MATHEMATICS AND LOGIC. 151 



relations that unite them, which make it possible to 

 demonstrate all their other properties : these relations 

 are postulates . . ." 



If we have previously defined all these notions 

 with one exception, then this last will be by defini- 

 tion the object which verifies these postulates. 



Thus certain indemonstrable axioms of mathe- 

 matics would be nothing but disguised definitions. 

 This point of view is often legitimate, and I have 

 myself admitted it, for instance, in regard to Euclid's 

 postulate. 



The other axioms of geometry are not sufficient to 

 define distance completely. Distance, then, will be 

 by definition, the one among all the magnitudes 

 which satisfy the other axioms, that is of such a 

 nature as to make Euclid's postulate true. 



Well, the logicians admit for the principle of com- 

 plete induction what I admit for Euclid's postulate, 

 and they see nothing in it but a disguised definition. 



But to give us this right, there are two conditions 

 that must be fulfilled. John Stuart Mill used to say 

 that every definition implies an axiom, that in which 

 we affirm the existence of the object defined. On 

 this score, it would no longer be the axiom that 

 might be a disguised definition, but, on the contrary, 

 the definition that would be a disguised axiom. 

 Mill understood the word existence in a material 

 and empirical sense ; he meant that in defining a 

 circle we assert that there are round things in 

 nature. 



In this form his opinion is inadmissible. Mathe- 

 matics is independent of the existence of material 

 objects. In mathematics the word exist can only 



