possible forms of the universe, now we see that he is omniscient. He has solved the final problems of science and has nailed up tight all future knowledge.
Herr Duehring considers that he can, as with the fundamental forms of existence, produce aprioristically by means of his own cogitations the whole of pure mathematics without making any use of the experience which is afforded us in the objective world. In pure mathematics the understanding is engaged "in its own free creations and imaginations"; the concepts of number and form are "self-sufficient objects proceeding from themselves" and so have "a value independent of individual experience and actual objective reality."
That pure mathematics has a significance independent of particular individual experience is quite true as are also the established facts of all the sciences and indeed of all facts. The magnetic poles, the formation of water from oxygen and hydrogen, the fact that Hegel is dead and that Herr Duehring is alive, are facts independent of my experience or that of any other single individual, and will be independent of that of Herr Duehring himself, as soon as he shall sleep the sleep of the just. But in pure mathematics the mind is not by any means engaged with its own creations and imaginings. The concepts of number and form have only come to us by the way of the real world. The ten fingers on which men count and thereby performed the first arithmetical calculations are anything but a free creation of the mind. To count not only requires objects capable of being counted but the ability, when these objects are regarded, of subtracting all qualities from them except number and this ability is the product of long historical development of actual experience. The concept form is, like that of number, derived exclusively
