§ 6.
Here is a great and established branch of knowledge, encompassing even
now a wonderfully large domain and promising an unlimited extension in
the future. Yet it carries with it thoroughly apodeictical certainty,
i.e., absolute necessity, which therefore rests upon no empirical
grounds. Consequently it is a pure product of reason, and moreover is
thoroughly synthetical. [Here the question arises:]
"How then is it possible for human reason to produce a cognition of
this nature entirely a priori?"
Does not this faculty [which produces mathematics], as it neither is
nor can be based upon experience, presuppose some ground of cognition
a priori, which lies deeply hidden, but which might reveal itself by
these its effects, if their first beginnings were but diligently
ferreted out?
§ 7. But we find that all mathematical cognition has this
peculiarity: it must first exhibit its concept in a visual form
(Anschauung) and indeed a priori, therefore in a visual form which is
not empirical, but pure. Without this mathematics cannot take a single
step; hence its judgments are always visual, viz., "intuitive";
whereas philosophy must be satisfied with discursive judgments from
mere concepts, and though it may illustrate its doctrines through a
visual figure, can never derive them from it. This observation on the
nature of mathematics gives us a clue to the first and highest
condition of its possibility, which is, that some non-sensuous
visualisation (called pure intuition, or reine Anschauung) must form
its basis, in which all its concepts can be exhibited or constructed,
in concreto and yet a priori. If we can find out this pure intuition
and its possibility, we may thence easily explain how synthetical
propositions a priori are possible in pure mathematics, and
consequently how this science itself is possible. Empirical intuition
[viz., sense-perception] enables us without difficulty to enlarge the
concept which we frame of an object of intuition [or
sense-perception], by new predicates, which intuition [i.e.,
sense-perception] itself presents synthetically in experience. Pure
intuition [viz., the visualisation of forms in our imagination, from
which every thing sensual, i.e., every thought of material qualities,
is excluded] does so likewise, only with this difference, that in the
latter case the synthetical judgment is a priori certain and
apodeictical, in the former, only a posteriori and empirically
certain; because this latter contains only that which occurs in
contingent empirical intuition, but the former, that which must
necessarily be discovered in pure intuition. Here intuition, being an
intuition a priori, is before all experience, viz., before any
perception of particular objects, inseparably conjoined with its
concept.
§ 8. But with this step our perplexity seems rather to increase than
to lessen. For the question now is, "How is it possible to intuite [in
a visual form] anything a priori?" An intuition [viz., a visual
sense-perception] is such a representation as immediately depends upon
the presence of the object. Hence it seems impossible to intuite from
the outset a priori, because intuition would in that event take place
without either a former or a present object to refer to, and by
consequence could not be intuition. Concepts indeed are such, that we
can easily form some of them a priori, viz., such as contain nothing
but the thought of an object in general; and we need not find
ourselves in an immediate relation to the object. Take, for instance,
the concepts of Quantity, of Cause, etc. But even these require, in
order to make them under stood, a certain concrete use—that is, an
application to some sense-experience (Anschauung), by which an object
of them is given us. But how can the intuition of the object [its
visualisation] precede the object itself?
§ 9. If our intuition [i.e., our sense-experience] were perforce of
such a nature as to represent things as they are in themselves, there
would not be any intuition a priori, but intuition would be always
empirical. For I can only know what is contained in the object in
itself when it is present and given to me. It is indeed even then
incomprehensible how the visualising (Anschauung) of a present thing
should make me know this thing as it is in itself, as its properties
cannot migrate into my faculty of representation. But even granting
this possibility, a visualising of that sort would not take place a
priori, that is, before the object were presented to me; for without
this latter fact no reason of a relation between my representation and
the object can be imagined, unless it depend upon a direct
inspiration.
Therefore in one way only can my intuition (Anschauung) anticipate the
actuality of the object, and be a cognition a priori, viz.: if my
intuition contains nothing but the form of sensibility, antedating in
my subjectivity all the actual impressions through which I am affected
by objects.
For that objects of sense can only be intuited according to this form
of sensibility I can know a priori. Hence it follows: that
propositions, which concern this form of sensuous intuition only, are
possible and valid for objects of the senses; as also, conversely,
that intuitions which are possible a priori can never concern any
other things than objects of our senses.{10}
===================================
{10} This whole paragraph (§ 9) will be better understood when compared
with Remark I., following this section, appearing in the present
edition on page 40.—Ed.
===================================
§ 10. Accordingly, it is only the form of the sensuous intuition by
which we can intuite things a priori, but by which we can know objects
only as they appear to us (to our senses), not as they are in
themselves; and this assumption is absolutely necessary if synthetical
propositions a priori be granted as possible, or if, in case they
actually occur, their possibility is to be comprehended and determined
beforehand.
