MATHEMATICS AND METAPHYSICIANS 77 



the old logic, though they form the chief contents of 

 mathematics. 



It is not easy for the lay mind to realise the importance 

 of symbolism in discussing the foundations of mathe- 

 matics, and the explanation may perhaps seem strangely 

 paradoxical. The fact is that symbolism is useful because 

 it makes things difficult. (This is not true of the advanced 

 parts of mathematics, but only of the beginnings.) What 

 we wish to know is, what can be deduced from what. 

 Now, in the beginnings, everything is self-evident ; and 

 it is very hard to see whether one self-evident proposition 

 follows from another or not. Obviousness is always the 

 enemy to correctness. Hence we invent some new and 

 difficult symbolism, in which nothing seems obvious. 

 Then we set up certain rules for operating on the symbols, 

 and the whole thing becomes mechanical. In this way 

 we find out what must be taken as premiss and what can 

 be demonstrated or defined. For instance, the whole of 

 Arithmetic and Algebra has been shown to require three 

 indefinable notions and five indemonstrable propositions. 

 But without a symbolism it would have been very hard 

 to find this out. It is so obvious that two and two are four, 

 that we can hardly make ourselves sufficiently sceptical 

 to doubt whether it can be proved. And the same holds 

 in other cases where self-evident things are to be proved. 



But the proof of self-evident propositions may seem, to 

 the uninitiated, a somewhat frivolous occupation. To 

 this we might reply that it is often by no means self- 

 evident that one obvious proposition follows from another 

 obvious proposition ; so that we are really discovering 

 new truths when we prove what is evident by a method 

 which is not evident. But a more interesting retort is, 

 that since people have tried to prove obvious propositions, 

 they have found that many of them are false. Self- 



