376 



NA TURE 



[August i8, 1904 



traditional prfsentment. The consistency and completeness 

 of the usual system of definitions, axioms, and postulates 

 have often been questioned;- and quite recently a more 

 thorough-going analysis of the logical elements of the sub- 

 ject than has ever before been attempted has been made by 

 Hilbert. The matter is a subtle one, and a general agree- 

 ment on such points is as yet hardly possible. The basis 

 for such an agreement may perhaps ultimately be found in 

 a more explicit recognition of the empirical source of the 

 fundamental conceptions. This would tend, at all events, 

 to mitigate the rigour of the demands which are sometimes 

 made for logical perfection. 



Even more important in some respects are the questions 

 which have arisen in connection with the applications of 

 Geometry to purposes of graphical representation. It is 

 not necessary to dwell on the great assistance which this 

 method has rendered in such subjects as Physics and 

 Engineering. The pure mathematician, for his part, will 

 freely testify to the influence which it has exercised in the 

 development of most branches of Analysis ; for example, we 

 owe to it all the leading ideas of the Calculus. Modern 

 analysts have discovered, however, that Geometry may be 

 a snare as well as a guide. In the mere act of drawing a 

 curve to represent an analytical function we make un- 

 consciously a host of assumptions which are difficult not 

 merely to prove, but even to formulate preciselv. It is now 

 sought to establish the whole fabric of mathematical analysis 

 on a strictly arithmetical basis. To those who were trained 

 in an earlier school, the results so far are in appearance 

 somewhat forbidding. If the shade of one of the great 

 analysts of a century ago could revisit the glimpses of the 

 moon, his feelings would, I think, be akin to those of the 

 traveller to some mediaeval town, who finds the buildings 

 he came to see obscured by scaffolding, and is told that the 

 ancient monuments are all in process of repair. It is to be 

 hoped that a good deal of this obstruction is onlv temporary, 

 that most of the scaffolding will eventualiv be cleared away', 

 and that the edifices when they reappear will not be entirely 

 transformed, but will still retain something of their historic 

 outlines. It would be contrary to the spirit of this Address 

 to undervalue in any way the critical examination and re- 

 vision of principles; we must acknowledge that it tends 

 ultmiately to simplification, to the clearing up of issues, 

 and the reconciliation of apparent contradictions. But it 

 would be a misfortune if this process were to absorb too 

 large a share of the attention of mathematicians, or were 

 allowed to set too high a standard of logical completeness. 

 In this particular matter of the " arithmetisation of Mathe- 

 matrcs " there is, I think, a danger in these respects. As 

 resrards the latter point, a traveller who refuses to pass over 

 a bridge until he has personally tested the soundness of 

 every part of it is not likely to go verv far ; something must 

 be risked, even in Mathematics. It 'is notorious that even 

 in this realm of " exact " thought discovery has often been 

 in advance of strict logic, as in the theory of imaginaries, 

 for example, and in the whole province of analysis of which 

 Fourier's theorem is the type. And it might even be claimed 

 that the services which Geometry has rendered to other 

 sciences have been almost as great' in virtue of the questions 

 which It implicitly begs as of those which it resolves. 



I would venture, with some trepidation, to go one step 

 further. Mathematicians love to build on as definite a foun- 

 dation as possible, and from this point of view the notion 

 of the integral number, on which (we are told) the Mathe- 

 matics of the future are to be based, is very attractive. 

 But, as an instrument for the studv of Nature, is it 

 really more fundamental than the geometrical notions which 

 it is to supersede? The accounts of primitive peoples 

 would seem to show that, in the generality which is a 

 necessary condition for this purpose, it is in no less degree 

 artificial and acquired. Moreover, does not the act of 

 enumeration, as applied to actual things, involve the very 

 same process of selection and idealisation which we have 

 already met with in other cases? As an illustration, 

 suppose we were to try to count the number of drops of 

 water in a cloud. I am not thinking of the mere practical 

 difficulties of enumeration, or even of the more pertinent 

 fact that it is hard to say where the cloud begins or ends. 

