Adams, Classical Physical Abstraction [23 pgs]

7/28/2019 Adams, Classical Physical Abstraction [23 pgs]

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E R N E S T W . A D A M S

C L A S S I C A L P H Y S I C A L A B S T R A C T I O N

A B S T R A C T . A n i n f o r m a l t h e o r y i s s e t f o r t h o f r e l a t i o n s b e t w e e n a b s t r a c t e n t i t i e s ,i n c l u d i n g colors, physical quantities, times, a n d places in space, a n d t h e c o n c r e t e t h i n g st h a t have t h e m , o r a r e at o r in t h e m , b a s e d o n t h e a s s u m p t i o n t h a t t h e r e a r e c l o s ea n a l o g i e s b e t w e e n t h e s e r e l a t i o n s a n d r e l a t i o n s b e t w e e n a b s t r a c t sets a n d t h e c o n c r e t et h i n g s t h a t a r e members o f t h e m . I t is s u g g e s t e d t h a t e v e n s t a n d a r d s c ie n t if i c u s a g e o ft h e s e a b s t r a c t i o n s p r e s u p p o s e s p r i n c ip l e s th a t a r e a n a l o g o u s t o p o s t u l a te s o f a b s t r a c ti o n ,i d e n t i t y , a n d o t h e r f u n d a m e n t a l p r i n c i p l e s o f s e t t h e o r y . A l s o d i s c u s s e d i s t h e s i g n i f ic a n c eo f i m p o r t a n t d i s a n a l o g i e s b e t w e e n s e t s a n d p h y s i c a l a b s t r a c t i o n s , i n c l u d i n g e s p e c i a l l ymodal a n d temporal a s p e c t s o f p h y s i c a l a b s t r a c t i o n s , w h i c h i s r e l a t e d t o t h e p r o b l e m o ft h e c h a r a c t e r i z i n g constancy, o f c o l o r s , p h y s i c a l a t t r i b u t e s , a n d l o c a t i o n s i n s p a c e .

1. INTRODUCTIONThe theme of this paper is that in spite of significant differences thereare important logical resemblances between abstract s e t s in their re-lations to the concrete things that are in them, and abstract things suchas c o l o r s , p h y s i c a l q u a n t i t i e s (e.g., lengths and weights), t i m e s and evenp l a c e s , in their relations to the concrete things that h a v e them or area t them. A fundamental resemblance has to do with the fact thatordinary usage, including ordinary scientific usage , presupposes rough'postulates' of existence and identity for colors, physical quantities,times and places, that are analogous to the abstraction and identitypostulates of set theory. These postulates not only describe the way inwhich abstract entities of given kinds are related to concrete things,but they are 'constitutive' of the abstract categories in the way that theset-theoretical abstraction and identity principles are constitutive ofabstract sets. 1 Specifically, existence and identity and related criteriafor abstract things are characterizable in terms of their relations to theconcrete things that 'instant iate' them. 2

Given the foregoing, analyzing an abstract category becomes a jobof formulating explicitly the existence, identity, and related postulatesthat are implicit in ordinary usage relating to the category, and studyingtheir implications. Doing this would follow the example of Cantor andhis successors in formulating principles that are implicit in 'we-analytic 'Erkennmis 38: 145-167 , 1993 . 1993 Kluwer Academic Publishers. Printed in the Netherlands.


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146 E R N E S T W . A D A M Su s e s o f se t c o n c e p t s , a n d i n v e s t ig a t in g t h e i r c o n s e q u e n c e s . T h i s p a p e rw i ll s k e t c h a s p e c t s o f t h is k i n d o f a n a l y s is , t h o u g h w i t h f a r m o r e m o d e s to b j e c ti v e s t h a n C a n t o r ' s . A l o n g t h e w a y , t h o u g h , w e w il l p o i n t o u ti m p o r t a n t d i s s im i l a r it ie s b e t w e e n s e t s a s t h e y a r e c o n c e i v e d in m a t h e -m a t i c a l s e t t h e o r y a n d p h y s i c a l a b s t r a c t i o n s l i k e c o l o r s , p h y s ic a l q u a n t i -t ie s , t im e s , a n d p l a c es . L e t u s b e g i n w i t h so m e d o w n - t o - e a r t h e x a m p l e s .

2 . E X A M P L E ST h i n k o f a p e r s o n m a k i n g a n o m e l e t w h o h a s b r o k e n f o u r eg g s f r o m ac a r t o n i n t o a p a n , a n d i s a b o u t t o b r e a k t h e f i ft h . H e r e a r e f iv e t h i n g st h a t t h e p e r s o n m i g h t s ay a b o u t t h e e g g:

S se, = T h e e g g s t h a t w e r e i n t h e c a r t o n i n c l u d e d t h i s o n e .S et ,o r = T h e c o l o r o f t h is e g g is n o t p e r f e c t l y w h i t e .S g r a r n s = T h e w e i g h t o f t h is e g g is n e a r l y 2 5 g r a m s .S tim e = T h e t i m e o f t h is e g g ' s b r e a k i n g w i ll b e b e f o r e n o o nt o d a y .S place = T h e p l a c e w h e r e t h i s e g g w a s i n t h e c a r t o n i s n o we m p t y .

S s e t - S p l a c e c o u l d h a v e b e e n e x p r e s s e d l e ss a b s t r a c t l y o f c o u r s e ; e . g . , S~etc o u l d h a v e b e e n m o r e s i m p l y e x p r e s s e d a s :

S ' se t --- Th i s e gg wa s in t he c a r to n .B u t t h e l e s s a n d t h e m o r e a b s t r a c t f o r m s w o u l d b e r e c o g n i z e d a se q u i v a l e n t b y t h e p e r s o n - i n - t h e - s t r e e t . W h a t is s i g n if i ca n t a b o u t a b s t r a c tf o r m s l i k e S ~e t i s t h a t i n s a y i n g t h a t a n a b s t r a c t t h i n g - a s e t , c o l o r ,w e i g h t , t i m e o r p la c e - h a s a p r o p e r t y o r s t a n d s i n a r e l a t i o n t os o m e t h i n g , t h e a b s t r a c t t h i n g i s r e f e r r e d t o b y a d e f i n i t e d e s c r i p t i o nt h a t ' d e f i n e s ' i t i n t e r m s o f it s r e l a t i o n t o c o n c r e t e t h i n g s . T h u s , S so td e s c r i b e s the eggs that were in the carton i n t e r m s o f it s r e l a t i o n t o t h ee g g s t h a t w e r e i n t h e c a r t o n , S eolou~ d e s c r i b e s the color of the egg b y i tsr e l a t i o n t o t h e e g g , a n d s o o n . T h e r e a r e t w o t h in g s t o n o t e a b o u t t hi s.

F i r s t, a s s u m i n g t h e R u s s e l l i a n a n a l y si s , t h e u s e o f t h e d e f i n i t e ar t ic l ei n r e f e r r i n g e x p r e s s i o n s l i k e ' t h e e g g s t h a t w e r e i n t h e c a r t o n ' , ' t h ec o l o r o f t h e e g g ' , a n d s o o n i m p l i e s t h e e x i s t e n c e o f u n i q u e ' e n t i t ie s 's a t is f y in g t h e c o n d i t i o n s o f b e i n g eggs that were in the carton, color ofthis egg, e t c. T h e r e f o r e w h a t e v e r t h e s e e n t it ie s a r e , t h e r e m u s t b ee x i s t e n c e a n d i d e n t i t y c r i t e r i a t h a t a r e a p p l i c a b l e t o t h e m , w h i c h l o g i c a l


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C L A S S I C A L P H Y S I C A L A B S T R A C T I O N 147a n a l y s is s h o u l d a i m t o f o r m u l a t e e x p l ic i tl y . 3 T h e s e w i ll b e c o m m e n t e do n i n t h e n e x t t w o s e c t i o n s .

S e c o n d , t h e f a c t t h a t p r o p e r t i e s a r e a s c r i b e d t o t h i n g s l ik e t h e e g g st h a t w e r e i n t h e c a r t o n , t h e c o l o r o f t h e e g g a n d s o o n , i s a f i r s ts u g g e s ti o n o f t h e o r y . T h e r e a r e m a n y s e t s , c o l o r s , w e i g h t s , t i m e s a n dp l a c e s , a n d t h e o r i e s o f s e ts , c o l o rs , a n d s o o n g e n e r a l i z e a b o u t t h e s ea b s t r a c t t h in g s a n d t h e i r p r o p e r t i e s a n d t h e i r r e l a t io n s t o o t h e r a b s t ra c -t i o n s o f li k e k i n d s . H o w e v e r , i t is n o t e w o r t h y t h a t t h e s e t h e o r i e s d on o t u s u a l l y e x p l ic i tl y f o r m u l a t e l a w s t h a t d e s c r i b e r e l a t io n s b e t w e e nc o n c r e t e t h i n gs a n d t h e a b s t r a c ti o n s t h e y g e n e ra l iz e a b o u t . T h e t h e o r yo f se ts , b a s e d o n t h e r e l a t i o n o f m e m b e r s h i p , s e e m s t o b e a r a r ee x c e p t i o n t o t h i s r u l e . 4 O t h e r t h e o r i e s s u c h a s o f c o l o r , w e i g h t a n do t h e r q u a n t i t i e s o r m a g n i t u d e s g i v e o n l y t h e r o u g h e s t h i n t s a t t h ep r i n c ip l e s t h a t s e c u r e t h e e x i s t e n c e o f t h e a b s t r a c t i o n s i n q u e s t i o n . 5

B u t n o w w e w i ll h a v e a f ir st g o a t f o r m u l a t i n g p r in c i p le s o f a b s t r a c t i o nt h a t a r e , a r g u a b l y , i m p l ic i t i n a b s t ra c t m o d e s o f e x p r e s s i o n l ik e t h o s ei n t h e e x a m p l e s .3. A B S T R A C T I O N P R I N C I P L E S ; F O U R D I S S I M I L A R I T I E S T O T H E

S E T - T H E O R E T I C A L

C o n s i d e r w h a t w o u l d b e p r e s u p p o s e d i n a R u s s e l l i a n a n a l y s i s o f ' t h ep l a c e w h e r e t h e e g g w a s i n t h e c a r t o n ' . T h i s c le a r l y p r e s u p p o s e s t h a tt h e r e is a p l a c e w h e r e t h e e g g w a s i n t h e c a r t o n , a n d a f i r s t s h o t a t a na b s t r a c t i o n p r i n c i p l e t h a t w o u l d g u a r a n t e e t h i s i s t h e f o l l o w i n g :

Place Abstract ion Principle . F o r a n y c o n c r e t e th i n g a n d a n yt i m e , t h e r e i s a p l a c e w h e r e t h e t h i n g w a s , i s , o r w i l l b e a tt h a t t i m e .

W h a t i s a f f i r m e d is t h e e x i s t e n c e o f a p la c e s t a n d i n g i n th e r e l a t i o n o fb e i n g where a c o n c r e t e t h i n g is , w a s o r w i ll b e , a n d i t o b v i o u s l y g u a r a n -t e e s t h e e x i s t e n c e o f a p l a c e w h e r e t h e e g g w a s i n t h e c a r t o n . T h i s i sa n a l o g o u s t o s e t t h e o r y ' s p o s t u l a t e t h a t t h e r e a r e s e t s w i t h c e r t a i n th i n g sin t h e m , a n d a r g u a b l y t o a s s u m p t i o n s t h a t t h e r e a r e c o l o r s , w e i g h t s ,e t c . , t h a t t h i n g s have , w h i c h a r e i m p l i c it in t h e u s e o f e x p r e s s io n s l i k eS o lo r, S g . . . . a n d s o o n . I t i s s i g n i f i c a n t t h a t t h e A b s t r a c t i o n P r i n c i p l en e i t h e r f o r m u l a t e s c r i t e r ia o f i d e n t i t y f o r p l a c e s ( it d o e s n o t a f f ir m t h a tt h e r e i s a unique p l a c e w h e r e t h e e g g w a s ) n o r d o e s i t s p e c if y w h a t ap l a c e is . I d e n t i t y a n d u n i q u e n e s s w i ll b e r e t u r n e d t o b e l o w , b u t t h e r e


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148 E R N E S T W . A D A M Sa r e t w o o b s e r v a t i o n s t o m a k e f i r s t o n t h e na ture o f t h e p l a c e s w h o s ee x i s t e n c e i s p o s t u l a t e d i n t h e A b s t r a c t i o n P r i n c i p l e , a n d o n t h e s t a t u so f t h e p r i n c i p l e i t s e l f .