Now, the intuitions which pure mathematics lays at the foundation of
all its cognitions and judgments which appear at once apodeictic and
necessary are Space and Time. For mathematics must first have all its
concepts in intuition, and pure mathematics in pure intuition, that
is, it must construct them. If it proceeded in any other way, it would
be impossible to make any headway, for mathematics proceeds, not
analytically by dissection of concepts, but synthetically, and if pure
intuition be wanting, there is nothing in which the matter for
synthetical judgments a priori can be given. Geometry is based upon
the pure intuition of space. Arithmetic accomplishes its concept of
number by the successive addition of units in time; and pure mechanics
especially cannot attain its concepts of motion without employing the
representation of time. Both representations, however, are only
intuitions; for if we omit from the empirical intuitions of bodies and
their alterations (motion) everything empirical, or belonging to
sensation, space and time still remain, which are therefore pure
intuitions that lie a priori at the basis of the empirical. Hence they
can never be omitted, but at the same time, by their being pure
intuitions a priori, they prove that they are mere forms of our
sensibility, which must precede all empirical intuition, or perception
of actual objects, and conformably to which objects can be known a
priori, but only as they appear to us.
§ 11. The problem of the present section is therefore solved. Pure
mathematics, as synthetical cognition a priori, is only possible by
referring to no other objects than those of the senses. At the basis
of their empirical intuition lies a pure intuition (of space and of
time) which is a priori. This is possible, because the latter
intuition is nothing but the mere form of sensibility, which precedes
the actual appearance of the objects, in that it, in fact, makes them
possible. Yet this faculty of intuiting a priori affects not the
matter of the phenomenon (that is, the sense-element in it, for this
constitutes that which is empirical), but its form, viz., space and
time. Should any man venture to doubt that these are determinations
adhering not to things in themselves, but to their relation to our
sensibility, I should be glad to know how it can be possible to know
the constitution of things a priori, viz., before we have any
acquaintance with them and before they are presented to us. Such,
however, is the case with space and time. But this is quite
comprehensible as soon as both count for nothing more than formal
conditions of our sensibility, while the objects count merely as
phenomena; for then the form of the phenomenon, i.e., pure intuition,
can by all means be represented as proceeding from ourselves, that is,
a priori.
§ 12. In order to add something by way of illustration and
confirmation, we need only watch the ordinary and necessary procedure
of geometers. All proofs of the complete congruence of two given
figures (where the one can in every respect be substituted for the
other) come ultimately to this that they may be made to coincide;
which is evidently nothing else than a synthetical proposition resting
upon immediate intuition, and this intuition must be pure, or given a
priori, otherwise the proposition could not rank as apodeictically
certain, but would have empirical certainty only. In that case, it
could only be said that it is always found to be so, and holds good
only as far as our perception reaches. That everywhere space (which
[in its entirety] is itself no longer the boundary of another space)
has three dimensions, and that space cannot in any way have more, is
based on the proposition that not more than three lines can intersect
at right angles in one point; but this proposition cannot by any means
be shown from concepts, but rests immediately on intuition, and indeed
on pure and a priori intuition, because it is apodeictically certain.
That we can require a line to be drawn to infinity (in indefinitum),
or that a series of changes (for example, spaces traversed by motion)
shall be infinitely continued, presupposes a representation of space
and time, which can only attach to intuition, namely, so far as it in
itself is bounded by nothing, for from concepts it could never be
inferred. Consequently, the basis of mathematics actually are pure
intuitions, which make its synthetical and apodeictically valid
propositions possible. Hence our transcendental deduction of the
notions of space and of time explains at the same time the possibility
of pure mathematics. Without some such deduction its truth may be
granted, but its existence could by no means be understood, and we
must assume "that everything which can be given to our senses (to the
external senses in space, to the internal one in time) is intuited by
us as it appears to us, not as it is in itself."
§ 13. Those who cannot yet rid themselves of the notion that space
and time are actual qualities inhering in things in themselves, may
exercise their acumen on the following paradox. When they have in vain
attempted its solution, and are free from prejudices at least for a
few moments, they will suspect that the degradation of space and of
time to mere forms of our sensuous intuition may perhaps be well
founded.
