66 MYSTICISM AND LOGIC 



what the theory of infinity very aptly illustrates that 

 many propositions seem self-evident to the untrained 

 mind which, nevertheless, a nearer scrutiny shows to be 

 false. By this means they will be led to a sceptical 

 inquiry into first principles, an examination of the 

 foundations upon which the whole edifice of reasoning is 

 built, or, to take perhaps a more fitting metaphor, the 

 great trunk from which the spreading branches spring. 

 At this stage, it is well to study afresh the elementary 

 portions of mathematics, asking no longer merely whether 

 a given proposition is true, but also how it grows out of 

 the central principles of logic. Questions of this nature 

 can now be answered with a precision and certainty 

 which were formerly quite impossible ; and in the chains 

 of reasoning that the answer requires the unity of all 

 mathematical studies at last unfolds itself. 



In the great majority of mathematical text-books there 

 is a total lack of unity in method and of systematic 

 development of a central theme. Propositions of very 

 diverse kinds are proved by whatever means are thought 

 most easily intelligible, and much space is devoted to 

 mere curiosities which in no way contribute to the main 

 argument. But in the greatest works, unity and in 

 evitability are felt as in the unfolding of a drama ; in the 

 premisses a subject is proposed for consideration, and in 

 every subsequent step some definite advance is made 

 towards mastery of its nature. The love of system, of 

 interconnection, which is perhaps the inmost essence of 

 the intellectual impulse, can find free play in mathematics 

 as nowhere else. The learner who feels this impulse 

 must not be repelled by an array of meaningless examples 

 or distracted by amusing oddities, but must be encouraged 

 to dwell upon central principles, to become familiar with 

 the structure of the various subjects which are put before 



