Nor again can we consider to be mathematicism the attempt made by some philosophers to expound their own ideas by a mathematical, algebraical or geometrical method. If their ideas were ideas and not numbers, the method to which they had recourse necessarily remained extrinsic, and possessed no mathematical character beyond the verbal complacency with which they adopted certain formulae of definitions, axioms, theorems, lemmas, corollaries and certain numerical symbols, These formulas and symbols could always be replaced by others, without any inconvenience whatever. It is possible to discuss, it has indeed been discussed, whether such modes of exposition are in good or bad literary taste, or of greater or less didactic convenience. They can be condemned, as they have been condemned, and caused to fall into disuse, as they have fallen; but the quality of the philosophic truth thus expressed, remains unaltered and is never changed into mathematics. Neither the system of Spinoza, who employed the geometrical method, nor that of Leibnitz, who desired the universal calculus, are mathematical systems. If they were so, modern philosophy would not owe some of its most important idealist concepts to those two systems. [Sidenote: _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.