154 NORMATIVE SCIENCE 



upon the human mind by the constitution of any primal form of 

 intuition, nor yet a branch of physical science, limited to describing 

 the spatial arrangement of phenomena in the external world. Pure 

 geometry is the theory of the consequences of certain postulates 

 which the geometer is at liberty consistently to make; so that there 

 are as many types of geometry as there are consistent systems of 

 postulates of that generic type of which the geometer takes account. 

 As is also now well known, it has long been impossible to define pure 

 mathematics as the science of quantity, or to limit the range of the 

 exactly stateable hypotheses or postulates with which the mathema- 

 tician deals to the world of those objects which, ideally speaking, 

 can be viewed as measurable. For the ideally defined measurable 

 objects are by no means the only ones whose properties can be stated 

 in the form of exact postulates or hypotheses; and the possible range 

 of pure mathematics, if taken in the abstract, and viewed apart from 

 any question as to the value of given lines of research, appears to be 

 identical with the whole realm of the consequences of exactly state- 

 able ideal hypotheses of every type. 



One limitation must, however, be mentioned, to which the asser- 

 tion just made is, in practice, obviously subject. And this is, indeed, 

 a momentous limitation. The exactly stated ideal hypotheses whose 

 consequences the mathematician develops must possess, as is some- 

 times said, sufficient intrinsic importance to be worthy of scientific 

 treatment. They must not be trivial hypotheses. The mathema- 

 tician is not, like the solver of chess problems, merely displaying 

 his skill in dealing with the arbitrary fictions of an ideal game. His 

 truth is, indeed, ideal; his world is, indeed, treated by his science as 

 if this world were the creation of his postulates a " freie Schopfung." 

 But he does not thus create for mere sport. On the contrary, he re- 

 ports a significant order of truth. As a fact, the ideal systems of the 

 pure mathematician are customarily defined with an obvious, even 

 though often highly abstract and remote, relation to the structure 

 of our ordinary empirical world. Thus the various algebras which 

 have been actually developed have, in the main, definite relations 

 to the structure of the space world of our physical experience. The 

 different systems of ideal geometry, even in all their ideality, still 

 cluster, so to speak, about the suggestions which our daily experi- 

 ence of space and of matter give us. Yet I suppose that no mathe- 

 matician would be disposed, at the present time, to accept any brief 

 definition of the degree of closeness or remoteness of relation to or- 

 dinary experience which shall serve to distinguish a trivial from 

 a genuinely significant branch of mathematical theory. In general, a 

 mathematician who is devoted to the theory of functions, or to group 

 theory, appears to spend little time in attempting to show why the 

 development of the consequences of his postulates is a significant 



