MATHEMATICS VERSUS PSYCHOLOGY 9 



further inference, in which it may be difficult for us to follow 

 them; namely, that the relations just described as everywhere 

 parallel are in fact identical. The necessity with which the cause 

 produces its effect means that a mind possessed of complete 

 knowledge of the former must be able to predict the latter, that 

 is to say, deduce it from the cause as premise. Thus the funda- 

 mental nature of the circle, conceived as produced by a rotating 

 line, is the cause of all its other properties for example, of the 

 fact that every radius is perpendicular to the tangent at its 

 extremity. From this extreme form of the doctrine, Hobbes is 

 saved by his nominalism; while Leibniz is distinguished by his 

 'principle of sufficient reason/ according to which the determi- 

 nation of an effect involves not only logical necessity but the 

 selection of the best out of an infinite number of logically pos- 

 sible alternatives. 



The keystone of continental rationalism is the doctrine of 

 substance. While the provinces of reason and sense-perception 

 are wholly distinct, a certain connection arises from the obvious 

 consideration, that when a fact is attested by perception a number 

 of consequences may logically follow from it. Indeed, every 

 observed fact, no matter how irrelevant it may appear from the 

 standpoint of pure science, is known by the law of causality to be 

 absolutely determined by, and thus deducible from, a series of 

 previous facts. Unless, then, some one or more facts could be 

 conceived as eternally necessary on their own account, and thus 

 as serving to support all other facts, the whole chain of facts, 

 taken in its entirety, must be thought of as hanging in mid-air 

 which appeared to be inconceivable. Such necessary fact or 

 facts could, however, be attested by no act of perception; the 

 only adequate witness is reason itself. The entity whose exist- 

 ence isimplied in any such eternal fact is called a substance; and 

 those philosophers who believe in the existence of but a single 

 substance call it God. In the nominalistic theory this entity is 

 an unknowable, to which, however, the name of God is also given . 

 Those, too, who accept the existence of a plurality of substances, 



