the theorems. Thus he recommends that in the special mathematics of the secondary school and in the first year of the Ecole Polytechnique, there should not be introduced the notion of functions with no derivatives. 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 definition should be given later, after the student has found many integrals. He says:[1]
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 Poincaré 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 combinations, 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. Poincaré 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 mathematical 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. However, it led eventually to things that were true. The direct conclusion from Poincaré'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
- ↑ L'Enseignement Math., 1899, p. 157.