 Waiving these points, it is obvious that there must be 

 transitional stages between a more or less dense group of 

 molecules and a drop, and in the case of some of these 



NO. t8i6, vol. 70] 



aggregates it would only be by an arbitrary exercise of 

 judgment that they would be assigned to one category 

 rather than to the other. In whatever form we meet with 

 it, the very notion of counting involves the highly artificial 

 conception of a number of objects which for some purposes 

 are treated as absolutely alike, whilst yet thev can be 

 distinguished. 



The net result of the preceding survey is that the systems 

 of Geometry, of Mechanics, and even of Arithmetic, on 

 which we base our study of Nature, are all contrivances 

 of the same general kind : they consist of series of 

 abstractions and conventions devised to represent, or rather 

 to symbolise, what is most interesting and most accessible 

 to us in the world of phenomena. And the progress of 

 science consists in a great measure in the improvement, 

 the development, and the simplification of these artificial 

 conceptions, so that their scope may be wider and the 

 representation more complete. The best in this kind are 

 but shadows, but we may continually do something to 

 amend them. 



As compared with the older view, the function of 

 physical science is seen to be much more modest than was 

 at one time supposed. We no longer hope by levers and 

 screws to pluck out the heart of the mystery of the 

 universe. But there are compensations. The conception of 

 the physical world as a mechanism, constructed on a rigid 

 mathematical plan, whose most intimate details might 

 possibly some day be guessed, was, I think, somewhat 

 depressing. We have been led to recognise that the formal 

 and mathematical element is of our own introduction ; that 

 it is merely the apparatus by which we map out our 

 knowledge, and has no more objective reality than the 

 circles of latitude and longitude on the sun. A distinguished 

 writer not very long ago speculated on the possibility of 

 the scientific mine being worked out within no distant 

 period. Recent discoveries seem to have put back this 

 possibility indefinitely ; and the tendency of modern specu- 

 lation as to the nature of scientific knowledge should be 

 to banish it altogether. The world remains a more 

 wonderful place than ever ; we may be sure that it abounds 

 in riches not yet dreamed of ; and although we cannot hope 

 ever to explore its innermost recesses, we may be confident 

 that it will supply tasks in abundance for the scientific mind 

 for ages to come. 



One significant result of the modern tendency is that we 

 no longer with the same obstinacy demand a mechanical 

 explanation of the phenomena of Light and Electricity, 

 especially since it has been made clear that if one 

 mechanical explanation is possible, there will be an infinity 

 of others. Some minds, indeed, revelling in their new- 

 found freedom, have attempted to disestablish ordinary or 

 " vulgar " matter altogether. I may refer to a certain 

 treatise which, by some accident, does not bear its proper 

 title of " j^ither and no Matter," and to the elaborate 

 investigations of Prof. Osborne Reynolds, which present 

 the same peculiarity, although the basis is different. 

 Speculations of this nature have, however, been so 

 recently and (if I may say it) so brilliantly dealt with by 

 Prof. Povnting before this Section that there is little 

 excuse for dwelling further on them now. I will only 

 advert to the question whether, as some suggest, physical 

 science should definitely abandon the attempt to construct 

 mechanical theories in the older sense. The question would 

 appear to be very similar to this, whether we should 

 abandon the use of graphical methods in analysis. In 

 either case we run the risk of introducing extraneous 

 elements, possibly of a misleading character ; but the gain 

 in vividness of perception and in suggestiveness is so great 

 that we are not likely altogether to forego it, by excess 

 of prudence, in one case more than in the other. 



We have travelled some distance from Stokes and the 

 mathematical physics of half a century ago. May I add 

 a few observations which might perhaps have claimed 

 his sympathy? They are in substance anything but new, 

 although I do not find them easy to express. We have 

 most of us frankly adopted the empirical attitude in 

 physical science ; it has justified itself abundantly in the 

 past, and has more and more forced itself upon us. We 

 have given up the notion of causation, except as a 

 convenient phrase ; what were once called laws of Nature 

 are now simply rules by which we can tell more or less 



