THE NEW LOGICS. 167 



all paradox, to develop the principles of logic and of 

 arithmetic simultaneously." 



We have seen above that what Mr. Hilbert says 

 of the principles of logic, as they are conwionly pre- 

 sented, applies equally to Mr. Russell's logic. For 

 Mr. Russell logic is anterior to arithmetic, and for 

 Mr. Hilbert they are "simultaneous." Further on we 

 shall find other and yet deeper differences ; but we 

 will note them as they occur. I prefer to follow the 

 development of Hilbert's thought step by step, quoting 

 the more important passages verbatim. 



" Let us first take into consideration the object i." 

 We notice that in acting thus we do not in any way 

 imply the notion of number, for it is clearly understood 

 that I here is nothing but a symbol, and that we do 

 not in any way concern ourselves with knowing its 

 signification. " The groups formed with this object, 

 two, three, or several times repeated . . ." This 

 time the case is quite altered, for if we introduce the 

 words two, three, and, above all, several, we introduce 

 the notion of number ; and then the definition of the 

 finite whole number that we find later on comes a 

 trifle late. The author was much too wary not to 

 perceive this petitio principii. And so, at the end of 

 his work, he seeks to effect a real patching-up. 



Hilbert then introduces two simple objects, I and 

 =, and pictures all the combinations of these two 

 objects, all the combinations of their combinations, 

 and so on. It goes without saying that we must 

 forget the ordinary signification of these two signs, 

 and not attribute any to them. He then divides these 

 combinations into two classes, that of entities and that 

 of nonentities, and, until further orders, this partition 



