MATHEMATICS AND METAPHYSICIANS 79 



three. Even these three can be explained by means of 

 the notions of relation and class ; but this requires the 

 Logic of Relations, which Professor Peano has never 

 taken up. It must be admitted that what a mathe- 

 matician has to know to begin with is not much. There 

 are at most a dozen notions out of which all the notions 

 in all pure mathematics (including Geometry) are com- 

 pounded. Professor Peano, who is assisted by a very 

 able school of young Italian disciples, has shown how 

 this may be done ; and although the method which he 

 has invented is capable of being carried a good deal 

 further than he has carried it, the honour of the pioneer 

 must belong to him. 



Two hundred years ago, Leibniz foresaw the science 

 which Peano has perfected, and endeavoured to create it. 

 He was prevented from succeeding by respect for the 

 authority of Aristotle, whom he could not believe guilty 

 of definite, formal fallacies ; but the subject which he 

 desired to create now exists, in spite of the patronising 

 contempt with which his schemes have been treated by all 

 superior persons. From this " Universal Characteristic," 

 as he called it, he hoped for a solution of all problems, 

 and an end to all disputes. " If controversies were to 

 arise," he says, " there would be no more need of dis- 

 putation between two philosophers than between two 

 accountants. For it would suffice to take their pens in 

 their hands, to sit down to their desks, and to say to 

 each other (with a friend as witness, if they liked), ' Let 

 us calculate.' ' This optimism has now appeared to be 

 somewhat excessive ; there still are problems whose 

 solution is doubtful, and disputes which calculation 

 cannot decide. But over an enormous field of what was 

 formerly controversial, Leibniz's dream has become sober 

 fact. In the whole philosophy of mathematics, which 



