THE PHILOSOPHY OF MATHEMATICS 239 



a confusion still exists and has survived from bygone times 

 when the psychology of education was not understood so well 

 as it is at present. It seems natural that the learner should 

 begin at the beginning of the subject. The truth is entirely 

 the opposite. The learner's natural beginning is the study of 

 the complex facts ot some application of the subject. The 

 simple abstract ideas are very difficult to grasp, because in 

 practical life we never consider them directly ; for beginners 

 they require some particular embodiment. 



In one instance, however, Mr. Berkeley has not made this 

 error — namely when he criticises parts of the introductory 

 chapter of my Universal Algebra. I am glad to be able to 

 close this article on a note of agreement with him. I think 

 the formalist position adopted in that chapter, whilst it has the 

 merit of recognising an important problem, does not give a 

 true solution, which is to be found, as explained above, in 

 the doctrine of the variable. To that extent I agree with Mr. 

 Berkeley's criticisms, though I hardly think that he always 

 interprets correctly the meaning of my statements. His own 

 solutions of the group of problems presented by the ideas of 

 algebra, which ignore the doctrine of the variable, appear to 

 me to be entirely inadequate, lacking in coherence, clearness 

 and definition. 



