Mathematics, then, opens up ever new horizons, and its achievements
during the past one hundred years give to thought the very freedom
it seeks. But if science is dispassionate, mathematics is even more
austere and impersonal. It cares not for teeming worlds and hearts
insurgent, so long as in the pure clarity of space, relationships
exist. Indeed, it requires neither time nor space, number nor
quantity. As the mathematician approaches the limits already
achieved by study, the colder and thinner becomes the air and the
fewer the contacts with the affairs of every day. The Promethean
fire of pure mathematics is perhaps the greatest of all in man's
catalogue of gifts; but it is not most itself, but least so, when,
immersed in the manifoldness of phenomenal life, it is made to serve
purely utilitarian ends.


INTUITION

Common sense, immersed in the mere business of living, knows no more
about life than a fish knows about water. The play of reason upon
phenomena dissects life, and translates it in terms of inertia. The
pure logic of mathematics ignores life and disdains its limitations,
leading away into cold, free regions of its own. Now our desire for
freedom is not to vibrate in a vacuum, but to live more abundantly.
_Intuition_ deals with life directly, and introduces us into
life's own domain: it is related to reason as flame is related to
heat. All of the great discoveries in science, all of the great
solutions in mathematics, have been the result of a _flash_ of
intuition, after long brooding in the mind. _Intuition illumines_.
Intuition is therefore the light which must guide us into that
undiscovered country conceded by mathematics, questioned by science,