Page:Popular Science Monthly Volume 80.djvu/370

366, 1844. Schopenhauer's views must have attracted considerable attention in Germany, for as late as 1894 Alfred Pringsheim thought it necessary to refute his argument, and only four years ago Felix Klein referred to him at some length in a mathematical lecture at the University of Goettingen. Schopenhauer had read Sir William Hamilton, as appears from the following passage:

These words of Schopenhauer are an unqualified endorsement of Hamilton, the only such endorsement with which I happen to be familiar.

Schopenhauer's own argument is mainly directed against Euclid and his geometrical demonstrations. Schopenhauer had his own ideas as to how absolute truth can be reached; these ideas did not agree with the method of Euclid. Our German philosopher says:

If now our conviction that perception is the primary source of all evidence, and that only direct or indirect connection with it is absolute truth; and further, that the shortest way to this is always the surest, as every interposition of concepts means exposure to many deceptions; if, I say, we now turn with this conviction to mathematics, as it was established as a science by Euclid, and has remained as a whole to our own day, we can not help regarding the method it adopts as strange and indeed perverted. We ask that every logical proof shall be traced back to an origin in perception; but mathematics, on the contrary, is at great pains deliberately to throw away the evidence of perception which is peculiar to it, and always at hand, that it may substitute for it logical demonstration. This must seem to us like the action of a man who cuts off his legs in order to go on crutches. . . (page 92). We are compelled by the principle of contradiction to admit that what Euclid demonstrates is true, but we do not comprehend why it is so. We have therefore almost the same uncomfortable feeling that we experience after a juggling trick, and, in fact, most of Euclid's demonstrations are remarkably like such feats. The truth almost always enters by the back door, for it manifests itself par accidens through some contingent circumstance.