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The Critique of Pure Reason by Immanuel Kant

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must be contemplated always as subject and never as mere predicate.
And so with all the other categories.

SS 11. SECTION II Transcendental Deduction of the pure Conceptions of
the Understanding.

Of the Possibility of a Conjunction of the manifold representations
given by Sense.

The manifold content in our representations can be given in an
intuition which is merely sensuous--in other words, is nothing but
susceptibility; and the form of this intuition can exist a priori in
our faculty of representation, without being anything else but the
mode in which the subject is affected. But the conjunction
(conjunctio) of a manifold in intuition never can be given us by the
senses; it cannot therefore be contained in the pure form of
sensuous intuition, for it is a spontaneous act of the faculty of
representation. And as we must, to distinguish it from sensibility,
entitle this faculty understanding; so all conjunction whether
conscious or unconscious, be it of the manifold in intuition, sensuous
or non-sensuous, or of several conceptions--is an act of the
understanding. To this act we shall give the general appellation of
synthesis, thereby to indicate, at the same time, that we cannot
represent anything as conjoined in the object without having
previously conjoined it ourselves. Of all mental notions, that of
conjunction is the only one which cannot be given through objects,
but can be originated only by the subject itself, because it is an
act of its purely spontaneous activity. The reader will easily enough
perceive that the possibility of conjunction must be grounded in the
very nature of this act, and that it must be equally valid for all
conjunction, and that analysis, which appears to be its contrary,
must, nevertheless, always presuppose it; for where the
understanding has not previously conjoined, it cannot dissect or
analyse, because only as conjoined by it, must that which is to be
analysed have been given to our faculty of representation.

But the conception of conjunction includes, besides the conception
of the manifold and of the synthesis of it, that of the unity of it
also. Conjunction is the representation of the synthetical unity of
the manifold.* This idea of unity, therefore, cannot arise out of that
of conjunction; much rather does that idea, by combining itself with
the representation of the manifold, render the conception of
conjunction possible. This unity, which a priori precedes all
conceptions of conjunction, is not the category of unity (SS 6); for
all the categories are based upon logical functions of judgement,
and in these functions we already have conjunction, and consequently
unity of given conceptions. It is therefore evident that the
category of unity presupposes conjunction. We must therefore look
still higher for this unity (as qualitative, SS 8), in that, namely,
which contains the ground of the unity of diverse conceptions in
judgements, the ground, consequently, of the possibility of the
existence of the understanding, even in regard to its logical use.

[*Footnote: Whether the representations are in themselves identical,
and consequently whether one can be thought analytically by means of
and through the other, is a question which we need not at present
consider. Our Consciousness of the one, when we speak of the manifold,
is always distinguishable from our consciousness of the other; and
it is only respecting the synthesis of this (possible) consciousness
that we here treat.]

SS 12. Of the Originally Synthetical Unity of Apperception.

The "I think" must accompany all my representations, for otherwise
something would be represented in me which could not be thought; in
other words, the representation would either be impossible, or at
least be, in relation to me, nothing. That representation which can
be given previously to all thought is called intuition. All the diversity
or manifold content of intuition, has, therefore, a necessary relation
to the "I think," in the subject in which this diversity is found.
But this representation, "I think," is an act of spontaneity; that
is to say, it cannot be regarded as belonging to mere sensibility.
I call it pure apperception, in order to distinguish it from empirical;
or primitive apperception, because it is self-consciousness which,
whilst it gives birth to the representation "I think," must necessarily
be capable of accompanying all our representations. It is in all acts
of consciousness one and the same, and unaccompanied by it, no
representation can exist for me. The unity of this apperception I call
the transcendental unity of self-consciousness, in order to indicate
the possibility of a priori cognition arising from it. For the
manifold representations which are given in an intuition would not
all of them be my representations, if they did not all belong to one
self-consciousness, that is, as my representations (even although I
am not conscious of them as such), they must conform to the condition
under which alone they can exist together in a common
self-consciousness, because otherwise they would not all without
exception belong to me. From this primitive conjunction follow many
important results.

For example, this universal identity of the apperception of the
manifold given in intuition contains a synthesis of representations
and is possible only by means of the consciousness of this
synthesis. For the empirical consciousness which accompanies different
representations is in itself fragmentary and disunited, and without
relation to the identity of the subject. This relation, then, does
not exist because I accompany every representation with consciousness,
but because I join one representation to another, and am conscious
of the synthesis of them. Consequently, only because I can connect
a variety of given representations in one consciousness, is it
possible that I can represent to myself the identity of
consciousness in these representations; in other words, the analytical
unity of apperception is possible only under the presupposition of
a synthetical unity.* The thought, "These representations given in
intuition belong all of them to me," is accordingly just the same
as, "I unite them in one self-consciousness, or can at least so
unite them"; and although this thought is not itself the consciousness
of the synthesis of representations, it presupposes the possibility
of it; that is to say, for the reason alone that I can comprehend the
variety of my representations in one consciousness, do I call them
my representations, for otherwise I must have as many-coloured and
various a self as are the representations of which I am conscious.
Synthetical unity of the manifold in intuitions, as given a priori,
is therefore the foundation of the identity of apperception itself,
which antecedes a priori all determinate thought. But the conjunction
of representations into a conception is not to be found in objects
themselves, nor can it be, as it were, borrowed from them and taken
up into the understanding by perception, but it is on the contrary
an operation of the understanding itself, which is nothing more than
the faculty of conjoining a priori and of bringing the variety of
given representations under the unity of apperception. This
principle is the highest in all human cognition.

[*Footnote: All general conceptions--as such--depend, for their existence,
on the analytical unity of consciousness. For example, when I think
of red in general, I thereby think to myself a property which (as a
characteristic mark) can be discovered somewhere, or can be united
with other representations; consequently, it is only by means of a
forethought possible synthetical unity that I can think to myself
the analytical. A representation which is cogitated as common to
different representations, is regarded as belonging to such as,
besides this common representation, contain something different;
consequently it must be previously thought in synthetical unity with
other although only possible representations, before I can think in
it the analytical unity of consciousness which makes it a conceptas
communis. And thus the synthetical unity of apperception is the
highest point with which we must connect every operation of the
understanding, even the whole of logic, and after it our
transcendental philosophy; indeed, this faculty is the understanding

This fundamental principle of the necessary unity of apperception is
indeed an identical, and therefore analytical, proposition; but it
nevertheless explains the necessity for a synthesis of the manifold
given in an intuition, without which the identity of
self-consciousness would be incogitable. For the ego, as a simple
representation, presents us with no manifold content; only in
intuition, which is quite different from the representation ego, can
it be given us, and by means of conjunction it is cogitated in one
self-consciousness. An understanding, in which all the manifold should
be given by means of consciousness itself, would be intuitive; our
understanding can only think and must look for its intuition to sense.
I am, therefore, conscious of my identical self, in relation to all
the variety of representations given to me in an intuition, because
I call all of them my representations. In other words, I am
conscious myself of a necessary a priori synthesis of my
representations, which is called the original synthetical unity of
apperception, under which rank all the representations presented to
me, but that only by means of a synthesis.

SS 13. The Principle of the Synthetical Unity of Apperception is
the highest Principle of all exercise of the Understanding.

The supreme principle of the possibility of all intuition in
relation to sensibility was, according to our transcendental
aesthetic, that all the manifold in intuition be subject to the formal
conditions of space and time. The supreme principle of the possibility
of it in relation to the understanding is that all the manifold in
it be subject to conditions of the originally synthetical unity or
apperception.* To the former of these two principles are subject all
the various representations of intuition, in so far as they are
given to us; to the latter, in so far as they must be capable of
conjunction in one consciousness; for without this nothing can be
thought or cognized, because the given representations would not
have in common the act Of the apperception "I think" and therefore
could not be connected in one self-consciousness.

[*Footnote: Space and time, and all portions thereof, are intuitions;
consequently are, with a manifold for their content, single
representations. (See the Transcendental Aesthetic.) Consequently,
they are not pure conceptions, by means of which the same
consciousness is found in a great number of representations; but, on
the contrary, they are many representations contained in one, the
consciousness of which is, so to speak, compounded. The unity of
consciousness is nevertheless synthetical and, therefore, primitive.
From this peculiar character of consciousness follow many important
consequences. (See SS 21.)]

Understanding is, to speak generally, the faculty Of cognitions.
These consist in the determined relation of given representation to
an object. But an object is that, in the conception of which the manifold
in a given intuition is united. Now all union of representations
requires unity of consciousness in the synthesis of them.
Consequently, it is the unity of consciousness alone that
constitutes the possibility of representations relating to an
object, and therefore of their objective validity, and of their
becoming cognitions, and consequently, the possibility of the
existence of the understanding itself.

The first pure cognition of understanding, then, upon which is
founded all its other exercise, and which is at the same time
perfectly independent of all conditions of mere sensuous intuition,
is the principle of the original synthetical unity of apperception.
Thus the mere form of external sensuous intuition, namely, space,
affords us, per se, no cognition; it merely contributes the manifold
in a priori intuition to a possible cognition. But, in order to
cognize something in space (for example, a line), I must draw it,
and thus produce synthetically a determined conjunction of the given
manifold, so that the unity of this act is at the same time the
unity of consciousness (in the conception of a line), and by this
means alone is an object (a determinate space) cognized. The
synthetical unity of consciousness is, therefore, an objective
condition of all cognition, which I do not merely require in order
to cognize an object, but to which every intuition must necessarily
be subject, in order to become an object for me; because in any other
way, and without this synthesis, the manifold in intuition could not
be united in one consciousness.

This proposition is, as already said, itself analytical, although it
constitutes the synthetical unity, the condition of all thought; for
it states nothing more than that all my representations in any given
intuition must be subject to the condition which alone enables me to
connect them, as my representation with the identical self, and so
to unite them synthetically in one apperception, by means of the
general expression, "I think."

But this principle is not to be regarded as a principle for every
possible understanding, but only for the understanding by means of
whose pure apperception in the thought I am, no manifold content is
given. The understanding or mind which contained the manifold in
intuition, in and through the act itself of its own
self-consciousness, in other words, an understanding by and in the
representation of which the objects of the representation should at
the same time exist, would not require a special act of synthesis of
the manifold as the condition of the unity of its consciousness, an
act of which the human understanding, which thinks only and cannot
intuite, has absolute need. But this principle is the first
principle of all the operations of our understanding, so that we
cannot form the least conception of any other possible
understanding, either of one such as should be itself intuition, or
possess a sensuous intuition, but with forms different from those of
space and time.

SS 14. What Objective Unity of Self-consciousness is.

