| 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 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. |