第 77 页

SECTION I. The Discipline of Pure Reason in the Sphere

of Dogmatism.

The science of mathematics presents the most brilliant example of

the extension of the sphere of pure reason without the aid of

experience. Examples are always contagious; and they exert an especial

influence on the same faculty, which naturally flatters itself that it

will have the same good fortune in other case as fell to its lot in

one fortunate instance. Hence pure reason hopes to be able to extend

its empire in the transcendental sphere with equal success and

security, especially when it applies the same method which was

attended with such brilliant results in the science of mathematics. It

is, therefore, of the highest importance for us to know whether the

method of arriving at demonstrative certainty, which is termed

mathematical, be identical with that by which we endeavour to attain

the same degree of certainty in philosophy, and which is termed in

that science dogmatical.

Philosophical cognition is the cognition of reason by means of

conceptions; mathematical cognition is cognition by means of the

construction of conceptions. The construction of a conception is the

presentation a priori of the intuition which corresponds to the

conception. For this purpose a non-empirical intuition is requisite,

which, as an intuition, is an individual object; while, as the

construction of a conception (a general representation), it must be

seen to be universally valid for all the possible intuitions which

rank under that conception. Thus I construct a triangle, by the

presentation of the object which corresponds to this conception,

either by mere imagination, in pure intuition, or upon paper, in

empirical intuition, in both cases completely a priori, without

borrowing the type of that figure from any experience. The

individual figure drawn upon paper is empirical; but it serves,

notwithstanding, to indicate the conception, even in its universality,

because in this empirical intuition we keep our eye merely on the

act of the construction of the conception, and pay no attention to the

various modes of determining it, for example, its size, the length

of its sides, the size of its angles, these not in the least affecting

the essential character of the conception.

Philosophical cognition, accordingly, regards the particular only in

the general; mathematical the general in the particular, nay, in the

individual. This is done, however, entirely a priori and by means of

pure reason, so that, as this individual figure is determined under

certain universal conditions of construction, the object of the

conception, to which this individual figure corresponds as its schema,

must be cogitated as universally determined.

The essential difference of these two modes of cognition consists,

therefore, in this formal quality; it does not regard the difference

of the matter or objects of both. Those thinkers who aim at

distinguishing philosophy from mathematics by asserting that the

former has to do with quality merely, and the latter with quantity,

have mistaken the effect for the cause. The reason why mathematical

cognition can relate only to quantity is to be found in its form

alone. For it is the conception of quantities only that is capable

of being constructed, that is, presented a priori in intuition;

while qualities cannot be given in any other than an empirical

intuition. Hence the cognition of qualities by reason is possible only

through conceptions. No one can find an intuition which shall

correspond to the conception of reality, except in experience; it

cannot be presented to the mind a priori and antecedently to the

empirical consciousness of a reality. We can form an intuition, by

means of the mere conception of it, of a cone, without the aid of

experience; but the colour of the cone we cannot know except from

experience. I cannot present an intuition of a cause, except in an

example which experience offers to me. Besides, philosophy, as well as

mathematics, treats of quantities; as, for example, of totality,

infinity, and so on. Mathematics, too, treats of the difference of

lines and surfaces- as spaces of different quality, of the

continuity of extension- as a quality thereof. But, although in such

cases they have a common object, the mode in which reason considers

that object is very different in philosophy from what it is in

mathematics. The former confines itself to the general conceptions;

the latter can do nothing with a mere conception, it hastens to

intuition. In this intuition it regards the conception in concreto,

not empirically, but in an a priori intuition, which it has

constructed; and in which, all the results which follow from the

general conditions of the construction of the conception are in all

cases valid for the object of the constructed conception.

Suppose that the conception of a triangle is given to a

philosopher and that he is required to discover, by the

philosophical method, what relation the sum of its angles bears to a

right angle. He has nothing before him but the conception of a

figure enclosed within three right lines, and, consequently, with

the same number of angles. He may analyse the conception of a right

line, of an angle, or of the number three as long as he pleases, but

he will not discover any properties not contained in these

conceptions. But, if this question is proposed to a geometrician, he

at once begins by constructing a triangle. He knows that two right

angles are equal to the sum of all the contiguous angles which proceed

from one point in a straight line; and he goes on to produce one

side of his triangle, thus forming two adjacent angles which are

together equal to two right angles. He then divides the exterior of

these angles, by drawing a line parallel with the opposite side of the

triangle, and immediately perceives that be has thus got an exterior

adjacent angle which is equal to the interior. Proceeding in this way,

through a chain of inferences, and always on the ground of

intuition, he arrives at a clear and universally valid solution of the

question.

But mathematics does not confine itself to the construction of

quantities (quanta), as in the case of geometry; it occupies itself

with pure quantity also (quantitas), as in the case of algebra,

where complete abstraction is made of the properties of the object

indicated by the conception of quantity. In algebra, a certain

method of notation by signs is adopted, and these indicate the

different possible constructions of quantities, the extraction of

roots, and so on. After having thus denoted the general conception

of quantities, according to their different relations, the different

operations by which quantity or number is increased or diminished

are presented in intuition in accordance with general rules. Thus,

when one quantity is to be divided by another, the signs which

denote both are placed in the form peculiar to the operation of

division; and thus algebra, by means of a symbolical construction of

quantity, just as geometry, with its ostensive or geometrical

construction (a construction of the objects themselves), arrives at

results which discursive cognition cannot hope to reach by the aid

of mere conceptions.

