THE VALUE OF SCIENCE 195 



of the scientist, susceptible of infinite variation? Volumes could be 

 written without exhausting this subject; I, in a few brief pages, have 

 only touched it lightly. That the geometer's mind is not like the 

 physicist's or the naturalist's, all the world would agree; but mathe- 

 maticians themselves do not resemble each other; some recognize only 

 implacable logic, others appeal to intuition and see in it the only source 

 of discovery. And this would be a reason for distrust. To minds so 

 unlike can the mathematical theorems themselves appear in the same 

 light? Truth which is not the same for all, is it truth? But look- 

 ing at things more closely, we see how these very different workers 

 collaborate in a common task which could not be achieved without 

 their cooperation. And that already reassures us. 



Next must be examined the frames in which nature seems enclosed 

 and which are called time and space. In ' Science and Hypothesis ' 

 I have already shown how relative their value is; it is not nature 

 which imposes them upon us, it is we who impose them upon nature 

 because we find them convenient. But I have spoken of scarcely more 

 than space, and particularly quantitative space, so to say, that is of the 

 mathematical relations whose aggregate constitutes geometry. I 

 should have shown that it is the same with time as with space and 

 still the same with ' qualitative space ' ; in particular, I should have 

 investigated why we attribute three dimensions to space. I may be 

 pardoned then for taking up again these important questions. 



Is mathematical analysis then, whose principal object is the study 

 of these empty frames, only a vain play of the mind? It can give to 

 the physicist only a convenient language ; is this not a mediocre service, 

 which, strictly speaking, could be done without; and even is it not to 

 be feared that this artificial language may be a veil interposed between 

 reality and the eye of the physicist? Far from it; without this 

 language most of the intimate analogies of things would have re- 

 mained forever unknown to us ; and we should forever have been igno- 

 rant of the internal harmony of the world, which is, we shall see, the 

 only true objective reality. 



The best expression of this harmony is law. Law is one of the most 

 recent conquests of the human mind; there still are people who live 

 in the presence of a perpetual miracle and are not astonished at it. 

 On the contrary, we it is who should be astonished at nature's regu- 

 larity. Men demand of their gods to prove their existence by miracles; 

 but the eternal marvel is that there are not miracles without cease. 

 The world is divine because it is a harmony. If it were ruled by 

 caprice, what could prove to us it was not ruled by chance? 



This conquest of law we owe to astronomy, and just this makes 

 the grandeur of the science rather than the material grandeur of the 

 objects it considers. It was altogether natural then that celestial 



