u6 MYSTICISM AND LOGIC 



The physical problem may be stated as follows : to find 

 in the physical world, or to construct from physical 

 materials, a space of one of the kinds enumerated by the 

 logical treatment of geometry. This problem derives 

 its difficulty from the attempt to accommodate to the 

 roughness and vagueness of the real world some system 

 possessing the logical clearness and exactitude of pure 

 mathematics. That this can be done with a certain 

 degree of approximation is fairly evident If I see three 

 people A, B, and C sitting in a row, I become aware of 

 the fact which may be expressed by saying that B is be- 

 tween A and C rather than that A is between B and C, 

 or C is between A and B. This relation of ' ' between ' 

 which is thus perceived to hold has some of the abstract 

 logical properties of those three-term relations which, 

 we saw, give rise to a geometry, but its properties fail to 

 be exact, and are not, as empirically given, amenable 

 to the kind of treatment at which geometry aims. In 

 abstract geometry we deal with points, straight lines, and 

 planes ; but the three people A, B, and C whom I see 

 sitting in a row are not exactly points, nor is the row 

 exactly a straight line. Nevertheless physics, which 

 formally assumes a space containing points, straight 

 lines, and planes, is found empirically to give results 

 applicable to the sensible world. It must therefore be 

 possible to find an interpretation of the points, straight 

 lines, and planes of physics in terms of physical data, or 

 at any rate in terms of data together with such hypo- 

 thetical additions as seem least open to question. Since 

 all data suffer from a lack of mathematical precision 

 through being of a certain size and somewhat vague in 

 outline, it is plain that if such a notion as that of a point 

 is to find any application to empirical material, the point 

 must be neither a datum nor a hypothetical addition tc 



