MATHEMATICS 53 



fitting that a philosopher should explain the fundamental 

 importance of this phase of modern mathematical research 

 for mathematics itself, for philosophy and for other sciences. 

 A similar spirit has lead to fundamental revision of all our 

 previous mathematical knowledge, to its reduction to such 

 certainty that we can express our ideas practically in terms 

 of elementary arithmetic to a degree of certainty approximat- 

 ing two plus two equals four. This movement has been 

 called indeed the arithmetization of mathematics. Finally a 

 new and bewildering theory, the so-called Cantor Theory of 

 "sets of points" has arisen in the last twenty-five years. It 

 is comparable in every way in its effect upon mathematics 

 to the new theory of radio-activity in its effect on the funda- 

 mental conceptions of physics and of chemistry. The revi- 

 sion of ideas caused in those sciences by the discovery of 

 radio-activity of matter is no less startling than is the funda- 

 mental revision of mathematics caused by the Cantor Theory 

 of Sets, a theory which after all is nothing but a cunningly pre- 

 cise study of numbers of the simplest possible kind, thus going, 

 as does radio-activity in its influence, into the very essential 

 foundation of the science in which it arose. Again, I cannot 

 more than present to you the notion of a new mathematical 

 chapter of fundamental importance and living meaning, in- 

 vented within 25 years, and now known throughout the civi- 

 lized world, and affecting practically every phase of mathe- 

 matical research. Indeed I cannot explain its details. The 

 very mention of it will perhaps show you more clearly than 

 anything else that I could say that mathematics is not finished, 

 that it is distinctly in the process of making, and that the 

 most of knowledge lies beyond us. Indeed a science which 

 is complete has lost its attraction. No man except a wage 

 earner will interest himself in a subject which is dead. An- 

 other proof patent to all that mathematics is not complete is 

 the existence of a large body of men throughout the world 

 who are giving their lives to the study and elaboration of its 

 theories. 



