i68 SCIENCE AND METHOD. 



is entirely arbitrary. Every affirmative proposition 

 teaches us that a combination belongs to the class of 

 entities, and every negative proposition teaches us 

 that a certain combination belongs to the class ot 

 nonentities. 



VII. 



We must now note a difference that is of the 

 highest importance. For Mr. Russell a chance object, 

 which he designates by ;tr, is an absolutely indeterminate 

 object, about which he assumes nothing- For Hilbert 

 it is one of those combinations formed with the symbols 

 I and = ; he will not allow the introduction of any- 

 thing but combinations of objects already defined. 

 Moreover, Hilbert formulates his thought in the most 

 concise manner, and I think I ought to reproduce 

 his statement in extenso : " The indeterminates which 

 figure in the axioms (in place of the 'some' or the 

 ' all ' of ordinary logic) represent exclusively the whole 

 of the objects and combinations that we have already 

 acquired in the actual state of the theory, or that we 

 are in course of introducing. Therefore, when we 

 deduce propositions from the axioms under considera- 

 tion, it is these objects and these combinations alone 

 that we have the right to substitute for the indeter- 

 minates. Neither must we forget that when we 

 increase the number of the fundamental objects, the 

 axioms at the same time acquire a new extension, and 

 must, in consequence, be put to the proof afresh and, 

 if necessary, modified." 



The contrast with Mr. Russell's point of view is 

 complete. According to this latter philosopher, wc 

 may substitute in place of x not only objects already 



