carnap and weyl on the foundations of geometry and relativity theory

8/10/2019 CARNAP AND WEYL ON THE FOUNDATIONS OF GEOMETRY AND RELATIVITY THEORY

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M IC HAEL FR IEDM AN

C A R N A P A N D W E Y L O N T H E F O U N D A T I O N S

O F G E O M E T R Y A N D R E L A T I V I T Y T H E O R Y *

At the end of the nineteenth century and, even more, in the early years

of the twentieth century the philosophy of geometry experienced unpre-

cedented pressures and tensions. For the revolutionary new developments

in the mathematical foundations of geometry and, even more, the appli-

cation of many of these new mathematical ideas to nature in Einstein's

theory of relativity seemed to suggest irresistibly that all earlier attempts

to comprehend philosophically the relationship between geometry on the

one hand and our experience of nature on the other were radically mis-

taken. In particular, the Kantian understanding of this relationship -

according to which geometry functions as an a priori transcendental

condition of the possibility of our scientific experience of nature, and

space is correspondingly viewed as a pure form of our sensible intuition

- seemed to be wholly undermined by the new mathematical-physical

developments. The question then - for philosophers, mathematicians, and

physicists alike - was what new understanding of the relationship between

geometry on the one hand and our experience of nature on the other was

to be put in its place.

The variety of mutually incompatible answers that were given to this

question is remarkable, in that precisely this variety reflects the true

complexity - the manifold pressures and tensions - engendered by the

radically new philosophical, mathematical, and physical situation. It is

especially remarkable, in particular, how seldom a straightforwardly em-

piricist understanding of the relationship between geometry and experi-

ence - according to which geometry is an empirical theory like any other

whose validity is straightforwardly verified or falsified by experience - was

represented. Pure mathematicians and logicians, for example, who were

most concerned to reject the Kantian conception that geometrical reason-

ing requires construction in pure intui tion (so that pure geometry is a

synthetic science), argued that geometry is first and foremost a purely

formal or analytic science having no intrinsic relation to experience what-

soever. Neokantian philosophers, on the other hand, argued that, whereas

the new mathematical-physical developments indeed undermined a strictly

Kantian conception of the intuitive certainty and experience-constituting

character of Euclidean geometry in particular, Kant's most basic and

fundamental insight into the transcendental function of space and geo-

metry within physics necessarily remained valid. Indeed, even for more

physically and empirically oriented thinkers such as Helmholtz it seemed

that there must be

som

important respect in which Kant's conception of

space and/or geometry as presupposed by empirical physics rather than

127

Erkenntnis

42: 247-260 1995.

1995 Kluwer Academic Publishers Printed in the Netherlands


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248 M I C H A E L F R I E D M A N

suppor t e d o r r e f u t e d by e mpi r i c a l phys i c s r e ma ins t r ue . I t s t i l l s e e me d

t h a t t h e r e m u s t b e s o m e t h i n g t o t h e i d e a t h a t w e m u s t f i r s t a s c r i b e o r

c o n t r i b u t e g e o m e t r ic a l s t r u c tu r e t o n a t u r e b e f o r e w e c a n e x t r a c t e m p i r ic a l

m a t h e m a t i c a l l a w s fr o m o u r e x p e r i e n c e o f n a t u r e . P o i n c a r 6 ' s c o n c e p t i o n

o f g e o m e t r y a s a f r e e s t i p u l a t i o n o r c o n v e n t i o n w a s s i m p l y o n e o f t h e

f o r m s i n w h i c h t h i s f u n d a m e n t a l K a n t i a n i d e a w a s p r e s e r v e d .

R ud o l f C a r na p ' s d oc to r a l d i s se r t a t ion , C a r n a p ( 1922) , i s a pa r t i c u l a rly

in t e r e s t i ng a t t e mp t t o i n j e c t o r de r i n to t h i s r a the r c ha o t i c s i t ua t ion . The

d i s s e rt a ti o n w a s w r i tt e n a t J e n a , w h e r e C a r n a p s t u d i e d th e n e w m a t h e m a -

t ic a l l o gi c w i t h F r e g e , p h i l o s o p h y w i t h th e N e o k a n t i a n B r u n o B a u c h , a n d

a l so e xpe r ime n ta l a nd the o r e t i c a l phys i c s . Af t e r f i r s t a t t e mp t ing to wr i t e

a d i s se rt a t i on in t he phys i c s de p a r tm e n t on the a x iom a t i c ba s is o f r e l a t iv i t y

t h e o r y ( w h i c h t h e p h y s i c s d e p a r t m e n t f o u n d t o o p h i l o s o p h i c a l ) C a r n a p

e n d e d u p r e c e i v i n g h i s d o c t o r a t e u n d e r B a u c h i n t h e p h i l o s o p h y d e p a r t -

m e n t . I n t h e r e su l ti n g d i s s e r ta t io n C a r n a p a t t e m p t s t o r e s o l v e th e c o n t e m -

po r a r y c on f l ic t s a nd t e ns ions i n t he f ound a t ions o f ge o m e t r y - c on f li c ts

invo lv ing m a the m a t i c i a ns , ( Ne oka n t i a n ) ph i lo soph e r s , a nd phys i ci s ts a l ike

- by c a r e f u l ly d i s ti ngu i sh ing a m ong th r e e d i s ti nc t type s o f spa c e : f o rma l

intuitive

a n d

physical

spa c e . C a r na p a r gue s t ha t t he d i f f e r e n t pa r t i e s

invo lve d in t he va r ious ma the ma t i c a l , ph i lo soph ic a l , a nd phys i c a l d i spu te s

a r e i n f a c t r e f e r r ing to d i f f e r e n t t ype s o f spa c e , a nd , i n t h i s wa y , t he r e i s

r e a l ly no c on t r a d i c t i on a f t e r a ll: [ a ll ] pa r t i e s we r e c o r r e c t a nd c ou ld

ha ve e a s i l y be e n r e c onc i l e d i f c l a r i t y ha d p r e va i l e d c onc e r n ing the t h r e e

d i f f e r e n t me a n ings o f spa c e ( 1922, p . 64) .

C a r n a p ' s c onc lus ion i s t ha t h i s t h i r d t y pe o f spa c e , i n tu i ti ve sp a c e , i s

syn the t i c a p r io r i a nd e xpe r i e nc e - c on s t i t u t i ng in p r e c i se ly K a n t ' s o r ig ina l

se nse . I t i s j u s t tha t w e ne e d a mo r e ge n e r a l s t r uc tu r e t ha n Ka n t ' s o r ig ina l

t h r e e d i m e n s i o n a l E u c l i d e a n s p a c e :

I t h a s a l r e a d y b e e n e x p l a i n e d m o r e t h a n o n c e , f r o m b o t h m a t h e m a t i c a l a n d p h i l o s o p h i c a l

p o i n t s o f v i e w , t h a t K a n t s c o n t e n t i o n c o n c e r n i n g t h e s i gn i f ic a n c e o f s p a c e f o r e x p e r i e n c e i s

n o t s h a k e n b y t h e t h e o r y o f n o n - E u c l i d e a n s p a c e s , b u t m u s t b e t r a n s f e r r e d f r o m t h e t h r e e

d i m e n s i o n a l E u c l i d e a n s t r u c t u r e , w h i c h w a s a l o n e k n o w n t o h i m , t o a m o r e g e n e r a l st r u c t u r e .

H o w e v e r , t o t h e q u e s t i o n o f w h a t t h i s l a t t e r i s n o w t o b e , t h e a n s w e r s a r e p a r t l y in d e t e r m i -

n a t e , i n t h a t o n l y i s o l a te d c h a r a c t e ri s t ic s o f th e t h r e e d i m e n s i o n a l E u c l i d e a n s t r u c t u r e a r e

p r o p o s e d a s r e q u i r i n g g e n e r a l i z a ti o n , a n d p a r t ly c o n t r a d i c to r y , c h i e fl y b e c a u s e o f a f a i l u re

t o d i s t i n g u i sh t h e d i f f e r e n t m e a n i n g s o f s p a c e a n d i n s u f f i ci e n t c l a ri ty a b o u t t h e c o n c e p t u a l

r e l a t i onsh i p o f t he spa c e - t ype s t he m se l ve s - e spe c i a ll y t he r e l a t i on o f t he me t r i c a l t o t he

s u p e r o r d i n a t e t o p o l o g i ca l o n e s . A c c o r d i n g t o t h e f o r e g o i n g r e f l ec t io n s , t h e K a n t i a n c o n c e p -

t i o n m u s t b e a c c e p te d . A n d , i n d e e d , t h e s p a t i a l s t r u c t u r e p o s s e ss i n g e x p e r ie n c e - c o n s t i tu t i n g

s i gn i f i c a nc e ( i n p l a c e o f t ha t suppose d by Ka n t ) c a n be p re c i s e l y spe c i f i e d a s t opo l og i c a l

i n t u i t iv e s p a c e w i t h in d e f i n it e ly m a n y d i m e n s i o n s . W e t h e r e b y d e c l a r e , n o t o n l y t h e d e t e r m i -

n a t i o n s o f t h i s s t r u c t u r e , b u t a t t h e s a m e t i m e t h o s e o f i t s f o r m o f o r d e r i n n - d i m e n s i o n a l

t opo l og i c a l

form l

spa c e ] t o be c ond i t i ons o f t he pos s i b i l i t y o f a ny ob j e c t o f e xpe r i e nc e

wha t soe ve r . (1922 , p . 67 )

B u t h o w a r e w e t o u n d e r s t a n d t h i s r e m a r k a b l e c o n c l u s i o n ?

