The essential difference of these two modes of cognition consists, therefore, in this formal quality; it does not regard the difference of the matter or objects of both.Those thinkers who aim at distinguishing philosophy from mathematics by asserting that the former has to do with quality merely, and the latter with quantity, have mistaken the effect for the cause.The reason why mathematical cognition can relate only to quantity is to be found in its form alone.For it is the conception of quantities only that is capable of being constructed, that is, presented a priori in intuition;while qualities cannot be given in any other than an empirical intuition.Hence the cognition of qualities by reason is possible only through conceptions.No one can find an intuition which shall correspond to the conception of reality, except in experience; it cannot be presented to the mind a priori and antecedently to the empirical consciousness of a reality.We can form an intuition, by means of the mere conception of it, of a cone, without the aid of experience; but the colour of the cone we cannot know except from experience.I cannot present an intuition of a cause, except in an example which experience offers to me.Besides, philosophy, as well as mathematics, treats of quantities; as, for example, of totality, infinity, and so on.Mathematics, too, treats of the difference of lines and surfaces- as spaces of different quality, of the continuity of extension- as a quality thereof.But, although in such cases they have a common object, the mode in which reason considers that object is very different in philosophy from what it is in mathematics.The former confines itself to the general conceptions;the latter can do nothing with a mere conception, it hastens to intuition.In this intuition it regards the conception in concreto, not empirically, but in an a priori intuition, which it has constructed; and in which, all the results which follow from the general conditions of the construction of the conception are in all cases valid for the object of the constructed conception.
Suppose that the conception of a triangle is given to a philosopher and that he is required to discover, by the philosophical method, what relation the sum of its angles bears to a right angle.He has nothing before him but the conception of a figure enclosed within three right lines, and, consequently, with the same number of angles.He may analyse the conception of a right line, of an angle, or of the number three as long as he pleases, but he will not discover any properties not contained in these conceptions.But, if this question is proposed to a geometrician, he at once begins by constructing a triangle.He knows that two right angles are equal to the sum of all the contiguous angles which proceed from one point in a straight line; and he goes on to produce one side of his triangle, thus forming two adjacent angles which are together equal to two right angles.He then divides the exterior of these angles, by drawing a line parallel with the opposite side of the triangle, and immediately perceives that be has thus got an exterior adjacent angle which is equal to the interior.Proceeding in this way, through a chain of inferences, and always on the ground of intuition, he arrives at a clear and universally valid solution of the question.
But mathematics does not confine itself to the construction of quantities (quanta), as in the case of geometry; it occupies itself with pure quantity also (quantitas), as in the case of algebra, where complete abstraction is made of the properties of the object indicated by the conception of quantity.In algebra, a certain method of notation by signs is adopted, and these indicate the different possible constructions of quantities, the extraction of roots, and so on.After having thus denoted the general conception of quantities, according to their different relations, the different operations by which quantity or number is increased or diminished are presented in intuition in accordance with general rules.Thus, when one quantity is to be divided by another, the signs which denote both are placed in the form peculiar to the operation of division; and thus algebra, by means of a symbolical construction of quantity, just as geometry, with its ostensive or geometrical construction (a construction of the objects themselves), arrives at results which discursive cognition cannot hope to reach by the aid of mere conceptions.
