366 THE POPULAR SCIENCE MONTHLY 



tion^ 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 Univer- 

 sity of Goettingen. Schopenhauer had read Sir William Hamilton, as 

 appears from the following passage :-* 



I rather recommend, as an investigation of the influence of mathematics 

 upon our mental powers, ... a very thorough and learned discussion, in the form 

 of a review of a book by Whewell in the Edinburgh Beview of .January, 1836. Its 

 author, who afterwards published it with some other discussions, with his name, 

 is Sir W. Hamilton, Professor of Logic and Metaphysics in Scotland. This work 

 has also found a German translator, and has appeared by itself under the title, 

 "Ueber den Werth und Unwerth der Mathematik, aus dem Englisehen, " 1836. 

 The conclusion the author arrives at is that the value of mathematics is only 

 indirect, and lies in the application to ends which are only attainable through 

 them; but in themselves mathematics leave the mind where they find it, and are 

 by no means conducive to its general culture and development, nay, even a 

 decided hindrance. This conclusion is not only proved by thorough dianoiological 

 investigation of the mathematical activity of the mind, but is also confirmed by 

 a very learned accumulation of examples and authorities. The only direct use 

 which is left to mathematics is that it can accustom restless and unsteady minds 

 to fix their attention. Even Descartes, who was yet himself famous as a mathe- 

 matician, held the same opinion with regard to mathematics. 



These words of Schopenhauer are an unqualified endorsement of 

 Hamilton, the only such endorsement with which I happen to be fa- 

 miliar. 



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 con- 

 cepts means exposure to many deceptions; if, I say, we now turn with this convic- 

 tion 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. 



" A. Schopenhauer, ' ' The World as Will and Idea, ' ' translated by E. B. 

 Haldane and J. Kemp, Vol. II., London, 1891, p. 323. 

 ''^Loc. cit., Vol. I., 1891, par. 15, p. 90. 



