86 THE FOUNDATIONS OF SCIENCE 



We now pass to the consideration of a different 

 class of knowledge, where we come in contact with 

 the principles of logic and of pure mathematics or 

 symbolic logic. One school of philosophers the 

 " empiricists " hold that here too all our know- 

 ledge is acquired by experience. An opposing school, 

 the rationalists, hold that logical principles are grasped 

 by an intuitive action of the mind, and are not proved 

 by experience, though they may be and usually are 

 suggested by experience. 



Thus the truth that two and two are four or the 

 axiom that things that are equal to the same thing 

 are equal to each other, may be suggested to a be- 

 ginner in mathematics by the process of counting 

 four real objects or by the actual measurement of 

 the equal lengths. As soon, however, as he has 

 grasped the meaning of these propositions, he recog- 

 nises general truths which are true not only of the 

 objects or lines used to illustrate them, but must be 

 true of all things at all times and in all conditions. 



This result is different in kind to the propositions 

 which need the support of all possible instances for 

 their belief, such propositions as we meet with in 

 the subject-matter of natural science. The state- 

 ment that all chemical elements were unchangeable 

 was gradually established by a long series of failures 

 to transmute them one into the other, till it became 

 a principle universally accepted : the probability in 

 its favour was enormous. Yet the discovery of the 



