400 



SCIENCE 



[N. S. Vol. XLII. No. 1082 



Chester; 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 honor in our Town Hall 

 has been given to Dalton and Joule. 



When we glance at the various occupa- 

 tions of the working parts of a nation, 

 comprising the student who accumulates or 

 extends knowledge, the engineer who ap- 

 plies that knowledge, the geologist or agri- 

 culturist 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 formid- 

 able 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 vari- 

 eties of the attributes which can be assigned 

 to the different occupations. 



Among all subjects mathematics is per- 

 haps the one that appears most definitely to 

 require a special and uncommon faculty. 

 Yet, Poincare — himself one of the clearest 

 thinkers and most brilliant exponents of the 

 subject — almost failed when he attempted 

 to fix the distinguishing intellectual qual- 

 ity of the mathematician. Starting from 

 the incontrovertible proposition 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 correctly is not also a mathema- 

 tician, and he is led to the conclusion that 

 the characterizing feature is a peculiar type 

 of memory. It is not a better memory, for 

 some mathematicians are very forgetful, 

 and many of them can not 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 without nec- 

 essarily retaining the details of the indi- 

 vidual steps. This Poincare illustrates by 

 contrasting the memory of a chess-player 

 with that of a mathematician. "When I 

 play chess," he says, "I reason out cor- 

 rectly 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 difiicult mathe- 

 matical reasoning in which the majority of 

 chess-players would be entirely lost? It is 

 because a mathematical demonstration is 

 not a juxtaposition of syllogisms, but con- 

 sists 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 intui- 

 tion — so to speak — of the order, so as to per- 

 ceive at one glance the whole of the rea- 

 soning, I need not fear to forget its ele- 

 ments: each of these will take its right 

 place of its own accord without making any 

 call on my memory. ' ' " 



Poincare next discusses the nature of the 

 intellectual gift distinguishing those who 

 can enrich knowledge with new and fertile 

 ideas of discovery. Mathematical inven- 

 tion, according to him, does not consist in 

 forming new combinations of known mathe- 

 matical entities, because the number of 

 combinations one could form are infinite, 

 and most of them would possess no interest 

 whatever. Inventing consists, on the con- 

 trary, in excluding useless combinations, 

 and therefore: "To invent is to select — to 

 choose." . . . "The expression 'choose' 



6 ' ' Science et M^thode, ' ' pp. 46 and 47. 



