Four-Dimensional Vistas | Page 3

movements of the stars, and tells of their
constitution; but the fact of their singing together, and that "such
harmony is in immortal souls," it leaves to poet and philosopher. The
timbre, loudness, pitch, of musical tones, is a concern of science; but
for this a Beethoven symphony is no better than the latest ragtime air
from the music halls. In brief, science deals only with phenomena, and
its gift to man is power over his material environment.
MATHEMATICS
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.
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,
denied by common sense--The Fourth Dimension of Space.
OUR SENSE OF SPACE
Space has been defined as "room to move about." Let us accord to this
definition the utmost liberty of interpretation. Let us conceive of space
not alone as room to move ponderable bodies in, but as room to think,
to feel, to strike out in unimaginable directions, to overtake felicities
and knowledges unguessed by experience and preposterous to common
sense. Space is not measurable: we attribute dimensionality to space
because such is the method of the mind; and that dimensionality we
attribute to space is progressive because progression is a law of the
mind. The so-called dimensions of space are to space itself as the steps

that a climber cuts in the face of a cliff are to the cliff itself. They are
not necessary to the cliff: they are necessary only to the climber.
Dimensionality is the mind's method of mounting to the idea of the
infinity of space. When we speak of the fourth dimension, what we
mean is the fourth stage in the apprehension of that infinity. We might
as legitimately speak of a fifth dimension, but the profitlessness of any
discussion of a fifth and higher stages lies in the fact that they can be
intelligently approached only through the fourth, which is still largely
unintelligible. The case is like that of a man promised an increase of
wages after he had worked a month, who