F o r m a l s p a c e , f o r C a r n a p , i s a n a b s t r a c t r e l a t i o n a l s t r u c t u r e w h o s e

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FOUNDATIONS OF GEOMETRY AND REL ATIV ITY THEORY 249

i n i ti a l ly u n i n t e r p r e t e d p r i m i t i v e t e r m s r e c e i v e a s p e c i fi c a ll y s p a t i al i n t e r -

p r e t a t i o n i n i n t u i t i v e s p a c e . A n d i t i s h e r e , i n f a c t , t h a t t h e p r o p e r l y

s pa t i a l c ha r a c t e r o f s pa c e - t ha t i s, i ts i n t u i t i ve spatiality - i s d i s t i ng u i s h e d

f r o m a l l o t h e r r e l a t i o n a l s y s t e m s h a v i n g t h e s a m e f o r m a l s t r u c t u r e . T h i s

i n tu i t iv e i n t e r p r e t a t i o n o f f o r m a l o r p u r e l y m a t h e m a t i c a l s p a ce i s n o t y e t

a p h y si c a l o r e m p i r i c a l i n t e r p r e t a t i o n , h o w e v e r . W e o b t a i n t h e l a t te r o n l y

b y t a k i n g t h e f u r t h e r s t e p , i n C a r n a p ' s l a n g u a g e , o f " s u b o r d i n a t i n g "

[Unterordnung, Subsum tion] a c t u a l l y e x p e r i e n c e d p h y s i c a l p h e n o m e n a t o

t h e s y n t h e t i c a p r i o r i f o r m o f i n t u i t iv e s p a c e . ( B y c o n t r a s t , C a r n a p c a ll s

t h e m o r e f a m i l i a r r e l a t i o n b e t w e e n f o r m a l a n d i n t u i t i v e s p a c e " s p e c i f i -

c a t i o n "

[Einsetzung, Substitution].

C a r n a p s u m s u p t h e d i s t in c t io n b e-

t w e e n t h e s e t w o d i f f e r e n t " a p p l i c a t i o n " o r " i n t e r p r e t a t i o n " r e l a ti o n s th u s :

The relation of [form al space] to [intuitive space] is that of the species of structures with

determ inate order-properties but unde termine dobjects to a structure with these sam e order-

properties but determ inate objects - viz., intuitivelyspatial form s. The relation of [intuitive

space] to [physical space] is that of a fo rm of intuition to a structure with this form m ade

up of r eal ob jects of ex perience. (1922, p. 61)

T h e r e i s n o d o u b t , t h e n , t h a t C a r n a p i n t e n d s h is n o t i o n o f in t u i ti v e sp a c e

t o b e a g e n e r a l i z e d i n t e r p r e t a t i o n - a p p r o p r i a t e t o t h e n e w m a t h e m a t i c a l ,

p h i l o s o p h i ca l , a n d p h y s i c a l c o n t e x t - o f K a n t ' s c o n c e p t i o n o f sp a c e a s a n

a p r i o r i f o r m o f i n t u i t i o n .

C a r n a p e x p li c it ly m o d e l s h is n o t i o n o f s p a ti a l i n t u i ti o n o n E d m u n d

H u s s e r l ' s c o n c e p t o f " e s s e n t i a l i n s i g h t "

[Wesenserschauung],

a s d e v e l o p e d

e s p e c i a l l y i n H u s s c r l ( 1 9 1 3 ) . A s e x p l a i n e d b y H u s s e r l Wesenserschauung

f u n c t i o n s a s f o l l o w s . J u s t a s i n s e n s e p e r c e p t i o n w e a r e i m m e d i a t e l y

p r e s e n t e d w i t h o r i m m e d i a t e l y g iv e n a s e n s ib l e p a r t ic u l a r o r i n d i v i d u a l

( a p a r t i c u l a r c o l o r s p o t , a p a r t i c u l a r t o n e , a n i n d i v i d u a l s p a t ia l f ig u r e ) ,

s o i n Wesenserschauung w e c a n i m m e d i a t e l y g ra s p t h e u n i v e r s a l f e a tu r e s

t h a t s u c h g i v e n s e n s i b l e p a r t i c u l a r s e x e m p l i f y ( t h e g e n e r a l c o l o r , t h e g e n -

e r a l t o n e , t h e g e n e r a l s p a t i a l f i g u r e ) . S i n c e

Wesenserschauung

i s t hus

d i r e c t e d a t u n i v e r s a l f e a t u r e s r a t h e r t h a n p a r t i c u l a r i n d i v i d u a l s , i t i s

e n t i r e l y i n d e p e n d e n t o f t h e p a r t i c u l a r i n d i v i d u a l s t h a t a c t u a l l y e x i s t i n

t h e r e a l w o r l d . I t c a n , f o r e x a m p l e , f u n c t i o n ju s t a s w e l l w i t h a n i m a g i n e d

i n d i v i d u a l o r o b j e c t o f f a n t a s y a s w i t h a r e a l o r a c t u a l i n d i v i d u a l . I n d e e d ,

f o r c e rt a i n p u r p o s e s , a s i n p u r e g e o m e t r y f o r e x a m p l e , Wesenserschauung

f u n c t i o n s e v e n b e t t e r w i t h p u r e l y i m a g i n e d i n d i v i d u a l s . A n d t h e k e y

c o n c l u s i o n i s n o w t h i s : s i n c e

Wesenserschauung

i s i n d e p e n d e n t o f t h e

p a r t i c u l a r i n d i v i d u a l s t h a t a c t u a l l y e x i s t i n t h e r e a l e m p i r i c a l w o r l d , i t i s

a s o u r c e o f a p r i o r i r a t h e r t h a n e m p i r i c a l k n o w e d g e . T h u s , f o r e x a m p l e ,

t h r o u g h

Wesenserschauung

w e o b t a i n a p r i o ri k n o w l e d g e o f t h e s t r u c t u r e

o f c o l o r s p a ce , o f t o n e s p a c e , o r ( i n p u r e g e o m e t r y ) o f p u r e g e o m e t r i c a l

s p a c e . T h i s k n o w l e d g e i s a l s o s y n t h e t i c a p r i o r i r a t h e r t h a n a n a l y t i c a

p r i o r i, f o r i t d o e s n o t f o ll o w s f r o m t h e m o s t g e n e r a l t r u t h s o f p u r e f o r m a l

l o g i c a l o n e . P u r e f o r m a l l o g i c , f o r H u s s e r l , e x p r e s s e s t h e " e s s e n c e "

129


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250

M I C H A E L F R I E D M A N

[Wesen Eidos] c o m m o n t o a l l o b j e ct s o f t h o u g h t w h a t s o e v e r . T h e a p r io r i

e i d e t i c s c ie n c e s o f c o l o r , o f to n e , o r o f g e o m e t r ic a l s p a c e , h o w e v e r ,

h o l d o n l y f o r p a r t i c u l a r s u b s p e c i e s o f s u c h o b j e c t s .

C a r n a p h i m s e l f e x p l a i n s th e i d e a a s f o ll o w s :

[ H ] e r e , a s H u s s e r l h a s s h o w n , w e a r e c e r t a i n l y n o t d e a l i n g w i t h f a c ts in t h e s e n s e o f

e x p e r i e n t i a l re a l it y , b u t r a t h e r w i t h t h e e s s e n c e ( E i d o s ) o f c e r ta i n d a t a w h i c h c a n a l r e a d y

b e g r a s p e d i n it s p a r t i c u l a r n a t u r e b y b e i n g g i v e n in a s i n g l e in s t a n c e . T h u s , j u s t a s I c a n

e s t a b l is h i n o n l y a s i n g le p e r c e p t i o n - o r e v e n m e r e i m a g i n a t i o n - o f th r e e p a r t i c u l a r c o l o r s ,

d a r k g r e e n , b l u e , a n d r e d , t h a t t h e f i rs t i s b y i ts n a t u r e m o r e a k i n t o t h e s e c o n d th a n t o t h e

t h i r d , s o I f i n d b y i m a g i n i n g s p a ti a l f o r m s t h a t s e v e r a l c u r v e s p a s s t h r o u g h t w o p o i n t s , t h a t

o n e a c h s u c h c u r v e s ti ll m o r e p o i n t s l i e, t h a t a s i m p l e l i n e - s e g m e n t , b u t n o t a s u r f a c e -

e l e m e n t , i s d i v i d e d i n t w o p i e c e s b y a n y p o i n t l y in g o n i t, a n d s o o n . B e c a u s e w e a r e n o t

f o c u s s in g h e r e o n t h e i n d i v i d u a l f a c t - s h a d e o f c o l o r s e e n h e r e - n o w - b u t o n i t s a t e m p o r a l

n a t u r e , it s e s s e n c e , i t i s i m p o r t a n t t o d is t in g u i s h t h is m o d e o f a p p r e h e n s i o n f r o m i n t u i ti o n

i n t h e n a r r o w e r s e n s e , w h i c h i s f o c u s s e d o n t h e f a c t it s e lf , b y c a l l in g it e s s e n t i a l in s i g h t

[ esenserschauung] ( H u s s e r l ) . I n g e n e r a l , h o w e v e r , t h e t e r m i n t u i t i o n m a y a l so i n c l u d e

e s s e n t i a l in s i g h t , s i n c e i t i s a l r e a d y u s e d i n t h i s w i d e r s e n s e s i n c e K a n t . ( 1 9 2 2 , p p . 2 2 - 2 3 )

C a r n a p ' s d e b t t o H u s s e r l i s t h e r e f o r e c l e a r .

