i66 SCIENCE AND METHOD. 



ing them for all, and so we must give up the 

 demonstration by example. 



It is necessary, then, to take all the consequences 

 of our axioms and see whether they contain any 

 contradiction. If the number of these consequences 

 were finite, this would be easy ; but their number 

 is infinite — they are the whole of mathematics, or at 

 least the whole of arithmetic. 



What are we to do, then ? Perhaps, if driven to 

 it, we might repeat the reasoning of Section III. 

 But, as I have said, this reasoning is complete induction., 

 and it is precisely the principle of complete induction 

 that we are engaged in justifying. 



VI. 



Hilbert's Logic. 



I come now to Mr. Hilbert's important work, 

 addressed to the Mathematical Congress at Heidelberg, 

 a French translation of which, by M. Pierre Boutroux, 

 appeared in V Enseignement Math'eviatique, while an 

 English translation by Mr. Halsted appeared in The 

 Mofiist. In this work, in which we find the most 

 profound thought, the author pursues an aim similar 

 to Mr. Russell's, but he diverges on many points from 

 his predecessor. 



" However," he says, " if we look closely, we recog- 

 nize that in logical principles, as they are com- 

 monly presented, certain arithmetical notions are 

 found already implied ; for instance, the notion of 

 whole, and, to a certain extent, the notion of number. 

 Thus we find ourselves caught in a circle, and that 

 is why it seems to me necessary, if we wish to avoid 



