520 TRANSACTIONS OF SECTION A. 



ideal of logical completeness ; but every great mathematician possesses the rarer 

 faculty of constructive imagination. 



The actual evolution of mathematical theories proceeds by a process of induc- 

 tion strictly analogous to the method of induction employed in building up the 

 physical sciences ; observation, comparison, classification, trial, and generalisation 

 are essential in both cases. Not only are special results, obtained independently 

 of one another, frequently seen to be really included in some generalisation but 

 branches of the subject which have been developed quite independently of one 

 another are sometimes found to have connections which enable them to be 

 synthesised in one single body of doctrine. The essential nature of mathematical 

 thought manifests itself in the discernment of fundamental identity in the mathe- 

 matical aspects of what are superficially very different domains. A striking 

 example of this species of immanent identity of mathematical form was exhibited 

 by the discovery of that distinguished mathematician, our General Secretary, 

 Major Macmahon, that all possible Latin squares are capable of enumeration by 

 the consideration of certain differential operators. Here we have a case in which 

 an enumeration, which appears to be not amenable to direct treatment, can 

 actually be carried out in a simple manner when the underlying identity of the 

 operation is recognised with that involved in certain operations due to differential 

 operators, the calculus of which belongs superficially to a wholly different region 

 of thought from that relating to Latin squares. The modern abstract theory of 

 groups affords a very important illustration of this point; all sets of operations, 

 whatever be their concrete character, which have the same group, are from the 

 point of view of the abstract theory identical, and an analysis of the properties of 

 the abstract group gives results which are applicable to all the actual sets of 

 operations, however diverse their character, which are dominated by the one 

 group. The characteristic feature of any special geometrical scheme is known 

 when the group of transformations which leave unaltered certain relations of 

 figures has been assigned. Two schemes in which the space elements may be quite 

 different have this fundamental identity, provided they have the same group; 

 every special theorem is then capable of interpretation as a property of figures 

 either in the one or in the other geometry. The mathematical physicist is familiar 

 with the fact that a single mathematical theory is often capable of interpretation 

 in relation to a variety of physical phenomena. In some instances a mathematical 

 formulation, as in some fashion representing observed facts, has survived the 

 physical theory it was originally devised to represent. In the case of electro- 

 magnetic and optical theory, there appears to be reason for trusting the equations, 

 even when the proper physical interpretation of some of the vectors appearing in 

 them is a matter of uncertainty and gives rise to much difference of opinion ; 

 another instance of the fundamental nature of mathematical form. 



One of the most general mathematical conceptions is that of functional rela- 

 tionship, or 'functionality.' Starting originally from simple cases such as a 

 function represented by a power of a variable, this conception has, under the 

 pressure of the needs of expanding mathematical theories, gradually attained the 

 completeness of generality which it possesses at the present time. The opinion 

 appears to be gaining ground that this very general conception of functionality, 

 born on mathematical ground, is destined to supersede the narrower notion of 

 causation, traditional in connection with the natural sciences. As an abstract 

 formulation of the idea of determination in its most general sense, the notion of 

 functionality includes and transcends the more special notion of causation as a 

 one-sided determination of future phenomena by means of present conditions ; it 

 can be used to express the fact of the subsumption under a general law of past, 

 present, and future alike, in a sequence of phenomena. From this point of view 

 the remark of Huxley that Mathematics ' knows nothing of causation ' could only 

 be taken to express the whole truth, if by the term ' causation ' is understood 

 'efficient causation.' The latter notion has, however, in recent times been to an 

 increasing extent regarded as just as irrelevant in the natural sciences as it is in 

 Mathematics; the idea of thorough-going determinancy, in accordance with 

 formal law, being thought to be alone significant in either domain. 



The observations I have made in the present address have, in the main, had 

 reference to Mathematics as a living and growing science related to and per- 

 meating other great departments of knowledge. The small remaining space at 



