Structuralism and Structures
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Structuralism and Structures
Published b y World Scientific Publishing C o . Pie. Ltd. P O Box 128, Farter Road, Singapore 9128 USA UK
office: office:
Suite I B , 1060 Main Street, River Edge, NJ 07661 57 Shelton Street, Covent Garden, London W C 2 H 9 H E
L i b r a r y of Congress Cataloging-in-Publication D a l a Rickart, C . E . (Charles Earl), 1913Structuralism and structures / Charles E . Rickart p.
cm. — {Series in pure mathematics ; v. 21)
Includes bibliographical references and index ISBN 9810218605 I. Mathematics. QA39.2.R535
2. Structuralism.
I. Title.
II. Series.
1995
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94-28563 CIP
Copyright © 1995 by World Scientific Publishing C o . Pte. Ltd. A l l rights
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P r i n t e d i n S i n g a p o r e by U t o - P r i n t
Series in Pure Mathematics - Volume 21
STRUCTURALISM AND STRUCTURES
A Mathematical Perspective
Charles E Rickart Department of Mathematics Yale University USA
Y ( b World Scientific w l Singapore • New Jersey • London • Hong Kong
A l b c r s . M U L T I P L E X B . 1948 Yale University A r t Gallery
" I m a g i n a t i o n is m o r e powerful t h a n knowledge.™ - Albert Einstein.
PREFACE
I w i s h to emphasize at the outset t h a t the t i t l e phrase, " A M a t h e m a t i c a l P e r s p e c t i v e , " d o e s n o t m e a n a f o r m a l m a t h e m a t i c a l t r e a t m e n t of the s u b j e c t . W h a t i t does m e a n is t h a t m a n y of the ideas c o n c e r n i n g s t r u c t u r e s a n d s t r u c t u r a l i s m that are developed here were suggested i n one way or a n other by m a t h e m a t i c s , a l t h o u g h the connection is u s u a l l y not spelled o u t . M a t h e m a t i c s , i n other words, generally serves as a m o d e l not as a t o o l . T h e t i t l e m i g h t also suggest a discussion of various examples i l l u s t r a t i n g a p p l i cations of m a t h e m a t i c s to diverse fields. T h e r e exist, of course, m a n y such a p p l i c a t i o n s , a n d they are indeed very m u c h concerned w i t h s t r u c t u r e s . A t the s a m e t i m e , their t r e a t m e n t w o u l d require an e x p l a n a t i o n of technical m a t e r i a l f r o m b o t h m a t h e m a t i c s a n d the involved field, so w o u l d c o n s t i t u t e a m a j o r digression, f r o m our m a i n goal to expose the n a t u r e of s t r u c t u r e s themselves. Therefore, i n d i v i d u a l a p p l i c a t i o n s o f m a t h e m a t i c s are a v o i d e d , a l t h o u g h the general character of such a p p l i c a t i o n s is discussed i n C h a p t e r VII. W e w i l l n o r m a l l y use the t e r m " s t r u c t u r a l i s m " to m e a n "any m e t h o d o f a n a l y z i n g a b o d y of i n f o r m a t i o n w i t h respect to its inherent s t r u c t u r e " (p. 1). A t the same t i m e , the t e r m often refers to a s p e c i a l " i n t e l l e c t u a l m o v e m e n t " t h a t emerged i n the 1950's a n d developed r a p i d l y i n t o the 1970's. T h e l a t t e r was based o n the use of s t r u c t u r e notions a p p l i e d most often to a s t u d y of c e r t a i n social science a n d h u m a n i t i e s subjects, a n d is c o m m o n l y associated w i t h the two names, C l a u d e L e v i - S t r a u s s ( a n t h r o p o l o g i s t ) a n d J e a n P i a g e t (psychologist, philosopher). T h e r e were, of course, m a n y other c o n t r i b u t o r s , i n c l u d i n g numerous workers i n a variety of fields r a n g i n g f r o m a n t h r o p o l o g y to poetry. L i n g u i s t i c s , i n p a r t i c u l a r , p l a y e d a key role, because s t r u c t u r e is so basic and also accessible i n the s t u d y of languages. In t h i s c o n n e c t i o n , the work of F e r d i n a n d de Saussure (done early i n the 1900's) was especially i m p o r t a n t . A n o t h e r early c o n t r i b u t o r was the s o c i a l a n t h r o p o l o g i s t A . R . R a d c l i f f e - B r o w n , who stands out i n the context of the present work m a i n l y because of his s y s t e m a t i c a n d u n u s u a l l y e x p l i c i t a p p e a l to s t r u c t u r e , based on a s t r a i g h t f o r w a r d definition t h a t is not essentially
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different f r o m the one we have chosen (Section 7). P e r h a p s as a result of r a p i d g r o w t h , some of the c o n t r i b u t i o n s to the s t r u c t u r a l i s t movement began to show an increase i n s u p e r f i c i a l i t y a n d a decrease i n awareness o f genuine s t r u c t u r e , t h u s suggesting a developing f a d r a t h e r t h a n a serious d i s c i p l i n e . I n the e n d , the m o v e m e n t receded i n p o p u l a r i t y a l m o s t as fast as it had g r o w n , a n d was displaced i n c e r t a i n areas by other even more transient " i s m s . " Nevertheless, the o v e r a l l c o n t r i b u t i o n s are i m p o r t a n t and the general s t r u c t u r a l i s t a p p r o a c h r e m a i n s v a l i d . A f t e r a l l , the basic ideas are not new but e x t e n d at least as far back as P l a t o w i t h his e m p h a s i s o n f o r m a n d p a t t e r n , so the m a i n ideas continue, at least i n d i r e c t l y , t o exert their influence. F o r some general accounts of s t r u c t u r a l i s m , the reader is referred to b o o k s by C a w s [C2], D e G e o r g e [D3], G a r d n e r [G2], a n d P i a g e t [P3]. T h e e m p h a s i s is quite different i n each, but together they p r o v i d e a g o o d p i c t u r e of the subject a n d its o r i g i n s . T h e b o o k by C a w s , " S t r u c t u r a l i s m : the A r t of the I n t e l l i g i b l e , " is the most recent a n d is especially g o o d , because, i n a d d i t i o n to i n c l u d i n g a t h o r o u g h b a c k g r o u n d discussion of the m o v e m e n t , it contains an extensive p h i l o s o p h i c a l account of the s u b j e c t . A n o t h e r b o o k , by R o b e r t Scholes o n " S t r u c t u r a l i s m i n L i t e r a t u r e " [S2], deals w i t h one of the m o s t conspicuous, a n d perhaps the most c o m p l e x , areas i n w h i c h s t r u c t u r a l i s m t h r i v e d i n its heyday. A t the same t i m e , despite its i m p o r t a n c e i n the development o f s t r u c t u r a l i s m , l i t e r a t u r e w i l l not play a direct role i n w h a t follows. T h i s is m a i n l y due to the fact t h a t l i t e r a t u r e , as c o m p a r e d t o subjects less removed f r o m the n a t u r a l sciences, does not e x h i b i t very clearly some o f the basic s t r u c t u r e properties t h a t are essential to our p o i n t of v i e w . T h e r e is m u c h v a r i a t i o n i n the degree t o w h i c h s t r u c t u r e s are e x p l i c i t l y recognized i n different fields. T h e y are u s u a l l y rather easy to detect i n m a t h e m a t i c s a n d the n a t u r a l sciences, b u t i n m a n y other fields the dependence o n structures is not so clear. In fact, the p r o b l e m o f e x p o s i n g the role of s t r u c t u r e s i n some o f these fields was the d r i v i n g force b e h i n d the s t r u c t u r a l i s t movement. Q u i t e a p a r t f r o m the s t r u c t u r a l i s t m o v e m e n t , it is o b v i o u s t h a t an u n d e r s t a n d i n g of any b o d y of i n f o r m a t i o n m u s t i n e v i t a b l y involve the u n d e r l y i n g s t r u c t u r e i n some f o r m or other. In other words, a necessary c o n d i t i o n for dealing i n t e l l i g e n t l y w i t h i n f o r m a t i o n is to organize i t i n a w a y t h a t recognizes its essential s t r u c t u r e . T h i s fact is reflected, b o t h i n the t i t l e o f the C a w s b o o k a n d i n its chapter 8, w h i c h is called " S t r u c t u r e as a Necessary and Sufficient C o n d i t i o n o f I n t e l l i g i b i l i t y . " It follows t h a t s t r u c t u r e s are not
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o n l y essential to the u n d e r s t a n d i n g of any subject at whatever level, b u t t h a t anyone w h o has h a d a conscious experience of u n d e r s t a n d i n g s o m e t h i n g of substance w i l l also have h a d a significant experience w i t h s t r u c t u r e . O n the other h a n d , a n awareness o f the s t r u c t u r e s themselves as objects to be u n d e r s t o o d is m u c h less c o m m o n . In fact, because s t r u c t u r e s are so ever present a n d enter so a u t o m a t i c a l l y i n the process o f u n d e r s t a n d i n g , they tend to be neglected, eclipsed by whatever topic happens to be the center o f a t t e n t i o n . T h i s occurs regularly even i n the s t r u c t u r a l i s t l i t e r a t u r e , a n d suggests t h a t m a n y s t r u c t u r a l i s t s , despite a perception of s t r u c t u r e w i t h i n their special fields, s t i l l do not t h i n k of a s t r u c t u r e as an independent e n t i t y . W h i l e naive encounters w i t h structures are n o r m a l l y u n s y s t e m a t i c and q u i t e unconscious, s t r u c t u r a l i s m proper is a d i s c i p l i n e d a p p r o a c h i n w h i c h s t r u c t u r a l a n a l y s i s is used as a t o o l to discover a n d u n d e r s t a n d f u n d a m e n t a l p r i n c i p l e s w i t h i n a subject. A t the same t i m e , a closer look at the w a y notions of s t r u c t u r e enter i n t o even o u r everyday t h i n k i n g , conscious or unconscious, suggests t h a t w h a t a c t u a l l y occurs is neither o b v i o u s nor simple. A s suggested by the above r e m a r k s , the a p p r o a c h t o s t r u c t u r a l i s m i n w h a t follows is p r i m a r i l y t h r o u g h the structures themselves. Therefore, the m a i n emphasis tends t o be o n general structures a n d t h e i r properties. T h e i n v e s t i g a t i o n , however, goes beyond the structures proper t o the way they evolve a n d relate t o other structures. T h e result is a conceptual basis for d e a l i n g more e x p l i c i t l y w i t h the special structures w i t h i n a p a r t i c u l a r s u b j e c t . A prerequisite for a l l of t h i s is a general awareness of s t r u c t u r e , w h i c h must be developed by an e x t r a c t i o n f r o m f a m i l i a r experiences some i d e a o f the n a t u r e a n d properties of structures. A s m i g h t be expected, examples p l a y a c e n t r a l role t h r o u g h o u t . In C h a p t e r I I , a general d e f i n i t i o n of s t r u c t u r e is a b s t r a c t e d f r o m p r o p erties o f a very s i m p l e example, a n d , i n C h a p t e r I I I , some of the general properties o f structures are i l l u s t r a t e d t h r o u g h a variety of carefully chosen examples. These examples, as well as the various topics discussed t h r o u g h out the b o o k , o b v i o u s l y reflect the personal interests ( a n d biases) of one professional m a t h e m a t i c i a n . A n o t h e r person, especially one t r a i n e d i n a different field, w o u l d no d o u b t make very different choices for the same purposes. T h e discussion of s t r u c t u r e s necessarily varies greatly i n d e p t h , r a n g i n g f r o m a relatively precise t r e a t m e n t of a few special topics, i n order to b r i n g out some of the f u n d a m e n t a l properties of structures, t o a c e r t a i n a m o u n t of " b a n d w a v i n g " over structures i n general. S o m e degree o f vagueness and
