268 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



essentially as the field of mathematical operations, i.e., num- 

 bers are any entities which may be operated upon in spec- 

 ifiable ways to produce other entities of a similar kind. But 

 if the operations are such as cannot be physically performed 

 upon empirical groups, there is no reason to suppose that the 

 resultant numbers will be empirically descriptive. Hence 

 mathematics becomes by this act of generalization more or 

 less removed from the empirically given. It is due to the 

 important part which acts of this kind play in mathematics 

 that it is commonly characterized as a strictly formal, or 

 non-existential, science. With the breaking away from 

 empirical groups the way is open for the development of 

 meaning along intensional rather than extensional lines; 

 hence the search is for formal postulate systems. 



Clearly, the essence of the formal definition of number lies 

 in the notion of operation. As a result there arise various 

 levels of generality. One may define merely the positive 

 integers; or one may define the entire class of whole numbers, 

 positive and negative; or one may define the rational num- 

 bers; or one may generalize so as to include the irrational 

 numbers as well, and thus define the real numbers; or one 

 may reach a still higher stage and formulate postulates 

 descriptive of numbers as including also the imaginary 

 numbers. Each of these represents a higher stage of gen- 

 eralization, and each includes a wider range of different types 

 of number. For the purposes of illustration the set chosen 

 will be that which defines number in the highly general 

 sense which includes the real and the imaginary numbers. 



In this connection the notion of group is important. A 

 group, in this technical sense, is a class of a certain kind, 

 viz., a class whose members satisfy certain laws with refer- 

 ence to operations which may be performed upon them. 

 Operations serve to correlate members of the group with one 

 another. Hence if one specifies a certain class of elements 

 whose natures are undefined except for the operations which 

 may be performed on them and the laws according to which 

 such operations must be performed, one has a group. One 



