MATHEMATICAL PHYSICS. 519 



very attractive. But, as an instrument for the study 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 idealization 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 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 count- 

 ing involves the highly artificial conception of a number of objects 

 which for some purposes are treated as absolutely alike, whilst yet 

 they can be distinguished. 



The net result of the preceding survey is that the systems of geom- 

 etry, 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 symbolize, 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 mys- 

 tery of the universe. But there are compensations. The conception 

 of the physical world as a mechanism, constructed on a rigid mathe- 

 matical plan, whose most intimate details might possibly some day be 

 guessed, was, I think, somewhat depressing. We have been led to 

 recognize 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 lati- 

 tude 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. Eecent discoveries seem to have put back 

 this possibility indefinitely; and the tendency of modern speculation as 

 to the nature of scientific knowledge should be to banish it altogether. 