Now, what is the cause of this difference in the fortune of the

philosopher and the mathematician, the former of whom follows the path

of conceptions, while the latter pursues that of intuitions, which

he represents, a priori, in correspondence with his conceptions? The

cause is evident from what has been already demonstrated in the

introduction to this Critique. We do not, in the present case, want to

discover analytical propositions, which may be produced merely by

analysing our conceptions- for in this the philosopher would have

the advantage over his rival; we aim at the discovery of synthetical

propositions- such synthetical propositions, moreover, as can be

cognized a priori. I must not confine myself to that which I

actually cogitate in my conception of a triangle, for this is

nothing more than the mere definition; I must try to go beyond that,

and to arrive at properties which are not contained in, although

they belong to, the conception. Now, this is impossible, unless I

determine the object present to my mind according to the conditions,

either of empirical, or of pure, intuition. In the former case, I

should have an empirical proposition (arrived at by actual measurement

of the angles of the triangle), which would possess neither

universality nor necessity; but that would be of no value. In the

latter, I proceed by geometrical construction, by means of which I

collect, in a pure intuition, just as I would in an empirical

intuition, all the various properties which belong to the schema of

a triangle in general, and consequently to its conception, and thus

construct synthetical propositions which possess the attribute of

universality.

It would be vain to philosophize upon the triangle, that is, to

reflect on it discursively; I should get no further than the

definition with which I had been obliged to set out. There are

certainly transcendental synthetical propositions which are framed

by means of pure conceptions, and which form the peculiar

distinction of philosophy; but these do not relate to any particular

thing, but to a thing in general, and enounce the conditions under

which the perception of it may become a part of possible experience.

But the science of mathematics has nothing to do with such

questions, nor with the question of existence in any fashion; it is

concerned merely with the properties of objects in themselves, only in

so far as these are connected with the conception of the objects.

In the above example, we merely attempted to show the great

difference which exists between the discursive employment of reason in

the sphere of conceptions, and its intuitive exercise by means of

the construction of conceptions. The question naturally arises: What

is the cause which necessitates this twofold exercise of reason, and

how are we to discover whether it is the philosophical or the

mathematical method which reason is pursuing in an argument?

All our knowledge relates, finally, to possible intuitions, for it

is these alone that present objects to the mind. An a priori or

non-empirical conception contains either a pure intuition- and in this

case it can be constructed; or it contains nothing but the synthesis

of possible intuitions, which are not given a priori. In this latter

case, it may help us to form synthetical a priori judgements, but only

in the discursive method, by conceptions, not in the intuitive, by

means of the construction of conceptions.

The only a priori intuition is that of the pure form of phenomena-

space and time. A conception of space and time as quanta may be

presented a priori in intuition, that is, constructed, either alone

with their quality (figure), or as pure quantity (the mere synthesis

of the homogeneous), by means of number. But the matter of

phenomena, by which things are given in space and time, can be

presented only in perception, a posteriori. The only conception

which represents a priori this empirical content of phenomena is the

conception of a thing in general; and the a priori synthetical

cognition of this conception can give us nothing more than the rule

for the synthesis of that which may be contained in the

corresponding a posteriori perception; it is utterly inadequate to

present an a priori intuition of the real object, which must

necessarily be empirical.

Synthetical propositions, which relate to things in general, an a

priori intuition of which is impossible, are transcendental. For

this reason transcendental propositions cannot be framed by means of

the construction of conceptions; they are a priori, and based entirely

on conceptions themselves. They contain merely the rule, by which we

are to seek in the world of perception or experience the synthetical

unity of that which cannot be intuited a priori. But they are

incompetent to present any of the conceptions which appear in them

in an a priori intuition; these can be given only a posteriori, in

experience, which, however, is itself possible only through these

synthetical principles.

If we are to form a synthetical judgement regarding a conception, we

must go beyond it, to the intuition in which it is given. If we keep

to what is contained in the conception, the judgement is merely

analytical- it is merely an explanation of what we have cogitated in

the conception. But I can pass from the conception to the pure or

empirical intuition which corresponds to it. I can proceed to

examine my conception in concreto, and to cognize, either a priori

or a posterio, what I find in the object of the conception. The

former- a priori cognition- is rational-mathematical cognition by

means of the construction of the conception; the latter- a

posteriori cognition- is purely empirical cognition, which does not

possess the attributes of necessity and universality. Thus I may

analyse the conception I have of gold; but I gain no new information

from this analysis, I merely enumerate the different properties

which I had connected with the notion indicated by the word. My

knowledge has gained in logical clearness and arrangement, but no

addition has been made to it. But if I take the matter which is

indicated by this name, and submit it to the examination of my senses,

I am enabled to form several synthetical- although still empirical-

propositions. The mathematical conception of a triangle I should

construct, that is, present a priori in intuition, and in this way

attain to rational-synthetical cognition. But when the

transcendental conception of reality, or substance, or power is

presented to my mind, I find that it does not relate to or indicate

either an empirical or pure intuition, but that it indicates merely

the synthesis of empirical intuitions, which cannot of course be given

a priori. The synthesis in such a conception cannot proceed a

priori- without the aid of experience- to the intuition which