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 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 à 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 à 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 à 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. À 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 à 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 à 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 à 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 à 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