PSYCHOLOGICAL SCIENCE 89 



first. This law is expressed not only more concisely 

 but also more clearly by the symbolic equation 



xy = yx. 



In this way we can perform operations of reason- 

 ing almost mechanically, which otherwise would 

 require careful and tiring thought. 



" Civilisation advances by extending the number 

 of important operations which we can perform with- 

 out thinking about them. Operations of thought 

 are like cavalry charges in a battle they are strictly 

 limited in number, they require fresh horses, and must 

 only be made at decisive moments." x 



Now, all this intuitive knowledge, whether ex- 

 pressed in words or symbols, is not knowledge about 

 things. It is knowledge about the relations between 

 universals. In our equation, x and y stand for any 

 two quantities whatever any number of apples or 

 shillings or feet or hours. The ideas of multiplica- 

 tion also, and of equality, are general ideas, not con- 

 fined to a particular operation or a particular identity. 



Thus the knowledge we gain intuitively is not 

 knowledge about natural objects or natural phe- 

 nomena. It is concerned with a purely ideal world, 

 a world of universals. 



If we wish to apply this knowledge to natural 

 processes, we must turn to experience ; and here 

 we touch the experimental sciences with which we 



1 Dr. A. N. Whitehead, Introduction to Mathematics. 