If two things are quite equal in all respects as much as can be
ascertained by all means possible, quantitatively and qualitatively,
it must follow, that the one can in all cases and under all
circumstances replace the other, and this substitution would not
occasion the least perceptible difference. This in fact is true of
plane figures in geometry; but some spherical figures exhibit,
notwithstanding a complete internal agreement, such a contrast in
their external relation, that the one figure cannot possibly be put in
the place of the other. For instance, two spherical triangles on
opposite hemispheres, which have an arc of the equator as their common
base, may be quite equal, both as regards sides and angles, so that
nothing is to be found in either, if it be described for itself alone
and completed, that would not equally be applicable to both; and yet
the one cannot be put in the place of the other (being situated upon
the opposite hemisphere). Here then is an internal difference between
the two triangles, which difference our understanding cannot describe
as internal, and which only manifests itself by external relations in
space.
But I shall adduce examples, taken from common life, that are more
obvious still.
What can be more similar in every respect and in every part more alike
to my hand and to my ear, than their images in a mirror? And yet I
cannot put such a hand as is seen in the glass in the place of its
archetype; for if this is a right hand, that in the glass is a left
one, and the image or reflexion of the right ear is a left one which
never can serve as a substitute for the other. There are in this case
no internal differences which our understanding could determine by
thinking alone. Yet the differences are internal as the senses teach,
for, notwithstanding their complete equality and similarity, the left
hand cannot be enclosed in the same bounds as the right one (they are
not congruent); the glove of one hand cannot be used for the other.
What is the solution? These objects are not representations of things
as they are in themselves, and as the pure understanding would cognise
them, but sensuous intuitions, that is, appearances, the possibility
of which rests upon the relation of certain things unknown in
themselves to something else, viz., to our sensibility. Space is the
form of the external intuition of this sensibility, and the internal
determination of every space is only possible by the determination of
its external relation to the whole space, of which it is a part (in
other words, by its relation to the external sense). That is to say,
the part is only possible through the whole, which is never the case
with things in themselves, as objects of the mere understanding, but
with appearances only. Hence the difference between similar and equal
things, which are yet not congruent (for instance, two symmetric
helices), cannot be made intelligible by any concept, but only by the
relation to the right and the left hands which immediately refers to
intuition.
Remark I.
Pure Mathematics, and especially pure geometry, can only have
objective reality on condition that they refer to objects of sense.
But in regard to the latter the principle holds good, that our sense
representation is not a representation of things in themselves, but of
the way in which they appear to us. Hence it follows, that the
propositions of geometry are not the results of a mere creation of our
poetic imagination, and that therefore they cannot be referred with
assurance to actual objects; but rather that they are necessarily
valid of space, and consequently of all that may be found in space,
because space is nothing else than the form of all external
appearances, and it is this form alone in which objects of sense can
be given. Sensibility, the form of which is the basis of geometry, is
that upon which the possibility of external appearance depends.
Therefore these appearances can never contain anything but what
geometry prescribes to them.
It would be quite otherwise if the senses were so constituted as to
represent objects as they are in themselves. For then it would not by
any means follow from the conception of space, which with all its
properties serves to the geometer as an a priori foundation, together
with what is thence inferred, must be so in nature. The space of the
geometer would be considered a mere fiction, and it would not be
credited with objective validity, because we cannot see how things
must of necessity agree with an image of them, which we make
spontaneously and previous to our acquaintance with them. But if this
image, or rather this formal intuition, is the essential property of
our sensibility, by means of which alone objects are given to us, and
if this sensibility represents not things in themselves, but their
appearances: we shall easily comprehend, and at the same time
indisputably prove, that all external objects of our world of sense
must necessarily coincide in the most rigorous way with the
propositions of geometry; because sensibility by means of its form of
external intuition, viz., by space, the same with which the geometer
is occupied, makes those objects at all possible as mere appearances.
It will always remain a remarkable phenomenon in the history of
philosophy, that there was a time, when even mathematicians, who at
the same time were philosophers, began to doubt, not of the accuracy
of their geometrical propositions so far as they concerned space, but
of their objective validity and the applicability of this concept
itself, and of all its corollaries, to nature. They showed much
concern whether a line in nature might not consist of physical points,
and consequently that true space in the object might consist of simple
[discrete] parts, while the space which the geometer has in his mind
[being continuous] cannot be such. They did not recognise that this
mental space renders possible the physical space, i.e., the extension
of matter; that this pure space is not at all a quality of things in
themselves, but a form of our sensuous faculty of representation; and
that all objects in space are mere appearances, i.e., not things in
themselves but representations of our sensuous intuition. But such is
the case, for the space of the geometer is exactly the form of
sensuous intuition which we find a priori in us, and contains the
ground of the possibility of all external appearances (according to
their form), and the latter must necessarily and most rigidly agree
with the propositions of the geometer, which he draws not from any
fictitious concept, but from the subjective basis of all external
phenomena, which is sensibility itself. In this and no other way can
geometry be made secure as to the undoubted objective reality of its
propositions against all the intrigues of a shallow Metaphysics, which
is surprised at them [the geometrical propositions], because it has
not traced them to the sources of their concepts.