B u t w h i c h e x a c t l y a r e t h e t r u t h s o f g e o m e t r y r e v e a l e d b y Wesenser-

schauung?

H u s s e r l h i m s e l f s e e m s t o h a v e i n m i n d t h e t o t a l it y o f t r u th s

o f E u c l i d e a n g e o m e t r y - a s p r e s e n t e d i n H i l b e r t ' s w e l l -k n o w n a x io m a t i z -

a t io n , f o r e x a m p l e . F o r C a r n a p , b y c o n t r a s t , th e w h o l e p r o b l e m , a s i t

w e r e , is t o a d a p t K a n t ' s n o t i o n o f a f o r m o f i n t u i ti o n to t h e g e n e r a l t h e o r y

o f r e la t iv i ty . A n d h e r e C a r n a p i n t r o d u c e s a m o s t in g e n i o u s i d e a. E u c l i d e a n

g e o m e t r y is i n d e e d u n i q u e l y p r e s e n t e d t o u s in i n t u i ti o n - m o r e p r e c i s e ly ,

t h r o u g h H u s s e r l i a n Wesenserschauung. B u t t h i s v e r y E u c l i d e a n i n t u i t i o n

i s v a l i d o n l y i n s m a l l o r l i m i t e d s p a t i a l r e g i o n s , s o t h a t w e a r e i n n o w a y

i n t u i t i v e l y g i v e n , f o r e x a m p l e , t h e v a l i d i ty o f t h e p a r a l l e l p o s t u l a t e i n

g l o b a l s p a c e. R a t h e r , o u r i n t u i ti o n te ll s u s o n l y th a t t h e E u c l i d e a n a x i o m s

- w h i c h , f o r C a r n a p , a r e d e f i n i t iv e l y g i v e b y H i l b e r t - a r e s a ti s fi e d i n

t h e s m a l l e s t p a r t s o f s p a c e in R i e m a n n ' s s e n s e . I n t u i t i o n t e ll s u s , t h a t i s,

t ha t s pa c e i s

infinitesimally

E u c l i d e a n ; a n d i t i s t h i s p r o p o s i t i o n t h a t e x -

p r e s s e s t h e s y n t h e t i c a p r i o r i k n o w l e d g e c h a r a c t e r i z i n g i n t u i t i v e s p a c e .

T h e p o i n t o f p h y s ic a l s p a c e , f o r C a r n a p , is t h e n t o o r d e r a n d a r r a n g e

t h e o b j e c t s o f o u r a c t u a l e x p e r i e n c e o f n a t u r e i n t h e i n t u i ti v e sp a c e w e

h a v e a l r e a d y c o n s t r u c t e d c o m p l e t e l y a p r io r i . T h i s m a k e s s e n s e , f o r C a r -

n a p , b e c a u s e t h e H u s s e r l i a n Wesenserschauung y i e l d i n g t h e b a s i s o f o u r

a p r i o r i k n o w l e d g e o f i n t u i ti v e s p a c e a r i s e s, i n t h e f i rs t i n s t a n c e , f r o m o u r

s e n s i b l e e x p e r i e n c e o f a c t u a l s p a t i a l n a t u r a l o b j e c t s . B y c o n s i d e r i n g t h i s

e x p e r i e n c e i n i ts g e n e r a l , e s s e n t ia l , o r e i d e t i c a s p e c t s w e t h e n a r r iv e a t

a p r i o r i la w s g o v e r n i n g t h e s t r u c t u r e o f in t u i t i v e s p a c e . P h y s i c a l s p a c e -

t h e s p a c e o f p h y s ic a l t h e o r y - is m u c h m o r e t h a n a m e r e a g g r e g a t e o f

p a r t i c u l a r i n t u i t i v e l y s p a t i al e x p e r i e n c e s , h o w e v e r ; i t is r a t h e r a p r e c is e

a n d c o n s i s t e n t o r d e r i n g o f s p a t i a l o b j e c t s i n a s i n g l e m a t h e m a t i c a l s t r u c -

t u r e . A n d t h e p o i n t o f s u c h a s t r u c t u r e i s t o a s s ig n m a t h e m a t i c a l l y p re c i s e

s p a t i a l r e l a t i o n s h i p s t o n a t u r a l o b j e c ts - d e t e r m i n a t e m u t u a l a n g l e s a n d

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251

distances, for example - so that precise mathematical laws of nature can

then be formulated.

Here, however, a problem arises for Carnap. Intuitive space, as he

understands it, has only what he calls topological structure. More pre-

cisely, it has only infinitesimally Euclidean structure, so that we are given

a priori, as it were, only the entire class of all possible Riemannian

manifolds. But the kind of mathematical structure required by physical

space is a full (local and global) metrical structure. More precisely, one

particular Riemannian manifold must somehow be singled out. How is

this to be done? For Carnap, such metrical structure is introduced by a

freely stipulated convention. We can, for example, stipulate the Riemann-

Jan structure in question directly (as Euclidean, say, in the context of pre-

relativistic physics or via the Schwarzschild solution, say, in the context

of the general theory of relativity). Alternatively, we can, if we prefer,

begin from what Carnap calls a measure-st ipulation

[M a B s e t z u n g ]

roughly, the stipulation that a particular physical body is rigid - and then

indirectly determine the metrical structure through measurement. The

important point, for Carnap, is that, on either alternative, the particular

metrical structure one arrives at is in no way inherent in the actual empiri-

cal facts of nature; it expresses instead merely the outcome of our own

free choice - a choice, to be sure, which is nonetheless absolutely essential,

for otherwise we simply could not make precise mathematical determina-

tions at all.

But why exactly is metrical structure thus independent of the actual

empirical facts of nature? Carnap explains this idea through the concept

of a matt er of fact

[ T a t b e s t a n d ]

of experience. We know, from our

consideration of the

W e s e n s e r s c h a u u n g

underlying intuitive space, that

the natural objects actually given to us in experience (or even in mere

imagination) have certain necessary or a priori spatial features - just those

necessary features expressed in the structure of intuitive space. These -

and only these - are the spatial features empirically given natural objects

have according to their very nature, as it were. These formal features -

and only these - belong to what Carnap calls the necessary form of

spatial objects of experience. But, as we have seen, the formal features

in question are exhausted by what Carnap calls topological structure; no

metrical structure (beyond the

i n f i n i t e s i m a l

metrical structure) is in fact

to be found here. Such further (metrical) spatial structure is thus not

inherent in the actual given facts - the matters of fact - of experience

according to their (spatial) nature, and must instead be conventionally

imposed on these facts by us in the form of a freely chosen stipulation:

N o w , w e h a v e c a l l e d e x p e r i e n c e , i n so f a r as it i s p r e s e n t e d o n l y i n t h e u n i q u e l y n e c e s s a r y

f o r m t h a t c o n t a i n s n o f r ee l y c h o s e n s t i p u la t io n s w h a t s o e v e r , m a t t e r o f f a ct

[Ta tbes t and]

T h e r e f o r e , o n l y t h e s p a ti a l d e t e r m i n a t i o n s c o n t a i n e d i n m a t t e r s o f f ac t c a n b e c o n d i t i o n s o f

t h e p o s s i b i li t y o f e x p e r i e n c e . A n d t h e s e , a s w e h a v e s e e n , a r e o n l y t h e t o p o l o g i c a l , b u t n o t

t h e p r o j e c t i v e a n d a b o v e a l l n o t t h e m e t r i c a l r e l a ti o n s . ( 19 2 2 , p . 6 5 )

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2 5 2 M I C H A E L F R I E D M A N

C a r n a p ' s m e t r i c a l c o n v e n t i o n a l i s m i n

e r R a u m

i s t he r e f o r e e n t i r e ly un -

i q u e a n d s h o u l d n o t b e a s s im i l a te d t o a n y o f th e o t h e r t h e n c u r r e n t f o r m s

o f c o n v e n t i o n a li s m : t h a t o f D i n g l e r, P o i n c a r r , S c h li ck , o r R e i c h e n b a c h .

