| If the conception is based upon empirical conditions, if, for example, I have a conception of a clock for a ship, this arbitrary conception does not assure me of the existence or even of the possibility of the object. |
| My definition of such a conception would with more propriety be termed a declaration of a project than a definition of an object. |
| There are no other conceptions which can bear definition, except those which contain an arbitrary synthesis, which can be constructed a priorI. Consequently, the science of mathematics alone possesses definitions. |
| For the object here thought is presented a priori in intuition; and thus it can never contain more or less than the conception, because the conception of the object has been given by the definition--and primarily, that is, without deriving the definition from any other source. |
| Philosophical definitions are, therefore, merely expositions of given conceptions, while mathematical definitions are constructions of conceptions originally formed by the mind itself; the former are produced by analysis, the completeness of which is never demonstratively certain, the latter by a synthesis. |
| In a mathematical definition the conception is formed, in a philosophical definition it is only explained. |
| From this it follows: |
| [*Footnote; The definition must describe the conception completely that is, omit none of the marks or signs of which it composed; within its own limits, that is, it must be precise, and enumerate no more signs than belong to the conception; and on primary grounds, that is to say, the limitations of the bounds of the conception must not be deduced from other conceptions, as in this case a proof would be necessary, and the so-called definition would be incapable of taking its place at the bead of all the judgements we have to form regarding an object.] |
| (a) That we must not imitate, in philosophy, the mathematical usage of commencing with definitions--except by way of hypothesis or experiment. |
| For, as all so-called philosophical definitions are merely analyses of given conceptions, these conceptions, although only in a confused form, must precede the analysis; and the incomplete exposition must precede the complete, so that we may be able to draw certain inferences from the characteristics which an incomplete analysis has enabled us to discover, before we attain to the complete exposition or definition of the conception. |
| In one word, a full and clear definition ought, in philosophy, rather to form the conclusion than the commencement of our labours.* In mathematics, on the contrary, we cannot have a conception prior to the definition; it is the definition which gives us the conception, and it must for this reason form the commencement of every chain of mathematical reasoning. |
| [*Footnote; Philosophy abounds in faulty definitions, especially such as contain some of the elements requisite to form a complete definition. |
| 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. |