Hilbert and Physics

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Arch . H i s t . Ex ac t Sc i . 51 (1997) 83 -198 . 9 Sp r ing e r -V er l ag 1997

D av id H ilbert an d the Axiomatization o f Ph ysics1894-1905)L E O C O R R Y

C o m m u n i c a t e d b y J . N O R T O N

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1. I n tr o d uc t io n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832. H i l b e rt a s S t u de n t a n d T e a c he r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853. T h e B a c k g r o u n d t o H i l b e r t ' s A x i o m a t i c A p p r o a c h : G e o m e t r y a n d P h y s i c s 8 94 . A x i o m a t i c s, G e o m e t r y a n d P h y s i c s i n H i l b e r t ' s E a r l y L e c t u r e s . . . . . . . . 1 045 G r u n d l a g e n d e r G e o m e t r i e 1096. T h e F r e g e -H i l b e rt C o r r es p o n d en c e . . . . . . . . . . . . . . . . . . . . . . . . . . 1167. T h e 1900 L i st o f P r o b l em s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198. H i l b e r t ' s 19 05 L e c tu r e s o n t h e A x i o m a t i c M e t h o d . . . . . . . . . . . . . . . . 123A r it hm e t ic a n d G e o m e t r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

M e c h an ic s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131T h e rm o d y n a m ic s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148P r o b a b i l i t y C a lc u lu s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158K i n e ti c T h e o r y o f G a s e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162I n su r a nc e M a t h e m a t i cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171E l ec tr od y na m i cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172P s yc h op h ys ic s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

9. C o n c lu d i ng R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183B i b li og r ap h y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

1 I n t r o d u c t i o n

I n 1 9 00 , a t a t i m e w h e n h is i n t e r n a t i o n a l p r o m i n e n c e a s a l e a d i n g m a t h e m a -t i c i a n w a s j u s t b e c o m i n g f i r m l y e s t a b l i s h e d , D AV ID H I LB E R T ( 18 6 2- -1 9 43 ) d e -l i v e re d o n e o f t h e c e n t r a l in v i t e d l e c t u r es a t t h e S e c o n d I n t e r n a t i o n a l C o n g r e s so f M a t h e m a t i c i a n s , h e l d in P a r i s . T h e l e c t u re b o r e t h e t it le " M a t h e m a t i c a lP r o b l e m s " . A t t h i s v e r y s i g n i f i c a n t o p p o r t u n i t y H IL BE RT a t t e m p t e d t o " l if t t h ev e il " a n d p e e r i n to t h e d e v e l o p m e n t o f m a t h e m a t i c s o f t h e c e n t u r y t h a t w a sa b o u t t o b e g i n (H IL BE RT 1 9 02 , 4 3 8) . H e c h o s e t o p r e s e n t a l is t o f t w e n t y - t h r e ep r o b l e m s t h a t i n h i s o p i n i o n w o u l d a n d s h o u l d o c c u p y t h e e ff o r ts o f m a t h e m a -t i c i a n s i n th e y e a r s t o c o m e . T h i s f a m o u s l i s t h a s e v e r si n c e b e e n a n o b j e c t o f


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84 L. CORRYm a t h e m a t i c a l a n d h i s t o r i c a l in t e re s t . M a t h e m a t i c i a n s o f a l l s p e c ia l ti e s a n d o f a llc o u n t r i e s h a v e t a k e n u p i t s c h a l le n g e s . S o l v i n g a n y i t e m o n t h e l i st c a m e t o b ec o n s i d e r e d a s i g n i f i c a n t m a t h e m a t i c a l a c h i e v e m e n t .

T h e s i x t h p r o b l e m o f t h e l is t d e a l s w i t h t h e a x i o m a t i z a t i o n o f p h y s i c s. I t w a ss u g g e s t e d b y h is o w n r e c e n t r e s e a r c h 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 ; H IL BE RTp r o p o s e d t o t r e a t in t h e s a m e m a n n e r [ -a s g e o m e t r y ] , b y m e a n s o f a x i o m s ,t h o s e p h y s i c a l s c i e n c e s i n w h i c h m a t h e m a t i c s p l a y s a n i m p o r t a n t p a r t ( H I L B E R T1902, 454) .

T h i s p r o b l e m d i f f e r s i n a n e s s e n t i a l w a y f r o m m o s t o t h e r s i n t h e l i s t , a n d i t si n c l u s i o n r a i s e s m a n y i n t r i g u i n g q u e s t i o n s . I n t h e f ir st p l a c e, as f o r m u l a t e d b yH IL BE RT , i t is m o r e o f a g e n e r a l t a s k t h a n a s p e c i fi c m a t h e m a t i c a l p r o b l e m . I t i sf a r f r o m e v i d e n t u n d e r w h a t c o n d i t i o n s t h is p r o b l e m m a y b e c o n s i d e r e d t o h a v eb e e n s o l v ed . I n fa c t, f r o m r e p o r t s t h a t h a v e o c c a s i o n a l l y b e e n w r i t t e n a b o u t t h ec u r r e n t s t a te o f r e s e a rc h o n t h e t w e n t y - t h r e e p r o b l e m s , n o t o n l y is it h a r d t od e c i d e to w h a t e x t e n t t h is p r o b l e m h a s a c t u a l l y b e e n s o lv e d , b u t m o r e o v e r , o n eg e t s t h e i m p r e s s i o n t h a t , f r o m a m o n g a l l t h e p r o b l e m s i n t h e l i s t , t h i s o n e h a sr e c ei v e d t h e l e as t a t t e n t i o n f r o m m a t h e m a t i c i a n s . 1

F r o m t h e p o i n t o f v i e w o f H IL B~R T'S o w n m a t h e m a t i c a l w o r k , a d d i t i o n a lh i s t o r i c a l q u e s t i o n s m a y b e a s k e d . A m o n g t h e m a r e t h e f o l l o w i n g : W h y w a s t h i sp r o b l e m s o c e n t r a l f o r H IL BE RT t h a t h e i n c l u d e d i t i n t h e l is t? W h a t c o n t a c t , i fa n y , h a d h e h i m s e l f h a d w i t h t h is p r o b l e m d u r i n g h i s m a t h e m a t i c a l c a r ee r ?W h a t w a s t h e a ct u a l c o n n e c t i o n b e t w e e n h is w o r k o n t h e f o u n d a t io n s o fg e o m e t r y a n d t h i s p r o b l e m ? W h a t e f fo r ts , if a n y , d i d H IL BE RT h i m s e l f d i r e c ta f t e r 1 9 0 0 t o i t s s o l u t i o n ?

T h e s e q u e s t io n s a r e p a r t i c u l a r ly p r e s s i n g b e c a u s e o f th e i r b e a r in g o n t h eo f t e n a c c e p t e d i d e n t i f ic a t i o n b e t w e e n H IL BE RT a n d t h e f o r m a l i s t a p p r o a c h t o t h ef o u n d a t i o n s o f m a t h e m a t i c s . H IL BE RT 'S m a i n a c h i e v e m e n t c o n c e r n i n g t h e f o u n -d a t i o n s o f g e o m e t r y w a s - - a c c o r d i n g t o a w i d e ly - h e l d v i e w - - t o p r e s e n t th i sm a t h e m a t i c a l d o m a i n a s a n a x i o m a t i c s y s t e m d e v o i d o f a n y s p ec if ic i n tu i t iv em e a n i n g , i n w h i c h t h e c e n t r a l c o n c e p t s ( p o i n t s , l i n e s , p l a n e s ) c o u l d w e l l b er e p l a c e d b y t a b l e s, c h a i rs a n d b e e r - m u g s , o n c o n d i t i o n t h a t t h e l a t t e r a rep o s t u l a t e d t o s a t is fy t h e r e l a ti o n s e s t a b l i sh e d b y t h e a x i o m s . T h e w h o l e s y s t e mo f g e o m e t r y s h o u l d r e m a i n u n a f f e c t e d b y s u c h a c h a n g e . T h e r e f o r e , it is o f t e ns ai d , HIL BE RT p r o m o t e d a v i e w o f m a t h e m a t i c s a s a n e m p t y f o r m a l g a m e , inw h i c h i n f e r e n c e r u l e s a r e p r e s c r i b e d i n a d v a n c e , a n d d e d u c t i o n s a r e d r a w n ,f o l lo w i n g t h o s e r u le s , f r o m a r b i t r a r i l y g i v e n s y s t e m s o f p o s t u l a t e s . 2 I f t h i s w a s

1 See , e .g . , WIGHTMAN 1976 , GN EDENKO 1979.2 Such a v iew has b een pu t fo rwa rd by , e .g . , the F rench ma them at ic ian JEANDIEUDONNI~ (19 06 1992). In a w idely re ad ex po sito ry ar ticle, DIEUDONNI~expla ined theessence o f HILBERT's ma them at ica l concep t ions by ana lo gy w i th a gam e of chess. A f te rexpla in ing tha t in the l a t t e r one does no t speak about t r u ths bu t r a the r about fo l lowingcorrect ly a se t of s t ipula ted rules , he adde d DIEUDONNI~ 1962 , 551 . I t a li c s in theor ig ina l ) : T ran spo son s cela en math6m at iques , e t nous auron s l a concep t ion de HIL-BERT: les m ath~ m atiqu es d evien nen t un j e u dont les pi~ces sont des s ignes graphiques sed is t inguant l e s uns des au t r es pa r l eur fo rme .


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Hilbert and the Axiomatization of Physics 85indeed HILBERT S view of mathematics, then in what sense could he haveintended to apply such a view to physics, as stated in the sixth problem? Byasking what HILBERT was aiming at when addressing the question of theaxiomatization of physics, we are thus asking what role HILRERT ascribed toaxiomatization in mathematics and in science in general (especially physics), andhow he conceived the relation between mathematics and physics. Answering thisquestion will help to clarify many aspects of HILBERT S overall conception ofmathematics.

The first part of the present article describes the roots of HILBERT S earlyconception of axiomatics, putting special emphasis on the analogies he drewbetween geometry and the physical sciences. In this light, HILBERT S axiomaticapproach is presented as an endeavor with little connection to the view ofmathematical theories as empty formal games, devoid of concrete content-- a view that became dominant in wide mathematical circles after the 1930s.Rather, it appears as the opposite: as a method for enhancing our understand-ing of the mathematical content of theories and for excluding possible contra-dictions or superfluous assertions that may appear in them. This understandingof HILBERT S axiomatics also explains the place of the sixth problem in hismathematical world. The second part of the article addresses in a moredetailed manner the question of how HILBERT conceived the specific applicationof the axiomatic approach to particular branches of science, and what imageof science emerges from that approach. Using the manuscript of a coursetaught by HILBERT in G6ttingen in 1905, I discuss HILBERT S axiomatictreatment of various scientific disciplines and his conception of the conceptualand methodological connections among the latter. This account is also intendedto open the way to a broader understanding of HILBERT S later works onphysics and, in particular, to a detailed analysis -- which I plan to undertake inthe near future - - of the path that led HILBERT to his research on generalrelativity.

2 H i l b e r t a s S t u d e n t an d T e a c h e r

Physics was not a side issue that occupied HILBERT S thought only sporadi-cally. At least since the mid-1890s HILBERT had been interested in currentprogress in physics, and this interest gradually became a constitutive feature ofhis overall conception of mathematics. In order to describe this properly, onehas to consider HILBERT S biography. HILBERT S studies and early mathematicalcareer between 1880 and 1895 took place in his native city of K6nigsberg,except for a short trip in 1885 - - after finishing his dissertation - - to FELIXKLEIN (1849--1925) in Leipzig and to CHARLES HER~ITE (1822--1901) in Paris.K6nigsberg had a small university, with a very respectable tradition of researchand education in mathematics and physics that had been established during thefirst half of the nineteenth century by CARL GUSTAV JACOBI (1804--1851) and


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86 L. CORRYFRANZ ERN ST NEUMANN (1798--1895). 3 D ur in g h i s f i r s t yea rs as a s tud en t ,H IL BE RT w a s a b l e t o a t t e n d t h e l e c t u r e s o f t h e d i s t i n g u i s h e d m a t h e m a t i c i a nHEINRIC H W EB ER (18 42 --19 13 ), 4 wh o s e i n t e r e s t s co v e red an a s t o n i s h i n g v a r i e t y o fi ss u es r a n g i n g f r o m t h e t h e o r y o f p o l y n o m i a l e q u a t i o n s , t o e l l ip t ic f u n c ti o n s , t om a t h e m a t i c a l p h y s ic s . T h e c o n g e n i a l e n v i r o n m e n t W E B ER f o u n d i n K 6 n i g s b e r gf o r p u r s u i n g h i s m a n i f o l d m a t h e m a t i c a l i n t e r e s t s w a s t h e o n e w i t h i n w h i c hH IL BE RT 'S e a r l y m a t h e m a t i c a l o u t l o o k w a s f o r m e d . H o w e v e r , W E BE R n e v e r d e -v e l o p e d a c i r c le o f s t u d e n t s a r o u n d h i m , a n d i t is u n li k e l y t h a t - - p r i o r t oW E BE R'S d e p a r t u r e f o r Z i i r i c h i n 1 88 3 - - t h e y o u n g H IL B ER T b e n e f i t e d f r o md i r e c t c o n t a c t w i t h h i m o r h i s c u r r e n t r e s e a r c h i n t e r e s t s .

