October 7, 1904.] 



SCIENCE. 



451 



mathematics shares in common with philos- 

 ophy this type of scientific interest in ideal, 

 as distinct from physical or phenomenal 

 truth. There is, to be sure, a marked con- 

 trast between the ways in which the mathe- 

 matician and the philosopher approach, 

 select and elaborate their respective sorts 

 of problems. But there is also a close re- 

 lation between the two types of investiga- 

 tion in question. Let us next consider both 

 the contrast and the analogy in some of 

 their other most general features. 



Pure mathematics is concerned with the 

 investigation of the logical consequences of 

 certain exactly statable postulates or hy- 

 potheses — such, for instance, as the postu- 

 lates upon which arithmetic and analysis 

 are founded, or such as the postulates that 

 lie at the basis of any type of geometry. For 

 the pure mathematician, the truth of these 

 hypotheses or postulates depends, not upon 

 the fact that physical nature contains phe- 

 nomena answering to the postulates, but 

 solely upon the fact that the mathematician 

 is able, with rational consistency, to state 

 these assumed first principles, and to de- 

 velop their consequences. Dedekind, in 

 his famous essay, ' Was Sind und Was 

 SoUen die Zahlen,' called the whole num- 

 bers 'freie Schopfungen des Menschlichen 

 Geistes'; and, in fact, we need not enter 

 into any discussion of the psychology of 

 our number concept in order to be able to 

 assert that, however we men first came by 

 our conception of the whole numbers, for 

 the mathematician the theory of numerical 

 truth must appear simply as the logical 

 development of the consequences of a few 

 fundamental first principles, such as those 

 which Dedekind himself, or Peano, or other 

 recent writers upon this topic, have, in 

 various forms, stated. A similar formal 

 freedom marks the development of any 

 other theory in the realm of pure mathe- 

 matics. Pure geometry, from the modern 

 point of view, is neither a doctrine forced 



upon the human mind by the constitution 

 of any primal form of intuition, nor yet a 

 branch of physical science, limited to de- 

 scribing the spatial arrangement of phe- 

 nomena in the external world. Pure geom- 

 etry 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 statable 

 hypotheses or postulates with which the 

 mathematician deals to the world of those 

 objects which, ideally speaking, can be 

 viewed as measurable. For the ideally de- 

 fined measurable objects are by no means 

 the only ones whose properties can be stated 

 in the form of exact postulates or hypoth- 

 eses ; and the possible range of pure mathe- 

 matics, if taken in the abstract, and viewed 

 apart from any question as to the value of 

 given lines of research, appears to be iden- 

 tical with the whole realm of the conse- 

 quences of exactly statable ideal hypoth- 

 eses of every type. 



One limitation must, however, be men- 

 tioned, to which the assertion just made is, 

 in practise, obviously subject. And this 

 is, indeed, a momentous limitation. The 

 exactly stated ideal hypotheses whose con- 

 sequences the mathematician develops must 

 possess, as is sometimes said, sufficient in- 

 trinsic importance to be worthy of scien- 

 tific treatment. They must not be trivial 

 hypotheses. The mathematician 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 Schop- 

 fung. ' But he does not thus create for 