T a k i n g s e t t h e o r y a s a m o d e l , w e c a n n o t e x p e c t a b s t r a c t i o n p r in c i p le st o t e l l u s a n y m o r e a b o u t t h e e n t i t i e s t h e y a s s u r e t h e e x i s t e n c e o f t h a nt h e b a r e f a c t t h a t t h e y do e x i s t , a n d t h a t c o n c r e t e t h i n g s s t a n d i nc o n s t i t u t i v e r e l a t i o n s t o t h e m . I p s o f a c t o , t h e y c a n n o t t e l l u s w h a t t h ea b s t r a c t t h i n g s a r e l ike . T h e y c a n n o t t e l l u s t h a t a p l a c e i s a s u r f a c e ,o r t h a t i t h a s a p a r t i c u l a r w e i g h t o r c o l o r. I f w e g o f a r t h e r a n d m a i n t a i nt h a t a b s t r a c t t h i n g s h a v e c e r t a i n p r o p e r t i e s o r t h a t t h e y s t a n d i n c e r t a i nr e l a t i o n s t o o n e a n o t h e r , e v e n o f i d e n t i t y , th i s m u s t b e j u s ti f ie d b yp r i n c ip l e s t h a t s ti ll r e m a i n t o b e f o r m u l a t e d . 6

A s t o t h e s t a tu s o f t h e A b s t r a c t i o n P r i n c i p le i t s el f , I w o u l d h a z a r dt h e f o l l o w i n g : n a m e l y t h a t n o o r d i n a r y j u s t if i c a t io n c a n b e o f f e r e d f o ri t , e i t h e r a p r i o r i o r a p o s t e r i o r i . A g a i n t a k i n g s e t t h e o r y a s a g u i d e ,t h e p r i n c i p l e m u s t b e pos tu la t ed a n d i t c a n n o t b e d e r i v e d f r o m m o r e' f u n d a m e n t a l ' p r i n c ip l e s . 7 N o r c a n w e ' te l l b y l o o k i n g ' t h a t t h e r e is a na b s t r a c t p l a c e w h e r e a c o n c r e t e o b j e c t i s, s in c e in l o o k i n g a t t h e p l a c ea l l w e s e e a r e o b j e c t s , s

W e ~w i l l e n d t h i s s e c t i o n b y n o t i n g f o u r i m p o r t a n t d i s s i m i l a r i t i e sb e t w e e n t h e s e t - t h e o r e t i c a l c a s e a n d t h o s e o f t h e p h y s i c a l a b s t r a c t i o n sinv o lv ed i n S color-Sp~ . . . T he f i r st r e l a t e s t o t he k in ds o f co nc re t e t h ing st h a t c a n i n s t a n t i a t e a b s t r a c t io n s o f t h e d i f f e r e n t k in d s . M a t e r i a l b o d i e si n s t a n ti a t e w e i g h t s a n d p l a ce s , a n d c o l o r e d ( n o n - t r a n s p a r e n t ) o n e s h a v ec o l o r s , b u t o n l y even t s i n s t a n t i a t e t i m e s . 9 T h e s e a r e a ll to s o m e d e g r e ep h y s i c a l , a n d t h e r e f o r e i t i s a p p r o p r i a t e t o c a ll c o l o r s a n d s o o n phys i ca la b s t ra c t io n s - t h o u g h t h e r e a r e d i f f e re n c e s a m o n g t h e m t h a t c a n m a k ei t d i ff ic u l t t o s p e c i fy e x a c t l y w h a t c a n i n s t a n t i a t e p h y s i c a l a b s t r a c t io n so f g i v e n k in d s . 1 O n t h e o t h e r h a n d a n y t h i n g c a n b e l o n g t o a s e t ( a tl e a s t i n t y p e l e s s s e t t h e o r i e s ) , a n d t h a t i s w h a t m a k e s s e t s s u i t a b l e f o ra p p l i c a t i o n t o a b s t r a c t m a t h e m a t i c a l o b j e c t s .

A n o t h e r d i s si m i l ar i ty a p p l i es , p e r h a p s , o n l y t o p laces a n d t h e spacest h e y a r e i n . T h e P l a c e A b s t r a c t i o n P r i n c i p le i m p l ic i tl y a s s u m e s t h a te v e r y t h i n g h a s a p la c e i n a s p a c e , b u t t h e r e a r e a c t u a l ly ma n y s p a c e s ,a n d w e s h o u l d f o r m u l a t e ' r e l a ti v e ' p la c e e x i s t e n c e p o s t u l a t e s t h a t a f f i rmt h e e x i s t e n c e o f p l a c e s w h e r e t h i n g s a r e i n t h e m . T h u s t h e e g g c a r t o nf u r n i s h e s a s p ac e o r ' f r a m e o f r e f e r e n c e ' o f li m i t e d e x t e n t w h i c h h a sp l a c e s in w h i c h s m a l l s iz e d t h in g s c a n b e a t t im e s , b u t t h e s e p l a c e s a n ds p a c e s a r e s u r e l y n o t t h e o n l y o n e s w h o s e e x i s t e n c e i s t a c i t l y a s s u m e d


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C L A S S I C A L P H Y S I C A L A B S T R A C T I O N 149

in ordinary usage. However, we cannot in this brief sketch enter intothe complications that multiple frames of reference introduce, or intothe problem of characterizing connections between places in differentframes. 11The third dissimilarity has to do with the time-dependence of inci-dence relations involving physical abstractions. Obviously colors,weights and places, though not times, are things that concrete thingscan have or be at at one time and not at another, but set theory hasno temporal dimension. 12 This means that the analysis of physicalabstractions must face a difficulty that does not arise in the case of sets:namely that of specifying what it is for concrete things to instantiatethe same abstract things at different times. This generalizes the problemof absolute space to colors and magnitudes, and it is no easier to solvein the latter cases than it is in the former. We will return to this brieflyin section 7.

The final dissimilarity is related to temporality. The egg was in aplace in the carton but it is no longer there, and it is conceivable thatthere might have been places in the carton where nothing ever was orwill be. The existence of such places cannot be deduced from the placeabstraction principle formulated above, because that only guaranteesthe existence of places where concrete things have been or will be atsome time. I will go out on a limb and postulate a more generalprinciple that would assure the existence of these other places.

Modal Place Abstraction Principle. For any possible thingexisting at any time, there is a place where it would be atthat time.

Again, I would argue that this principle is presupposed in ordinary use,this time of counterfactuals involving referring expressions like 'Theplace where the egg would hit the ground if it were dropped'. The eggdoesn't have to be dropped, ordinary counterfactual use presupposes,for there to be a place where it would hit the ground if it were dropped,and the existence of that place does not require that anything shouldever actually be at it. Places are in 'container spaces', though the sizeof the container and what could be in it varies with the space. It isplausible that something similar applies to the other physical abstrac-tions that we have been considering which, unlike sets, have a non-extensional aspect. Thus, we speak freely of the colors and weights thatthings would have under imaginable circumstances, and of the times of


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1 5 0 E R N E ST W . A D A M S

i m a g i n a b l e o c c u r r e n c e s , a n d t h e o r i e s o f t h e s e th i n gs a r e c o n c e r n e d w i t ht h e c o l o r s , p l a c e s , t im e s , e t c . o f t h i n g s a n d e v e n t s t h a t c o u l d p o s s i b l ye x is t o r h a p p e n . T h u s , t h e y a r e th e o r i e s o f phy s i c a l pos s i b i l i t i e s . ~ 3

W e w i ll r e t u r n t o t h i s, b u t w e w il l f ir st c o n s i d e r i de n t i t y a n d u n i q u e -ne s s o f p h y s i c a l a b s t r a c t io n s , w h i c h w i l l b r i n g i n a n o t h e r i m p o r t a n td i s a n a l o g y b e t w e e n t h e m a n d s e t s .

4 . I D E N T I T Y , U N I Q U E N E S S A N D P H Y S IC A L C O I N C I D E N C EA n o b v i o u s p r in c i p l e o f i d e n t i t y f o r a b s t r a c t p l a c e s is a m o d a l a n a l o g u eo f t h e s e t - i d e n t i t y c r i t e ri o n :

M o d a l P l a c e I d e n t i ty P r i n c ip l e . T w o p l a c e s a r e t h e s a m e i fa n d o n l y i f n o c o n c r e t e t h in g c o u l d b e a t o n e w i t h o u t b e i n ga t t h e o t h e r .

T h e r e a r e m a n y p l a c e s i n th e e g g c a r to n , a n d t h e m o d a l f o r m u l a t io ne v e n a l l o w s f o r a m u l t i p l i c i t y o f e m p t y p l a c e s, i n d i s a n a l o g y t o e x t e n -s i o n a l s e t t h e o r y ' s s in g l e e m p t y s e t. A p l a c e i s e m p t y i f t h e r e is n o t h i n gi n it , a n d t h e r e i s m o r e t h a n o n e e m p t y p l a c e i f i t i s p o s s i b l e t o p u ts o m e t h i n g i n to o n e o f t h e m w h i c h is n o t i n o t h e r e m p t y p l a c es .A s w i t h a b s t r a c t e x i s t e n c e , i t is p l a u s i b l e t h a t t h e c a t e g o r i e s o f c o l o r ,p h y s i c a l q u a n t i ti e s , a n d o f t i m e s a l so p r e s u p p o s e i d e n t i t y p r in c i p l e sa n a l o g o u s t o t h o s e p r e s u p p o s e d b y p la c e s , a n d t h a t t h e s e a l lo w f o r t h ee x i s t e n c e o f a m u l ti p l ic i t y o f u n i n s t a n t i a t e d c o l o r s , l e n g t h s , w e i g h t s ,t i m e s , a n d s o o n .

