664 



SCIENCE 



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



women who avail themselves of its exhibi- 

 tions and lessons, an exposition of human- 

 ity: an exposition, no doubt, of the activi- 

 ties and aspirations and prowess of individ- 

 ual men and women, but of men and 

 women, not in their capacity as individuals, 

 but as representatives of humankind. In- 

 dividual achievements are not the object, 

 they are the means, of the exposition. The 

 object is humanity. 



What is the human significance — what 

 Is the significance for humanity — of "the 

 mother of the sciences"? And how may 

 the matter be best set forth, not for the 

 special advantage of professional mathe- 

 maticians, for I shall take the liberty of 

 having these but little in mind, but for the 

 advantage and understanding of educated 

 men and women in general? I am unable 

 to imagine a more difficult undertaking, so 

 technical, especially in its language, and 

 so immense is the subject. It is clear that 

 the task is far beyond the resources of an 

 hour's discourse, and so it is necessary to 

 restrict and select. This being the case, 

 what is it best to choose ? The material is 

 superabundant. What part of it or aspect 

 of it is most available for the end in view ? 

 "In abundant matter to speak a little with 

 elegance, ' ' says Pindar, " is a thing for the 

 wise to listen to." It is not, however, a 

 question of elegance. It is a question of 

 emphasis, of clarity, of effectiveness. What 

 shall be our major theme ? 



Shall it be the history of the subject? 

 Shall it be the modern developments of 

 mathematics, its present status and its fu- 

 ture outlook? Shall it be the utilities of 

 the science, its so-called applications, its 

 service in practical affairs, in engineering 

 and in what it is customary to call the sci- 

 ences of nature? Shall it be the logical 

 foundations of mathematics, its basic prin- 

 ciples, its inner nature, its characteristic 

 processes and structure, the differences and 



similitudes that come to light in comparing 

 it with other forms of scientific and philo- 

 sophic activity? Shall it be the bearings 

 of the science as distinguished from its 

 applications — the bearings of it as a spirit- 

 ual enterprise upon the higher concerns of 

 man as man ? It might be any one of these 

 things. They are all of them great and in- 

 spiring themes. 



It is easy to understand that a historian 

 would choose the first. The history of 

 mathematics is indeed impressive, but is it 

 not too long and too technical? And is it 

 not already accessible in a large published 

 literature of its own? I grant, the histor- 

 ian would say, that its history is long, for 

 in respect of antiquity mathematics is a 

 rival of art, surpassing nearly all branches 

 of science and by none of them surpassed. 

 I grant that, for laymen, the history is 

 technical, frightfully technical, requiring 

 interpretation in the interest of general in- 

 telligence. I grant, too, that the history 

 owns a large literature, but this, the his- 

 torian would say, is not designed for the 

 general reader, however intelligent, the nu- 

 merous minor works, no less than the major 

 ones, including that culminating monu- 

 mental work of Moritz Cantor, being, all of 

 them, addressed to specialists and intel- 

 ligible to them alone. And yet it would be 

 possible to tell in one hour, not indeed the 

 history of mathematics, but a true story jf 

 it that would be intelligible to all and 

 would show its human significance to be 

 profound, manifold and even romantic. It 

 would be possible to show historically that 

 this science, which now carries its head so 

 high in the tenuous atmosphere of pure 

 abstractions, has always kept its feet upon 

 the solid earth ; it would be possible to show 

 that it owns indeed a lowly origin, in the 

 familiar needs of common life, in the 

 homely necessities of counting herds and 

 measuring lands; it would be possible to 



