36 SCIENCE AND METHOD. 



We cannot forget what our aim should be, and in my 

 opinion this aim is a double one. Our science borders 

 on both philosophy and physics, and it is for these 

 two neighbours that we must work. And so we have 

 always seen, and we shall still see, mathematicians 

 advancing in two opposite directions. 



On the one side, mathematical science must reflect 

 upon itself, and this is useful because reflecting upon 

 itself is reflecting upon the human mind which has 

 created it ; the more so because, of all its creations, 

 mathematics is the one for which it has borrowed 

 least from outside. This is the reason for the utility 

 of certain mathematical speculations, such as those 

 which have in view the study of postulates, of un- 

 usual geometries, of functions with strange behaviour. 

 The more these speculations depart from the most 

 ordinary conceptions, and, consequently, from nature 

 and applications to natural problems, the better will 

 they show us what the human mind can do when it 

 is more and more withdrawn from the tyranny of 

 the exterior world ; the better, consequently, will they 

 make us know this mind itself. 



But it is to the opposite side, to the side of nature, 

 that we must direct our main forces. 



There we meet the physicist or the engineer, who 

 says, " Will you integrate this differential equation for 

 me ; I shall need it within a week for a piece of 

 construction work that has to be completed by a 

 certain date ? " " This equation," we answer, " is not 

 included in one of the types that can be integrated, 

 of which you know there are not very many." " Yes, 

 I know ; but, then, what good are you ? " More often 

 than not a mutual understanding is sufficient. The 



