LAST EFFORTS OF LOGISTICIANS. 189 



However that may be, we understand Mr. Russell's 

 hesitation at the modifications to which he is about 

 to submit the fundamental principles he has hitherto 

 adopted. Criteria will be necessary to decide whether 

 a definition is too complicated or too extensive, and 

 these criteria cannot be justified except by an appeal 

 to intuition. 



It is towards the no classes theory that Mr. Russell 

 eventually inclines. 



However it be, Logistic must be refashioned, and it 

 is not yet known how much of it can be saved. It is 

 unnecessary to add that it is Cantorism and Logistic 

 alone that are in question. The true mathematics, the 

 mathematics that is of some use, may continue to 

 develop according to its own principles, taking no 

 heed of the tempests that rage without, and step 

 by step it will pursue its wonted conquests, which are 

 decisive and have never to be abandoned. 



VII. 



The True Solution. 



How are we to choose between these different 

 theories? It seems to me that the solution is con- 

 tained in M. Richard's letter mentioned above, which 

 will be found in the Revue Generate des Sciences of June 

 30, 1905. After stating the antinomy that I have called 

 Richard's antinomy, he gives the explanation. 



Let us refer to what was said of this antinomy in 

 Section V. E is the aggregate of rt// the numbers that 

 can be defined by a finite number of words, without 

 introducing the notion 0/ the aggregate E itself, otherwise 