It is by means of the transcendental unity of apperception that
all the manifold, given in an intuition is united into a conception
of the object. On this account it is called objective, and must be
distinguished from the subjective unity of consciousness, which is
a determination of the internal sense, by means of which the said
manifold in intuition is given empirically to be so united. Whether
I can be empirically conscious of the manifold as coexistent or as
successive, depends upon circumstances, or empirical conditions. Hence
the empirical unity of consciousness by means of association of
representations, itself relates to a phenomenal world and is wholly
contingent. On the contrary, the pure form of intuition in time,
merely as an intuition, which contains a given manifold, is subject
to the original unity of consciousness, and that solely by means of
the necessary relation of the manifold in intuition to the "I think,"
consequently by means of the pure synthesis of the understanding,
which lies a priori at the foundation of all empirical synthesis.
The transcendental unity of apperception is alone objectively valid;
the empirical which we do not consider in this essay, and which is
merely a unity deduced from the former under given conditions in
concreto, possesses only subjective validity. One person connects
the notion conveyed in a word with one thing, another with another
thing; and the unity of consciousness in that which is empirical,
is, in relation to that which is given by experience, not
necessarily and universally valid.

SS 15. The Logical Form of all Judgements consists in the Objective
Unity of Apperception of the Conceptions contained therein.

I could never satisfy myself with the definition which logicians
give of a judgement. It is, according to them, the representation of
a relation between two conceptions. I shall not dwell here on the
faultiness of this definition, in that it suits only for categorical
and not for hypothetical or disjunctive judgements, these latter
containing a relation not of conceptions but of judgements themselves--
a blunder from which many evil results have followed.* It is more
important for our present purpose to observe, that this definition
does not determine in what the said relation consists.

[*Footnote: The tedious doctrine of the four syllogistic figures concerns
only categorical syllogisms; and although it is nothing more than an
artifice by surreptitiously introducing immediate conclusions
(consequentiae immediatae) among the premises of a pure syllogism,
to give ism' give rise to an appearance of more modes of drawing a
conclusion than that in the first figure, the artifice would not
have had much success, had not its authors succeeded in bringing
categorical judgements into exclusive respect, as those to which all
others must be referred--a doctrine, however, which, according to SS
5, is utterly false.]

But if I investigate more closely the relation of given cognitions
in every judgement, and distinguish it, as belonging to the
understanding, from the relation which is produced according to laws
of the reproductive imagination (which has only subjective
validity), I find that judgement is nothing but the mode of bringing
given cognitions under the objective unit of apperception. This is
plain from our use of the term of relation is in judgements, in
order to distinguish the objective unity of given representations from
the subjective unity. For this term indicates the relation of these
representations to the original apperception, and also their necessary
unity, even although the judgement is empirical, therefore contingent,
as in the judgement: "All bodies are heavy." I do not mean by this,
that these representations do necessarily belong to each other in
empirical intuition, but that by means of the necessary unity of
appreciation they belong to each other in the synthesis of intuitions,
that is to say, they belong to each other according to principles of
the objective determination of all our representations, in so far as
cognition can arise from them, these principles being all deduced from
the main principle of the transcendental unity of apperception. In
this way alone can there arise from this relation a judgement, that
is, a relation which has objective validity, and is perfectly distinct
from that relation of the very same representations which has only
subjective validity--a relation, to wit, which is produced according
to laws of association. According to these laws, I could only say:
"When I hold in my hand or carry a body, I feel an impression of
weight"; but I could not say: "It, the body, is heavy"; for this is
tantamount to saying both these representations are conjoined in the
object, that is, without distinction as to the condition of the
subject, and do not merely stand together in my perception, however
frequently the perceptive act may be repeated.

SS 16. All Sensuous Intuitions are subject to the Categories, as
Conditions under which alone the manifold Content of them
can be united in one Consciousness.

The manifold content given in a sensuous intuition comes necessarily
under the original synthetical unity of apperception, because
thereby alone is the unity of intuition possible (SS 13). But that
act of the understanding, by which the manifold content of given
representations (whether intuitions or conceptions) is brought under
one apperception, is the logical function of judgements (SS 15). All
the manifold, therefore, in so far as it is given in one empirical
intuition, is determined in relation to one of the logical functions
of judgement, by means of which it is brought into union in one
consciousness. Now the categories are nothing else than these
functions of judgement so far as the manifold in a given intuition
is determined in relation to them (SS 9). Consequently, the manifold
in a given intuition is necessarily subject to the categories of the

SS 17. Observation.

The manifold in an intuition, which I call mine, is represented by
means of the synthesis of the understanding, as belonging to the
necessary unity of self-consciousness, and this takes place by means
of the category.* The category indicates accordingly that the
empirical consciousness of a given manifold in an intuition is subject
to a pure self-consciousness a priori, in the same manner as an
empirical intuition is subject to a pure sensuous intuition, which
is also a priori. In the above proposition, then, lies the beginning
of a deduction of the pure conceptions of the understanding. Now, as
the categories have their origin in the understanding alone,
independently of sensibility, I must in my deduction make
abstraction of the mode in which the manifold of an empirical
intuition is given, in order to fix my attention exclusively on the
unity which is brought by the understanding into the intuition by
means of the category. In what follows (SS 22), it will be shown, from
the mode in which the empirical intuition is given in the faculty of
sensibility, that the unity which belongs to it is no other than
that which the category (according to SS 16) imposes on the manifold
in a given intuition, and thus, its a priori validity in regard to
all objects of sense being established, the purpose of our deduction
will be fully attained.

[*Footnote: The proof of this rests on the represented unity of intuition,
by means of which an object is given, and which always includes in
itself a synthesis of the manifold to be intuited, and also the relation
of this latter to unity of apperception.]

But there is one thing in the above demonstration of which I could
not make abstraction, namely, that the manifold to be intuited must
be given previously to the synthesis of the understanding, and
independently of it. How this takes place remains here undetermined.
For if I cogitate an understanding which was itself intuitive (as,
for example, a divine understanding which should not represent given
objects, but by whose representation the objects themselves should
be given or produced), the categories would possess no significance
in relation to such a faculty of cognition. They are merely rules for
an understanding, whose whole power consists in thought, that is, in
the act of submitting the synthesis of the manifold which is presented
to it in intuition from a very different quarter, to the unity of
apperception; a faculty, therefore, which cognizes nothing per se,
but only connects and arranges the material of cognition, the intuition,
namely, which must be presented to it by means of the object. But to
show reasons for this peculiar character of our understandings, that
it produces unity of apperception a priori only by means of
categories, and a certain kind and number thereof, is as impossible
as to explain why we are endowed with precisely so many functions of
judgement and no more, or why time and space are the only forms of
our intuition.

SS 18. In Cognition, its Application to Objects of Experience is
the only legitimate use of the Category.

To think an object and to cognize an object are by no means the same
thing. In cognition there are two elements: firstly, the conception,
whereby an object is cogitated (the category); and, secondly, the
intuition, whereby the object is given. For supposing that to the
conception a corresponding intuition could not be given, it would
still be a thought as regards its form, but without any object, and
no cognition of anything would be possible by means of it, inasmuch
as, so far as I knew, there existed and could exist nothing to which
my thought could be applied. Now all intuition possible to us is
sensuous; consequently, our thought of an object by means of a pure
conception of the understanding, can become cognition for us only in
so far as this conception is applied to objects of the senses.
Sensuous intuition is either pure intuition (space and time) or
empirical intuition--of that which is immediately represented in space
and time by means of sensation as real. Through the determination of
pure intuition we obtain a priori cognitions of objects, as in
mathematics, but only as regards their form as phenomena; whether
there can exist things which must be intuited in this form is not
thereby established. All mathematical conceptions, therefore, are
not per se cognition, except in so far as we presuppose that there
exist things which can only be represented conformably to the form
of our pure sensuous intuition. But things in space and time are given
only in so far as they are perceptions (representations accompanied
with sensation), therefore only by empirical representation.
Consequently the pure conceptions of the understanding, even when they
are applied to intuitions a priori (as in mathematics), produce
cognition only in so far as these (and therefore the conceptions of
the understanding by means of them) can be applied to empirical
intuitions. Consequently the categories do not, even by means of
pure intuition afford us any cognition of things; they can only do
so in so far as they can be applied to empirical intuition. That is
to say, the, categories serve only to render empirical cognition
possible. But this is what we call experience. Consequently, in
cognition, their application to objects of experience is the only
legitimate use of the categories.

SS 19.

The foregoing proposition is of the utmost importance, for it
determines the limits of the exercise of the pure conceptions of the
understanding in regard to objects, just as transcendental aesthetic
determined the limits of the exercise of the pure form of our sensuous
intuition. Space and time, as conditions of the possibility of the
presentation of objects to us, are valid no further than for objects
of sense, consequently, only for experience. Beyond these limits
they represent to us nothing, for they belong only to sense, and
have no reality apart from it. The pure conceptions of the
understanding are free from this limitation, and extend to objects
of intuition in general, be the intuition like or unlike to ours,
provided only it be sensuous, and not intellectual. But this extension
of conceptions beyond the range of our intuition is of no advantage;
for they are then mere empty conceptions of objects, as to the
possibility or impossibility of the existence of which they furnish
us with no means of discovery. They are mere forms of thought, without
objective reality, because we have no intuition to which the
synthetical unity of apperception, which alone the categories contain,
could be applied, for the purpose of determining an object. Our
sensuous and empirical intuition can alone give them significance
and meaning.

If, then, we suppose an object of a non-sensuous intuition to be
given we can in that case represent it by all those predicates which
are implied in the presupposition that nothing appertaining to
sensuous intuition belongs to it; for example, that it is not
extended, or in space; that its duration is not time; that in it no
change (the effect of the determinations in time) is to be met with,
and so on. But it is no proper knowledge if I merely indicate what
the intuition of the object is not, without being able to say what
is contained in it, for I have not shown the possibility of an object
to which my pure conception of understanding could be applicable,
because I have not been able to furnish any intuition corresponding
to it, but am only able to say that our intuition is not valid for
it. But the most important point is this, that to a something of this
kind not one category can be found applicable. Take, for example, the
conception of substance, that is, something that can exist as subject,
but never as mere predicate; in regard to this conception I am quite
ignorant whether there can really be anything to correspond to such
a determination of thought, if empirical intuition did not afford me
the occasion for its application. But of this more in the sequel.

SS 20. Of the Application of the Categories to Objects of the
Senses in general.

