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

HYPEUSPACES 838
as if the hyperspaces in question exist, without thereby
intending either to affirm or to deny that they do exist
in fact. After much reflection I have myself no longer
any doubt in the premises, and in my Human Worth of
Rigorous Thinking, p. 256, I have stated my conviction
in the words: hyperspaces have every kind of existence that
may be warrantably attributed to the space of ordinary
geometry. The considerations that have led me to that
conclusion are set forth at sufficient length in the work
cited and need not be restated here.
In relation to the matter in hand, note carefully the
sharp difference of temper, attitude and interest between
the following two classes of mathematicians: those of the
one class, primarily interested in geometry, affirm the
existence of a point-space of n dimensions and then investigate its properties-build up its geometry-by the
algebraic or analytic method-by applying, that is, the
theory of n independent numerical variables (say, x1, x2,
x,... Xn) to the postulated space; those of the other
class, primarily interested in algebra or analysis, employ,
in their discourse about the system of n variables, the
nomenclature of the geometry of a point-space of n dimensions as if there were such a space, but do not affirm its
existence. (Of course the former class are not obliged
to employ the analytic method nor are the latter class
obliged to employ geometric speech.) Note the " as if"
in the following extract from J. J. Sylvester's "A Plea
for the Mathematician" (Mathematical Papers, Vol. II):
Dr. Salmon in his extension of Chasles' theory of
characteristics to surfaces, Mr. Clifford in a question
of probability, and myself in my theory of partitions,
and also in my paper on barycentric projection, have all
felt and given evidence of the practical utility of