F o r C a r n a p ' s v e r s i o n r es t s in t h e e n d o n h i s o w n p e c u l i ar l y h y b r id c o n c e p -

t i on o f spa c e a s a f o r m o f in tu i t i on - a f o r m o f in tu i t i on ne c e s sa r i l y l a c k ing

a fu l l ( loca l and g loba l ) me t r ica l s t ruc ture .

W e c a n g a in a d e e p e r u n d e r s t a n d i n g o f C a r n a p ' s p o s i t io n b y j u x t a p o s i n g

i t w i t h t h e i n i m p o r t a n t r e s p e c t s q u i t e s im i l ar p o s i t i o n d e v e l o p e d a t r o u g h l y

t h e s a m e t i m e b y H e r m a n n W e y l. W e y l w a s o f c o u r se o n e o f t h e m o s t

p e n e t r a t i n g a n d r e m a r k a b l e c o n t r i b u t o r s t o t h e d e b a t e f o c u s s e d o n t h e

n e w s i tu a t io n i n t h e f o u n d a t i o n s o f g e o m e t r y a n d t h e t h e o r y o f re l a ti v it y

- e s p e c ia l ly r e m a r k a b l e b e c a u s e h e c o n t r i b u t e d t o t h e m a t h e m a t i c a l , t o

the phys i c a l , a nd to t he ph i lo soph ic a l a spe c t s o f t h is de ba t e . I t a ppe a r s ,

m o r e o v e r , t h a t W e y l ' s w o r k w a s c e n t ra l ly i m p o r t a n t t o C a r n a p ' s t h in k i n g

in h i s d i s se r t a t ion . W e y l ( 1918) i s r e c om m e n de d in t he fi rs t p l a c e a s a

r e f e r e n c e f o r t he ge ne r a l t he o r y o f r e l a ti v i t y , a nd , f o r the i de a tha t

E u c l i d e a n g e o m e t r y i s v a l i d i n t h e s m a l l ( p a r t i c u l a r l y i n t h e c o n t e x t o f

ge ne r a l r e l a t i v i t y ) , t he r e a de r i s r e f e r r e d to W e y l ' s Erl i iu terungen t o

R ie ma nn ( 1919) : C a r na p ( 1922 , pp . 81 , 84 ) . I t i s a l so wor th no t ing , i n

a dd i t i on , t ha t C a r na p ( 1922) is on e o f t he ve r y fe w ph i lo soph ic a l wo r ks

in t he l i t e r a tu r e on ge n e r a l r e l a t i v i ty c i t e d in W e y l ( 19225) - a long w i th

Schl ick (1917) and Cass i re r (1921) .

T h e s i m i l a r i t i e s b e t w e e n C a r n a p ' s c o n c e p t i o n a n d W e y l ' s a r e s t r i k i n g

inde e d . F o r W e y l a l so a r t i c u l a t e s a pos i t i on a c c o r d ing to wh ic h spa c e i s

e s s e n t i a l ly o r b y i ts v e r y n a t u r e infinitesimally E u c l i d e a n , b u t n o t ,

o f c o u r s e , e i t h e r locally o r globa l ly E u c l i d e a n . W e y l , l i k e C a r n a p , d e f e n d s

the ide a tha t sp a c e is a p r io r i r e p r e s e n ta b l e by the e n t i r e c l a s s o f R ie m a nn -

i a n ma n i f o ld s . Thus , a c c o r d ing to W e y l , wha t he c a l l s t he nature o f s p a c e

i s g ive n by the c i r c ums ta nc e tha t a t e a c h po in t o f t he ma n i f o ld t he t a nge n t

s p a c e b e a r s t h e s a m e E u c l i d e a n m e t r i c a l s t r u c t u r e . W h a t W e y l c a l l s t h e

m u t u a l orientat ion o f t h e m e t r i c a l s t r u c t u r e a s w e m o v e f r o m p o i n t t o

po in t , f r om t a nge n t spa c e to t a nge n t sp a c e , is , by c on t r a s t , e n t i r e ly a cc i -

de n ta l a nd c on t inge n t :

T h e na tu r e o f t h e m e t r i c c h a r a c t e r i z e s t h e a p r i o r i e s s e n c e [Wesen ] o f s p a c e in r e g a r d t o t h e

me t r i c a l ; i t i s o n e a n d i t i s t h e r e f o r e a l s o a b s o l u t e l y d e t e r m i n e d a n d d o e s n o t p a r t i c i p a t e i n

t h e u n a v o i d a b l e v a g u e n e s s o f t h a t w h i c h o c c u p i e s a v a r i a b l e p o s i t i o n in a c o n t i n u o u s s c a l e.

W h a t i s n o t d e t e r m i n e d t h r o u g h t h e e s s e n c e o f s p a c e b u t r a t h e r a p o s t e r i o r i - i . e . , a c c i d e n t a l

a n d c a p a b l e i n i t s e lf o f f r e e a n d a r b i t r a r y v i r tu a l v a r i a t i o n s - i s t h e m u t u a l or ien ta t ion o f t h e

m e t r i c s a t d i f f e r e n t p o i n t s ; in r e a l i ty i t s t a n d s in c a u s a l d e p e n d e n c e w i t h m a t t e r - a n d c a n -

p a r t i c i p a t i n g i n t h e v a g u e n e s s o f c o n t i n u o u s l y v a r ia b l e m a g n i t u d e s - n e v e r b e f ix e d e x a c tl y

i n a r a t i o n a l w a y , b u t a l w a y s o n l y v i a a p p r o x i m a t i o n a n d a l so n e v e r w i t h o u t t h e h e l p o f

i m m e d i a t e ly i n tu i ti v e r e f e r e n c e s to r e a l it y . O n e s e e s t h a t t h e R i e m a n n i a n c o n c e p t i o n d o e s

n o t d e n y t h e e x i s t e n c e o f a n a p r i o r i e l e m e n t i n t h e s t r u c t u r e o f s p a c e ; i t i s o n l y t h a t t h e

b o u n d a r y b e t w e e n t h e a p r i o ri a n d t h e a p o s t e r i o r i h a s b e e n s h i f t e d . 1 9 1 8, 13 )

A n d i t i s c l ea r , m o r e o v e r , t h a t W e y l , li k e C a r n a p , c o n c e i v e s t h is n e w

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F O U N D T I O N S O F G E O M E T R Y N D R E L T I V I T Y T H E O R Y 53

R ie m a nn ia n c on c e p t io n o f t he spa t i a l a p r io r i a s a ge ne r a l iz a t ion o f t he

Ka n t i a n c onc e p t ion - a ge ne r a l i z a t ion a da p te d to the ne w sci e n ti fi c

s i t ua t ion c r e a t e d by r e l a t i v i t y t he o r y .1

T h e k i n sh i p b e t w e e n t h e t w o t h i n k e r s b e c o m e s e s p e c ia l ly i n tr ig u i n g

w h e n w e n o t e t h a t W e y l , l i k e C a r n a p , b e a r s a s u b s t a n t i a l p h i l o s o p h i c a l

d e b t t o E d m u n d H u s s e r l . T h u s , i n t h e I n t r o d u c t i o n t o W e y l ( 1 91 8 ) , al-

t hough W e y l e xp l i c i t l y s t a t e s t ha t he w i l l r e f r a in f r om e n te r ing mor e

d e e p l y i n t o t h e p u r e l y p h i l o s o p h ic a l a s p e c t s o f t h e p r o b l e m i n t h e p r e s e n t

b o o k , h e n o n e t h e l e s s a d d s s e v e r a l p a r ag r a p h s o f a s e l f- c o n sc i o u s ly p h i lo -

soph ic a l na tu r e i n wh ic h he p r ov ide s a ge ne r a l ou t l i ne o f t he r e l a t i onsh ip

b e t w e e n m a t h e m a t i ca l - p h y s ic a l t h e o r i z in g a n d t h e r e a lm o f i m m e d i a t e

s u b j e c ti v e e x p e r i e n c e - t h e re a l m o f p h e n o m e n o l o g i c a l p u r e co n s c i o u s -

n e s s

[reines B ewul3tsein].

A n e n d n o t e t o t h e s e p h i l o s o p h i c a l p a r a g r a p h s

t h e n e x p l a in s t h at t h e p r e c i s e v e r s i o n o f t h e s e t h o u g h t s d e p e n d s m o s t

c l o s el y o n H u s s e r l ' s

Id een z u e i n e r r e i n en P h i i n o m en o l o g i e

(1918, no te 1

t o I n t r o d u c t i o n ) . M o r e o v e r , a t t h e e n d o f C h a p t e r I I o f W e y l 1 9 2 0 4 ) -

t ha t i s , a t t he c onc lus ion o f h i s ow n p r e se n ta t ion o f pu r e ly i n fin i te s ima l

E u c l i d e a n g e o m e t r y a s c o n s t it u t in g t h e e s s e n c e o r n a t u r e o f s p a c e - W e y l

e x p li c it ly c o n n e c t s h is a n a ly s is w i t h H u s s e r li a n p h e n o m e n o l o g y i n a n e v e n

m o r e s tr ik i n g f a sh i o n : T h e i n v e s ti g a ti o n s u n d e r t a k e n i n C h a p t e r I I c o n -

c e r n i n g s p a c e s e e m t o m e t o b e a g o o d e x a m p l e o f t h e e s s e n t i a l a n a l y s i s

[Wesensanalyse] s t r i v e n a f t e r b y t h e p h e n o m e n o l o g i c a l p h i l o s o p h y ( H u s -

se r l ) (19204, 19 ). C o u ld i t be , t he n , t ha t W e y l ' s unde r ly ing ( i f no t

e xp l i c i t l y e xp r e s se d ) ph i lo soph ic a l c onc e p t ion i s a l so c lo se t o C a r na p ' s ?

