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

[4]

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