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indicated by the word curved is superfluous. For there ought to be a

particular theorem, which may be easily proved from the definition, to

the effect that every line, which has all its points at equal

distances from another point, must be a curved line- that is, that not

even the smallest part of it can be straight. Analytical

definitions, on the other hand, may be erroneous in many respects,

either by the introduction of signs which do not actually exist in the

conception, or by wanting in that completeness which forms the

essential of a definition. In the latter case, the definition is

necessarily defective, because we can never be fully certain of the

completeness of our analysis. For these reasons, the method of

definition employed in mathematics cannot be imitated in philosophy.

2. Of Axioms. These, in so far as they are immediately certain,

are a priori synthetical principles. Now, one conception cannot be

connected synthetically and yet immediately with another; because,

if we wish to proceed out of and beyond a conception, a third

mediating cognition is necessary. And, as philosophy is a cognition of

reason by the aid of conceptions alone, there is to be found in it

no principle which deserves to be called an axiom. Mathematics, on the

other hand, may possess axioms, because it can always connect the

predicates of an object a priori, and without any mediating term, by

means of the construction of conceptions in intuition. Such is the

case with the proposition: Three points can always lie in a plane.

On the other hand, no synthetical principle which is based upon

conceptions, can ever be immediately certain (for example, the

proposition: Everything that happens has a cause), because I require a

mediating term to connect the two conceptions of event and cause-

namely, the condition of time-determination in an experience, and I

cannot cognize any such principle immediately and from conceptions

alone. Discursive principles are, accordingly, very different from

intuitive principles or axioms. The former always require deduction,

which in the case of the latter may be altogether dispensed with.

Axioms are, for this reason, always self-evident, while

philosophical principles, whatever may be the degree of certainty they

possess, cannot lay any claim to such a distinction. No synthetical

proposition of pure transcendental reason can be so evident, as is

often rashly enough declared, as the statement, twice two are four. It

is true that in the Analytic I introduced into the list of

principles of the pure understanding, certain axioms of intuition; but

the principle there discussed was not itself an axiom, but served

merely to present the principle of the possibility of axioms in

general, while it was really nothing more than a principle based

upon conceptions. For it is one part of the duty of transcendental

philosophy to establish the possibility of mathematics itself.

Philosophy possesses, then, no axioms, and has no right to impose

its a priori principles upon thought, until it has established their

authority and validity by a thoroughgoing deduction.

3. Of Demonstrations. Only an apodeictic proof, based upon

intuition, can be termed a demonstration. Experience teaches us what

is, but it cannot convince us that it might not have been otherwise.

Hence a proof upon empirical grounds cannot be apodeictic. A priori

conceptions, in discursive cognition, can never produce intuitive

certainty or evidence, however certain the judgement they present

may be. Mathematics alone, therefore, contains demonstrations, because

it does not deduce its cognition from conceptions, but from the

construction of conceptions, that is, from intuition, which can be

given a priori in accordance with conceptions. The method of

algebra, in equations, from which the correct answer is deduced by

reduction, is a kind of construction- not geometrical, but by symbols-

in which all conceptions, especially those of the relations of

quantities, are represented in intuition by signs; and thus the

conclusions in that science are secured from errors by the fact that

every proof is submitted to ocular evidence. Philosophical cognition

does not possess this advantage, it being required to consider the

general always in abstracto (by means of conceptions), while

mathematics can always consider it in concreto (in an individual

intuition), and at the same time by means of a priori

representation, whereby all errors are rendered manifest to the

senses. The former- discursive proofs- ought to be termed acroamatic

proofs, rather than demonstrations, as only words are employed in

them, while demonstrations proper, as the term itself indicates,

always require a reference to the intuition of the object.

It follows from all these considerations that it is not consonant

with the nature of philosophy, especially in the sphere of pure

reason, to employ the dogmatical method, and to adorn itself with

the titles and insignia of mathematical science. It does not belong to

that order, and can only hope for a fraternal union with that science.

Its attempts at mathematical evidence are vain pretensions, which

can only keep it back from its true aim, which is to detect the

illusory procedure of reason when transgressing its proper limits, and

by fully explaining and analysing our conceptions, to conduct us

from the dim regions of speculation to the clear region of modest

self-knowledge. Reason must not, therefore, in its transcendental

endeavours, look forward with such confidence, as if the path it is

pursuing led straight to its aim, nor reckon with such security upon

its premisses, as to consider it unnecessary to take a step back, or

to keep a strict watch for errors, which, overlooked in the

principles, may be detected in the arguments themselves- in which case

it may be requisite either to determine these principles with

greater strictness, or to change them entirely.

I divide all apodeictic propositions, whether demonstrable or

immediately certain, into dogmata and mathemata. A direct

synthetical proposition, based on conceptions, is a dogma; a

proposition of the same kind, based on the construction of

conceptions, is a mathema. Analytical judgements do not teach us any

more about an object than what was contained in the conception we

had of it; because they do not extend our cognition beyond our

conception of an object, they merely elucidate the conception. They

cannot therefore be with propriety termed dogmas. Of the two kinds

of a priori synthetical propositions above mentioned, only those which

are employed in philosophy can, according to the general mode of

speech, bear this name; those of arithmetic or geometry would not be

rightly so denominated. Thus the customary mode of speaking confirms

the explanation given above, and the conclusion arrived at, that

only those judgements which are based upon conceptions, not on the

construction of conceptions, can be termed dogmatical.