T h a t t h is is n o t t h e c a s e e m e r g e s w h e n w e re f le c t m o r e c l o s el y o n h o w

W e yl a r r ive s a t t he i de a tha t i n f in i t e s ima l ly Euc l ide a n ge ome t r y e xp r e s se s

t h e a p r i o r i e s s e n c e o r n a t u r e s p a c e . D o e s W e y l , l i k e C a r n a p , a r r i v e a t

t hi s i d e a b y c o n s i d e ri n g o u r i m m e d i a t e , q u a s i - p e r c e p t u a l i n t u it iv e c o n -

sc iousne s s o f ve r y sma l l spa t i a l r e g ions? N o t a t a ll ; he i n s t e a d p r e s e n t s a

s u b t l e a n d e x t r e m e l y c o m p l i c a t e d m a t h e m a t i c a l a n a ly s is t h a t is i n t e n d e d

t o g e n e r a li z e t h e g r o u p - t h e o r e t i c s o l u t i o n t o t h e s p a c e p r o b l e m o f

H e l m h o l t z a n d L i e t o m a n i f o l d s o f v a r i a b l e c u r v a t u r e , t h a t i s, to R i e m a n n -

i a n m a n i f o l d s i n g e n er a l. J u s t a s H e l m h o l t z a n d L i e d e r i v e d th e P y t h a g o -

r e a n ( tha t is , i n f in i t es ima l ly Eu c l ide a n ) c ha r a c t e r o f t he R ie m a nn ia n

m e t r i c f ro m a p o s t u l a t e o f f r e e m o b i l i ty - w h i c h , h o w e v e r , t h e r e b y l i m i t e d

t h e ir s o l u t io n t o R i e m a n n i a n m a n i f o l d s o f c o n s t a n t c u r v a t u r e - W e y l

d e r i v e s t h e P y t h a g o r e a n c h a r a c t e r o f t h e R i e m a n n i a n m e t r i c fr o m a m o r e

ge ne r a l g r ou p - the o r e t i c a r gu m e n t o f a s t ri c tl y i n f in i te s ima l c ha r a c t e r .

W e y l ' s m o r e g e n e r a l g r o u p - t h e o r e t i c c o n d i t i o n s t h u s y ie l d t h e P y t h a g o r e a n

( tha t is , i n f in i t e sima l ly Eu c l ide a n ) na tu r e o f t he R ie m a nn ia n m e t r i c w i th -

o u t r e s t r i c t i n g u s t o h o m o g e n e o u s s p a c e s . T h e i m p o r t a n t p o i n t f o r o u r

p u r p o s e s , h o w e v e r , i s t h a t W e y l ' s d e e p m a t h e m a t i c a l a n a l y si s o f t h e

s p a c e p r o b l e m - a n a na l ys i s h e w a s o n l y a b l e t o b r in g~ to successfu l

f r u i t ion b y in t e ns ive e f f o r t s ove r s e ve r a l ye a r s - ha s~ a l m o s t n o t h i n g i n

c o m m o n w i t h C a r n a p ' s c o n c e p t i o n o f o u r a p ri o r i k n o w l e d g e o f s p a c e .

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54

M I C H E L F R I E D M N

This stands out part icularly clearly, in fact , in the continuat ion of the

passage, ci ted above, in which Weyl compares his analysis of space to

Husserl ian Wesensanalyse:

T h e i n v e st i g at i o ns u n d e r t a k e n i n C h a p t e r I I c o n c e r n i n g s p a ce s e e m t o m e t o b e a g o o d

exam ple o f the es s en t ia l ana lys i s

[ esensanaiyse]

s t r iven a f te r by the phenom eno log ica l

ph i los ophy (Hus s e r l ) - an exa m ple tha t i s typ ica l fo r such cas es where i t is a m at te r o f

n o n i m m a n e n t e s s e n c e s . W e s e e h e r e i n t h e h i s t or i ca l d e v e l o p m e n t o f t h e s p a c e - p r o b l e m h o w

diff icul t i t i s for us in real i ty prejudiced humans to d iscover what is t ru ly decis ive. A long

m athem at ica l deve lopm en t was r equ i r ed : the g rea t un fo ld ing o f geom et r ica l s tud ies f rom

E ucl id to R iem ann , the phys ica l pene t r a t ion o f na tu re and i t s l aws s ince Ga l i l eo w i th a l l o f

i t s ever r ene wed im pe tus f rom th e em pi r ica l r ea lm , and f ina lly the gen ius o f ind iv idua l g rea t

m inds - N ewto n , Gaus s , R i em a nn , E ins te in - a ll o f th i s was r equ i r ed in o rder to t ea r u s

away f rom the ex te rna l , acc iden ta l , non-es s en t ia l charac te r i s ti c s to wh ich we wo u ld have

o therwis e r em ained a t t ached . Cer ta in ly , i f the t rue s t and po in t i s once a t t a ined , a l igh t dawns

on our r e as on , and i t then know s and r ecogn izes wha t i t can under s t and ou t o f i ts own s e l f.

N e v e r t h e l e s s , o u r r e a s o n d i d n o t h a v e t h e p o w e r ( a l t h o u g h d u r i n g t h e e n t i r e d e v e l o p m e n t

o f t h e p r o b l e m i t o f c o u r s e al w a ys w a s n e a r b y ) t o s e e t h r o u g h t h e p r o b l e m i n o n e s t r o k e .

T h is m us t be he ld over aga ins t the im pa t ience o f the ph i los opher s , who be l i eve tha t they

are ab le adeq ua te ly to des c r ibe the e s s ence on the bas i s o f a s ing le ac t o f exem pla ry m ak ing-

p res en t . In p r inc ip le they a re co r rec t , bu t in hu m an te rm s inco r rec t . (1904, 19)

For W eyl, then, the essent ial structure of space can in no way be discerned

on the basis of a single act of exemplary making-present [auf Grund

eines einzigen Aktes exemplarischer Vergegenwdrtigung]. On the cont rary ,

only a lengthy historical evolut ion of mathematical , phi losophical and

physical ideas can reveal this essent ial st ructure to us.

Why then does Weyl, l ike Carnap, say that his conception of the a

pr iori essenc e of space is based on the Husser l ian proce dure of Wesens

analyse? To understand this we must dist inguish two very different ways

in which space and geom etry may be subjec t to phenomen ologica l Wesens

analyse. On e the on e hand, geom etry - taken mere ly as a fac tual ly given

science - may be used to exempl i fy and mot iva te the ideas of Wesenser

schauung and Wesensanalyse in the fi rst place. Here we simply take i t for

granted that a priori geometrical knowledge is based on an intui t ive grasp

of immediately presented spat ial forms (forms that can be intui t ively given

in fantasy or imaginat ion just as well as in actual spat ial perception), and

we then use this idea to motivate the possibl i ty of analogous Wesenser

schauung in o ther rea lms - in the rea lm of phenome nologica l pure con-

sciousness in part icular. This use of geo me try is part icularly evide nt in

Hus serl (1913, 70) wh ere Husser l i l lustrates the idea that imm ediate ly

in tu i t ive grasp of essences . . , can be achieved on the basis of the mere

making present of exemp lary par t icu lars [auf Grund bloBer Vergegenwiir.

tigung von exemplarischen Einzelheiten]

precise ly by the example of geo-

metry. It is clearly this way of thinking about geometrical Wesensanalyse

that inspires Carnap's conception in his dissertat ion; and i t is this way,

too, that appears to be the target of W eyl 's complaint regarding the

impat ience of the phi losophers quote d above,

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55

On the other hand, however, space and geometry may be subject to

phenomenological Wesensanalysein a more subtle and, as it were, more

constructive fashion. When we make the knowing subject or phenomeno-

logical pure consciousness itself into an object of

Wesenserschauung

and

Wesensanalyse we eventually arrive at the point where we consider the

phenomenological constitution of the cognitive spatial structure - that is,

the phenomenological constitution of

physical

space - in which and

through which the knowing subject looks out at the physical world.

And, as Husserl himself makes clear in (1913, 136-153), the phenomen-

ological constitution of space in this sense (that is, of physical space)

cannot he achieved on the basis of a single intuitive act of

Vergegenwiirti-

gung.

