510 TRANSACTIONS OF SECTION A. 



great importance to 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 to the more technical aspects of the various departments of science. 

 Ample opportunities for the full discussion of all the detailed problems, the solu- 

 tion 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 make 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 depart- 

 ment 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 sense 

 of solidarity of men of science, to do something to counteract that tendency to 

 narrowness of view which is a. danger arising from increasing specialisation, are 

 functions the due performance of which may do much -to further that supreme 

 object, the advancement of science, for which the British Association exists. 



I propose to address to you a few remarks, necessarily fragmentary and incom- 

 plete, 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 Mathematics. Some of its more elementary concep- 

 tions and methods have become part of the common heritage of our civilisation, 

 interwoven in the everyday life of the people. Perhaps the greatest labour- 

 saving invention that the world has seen belongs to the formal side of Mathe- 

 matics; I allude to our system of numerical notation. This system which, when 

 scrutinised, affords the simplest illustration of the importance of Mathematical 

 form, has become so much an indispensable part of our mental furniture that 

 some effort is required to realise that an apparently so obvious idea embodies 

 a great invention ; one to which the Greeks, with their unsurpassed capacity for 

 abstract thinking, never attained. An attempt to do a multiplication sum in 

 Roman numerals is perhaps the readiest road to an appreciation of the advan- 

 tages of this great invention. In a large groun of sciences, the formal element, 

 the common language, so to speak, is supplied by Mathematics ; the range of the 

 application of mathematical methods and symbolism is ever increasing. Without 

 taking too literally (lie celebrated dictum of the great philosopher Kant, that 

 the amount of real science to be found in any special subject is the amount of 

 Mathematics contained therein, it must be admitted I hat each branch of science 

 which is concerned with natural phenomena, when it has reached a certain stage 

 of development, becomes 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 

 mathematical methods, at least on their statistical side. As a science emerges 

 from the stages in which it consists solely of more or less systematised descrip- 

 tions of the phenomena with which it is concerned in their more superficial 

 aspect ; when the intensive magnitudes discerned in the phenomena become re- 

 presentable 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 applicable to the sub- 

 ject to a still greater extent. 



Mathematics shares with the closely allied subject of Astronomy the honour 

 of being 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 altogether surprising. The com- 

 paratively 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. 



