MATHEMATICS AND METAPHYSICIANS 91 



that the number of days in all time is no greater than the 

 number of years. 



Thus on the subject of infinity it is impossible to avoid 

 conclusions which at first sight appear paradoxical, and 

 this is the reason why so many philosophers have supposed 

 that there were inherent contradictions in the infinite. 

 But a little practice enables one to grasp the true prin- 

 ciples of Cantor's doctrine, and to acquire new and 

 better instincts as to the true and the false. The oddities 

 then become no odder than the people at the antipodes, 

 who used to be thought impossible because they would 

 find it so inconvenient to stand on their heads. 



The solution of the problems concerning infinity has 

 enabled Cantor to solve also the problems of continuity. 

 Of this, as of infinity, he has given a perfectly precise 

 definition, and has shown that there are no contradictions 

 in the notion so defined. But this subject is so technical 

 that it is impossible to give any account of it here. 



The notion of continuity depends upon that of order, 

 since continuity is merely a particular type of order. 

 Mathematics has, in modern times, brought order into 

 greater and greater prominence. In former days, it was 

 supposed (and philosophers are still apt to suppose) that 

 quantity was the fundamental notion of mathematics. 

 But nowadays, quantity is banished altogether, except 

 Irom.one little corner of Geometry, while order more and 

 more reigns supreme. The investigation of different 

 kinds of series and their relations is now a very large part 

 of mathematics, and it has been found that this investiga- 

 tion can be conducted without any reference to quantity, 

 and, for the most part, without any reference to number. 

 All types of series are capable of formal definition, and 

 their properties can be deduced from the principles of 

 symbolic logic by means of the Algebra of Relatives. 



