398 



SCIENCE 



[N. S. Vol. XXXII. No. 821 



more identical with a mere complex of 

 syllogisms than a melody is identical with 

 the mere sum of the musical notes em- 

 ployed in its composition. In each case 

 the whole is something more than merely 

 the sum of its parts; it has a unity of its 

 own, and that imity must be, in some meas- 

 ure at least, discerned by its creator before 

 the parts fall completely into their places. 

 Logic is, so to speak, the grammar of 

 mathematics ; but a knowledge of the rules 

 of grammar and the letters of the alphabet 

 would not be sufficient equipment to enable 

 a man to write a book. There is much 

 room for individuality in the modes of 

 mathematical discovery. Some great 

 mathematicians have employed largely 

 images derived from spatial intuition as a 

 guide to their results; others appear 

 wholly to have discarded such aids, and 

 were led by a fine feeling for algebraic and 

 other species of mathematical form. A 

 certain tentative process is common, in 

 which, by the aid of results known or ob- 

 tained in special cases, generalizations are 

 perceived and afterwards established, 

 which take up into themselves all the 

 special cases so employed. Most mathe- 

 maticians leave some traces, in the final 

 presentation of their work, of the scaffold- 

 ing they have employed in building their 

 edifices: some much more than others. 



The difference between a mathematical 

 theory in the making and as a finished 

 product is, perhaps, most strikingly illus- 

 trated by the case of geometry, as pre- 

 sented in its most approved modern shape. 

 It is not too much to say that geometry, re- 

 duced to a purely deductive form — as pre- 

 sented, for example, by Hilbert, or by some 

 of the modern Italian school — has no neces- 

 sary connection with space. The words 

 "point," "line," "plane" are employed 

 to denote any entities whatever which sat- 

 isfy certain prescribed conditions of rela- 



tionship. 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 as- 

 sumed which, by their purely abstract 

 character, avoid the very real difficulties 

 that arise in this regard in reducing per- 

 ceptual 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 rec- 

 ognition. But what I want to call atten- 

 tion 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 hav- 

 ing continual recourse to the guidance 

 given them by spatial intuition. If one at- 

 tempts to follow one of the demonstra- 

 tions 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 sugges- 

 tions which have been abstracted. This 

 is perhaps sufficiently warranted by the 

 fact that writers of this school find it nec- 

 essary to provide their readers with figures, 

 in order to avoid complete bewilderment 

 in following the demonstrations, although 

 the processes, being purely logical deduc- 

 tions 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 mathemati- 

 cians of our time, M. Henri Poincare, of 

 the way in which he was led to some of 



