44 SCIENCE AND METHOD. 



this. We must succeed in constructing it completely 

 in the higher spaces, and we shall then have an instru- 

 ment which will enable us really to see into hyperspace 

 and to supplement our senses. 



The problems of Geometry of Position would perhaps 

 not have presented themselves if only the language of 

 analysis had been used. Or rather I am wrong, for 

 they would certainly have presented themselves, since 

 their solution is necessary for a host of questions of 

 analysis, but they would have presented themselves 

 isolated, one after the other, and without our being 

 able to perceive their common link. 



Cantorism. 



I have spoken above of the need we have of 

 returning continually to the first principles of our 

 science, and of the advantage of this process to the 

 study of the human mind. It is this need which has 

 inspired two attempts which have held a very great 

 place in the most recent history of mathematics. The 

 first is Cantorism, and the services it has rendered to 

 the science are well known. Cantor introduced into 

 the science a new method of considering mathematical 

 infinity, and I shall have occasion to speak of it again 

 in Book II., chapter iii. One of the characteristic 

 features of Cantorism is that, instead of rising to the 

 general by erecting more and more complicated con- 

 structions, and defining by construction, it starts with 

 the genus supreinu^n and only defines, as the scholastics 

 would have said, per genus proximuin et differentiam 

 specificam. Hence the horror he has sometimes in- 

 spired in certain minds, such as Hermitte's, whose 

 favourite idea was to compare the mathematical with 



