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corresponds to the conception; and, for this reason, none of these

conceptions can produce a determinative synthetical proposition,

they can never present more than a principle of the synthesis* of

possible empirical intuitions. A transcendental proposition is,

therefore, a synthetical cognition of reason by means of pure

conceptions and the discursive method, and it renders possible all

synthetical unity in empirical cognition, though it cannot present

us with any intuition a priori.

*In the case of the conception of cause, I do really go beyond the

empirical conception of an event- but not to the intuition which

presents this conception in concreto, but only to the time-conditions,

which may be found in experience to correspond to the conception. My

procedure is, therefore, strictly according to conceptions; I cannot

in a case of this kind employ the construction of conceptions, because

the conception is merely a rule for the synthesis of perceptions,

which are not pure intuitions, and which, therefore, cannot be given a

priori.

There is thus a twofold exercise of reason. Both modes have the

properties of universality and an a priori origin in common, but

are, in their procedure, of widely different character. The reason

of this is that in the world of phenomena, in which alone objects

are presented to our minds, there are two main elements- the form of

intuition (space and time), which can be cognized and determined

completely a priori, and the matter or content- that which is

presented in space and time, and which, consequently, contains a

something- an existence corresponding to our powers of sensation. As

regards the latter, which can never be given in a determinate mode

except by experience, there are no a priori notions which relate to

it, except the undetermined conceptions of the synthesis of possible

sensations, in so far as these belong (in a possible experience) to

the unity of consciousness. As regards the former, we can determine

our conceptions a priori in intuition, inasmuch as we are ourselves

the creators of the objects of the conceptions in space and time-

these objects being regarded simply as quanta. In the one case, reason

proceeds according to conceptions and can do nothing more than subject

phenomena to these- which can only be determined empirically, that is,

a posteriori- in conformity, however, with those conceptions as the

rules of all empirical synthesis. In the other case, reason proceeds

by the construction of conceptions; and, as these conceptions relate

to an a priori intuition, they may be given and determined in pure

intuition a priori, and without the aid of empirical data. The

examination and consideration of everything that exists in space or

time- whether it is a quantum or not, in how far the particular

something (which fills space or time) is a primary substratum, or a

mere determination of some other existence, whether it relates to

anything else- either as cause or effect, whether its existence is

isolated or in reciprocal connection with and dependence upon

others, the possibility of this existence, its reality and necessity

or opposites- all these form part of the cognition of reason on the

ground of conceptions, and this cognition is termed philosophical. But

to determine a priori an intuition in space (its figure), to divide

time into periods, or merely to cognize the quantity of an intuition

in space and time, and to determine it by number- all this is an

operation of reason by means of the construction of conceptions, and

is called mathematical.

The success which attends the efforts of reason in the sphere of

mathematics naturally fosters the expectation that the same good

fortune will be its lot, if it applies the mathematical method in

other regions of mental endeavour besides that of quantities. Its

success is thus great, because it can support all its conceptions by a

priori intuitions and, in this way, make itself a master, as it

were, over nature; while pure philosophy, with its a priori discursive

conceptions, bungles about in the world of nature, and cannot accredit

or show any a priori evidence of the reality of these conceptions.

Masters in the science of mathematics are confident of the success

of this method; indeed, it is a common persuasion that it is capable

of being applied to any subject of human thought. They have hardly

ever reflected or philosophized on their favourite science- a task

of great difficulty; and the specific difference between the two modes

of employing the faculty of reason has never entered their thoughts.

Rules current in the field of common experience, and which common

sense stamps everywhere with its approval, are regarded by them as

axiomatic. From what source the conceptions of space and time, with

which (as the only primitive quanta) they have to deal, enter their

minds, is a question which they do not trouble themselves to answer;

and they think it just as unnecessary to examine into the origin of

the pure conceptions of the understanding and the extent of their

validity. All they have to do with them is to employ them. In all this

they are perfectly right, if they do not overstep the limits of the

sphere of nature. But they pass, unconsciously, from the world of

sense to the insecure ground of pure transcendental conceptions

(instabilis tellus, innabilis unda), where they can neither stand

nor swim, and where the tracks of their footsteps are obliterated by

time; while the march of mathematics is pursued on a broad and

magnificent highway, which the latest posterity shall frequent without

fear of danger or impediment.

As we have taken upon us the task of determining, clearly and

certainly, the limits of pure reason in the sphere of

transcendentalism, and as the efforts of reason in this direction

are persisted in, even after the plainest and most expressive

warnings, hope still beckoning us past the limits of experience into

the splendours of the intellectual world- it becomes necessary to

cut away the last anchor of this fallacious and fantastic hope. We

shall, accordingly, show that the mathematical method is unattended in

the sphere of philosophy by the least advantage- except, perhaps, that

it more plainly exhibits its own inadequacy- that geometry and

philosophy are two quite different things, although they go band in

hand in hand in the field of natural science, and, consequently,

that the procedure of the one can never be imitated by the other.

The evidence of mathematics rests upon definitions, axioms, and

demonstrations. I shall be satisfied with showing that none of these

forms can be employed or imitated in philosophy in the sense in

which they are understood by mathematicians; and that the

geometrician, if he employs his method in philosophy, will succeed

only in building card-castles, while the employment of the

philosophical method in mathematics can result in nothing but mere

verbiage. The essential business of philosophy, indeed, is to mark out

the limits of the science; and even the mathematician, unless his

talent is naturally circumscribed and limited to this particular

department of knowledge, cannot turn a deaf ear to the warnings of

philosophy, or set himself above its direction.

