MATHEMATICS AND METAPHYSICIANS 95 



scandal that he should still be taught to boys in England. 1 

 A book should have either intelligibility or correctness ; 

 to combine the two is impossible, but to lack both is to 

 be unworthy of such a place as Euclid has occupied in 

 education. 



The most remarkable result of modern methods in 

 mathematics is the importance of symbolic logic and of 

 rigid formalism. Mathematicians, under the influence of 

 Weierstrass, have shown in modern times a care for 

 accuracy, and an aversion to slipshod reasoning, such as 

 had not been known among them previously since the time 

 of the Greeks. The great inventions of the seventeenth 

 century Analytical Geometry and the Infinitesimal 

 Calculus were so fruitful in new results that mathe- 

 maticians had neither time nor inclination to examine 

 their foundations. Philosophers, who should have taken 

 up the task, had too little mathematical ability to invent 

 the new branches of mathematics which have now been 

 found necessary for any adequate discussion. Thus 

 mathematicians were only awakened from their " dog- 

 matic slumbers " when Weierstrass and his followers 

 showed that many of their most cherished propositions 

 are in general false. Macaulay, contrasting the certainty 

 of mathematics with the uncertainty of philosophy, asks 

 who ever heard of a reaction against Taylor's theorem ? 

 If he had lived now, he himself might have heard of such 

 a reaction, for this is precisely one of the theorems which 

 modern investigations have overthrown. Such rude 

 shocks to mathematical faith have produced that love of 

 formalism which appears, to those who are ignorant of 

 its motive, to be mere outrageous pedantry. 



1 Since the above was written, he has ceased to be used as a text- 

 book. But I fear many of the books now used are so bad that the 

 change is no great improvement. [Note added in 1917.] 



