Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

NON-EUCLIDEAN GEOMETRIES 359
perfect, and eternal. But it is not this aspect of logical
consistency that the "true" of your question is designed
to mean.
Perhaps the question you intend to ask is this: Are
the geometries true in the sense of giving an exact account of Space? Or, better, since they contradict one
another in cardinal matters, is one of them true in the
indicated sense? I have already pointed out that the
term "space" does not occur in Euclid's Elements, and
I may add that there is no necessity for its occurrence in
either of the non-Euclidean geometries. Since it is nevertheless customary to use the term in philosophic discussions of geometry, we, too, must do so here. If we are
to do so profitably, we must make and keep steadily in
mind a fundamental distinction, which is indeed a pretty
obvious one but is commonly neglected; and the neglect
of it is always attended with utter confusion. We must,
I mean, not fail to distinguish sharply between perceptual
space and spaces of conception; that is, between space in
which points, lines, planes, circles, spheres, and so on, are
material or physical dots, rods, or ropes, slabs or rough
irregular "surfaces" thereof, hoops or rings, globes or
balls (of wood or gold or marble), and so on, and a
space in which the terms point, line,... denote pure
concepts of which no instance is found, for no instance
exists, in perceptual space. Unless we make and keep
that radical distinction, we might better abandon the
discussion; but if we make the distinction clearly and do
not lose it, the question you have put can be answered
clearly and rightly.
And the answer? It is found in the following considerations.
The space of a geometry is always a conceptual space.