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SCIENCE 



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



sheer ideality, on the ground of its intel- 

 lectual harmony, on the ground of its 

 beauty, "free from the gorgeous trap- 

 pings" of sense, pure, austere, supreme. 

 To do this, which ought, it seems, to be easy, 

 experience has shown to be exceedingly 

 difficult. For the multitude of men and 

 women, even the educated multitude, are 

 wont to cry. 



Such knowledge is too wonderful for me. 

 It 13 too high, I can not attain unto it, 



thus meaning to imply. What, then, or 

 where is its human significance? Their 

 voice is heard in the challenge once put to 

 me by the brilliant author of "East Lon- 

 don Visions." What, said he, can be the 

 human significance of "this majestic intel- 

 lectual cosmos of yours, towering up like a 

 million-lustred iceberg into the arctic 

 night," seeing that, among mankind, none 

 is permitted to behold its more resplendent 

 wonders save the mathematician alone? 

 What response will our pure mathemati- 

 cian make to this challenge? Make, I 

 mean, if he be not a wholly naive devotee 

 of his science and so have failed to reflect 

 upon the deeper grounds of its justifica- 

 tion. He may say, for one thing, what Pro- 

 fessor Klein said on a similar occasion : 



Apart from the fact that pure mathematics can 

 not be supplanted by anything else as a means 

 for developing the purely logical faculties of the 

 mind, there must be considered here, as elsewhere, 

 the necessity of the presence of a few individuals 

 in each country developed in far higher degree 

 than the rest, for the purpose of keeping up and 

 gradually raising the general standard. Even a 

 slight raising of the general level can be accom- 

 plished only when some few minds have progressed 

 far ahead of the average. 



That is doubtless a weighty considera- 

 tion. But is it all or the best that may be 

 said? It is just and important but it does 

 not go far enough; it is not, I fear, very 

 convincing; it is wanting in pungence and 

 edge ; it does not touch the central nerve of 



the challenge. Our pure mathematician 

 must rally his sceptics with sharper con- 

 siderations. He may say to them: You 

 challenge the human significance of the 

 higher developments of pure mathematics 

 because they are inaccessible to all but a 

 few, because their charm is esoteric, because 

 their deeper beauty is hid from nearly all 

 mankind. Does that consideration justify 

 your challenge? You are individuals, but 

 you are also members of a race. Have you 

 as individuals no human interest nor hu- 

 man pride in the highest achievements of 

 your race? Is nothing human, is nothing 

 humane, except mediocrity and the com- 

 monplace ? Was Phidias or Michel Angelo 

 less human than the carver and painter of 

 a totem-pole? Was Euclid or Gauss or 

 Poincare less representative of man than 

 the countless millions for whom mathe- 

 matics has meant only the arithmetic of 

 the market place or the rude geometry of 

 the carpenter? Does the quality of hu- 

 manity in human thoughts and deeds de- 

 crease as they ascend towards the peaks of 

 achievement, and increase in proportion as 

 they become vulgar, attaining an upper 

 limit in the beasts ? Do you not know that 

 precisely the reverse is true? Do you not 

 count aspiration humane? Do you not see 

 that it is not the common things that every 

 one may reach, but excellences high-dwell- 

 ing among the rocks — do you not know 

 that, in respect of human worth, these 

 things, which but few can attain, are sec- 

 ond only to the supreme ideals attainable 

 by none? 



How very different and how very much 

 easier the task of one who sought to vindi- 

 cate the human significance of mathematics 

 on the ground of its applications! In re- 

 spect of temperamental interest, of atti- 

 tude and outlook, the difference between 

 the pure and the applied mathematician is 

 profound. It is — if we may liken spiritual 



