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

LECTURE XVI
Hyperspaces
MEANING OF DIMENSIONALITY-SPACES OF FOUR OR
MORE DIMENSIONS-THE MODE OF THEIR EXISTENCE
-DISTINCTION OF IMAGINATION AND CONCEPTION
-LOGICAL EXISTENCE AND SENSUOUS EXISTENCEOPEN AVENUES TO UNIMAGINABLE WORLDS.
IT is the aim of this lecture to explain in simple ways
what mathematicians mean by Hyperspace and to convey
some sense of the scientific and philosophic importance of
the concept which the term denotes. Let us understand
in the first place that the terms,-"hyperspace," "multidimensional space," "space of n dimensions," "n-dimensional space," and some other readily recognized variants
upon them,-are but different names for one and the
same idea; they are employed interchangeably as equivalents; and the like may be said of the terms,-"geometry
of hyperspace," "multi-dimensional geometry," "geometry of n dimensions," and their variants,-each of which
simply denotes the geometry, or the science, of a space
having more than three dimensions.
The concept of hyperspace, though it is a modern
notion, is not strictly new,-it goes back three or four
generations and is now, among enlightened mathematicians, as classic and orthodox as the ordinary multiplication table. Though only a short while ago it was re317