672 



SCIENCE 



[N. S, Vol. XLII. No. 1089 



centuries immediately preceding and fol- 

 lowing the great date of Euclid. Indeed 

 but for that general declension of Greek 

 spirit which Professor Gilbert Murray in 

 his "Four Stages in Greek Religion" has 

 happily characterized as "the failure of 

 nerve," what we know as the modern crit- 

 ical movement in mathematics might well 

 have come to its present culmination, so 

 far at least as piire geometry is concerned, 

 fifteen hundred or more years ago. It is 

 a pity that the deeper and stabler things 

 of science and the profounder spirit of 

 man can not be here disclosed in a manner 

 commensurate with the great exposition, 

 surrounding us, of the manifold practical 

 arts and industries of the world. It is a 

 pity there is no means by which our speaker 

 might, in a manner befitting the subject 

 and the occasion, exhibit intelligibly to his 

 fellow men and women the ways and re- 

 sults of the last hundred years of research 

 into the groundwork of mathematical sci- 

 ence and therewith the highly important 

 modern developments in logic and the 

 theory of knowledge. How astonished the 

 beholders would be, how delighted, too, 

 and proud to belong to a race capable of 

 such patience and toil, of such disinterested 

 devotion, of such intellectual finesse and 

 depth of penetration. I can think of no 

 other spectacle quite so impressive as the 

 inner vision of all the manifold branches of 

 rigorous thought seen to constitute one 

 immense structure of autonomous doctrine 

 reposing upon the spiritual basis of a few 

 select ideas and, superior to the fading 

 beauties of time and sense, shining there 

 like a celestial city, in "the white radiance 

 of eternity." That is the vision of mathe- 

 matics that a student of its philosophy 

 would, were it possible, present to his fel- 

 low men and women. 



In view of the foregoing considerations 

 it evidently is, I think, in the nature of the 



case impossible to give an adequate sense 

 of the human worth of mathematics if one 

 choose to devote the hour to any one of the 

 great aspects of it with which we have been 

 thus far concerned. Neither the history 

 of the subject nor its present estate nor its 

 applications nor its logical foundations — 

 no one of these themes lends itself well to 

 the purpose of such an exposition, and still 

 less do two or more of them combined. 

 Even if such were not the ease I should yet 

 feel bound to pursue another course; for I 

 have been long persuaded that, in respect 

 of its human significance, mathematics in- 

 vites to a point of view which, unless I am 

 mistaken, has not been taken and held in 

 former attempts at appreciation. I have 

 already alluded to bearings of mathematics 

 as distinguished from applications. It is 

 with its bearings that I wish to deal. I 

 mean its bearings upon the higher con- 

 cerns of man as man — those interests, 

 namely, which have impelled him to seek, 

 over and above the needs of raiment and 

 shelter and food, some inner adjustment of 

 life to the poignant limitations of life in 

 our world and which have thus drawn him 

 to manifold forms of wisdom, not only to 

 mathematics and natural sciences, but also 

 to literature and philosophy, to religion 

 and art, and theories of righteousness. 

 What is the role of mathematics in this per- 

 petual endeavor of the human spirit every- 

 where to win reconciliation of its dreams 

 and aspirations with the baffling conditions 

 and tragic facts of life and the world? 

 What is its relation to the universal quest 

 of man for some supreme and abiding good 

 that shall assuage or annul the discords 

 and tyrannies of time and limitation, with- 

 holding less and less, as time goes by, the 

 freedom and the peace of an ideal harmony 

 infinite and eternal? 



In endeavoring to suggest, in the time 

 remaining for this address, a partial answer 



