150 SCIENCE AND METHOD. 



Examining these arguments somewhat closely, we 

 should discover the application of many other similar 

 principles, offering the same essential characteristics. 

 In this category of principles, that of complete induc- 

 tion is only the simplest of all, and it is for that 

 reason that I selected it as a type. 



The term principle of complete induction which 

 has been adopted is not justifiable. This method 

 of reasoning is none the less a true mathematical 

 induction itself, which only differs from the ordinary 

 induction by its certainty. 



IV. 



Definitions and Axioms. 



The existence of such principles is a difficulty for 

 the inexorable logicians. How do they attempt to 

 escape it? The principle of complete induction, they 

 say, is not an axiom properly so called, or an a 

 priori synthetic judgment ; it is simply the defini- 

 tion of the whole number. Accordingly it is a mere 

 convention. In order to discuss this view, it will be 

 necessary to make a close examination of the rela- 

 tions between definitions and axioms. 



We will first refer to an article by M. Couturat 

 on mathematical definitions which appeared in 

 l" Enseignement Mathiniatiqiie, a review published by 

 Gauthier-Villars and by Georg in Geneva. We find 

 a distinction between direct definition and definition 

 by postulates. 



"Definition by postulates," says M. Couturat, 

 " applies not to a single notion, but to a system of 

 notions ; it consists in enumerating the fundamental 