Thus, pure reason, in the sphere of speculation, does not contain

a single direct synthetical judgement based upon conceptions. By means

of ideas, it is, as we have shown, incapable of producing

synthetical judgements, which are objectively valid; by means of the

conceptions of the understanding, it establishes certain indubitable

principles, not, however, directly on the basis of conceptions, but

only indirectly by means of the relation of these conceptions to

something of a purely contingent nature, namely, possible

experience. When experience is presupposed, these principles are

apodeictically certain, but in themselves, and directly, they cannot

even be cognized a priori. Thus the given conceptions of cause and

event will not be sufficient for the demonstration of the proposition:

Every event has a cause. For this reason, it is not a dogma;

although from another point of view, that of experience, it is capable

of being proved to demonstration. The proper term for such a

proposition is principle, and not theorem (although it does require to

be proved), because it possesses the remarkable peculiarity of being

the condition of the possibility of its own ground of proof, that

is, experience, and of forming a necessary presupposition in all

empirical observation.

If then, in the speculative sphere of pure reason, no dogmata are to

be found; all dogmatical methods, whether borrowed from mathematics,

or invented by philosophical thinkers, are alike inappropriate and

inefficient. They only serve to conceal errors and fallacies, and to

deceive philosophy, whose duty it is to see that reason pursues a safe

and straight path. A philosophical method may, however, be

systematical. For our reason is, subjectively considered, itself a

system, and, in the sphere of mere conceptions, a system of

investigation according to principles of unity, the material being

supplied by experience alone. But this is not the proper place for

discussing the peculiar method of transcendental philosophy, as our

present task is simply to examine whether our faculties are capable of

erecting an edifice on the basis of pure reason, and how far they

may proceed with the materials at their command.

SECTION II. The Discipline of Pure Reason in Polemics.

Reason must be subject, in all its operations, to criticism, which

must always be permitted to exercise its functions without

restraint; otherwise its interests are imperilled and its influence

obnoxious to suspicion. There is nothing, however useful, however

sacred it may be, that can claim exemption from the searching

examination of this supreme tribunal, which has no respect of persons.

The very existence of reason depends upon this freedom; for the

voice of reason is not that of a dictatorial and despotic power, it is

rather like the vote of the citizens of a free state, every member

of which must have the privilege of giving free expression to his

doubts, and possess even the right of veto.

But while reason can never decline to submit itself to the

tribunal of criticism, it has not always cause to dread the

judgement of this court. Pure reason, however, when engaged in the

sphere of dogmatism, is not so thoroughly conscious of a strict

observance of its highest laws, as to appear before a higher

judicial reason with perfect confidence. On the contrary, it must

renounce its magnificent dogmatical pretensions in philosophy.

Very different is the case when it has to defend itself, not

before a judge, but against an equal. If dogmatical assertions are

advanced on the negative side, in opposition to those made by reason

on the positive side, its justification kat authrhopon is complete,

although the proof of its propositions is kat aletheian

unsatisfactory.

By the polemic of pure reason I mean the defence of its propositions

made by reason, in opposition to the dogmatical counter-propositions

advanced by other parties. The question here is not whether its own

statements may not also be false; it merely regards the fact that

reason proves that the opposite cannot be established with

demonstrative certainty, nor even asserted with a higher degree of

probability. Reason does not hold her possessions upon sufferance;

for, although she cannot show a perfectly satisfactory title to

them, no one can prove that she is not the rightful possessor.

It is a melancholy reflection that reason, in its highest

exercise, falls into an antithetic; and that the supreme tribunal

for the settlement of differences should not be at union with

itself. It is true that we had to discuss the question of an

apparent antithetic, but we found that it was based upon a

misconception. In conformity with the common prejudice, phenomena were

regarded as things in themselves, and thus an absolute completeness in

their synthesis was required in the one mode or in the other (it was

shown to be impossible in both); a demand entirely out of place in

regard to phenomena. There was, then, no real self-contradiction of

reason in the propositions: The series of phenomena given in

themselves has an absolutely first beginning; and: This series is

absolutely and in itself without beginning. The two propositions are

perfectly consistent with each other, because phenomena as phenomena

are in themselves nothing, and consequently the hypothesis that they

are things in themselves must lead to self-contradictory inferences.

But there are cases in which a similar misunderstanding cannot be

provided against, and the dispute must remain unsettled. Take, for

example, the theistic proposition: There is a Supreme Being; and on

the other hand, the atheistic counter-statement: There exists no

Supreme Being; or, in psychology: Everything that thinks possesses the

attribute of absolute and permanent unity, which is utterly

different from the transitory unity of material phenomena; and the

counter-proposition: The soul is not an immaterial unity, and its

nature is transitory, like that of phenomena. The objects of these

questions contain no heterogeneous or contradictory elements, for they

relate to things in themselves, and not to phenomena. There would

arise, indeed, a real contradiction, if reason came forward with a

statement on the negative side of these questions alone. As regards

the criticism to which the grounds of proof on the affirmative side

must be subjected, it may be freely admitted, without necessitating

the surrender of the affirmative propositions, which have, at least,

the interest of reason in their favour- an advantage which the

opposite party cannot lay claim to.

I cannot agree with the opinion of several admirable thinkers-

Sulzer among the rest- that, in spite of the weakness of the arguments

hitherto in use, we may hope, one day, to see sufficient

demonstrations of the two cardinal propositions of pure reason- the

existence of a Supreme Being, and the immortality of the soul. I am

certain, on the contrary, that this will never be the case. For on

what ground can reason base such synthetical propositions, which do

not relate to the objects of experience and their internal