NOVBMBEK 12, 1915] 



SCIENCE 



665 



show that, notwithstanding its birth in the 

 concrete things of sense and raw reality, it 

 yet so appealed to sheer intellect — and we 

 must not forget that creative intellect is the 

 human faculty par excellence — it so ap- 

 pealed to this distinctive and disinterested 

 faculty of man that, long before the science 

 rose to the level of a fine art in the great 

 days of Euclid and Archimedes, Plato in 

 the wisdom of his maturer years judged It 

 essential to the education of freemen be- 

 cause, said he, there is in it a necessary 

 something against which even God can not 

 contend and without which neither gods 

 nor demi-gods can wisely govern mankind; 

 it would be possible, our historian could 

 say, to show historically to educated lay- 

 men that, even prior to the inventions of 

 analytical geometry and the infinitesimal 

 calculus, mathematics had played an in- 

 dispensable role in the "Two New Sci- 

 ences" of physics and mechanics in which 

 Galileo laid the foundations of our modern 

 knowledge of nature; it would be possible 

 to show not only that the analytical geom- 

 etry of Descartes and Fermat and the cal- 

 culus of Leibnitz and Newton have been 

 and are essential to our still advancing 

 conquest of the sea, but that it is owing to 

 the power of these instruments that the 

 genius of such as Newton, Laplace and La- 

 grange has been enabled to create for us a 

 new earth and a new heavens compared 

 with which the Mosaic cosmogony or the 

 sublimest creation of the Greek imagination 

 is but " as a cabinet of brilliants, or rather 

 a little jewelled cup found in the ocean or 

 the wilderness"; it would be possible to 

 show historically that, just because the pur- 

 suit of mathematical truth has been for the 

 most part disinterested — led, that is, by 

 wonder, as Aristotle says, and sustained by 

 the love of beauty with the joy of discov- 

 ery — it would be possible to show that, just 

 because of the disinterestedness of mathe- 



matical research, this science has been so 

 well prepared to meet everywhere and al- 

 ways, as they have arisen, the mathematical 

 exigencies of natural science and engineer- 

 ing ; above all, it would be feasible to show 

 historically that to the same disinterested- 

 ness of motive operating through the cen- 

 turies we owe the upbuilding of a body of 

 pure doctrine so towering to-day and vast 

 that no man, even though he have the 

 "Andean intellect" of a Poincare, can em- 

 brace it all. This much, I believe, and per- 

 haps more, touching the human significance 

 of mathematics, a historian of the science 

 might reasonably hope to demonstrate in 

 one hour. 



More difficult, far more difficult, I think, 

 would be the task of a pure mathematician 

 who aimed at an equivalent result by ex- 

 pounding, or rather by delineating, for he 

 could not in one hour so much as begin to 

 expound, the modern developments of the 

 subject. Could he contrive even to delin- 

 eate them in a way to reveal their relation 

 to what is essentially humane? Do but 

 consider for a moment the nature of such 

 an enterprise. Mathematics may be legiti- 

 mately pursued for its own sake or for the 

 sake of its applications or with a view to 

 understanding its logical foundations and 

 internal structure or in the interest of mag- 

 nanimity or for the sake of its bearings 

 upon the supreme concerns of man as man 

 or from two or more of these motives com- 

 bined. Our supposed delineator is actuated 

 by the first of them : his interest in mathe- 

 matics is an interest in mathematics for 

 the sake of mathematics; for him the sci- 

 ence is simply a large and growing body of 

 logical consistencies or compatibilities; he 

 derives his inspiration from the muse of 

 intellectual harmony; he is a pure mathe- 

 matician. He knows that pure mathe- 

 matics is a house of many chambers; he 

 knows that its foundations lie far beneath 



