PRESIDENTIAL ADDRESS. 511 



I have said that Mathematics is the oldest of the sciences ; a glance at its 

 more recent history will show that it has the energy of perpetual youth. The 

 output of contributions to the advance of the science during the last century and 

 more has been so enormous that it is difficult to say whether pride in the great- 

 ness of achievement in his subject, or despair at his inability to cope with the 

 multiplicity of its detailed developments, should be the dominant feeling of the 

 mathematician. Few people outside the small circle of mathematical specialists 

 have any idea of the vast growth of mathematical literature. The Royal Society 

 Catalogue contains a list of nearly thirty-nine thousand papers on subjects of Pure 

 Mathematics alone, which have appeared in seven hundred serials during the nine- 

 teenth century. This represents only a portion of the total output; the very large 

 number of treatises, dissertations, and monographs published during the century 

 being omitted. During the first decade of the twentieth century this activity has 

 proceeded at an accelerated rate. Mathematical contributions to Mechanics, 

 Physics, and Astronomy would greatly swell the total. A notion of the range of 

 the literature relating not only to 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 

 Wissenschaften ' — when it is completed. 



The concepts of the pure mathematician, no less than those of the physicist, 

 had their origin in physical experience analysed 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 forma- 

 tion. Those of the mathematician are more remote from actual unanalysed per- 

 cepts than are those of the physicist, having undergone in their formation a more 

 complete idealisation and removal of elements inessential in regard to the purposes 

 for which they are constructed. This difference in the planes of thought fre- 

 quently gives rise to a certain misunderstanding 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 par- 

 ticular mathematicians that they are lacking in the physical instinct; and on the 

 other hand a certain lack of sympathy is frequently manifested on the part of 

 physicists for the aims and ideals of the mathematician. The habits of mind 

 and the ideals of the mathematician and of the physicist cannot be of an identical 

 charac'er. 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 mathe- 

 matical standard of exactitude is one of which the physicist can make no direct 

 use. The calculations in Mathematics are directed towards ideal precision, those 

 in Physics consist of approximations 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 purpose. 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 physical counterpart of mathematical 

 discontinuity is very rapid change through a thin layer of transition, or during 

 a very short time. Much of the skill of the true mathematical physicist and of 

 the mathematical astronomer consists in the power of adapting methods and 

 results carried out on an exact mathematical basis to obtain approximations suffi- 

 cient for the purposes of physical measurement. It might perhaps be thought 

 that a scheme of Mathematics on a frankly approximative basis would be suffi- 

 cient 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, how- 

 ever, involve an intolerable awkwardness and prolixity in the statement of results, 

 especially in view of the fact that the degrees of approximation necessary for 

 various purposes are very different, and thus that unassigned grades of approxi- 

 mation would have to be provided for. Moreover the mathematician working 

 on these lines would be cut off from his chief sources of inspiration, the ideals 

 of exactitude and logical rigour, as well as from one of his most indispensable 

 guides to discovery, symmetry and permanence of mathematical form. The 



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