DEVELOPMENT OF MATHEMATICAL THOUGHT. 635 



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

 and Weieistrass, 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." ^ 



^ See ' The Evanston Colloquium, 

 Lectures on Mathematics delivered 

 in August and September 1893,' by 

 Felix Klein, notably Lecture vi. 

 In this lecture Prof. Klein ex]>lains 

 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 

 upon 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 

 everj- ca.se, of a velocity in a mov- 

 ing point, without troubling himself 

 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 nut capable of being verified by 

 intuition, no mental image being 

 possible. Thus, for instance, the 

 abstract geometry of Lobatchev.sky 

 and Kiemann led Beltrami to the 

 logical conception of the pseudo- 

 sjdiere 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 I'Hypoth^se ' (Paris, 

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

 "... L'exp(5rience joue un role 

 indispensable dans la gencse de la 

 geometric ; mais ce serait une 

 erreur d'en conclure que la geo- 

 metric est une science experi- 

 mentale, meme en partie. . . . La 

 g(5ometrie ue serait que I'etude de.* 

 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 n'en sont qu'une image 

 simplifiee et bien lointaine. . . . Ce 

 qui est I'objet de la gcomclrie c'est 

 I'etude d'un ' groupe ' particulier ; 

 mais le concept gdndral de groupe 

 preexiste dans notre esprit au 

 moins en puissance. . . . Seule- 

 ment, parmi tons les groupes 

 jiossibles, il faut choisir celui qui 

 sera pour ain.^i dire I'dtalon autiuel 

 nous rapporterons les phenomenes 

 naturels." This distinction be- 

 tween tiie matliematics of intuition 

 and the mathematics of logic has 

 also been forced upon us from quite 

 a different quarter. The complica- 