The pure conceptions of the understanding apply to objects of
intuition in general, through the understanding alone, whether the
intuition be our own or some other, provided only it be sensuous,
but are, for this very reason, mere forms of thought, by means of
which alone no determined object can be cognized. The synthesis or
conjunction of the manifold in these conceptions relates, we have
said, only to the unity of apperception, and is for this reason the
ground of the possibility of a priori cognition, in so far as this
cognition is dependent on the understanding. This synthesis is,
therefore, not merely transcendental, but also purely intellectual.
But because a certain form of sensuous intuition exists in the mind
a priori which rests on the receptivity of the representative
faculty (sensibility), the understanding, as a spontaneity, is able
to determine the internal sense by means of the diversity of given
representations, conformably to the synthetical unity of apperception,
and thus to cogitate the synthetical unity of the apperception of
the manifold of sensuous intuition a priori, as the condition to which
must necessarily be submitted all objects of human intuition. And in
this manner the categories as mere forms of thought receive
objective reality, that is, application to objects which are given
to us in intuition, but that only as phenomena, for it is only of
phenomena that we are capable of a priori intuition.

This synthesis of the manifold of sensuous intuition, which is
possible and necessary a priori, may be called figurative (synthesis
speciosa), in contradistinction to that which is cogitated in the mere
category in regard to the manifold of an intuition in general, and
is called connection or conjunction of the understanding (synthesis
intellectualis). Both are transcendental, not merely because they
themselves precede a priori all experience, but also because they form
the basis for the possibility of other cognition a priori.

But the figurative synthesis, when it has relation only to the
originally synthetical unity of apperception, that is to the
transcendental unity cogitated in the categories, must, to be
distinguished from the purely intellectual conjunction, be entitled
the transcendental synthesis of imagination. Imagination is the
faculty of representing an object even without its presence in
intuition. Now, as all our intuition is sensuous, imagination, by
reason of the subjective condition under which alone it can give a
corresponding intuition to the conceptions of the understanding,
belongs to sensibility. But in so far as the synthesis of the
imagination is an act of spontaneity, which is determinative, and not,
like sense, merely determinable, and which is consequently able to
determine sense a priori, according to its form, conformably to the
unity of apperception, in so far is the imagination a faculty of
determining sensibility a priori, and its synthesis of intuitions
according to the categories must be the transcendental synthesis of
the imagination. It is an operation of the understanding on
sensibility, and the first application of the understanding to objects
of possible intuition, and at the same time the basis for the exercise
of the other functions of that faculty. As figurative, it is
distinguished from the merely intellectual synthesis, which is
produced by the understanding alone, without the aid of imagination.
Now, in so far as imagination is spontaneity, I sometimes call it also
the productive imagination, and distinguish it from the
reproductive, the synthesis of which is subject entirely to
empirical laws, those of association, namely, and which, therefore,
contributes nothing to the explanation of the possibility of a
priori cognition, and for this reason belongs not to transcendental
philosophy, but to psychology.

We have now arrived at the proper place for explaining the paradox
which must have struck every one in our exposition of the internal
sense (SS 6), namely--how this sense represents us to our own
consciousness, only as we appear to ourselves, not as we are in
ourselves, because, to wit, we intuite ourselves only as we are
inwardly affected. Now this appears to be contradictory, inasmuch as
we thus stand in a passive relation to ourselves; and therefore in
the systems of psychology, the internal sense is commonly held to be
one with the faculty of apperception, while we, on the contrary, carefully
distinguish them.

That which determines the internal sense is the understanding, and
its original power of conjoining the manifold of intuition, that is,
of bringing this under an apperception (upon which rests the
possibility of the understanding itself). Now, as the human
understanding is not in itself a faculty of intuition, and is unable
to exercise such a power, in order to conjoin, as it were, the
manifold of its own intuition, the synthesis of understanding is,
considered per se, nothing but the unity of action, of which, as such,
it is self-conscious, even apart from sensibility, by which, moreover,
it is able to determine our internal sense in respect of the
manifold which may be presented to it according to the form of
sensuous intuition. Thus, under the name of a transcendental synthesis
of imagination, the understanding exercises an activity upon the
passive subject, whose faculty it is; and so we are right in saying
that the internal sense is affected thereby. Apperception and its
synthetical unity are by no means one and the same with the internal
sense. The former, as the source of all our synthetical conjunction,
applies, under the name of the categories, to the manifold of
intuition in general, prior to all sensuous intuition of objects.
The internal sense, on the contrary, contains merely the form of
intuition, but without any synthetical conjunction of the manifold
therein, and consequently does not contain any determined intuition,
which is possible only through consciousness of the determination of
the manifold by the transcendental act of the imagination (synthetical
influence of the understanding on the internal sense), which I have
named figurative synthesis.

This we can indeed always perceive in ourselves. We cannot
cogitate a geometrical line without drawing it in thought, nor a
circle without describing it, nor represent the three dimensions of
space without drawing three lines from the same point perpendicular
to one another. We cannot even cogitate time, unless, in drawing a
straight line (which is to serve as the external figurative
representation of time), we fix our attention on the act of the
synthesis of the manifold, whereby we determine successively the
internal sense, and thus attend also to the succession of this
determination. Motion as an act of the subject (not as a determination
of an object),* consequently the synthesis of the manifold in space,
if we make abstraction of space and attend merely to the act by
which we determine the internal sense according to its form, is that
which produces the conception of succession. The understanding,
therefore, does by no means find in the internal sense any such
synthesis of the manifold, but produces it, in that it affects this
sense. At the same time, how "I who think" is distinct from the "I"
which intuites itself (other modes of intuition being cogitable as
at least possible), and yet one and the same with this latter as the
same subject; how, therefore, I am able to say: "I, as an intelligence
and thinking subject, cognize myself as an object thought, so far as
I am, moreover, given to myself in intuition--only, like other
phenomena, not as I am in myself, and as considered by the
understanding, but merely as I appear"--is a question that has in it
neither more nor less difficulty than the question--"How can I be an
object to myself?" or this--"How I can be an object of my own
intuition and internal perceptions?" But that such must be the fact,
if we admit that space is merely a pure form of the phenomena of
external sense, can be clearly proved by the consideration that we
cannot represent time, which is not an object of external intuition,
in any other way than under the image of a line, which we draw in
thought, a mode of representation without which we could not cognize
the unity of its dimension, and also that we are necessitated to
take our determination of periods of time, or of points of time, for
all our internal perceptions from the changes which we perceive in
outward things. It follows that we must arrange the determinations
of the internal sense, as phenomena in time, exactly in the same
manner as we arrange those of the external senses in space. And
consequently, if we grant, respecting this latter, that by means of
them we know objects only in so far as we are affected externally,
we must also confess, with regard to the internal sense, that by means
of it we intuite ourselves only as we are internally affected by
ourselves; in other words, as regards internal intuition, we cognize
our own subject only as phenomenon, and not as it is in itself.*[2]

[*Footnote: Motion of an object in space does not belong to a pure
science, consequently not to geometry; because, that a thing is movable
cannot be known a priori, but only from experience. But motion,
considered as the description of a space, is a pure act of the
successive synthesis of the manifold in external intuition by means
of productive imagination, and belongs not only to geometry, but even
to transcendental philosophy.]

[*[2]Footnote: I do not see why so much difficulty should be found
in admitting that our internal sense is affected by ourselves. Every
act of attention exemplifies it. In such an act the understanding
determines the internal sense by the synthetical conjunction which
it cogitates, conformably to the internal intuition which
corresponds to the manifold in the synthesis of the understanding.
How much the mind is usually affected thereby every one will be able
to perceive in himself.]

SS 21.

On the other hand, in the transcendental synthesis of the manifold
content of representations, consequently in the synthetical unity of
apperception, I am conscious of myself, not as I appear to myself,
nor as I am in myself, but only that "I am." This representation is
a thought, not an intuition. Now, as in order to cognize ourselves,
in addition to the act of thinking, which subjects the manifold of
every possible intuition to the unity of apperception, there is
necessary a determinate mode of intuition, whereby this manifold is
given; although my own existence is certainly not mere phenomenon
(much less mere illusion), the determination of my existence* Can only
take place conformably to the form of the internal sense, according
to the particular mode in which the manifold which I conjoin is given
in internal intuition, and I have therefore no knowledge of myself
as I am, but merely as I appear to myself. The consciousness of self
is thus very far from a knowledge of self, in which I do not use the
categories, whereby I cogitate an object, by means of the
conjunction of the manifold in one apperception. In the same way as
I require, for the sake of the cognition of an object distinct from
myself, not only the thought of an object in general (in the
category), but also an intuition by which to determine that general
conception, in the same way do I require, in order to the cognition
of myself, not only the consciousness of myself or the thought that
I think myself, but in addition an intuition of the manifold in
myself, by which to determine this thought. It is true that I exist
as an intelligence which is conscious only of its faculty of
conjunction or synthesis, but subjected in relation to the manifold
which this intelligence has to conjoin to a limitative conjunction
called the internal sense. My intelligence (that is, I) can render
that conjunction or synthesis perceptible only according to the
relations of time, which are quite beyond the proper sphere of the
conceptions of the understanding and consequently cognize itself in
respect to an intuition (which cannot possibly be intellectual, nor
given by the understanding), only as it appears to itself, and not
as it would cognize itself, if its intuition were intellectual.

[*Footnote: The "I think" expresses the act of determining my own
existence. My existence is thus already given by the act of consciousness;
but the mode in which I must determine my existence, that is, the mode
in which I must place the manifold belonging to my existence, is not
thereby given. For this purpose intuition of self is required, and
this intuition possesses a form given a priori, namely, time, which
is sensuous, and belongs to our receptivity of the determinable. Now,
as I do not possess another intuition of self which gives the
determining in me (of the spontaneity of which I am conscious),
prior to the act of determination, in the same manner as time gives
the determinable, it is clear that I am unable to determine my own
existence as that of a spontaneous being, but I am only able to
represent to myself the spontaneity of my thought, that is, of my
determination, and my existence remains ever determinable in a
purely sensuous manner, that is to say, like the existence of a
phenomenon. But it is because of this spontaneity that I call myself
an intelligence.]

SS 22. Transcendental Deduction of the universally possible employment
in experience of the Pure Conceptions of the Understanding.

In the metaphysical deduction, the a priori origin of categories was
proved by their complete accordance with the general logical of
thought; in the transcendental deduction was exhibited the possibility
of the categories as a priori cognitions of objects of an intuition
in general (SS 16 and 17).At present we are about to explain the
possibility of cognizing, a priori, by means of the categories, all
objects which can possibly be presented to our senses, not, indeed,
according to the form of their intuition, but according to the laws
of their conjunction or synthesis, and thus, as it were, of prescribing
laws to nature and even of rendering nature possible. For if the
categories were inadequate to this task, it would not be evident to
us why everything that is presented to our senses must be subject to
those laws which have an a priori origin in the understanding itself.