I n s p i te o f h a v in g f o r m u l a t e d c r i te r i a o f i d e n t i t y f o r p l a c e s, c o l o r sa n d o t h e r p h y s i c a l a b s t r a c t i o n s , w e a r e n o t y e t in a p o s i t i o n t o a c c o u n tf o r u n i q u e n e s s , f o r in s t a n c e , o f w h a t i s r e f e r r e d t o a s ' t he p l a c e w h e r et h e e g g w a s ' . I n o n e s e n s e , i n fa c t , th e e g g w a s n o t i n a u n i q u e p l a c ei n t h e c a r t o n b e c a u s e i t w a s ' a t ' a l l o f t h e p l a c e s i n t h e e x t e n d e d r e g i o ni t o c c u p i e d . 14 P l a u s i b l y , i n s p e a k i n g o f th e p l a c e w h e r e t h e e g g w a s ,w h a t w e h a v e i n m i n d is a s p ec i al o n e o f t h e s e p l a c e s , w h i c h w e m u s tn o w s e e k t o c h a r a c t e ri z e . I h a z a r d t h a t t h i s i s t h e p l a c e t h a t i s c o e x t e n -s i v e w i t h t h e e g g , w h i c h i t f i l l s , w h i c h c a n b e i n d i r e c t l y c h a r a c t e r i z e db y g i v i n g t h e c o n d i t i o n s f o r a n y o b j e c t t o b e a t t h a t p l a c e :

P l a c e C o e x t e n s i v e n e s s C h a r a c t e r i z a t i o n . O n e o b j e c t i s a t ap l a c e f il le d b y a n o t h e r o b j e c t a t a t i m e i f a n d o n l y i f i tt o u c h e s t h e o t h e r o b j e c t a t t h e t i m e . 15


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C L A S S I C A L P H Y S I C A L A B S T R A C T I O N 151I f w e a s s u m e a s a m a t t e r o f c o n v e n t i o n t h a t a n o b j e c t m u s t t o u c h i t se l fa t a n y ti m e , t h e n t h e c o e x t e n s i v e n e s s c h a r a c t e r i z a t i o n im p l i e s t h a t t h e r em u s t b e a p l a c e f il le d b y t h e o b j e c t a t t h a t t i m e , a n d t h e o b j e c t m u s ti ts e l f b e a t t h a t p l a c e . M o r e o v e r t h is p la c e m u s t b e u n i q u e , f o r if t h eo b j e c t f il le d t w o d i f f e r e n t p l a c e s t h e r e c o u l d b e a n o t h e r o b j e c t a t o n eo f th e s e p l a c e s b u t n o t a t th e o t h e r , a n d t h a t w o u l d i m p l y t h a t t h es e c o n d o b j e c t b o t h t o u c h e d a n d d i d n o t t o u c h t h e f i rs t o n e .

A v e r y i m p o r t a n t t h i n g a b o u t t h e c o e x t e n s i v e n e s s c h a r a c te r i z a ti o n ist h a t , u n l i k e t h e a b s t r a c t i o n a n d i d e n t i t y p r i n c ip l e s , it h a s a n e m p i r i c a la s p e c t t h a t t h e o t h e r p r i n c ip l e s l a c k b e c a u s e i t r e f e r s to t h i n g s touchingo n e a n o t h e r , w h i c h i s s o m e t h i n g t h a t m u s t b e e s t a b l i s h e d b y s e n s eo b s e r v a t i o n . O t h e r e m p i r i c a l r e l a t i o n s p l a y a n a l o g o u s r 6 t e s i n t h e o r i e so f c o l o r, p h y s i c a l m a g n i t u d e s a n d t i m e , s in c e w e h a v e o b s e r v a t i o n a lm e a n s f o r d e t e r m i n i n g w h e n t w o t h i n g s m a t c h i n c o l o r , w e i g h t o rl e n g t h , a n d w h e n t w o e v e n t s o v e r l a p i n t im e . 16 W h a t i s s i g n if i ca n t a b o u tt h e s e r e l a t i o n s is th a t w h i le s e n s e o b s e r v a t i o n e s t a b li s h e s t h a t t h e y h o l db e t w e e n c o n c r e t e t h i n g s , i t a l s o e s t a b l i s h e s t h a t t h e s a m e abstract t h i n gis i n c i d e n t i n b o t h o f t h e m : i . e ., i t e s t a b l is h e s t h a t t h e y a r e coincident.T h u s , t h i n g s t h a t t o u c h a t s o m e p l a c e , t h i n g s t h a t m a t c h i n c o l o r h a v ea n a b s t r a c t c o l o r i n c o m m o n , t h i n g s t h a t m a t c h i n w e i g h t o r l e n g t hh a v e a n a b s t r a c t w e i g h t o r l e n g t h i n c o m m o n , a n d o v e r l a p p i n g e v e n t so v e r l a p a t a t i m e . T h i s ' d u a l i n t e r p r e t a t i o n ' o f c o i n c i d e n c e , e i t h e r a s ab i n a r y e m p i r i c a l r e l a t i o n b e t w e e n c o n c r e t e t h i n g s , o r a s a t e r n a r y a n dp a r t l y a b s t r a c t r e l a t i o n b e t w e e n t h e c o n c r e t e t h i n g s a n d a n a b s t r a c tt h i n g t h a t t h e y h a v e i n c o m m o n , 17 g i ve s t o t h e o r i e s o f p h y s i c a l a b s tr a c -t io n s a n e m p i r i c a l a s p e c t t h a t s e t t h e o r y l a c k s , s in c e t h e s a m e s e t b e i n gi n c i d e n t i n t w o c o n c r e t e t h i n g s h a s n o e m p i r i c a l ' m e a n i n g ' .

T h e r e a r e t w o o t h e r t h i n g s t o n o t e a b o u t c o i n c i d e n c e . O n e i s t h a t ,a s i s i n c i d e n c e , m o s t c o i n c i d e n c e r e l a t i o n s a r e t i m e - d e p e n d e n t , a n d t h ep r o b l e m o f a n a l y z i n g diachronic coincidence ( t h e s a m e a b s t r a c t t h i n gb e i n g i n c i d e n t i n d i f f e r e n t c o n c r e t e t h i n g s a t d i f f e r e n t t im e s ) is e ss e n -t i a l l y t h e s a m e a s t h e p r o b l e m o f a n a l y z i n g ' a b s o l u t e ' c o n s t a n c y a n dc h a n g e t h a t w a s r e f e r r e d t o e a r l i e r , a n d w h i c h w ill b e r e t u r n e d t o ins e c t i o n 7 .

T h e o t h e r p o i n t i s t h a t c o i n c i d e n c e r e l a t i o n s h a v e t o s a t i s f y o b s e r -v a t i o n a l l y t e s ta b l e l a w s t h a t i t i s p a r t o f t h e j o b o f a n a l y s i s t o i n v e s t i g a te .O b v i o u s l y b e i n g a t t h e s a m e p l a c e - t o u c h i n g - is s y m m e t r i c , a s a r ec o l o r , w e i g h t a n d t i m e m a t c h i n g . W e i g h t a n d l e n g t h c o i n c i d e n c e a r ea l s o s u p p o s e d t o b e transitive - t h e y a r e equivalence relations. P l a c e ,


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152 E R N E S T W . A D A M Sc o l o r , a n d t i m e - m a t c h i n g a r e n o t a l w a y s t r a n s i t i v e , b u t t h e r e i s r e a s o nt o t h i n k t h a t t h e s e a r e i n v o l v e d i n e v e n m o r e c o m p l e x l a w s w h i c h i ns o m e c a s e s st il l r e m a i n t o b e s t u d i e d .

H o w e v e r t h e f a i lu r e o f t r a n s it i v it y is it s e lf i m p o r t a n t , a s w e w i ll n o ws e e .

5 . U N E X T E N D E D E N T IT IE S A N D G E N E R A L IZ E D C O I N C I D E N C E

T h e n o n - t ra n s i ti v it y o f to u c h i n g m e a n s t h a t t w o t h in g s c a n t o u c h a t h i r dt h in g w i t h o u t t o u c h i n g e a c h o t h e r , t h e n o n - t r a n s it iv i t y c o l o r m a t c h i n go f m e a n s t h a t t w o t h i n g s c a n m a t c h a t h i r d t h in g w i t h o u t m a t c h i n g e a c ho t h e r i n c o lo r , a n d t h e n o n - t r a n s it iv i t y o f ti m e - o v e r l a p m e a n s t h a t t w oe v e n t s c a n o v e r l a p a t h i r d o n e i n t i m e w i t h o u t o v e r l a p p i n g e a c h o t h e r .B u t t h e r e a r e t h i n g s t h a t m i g h t s a t is f y a k i n d o f t r a n s i t i v i ty c o n d i t i o n :t h a t o t h e r t h i n g s t h a t t o u c h e d t h e m , m a t c h e d t h e m i n c o l o r , o r o v e r -l a p p e d t h e m i n t i m e w o u l d n e c e s s a ri ly h a v e t o t o u c h e a c h o t h e r , m a t c he a c h o t h e r i n c o l o r , o r o v e r l a p i n t i m e . T h e s e a r e unextended , p e r f e c t l yp o i n t - l i k e t h i n g s : s p a t i a l p o i n t s , i n s t a n t a n e o u s e v e n t s , a n d , i n t h e c a s eo f c o lo r s , p e r f e c tl y u n i f o r m l y c o l o r e d o b je c ts . T w o t h in g s t h a t t o u c h e da n o b j e c t t h a t o c c u p i e d o n l y o n e s p a t ia l p o in t w o u l d h a v e t o t o u c h e a c ho t h e r , t w o e v e n t s t h a t o v e r l a p p e d a n i n s t a n t a n e o u s e v e n t in t im e w o u l dh a v e t o o v e r l a p e a c h o t h e r , a n d t w o t h i n g s t h a t m a t c h e d a p e r f e c t l yu n i f o r m l y c o l o r e d o r m onochrom at i c t h i n g w o u l d h a v e t o m a t c h e a c ho t h e r . O f c o u r s e w e d o n ' t t h i n k t h a t t h e r e are a n y o b j e c t s s o s m a l l a st o o c c u p y a s in g le s p a ti a l p o i n t , a n d w e m a y b e s k e p t i c a l o f t h e e x i s t e n c eo f i n s ta n t a n e o u s e v e n t s o r o f p e r f e c t ly m o n o c h r o m a t i c o b je c ts , a n d t h i sl e a d s t o a p r o b l e m .

W e c o n c e iv e o f s p a c e a s f il le d w i t h u n e x t e n d e d poin t s t h a t s a t is f y t h ef o l l o w i n g :

Unextendedness condit ion. A p l a c e i s unextended i f a n d o n l yi f a n y t w o t h i n g s a t t h a t p l a c e a t a t i m e w o u l d h a v e t o t o u c he a c h o t h e r a t t h a t t im e .

S i m i l a r l y , w e t h i n k o f t i m e a s f i l l e d b y t e m p o r a l instants, a n d p e r h a p so f ' c o lo r s p a c e ' a s c o m p o s e d o f p e rf e c t m o n o c h r o m e s , w h i c h sa ti sf yu n e x t e n d e d n e s s c o n d i t i o n s a n a l o g o u s t o t h e a b o v e . B u t w e n o t e d t h a tt h e r e i s n o r e a s o n t o t h i n k t h a t t h e r e a r e a n y i n s t a n t a n e o u s e v e n t s ,p e r f e c t ly m o n o c h r o m a t i c b o d ie s , o r u n e x t e n d e d c o n c r e t e t h in g s ( if a


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c o n c r e t e t h in g w e r e u n e x t e n d e d t h e n a n y o t h e r c o n c r e t e th i n g s t o u c h i n gi t w o u l d h a v e t o t o u c h e a c h o t h e r , a n d t h a t w o u l d i m p l y t h a t t h e f ir stt h i n g h a d t o b e ' in f i n i te s i m a l l y s m a l l ' ) , i s T h e e x i s t e n c e o f u n e x t e n d e da b s t r a c t p l a c e s , t i m e s a n d c o l o r s i s n o t g u a r a n t e e d b y t h e a b s t r a c t i o np r i n c ip l e s f o r m u l a t e d p r e v i o u s l y , a n d t h e r e f o r e i f w e c o n c e i v e t h es p a c e s o f p l a c e , t i m e a n d c o l o r to b e f i ll ed b y t h e s e u n e x t e n d e d a b s t r a c tt h i n g s t h e r e m u s t b e o t h e r p r i n c i p le s i m p l i c it i n o r d i n a r y d i s c o u r s e t h a tg u a r a n t e e t h e i r e x i st e n c e . A s s u m i n g t h is , a n a ly s is s h o u l d s e e k t o m a k et h e b a s i s o f th e s e ' li m i t m y t h s ' c l e a r, a n d w e w i ll m a k e v e r y b r i e fo b s e r v a t i o n s o n t h i s .