HILBERT'S d o c to ra l ad vi se r w as FERDINAND LINDEMANN (1852--1939), a fo rm ers t u d e n t o f F E L I X K L E I N . L IN DE MA NN 'S m a t h e m a t i c a l a c h i e v e m e n t s - - h e is r e -m e m b e r e d t o d a y m o s t ly f o r h is p r o o f o f th e t r a n s c e n d e n c e o f ~c - - w e r e n o to u t s t a n d i n g , b u t h e c e r t a i n l y e x e r t e d a n i m p o r t a n t i n f l u e n c e o n H I L B E R T ' Sm a t h e m a t i c a l f o r m a t i o n . B u t p e r h a p s t h e f o r e m o s t i n f lu e n c e o n s h a p i n g H IL -BERT'S i n t e l l e c t u a l h o r i z o n i n K 6 n i g s b e r g c a m e f r o m h i s e x c e p t i o n a l r e l a t i o n s h i pw i t h t w o o t h e r y o u n g m a t h em a t i c i an s : ADOLF HUR W ITZ (18 59 --19 19 ), f i r s t HIL-B ER T'S t ea ch e r an d l a t e r h i s co l l eag u e , an d HERM ANN M INKOW SKI (1 8 6 4 1 9 0 9 ) .B e f o r e a c c e p t i n g i n 1 8 8 4 a n e w c h a i r e s p e c i a l l y c r e a t e d f o r h i m i n K 6 n i g s b e r g ,H U RW IT Z h a d s t u d i e d f i rs t w i t h K L EIN i n L e i p z i g a n d t h e n i n B e r l i n , a n d h a dl a t e r h a b i l i t a t e d i n G 6 t t i n g e n i n 1 88 2. H U RW IT Z w a s t h u s w e l l a w a r e o f t h e k i n do f m a t h e m a t i c a l i n t e r e s ts a n d t e c h n i q u e s d o m i n a t i n g c u r r e n t r e s e a r c h in e a c h o ft h e s e i m p o r t a n t c e n te r s . H UR W IT Z t a u g h t f o r e i g h t y e a r s i n K 6 n i g s b e r g b e f o r em o v i n g t o Z ~ r i c h , a n d h i s i n f l u e n c e d u r i n g t h i s t im e w a s d e c i s i v e i n s h a p i n gH IL BE RT 'S v e r y w i d e s p e c t r u m o f m a t h e m a t i c a l i n t e r e s ts , b o t h a s a s t u d e n t a n da s a y o u n g r e s e a r c h e r .

M IN KO W SK I'S m a i n i n t e r e s t s a l s o l a y i n p u r e m a t h e m a t i c s , b u t t h e y b y n om e a n s r e m a i n e d c o n f i n e d t o it . A s a s t u d e n t , M I NK O W SK I s p e n t t h r e e s e m e s t e r si n B o n n b e f o r e r e c e iv i n g h is d o c t o r a t e i n K 6 n i g s b e r g i n 1 885 . H e r e t u r n e d t oB o n n a s a rivatdozent a n d r e m a i n e d t h e r e u n t i l 1 8 9 4 , w h e n h e m o v e d t oZ i i r ic h . N o t u n t i l 1 9 02 d i d h e j o i n H IL B ER T i n G 6 t t i n g e n , f o l l o w i n g K L E IN 'Ss u c c e s s i n p e r s u a d i n g t h e P r u s s i a n e d u c a t i o n a l a u t h o r i t i e s t o c r e a t e a t h i r dc h a i r o f m a t h e m a t i c s e s p e c ia l ly f o r h i m . D u r i n g a ll t h o s e y e a r s t h e f r i e n d s h ipb e t w e e n MINKOWSKIHUR W ITZ an d HILB ER T rem a i n ed c l o s e . M INKOW SKI v i s i t edK / S n i g s b e r g e a c h s u m m e r , a n d t h e t h r e e m a t h e m a t i c i a n s w o u l d m e e t d a i l y f o rm a t h e m a t i c a l w a l k s . D u r i n g t h e C h r i s t m a s h o l i d a y s o f 1 89 0 MIN K O W SK I r e -m a i n e d i n B o n n , a n d i n a l e t t e r t o H I L BE R T h e d e s c r i b e d h i s c u r r e n t i n t e r e s t i np h y s i c s . I n h i s o b i t u a r y o f MINKOWSKIH IL BE RT r e p o r t e d - - i n a n o f t e n - q u o t e d

a O n the K6n igsb erg sch ool see KL EIN 1926--7 Vol. 1, 112-115 216-221; VOLK1967. The workings of the K 6nigsberg physics seminar - - in i t ia ted in 1834 by FRANZNEUMANN and its eno rm ous influence on ninetee nth-cen tury physics educ at ion inG erm any are described in great detai l in OLESKO 1991.4 F or m ore details on WEBER (especially conce rning his con tribu tions to algebra) seeCORRY 19 96 , w167 2.2.4.


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H i l b e r t a n d t h e A x i o m a t i za t i o n o f P h y s ic s 8 7p a s s a g e - - t h a t u p o n h is i n si st e n ce t h a t M IN KO W SK I c o m e t o G 6 t t i n g e n t o j o i nh i m a n d H u R w I x z , M IN K O W SK I h a d d e s c r i b e d h i m s e l f a s b e i n g n o w c o n -t a m i n a t e d w i t h p hy s ic s , a n d i n n e e d o f a t e n - d a y q u a r a n t i n e b e f o r e b e i n g a b let o r e t u r n t o t h e p u r e l y m a t h e m a t i c a l a t m o s p h e r e o f K 6 n i g s b e r g . H IL BE RT a l soq u o t e d M IN K O W S K I'S l e t t e r a s f o l l o w s :

I have devo ted m yse l f fo r the t ime be ing com ple te ly to m agic , tha t i s to s ay , tophysics . 1 hav e m y pract ica l exerc ises a t the p hys ics ins t i tute , and a t h om e I s tud yTho mso n, He lm hol tz and the i r accomplices. S ta r t ing nex t weekend , I ' l l wo rk som edays eve ry week in a b lue smock in an ins t i tu te tha t p roduces phys ica l ins t ruments ;th is is a k ind of p rac ti ca l tra in ing than which you co uld no t ev en imag ine a m oresham eful one. 5

M IN K O W SK I'S i n t e r e s t i n p h y s i c s c a n c e r t a i n l y b e d a t e d e v e n e a r l i e r t h a n t h is ; i n1 88 8 h e h a d a l r e a d y p u b l i s h e d a n a r ti c le o n h y d r o d y n a m i c s , s u b m i t t e d t o t h eB e r l i n A c a d e m y b y H E RM A N N YO N H E LM H O L TZ (M IN K O W S K I 1 88 8 ). L a t e r , d u r i n gh i s Z / i r i c h y e a r s , M IN I(O W S K~ 'S n t e r e s t i n p h y s i c s r e m a i n e d a l iv e , a n d s o d i d h i sc o n t a c t w i t h H I LB E RT . F r o m t h e i r c o r r e s p o n d e n c e w e l e a r n t h a t M IN K OW S KId e d i c a t e d p a r t o f h i s e f fo r ts to m a t h e m a t i c a l p h y s i cs , a n d i n p a r t i c u l a r t o t h e r m o -d y n a m i c s . 6 F i n a l l y , M IN K O W S KI'S l a s t y e a r s i n G 6 t t i n g e n w e r e i n t e n s i v e l y d e d i -c a t e d t o p h y s i c s . D u r i n g t h o s e y e a r s H m B E RT 'S i n t e r e s t i n p h y s i c s b e c a m e m o r ev i g o r o u s t h a n e v e r b e f o re ; h e a n d M I NK O W S K I, i n f ac t , c o n d u c t e d a d v a n c e ds e m i n a r s o n p h y s i c a l i ss ue s. 7 A t t e n t i o n t o c u r r e n t d e v e l o p m e n t s i n p h y s i cs w a sne ve r fo re ign to HmBERT'S an d MINKOWSKI'Sm a i n c o n c e r n s w i t h p u r e m a t h e m a t i c s .

A b a l a n c e d u n d e r s t a n d i n g o f HIL BE RT 'S m a t h e m a t i c a l w o r l d c a n n o t b ea c h i e v e d w i t h o u t p a y i n g c lo s e a t t e n t i o n t o h i s t e a c h i n g , f ir s t a t K 6 n i g s b e r g a n de s p e c i a l l y a t G 6 t t i n g e n b e g i n n i n g i n 1 89 5. H ~ LB ER T d i r e c t e d n o l es s t h a n s i x t y -e i g h t d o c t o r a l d i s s e r t a t io n s , s i x t y o f t h e m i n t h e r e l a t iv e l y s h o r t p e r i o d b e t w e e n1 89 8 a n d 1 91 4. A s is w e l l - k n o w n , a t t h e m a t h e m a t i c a l i n s t it u t e c r e a t e d i nG 6 t t i n g e n b y F E L I X K L EIN , H IL B ER T b e c a m e t h e l e a d e r o f a u n i q u e s c i e n ti f icc e n t e r t h a t b r o u g h t t o g e t h e r a g a l l e ry o f w o r l d - c l as s r e s e a rc h e r s in m a t h e m a t i c sa n d p h y s i c s, s I t i s h a r d t o e x a g g e r a t e t h e i n f lu e n c e o f H tL BE RT 'S t h i n k i n g a n dp e r s o n a l i t y o n a l l t h a t c a m e o u t o f t h e i n s ti t u t e u n d e r h i s d i r e c ti o n . F o r t u n a t e -ly , w e c a n d o c u m e n t w i t h g r e a t a c c u r a c y t h e c o n t e n t s o f Hm BE RT 'S G 6 t t i n g e nl e c tu r e s , w h i c h i n t e r e s t i n g l y i l l u m i n a t e t h e e v o l u t i o n o f h i s i d e a s o n m a n yi ss u es . T h e s e l e c t u r e s w e r e f a r f r o m b e i n g o r g a n i z e d p r e s e n t a t i o n s o f w e l l-k n o w n r e s u l t s a n d e s t a b l i s h e d t h e o r i e s . R a t h e r , h e u s e d h i s l e c t u r e s t o e x p l o r e

5 F or the or iginal le tter , f ro m wh ich this passa ge is t rans la ted, see R~DENBERG& ZASSENHAVS (eds . ) 197 3, 39-42, on pp. 39-40. Fo r HILBERT's qu ota t io n see GVol. 3 , 355. Un less otherw ise s ta ted in this ar ticle, a l l t rans la t ions into Engl ish a re mine.6 See RODEN BERG ZASSENHAUS e d s . ) 1973 , 110-114.7 On MINKOWSKI'Syears in G6 t t ingen, see CORRY 1997a; GALISON 1977; PYENSON1977, 1979.

s Accou nts o f G 6t t ingen as the wo r ld lead ing cen te r o f ma themat ics , and the ro les o fKLE IN and HILBERT in fos te r ing th i s cen t ra l i ty ap pea r in RE ID 1970; ROW E 1989;PARSHALL & ROW E 1994 , 150-154.


