76 MYSTICISM AND LOGIC 



and Geometry is built up by combinations of the primi- 

 tive ideas of logic, and its propositions are deduced from 

 the general axioms of logic, such as the syllogism and the 

 other rules of inference. And this is no longer a dream 

 or an aspiration. On the contrary, over the greater and 

 more difficult part of the domain of mathematics, it has 

 been already accomplished ; in the few remaining cases, 

 there is no special difficulty, and it is now being rapidly 

 achieved. Philosophers have disputed for ages whether 

 such deduction was possible ; mathematicians have sat 

 down and made the deduction. For the philosophers 

 there is now nothing left but graceful acknowledg- 

 ments. 



The subject of formal logic, which has thus at last 

 shown itself to be identical with mathematics, was, as 

 every one knows, invented by Aristotle, and formed the 

 chief study (other than theology) of the Middle Ages. 

 But Aristotle never got beyond the syllogism, which is a 

 very small part of the subject, and the schoolmen never 

 got beyond Aristotle. If any proof were required of our 

 superiority to the mediaeval doctors, it might be found in 

 this. Throughout the Middle Ages, almost all the best 

 intellects devoted themselves to formal logic, whereas in 

 the nineteenth century only an infinitesimal proportion of 

 the world's thought went into this subject. Nevertheless, 

 in each decade since 1850 more has been done to advance 

 the subject than in the whole period from Aristotle to 

 Leibniz. People have discovered how to make reasoning 

 symbolic, as it is in Algebra, so that deductions are 

 effected by mathematical rules. They have discovered 

 many rules besides the syllogism, and a new branch of 

 logic, called the Logic of Relatives, 1 has been invented 

 to deal with topics that wholly surpassed the powers of 

 1 This subject is due in the main to Mr. C. S. Peirce. 