F i r s t, o u r p r o g r a m is t o f o r m u l a t e e x i s t e n c e p ri n c ip l e s t h a t a r e a l r e a d yi m p l i c i t i n ' p r e - t h e o r e t i c a l ' c o n c e p t u a l i z a t i o n s , a n d n o t n e c e s s a r i l y t oc o n s t r u c t i d e a l s l i k e p o i n t s a n d t e m p o r a l i n s t a n t s b y m a t h e m a t i c a lm e a n s , s u c h a s W h i t e h e a d ' s m e t h o d o f E x t e n s iv e A b s t r a c t io n . 19 A s am a t t e r o f f a c t , o n e i m p l i c i t e x i s t e n c e p r i n c i p l e h a s a l r e a d y b e e n a s -s u m e d , n a m e l y a

P l a c e c o i n c i d e n c e p r i n c i p l e . T w o c o n c r e t e t h i n g s t o u c h a t at im e i f a n d o n l y i f t h e y a r e a t t h e s a m e u n e x t e n d e d p l a c e a tt h a t t i m e . 2

T h e i d e a i s t h a t w e c o n c e i v e o f p h y s i c a l o b j e c t s a s t o u c h i n g i f t h e yt o u c h o r ' m e e t ' a t a p o i n t . S i m i l a rl y , o b j e c t s m a t c h i n c o l o r i f t h e ys h a re a m o n o c h r o m a t i c c o lo r , a n d e v e n t s a r e c o n t e m p o r a n e o u s i f t h e yo v e r l a p a t a c o m m o n t e m p o r a l i n s ta n t . A s s u m i n g t h e s e c o in c i d e n c ep r i n c i p le s , i t f o l lo w s f r o m t h e f a c t t h a t c o n c r e t e t h i n g s t o u c h , m a t c h i nc o l o r o r o v e r l a p i n t i m e t h a t u n e x t e n d e d p l a c e s , c o l o r s a n d t i m e s e x i s t .

A m o n g m a n y q u e s t io n s t h a t c an b e r a is e d a b o u t u n e x t e n d e d i d e al s,w e m a y a sk w h y p h y s i c a l c o i n c i d e n c e s s h o u l d b e c o n c e i v e d i n t e r m s o ft h e m , a n d w h y w e s h o u l d p o p u l a t e t h e w o r l d n o t o n l y w i t h pl a ce s ,c o l o r s a n d t i m e s , b u t w i t h u n e x t e n d e d , i d e a l o n e s t h a t a r e n o t i n -s t a n t i a t e d b y a n y t h i n g c o n c r e t e . T h i s i s a v e r y d i f fi c u lt q u e s t i o n , b u tt h e a n s w e r m a y l ie in t h e f o l l o w i n g d i r e c t i o n . W e o f t e n a s s e r t t h a t m o r et h a n t w o c o n c r e t e t h i n g s ' m e e t ' i n a n u n e x t e n d e d a b s t r a c t p o i n t , a sw h e n w e s a y t h a t t h e s ta t e s o f A r i z o n a , N e w M e x i c o , C o l o r a d o a n dU t a h m e e t a t t h e F o u r C o r n e r s . H o w e v e r it is h a r d e r t o e x p la i n w h a tw e se e w h e n w e o b s e r v e t h a t m o r e t h a n t w o t h i n g s a r e c o i n c i d e n t t h a ni t i s t o e x p l a i n t w o - b o d y c o i n c i d e n c e ( t o u c h i n g ) . F o r i n s t a n c e , t h er e a d e r m i g h t a s k h e r o r h i m s e l f w h a t ' t e s t ' w o u l d s h o w t h a t t h e t h r e es e g m e n t s f o r m i n g t h e f i gu r e ' ~ ' m e e t i n a c o m m o n p o i n t , w h i l e t h o s e


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1 5 4 E R N E S T W . A D A M S

fo rm ing the f igu re ' _~_' do n o t . I t is ev id en t th a t we h a v e o b s e r v a t i o n a lm e a n s o f d e t e r m i n i n g t h e s e t h i ng s , w h i c h is p a r t o f t h e j o b o f c o n c e p t u a la n a ly s is t o a n a l y z e , b u t t h e c o m p l e x i t y o f t h is a n a l y s is m a y e x p l a i n w h yw e u s e th e m e e t - i n -a - c o m m o n - p o i n t i d i o m t o ' r e p r e s e n t ' t hi s c o n c e p tu -a l l y . T h i s i s s p e c u l a t i v e o f c o u r s e , b u t w e c a n n o t p u r s u e t h i s m a t t e rf a r th e r he re . 21 Th e f ina l s ec t ions o f th i s pap er w i ll com m en t sp ecu la -t i v e ly o n t h r e e o t h e r m a t t e r s .

6. C O N S T R U C T A B I L I T Y A N D T H E L I M I T S O F A B S T R A C TE X I S T E N C E

H a v i n g f o r m u l a t e d p r i n c i p le s th a t g u a r a n t e e t h e e x i s t e n c e o f u n e x -t e n d e d p l a c e s , c o l o r s , a n d t i m e s , w h o s e e x i s t e n c e i s n o t g u a r a n t e e d b yt h e ' b a s i c ' a b s t r a c t i o n p r i n c i p l e s f o r m u l a t e d i n s e c t i o n 3 , w e m a y a s k :a r e t h e r e s t i l l o t h e r a b s t r a c t p l a c e s , c o l o r s , q u a n t i t i e s a n d s o o n a n do t h e r p r in c i p l e s g u a r a n t e e i n g t h e i r e x i s te n c e , w h i c h a r e t a c it l y a s s u m e di n e v e r y d a y l if e o r i n s c i e n c e , a n d h o w l a r g e a r e t h e c l a ss e s o f a b s t r a c te n t i t i e s w h o s e e x i s t e n c e i s t h u s g u a r a n t e e d ? T h i s s e c t i o n w i l l s u g g e s tt h a t w h i l e th e r e m a y b e n o p r e c i s e a n s w e r t o t hi s q u e s t i o n , t h e s e c l a ss e sa r e l im i t e d b y t h e f a c t t h a t t h e i r m e m b e r s m u s t b e ' d i s t in g u i s h a b l e ' i nt e r m s o f th e i r r e l a t i o n s t o c o n c r e t e t h i n g s w h o s e e x i s t e n c e i s p h y s i c a ll yp o s s i b l e , a n d w h i c h i n m a n y c a s e s i t i s p o s s i b l e t o cons t ruc t . R o u g h l y :t he cons t ruc tab l e concre t e l imi t s t he abs t rac t ex i s t en t i n t h e s e n s e t h a t a n ytwo d i s t inc t abs t r ac t en t i t i e s mus t s t and in d i s t inc t inc idence r e la t ions toc o n s t r u c t a b l e t h i n g s .

T h e f o r e g o i n g a p p l i e s t o p o i n t s i n t h e E u c l i d e a n p l a n e . T h o u g h t h e ym a y n o t b e c o n s t r u c ta b l e i n a n y o r d i n a r y s e n s e , = n e v e r t h e l e s s a n y t w oo f t h e m a r e d i f f e r e n t i a t e d i n a m a n n e r a n a l o g o u s t o th a t i n w h i c h r e a ln u m b e r s a r e d i f f e r e n t i a t e d b y t h e i r re l a t io n s t o ra t i o n a l n u m b e r s . I np a r t i c u la r , a n y tw o p o i n t s a r e ' s e p a r a t e d ' b y li n es t h a t c a n b e p r o d u c e db y s t a n d a r d g e o m e t r i c a l m e t h o d s . 23 W h i l e t h is ' d is t in g u i s h a b i li t y r e -q u i r e m e n t ' d o e s n o t def ine the c lass o f po in t s in the p lan e , i t limi t s i t . 24N e x t c o n s id e r l ines . S t ra ight l in e s a r e s i m p l e s t , s i n c e t h e y a r e u n i q u e l yd e t e r m i n e d b y p a i r s o f p o i n t s o n t h e m , a n d p o i n t - p a i r s c a n b e d i s ti n -g u i s h e d ' c o n s t r u c t iv e l y ' .22 G i v e n A n c i e n t G r e e k c o n t r o v e r s i e s c o n c e r n -i n g t h e g e n e r a l c o n c e p t o f a l i n e ( c i r c l e s , p a r a b o l a s , e t c . , c f . H e a t h ,1 95 6, V o l . I , p p . 1 5 8 - 1 6 5 ) a n d t h e l o n g c o n t r o v e r s y o v e r t h e n a t u r e o fa f u n c t i o n , i t m a y b e i m a g i n e d t h a t l i n e s i n g e n e r a l a r e m u c h m o r ed i f fi cu lt than po in t s . In f ac t , i f the genu s l ine i n c lu d e d g e o m e t r i c a l loc i


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such as the class of points on a straight line at algebraic distances froma given origin, there would be no way to distinguish that point set'constructively' from the set of all points on the line. But it is intuitivelydoubtful that we would consider such an irregular point set as a line,and, as said, I speculate that loci that would be recognized to be linesare all constructively distinguishable. 26 This technically difficult topiccannot be pursued here, and we will end this section with brief remarkson the other physical abstractions that we have been concerned with.That there are constructive methods of circumscribing quantitativeabstractions like weights and lengths is guaranteed by Archimedeanconditions, which postulate that these quantities can be added, and ifthey are added sufficiently many times they will exceed any givenmagnitude - there are neither infinitesimal nor infinite magnitudes. 27

I suggest that the producibility operations for colors are color mixingprocesses, primarily of pigments but also of lights. It is possible tomatch any given hue by a suitable, measurable mixture of pigments ofthe primary hues, 28 and the possibility of matching by measurablepigment mixtures delimits the class of possible hues and thereby locatesthem in a 'color space'.

As to time, the most obvious time-locating processes involve naturalperiodic phenomena such as the succession of days and nights, seasons,etc. Of course these periods do not encompass 'all of time', and therewere events that preceded the formation of the solar system and nodoubt there will be ones that succeed its dissolution. But oscillationsof cesium atoms allow the dating of events beyond the compass of solarphenomena, and it is plausible that in even in the 'high energy soup'of the Big Bang there were periodic processes that furnished a potentialtime-frame. 29 In short, though it appears that it is Nature and notpeople that 'constructs' them, regular successions of events 'measure ' allof time and thereby delimit the class of possible temporal abstractions. 3

As with other difficult topics, we cannot go farther into the limits ofthe abstract and its relation to constructability here, and we now turnbriefly to a topic introduced in section 3: namely that of constancy andchange.

7. C O N S T A N C Y A N D C H A N G EThis section will suggest that the analogy between change in color orphysical magnitude and change of place goes deeper than has often


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156 E R N E S T W . A D A M S

been supposed. Leibniz argued in section 47 of the F i f t h L e t t e r "S a m u e l C l a r k e that change of place is a matter of something changingits position with respect to objects that maintain constant positionsrelative to each other, and a case can be made that change of color ormagnitude is similar: it is essentially a matter of a thing chariging itscoincidence relations to 'standards' whose coincidence relations to oneanother are invariable.First consider colors and color-fastness: the property of a thing thatdoes not change its color over time. What distinguishes these thingsfrom others is that if they match other color-fast things at one timethen they do so at other times, while non-color-fast things change incolor relative to one another. This is evident in the case of Munsellcharts, which are used as reference standards when 'objective' colordeterminations are required. Color-fastness is relative of course, sincecolor standards are occasionally harmed or destroyed, but in generalthey are more permanent in their matching relations than other thingsare. 31

Is there any more to color-constancy than the sort of relative con-stancy that color-standards exhibit? Following the example of spatialconstancy (being at rest in space), I will hazard that to a first approxi-mation there are no other 'facts of the matter' that make standards thatare unchanging relative to each other the r i g h t criterion of color con-stancy and change. This is, in effect, to espouse a r e l a t i o n a l account ofcolor. More will be said about this at the end of this section, but fornow it is at least supported by the analogy to relational theories ofspace. 32

As to physical magnitudes, it is clear that standards like measuringrods and standard weights maintain quite constant length and weightcoincidence relations. Marks on measuring rods that 'fit' at one timealso fit at others, and standard weights that balance at one time do soat other times too. And, these standards are much more constant thannon-standards. For instance, people are measured in height and weightby comparison with standards, but they change in height and weightrelative to each other while the standards do not. Whether there aremore ultimate facts of height and weight constancy will be returned to,but again the parallel to relativity of motion makes a prima facie casefor the relativity of constancy and change in physical magnitudes.

