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

860 MATHEMATICAL PHILOSOPHY
What is the space? It is the class of conceptual entities
(no matter what we call them, points, lines, and so on,
or jabberwockies) which satisfy or verify the geometry's
postulates and about which the geometry is therefore a
reasoned discourse. The geometry is, therefore, true in
the sense of giving an exact account of space, where by
"space" is meant the space of that geometry. I say "of
that geometry" for, you see, two geometries which contradict one another in one or more respects have different spaces. The answer to your question is, then, this:
Euclidean geometry, Lobachevskian geometry, and Riemannian geometry are each of them true in the sense that
each of them gives an exact account of its own space,
which is a conceptual space.
But what of perceptual space? What, I mean, of
that all-enveloping region or room or spread which is
revealed to us by touch and sight and hearing and the
sense of muscular movement? Is one of the three geometries true in the sense of giving an exact account of this
space? The question is a fallacy of interrogation; it implies, that is, that our perceptual space is a thing of which
an exact geometric account is possible; but it is not such
a thing-perceptual space is not, rightly speaking, geometrizable. Wherein it fails to be so, is easy to make
clear. Consider, for example, three of its "lines," say
pencils or rods, 1I, 3, l. Compare their "lengths" perceptually, which is the only way in which such "lengths"
can be compared. Compare 1, with 1, then 12 with 13,
and then li with 13. You know what may happen, for it
is a common phenomenon of such comparison of perceptual things. It may happen that the "length" of hi
is equal to (indistinguishable from) the "length" of 1,,
that the "length" of 12 is equal to the "length" of 4s,