Remark II.
Whatever is given us as object, must be given us in intuition. All our
intuition however takes place by means of the senses only; the
understanding intuites nothing, but only reflects. And as we have just
shown that the senses never and in no manner enable us to know things
in themselves, but only their appearances, which are mere
representations of the sensibility, we conclude that 'all bodies,
together with the space in which they are, must be considered nothing
but mere representations in us, and exist nowhere but in our
thoughts.' You will say: Is not this manifest idealism?
Idealism consists in the assertion, that there are none but thinking
beings, all other things, which we think are perceived in intuition,
being nothing but representations in the thinking beings, to which no
object external to them corresponds in fact. Whereas I say, that
things as objects of our senses existing outside us are given, but we
know nothing of what they may be in themselves, knowing only their
appearances, i.e., the representations which they cause in us by
affecting our senses. Consequently I grant by all means that there are
bodies without us, that is, things which, though quite unknown to us
as to what they are in themselves, we yet know by the representations
which their influence on our sensibility procures us, and which we
call bodies, a term signifying merely the appearance of the thing
which is unknown to us, but not therefore less actual. Can this be
termed idealism? It is the very contrary.
Long before Locke's time, but assuredly since him, it has been
generally assumed and granted without detriment to the actual
existence of external things, that many of their predicates may be
said to belong not to the things in themselves, but to their
appearances, and to have no proper existence outside our
representation. Heat, color, and taste, for instance, are of this
kind. Now, if I go farther, and for weighty reasons rank as mere
appearances the remaining qualities of bodies also, which are called
primary, such as extension, place, and in general space, with all that
which belongs to it (impenetrability or materiality, space, etc.)—no
one in the least can adduce the reason of its being inadmissible. As
little as the man who admits colors not to be properties of the object
in itself, but only as modifications of the sense of sight, should on
that account be called an idealist, so little can my system be named
idealistic, merely because I find that more, nay,
All the properties which constitute the intuition of a body belong
merely to its appearance.
The existence of the thing that appears is thereby not destroyed, as
in genuine idealism, but it is only shown, that we cannot possibly
know it by the senses as it is in itself.
I should be glad to know what my assertions must be in order to avoid
all idealism. Undoubtedly, I should say, that the representation of
space is not only perfectly conformable to the relation which our
sensibility has to objects—that I have said—but that it is quite
similar to the object,—an assertion in which I can find as little
meaning as if I said that the sensation of red has a similarity to the
property of vermilion, which in me excites this sensation.
Remark III.
Hence we may at once dismiss an easily foreseen but futile objection,
"that by admitting the ideality of space and of time the whole
sensible world would be turned into mere sham." At first all
philosophical insight into the nature of sensuous cognition was
spoiled, by making the sensibility merely a confused mode of
representation, according to which we still know things as they are,
but without being able to reduce everything in this our representation
to a clear consciousness; whereas proof is offered by us that
sensibility consists, not in this logical distinction of clearness and
obscurity, but in the genetical one of the origin of cognition itself.
For sensuous perception represents things not at all as they are, but
only the mode in which they affect our senses, and consequently by
sensuous perception appearances only and not things themselves are
given to the understanding for reflexion. After this necessary
corrective, an objection rises from an unpardonable and almost
intentional misconception, as if my doctrine turned all the things of
the world of sense into mere illusion.
When an appearance is given us, we are still quite free as to how we
should judge the matter. The appearance depends upon the senses, but
the judgment upon the understanding, and the only question is, whether
in the determination of the object there is truth or not. But the
difference between truth and dreaming is not ascertained by the nature
of the representations, which are referred to objects (for they are
the same in both cases), but by their connexion according to those
rules, which determine the coherence of the representations in the
concept of an object, and by ascertaining whether they can subsist
together in experience or not. And it is not the fault of the
appearances if our cognition takes illusion for truth, i.e., if the
intuition, by which an object is given us, is considered a concept of
the thing or of its existence also, which the understanding can only
think. The senses represent to us the paths of the planets as now
progressive, now retrogressive, and herein is neither falsehood nor
truth, because as long as we hold this path to be nothing but
appearance, we do not judge of the objective nature of their motion.
But as a false judgment may easily arise when the understanding is not
on its guard against this subjective mode of representation being
considered objective, we say they appear to move backward; it is not
the senses however which must be charged with the illusion, but the
understanding, whose province alone it is to give an objective
judgment on appearances.
