an empirical nor a pure intuition, but merely the
synthesis of empirical intuitions, which, being em-
pirical, cannot be given a priori. No determining
synthetical proposition therefore can spring from it,
because the synthesis cannot a priori pass beyond to
DISCIPLINE OF PURE REASON. 619
the intuition that corresponds to it, but only a prin-
ciple of the synthesis 1 of possible empirical in-
tuitions.
A transcendental proposition, therefore, is synthet-
ical knowledge acquired by reason, according to mere
concepts; and it is discursive, because through it
alone synthetical unity of empirical knowledge be-
comes possible, while it cannot give us any intuition
a priori.
We see, therefore, that reason is used in [p. 723]
two ways which, though they share in common the
generality of their knowledge and its production a
priori, yet diverge considerably afterwards, because
in each phenomenon (and no object can be given us,
except as a phenomenon), there are two elements, the
form of intuition (space and time), which can be
known and determined entirely a priori, and the
matter (the physical) or the contents, something
which exists in space and time, and therefore contains
an existence corresponding to sensation. As regards
the latter, which can never be given in a definite form
except empirically, we can have nothing a priori ex-
cept indefinite concepts of the synthesis of possible
sensations, in so far as they belong to the unity of
apperception (in a possible experience). As regards
1 In the concept of cause I really pass beyond the empirical
concept of an event, but not to the intuition which represents the
concept of cause in concrete, but to the conditions of time in
general, which in experience, might be found in accordance with
the concept of cause. I therefore proceed here, according to con-
cepts only, but cannot proceed by means of the construction of
concepts, because the concept is only a rule for the synthesis of
perceptions, which are not pure intuitions, and therefore cannot be
given a 2>riori.
620 DISCIPLINE OF PURE REASON.
the former, we can determine a priori our concepts
in intuition, by creating to ourselves in space and
time, through a uniform synthesis, the objects them-
selves, considering them simply as quanta. The
former is called the use of reason according to con-
cepts; and here we can do nothing more than to bring
phenomena under concepts, according to their real
contents, which therefore can be determined empiri-
cally only, that is a posteriori (though in accordance
with those concepts as rules of an empirical synthesis).
The latter is the use of reason through the [p. 7 2 4]
construction of concepts, which, as they refer to an
intuition a priori, can for that reason be given a priori,
and defined in pure intuition, without any empirical
data. To consider everything which exists (every-
thing in space or time) whether, and how far, it is a
quantum or not ; to consider that we must represent in
it either existence, or absence of existence ; to consider
how far this something which fills space or time is a
primary substratum, or merely determination of it;
to consider again whether its existence is related to
something else as cause or effect, or finally, whether it
stands isolated or in reciprocal dependence on others,
with reference to existence, this and the possibility,
reality, and necessity of its existence, or their oppo-
sites, all belong to that knowledge of reason, derived
from concepts, which is called philosophical. But to
determine a priori an intuition in space (figure), to
divide time (duration), or merely to know the general
character of the synthesis of one and the same thing
in time and space, and the quantity of an intuition
in general which arises from it (number), all this is
DISCIPLINE OF PUBE REASON. 62 1
the work of reason by means of the construction of
concepts, and is called mathematical
The great success which attends reason in its
mathematical use produces naturally the expectation
that it, or rather its method, would have the same
success outside the field of quantities also, by reducing
all concepts to intuitions which may be given [p-V^S]
a priori, and by which the whole of nature might be
conquered, while pure philosophy, with its discursive
concepts a priori, does nothing but bungle m every
part of nature, without being able to render the
reality of those concepts intuitive a prion, and
thereby legitimatised. Nor does there seem to be
any lack of confidence on the part of those who are
masters in the art of mathematics, or of high expec-
tations on the part of the public at large, as to their
ability of achieving success, if only they would try it.
