DOC. 52 GEOMETRY AND EXPERIENCE 217
GEOMETRY AND EXPERIENCE 241
The usual
answer
to
this
question
is
“No,”
but that
is not
the
right
answer.
The
purpose
of the
following
remarks
is to
show
that
the
answer
should be
“Yes.’’
I
want to
show
that without
any
extraordinary
difficulty
we can
illustrate
the
theory
of
a
finite universe
by
means
of
a
mental
picture
to
which,
with
some
practice, we
shall
soon
grow
accustomed.
First of
all,
an
observation of
epistemological nature.
A
geometrical-physical theory
as
such
is
incapable
of
being
directly
pictured, being merely
a
system
of
concepts.
But these
concepts
serve
the
purpose
of
bringing
a
multiplicity
of real
or imaginary
sensory
experiences
into
connection in the mind. To “visual-
ize”
a theory
therefore
means
to
bring to
mind
that
abundance
of
sensible
experiences
for which the
theory supplies
the
sche-
matic
arrangement.
In the
present
case we
have
to
ask ourselves
how
we can
represent
that behavior of solid
bodies
with
respect
to
their mutual
disposition
(contact)
which
corresponds
to
the
theory
of
a
finite universe. There
is
really
nothing
new
in what
I have
to
say
about
this;
but innumerable
questions
addressed
[32]
to
me prove
that
the
curiosity
of those who
are
interested
in
these
matters
has
not yet
been
completely
satisfied.
So,
will
the
initiated
please
pardon
me,
in that
part
of what
I
shall
say
has
long
been
known?
What
do
we
wish
to
express
when
we say
that
our space
is
infinite?
Nothing
more
than that
we
might
lay any
number
of
bodies of
equal
sizes
side
by
side
without
ever
filling
space. Sup-
pose
that
we are
provided
with
a
great
many
cubic boxes all of
the
same
size.
In accordance with Euclidean
geometry
we can
place
them
above,
beside,
and
behind
one
another
so as
to
fill
an
arbitrarily
large part
of
space;
but this
construction would
never
be
finished;
we
could
go on
adding
more
and
more
cubes with-
out
ever
finding
that
there
was no more
room.
That
is
what
we
wish
to
express
when
we say
that
space
is
infinite.
It
would
be
better
to
say
that
space
is
infinite
in relation
to
practically-rigid
bodies,
assuming
that
the laws of
disposition
for these bodies
are
given by
Euclidean
geometry.
Another
example
of
an
infinite continuum
is
the
plane.
On
a
plane
surface
we may lay squares
of
cardboard
so
that
each