Thus, even if we did not at all reflect on the origin of our
representations, whenever we connect our intuitions of sense (whatever
they may contain), in space and in time, according to the rules of the
coherence of all cognition in experience, illusion or truth will arise
according as we are negligent or careful. It is merely a question of
the use of sensuous representations in the understanding, and not of
their origin. In the same way, if I consider all the representations
of the senses, together with their form, space and time, to be nothing
but appearances, and space and time to be a mere form of the
sensibility, which is not to be met with in objects out of it, and if
I make use of these representations in reference to possible
experience only, there is nothing in my regarding them as appearances
that can lead astray or cause illusion. For all that they can
correctly cohere according to rules of truth in experience. Thus all
the propositions of geometry hold good of space as well as of all the
objects of the senses, consequently of all possible experience,
whether I consider space as a mere form of the sensibility, or as
something cleaving to the things themselves. In the former case
however I comprehend how I can know a priori these propositions
concerning all the objects of external intuition. Otherwise,
everything else as regards all possible experience remains just as if
I had not departed from the vulgar view.
But if I venture to go beyond all possible experience with my notions
of space and time, which I cannot refrain from doing if I proclaim
them qualities inherent in things in themselves (for what should
prevent me from letting them hold good of the same things, even though
my senses might be different, and unsuited to them?), then a grave
error may arise due to illusion, for thus I would proclaim to be
universally valid what is merely a subjective condition of the
intuition of things and sure only for all objects of sense, viz., for
all possible experience; I would refer this condition to things in
themselves, and do not limit it to the conditions of experience.
My doctrine of the ideality of space and of time, therefore, far from
reducing the whole sensible world to mere illusion, is the only means
of securing the application of one of the most important cognitions
(that which mathematics propounds a priori) to actual objects, and of
preventing its being regarded as mere illusion. For without this
observation it would be quite impossible to make out whether the
intuitions of space and time, which we borrow from no experience, and
which yet lie in our representation a priori, are not mere phantasms
of our brain, to which objects do not correspond, at least not
adequately, and consequently, whether we have been able to show its
unquestionable validity with regard to all the objects of the sensible
world just because they are mere appearances.
Secondly, though these my principles make appearances of the
representations of the senses, they are so far from turning the truth
of experience into mere illusion, that they are rather the only means
of preventing the transcendental illusion, by which metaphysics has
hitherto been deceived, leading to the childish endeavor of catching
at bubbles, because appearances, which are mere representations, were
taken for things in themselves. Here originated the remarkable event
of the antimony of Reason which I shall mention by and by, and which
is destroyed by the single observation, that appearance, as long as it
is employed in experience, produces truth, but the moment it
transgresses the bounds of experience, and consequently becomes
transcendent, produces nothing but illusion.
Inasmuch, therefore, as I leave to things as we obtain them by the
senses their actuality, and only limit our sensuous intuition of these
things to this, that they represent in no respect, not even in the
pure intuitions of space and of time, anything more than mere
appearance of those things, but never their constitution in
themselves, this is not a sweeping illusion invented for nature by me.
My protestation too against all charges of idealism is so valid and
clear as even to seem superfluous, were there not incompetent judges,
who, while they would have an old name for every deviation from their
perverse though common opinion, and never judge of the spirit of
philosophic nomenclature, but cling to the letter only, are ready to
put their own conceits in the place of well-defined notions, and
thereby deform and distort them. I have myself given this my theory
the name of transcendental idealism, but that cannot authorise any one
to confound it either with the empirical idealism of Descartes,
(indeed, his was only an insoluble problem, owing to which he thought
every one at liberty to deny the existence of the corporeal world,
because it could never be proved satisfactorily), or with the mystical
and visionary idealism of Berkeley, against which and other similar
phantasms our Critique contains the proper antidote. My idealism
concerns not the existence of things (the doubting of which, however,
constitutes idealism in the ordinary sense), since it never came into
my head to doubt it, but it concerns the sensuous representation of
things, to which space and time especially belong. Of these [viz.,
space and time], consequently of all appearances in general, I have
only shown, that they are neither things (but mere modes of
representation), nor determinations belonging to things in themselves.
But the word "transcendental," which with me means a reference of our
cognition, i.e., not to things, but only to the cognitive faculty, was
meant to obviate this misconception. Yet rather than give further
occasion to it by this word, I now retract it, and desire this
idealism of mine to be called critical. But if it be really an
objectionable idealism to convert actual things (not appearances) into
mere representations, by what name shall we call him who conversely
changes mere representations to things? It may, I think, be called
"dreaming idealism," in contradistinction to the former, which may be
called "visionary," both of which are to be refuted by my
transcendental, or, better, critical idealism.