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o v e r - s i m p l i f i c a t i o n here is a l m o s t u n a v o i d a b l e , because m a n y structures of interest are so e x t r e m e l y c o m p l e x t h a t a detailed t r e a t m e n t is v i r t u a l l y ruled o u t . T h e r e are also s u b s t a n t i a l v a r i a t i o n s i n technical level of the m a t e r i a l u n d e r d i s c u s s i o n . T h o u g h perhaps a b i t u n u s u a l , these are a c t u a l l y quite a p p r o p r i a t e to the s u b j e c t , because they reflect the i m p o r t a n t fact t h a t m a n y o f the properties o f structures are manifest i n a w i d e v a r i e t y of contexts, r a n g i n g f r o m the h i g h l y technical to the c o m m o n p l a c e . F i n a l l y , c o n c e r n i n g the question of j u s t how structures m i g h t be i n v o l v e d i n c e r t a i n m e n t a l processes, such as those i n v o l v e d i n u n d e r s t a n d i n g or c r e a t i v i t y , eve r y t h i n g becomes q u i t e speculative, m a i n l y because there are so few details k n o w n a b o u t the way structures are a c t u a l l y recorded a n d processed i n the brain. A l t h o u g h an independent t r e a t m e n t of s t r u c t u r a l i s m i l l u m i n a t e s m a n y aspects o f the subject, it cannot serve as a "how t o " m a n u a l for a c t u a l a p p l i c a t i o n s . T h e l a t t e r can be very s u b t l e a n d require e x p e r t knowledge of the target field. F o r this reason, no a t t e m p t is m a d e t o offer a significant s t r u c t u r a l analysis of any p a r t i c u l a r subject. I n fact, the m a i n purpose of c o n s i d e r i n g s p e c i a l examples is almost always to b r i n g out c e r t a i n properties of general s t r u c t u r e s rather t h a n to i l l u m i n a t e the e x a m p l e itself, a l t h o u g h the result m a y expose an u n c o n v e n t i o n a l view of the s u b j e c t . T h e r e are also subjects, such as m u s i c , t h a t are replete w i t h s t r u c t u r e b u t are not t o u c h e d u p o n at a l l , m a i n l y because of a personal lack o f the expertise needed to deal adequately w i t h t h e m . A t the same t i m e , m a n y of the features o f s t r u c t u r a l i s m present i n a l m o s t any a p p l i c a t i o n are made e x p l i c i t i n one way or another by our t r e a t m e n t . Despite the general avoidance of technical m a t h e m a t i c s , a n e x a m i n a t i o n of a few genuine m a t h e m a t i c a l i l l u s t r a t i o n s is desirable, especially i n a work where m a t h e m a t i c s p l a y s a definite, i f largely i n d i r e c t , role. T h e m a t e r i a l chosen for t h i s purpose constitutes a r e l a t i v e l y s m a l l p o r t i o n of the w h o l e , and is concentrated i n the last chapter plus three short sections (10, 2 1 , 28) c o n c e r n i n g groups. C h a p t e r V I I , o n " M a t h e m a t i c a l S t r u c t u r e s , " is a n o n t e c h n i c a l c o m m e n t a r y a b o u t m a t h e m a t i c s a n d is definitely not m a t h e m a t i c s proper. Readers w i t h a m i n i m a l knowledge of e l e m e n t a r y m a t h e m a t i c s s h o u l d be able t o e x t r a c t the m a i n ideas out of the m o r e t e c h n i c a l m a t e r i a l w i t h out b e c o m i n g bogged d o w n i n the details. In order to ease the process, an a t t e m p t is m a d e to i n d i c a t e where feasible w h a t the m a i n ideas are a n d to suggest how they are established. O n the other h a n d , some m a y w i s h to scan or even o m i t the f o r m a l details altogether. E v e n m a t h e m a t i c i a n s
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r o u t i n e l y m u s t decide i n a given case j u s t how deeply they need t o delve i n t o t e c h n i c a l m a t e r i a l . Nevertheless, w o r k i n g t h r o u g h such m a t e r i a l m a y deepen ones u n d e r s t a n d i n g of a topic a n d perhaps settle questions t h a t m i g h t otherwise r e m a i n unclear. Despite the i n c l u s i o n of a few i t e m s t h a t some m a y f i n d d i f f i c u l t , I sincerely hope ( a n d also intend) t h a t a l l o f the m a i n ideas w i l l be accessible to every interested reader, w i t h or w i t h o u t benefit o f s p e c i a l m a t h e m a t i c a l s k i l l s . S o m e readers, i n order to o b t a i n a clearer idea o f the character of serious m a t h e m a t i c s , m a y w i s h t o read a n excellent article on the subject b y P a u l H a l m o s [H2]. It is called " M a t h e m a t i c s a s a C r e a t i v e A r t " a n d is q u i t e accessible to the general reader. A n y s t u d y of general structures is b o u n d t o be rather abstract. M o r e o v e r , because it is necessary to deal early w i t h the general concepts, the a b s t r a c t m a t e r i a l a l r e a d y occurs i n p a r t s o f C h a p t e r s II a n d III. A l t h o u g h the s u b ject is a m p l y i l l u s t r a t e d i n a v a r i e t y of concrete e x a m p l e s t h a t s h o u l d be accessible to everyone, the fact r e m a i n s t h a t m a n y w h o have t r o u b l e w i t h m a t h e m a t i c s w i l l consistently (though often needlessly!) shy away f r o m a n y t h i n g a b s t r a c t . O n the other h a n d , this m a t e r i a l is n o n t y p i c a l because the r o a d t o u n d e r s t a n d i n g is not o b s t r u c t e d b y an u n a v o i d a b l e technical b a r r i e r , a n d t h a n k s a g a i n to the u n i v e r s a l s t r u c t u r e experience, anyone w i l l i n g t o m a k e a reasonable effort s h o u l d be able t o u n d e r s t a n d it despite the abstractness. A l t h o u g h there is not a lot of discussion devoted specifically to p h i l o s o p h i c a l questions, it m u s t be a d m i t t e d t h a t a general t r e a t m e n t o f s t r u c t u r e s , m o s t l y because of t h e i r abstract character, does have s o m e t h i n g i n c o m m o n w i t h a t y p i c a l p h i l o s o p h i c a l discussion: N e i t h e r one "bakes any b r e a d . " I n fact, as i l l u m i n a t i n g as a general s t r u c t u r a l p o i n t of view m i g h t be, i t does not b e g i n to suggest the difficult t e c h n i c a l p r o b l e m s dealt w i t h b y e x p e r t s i n a p a r t i c u l a r f i e l d . T h i s is especially true of fields such as m a t h e m a t i c s and the sciences. Y e t , an awareness of structures and some i d e a of the m a n ner i n w h i c h they enter i n t o a subject adds an element o f u n d e r s t a n d i n g t h a t extends w e l l beyond the technicalities. F u r t h e r m o r e , because of the u n i v e r s a l occurrence of structures and the fact t h a t an abstract s t r u c t u r e is essentially independent o f a p a r t i c u l a r r e a l i z a t i o n , structures c a n p r o v i d e a bridge between fields u s u a l l y regarded as unrelated a n d also give a deeper u n d e r s t a n d i n g of their a c t u a l differences. One venture e r y " of danger
of the p i t f a l l s t h a t lies i n the way o f an i n d i v i d u a l w h o dares to o u t s i d e the security of his o w n area of competence, is the " d i s c o v ideas t h a t are already obvious or w e l l - k n o w n to the e x p e r t s . T h e is especially great for m a t h e m a t i c i a n s , w h o , because of the v i v i d -
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ness a n d p u r i t y o f their o w n creative experiences, often i m a g i n e t h a t the G o d s have given t h e m a s p e c i a l glimpse of the T r u t h . A t the same t i m e , the m a t h e m a t i c a l experience not o n l y abounds i n s t r u c t u r e s but is unique i n i t s way, so m a y cast a b i t of new l i g h t even o n certain t h i n g s t h a t are already k n o w n . I take this o p p o r t u n i t y t o t h a n k colleagues, f a m i l y , a n d friends for their advice, c r i t i c i s m s , a n d encouragement. O f the m a n y i n d i v i d u a l s w i t h w h o m I have discussed ideas developed here, I w i s h t o single o u t several of m y present a n d former Y a l e colleagues. W e have first, a n d perhaps most i m p o r t a n t , the late Professors R o b e r t B r u m b a u g h ( P h i l o s o p h y ) a n d G . E v e l y n H u t c h i n s o n ( B i o l o g y ) . N e x t there are Professors S a m u e l E . M a r t i n ( L i n guistics), Robert J . Sternberg (Psychology), and T h o m a s Schatt (Sociolo g y ) . I a m i n d e b t e d t o Professor S c h a t t , w h o is now at the U n i v e r s i t y of P i t t s b u r g , for c a l l i n g m y a t t e n t i o n to the work of A . R . R a d c l i f f e - B r o w n , m e n t i o n e d earlier. I also w a n t to give s p e c i a l t h a n k s t o t w o of m y sons, M a r k ( w h o figures i n the two personal examples i n c l u d e d i n Sections 26 a n d 39) a n d E r i c , b o t h of w h o m read p o r t i o n s of the m a n u s c r i p t at v a r ious stages a n d offered valuable suggestions for i m p r o v e m e n t . C o m m e n t s b y E r i c , w h o is a biologist b y profession, were especially h e l p f u l i n C h a p t e r V I I I . H e , o f course, cannot be held responsible for any of the errors or other defects. F i n a l l y , I w a n t t o express m y a d m i r a t i o n a n d g r a t i t u d e to D o n n a B e l l i for her s p e c i a l skills i n p u t t i n g this m a n u s c r i p t i n t o A M S T e X a n d her great patience i n d e a l i n g w i t h the m a n y changes I p l a g u e d her w i t h d u r i n g the process. T h e m a t e r i a l is o r g a n i z e d i n t o seventy sections of v a r y i n g lengths, w h i c h are g r o u p e d i n t o nine separate chapters. T e x t references to the b i b l i o g r a p h y at the end o f the b o o k are enclosed i n square brackets. Yale University M a y , 1994
CONTENTS v
PREFACE I.
INTRODUCTION 1. T h e S t r u c t u r a l i s t A p p r o a c h 2. T h e S p e c i a l R o l e of M a t h e m a t i c s 3. P l a t o ' s L e c t u r e on T h e G o o d
1 . 7 8
II. G E N E R A L S T R U C T U R E C O N C E P T S 4. 5. 6. 7. 8. 9. 10.
T h e Definition Problem S t r u c t u r a l i s t N o t i o n s of S t r u c t u r e A Simple Example T h e Basic Definitions I s o m o r p h i s m s of S t r u c t u r e s Analogies and Isomorphisms A n Analysis of P o i n t - L i n e Structures
11 11 15 17 -21 23 27
11. S p e c i a l K i n d s of R e l a t i o n s 12. S t r u c t u r a l S t a b i l i t y
29 30
13. S t r u c t u r a l I n f o r m a t i o n 14. O n A b s t r a c t S t r u c t u r e s
33 35
III. S O M E E X A M P L E S O F S T R U C T U R E S 15. 16. 17. 18. 19. 20. 21. 22.
Introduction A t o m s and M a c h i n e s L i n e D r a w i n g s by Josef A l b e r s Configurations T h e Pascal Configuration The Triangle G r o u p G r o u p Structures The Real Number System
39 40 42 44 46 48 50 54
IV. M A N A G E M E N T O F C O M P L E X S T R U C T U R E S 23. T h e A n a l y s i s of S t r u c t u r e s
57
24. 25. 26. 27. 28.
58 58 60 65 71
T h e A p p r o x i m a t i o n of S t r u c t u r e s Axiomatics and Approximation Structural Determinism and Reductionism Contractions C o n t r a c t i o n of G r o u p S t r u c t u r e s xi
CONTENTS
V. L A N G U A G E AND S T R U C T U R E 29. T h e R o l e of L a n g u a g e 30. S i m p l e C o m m u n i c a t i o n
73 75
31. Structural Linguistics 32. S e m i o t i c s
77 82
33. T h e L a n g u a g e F a c u l t y
88
V L S T R U C T U R E S IN M E N T A L P H E N O M E N A 34. 35. 36. 37. 38.
