270 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



tween the scientific content of the notion and the empirical 

 content. There are two main differences — one pertaining 

 to the order of the elements, and the other applying to the 

 nature of the operations. The former is indicated by the 

 fact that that which is psychologically primitive at the 

 empirical level is the collection of numbered groups, 2, 5, 6, 

 4, 3, from which 1 and are considered to be psychologically 

 derivative; but that which is logically primitive at the 

 scientific level is the class of elements, of which 1 and are 

 given, and from which 2, 5, 6, 4, 3, and the rest of the num- 

 bers follow by logical derivation. Hence the number 5, say, 

 is psychologically primitive but logically derivative, while 

 the number is psychologically derivative but logically 

 primitive. The distinctions in the meaning of the term 

 "operation" are illustrated in the fact that at the empirical 

 level an operation is a method for manipulating groups and 

 collections, while at the scientific level it is a rule for manipu- 

 lating symbols. In the present case this is not quite true, 

 since the operations have not been taken at the highest 

 possible level of abstraction; they retain a tinge of their 

 empirical reference which permits one still to speak of the 

 things which are operated upon as numbers. But the tend- 

 ency to the sharp separation between the extensional and 

 the intensional is clearly indicated. At a high level of gen- 

 erality emphasis shifts from extension to intension. Ab- 

 stractions are elusive features of the given, and cannot be 

 readily pointed to; hence symbols which describe them can- 

 not be defined by the denotative method but require the 

 connotative method instead. This leads to a formalization 

 of the concepts in question, and to a gradual elimination of 

 empirical content. It results in the conception that mathe- 

 matics is not an existential science but is concerned solely 

 with ideal objects. A recognition of the empirical content 

 of the notion, and of the operational routes involved in the 

 passage to the scientific content dulls the sharpness of this 

 dualism, and results in the conception that mathematics is 

 the same kind of study as empirical science, except that 



