264 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



tion. In a country where neither polyandry nor polygamy 

 is permitted, the number of husbands living at any moment 

 is the same as the number of wives. By means of this tech- 

 nique all pairs could be correlated, all triads, and so on. 

 The number 2 could then be empirically defined as the prop- 

 erty possessed in common by all classes similar to some given 

 class, say a pair of shoes; and the number 3 could be defined 

 as the property possessed in common by all classes similar to 

 another given class, say the sides of a triangle. 1 



This should suffice as an empirical definition of number. 

 By means of such a criterion one should be able to distin- 

 guish the various types of complex from one another, group 

 together all similar complexes, and devise the proper numeri- 

 cal symbol for the characterization of each type. But there 

 is also involved in the empirical conception of number a 

 feature which, I am convinced, must be considered as part 

 and parcel of it. The different types of complex are recog- 

 nized not only as being different in number from one another 

 but as being derivable from one another by certain mathe- 

 matical operations. It seems impossible that there should be 

 an awareness of the difference between a 3-group and a 

 5-group without a simultaneous awareness of the fact that 

 the 3-group would be similar to the 5-group if it had two more 

 members, i.e., if it were combined with a 2-group. This seems 

 to be the foundation for the notion of mathematical opera- 

 tion, which, in its general form, is a concept of great im- 

 portance for science. Reference has already been made to 

 the physical, biological, and psychological operations in- 

 volved in the perceptual act. 2 In general an operation may be 

 defined as an act performed upon something given to produce 

 something else. Mathematically, the notion of operation is 

 somewhat more specific; "an operation upon the elements 



1 This is in essence Russell's definition. He defines number as follows: "The 

 number of a class is the class of all those classes that are similar to it." The main 

 difference is that Russell places more emphasis on extension; since there may not be 

 any such property as number, while there obviously is such a thing as a class of 

 classes, he considers that the safest procedure is to adopt the extensional definition. 

 Ibid., p. 18. 



2 Chapters V, VI. 



