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

242 INTRODUCTION TO MATHEMATICS
which abstractness is ascribed to it in
Chapter I.
This, however, is a mistake; the truth
being that the "spaciness" of space does not
enter into our geometrical reasoning at all.
It enters into the geometrical intuitions of
mathematicians in ways personal and peculiar to each individual. But what enter into
the reasoning are merely certain properties of
things in space, or of things forming space,
which properties are completely abstract in
the sense in which abstract was defined in
Chapter I; these properties do not involve
any peculiar space-apprehension or spaceintuition or space-sensation. They are on
exactly the same basis as the mathematical
properties of number. Thus the space-intuition which is so essential an aid to the study
of geometry is logically irrelevant: it does
not enter into the premisses when they are
properly stated, nor into any step of the reasoning. It has the practical importance of an
example, which is essential for the stimulation
of our thoughts. Examples are equally necessary to stimulate our thoughts on number.
When we think of "two" and "three" we
see strokes in a row, or balls in a heap, or
some other physical aggregation of particular
things. The peculiarity of geometry is the
fixity and overwhelming importance of the
one particular example which occurs to our