92 HUMANISM v 



able. In short, as applied, a geometry is not certain, but 

 useful. 1 



Again, the necessity of geometry is simply the necessity 

 of a logical inference hypothetical, and in no wise 

 peculiar to geometry. Similarly, the universality of 

 geometrical judgments is by no means peculiar to them, 

 but may be explained as arising out of the methodological 

 character of the assumptions on which they rest. If we 

 decide to make certain assumptions because they are the 

 most serviceable, we can certainly know beforehand that we 

 shall always and under all circumstances judge accordingly. 

 To expect us to do otherwise, would be to expect us to 

 stultify ourselves. And certainly we have a great in 

 terest in upholding the universal validity of geometrical 

 judgments. Is it a small thing to be able to draw a 

 figure on paper in one s study, and on the strength of it, 

 and by virtue of the homogeneity of space, to draw 

 inferences about what happens beyond the path of the 

 outmost sun ? Should we not be incredible idiots, if we 

 allowed any cheat of appearances to cajole us into a 

 moment s doubt of so precious an organon of knowledge ? 

 It would seem, then, that the chief result of metageometry 

 is to raise into clearer consciousness the nature of the 

 complex processes whereby we organize our experiences, 

 and to assimilate the case of space to our procedure 

 elsewhere. 2 



But it has already become abundantly evident that a 

 view of Space, such as that propounded, provokes conflicts 

 with ancient and venerable views that have long adorned 

 the histories of Philosophy. Among them Kant s con 

 ception of the apriority of Space is pre-eminent. 



At a cursory glance it might indeed seem as though 

 the new geometry afforded a welcome support to the 

 Kantian position. If Euclidean geometry alone could 

 prove the possibility of synthetic judgments a priori, could 

 enrich us with absolutely certain knowledge absolutely 

 independent of experience, could sustain an all-embracing, 



1 Cp. Poincar^ s La Science et r Hypothese, pp. 66-7. 

 2 Cp. Axioms as Postulates, 40-43. 



