266 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



in other words, one does not yet have the series 2, 3, 4, 5, 6, 

 but merely the collection 2, 5, 4, 6, 3. One is, therefore, not 

 yet able to count. 



number: operational derivation 



The concept of number, like the concept of order, accom- 

 plishes its task in science through generalization of its 

 empirical foundation. But whereas this was attained in the 

 case of order through a direct generalization of the concept 

 itself, it is brought about in the case of number through a 

 generalization of the notion of operation. 



It should be apparent that on the empirical level the no- 

 tion of mathematical operation involves the conception of 

 illegitimate operations. While addition may always be per- 

 formed, since it produces a further complex which may then 

 be presumed to exhibit number, subtraction often cannot be 

 performed; for example, a 2-group cannot be subtracted 

 from a 3-group, for what would remain would not be a group 

 at all, hence could not be numbered; furthermore, a 3-group 

 could not be subtracted from a 2-group for the act could 

 not be performed. Similarly, though multiplication is 

 always possible, division is generally impossible. Hence a 

 proper formulation of the empirical conception would re- 

 quire a classification of operations on the basis of legitimacy 

 and illegitimacy. It is this distinction which is abandoned 

 at the scientific level. The generalization of the concept of 

 number is accomplished primarily by the elimination of 

 illegitimate operations. If mathematical operations can 

 always be performed, the outcome of an operation will 

 always be a number. As a result of this act of generalization 

 place must be found for 1, and the negative numbers, 

 which arise through subtraction; for the fractions, which 

 arise through division; and for the irrationals and the 

 imaginaries, which arise through extraction of roots. This is 

 accomplished, as will be seen immediately, through the 

 introduction of the notion of group. 



But a further transformation is sometimes introduced 



