DEVELOPMENT OF MATHEMATICAL THOUGHT. 739 



also such apparently algebraical notions as those of 

 irrational and complex quantities. This attempt is an 

 outcome of the school of Weierstrass, which has done 

 so much to banish vagueness and introduce precision 

 into modern text-books. 



Opposed to this so-called arithmetising l tendency is 

 the equally emphatic view, strongly urged by the late 

 Prof. Paul Du Bois-Eeymond in his general theory of 

 Functions, that the separation of the operations of 

 counting and measuring is impossible, and, if it were 

 possible (as, since the publication of his work, the fuller 

 expositions of Kronecker and his followers have tried to 

 show that it is), would degrade mathematics to a mere 

 play with symbols. 2 He tries to show that such is philo- 

 sophically impossible, and finds a support for his view in 

 the historical genesis of the idea of irrational numbers in 

 the incommensurable magnitudes of Euclid and ancient 

 geometry. Prof. Klein in his address favours the 

 arithmetical tendency as destined to introduce logical 



1 The term seems to have been be attached to the signs as if they 



coined by Kronecker. See Prof. were the figures tin the chessboard 



Pringsheim in the ' Encyklop. or on playing-cards. However amus- 



Math. Wiss.,' vol. i. p. 58, note 40. ! ing such a play might be, nay, 



Krouecker's position is set forth ; however useful for analytical pur- 



in Journal fiir Math., vol. ci. pp. ' poses the solution would be of the 



337-355, 1887. problem, to follow up the rules of 



"The separation of the con- 

 ception of number and of the 

 analytical symbols from the con- 

 ception of magnitude would reduce 

 analysis to a mere formal and 



the signs which emanated from the 

 conception of magnitude into their 

 last formal consequences, such a 

 literal mathematics would soon 

 exhaust itself in fruitless efforts ; 



literal skeleton. It would degrade I whereas the science which Gauss 



this science, which in truth is a called with so much truth the 



natural science, although it only I science of magnitude possesses an 



admits the most general properties inexhaustible source of new ma- 



of what we perceive into the domain ,, ! terial in the ever-increasing field 



of its researches ultimately to the : of actual perceptions," &c., &c. 



rank of a mere play with symbols, ; ( ' Allgenieine Fuuctionen-Theorie,' 



wherein arbitrary meanings would 1882, p. 54). 



