MATHEMATICS AND LOGIC. 149 



a sure instinct, or by some vague consciousness of 

 I know not what profounder and more hidden geom- 

 etry, which alone gives a value to the constructed 

 edifice. 



To seek the origin of this instinct, and to study 

 the laws of this profound geometry which can be 

 felt but not expressed, would be a noble task for 

 the philosophers who will not allow that logic is 

 all. But this is not the point of view I wish to 

 take, and this is not the way I wish to state 

 the question. This instinct I have been speaking 

 of is necessary to the discoverer, but it seems at 

 first as if we could do without it for the study of 

 the science once created. Well, what I want to find 

 out is, whether it is true that once the principles of 

 logic are admitted we can, I will not say discover, 

 but demonstrate all mathematical truths without 

 making a fresh appeal to intuition. 



III. 



To this question I formerly gave a negative answer. 

 (See " Science et Hypothese," Chapter I.) Must our 

 answer be modified by recent works ? I said no, 

 because " the principle of complete induction " ap- 

 peared to me at once necessary to the mathematician, 

 and irreducible to logic. We know the statement of 

 the principle : " If a property is true of the number 

 I, and if it is established that it is true of ;/+ i pro- 

 vided it is true of «, it will be true of all whole 

 numbers." I recognized in this the typical mathe- 

 matical argument. I did not mean to say, as has 

 been supposed, that all mathematical arguments can 

 be reduced to an application of this principle. 



