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SCIENCE 



[N. S. Vol. XXXII. No. 821 



pure mathematics but also to all branches 

 of science to which mathematical methods 

 have been applied will be best obtained by 

 an examination of that monumental work, 

 the "Encyclopadie der mathematischen 

 Wissenschaf ten " — when it is completed. 

 The concepts of the pure mathematician, 

 no less than those of the physicist, had their 

 origin in physical experience analyzed and 

 clarified by the reflective activities of the 

 human mind; but the two sets of concepts 

 stand on different planes in regard to the 

 degree of abstraction which is necessary in 

 their formation. Those of the mathemati- 

 cian are more remote from actual analyzed 

 precepts than are those of the physicist, 

 having undergone in their formation a 

 more complete idealization and removal of 

 elements inessential in regard to the pur- 

 poses for which they are constructed. This 

 ■difference in the planes of thought fre- 

 quently gives rise to a certain misunder- 

 standing between the mathematician and 

 the physicist, due in the case of either to 

 an inadequate appreciation of the point of 

 view of the other. On the one hand it is 

 frequently and truly said of particular 

 mathematicians that they are lacking in 

 the physical instinct; and on the other 

 hand a certain lack of sympathy is fre- 

 quently manifested on the part of physi- 

 cists for the aims and ideals of the mathe- 

 matician. The habits of mind and the 

 ideals of the mathematician and of the 

 physicist can not be of an identical char- 

 acter. The concepts of the mathematician 

 necessarily lack, in their pure form, just 

 that element of concreteness which is an 

 essential condition of the success of the 

 physicist, but which to the mathematician 

 would often only obscure those aspects of 

 things which it is his province to study. 

 The abstract mathematical standard of 

 exactitude is one of which the physicist 

 can make no direct use. The calculations in 



mathematics are directed towards ideal pre- 

 cision, those in physics consist of approxi- 

 mations within assigned limits of error. 

 The physicist can, for example, make no 

 direct use of such an object as an irrational 

 number; in any given case a properly 

 chosen rational number approximating to 

 the irrational one is sufficient for his pur- 

 pose. Such a notion as continuity, as it 

 occurs in mathematics, is, in its purity, 

 unknown to the physicist, who can make 

 use only of sensible continuity. The phys- 

 ical counterpart of mathematical discon- 

 tinuity is very rapid change through a thin 

 layer of transition, or during a very short 

 time. Much of the skill of the true mathe- 

 matical physicist and of the mathematical 

 astronomer consists in the power of adapt- 

 ing methods and results carried out on an 

 exact mathematical basis to obtain approx- 

 imations sufficient for the purposes of 

 physical measurement. It might perhaps 

 be thought that a scheme of mathematics 

 on a frankly approximate basis would be 

 sufficient for all the practical purposes of 

 application in physics, engineering science 

 and astronomy; and no doubt it would be 

 possible to develop, to some extent at least, 

 a species of mathematics on these lines. 

 Such a system would, however, involve an 

 intolerable awkwardness and prolixity in 

 the statement of results, especially in view 

 of the fact that the degrees of approxima- 

 tion necessary for various purposes are very 

 different, and thus that unassigned grades 

 of approximation would have to be pro- 

 vided for. Moreover the mathematician 

 working on these lines would be cut off 

 from his chief sources of inspiration, the 

 ideals of exactitude and logical rigor, as 

 well as from one of his most indispensable 

 guides to discovery, symmetry and perma- 

 nence of mathematical form. The history 

 of the actual movements of mathematical 

 thought through the centuries shows that 