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88 L. CORRYn e w i d e a s a n d t o t h i n k a l o u d a b o u t t h e i s s u e s t h a t c u r r e n t l y o c c u p i e d h i m .F o l l o w i n g a t r a d i t i o n i n i t i a te d b y K L EIN in G 6 t t i n g e n , H IL BE RT S l e c t u r e n o t e sw e r e m a d e a v a i l a b l e t o a ll s t u d e n t s w h o w i s h e d t o c o n s u l t t h e m a t t h e Lesezim-mer t h e h e a r t o f t h e m a t h e m a t i c a l i n s t it u t e . A t l e a s t si n ce 1 90 2, i n e v e r y c o u r s eh e t a u g h t , H m BE RT c h o s e a s t u d e n t t o t a k e n o t e s d u r i n g t h e le c tu r e s . T h es t u d e n t w a s e x p e c t e d t o w r i t e u p t h e s e n o t e s c o h e r e n t l y , w h e r e u p o n H I L B E R Tw o u l d g o t h r o u g h t h e m , a d d i n g h is o w n c o r r e ct i o n s a n d r e m a r k s . 9 T o d a y t hec o l l e c t i o n o f t h e s e n o t e s o f f e r s a n i n v a l u a b l e s o u r c e f o r t h e h i s t o r i a n i n t e r e s t e di n u n d e r s t a n d i n g H IL BE RT S t h o u g h t .

L a t e i n l if e H IL BE RT v i v i d l y re c a l l e d t h a t t h e s e l e c t u r e s p r o v i d e d i m p o r t a n to c c a s i o n s f o r t h e f r ee e x p l o r a t i o n o f u n t r i e d i d e a s. H e t h u s s ai d:

The c loses t conce ivab le connec t ion be tween resea rch and teach ing became a dec i s ivefea ture o f my m athem at ica l ac t iv ity . The in te rchange of sc ien ti fi c ideas, the co m m un-i c a ti o n o f w h a t o n e fo u n d b y h i m s e lf a n d t h e e l a b o ra t i o n o f w h a t o n e h a d h e a rd ,w a s f ro m m y e a r ly y e a rs a t K 6 n i g s b e rg a p i v o t a l a sp e c t o f m y s ci en tif ic w o r k . . . I nmy lec tures , and above a l l in the seminars , my gu id ing pr inc ip le was no t to p resen tma te r ia l in a s t and ard and as smo oth as poss ib le way , jus t to he lp the s tuden tkeep ing c lean and orde red no tebooks . Above a l l , I a lways t r i ed to i l lumina te theprob lem s and d if fi cu lt ie s an d to o f fe r a b r idge l ead ing to cu r ren t ly ope n ques t ions. I to f te n h a p p e n e d t h a t i n t h e c o u r s e o f a s e m e s t e r t h e p ro g ra m o f a n a d v a n c e d l e c tu r ewas com ple te ly changed , because I w anted to d i scuss i ssues in which I w as cur ren t lyinvolved as a re sea rcher and which had no t ye t by any means a t t a ined the i r de f in i t eformu la t ion . (T rans la ted f rom HILBERT 1971 , 79)R e c o g n i z i n g t h e c e n t r a l i t y o f h i s te a c h i n g a c t iv i t ie s a n d t h e e x t e n t t o w h i c h

h i s l e c t u r e s r e f l e c te d h is c u r r e n t m a t h e m a t i c a l i n t e r e st s , o n e i s l e d t o r e a s s e s sl o n g - e s t a b l i sh e d a s s u m p t i o n s a b o u t t h e p e r i o d i z a t i o n o f HIL BE RT S w o r k . I n a no f t e n - q u o t e d p a s s a g e , H E RM A N N W E Y L ( 1 94 4 , 6 1 9 ) a s s e r t e d t h a t I-tIL BE RT Sw o r kc o m p r i s e d f iv e s e p a r a t e , a n d c l e a r l y d i s c e r n i b le m a i n p e r i o d s : (1 ) T h e o r y o fi n v a r i a n t s ( 1 8 8 5 -1 8 9 3 ) ; ( 2) T h e o r y o f a l g e b r a i c n u m b e r f i e ld s (1 8 9 3 - 18 9 8 ) ; (3 )F o u n d a t i o n s , ( a) o f g e o m e t r y ( 1 8 9 8- 1 90 2 ), ( b) o f m a t h e m a t i c s i n g e n e r a l( 1 9 2 2 - 1 9 3 0 ) ; ( 4 ) I n t e g r a l e q u a t i o n s ( 1 9 0 2 - 1 9 1 2 ) ; ( 5 ) P h y s i c s ( 1 9 1 0 - 1 9 2 2 ) . T h i sp e r i o d i z a t i o n r e fl e ct s f a i t h fu l l y t h e d i v i s i o n o f H I LB ER T S p u b l i s h e d w o r k , a n dw h a t c o n s t i t u t e d h i s c e n t r a l d o m a i n o f i n t e r e s t a t d i f fe r e n t t im e s . I t s a y s m u c hle ss , h o w e v e r , a b o u t t h e e v o l u t i o n o f h i s th o u g h t , a n d a b o u t t h e e ff o rt s h ed e d i c a t e d t o o t h e r f ie ld s s i m u l t a n e o u s l y w i t h h i s m a i n c u r r e n t i n t e re s t s. 1 ~ A sw i l l b e s e e n i n w h a t f o l l o w s , t h e l is t o f HIL B ER T S l e c t u r e s d u r i n g t h o s e y e a r ss h o w s a m o r e c o m p l e x p i c t u r e t h a n W E Y L S p e r i o d i z a t i o n s u g ge s ts . I n p a r t i c u -l a r , i t w i ll b e s e e n t h a t H IL B ER T S c o n c e r n w i t h t h e p h y s i c a l s c i e n c es w a sa s u s t a in e d o n e , w h i c h c a n b e d o c u m e n t e d t h r o u g h o u t h i s c a r ee r .

9 See BORN 1978 , 81- -85 , fo r a re t rospec t ive acco unt o f BORN S ow n exper ience asHILBERT S s tud ent .lo In fac t, no one w as in a be t t e r pos i t ion than W EYL h imse l f to apprec ia te theim pac t o f HILBERT S doc ent act iv it ie s, a s he m ade c lea r in va r ious oppor tun i t i e s .O n WEY L S (som etimes changing ) assessm ents of HILBERT S nfluence as a teacher , seeSIGUROSSON 1994, 356--358.


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Hilber t and the Axiom at iza tion o f Phys ics 893. Th e Background to Hi lbert s A xiom atic Approach:Geometry and Physics

H IL BE RT S f ir st p u b l is h e d , c o m p r e h e n s i v e p r e s e n t a t i o n o f a n a x i o m a t i z e dd i s c i p l i n e ap p ea red i n 1 8 9 9 , i n t h e ev e r s i n ce f am o u s Grundlagen der Geometrie.T h e r o o t s o f H IL BE RT S a x i o m a t i c c o n c e p t i o n a c c o r d i n g l y a n d o b v i o u s l y l ie i nc o n t e m p o r a r y d e v e l o p m e n t s i n g e o m e t r y . I n w h a t f o ll o w s I w il l b r ie f ly d e s c ri b es o m e o f t h e s e d e v e l o p m e n t s , o f w h i c h s e v e r a l tr a d i t i o n a l a c c o u n t s e x i st .O n l y r e l a t iv e l y r e c e n t l y , h o w e v e r , h a s t h e r e l e v a n t h i s t o r i c a l e v i d e n c e b e e nt h o r o u g h l y s t u d ie d . 11 M o r e t o t h e p o i n t f o r m y p r e s e n t p u r p o s e s , l w il l s h o wt h a t H IL BE RT S u r g e t o a x i o m a t i z e p h y s i c a l t h e o r i e s , a s w e l l as h i s c o n c e p t i o n o fh o w t h is s h o u l d b e d o n e , a r o s e s i m u l t a n e o u s l y w i th t h e c o n s o l i d a t i o n o f h isa x i o m a t i c t r e a t m e n t o f g e o m e t r y . C e r t a i n l y t o a l es se r d e g r e e t h a n g e o m e t r y ,b u t s ti ll i n s i g n i f i can t w ay s , HILBER T S i n c rea s i n g i n t e r e s t i n p h y s i c s p l ay s ani m p o r t a n t r o l e i n u n d e r s t a n d i n g t h e e v o l u t i o n o f h is t h o u g h t s o n t h e a x i o m a t i cm e t h o d .

D u r i n g t h e n i n e t e e n t h c e n t u r y , f o l l o w i n g t h e w o r k o f J EA N V IC TO R PO NC EL ET(17 88 --18 67 ) i n 1 82 2, p ro j e c t i v e g eo m e t r y b e ca m e an ac t i v e f i e ld o f r e s e a rcht h a t a t t r a c t e d t h e a t t e n t i o n o f m a n y m a t h e m a t i c i a n s , e s p ec i a ll y i n G e r m a n y .H IL BE RT S o w n i n t e r e s t i n f o u n d a t i o n a l q u e s t i o n s o f g e o m e t r y a r o s e i nc o n n e c t i o n w i th l o n g - s t a n d i n g o p e n i ss ue s i n th i s d o m a i n - - m a i n l y h a v i n gt o d o w i t h t h e r o l e o f c o n t i n u i t y c o n s i d e r a t i o n s i n t h e s u b je c t s f o u n d a t i o n s .A m a j o r c o n t r i b u t i o n h e r e c a m e f r o m t h e e a r l y a t t e m p t s o f F ~ L I X K L E INt o e x p l a in t h e i n t e r re l a t i o n s a m o n g t h e v a r i o u s k in d s o f g e o m e t r y a n d t os h o w t h a t E u c l i d e a n a n d n o n - E u c l i d e a n g e o m e t r i e s a r e i n s o m e s e n s e d e r i v a t i v ec a s e s o f p r o j e c t i v e g e o m e t r y . A c r u c i a l st e p i n t h is p r o j e c t w a s t h e i n t r o d u c -t i o n o f a t y p e o f d i s t a n c e , o r m e t r i c , i n t o n o n - E u c l i d e a n s t r u c t u r e s , w i t h -o u t u s in g c o n c e p t s d e r i v e d f r o m t h e E u c l i d e a n c a se . K L E I N i n t r o d u c e d o n es u c h m e t r i c u s i ng t h e c o n c e p t o f th e c r o s s - r a ti o o f f o u r p o i n t s, w h i c his i n v a r i a n t u n d e r p r o j e c t i v e t r a n s f o r m a t i o n s . H e r e l i e d o n i d e a s o r i g i n a l l yi n t ro d u ce d b y AR THUR C AYLEY (18 21 --18 95 ) i n h i s w o r k o n q u ad ra t i c i n v a r i -a n t s , 12 b u t e x t e n d e d t h e m t o c o v e r t h e n o n - E u c l i d e a n c a s e, w h i c h CA YLEY h a de x p r e s s l y a v o i d e d i n h i s o w n w o r k . I n o r d e r t o d e f i n e c r o s s - r a t i o s i n p u r e l yp r o j e c t i v e t e r m s , K L E I N a p p e a l e d t o a r e s u l t o f V ON S TA U D T, a c c o r d i n gt o w h i c h o n e c o u l d i n t r o d u c e c o o r d i n a t e s i n t o p r o j e c t iv e g e o m e t r y , i n d e p e n -d e n t l y o f m e t r i c a l n o t i o n s a n d o f t h e p a r a l l e l p o s t u l a t e . I n f a c t , K L EIN

f a i l e d t o e x p l a i n i n d e t a i l h o w t h is c o u l d b e e f fe c te d , b u t i n a n y c a s e

11 Based on the manuscripts of HILBERT s early lectures, MICHAEL M. TOEPELL(1986) has analyzed in considerable detai l the developm ent of HILBERT s ideas previousto the publ icat ion of the Grundlagen and his encounters with the foundat ions ofgeo m etry since his K 6nig sberg years. In this section I p art ly rely on TOEPELL S llumina-t ing account .12 Fo r an acc ou nt of CAYLEY S con tributio ns see K LE IN 1926--7 Vol. 1, 147-151.