On the contrary, since we are here dealing with the constitution of

reality,

we are necessarily involved with a never-ending approximation to

a limiting idea

[Grenzidee]

in the sense of Kant. Essential knowledge of

the nature of space in this sense - that is, of the physical space encountered

at a definite stage of transcendental const itut ion in the pure conscious-

ness of the phenomenological subject - can and indeed must be the

outcome of a lengthy constructive process rather than an immediately

intuitive act.

This latter kind of approach to a phenomenological Wesensanalyseof

space - an approach only suggested in Husserl (1913) - was in fact

developed in great detail by Husserl's student Oskar Becker in Becker

(1923). Becker's idea is to remove the a priori contingency that appears

to attach to Euclidean geometry considered merely as a factually given

material

eidetic science by actually deriving or constructing Euclidean

space on the basis of a phenomenological

Wesensanalyse

of the knowing

subject. For, if we consider Euclidean geometry merely as a factually

given eidetic science, then, since geometry so understood is not a

for-

mal

eidetic science in Husserl's sense (that is, not a branch of pure

formal logic), it appears utterly accidental and thus contingent that

Euclidean geometry - as opposed to some other possible geometry -

actually describes the essential structure of space. Becker then attempts

to remove this a priori contingency by constituting specifically Euclidean

geometry phenomenologically on the basis of group-theoretic consider-

ations - in an argument that goes roughly as follows: The phenomenolog-

ical subject is located in a space, in which and through which it perceives

the surrounding physical world. The subject must be able to move freely

through this space; therefore, by the Helmholtz-Lie theorem, the space

must have constant curvature. Moreover, it must be possible to distinguish

rotations (by which the subject changes its orientation without changing

its position) and translations (by which the subject merely changes its

position); therefore, the group of rigid motions must possess a distin-

guished subgroup of translations, and hence the space must be Euclidean.

Now this phenomenological analysis of Becker's was of course published

after Weyl's own group-theoretic work on the space problem . Neverthe-

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56

M I C H E L F R I E D M N

l es s, it a p p e a r s l i k e ly t h a t W e y l a n d B e c k e r c o m m u n i c a t e d a b o u t t h e s e

q u e s t io n s m u c h e a r l i e r, t h r o u g h t h e ir c o m m o n p a r t i c i p a t io n i n t h e p h e n o -

m e n o l o g i c a l ci rc le a r o u n d H u s s e r l in G o t t i n g e n ; a n d , i n a n y c a s e , th e t w o

t h i n k e r s B e c k e r t h a n k s i n h is I n t r o d u c t i o n a r e p r e ci s e ly H u s s e r l a n d W e y l .

M o r e o v e r , i t s e e m s c l e a r t h a t W e y l c o n c e i v e s h is o w n a n a ly s is o f t h e

s p a c e p r o b l e m a s a g e n e r a l i z a t i o n a n d r e f i n e m e n t o f B e c k e r ' s - a g e n e r -

a l iz a t io n a p p r o p r i a t e t o th e v a r i a b l y c u r v e d , n o n - E u c l i d e a n s p a c e ( - t i m e )

o f g e n e r a l r e l a ti v i ty , z F o r W e y l , t o o , t h e p h e n o m e n o l o g i c a l s u b j e c t is

l o c a t e d i n a s p a c e , i n w h i c h a n d t h r o u g h w h i c h i t p e r c e i v e s t h e s u r r o u n d i n g

p h y s i ca l w o r l d . B u t W e y l d o e s n o t a s s u m e t h a t f r e e m o t i o n is p o s s i b le

a n d t h u s t h a t th is s p a c e m u s t h a v e c o n s t a n t c u r v a t u r e . I n s t e a d , W e y l

b e g i n s w i t h th e i d e a o f a n i n f in i t es i m a l r o t a t i o n g r o u p a t e v e r y p o i n t - a

g r o u p t h a t is a s s u m e d t o b e t h e s a m e ( i s o m o r p h i c ) a t e v e r y p o i n t , b u t is

o t h e r w i s e s o fa r u n d e t e r m i n e d . W e t h e n f ix t h is in f i n it e s im a l r o t a t i o n

g r o u p a s t h e E u c l i d e a n - P y t h a g o r e a n g r o u p b y p o s tu l a ti n g t h a t th e t h e r e b y

i n d u c e d m e t r i c - w h a t e v e r it is - m u s t d e t e r m i n e a n a s s o c i a t e d a f fi n e

c o n n e c t i o n uniquely F r o m a p h e n o m e n o l o g i c a l p o i n t o f v i e w , t h e n , t h e

s u b j e c t i s l o c a t e d a t a h e r e a n d n o w - a b o u t w h i c h i t m u s t h e a b le t o

c h a n g e i t s o r i e n t a t i o n . B y t h e n p o s t u l a t i n g t h a t i n f in i t e si m a l t r a n s l a t io n

( b u t n o t n e c e s s a ri ly f r e e m o t i o n ) fr o m th is h e r e a n d n o w is t h e r e b y

u n i q u e l y d e t e r m i n e d , w e g u a r a n t e e t h a t th e s p a c e o f o u r s u b j e c t is

in f ini tes imal ly E u c l i d e a n ( b u t n o t n e c e s s a r i l y e i t h e r l o c a l ly o r g l o b a l l y

E u c l i d e a n ) .

I n W e y l ' s l a te r d i s cu s s io n o f sp a c e a n d g e o m e t r y i n W e y l ( 19 2 7) t h e

c o n n e c t i o n b e t w e e n h is o w n a n a ly s is o f t h e s p a c e p r o b l e m a n d t h e

i d e a o f p h e n o m e n o l o g i c a l c o n s t i t u t i o n i s m a d e f u l ly e x p l ic i t. I n a s e c t io n

r e f e r r i n g t o H u s s e r l ( 1 9 1 3) a n d B e c k e r ( 1 9 2 3) - a n d a l s o , i n t e r e s t in g l y

e n o u g h , t o C a r n a p ( 1 9 2 2) - W e y l w r i t es :

A way for understanding the Ph ythagorean nature of the metric expressed in the Euclidean

rotation gro up precisely on the basis of the sep aration of a priori and a posteriori has been

given by the author: On ly in the case of this g rou p does the intrinsically accidental quantitat-

ive distribution o f the m etric field unique ly determine in all circumstan ces (how ever it ma y

have been forme d in the context o f its a priori fix ed nature) the infinitesimal parallel

displacement: the non-rotational progression from a p oint into the wo rld. Th is assertion

involves a deep m athematical theorem of group theory that I have proved. I believe that

this solution of the space-problem play s the sam e role in the co ntext of the Rie m ann -

Einstein theory that the He lmho ltz-Lie solution (Section 14) plays for rigid Euclidean spa ce.

Perhaps the postulate of the uniq ue determination of straight-progression can be also

justified from the requirements of the phenom enological constitution of space; Becket would

still l ike to gro un d the significance of the Euclidean rotation group for intuitive s pac e on

Helmho ltz's postulate of free mob ility. (1927, 18, pp. 99-100)

T h i s p a s s a g e s t r o n g l y c o n f i r m s t h e i d e a t h a t W e y l c o n c e i v e s h i s g r o u p -

t h e o r e t i c a n a ly s is o f t h e s p a c e p r o b l e m a s a g e n e r a l i z a t io n a n d re f in e -

m e n t s o f B e c k e r ' s - a n a n a l y si s w h i c h t h e r e b y f in d s i ts p r o p e r p h i l o so -

p h i c a l h o m e w i t h in t h e p h e n o m e n o l o g i c a l t h e o r y o f t h e c o n s t i t u t io n o f

( p h y s i c a l ) s p a c e .

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FOUNDATIONS OF GEOMETRY AND RELATIVITY THEORY

57

B o t h C a r n a p a n d W e y l th u s r e a c t to t h e n e w s i tu a t io n c r e a t e d b y t h e

ge ne r a l t h e o r y o f r e l a t iv i t y , no t by a dop t ing a s t r a igh t f o r wa r d ly e m pi r i c is t

c o n c e p t i o n o f t h e f o u n d a t i o n s o f g e o m e t r y , b u t r a t h e r b y ge n e ra l iz i n g

t h e K a n t i a n n o t i o n o f th e s y n t h e t i c a p r i o ri t o t h e

infinitesimally

E u c l i d e a n

c ha r a c t e r o f spa c e : a n in tu it i ve a nd /o r c on s t r uc t ive a p r io r i p r o c e d ur e

t e l l s u s t ha t spa c e i s one o r a no the r o f t he i n f in i t e ly ma ny poss ib l e

R i e m a n n i a n m a n i fo l d s. M o r e o v e r , b o t h C a r n a p a n d W e y l b a s e th e ir g e n-

e r a l i za t ion o f t he syn th e t i c a p r io r i on the Huss e r l i a n c on c e p t io n o f

Wesensanalyse. T h e t w o t h i n k e r s t h e n d i v e r g e , h o w e v e r , o n h o w s u c h

spa t ia l

Wesensanalyse

i s t o be unde r s tood . C a r na p t a ke s ( i n f in i t e s ima l ly

E u c l i d e a n ) g e o m e t r y , i n H u s s e r l i a n t e r m s , a s s i m p l y a f ac t u a ll y g i v e n

( m a t e r i a l ) e i d e t ic s c ie n c e w h o s e e s s e n c e is r e v e a l e d t o u s i n i m m e -

d ia t e ly i n tu i t i ve , qua s i - pe r c e p tua l a c t s o f

Vergegenwiirt igung

o r

Wesenser

schauung.

