The success which attends the efforts of reason in the sphere of mathematics naturally fosters the expectation that the same good fortune will be its lot, if it applies the mathematical method in other regions of mental endeavour besides that of quantities.Its success is thus great, because it can support all its conceptions by a priori intuitions and, in this way, make itself a master, as it were, over nature; while pure philosophy, with its a priori discursive conceptions, bungles about in the world of nature, and cannot accredit or show any a priori evidence of the reality of these conceptions.
Masters in the science of mathematics are confident of the success of this method; indeed, it is a common persuasion that it is capable of being applied to any subject of human thought.They have hardly ever reflected or philosophized on their favourite science- a task of great difficulty; and the specific difference between the two modes of employing the faculty of reason has never entered their thoughts.
Rules current in the field of common experience, and which common sense stamps everywhere with its approval, are regarded by them as axiomatic.From what source the conceptions of space and time, with which (as the only primitive quanta) they have to deal, enter their minds, is a question which they do not trouble themselves to answer;and they think it just as unnecessary to examine into the origin of the pure conceptions of the understanding and the extent of their validity.All they have to do with them is to employ them.In all this they are perfectly right, if they do not overstep the limits of the sphere of nature.But they pass, unconsciously, from the world of sense to the insecure ground of pure transcendental conceptions (instabilis tellus, innabilis unda), where they can neither stand nor swim, and where the tracks of their footsteps are obliterated by time; while the march of mathematics is pursued on a broad and magnificent highway, which the latest posterity shall frequent without fear of danger or impediment.
As we have taken upon us the task of determining, clearly and certainly, the limits of pure reason in the sphere of transcendentalism, and as the efforts of reason in this direction are persisted in, even after the plainest and most expressive warnings, hope still beckoning us past the limits of experience into the splendours of the intellectual world- it becomes necessary to cut away the last anchor of this fallacious and fantastic hope.We shall, accordingly, show that the mathematical method is unattended in the sphere of philosophy by the least advantage- except, perhaps, that it more plainly exhibits its own inadequacy- that geometry and philosophy are two quite different things, although they go band in hand in hand in the field of natural science, and, consequently, that the procedure of the one can never be imitated by the other.
The evidence of mathematics rests upon definitions, axioms, and demonstrations.I shall be satisfied with showing that none of these forms can be employed or imitated in philosophy in the sense in which they are understood by mathematicians; and that the geometrician, if he employs his method in philosophy, will succeed only in building card-castles, while the employment of the philosophical method in mathematics can result in nothing but mere verbiage.The essential business of philosophy, indeed, is to mark out the limits of the science; and even the mathematician, unless his talent is naturally circumscribed and limited to this particular department of knowledge, cannot turn a deaf ear to the warnings of philosophy, or set himself above its direction.
I.Of Definitions.A definition is, as the term itself indicates, the representation, upon primary grounds, of the complete conception of a thing within its own limits.* Accordingly, an empirical conception cannot be defined, it can only be explained.For, as there are in such a conception only a certain number of marks or signs, which denote a certain class of sensuous objects, we can never be sure that we do not cogitate under the word which indicates the same object, at one time a greater, at another a smaller number of signs.Thus, one person may cogitate in his conception of gold, in addition to its properties of weight, colour, malleability, that of resisting rust, while another person may be ignorant of this quality.We employ certain signs only so long as we require them for the sake of distinction; new observations abstract some and add new ones, so that an empirical conception never remains within permanent limits.It is, in fact, useless to define a conception of this kind.