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90 L. CORRYh is a r g u m e n t s e x p li c it ly p r e s u p p o s e d t h e n e e d t o a d d a c o n t i n u i t y a x i o m t o VONSTAVDT'S re su lts . 13

T h e u n c e r t a i n t i e s a s s o c i a t e d w i t h K L E I N 'S r e s u lt s , a s w e l l a s w i t h o t h e rc o n t e m p o r a r y w o r k s , i n d ic a t e d t o s o m e m a t h e m a t i c i a n s t h e n e e d t o r e - e x a m i n ew i t h g r e a t e r c a r e t h e d e d u c t i v e s t r u c t u r e s o f th e e x i s ti n g b o d y o f k n o w l e d g e i np r o j e c t i v e g e o m e t r y . T h e f i rs t e l a b o r a t e a t t e m p t t o d o s o a p p e a r e d i n 18 82,w h e n M o R ITZ PASCH (1 84 3 --1 93 0 ) p u b l i s h ed h i s b o o k Vorlesungen iiber neuereGeometrie, p r e s e n t i n g p r o j e c t i v e g e o m e t r y in w h a t h e s a w a s a n i n n o v a t i v e ,t h o r o u g h l y a x i o m a t i c f a s h i o n . 14 P A SC H u n d e r t o o k a r e v i s i o n o f E uc LIO 'S b a s i ca s s u m p t i o n s a n d r u le s o f i n fe r e n c e, a n d c a r e f u ll y c l o se d s o m e f u n d a m e n t a ll o g i c a l g a p s a f f e c t in g t h e l a t te r . I n P A SC H 'S r e c o n s t r u c t i o n o f p r o j e c t i v e g e -o m e t r y , o n c e t h e a x i o m s a r e d e t e r m i n e d , a ll o t h e r r e s u lt s o f g e o m e t r y w e r e t ob e a t t a i n e d b y s t r i c t l o g i c a l d e d u c t i o n , a n d w i t h o u t a n y a p p e a l t o d i a g r a m s o rt o p r o p e r t i e s o f t h e f i g u r e s i n v o l v e d . Y e t i t is i m p o r t a n t t o s t re s s t h a t PA SC Ha l w a y s c o n c e i v e d g e o m e t r y a s a n a t u r a l s c ie n c e , h a v i n g a s it s s u b j e c t m a t t e rt h e s t u d y o f t h e e x t e r n a l s h a p e o f th i ng s , a n d w h o s e t r u t h s c a n b e o b t a i n e df r o m a h a n d f u l o f c o n c e p t s a n d b a s ic l a w s ( t h e a x io m s ) , t h a t a r e d i r e c t l y d e r i v e df r o m e x p e r ie n c e . F o r P AS CH , t h e m e a n i n g o f th e a x i o m s t h e m s e l ve s i s p u r e l yg e o m e t r i c a l a n d c a n n o t b e g r a s p e d w i t h o u t a p p e a l t o t h e d ia g r a m s f r o m w h i c ht h e y a r e d e r i v e d . P A SC H , f o r i n s t a n c e , c o n s i d e r e d t h a t t h e c o n t i n u i t y a x i o m f o rg e o m e t r y w a s n o t c o n v i n c i n g l y s u p p o r t e d b y e m p i r i c a l e v i d en c e . 15

T h o u g h P AS CH s u b s t a n t i a ll y c o n t r i b u t e d t o c l a ri f y in g m a n y a s p e c ts o f th el o g ic a l s t r u c t u r e o f p r o j e c t i v e g e o m e t r y , t h e t r u e s t a tu s o f c o n t i n u i t y a s s u m p -t i o n s in p r o j e c t i v e g e o m e t r y , r e m a i n e d u n c l e a r. T h i s i s p a r t i c u l a r l y t r u e c o n -c e r n i n g t h e p o s s i b i l it y o f e s t a b l i s h i n g a l i n k b e t w e e n t h is g e o m e t r y a n d a s y s t e mo f r e a l - n u m b e r c o o r d i n a t e s ( c o o r d i n a t i z a t i o n ) a s w e l l as d e f i n in g a m e t r i c s f o r it( m e t r i z a t i o n ) . T h e q u e s t i o n w a s o p e n w h e t h e r c o n t i n u i t y s h o u l d b e c o n s i d e r e dt o b e g i v e n w i t h t h e v e r y id e a o f s p ac e , o r w h e t h e r i t s h o u l d b e r e d u c e d t om o r e e l e m e n t a r y c o n c e p t s . K L EIN a n d W I LH E LM K IL L IN G ( 18 4 7- -1 9 23 ) e l a b o r a t e dt h e f i r s t o f t h e s e a l t e rn a t i v e s , wh i l e HER M ANN LUDW IG W IENER (1 85 7 --1 93 9 ) a n dFRIEDR IC H SCHUR (1 8 5 6 -1 9 3 2 ) w o r k e d o u t t h e s eco n d . W IENER p u t fo rw ard h i sp o i n t o f vi e w i n 1 89 1, i n a l e c t u r e o n f o u n d a t i o n a l q u e s t i o n s o f g e o m e t r yd e l iv e r ed at t h e a n n u a l m e e t i n g o f t h e G e r m a n M a t h e m a t i c i a n s ' A s s o c ia t io nD M V ) i n H a l l e ( W IE N ER 1 89 1). W i e n e r c l a i m e d t h a t s t a r t i n g s o l e ly w i t h t h et h e o r em s o f DESAR GUES an d PAPPUS (o r PASCAL'S t h eo re m fo r t w o l in e s , a sW IENER , an d l a t e r a l s o HIL~ ZR T ca l l ed i t) , i t i s p o s s i b l e t o p r o v e t h e f u n d a m en t a lt h e o r e m o f p r o j e c t i v e g e o m e t r y , n a m e l y , t h a t f o r t w o g i v en li ne s t h e r e e x is ts o n ea n d o n l y o n e p r o j e c t i v e m a p p i n g t h a t c o r r e l a t e s a n y t h r e e g i v e n p o i n t s o f t h ef i rs t t o a n y t h r e e g i v e n p o i n t s o f t h e s e c o n d i n a g i v e n o r d e r . T h e c l a s s i c a l p r o o fo f t h is t h e o r e m w a s b a s e d o n t h e p r o j e c t i v e in v a r i a n c e o f t h e c r o s s - ra t i o ; th i s

13 KL EIN 1871 & 1873. F or com me nts o n these contr ibu t ions of KLEIN see ROWE1994, 194-195; TOEP ELL 1986, 4-6; TORRETTI 1978 , 110-152, O n VON STAUDT'S con tri-bu tion see FREUDENTHAL 1974.14 O n PASCH'S bo ok see , e .g . , TORRETTI 1978, 44-53.15 See CONTRO 1976, 284-289; NAG EL 1939, 193 199 ; TORRETTI 1978 , 210-218.


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Hilbe rt and the Ax iomatizat ion of Physics 91i n v a r i a n c e i m p l i e s t h a t t h e i m a g e o f a f o u r t h p o i n t i n t h e f i rs t l in e is u n i q u e l yd e t e r m i n e d u n d e r t h e g i v e n p r o j e c t iv e m a p p i n g , b u t t h e e x i st e n c e o f t h e f o u r t hp o i n t o n t h e s e c o n d l i n e ty p i c a l ly c a ll s fo r t h e i n t r o d u c t i o n o f s o m e k i n d o fc o n t i n u i t y a r g u m e n t . 16 W I EN E R S i d e a s s e e m e d t o o p e n t h e p o s s i b i l it y o f d e v e l -o p i n g p r o j e c t i v e g e o m e t r y f r o m a n e w p e r s p e c t iv e w i t h o u t a n y u s e o f c o n t i n u i t yc o n s i d e r a t i o n s . L a t e r , i n 1 8 98 , S CI~U R f u r t h e r p r o v e d P m , P US S t h e o r e m w i t h o u tu s i n g a n y c o n t i n u i t y a s s u m p t i o n s (S CH U R 1 89 8). T h i s w h o l e i s s u e o f th e p r e c i s er o l e o f c o n t i n u i t y 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 l a t e r b e c a m e , a s w e w i ll se e,a m a j o r s t i m u l u s f o r H IL BE RT S a c t i v e i n v o l v e m e n t i n t h is d o m a i n .

PA SC H S a x i o m a t i c t r e a t m e n t o f p r o j e c t iv e g e o m e t r y h a d c o n s i d e r a b l e i n fl u -e n c e a m o n g I t a l i a n m a t h e m a t i c i a n s , a n d i n t h e f i r st p l a c e o n G IU S EP PE PE AN O(1 85 8--1 93 0). P E AN O w a s a c o m p e t e n t m a t h e m a t i c i a n , w h o m a d e s i g n i fi c a n tc o n t r i b u t i o n s i n a n a l y s i s a n d w r o t e i m p o r t a n t t e x t b o o k s i n t h i s f ie ld . 17 B u tb e s i d e s th e s e s t a n d a r d m a t h e m a t i c a l a c t i v it ie s , P E AN O i n v e s t e d m u c h o f hi se f fo r ts t o a d v a n c e t h e c a u s e o f i n t e r n a t i o n a l l a n g u a g e s - - h e d e v e l o p e d o n es u c h l a n g u a g e c a l l e d I n t e r l i n g u a - - a n d t o d e v e l o p a n a r t i f i c i a l c o n c e p t u a ll a n g u a g e t h a t w o u l d a l l o w c o m p l e t e l y f o r m a l t r e a t m e n t s o f m a t h e m a t i c a lp r o o f s . I n 1 8 89 h is s u c c e ss f u l a p p l i c a t i o n o f s u c h a c o n c e p t u a l l a n g u a g e t oa r i t h m e t i c , y i e l d e d h i s f a m o u s p o s t u l a t e s f o r t h e n a t u r a l n u m b e r s . PA SC H Ss y s t e m s o f a x i o m s f o r p r o j e c t i v e g e o m e t r y p o s e d a c h a l l e n g e t o P EA N O S a r t if i -c i a l la n g u a g e . I n a d d r e s s i n g t h is c h a l l e n g e , PE A NO w a s i n t e r e s t e d i n t h e r e l a t i o n -s h ip b e t w e e n t h e l o g ic a l a n d t h e g e o m e t r i c a l t e r m s i n v o l v e d i n t h e d e d u c t i v es t r u c t u r e o f g e o m e t r y , a n d i n t h e p o s s i b i li t y o f c o d i f y i n g th e l a t t e r i n h i s o w na r t if i c ia l l a n g u a g e . T h i s i n t e r e s t l e d P E AN O t o i n t r o d u c e t h e i d e a o f a n i n d e p e n -d e n t s e t o f a x i o m s , n a m e l y , a s et n o n e o f w h o s e a x i o m s i s a l o g i c a l c o n s e q u e n c eo f t h e o t h e r s . H e a p p l i e d t h is c o n c e p t t o h i s o w n s y s t e m o f a x i o m s f o r p r o j e c -t i v e g e o m e t ry , w h i ch w ere a s l ig h t m o d i f i ca t i o n o f PASC H S . PEANO S s p ec i fi c wayo f d e a l i n g w i th s y s t e m s o f a x i o m s , a n d t h e i m p o r t a n c e h e a t t r i b u t e d t o t h es e a r c h f o r i n d e p e n d e n t s e t s o f p o s t u l a t e s , is s i m i la r in m a n y r e s p e c t s t o t h ep e r s p e c t i v e d e v e l o p e d l a t e r b y H I L ~E R T; y e t P E A NO n e v e r u n d e r t o o k t o p r o v et h e i n d e p e n d e n c e o f w h o l e s y s t e m s o f p o s t u l a t e s , x s F o r a ll o f hi s i n s is t e n c e o nt h e l o g i c a l a n a ly s i s o f th e d e d u c t i v e s t r u c t u r e o f m a t h e m a t i c a l t h e o r i e s , PE AN O So v e r a l l v ie w o f m a t h e m a t i c s - - l ik e P ASC H S b e f o r e h i m - - w a s n e i t h e r f o r m a l i s tn o r l o g i c i st in t h e s e n se l a t e r a t t r i b u t e d t o t h e s e t e r m s . P EA N O c o n c e i v e dm a t h e m a t i c a l i d e a s a s b e i n g d e r i v e d f r o m o u r e m p i r i c a l e x p e r ie n c e . 19

16 Obviously the theorem can be dual ly formulated for two penci ls of l ines . Fora m ore o r l es s con temp orary fo rmula t ion o f the theorem see NRIQU S 1903. Interesting -ly, ENRIQUES explici tly rem arked in the in trodu ct ion to the G erm an vers ion of h is bo ok(p. v i i tha t h e w as following th e classical app roa ch intro duc ed by VON STAUOT, andfo llowed by K LEIN and o thers , ra ther than to the m ore mo dern one deve loped recen tlyby PASCH and HILBERT.

17 A brief account of PEANO s ma thema tical w ork appears in KENNEDY 1981. Fo rm ore elabora te accoun ts see KENNEDY 1980; S GR 1994.is Cf. TORRETTI 1978 , 221.19 See KENN EDY 198 1, 443.


