carnap and weyl on the foundations of geometry and relativity theory
TRANSCRIPT
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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 "
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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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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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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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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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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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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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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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