96 MYSTICISM AND LOGIC 



The proof that all pure mathematics, including 

 Geometry, is nothing but formal logic, is a fatal blow to 

 the Kantian philosophy. Kant, rightly perceiving that 

 Euclid's propositions could not be deduced from Euclid's 

 axioms without the help of the figures, invented a theory 

 of knowledge to account for this fact ; and it accounted 

 so successfully that, when the fact is shown to be a mere 

 defect in Euclid, and not a result of the nature of geo- 

 metrical reasoning, Kant's theory also has to be aban- 

 doned. The whole doctrine of a priori intuitions, by which 

 Kant explained the possibility of pure mathematics, is 

 wholly inapplicable to mathematics in its present form. 

 The Aristotelian doctrines of the schoolmen come nearer 

 in spirit to the doctrines which modern mathematics 

 inspire ; but the schoolmen were hampered by the fact 

 that their formal logic was very defective, and that the 

 philosophical logic based upon the syllogism showed a 

 corresponding narrowness. What is now required is to 

 give the greatest possible development to mathematical 

 logic, to allow to the full the importance of relations, and 

 then to found upon this secure basis a new philosophical 

 logic, which may hope to borrow some of the exactitude 

 and certainty of its mathematical foundation. If this 

 can be successfully accomplished, there is every reason 

 to hope that the near future will be as great an epoch in 

 pure philosophy as the immediate past has been in the 

 principles of mathematics. Great triumphs inspire great 

 hopes ; and pure thought may achieve, within our 

 generation, such results as will place our time, in this 

 respect, on a level with the greatest age of Greece. 1 



1 The greatest age of Greece was brought to an end by 

 Peloponnesian War. [Note added in 1917.] 



