At this point an enigma presents itself which in all ages has agitated
inquiring minds. How can it be that mathematics, being after all
a product of human thought which is independent of experience, is
so admirably appropriate to the objects of reality? Is human reason,
then, without experience, merely by taking thought, able to fathom
the properties of real things.

In my opinion the answer to this question is, briefly, this:--As far
as the laws of mathematics refer to reality, they are not certain;
and as far as they are certain, they do not refer to reality.
It seems to me that complete clearness as to this state of things
first became common property through that new departure in mathematics
which is known by the name of mathematical logic or "Axiomatics."
The progress achieved by axiomatics consists in its having neatly
separated the logical-formal from its objective or intuitive
content; according to axiomatics the logical-formal alone forms
the subject-matter of mathematics, which is not concerned with the
intuitive or other content associated with the logical-formal.

Let us for a moment consider from this point of view any axiom of
geometry, for instance, the following:--Through two points in space
there always passes one and only one straight line. How is this
axiom to be interpreted in the older sense and in the more modern
sense?

The older interpretation:--Every one knows what a straight line
is, and what a point is. Whether this knowledge springs from an
ability of the human mind or from experience, from some collaboration
of the two or from some other source, is not for the mathematician
to decide. He leaves the question to the philosopher. Being based