78 MYSTICISM AND LOGIC 



evidence is often a mere will-o -the-wisp, which is sure to 

 lead us astray if we take it as our guide. For instance, 

 nothing is plainer than that a whole always has more 

 terms than a part, or that a number is increased by add 

 ing one to it. But these propositions are now known to 

 be usually false. Most numbers are infinite, and if a 

 number is infinite you may add ones to it as long as you 

 like without disturbing it in the least. One of the merits 

 of a proof is that it instils a certain doubt as to the result 

 proved ; and when what is obvious can be proved in 

 some cases, but not in others, it becomes possible to sup 

 pose that in these other cases it is false. 



The great master of the art of formal reasoning, among 

 the men of our own day, is an Italian, Professor Peano, 

 of the University of Turin. 1 He has reduced the greater 

 part of mathematics (and he or his followers will, in time, 

 have reduced the whole) to strict symbolic form, in which 

 there are no words at all. In the ordinary mathematical 

 books, there are no doubt fewer words than most readers 

 would wish. Still, little phrases occur, such as therefore, 

 let us assume, consider, or hence it follows. All these, how 

 ever, are a concession, and are swept away by Professor 

 Peano. For instance, if we wish to learn the whole of 

 Arithmetic, Algebra, the Calculus, and indeed all that is 

 usually called pure mathematics (except Geometry), we 

 must start with a dictionary of three words. One symbol 

 stands for zero, another for number, and a third for next 

 after. What these ideas mean, it is necessary to know if 

 you wish to become an arithmetician. But after symbols 

 have been invented for these three ideas, not another 

 word is required in the whole development. All future 

 symbols are symbolically explained by means of these 



1 I ought to have added Frege, but his writings were unknown to 

 me when this article was written. [Note added in 1917.] 