Introduction T h e C e n t r a l R o l e of S t r u c t u r e s T h e D r i v e for I n t e l l i g i b i l i t y Philosophical Questions T h e B a c k g r o u n d S t r u c t u r e and U n d e r s t a n d i n g
39. T e a c h i n g and L e a r n i n g
93 94 97 100 105 108
VTI. M A T H E M A T I C A L S T R U C T U R E S 40. I n t r o d u c t i o n 41. M a t h e m a t i c a l Language 42. H o w to Recognize a M a t h e m a t i c a l S t r u c t u r e
115 116 119
4 3 . Research a n d D e v e l o p m e n t o f M a t h e m a t i c s
120
44. 45. 46. 47. 48.
122 128 131 133 138
T h e R o l e o f Insight i n Research A S t r u c t u r a l I n t e r p r e t a t i o n of C r e a t i v i t y H o w M a t h e m a t i c s is A p p l i e d T h e Effectiveness of M a t h e m a t i c s i n P h y s i c s Other Applications of Mathematics
VIII. B I O L O G I C A L S T R U C T U R E S 49. 50. 51. 52.
Introduction C l a s s i f i c a t i o n of O r g a n i s m s T h e Genetic Structure T h e E n v i r o n m e n t of a S t r u c t u r e
145 146 148 152
53. 54. 55. 56. 57. 58. 59.
T h e E v o l u t i o n a r y Process Complexity in Evolution Multiple Function Biological Catastrophes Determining Structures Convergent E v o l u t i o n Anthropomorphism
153 157 163 168 173 174 175
CONTENTS
IX. S P A C E S T R U C T U R E S A N D
Jtiii
STABILITY
60. 61. 62. 63. 64. 65. 66.
Introduction E u c l i d e a n Spaces S u b s t r u c t u r e s o f E u c l i d e a n Space T h e C o n i c Sections S t a b i l i t y in a F a m i l y of C o n i e s Catastrophe Theory Zeeman's Catastrophe Machine
179 180 181 182 186 188 190
67. 68. 69. 70.
A Mathematical Example A t t a c k or R e t r e a t M e t r i c Spaces S t a b i l i t y of P o i n t - L i n e S t r u c t u r e s
191 196 199 201
BIBLIOGRAPHY
207
INDEX
211
CHAPTER
I
INTRODUCTION
1. T h e S t r u c t u r a l i s t A p p r o a c h For our purposes, " s t r u c t u r a l i s m " m a y be defined f o r m a l l y as a m e t h o d of a n a l y z i n g a b o d y of i n f o r m a t i o n w i t h r e s p e c t t o i t s i n h e r e n t s t r u c t u r e . T h i s d e f i n i t i o n is somewhat more a b b r e v i a t e d (and less specific) t h a n the ones u s u a l l y encountered i n discussions o f s t r u c t u r a l i s m . C o n s i d e r , for e x a m p l e , the f o l l o w i n g statement by H o w a r d G a r d n e r i n his i n f o r m a t i v e b o o k , " T h e Quest for M i n d " , [G2, p. 170]. A m e t h o d or a p p r o a c h rather t h a n a carefully f o r m u l a t e d c a t e c h i s m , s t r u c t u r a l i s m is a n a t t e m p t to discern the arrangements of elements u n d e r l y i n g a g i v e n d o m a i n isolated by an a n a l y s t . T h e s t r u c t u r a l i s t notes v a r i a t i o n s i n these arrangements; he then a t t e m p t s to relate the v a r i a t i o n s b y specifying rules whereby one can be t r a n s f o r m e d to another. T h e first sentence does not differ essentially f r o m the d e f i n i t i o n given above, since an " a r r a n g e m e n t of elements" is j u s t another i n f o r m a l expression for the i d e a o f " s t r u c t u r e " . T h e second sentence refers to the ways in w h i c h the perceived structures change a n d the i n t e r r e l a t i o n s h i p s a m o n g these changes. It is influenced by the L e v i - S t r a u s s definition w h i c h is stated in Section 5. N o m a t t e r h o w a d e f i n i t i o n is f o r m u l a t e d , the s t r u c t u r a l i s t o b jective is t o identify a n d u n d e r s t a n d u n d e r l y i n g structures w i t h i n a g i v e n field of interest, a n d so p r o v i d e a unified a p p r o a c h to a v a r i e t y o f phen o m e n a t h a t otherwise w o u l d be treated more or less i n d e p e n d e n t l y w i t h i n t h e i r special c o n t e x t s . E x a c t l y w h a t a l l o f this means i n a c t u a l practice w i l l become clearer as we proceed. T h e m a n n e r i n w h i c h structures are dealt w i t h m a y change d r a s t i c a l l y as one passes f r o m one field of i n v e s t i g a t i o n to another, as for e x a m p l e f r o m a p h y s i c a l science to one of the social sciences. F u r t h e r m o r e , a s t r u c t u r a l a n a l y s i s i n a g i v e n field m a y take place at several levels r a n g i n g f r o m a d i rect analysis o f the g i v e n i n f o r m a t i o n (perhaps i n v o l v i n g o n l y a s u p e r f i c i a l o r g a n i z a t i o n of the m a t e r i a l ) to the i d e n t i f i c a t i o n of d e e p - l y i n g s t r u c t u r e s w h i c h m a y b e q u i t e abstract a n d not at a l l i n t u i t i v e . T h e p r o b l e m of u n c o v e r i n g n o n t r i v i a l s t r u c t u r e is d o u b l y difficult i n areas where the t r a d i t i o n a l
]
2
STRUCTURALISM
AND STRUCTURES
e m p h a s i s is o n other t h i n g s . In most cases, an i d e n t i f i c a t i o n of genuinely significant s t r u c t u r e w i t h i n a field w i l l require expert knowledge a n d u n d e r s t a n d i n g of t h a t field. A n y o n e w h o t h i n k s seriously a b o u t structures cannot avoid b e i n g i m pressed b y the o v e r w h e l m i n g c o m p l e x i t y of c o m m o n p l a c e s t r u c t u r e s t h a t o r d i n a r y i n d i v i d u a l s r o u t i n e l y process w i t h o u t even b e i n g aware t h a t they are d o i n g so. A l t h o u g h the m i n d is somehow able to manage these s t r u c tures, the c o m p l e x i t y is frequently so great t h a t m o s t of w h a t one m i g h t say c o n c e r n i n g t h e m is b o u n d t o be an o v e r s i m p l i f i c a t i o n of w h a t a c t u a l l y is true. Nevertheless, the g o a l here is to p r o v i d e an a p p r o a c h t o the s u b ject t h a t w i l l help one to deal i n t e l l i g e n t l y w i t h general s t r u c t u r e s , some o f w h i c h m a y be far too c o m p l e x to a d m i t a detailed d e s c r i p t i o n or a n a l y s i s . T h e b r a i n ( h u m a n or otherwise) a u t o m a t i c a l l y s t r u c t u r e s i n some way or other i n f o r m a t i o n c o n c e r n i n g every object t h a t is perceived by i t , a n d the c o r r e s p o n d i n g structures are recorded i n m e m o r y to represent t h a t o b ject. T h e basic s t r u c t u r e p r o b l e m here is the q u e s t i o n of j u s t how s t r u c t u r e s are a c t u a l l y r e c o r d e d a n d p r o c e s s e d i n the b r a i n . T h i s is o b v i o u s l y an exceedingly c o m p l e x p h e n o m o n e n t h a t is s t i l l very p o o r l y u n d e r s t o o d , despite m u c h work done o n related issues. F o r e x a m p l e , neurobiologists a n d psychologists have devoted a great deal of research to the s t u d y of b r a i n a c t i v i t y associated w i t h certain p e r c e p t u a l p h e n o m e n a , m u c h of it i n v o l v i n g v i s i o n [Z2]. T h e last reference, w h i c h emphasizes v i s i o n , is to an a r t i c l e b y S e m i r Z e k i t h a t appeared i n Scientific A m e r i c a n , V o l u m e 267, N u m b e r 3. T h i s was a s p e c i a l issue of the magazine devoted to " m i n d a n d b r a i n " , a n d contains a n u m b e r of other very interesting articles relevant to the general p r o b l e m . Despite their general interest, such c o n t r i b u t i o n s t h r o w l i t t l e l i g h t o n the basic s t r u c t u r e p r o b l e m itself. F u r t h e r m o r e , the enormous c o m p l e x i t y of the b r a i n itself presents a f o r m i d a b l e b a r r i e r t o a n u n d e r s t a n d i n g o f the p r o b l e m . A t t e m p t s to penetrate it . a n g e f r o m s t u d ies o f a c t u a l n e u r a l systems {or c o m p u t e r s i m u l a t i o n s of such), t h a t m a y be observed i n r e l a t i v e l y s i m p l e o r g a n i s m s , to s o p h i s t i c a t e d f o r m a l m a t h e m a t i c a l treatments of c o m p l e x systems presumed t o resemble the b i o l o g i c a l case [S3]. W h a t e v e r the u l t i m a t e e x p l a n a t i o n t u r n s out t o be, it w i l l surely involve a deeper a n d more e x p l i c i t t r e a t m e n t o f general structures t h a n is u s u a l l y f o u n d i n such discussions. Because o f the c e n t r a l role t h a t structures m u s t p l a y i n the m e n t a l p r o cesses o f a l l i n d i v i d u a l s , it is reasonable to assume t h a t an u n d e r s t a n d i n g of s t r u c t u r e s h o u l d be more or less accessible to v i r t u a l l y everyone. T h i s a s s u m p t i o n , i n fact, is i n v o l v e d directly or i n d i r e c t l y i n a large p a r t of eve r y t h i n g t h a t follows. S t r u c t u r e is a n o t i o n of w h i c h every t h i n k i n g person is at least p o t e n t i a l l y aware, a n d a m a j o r o b j e c t i v e o f t h i s work is t o b r i n g out t h a t awareness.
