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

HYPERSPACES 335
critical survey. An excellent clue to Vaihinger's work is
found in Dr. W. B. Smith's penetrating review of it in
The Literary Review (N. Y. Evening Post), July 9, I92I.
The hyperspaces of points are unimaginable worldsunimaginable for us humans, 1 mean, in our present stage
of development-but they are thoroughly conceivable
worlds; and for mathematical purposes nothing is
demanded but thorough conceivability. The importance
of that fact is fundamental. Experience has taught me
that it is hard to drive the fact home to the average
understanding. Wherever the distinction involved in the
fact is not understood by "critics," whether scientific
or literary or philosophic, criticism is blind and worse
than futile, being at once misled and misleading, confused
and confusing. It is important to observe and to bear
in mind that, with respect to the great powers, or types,
of mental activity,-Sensibility (or Sense-perception),
Imagination, Conception,-we humans fall into three
classes: there are those who have the first power but
little of the second; there are those who have the first and
the second powers but little of the third; and there are
those who have in good measure the three powers. The
second class is related to the third very much as the first
is related to the second. Beware of the first two classes,
-they can give you neither science nor genuine philosophy
nor,-properly speaking,-criticism. Are they aware of
their limitations? No; at all events not keenly. How
could they be?
There are various avenues by which beginners may
approach those unimaginable worlds and enter them; and
that is a blessing, for the worlds are replete with wonders.
One of the ways is that followed by Professor H. P. Manning, for example, in his Geometry of Four Dimensions,