GENERAL PAPERS 41 



number of values necessary to consider becomes relatively 

 large, though still finite, as, for instance, 10 24 , the limitations 

 of humanity make it impracticable to utilize a table of values 

 to ascertain 37 when x is given. In this case and in the case 

 in which there is an infinity of values of x, the values of y 

 must be given by some kind of an expression which can be 

 computed in at least a reasonable time. 



In determining the laws of science, we find that in some 

 cases these are worked out from a given set of observations, 

 necessarily finite in number, and indeed relatively few. In 

 order to determine a general formula, then, which will hold 

 good for an infinity of cases or for a relatively large number of 

 cases, it becomes necessary to supplement the observation with 

 various hypotheses, or assumptions. The most common of 

 these is the assumption of continuity, which means that if we 

 change the value of the independent variable by a variable in- 

 crement, the change in y will be a variable increment (includ- 

 ing the case when this variable assumes equal values for all its 

 range) and the two increments decrease together, indefinitely. 

 That this assumption is the most natural one for the mind to 

 make would be quite evident if we accept C. S. Peirce's analysis 

 of mind, in which he finds the great characteristic of mind is 

 its continuity, which, indeed, is Bergson's conclusion. In any 

 case it seemed for a long time that if for every value of x be- 

 tween two given values, x x and x 2 , y must assume every value 

 intermediate between y x corresponding to x, and y 2 corre- 

 sponding to x 2 , then y would have to be a continuous function 

 of x. But in the progress of mathematics Darboux invented a 

 function which does assume between y x and y 2 every inter- 

 mediate value and yet is discontinuous everywhere. The sig- 

 nificance of this invention for science is that science is no longer 

 compelled to assume continuity in order to have the property 

 cited. It is no longer necessary to depend upon actual con- 

 tinuity of values, that is to say, states may change instantane- 

 ously by finite amounts and yet assume every value between 

 two given states. 



Another common assumption is that of derivative. In many 

 investigations it is assumed that if we are concerned with the 

 ratio of two increments which decrease together, we must 

 substitute for the limit of the ratio a derivative. For ex- 