I. I N T R O D U C T I O N
3
T h e thesis t h a t everyone is p o t e n t i a l l y aware of the general n o t i o n o f s t r u c t u r e , is s u p p o r t e d d i r e c t l y by the fact t h a t p e r c e p t i o n at any level is inconceivable w i t h o u t some o r g a n i z a t i o n of m a t e r i a l . It is also s u p p o r t e d b y m a n y specific examples, some of t h e m so c o m m o n p l a c e t h a t t h e i r s i g nificance is easily overlooked. Here we w i l l m e n t i o n o n l y t w o . T h e first concerns the general a b i l i t y to recognize v a r i o u s categories of o r d i n a r y objects. F o r e x a m p l e , a c h i l d q u i c k l y learns t o recognize a l l k i n d s of dogs, i n c l u d i n g breeds t h a t he has never seen before, a n d also t o d i s t i n guish t h e m f r o m other four-legged a n i m a l s . H e is also able t o i d e n t i f y dogs i n pictures or cartoons, a n d even i n crude d r a w i n g s . T h i s is a r e m a r k a b l e feat, a n d one is at a loss t o e x p l a i n e x a c t l y how it is a c c o m p l i s h e d . R e g a r d less of details, however, the process o b v i o u s l y m u s t involve the recognition of a " d o g s t r u c t u r e " c o m m o n to the various d o g e x a m p l e s . T h e mystery r e m a i n s as t o how such structures are perceived, a p r o b l e m t h a t w i l l be touched u p o n a g a i n i n C h a p t e r V I . T h e second e x a m p l e , w h i c h also involves c o m m o n s t r u c t u r e s , concerns the p e r c e p t i o n of r e l a t i o n s h i p s between two or more t h i n g s (or systems, or s i t u a t i o n s ) w h i c h are deemed to be s i m i l a r or t o resemble one another. T h e first q u e s t i o n i n each case is, " W h a t does it m e a n for t w o t h i n g s t o resemble or be s i m i l a r to one a n o t h e r " ? A n obvious answer, w h i c h is general enough to cover a l l cases, is t h a t they possess some " c o m m o n s t r u c t u r e " . T h a t this is a viable answer w i l l become clearer as we proceed. A l t h o u g h one c o u l d m e n t i o n m a n y c o m p l e x a n d subtle examples of such comparisons, we w i l l concentrate o n the case of s i m p l e analogies, w h i c h are o b v i o u s l y based on a perceived s i m i l a r i t y . A n a l o g i e s , as w i t h so m a n y other m e n t a l p h e n o m e n a , have been subjected to considerable s t u d y and analysis by psychologists (see, for e x a m p l e , [9]). It seems to be t y p i c a l , however, t h a t such studies n o r m a l l y do not e x a m i n e the u n d e r l y i n g basic r e c o r d i n g a n d processing o f s t r u c t u r e s , t h a t interest us, b u t concentrate instead o n higher level m e n t a l s t r u c t u r e p h e n o m e n a l o n g s t u d i e d i n psychology. T h e fact a b o u t analogies, t h a t bears o n our thesis, is t h a t they are a regular part of everyday exchanges between o r d i n a r y people. Moreover, they are not o n l y easy to f o r m u l a t e b u t are also i m m e d i a t e l y u n d e r s t o o d by v i r t u a l l y everyone t o w h o m they are presented. In other words, the shared s t r u c t u r e s w i l l u s u a l l y be perceived a l m o s t i n s t a n t l y and w i t h essentially no effort. T h i s fact appears even more s t r i k i n g w h e n one notices t h a t m a n y analogies involve objects t a k e n f r o m entirely different c o n t e x t s , so the c o m m o n s t r u c t u r e is necessarily quite a b s t r a c t . Despite the ease w i t h w h i c h we deal w i t h analogies, the subtle m e n t a l a c t i v i t y i n the process is n e a r l y i m p o s s i b l e t o capture because i t is so r a p i d a n d m u c h of i t is u n c o n scious. A l s o , the exposure a n d description of the c o m m o n s t r u c t u r e is often difficult, p a r t l y because an analysis may u s u a l l y be m a d e i n several ways
STRUCTURALISM AND STRUCTURES
and there does not exist a s t a n d a r d m e t h o d o f d e s c r i p t i o n . A m o d e r a t e l y c o m p l e x e x a m p l e of an analogy w i l l be a n a l y z e d completely i n Section 8 of the next c h a p t e r , after some o f the e l e m e n t a r y ideas a b o u t s t r u c t u r e s are i n t r o d u c e d . T h e o v e r a l l picture w i l l become progressively clearer as our s t u d y o f structures a n d t h e i r properties develops. A p r i m a r y m o t i v e b e h i n d the s t r u c t u r a l i s t m o v e m e n t , at least i n the key fields, e v i d e n t l y was to develop a more scientific a p p r o a c h to the s u b j e c t s i n v o l v e d . Since some o f the m a i n p a r t i c i p a n t s were d i r e c t l y influenced b y science a n d m a t h e m a t i c s , one w o u l d expect considerable i n t e r a c t i o n w i t h scientists. Nevertheless, there appears a c t u a l l y t o have been very l i t t l e gene r a l c o n t a c t between most s t r u c t u r a l i s t s a n d n a t u r a l scientists. I n fact, the m e n t i o n of " s t r u c t u r a l i s m " to the scientist, or a m a t h e m a t i c i a n , u s u a l l y d r a w s a complete b l a n k , followed b y the q u e s t i o n , " W h a t do y o u m e a n b y s t r u c t u r a l i s m ? " T h a t the expected contacts a p p a r e n t l y d i d not occur is p r o b a b l y due, i n a d d i t i o n to the u s u a l p a r o c h i a l i s m , t o the fact t h a t s t r u c t u r e is so r o u t i n e l y a p a r t of science t h a t the p r a c t i t i o n e r s use a s t r u c t u r a l i s t a p p r o a c h w i t h o u t h a v i n g to t h i n k a b o u t i t . M o s t scientists w o u l d p r o b a b l y regard the s t r u c t u r a l i s t m o v e m e n t , i f it c a m e u p , as " m u c h a d o a b o u t the o b v i o u s , " so w o u l d have l i t t l e reason t o consider it seriously. S o m e such d e s c r i p t i o n w o u l d c e r t a i n l y a p p l y to m y o w n first impressions of the subject. T h e r e is another " c u l t u r a l " b a r r i e r t h a t tends to t u r n scientists away f r o m s t r u c t u r a l i s m . It is the s i m p l e fact t h a t so m u c h of the basic m a t e r i a l was w r i t t e n by nonscientists, w h o are prone t o adopt a l i t e r a r y style t h a t exploits the flexibility a n d richness of content of the language. Ideas m a y a c c o r d i n g l y be suggested b y the use of association a n d l i t e r a r y reference a l o n g w i t h the sounds and c o n n o t a t i o n s , as w e l l as the u s u a l m e a n i n g s of words. T h e result contrasts w i t h the more f o r m a l (and sometimes r a t h e r pedestrian) s t y l e n o r m a l l y adopted b y m a t h e m a t i c i a n s a n d scientists, even when d e a l i n g w i t h n o n t e c h n i c a l subjects. T h i s is not to say, o f course, t h a t a n i d e a developed i n the l i t e r a r y style necessarily lacks p r e c i s i o n , t h o u g h t o e x t r a c t t h a t p r e c i s i o n f r o m the ambient verbiage is sometimes rather difficult. D e s p i t e such differences, the fact remains t h a t s t r u c t u r a l i s m does represent a l e g i t i m a t e a t t e m p t t o a p p l y scientific m e t h o d to c e r t a i n fields t h a t are u s u a l l y regarded (at least b y m a n y scientists) as nonscientific i n character. M a n y of the t r a d i t i o n a l a t t e m p t s to i m i t a t e the scientific m e t h o d are based o n the n o t i o n , advanced for e x a m p l e by Descarte and K a n t , t h a t the c r i t e r i o n of t r u e science lies i n its r e l a t i o n t o m a t h e m a t i c s . Therefore, the u l t i m a t e g o a l is often to involve i n one way or another some m a t h e m a t ics, the ideal m o d e l b e i n g physics. F u r t h e r m o r e , a c o m m o n i n t e r p r e t a t i o n of this p o i n t of v i e w is t h a t a true science m u s t first of a l l be based o n n u m e r i c a l measurements. O n e response t o this i n t e r p r e t a t i o n has been
I. I N T R O D U C T I O N
5
the extensive use of statistics i n the a n a l y s i s and presentation of results. A l t h o u g h statistics is a n i m p o r t a n t a n d useful t o o l i n d e a l i n g w i t h large a m o u n t s of c e r t a i n k i n d s of n u m e r i c a l d a t a a n d m a y help i n the i d e n t i fication of s t r u c t u r e , i t is not a s u b s t i t u t e for the a c t u a l i n t r o d u c t i o n of m a t h e m a t i c s . I n any case, a subject does not become m a t h e m a t i c a l , a n d hence more " s c i e n t i f i c " , s i m p l y t h r o u g h the measurement of certain of its parameters. T h e c h a r a c t e r i z a t i o n of key properties of a s y s t e m i n t e r m s o f the values of a few parameters is o b v i o u s l y very i m p o r t a n t whenever it is possible. Nevertheless, the emphasis o n n u m b e r a n d measurement has a tendency to d i s t r a c t a t t e n t i o n f r o m a more f u n d a m e n t a l m a t t e r : the parameters t h e m selves a n d their i n t e r r e l a t i o n s h i p s . In other words, i t is the s t r u c t u r e of the set of parameters t h a t is i m p o r t a n t . It is here t h a t m a t h e m a t i c a l s t r u c t u r e m a y i n some cases be i n t r o d u c e d . T h e n u m e r i c a l values o f the p a r a m e t e r s , however i m p o r t a n t they m i g h t be i n specific instances, do not represent the essence of the subject. T h e preceding r e m a r k s i n d i c a t e why a serious s t r u c t u r a l i s t a p p r o a c h to any subject has s o m e t h i n g i n c o m m o n w i t h a general scientific a p p r o a c h . A first goal for b o t h is to search out and expose the essential s t r u c t u r e (or structures) i m p l i c i t i n the g i v e n subject i n f o r m a t i o n . T h i s is genuinely scientific i n s p i r i t even when the structures o b t a i n e d are not m a t h e m a t i c a l i n character. If the exposed structures are indeed essential, they w i l l p r o vide a basis for o r g a n i z i n g a n d u n d e r s t a n d i n g properties of the s u b j e c t a n d perhaps also suggest (or predict) new properties as w e l l . T h e latter role, i n c i d e n t a l l y , is often regarded as a n essential feature o f a science. In special cases, such as i n m u c h of physics, the structures w i l l a d m i t a m a t h e m a t i c a l d e s c r i p t i o n a n d some of their c r u c i a l properties may be expressible i n n u m e r i c a l terms. O n the other h a n d , there are m a n y m a t h e m a t i c a l s t r u c t u r e s t h a t do not depend o n n u m e r i c a l measurements (groups, for e x a m p l e ) , but are no less m a t h e m a t i c a l because of this fact. W h e t h e r or not n u m b e r s are i n v o l v e d , the power of the m a t h e m a t i c s is t h a t its f o r m a l i s m provides a t o o l for m a n i p u l a t i n g the s t r u c t u r e , e n a b l i n g one, for e x a m p l e , t o m a k e precise predictions c o n c e r n i n g the subject i n q u e s t i o n . A m o r e t h o r o u g h description of how m a t h e m a t i c s is a p p l i e d w i l l be found i n Sections 46-48. A l t h o u g h the ideal m o d e l for s t r u c t u r a l i s m m i g h t be the a p p l i c a t i o n of m a t h e m a t i c s , the h a r d fact is t h a t m a t h e m a t i c a l s t r u c t u r e s a p p r o p r i a t e to m a n y fields s i m p l y do not exist. F o r t u n a t e l y , an independent s t r u c t u r a l a n a l y s i s of languages was already i n an advanced stage of development w h e n the m o d e r n s t r u c t u r a l i s t movement arose. A t the same t i m e , it a p pears t h a t a l l social p h e n o m e n a m a y to some degree be s t r u c t u r e d like a language (Section 31). T h i s , a l o n g w i t h the fact t h a t l i n g u i s t i c s t r u c tures are generally more accessible t h a n m a t h e m a t i c a l ones, e x p l a i n s why
6
STRUCTURALISM
AND STRUCTURES
