SCIENTIFIC METHOD IN PHILOSOPHY 115 



though it is difficult to find any kind of which the space 

 of physics may be an example, and it is impossible to 

 find any kind of which the space of physics is certainly 

 an example. As an illustration of one possible logical 

 system of geometry we may consider all relations of 

 three terms which are analogous in certain formal respects 

 to the relation " between " as it appears to be in actual 

 space. A space is then defined by means of one such 

 three-term relation. The points of the space are all the 

 terms which have this relation to something or other, 

 and their order in the space in question is determined 

 by this relation. The points of one space are necessarily 

 also points of other spaces, since there are necessarily 

 other three-term relations having those same points for 

 their field. The space in fact is not determined by the 

 class of its points, but by the ordering three-term rela- 

 tion. When enough abstract logical properties of such 

 relations have been enumerated to determine the resulting 

 kind of geometry, say, for example, Euclidean geometry, 

 it becomes unnecessary for the pure geometer in his ab- 

 stract capacity to distinguish between the various relations 

 which have all these properties. He considers the whole 

 class of such relations, not any single one among them. 

 Thus in studying a given kind of geometry the pure 

 mathematician is studying a certain class of relations 

 defined by means of certain abstract logical properties 

 which take the place of what used to be called axioms. 

 The nature of geometrical reasoning therefore is purely 

 deductive and purely logical ; if any special epistemolo- 

 gical peculiarities are to be found in geometry, it must 

 not be in the reasoning, but in our knowledge concerning 

 the axioms in some given space. 



(2) The physical problem of space is both more in- 

 teresting and more difficult than the logical problem. 