W e yl , on the o th e r ha n d , e xp l i c it l y r e j e c t s t h is k ind o f p i c tu r e

a n d i n s t e a d v i e w s ( in f in i te s im a l l y E u c l i d e a n ) g e o m e t r y a s th e o u t c o m e o f

a c o m p l i c a t e d t r a n s c e n d e n t a l c o n s t i t u t i o n b y t h e p h e n o m e n o l o g i c a l

s u b j e ct : o u r s y n t h e t ic a p r io r i g e o m e t r i c a l k n o w l e d g e is i n n o w a y i m m e -

d i a t e ly g iv e n in m e r e i n t u i ti o n b u t r a t h e r e x p r e s s e s a p h e n o m e n o l o g i c a l l y

b a s e d g r o u p - t h e o r e t i c c o n s t r u c t io n r e q u i ri n g a ll t h e r e s o u r c e s o f h ig h e r

m a t h e m a t i c s . 3

T h i s l a s t i d e a u l t i m a t e l y l e a d s W e y l t o d i s t a n c e h i m s e l f e v e n f u r t h e r

f r o m a n i m m e d i a t e l y i n tu i t iv e c o n c e p t i o n o f g e o m e t r y - a n d t h u s to

d i s t a n c e h i m s e l f e v e n f u r t h e r f r o m b o t h C a r n a p ( 1 9 2 2 ) a n d f r o m H u s s e r l

( 19 1 3 ). F o r W e y l c o m e s m o r e a n d m o r e t o d e f e n d t h e v i e w t h a t , a l t h o u g h

o u r m a t h e m a t i ca l - p h y s i c al u n d e r s t a n d i n g o f s p a c e m u s t in d e e d begin wi th

im m e d ia t e ly i n tu i t i ve a c t s ( by wh ic h the su b je c t loc a t e s a nd o r i e n t s i t se l f

i n t h e h e r e a n d n o w ) , t h e n e c e s s a r y ou t come of t h i s p r oc e dur e i s a

w h o l l y n o n - i n tu i t iv e , p u r e l y c o n c e p t u a l o r s y m b o l i c c o n s t r u c t io n b y

w h i c h w e r e p r e s e n t t h e p h y s i c a l - s p a t i a l w o r l d b y a b s t r a c t m a t h e m a t i c a l

s y m b o l s h a v i n g n o in t u it iv e c o n t e n t . A n d W e y l c o m e s t o e m p h a s i z e m o r e

a n d m o r e t h a t i t is o n l y th r o u g h s u c h p u r e l y s y m b o l i c c o n s t r u c t io n , i n

f a c t , t ha t we c a n ob ta in t r u ly

objective

k n o w l e d g e o f t h e p h y s i c a l w o r l d .

Th i s be c ome s e spe c i a l l y c l e a r i n a pa s sa ge f r om W e y l ( 1927) whe r e , a f t e r

m e n t i o n i n g K a n t ' s v i e w o f s p a c e a n d t i m e a s m e r e f o r m s o f o u r in t u i ti o n ,

W e y l c o n t i n u e s:

Intuitive space and intuitive time can therefore not serve as medium in which physics

constructs the external world; [we need] ra ther a four-dimensional continuum in the abstract-

arithmetical sense. As colors for Huygens were in reality vibrations in the aether, so they

now appear only as mathematical functional distributions of a periodic character, whereby

four independent variables enter into the functions as representatives of the space-time

medium via coordinates. What remains is thus finally a symbol ic construc t ion in precisely

the sense Hilbert carries through in mathematics.

The construction of this objective world, which is only presentable in symbols, from what

is immediately given to me in intuition is completed in various levels whereby the progress

from level to level is determined by the condition that what is present at one level always

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258 M I C H A E L F R I E D M A N

r e v e a l s i t s el f a s a n a p p e a r a n c e o f a h i g h e r re a l i t y - t h e r e a l i t y o f t h e n e x t l e ve l . 1 9 2 7, 1 7 ,

p . 8 0 )

Thi s concep t i on o f a s t ep -wi se

object i f ic t ion

o f i n t u i t i v e e x p e r i e n c e

t h r o u g h e v e r m o r e a b s t r a c t p u r e l y c o n c e p t u a l c o n s t r u c t i o n h a s m u c h

m o r e i n c o m m o n w i th t h e N e o k a n t i a n i s m o f C a ss i re r - a s e x p r e s s e d , fo r

exa m ple , in Cas s i r e r (1921) - t ha n wi th e i t he r Ca rnap (1922) o r Hu sse r l i an

p h e n o m e n o l o g y .

I t i s a g a i n s t t h i s b a c k d r o p , m o r e o v e r , t h a t w e s h o u l d u n d e r s t a n d t h e

s tr ik in g d i s a g r e e m e n t b e t w e e n W e y l a n d C a r n a p o v e r t h e c o n v e n t io n a l it y

o f p h y s i c a l m e t r i c a l s t r u c t u r e . 4 F o r W e y l h i m s e l f e n t i re l y r e j e c t s t h e i d e a

tha t phys i ca l me t r i ca l s t ruc tu re i s conven t i ona l l y s t i pu l a t ed , and i n s t ead

s e e s i n t h e g e n e r a l t h e o r y o f r e l a t i v i t y t h e c u l m i n a t i o n o f R i e m a n n ' s

s u g g e s t i v e r e m a r k s a c c o r d i n g t o w h i c h t h e m e t r i c o f p h y s i c a l s p a c e i s

empiric l ly

d e t e r m i n e d b y b i n d in g fo r c e s a c ti n g o n t h e u n d e r l y in g m e t r -

i ca l c o n t i n u u m . 5 T h i s d i s a g r e e m e n t b e t w e e n W e y l a n d C a r n a p s h o u l d , I

t h i n k , b e u n d e r s t o o d i n t h e f o l lo w i n g w a y . O n W e y l ' s a c c o u n t , sp a c e i n

re l a t iv i t y t heo r y has de c i s ive ly t r ans cen ded ou r spa t i a l i n tu i ti on . I t i s

r a t h e r a p u r e l y a b s t ra c t , p u r e l y c o n c e p t u a l s t r u c t u r e w h o s e e s s e n c e o r

n a t u r e is e x p r e s s e d b y t h e c i r c u m s t a n c e t h a t t h e ta n g e n t sp a c e a t e a c h

p o i n t o f t h e m a t h e m a t i c a l m a n i f o l d is E u c l i d e a n . I t t h e n m a k e s p e r f e c t l y

g o o d s e n s e , f o r W e y l , t o a s s e r t t h a t t h e n o n - e s s e n t ia l , e n t i r e ly c o n t i n g e n t

m u t u a l o r i e n t a t i o n o f t h e s e t a n g e n t s p a c es is e m p i r i c al ly d e t e r m i n e d b y

Eins t e in ' s f i e ld equa t i on ; fo r i t is E ins t e in ' s t he o ry a lone t ha t p rov ides u s

w i t h a n a d e q u a t e

symbol ic

cons t ruc t i on o f ob j ec t i ve phys i ca l r ea l i t y . On

Carn ap ' s acco un t , by con t ras t , t he a p r i o r i i n f in i t e s ima l ly Eu c l i de an na tu re

o f s p a c e i s c o n c e i v e d o f a s a d i r e c t r e fl e c t io n o f t h e n e c e s s a r y s t r u c t u r e o f

ou r spa t i al i n tu i t ion . A nd i t fo l l ows , fo r C arna p , t ha t spec if i ca l ly spa t ia l

s t r u c t u r e ( a s o p p o s e d t o p u r e l y f o r m a l lo g i c a l - m a t h e m a t i c a l s t r u c t u r e ) c a n

o n l y b e

intuit ively

spa t i a l s t ruc tu re . Tha t t h i s i n tu i t i ve ly spa t i a l s t ruc tu re

t h e n n e c e s s a r il y l a c ks w h a t W e y l c a ll s a d e t e r m i n a t e o r i e n t a t i o n o f t h e

p u r e l y i n f i n i t e s i m a l m e t r i c s , c a n o n l y m e a n , f o r C a r n a p , t h a t n o s u c h

or i en t a t i o n i nhere s i n phys i ca l ly spa t ia l r ea l i ty a t a ll - i n o th e r words ,

t ha t t h e fu l l me t r i ca l s t ruc tu re o f phys ica l space can o n ly be con ven t i on a l .