s t r u c t u r a l l i n g u i s t i c s has h a d a m o r e direct influence o n the development of s t r u c t u r a l i s m t h a n has m a t h e m a t i c s . Some s t r u c t u r a l i s t s , for e x a m p l e L e v i - S t r a u s s [L6] a n d the French psychoanalyst Jacques L a c a n (see [ D 3 , C h . 3], [ L I ] , a n d [L3]), place great e m p h a s i s u p o n l i n g u i s t i c s . P i a g e t , however, tended to m i n i m i z e its i m p o r t a n c e i n his work [P3], preferring to emphasize mathematics instead. L i n g u i s t i c structures are i n v o l v e d directly and i n d i r e c t l y w i t h s t r u c t u r a l i s m i n a variety of ways. T h i s r e l a t i o n s h i p is b o t h i n t e r e s t i n g a n d i n s t r u c t i v e , a n d w i l l be m u c h easier to u n d e r s t a n d after a f o r m a l d e f i n i t i o n of s t r u c t u r e a n d some f u n d a m e n t a l s of the theory of general structures have been developed i n the next several chapters. It w i l l be dealt w i t h i n C h a p t e r V , w h i c h is concerned w i t h the way language enters i n t o the m a n a g e m e n t a n d c o m m u n i c a t i o n of s t r u c t u r e s , a n d w i t h c e r t a i n aspects o f language s t r u c t u r e itself. A l t h o u g h structures are necessarily involved i n a n y t h i n g concerned w i t h i n t e l l i g i b i l i t y , a n a c t u a l i d e n t i f i c a t i o n and d e s c r i p t i o n of the structures t h e m selves m a y be difficult to o b t a i n . A molecule, or a l i v i n g o r g a n i s m , or a k i n s h i p s y s t e m does not e x h i b i t i n any obvious way its c h a r a c t e r i s t i c s t r u c t u r e . T h e s i t u a t i o n is further c o m p l i c a t e d by the fact t h a t an o b j e c t often m a y be a n a l y z e d i n m o r e t h a n one way w i t h respect t o s t r u c t u r e . These p r o b l e m s m a y arise even i n a science, where s t r u c t u r e s tend to lie rather close to the surface, a n d they are m u c h more prevalent i n certain other fields, where a search for u n d e r l y i n g s t r u c t u r e m a y be u n c o n v e n t i o n a l . D e s p i t e a l l of t h i s , a s t r u c t u r a l approach is so characteristic of t r a d i t i o n a l science t h a t s t r u c t u r a l i s m , t h o u g h perhaps m i s u n d e r s t o o d a n d sometimes m i s u s e d , represents a l e g i t i m a t e a t t e m p t to i n t r o d u c e scientific m e t h o d s i n t o nonscientific fields. A t the same t i m e , a n a t u r a l science, because o f its special r e l a t i o n s h i p t o the real w o r l d t h r o u g h e x p e r i m e n t a n d p r e d i c t i o n , o b v i o u s l y involves m o r e t h a n j u s t the i d e n t i f i c a t i o n of s t r u c t u r e . T h e most basic question t h a t must be faced i n d e a l i n g w i t h s t r u c t u r e s is the obvious one, " W h a t a c t u a l l y is a s t r u c t u r e ! " A l t h o u g h a s i m p l e w o r k i n g definition is offered i n Section 7 of the next chapter, the concept itself covers such a wide variety o f objects t h a t m u c h discussion a n d analysis of examples is needed t o expose a reasonably adequate idea of w h a t is i n v o l v e d . I n fact, the next three chapters m a y be regarded as an extended answer to the a b o v e question. A s is p o i n t e d out i n the next s e c t i o n , m a t h e m a t i c s occupies a u n i q u e l y c e n t r a ! p o s i t i o n a m o n g a l l other fields w i t h respect to the s t u d y of s t r u c t u r e . Therefore, m u c h of w h a t we have to say is based ( d i r e c t l y or i n d i r e c t l y ) o n various s t r u c t u r e notions f r o m m a t h e m a t i c s . A l t h o u g h m a t h e m a t i c s is a key source for structures and t h e i r properties, it is o b v i o u s l y not the o n l y one. I n fact, we w i l l have occasion t o e x a m i n e i n some d e t a i l s t r u c t u r e s as they o c c u r i n other fields, especially i n l i n g u i s t i c s
I. I N T R O D U C T I O N
7
a n d biology. In a l l cases, however, the purpose is s t r i c t l y t o expose c e r t a i n general s t r u c t u r e ideas a n d is not to give a s t r u c t u r a l analysis of the field itself. A t the same t i m e , a c o n c e n t r a t i o n o n structures sometimes h i g h l i g h t s c e r t a i n features o f a s u b j e c t t h a t are not u s u a l l y e m p h a s i z e d . 2. T h e S p e c i a l R o l e o f
Mathematics
T h e most o b v i o u s feature of m a t h e m a t i c s , to n o n m a t h e m a t i c i a n s , is the general use o f a f o r m a l "language" of s y m b o l s . C o n s e q u e n t l y , more often t h a n n o t , the casual observer w i l l identify m a t h e m a t i c s w i t h its f o r m a l i s m . T h e r e are also a few m a t h e m a t i c i a n s and logicians w h o , for very t e c h n i c a l p h i l o s o p h i c a l reasons, make the same i d e n t i f i c a t i o n . T h e s e are the f o r m a l ists. But most m a t h e m a t i c i a n s are not f o r m a l i s t s a n d regard m a t h e m a t i c s as h a v i n g a content independent o f the language. T h i s is also the p o i n t o f view i n a l l t h a t follows. It is a c c o r d i n g l y assumed t h a t , a l t h o u g h a s p e c i a l language (or l a n guages) does indeed p l a y a v i t a l role i n m a t h e m a t i c s , the a c t u a l content of m a t h e m a t i c s consists of special structures representing m a t h e m a t i c a l concepts. T h i s is w h a t sets m a t h e m a t i c s a p a r t f r o m other fields of s t u d y . In most areas, the p r o b l e m is first to i d e n t i f y u n d e r l y i n g s t r u c t u r a l p r o p erties of the g i v e n subject m a t t e r , w h i c h is then s t u d i e d i n the l i g h t of these s t r u c t u r e s , w h i l e i n m a t h e m a t i c s the subject m a t t e r already consists of s t r u c t u r e s . F r o m t h i s p o i n t of v i e w , the f o r m a l i s m is j u s t an e x t r e m e l y efficient language for representing a n d m a n i p u l a t i n g m a t h e m a t i c a l s t r u c tures. T h e s p e c i a l character o f these s t r u c t u r e s , d e t e r m i n e d i n p a r t by their s u s c e p t i b i l i t y t o f o r m a l t r e a t m e n t , w i l l be discussed i n C h a p t e r V I I . Since the s t r u c t u r e s t h a t constitute the subject m a t t e r of m a t h e m a t i c s occur i n a r e l a t i v e l y pure f o r m , unencumbered b y extraneous i n f o r m a t i o n , they m a y be s t u d i e d a n d u n d e r s t o o d (as structures!) to a degree difficult t o a t t a i n i n other fields. F u r t h e r m o r e , m a t h e m a t i c a l s t r u c t u r e s , despite t h e i r special character, o c c u r w i t h great variety a n d c o m p l e x i t y , e x h i b i t i n g m a n y i m p o r t a n t properties c o m m o n to a l l s t r u c t u r e s . It is for these reasons t h a t m a t h e m a t i c a l structures p r o v i d e an especially good a p p r o a c h to the s t u d y of general s t r u c t u r e s a n d their properties. T h e i n v o l v e m e n t of m a t h e m a t i c s w i t h other fields, i m p l i e d here, is different f r o m the u s u a l a p p l i c a t i o n s of m a t h e m a t i c s , as for e x a m p l e i n physics. T h e l a t t e r depend on the i d e n t i f i c a t i o n of a p o r t i o n of the target subject as h a v i n g s t r u c t u r e s i m i l a r to a k n o w n m a t h e m a t i c a l s t r u c t u r e , so t h a t i t m a y a c c o r d i n g l y be s t u d i e d using m a t h e m a t i c a l techniques. (See S e c t i o n 47.) Here, o n the other h a n d , the idea is to extract or generalize f r o m m a t h e m a t i c a l structures c e r t a i n characteristics t h a t w i l l c a r r y over t o , a n d t h u s help to u n d e r s t a n d , structures of a l l k i n d s . A s suggested e a r l i e r , a s i m i l a r t h o u g h less specific g o a l is i m p l i c i t i n o u r consideration o f other subjects
8
STRUCTURALISM
AND STRUCTURES
as w e l l . S o m e o f the f o l l o w i n g discussion of s t r u c t u r e s is influenced by m a t h e m a t i c a l m a t e r i a l w h i c h is p r o b a b l y not f a m i l i a r to m a n y readers w h o m i g h t be interested i n the subject. A s a rule, however, such m a t e r i a l of a nonelem e n t a r y character is presented i n f o r m a l l y or enters only i n d i r e c t l y t h r o u g h the a u t h o r ' s o w n experience as a m a t h e m a t i c i a n , so s h o u l d not cause i n s u r m o u n t a b l e difficulties. M o s t readers w i l l be able, i f necessary, t o d r a w f r o m a l t e r n a t e sources most of the knowledge a n d experience of s t r u c t u r e s required to follow the discussion. W e a c c o r d i n g l y believe t h a t a perceptive reader, despite m a t h e m a t i c a l deficiencies, w i l l w i n d u p w i t h a m u c h better i d e a , not o n l y of s t r u c t u r e , but also of the nature of m a t h e m a t i c s a n d the way it develops. T h e i d e a t h a t a knowledge of m a t h e m a t i c s m a y f a c i l i t a t e one's unders t a n d i n g of another q u i t e different subject is very o l d , g o i n g back at least to P l a t o . F u r t h e r m o r e , the connection also t u r n s out to be t h r o u g h s t r u c t u r e ! 3. P l a t o ' s L e c t u r e o n T h e
Good
P l a t o is reported to have delivered i n A t h e n s a lecture (or lectures) o n " T h e N o t i o n of T h e G o o d " . A r i s t o t l e , w h o a t t e n d e d the lecture, discussed it later i n his w r i t i n g s on the same subject. U n f o r t u n a t e l y , this p o r t i o n of A r i s t o t l e ' s work has not s u r v i v e d , so the report o n P l a t o ' s lecture is second h a n d t h r o u g h A r i s t o t l e ' s o w n students. Nevertheless, it seems t o be generally agreed t h a t P l a t o devoted most of the lecture to a discussion of m a t h e m a t i c s , a n d a p p a r e n t l y took the p o s i t i o n t h a t the n a t u r e o f T h e G o o d c o u l d be u n d e r s t o o d t h r o u g h m a t h e m a t i c s . T h i s unexpected thesis caused a great deal of confusion, a n d over the years has given rise t o m u c h controversy a m o n g philosophers as to w h a t P l a t o a c t u a l l y m e a n t . S o m e even went so far as to conjecture t h a t A r i s t o t l e ' s account of the lecture was incorrect. A l f r e d N o r t h W h i t e h e a d , one of those philosophers w h o accepted the r e p o r t e d content of P l a t o ' s lecture, discussed the question i n an article entitled " M a t h e m a t i c s a n d T h e G o o d " , where he makes the following c o m ments c o n c e r n i n g the famous lecture [ W 4 , p. 75]; B u t u n d o u b t e d l y his lecture was a failure; for he d i d not succeed i n m a k i n g evident to future generations his i n t u i t i o n of m a t h e m a t i c s as e l u c i d a t i n g T h e G o o d . M a n y m a t h e m a t i c i a n s have been g o o d men for e x a m p l e , P a s c a l a n d N e w t o n . A l s o m a n y philosophers have been m a t h e m a t i c i a n s . B u t the peculiar associations of m a t h e m a t i c s a n d T h e G o o d r e m a i n s an undeveloped t o p i c , since its first i n t r o d u c t i o n by P l a t o . T h e r e have been researches i n t o the topic conceived as an interesting characteristic of P l a t o ' s m i n d . B u t the d o c t r i n e conceived as a basic t r u t h of philosophy, faded f r o m active thought after the first
I. I N T R O D U C T I O N
9
i m m e d i a t e P l a t o n i c epoch. T h r o u g h o u t the various ages of E u r o p e a n c i v i l i z a t i o n , m o r a l p h i l o s o p h y and m a t h e m a t i c s have been assigned to separate departments of u n i v e r s i t y life. W h i t e h e a d goes on to p o i n t out t h a t it is possible, i n the l i g h t of our m o d e r n knowledge, to clarify "ideas w h i c h P l a t o could o n l y express w i t h obscure sentences a n d m i s l e a d i n g m y t h s " . T h e m a i n topic i n the a r t i c l e is "the c o n n e c t i o n between m o d e r n m a t h e m a t i c s a n d the n o t i o n of T h e G o o d " , and he u l t i m a t e l y makes the point t h a t " m a t h e m a t i c s is now b e i n g t r a n s f o r m e d i n t o the i n t e l l e c t u a l analysis of types of p a t t e r n " . (Note t h a t we w o u l d s u b s t i t u t e " s t r u c t u r e " for the w o r d " p a t t e r n " t h r o u g h o u t these remarks.) T h e r e follows W h i t e h e a d ' s c l a r i f i c a t i o n of P l a t o ' s association of mathematics with The G o o d : T h e n o t i o n of the i m p o r t a n c e of p a t t e r n is as o l d as c i v i l i z a t i o n . E v ery art is founded on the study of p a t t e r n . A l s o the cohesion of s o c i a l systems depends on the maintenance of patterns of b e h a v i o r ; a n d a d vances i n c i v i l i z a t i o n depend on the fortunate m o d i f i c a t i o n of such b e h a v i o r patterns. T h u s the infusion of p a t t e r n i n t o n a t u r a l occurrences, and the s t a b i l i t y of such patterns, and the m o d i f i c a t i o n of such p a t t e r n s , is the necessary c o n d i t i o n for the r e a l i z a t i o n of T h e G o o d . M a t h e m a t i c s is the most powerful technique for the u n d e r s t a n d i n g of p a t t e r n , a n d for the a n a l y s i s of the r e l a t i o n s h i p s of patterns. Here we reach the f u n d a m e n t a l j u s t i f i c a t i o n for the topic of P l a t o ' s lecture. H a v i n g regard to the i m m e n s i t y of its s u b j e c t - m a t t e r m a t h e m a t i c s , even m o d e r n m a t h e m a t i c s , is a science i n its b a b y h o o d . If c i v i l i z a t i o n continues to advance, i n the next two t h o u s a n d years the o v e r w h e l m ing novelty i n h u m a n thought w i l l be the d o m i n a n c e of m a t h e m a t i c a l understanding. T h e essence of t h i s generalized m a t h e m a t i c s is the s t u d y of the most observable examples of the relevant p a t t e r n s ; and a p p l i e d m a t h ematics is the transference of this s t u d y to other e x a m p l e s of the r e a l i z a t i o n of these p a t t e r n s . [ W 4 , pp. 83,84] In these comments, W h i t e h e a d observes the u n i v e r s a l occurrence of p a t terns, or s t r u c t u r e s , and identifies t h e m as the n a t u r a l d o m a i n of m a t h ematics, thereby c h a l l e n g i n g future m a t h e m a t i c i a n s w i t h the t r u l y enorm o u s task of g i v i n g a m a t h e m a t i c a l t r e a t m e n t of structures as they arise i n m a n y different areas. W h e t h e r or not this w i l l h a p p e n m a y d e p e n d o n how " s t r u c t u r e " is a c t u a l l y defined a n d on the i n t e r p r e t a t i o n of " m a t h e m a t i c a l t r e a t m e n t " . If the o b j e c t i v e is to involve m a t h e m a t i c s i n a s u b s t a n t i a l way, as opposed to a use of m a t h e m a t i c a l language i n a purely d e s c r i p t i v e role, then it is difficult to v i s u a l i z e how the goal m i g h t be a t t a i n e d , at least w i t h out rather severe restrictions on the a d m i t t e d s t r u c t u r e s , or some presently
10
unpredictable t a n t question w i l l be t a k e n the n a t u r e of
STRUCTURALISM
AND STRUCTURES
developments i n m a t h e m a t i c s . T h i s raises a g a i n the i m p o r of w h a t it means to a p p l y m a t h e m a t i c s t o other fields, w h i c h up i n C h a p t e r V I I f o l l o w i n g a more careful e x a m i n a t i o n of m a t h e m a t i c a l structures a n d how they are dealt w i t h .
