THE NEW LOGICS. 169 



known, but anything whatsoever. Russell is faithful 

 to his point of view, which is that of comprehension. 

 He starts with the general idea of entity, and enriches 

 it more and more, even while he restricts it, by adding 

 to it new qualities. Hilbert, on the contrary, only 

 recognizes as possible entities combinations of objects 

 already known ; so that (looking only at one side of 

 his thought) we might say that he takes the point 

 of view of extension. 



VHI. 



Let us proceed with the exposition of Hilbert's 

 ideas. He introduces two axioms which he enunciates 

 in his symbolical language, but which signify, in the 

 language of the uninitiated like us, that every quantity 

 is equal to itself, and that every operation upon two 

 identical quantities gives identical results. So stated 

 they are evident, but such a presentation of them 

 does not faithfully represent Hilbert's thought. For 

 him mathematics has to combine bnly pure symbols, 

 and a true mathematician must base his reasoning 

 upon them without concerning himself with their 

 meaning. Accordingly, his axioms are not for him 

 what they are for the ordinary man. 



He considers them as representing the definition by 

 postulates of the .symbol =, up to this time devoid 

 of all signification. But in order to justify this defini- 

 tion, it is necessary to show that these two axioms do 

 not lead to any contradiction. 



For this purpose Hilbert makes use of the reasoning 

 of Section HI., without apparently perceiving that he 

 is making complete induction. 