I. Of Definitions. A definition is, as the term itself indicates,

the representation, upon primary grounds, of the complete conception

of a thing within its own limits.* Accordingly, an empirical

conception cannot be defined, it can only be explained. For, as

there are in such a conception only a certain number of marks or

signs, which denote a certain class of sensuous objects, we can

never be sure that we do not cogitate under the word which indicates

the same object, at one time a greater, at another a smaller number of

signs. Thus, one person may cogitate in his conception of gold, in

addition to its properties of weight, colour, malleability, that of

resisting rust, while another person may be ignorant of this

quality. We employ certain signs only so long as we require them for

the sake of distinction; new observations abstract some and add new

ones, so that an empirical conception never remains within permanent

limits. It is, in fact, useless to define a conception of this kind.

If, for example, we are speaking of water and its properties, we do

not stop at what we actually think by the word water, but proceed to

observation and experiment; and the word, with the few signs

attached to it, is more properly a designation than a conception of

the thing. A definition in this case would evidently be nothing more

than a determination of the word. In the second place, no a priori

conception, such as those of substance, cause, right, fitness, and

so on, can be defined. For I can never be sure, that the clear

representation of a given conception (which is given in a confused

state) has been fully developed, until I know that the

representation is adequate with its object. But, inasmuch as the

conception, as it is presented to the mind, may contain a number of

obscure representations, which we do not observe in our analysis,

although we employ them in our application of the conception, I can

never be sure that my analysis is complete, while examples may make

this probable, although they can never demonstrate the fact. instead

of the word definition, I should rather employ the term exposition-

a more modest expression, which the critic may accept without

surrendering his doubts as to the completeness of the analysis of

any such conception. As, therefore, neither empirical nor a priori

conceptions are capable of definition, we have to see whether the only

other kind of conceptions- arbitrary conceptions- can be subjected

to this mental operation. Such a conception can always be defined; for

I must know thoroughly what I wished to cogitate in it, as it was I

who created it, and it was not given to my mind either by the nature

of my understanding or by experience. At the same time, I cannot say

that, by such a definition, I have defined a real object. If the

conception is based upon empirical conditions, if, for example, I have

a conception of a clock for a ship, this arbitrary conception does not

assure me of the existence or even of the possibility of the object.

My definition of such a conception would with more propriety be termed

a declaration of a project than a definition of an object. There

are no other conceptions which can bear definition, except those which

contain an arbitrary synthesis, which can be constructed a priori.

Consequently, the science of mathematics alone possesses

definitions. For the object here thought is presented a priori in

intuition; and thus it can never contain more or less than the

conception, because the conception of the object has been given by the

definition- and primarily, that is, without deriving the definition

from any other source. Philosophical definitions are, therefore,

merely expositions of given conceptions, while mathematical

definitions are constructions of conceptions originally formed by

the mind itself; the former are produced by analysis, the completeness

of which is never demonstratively certain, the latter by a

synthesis. In a mathematical definition the conception is formed, in a

philosophical definition it is only explained. From this it follows:

*The definition must describe the conception completely that is,

omit none of the marks or signs of which it composed; within its own

limits, that is, it must be precise, and enumerate no more signs

than belong to the conception; and on primary grounds, that is to say,

the limitations of the bounds of the conception must not be deduced

from other conceptions, as in this case a proof would be necessary,

and the so-called definition would be incapable of taking its place at

the bead of all the judgements we have to form regarding an object.

(a) That we must not imitate, in philosophy, the mathematical

usage of commencing with definitions- except by way of hypothesis or

experiment. For, as all so-called philosophical definitions are merely

analyses of given conceptions, these conceptions, although only in a

confused form, must precede the analysis; and the incomplete

exposition must precede the complete, so that we may be able to draw

certain inferences from the characteristics which an incomplete

analysis has enabled us to discover, before we attain to the

complete exposition or definition of the conception. In one word, a

full and clear definition ought, in philosophy, rather to form the

conclusion than the commencement of our labours.* In mathematics, on

the contrary, we cannot have a conception prior to the definition;

it is the definition which gives us the conception, and it must for

this reason form the commencement of every chain of mathematical

reasoning.

*Philosophy abounds in faulty definitions, especially such as

contain some of the elements requisite to form a complete

definition. If a conception could not be employed in reasoning

before it had been defined, it would fare ill with all philosophical

thought. But, as incompletely defined conceptions may always be

employed without detriment to truth, so far as our analysis of the

elements contained in them proceeds, imperfect definitions, that is,

propositions which are properly not definitions, but merely

approximations thereto, may be used with great advantage. In

mathematics, definition belongs ad esse, in philosophy ad melius esse.

It is a difficult task to construct a proper definition. Jurists are

still without a complete definition of the idea of right.

(b) Mathematical definitions cannot be erroneous. For the conception

is given only in and through the definition, and thus it contains only

what has been cogitated in the definition. But although a definition

cannot be incorrect, as regards its content, an error may sometimes,

although seldom, creep into the form. This error consists in a want of

precision. Thus the common definition of a circle- that it is a curved

line, every point in which is equally distant from another point

called the centre- is faulty, from the fact that the determination