N ow Ca rnap , i n h is l a t e r wr i t ings , soon l eaves t he i n tu i ti ve space o f h i s

d i s se r t a t ion com ple t e ly beh in d . Ind ee d , Ca rnap (1928) adop t s a s tep -wi se

c o n s t i t u t i o n o f o b je c t iv i ty v i a p u r e l y c o n c e p t u a l o r l o gi ca l m e a n s t h a t

is p a r a ll e l to b o t h t h e N e o k a n t i a n i s m o f C a s s ir e r a n d t h e s y m b o l ic c o n -

s t ruc t i on o f W ey l . P hys i ca l space , i n pa r t i cu l a r , beco m es a pu re ly abs t r ac t

m a t h e m a t i c a l o b j e c t ( t h e s e t o f q u a d r u p l e s o f r e a l n u m b e r s R 4) w h i c h

i s d i s t i n g u i s h e d f r o m o t h e r i s o m o r p h i c r e l a t i o n a l s t r u c t u r e s o n l y b y t h e

c i r c u m s t a n c e t h a t o u r p r e v i o u s ly c o n s t i t u t e d e p i s t e m i c s u b j ec t is p e r c e p t u -

a ll y e m b e d d e d w i t h in i t - a t a d ef i n it e p o i n t o f v i e w , a s i t w e r e . I n d e e d ,

C a r n a p p u s h e s s u c h l o g ic a l - c o n c e p tu a l o b je c t if ic a t io n f a r b e y o n d a n y t h i n g

env i s i one d by W e y l ; fo r C arna p (1928) exp l i ci tl y a s se r ts t ha t t he ob j ec t i v it y

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F O U N D A T I O N S O F G E O M E T R Y A N D R E L A T I V I T Y T H E O R Y 259

o f s c i e n c e r e q u i r e s t h a t a l l p u r e l y i n t u i t i v e o r o s t e n s i v e e l e m e n t s m u s t b e

c o m p l e t e l y a n d d e f i n i t iv e l y e x p u n g e d . 6 A c c o r d i n g l y , C a r n a p t h e r e a d o p t s

a s t r a te g y o f p u r e l y s t r u ct u r a l d e f i n i t e d e s c r i p t i o n s t h a t a i m s t o i n d i v id u -

a t e a l l o b j e c t s o f s c i e n c e s o l e l y o n t h e b a s i s o f t h e i r f o r m a l o r s t r u c t u r a l

p r o p e r t i e s w i t h i n t h e l o g i c o f Pr i n c i p i a M a thema t i c a and i t i s in th i s way ,

i n f a c t , t h a t C a r n a p f i n a l l y b r e a k s e n t i r e l y w i t h b o t h N e o k a n t i a n i s m a n d

H u s s e r l ia n p h e n o m e n o l o g y . B u t t h i s i s a s to r y fo r a n o t h e r o c c a s io n .

N O T E S

* A n e a r l i e r v e r s i o n o f t h i s p a p e r w a s p r e s e n t e d a t a w o r k s h o p o n S e m a n t i c a l A s p e c t s o f

S pa c e - T i m e The or i e s a t t he Z e n t ru m f t i r i n t e rd i s z ip l i n~ ire F o r sc hu ng a t t he Un i ve r s i t ~ it

B i e l e f e l d i n M a y 1 9 9 4 . I a m i n d e b t e d t o t h e p a r t i c i p a n t s f o r h e l p f u l c o m m e n t s . I a m a l s o ,

a n d e s p e c ia l ly , i n d e b t e d t o c o m m e n t s f r o m a n d d i sc u s s io n s w i t h T h o m a s R y c k m a n . A l l

t r a n sl a ti o n s f r o m t h e G e r m a n a r e m y o w n .

1 A l t h o u g h W e y l d o e s n o t e x p l i c it ly m e n t i o n K a n t i n t h e a b o v e - c i t e d p a ss a g e , c l o s el y p a r a l le l

p a s s a g e s i n o t h e r w o r k s m a k e t h e K a n t i a n c o n t e x t p e r f e ct l y c le a r . S e e e s p e c i al l y W e y l ( 1 9 2 2 ,

p . 116) whe re t he i n f i n i t e s i ma l l y Eu c l i de a n na t u re o f spa c e i s s a i d to be characteristic o f

space as form of appearance ,

a nd W e y l (1927 , 18 , p . 97 ) whe r e K a n t i s e xp l i c it l y m e n t i o ne d .

z B e c k e r ( 1 92 3 ) a l so c o n s i d e r s t h e a p p l i c a ti o n o f n o n - E u c l i d e a n , v a r i a b l y c u r v e d s p a c e s

i n g e n e r a l r e l a t i v i t y . A c c o r d i n g t o B e c k e r , h o w e v e r , t h i s a p p l i c a t i o n m u s t b e c o n c e i v e d

i n s t r u m e n t a l i s ti c a l ly w i t h r e s p e c t t o t h e u n i q u e l y r e a l ( E u c l i d e a n ) s p a c e o f e x p e r i e n c e - a s

a d e v i c e f o r s i m p l i f y i n g t h e p r e s e n t a t i o n o f p h y s i c a l l a w s . F o r W e y l , b y c o n t r a s t , o n l y t h e

v a r i a b l y c u r v e d s p a c e ( - t i m e ) o f g e n e r a l r e l a ti v i t y i s th e u n i q u e l y r e a l o r a c t u a l s p a c e ( -t i m e ) .

3 T h u s , t h e p h i l o s o p h i c a l p a s s a g e f r o m t h e I n t r o d u c t i o n t o W e y l ( 1 9 1 8 ) n o t e d a b o v e p u t s

mos t s t r e s s on t he i de a o f phys i c a l r e a l i t y a s a l i m i t i ng i de a

[Grenzidee]

- i n c o m p l e t e

a g r e e m e n t w i t h t h e c o n c e p t i o n o f p h y s i c a l s p a c e s k e t c h e d i n H u s s e r l ( 1 9 1 3 , 1 3 6 - 1 5 3 ) .

T h e s e s e c t io n s f r o m H u s s e r l ( 19 1 3 ) - a l o n g w i t h B e c k e t ( 1 9 2 3) - a r e in t u r n a p p r o v i n g l y

c i ted in Weyl (1927, 18, p . 92) .

4 T h i s d i s a g r e e m e n t c a n b e p i n p o i n t e d v e r y p r e c is e ly : C a r n a p ( 1 9 22 , p p . 5 6 - 5 9 ) u s e s a

p a r t i c u l a r f o r m o f t h e S c h w a r z s c h i l d m e t r i c d u e t o L . F l a m m t o g e n e r a t e a n a l t e r n a t i v e

E u c l i d e a n d e s c r i p t i o n o f t h e g r a v i t a t i o n a l f i el d in v o l v i n g c o m p e n s a t i n g c o n t r a c t i o n s in o u r

m e a s u r i n g r o d s - t h e r e b y d e f e n d i n g c o n v e n t i o n a l i s m . W e y l ( 1 9 1 8 , 3 3 ) u s e s e x a c t l y t h e

s a m e f o r m o f t h e S c h w a r z s c h il d m e t r i c t o a r g u e t h a t s u c h a l t e r n a t i v e E u c l i d e a n d e s c r i p t i o n s

- a l t h o u g h m a t h e m a t i c a l l y p e r f e c t ly p o s s ib l e - a r e c o n c e p t u a l ly a r b i t r a r y a n d t h e r e f o r e

de f i n i t e l y in fe r i o r t o t he r e l a t i v i s ti c de sc r i p t i on .

5 C o m p a r e W e y l ' s w e l l - k n o w n s i x t h a n d f i n a l

Erli~uterung

t o t h e f i n a l s e c t i o n o f R i e m a n n

(1919) .

6 F o r W e y l , b y c o n t r a s t , t h e origin o f o u r s y m b o l i c c o n s t r u c t i o n o f t h e o b j e c t i v e w o r l d i n

i m m e d i a t e l y i n tu i t i v e e x p e r i e n c e c a n n e v e r b e c o m p l e t e l y o v e r c o m e , f o r s e t t i n g u p a c o o r d i n -

a t e sys t e m i n t he f i r s t p l a c e ne c e s sa r i l y r e qu i re s a n

ostensive

r e f e r e n c e t o t h e h e r e a n d

n o w . S e e , f o r e x a m p l e , t h e I n t r o d u c t i o n t o W e y l ( 1 9 18 ) : B u t t h is o b j e c t if i c a ti o n t h r o u g h

t h e e x c l u s i o n o f t h e I a n d i t s i m m e d i a t e l if e o f i n t u i t i o n d o e s n o t s u c c e e d w i t h o u t r e m a i n d e r ;

t h e c o o r d i n a t e s y s t e m , w h i c h c a n o n l y b e i n d i c a t e d b y a n i n d i v i d u a l a c t i o n ( a n d o n l y

a p p r o x i m a t e l y ) , r e m a i n s a s t h e n e c e s s a r y r e s i d u e o f t h i s I - a n n i h i l a t i o n .

R E F E R E N C E S

B e c k e r , O s k a r : 1 92 3, ' B ei tr ~ ig e z u r p h a n o m e n o t o g i s c h e n B e g r t i n d u n g d e r G e o m e t r i e u n d

i h r e r p h y s i k a l i s c h e n A n w e n d u n g e n ' ,

Yahrbuch fo r Philosophie und phiinomenologische

Forschung

V I , 3 8 5 - 5 6 0 .

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carnap and weyl on the foundations of geometry and relativity theory

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