Septembee 23, 1910] 



SCIENCE 



389 



these ideals are the very life-blood of the 

 science, and warrants the conclusion that 

 a constant striving towards their attain- 

 ment is an absolutely essential condition of 

 vigorous growth. These ideals have their 

 roots in irresistible impulses and deep- 

 seated needs of the human mind, mani- 

 fested in its efforts to introduce intelligi- 

 bility into certain great domains of the 

 world of thought. 



There exists a wide-spread impression 

 amongst physicists, engineers and other 

 men of science that the effect of recent de- 

 velopments of pure mathematics, by ma- 

 king it more abstract than formerly, has 

 been to remove it further from the order 

 of ideas of those who are primarily con- 

 cerned with the physical world. The 

 prejudice that pure mathematics has its 

 sole raison d'etre in its function of provid- 

 ing useful tools for application in the 

 physical sciences, a prejudice which did 

 much to retard the due development of 

 pure mathematics in this country during 

 the nineteenth century, is by no means ex- 

 tinct. It is not infrequently said that the 

 present devotion of many mathematicians 

 to the interminable discussion of purely 

 abstract questions relating to modern de- 

 velopments of the notions of number and 

 function, and to theories of algebraic form, 

 serves only the purpose of deflecting them 

 from their proper work into paths which 

 lead nowhere. It is considered that mathe- 

 maticians are apt to occupy themselves too 

 exclusively with ideas too remote from the 

 physical order in which mathematics had 

 its origin and in which it should still find 

 its proper applications. A direct answer 

 to the question ciii bono? when it is raised 

 in respect of a department of study such as 

 pure mathematics, seldom carries convic- 

 tion, in default of a standard of values 

 common to those who ask and to those who 

 answer the question. To appreciate the im- 



portance of a sphere of mental activity 

 different from our own always requires 

 some effort of the sympathetic imagination, 

 some recognition of the fact that the abso- 

 lute value of interests and ideals of a par- 

 ticular class may be much greater than the 

 value which our own mentality inclines us 

 to attach to them. If a defense is needed 

 of the expenditure of time and energy on 

 the abstract problems of pure mathematics, 

 that defense must be of a cumulative char- 

 acter. The fact that abstract mathematical 

 thinking is one of the normal forms of ac- 

 tivity of the human mind, a fact which the 

 general history of thought fully estab- 

 lishes, will appeal to some minds as a 

 ground of decisive weight. A great de- 

 partment of thought must have its own 

 inner life, however transcendent may be 

 the importance of its relations to the out- 

 side. No department of science, least of all 

 one requiring so high a degree of mental 

 concentration as mathematics, can be de- 

 veloped entirely, or even mainly, with a 

 view to applications outside its own range. 

 The increased complexity and specializa- 

 tion of all branches of knowledge makes it 

 true in the present, however it may have 

 been in former times, that important ad- 

 vances in such a department as mathe- 

 matics can be expected only from men who 

 are interested in the subject for its own 

 sake, and who, whilst keeping an open 

 mind for suggestions from outside, allow 

 their thought to range freely in those lines 

 of advance which are indicated by the 

 present state of their subject, untram- 

 melled by any preoccupation as to applica- 

 tions to other departments of science. 

 Even with a view to applications, if mathe- 

 matics is to be adequately equipped for the 

 purpose of coping with the intricate prob- 

 lems which will be presented to it in the 

 future by physics, chemistrj- and other 

 branches of physical science, many of these 