It is worth making a further comment on change of place, eventhough that is the example that inspires our relational approach to


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C L A S S I C A L P H Y S I C A L A B S T R A C T I O N 157c o n s t a n c y a n d c h a n g e i n o t h e r p h y s ic a l a b s t ra c t io n s . L e i b n i z a d v a n c e dt h e ' r e l a t i v i ty th e s i s ' a s a n a c c o u n t o f c h a n g e o f p l a c e i n S p a c e , a n dn o t n e c e s s a r il y o f c h a n g e o f p la c e i n o t h e r f r a m e s o f r e f e r e n c e s u c h a st h e s u r f a c e o f t h e e a r t h o r t h e m a t t e r o f w h i c h i t is f o r m e d . O n t h ec o n t r a r y , w e o f t e n t h i n k o f t h e s u r f a c e o f t h e e a r t h a s a fi x e d b a c k -g r o u n d s p a c e o r ' s t a g e ', o n w h i c h th e e v e n t s o f h is t o r y a r e p l a y e d o u t .B u t W e y l m a d e a n i m p o r t a n t o b s e rv a t io n :

i n a c o m p l e t e ly h o m o g e n e o u s s u b s t a n c e w i t h o u t a n y q u a l i ty t h e r e c o g n i t i o n o f t h e s a m ep l a c e i s a s i m p o s s i b l e a s t h a t o f t h e s a m e p o i n t i n h o m o g e n e o u s s p a c e . ( W e y l , 1 9 4 9 , p .165)

I i n t e r p r e t t h i s a s s u g g e s t in g t h a t w h a t is t r u e o f S p a c e is j u s t a s t r u eo f m a t e r i a l b o d i e s a n d t h e i r su r fa c e s . W e r e q u i r e s t a n d a r d ' r e f e r e n c ef e a t u r e s ' s u c h a s r i v e rs , h il ls , s u r v e y m a r k e r s , a n d t h e l ik e to d e t e r m i n es a m e n e s s o f p l a c e o n t h e e a r t h ' s s u r f ac e j u s t a s m u c h a s w e d o t od e t e r m i n e s a m e n e s s o f p l a c e in S p a c e , a n d s a m e n e s s o f c o l o r o r m a g n i -t u d e . 33B u t t h e r e is a p r o b l e m w i t h t h e r e l a t i o n a l t h e o r y o f p l a ce , c o l o r a n dm a g n i t u d e t h a t m u s t b e n o t e d , e v e n th o u g h w e c a n n o t a t t e m p t t or e s o l v e it . T h e r e c o g n i t i o n t h a t s t a n d a r d s m a i n t a i n i n v a r i a b l e c o in c i-d e n c e r e l a t io n s t o o n e a n o t h e r r e q u i r e s u s t o ' r e - id e n t i f y ' t h e s e s t a n -d a r d s o n d i f f e r e n t o c c a s i o n s , a n d i f W e y l ' s s u g g e s t i o n i s r i g h t w e c a no n l y d o t h a t i f t h e y h a v e s o m e o f t h e s a m e q u a l i ti e s o n t h e s e o c c a s io n s( s a m e n e s s o f s u b s t a n c e d e t e r m i n e d b y s a m e n e s s o f q u a li ti e s) . 34 B u t w ec a n n o t a p p e a l t o t h e i n v a r i a b il i ty o f c o i n c i d e n c e re l a t io n s t o o t h e rs t a n d a r d s t o d e t e r m i n e t h e q u a l i ta t i v e in v a r ia b i li t y o f t h e s t a n d a r d st h e m s e l v e s , a n d i t c o u l d s e e m t h a t a t s o m e s ta g e w e m u s t b e a b l e tor e c o g n i z e t h e ' in t r in s i c ' i n v a r i a b il i ty o f th e s t a n d a r d s , i n d e p e n d e n t l y o ft h e i r c o i n c i d e n c e s w i t h o t h e r s t a n d a r d s . M y i m p r e s s i o n i s t h a t t h i s i st o o s i m p l e , b u t i t d o e s s h o w t h a t t h e p r o b l e m o f an a l y z i n g q u a l i ta t i v ea n d q u a n t i t a t i v e i n v a r i a b i l i t y a n d t h a t o f a n a l y z i n g i n d i v i d u a l i d e n t i t yo v e r t i m e m a y u l t i m a t e l y b e i n s e p a r a b l e .

W e c o n c l u d e t h i s p a p e r w i t h c o m m e n t s o n t w o w a y s i n w h i c h t h ec o n c e p t u a l s y s te m s o f m o d e r n s c ie n c e a p p e a r t o b e e v o l vi n g aw a y f r o mt h e ' cl as s ic a l m o d e l ' t h a t w e h a v e a t t e m p t e d t o d e s c ri b e i n t h e p r e c e d i n gs e c t i o n s .


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158 ERNE ST W. ADAM S

8 . T W O NON- C L AS SI C AL T R E NDS

In the last century scientific concepts, especially quantitative ones,have come to be learned more in the laboratory than from ~textbookdefinitions' in terms of familiar concepts, and we might expect this tobe reflected in the principles of application of the concepts in question.In one respect this is certainly the case. The standards employed andthe methods of determining coincidence relations to them are far morerefined now than they were one or two centuries ago, and it is doubtfulthat these methods can be taught by verbal instruction alone. 35 Thiscould explain the trend in modern science towards not giving 'ordinarylanguage definitions' of fundamental concepts like length and mass:i.e., of not making their principles of application explicit. Genuinelynew ideas cannot be fully explicated in terms of old concepts, and it ispossible that classical principles of application do not apply to them.

On the other hand we use old words like 'five pounds ' to express thenew concepts, and this may reflect the fact that the logic of theirapplication is similar to the classical form that we have been sketching.We still assume that concrete objects have abstract metrical propertieslike weight and length, which are referred to by phrases of the forms'the weight of... ' , ' ... the length of... ' , etc., and therefore we stillpresuppose principles of abstraction relating them to these concretethings, though now knowledge of these things and of the coincidencerelations between them may only be acquired through specialized train-ing. But so long as the logic remains the same, something like theclassical abstract metrical ontology is still with us. In fact, tha t ontologyis now augmented by quantities like pound - f e e t that were not recognizedin earlier times, and which have their own principles of application.The classical abstract conceptual scheme has evolved, but so far it hasnot been totally discarded. 36However, more radical changes have been advocated. There havebeen attempts to carry out programmes in science similar to the pro-gram of reducing the disparate pre-Cantorian ontology of mathematicsto one of sets alone. It is a step in this direction to describe spatialpositions in terms of distances from coordinate axes, for if the axesthemselves can be considered to be concrete things and distances fromthem can be considered as 'pure numbers', we have eliminated placeas an abstract category in favor of number . If numbers are reduced tosets in any one of a number of familiar ways, everything is apparentlyreduced to a thing-and-set ontology.37


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C L A S S I C A L P H Y S I C A L A B S T R A C T I O N 159The foregoing fits in with 'representationalist' approaches to funda-

mental measurement (e.g., Krantz e t a l . , 1970), which has much torecommend it. In this approach a statement like 'the distance from xto y is 25 feet', which refers to the abstract quantity 2 5 f e e t , is replacedby 'the value of the distance-in-feet function for the arguments x andy is 25', which refers the pure number 2 5 , that differs from the 'denomi-nate number' 2 5 f e e t . This has the advantage of obviating the need for'denominate number arithmetics' that in many ways reduplicate purenumber arithmetic, but which involve such curious rules as 'like canonly be added to like' (e.g., feet can only be added to feet) but some-times 'like can be m u l t i p l i e d by unlike' (e.g., feet can be multiplied bypounds) .38

The reductionist program just noted has only met with limited successto date, possibly because conceptual inertia militates against learningnew ways of speaking when old ones will serve. But it is also possiblethat this or a similar program will ultimately be accepted by the scientificcommunity, which will mean tha t the classical abstract ontology will belargely abandoned, just as the disparate pre-Cantorian ontology ofmathematics has largely been abandoned.

However, I cannot refrain from ending with a comment on somethingthat I think must be faced when and if the classical abstract ontologyis finally replaced. C o n s t r u c t a b i l i t y is part and parcel of the conceptualscheme that I have tr ied to out line, and I suggest that nothing that failsto do justice to it can be regarded as an acceptable reformulation ofscientific doctrine. If this is so it means that scope must be given towhat transcends the actual existent, and it is implausible that that canbe described in terms of purely extensional abstractions like sets. Onthe other hand, the extent to which science will in the future make useof 'classical' non-extensional abstractions like places, colors, magni-tudes and times cannot be foretold. 39

N O T E S1 T h e a n a l o g y b e t w e e n p r i n c i p l e s o f a b s t r a c t a p p l i c a ti o n t o t h e c o n c r e t e a n d s e t - t h e o r e ti -c a l a b s t r a c t i o n i s o n e i n d i c a t i o n o f h o w f a r t h e ' s e t - t h e o r e t i c a l m o d e l ' o f t h e a b s t r a c td e p a r t s f r o m o n c e p o p u l a r v i e w s o f a p p l i e d t h e o r i e s a s 'i n t e r p r e t e d f o r m a l s y s t e m s ' o ra s fo r m a l c a lc u li a u g m e n t e d b y ' s e m a n t i c r u le s '. O n t h e o t h e r h a n d o u r d e p a r t u r e f r o mP o s i t iv i s t v i e w s is a t t h e f a r t h e s t r e m o v e f r o m r e c e n t a n t i - f o r m a l i s t, ' h i s t o r ic i s t' o u t l o o k s .2 W e s h a l l u s e t h e g e n e r i c t e r m i n s t an t i a t e f o r t h e r e l a t i o n b e t w e e n t h e c o n c r e t e o b j e c ta n d t h e a b s t r a c t p l a c e , c o l o r , w e i g h t , o r w h a t - h a v e - y o u t h a t i t is a t o r it h a s , a n d w e


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160 E R N E S T W . A D A M Ss h a ll c a ll t h e c o n v e r s e r e l a t i o n incidence. T h u s , t h e a b s t r a c t c o l o r , w e i g h t , e t c. i s i n c i d e n ti n t h e c o n c r e t e t h i n g t h a t i n s t a n t i a t e s i t .3 C a n t h e R u s s e l l ia n a n a l y s i s a p p l y t o a ll s i m i la r u s e s o f t h e d e f i n i t e d e s c r i p ti v e f o r m , a si n ' H e i s r e p a i r i n g the hole in the roof'? T o a s s u m e t h is w o u l d b e t o p o p u l a t e t h e w o r l dw i t h a s u p e r a b u n d a n c e o f d u b i o u s e n t i ti e s s u c h a s holes. W e w o n ' t a d d r e s s t h i s m a t t e r ,b u t o n l y c o n f i n e o u r a t t e n t i o n t o e n t i t i e s t h a t a t l e a st h a v e t h e r e s p e c t a b i l i t y o f h a v i n gb e e n t h e o r i z e d a b o u t - s u c h a s s e t s , c o l o r s , p l a c e s , a n d s o o n , a s b e l o w . H o w e v e r , t h ist e s t is e q u iv o c a l, s i n c e b e f o r e C a n t o r ( m a y b e B o o l e ) n o o n e t h e o r i z e d a b o u t s e ts , y e t t h ef o u n d a t i o n i n e v e r y d a y u s a g e w a s t h e r e , w a i t in g t o b e s e i ze d u p o n f o r t h e m a t h e m a t i c i a n ' sp u r p o s e s .