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92 L. CORRYS e v e r a l I t a l i a n m a t h e m a t i c i a n s , i n f l u e n c e d b y P EA N O'S i d e a s, p u b l i s h e d s i m -

i la r w o r k s i n w h i c h t h e l o g i c a l s t r u c t u r e 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 w a si n v e s t i g a t e d . A m o n g t h e m o n e s h o u l d m e n t i o n M A R IO P I ER I (1 86 0--1 91 3), 2 ~ w h os t r o n g l y p r o m o t e d t h e i d e a o f g e o m e t r y a s a h y p o t h e t i c o - d e d u c t i v e sy s te m , a n di n t r o d u c e d f o r h is s y s te m s o f p o s t u l a t e s a k i n d o f o r d i n a l in d e p e n d e n c e ,s o m e w h a t m o r e l i m i te d t h a n t h e o n e d e f i n e d b y PEA NO .2 1 Of s p ec i a l i n t e re s t i sthe o r ig ina l w or k o f GIUSEPPE VERONESE (1845--1917), w ho in 1891 p ub l i sh ed thef ir st s y s t e m a t i c s t u d y o f t h e p o s s i b i li t y o f a n o n - A r c h i m e d e a n g e o m e t r y , 22 a n dp r o v e d t h e i n d e p e n d e n c e o f t h e A r c h i m e d e a n p o s t u l a t e f r o m t h e o t h e r p o s t u -l a t e s o f g eo m e t ry . 23 HIL B ER T tOO w o u ld ev e n tu a l l y d ea l w i th t h e s e i s s u es i n h i sa x i o m a t i c s t u d y o f g e o m e t r y . 2~

S o m u c h f o r t h e g e o m e t r i c b a c k g r o u n d a g a i n s t w h i c h H I LB ER T'S a x i o m a t i cm e t h o d a r o s e . I w i ll r e t u r n t o i t i n t h e n e x t s e ct i o n . B u t a s I h a v e a l r e a d ys u g g es te d , w e m u s t a l so l o o k a t c e r t a i n d e v e l o p m e n t s in p h y s i c s in t h e n i n e -t e e n t h c e n t u r y , in w h i c h n e w a x i o m a t i c t r e a t m e n t s o f o l d b o d i e s o f k n o w l e d g ew e r e a l s o b e i n g p u r s u e d , i n d e p e n d e n t l y o f t h e d e v e l o p m e n t s in g e o m e t r y di s-c u s s e d a b o v e . T h e a x i o m a t i c t r e a t m e n t o f m e c h a n i c s p u t f o r w a r d b y H EIN RIC HHE R TZ (1 85 7 --1 89 4 ) h a s b ee n m u c h l e ss a s s o c i a t e d w i th HILB E R T'S ax io m a t i c st h a n t h e a b o v e m e n t i o n e d w o r k o f PA SC H a n d t h e t r a d i t i o n t o w h i c h i t be l o ng s .Ye t , a s w i l l b e s een i n wh a t fo l l o ws , HE R T Z 'S P r i n c i p l e s o f M ech a n i c s m a d ea s t r o n g i m p r e s s i o n on HILBERT w h i c h c a n b e c o u n t e d a m o n g t h e s t i m u l i f o rt h e c o n s o l i d a t i o n o f h is a x i o m a t i c c o n c e p t i o n .I n 1 89 1 H E R T Z b e g a n t o w o r k f o r t h e f i r st ti m e i n h i s c a r e e r o n m e c h a n i c s .T h i s w o r k , t o w h i c h a l l h i s e f f o r t s w e r e d i r e c t e d d u r i n g t h e l a s t t h r e e y e a r s o fh i s l if e, l e d t o t h e p o s t h u m o u s p u b l i c a t i o n i n 1 8 94 o f Th e P r i n c ip l e s o f M ech a ni c s P r es en t ed i n a New F o r m . H ER TZ u n d e r t o o k t h i s w o r k m o t i v a t e d b y t h e t h e nw i d e ly - - t h o u g h n o t u n a n i m o u s l y - - a c c e p t e d c o n c e p t i o n t h a t m e c h a n i csc o n s t i t u t e s t h e m o s t b a s i c d i s c ip l i n e o f p h y s i c s, a n d a t t h e s a m e t i m e , b y h isf e el in g t h a t a ll a c c e p t e d p r e s e n t a t i o n s o f m e c h a n i c s h a d s e r io u s s h o r t c o m i n g s .I n p a r t i c u l a r , H E R T Z w a s d e e p l y d i s sa t i sf i e d w i t h t h e c e n t r a l r o l e p l a y e d b y t h ec o n c e p t o f f o rc e , a c o n c e p t w h i c h h e s e t o u t t o e x c l u d e f r o m h i s o w n p r e s e n t a -t i o n . T h i s p r e s e n t a t i o n i s u s u a l l y d e s c r i b e d a s ' a x i o m a t i c ' , a t e r m w h i c h , h o w -e v e r , H E R T Z h i m s e l f n e v e r u s e d i n d e s c r i b i n g h i s o w n w o r k . I n t h e f o l l o w i n gp a r a g r a p h s I w i l l a t t e m p t t o c l a r if y i n w h a t s e n s e t h i s t e r m c a n u s e f u l ly b ea p p l i e d t o H E RT Z'S w o r k , i n o r d e r t o t r a c e H E RT Z'S i n f l u e n c e o n t h e e m e r g e n c eo f H IL BE RT 'S a x i o m a t i c a p p r o a c h . T h i s i n f lu e n c e , as w i ll b e s e en , c a n b e f o u n db o t h i n t h e g e n e r a l c o n c e p t i o n o f t h e r o l e o f a x i o m a t i z a t i o n i n s c ie n c e a n d i nH IL BE RT 'S s p e c if i c a x i o m a t i c t r e a t m e n t o f m e c h a n i c s .

20 On PIERI see KENNEDY 1981a.21 Cf TORRETTI 197 8, 225-226.22 In VERONESE 189 1. See TR ICOM I 1981.23 On crit ic ism directed a t VERONESE'S w ork by G erm an mathem aticians seeTOEPELL 1986, 56.24 Fo r a concise con tem por ary acco unt o f the p lace of HILBERT'S contrib ut ion inconn ect ion with these deve lopm ents see SCHUR 1909, iv vi.


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94 L. CORRYnumber. When these painful contradict ions are removed, the quest ion as to thenature of force will not have been answered; but our minds, no longer vexed, wil lcease to as k illegitima te questions. 27H ERTZ d es c r i b e d th eo r i e s a s i m ag e s B i l d e r ) t h a t w e f o r m f o r o u r s e l v e s o f

n a t u r a l p h e n o m e n a . H e p r o p o s e d t h r e e c r i t e r i a t o e v a l u a t e a m o n g s e v e r a lp o s s i b l e i mag es o f o n e t h i n g : p e rmi s s i b i l it y , co r r ec t n es s , an d ap p ro p r i a t en e s s .A n i m a g e i s p e r m i ss ib l e , a c c o r d i n g t o H E RT Z, i f i t d o e s n o t c o n t r a d i c t t h e l a w so f t h o u g h t . T h i s r e q u i r e m e n t a p p e a r s , e v e n a t t h e m o s t i m m e d i a t e l ev e l, a ss i mi l a r t o H ~LBERT'S r eq u i r em en t o f co n s i s t en cy . Bu t i n f ac t t h i s p a r a l l e l i s ev end e e p e r , in t h e s e n se t h a t , i n s p e a k i n g a b o u t t h e l a w s o f t h o u g h t , H E R TZ i m p l i-c i t l y t o o k l o g i c t o b e g i v e n a p r i o r i , i n K A N T 'S s ens e , an d t h e re fo re t o b eu n p ro b l e m a t i c i n t h i s co n t ex t . T h i s w as a l s o t h e ca s e i n H ~LBERT'S ea r l y ax i o -m a t i c c o n c e p t i o n a l t h o u g h , a s w i l l b e s e e n b e l o w , h i s c o n c e p t i o n l a t e r c h a n g e di n t h e f a c e o f l o g i ca l p a r a d o x e s .

A p e rm i s s i b l e i ma g e i s co r r ec t f o r H E RT Z i f i ts e s s en t i a l r e l a t i o n s d o n o tc o n t r a d i c t t h e r e l a t io n s o f e x t e r n a l t h i n g s. I n f a ct , HE R TZ a c t u a l l y d e f i n e d a ni m a g e b y m e a n s o f t he r e q u ir e m e n t th a t its n e c e ss a ry c o n s e q u e n t s . . , i nt h o u g h t a r e a l w a y s t h e i m a g e s o f t h e n e c e s s a ry c o n s e q u e n t s i n n a t u r e o f th et h i n g s p i c t u r ed . ( p . 1 ) O n e a l s o f i n d s a p a ra l l e l t o t h i s i n H IL BE RT 'S r eq u i r em en tt h a t a ll th e k n o w n f a c ts o f a m a t h e m a t i c a l t h e o r y m a y b e d e r i v e d f r o m it ss y s t e m o f p o s t u l a t e s .

B u t g i v e n t w o p e r m i s si b le a n d c o r r e c t i m a g e s o f o n e a n d t h e s a m e t h i n g , itis b y c o n s i d e r i n g t h e a p p r o p r i a t e n e s s o f e a c h t h a t H E RT Z p r o p o s e d t o a s se s st h e i r r e la t iv e v a lu e . T h e a p p r o p r i a t e n e s s o f a n i m a g e c o m p r i se s t w o e l e m e n t s :d i s ti n c t n e s s a n d s i m p l ic i ty . B y t h e f o r m e r , H E R TZ u n d e r s t o o d t h e a b i l it y t op i c t u r e th e g r e a t e s t p o s s ib l e a m o u n t o f t h e e s s e n ti a l r e l a t i o n s o f t h e o b j e c t .A m o n g v a r i o u s p ic t u r e s o f t h e sa m e o b j ec t , t h e s i m p l e r o n e i s t h a t w h i c ha t t a i n s t h i s d i s t in c t n e s s w h i le i n c l u d i n g t h e s m a l le r n u m b e r o f e m p t y r e l a t io n s .H E RT Z d e em ed s i m p l e r i mag e s m o re ap p ro p r i a t e (p . 2); h e u s ed t h i s l a s t c ri -t e r io n d i r e c t ly to a r g u e t h a t h i s o w n p r e s e n t a t io n o f m e c h a n i c s w a s b e t te r t h a ne x i s ti n g on e s , si nc e , b y r e n o u n c i n g t h e c o n c e p t o f f o rc e , i t p r o v i d e d a s i m p l e ri m a g e . I n g e n e r a l , h o w e v e r , b o t h d i s t i n c t n e s s a n d s i m p l i c i t y a r e f a r f r o m b e i n gs t r a i g h t fo rw ard l y ap p l i cab l e c r i te r i a . H ~L BE RT 'S r eq u i r e m en t o f i n d ep e n d en ce ,a l t h o u g h n o t i d e n t i c a l t o t h i s , c a n b e s e e n a s a m o r e p r e c i s e a n d w o r k a b l ef o r m u l a t i o n o f H E R TZ 'S c r i t e ri o n o f a p p r o p r i a t e n e s s .

T h e p e r m i s s i b i li t y a n d t h e c o r r e c t n e ss o f a n i m a g e c o n n e c t th e l a t t e r t o t w od i f f e r en t s o u rces o f k n o w l ed g e : t h e m i n d an d ex p e r i en ce r es pectiv elY . T h e p e r -m i s s i b il i ty o f a n i m a g e , t h o u g h t H E RT Z, c a n t h e r e f o r e b e u n a m b i g u o u s l y e s t a b -l i s h ed o n ce an d fo r all. I t s co r r ec t n es s i s a f u n c t i o n o f t h e p r e s en t s t a t e o fk n o w l e d g e , a n d i t m a y v a r y a s t h e la t t e r c h a n g e s . A s t o t h e a p p r o p r i a t e n e s s o fa n i m a g e , H ER T Z c o n c e d e d t h a t i t m a y b e a m a t t e r o f o p i n i o n .

H E RT Z a l so m a d e c l e a r w h a t h e u n d e r s t o o d b y p r i n c ip l e s i n h i s w o r k .A l t h o u g h t h e w o r d h a d b e e n u s e d w i t h v a r i o u s m e a n i n g s , h e m e a n t b y i t a n y

27 HER TZ 1956, 7-8. In w ha t follows, all quo tatio ns refer to this E nglish trans lation .


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Hilber t and the Axiomat iza t ion o f Phys ics 95p r o p o s i t i o n s o r s y s te m s o f p r o p o s i t i o n s f r o m w h i c h t h e w h o l e o f m e c h a n i c s c a nb e d e v e l o p e d b y p u r e l y d e d u c t i v e r e a s o n i n g w i t h o u t a n y f u r t h e r a p p e a l toe x p e r i e n c e ( p . 4 ) . D i f f e r e n t c h o i c e s o f p r i n c i p l e s w o u l d y i e l d d i f f e r e n t i m a g e s o fm e c h a n i c s .

