NOVEMBEE 12, 1915] 



SCIENCE 



667 



have them convinced that mathematics 

 stands for an immense body of human 

 achievements, for a diversified continent 

 of pure doctrine, for a discovered world of 

 intellectual harmonies. He can not shotv 

 it to them as a painter displays a canvas 

 or as an architect presents a cathedral. 

 He can not give them an immediate vision 

 of it, but he can give them intimations ; by 

 appealing to their fantasie and, through 

 analogy with what they know, to their 

 understanding, not only can he convince 

 them that his world exists, but he can give 

 them an intuitive apprehension of its liv- 

 ing presence and its meaning for human- 

 kind. This is possible because, like him, 

 they, too, are idealists, dreamers and poets 

 — such essentially are all men and women. 

 His auditors or his readers have all had 

 some experience of ideas and of truth, they 

 have all had inklings of more beyond, they 

 have all been visited and quickened by a 

 sense of the limitless possibilities of further 

 knowledge in every direction, they have 

 all dreamed of the perfect and have felt 

 its lure. They are thus aware that the 

 small implies the large; having seen hills, 

 they can believe in mountains; they know 

 that Euripides, Shakespeare, Dante, Goethe, 

 are but fulfillments of prophecies heard in 

 peasant tales and songs; they know that 

 the symphonies of Beethoven or the dramas 

 of "Wagner are harbingered in the melodies 

 and the sighs of those who garner grain and 

 in their hearts respond to the music of the 

 winds or the "solemn anthems of the sea"; 

 they sense the secret by which the astron- 

 omy of Newton and Laplace is foretokened 

 in the shepherd's watching of the stars; 

 and knowing thus this plain spiritual law 

 of progressiveness and implication, they 

 are prepared to grasp the truth that mod- 

 ern mathematics, though they do not under- 

 stand it, is, like the other great things, but 

 a sublime fulfillment, the realization of 



prophecies involved in what they them- 

 selves, in common with other educated folk, 

 know of the rudiments of the science. In- 

 deed, they would marvel if upon reflection 

 it did not seem to be so. Our pure mathe- 

 matician in speaking to his fellow men and 

 women of his science will have no difficulty 

 in persuading them that he is speaking of 

 a subject immense and eternal. As born 

 idealists they have intimations of their 

 own — the evidence of intuition, if you 

 please — or a kind of insight resembling 

 that of the mystic — that in the world of 

 mind there must be something deeper and 

 higher, stabler and more significant, than 

 the pitiful ideas in life's routine and the 

 familiar vocations of men. They are thus 

 prepared to believe, before they are told, 

 that behind the veil there exists a univer- 

 sum of exact thought, an everlasting cos- 

 mos of ordered ideas, a stable world of con- 

 catenated truth. In their study of the ele- 

 ments, in school or college, they may have 

 caught a shimmer of it or, in rare moments 

 of illumination, even a gleam. Of the ex- 

 istence, the reality, the actuality, of our 

 pure mathematician's world they will have 

 no doubt, and they will have no doubt of 

 its grandeur. They may even, in a vague 

 way, magnify it overmuch, feeling that it 

 is, in some wise, more than human, signifi- 

 cant only for the rarely gifted spirit that 

 dwells, like a star, apart. The pure mathe- 

 matician 's difficulty lies in showing, in his 

 way, that such is not the ease. For he does 

 not wish to adduce utilities and applica- 

 tions. He is well aware of these. He 

 knows that if he "would tell them they are 

 more in number than the sands." Neither 

 does he despise them as of little moment. 

 On the contrary, he values them as pre- 

 cious. But he wishes to do his subject and 

 his auditors the honor of speaking from a 

 higher level: he desires to vindicate the 

 worth of mathematics on the ground of its 



