368 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



Hence the main difference between scientific laws and em- 

 pirical laws lies in the fact that functional correlations con- 

 stitute an important type of the former, while they occur 

 rarely or not at all as instances of the latteT. For this reason, 

 an analysis of the notion of function is important in the 

 understanding of scientific law. One may define a function, 

 of which the simplest type is a single-valued function, as 

 follows: "A single- valued function of a variable # is a second 

 variable y so related to x that whenever a value is assigned 

 to x from the z-range, a corresponding value of y is uniquely 

 determined in the y-range." 1 A variable represents any 

 qualitative event which is capable of quantitative variation. 

 Hence a scientific law, from this point of view, states not 

 mere presence or absence but degree of presence or absence; 

 it states not merely that two events are found frequently 

 together, but how much of one is found with how much of 

 the other. This involves recognition of the dependence of 

 the law upon measured values, to which reference was made 

 in the previous section. A law states a functional relation 

 between variables which are qualitatively distinct from one 

 another, and which are qualitatively distinct from variables 

 symbolized in other ways and occurring in other laws. The 

 fact that kinetic energy is equivalent to i mv 2 and the space 

 of a falling body is equal to \gt 2 does not make the laws 

 identical; for m, v, g, and t represent different variables 

 whose values are determined by different techniques of 

 measurement. Though the general form of the function is 

 the same, neither the entities measured nor the techniques 

 employed are the same; hence the laws mean something 

 different in the two cases. This fact is often lost from view 

 in theories which attempt to reduce science to mathematics. 

 By virtue of the establishment of functional relationships 

 the distinction between laws of coexistence and laws of 

 succession becomes obliterated. Mathematical laws state 

 merely the logical dependence of one measured value upon 



1 Quoted by G. A. Bliss from Dirichlet in J. W. A. Young, Monographs on Modern 

 Mathematics (New York: Longmans, Green, 1924), p. 266. 