For as they have hardly ever philosophised on mathe-
matics (which is indeed no easy task), they never
think of the specific difference between the two uses
of reason which we have just explained. Current
and empirical rules, borrowed from the ordinary
operations of reason, are then accepted instead ot
axioms. From what quarter the concepts of space
and time with which alone (as the original quanta)
they have to deal, may have come to them, they do
not care to enquire, nor do they see any use m in-
vestigating the origin of the pure concepts of the
understanding, and with it the extent of their validity,
being satisfied to use them as they are. In all this
no blame would attach to them, if only they did not
overstep their proper limits, namely those of nature.
622 DISCIPLINE OF PURE REASON.
But as it is, they lose themselves, without being aware
of it, away from the field of sensibility on the uncer-
tain ground of pure and even transcendental concepts
(instabilis tellus, innabilis unda) where they are [p. 7 2 6]
neither able to stand nor to swim, taking only a few
hasty steps, the vestiges of which, are soon swept
away, while their steps in mathematics become a
highway, on which the latest posterity may march on
with perfect confidence.
We have chosen it as our duty to determine with
accuracy and certainty the limits of pure reason in
its transcendental use. These transcendental efforts,
however, have this peculiar character that, in spite
of the strongest and clearest warnings, they continue
to inspire us with new hopes, before the attempt is
entirely surrendered at arriving beyond the limits
of experience at the charming fields of an intel-
lectual world. It is necessary therefore to cut away
the last anchor of that fantastic hope, and to show
that the employment of the mathematical method
cannot be of the slightest use for this kind of know-
ledge, unless it be in displaying its own deficiencies ;
and that the art of measuring and philosophy are
two totally different things, though they are mutually
useful to each other in natural science, and that the
method of the one can never be imitated by the
other.
The exactness of mathematics depends on defini-
tions, axioms, and demonstrations. I shall content
myself with showing that none of these can be
achieved or imitated by the philosopher in the
sense in which they are understood by the mathe-
DISCIPLINE OF PURE REASON. 62
J
matician. I hope to show at the same time [p. 727]
that the art of measuring, or geometry, will by its
method produce nothing in philosophy but card-
houses, while the philosopher with his method pro-
duces in mathematics nothing but vain babble. It
is the very essence of philosophy to teach the limits
of knowledge, and even the mathematician, unless
his talent is limited already by nature and restricted
to its proper work, cannot decline the warnings of
philosophy or altogether defy them.
I. Of Definitions. To define, as the very name
implies, means only to represent the complete con-
concept of a thing within its limits and in its
primary character 1 . From this point of view, an em-
pirical concept cannot be defined, but can be explained
only. For, as we have in an empirical concept
some predicates only belonging to a certain class of
sensuous objects, we are never certain whether
by the word which denotes one and the same
object, we do not think at one time a greater, at
another a smaller number of predicates, [p. 728]
Thus one man may by the concept of gold think,
in addition to weight, colour, malleability, the quality
of its not rusting, while another may know nothing
of the last. We use certain predicates so long only
as they are required for distinction. New observa-
1 Completeness means clearness and sufficiency of predicates;
limits mean precision, so that no more predicates are given than
belong to the complete concept ; in its primary acceptation means
that the determination of these limits is not derived from anything
else, and therefore in need of any proof, because this would render
the so-called definition incapable of standing at the head of all the
judgments regarding its object.
O24 DISCIPLINE OF PURE REASON.
