NOYEMBEE 12, 1915] 



SCIENCE 



679 



the spiritual needs of humanity. What 

 are the responses worth? What are their 

 values, joint and several, absolute and 

 relative? And what, especially, is the hu- 

 man worth of the response of mathematics ? 

 It is, he would say, not only our privilege, 

 but, as educated individuals and especially 

 as representatives of our race, it is our 

 duty, to ponder the matter and reach, if 

 we can, a right appraisement. For the 

 proper study of mankind is man, and it is 

 essential to remember that "La vie de la 

 science est la critique." I have, he would 

 say, tried to make it clear that mathematics 

 is not an isolated science. I have tried to 

 show that it is not an antagonist, nor a rival, 

 but is the comrade and ally of the other 

 great forms of spiritual activity, all aim- 

 ing at the same high end. I have re- 

 minded you of the principal answers made 

 by these to the spiritual needs of man, and 

 I do not, he would say, desire to underrate 

 or belittle them. They are a precious in- 

 heritance. Many of them have not, indeed, 

 stood the test of time ; others will doubtless 

 endure for aye; all of them, for a longer 

 or shorter period, have softened the ways 

 of life to millions of men and women. 

 Neither do I desire, he would say, to ex- 

 aggerate the contributions of mathematics 

 to the spiritual weal of humanity. What 

 I desire is a fair comparative estimate of 

 its claims. "Truth is the beginning of 

 every good thing, both to gods and men." 

 I am asking you to compare, consider and 

 judge for yourselves. The task is arduous 

 and long. 



There are, our critic would say, certain 

 paramount considerations that every one 

 in such an enterprise must weigh, and a 

 few of them may, in the moments that re- 

 main, be passed in brief review. Con- 

 sider, for example, our human craving for 

 a world of stable reality. Where is it to be 

 found? We know the answer of theology, 



of philosophy, of natural science and the 

 rest. We know, too, the answer of litera- 

 ture and general thought: 



The cloud-capped towers, the gorgeous palaces. 

 The solemn temples, the great globe itself, 

 Yea, all which it inherit, shall dissolve. 

 And, like the baseless fabric of this vision, 

 Leave not a rock behind. 



And now what, he would ask, is the answer 

 of mathematics ? The answer, he would have 

 to say, is this: Transcending the flux of 

 the sensuous universe, there exists a stable 

 world of pure thought, a divinely ordered 

 world of ideas, accessible to man, free from 

 the mad dance of time, infinite and eternal. 

 Consider our human craving for free- 

 dom. Of freedom there are many kinds. 

 Is it the freedom of limitless room, where 

 our passion for outward expression, for 

 externalization of thought, may attain its 

 aim? It is to mathematics, our critic 

 would say, that man is indebted for that 

 priceless boon; for it is the cunning of 

 this science that has at length contrived to 

 release our long imprisoned thought from 

 the old confines of our three-fold world of 

 sense and opened to its wing the intermin- 

 able skies of hyperspace. But if it be a 

 more fundamental freedom that is meant, 

 if it be freedom of thought proper — free- 

 dom, that is, for the creative activity of 

 intellect — then again it is to mathematics 

 that our faculties must look for the defi- 

 nition and a right estimate of their pre- 

 rogatives and power. For, regarding this 

 matter, we may indeed acquire elsewhere 

 a suspicion or an inkling of the truth, but 

 mathematics, and nothing else, is qualified 

 to give us knowledge of the fact that our 

 intellectual freedom is absolute save for a 

 single limitation — the law of non-contra- 

 diction, the law of logical compatibility, 

 the law of intellectual harmony — sole re- 

 striction imposed by "the nature of 



