MATHEMATICS AND METAPHYSICIANS 85 



have wandered into a " cul-de-sac." This difficulty led 

 to Kant's antinomies, and hence, more or less indirectly, 

 to much of Hegel's dialectic method. Almost all current 

 philosophy is upset by the fact (of which very few philo- 

 sophers are as yet aware) that all the ancient and respect- 

 able contradictions in the notion of the infinite have been 

 once for all disposed of. The method by which this has 

 been done is most interesting and instructive. In the 

 first place, though people had talked glibly about infinity 

 ever since the beginnings of Greek thought, nobody had 

 ever thought of asking, What is infinity ? If any 

 philosopher had been asked for a definition of infinity, he 

 might have produced some unintelligible rigmarole, but he 

 would certainly not have been able to give a definition 

 that had any meaning at all. Twenty years ago, roughly 

 speaking, Dedekind and Cantor asked this question, and, 

 what is more remarkable, they answered it. They found, 

 that is to say, a perfectly precise definition of an infinite 

 number or an infinite collection of things. This was the 

 first and perhaps the greatest step. It then remained to 

 examine the supposed contradictions in this notion. 

 Here Cantor proceeded in the only proper way. He took 

 pairs of contradictory propositions, in which both sides 

 of the contradiction would be usually regarded as demon- 

 strable, and he strictly examined the supposed proofs. He 

 found that all proofs adverse to infinity involved a certain 

 principle, at first sight obviously true, but destructive, 

 in its consequences, of almost all mathematics. The 

 proofs favourable to infinity, on the other hand, involved 

 no principle that had evil consequences. It thus appeared 

 that common sense had allowed itself to be taken in by a 

 specious maxim, and that, when once this maxim was 

 rejected, all went well. 

 The maxim in question is, that if one collection is part 