I premise that by the term synthesis of apprehension I understand
the combination of the manifold in an empirical intuition, whereby
perception, that is, empirical consciousness of the intuition (as
phenomenon), is possible.

We have a priori forms of the external and internal sensuous
intuition in the representations of space and time, and to these
must the synthesis of apprehension of the manifold in a phenomenon
be always comformable, because the synthesis itself can only take
place according to these forms. But space and time are not merely
forms of sensuous intuition, but intuitions themselves (which
contain a manifold), and therefore contain a priori the
determination of the unity of this manifold.* (See the Transcendent
Aesthetic.) Therefore is unity of the synthesis of the manifold
without or within us, consequently also a conjunction to which all
that is to be represented as determined in space or time must
correspond, given a priori along with (not in) these intuitions, as
the condition of the synthesis of all apprehension of them. But this
synthetical unity can be no other than that of the conjunction of
the manifold of a given intuition in general, in a primitive act of
consciousness, according to the categories, but applied to our
sensuous intuition. Consequently all synthesis, whereby alone is
even perception possible, is subject to the categories. And, as
experience is cognition by means of conjoined perceptions, the
categories are conditions of the possibility of experience and are
therefore valid a priori for all objects of experience.

[*Footnote: Space represented as an object (as geometry really requires
it to be) contains more than the mere form of the intuition; namely,
a combination of the manifold given according to the form of sensibility
into a representation that can be intuited; so that the form of the
intuition gives us merely the manifold, but the formal intuition gives
unity of representation. In the aesthetic, I regarded this unity as
belonging entirely to sensibility, for the purpose of indicating
that it antecedes all conceptions, although it presupposes a synthesis
which does not belong to sense, through which alone, however, all
our conceptions of space and time are possible. For as by means of
this unity alone (the understanding determining the sensibility) space
and time are given as intuitions, it follows that the unity of this
intuition a priori belongs to space and time, and not to the
conception of the understanding (SS 20).]

When, then, for example, I make the empirical intuition of a house
by apprehension of the manifold contained therein into a perception,
the necessary unity of space and of my external sensuous intuition
lies at the foundation of this act, and I, as it were, draw the form
of the house conformably to this synthetical unity of the manifold
in space. But this very synthetical unity remains, even when I
abstract the form of space, and has its seat in the understanding,
and is in fact the category of the synthesis of the homogeneous in
an intuition; that is to say, the category of quantity, to which the
aforesaid synthesis of apprehension, that is, the perception, must
be completely conformable.*

[*Footnote: In this manner it is proved, that the synthesis of
apprehension, which is empirical, must necessarily be conformable to
the synthesis of apperception, which is intellectual, and contained
a priori in the category. It is one and the same spontaneity which
at one time, under the name of imagination, at another under that of
understanding, produces conjunction in the manifold of intuition.]

To take another example, when I perceive the freezing of water, I
apprehend two states (fluidity and solidity), which, as such, stand
toward each other mutually in a relation of time. But in the time,
which I place as an internal intuition, at the foundation of this
phenomenon, I represent to myself synthetical unity of the manifold,
without which the aforesaid relation could not be given in an
intuition as determined (in regard to the succession of time). Now
this synthetical unity, as the a priori condition under which I
conjoin the manifold of an intuition, is, if I make abstraction of
the permanent form of my internal intuition (that is to say, of time),
the category of cause, by means of which, when applied to my
sensibility, I determine everything that occurs according to relations
of time. Consequently apprehension in such an event, and the event
itself, as far as regards the possibility of its perception, stands
under the conception of the relation of cause and effect: and so in
all other cases.

Categories are conceptions which prescribe laws a priori to
phenomena, consequently to nature as the complex of all phenomena
(natura materialiter spectata). And now the question arises--
inasmuch as these categories are not derived from nature, and do not
regulate themselves according to her as their model (for in that
case they would be empirical)--how it is conceivable that nature
must regulate herself according to them, in other words, how the
categories can determine a priori the synthesis of the manifold of
nature, and yet not derive their origin from her. The following is
the solution of this enigma.

It is not in the least more difficult to conceive how the laws of the
phenomena of nature must harmonize with the understanding and with its
a priori form--that is, its faculty of conjoining the manifold--than
it is to understand how the phenomena themselves must correspond with
the a priori form of our sensuous intuition. For laws do not exist in
the phenomena any more than the phenomena exist as things in
themselves. Laws do not exist except by relation to the subject in
which the phenomena inhere, in so far as it possesses understanding,
just as phenomena have no existence except by relation to the same
existing subject in so far as it has senses. To things as things in
themselves, conformability to law must necessarily belong independently
of an understanding to cognize them. But phenomena are only
representations of things which are utterly unknown in respect to what
they are in themselves. But as mere representations, they stand under
no law of conjunction except that which the conjoining faculty
prescribes. Now that which conjoins the manifold of sensuous intuition
is imagination, a mental act to which understanding contributes unity
of intellectual synthesis, and sensibility, manifoldness of
apprehension. Now as all possible perception depends on the synthesis
of apprehension, and this empirical synthesis itself on the
transcendental, consequently on the categories, it is evident that all
possible perceptions, and therefore everything that can attain to
empirical consciousness, that is, all phenomena of nature, must, as
regards their conjunction, be subject to the categories. And nature
(considered merely as nature in general) is dependent on them, as the
original ground of her necessary conformability to law (as natura
formaliter spectata). But the pure faculty (of the understanding) of
prescribing laws a priori to phenomena by means of mere categories, is
not competent to enounce other or more laws than those on which a
nature in general, as a conformability to law of phenomena of space and
time, depends. Particular laws, inasmuch as they concern empirically
determined phenomena, cannot be entirely deduced from pure laws,
although they all stand under them. Experience must be superadded in
order to know these particular laws; but in regard to experience in
general, and everything that can be cognized as an object thereof,
these a priori laws are our only rule and guide.

SS 23. Result of this Deduction of the Conceptions of the

We cannot think any object except by means of the categories; we
cannot cognize any thought except by means of intuitions corresponding
to these conceptions. Now all our intuitions are sensuous, and our
cognition, in so far as the object of it is given, is empirical. But
empirical cognition is experience; consequently no a priori
cognition is possible for us, except of objects of possible

[Footnote: Lest my readers should stumble at this assertion, and the
conclusions that may be too rashly drawn from it, I must remind them
that the categories in the act of thought are by no means limited by
the conditions of our sensuous intuition, but have an unbounded sphere
of action. It is only the cognition of the object of thought, the
determining of the object, which requires intuition. In the absence
of intuition, our thought of an object may still have true and useful
consequences in regard to the exercise of reason by the subject. But
as this exercise of reason is not always directed on the determination
of the object, in other words, on cognition thereof, but also on the
determination of the subject and its volition, I do not intend to
treat of it in this place.]

But this cognition, which is limited to objects of experience, is
not for that reason derived entirely, from, experience, but--and
this is asserted of the pure intuitions and the pure conceptions of
the understanding--there are, unquestionably, elements of cognition,
which exist in the mind a priori. Now there are only two ways in which
a necessary harmony of experience with the conceptions of its
objects can be cogitated. Either experience makes these conceptions
possible, or the conceptions make experience possible. The former of
these statements will not bold good with respect to the categories
(nor in regard to pure sensuous intuition), for they are a priori
conceptions, and therefore independent of experience. The assertion
of an empirical origin would attribute to them a sort of generatio
aequivoca. Consequently, nothing remains but to adopt the second
alternative (which presents us with a system, as it were, of the
epigenesis of pure reason), namely, that on the part of the
understanding the categories do contain the grounds of the possibility
of all experience. But with respect to the questions how they make
experience possible, and what are the principles of the possibility
thereof with which they present us in their application to
phenomena, the following section on the transcendental exercise of
the faculty of judgement will inform the reader.

It is quite possible that someone may propose a species of
preformation-system of pure reason--a middle way between the two--to
wit, that the categories are neither innate and first a priori
principles of cognition, nor derived from experience, but are merely
subjective aptitudes for thought implanted in us contemporaneously
with our existence, which were so ordered and disposed by our Creator,
that their exercise perfectly harmonizes with the laws of nature which
regulate experience. Now, not to mention that with such an
hypothesis it is impossible to say at what point we must stop in the
employment of predetermined aptitudes, the fact that the categories
would in this case entirely lose that character of necessity which
is essentially involved in the very conception of them, is a
conclusive objection to it. The conception of cause, for example,
which expresses the necessity of an effect under a presupposed
condition, would be false, if it rested only upon such an arbitrary
subjective necessity of uniting certain empirical representations
according to such a rule of relation. I could not then say--"The
effect is connected with its cause in the object (that is,
necessarily)," but only, "I am so constituted that I can think this
representation as so connected, and not otherwise." Now this is just
what the sceptic wants. For in this case, all our knowledge, depending
on the supposed objective validity of our judgement, is nothing but
mere illusion; nor would there be wanting people who would deny any
such subjective necessity in respect to themselves, though they must
feel it. At all events, we could not dispute with any one on that
which merely depends on the manner in which his subject is organized.

Short view of the above Deduction.

The foregoing deduction is an exposition of the pure conceptions
of the understanding (and with them of all theoretical a priori
cognition), as principles of the possibility of experience, but of
experience as the determination of all phenomena in space and time
in general--of experience, finally, from the principle of the original
synthetical unity of apperception, as the form of the understanding
in relation to time and space as original forms of sensibility.

I consider the division by paragraphs to be necessary only up to
this point, because we had to treat of the elementary conceptions.
As we now proceed to the exposition of the employment of these, I
shall not designate the chapters in this manner any further.


Analytic of Principles.

General logic is constructed upon a plan which coincides exactly
with the division of the higher faculties of cognition. These are,
understanding, judgement, and reason. This science, accordingly,
treats in its analytic of conceptions, judgements, and conclusions
in exact correspondence with the functions and order of those mental
powers which we include generally under the generic denomination of

As this merely formal logic makes abstraction of all content of
cognition, whether pure or empirical, and occupies itself with the
mere form of thought (discursive cognition), it must contain in its
analytic a canon for reason. For the form of reason has its law,
which, without taking into consideration the particular nature of
the cognition about which it is employed, can be discovered a
priori, by the simple analysis of the action of reason into its

Transcendental logic, limited as it is to a determinate content,
that of pure a priori cognitions, to wit, cannot imitate general logic
in this division. For it is evident that the transcendental employment
of reason is not objectively valid, and therefore does not belong to
the logic of truth (that is, to analytic), but as a logic of illusion,
occupies a particular department in the scholastic system under the
name of transcendental dialectic.

