An introduction to mathematics, by A. N. Whitehead.

GEOMETRY 243
minds. The abstract logical form of the
propositions when fully stated is, "If any
collections of things have such and such
abstract properties, they also have such and
such other abstract properties." But what
appears before the mind's eye is a collection
of points, lines, surfaces, and volumes in the
space: this example inevitably appears, and
is the sole example which lends to the proposition its interest. However, for all its overwhelming importance, it is but an example.
Geometry, viewed as a mathematical
science, is a division of the more general
science of order. It may be called the
science of dimensional order; the qualification "dimensional" has been introduced
because the limitations, which reduce it to
only a part of the general science of order,
are such as to produce the regular relations
of straight lines to planes, and of planes to
the whole of space.
It is easy to understand the practical importance of space in the formation of the
scientific conception of an external physical
world. On the one hand our space-perceptions are intertwined in our various sensations and connect them together. We normally judge that we touch an object in the
same place as we see it; and even in abnormal cases we touch it in the same space
as we see it, and this is the real fundamental