H ER TZ 'S o w n p r e s e n t a t i o n o f m e c h a n i c s , a s i t i s w e ll k n o w n , u s e s o n l y t h r e eb as i c co n cep t s : t im e , s p ace , m as s ; HE R T Z was t ry in g t o e l im in a t e fo rce s f ro m h i sa c c o u n t o f m e c h a n i c s . H e t h o u g h t t h a t t h is c o n c e p t , e s p ec i al ly as i t c o n c e r n sf o r c e s t h a t a c t a t a d i s t a n c e , w a s a r t if i c ia l a n d p r o b l e m a t i c . H e t h o u g h t , m o r e -o v e r , t h a t m a n y p h y s i c i s t s , f r o m N e w t o n o n , h a d e x p r e s s e d t h e i r e m b a r r a s s m e n tw h e n i n t r o d u c i n g i t i n t o m e c h a n i c a l r e a s o n i n g , t h o u g h n o o n e h a d d o n e a n y -t h i n g t o o v e r c o m e th i s s i t u a t i o n ( p p . 6 7 ). I n h i s p r e s e n t a t i o n , H E R TZ w a s a b l et o e l im i n a t e f o rc e s b y i n t r o d u c i n g c o n c e a l e d m a s s e s a n d c o n c e a l e d m o t i o n s .B a s e d o n t h e c r i t e r i a d i s c u ss e d i n h i s i n t r o d u c t i o n , H E R T Z c r i ti c i z e d th e t w om a i n e x i s t i n g p r e s e n t a t i o n s o f m e c h a n i c s : t h e tr a d i t i o n a l o n e , b a s e d o n t h ec o n c e p t s o f t im e , s p a c e, m a s s a n d f o r c e , a n d t h e e n e r g e t i c o n e , b a s e d o n t h e u s eo f H a m i l t o n ' s p r in c ip l e. H e t h e n e x p l a i n e d hi s o w n v i e w a n d - - b a s e d a g a i n o nt h e s a m e c r i te r i a - - e s t a b li s h e d t h e s u p e r i o r i t y o f h is p r e s e n t a t i o n o f m e c h a n i c s .

T h i s i s n o t t h e p l a c e t o g i v e a f u ll a c c o u n t o f H E ~ T Z 'S c r i t ic i s m o f t h ee x i s ti n g p r e s e n t a t i o n s o f m e c h a n i c s n o r t o d i sc u s s h i s o w n i n d e t a i l. 2 s I w i llo n l y f o c u s o n s o m e o f H ER TZ 'S r e m a r k s c o n c e r n i n g t h e b a s i c p r i n c i p l e s o f h isa p p r o a c h . T h e s e w i ll h e l p u s i n u n d e r s t a n d i n g H IL BE RT 'S a x i o m a t i c c o n c e p t i o na n d w i ll a l s o a l l o w u s i d e n t i f y th e r o o t s o f t h is c o n c e p t i o n i n H E R TZ 'S w o r k .I n p r i n c ip l e , H E RT Z'S c r i ti c i sm o f th e t r a d i t i o n a l a p p r o a c h t o m e c h a n i c sc o n c e r n e d n e i t h e r i t s c o r r e c t n e s s n o r i t s p e r m i s s i b i l i t y , b u t o n l y i t s a p p r o p r i a t e -n e ss . Y e t h e a l so a l l o w e d r o o m f o r c h a n g e s i n t h e s t a t u s o f c o r r e c t n e s s i n t h ef u t u r e . I n c r i t ic i z i n g t h e r o l e p l a y e d b y f o r c e i n t h e t r a d i t i o n a l i m a g e o f m e c h a n -i cs , H E RT Z s t r e ss e d t h a t t h e p r o b l e m s r a i s e d b y t h e u s e o f th i s c o n c e p t a r e p a r to f o u r r e p r e s e n t a t i o n o f t h is i m a g e , r a t h e r t h a n o f th e e s s e n ce o f th e i m a g ei ts e lf . T h i s r ep re s en t a t i o n h ad s im p ly n o t a t t a i n ed , i n HE R T Z'S v i ew, s c ien t i fi cc o m p l e t e n e s s ; i t f a i le d t o d i s t in g u i s h t h o r o u g h l y a n d s h a r p l y b e t w e e n t h ee l e m e n t s in t h e i m a g e w h i c h a r i s e f r o m t h e n e c e s s i t y o f t h o u g h t , f r o m e x p e r i -e n c e , a n d f r o m a r b i t r a r y c h o i c e ( p. 8 ). A s u i t a b l e a r r a n g e m e n t o f d e f i n it i o n s,n o t a t i o n s , a n d b a s ic c o n c e p t s w o u l d c e r t a i n l y l e a d t o a n e s s e n ti a l i m p r o v e m e n ti n th is s i t u a ti o n . T h i s i m p r o v e m e n t i n p r e s e n t a t io n , h o w e v e r , w o u l d a ls o a l l o wt h e c o r r e c t n e s s o f th e t h e o r y t o b e e v a l u a t e d i n t h e f a c e o f l a t e r c h a n g e s i n t h es t a t e o f k n o w l e d g e . H E R T Z t h u s w r o t e :

O ur assurance, of course, is restricted to the rang e o f prev ious experience: as far asfuture experience is concerned, there w il l be yet occ asion to re turn to the q uest ion ofcorrectness. To m any this w il l seem to be excessive and a bsurd caut ion: to m an yphysicists i t ap pears s imply inconceivable that any further experience wh atevershou ld find any thing to alter in the firm fou nda tion s of mechanics. N evertheless, tha twhich is derived from experience can again be annul led by experience. Th is over-favorable opinion of the fundamental laws must obviously arise from the fact that

28 For one such accoun t see L TZEN 1995.


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96 L. CORRYthe dem ents o f exper ience a re to a ce r ta in ex ten t h idden in them and b lended w i ththe una l te rab le e lem ents which a re necessa ry consequences o f ou r though t . Thu s thelogical indefini teness of the rep resen ta t ion, wh ich we have jus t censured, has o nea d v a n t a g e . I t g iv es t h e fo u n d a t i o n a n a p p e a ra n c e o f im m u t a b il i ty ; a n d p e rh a p s i twas wise to in t rod uce i t in the b eg inn ings o f s cience and to a l low i t to rem ain fora whi le . The correctness o f the im age in a l l cases was careful ly pro vid ed for bym akin g the re se rva t ion tha t , i f need be , fac t s de r ived f rom exper ience should de te r -mine de f in it ions o r v iceversa . In a pe r fec t sc ience such grop ing , such an app earanc eof ce r ta in ty , is inadmissib le . M ature know ledge rega rds log ica l c learness a s o f p r imeimpor tance : on ly log ica l ly c lea r images does i t t e s t a s to cor rec tness ; on ly cor rec timages i t comp ares a s to approp r ia teness . By pres sure o f c i rcums tances the p rocess isof ten reve rsed . Images a re found to be su i t ab le fo r a ce r ta in purpose ; a re nex t t e s tedin the i r cor rec tness ; and on ly in the l a s t p lace purged of impl ied con t rad ic t ions .(HERTZ 19 56, 10)I t s e e m s n a t u r a l t o a s s u m e t h a t b y m a t u r e s c ie n c e H E R T Z w a s r e fe r r in g

h e r e t o E u c l i d e a n g e o m e t r y . B u t a s H IL BE RT n o t i c e d i n 1 89 4 w h e n p r e p a r i n g h i sK / S n i g s b e rg l e c t u r e s 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 ( d i sc u s s e d b e lo w ) , t h es i t u a t i o n i n t h i s d i s c i p l i n e , a l t h o u g h p e r h a p s m u c h b e t t e r t h a n i n m e c h a n i c s ,w a s a l s o b e g g i n g f o r f u r t h e r i m p r o v e m e n t . T h e n i n 1 89 9, H IL BE RT f e lt p r e p a r e dt o a d d r e s s t h o s e f o u n d a t i o n a l p r o b l e m s o f g e o m e t r y t h a t h a d r e m a i n e d e s se n t ia l lyu n a n s w e r e d s in c e K L EIN 'S a t t e m p t s t o d e f in e a m e t r i c f o r p r o j e c t iv e g e o m e t r y . T h em e t h o d o l o g i c a l a p p r o a c h H IL RE RT a d o p t e d f o r t h is t a s k r e s e m b l e d v e r y m u c h , a sw i ll b e s e e n b e l o w , H E R T Z'S s t i p u l a t i o n s f o r m e c h a n i c s a s m a n i f e s t in t h e a b o v eq u o t e d p a s s a g e : t o a t t a i n l o g i c a l c l e a r n e s s , t o t e s t f o r c o r r e c t n e s s , t o c o m p a r e a st o a p p r o p r i a t e n e s s , a n d t o m a k e s u r e t h a t i m p l i e d c o n t r a d i c t i o n s h a d b e e np u r g e d . M o r e o v e r , l ik e H E R TZ b e f o r e h i m , H I LB ER T t h o u g h t t h a t s u c h a p r o c e d u r es h o u l d b e a p p l i e d t o a ll o f n a t u r a l s c ie n c e a n d n o t t o g e o m e t r y a l o ne .

I n H E R TZ 'S p r e s e n t a t i o n o f m e c h a n i c s , e v e r y n e w s t a t e m e n t is d e d u c e d o n l yf r o m a l r e a d y e s t a b l i s h e d o n e s . T h i s i s w h a t h a s b e e n c a l l e d h i s a x i o m a t i ca p p r o a c h . A l t h o u g h t h is i n it s el f i s n o g u a r a n t e e a g a i n s t e r r o r , H E R T Z c o n c e d e d ,i t h a s t h e v i r t u e t h a t i t a ll o w s t h e l o g i c al v a lu e o f e v e r y i m p o r t a n t s t a t e m e n t t ob e u n d e r s t o o d , a n d a n y m i s t a k e t o b e e a si ly i d en t if ie d a n d r e m o v e d . I n th es e c o n d p a r t o f t h e b o o k , H E R T Z i n v e s t ig a t e d t h e l o g i c a l r e la t i o n b e t w e e n v a r i o u sp r i n c i p l e s o f m e c h a n i c s . H e w a s a b l e t o s p e c i fy w h i c h s t a t e m e n t s a r e e q u i v a l e n tt o t h e f u n d a m e n t a l l a w s o f m o t i o n , a n d w h i c h s t a t em e n t s o f t h e f u n d a m e n t a ll a w s a r e n o t i m p l i e d b y a g i v e n p r i n c ip l e . B u t t o w h a t e x t e n t is m e c h a n i c s t h u sp r e s e n t e d c o r r e c t , i n H E RT Z 'S s e n se o f t h e w o r d ? A l t h o u g h n o k n o w n f a c t o fe x p e r i e n c e w a s t h e n c o n s i d e r e d t o c o n t r a d i c t t h e r e s u lt s o f m e c h a n i c s , H ER TZa d m i t t e d t h a t t h e l a tt e r c o u l d n o t b e f ul ly c o n f r o n t e d w i t h a l l p o s s i b le p h e -n o m e n a . T h u s , m e c h a n i cs h a d b e e n b u i lt o n s o m e f a r - re a c h in g a s s u m p t i o n s t h a tc o u l d c o n c e i v a b l y b e q u e s t i o n e d . F o r i n s t a n c e : is t h e r e a f ul l j u s t i f i c a t io n f o ra s s u m i n g t h e c e n t r a l i t y o f l i n e a r d i f f e re n t ia l e q u a t i o n s o f t h e f ir s t o r d e r i nd e s c r i b i n g m e c h a n i c a l p r o c e s s e s ? A n o t h e r c e n t r a l , b u t p e r h a p s n o t f u l ly j u s t if i e da s s u m p t i o n is th a t o f t h e c o n t i n u i t y o f n a t u r e . H E R T Z d e s c r ib e d i t a s a ne x p e r i en c e o f t h e m o s t g e n e r al k i n d . . . a n e x p er i e n ce w h i c h h a s c r y s ta l li z e di n t o f i r m c o n v i c t i o n i n t h e o l d p r o p o s i t i o n - - a t u r a n o n f a c i t s a l tu s


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Hilber t and the Axiomat iza t ion o f Phys ics 97(pp. 36--37). HILBERT i n h is t r e a t m e n t o f p h y s i c a l th e o r i e s , w o u l d n o t o n l ya c c e p t th i s a s s u m p t i o n , b u t a ls o a t t e m p t t o g i ve i t a m o r e m a t h e m a t i c a l l yc o n s i s t e n t f o r m u l a t i o n .

