250
DOC.
42
SPECIAL AND GENERAL RELATIVITY
4
Relativity
those
axioms, i.e. they
are
proven.
A
proposition
is
then
correct
("true") when it
has
been
derived
in
the
recognised
manner
from
the
axioms.
The
question
of
the
"truth"
of
the
individual
geometrical propositions
is
thus reduced
to
one
of the
"truth"
of
the
axioms.
Now it
has
long
been
known that the last
ques-
tion
is not only
unanswerable
by
the methods of
geometry,
but
that
it
is
in
itself
entirely
without
meaning.
We
cannot
ask
whether it
is true
that
only
one
straight
line
goes
through two
points.
We
can
only
say
that
Euclidean
geometry
deals with
things
called
"straight lines," to
each of which
is
ascribed the
property
of
being uniquely
determined
by two points
situated
on
it.
The
concept
"true"
does
not tally
with the
assertions of
pure geometry,
because
by
the
word
"true"
we are
eventually
in
the habit of
designating
always
the
correspondence
with
a
"real"
object; geometry,
however, is
not
concerned with the
relation
of
the
ideas involved in it
to objects
of
experience,
but
only
with the
logical
connection of these ideas
among
them-
selves.
It
is not
difficult
to
understand
why,
in
spite
of
this,
we
feel
constrained
to
call
the
propositions
of
geometry
"true."
Geo-
metrical ideas
correspond
to
more or
less
exact
objects
in
nature,
and these last
are
undoubtedly
the exclusive
cause
of
the
genesis
of those ideas.
Geometry ought
to
refrain from
such
a
course,
in order
to give to
its
structure
the
largest
possible
logical
unity.
The
practice,
for
example,
of
seeing
in
a
"distance"
two
marked
positions
on a
practically
rigid
body
is
something
which
is
lodged deeply
in
our
habit of
thought.
We
are
accustomed further
to regard
three
points
as
being
situated
on a
straight
line,
if their
apparent positions
can
be