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

418 MATHEMATICAL PHILOSOPHY
purpose it will be sufficient, to begin with the simple case
where n =2 and to observe what happens when n is taken
larger and larger.
It is essential to note the fact that the above-given
symmetric interpretations are conceptual-interpretations
by, in and for pure thought. It is equally essential to
note that our "spatial" imagination or intuition or
picturing power attempts to imitate them,-attempts
to make in its way parallel interpretations,-interpretations, that is, which correspond to or match in detailed
one-to-one fashion the thought interpretations. In other
words, imagination endeavors to find in its own domain
images, pictures or objects to match the conceptual
objects-points, S_-1's, lines, S,_2's, and so on-which
figure in the interpretations by thought. We are going to
see that this enterprise of imagination succeeds fairly
well if n be small, that its prosperity decreases as n
increases, and that its failure is well-nigh complete when
n is taken very large. At the same time, we shall see
that, in the case of interpretations by thought, symmetry
never fails in even the least degree, no matter how high
the dimensionality of the space in which we are operating.
Let us for convenience denote any two reciprocal
thought-interpretations by the symbols T(u) and T(x),
the former when the u's are fixed and the x's are variable,
and the latter when the x's are fixed and the u's are
variable; and let I(u) and I(x) denote the corresponding
pair of interpretations essayed by imagination.
Consider first the simple case where n=2; S2, the
space of operation, is a plane; equation (I) now is:
(I) UlX1+U2X2+I =0. What are T(u) and T(x)? The
former is a conceptual range of points; the latter, a
conceptual pencil of lines. What are I(u) and I(x)?