F i n a l l y , i n e x p l a i n i n g t h e s e n s e in w h i c h h i s n e w i m a g e o f m e c h a n i c s w a ss i m p l e r t h a n t h e o t h e r e x i s t i n g tw o , H E R T Z s t r e ss e d t h a t t h is s i m p l i c it y ( a n dt h e r e f o r e a p p r o p r i a te n e s s ) d i d n o t c o n c e r n t h e p r a c t i c a l s i de o f m e c h a n i c s , b u tr a t h e r t h e e p i s t e m o l o g i c a l o n e :

We have on ly spoken o f appropr ia teness i n . . . the sense o f a mind which endeavorsto em brace o bject ively the w hole of our physical know ledge without considering theacc iden ta l pos i tion o f ma n in na tu r e . . . The appropr ia teness o f which we havespoken has n o reference to the pract ical appl icat ion of the needs of mankind. (HERTZ1956, p. 40)H ~ RT Z 'S b o o k w a s w i d e l y p r a i s e d f o l l o w i n g it s p u b l i c a t i o n i n 1 89 4. T h e

i n t e re s t it a r o u s e d c o n c e r n e d h i s c o n s t r u c t i o n o f m e c h a n i c s w h i l e a v o i d i n g t h eu s e o f fo rce s ac t i n g a t a d i s t an ce , a s we l l a s it s p h i l o s o p h i c a l a s p ec t s an d i t sm a t h e m a t i c a l e l a b o r a t i o n . T h e a c t u a l i m p a c t o f H ER TZ 'S a p p r o a c h o n p h y s i c a lr e s e a r c h , h o w e v e r , w a s f a r l es s t h a n t h e i n t e r e s t i t a r o u s e d . 29 O n t h e o t h e rh an d , HER TZ 'S i n f l u en ce o n HILB ERT was , a s I w i ll s h o w b e l o w , m o r e s i g n i f i can tt h a n h as u s u a l l y b ee n p o i n t e d o u t . LUDW t G BOLTZM ANN (1 8 4 4- -19 0 6 ) s h o u l d b em e n t i o n e d h e r e a m o n g t h o s e p h y s i c i s t s w h o w e r e s t r o n g l y i m p r e s s e d b yH E RT Z 'S t r e a t m e n t . I n 1 8 97 h e p u b l i s h e d h i s o w n t e x t b o o k o n m e c h a n i c s ,m o d e l e d i n m a n y r e s p e c t s a f t e r H E RT Z 'S . T h i s b o o k h a d a l e s se r i m p a c t o nHILBERT'S g en e ra l co n cep t i o n s ; y e t i t s t r ea t m en t o f m e ch an i c s , a s we w i ll s eeb e l o w, was a l s o h i g h l y ap p rec i a t ed b y HILB ER T.

T h e p o s i t i v e re a c t i o n s o f t e n a s s o c i a t e d w i t h t h e p u b l i c a t i o n o f H E RT Z'SPrinciples s h o u l d n o t m i s l e a d u s t o b e l ie v e t h a t t h e i d ea o f a x io m a t i z in gp h y s i c a l d i s c i p l i n e s w a s a w i d e l y a c c e p t e d o n e , o r b e c a m e s o a f t e r H E R T Z .A l t h o u g h a n o v e r a l l a c c o u n t o f t h e e v o l u t i o n o f t h is i d e a t h r o u g h o u t t h en i n e t e e n t h c e n t u r y a n d i ts p l a c e in t h e h i s t o r y o f p h y s i c s s e em s y e t t o b eu n w r i t t e n , o n e s h o u l d s t r e s s h e r e t h a t a x i o m a t i z a t i o n w a s s e l d o m c o n s i d e r e da m a i n t a s k o f th e d i sc i p li n e . N e v e r t h e l e s s , i t is w o r t h d i s c u ss i n g h e r e b r i e f ly t h ei d eas o f t w o o t h e r G e r m an p ro fe s s o r s , C AR L NEUM ANN an d PAU L VOLKM ANN,w h o r a i s e d i n t e r e s t i n g is s ue s c o n c e r n i n g t h e r o l e o f a x i o m s i n p h y s i c a l s c i en c e( o n e o f t h e m w r i t i n g b e f o r e H E R TZ 'S Principles t h e s e c o n d o n e a f t e r ) . S i n c e t h e i rideas a re v i s ib ly echoed in H1LBERT'S w o rk , a b r i e f d i s cu s s i o n o f NEUM ANN'S an dVOLKMANN'S wr i t i n g s w i l l h e l p s e t u p t h e b a ck g r o u n d ag a i n s t w h i ch HILBERT'Si d ea s c o n c e r n i n g t h e a x i o m a t i z a t i o n o f p h y s ic s a r o s e.

CARL NEUMANN (1832--1925) wa s the son o f the K 6n ig sb er g p hys ic i s t FRANZN EU MA NN . A t v a r i a n c e w i t h t h e m o r e e x p e r i m e n t a l l y - o r i e n t e d sp i r it o f h isf a t h e r ' s w o r k , C A R L N E UM A N N'S c o n t r i b u t i o n s f o c u s e d o n t h e m a t h e m a t i c a la s p e c ts o f p h y si cs , p a r t i c u l a r l y o n p o t e n t i a l t h e o r y , t h e d o m a i n w h e r e h em a d e h i s m o s t i m p o r t a n t c o n t r i b u t i o n s . H i s c a r e e r a s p r o f e s s o r o f m a t h e m a t i c s

29 S ee LI~TZEN 199 5, 76--83.


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98 L. CORRYe v o l v e d i n H a l l e , B a s e l, T f i b i n g e n a n d L e i p z i g . 3 ~ N EU M AN N'S i n a u g u r a l l e c t u r ei n L e i p z i g i n 1 86 9 d is c u s s e d t h e q u e s t i o n o f t h e p r i n c i p l e s u n d e r l y i n g t h eGALIL EO-NEW T ON th eo ry o f m o v em en t . NE UM ANN ad d re s s ed t h e c l a s s i ca l q u es -t i o n o f a b s o l u t e v s . r e l a ti v e m o t i o n , e x a m i n i n g i t f r o m a n e w p e r s p e c t iv ep r o v i d e d b y a p h i l o s o p h i c a l a n a l y si s o f t h e b a s ic a s s u m p t i o n s b e h i n d t h e l aw o fi n e r t ia . T h e i d e a s i n t r o d u c e d b y N E U M AN N i n t h i s l e c t u r e , a n d t h e e n s u i n gc r i ti c is m o f t h e m , i n a u g u r a t e d a n i m p o r t a n t t r e n d o f c r it ic a l e x a m i n a t i o n o f th eb a s ic c o n c e p t s o f d y n a m i c s - - a t r e n d o f w h i c h E RN ST M A Clt w a s a l so p a r t- - w h i c h h e l p e d to p r e p a r e t h e w a y f o r t h e f u n d a m e n t a l c h a n g e s t h a t a f f e c te dt h e p h y s i c a l s c i e n c e s a t t h e b e g i n n i n g o f th i s c e n t u r y ? 1

N EU MA NN o p e n e d h is i n a u g u r a l l e c t u r e o f 1 8 69 b y f o r m u l a t i n g w h a t h ec o n s i d e r e d to b e t h e u n i v e r s a ll y a c k n o w l e d g e d g o a l o f t h e m a t h e m a t i c a ls c ie n c e s: t h e d i s c o v e r y o f t h e l e a s t p o s s i b l e n u m b e r s o f p r i n c i p l e s ( n o t a b l yp r i n c ip l e s t h a t a r e n o t f u r t h e r e x p li c a b le ) f r o m w h i c h t h e u n i v e r s a l la w s o fe m p i r i c a l l y g i v e n f a c ts e m e r g e w i t h m a t h e m a t i c a l n e c e s s i t y , a n d t h u s t h e d i s -c o v e r y o f p r i n c ip l e s equivalent t o t h o s e em p i r i ca l f ac t s . ' 32 NEUM ANN in t e n d ed t os h o w t h a t t h e p r i n c ip l e o f i n e rt ia , a s u s u a ll y f o r m u l a t e d , c o u l d n o t c o u n t a s o n es u c h b a s ic p r i n c i p l e f o r m e c h a n i c s . R a t h e r i t m u s t b e d i s s o l v e d i n t o a fa i r lyl a r g e n u m b e r o f p a r t l y f u n d a m e n t a l p r in c i pl e s, p a r t l y d e fi n i ti o n s d e p e n d e n t o nt h e m . T h e l a t t e r i n c l u d e t h e d e f i n i ti o n o f r e s t a n d m o t i o n a n d a l so t h e d e fi n i-t i o n o f equal ly long t ime in tervals . N EU M AN N 'S r e c o n s i d e r a t i o n o f t h e s e f u n d a -m e n t a l i d e as o f N e w t o n i a n m e c h a n i c s w a s p r e s e n t e d a s p a r t o f a m o r e g e n e r ald i s c u ss i o n o f t h e a i m s a n d m e t h o d s o f t h e o r e t i c a l p h ys ic s .

E c h o i n g s o m e i d e as o r i g in a l l y f o r m u l a t e d a s e a r l y as t h e m i d d l e a ge s, a n drece n t l y r ev iv ed b y p h y s i c i s t s l i k e KIR C HFIOFF an d M AC H, NE UM ANN c l a im edt h a t p h y s i c a l t h e o r ie s , r a t h e r t h a n e x p l a i n i n g p h e n o m e n a , a m o u n t e d t o a re d u c -t i o n o f in f in i te l y m a n y p h e n o m e n a o f l ik e k i n d t o a f in i te s e t o f u n e x p l a i n e d ,m o r e b a s ic o n e s . T h e b e s t k n o w n e x a m p l e o f t h is w a s t h e r e d u c t i o n o f a llp h e n o m e n a o f c e le s ti a l m o t i o n t o i n e r t i a a n d g r a v i t a t i o n a l a t tr a c t i o n . T h el a t t e r , w h i l e f ul f il li n g t h e i r r e d u c t i o n i s t t a s k p r o p e r l y , r e m a i n e d i n t h e m s e l v e su n e x p l a i n e d , NE UM A NN a r g u e d . B u t N E U M A ~ w e n t o n , a n d c o m p a r e d t hi sr e d u c t i o n t o t h e o n e k n o w n i n g e o m e t r y , w h e r e i n t h e s c i e nc e o f t ri a ng l es , c ir c le sa n d c o n i c s e ct io n s h a s g r o w n i n m a t h e m a t i c a l r i g o r o u t o f a fe w p r in c i pl e s, o fa x i o m s , t h a t a r e n o t f u r t h e r e x p l i c a b l e a n d t h a t a r e n o t a n y f u r t h e r d e m o n -s t r a b l e . N EU M AN N w a s t h u s p l a c i n g m e c h a n i c s a n d g e o m e t r y ( l ik e HrL BE ~T d i dl a te r ) o n t h e s a m e s id e o f a c o m p a r i s o n , t h e s e c o n d s id e o f w h i c h w a s r e p r e s e n t -e d b y l o g i c a n d a r i t h m e t i c ; t h e r e s u l t s a t t a i n e d i n t h e s e l a t t e r d o m a i n s - - a so p p o s e d to t ho s e o f g e o m e t r y a n d m e c h a n i c s - - b e a r t h e s t a m p o f irrevocablecer ta in ty , t h a t p r o v i d e s t h e g u a r a n t e e o f a n unassai lable t ru th . T h e n o n -e x p l a n a t o r y c h a r a c t e r o f m e c h a n i c s a n d g e o m e t r y , NEU MA NN s tr e ss e d , c a n n o t b e

30 See DISALLE 199 3, 34 5; JUNGNICKEL McCORMM ACH 198 6, Vol. 1, 181-185.31 Th is tr en d is discussed in BARBOUR 19 89 , Chp. 12.32 NEUMANN 187 0, 3. I will refer here to the tran slatio n NEUMANN 1993.


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Hilbert and the Axiom atizat ion of Physics 99c o n s i d e r e d a s a f l a w o f t h e s e s c ie n c es . R a t h e r , i t is a c o n s t r a i n t i m p o s e d b yh u m a n c a p a c i t i e s .