CHAPTER
GENERAL
II
STRUCTURE
CONCEPTS
4. T h e D e f i n i t i o n P r o b l e m A l t h o u g h c e r t a i n special k i n d s of structures are reasonably m a n a g e a b l e , it is difficult t o p i n d o w n the general n o t i o n because i t appears i n so m a n y guises a n d contexts. T h i s is already i n d i c a t e d by the variety of words t h a t are c o m m o n l y used t o suggest s t r u c t u r e . These i n c l u d e , for e x a m ple, " c o m p l e x " , " c o n s t r u c t i o n " , " f i g u r e " , " f o r m " , " f r a m e w o r k " , " m o d e l " , " o r g a n i s m " , " p a t t e r n " , " p l a n " , " s y s t e m " , a n d m a n y more. O n e of the p r o b l e m s i n d e a l i n g w i t h a concept as general a n d i n c l u s i v e as t h a t of a s t r u c t u r e is t h a t no single e x a m p l e can suggest more t h a n a fragment of the f u l l concept, so any g o o d e x a m p l e is i n danger of b e i n g perceived as more representative t h a n i t possibly c a n be. T h e r e is a c c o r d i n g l y not m u c h hope for s t a t i n g i n a few lines a precise and complete d e f i n i t i o n of s t r u c t u r e . O n the other h a n d , there is a n a l t e r n a t i v e a p p r o a c h to p r o b l e m s o f this k i n d , more c o m m o n i n the h u m a n i t i e s t h a n i n the sciences, t h a t emphasizes a n extended discussion of the subject rather t h a n a f o r m a l t r e a t m e n t . I n the present case, i t involves the f o r m u l a t i o n of an a d m i t t e d l y imprecise a p p r o x i m a t e d e f i n i t i o n , w h i c h is then e l a b o r a t e d a n d made i n c r e a s i n g l y more complete t h r o u g h e x a m p l e s a n d e x p l a n a t i o n s . A t the same t i m e , the concept suggested by the d e f i n i t i o n , perhaps rather vague a n d l i m i t e d at the outset, becomes progressively sharper and m o r e i n c l u s i v e as the discussion proceeds. Before f o r m u l a t i n g our s t a r t i n g d e f i n i t i o n for " s t r u c t u r e " i n Sect i o n 7, we consider some definitions i n the next section t h a t have appeared i n the s t r u c t u r a l i s t l i t e r a t u r e , and e x a m i n e carefully i n Section 6 a s i m p l e object t h a t everyone w i l l no doubt accept as an e x a m p l e of a s t r u c t u r e . 5. S t r u c t u r a l i s t N o t i o n s o f S t r u c t u r e S t r u c t u r a l i s t w r i t i n g s n a t u r a l l y contain n u m e r o u s references to s t r u c tures b u t s e l d o m deal e x p l i c i t l y , let alone s y s t e m a t i c a l l y , w i t h the n o t i o n of s t r u c t u r e itself. E v e n when a d e f i n i t i o n of s t r u c t u r e is offered, i t tends t o be t a i l o r e d t o the subject being s t u d i e d a n d often p l a y s o n l y an i n d i r e c t role i n the w o r k . F o u r representative definitions are reviewed below. T h e w i d e differences a m o n g the definitions emphasize further the very b r o a d character of the s t r u c t u r e concept. 11
12
STRUCTURALISM
AND STRUCTURES
W e begin w i t h a definition by A . R . R a d c l i f f e - B r o w n , one a m o n g several a n t h r o p o l o g i s t s whose use of s t r u c t u r e ideas a n t i c i p a t e d w h a t is u s u a l l y regarded as the s t r u c t u r a l i s t m o v e m e n t . T h e f o l l o w i n g d e f i n i t i o n appears i n the i n t r o d u c t i o n to his b o o k , " S t r u c t u r e a n d F u n c t i o n i n P r i m i t i v e S o c i e t y " [ R l ] , w h i c h is a collection of essays and lectures. W h e n we use the t e r m s t r u c t u r e we are referring to some sort o f ordered arrangement of parts or components. A m u s i c a l c o m p o s i t i o n has a s t r u c t u r e , a n d so does a sentence. A b u i l d i n g has a s t r u c t u r e , so does a molecule or an a n i m a l . T h e c o m p o n e n t s or u n i t s of social s t r u c t u r e are persons, and a person is a h u m a n b e i n g considered not as an o r g a n i s m b u t as o c c u p y i n g p o s i t i o n i n a s o c i a l s t r u c t u r e , [p. 9] T h e purpose of t h i s d e f i n i t i o n was to help c l a r i f y some of the a u t h o r ' s ideas o u t l i n e d earlier i n a presidential address delivered to the R o y a l A n t h r o p o l o g i c a l I n s t i t u t e . It is not only by far the simplest of the four, but is closest to our general d e f i n i t i o n given i n Section 7. T h e address, " O n S o c i a l S t r u c t u r e " , first p u b l i s h e d i n 1940, is C h a p t e r X of his b o o k . It exp l a i n s the a u t h o r ' s v i e w of social a n t h r o p o l o g y "as the t h e o r e t i c a l n a t u r a l science of h u m a n society, t h a t is, the i n v e s t i g a t i o n of s o c i a l p h e n o m e n a by m e t h o d s essentially s i m i l a r to those used i n the p h y s i c a l a n d b i o l o g i c a l s c i ences" , and is not o n l y a c o n v i n c i n g defense of the a u t h o r ' s p o s i t i o n b u t also a r e m a r k a b l y clear statement of w h a t s t r u c t u r a l i s m is a l l a b o u t . T h e other three definitions are m u c h less clear as t o j u s t w h a t the a u t h o r s h a d in m i n d . T h e next d e f i n i t i o n , by L e v i - S t r a u s s , is quoted f r o m his book on " S t r u c t u r a l A n t h r o p o l o g y " [L6, p. 279], It was offered i n the course of a discussion of "social s t r u c t u r e s " as an answer to the q u e s t i o n , " W h a t k i n d of m o d e l deserves the name ' s t r u c t u r e ' ? " He also points out t h a t " T h i s is not an a n t h r o p o l o g i c a l question, b u t one w h i c h belongs to the m e t h o d o l o g y of science i n g e n e r a l " . . . . a s t r u c t u r e consists of a m o d e l meeting w i t h several requirements. F i r s t , the structure e x h i b i t s the characteristics of a s y s t e m . It is m a d e u p of several elements, none of w h i c h can undergo a change w i t h o u t effecting changes i n a l l of the other elements. Second, for any given m o d e l there should be a p o s s i b i l i t y of o r d e r i n g a series of transform a t i o n s r e s u l t i n g i n a group of models of the same type. T h i r d , the above properties make it possible to predict how the m o d e l w i l l react if one or more of its elements are s u b m i t t e d to certain m o d i f i c a t i o n s . F i n a l l y , the models should be c o n s t i t u t e d so as to m a k e i m m e d i a t e l y intelligible a l l of the observed facts. T h e t h i r d d e f i n i t i o n is due to P i a g e t . It is quoted f r o m his b o o k , " S t r u c t u r a l i s m " [P3,p.5], a n d is o b v i o u s l y colored by his intent to confine a t t e n t i o n
II. G E N E R A L
STRUCTURE CONCEPTS
to "the k i n d s o f structures t h a t are to be met i n m a t h e m a t i c s a n d the several e m p i r i c a l sciences". T h i s r e s t r i c t i o n does not m e a n , of course, t h a t his a t t e n t i o n was confined to these subjects, since a m a j o r o b j e c t i v e was to identify such structures i n other areas. A s a first a p p r o x i m a t i o n , we m a y say t h a t a s t r u c t u r e is a s y s t e m of t r a n s f o r m a t i o n s . In as m u c h as i t is a s y s t e m a n d not a mere collect i o n of elements a n d their properties, these t r a n s f o r m a t i o n s involve laws: the s t r u c t u r e is preserved or enriched by the i n t e r p l a y of its t r a n s f o r m a t i o n laws, w h i c h never y i e l d results e x t e r n a l t o the s y s t e m nor e m p l o y elements t h a t are e x t e r n a l to i t . In short, the n o t i o n of s t r u c t u r e is comprised of three key ideas: the i d e a of wholeness, the idea of t r a n s f o r m a t i o n , a n d the idea of self-regulation. T h e f o u r t h d e f i n i t i o n , w h i c h is m u c h more recent t h a n the others, is by Peter C a w s , and is t a k e n f r o m his b o o k o n " S t r u c t u r a l i s m " [C2, p p . 1 2 , 13]. H i s a p p r o a c h to s t r u c t u r a l i s m is different i n t h a t it includes considerable discussion of related p h i l o s o p h i c a l questions. W e w i l l r e t u r n to some of these m a t t e r s i n Section 14 at the end of this chapter a n d i n Section 37 i n C h a p t e r V I . H i s d e f i n i t i o n of " s t r u c t u r e " also depends on a p r e l i m i n a r y n o t i o n of a " s y s t e m " . B y a s y s t e m I s h a l l u n d e r s t a n d a set of entities (called the e l e m e n t s of the system) m u t u a l l y related i n such a way t h a t the state of each element determines a n d / o r is d e t e r m i n e d b y the state of some other element or elements, and every element is connected to every other by a c h a i n of such d e t e r m i n a t i o n s , t h a t is, the s y s t e m has no isolated elements [p. 12]. B y a s t r u c t u r e , finally, I s h a l l u n d e r s t a n d a set of r e l a t i o n s entities t h a t f o r m the elements of a s y s t e m ; the s t r u c t u r e w i l l to be c o n c r e t e i f the relations are a c t u a l l y e m b o d i e d i n some a b s t r a c t i f they are m e r e l y specified b u t not so e m b o d i e d [p.
among be s a i d system, 13].
