64 MYSTICISM AND LOGIC 



and presently, when he is able to obtain the answer that 

 the teacher desires, he feels that he has mastered the 

 difficulties of the subject. But of inner comprehension 

 of the processes employed he has probably acquired 

 almost nothing. 



When algebra has been learnt, all goes smoothly until 

 we reach those studies in which the notion of infinity is 

 employed the infinitesimal calculus and the whole of 

 higher mathematics. The solution of the difiiculties 

 which formerly surrounded the mathematical infinite is 

 probably the greatest achievement of which our own age 

 has to boast. Since the beginnings of Greek thought 

 these difficulties have been known ; in every age the finest 

 intellects have vainly endeavoured to answer the appar- 

 ently unanswerable questions that had been asked by 

 Zeno the Eleatic. At last Georg Cantor has found the 

 answer, and has conquered for the intellect a new and 

 vast province which had been given over to Chaos and 

 old Night. It was assumed as self-evident, until Cantor 

 and Dedekind estabhshed the opposite, that if, from any 

 collection of things, some were taken away, the number 

 of things left must always be less than the original 

 number of things. This assumption, as a matter of fact, 

 holds only of finite collections ; and the rejection of it, 

 . where the infinite is concerned, has been shown to remove 

 all the difficulties that had hitherto baffled human reason 

 in this matter, and to render possible the creation of 

 an exact science of the infinite. This stupendous fact 

 ought to produce a revolution in the higher teaching 

 of mathematics ; it has itself added immeasurably to 

 the educational value of the subject, and it has at last 

 given the means of treating with logical precision many 

 studies which, until lately, were wrapped in fallacy 

 and obscurity. By those who were educated on the 



