ignorant of points, lines, surfaces, and solids. We are all aware that
a short line is not a point, a narrow surface is not a line, and a thin
solid is not a mere surface. A door so thin as to have only one side
would be repudiated by every man of sense as a monstrosity. When the
geometrician defines for us the point, the line, the surface, and the
solid, and when he sets before us an array of axioms, or self-evident
truths, we follow him with confidence because he seems to be telling us
things that we can directly see to be reasonable; indeed, to be telling
us things that we have always known.

The truth is that the geometrician does not introduce us to a new world
at all. He merely gives us a fuller and a more exact account than was
before within our reach of the space relations which obtain in the
world of external objects, a world we already know pretty well.

Suppose that we say to him: You have spent many years in dividing up
space and in scrutinizing the relations that are to be discovered in
that realm; now tell us, what is space? Is it real? Is it a thing, or
a quality of a thing, or merely a relation between things? And how can
any man think space, when the ideas through which he must think it are
supposed to be themselves non-extended? The space itself is not
supposed to be in the mind; how can a collection of non-extended ideas
give any inkling of what is meant by extension?

Would any teacher of mathematics dream of discussing these questions
with his class before proceeding to the proof of his propositions? It
is generally admitted that, if such questions are to be answered at
all, it is not with the aid of geometrical reasonings that they will be
answered.