shown, a conclusion, upon the correctness of which all logic depends, and which is demonstrated therefore outside of pure mathematics. The remaining axioms with regard to equality and inequality are merely logical extensions of this conclusion. Such barren statements are not enticing either in mathematics or anywhere else. To proceed we must have realities, conditions and forms taken from real material things; representations of lines, planes, angles, polygons, spheres, etc., are all borrowed from reality, and it is just naive ideology to believe the mathematicians, who assert that the first line was made by causing a point to progress through space, the first plane by means of the movement of a line, and the first solid by revolving a plane, etc. Even speech rebels against this idea. A mathematical figure of three dimensions is called a solid—corpus solidum—and hence, according to the Latin, a body capable of being handled. It has a name derived, therefore, by no means from the independent play of imagination but from solid reality.
But to what purpose is all this prolixity? After Herr Duehring has enthusiastically proclaimed the independence of pure mathematics of the world of experience, their apriorism, their connection with free creation and imagination, he says "it will be readily seen that these mathematical elements (number, magnitude, time, space, geometric progression), are therefore ideal forms with relation to absolute magnitudes and therefore something quite empiric, no matter to what species they belong." But "mathematical general notions are, apart from experience, nevertheless capable of sufficient characterization," which latter proceeds, more or less, from each abstraction, but does not by any means prove that it is not deprived from the actual. In the scheme of the
