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