222 THE POPULAR SCIENCE MONTHLY 



the theorems. Thus he recommends that in the special mathematics 

 of the secondary school and in the first year of the Ecole Poly technique, 

 there should not be introduced the notion of functions with no deriva- 

 tives. At most we should content ourselves with saying " there are 

 such, but we are not concerned with them now." When integrals are 

 first spoken of, they should be defined as areas, and the rigorous defini- 

 tion should be given later, after the student has found many integrals. 

 He says: 10 



The chief end of mathematical instruction is to develop certain powers of 

 the mind, and among these the intuition is not the least precious. By it the 

 mathematical world comes in contact with the real world, and even if pure 

 mathematics could do without it, it would always be necessary to turn to it to 

 bridge the gulf between symbol and reality. The practician will always need it, 

 and for one mathematician there are a hundred practicians. However, for the 

 mathematician himself the power is necessary, for while we demonstrate by 

 logic, we create by intuition; and we have more to do than to criticize others' 

 theorems, we must invent new ones, this art, intuition teaches us. 



We turn finally to the research student. How is he to bring the 

 intuition to bear on his problem effectively ? If creative work is to be 

 hoped for only through this agency, how do we set it to work? This 

 question Poincare answers in his analysis of his creation of the fuchsian 

 functions. He holds that the intuition does its work unconsciously. 

 We can not use the term " subconsciously," for he had a repugnance to 

 the doctrine of the superiority of the subliminal self. He points out 

 that our unconscious activity forms large numbers of mental combina- 

 tions, as an architect, we will say, makes many trial sketches, and of 

 these combinations some are brought into consciousness. These are 

 selected, he believes, by their appeal to the sentiment of beauty, the 

 intellectual esthetic sense of the fitness of things, the unity of ideas, 

 just as the architect from his haphazard sketches selects the right one 

 finally by its appeal to his sense of beauty. Poincare admits that this 

 explanation of the facts is a hypothesis, but he finds many things to 

 confirm it. One is the fact that the theorems thus suggested in mathe- 

 matical creation are not always true, yet their elegance, if they were 

 true, has opened the door of consciousness to them. It was Sylvester 

 who used to declare : 



Gentlemen, I am certain my conclusion is correct. I will wager a hundred 

 pounds to one on it; yes, I will wager my life on it. 



But it often turned out the next day that it was not true. How- 

 ever, it led eventually to things that were true. The direct conclusion 

 from Poincare' s hypothesis would be that we must conserve and develop 

 the esthetic sense of our field, whether mathematics, physics, chemistry, 

 or what not. And we may well pause to consider whether the young 



10 L'Enseignement Math., 1899, p. 157. 