tions add and remove certain predicates, so that the
concept never stands within safe limits. And of
what use would it be to define an empirical concept,
as for instance that of water, because, when we speak
of water and its qualities, we do not care much what
is thought by that word, but proceed at once to ex-
periments % the word itself with its few predicates
being a designation only and not a concept, so that
a so-called definition would be no more than a de-
termination of the word. Secondly, if we reasoned
accurately, no a priori given concept can be defined,
such as substance, cause, right, equity, &c. For I
can never be sure that the clear representation of
a given but still confused concept has been com-
pletely analysed, unless I know that such repre-
sentation is adequate to the object. As its concept,
however, such as it is given, may contain many
obscure representations which we pass by in our
analysis, although we use them always in the prac-
tical application of the concept, the completeness of
the analysis of my concept must always remain
doubtful, and can only be rendered probable by means
of apt examples, although never apodictically [p. 729]
certain. I should therefore prefer to use the term ex-
position rather than definition, as being more modest,
and more likely to be admitted to a certain extent
by a critic who reserves his doubts as to its com-
pleteness. As therefore it is impossible to define
either empirically or a priori given concepts, there
remain arbitrary concepts only on which such an ex-
periment may be tried. In such a case I can always
define my concept, because I ought certainly to know
DISCIPLINE OF PURE REASON. 625
what I wish to think, the concept being made
intentionally by myself, and not given to me either
by the nature of the understanding or by expe-
rience. But I can never say that I have thus de-
fined a real object. For if the concept depends on
empirical conditions, as, for instance, a ship's chro-
nometer, the object itself and its possibility are not
given by this arbitrary concept ; it does not even tell
us whether there is an object corresponding to it,
so that my explanation should be called a declara-
tion (of my project) rather than a definition of an
object. Thus there remain no concepts fit for de-
finition except those which contain an arbitrary
synthesis that can be constructed a priori. It fol-
lows, therefore, that mathematics only can possess
definitions, because it is in mathematics alone that
we represent a priori in intuition the object which
we think, and that object cannot therefore contain
either more or less than the concept, because [p. 730]
the concept of the object was given by the definition
in its primary character, that is, without deriving
the definition from anything else. The German
language has but the one word ErMarung (literally
clearing up) for the terms exposition, explication, de-
claration, and definition ; and we must not therefore
be too strict in our demands, when denying to the
different kinds of a philosophical clearing up the
honourable name of definition. What we really
insist on is this, that philosophical definitions are
possible only as expositions of given concepts, mathe-
matical definitions as constructions of concepts,
originally framed by ourselves, the former there-
VOL. II. S S
626 DISCIPLINE OF PURE REASON.
fore analytically (where completeness is never apo-
dictically certain), the latter synthetically. Mathe-
matical definitions make the concept, philosophical
definitions explain it only. Hence it follows,
a. That we must not try in philosophy to imitate
mathematics by beginning with definitions, except
it be by way of experiment. For as they are meant
to be an analysis of given concepts, these concepts
themselves, although as yet confused only, must
come first, and the incomplete exposition must precede
the complete one, so that we are able from some
characteristics, known to us from an, as yet, incom-
plete analysis, to infer many things before we come
to a complete exposition, that is, the definition of the
concept. In philosophy, in fact, the definition [p. 731]
in its complete clearness ought to conclude rather
than begin our work *; while in mathematics we really
have no concept antecedent to the definition by which
the concept itself is first given, so that in mathe-
matics no other beginning is necessary or possible.
b. Mathematical definitions can never be erroneous,
because, as the concept is first given by the de-
1 Philosophy swarms with faulty definitions, particularly such
as contain some true elements of a definition, hut not all. If,
therefore, it were impossible to use a concept until it had been com-
pletely defined, philosophy would fare very ill. As, however, we
may use a definition with perfect safety, so far at least as the ele-
ments of the analysis will carry us, imperfect definitions also, that
is, propositions which are not yet properly definitions, but are yet
true, and, therefore, approximations to a definition, may be used
with great advantage. In mathematics definitions belong ad esse,
in philosophy ad melius esse. It is desirable, but it is extremely
difficult to construct a proper definition. Jurists are without a
definition of right to the present day.
DISCIPLINE OF PURE REASON. 627
finition, it contains neither more nor less than what
the definition wishes should be conceived by it.
But although there can be nothing wrong in it, so
far as its contents are concerned, mistakes may some-
times, though rarely, occur in the form or wording,
particularly with regard to perfect precision. Thus
the common definition of a circle, that it is a curved
line, every point of which is equally distant from
one and the same point (namely the centre), [p. 732]
is faulty, because the determination of curved is in-
troduced unnecessarily. For there must be a par-
ticular theorem, derived from the definition, and
easily proved, viz. that every line, all points of which
are equidistant from one and the same point, must
be curved (no part of it being straight). Analytical
definitions, however, may be erroneous in many
respects, either by introducing characteristics which
do not really exist in the concept, or by lacking
that completeness which is essential to a definition,
because we can never be quite certain of the com-
pleteness of our analysis. It is on these accounts
that the method of mathematics cannot be imitated
in the definitions of philosophy.