T h e p r i n c i p l e s t o w h i c h p h y s i c a l t h e o r i e s a r e r e d u c e d n o t o n l y r e m a i nu n e x p l a i n e d , s a i d N EU M AN N, b u t i n f a c t o n e c a n n o t s p e a k o f t h e i r b e i n g c o r r e c to r i n c o r r e c t, o r e v e n o f t h e i r b e i n g p r o b a b l e o r i m p r o b a b l e . T h e p r i n c ip l e s o fa n y p h y s i c a l t h e o r y - - e .g ., F RE SN EL a n d Y O UN G 'S t h e o r y o f li g h t - - c a n o n l y b es a i d t o h a v e t e m p o r a r i l y b e e n c o n f i r m e d ; t h e y a r e i n c o m p r e h e n s i b l e u n b e g r e i f -l i ch ) a n d a r b i t r a r y w i l l k i i r l i c h ) . NEUM ANN q u o t ed LEIB NIZ , i n o rd e r t o ex p l a i n h i sp o i n t: n a t u r e s h o u l d i n d e e d b e e x p l a i n e d f r o m e s t a b l is h e d m a t h e m a t i c a l a n dp h y s i c a l p r i n c ip l e s , b u t t h e p r i n c i p l e s t h e m s e l v e s c a n n o t b e d e d u c e d f r o m t h el a w s o f m a t h e m a t i c a l n e c e s si ty . '3 3 T h u s , i n u s i n g t h e t e r m s a r b i t r a r y a n di n c o m p r e h e n s i b l e , N EU M AN N w a s r e f e r r i n g t o t h e l i m i t a t i o n s o f o u r p o w e r o fr e a s o n i n g . A l w a y s r e l y i n g o n b a s i c a l l y K a n t i a n c o n c e p t i o n s , h e c o n t r a s t e d t h es t a t u s o f t h e c h o i c e o f th e p r i n c i p l e s in t h e p h y s i c a l s c i e n c es t o t h e k i n d o fn e c e s s i t y t h a t g u i d e s t h e c h o i c e o f m a t h e m a t i c a l o n e s . T h i s i s w h a t t h e i ra r b i t r a r i n e s s m e a n s . N EU M AN N w a s c l e a r l y n o t i m p l y i n g t h a t p h y s i c a l t h e o r i e sa r e s i m p l y f o r m a l d e d u c t i o n s o f a n y a r b i t r a r i l y g i ve n , c o n s is t e n t s y s t em o fa x i o m s d e v o i d o f d i r e c tl y i n tu i ti v e c o n t e n t . R a t h e r t h e y h a v e v e r y c o n c r e t ee m p i r i c a l o ri g in s a n d i n t e r p r e t a ti o n s , b u t , g iv e n th e l i m i t a t io n s o f h u m a n m e n -t a l c ap ac i t i e s , t h e i r s t a t u s i s n o t a s d e f i n i t i v e a s t h a t o f t h e p r i n c i p l e s o f l o g i ca n d a r i t h m e t i c .

N EU MA NN c o n c l u d e d t h e p h i l o s o p h i c a l s e c t i o n o f h i s l e c t u r e b y r e f o r m u l a t -i n g t h e t a s k o f t h e p h y s i c i s t i n t h e t e r m s d i s c u s se d b e f o r e : t o r e d u c e p h y s i c a lp h e n o m e n a

. . . to the fewest poss ible arbi t rar i ly chosen principles -- in o ther words , to reducethem to the fewest possible things remaining incomprehens ib le . The g rea te r thenum ber of phen om ena encom passed by a physical theory , and the smal ler thenum ber o f inexplicable i tems to which the phe nom ena a re reduced, the m ore perfectis the theory to be judged.F r o m h e r e h e w e n t o n t o a n a l y z e t h e c o n c e p t u a l d i ff ic u lt ie s i n v o l v e d in t h ep r i n c i p l e o f i n e r ti a , u s u a l l y f o r m u l a t e d a s f o ll o w s :

A m aterial point th at was set in motio n wil l m ove on - - i f no foreign cause affects i t,if i t is entirely left to i tself - - in a s t ra ight l ine and i t wi ll t raverse in equal t ime equaldistances. 34T h e f ir st p r o b l e m p o i n t e d o u t b y N EU MA NN c o n c e r n i n g t hi s f o r m u l a t i o n h a s

t o d o w i t h t h e c o n c e p t o f s t r a i g h t l in e . R e c o g n i z i n g a s t r a i g h t l in e in p h y s i c a ls p a c e r a i s e s t h e d i f f ic u l ti e s t r a d i t i o n a l l y a s s o c i a t e d w i t h t h e q u e s t i o n o f re l a t i v ev s. a b s o l u t e s p a c e. I n a d d r e s s i n g t h is q u e s t i o n , N E U NA N N i n t r o d u c e d t h e i d e a o ft h e B o d y A l p h a : a r i g i d o b j e c t l o c a t e d s o m e w h e r e i n t h e u n i v e r s e , t o w h i c h a l lm o t i o n s r e fe r . T h u s , t h e p r i n c i p l e o f in e r t i a i s a n a l y z e d , i n t h e f ir s t p l a c e , i n t o

33 NEUMAN N 1993, 361 . T he reference is to LEIBNIZMathemat i sche Schr i f ten , part 2 ,Vol. 2 (Halle 18 60, p. 135.)34 NEUMANN 1870, 14 (1993 , 361).


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100 L. CORRYtW O s i m p l e r c o m p o n e n t s : t h e f i r st a s s e r ts t h e e x i s t e n c e o f t h e B o d y A l p h a , t h es e c o n d a s s e r t s t h a t e v e r y m a t e r i a l p o i n t l e ft to i ts e l f w i ll m o v e i n a s t r a i g h t l i ne ,i.e ., i n a p a t h r e c t i l i n e a r i n r e l a t i o n t o t h is B o d y A l p h a . T h i s w a y o f a n a l y z i n gt h e p r i n c i p l e o f i n e r ti a e m b o d i e d N EU MA NN 'S p r e s c r i p t i o n o f i n c o m p r e h e n s i b l ea n d a r b i t r a r y p r i n c ip l e s w h i c h h e l p e d t o m a k e s e n s e o f a p h y si c a l t h e o r y . T h i si d e a a t t r a c t e d m u c h a t t e n t i o n a n d c r it ic i sm , a n d N EU MA NN h i m s e l f r e f o r m u l a t e di t s ev e ra l t i m es . Th i s i s n o t , h o wev e r , t h e p l ace t o d i s cu s s t h e i d ea an d i t s c r i t i c sin detai l .35

M o r e d i r e c t l y p e r t i n e n t t o o u r a c c o u n t , s i n c e i t w il l r e a p p e a r i n H IL BE RT 'Sl e c t u r e s o n p h y s ic s , w a s N EU M AN N 'S t r e a t m e n t o f t h e s e c o n d p a r t o f t h e p r i n -c i p le o f i n e r ti a : t h e c o n c e p t o f e q u a l v e l o c it i e s . A n a p p r o p r i a t e e l u c i d a t i o n o ft h is c o n c e p t is c l e a r l y r e l a t e d t o t h e p r o b l e m o f r e l a t i v e v s. a b s o l u t e t im e .N EU MA NN d i s c u ss e d i n h i s l e c t u r e t h e p r o b l e m o f t h e m e a s u r e m e n t o f ti m e a n do f t h e d e t e r m i n a t i o n o f t w o e q u a l t i m e - in t e rv a l s. H e p r o p o s e d t o r e d u c e t i m e t om o v e m e n t i n o r d e r t o e x p l a i n t h e f o r m e r . I n h i s v i e w , t h e c o r r e c t f o r m u l a t i o n o ft h e t h i r d c o m p o n e n t o f t h e p r in c i p l e o f i n e r t i a s h o u l d r e a d a s f o ll ow s : T w om a t e r i a l p o i n t s , e a c h l e ft t o i ts el f, m o v e in s u c h a w a y t h a t t h e e q u a l p a t h s o fo n e o f t h e m a l w a y s c o r r e s p o n d t o t h e e q u a l p a t h s o f t he o t h e r . F r o m h e r e o n ea l s o g e t s t h e d e f i n i t i o n o f e q u a l t i m e i n t e r v a l s a s t h o s e i n w h i c h a p o i n t l e f t t oi t se l f c o v e r s e q u a l p a t h s .

T h i s p a r t o f N EU M AN N 'S a n a l y s i s a l s o a t t r a c t e d a t t e n t i o n a n d g a v e r i s e t oc r i ti c i sm s a n d i m p r o v e m e n t s . O f s p e c i a l i n t e r e s t i s t h e c o n c e p t o f i n e r t i a ls y s t e m , i n t r o d u c e d i n t h is c o n t e x t b y L U D W IG L A N G E i n 1 8 86 , w h i c h b e c a m es t a n d a r d a n d h a s r e m a i n e d s o e v e r s i nc e . 36

I n h i s c l o s i n g r e m a r k s N EU M AN N e x p r e s s e d t h e h o p e t h a t h i s a n a ly s i s m a yh a v e s h o w n t h a t m a t h e m a t i c a l p h y s i c a l t h e o r ie s i n g e n e r a l m u s t b e s e e n a ss u b j e c t i v e c o n s t r u c t i o n s , o r i g i n a t i n g w i t h u s , w h i c h ( s t a r t i n g f r o m a r b i t r a r i l yc h o s e n p r i n c ip l e s a n d d e v e l o p e d i n a st r ic t ly m a t h e m a t i c a l m a n n e r ) a r ei n t e n d e d t o s u p p l y u s w i t h t h e m o s t f a i t h f u l p i c t u r e s p o s s i b l e o f t h e p h e -n o m e n a . '3 ~ F o l l o w i n g H E LM H O LT Z, N E UM A N N c l a i m e d t h a t a n y s u c h t h e o r yc o u l d o n l y c l a im o b j e c ti v e r e a l it y o r a t l e a st g e n e r a l n e c e s s it y - - i f o n e c o u l ds h o w t h a t it s p r i n c i p l e s a r e th e on l y pos s i b l e one s t h a t n o o t h e r t h e o r y t h a nt h is o n e is c o n c e i v a b l e w h i c h c o n f o r m s t o th e p h e n o m e n a . H o w e v e r , h ed e e m e d s u c h a p o s s i b i l i t y a s l y i n g b e y o n d h u m a n c a p a b i l i t i e s . N e v e r t h e l e s s

an d t h i s i s a p o i n t t h a t HILB ER T wi l l a l s o s t r e s s t i m e an d ag a i n i n h i s o w na t t e m p t s t o a x i o m a t i z e ph y s ic a l d o m a i n s - - t h e c o n s ta n t r e - e x a m i n a t io n o fp r i n c i p l e s a n d o f t h e i r s p e c i fi c c o n s e q u e n c e s f o r t h e t h e o r y i s v i t a l t o t h e f u r t h e rp r o g r e s s o f s c i e n ce . N E UM A NN t h u s c o n c l u d e d :

High and m ighty as a theory may appear, we shall a lways be forced to ren dera precise ac co un t of i ts principles. W e mu st always b ea r in mind tha t these principlesare someth ing arbitrary and therefo re someth ing mutable. This is necessary in order

35 See BARBOUR 19 89 , 646-653; DISA LLE 199 3, 348--349.36 LANGE'S ideas are discussed in BARBOUR 19 89 , 655--662.37 NEUM ANN 187 0, 22 (199 3, 367).


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Hilbe r t and the Ax ioma t iza t ion of Phys ics 101to survey wherever poss ib le wha t e f f ec t a change of these pr inc ip les would have onthe entire edif ice GestaltUng) of a theory , and to be ab le to in t roduce such a changea t the r igh t t ime , and ( in a word) tha t we may be in a pos i t ion to p r ese rve the theoryf r o m a petrification, f r o m a n ossification t h a t c a n only be de le te r ious and a hindranceto the adv an cem en t o f science. 3sH IL B ER T n e v e r d i r e c t l y c i t e d N Z U M A N N'S i n a u g u r a l l e c t u re , o r a n y o t h e r o f

h i s p u b l i c a t i o n s , b u t i t se e m s fa i r t o a s s u m e t h a t H I L B ER T k n e w a b o u tNE UM ANN'S i d e a s f r o m v e r y e a r l y o n . T o g e t h e r w i t h R UDOL V AL VRE O C LE BS CH(1833- -1872) , NEUMA NN fo un de d the M a t h e m a t i s c h e A n n a l e n i n 1 8 6 8 a n d c o -e d i t e d i t u n t i l 1 87 6, 39 a n d w a s s u r e l y a w e l l - k n o w n m a t h e m a t i c i a n . M o r e o v e r ,i n 1 88 5, w h e n H m B E R T s p e n t a s e m e s t e r i n L e i p z i g ,

Hilbert and Physics

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