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