DOC. 52 GEOMETRY AND EXPERIENCE 209
GEOMETRY AND
EXPERIENCE
233
another
department
of science would
not
need
to envy
the
mathematician
if the
propositions
of mathematics referred
to
objects
of
our
mere
imagination,
and
not to objects
of
reality.
For
it
cannot
occasion
surprise
that different
persons
should
arrive
at
the
same
logical
conclusions when
they
have
already
agreed upon
the
fundamental
propositions
(axioms), as
well
as
the methods
by
which other
propositions
are
to
be
deduced
therefrom.
But
there
is
another
reason
for the
high repute
of
mathematics, in
that it
is
mathematics which affords
the
exact
natural sciences
a
certain
measure
of
certainty, to
which with-
out
mathematics
they
could
not
attain.
At this
point
an enigma presents
itself which in all
ages
has
[p.
124]
agitated
inquiring
minds.
How
can
it
be
that
mathematics,
be-
ing
after
all
a
product
of
human
thought
which
is
independent
of
experience,
is
so
admirably appropriate
to
the
objects
of
reality?
Is
human
reason,
then,
without
experience,
merely by
taking thought,
able
to
fathom
the
properties
of
real
things?
In
my opinion
the
answer
to
this
question is,
briefly,
this:
as
far
as
the
propositions
of
mathematics refer
to reality, they
are
not
certain; and
as
far
as
they
are
certain,
they
do
not
refer
to
reality.
It
seems
to
me
that
complete clarity
as
to
this
state
of
things
became
common
property only through
that trend in
mathematics which
is
known
by
the
name
of
“axiomatics.” The
progress
achieved
by
axiomatics consists
in
its
having
neatly
separated
the
logical-formal
from its
objective or
intuitive
con-
tent; according to
axiomatics the
logical-formal
alone forms the
subject matter
of
mathematics,
which
is
not
concerned with
the intuitive
or
other
content
associated with the
logical-formal.
Let
us
for
a
moment
consider from this
point
of
view
any
axiom of
geometry,
for
instance,
the
following:
through
two
points
in
space
there
always
passes
one
and
only
one
straight
line. How
is
this axiom
to
be
interpreted
in the older
sense
and in the
more
modern
sense?
The older
interpretation:
everyone
knows what
a
straight
line
is,
and what
a
point is.
Whether
this
knowledge springs
from
an
ability
of the human mind
or
from
experience,
from
some
cooperation
of the
two
or
from
some
other
source,
is
not
for
the
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