36 



NATURE 



[September 9, 191 5 



repeated efforts were made since the year 1640 to 

 establish a University in Manchester; it was not for 

 reasons of material gain that the Royal Institution and 

 Owens College were founded ; nor was it because they 

 increased the wealth of the district that the place of 

 honour in our Town Hall has been given to Dalton 

 and Joule. 



When we glance at the various occupations of the 

 working parts of a nation, comprising the student who 

 accumulates or extends knowledge, the engineer who 

 applies that knowledge, the geologist or agriculturist 

 who discloses the store of wealth hidden in the soil, 

 the commercial man who distributes that wealth, it 

 seems as if we ought to be able to name the qualities 

 of intellect and temperament which in each pursuit are 

 most needed to carry out the work successfully. But 

 on trying to define these qualities we soon discover 

 the formidable nature of the task. Reasoning power, 

 inventive power, and sound balance of judgment are 

 essential attributes in all cases, and the problem is 

 reduced to the question whether there are different 

 varieties of the attributes which can be assigned to 

 the different occupations. 



Among all subjects mathematics Is perhaps the one 

 that appears most definitely to require a special and 

 uncommon faculty. Yet, Polncar^ — himself one of the 

 clearest thinkers and most brilliant exponents of the 

 subject — almost failed when he attempted to fix the 

 distinguishing Intellectual quality of the mathe- 

 matician. Starting from the Incontrovertible proposi- 

 tion that there is only one kind of correct reasoning, 

 which is logical reasoning, he raises the question why 

 it is that everybody who is capable of reasoning cor- 

 rectly Is not also a mathematician, and he is led to 

 the conclusion that the characterising feature is a 

 peculiar type of memory. It Is not a better memory, 

 for some mathematicians are very forgetful, and many 

 of them cannot add a column of figures correctly; but 

 It Is a memory which fixes the order in which the 

 successive steps of reasoning follow each other with- 

 out necessarily retaining the details of the individual 

 steps. This Polncare illustrates by contrasting the 

 memory of a chess-player with that of a mathe- 

 matician. "When I play chess," he says, "I reason 

 out correctly that If I were to make a certain move, 

 I should expose myself to a certain danger. I should, 

 therefore, consider a number of other moves, and, 

 after rejecting each of them in turn, I should end 

 by making the one which I first contemplated and dis- 

 missed, having forgotten in the meantime the ground 

 on which I had abandoned it." "Why, then," he 

 continues, "does my memory not fail me in a difficult 

 mathematical reasoning in which the majority of chess- 

 players would be entirely lost? It Is because a mathe- 

 matical demonstration is not a juxtaposition of syllog- 

 isms, but consists of syllogisms placed in a certain 

 order; and the order in which Its elements are placed 

 is much more important than the elements themselves. 

 If I have this intuition — so to speak — of the order, so 

 as to perceive at one glance the whole of the reason- 

 ing, I need not fear to forget its elements : each of 

 these will take its right place of Its own accord 

 without making any call on my memory."* 



Polncar^ next discusses the nature of the intellectual 

 gift distinguishing those who can enrich knowledge 

 with new and fertile ideas of discovery. Mathematical 

 invention, according to him, does not consist in form- 

 ing new combinations of known mathematical entities, 

 because the number of combinations one could form 

 are Infinite, and most of them would possess no in- 

 terest whatever. Inventing consists, on the contrary, 

 in excluding useless combinations, and therefore : — 

 "To invent is to select — to choose., . . .The ex- 



5 " Science et Mithode." pp. 46 and 47. 



NO. 2393, VOL. 96] 



pression ' choose ' perhaps requires qualifying, because 

 it recalls a buyer to whom one offers a large number 

 of samples which he examines before making his 

 choice. In our case the samples would be so numer- 

 ous that a lifetime would not suffice to complete the 

 examination. That Is not the way things are done. 

 The sterile combinations never present themselves to 

 the mind of the inventor, and even those which 

 momentarily enter his consciousness, only to be re- 

 jected, partake something of the character of useful 

 combinations. The inventor Is therefore to be com- 

 pared with an examiner who has only to deal with 

 candidates who have already passed a previous test 

 of competence." 



All those who have attempted to add something to 

 knowledge must recognise that there is a profound 

 truth in these remarks. New ideas may float across 

 our consciousness, but, selecting the wrong ones for 

 more detailed study, we waste our time fruitlessly. 

 We are bewildered by the multitude of roads which 

 open out before us, and, like Polncare when he tries 

 to play chess, lose the game because we make the 

 wrong move. Do we not all remember how, after 

 the announcement of a new fact or generalisation, 

 there are always many who claim to have had, and 

 perhaps vaguely expressed, the same idea? They put 

 It down to bad luck that they have not pursued it, but 

 they have failed precisely In what, according to 

 Polncar^, Is the essence of Inventive power. It may 

 be bad luck not to have had a good idea, but to have 

 had It and failed to appreciate its importance is down- 

 right Incapacity. 



An objection may be raised that before a selection 

 can be made the ideas themselves must appear, and 

 that, even should they arrive in sufficient numbers, 

 the right one may not be among them. It may even 

 be argued that Poincar^ gives his case away by saying 

 that " the sterile combinations do not even present 

 themselves to the mind of the Inventor," expressing 

 In a , negative . form what may be the essence of the 

 matter. Moreover, a fertile mind like that of 

 Polncare would be apt to place too low a value on 

 his own exceptional gifts. Nevertheless, If Poincar^'s 

 more detailed exposition be read attentively, and more 

 especially the description of how the discoveries which 

 made him famous among mathematicians originated 

 In his mind, it will be found that his judgment is well 

 considered and should not be lightly set aside. New 

 Ideas seldom are born out of nothing. They most 

 frequently are based on analogies, or the recollection 

 of a sequence of thoughts suggested by a different 

 branch of the subject, or perhaps by a different subject 

 altogether. It is here that the memory comes In, 

 which is not a memory of detail, but a memory of 

 premises with their conclusions, detached from the 

 particular case to which they were originally applied. 

 Before we pronounce an adverse opinion on Polncar^'s 

 judgment, we must Investigate what constitutes 

 novelty in a new idea; but the subject is too vast to 

 be dealt with here, nor can I attempt to discuss 

 whether an essential distinction exists between 

 mathematical Invention and that more practical form 

 of invention with which, for instance, the engineer 

 has to deal. 



If Poincar^, by this introspective analysis of his 

 own powers, has dimmed the aureole which. In the 

 eyes of the public, surrounds the mathematician's 

 head, he removes It altogether by his definition of 

 mathematics. According to him, "mathematics is 

 the art of calling two different things by the same 

 name." It would take me too far were I to try to 

 explain the deep truth expressed in this apparently 

 'flippant form: physicists, at any rate, will remember 

 the revolution created in the fundamental outlook of 

 science by the application-, pf the 4:erm. "energy" to 