Understanding and judgement accordingly possess in transcendental
logic a canon of objectively valid, and therefore true exercise, and
are comprehended in the analytical department of that logic. But
reason, in her endeavours to arrive by a priori means at some true
statement concerning objects and to extend cognition beyond the bounds
of possible experience, is altogether dialectic, and her illusory
assertions cannot be constructed into a canon such as an analytic
ought to contain.

Accordingly, the analytic of principles will be merely a canon for
the faculty of judgement, for the instruction of this faculty in its
application to phenomena of the pure conceptions of the understanding,
which contain the necessary condition for the establishment of a
priori laws. On this account, although the subject of the following
chapters is the especial principles of understanding, I shall make
use of the term Doctrine of the faculty of judgement, in order to define
more particularly my present purpose.

INTRODUCTION. Of the Transcendental Faculty of judgement in General.

If understanding in general be defined as the faculty of laws or
rules, the faculty of judgement may be termed the faculty of
subsumption under these rules; that is, of distinguishing whether this
or that does or does not stand under a given rule (casus datae legis).
General logic contains no directions or precepts for the faculty of
judgement, nor can it contain any such. For as it makes abstraction
of all content of cognition, no duty is left for it, except that of
exposing analytically the mere form of cognition in conceptions,
judgements, and conclusions, and of thereby establishing formal
rules for all exercise of the understanding. Now if this logic
wished to give some general direction how we should subsume under
these rules, that is, how we should distinguish whether this or that
did or did not stand under them, this again could not be done
otherwise than by means of a rule. But this rule, precisely because
it is a rule, requires for itself direction from the faculty of
judgement. Thus, it is evident that the understanding is capable of
being instructed by rules, but that the judgement is a peculiar
talent, which does not, and cannot require tuition, but only exercise.
This faculty is therefore the specific quality of the so-called mother
wit, the want of which no scholastic discipline can compensate.

For although education may furnish, and, as it were, engraft upon
a limited understanding rules borrowed from other minds, yet the power
of employing these rules correctly must belong to the pupil himself;
and no rule which we can prescribe to him with this purpose is, in
the absence or deficiency of this gift of nature, secure from misuse.*
A physician therefore, a judge or a statesman, may have in his head
many admirable pathological, juridical, or political rules, in a degree
that may enable him to be a profound teacher in his particular
science, and yet in the application of these rules he may very
possibly blunder--either because he is wanting in natural judgement
(though not in understanding) and, whilst he can comprehend the
general in abstracto, cannot distinguish whether a particular case
in concreto ought to rank under the former; or because his faculty
of judgement has not been sufficiently exercised by examples and
real practice. Indeed, the grand and only use of examples, is to
sharpen the judgement. For as regards the correctness and precision
of the insight of the understanding, examples are commonly injurious
rather than otherwise, because, as casus in terminis they seldom
adequately fulfil the conditions of the rule. Besides, they often
weaken the power of our understanding to apprehend rules or laws in
their universality, independently of particular circumstances of
experience; and hence, accustom us to employ them more as formulae
than as principles. Examples are thus the go-cart of the judgement,
which he who is naturally deficient in that faculty cannot afford to
dispense with.

[*Footnote: Deficiency in judgement is properly that which is called
stupidity; and for such a failing we know no remedy. A dull or
narrow-minded person, to whom nothing is wanting but a proper degree
of understanding, may be improved by tuition, even so far as to deserve
the epithet of learned. But as such persons frequently labour under
a deficiency in the faculty of judgement, it is not uncommon to find
men extremely learned who in the application of their science betray
a lamentable degree this irremediable want.]

But although general logic cannot give directions to the faculty
of judgement, the case is very different as regards transcendental
logic, insomuch that it appears to be the especial duty of the
latter to secure and direct, by means of determinate rules, the
faculty of judgement in the employment of the pure understanding. For,
as a doctrine, that is, as an endeavour to enlarge the sphere of the
understanding in regard to pure a priori cognitions, philosophy is
worse than useless, since from all the attempts hitherto made,
little or no ground has been gained. But, as a critique, in order to
guard against the mistakes of the faculty of judgement (lapsus
judicii) in the employment of the few pure conceptions of the
understanding which we possess, although its use is in this case
purely negative, philosophy is called upon to apply all its
acuteness and penetration.

But transcendental philosophy has this peculiarity, that besides
indicating the rule, or rather the general condition for rules,
which is given in the pure conception of the understanding, it can,
at the same time, indicate a priori the case to which the rule must
be applied. The cause of the superiority which, in this respect,
transcendental philosophy possesses above all other sciences except
mathematics, lies in this: it treats of conceptions which must
relate a priori to their objects, whose objective validity
consequently cannot be demonstrated a posteriori, and is, at the
same time, under the obligation of presenting in general but
sufficient tests, the conditions under which objects can be given in
harmony with those conceptions; otherwise they would be mere logical
forms, without content, and not pure conceptions of the understanding.

Our transcendental doctrine of the faculty of judgement will contain
two chapters. The first will treat of the sensuous condition under
which alone pure conceptions of the understanding can be employed--
that is, of the schematism of the pure understanding. The second
will treat of those synthetical judgements which are derived a
priori from pure conceptions of the understanding under those
conditions, and which lie a priori at the foundation of all other
cognitions, that is to say, it will treat of the principles of the
pure understanding.


CHAPTER I. Of the Schematism at of the Pure Conceptions
of the Understanding.

In all subsumptions of an object under a conception, the
representation of the object must be homogeneous with the
conception; in other words, the conception must contain that which
is represented in the object to be subsumed under it. For this is
the meaning of the expression: "An object is contained under a
conception." Thus the empirical conception of a plate is homogeneous
with the pure geometrical conception of a circle, inasmuch as the
roundness which is cogitated in the former is intuited in the latter.

But pure conceptions of the understanding, when compared with
empirical intuitions, or even with sensuous intuitions in general,
are quite heterogeneous, and never can be discovered in any intuition.
How then is the subsumption of the latter under the former, and
consequently the application of the categories to phenomena,
possible?--For it is impossible to say, for example: "Causality can
be intuited through the senses and is contained in the phenomenon."--This
natural and important question forms the real cause of the necessity
of a transcendental doctrine of the faculty of judgement, with the
purpose, to wit, of showing how pure conceptions of the
understanding can be applied to phenomena. In all other sciences,
where the conceptions by which the object is thought in the general
are not so different and heterogeneous from those which represent
the object in concreto--as it is given, it is quite unnecessary to
institute any special inquiries concerning the application of the
former to the latter.

Now it is quite clear that there must be some third thing, which
on the one side is homogeneous with the category, and with the
phenomenon on the other, and so makes the application of the former
to the latter possible. This mediating representation must be pure
(without any empirical content), and yet must on the one side be
intellectual, on the other sensuous. Such a representation is the
transcendental schema.

The conception of the understanding contains pure synthetical
unity of the manifold in general. Time, as the formal condition of
the manifold of the internal sense, consequently of the conjunction
of all representations, contains a priori a manifold in the pure intuition.
Now a transcendental determination of time is so far homogeneous
with the category, which constitutes the unity thereof, that it is
universal and rests upon a rule a priori. On the other hand, it is
so far homogeneous with the phenomenon, inasmuch as time is
contained in every empirical representation of the manifold. Thus an
application of the category to phenomena becomes possible, by means
of the transcendental determination of time, which, as the schema of
the conceptions of the understanding, mediates the subsumption of
the latter under the former.

After what has been proved in our deduction of the categories, no
one, it is to be hoped, can hesitate as to the proper decision of
the question, whether the employment of these pure conceptions of
the understanding ought to be merely empirical or also transcendental;
in other words, whether the categories, as conditions of a possible
experience, relate a priori solely to phenomena, or whether, as
conditions of the possibility of things in general, their
application can be extended to objects as things in themselves. For
we have there seen that conceptions are quite impossible, and utterly
without signification, unless either to them, or at least to the
elements of which they consist, an object be given; and that,
consequently, they cannot possibly apply to objects as things in
themselves without regard to the question whether and how these may
be given to us; and, further, that the only manner in which objects
can be given to us is by means of the modification of our sensibility;
and, finally, that pure a priori conceptions, in addition to the
function of the understanding in the category, must contain a priori
formal conditions of sensibility (of the internal sense, namely),
which again contain the general condition under which alone the
category can be applied to any object. This formal and pure
condition of sensibility, to which the conception of the understanding
is restricted in its employment, we shall name the schema of the
conception of the understanding, and the procedure of the
understanding with these schemata we shall call the schematism of
the pure understanding.

The schema is, in itself, always a mere product of the
imagination. But, as the synthesis of imagination has for its aim no
single intuition, but merely unity in the determination of
sensibility, the schema is clearly distinguishable 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 think a number in
general, which may be either five or a hundred, this thought is rather
the representation of a method of representing in an image a sum
(e.g., a thousand) in conformity with a conception, than the image
itself, an image which I should find some little difficulty in
reviewing, and comparing with the conception. Now this
representation of a general procedure of the imagination to present
its image to a conception, I call the schema of this conception.

In truth, 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 the
generalness of the conception it never could attain to, as this
includes under itself all triangles, whether right-angled,
acute-angled, etc., whilst the image would always be limited to a
single part of this sphere. The schema of the triangle can exist
nowhere else than in thought, and it indicates a rule of the synthesis
of the imagination in regard to pure figures in space. Still less is
an object of experience, or an image of the object, ever to the
empirical conception. On the contrary, the conception always relates
immediately to the schema of the imagination, as a rule for the
determination of our intuition, in conformity with a certain general
conception. The conception of a dog indicates a rule, according to
which my imagination can delineate the figure of a four-footed
animal in general, without being limited to any particular
individual 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 shall only with difficulty discover and
unveil. Thus much only can we say: "The image is a product of the
empirical faculty of the productive imagination--the schema of
sensuous conceptions (of figures in space, for example) is a
product, and, as it were, a monogram of the pure imagination a priori,
whereby and according to which images first become possible, which,
however, can be connected with the conception only mediately by
means of the schema which they indicate, and are in themselves never
fully adequate to it." On the other hand, the schema of a pure
conception of the understanding is something that cannot be reduced
into any image--it is nothing else than the pure synthesis expressed
by the category, conformably, to a rule of unity according to
conceptions. It is a transcendental product of the imagination, a
product which concerns the determination of the internal sense,
according to conditions of its form (time) in respect to all
representations, in so far as these representations must be
conjoined a priori in one conception, conformably to the unity of

Without entering upon a dry and tedious analysis of the essential
requisites of transcendental schemata of the pure conceptions of the
understanding, we shall rather proceed at once to give an
explanation of them according to the order of the categories, and in
connection therewith.

