DEVELOPMENT OF MATHEMATICAL THOUGHT. 635 



our century, among whom I only mention Gauss, Cauchy, 

 and Weierstrass, attempted to do for the new science 

 which was created during the two preceding centuries. 

 As Prof. Klein says, " We are living in a critical period, 

 similar to that of Euclid." l 



1 See ' The Evanston Colloquium, 

 Lectures on Mathematics delivered 

 in August and September 1893,' by 

 Felix Klein, notably Lecture vi. 

 In this lecture Prof. Klein explains 

 his view (to which he had given 

 utterance in his address before the 

 Congress of Mathematics at Chicago : 

 ' Papers published by the American 

 Mathematical Society,' vol. i. p. 

 133. New York, 1896) on the 

 relation of pure mathematics to 

 applied science. This view is based 

 upou the distinction between what 

 he calls the " naive and the refined 

 intuition." . . . " It is the latter 

 that we find in Euclid ; he carefully 

 develops his system on the basis of 

 well - formulated axioms, is fully 

 conscious of the necessity of exact 

 proofs, clearly distinguishes be- 

 tween the commensurable and the 

 incommensurable, and so forth. . . . 

 The naive intuition, on the other 

 hand, was especially active during 

 the period of the genesis of the 

 differential and integral calculus. 

 Thus we see that Newton assumes 

 without hesitation the existence, in 

 every case, of a velocity in a mov- 

 ing point, without troubling him- 

 self with the inquiry whether 

 there might not be continuous 

 functions having no derivative." 



In the opinion of Prof. Klein 

 " the root of the matter lies in the 

 fact that the naive intuition is not 

 exact, while the refined intuition is 

 not properly intuition at all, but 

 arises through the logical develop- 

 ment from axioms considered as 

 perfectly exact." 



In the sequel Prof. Klein shows 

 that the naive intuition imports 



into the elementary conceptions 

 elements which are left out in the 

 purely logical development, and that 

 this again leads to conclusions which 

 are not capable of being verified by 

 intuition, no mental image being 

 possible. Thus, for instance, the 

 abstract geometry of Lobatchevsky 

 and Biemann led Beltrami to the 

 logical conception of the pseudo- 

 sphere of which we cannot form 

 any mental image. Similar views 

 to those of Prof. Klein have been 

 latterly expressed by H. Poincar^ 

 in his suggestive volume ' La 

 Science et 1'Hypothese ' (Paris, 

 1893). He there says (p. 90) :. 

 "... L'expe>ience joue un role 

 indispensable dans la genese de la 

 geometric ; mais ce serait une 

 erreur d'en conclure que la geo- 

 me'trie est une science experi- 

 mentale, merne en partie. ... La 

 ge'om^trie ne serait que I'e'tude des 

 mouvements des solides ; mais elle 

 ne s'occupe pas en realite" des solides 

 naturels, elle a pour objet certains 

 solides ideaux, absolument invari- 

 ables, qui ii'en sont qu'une image 

 simplifi^e et bien lointaine. . . . Ce 

 qui est 1' objet de la geometric c'est 

 1'^tude d'un ' groupe ' particulier ; 

 mais le concept gtSneYal de groupe 

 pre'existe dans notre esprit au 

 mojns en puissance. . . . Seule- 

 ment, parmi tous les groupes 

 possibles, il faut choisir celui qui 

 sera pour ainsi dire 1'etalon auquel 

 nous rapporterons les phdnomenes 

 naturels." This distinction be- 

 tween the mathematics of intuition 

 and the mathematics of logic has 

 also been forced upon us from quite 

 a different quarter. The complica- 



