METHODS OF SCIENCE; NUMBERS 23 



method, we finally reach the conclusion that the 

 method of counting can only be applied to 

 things which can be counted, and we think of 

 this as a sort of vicious circle; but it is not 

 vicious nor is it quite a circle. It is the sort of 

 circle that a spider makes in weaving its web. 

 The growth of living thought is not to be re- 

 pressed; and a cyclic thought is not a circle, 

 but rather an ascending spiral which, with 

 every turn, leads to greater heights. The vicious 

 circle is only the shadow of such an ascending 

 spiral cast upon the flat surface of formal logic. 



As we proceed with these lectures we shall 

 develop the thesis that all the branches of 

 mathematics are but dummies upon which we 

 attempt to fit our observations. To the scientist, 

 whose life is spent in trial and error (I will not 

 say in trials and errors), it seems a natural 

 thing to try one kind of mathematics after an- 

 other in order to find one that fits his imme- 

 diate problem. 



Let us now go back to our arithmetic opera- 

 tions. Having become familiar with addition 

 and multiplication, we find in the course of our 

 figuring that what we have just done must be 

 undone. Thus having added two we may undo 

 the operation by subtracting two ; so the inverse 



