DEFINITIONS AND EDUCATION. 131 



in correct reasoning in those parts of mathematics in 

 which the disadvantages I have mentioned do not 

 occur. We have long series of theorems in which 

 absolute logic has ruled from the very start and, so to 

 speak, naturally, in which the first geometricians have 

 given us models that we must continually imitate and 

 admire. 



It is in expounding the first principles that we must 

 avoid too much subtlety, for there it would be too 

 disheartening, and useless besides. We cannot prove 

 everything, we cannot define everything, and it will 

 always be necessary to draw upon intuition. What 

 does it matter whether we do this a little sooner or a 

 little later, and even whether we ask for a little more 

 or a little less, provided that, making a correct use 

 of the premises it gives us, we learn to reason 

 accurately ? 



ir. Is it possible to satisfy so many opposite 

 conditions? Is it possible especially when it is a 

 question of giving a definition } How are we to find 

 a statement that will at the same time satisfy the 

 inexorable laws of logic and our desire to understand 

 the new notion's place in the general scheme of the 

 science, our need of thinking in images? More often 

 than not we shall not find it, and that is why the 

 statement of a definition is not enough ; it must be 

 prepared and it must be justified. 



What do I mean by this ? You know that it has 

 often been said that every definition imph'es an axiom, 

 since it asserts the existence of the object defined. 

 The definition, then, will not be justified, from the 

 purely logical point of view, until we have proved that ^ 



