SCIENTIFIC METHOD IN PHILOSOPHY 119 



problem : " How do we come to have knowledge of 

 geometry a priori ? " By the distinction between the 

 logical and physical problems of geometry, the bearing 

 and scope of this question are greatly altered. Our 

 knowledge of pure geometry is a priori but is wholly 

 logical. Our knowledge of physical geometry is synthetic, 

 but is not a priori. Our knowledge of pure geometry 

 is hypothetical, and does not enable us to assert, for 

 example, that the axiom of parallels is true in the physical 

 world. Our knowledge of physical geometry, while it 

 does enable us to assert that this axiom is approximately 

 verified, does not, owing to the inevitable inexactitude 

 of observation, enable us to assert that it is verified 

 exactly. Thus, with the separation which we have made 

 between pure geometry and the geometry of physics, the 

 Kantian problem collapses. To the question, " How 

 is synthetic a priori knowledge possible ? " we can 

 now reply, at any rate so far as geometry is concerned, 

 "It is not possible," if "synthetic" means "not de- 

 ducible from logic alone." Our knowledge of geometry, 

 like the rest of our knowledge, is derived partly from 

 logic, partly from sense, and the peculiar position which 

 in Kant's day geometry appeared to occupy is seen now 

 to be a delusion. There are still some philosophers, it is 

 true, who maintain that our knowledge that the axiom of 

 parallels, for example, is true of actual space, is not to 

 be accounted for empirically, but is as Kant maintained 

 derived from an a priori intuition. This position is not 

 logically refutable, but I think it loses all plausibility as 

 soon as we realise how complicated and derivative is 

 the notion of physical space. As we have seen, the 

 application of geometry to the physical world in no way 

 demands that there should really be points and straight 

 lines among physical entities. The principle of economy, 