For the external sense the pure image of all quantities
(quantorum) is space; the pure image of all objects of sense in
general, is time. But the pure schema of quantity (quantitatis) as
a conception of the understanding, is number, a representation which
comprehends the successive addition of one to one (homogeneous
quantities). Thus, number is nothing else than the unity of the
synthesis of the manifold in a homogeneous intuition, by means of my
generating time itself in my apprehension of the intuition.

Reality, in the pure conception of the understanding, is that
which corresponds to a sensation in general; that, consequently, the
conception of which indicates a being (in time). Negation is that
the conception of which represents a not-being (in time). The
opposition of these two consists therefore in the difference of one
and the same time, as a time filled or a time empty. Now as time is
only the form of intuition, consequently of objects as phenomena, that
which in objects corresponds to sensation is the transcendental matter
of all objects as things in themselves (Sachheit, reality). Now
every sensation has a degree or quantity by which it can fill time,
that is to say, the internal sense in respect of the representation
of an object, more or less, until it vanishes into nothing (= 0 =
negatio). Thus there is a relation and connection between reality
and negation, or rather a transition from the former to the latter,
which makes every reality representable to us as a quantum; and the
schema of a reality as the quantity of something in so far as it fills
time, is exactly 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 quantity thereof.

The schema of substance is the permanence of the real in time;
that is, the representation of it as a substratum of the empirical
determination of time; a substratum which therefore remains, whilst
all else changes. (Time passes not, but in it passes the existence
of the changeable. To time, therefore, which is itself unchangeable
and permanent, corresponds that which in the phenomenon is
unchangeable in existence, that is, substance, and it is only by it
that the succession and coexistence of phenomena can be determined
in regard to time.)

The schema of cause and of the causality of a thing is the real
which, when posited, is always followed by something else. It
consists, therefore, in the succession of the manifold, in so far as
that succession is subjected to a rule.

The schema of community (reciprocity of action and reaction), or the
reciprocal causality of substances in respect of their accidents, is
the coexistence of the determinations of the one with those of the
other, according to a general rule.

The schema of possibility is the accordance of the synthesis of
different representations with the conditions of time in general
(as, for example, opposites cannot exist together at the same time
in the same thing, but only after each other), and is therefore the
determination of the representation of a thing at any time.

The schema of reality is existence in a determined time.

The schema of necessity is the existence of an object in all time.

It is clear, from all this, that the schema of the category of
quantity contains and represents the generation (synthesis) of time
itself, in the successive apprehension of an object; the schema of
quality the synthesis of sensation with the representation of time,
or the filling up of time; the schema of relation the relation of
perceptions to each other in all time (that is, according to a rule
of the determination of time): and finally, the schema of modality
and its categories, time itself, as the correlative of the determination
of an object--whether it does belong to time, and how. The schemata,
therefore, are nothing but a priori determinations of time according
to rules, and these, in regard to all possible objects, following
the arrangement of the categories, relate to the series in time, the
content in time, the order in time, and finally, to the complex or
totality in time.

Hence it is apparent that the schematism of the understanding, by
means of the transcendental synthesis of the imagination, amounts to
nothing else than the unity of the manifold of intuition in the
internal sense, and thus indirectly to the unity of apperception, as
a function corresponding to the internal sense (a receptivity). Thus,
the schemata of the pure conceptions of the understanding are the true
and only conditions whereby our understanding receives an
application to objects, and consequently significance. Finally,
therefore, the categories are only capable of empirical use,
inasmuch as they serve merely to subject phenomena to the universal
rules of synthesis, by means of an a priori necessary unity (on
account of the necessary union of all consciousness in one original
apperception); and so to render them susceptible of a complete
connection in one experience. But within this whole of possible
experience lie all our cognitions, and in the universal relation to
this experience consists transcendental truth, which antecedes all
empirical truth, and renders the latter possible.

It is, however, evident at first sight, that although the schemata
of sensibility are the sole agents in realizing the categories, they
do, nevertheless, also restrict them, that is, they limit the
categories by conditions which lie beyond the sphere of understanding--
namely, in sensibility. Hence the schema is properly only the
phenomenon, or the sensuous conception of an object in harmony with
the category. (Numerus est quantitas phaenomenon--sensatio realitas
phaenomenon; constans et perdurabile rerum substantia phaenomenon--
aeternitas, necessitas, phaenomena, etc.) Now, if we remove a
restrictive condition, we thereby amplify, it appears, the formerly
limited conception. In this way, the categories in their pure
signification, free from all conditions of sensibility, ought to be
valid of things as they are, and not, as the schemata represent
them, merely as they appear; and consequently the categories must have
a significance far more extended, and wholly independent of all
schemata. In truth, there does always remain to the pure conceptions
of the understanding, after abstracting every sensuous condition, a
value and significance, which is, however, merely logical. But in this
case, no object is given them, and therefore they have no meaning
sufficient to afford us a conception of an object. The notion of
substance, for example, if we leave out the sensuous determination
of permanence, would mean nothing more than a something which can be
cogitated as subject, without the possibility of becoming a
predicate to anything else. Of this representation I can make nothing,
inasmuch as it does not indicate to me what determinations the thing
possesses which must thus be valid as premier subject. Consequently,
the categories, without schemata are merely functions of the
understanding for the production of conceptions, but do not
represent any object. This significance they derive from
sensibility, which at the same time realizes the understanding and
restricts it.

CHAPTER II. System of all Principles of the Pure Understanding.

In the foregoing chapter we have merely considered the general
conditions under which alone the transcendental faculty of judgement
is justified in using the pure conceptions of the understanding for
synthetical judgements. Our duty at present is to exhibit in
systematic connection those judgements which the understanding
really produces a priori. For this purpose, our table of the
categories will certainly afford us the natural and safe guidance.
For it is precisely the categories whose application to possible
experience must constitute all pure a priori cognition of the
understanding; and the relation of which to sensibility will, on
that very account, present us with a complete and systematic catalogue
of all the transcendental principles of the use of the understanding.

Principles a priori are so called, not merely because they contain
in themselves the grounds of other judgements, but also because they
themselves are not grounded in higher and more general cognitions.
This peculiarity, however, does not raise them altogether above the
need of a proof. For although there could be found no higher
cognition, and therefore no objective proof, and although such a
principle rather serves as the foundation for all cognition of the
object, this by no means hinders us from drawing a proof from the
subjective sources of the possibility of the cognition of an object.
Such a proof is necessary, moreover, because without it the
principle might be liable to the imputation of being a mere gratuitous

In the second place, we shall limit our investigations to those
principles which relate to the categories. For as to the principles
of transcendental aesthetic, according to which space and time are
the conditions of the possibility of things as phenomena, as also the
restriction of these principles, namely, that they cannot be applied
to objects as things in themselves--these, of course, do not fall
within the scope of our present inquiry. In like manner, the
principles of mathematical science form no part of this system,
because they are all drawn from intuition, and not from the pure
conception of the understanding. The possibility of these
principles, however, will necessarily be considered here, inasmuch
as they are synthetical judgements a priori, not indeed for the
purpose of proving their accuracy and apodeictic certainty, which is
unnecessary, but merely to render conceivable and deduce the
possibility of such evident a priori cognitions.

But we shall have also to speak of the principle of analytical
judgements, in opposition to synthetical judgements, which is the
proper subject of our inquiries, because this very opposition will
free the theory of the latter from all ambiguity, and place it clearly
before our eyes in its true nature.


SECTION I. Of the Supreme Principle of all Analytical Judgements.

Whatever may be the content of our cognition, and in whatever manner
our cognition may be related to its object, the universal, although
only negative conditions of all our judgements is that they do not
contradict themselves; otherwise these judgements are in themselves
(even without respect to the object) nothing. But although there may
exist no contradiction in our judgement, it may nevertheless connect
conceptions in such a manner that they do not correspond to the
object, or without any grounds either a priori or a posteriori for
arriving at such a judgement, and thus, without being
self-contradictory, a judgement may nevertheless be either false or

Now, the proposition: "No subject can have a predicate that
contradicts it," is called the principle of contradiction, and is a
universal but purely negative criterion of all truth. But it belongs
to logic alone, because it is valid of cognitions, merely as
cognitions and without respect to their content, and declares that
the contradiction entirely nullifies them. We can also, however, make
a positive use of this principle, that is, not merely to banish
falsehood and error (in so far as it rests upon contradiction), but
also for the cognition of truth. For if the judgement is analytical,
be it affirmative or negative, its truth must always be recognizable
by means of the principle of contradiction. For the contrary of that
which lies and is cogitated as conception in the cognition of the
object will be always properly negatived, but the conception itself
must always be affirmed of the object, inasmuch as the contrary
thereof would be in contradiction to the object.

We must therefore hold the principle of contradiction to be the
universal and fully sufficient Principle of all analytical
cognition. But as a sufficient criterion of truth, it has no further
utility or authority. For the fact that no cognition can be at
variance with this principle without nullifying itself, constitutes
this principle the sine qua non, but not the determining ground of
the truth of our cognition. As our business at present is properly
with the synthetical part of our knowledge only, we shall always be
on our guard not to transgress this inviolable principle; but at the
same time not to expect from it any direct assistance in the
establishment of the truth of any synthetical proposition.

There exists, however, a formula of this celebrated principle--a
principle merely formal and entirely without content--which contains
a synthesis that has been inadvertently and quite unnecessarily mixed
up with it. It is this: "It is impossible for a thing to be and not
to be at the same time." Not to mention the superfluousness of the
addition of the word impossible to indicate the apodeictic
certainty, which ought to be self-evident from the proposition itself,
the proposition is affected by the condition of time, and as it were
says: "A thing = A, which is something = B, cannot at the same time
be non-B." But both, B as well as non-B, may quite well exist in
succession. For example, a man who is young cannot at the same time
be old; but the same man can very well be at one time young, and at
another not young, that is, old. Now the principle of contradiction
as a merely logical proposition must not by any means limit its
application merely to relations of time, and consequently a formula
like the preceding is quite foreign to its true purpose. The
misunderstanding arises in this way. We first of all separate a
predicate of a thing from the conception of the thing, and
afterwards connect with this predicate its opposite, and hence do
not establish any contradiction with the subject, but only with its
predicate, which has been conjoined with the subject synthetically--
a contradiction, moreover, which obtains only when the first and
second predicate are affirmed in the same time. If I say: "A man who
is ignorant is not learned," the condition "at the same time" must
be added, for he who is at one time ignorant, may at another be
learned. But if I say: "No ignorant man is a learned man," the
proposition is analytical, because the characteristic ignorance is
now a constituent part of the conception of the subject; and in this
case the negative proposition is evident immediately from the
proposition of contradiction, without the necessity of adding the
condition "the same time." This is the reason why I have altered the
formula of this principle--an alteration which shows very clearly
the nature of an analytical proposition.

