| 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. |