II. Of Axioms. These, so far as they are im-
mediately certain, are synthetical principles a priori.
One concept cannot, however, be connected syn-
thetically and yet immediately with another, because,
if we wish to go beyond a given concept, a third
connecting knowledge is required; and, as philosophy
is the knowledge of reason based on concepts, no
principle can be found in it deserving the name of
an axiom. Mathematics, on the other hand, may
s s 2
628 DISCIPLINE OF PURE REASON.
well possess axioms, because here, by means of the
construction of concepts in the intuition of their
object, the predicates may always be connected a
priori and immediately ; for instance, that three
points always lie in a plane. A synthetical prin-
ciple, on the contrary, made up of concepts [p. 733]
only, can never be immediately certain, as, for ex-
ample, the proposition, that everything which hap-
pens has its cause. Here I require something else,
namely, the condition of the determination by time
in a given experience, it being impossible for me
to know such a principle, directly and immediately,
from the concepts. Discursive principles are, there-
fore, something quite different from intuitive prin-
ciples or axioms. The former always require, in
addition, a deduction, not at all required for the
latter, which, on that very account, are evident,
while philosophical principles, whatever their cer-
tainty may be, can never pretend to be so. Hence
it is very far from true to say that any synthetical
proposition of pure and transcendental reason is
so evident (as people sometimes emphatically main-
tain) as the statement that twice two are four. It
is true that in the Analytic, when giving the table
of the principles of the pure understanding, I men-
tioned also certain axioms of intuition; but the
principle there mentioned was itself no axiom, but
served only to indicate the principle of the possibility
of axioms in general, being itself no more than a
principle based on concepts. It was necessary in
our transcendental philosophy to show the possibility
even of mathematics. Philosophy, therefore, is
DISCIPLINE OF PURE REASON. 629
without axioms, and can never put forward its
principles a priori with absolute authority, but
must first consent to justify its claims by a thorough
deduction. [p. 734]
III. Of Demonstrations. An apodictic proof only,
so far as it is intuitive, can be called demonstration.
Experience may teach us what is, but never that
it cannot be otherwise. Empirical arguments, there-
fore, cannot produce an apodictic proof. From con-
cepts a priori, however (in discursive knowledge),
it is impossible that intuitive certainty, that is,
evidence, should ever arise, however apodictically
certain the judgment may otherwise seem to be.
Demonstrations we get in mathematics only, because
here our knowledge is derived not from concepts, but
from their construction, that is, from intuition, which
can be given a priori, in accordance with the con-
cepts. Even the proceeding of algebra, with its
equations, from which by reduction both the correct
result and its proof are produced, is a construction
by characters, though not geometrical, in which, by
means of signs, the concepts, particularly those of
the relation of quantities, are represented in intuition,
and (without any regard to the heuristic method) all
conclusions are secured against errors by submitting
each of them to intuitive evidence. Philosophical
knowledge cannot claim this advantage, for here we
must always consider the general in the abstract (by
concepts), while in mathematics we may consider the
general in the concrete, in each single intuition, and
yet through pure representation a priori, where every
mistake becomes at once manifest. I should [p. 735]
63O DISCIPLINE OF PURE REASON.
prefer, therefore, to call the former acroamatic, or
audible (discursive) proofs, because they can be car-
ried out by words only (the object in thought),
rather than demonstrations, which, as the very term
implies, depend on the intuition of the object.
