40 ILLINOIS ACADEMY OF SCIENCE 



SCIENCE AND MATHEMATICS 

 James B. Shaw, University of Illinois 



That science has been dependent upon mathematics in many 

 ways is so well recognized that it needs no comment. Sooner 

 or later each branch of science develops a system of observa- 

 tional methods which assume numerical counting or the regis- 

 tration of numerical results of measure. These statistical 

 results or quantitative results must then be discussed and their 

 significance ascertained. It is true, of course, that there are 

 other mathematical conceptions than those of numbers which 

 enter largely into scientific work, but these cannot be discussed 

 in the present paper. Such conceptions as vector, vectorline, 

 dyadic, fields, whether scalar vector or dyadic, curl, divergence, 

 line-integral, may involve numerical elements, but the essence 

 of their characters is non-numerical. These, however, must 

 be passed by in order to remain within the limits of time, and 

 I desire to consider only one notion that science owes to mathe- 

 matics and which appears in one way or another in practically 

 every science. 



The notion of scientific law rests upon the mathematical 

 concept of functionality. In a law it is stated that a certain 

 effect is to be expected from a given cause, or to be rather 

 more technical and at the same time more exact, that a certain 

 phenomenon called the effect is a function of a certain phenom- 

 enon called the cause. If a quantitative measure can be ap- 

 plied to these two phenomena the law may then be stated in a 

 formula of the type y=f (x). We may leave to one side for 

 the sake of definiteness the functions that depend upon more 

 than one variable. The problem then in determining an exact 

 law is that of ascertaining the character of /. 



Now when we define a function we must by the definition 

 be able to calculate in some manner the value of the dependent 

 variable y, for any assigned value of the independent variable 

 x. If the independent variable can assume only a finite and in 

 fact a relatively small number of values, then our mode of 

 ascertaining y may be reference to a table of values giving 

 y for each x, as for example the farmer refers to the almanac 

 to find the time of sunrise for each day of the year. If the 



