September i, 1910] 



NATURE 



2^5 



sumptions against the change that has been suggested. 

 Meteorology, so far as it goes beyond the purely empirical 

 region, is, and must always remain, a branch of Physics. 

 No doubt the more technical problems which arise in 

 connection with these subjects, though of great importance 

 lo specialists, are often of little or no interest to workers 

 in cognate departments. It appears to me, however, that 

 it is unwise, in view of the general objects of the British 

 .Association, to give too much prominence in the meetings 

 lo the more technical aspects of the various departmentii 

 of science, .\mple opportunities for the full discussion of 

 all the detailed problems, the solution of which forms a 

 great and necessary part of the work of those who are 

 advancing science in its various branches, are afforded by 

 the special .Societies which malce those branches their 

 exclusive concern. The British .Association will, in my 

 view, be performing its functions most efficiently if it gives 

 much prominence to those aspects of each branch of science 

 which are of interest to a public at least in some degree- 

 larger than the circle of specialists concerned with the 

 particular branch. To afford an opportunity to workers 

 in any one department of obtaining some knowledge of 

 what is going on in other departments, to stimulate by 

 means of personal intercourse with workers on other lines 

 the s?ns° cf solidarity of men of science, lo do something 

 to counteract that tendency lo narrowness of view which 

 is a d.-nger arising from increasing specialisation, are 

 functions ;he due performance of which may do much to 

 further that supreme object, the advancement of science, 

 fc which th" Brit'sh .Association exists. 



I propose to address to you a few remarks, necessarily 

 fragmentary and incomplete, upon the scope and tendencies 

 of modern Mathematics. Not to transgress against the 

 canon I have laid down, I shall endeavour to make my 

 treatment of the subject as little technical as possible. 



Probably no other department of knowledge plays a 

 larger part outside its own narrower domain than Mathe- 

 matics. Some of its more elementary conceptions and 

 methods have become p;.rl of the common heritage of our 

 civilisation, interwoven in the evervday life of the people. 

 Perhaps the greatest labour-saving invention that the world 

 has seen belongs to the formal side of Mathematics ; I 

 allude to our system of numerical notation. This svstem, 

 which, when scrutinised, affords the simplest illustration 

 of the imoortance of Malhemalical form, has become so 

 much an indispensable part of our mental furniture that 

 some effort is required to realise Ih.at an apparently so 

 obvious idea embodies a great invention, one to which the 

 Greeks, with their unsurpassed capacity for abstract think- 

 ing, never attained. .\n attempt to do a multiplication 

 sum in Roman numerals is perhaps the readiest road to 

 an appreciation of the advantages of this great invention. 

 In a large group of sciences, the formal element, the 

 common language, so to speak, is supplied bv Mathe- 

 matics : the range of the application of mathematical 

 methods and symbolism is ever increasing. Without 

 taking too literally the celebrated dictum of the great 

 philosopher Kant, that the amount of real science to be 

 found in anv special subject is the amount of Mathematics 

 contained therein, it must l.>e admitted that each branch 

 .of science which is concerned with natural phenomena, 

 when it has reached a certain stage of development, be- 

 comes accessible to, and has need of, mathematical methods 

 and language ; this stage has, for example, been reached 

 in our time by parts of the science of Chemistry. Even 

 Biology and Economics have begun to require mathe- 

 matical methods, at least on their statistical side. As a 

 science emerges from the stages in which it consists solelv 

 of more or less systematised descriptions of the phenomena 

 with which it is concerned in their more superficial aspect ; 

 when the intensive magnitudes discerned in the phenomena 

 become representable as extensive magnitudes, then is the 

 beginning of the application of mathematical modes of 

 thought ; at a still later stage, when the phenomena become 

 accessible to dynamical treatment. Mathematics is applic- 

 able to the subject to a still greater extent. 



Mathematics shares with the closelv allied subject of 

 Astronomy the honour of behig the oldest of the sciences. 

 When we consider that it embodies, in an abstract form, 

 some of the more obvious, and yet fundamental, aspects 

 of our experience of the external world, this is not 



NO. 2 13 I, VOL. 84] 



altogether surprising. The comparatively high degree of 

 development which, as recent historical discoveries have 

 disclosed, it had attained amongst the Babylonians more 

 than five thousand years b.c, may well astonish us. 

 These times must have been preceded by still earlier ages,, 

 in which the mental evolution of man led him to the use 

 of the tally, and of simple modes of measurement, long 

 before the notions of number and of magnitude appeared 

 in an explicit form. 



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 greatness 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 out- 

 side 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 nineteenth century. This represents only a portion of 

 the total output, the very large number of treatises, dis- 

 sertations, 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 

 .\stronomy would greatly swell the total. .\ notion of the 

 range of the literature relating, not only to Pure Mathe 

 matics, but also to all branches of science to which mathe 

 matical 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 experi- 

 ence 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 abstr.action 

 which is necessarv in their formation. Those of the 

 mathematician are more rernote from actual unanalysed 

 precepts 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 frequently 

 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 particular mathematicians that they are lacking 

 in the physical instinct, and on the other hand a certain 

 lack of sympathy is frequenilv 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 character. 

 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 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 Duritv, unknown to the ohysicist. 

 who can make use only of sensible continuity. The 

 physical counterpart of mathematical discontinuity is verv 

 rapid change through a thin laver of transition, or during 

 a very short time. Much of the skill of the true mathe- 

 matical ohysicist and of the mathematical astronomer 

 consists in the power of adapting methods and resu'ts 

 carried out on an exact mathematical basis to obtain 

 approximations sufficient for the purposes of physical 



