210 DOC. 52 GEOMETRY AND EXPERIENCE

234

CONTRIBUTIONS TO SCIENCE

mathematician

to

decide. He leaves the

question to

the

philoso-

pher.

Being

based

upon

this

knowledge,

which

precedes

all

mathematics,

the

axiom stated above

is,

like all

other

axioms,

self-evident,

that

is,

it

is

the

expression

of

a

part

of

this

a

priori

knowledge.

The

more

modern

interpretation:

geometry treats

of

objects

which

are

denoted

by

the words

straight line,

point, etc.

No

knowledge

or

intuition

of these

objects

is

assumed

but

only

the

validity

of the

axioms,

such

as

the

one

stated

above,

which

are

to

be taken

in

a

purely

formal

sense,

i.e.,

as

void of all

content

of intuition

or experience.

These

axioms

are

free creations

of

the human mind.

All

other

propositions

of

geometry are

logical

inferences from the axioms

(which

are

to

be

taken in the nomi-

nalistic

sense only).

The

axioms

define

the

objects

of which

geometry treats.

Schlick in his book

on

epistemology

has there-

[5]

fore characterized axioms

very aptly as

“implicit

definitions.”

[p.

125]

This

view of

axioms,

advocated

by

modern

axiomatics,

purges

mathematics

of all

extraneous

elements,

and thus

dispels

the

mystic

obscurity

which

formerly

surrounded

the

basis of mathe-

matics. But such

an

expurgated exposition

of mathematics

makes it also

evident that

mathematics

as

such

cannot

predicate

anything

about

objects

of

our

intuition

or

real

objects.

In

axiomatic

geometry

the words

“point,” “straight

line,”

etc.,

stand

only

for

empty

conceptual

schemata.

That

which

gives

them

content is not

relevant

to

mathematics.

Yet

on

the

other

hand

it

is

certain that

mathematics

generally,

and

particularly

geometry, owes

its existence

to

the

need

which

was

felt of

learning

something

about the behavior

of real

ob-

jects.

The

very

word

geometry,

which,

of

course, means

earth-

measuring,

proves

this. For

earth-measuring

has

to

do with the

possibilities

of

the

disposition

of

certain

natural

objects

with

respect

to

one

another,

namely,

with

parts

of the

earth,

measur-

ing-lines, measuring-wands,

etc.

It

is

clear that the

system

of

concepts

of

axiomatic

geometry

alone

cannot

make

any asser-

tions

as

to

the

behavior

of real

objects

of this kind, which

we

will call

practically-rigid

bodies.

To be able

to

make such

asser-

tions,

geometry

must

be

stripped

of its

merely logical-formal

[6]

[7]