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NATURE 



[ShPTEMlJEK I, I910 



measurement. It might perhaps be thought that a scheme 

 of Mathematics on a franl<ly approximative basis would 

 be sufficient for all ihe practical purposes of application 

 in Physics, Engineering Science, and Astronomy, and no 

 doubt it would be possible lo develop, to some extent at 

 least, a species of Mathematics on these lines. Such a 

 system would, however, involve an intolerable awkward- 

 ness and proli.xity 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 approximation 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 dis- 

 covery, symmetry, and permanence of mathematical form. 

 The history of the actual movements of mathematical 

 thought through the centuries shows that these ideals are 

 the very life-blood of the science, and warrants the con- 

 clusion that a constant striving towards their attainment 

 is an absolutely essential condition of vigorous growth. 

 These ideals have their roots in irresistible impulses and 

 deep-seated needs of the humaji mind, manifested in its 

 efforts to introduce intelligibility into certain great domains 

 of the world of thought. 



There exists a widespread impression among physicists, 

 engineers, and other men of science that the effect of 

 recent developments of Pure Mathematics, by making jt 

 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 raisim d'etre in its function of 

 providing 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 extinct. It is not 

 infrequently said that the present devotion of many mathe- 

 maticians to the interminable discussion of purely abstract 

 questions relating to modern developments of the notions 

 of number and function, and to theories of algebraic form, 

 serves only the purpose of deflecting them from their 

 proper w-ork into paths which lead nowhere. It is con- 

 sidered- that mathematicians 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 

 lo the question cui bono? when it is raised in respect of 

 a department of study such as Pure Mathematics, seldom 

 carries conviction, in default of a standard of values 

 common to those who ask and to those who answer the 

 question. To appreciate the importance of a sphere of 

 mental activity different from our own always requires 

 some effort of the sympathetic imagination, some recogni- 

 tion of Ihe fact that the absolute value of interests and 

 ideals of a jiarticular class may be much greater than the 

 value which our own mentality inclines us lo attach to 

 them. If .-i defence is needed of the expenditure of time 

 and energy on the abstract problems of Pure Mathematics, 

 that defence must be of a cumulative character. The fact 

 that abstract mathematical thinking is one of the normal 

 forms of activity of the human mind, a fact which the 

 general history of thought fully establishes, will appeal 

 to some minds as a ground of decisive weight. .\ great 

 department of thought must have its own inner life, how- 

 ever transcendent may be the importance of its relations 

 to the outside. No department of science, least of all one 

 requiring so high a degree of mental concentration as 

 Mathematics, can be developed enlirrly, or even mainly, 

 with a view to applications outside its own range. The 

 increased complexity and specialisation of all branches of 

 knowledge makes it true in the present, however it may 

 have been in former times, that important advances in 

 .such a department as Mathematics can be exoected only 

 from men who are interested in the subject for its own 

 sake, and who, whilst keening an onen mind for sugges- 

 tions from outside, allow their thought to range freely in 

 those lines of advance which are indicated by the present 

 state of their subject, untrainmelled by any preoccupation 

 a.s to applications to other departments of .science. Even 

 with a view to applications, if Mathematics is to be 

 adequ.-itely eouipped for the purpose of coping with the 



NO. 2 13 I, VOL. 84] 



intricate problems which will be presented to it in the 

 future by Physics, Chemistry, and other branches of 

 physical science, many of these problems probably of a 

 character which we cannot at present forecast, it is 

 essential that Mathematics should be allowed to develop 

 jtself freely on its own lines. Even if much of our present 

 mathematical theorising turns out to be useless for external 

 purposes, it is wiser, for a well-known reason, lo allow 

 the wheat and the tares to grow together. It would be 

 easy lo establish in detail that many of the applications 

 which have been actually made of Mathematics were- 

 wholly unforeseen by those who first developed the methods 

 and ideas on which they rest. Recently, the more refined 

 matherrjatical methods which have been applied to gravita- 

 tional Astronomy by Delaunay, G. \V. Hill, Poincarc"', 

 E. W. Brown, and others, have thrown much light on 

 questions relating to the solar system, and have much 

 increased the accuracy of our knowledge of the motions 

 of the moon and the planets. Who know's what weapons 

 forged by the theories of functions, of differential equa- 

 tions, or of groups, may be required when the time comes 

 for such an empirical law as Mendeleeff's periodic law of 

 the elements lo receive its dynamical explanation bv means 

 of an analysis of the detailed possibilities of relatively 

 stable types of motion, the general schematic character of 

 which will have been indicated by the physicist? It is 

 undoubtedly true that the cleft between Pure Mathematics 

 and Physical Science is at the present time wider than 

 formerly. That is, however, a result of the natural 

 development, on their own lines, of both subjects. In the 

 classical period of the eighteenth century, the time of 

 Lagrange and Laplace, the nature of the physical investi- 

 gations, consisting largely of the detailed working out of 

 nroblems of gravitational Astronomy in accordance with 

 Newton's law, was such that the passage was easy from 

 the concrete problems to the corresponding abstract mathe- 

 matical ones. Later on, mathematical physicists were 

 much occupied with problems which lent themselves readily 

 to treatment by means of continuous analysis. In our 

 own time the effect of recent developments of Physics has 

 been to present problems of molecular and sub-molecular 

 Mechanics to which continuous analysis is not at least 

 directly applicable, and can only be made applicable by a 

 process of averaging the effects of great swarms of discrete 

 entities. The speculative and incomplete character of our 

 conceptions of the structure of the objects of investigation 

 has made the applications of Dynamics to their detailed 

 elucidation tentative and partial. The generalised 

 dynamical scheme developed by Lagrange and Hamilton, 

 with its power of dealing with systems, the detailed struc- 

 ture of which is partially unknown, has, however, proved 

 a powerful weapon of attack, and affords a striking 

 instance of the deep-rooted significance of mathematical 

 form. The wonderful and perhaps unprccedentedly rapid 

 discoveries in Physics which have been made in the last 

 two decades have given rise to many questions which are 

 as, yet hardly sufficiently definite in form to be ripe for 

 mathematical treatment, a necessary condition of which 

 tieatment consists in a certain kind of precision in the 

 data of the problems to be solved. 



The dilTiculty of obtaining an adequate notion of the 

 general scoop and aims of Mathematics, or even of special 

 branches of it, is perhaps greater than in the case of any 

 other science. Many persons, even such as have made a 

 serious and prolonged study of the subject, feel the diffi- 

 culty of seeing the w'ood for trees. The severe demand; 

 mode upon students by the labour of acquiring a difficu't 

 technique largely accounts for this; but teachers might 

 do much to facilitate the attainment of a wider outlook 

 by directing the attention of their students to the more 

 general and less technical aspects of the various narts of 

 the subject, and especially by the introduction into the 

 courses of instruction of more of the historical element 

 than has hitherto been usual. 



All attempts to characterise the domain of Mathematics 

 by means of a formal definition which shall not only bf 

 complete, but which shall also rigidly mark off thnt 

 domain from the adjacent provinces of Formal I^ogic on 

 the one side and of Physical Science on the other side, 

 are almost certain to meet with but doubtful success : 

 siich success as they may attain will probably be only 



