The gift of pure mathematics, on the other hand, is primarily to the
mind and spirit: the fact that man uses it to get himself out of his
physical predicaments is more or less by the way. Consider for a
moment this paradox. Mathematics, the very thing common sense swears
by and dotes on, contradicts common sense at every turn. Common
sense balks at the idea of _less than nothing_; yet the _minus_
quantity, which in one sense is less than nothing in that something
must be added to it to make it equal to nothing, is a concept
without which algebra would have to come to a full stop. Again, the
science of quaternions, or more generally, a vector analysis in
which the progress of electrical science is essentially involved,
embraces (explicitly or implicitly) the extensive use of _imaginary_
or _impossible_ quantities of the earlier algebraists. The very
words "imaginary" and "impossible" are eloquent of the defeat of
common sense in dealing with concepts with which it cannot
practically dispense, for even the negative or imaginary solutions
of imaginary quantities almost invariably have some physical
significance. A similar statement might also be made with regard to
_transcendental_ functions.
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.