SECTION II. Of the Supreme Principle of all Synthetical Judgements.

The explanation of the possibility of synthetical judgements is a task
with which general logic has nothing to do; indeed she needs not even
be acquainted with its name. But in transcendental logic it is the most
important matter to be dealt with--indeed the only one, if the question
is of the possibility of synthetical judgements a priori, the
conditions and extent of their validity. For when this question is
fully decided, it can reach its aim with perfect ease, the
determination, to wit, of the extent and limits of the pure

In an analytical judgement I do not go beyond the given conception, in
order to arrive at some decision respecting it. If the judgement is
affirmative, I predicate of the conception only that which was already
cogitated in it; if negative, I merely exclude from the conception its
contrary. But in synthetical judgements, I must go beyond the given
conception, in order to cogitate, in relation with it, something quite
different from that which was cogitated in it, a relation which is
consequently never one either of identity or contradiction, and by
means of which the truth or error of the judgement cannot be discerned
merely from the judgement itself.

Granted, then, that we must go out beyond a given conception, in
order to compare it synthetically with another, a third thing is
necessary, in which alone the synthesis of two conceptions can
originate. Now what is this tertium quid that is to be the medium of
all synthetical judgements? It is only a complex in which all our
representations are contained, the internal sense to wit, and its form
a priori, time.

The synthesis of our representations rests upon the imagination;
their synthetical unity (which is requisite to a judgement), upon
the unity of apperception. In this, therefore, is to be sought the
possibility of synthetical judgements, and as all three contain the
sources of a priori representations, the possibility of pure
synthetical judgements also; nay, they are necessary upon these
grounds, if we are to possess a knowledge of objects, which rests
solely upon the synthesis of representations.

If a cognition is to have objective reality, that is, to relate to
an object, and possess sense and meaning in respect to it, it is
necessary that the object be given in some way or another. Without
this, our conceptions are empty, and we may indeed have thought by
means of them, but by such thinking we have not, in fact, cognized
anything, we have merely played with representation. To give an
object, if this expression be understood in the sense of "to
present" the object, not mediately but immediately in intuition, means
nothing else than to apply the representation of it to experience,
be that experience real or only possible. Space and time themselves,
pure as these conceptions are from all that is empirical, and
certain as it is that they are represented fully a priori in the mind,
would be completely without objective validity, and without sense
and significance, if their necessary use in the objects of
experience were not shown. Nay, the representation of them is a mere
schema, that always relates to the reproductive imagination, which
calls up the objects of experience, without which they have no
meaning. And so it is with all conceptions without distinction.

The possibility of experience is, then, that which gives objective
reality to all our a priori cognitions. Now experience depends upon
the synthetical unity of phenomena, that is, upon a synthesis
according to conceptions of the object of phenomena in general, a
synthesis without which experience never could become knowledge, but
would be merely a rhapsody of perceptions, never fitting together into
any connected text, according to rules of a thoroughly united
(possible) consciousness, and therefore never subjected to the
transcendental and necessary unity of apperception. Experience has
therefore for a foundation, a priori principles of its form, that is
to say, general rules of unity in the synthesis of phenomena, the
objective reality of which rules, as necessary conditions even of
the possibility of experience can which rules, as necessary
conditions--even of the possibility of experience--can always be shown
in experience. But apart from this relation, a priori synthetical
propositions are absolutely impossible, because they have no third
term, that is, no pure object, in which the synthetical unity can
exhibit the objective reality of its conceptions.

Although, then, respecting space, or the forms which productive
imagination describes therein, we do cognize much a priori in
synthetical judgements, and are really in no need of experience for
this purpose, such knowledge would nevertheless amount to nothing
but a busy trifling with a mere chimera, were not space to be
considered as the condition of the phenomena which constitute the
material of external experience. Hence those pure synthetical
judgements do relate, though but mediately, to possible experience,
or rather to the possibility of experience, and upon that alone is
founded the objective validity of their synthesis.

While then, on the one hand, experience, as empirical synthesis,
is the only possible mode of cognition which gives reality to all
other synthesis; on the other hand, this latter synthesis, as
cognition a priori, possesses truth, that is, accordance with its
object, only in so far as it contains nothing more than what is
necessary to the synthetical unity of experience.

Accordingly, the supreme principle of all synthetical judgements is:
"Every object is subject to the necessary conditions of the
synthetical unity of the manifold of intuition in a possible

A priori synthetical judgements are possible when we apply the
formal conditions of the a priori intuition, the synthesis of the
imagination, and the necessary unity of that synthesis in a
transcendental apperception, to a possible cognition of experience,
and say: "The conditions of the possibility of experience in general
are at the same time conditions of the possibility of the objects of
experience, and have, for that reason, objective validity in an a
priori synthetical judgement."

SECTION III. Systematic Representation of all Synthetical
Principles of the Pure Understanding.

That principles exist at all is to be ascribed solely to the pure
understanding, which is not only the faculty of rules in regard to
that which happens, but is even the source of principles according
to which everything that can be presented to us as an object is
necessarily subject to rules, because without such rules we never
could attain to cognition of an object. Even the laws of nature, if
they are contemplated as principles of the empirical use of the
understanding, possess also a characteristic of necessity, and we
may therefore at least expect them to be determined upon grounds which
are valid a priori and antecedent to all experience. But all laws of
nature, without distinction, are subject to higher principles of the
understanding, inasmuch as the former are merely applications of the
latter to particular cases of experience. These higher principles
alone therefore give the conception, which contains the necessary
condition, and, as it were, the exponent of a rule; experience, on
the other hand, gives the case which comes under the rule.

There is no danger of our mistaking merely empirical principles
for principles of the pure understanding, or conversely; for the
character of necessity, according to conceptions which distinguish
the latter, and the absence of this in every empirical proposition,
how extensively valid soever it may be, is a perfect safeguard against
confounding them. There are, however, pure principles a priori,
which nevertheless I should not ascribe to the pure understanding--for
this reason, that they are not derived from pure conceptions, but
(although by the mediation of the understanding) from pure intuitions.
But understanding is the faculty of conceptions. Such principles
mathematical science possesses, but their application to experience,
consequently their objective validity, nay the possibility of such
a priori synthetical cognitions (the deduction thereof) rests entirely
upon the pure understanding.

On this account, I shall not reckon among my principles those of
mathematics; though I shall include those upon the possibility and
objective validity a priori, of principles of the mathematical
science, which, consequently, are to be looked upon as the principle
of these, and which proceed from conceptions to intuition, and not
from intuition to conceptions.

In the application of the pure conceptions of the understanding to
possible experience, the employment of their synthesis is either
mathematical or dynamical, for it is directed partly on the
intuition alone, partly on the existence of a phenomenon. But the a
priori conditions of intuition are in relation to a possible
experience absolutely necessary, those of the existence of objects
of a possible empirical intuition are in themselves contingent.
Hence the principles of the mathematical use of the categories will
possess a character of absolute necessity, that is, will be
apodeictic; those, on the other hand, of the dynamical use, the
character of an a priori necessity indeed, but only under the
condition of empirical thought in an experience, therefore only
mediately and indirectly. Consequently they will not possess that
immediate evidence which is peculiar to the former, although their
application to experience does not, for that reason, lose its truth
and certitude. But of this point we shall be better able to judge at
the conclusion of this system of principles.

The table of the categories is naturally our guide to the table of
principles, because these are nothing else than rules for the
objective employment of the former. Accordingly, all principles of
the pure understanding are:

of Intuition

2 3
Anticipations Analogies
of Perception of Experience
Postulates of
Empirical Thought
in general

These appellations I have chosen advisedly, in order that we might
not lose sight of the distinctions in respect of the evidence and
the employment of these principles. It will, however, soon appear
that--a fact which concerns both the evidence of these principles,
and the a priori determination of phenomena--according to the categories
of quantity and quality (if we attend merely to the form of these),
the principles of these categories are distinguishable from those of
the two others, in as much as the former are possessed of an
intuitive, but the latter of a merely discursive, though in both
instances a complete, certitude. I shall therefore call the former
mathematical, and the latter dynamical principles.* It must be
observed, however, that by these terms I mean just as little in the
one case the principles of mathematics as those of general
(physical) dynamics in the other. I have here in view merely the
principles of the pure understanding, in their application to the
internal sense (without distinction of the representations given
therein), by means of which the sciences of mathematics and dynamics
become possible. Accordingly, I have named these principles rather
with reference to their application than their content; and I shall
now proceed to consider them in the order in which they stand in the

[*Footnote: All combination (conjunctio) is either composition
(compositio) or connection (nexus). The former is the synthesis of a
manifold, the parts of which do not necessarily belong to each other.
For example, the two triangles into which a square is divided by a
diagonal, do not necessarily belong to each other, and of this kind is
the synthesis of the homogeneous in everything that can be
mathematically considered. This synthesis can be divided into those of
aggregation and coalition, the former of which is applied to extensive,
the latter to intensive quantities. The second sort of combination
(nexus) is the synthesis of a manifold, in so far as its parts do
belong necessarily to each other; for example, the accident to a
substance, or the effect to the cause. Consequently it is a synthesis
of that which though heterogeneous, is represented as connected a
priori. This combination--not an arbitrary one--I entitle dynamical
because it concerns the connection of the existence of the manifold.
This, again, may be divided into the physical synthesis, of the
phenomena divided among each other, and the metaphysical synthesis, or
the connection of phenomena a priori in the faculty of cognition.]


The principle of these is: All Intuitions are Extensive Quantities.


All phenomena contain, as regards their form, an intuition in
space and time, which lies a priori at the foundation of all without
exception. Phenomena, therefore, cannot be apprehended, that is,
received into empirical consciousness otherwise than through the
synthesis of a manifold, through which the representations of a
determinate space or time are generated; that is to say, through the
composition of the homogeneous and the consciousness of the
synthetical unity of this manifold (homogeneous). Now the
consciousness of a homogeneous manifold in intuition, in so far as
thereby the representation of an object is rendered possible, is the
conception of a quantity (quanti). Consequently, even the perception
of an object as phenomenon is possible only through the same
synthetical unity of the manifold of the given sensuous intuition,
through which the unity of the composition of the homogeneous manifold
in the conception of a quantity is cogitated; that is to say, all
phenomena are quantities, and extensive quantities, because as
intuitions in space or time they must be represented by means of the
same synthesis through which space and time themselves are determined.

