DOC. 52 GEOMETRY AND EXPERIENCE 215
GEOMETRY AND EXPERIENCE
239
of
course,
be
objected
that
a
construction
composed
of solid rods
departs
the
more
from
ideal
rigidity
the
greater
its
spatial ex-
tent.
But it
will
hardly
be
possible,
I think,
to assign
fundamen-
tal
significance
to
this
objection.
Therefore
the
question
whether the
universe
is
spatially
finite
or
not
seems
to
me an
entirely meaningful question
in
the
sense
of
practical
geometry.
I
do
not
even
consider
it
impossible
that
this
question
will be
[25]
answered before
long
by astronomy.
Let
us
call
to
mind what
the
general theory
of
relativity
teaches
in
this
respect.
It offers
two
possibilities:
1.
The
universe
is
spatially
infinite. This
is
possible
only
if
in the
universe the
average spatial density
of matter,
concen-
trated
in the
stars, vanishes, i.e.,
if the
ratio
of the total
mass
of
the
stars to
the
volume of
the
space
through
which
they are
scat-
tered
indefinitely
approaches zero as
greater
and
greater vol-
umes are
considered.
[26]
2.
The universe
is
spatially
finite.
This
must
be
so,
if there
exists
an average density
of the
ponderable
matter
in
the
uni-
verse
which
is
different
from
zero.
The smaller that
average
density,
the
greater
is
the volume of
the
universe.
I
must not
fail
to
mention that
a
theoretical
argument
can
be
adduced in
favor of
the
hypothesis
of
a
finite universe.
The
general
theory
of
relativity
teaches
that the
inertia
of
a
given
body
is greater
as
there
are more
ponderable
masses
in
prox-
imity
to
it;
thus
it
seems
very
natural
to
reduce
the
total inertia
of
a
body to
interaction between it and
the
other
bodies
in the
universe,
as
indeed,
ever
since Newton’s
time,
gravity
has
been
completely
reduced
to
interaction
between bodies. From
the
equations
of
the
general
theory
of
relativity
it
can
be
deduced
that
this
total reduction
of
inertia
to
interaction between
masses
-as
demanded
by
E.
Mach,
for
example-is
possible
only
if the
[27]
universe
is
spatially
finite.
Many physicists
and
astronomers
are
not
impressed
by
this
argument.
In
the
last
analysis,
experience
alone
can
decide
[28]
which
of the
two
possibilities
is realized
in
nature.
How
can
experience
furnish
an
answer?
At
first
it
might seem
possible
to
determine the
average
density
of
matter by
observation
of