654 PSYCHOLOGY. 



which all modes of grouping or combining simply give 

 us back unchanged. In short, combinations of numbers 

 are combinations of their units, which is the fundamental 

 axiom of arithmetic,* leading to such consequences as that 

 7 -|- 5 = 8 -f- 4 because both = 12. The general axiom of 

 mediate equality, that equals of equals are equal, conies in 

 here.t The princij)le of constancy in our meanings, when 

 applied to strokes of counting, also gives rise to the axiom 

 that the same number, operated on (interrupted, grouped) 

 in the same way will always give the same result or be the 

 same. How shouldn't it ? Nothing is supposed changed. 

 Arithmetic and its fundamental principles are thus in- 

 depenilent of our experiences or of the order of the world. 

 The matter of arithmetic is mental matter; its principles flow 

 from the fact that the matter forms a series, which can be cut 

 into by us wherever we like without the matter changing. 

 The empiricist school has strangely tried to interjjret the 

 truths of number as results of coexistences among out- 

 ward things. John Mill calls number a physical property 

 of things. ' One,' according to Mill, means one sort of 

 passive sensation which we receive, ' two ' another, ' three ' 

 a third. The same things, however, can give us different 

 number-sensations. Three things arranged thus, ^ ^ ^ , for 

 example, impress us differently from three things arranged 

 thus, °„°. Eat experience tells us that every real object-group 

 which can be arranged in one of these ways can always be 

 arranged in the other also, and that 2 + 1 and 3 are thus 

 modes of numbering things which ' coexist ' invariably with 

 each other. The indefeasibility of our belief iui their ' co- 

 existence ' (which is Mill's word for their equivalence) is 

 due solely to the enormous amount of experience we have 

 of it. For all things, whatever other sensations they may 

 give us, give us at any rate number-sensations. Those 

 number-sensations which the same thing may be suc- 

 cessively made to arouse are the numbers which we deem 



* Said to be expressed by Grassraaa in the fundamental Axiom of 

 Arithmetic (a + 6) -|- 1 = a + ( ^' + 1). 



f Compare Helmholtz's more technically expressed Essay 'ZShlen u. 

 Messeu,'in the Philosophische Aufsatze, Ed. Zeller gewidmet (Leipzig, 

 1887), p. 17. 



