THE VALVE OF SCIENCE 205 



of their demonstrations, we shall find it there at each instant beside the 

 classic syllogism of Aristotle. We, therefore, see already that the 

 analysts are not simply makers of syllogisms after the fashion of the 

 scholastics. 



Besides, do you think they have always marched step by step with 

 no vision of the goal they wished to attain? They must have divined 

 the way leading thither, and for that they needed a guide. This guide 

 is, first, analogy. For example, one of the methods of demonstration 

 dear to analysts is that founded on the employment of dominant func- 

 tions. We know it has already served to solve a multitude of problems; 

 in what consists then the role of the inventor who wishes to apply it to 

 a new problem? At the outset he must recognize the analogy of this 

 question with those which have already been solved by this method; 

 then he must perceive in what way this new question differs from the 

 others, and thence deduce the modifications necessary to apply to the 

 method. 



But how does one perceive these analogies and these differences? 

 In the example just cited they are almost always evident, but I could 

 have found others where they would have been much more deeply 

 hidden; often a very uncommon penetration is necessary for their 

 discovery. The analysts, not to let these hidden analogies escape them, 

 that is, in order to be inventors, must, without the aid of the senses and 

 imagination, have a direct sense of what constitutes the unity of a 

 piece of reasoning, of what makes, so to speak, its soul and inmost life. 



When one talked with M. Hermite, he never evoked a sensuous 

 image, and yet you soon perceived that the most abstract entities were 

 for him like living beings. He did not see them, but he perceived that 

 they are not an artificial assemblage, and that they have some principle 

 of internal unity. 



But, one will say, that still is intuition. Shall we conclude that 

 the distinction made at the outset was only apparent, that there is only 

 one sort of mind and that all the mathematicians are intuitionalists, 

 at least those who are capable of inventing? 



Xo, our distinction corresponds to something real. I have said 

 above that there are many kinds of intuition. I have said how much 

 the intuition of pure number, whence comes rigorous mathematical 

 induction, differs from sensible intuition to which the imagination, 

 properly so called, is the principal contributor. 



Is the abyss which separates them less profound than it at first 

 appeared? Could we recognize with a little attention that this pure 

 intuition itself could not do without the aid of the senses? This is 

 the affair of the psychologist and the metaphysician and I shall not 

 discuss the question. But the thing's being doubtful is enough to 

 justify me in recognizing and affirming an essential difference between 



