ORDER, NUMBER, QUANTITY 265 



a and b is defined, if, corresponding to the elements a and b 

 and to a certain order of those elements, there exists a certain 

 third thing c." 1 Thus the operation of addition upon a 

 3-group and a 2-group is defined, if, when the groups are 

 taken in the given order there is determined a third group, 

 viz., a 5-group; this operation is symbolized in the ordinary 

 way, 3 +2 = 5. It seems important to recognize that certain 

 elemental operations — addition, subtraction, and possibly 

 also multiplication — are intimately tied up with the recogni- 

 tion of the simple numbers. As such they come definitely 

 into the empirical picture, and are considered inseparable 

 from the descriptive concept of number. Unless this is 

 recognized a difficulty arises in the understanding of the 

 scientific concept, for the latter makes important use of the 

 concept of operation. 



On the empirical level, then, numbers may be considered 

 as collective properties of certain complex events which can 

 be correlated with one another. What seems important to 

 recognize is that the pluralities here considered have a 

 complexity of not more than six or seven terms, and not less 

 than two terms. At least groups of greater complexity can 

 probably not be recognized by a simple act of attention, 

 but must be built up by counting or by operations upon 

 smaller groups. And groups of lesser complexity could not, 

 according to the empirical conception, be groups. This sug- 

 gests that the notion 1 is somewhat more abstract than the 

 notions 2 ... 6, and hence must be defined by logical 

 extension from the latter group. The notion is clearly one 

 involving complicated acts, and arises only very late in the 

 psychological development. The notions of the negative 

 numbers, the fractions, and the irrationals arise by similar 

 extension, hence are not to be examined at this level of 

 analysis. Mention may be made further of the fact that the 

 aggregate of complexes which constitutes the empirical 

 reference of the concept of number has not yet been ordered; 



1 J. W. Young. Fundamental Concepts of Algebra and Geometry (New York: Mao 

 millan, 1927), p. 88. 



