Errors of mathematicist philosophy.
Better examples of mathematicism than the treatises and systems developed according to its rules are found in the unfulfilled programmes of such treatises and systems, or in the mathematicist treatment of certain philosophie problems. Such, for instance, is that concerning the infinity of the world in space and time, a problem which, treated mathematistically, becomes insoluble and makes many people's heads turn. It is impossible to comprehend the world in one's own mind with the mathematical infinite; and either to give or to refuse to it a beginning and an end. Hence the exclamations of terror before that infinite, and the sense of sublimity which seems to arise in the struggle joined between it, which is indomitable, and the human mind which wishes to dominate it. It has, however, already been observed with reason, that such sublimity is not only very near to the ridiculous, but falls into it with all its weight; and that such terror could not in truth be anything but terror of the ennui of having to count and recount in the void and to infinity. The mathematical infinite is nothing real; its appearance of reality is the shadow projected by the mathematical power which the human spirit possesses, of always adding a unit to any number. The true infinite is all before us, in every real fact, and it is only when the continuous unity of reality is divided into separate facts, and space and time are rendered abstract and mathematical, only then, if the complete operation be forgotten, that the desperate problem arises and the anguish of never being able to solve it. Another and more actual example of this mathematicist mode of treatment is that of the dimensions of space. Here, forgetting that space of three dimensions is nothing real that can be experienced, but is a mathematical construction, and on the other hand finding it convenient for mathematical reasons to construct spaces of less or more than three dimensions, or of n dimensions, they end by treating these constructions as conceivable realities, and seriously discuss bi-dimensional beings or four-dimensional worlds.
Dualism, agnosticism and superstition of mathematicism.
With affirmations such as those of infinites incomprehensible to thought, and of real but not experienceable spaces, mathematicism also creates a dualism of thought and of reality superior to thought, or (what amounts to the same thing) of thought which meets its equivalent in experience and thought without a corresponding experience. The unknowable here too lies in wait and falls upon the imprudent mathematicist philosopher, who feels himself lost before a second, third, fourth and infinite worlds, excogitated by himself, superior or inferior worlds to those of man, underworlds and overworlds and over-over worlds. He then becomes even spiritualist and asks with Zollner, why spiritualist facts should not possess reality and be produced in the fourth dimension of space, shut off from us. The contradiction of the mathematicist attempt, like that of the æsthetic and empiricist, is clearly revealed in the dualistic, agnostic and mystical consequences to which, as we shall see more clearly further on, all of them necessarily lead.
[III]
PHILOSOPHISM
Rupture of the unity of the a priori synthesis.
The three modes of error examined exhaust the possible combinations of the pure concept with the forms of the theoretic or theoretic-practical spirit, anterior or posterior to it. Other modes of error arise from the breaking up of the unity of the concept, from the separation of its constitutive elements. Each one of these elements, abstracted from the other, and finding that other before it, annuls, instead of recognizing the other as an organic part of itself; that is to say, substitutes for it its own abstract existence.