290 



NATURE 



[September i, 1910 



ployed to denotf any entities whatever which satisfy certain 

 prescribed conditions of relationship. Various premises are 

 postulated that would appear to be of a perfectly arbitrary 

 nature, if we did not know how they had been suggested. 

 In that division of the subject known as metric geometry, 

 for example, axioms of congruency are assumed which, by 

 their purely abstract character, avoid the very real diffi- 

 culties that arise in this regard in reducing perceptual 

 space-relations of measurements to a purely conceptual 

 form. Such schemes, triumphs of constructive thought at 

 its highest and most abstract level as they are, could never 

 have been constructed apart from the space-perceptions that 

 suggested them, although the concepts of spatial origin are 

 transformed almost out of recognition. But what I want 

 to direct attention to here is that, apart from the basis of 

 this geometry, mathematicians would never have been able 

 to find their way through the details of the deductions 

 without having continual recourse to the guidance given 

 them by spatial intuition. If one attempts to follow one 

 of the demonstrations of a particular theorem in the work 

 of writers of this school, one would find it quite impossible 

 to retain the steps of the process long enough to master the 

 whole, without the aid of the very spatial suggestions 

 which have been abstracted. This is perhaps sufficiently 

 warranted by the fact that writers of this school find it 

 necessary to provide their readers with figures, in order to 

 avoid complete bewilderment in following the demonstra- 

 tions, although the processes, being purely logical deductions 

 from premises of the nature I have described, deal only 

 with entities which have no necessary similarity to anything 

 indicated by the figures. 



A most interesting account has been written by one of the 

 greatest mathematicians of our time, M. Henri Poincar^, 

 of the way in which he was led to some of his most 

 important mathematical discoveries.' He describes the pro- 

 cess of discovery as consisting of three stages : the first of 

 these consists of a long effort of concentrated attention 

 upon the problem in hand in all its bearings ; during the 

 second stage he is not consciously occupied with the subject 

 at all, but at some quite unexpected moment the central 

 idea which enables him to surmount the difficulties, the 

 nature of which he had made clear to himself during the 

 first stage, flashes suddenly into his consciousness. " The 

 third stage consists of the work of carrying out in detail 

 and reducing to a connected form the results to which he 

 is led by the light of his central idea : this stage, like the 

 first, is one requiring conscious effort. This is, I think, 

 clearly not a description of a purely deductive process ; it 

 is assuredly more interesting to the psychologist than to 

 the logician. We have here the account of a complex of 

 mental processes in which it is certain that the reduction 

 to a scheme of precise logical deduction is the latest stage. 

 After all, a mathematician is a human being, not a logic- 

 engine. Who that has studied the works of such men as 

 Euler, Lagrange, Cauchy, Ricmann, Sophus Lie, and 

 Weierstrass, can doubt that a great mathematician is a 

 great artist? The faculties possessed bv such men, varying 

 greatly in kind and degree with the individual, are 

 analogous to those requisite for constructive art. Not 

 every great mathematician possesses in a specially high 

 degree that critical faculty which finds its employment in 

 the_ perfection of form, in conformity with the ideal of 

 logical completeness ; but every great mathematician pos- 

 sesses the rarer faculty of constructive imagination. 



The actual evolution of mathematical theories proceeds 

 by a process of induction strictly analogous to the method 

 of induction employed in building up the phvsical sciences ; 

 observation, comparison, classification, trial, and generalisa- 

 tion 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 svnthesised in one 

 single body of doctrine. The essential 'nature of mathe- 

 matical thought manifests itself in the discernment of 

 fundamental identity in the mathematical aspects of what 

 are superficially very different domains. A striking example 

 of tliis .species of immanent identity of mathematical form 

 was exhibited by the discoverv of that distinguished 



> See the 



1 90S. 



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 im- 

 portant 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 interpreta- 

 tion 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 electromagnetic 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 relationship, 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 deter- 

 mination of future phenomena by means of present con- 

 ditions ; it can be used to express the fact of the subsump- 

 tion 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 under- 

 stood "efficient causation." The latter notion has, 

 however, in recent times been to an increasing extent 

 regarded as jijst as irrelevant in the natural sciences 

 as it is in Mathematics ; the idea of thorough-going deter- 

 minancy, 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 permeating other great 

 departments of knowledge. The small remaining space at 

 my disposal I propose to devote to a few words about some 

 matters connected with the teaching of the more elementary 

 parts of Mathematics. Of late years a new spirit has 

 come over the mathematical teaching in many of our insti- 

 tutions, due in no small measure to the reforming zeal of 

 our General Treasurer, Prof. John Perry. The changes 

 that have been made followed a recognition of the fact that 

 the abstract mode of treatment of the subject that had 

 been traditional was not only wholly unsuitable as a train- 

 ing for physicists and engineers, but was also to a large 

 extent a failure in relation to general education, because 

 it neglected to bring out clearly the bearing of the subject 

 on the concrete side of things. With the general principle 

 that a much less abstract mode of treatment than was 



