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

THE PSYCHOLOGY OF MATHEMATICS 405
of sensible or perceptible things, must at all events be
concepts of things that we can imagine. The evidence
supporting my answer is unmistakable and conclusive. Let
us consider some of it. I am here greatly indebted to
Professor Manning's admirable "Introduction" to his
Geometry of Four Dimensions. In the following quotations from various authors cited by him, I shall take the
liberty, for it will be helpful, to italicise some of their
words.
In the De Caelo of Aristotle we are told that "The
line has magnitude in one way, the plane in two ways,
and the solid in three ways, and beyond these there is no
other magnitude because the three are all." It is plain
that the "all" is an "all" for imagination, not for conception. And we are further told that "There is no
transfer into another kind, like the transfer from length
to area and from area to solid." The statement is true
for perception and for imagination; but for thought or
conception, it is false.
For another instance in point consider the following
statement (of the sixth century, A. D.) found in the
Commentaries of Simplicius: "The admirable Ptolemy
in his book On Distance well proved that there are no
more than three dimensions, because of the necessity that
distances should be defined, and that the distances defined
should be taken along perpendicular lines, and because
it is possible to take only three lines that are mutually perpendicular, two by which the plane is defined and a third
measuring depth; so that if there were any other distance
after the third it would be entirely without measure and
without definition. Thus Aristotle seemed to conclude
from induction that there is no transfer into another
magnitude, but Ptolemy proved it." Here it is again