90 HUMANISM v 



M. Delbceuf 1 and M. Poincar6 2 stating the characteristics 

 of Euclidean space and its fundamental distinction from 

 perceptual space in almost identical terms. The former is 

 one, empty, homogeneous, continuous, infinite, infinitely 

 divisible, identical, invariable ; the latter is many, filled, 

 heterogeneous, continuous only for perception (if the 

 atomic view of matter holds), probably finite, not infinitely 

 divisible and variable. Both sides agree that our physical 

 world is neither in Euclidean nor in non-Euclidean space, 

 both of which are conceptual abstractions ; their dispute 

 is merely as to which furnishes the proper method for 

 calculating spatial phenomena. 3 Thus all the geometrical 

 spaces are grounded on the same experience of physical 

 space, which they interpret and idealize differently, while 

 seeking to simplify and systematize it by means of the 

 various postulates which define them. 



But if conceptual and perceptual space are so different, 

 have they anything in common but the name ? If the 

 former are abstracted from the latter, upon what principles 

 and by what methods does the abstraction proceed ? 



I conceive the answer to this important question to be, 

 by the same methods as those by which real or physical 

 space is developed out of the psychological spaces. For, 

 as M. Poincare 4 well shows, we form our notion of real 

 space by fusing together the data derived from visual, 

 tactile, and motor sensations. That fusion is largely 

 accomplished by ignoring the differences between their 

 several deliverances and by correcting the appearances to 

 one sense by another, in such a manner as to give the 

 most complete and trustworthy perception of the object. 

 We manipulate the data of the senses in order to perceive 

 things (in real space), and at a higher stage the same 

 purposive process yields conceptual space, of course at 

 first in its simplest form, the Euclidean. And (though I 

 have not found this stated) all the characteristics of 

 Euclidean space may be shown to have been constructed 



1 Rev. Phil, xviii. n. 2 Rev. de Mtt. iii. p. 632. 



3 Cp. Calinon, Rev. Phil, xviii. 12, &quot; Sur I ind^termination gomtrique de 

 1 univers.&quot; 4 Loc. cit. 



