subsumed under the categories, we see that each of these three
characteristics makes it impossible for them to fulfil this purpose.
For firstly, an individual manifold A has to be brought under a
category B. Since _ex hypothesi_ this cannot be effected directly,
there is needed a mediating conception C. C, therefore, it would seem,
must be at once a species of B and a conception of which A is an
instance. In any case C must be a conception relating to the reality
to be known, and not to any process of knowing on our part, and,
again, it must be more concrete than B. This is borne out by the list
of the schemata of the categories. But, although a schema may be said
to be more concrete than the corresponding conception, in that it
presupposes the conception, it neither is nor involves a more
concrete conception of an _object_ and in fact, as has been pointed
out, relates not to the reality to be known but to the process on our
part by which we construct or apprehend it.[10] In the second place,
the time in respect of which the category B has to be made more
concrete must relate to the object, and not to the successive process
by which we apprehend it, whereas the time involved in a schema
concerns the latter and not the former. In the third place, from the
point of view of the categories, the process of schematizing should be
a process whereby we combine the manifold into a whole A in accordance
with the conception C, and thereby render _possible_ the subsumption
of A under the category B. If it be a process which actually subsumes
the manifold under B, it will _actually_ perform that, the very
impossibility of which has made it necessary to postulate such a
process at all. For, according to Kant, it is just the fact that the
manifold cannot be subsumed directly under the categories that renders
schematism necessary. Yet, on Kant's general account of a schema, the
schematizing must actually bring a manifold under the corresponding
conception. If we present to ourselves an individual triangle by
successively joining three lines according to the conception of a
triangle, i. e. so that they enclose a space, we are directly bringing
the manifold, i. e. the lines, under the conception of a triangle.
Again, if we present to ourselves an instance of a group of 100 by
combining 10 groups of 10 units of any kind, we are directly bringing
the units under the conception of 100. If this consideration be
applied to the schematism of a category, we see that the process said
to be necessary because a certain other process is impossible is the
very process said to be impossible.
[10] It may be objected that, from Kant's point of view, the
thought of a rule of construction, and the thought of the
principle of the whole to be constructed, are the same thing
from different points of view. But if this be insisted on,
the schema and its corresponding conception become the same
thing regarded from different points of view; consequently
the schema will not be a more concrete conception of an
object than the corresponding conception, but it will be the
conception itself.
If, therefore, Kant succeeds in finding schemata of the categories in
detail in the sense in which they are required for the solution of his
problem, i. e. in the sense of more concrete conceptions involving the
thought of time and relating to objects, we should expect either that
he ignores his general account of a schema, or that if he appeals to
it, the appeal is irrelevant. This we find to be the case. His account
of the first two transcendental schemata makes a wholly irrelevant
appeal to the temporal process of synthesis on our part, while his
account of the remaining schemata makes no attempt to appeal to it at
all.
"The pure _schema_ of _quantity_, as a conception of the
understanding, is _number_, a representation which comprises the
successive addition of one to one (homogeneous elements). Accordingly,
number is nothing else than the unity of the synthesis of the manifold
of a homogeneous perception in general, in that I generate time itself
in the apprehension of the perception."[11]
[11] B. 182, M. 110.
It is clear that this passage, whatever its precise interpretation may
be,[12] involves a confusion between the thought of counting and that
of number. The thought of number relates to objects of apprehension
and does not involve the thought of time. The thought of counting,
which presupposes the thought of number, relates to our apprehension
of objects and involves the thought of time; it is the thought of a
successive process on our part by which we count the number of units
contained in what we already know to consist of units.[13] Now we must
assume that the schema of quantity is really what Kant says it is,
viz. number, or to express it more accurately, the thought of number,
and not the thought of counting, with which he wrongly identifies it.
For his main problem is to find conceptions which at once are more
concrete than the categories and, at the same time, like the
categories, relate to objects, and the thought of counting, though
more concrete than that of number, does not relate to objects. Three
consequences follow. In the first place, although the schema of
quantity, i. e. the thought of number, is more concrete than the
thought of quantity,[14] it is not, as it should be, more concrete in
respect of time; for the thought of number does not include the
thought of time. Secondly, the thought of time is only introduced into
the schema of quantity irrelevantly by reference to the temporal
process of _counting_, by which we come to apprehend the number of a
given group of units. Thirdly, the schema of quantity is only in
appearance connected with the nature of a schema in general, as Kant
describes it, by a false identification of the thought of number with
the thought of the process on our part by which we count groups of
units, i. e. numbers.
