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

TRUTH AND THE CRITIC'S ART 133
doctrines derivable from them, having them for matrices,
and owning their forms. The view, as you will readily
see upon a little reflection, is recommended and confirmed
by its harmony with many an insight similarly or otherwise gained and usually justified in other terms.
It accords, for exarnple, with the splendid mot of
Bertrand Russell that "mathematics is the science in
which one never knows what one is talking about nor
whether what one says is true"; for a doctrinal function,
as we have said so often, has no determinate subjectmatter and, without losing its integrity as a function,
might conceivably be not even verifiable by any of the
subject.matters in our world.
It accords with another just saying (before quoted)
of the same author that "pure logic, and pure mathematics (which is the same thing), aims at being true, in
Leibnizian phraseology, in all possible worlds, and not
only in this higgledy-piggledy job-lot of a world in which
chance has imprisoned us"; for the connection of the
theorems of a doctrinal function with its postulates,the logical lien binding the former to the latter as conclusions to premises indissolubly, forever,-depends iii
no manner or degree upon the content, the accidents, or
the vicissitudes of the "big buzzing blooming confusion"
which we call our universe.
It accords perfectly with the critical judgment, elsewhere 1 expressed, that "it is in implications and not in
applications that (pure) mathematics has its lair"; for
the very essence of a doctrinal function,-constituting of
its elements a single indestructible Form of forms,-is
that its postulates logically imply its theorems.
It accords with the often quoted definition of pure
'Human Worth of Riçorous Thinkitg, p. 303.