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

GEOMETRY 241
latter. The number of the archangels can be
counted just because they are things. When
we once know that their names are Raphael,
Gabriel, and Michael, and that these distinct
names represent distinct beings, we know
without further question that there are three
of them. All the subtleties in the world
about the nature of angelic existences cannot alter this fact, granting the premisses.
But we are still quite in the dark as to
their relation to space. Do they exist in
space at all? Perhaps it is equally nonsense
to say that they are here, or there, or anywhere, or everywhere. Their existence may
simply have no relation to localities in space.
Accordingly, while numbers must apply to
all things, space need not do so.
The perception of the locality of things
would appear to accompany, or be involved
in many, or all, of our sensations. It is independent of any particular sensation in the
sense that it accompanies many sensations.
But it is a special peculiarity of the things
which we apprehend by our sensations. The
direct apprehension of what we mean by the
positions of things in respect to each other
is a thing sui generis, just as are the apprehensions of sounds, colours, tastes, and smells.
At first sight therefore it would appear that
mathematics, in so far as it includes geometry
in its scope, is not abstract in the sense in