W h y n o t h a v e i n c l u d e d o t h e r a b s t r a c t th i n g s th a t c l e ar ly h a v e b e e n t h e o r i z e d a b o u t ,s u c h a s biological species, material substances, maladies o r shapes? S p e c u l a t i n g , m u c h o fw h a t w i ll b e s a i d a b o u t c o l o r s , p l a c e s a n d t h e l i k e d o e s a p p l y t o t h e s e ' n a t u r a l k i n d s ' ,b u t t h e y a r e l e ss w e l l d e f i n e d i n a w a y t h a t w i l l b e i m p o r t a n t t o u s h e r e : t h e r e a r e l e s sc l e a r ' l im i t s o f t h e p o s s i b l e ' , s u c h a s o f p o s s i b l e b i o l o g i c a l sp e c i e s . N o t e 3 0 a l s o a l lu d e so n t h i s .4 T h e t h e o r y o f ratios d e v e l o p e d in B o o k V o f E u c l i d 's Elements m a y b e a n o t h e re x c e p t i o n . S e e a l s o m e d i e v a l t h e o r i e s s u c h a s t h a t o f O r e s m e ( C l a g g e t t , 1 9 6 8 ) .5 T h u s N e w t o n , f o l lo w i n g P l a to , h e l d t h a t t h e A b s o l u t e s o f h i s t h e o r y c a n o n l y b e k n o w nb y ' a b s t r a c t i n g f r o m t h e s e n s e s ' (Schol ium t o f u n d a m e n t a l d e f i n it io n s B o o k I o f t h ePrincipia, p . 8 o f C a j o r i e d i t i o n , 1 9 3 4 ), w h i c h f o r m u l a t e s a m o d u s o p e r a n d i o f a b s t r a c tt h e o r i z i n g w h e n t h e o r y d o e s n o t s t a t e i ts p r i n c i p l e s o f a p p l i c a t i o n , e x i s t e n c e a n d i d e n t i t ye xp l i c i t l y .

S o m e t h i n g s im i l ar c an b e s a id f o r a b s t r a ct t h e o r i e s o f n u m b e r s . I t r e m a i n e d t o C a n t o rt o f o r m u l a t e p r i n c i p l e s o f a p p l i c a t io n a n d e x i s t e n c e f o r p u r e a r i t h m e t i c , a s a th e o r y o f' s e c o n d - l e v e l a b s t r a c t i o n s ' : o f e n t i t ie s , t h e p o s i t i v e n u m b e r s , t h a t s t a n d t o f i r s t- l e v ela b s t r a c t i o n s - i . e . , t o s e ts - a s a b s t r a c t io n s l ik e c o l o r a n d p l a c e s t a n d t o c o n c r e t et h in g s . K r o n e c k e r ' s I n t u i ti o n i s m c a n b e r e g a r d e d a s t h e a n a l o g u e o f P l a t o 's a n d N e w t o n ' sG e o m e t r i c a l I d e al i sm i n t h e r e a l m o f p u r e m a t h e m a t i c s .6 I t i s a m e r i t o f s e t t h e o r y ' s a b s t r a c t i o n p r i n c i p l e t o m a k e t h i s m a n i f e s t ; a n d a l s o , w h e r et h e o r y d o e s a s c r i b e p r o p e r t i e s a n d r e l a t i o n s to s e t s , to d e f i n e t h e m e x p l ic i tl y . I h y p o t h e -s i ze th a t a f u n d a m e n t a l f a u l t o f c la s s ic a l m e t a p h y s i c s w a s to s u p p o s e t h a t a n a c c o u n t o fa b s t r a c t i o n s l ik e c o l o r s , p l a c e s e t c . m u s t t e l l u s w h a t t h e s e t h i n g s a r e like.7 A g a i n , t h i s s e e m s t o m e b e t t e r t h a n , s a y , p o s i t i n g t h a t s p a c e a n d i ts p r o p e r t i e s a r ep s y c h o lo g i s ti c A n s c h a u u n g e n . T h e p r i n c ip l e s t h a t a p p l y t o t h e m m a y b e s y n t h e t i c a n d ap r i o r i , b u t t o g o b e y o n d t h a t i s g r a t u it o u s s p e c u l a t i o n w i t h o u t t h e ju s t i fi c a t io n o f b e i n gr o o t e d i n o r d in a r y u s a g e .

R u s s e l l ' s s l u r t h a t p o s t u l a t i o n ' h a s a l l t h e a d v a n t a g e s o v e r c o n s t r u c t i o n t h a t t h e f t h a so v e r h o n e s t to i l' s e e m s t o m e t o o s i m p l e . C o n s t r u c t io n a i m s , a m o n g o t h e r t h i n g s , a ts h o w i n g t h e c o n s i s t e n c y t h a t p o s t u l a t i o n o n l y p o s t u l a t e s , b u t i n f o r m u l a t i n g w h a t i sa l r e a d y i m p l i c i t i n o r d i n a r y u s e t h e A b s t r a c t i o n P o s t u l a t e g a i n s t h e p l a u s i b i l i t y t h a t t h a tu s e c o n f e r s o n i t . M o r e o v e r , i t is a n a d v a n t a g e o f p o s t u la t i o n o v e r c o n s t r u c t i o n n o t t oi m p u t e g r a t u i to u s f o r m t o t h a t w h i c h is p o s t u l a t e d - n o t t o m a k e n u m b e r s be s e t s , a n dSO on.

A l l o f t h is d o e s n o t m e a n t h a t O c c a m ' s R a z o r is t o b e d i s p e n s e d w i th . Theory n o r m a l l ys y s t e m a t i z e s , e x t e n d s , a n d i f n e c e s s a r y r e v i s e s p r e - t h e o r e t i e a l p r a c t ic e s , a n d i n t h e p r o c e s s


17/23

C L A S S I C A L P H Y S I C A L A B S T R A C T I O N 161t a k e s o n m o r e t h a n o r d i n a r y u s a g e i s c o m m i t t e d t o , i n c l u d i n g s u c h r e q u i r e m e n t s a sconsistency a n d frui t fu lness - i . e . , s u i t a b i l i t y t o t h e t h e o r e t i c i a n ' s p u r p o s e s . F a i l i n g t os a ti sf y th e s e r e q u i r e m e n t s , o r n o t s a ti s fy i n g t h e m a s w e ll a s a n o t h e r t h e o r y d o e s ( w h i c his s o m e t h i n g t h a t m a y o n l y b e c o m e a p p a r e n t o v e r a p e r i o d o f t im e ) , a t h e o r y ' c o n s t i tu t e d 'b y a n a b s t r a c t e x i s t e n c e p r i n c i p l e s h o u l d , r i g h t l y , b e d i s c a r d e d . B u t n o t o n a p r i o r im e t a p h y s i c a l g r o u n d s .s P e r h a p s w e m a y c a l l t h e A b s t r a c t i o n P r i n c i p l e a mea ning postulate b e c a u s e i t m a k e se x p l ic i t w h a t s e e m s t o b e i m p l i ci t i n t h e u s e o f e x p r e s s i o n s l ik e ' t h e p l a c e w h e r e t h e e g gw a s i n t h e c a r t o n ' . C e r t a i n l y t h e r e a d i n e s s w i t h w h i c h a b s t r a c t i o n p r i n c i p l e s f o r p l a c e s ,s e ts , a n d t h e l i ke a r e a c c e p t e d w o u l d s e e m t o b e a m a t t e r o f h o w ' c o m f o r t a b l e ' w e a rew i t h p r e - t h e o r e t i c a l u s e s o f t h e c o r r e s p o n d i n g d e f i n i te d e s c r i p t i o n s , w h i c h o n a R u s s e l l i a na n a l y si s p r e s u p p o s e t h e e x i s t e n c e o f t h e e n t i t i e s i n q u e s t i o n .9 T h e t e r m ' e v e n t ' m a y b e t o o t e m p o r a l l y r e s t r i c t e d : w e a l so w i sh t o in c l u d e t h i n g s l ik ew a r s a n d g l a ci a l e r a s , w h i c h c a n b e o f v e r y l o n g d u r a t i o n .10 T w o e x a m p l e s b r i n g t h i s o u t . F i r s t , n o t o n l y b o d i e s b u t b o d i e s ' s u r f a c e s , c av i t ie s int h e m , a n d a f a ir r a n g e o f o t h e r t h i n g s h a v e p l a c e s , a n d i t i s n o t e a s y t o c i r c u m s c r i b e t h ec l a s s o f ' p l a c e a b l e t h i n g s ' . S e c o n d , i t i s c u s t o m a r y t o s a y t h a t light i s c o l o r e d , y e t l i g h t sa r e v e r y d i f f e r e n t f r o m b o d i e s a n d i t i s o n l y w i t h d i f f ic u l ty t h a t w e m a t c h t h e i r c o l o r s t ot h o s e o f b o d i e s .11 N o r c a n w e e n t e r i n to t h e q u e s t i o n o f w h e t h e r o r d i n a r y u s a g e t a c it ly a s s u m e s t h ee x i s t e n c e o f a ' u n i v e r s a l s p a c e ' t h a t h a s a p l a c e f o r e v e r y t h i n g i n it . O f c o u r s e t h ee x i s te n c e o f a un ique u n i v e r s a l s p a c e w a s a s s u m e d i n c la s s ic a l N a t u r a l P h i l o s o p h y , ( e . g . ,i n N e w t o n ' s t h e o r y a n d i n t h e Crit ique of Pure Reason, p . 6 9 ) , t h o u g h t h i s c o n t r a s t s w i t ht h e p l u r a l i ty o f p h y s i ca l r e f e r e n c e f r a m e s t h a t c a m e t o b e a c c e p t e d i n l a te c l a s si c al p h y s i c sa n d i n R e l a t i v i t y ( t h o u g h i t i s w o r t h n o t i n g t h a t c l a s s i c a l E u c l i d e a n g e o m e t r y d i d n o tr e f e r e x p l ic i tl y t o S p a c e ) .

S o m e t h i n g a n a l o g o u s t o t h e r e l a t i v i t y o f p lace a r o s e i n c e r t a i n m e d i e v a l t h e o r i e s o fw e i g h t , a c c o r d i n g to w h i c h t h e r e w e r e ' f r a m e s o f w e i g h t r e f e r e n c e '. T h u s , w e i g h ts w e r ec o n c e i v e d o f a s b e i n g in media - e . g . , w e i g h t s in air, w e i g h t s in water, e t c ( s e e t h e 1 4 t hc e n t. p s e u d o - A r c l i i m e d e a n w o r k Liber Arch imed is de Ins ident ibus in H um idum i n M o o d ya n d C l a g g e t t , 1 9 52 ) . H o w c l o s e t h e a n a l o g y i s b e t w e e n t h e r e l a t iv i t y o f w e i g h t t o am e d i u m a n d t h e r e l a t i v i ty o f p l a c e t o a space i s s o m e t h i n g t h a t c a n n o t b e e n t e r e d i n t oh e r e .12 T h a t i s t o s a y t h a t mathemat ica l s e t t h e o r y h a s n o t e m p o r a l d i m e n s i o n , f o r th e o b v i o u sr e a s o n t h a t i t s o b j e c t s a r e t i m e l e s s t h i n g s li k e n u m b e r s ; i t i s n o t t o s a y t h a t e v e r y d a y' p r e t h e o r e t i c a l ' i d e a s o f c la ss a n d s e t h a v e n o t e m p o r a l a s p e c t . A d m i t t i n g t h a t t h i n g s c a nh a v e a t t r i b u t es a t o n e t i m e b u t n o t a t a n o t h e r a n d t h a t t h e e x t e n s i o n s o f a tt r ib u t e s a r es e ts s u g g e s ts t h a t n o n - m a t h e m a t i c a l s e ts m i g h t b e a l lo w e d t o a c q u i re a n d l o s e m e m b e r s .H o w e v e r , t h i s p o s si b i li t y i s i g n o r e d i n in f o r m a l s e t t h e o r i e s th a t p u r p o r t t o a p p l y t on o n m a t h e m a t i c a l t h in g s , a n d w e m i g h t b e i n c l in e d t o s a y th a t t h e s e t h e o r i e s d e a l w i t h ad i f f e r e n t s e t concep t t h a n t h e p e r s o n - i n - t h e - s t r e e t d o e s . B u t t o p u t i t t h a t w a y w o u l dd r a w a t t e n t i o n a w a y f r o m t h e f a c t t h a t l a t t e r ' s u s a g e s a r e w h a t w i n t h e t h e o r y ' s i n i t i a la c c e p t a n c e .13 A s s u c h t h e y r u n c o u n t e r t o a t r a d i t i o n t h a t d e n i e s e x i s t e n c e t o a l l b u t a c t u al l yi n s t a n t i a t e d p l a c e s , c o l o r s , t i m e s a n d s o o n . T h i s i n c l u d e s t h e d e n i a l o f t h e p o s s i b i li t y o ft h e vo id , o f s h a d e s o f c o l o r n e v e r p e r c e i v e d , a n d o f t im e s o u t s i d e a ll e x p e r i e n c e .