[12] The drift of the passage would seem to be this: 'If we
are to present to ourselves an instance of a quantity, we
must successively combine similar units until they form a
quantity. This process involves the thought of a successive
process by which we add units according to the conception of
a quantity. This thought is the thought of number, and since
by it we present to ourselves an instance of a quantity, it
is the schema of quantity.' But if this be its drift,
considerations of sense demand that it should be rewritten,
at least to the following extent: 'If we are to present to
ourselves an instance of a _particular_ quantity [which will
really be a particular number, for it must be regarded as
discrete, (cf. B. 212, M. 128 fin., 129 init.)] e. g. three,
we must successively combine units until they form _that_
quantity. This process involves the thought of a successive
process, by which we add units according to the conception of
_that_ quantity. This thought is the thought of a particular
number, and since by it we present to ourselves an instance
of _that_ quantity, this thought is the schema of _that_
quantity.' If this rewriting be admitted to be necessary, it
must be allowed that Kant has confused (_a_) the thoughts of
particular quantities and of particular numbers with those of
quantity and of number in general respectively, (_b_) the
thought of a particular quantity with that of a particular
number (for the process referred to presupposes that the
particular quantity taken is known to consist of a number of
equal units) and (_c_) the thought of counting with that of
number.
[13] This statement is, of course, not meant as a definition
of counting, but as a means of bringing out the distinction
between a process of counting and a number.
[14] For the thought of a number is the thought of a quantity
of a special kind, viz. of a quantity made up of a number of
similar units without remainder.
The account of the schema of reality, the second category, runs as
follows: "Reality is in the pure conception of the understanding that
which corresponds to a sensation in general, that therefore of which
the conception in itself indicates a being (in time), while negation
is that of which the conception indicates a not being (in time). Their
opposition, therefore, arises in the distinction between one and the
same time as filled or empty. Since time is only the form of
perception, consequently of objects as phenomena, that which in
objects corresponds to sensation is the transcendental matter of all
objects as things in themselves (thinghood, reality).[15] Now every
sensation has a degree or magnitude by which it can fill the same
time, i. e. the internal sense, in respect of the same representation
of an object, more or less, until it vanishes into nothing ( = 0 =
_negatio_). There is, therefore, a relation and connexion between
reality and negation, or rather a transition from the former to the
latter, which makes every reality representable as a _quantum_; and
the schema of a reality, as the quantity of something so far as it
fills time, is just this continuous and uniform generation of the
reality in time, as we descend in time from the sensation which has a
certain degree, down to the vanishing thereof, or gradually ascend
from negation to the magnitude thereof."[16]
[15] It is difficult to see how Kant could meet the criticism
that here, contrary to his intention, he is treating physical
objects as things in themselves. Cf. p. 265.
[16] B. 182-3, M. 110-11.
This passage, if it be taken in connexion with the account of the
anticipations of perception,[17] seems to have the following meaning:
'In thinking of something as a reality, we think of it as that which
corresponds to, i. e. produces, a sensation, and therefore as
something which, like the sensation, is in time; and just as every
sensation, which, as such, occupies time, has a certain degree of
intensity, so has the reality which produces it. Now to produce for
ourselves an instance of a reality in this sense, we must add units of
reality till a reality of the required degree is produced, and the
thought of this method on our part of constructing an individual
reality is the schema of reality.' But if this represents Kant's
meaning, the schema of reality relates only to our process of
apprehension, and therefore is not a conception which relates to
objects and is more concrete than the corresponding category in
respect of time. Moreover, it is matter for surprise that in the case
of this category Kant should have thought schematism necessary, for
time is actually included in his own statement of the category.
[17] B. 207-18, M. 125-32.
The account of the schemata of the remaining categories need not be
considered. It merely _asserts_ that certain conceptions relating
to objects and involving the thought of time are the schemata
corresponding to the remaining categories, without any attempt to
connect them with the nature of a schema. Thus, the schema of
substance is asserted to be the _permanence_ of the real _in time_,
that of cause the _succession_ of the manifold, in so far as that
succession is subjected to a rule, that of interaction the
_coexistence_ of the determinations or accidents of one substance
with those of another according to a universal rule.[18] Again, the
schemata of possibility, of actuality and of necessity are said to be
respectively the accordance of the synthesis of representations with
the conditions of time in general, existence in a determined time, and
existence of an object in all time.
[18] The italics are mine.
The main confusion pervading the chapter is of course that between
temporal relations which concern the process of apprehension and
temporal relations which concern the realities apprehended. Kant is
continually referring to the former as if they were the latter. The
cause of this confusion lies in Kant's reduction of physical realities
to representations. Since, according to him, these realities are only
our representations, all temporal relations are really relations of
our representations, and these relations have to be treated at one
time as relations of our apprehensions, and at another as relations of
the realities apprehended, as the context requires.