upon this knowledge, which precedes all mathematics, the axiom
stated above is, like all other axioms, self-evident, that is, it
is the expression of a part of this _a priori_ knowledge.

The more modern interpretation:--Geometry treats of entities which
are denoted by the words straight line, point, etc. These entities
do not take for granted any knowledge or intuition whatever, but
they presuppose only the validity of the axioms, such as the one
stated above, which are to be taken in a purely formal sense, i.e.
as void of all content of intuition or experience. These axioms are
free creations of the human mind. All other propositions of geometry
are logical inferences from the axioms (which are to be taken in
the nominalistic sense only). The matter of which geometry treats
is first defined by the axioms. Schlick in his book on epistemology has
therefore characterised axioms very aptly as "implicit definitions."

This view of axioms, advocated by modern axiomatics, purges mathematics
of all extraneous elements, and thus dispels the mystic obscurity
which formerly surrounded the principles of mathematics.

But a presentation of its principles thus clarified makes it also
evident that mathematics as such cannot predicate anything about
perceptual objects or real objects. In axiomatic geometry the words
"point," "straight line," etc., stand only for empty conceptual
schemata. That which gives them substance is not relevant to
mathematics.

Yet on the other hand it is certain that mathematics generally,
and particularly geometry, owes its existence to the need which
was felt of learning something about the relations of real things