It follows from all this that it is not in accordance
with the very nature of philosophy to boast of its
dogmatical character, particularly in the field of pure
reason, and to deck itself with the titles and ribands
of mathematics, an order to which it can never
belong, though it may well hope for co-operation
with that science. All those attempts are vain pre-
tensions which can never be successful, nay, which
can only prove an obstacle in the discovery of the
illusions of reason, when ignoring its own limits, and
which must mar our success in calling back, by means
of a sufficient explanation of our concepts, the con-
ceit of speculation to the more modest and thorough
work of self-knowledge. Eeason ought not, there-
fore, in its transcendental endeavours, to look forward
with such confidence, as if the path which it has
traversed must lead straight to its goal, nor depend
with such assurance on its premisses as to consider
it unnecessary to look back from time to time, to
find out whether, in the progress of its conclusions,
errors may come to light, which were overlooked
in the principles, and which render it neces- [p. 736]
sary either to determine those principles more ac-
curately or to change them altogether.
I divide all apodictic propositions, whether demon-
strable or immediately certain, into Dogmata and
Mathemata. A directly synthetical proposition, based
DISCIPLINE OF PURE REASON. 63 1
on concepts, is a Dogma ; a proposition of the same
kind, arrived at by the construction of concepts, is
a Mathema. Analytical judgments teach us really
no more of an object than what the concept which
we have of it contains in itself. They cannot en-
large our knowledge beyond the concept, but only
clear it. They cannot, therefore, be properly called
dogmas (a word which might perhaps best be trans-
lated by precepts, Lehrsprilche). According to our
ordinary mode of speech, we could apply that name
to that class only of the two above mentioned classes
of synthetical propositions a priori which refers to
philosophical knowledge, and no one would feel
inclined to give the name of Dogma to the proposi-
tions of arithmetic or geometry. In this way the
usage of language confirms our explanation that
those judgments only which are based on concep-
tions, and not those which are arrived at by the
construction of concepts, can be called dogmatic.
Now in the whole domain of pure reason, in its
purely speculative use, there does not exist a single
directly synthetical judgment based on concepts. We
have shown that reason, by means of ideas, is in-
capable of any synthetical judgments which could
claim objective validity, while by means of the con-
cepts of our understanding it establishes no [p. 737]
doubt some perfectly certain principles, but not di-
rectly from concepts, but indirectly only, by referring
such concepts to something purely contingent, namely,
possible experience. When such experience (anything
as an object of possible experience) is presupposed,
these principles are, no doubt, apodictically certain,
632 DISCIPLINE OF PURE REASON.
but in themselves (directly) they cannot even be
known a priori. Thus the proposition that every-
thing which happens has its cause, can never be
thoroughly understood by means of the concepts
alone which are contained in it ; hence it is no
dogma in itself, although, from another point of view,
that is, in the only field of its possible use, namely,
in experience, it may be proved apodictically. It
should be called, therefore, a principle, and not a
precept or a dogma (though it is necessary that it
should itself be proved), because it has this pecu-
liarity that it first renders its own proof, namely,
experience, possible, and has always to be presup-
posed for the sake of experience.
If, therefore, there are no dogmata whatever in
the speculative use of pure reason, with regard to
their contents also, all dogmatical methods, whether
borrowed from mathematics or invented on purpose,
are alike inappropriate. They only serve to hide
mistakes and errors, and thus deceive philosophy,
whose true object is to shed the clearest light on
every step which reason takes. The method may,
however, well be systematical ; for our reason (sub-
jectively) is itself a system, though in its pure [p. 738]
use, by means of mere concepts, a system intended
for investigation only, according to principles of
unity, to which experience alone can supply the
material. We cannot, however, dwell here on the
method of transcendental philosophy, because all we
have to do at present is to take stock in order to
find out whether we are able to build at all, and
how high the edifice may be which we can erect
DISCIPLINE OF PURE REASON. 633
with the materials at our command (the pure con-
cepts a priori).
METHOD OF TRANSCENDENTALISM.
Section II.
The Discipline of Pure Beason in its polemical use.
Keason in all her undertakings must submit to
criticism, and cannot attempt to limit the free exer-
cise of such criticism without injury to herself, and