V NON-EUCLIDEAN GEOMETRY AND THE KANTIAN A PRIORI¹ ARGUMENT Importance of geometry as a type of philosopbic method, and consequently of the metageometrical ideas. 1. 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. 85
