154 SCIENCE AND METHOD. 



this pure notion, but with a concrete object which is 

 often only a rough image of it. To say that this 

 object satisfies the definition, even approximately, is 

 to enunciate a new truth, which has no longer the 

 character of a conventional postulate, and that expe- 

 rience alone can establish beyond a doubt. 



But, without departing from pure mathematics, we 

 still meet with the same difficulty. You give a 

 subtle definition of number, and then, once the 

 definition has been given, you think no more about 

 it, because in reality it is not your definition that 

 has taught you what a number is, you knew it long 

 before, and when you come to write the word 

 number farther on, you give it the same meaning 

 as anybody else. In order to know what this 

 meaning is, and if it is indeed the same in this 

 phrase and in that, we must see how you have been 

 led to speak of number and to introduce the word 

 into the two phrases. I will not explain my point 

 any further for the moment, for we shall have occa- 

 sion to return to it. 



Thus we have a word to which we have explicitly 

 given a definition A. We then proceed to make use 

 of it in our text in a way which implicitly supposes 

 another definition B. It is possible that these two 

 definitions may designate the same object, but that 

 such is the case is a new truth that must either be 

 demonstrated or else admitted as an independent 

 axiom. 



We shall see further on that the logicians have not 

 fulfilled this second condition any better than the first. 



