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

NON-EUCLIDEAN GEOMETRIES 363
and serviceable; the reason is that, though the formulas
of any one of the geometries differ radically from their
correspondents in either of the other two, yet they differ
in such a way that the difference could not crop out in
any physical structure unless the latter were vastly larger
than any our small planet admits of. We may say,
then, that all three of the geometries are pragmatically,
or instrumentally, true of our perceptual space-the
space of sensuous experience, though, as we have seen,
no one of them and no other geometry is or can be,
rightly speaking, the geometry or a geometry of perceptual space.
Since all three of the geometries in question are pragmatically true of our perceptual space, why is it that in
practical work, like bridge building and the like, the
Euclidean variety is employed exclusively? It is because
the Euclidean formulas are simpler, easier to use than
the others. What, if any, is the epistemological significance of this fact? The question seems important. I do
not know the answer. Maybe one of you will discover it.
Is non-Euclidean geometry always 3-dimensional?
No; like the Euclidean variety, it may have any number
of dimensions from one up.
What of Einstein geometry? The answer is implicit
in what has been said. "Einstein geometry" is not geometry,-not yet, at all events,-it is a figure of speech, convenient for experts, misleading for laymen. That is not
a comment upon the doctrine of Relativity regarded as
being,-what it is,-a physical theory.
The advent of non-Euclidean geometry is, I have said,
one of the gravest events in the history of thought. It
has been tragic as well. The two facts are connected.
Thirty years ago, I visited a locally eminent professor of