o8 MYSTICISM AND LOGIC 



due order and proportion, with a new perfection in every 

 part. 



Until symbolic logic had acquired its present develop- 

 ment, the principles upon which mathematics depends 

 were always supposed to be philosophical, and discover- 

 able only by the uncertain, unprogressive methods 

 hitherto employed by philosophers. So long as this was 

 thought, mathematics seemed to be not autonomous, but 

 dependent upon a study which had quite other methods 

 than its own. Moreover, since the nature of the postulates 

 from which arithmetic, analysis, and geometry are to be 

 deduced was wrapped in all the traditional obscurities of 

 metaphysical discussion, the edifice built upon such 

 dubious foundations began to be viewed as no better 

 than a castle in the air. In this respect, the discovery 

 that the true principles are as much a part of mathe- 

 matics as any of their consequences has very greatly 

 increased the intellectual satisfaction to be obtained. 

 This satisfaction ought not to be refused to learners 

 capable of enjoying it, for it is of a kind to increase our 

 respect for human powers and our knowledge of the 

 beauties belonging to the abstract world. 



Philosophers have commonly held that the laws of 

 logic, which underlie mathematics, are laws of thought, 

 laws regulating the operations of our minds. By this 

 opinion the true dignity of reason is very greatly lowered : 

 it ceases to be an investigation into the very heart ana 

 immutable essence of all things actual and possible, be- 

 coming, instead, an inquiry into something more or less 

 human and subject to our limitations. The contemplation 

 of what is non-human, the discovery that our minds are 

 capable of dealing with material not created by them, 

 above all, the realisation that beauty belongs to the outer 

 world as to the inner, are the chief means of overcoming 



