September 23. 1910] 



SCIENCE 



399 



his most important mathematical discover- 

 ies.^ He describes the process of discovery 

 as consisting of three stages : the first of 

 these consists of a long effort of concen- 

 trated attention upon the problem in hand 

 in all its bearings; during the second 

 stage he is not conseioiisly occupied with 

 the subject at all, but at some quite unex- 

 pected moment the central idea which en- 

 ables him to surmount the difficulties, the 

 natui'e of which he had made clear to him- 

 self during the first stage, flashes suddenly 

 into his consciousness. The third stage 

 consists of the work of carrying out in de- 

 tail 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 de- 

 ductive process; it is assuredly more in- 

 teresting 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 be- 

 ing, not a logic-engine. Who that has 

 studied the works of such men as Buler, 

 Lagrange, Cauchy, Riemann, Sophus Lie 

 and Weierstrass can doubt that a great 

 mathematician is a great artist? The fac- 

 ulties possessed by such men, varying 

 greatly in kind and degree with the indi- 

 vidual, are analogous to those requisite for 

 constructive art. Not every great mathe- 

 matician possesses in a specially high de- 

 gree that critical faculty which finds its 

 employment in the perfection of form, in 

 conformity with the ideal of logical com- . 

 pleteness; but every great mathematician 

 possesses the rarer faculty of constructive 

 imagination. 



The actual evolution of mathematical 

 theories proceeds by a process of induction 



' See the Revue du Hois for 1908. 



strictly analogous to the method of induc- 

 tion employed in building up the physical 

 sciences; observation, comparison, classifi- 

 cation, trial and generalization are essential 

 in both cases. Not only are special results, 

 obtained independently of one another, 

 frequently seen to be really included in 

 some generalization, but branches of the 

 subject which have been developed quite 

 independently of one another are some- 

 times found to have connections which en- 

 able them to be synthesized in one single 

 body of doctrine. The essential nature of 

 mathematical thought manifests itself in 

 the discernment of fundamental identity 

 in the mathematical aspects of what are 

 superficially very different domains. A 

 striking example of this species of imma- 

 nent identity of mathematical form was 

 exhibited by the discovery of that distin- 

 guished mathematician, our general secre- 

 tary. 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 

 recognized 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 



