148 THE LIMITATIONS OF SCIENCE 



space, time, and matter, except as they are individually 

 and concretely measured. Kant, to be sure, main- 

 tained that we were endowed with an innate and inex- 

 plicable, but sufficient idea of pure space and time. 

 These qualities are, however, by themselves inappreci- 

 able to our senses. To make them sensible, we need 

 a third, which he calls the Ding an sich, corresponding 

 in the external world to what we call the entity, 

 matter. 



However vulnerable and unsatisfactory the doctrine 

 of innate ideas may be, yet it seems to me that by the 

 abstractive method as used in the science of mechanics 

 we do obtain a real and adequate idea of these three 

 fundamental postulates. If we have not such a power 

 of abstracting real ideas from our concrete observa- 

 tions of phenomena, then it is difficult to believe that 

 the conclusions of Euclidean geometry and of mathe- 

 matical analysis in general are rigorous. Thus, in 

 mathematical definitions, space is a mere volume en- 

 closed in an imagined boundary which may be con- 

 ceived as of any extent from the indefinitely small to 

 the indefinitely large. And this imagined bounding 

 surface bears no closer relation to a concrete material 

 envelope than does an image in a mirror to its object. 

 It seems to me that I have an adequate idea, for 

 instance, of the space in an empty room and that I 

 can abstract all the properties from my concrete per- 

 ception of the material walls of this room except the 