A l t h o u g h C a w s identifies a s t r u c t u r e w i t h a set of r e l a t i o n s , his d e f i n i t i o n o f a concrete s t r u c t u r e also suggests the one we give i n Section 7. A s already suggested, it is not feasible to require a great deal o f precision in any reasonably general d e f i n i t i o n of s t r u c t u r e . Nevertheless, w i t h o u t considerable a d d i t i o n a l subject i n f o r m a t i o n , it is difficult t o f o r m w i t h m u c h confidence a very clear n o t i o n of w h a t is b e i n g specified i n any o f the last three definitions, let alone to correlate t h e m . T h e p r o b l e m is t h a t the defi n i t i o n s e v i d e n t l y were abstracted f r o m rather specific e x a m p l e s t h a t the a u t h o r s h a d i n m i n d . T h i s is, i n fact, a c o m m o n a p p r o a c h t o a b s t r a c t i o n . It consists i n t a k i n g a description of a " t y p i c a l " concrete e x a m p l e a n d s y s t e m a t i c a l l y suppressing the concreteness by s u b s t i t u t i n g general t e r m i n o l o g y for the concrete. T h e idea seems to be t h a t the " a b s t r a c t " f o r m u l a t i o n so
14
STRUCTURALISM
AND STRUCTURES
o b t a i n e d w i l l c a p t u r e the "essence"of the s y s t e m . T h e a p p r o a c h m a y work, b u t i t is often difficult to see w h a t is intended w i t h o u t considerable k n o w l edge of the o r i g i n a l concrete o b j e c t . In other words, the desired abstract concept fails t o a t t a i n an independent existence. T h i s is not the place to a t t e m p t a detailed analysis of the s p e c i a l features of the above definitions, since t h a t w o u l d require a review of the subject m a t t e r w i t h w h i c h the a u t h o r s are concerned, a task t h a t w o u l d be a digression for us. T h e r e f o r e , the f o l l o w i n g r e m a r k s , directed o n l y to the L e v i - S t r a u s s and P i a g e t definitions, are restricted t o a few of the i m m e d i ately relevant features. Despite their obvious differences, the two definitions do involve some c o m m o n ideas. In the first place, each requires a s t r u c t u r e to be a s y s t e m . S i n c e , by c o m m o n usage, the word " s y s t e m " is almost s y n o n y m o u s w i t h " s t r u c t u r e " (though the former is perhaps somewhat more i n c l u s i v e ) , i t follows t h a t the definitions are intended to single out s p e c i a l classes of structures. Observe also t h a t the notion of a t r a n s f o r m a t i o n enters i n t o b o t h the L e v i - S t r a u s s and P i a g e t definitions, t h o u g h the m a n n e r i n w h i c h i t is i n volved is different. For L e v i - S t r a u s s , a t r a n s f o r m a t i o n is a p p a r e n t l y a m e t h o d of r e l a t i n g two models of the same type. In the t e r m i n o l o g y t h a t w i l l be i n t r o d u c e d i n Section 7, a m o d e l is a " r e p r e s e n t a t i o n " of an u n d e r l y i n g s t r u c t u r e , a n d two models w o u l d be of the same type i f they represent the s a m e s t r u c t u r e . T h e L e v i - S t r a u s s t r a n s f o r m a t i o n m i g h t a c c o r d i n g l y be interpreted as a process, associated w i t h the u n d e r l y i n g s t r u c t u r e , of passi n g f r o m one representing model to another. A d d i t i o n a l r e m a r k s c o n c e r n i n g t r a n s f o r m a t i o n s of this k i n d w i l l be found i n Section 8. In the P i a g e t d e f i n i t i o n , the system itself consists of t r a n s f o r m a t i o n s w h i l e L e v i - S t r a u s s ' s system consists of elements, so P i a g e t ' s t r a n s f o r m a tions correspond to L e v i - S t r a u s s ' s elements. T h u s , for P i a g e t the transform a t i o n s are, so to speak, i n t e r n a l to the s t r u c t u r e w h i l e for L e v i - S t r a u s s they are e x t e r n a l . P i a g e t also asserts [ P 3 , p. I l j t h a t " a l l k n o w n structures - f r o m m a t h e m a t i c a l groups t o k i n s h i p systems - are, w i t h o u t e x c e p t i o n , systems of t r a n s f o r m a t i o n s " ! In spite of (or perhaps, because of) this s t r o n g s t a t e m e n t , it is not very clear j u s t w h a t P i a g e t means by a " t r a n s f o r m a t i o n " . It is also not clear w h a t e x a c t l y is b e i n g " t r a n s f o r m e d " . He perhaps h a d i n m i n d a n o t i o n of t r a n s f o r m a t i o n analogous to t h a t associated w i t h the elements of a " g r o u p " i n m a t h e m a t i c s , where each element of the group may be regarded, v i a the group o p e r a t i o n , as a t r a n s f o r m a t i o n a c t i n g o n the set of a l l the group elements. P a r t of the difficulty i n b o t h definitions may be an a t t e m p t to incorporate i n t h e m more t h a n j u s t the n o t i o n of s t r u c t u r e itself. D o t h L e v i - S t r a u s s and P i a g e t were influenced i n a general way by m o d e r n
II. G E N E R A L
STRUCTURE CONCEPTS
15
m a t h e m a t i c s (as w e l l as n a t u r a l science), and P i a g e t was p a r t i c u l a r l y t a k e n b y m o d e r n a l g e b r a . T h e algebra influence is also evident i n his s t u d y of the m e n t a l development of c h i l d r e n , where he identifies a n d follows the developm e n t of m e n t a l processes t h a t suggest operations analogous t o group o p e r a t i o n s . T h e a p p r o a c h was developed i n some d e t a i l i n his b o o k o n " G e n e t i c E p i s t e m o l o g y " [P2] and underlies m u c h of the discussion i n " S t r u c t u r a l i s m " . S o m e o f the m a t h e m a t i c a l ideas t h a t a p p a r e n t l y influenced P i a g e t are discussed i n C h a p t e r I X o n space structures a n d i n Section 21 o n group s t r u c t u r e s . D e s p i t e the m a t h e m a t i c a l n o t i o n s t h a t color these two definit i o n s , neither one is adequate for our purposes. A s m a n y of the e x a m p l e s and the discussion below i n d i c a t e , a m u c h broader n o t i o n of s t r u c t u r e is needed even i n science and m a t h e m a t i c s . L e v i - S t r a u s s , i n c o m p a r i s o n t o P i a g e t , does not a t t e m p t e x p l i c i t use of special m a t h e m a t i c a l concepts i n his work (at least i n " S t r u c t u r a l A n t h r o p o l o g y " ) , a n d perhaps for t h i s reason is less v u l n e r a b l e to c r i t i c i s m . F u r t h e r m o r e , i n the f o l l o w i n g perceptive comment on s t r u c t u r e a n d measure [L6, p . 283], he offers an especially clear d e s c r i p t i o n of the p o t e n t i a l role of m a t h e m a t i c s i n the social sciences. H i s ideas mesh w i t h some of those offered i n S e c t i o n 1. However, one s h o u l d keep i n m i n d t h a t there is no necessary connection between m e a s u r e a n d s t r u c t u r e . S t r u c t u r a l studies are, i n the social sciences, the indirect o u t c o m e of m o d e r n developments i n m a t h e m a t i c s w h i c h have given increasing i m p o r t a n c e to the q u a l i t a t i v e p o i n t of view i n c o n t r a d i s t i n c t i o n to the q u a n t i t a t i v e p o i n t of v i e w of t r a d i t i o n a l m a t h e m a t i c s . It has become possible, therefore, i n fields such as m a t h e m a t i c a l logic, set theory, group theory, a n d t o p o l ogy, to develop a rigorous a p p r o a c h to p r o b l e m s w h i c h do not a d m i t of a metrical solution. T h e above definitions o f s t r u c t u r e b r i n g out a general p r o b l e m w i t h respect t o s t r u c t u r a l i s m . It is t h a t the accepted n o t i o n o f s t r u c t u r e w i t h i n a p a r t i c u l a r field is u s u a l l y so colored by the special features o f t h a t field t h a t one m a y have difficulty i n d i s c e r n i n g j u s t w h a t the structures i n one field have i n c o m m o n w i t h those i n another. T h e e l i m i n a t i o n of this p r o b l e m is a m a j o r benefit derived f r o m a s y s t e m a t i c s t u d y o f general s t r u c t u r e s a n d the a c c o m p a n y i n g development of a language for d e a l i n g w i t h t h e m . 6. A S i m p l e E x a m p l e T h e w o r d " s t r u c t u r e " by itself i m m e d i a t e l y calls t o m i n d s o m e t h i n g l i k e a b u i l d i n g f r a m e w o r k (already m e n t i o n e d b y R a d c l i f f e - B r o w n ) . I n fact, a c o m m o n t e r m for a f r a m e w o r k of this k i n d is " s t r u c t u r e " . T h o u g h everyone w i l l surely agree t h a t this is a s t r u c t u r e (or t h a t it h a s s t r u c t u r e ) , one m i g h t s t i l l ask j u s t w h i c h of the various properties of an a c t u a l framework are
16
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essential to i t s s t r u c t u r e . C e r t a i n l y the weights of the i n d i v i d u a l c o m p o n e n t s a n d the m a t e r i a l of w h i c h t h e y are m a d e are i r r e l e v a n t . T h e i r cross sectional shape, as well as the p a r t i c u l a r m a n n e r of f a s t e n i n g t h e m together, m u s t also be u n i m p o r t a n t . E l i m i n a t i o n o f other such properties leaves f i n a l l y the bare fact t h a t c e r t a i n girders or p i l l a r s are j o i n e d to c e r t a i n others. F u r t h e r m o r e , a scale m o d e l (constructed, say, o f wire segments) w i l l also be s a i d t o have the s a m e s t r u c t u r e as the f r a m e w o r k . A m o r e a b s t r a c t geometric m o d e l , also h a v i n g the same s t r u c t u r e , is o b t a i n e d b y representing the j o i n t s i n the wire m o d e l b y p o i n t s i n space, and representing the wires themselves b y l i n e segments c o n n e c t i n g these p o i n t s . T h i s s t r u c t u r e is c o m p l e t e l y d e t e r m i n e d as s o o n as the p o i n t s are g i v e n a n d the connections between t h e m are specified. A n e x a m p l e of such a s t r u c t u r e is i l l u s t r a t e d i n F i g u r e 6.1.
