250 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



cannot be easily located in space and time, and they do not 

 exhibit the ordinary sensible qualities. This is merely to 

 say that they are inherently obscure as empirical entities. 



DIFFICULTIES IN ASCERTAINMENT OF OPERATIONAL 



DERIVATION 



The main difficulty in the ascertainment of the operational 

 routes by which the scientific symbols are derived from their 

 empirical foundations is the necessity for a previous knowl- 

 edge of the main types of operation which may be per- 

 formed. The techniques which have been employed in a 

 specific case can be more readily recognized if one knows 

 what the most common techniques are. But no satisfactory 

 classification of the operations of symbolic derivation has 

 ever been made. The list given in Chapter IX was merely 

 illustrative and made no pretension to completeness. As 

 will be recalled, the main types there mentioned were 

 abstraction (including measurement), concretion, serial opera- 

 tions, cause and effect inferences, and operations of analysis 

 and synthesis. It will be the task in the following chapters to 

 illustrate the uses of some of these methods with reference 

 to certain of the concepts. A more adequate classification 

 would insure a more accurate analysis. 



Another difficulty is the fact that alternative operations 

 are often employed to derive concepts which are verbally 

 the same. The best known example of this is the two at- 

 tempts to derive the concept of the mathematical point. 

 According to the one method it is defined as the limit of a 

 series of gradually decreasing volumes; according to the 

 other it is defined as the enclosure property which is exhibited 

 by events through their capacity to extend over other 

 events and thus to generate a series. The former route is 

 that of serial extension, the latter is abstraction. Irrationals, 



such as v 2, possess similar alternative routes of derivation; 

 they may be defined either as the limits of certain series of 

 increasing fractions, or as the class of such fractions. The 

 concept of force may be derived either as the cause of motion 



