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

332 MATHEMATICAL PHILOSOPHY
there exists a plane (say) containing points for the dyads
(x, y) to represent? The answer is that some of them
suppose it and some of them do not; and in this fact is
the key to the two meanings of the term, "hyperspace
of points." According to one of the meanings, a pointspace of n-dimensions is, strictly speaking, not a space
at all, but is simply and purely an n-dimensional system
of number sets (each having n numbers); and the theory
or science of such a system, verbally geometrized as I have
indicated, is not genuine geometry, but is simply a species
of n-dimensional Algebra or Analysis conducted and
couched in geometric speech. Such,-to take an example
as early as I847,-is Cauchy's Mémoir sur les lieux
analytiques where he says, "We shall call a set of n
variables an analytic point, an equation or system of
equations an analytical locus," and so on. According to
the other meaning a hyperspace of points is held to be a
genuine space; the points constituting it, though representable by number sets of n numbers each, are distinct
from, and independent of, such sets as the points of
ordinary space are distinct from, and independent of, their
representative number triads (x, y, z); and the theory of
such a space, whether the theory be built up synthetically
or analytically, is genuine n-dimensional geometrygeniune geometry of a hyperspace of points.
I am sure that in this connection you are impatient to
raise the question whether hyperspaces of points may be
said to exist, and, if we allow that they may, in what
sense of the term "exist." The question evidently
involves nice matters both of psychology and of metaphysics. Many mathematicians have not carefully considered those "nice matters" and are quite content
(because of convenience as already explained) to speak