262 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



The notion of number as plurality seems to have a somewhat 

 more direct empirical reference than the notion of number as 

 distinguishability ; hence it seems best to suppose that the 

 immediate recognition of pluralities is possible without the 

 employment of counting. At least there are simple empirical 

 groups whose number seems to be discernible without a 

 conscious act of counting, and these may be taken at least 

 temporarily as on the primitive level of givenness. It may 

 be that counting can be shown to be logically more basic 

 than number, but this is an independent issue. The problem, 

 then, is to determine what kinds of events have numbers, 

 and what sort of thing number is when thus derived. 



To say that number is a property of complexes or plurali- 

 ties is merely to state a first approximation to an empirical 

 definition of number; what is required further is a concep- 

 tion of the kind of property which number is. An under- 

 standing of the distinction between collective and distributive 

 properties is helpful in this connection. For example, the 

 word "all" is ambiguous in the English language; it may 

 refer to an aggregate taken as a whole as in the proposition, 

 "All the angles in a triangle are equal to two right angles," 

 or it may refer to the individual members of an aggregate as 

 in the proposition, "All the angles in a triangle are less than 

 two right angles." This suggests that there are two kinds of 

 properties — collective, or "holistic," properties which are 

 applicable only to aggregates and not in general to the in- 

 dividual elements, and distributive, or elemental, properties 

 which belong to the members and not in general to the 

 totalities. Inference from a class to the members, as given 

 in the ordinary syllogism, is based upon distributive prop- 

 erties; the absurdity of an inference constructed after the 

 same pattern but with a collective property substituted for 

 the distributive property is illustrated in the following: 

 "Americans are numerous; since I am an American, I am 

 numerous." The general fact to be pointed out, then, is that 

 number is a collective property, applicable only to aggregates 

 or complexes. In this respect it belongs in the same category 



