that if, from any collection of things, some were taken away, the
number of things left must always be less than the original number of
things. This assumption, as a matter of fact, holds only of finite
collections; and the rejection of it, where the infinite is concerned,
has been shown to remove all the difficulties that hitherto baffled
human reason in this matter." And he adds in another place: "To
reconcile us, by the exhibition of its awful beauty, to the reign of
Fate ... is the task of tragedy. But mathematics takes us still
further from what is human, into the region of absolute necessity, to
which not only the actual world, but every possible world, must
conform; and even here it builds a habitation, or rather finds a
habitation eternally standing, where our ideals are fully satisfied
and our best hopes are not thwarted. It is only when we thoroughly
understand the entire independence of ourselves, which belongs to this
world that reason finds, that we can adequately realise the profound
importance of its beauty."

Mathematics seems to have a value for Mr. Russell akin to that of
religion. It affords a sanctuary to which to flee from the world, a
heaven suffused with a serene radiance and full of a peculiar
sweetness and consolation. "Real life," he writes, "is to most men a
long second-best, a perpetual compromise between the ideal and the
possible; but the world of pure reason knows no compromise, no
practical limitations, no barrier to the creative activity embodying
in splendid edifices the passionate aspiration after the perfect from
which all great work springs. Remote from human passions, remote even
from the pitiful laws of nature, the generations have gradually
created an ordered cosmos where pure thought can dwell as in its
natural home, and where one, at least, of our nobler impulses can
escape from the dreary exile of the actual world." This study is one