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

334 MATHEMATICAL PHILOSOPHY
handling space of four dimensions as if it were conceivable space.
By " conceivable " he here means actual.
It is noteworthy that in the difference between affirming the existence of hyperspaces of points and merely
speaking as if such spaces existed we have a striking
illustration of the Kantian distinction between postulating
and feigning-between hypothesis and fiction-a muchneglected distinction justly and stoutly insisted upon by
Vaihinger in his great Philosophie des Als Ob as being of
fundamental importance in the philosophy of science and
the philosophical history of thought in general. The
distinction is indeed very important and very wide in its
application. One must be pretty dull not to perceive
that the difference is radical between saying, for example,
"there is an infinite and all-wise God and hence we
ought to live so-and-so " and saying "we ought to live
as if there were such a God "; or between saying " there
is a universal ether having such-and-such properties and
that is why light behaves in such-and-such a way" and
saying " light behaves as if there were an ether having
such-and-such properties"; or between saying "the
human soul is immortal and hence we ought to live soand so " and saying " we ought to live as if the human soul
were immortal "; and so on throughout the whole range of
thought. A postulate or hypothesis, as here understood,
is a proposition and is true or false; but a fiction is not a
proposition and is neither true nor false. It would be
very enlightening to make a survey of scientific "hypotheses " with a view to ascertaining which of them are
genuine hypotheses and which ones are only fictionsonly as ifs. There can be no doubt that many a scientific
worker would be astonished at the results of such g