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162 E R N E S T W . A D A M ST h e B i g B a n g i s s a id t o m a r k t h e b e g i n n i n g o f t i m e , o n t h e g r o u n d s t h a t t h a t w a s t h e

b e g i n n i n g o f t h e p h y s i ca l u n i v e r s e . H o w e v e r , t h a t m a y m a r k a t h e o r e t i c a l l im i t t o t h ep o s s ib l e - t h e b o u n d s o f t h e c o n t a i n e r s p a ce o f ti m e - w h i c h w il l b e r e t u r n e d t o i nsec t i on 6 .1 4 P r o p e r l y , w e s h o u l d s a y t h a t t h e e g g s t o o d i n d i f f e r e n t i n c i d e n c e r e l a t i o n s t o t h e s ed i f f e r e n t p l a c e s ( i t f i l l e d t h e c o m p a r t m e n t a t t h e s a m e t i m e t h a t i t w a s a t t h e o t h e r p l a c e s ),a n d w e s h o u l d f o r m u l a t e d i f f e r e n t a b s t r a c t i o n p r i n c i p l e s c o r r e s p o n d i n g t o t h e d i f f e r e n tk i n d s o f in c i d e n c e. W e a r e n o t p r e p a r e d t o e n t e r i n t o t h e s e c o m p l i ca t io n s in t h i s b r i e fo v e r v i e w .1 5 T h i s i s t h e k i n d o f p l a c e t h a t A r i s t o t l e a t t e m p t e d t o d e s c r i b e o r ' c i r c u m s c r i b e ' a s ' t h eb o u n d a r y o f th e s u r r o u n d i n g m e d i u m ' . A m o n g o t h e r d r a w b a c k s, t h i s p r e s u p p o se s t h a tt h e r e is a s u r r o u n d i n g m e d i u m a n d t h a t t h a t m e d i u m d o e s n ' t i t s e l f h a v e a p l a c e t h a tc o u l d b e c i r c u m s c r ib e d in t h e s a m e w a y .1 6 I t m u s t n o t b e t h o u g h t t h a t a n y o f t h e s e r e l a t i o n s i s s i m p l e , s i n c e a n a l y s i s r e v e a l sm e s s y d e t a i l s i n a l l o f t h e m . C o l o r m a t c h i n g i s a v e r y c o m p l e x m a t t e r , M u n s e l l c h a r tc o m p a r i s o n s b e i n g p e r h a p s t h e m o s t r e f in e d v is u a l i n s p e c t io n m e t h o d , b u t n o t o n l y i st h a t c o m p l i c a t e d , b u t i t a p p l i e s o n l y t o t h e c o l o r s o f o b j e c t s a n d n o t t o t h o s e o f l ights ,w h i c h a r e t h e p r i m a r y t h i n g s c o m p a r e d b y s p e c t r o g r a p h i c a n a l y s i s . W e i g h t c o i n c i d e n c ei s d e f i n e d b y m e t h o d s l i k e c o m p a r i s o n o n t h e p a n s o f a c h e m i c a l b a l a n c e , b u t d e t a i ls a r ea g a i n m e s s y a n d t h e y a r e c o m p l i c a t e d s ti ll m o r e b y t h e f a c t th a t w e i g h t is n o w d e f i n edi n P h y s i c s a s t h e f o r c e o f t h e e a r t h ' s g r a v i t a t i o n a l a t t r a c t i o n f o r a n o b j e c t , w h i c h m a k e sw e i g h t c o m p a r i s o n a s p e ci a l c a s e o f f o r c e c o m p a r i s o n . C o n t e m p o r a n e i t y ( o v e r l a p p i n g i nt i m e ) w e n t e s s e n t i a l l y u n e x a m i n e d p r i o r t o E i n s t e i n ' s s t u n n i n g a n a l y s i s , w h i c h s h o w e dt h a t c e r t a i n e m p i r ic a l a s s u m p t i o n s in v o l v i n g i t h a d t o b e r e v is e d . N o n e o f t h e s e t h i n g sc a n b e e n t e r e d i n t o i n d e t a i l h e r e , b u t i t s h o u l d b e c l e a r t h a t c a r e f u l s t u d y o f a n yp a r t i c u l a r a b s t r a c t c a t e g o r y m u s t p a y e s p e c i a l l y c l o s e a t t e n t i o n t o t h e m . S e c t i o n 8 w i l lr e t u r n t o t h e e v o l u t i o n o f c o i n c i d e n c e d e t e r m i n a t i o n m e t h o d s .17 In the F o u n d a t io n s o f A r i t h m e t i c , p a r . 6 4 , F r e g e s t r e s s e d a si m i l a r p o i n t a b o u t t h er e l a ti o n o f p a r a l le l is m b e t w e e n f in e s , w h i c h c a n b e i n t e r p r e t e d e i t h e r a s a b i n a r y r e l a t i o nb e t w e e n l i n e s, o r a s a t e r n a r y r e l a t io n b e t w e e n t h e l i n e s a n d a t h i r d t h i n g , t h e d i r e c t i o nt h a t t h ey h a v e i n c o m m o n .18 A s m a l l d o t s u c h a s t h a t i n t h e q u o t e s ' . ' c o m e s c l o se t o ,s a t i s fy i n g t h e u n e x t e n d e d n e s sc o n d i t i o n , s in c e a n y t w o t h i n g s t o u c h i n g i t c o m e c lo s e to t o u c h i n g e a c h o t h e r . N o d o u b tt h a t i s w h y w e r epr es en t p o i n t s b y t h e s e p o i n t - li k e th i n g s , a n d i t is p r o b a b l y t h e i n s p i r a ti o nf o r c o n c e i v i n g i d e a l p o i n t s t o b e l im i t s . B u t t o b e p o i n t - l i k e i s t o h a v e a s m a l l e x t e n s i o na n d n o t t o b e u n e x t e n d e d , a n d o u r jo b is to e x p l a in u n e x t e n d e d a b s tr a c ti o n s . T h ef o ll o w i n g f o o t n o t e c o m m e n t s o n a t t e m p t s t o ' c o n s t r u c t ' p o i n t s a s o n e o r a n o t h e r k i n d o fl imi t .19 T h e l is t o f a t t e m p t e d c o n s t r u c t io n s o r r e p r e s e n t a t i o n s o f g e o m e t r ic a l a n d s p a ti o -t e m p o r a l m i n i m a , b e g i n n i n g w i t h B e r k e l e y ' s m in im a v i s ib i l ia , a n d p a s s i n g b y w a y o fc o n s t r u c t io n s s u g g e s t e d b y P o i n e a r 6 , R u s s e ll , N i co d , a n d C a r n a p t o G o o d m a n ' s ' t o p o l o -g y ' o f s u b j e c t i v e q u a l it i e s is t o o l o n g t o b e c o n s i d e r e d i n d e t a i l. T h e s e a r e a l l s u b j e c t t ot h e s a m e o b j e c t i o n : n a m e l y t h a t t h e t h i n g s c o n s t r u c t e d a r e n o t s h o w n t o h a v e a n yn e c e s s a r y c o n n e c t i o n w i t h t h e t h i n g s t h a t a r e o r d i n a r il y c o n c e i v e d t o b e p o i n t s .

A n o t h e r a p p r o a c h t o s p a t ia l m i n i m a c a n a ls o b e c r i ti c iz e d o n t h e s e g r o u n d s . T h e ym i g h t b e p o s t u l a te d t o b e t h e p l ac e s o f t h e u n e x t e n d e d ' m a s s p o i n t s ' t h a t p h y s i ca l t h e o r i s ts


