MATHEMATICS AND LOGIC. 153 



other. If the number of these propositions is finite, 

 a direct verification is possible ; but this is a case 

 that is not frequent, and, moreover, of little interest. 



If the number of the propositions is infinite, we 

 can no longer make this direct verification. We 

 must then have recourse to processes of demonstra- 

 tion, in which we shall generally be forced to invoke 

 that very principle of complete induction that we are 

 attempting to verify. 



I have just explained one of the conditions which 

 the logicians were bound to satisfy, and we shall see 

 further on that they have not done so. 



V. 



There is a second condition. When we give a 

 definition, it is for the purpose ol using it. 



Accordingly, we shall find the word defined in the 

 text that follows. Have we the right to assert, of 

 the object represented by this word, the postulate 

 that served as definition ? Evidently we have, if the 

 word has preserved its meaning, if we have not 

 assigned it a different meaning by implication. Now 

 this is what sometimes happens, and it is generally 

 difficult to detect it. We must see how the word 

 was introduced into our text, and whether the door 

 through which it came does not really imply a 

 different definition from the one enunciated. 



This difficulty is encountered in all applications of 

 mathematics. The mathematical notion has received 

 a highly purified and exact definition, and for the 

 pure mathematician all hesitation has disappeared. 

 But when we come to apply it, to the physical 

 sciences, for instance, we are no longer dealing with 