An extensive quantity I call that wherein the representation of
the parts renders possible (and therefore necessarily antecedes) the
representation of the whole. I cannot represent to myself any line,
however small, without drawing it in thought, that is, without
generating from a point all its parts one after another, and in this
way alone producing this intuition. Precisely the same is the case
with every, even the smallest, portion of time. I cogitate therein
only the successive progress from one moment to another, and hence,
by means of the different portions of time and the addition of them,
a determinate quantity of time is produced. As the pure intuition in
all phenomena is either time or space, so is every phenomenon in its
character of intuition an extensive quantity, inasmuch as it can
only be cognized in our apprehension by successive synthesis (from
part to part). All phenomena are, accordingly, to be considered as
aggregates, that is, as a collection of previously given parts;
which is not the case with every sort of quantities, but only with
those which are represented and apprehended by us as extensive.

On this successive synthesis of the productive imagination, in the
generation of figures, is founded the mathematics of extension, or
geometry, with its axioms, which express the conditions of sensuous
intuition a priori, under which alone the schema of a pure
conception of external intuition can exist; for example, "be tween
two points only one straight line is possible," "two straight lines
cannot enclose a space," etc. These are the axioms which properly relate
only to quantities (quanta) as such.

But, as regards the quantity of a thing (quantitas), that is to say,
the answer to the question: "How large is this or that object?"
although, in respect to this question, we have various propositions
synthetical and immediately certain (indemonstrabilia); we have, in
the proper sense of the term, no axioms. For example, the
propositions: "If equals be added to equals, the wholes are equal";
"If equals be taken from equals, the remainders are equal"; are
analytical, because I am immediately conscious of the identity of
the production of the one quantity with the production of the other;
whereas axioms must be a priori synthetical propositions. On the other
hand, the self-evident propositions as to the relation of numbers,
are certainly synthetical but not universal, like those of geometry,
and for this reason cannot be called axioms, but numerical formulae.
That 7 + 5 = 12 is not an analytical proposition. For neither in the
representation of seven, nor of five, nor of the composition of the
two numbers, do I cogitate the number twelve. (Whether I cogitate
the number in the addition of both, is not at present the question;
for in the case of an analytical proposition, the only point is
whether I really cogitate the predicate in the representation of the
subject.) But although the proposition is synthetical, it is
nevertheless only a singular proposition. In so far as regard is
here had merely to the synthesis of the homogeneous (the units), it
cannot take place except in one manner, although our use of these
numbers is afterwards general. If I say: "A triangle can be
constructed with three lines, any two of which taken together are
greater than the third," I exercise merely the pure function of the
productive imagination, which may draw the lines longer or shorter
and construct the angles at its pleasure. On the contrary, the number
seven is possible only in one manner, and so is likewise the number
twelve, which results from the synthesis of seven and five. Such
propositions, then, cannot be termed axioms (for in that case we
should have an infinity of these), but numerical formulae.

This transcendental principle of the mathematics of phenomena
greatly enlarges our a priori cognition. For it is by this principle
alone that pure mathematics is rendered applicable in all its
precision to objects of experience, and without it the validity of
this application would not be so self-evident; on the contrary,
contradictions and confusions have often arisen on this very point.
Phenomena are not things in themselves. Empirical intuition is
possible only through pure intuition (of space and time);
consequently, what geometry affirms of the latter, is indisputably
valid of the former. All evasions, such as the statement that
objects of sense do not conform to the rules of construction in
space (for example, to the rule of the infinite divisibility of
lines or angles), must fall to the ground. For, if these objections
hold good, we deny to space, and with it to all mathematics, objective
validity, and no longer know wherefore, and how far, mathematics can
be applied to phenomena. The synthesis of spaces and times as the
essential form of all intuition, is that which renders possible the
apprehension of a phenomenon, and therefore every external experience,
consequently all cognition of the objects of experience; and
whatever mathematics in its pure use proves of the former, must
necessarily hold good of the latter. All objections are but the
chicaneries of an ill-instructed reason, which erroneously thinks to
liberate the objects of sense from the formal conditions of our
sensibility, and represents these, although mere phenomena, as
things in themselves, presented as such to our understanding. But in
this case, no a priori synthetical cognition of them could be
possible, consequently not through pure conceptions of space and the
science which determines these conceptions, that is to say,
geometry, would itself be impossible.


The principle of these is: In all phenomena the Real, that
which is an object of sensation, has Intensive Quantity,
that is, has a Degree.


Perception is empirical consciousness, that is to say, a
consciousness which contains an element of sensation. Phenomena as
objects of perception are not pure, that is, merely formal intuitions,
like space and time, for they cannot be perceived in themselves.
[Footnote: They can be perceived only as phenomena, and some part of
them must always belong to the non-ego; whereas pure intuitions are
entirely the products of the mind itself, and as such are coguized
IN THEMSELVES.--Tr] They contain, then, over and above the intuition,
the materials for an object (through which is represented something
existing in space or time), that is to say, they contain the real of
sensation, as a representation merely subjective, which gives us merely
the consciousness that the subject is affected, and which we refer
to some external object. Now, a gradual transition from empirical
consciousness to pure consciousness is possible, inasmuch as the
real in this consciousness entirely vanishes, and there remains a
merely formal consciousness (a priori) of the manifold in time and
space; consequently there is possible a synthesis also of the
production of the quantity of a sensation from its commencement,
that is, from the pure intuition = 0 onwards up to a certain
quantity of the sensation. Now as sensation in itself is not an
objective representation, and in it is to be found neither the
intuition of space nor of time, it cannot possess any extensive
quantity, and yet there does belong to it a quantity (and that by
means of its apprehension, in which empirical consciousness can within
a certain time rise from nothing = 0 up to its given amount),
consequently an intensive quantity. And thus we must ascribe intensive
quantity, that is, a degree of influence on sense to all objects of
perception, in so far as this perception contains sensation.

All cognition, by means of which I am enabled to cognize and
determine a priori what belongs to empirical cognition, may be
called an anticipation; and without doubt this is the sense in which
Epicurus employed his expression prholepsis. But as there is in
phenomena something which is never cognized a priori, which on this
account constitutes the proper difference between pure and empirical
cognition, that is to say, sensation (as the matter of perception),
it follows, that sensation is just that element in cognition which
cannot be at all anticipated. On the other hand, we might very well
term the pure determinations in space and time, as well in regard to
figure as to quantity, anticipations of phenomena, because they represent
a priori that which may always be given a posteriori in experience.
But suppose that in every sensation, as sensation in general,
without any particular sensation being thought of, there existed
something which could be cognized a priori, this would deserve to be
called anticipation in a special sense--special, because it may seem
surprising to forestall experience, in that which concerns the
matter of experience, and which we can only derive from itself. Yet
such really is the case here.

Apprehension*, by means of sensation alone, fills only one moment,
that is, if I do not take into consideration a succession of many
sensations. As that in the phenomenon, the apprehension of which is
not a successive synthesis advancing from parts to an entire
representation, sensation has therefore no extensive quantity; the
want of sensation in a moment of time would represent it as empty,
consequently = 0. That which in the empirical intuition corresponds
to sensation is reality (realitas phaenomenon); that which corresponds
to the absence of it, negation = 0. Now every sensation is capable
of a diminution, so that it can decrease, and thus gradually disappear.
Therefore, between reality in a phenomenon and negation, there
exists a continuous concatenation of many possible intermediate
sensations, the difference of which from each other is always
smaller than that between the given sensation and zero, or complete
negation. That is to say, the real in a phenomenon has always a
quantity, which however is not discoverable in apprehension,
inasmuch as apprehension take place by means of mere sensation in
one instant, and not by the successive synthesis of many sensations,
and therefore does not progress from parts to the whole. Consequently,
it has a quantity, but not an extensive quantity.

[*Footnote: Apprehension is the Kantian word for preception, in the
largest sense in which we employ that term. It is the genus which
includes under i, as species, perception proper and sensation proper--Tr]

Now that quantity which is apprehended only as unity, and in which
plurality can be represented only by approximation to negation = O,
I term intensive quantity. Consequently, reality in a phenomenon has
intensive quantity, that is, a degree. If we consider this reality
as cause (be it of sensation or of another reality in the
phenomenon, for example, a change), we call the degree of reality in
its character of cause a momentum, for example, the momentum of
weight; and for this reason, that the degree only indicates that
quantity the apprehension of which is not successive, but
instantaneous. This, however, I touch upon only in passing, for with
causality I have at present nothing to do.

Accordingly, every sensation, consequently every reality in
phenomena, however small it may be, has a degree, that is, an
intensive quantity, which may always be lessened, and between
reality and negation there exists a continuous connection of
possible realities, and possible smaller perceptions. Every colour--
for example, red--has a degree, which, be it ever so small, is never
the smallest, and so is it always with heat, the momentum of weight,

This property of quantities, according to which no part of them is
the smallest possible (no part simple), is called their continuity.
Space and time are quanta continua, because no part of them can be
given, without enclosing it within boundaries (points and moments),
consequently, this given part is itself a space or a time. Space,
therefore, consists only of spaces, and time of times. Points and
moments are only boundaries, that is, the mere places or positions
of their limitation. But places always presuppose intuitions which
are to limit or determine them; and we cannot conceive either space
or time composed of constituent parts which are given before space
or time. Such quantities may also be called flowing, because synthesis
(of the productive imagination) in the production of these
quantities is a progression in time, the continuity of which we are
accustomed to indicate by the expression flowing.

All phenomena, then, are continuous quantities, in respect both to
intuition and mere perception (sensation, and with it reality). In
the former case they are extensive quantities; in the latter, intensive.
When the synthesis of the manifold of a phenomenon is interrupted,
there results merely an aggregate of several phenomena, and not
properly a phenomenon as a quantity, which is not produced by the mere
continuation of the productive synthesis of a certain kind, but by
the repetition of a synthesis always ceasing. For example, if I call
thirteen dollars a sum or quantity of money, I employ the term quite
correctly, inasmuch as I understand by thirteen dollars the value of
a mark in standard silver, which is, to be sure, a continuous
quantity, in which no part is the smallest, but every part might
constitute a piece of money, which would contain material for still
smaller pieces. If, however, by the words thirteen dollars I
understand so many coins (be their value in silver what it may), it
would be quite erroneous to use the expression a quantity of
dollars; on the contrary, I must call them aggregate, that is, a
number of coins. And as in every number we must have unity as the
foundation, so a phenomenon taken as unity is a quantity, and as
such always a continuous quantity (quantum continuum).

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