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C L A S S I C A L P H Y S I C A L A B S T R A C T I O N 163h a v e s o m e t i m e s t h e o r i z e d a b o u t ( c f. N e w t o n , Principia, B o o k I I , P r o p o s i t i o n X X I I I ,T h e o r e m V X I I I o r M a x w e l l , 1 87 7, p . 2 ) w h i c h w o u l d a l s o fi t i n w i t h E u c l i d ' s d e f i n i ti o n' A p o i n t i s t h a t w h i c h h a s n o p a r t ' (E lemen ts , B o o k I , D e f i n i t i o n i , H e a t h , V o l . 1 , p .1 55 ). B u t t h e c o n c e p t o f a m a s s p o i n t i s n o t g r o u n d e d i n e v e r y d a y u s a g e , a n d t h e r e f o r ej u s t if y i n g t h e e x i s t e n c e o f s p a ti a l p o i n t s b y r e f e r e n c e t o t h e m w o u l d b e t o ju s t i fy t h ef a m i l i a r i n t e r m s o f t h e o b s c u r e .2o T h i s p r i n c i p l e p r e s u p p o s e s t h a t t h e b o u n d a r i e s o f c o n c r e t e t h i n g s 'b e l o n g ' t o t h e t h in g st h e y a r e b o u n d a r i e s o f ; i .e . , t h a t t h e p o i n t s e t s t o w h i c h t h e t h in g s c o r r e s p o n d a r et o p o l o g i c a l l y closed ( K e l l e y , 1 9 5 5 , p . 4 0 ) . I f c o n c r e t e t h i n g s d i d n ' t c o n t a i n t h e i r b o u n d -a r i es - s a y t h e y w e r e l i k e t h e i n t e r i o r s o f s p h e r e s - t h e n t h e y c o u l d b e ' z e r o d i s t a n c ea p a r t ' w i t h o u t h a v i n g p o i n t s i n c o m m o n .21 A d a m s ( 1 9 7 4) a n d ( 1 98 6 ) p r o p o s e a n a l y s e s o f e m p i r i c a l m e t h o d s o f m u l t i - th i n g c o in c i -d e n c e d e t e r m i n a t i o n . T h e l a tt e r p a p e r a l s o s h o w s h o w t h e s e m e t h o d s a n d t h e i r p r i n c ip l e so f a b s t r a c t i o n n o t o n l y ju s t i fy t h e c l a i m t h a t u n e x t e n d e d s p a t i a l a b s t r a c t i o n s e x i s t, b u tt h a t th e r e a r e u n c o u n t a b l y m a n y - a c o n t i n u u m - o f t h e m .2 2 B u t t h e p o i n t p lac ing o p e r a t i o n s p o k e n o f i n P r o p o s i t i o n 2 , B o o k I o f The E lemen tsm i g h t b e i n t e r p r e t e d a s a k i n d o f c o n s t r u c t i o n . C f . t h e c o m m e n t s o n t h i s i n H e a t h 1 95 6,V o l . 1 , p . 24 .23 T h i s i s c l o s e l y r e l a t e d t o separability conditions f o r t o p o l o g i c a l s p a c e s ( c f. K e l l e y , 1 95 5,p . 4 8 ) . T h e s i m p l e s t s e p a r a b i l i t y c o n d i t i o n i s t h a t t h e r e s h o u l d b e a c o u n t a b l e s e t o fp o i n t s t h a t i s d e n s e i n t h e s p a c e i n t h e s e n s e t h a t e v e r y ' n e i g h b o r h o o d ' in t h e s p a c ec o n t a i n s a p o i n t o f t h is c o u n t a b l e s e t . C o u n t a b i l i t y i s c l o s e ly r e l a t e d t o c o n s t r u c t a b i li t y ,b e c a u s e i t is a p r o p e r t y o f i t e ra t e d o p e r a t i o n s o f c o n s t ru c t i o n .24 I n p a r t i c u l a r , i t li m i ts it to t h e c a r d i n a l i ty o f t h e c o n t i n u u m . H o w e v e r , w e a r e n o te n t i t l e d t o a s s u m e t h a t t h e c l a ss o f p o i n t s is m a x i m a l i n t h a t i t i n c l u d e s a m a x i m u mn u m b e r w h o s e e x i s te n c e w o u l d b e c o n s i s t e n t w i t h b e i n g d i s ti n g u is h a b le in t h e m a n n e rc o n s i d e r e d ( c f . H i l b e r t ' s Completeness Postulate). A s w i t h t h e c l a s s o f constructabiliai t s e l f , t h e c l a s s o f a b s t r a c t e x i s t e n t s m i g h t b e s t b e r e g a r d e d a s a primit ive o f g e o m e t r i ca lt h e o r y .2 s N o t e t h a t i t is a m e a s u r e o f t h e a b s t r a c t n e s s o f t h e s e l i n e s t h a t d i f f e r e n t ' d r a w i n g s ' o ft h e i r p a r t s a r e n o t c o n c e i v e d t o c o n s t r u c t d i f f e r e n t l in e s . I t is s ig n i f ic a n t t h a t e v e n E u c l i d ' sv e r y c o n s t r u c t i v e g e o m e t r y t h e o r i z e d a b o u t l in e s t h a t a r e c l e a rl y n o t c o n s t r u c t a b l e , a n ds p e c i f ic a l l y a b o u t infinite s t r a i g h t li n e s ( o f. P r o p o s i t i o n 1 2 o f B o o k I , ( H e a t h , 1 9 56 , V o l .I , p . 2 7 0 ). N o n e t h e l e s s , t h e s e n o n - c o n s t r u c t a b l e s c a n b e d i s t in g u i s h e d b y r e f e r e n c e t oe x t e n d e d a n d f i n it e c o n s t r u c t a b l e s .26 F o r i n s t a n c e , i t i s p l a u s i b l e t h a t l i n e s c o r r e s p o n d t o s e t s o f p o i n t s t h a t a r e closed int h e t o p o l o g y o f t h e p l a n e . T h e s e t o f p o i n t s o n a l i n e w h o s e d i s t a n ce s f r o m a n o r ig i n isa n a l g e b r a i c n u m b e r i s n o t c l o s e d , a n d i t i s p o s s i b l e t h a t a n y t w o d i f f e r e n t c l o s e d p o i n ts e t s a r e c o n s t r u c t i v e l y d i s t in g u i s h a b l e .27 B o o k V o f E u c l i d ' s Elements m a k e s t h i s c l e a r , a n d r e c e n t ' r e p r e s e n t a t i o n a l ' t h e o r i e so f e x t e n s i v e m e a s u r e m e n t a s s u m e e s s en t ia l ly t h e s a m e t h i n g ( cf . C o n d i t i o n 4 o f D e f i n it i o n1 , p . 7 3 , i n K r a n t z e t a l . , 1 9 70 ). T h e g e n e r i c t e r m n o w c o m m o n l y u s e d f o r p h y s i c a la d d i t i o n ( o f w e i g h t s , l e n g t h s , e t c . ) i s concatenation, b u t i t i s n o t e w o r t h y t h a t w h e r et r a d i t i o n a l f o r m u l a t i o n s a f f i r m e d t h a t w e i g h t s o r l e n g t h s ca n b e a d d e d , m o d e r n ' e x t e n -s i o n a l ' f o r m u l a t i o n s p r e s u p p o s e t h a t t h e y are a d d e d ( i . e . , t h a t t h e r e exists s o m e t h i n ge q u a l t o t h e p h y si c al su m o f t w o t h i n g s) . H e r e t h e m o d e r n s f o l l o w H e l m h o l t z ( ' N u m b e r i n g


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164 ERNEST W. ADAMSand Measuring', 1887), who, himself following earlier developments , had to contend withdistinguishing so called 'denominate numbers" like 5 . i pounds from "pure numbers' likethe number 5.1 - a problem that the ancient Greeks . lacking modern concepts of purenumbers, did not have to face.

Confusing producibility with existence results partly from conflating the producibleconcrete with the existent abstract. Another cause for this arises from the fact that forEuclidean constructions at least, constructabiliO, quan tifiers of the form 'It is possible todraw (produce, describe, etc.) something such that.." can be paraphrased as existentialquantifiers over classes of "constructable things', e .g.. as "There is a drawable (producible,describable, etc.) thing such tha t.. ', where the classes of constructabilia are best regardedas primitives of Euclidean theory. This representation of constructability as 'existentconstructability' is not trivial, since it excludes constructional conflict, as when the con-struction of a building would conflict with the construction of a parking lot in the samespace.2s In measurement theoretic terms this sort of "color location process" is a derivedmeasure, since it depends on fundamenta l measures of the quantities of the primarypigments (or monochromatic spectral lights) that are mixed. Interestingly, a color-spacetopology and its dimensionality may be characterizable independently of measurementsof quantities of primary pigments. Points in the space (perfect monochromata) arecharacterizable as above, and neighborhoo dscan be characterized in terms of a kind ofcontinuous color variation that can be ascertained by direct visual inspection, which isitself dependent on continuous spatial variation.29 This was pointed out to me by Paul Teller. Of course Relativity requires us to modifythis, either to temporally locate all events in any inertial frame, or else to locate themin an over-arching 'space-time'. But we will leave these matters aside here.30 It is the lack of constraints and means of producing possible biological species, physicalailments, personality types, and o ther natural kinds that sets them apart from the physicalabstractions that we are concerned with here. This is associated with the fact that thereare no uninstantiated ' empty spaces' in the 'species space' , analogous to places in physicalspace that have nothing in them.31 Perhaps there are no perfec tly color-fast things that always ~stay in step' in color, butwe are now able to use wave-lengths of light produced by heated substances for thispurpose.32 In a general discussion of measurement (Adams , 1966) I have argued that both classicaland modern theories of magnitude or 'quantity' that omit the temporal dimension shouldbe regarded as theories of ideal standards which are unchanging in relation to oneanother. The Appendix to this paper shows that under normal circumstances determiningconstancy and change by comparing objects with such standards is justified by practical'informational' considerations. We 'measure' colors, weights or what-have-you by com-paring them with standards at different times, but we use the information thus gained toestimate how the things measured will 'fit together' at one time. It is more difficult toargue, but I would also suggest that the assurance of fitting together is why it is desirableto obtain the interpersonal agreement which can be gained when observers comparethings with known standards.33 That most of the earth's surface is solid is what makes it capable of supporting such"features; that is what distinguishes the solid from the fluid.


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C L A S S I C A L P H Y S I C A L A B S T R A C T I O N 1653 4 I t m i g h t a l s o b e a r g u e d t h a t w e h a v e t h e o r e t i c a l g r o u n d s f o r h o l d i n g , s a y , t h a t t h ew a v e l e n g t h s o f l i g h t e m i t t e d b y c e r t a i n t h i n g s a r e c o n s t a n t , a n d t h a t c e r t a i n m o l e c u l a rl a t ti c e s a lw a y s h a v e t h e s a m e d i m e n s i o n s , a n d t h i s p r o v i d e s t h e u l t i m a t e ju s t i f ic a t i o n f o rd e t e r m i n i n g c o l o r a n d l e n g t h c o n s t a n c y b y r e f e r e n c e t o t h e s e t h e o r e t i c a l c o n s t a n t s . T h e r ec a n b e n o d o u b t t h a t w e d o u s e t h e o r y i n t h e d e v e l o p m e n t o f e v e r m o r e r e f in e d s t a n d a r d s ,b u t I w o u l d a r g u e t h a t t h e u l t i m a t e ' p r o o f ' o f t h i s p r o c e d u r e i s t h e actual re l a t i vec o n s t a n c y o f t h e i m p r o v e d s t a n d a r d s .35 W e m i g h t b e i n c li n e d t o s a y s c ie n c e e v o lv e s m o r e a n d m o r e r e f in e d concepts o f w e i g h t,c o l o r , a n d s o o n w h e n i t e v o l v e s m o r e r e f i n e d s ta n d a r d s a n d m e t h o d s o f m e a s u r i n gw e i g h t , l e n g t h , c o l o r , e t c . T h e r e i s s o m e t h i n g t o t h i s , b u t i t i s s i g n if i c a n t t h a t w h i l es t a n d a r d s a n d m e t h o d s o f m e a s u r e m e n t c h a n g e t h e r e i s s e l d o m a c h a n g e i n t h e o r y , a n dt h e o r e t i c a l c h a n g e i s w h a t w e c o m m o n l y a s s o c i at e w i t h c o n c e p t u a l c h a n g e . F o r i n s t a n c e ,t h o u g h i n c r ea s i n g ly ' e x a c t ' w a y s o f m e a s u r i n g l e n g t h s a n d t im e s w e r e e v o l v e d i n c l as s ic a lp h y s ic s , E i n s t e i n ' s t h e o r e t ic a l r e v o l u t i o n i s w h a t w e r e g a r d a s c h a n g i n g o u r c o n c e p t s o fl e n g t h a n d t i m e .3 6 O f c o u r s e R e l a t i v i s t i c a n d Q u a n t u m M e c h a n i c a l d e v e l o p m e n t s c a l l f o r m o r e r a d i c a la l t e ra t io n s i n t h i s s ch e m e o f t h in g s . F o r i n s t a n c e , w h e n w e a r e c a r e fu l w e d o n o t s p e a ko f the l e n g t h o f s u c h a n d s u c h a t h i n g , b u t r a t h e r o f t h e t h i n g ' s l e n g t h relative to ap a r t i c u la r f r a m e o f r e f e r e n c e , s o t h e c la s si ca l d i m e n s i o n a l o n t o l o g y b e c o m e s r e l at i v iz e dw h i l e a n e w n o n - r e l a t iv i z e d c o n c e p t is a d d e d : n a m e l y t h a t o f ' s p a t i o - t e m p o r a l d i s t a n c e '.M y v a g u e i m p r e s s i o n i s t h a t t h e r e t u r n s a r e n o t i n y e t a s f a r a s c o n c e r n s ' q u a n t i z e dq u a n t i t i e s ', w i t h o n l y v a g u e l y f o r m u l a t e d ' c o r r e s p o n d e n c e p r in c i p l e s ' c o n n e c t i n g q u a n t u mp h e n o m e n a w i t h t h e i r ' c la s s ic a l c o u n t e r p a r t s ' .37 C f . Q u i n e ( 1 9 6 0 , s e c t io n s 5 0 a n d 5 2 ) . I w i ll n o t c o n s i d e r h e r e t h e s ti ll m o r e r a d i c a ln o m i n a l i s t p r o g r a m o f H a r t r y F i e l

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Adams, Classical Physical Abstraction [23 pgs]