| 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. |