210 DOC. 52 GEOMETRY AND EXPERIENCE
234
CONTRIBUTIONS TO SCIENCE
mathematician
to
decide. He leaves the
question to
the
philoso-
pher.
Being
based
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
objects
which
are
denoted
by
the words
straight line,
point, etc.
No
knowledge
or
intuition
of these
objects
is
assumed
but
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 nomi-
nalistic
sense only).
The
axioms
define
the
objects
of which
geometry treats.
Schlick in his book
on
epistemology
has there-
[5]
fore characterized axioms
very aptly as
“implicit
definitions.”
[p.
125]
This
view of
axioms,
advocated
by
modern
axiomatics,
purges
mathematics
of all
extraneous
elements,
and thus
dispels
the
mystic
obscurity
which
formerly
surrounded
the
basis of mathe-
matics. But such
an
expurgated exposition
of mathematics
makes it also
evident that
mathematics
as
such
cannot
predicate
anything
about
objects
of
our
intuition
or
real
objects.
In
axiomatic
geometry
the words
“point,” “straight
line,”
etc.,
stand
only
for
empty
conceptual
schemata.
That
which
gives
them
content 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 behavior
of real
ob-
jects.
The
very
word
geometry,
which,
of
course, means
earth-
measuring,
proves
this. For
earth-measuring
has
to
do with the
possibilities
of
the
disposition
of
certain
natural
objects
with
respect
to
one
another,
namely,
with
parts
of the
earth,
measur-
ing-lines, measuring-wands,
etc.
It
is
clear that the
system
of
concepts
of
axiomatic
geometry
alone
cannot
make
any asser-
tions
as
to
the
behavior
of real
objects
of this kind, which
we
will call
practically-rigid
bodies.
To be able
to
make such
asser-
tions,
geometry
must
be
stripped
of its
merely logical-formal
[6]
[7]