V 



NON-EUCLIDEAN GEOMETRY AND THE 

 KANTIAN A PRIORI 1 



ARGUMENT 



Importance of geometry as a type of philosophic method, and consequently of 

 the metageometrical ideas. I. Fallacy of the fourth-dimension analogy. 

 Non-Euclidean three dimensional spaces, come with Euclidean under 

 the genus of general geometry. They form coherent and thinkable 

 systems analogous to Euclid s, but so far not useful because too com 

 plicated. II. Necessity of distinguishing between perceptual and 

 conceptual spaces. Geometrical spaces all alike conceptual constructions, 

 and the physical world not in any one of them. III. Philosophic im 

 portance of this. The certainty of geometry not peculiar, but 

 identical with the logical necessity of consistent assumptions elsewhere. 

 The real validity of geometry empirical and = its usefulness when 

 applied. Universality and necessity of geometrical judgments as results 

 of postulation. Kant s account of space vitiated by his failure to observe 

 the ambiguities of the term. 



FROM the days of Pythagoras and Plato down to those 

 of Kant and Herbart the mathematical sciences, and 

 especially geometry, have played so important a part in 

 the discussions of philosophers as models of method and 

 patterns of certitude, that philosophy cannot but be 

 extremely sensitive to any change or progress occurring 

 in the views of mathematicians. Accordingly the philo 

 sophic world was considerably startled, not so many years 

 ago, to hear that certain mathematicians and physicists 

 had had the audacity to question the assumptions con- 



1 From the Philosophical Review of March 1896, since when the subject has 

 not, of course, stood still. I am painfully aware that as an account of meta- 

 geometry this paper is quite inadequate, but as students of philosophy are still 

 obfuscated with the mystical mathematics of metaphysicians, and as even so able 

 and detailed a work as Mr. Russell s Foundations of Geometry has failed to make 

 clear the capital importance of the distinction of perceptual and conceptual space, 

 even so slight a treatment may retain some pedagogical value. 



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