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

408 MATHEMATICAL PHILOSOPHY
fled for over twenty centuries. Baffled by what? By a
psychology which recognized no space except our sensuous
space, which believed that our sensous space is geometrizable, that Euclid's axioms are "self-evident" truths
regarding it, and that his Elements embodies an exact
description of it; by a psychology which, therefore, could
not contemplate even the possibility of non-Euclidean
geometry and which, when such a geometry was at length
devised in spite of it, insisted upon trying its claims in the
courts of sensibility and perception and imagination, not
knowing that, in questions regarding the logical validity
of geometric science, those courts are entirely without
jurisdiction.
Let me allude briefly to another branch of modern
mathematics-projective geometry. It was invented, we
have seen, in the seventeenth century, lost, forgotten, and
re-invented in the nineteenth. Why not before-centuries
before? What was in the way? Logic? Not primarily;
it was psychology-a false psychology of mathematics.
The invention, as you know, required the conception of
infinitely distant points and the conception of lines and
planes such that, if parallel, they meet in those points. But
how could that happen? How could parallels meet?
They could not, it was said,-it was psychologically impossible,-the possibility was denied by sense, denied by
perception, and, most conclusive of all, denied by imagination.
I have just now mentioned "infinitely" distant points.
We are thus reminded of the modern concept of infinity,
-of infinite classes, ensembles, sets, or manifolds,-the
subject of Lecture XV. We have seen that this great
concept, though it is classic today, was born but yesterday.
Why not a thousand or two thousand years ago? Well,