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i
T\
1
1 ' p
1
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6.1
T h e assertion t h a t the b u i l d i n g f r a m e w o r k a n d the m o d e l s "have the s a m e s t r u c t u r e " suggests t h a t " s t r u c t u r e " is a c t u a l l y s o m e t h i n g associated w i t h a t h i n g r a t h e r t h a n the t h i n g itself. W e s h a l l continue, however, to refer t o a n y t h i n g h a v i n g s t r u c t u r e as " a s t r u c t u r e " , r e l y i n g o n context to m a k e the d i s t i n c t i o n wherever possible. W h e n i t is necessary t o a v o i d confusion t h e " t h i n g " w i t h s t r u c t u r e w i l l be c a l l e d a "concrete s t r u c t u r e " . A precise d e f i n i t i o n of w h a t i t means for t w o concrete structures t o have the " s a m e s t r u c t u r e " w i l l be g i v e n i n S e c t i o n 8. W e have considered here some p r o t o t y p e s of the s i m p l e s t a n d most i n t u i t i v e k i n d of s t r u c t u r e . M a n y m o r e such e x a m p l e s c o u l d be g i v e n , a n d there is m u c h m o r e t o be learned f r o m t h e m . T h e y are also very s p e c i a l , however, a n d do not b e g i n to suggest the great v a r i e t y a n d c o m p l e x i t y of structures t h a t occur i n v i r t u a l l y a l l areas of s t u d y . M o r e e x a m p l e s w i l l
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17
STRUCTURE CONCEPTS
be i n t r o d u c e d below and i n C h a p t e r III. W e r e t u r n now to the p r o b l e m o f definition. 7. T h e B a s i c D e f i n i t i o n s Despite their s i m p l i c i t y , the s t r u c t u r e s associated w i t h a b u i l d i n g framework already suggest a useful a p p r o x i m a t e d e f i n i t i o n of the general n o t i o n o f s t r u c t u r e , as w e l l as some associated concepts. Observe, for e x a m p l e , t h a t the s t r u c t u r e depicted i n F i g u r e 6.1 m a y be thought of as c o n s i s t i n g o f a collection of o b j e c t s (points i n space), c e r t a i n subsets of w h i c h are r e l a t e d because they lie o n a (designated) s t r a i g h t line. T h i s observation suggests the general d e f i n i t i o n of s t r u c t u r e given below. It is not essentially different f r o m one given by W . Hodges [H4] i n a purely m a t h e m a t i c a l context. A s we shall see, the definition is considerably more subtle t h a n its s i m p l e f o r m might indicate. A s t r u c t u r e is any set o f o b j e c t s (also called e l e m e n t s ) c e r t a i n r e l a t i o n s a m o n g those objects.
along with
A s u b s t r u c t u r e of a given s t r u c t u r e is any subset of the objects of t h a t s t r u c t u r e , plus restrictions of some or a l l of the given relations to the subset. In p a r t i c u l a r , the s t r u c t u r e itself is i n c l u d e d a m o n g its s u b s t r u c t u r e s . A l l other s u b s t r u c t u r e s are said to be p r o p e r . E v e r y s u b s t r u c t u r e is o b v i o u s l y a s t r u c t u r e i n its o w n r i g h t . A s t r u c t u r e is called an e x t e n s i o n of each of its substructures. A s t r u c t u r e may involve an infinity of b o t h objects a n d r e l a t i o n s . If, however, b o t h objects a n d relations are finite i n n u m b e r , the s t r u c t u r e itself is said to be f i n i t e . O b s e r v e t h a t a proper s u b s t r u c t u r e c o u l d consist of a l l the given objects and o n l y some of the relations. P e r h a p s the most n a t u r a l s u b s t r u c t u r e , however, consists of a subset of the objects plus a l l relations o b t a i n e d by r e s t r i c t i n g the given relations to t h a t subset. A n o b j e c t may be thought of as a n y t h i n g whatsoever a n d a r e l a t i o n as any " a s s o c i a t i o n " or " c o n n e c t i o n " i n v o l v i n g some of the objects. A n object, s t r i c t l y as a n element of the s t r u c t u r e , has o n l y those properties t h a t it derives f r o m the s t r u c t u r e . T h i s means t h a t a l l of its s t r u c t u r a l properties are u l t i m a t e l y expressed i n the relations t h a t involve i t . T h e r e f o r e , any independent q u a l i t i e s t h a t an o b j e c t m i g h t possess are irrelevant as far as the s t r u c t u r e is concerned. T h i s fact is u l t i m a t e l y the basis for the d e f i n i t i o n of a s t r u c t u r e s i m p l y as a collection of r e l a t i o n s . In such a d e f i n i t i o n , an object, as perceived i n our d e f i n i t i o n , w o u l d be regarded at most as i m p l i c i t in the relations. For our purposes, however, it w i l l be m o r e convenient t o deal e x p l i c i t l y w i t h the objects. A l t h o u g h there are m a n y more features of the d e f i n i t i o n t o be discussed, it w i l l be helpful to describe first, i n the l i g h t of the d e f i n i t i o n , a s i m p l e e x a m p l e q u i t e different f r o m a b u i l d i n g framework. T h e e x a m p l e is the
18
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s t r u c t u r e i n the real n u m b e r s y s t e m associated w i t h the concept o f one n u m b e r b e i n g less t h a n a n o t h e r . It w i l l i n c i d e n t a l l y i l l u s t r a t e a n i m p o r t a n t convention i n the way we describe r e l a t i o n s . T h e s t r u c t u r e consists o f i n d i v i d u a l r e a l n u m b e r s as objects (infinite i n n u m b e r ) a l o n g w i t h "less t h a n " relations i n w h i c h a n u m b e r x is r e l a t e d to a n u m b e r y i f i t is less t h a n y , w r i t t e n x < y . I n p a r t i c u l a r , 2 < 3. N o t i c e t h a t " x < y " here represents a n i n f i n i t y o f r e l a t i o n s , one for each a p p r o p r i a t e choice of values for x a n d y . A t the same t i m e , i t is convenient t o t h i n k of the expression " x < y " as s t a n d i n g for a l l of the relations a n d refer to i t i n the s i n g u l a r as " t h e less t h a n r e l a t i o n " . S i m i l a r conventions o c c u r i n other contexts. T h e set of a l l o r d e r e d p a i r s ( x , y ) such t h a t the n u m b e r x is less t h a n the n u m b e r y is called the " d o m a i n of definition of the r e l a t i o n x < y " . Because the d o m a i n consists of pairs of n u m b e r s , the r e l a t i o n is called a "binary" relation. T h e objects of the s t r u c t u r e (real n u m b e r s ) m a y be represented by p o i n t s o n the " n u m b e r l i n e " , as i l l u s t r a t e d below, where x < y i f x lies t o the left of y o n the n u m b e r l i n e . T h e "less t h a n " r e l a t i o n is a n e x a m p l e of an "order relation".
x< y
.
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7.1.
T h e characteristic properties of an order r e l a t i o n m a y be t r a n s l a t e d i n t o conditions o n its d o m a i n of d e f i n i t i o n . F o r e x a m p l e , the a n t i s y m m e t r y p r o p e r t y , w h i c h asserts t h a t b o t h x < y a n d y < x cannot h o l d (i.e., one cannot have b o t h " x r e l a t e d t o y " a n d " y related t o x " ) , translates i n t o the c o n d i t i o n t h a t b o t h ( x , y ) a n d ( y , x ) cannot b e l o n g to the d o m a i n . T h e t r a n s i t i v i t y p r o p e r t y , w h i c h asserts t h a t x < y a n d y < z i m p l y x < z , translates i n t o the c o n d i t i o n t h a t , i f ( x , y ) a n d ( y , z ) b e l o n g to the d o m a i n , t h e n ( x , z ) m u s t also belong. I n the case o f the r e a l n u m b e r s , there are also properties t h a t relate order t o a d d i t i o n a n d m u l t i p l i c a t i o n , b u t w h i c h we w i l l not b o t h e r n o w to t r a n s l a t e . T h e r a t i o n a l n u m b e r s , under the "less t h a n " r e l a t i o n , constitute a s u b s t r u c t u r e of the ordered reals. T h e integers i n t u r n c o n s t i t u t e a substructure of the r a t i o n a l s , a n d hence also of the reals. T h e n o t i o n of structure suggested by our d e f i n i t i o n is not essentially different f r o m t h a t u n d e r l y i n g the R a d c l i f f e - B r o w n d e f i n i t i o n quoted i n Section 5. A t the same t i m e , i t is m o r e i n c l u s i v e t h a n either of the L e v i - S t r a u s s or P i a g e t definitions. I n each of the l a t t e r , a s t r u c t u r e is defined as a s y s t e m p l u s restrictions suggested by the p a r t i c u l a r field of interest. A s already
II. G E N E R A L S T R U C T U R E C O N C E P T S
19
p o i n t e d o u t , the w o r d " s y s t e m " is a c t u a l l y a n i m p l i c i t reference t o a more i n c l u s i v e n o t i o n of s t r u c t u r e . It w i l l be convenient for our purposes, h o w ever, to m a k e a f o r m a l d i s t i n c t i o n between the n o t i o n of a " s t r u c t u r e " , as defined above, a n d a " s y s t e m " . T h e difference is i l l u s t r a t e d by the real n u m b e r s y s t e m , w h i c h possesses structures associated w i t h the o p e r a t i o n s of a d d i t i o n a n d m u l t i p l i c a t i o n as well as the order s t r u c t u r e . T h e f o l l o w i n g definition of a s y s t e m is o b v i o u s l y consistent w i t h o r d i n a r y usage o f the term. A s y s t e m is any collection of i n t e r r e l a t e d objects a l o n g w i t h a l l of the p o t e n t i a l structures t h a t m i g h t be identified w i t h i n i t . A s u b s y s t e m o f a g i v e n s y s t e m is any subset of the objects of t h a t s y s t e m a l o n g w i t h the p o t e n t i a l structures d e t e r m i n e d i n the subset by the s y s t e m . A s i n the case of s t r u c t u r e s , a s y s t e m is called a n e x t e n s i o n of each o f its subsystems. E v e r y s t r u c t u r e , a l o n g w i t h its s u b s t r u c t u r e s , is o b v i o u s l y a s y s t e m , b u t a s y s t e m is o n l y " p o t e n t i a l l y " s t r u c t u r e d . It w i l l e x h i b i t s t r u c t u r e as soon as any of its p o t e n t i a l structures are made e x p l i c i t . A s suggested b y the d e f i n i t i o n , a s y s t e m m a y be perceived i n more t h a n one way as h a v i n g s t r u c t u r e , d e p e n d i n g on w h i c h properties are singled out for a t t e n t i o n . I n the extreme case, when a l l p o t e n t i a l structures are identified, the s y s t e m is u n a m b i g u o u s l y a s t r u c t u r e a c c o r d i n g t o the general d e f i n i t i o n , hence the occasional confusion of the t e r m s . T y p i c a l l y , however, a s y s t e m m a y be recognized to possess m a n y properties t h a t are neither i n v o l v e d i n nor i m p l i e d by a p a r t i c u l a r one of its perceived s t r u c t u r e s . T h i s does not o c c u r i n a s t r u c t u r e proper, because a l l of its properties are d e t e r m i n e d i n one way or another by the specified objects and r e l a t i o n s . N o t e t h a t any concrete s t r u c t u r e may also have properties irrelevant t o its perceived s t r u c t u r e , b u t these are suppressed i n its role as a s t r u c t u r e . A l t h o u g h the specification of a s t r u c t u r e m a y ignore m u c h o f the a v a i l able i n f o r m a t i o n i n a s y s t e m , it m a y nevertheless involve the essential i n f o r m a t i o n . ( T h e m e a n i n g of "essential i n f o r m a t i o n " is, of course, a relative m a t t e r . ) T h e o b j e c t i v e of a s t r u c t u r a l i s t a p p r o a c h to a subject is to e x t r a c t the essential i n f o r m a t i o n f r o m the b a c k g r o u n d of irrelevant or u n i m p o r t a n t i n f o r m a t i o n . A n y loss of essential i n f o r m a t i o n i n this process w i l l i n d i c a t e a n inadequate s t r u c t u r a l analysis. W e m a y t h i n k of the relations i n a s t r u c t u r e as " b i n d i n g " the given objects i n t o a unified whole. T h e wholeness of any s t r u c t u r e w i l l depend u p o n the degree o f interrelatedness a m o n g its objects. It is by v i r t u e of "wholeness" t h a t one s t r u c t u r e m a y serve as an o b j e c t i n a second. Despite t h i s p o s s i b i l i t y , a specified structure u s u a l l y does not "recognize" e x p l i c i t l y any i n t e r n a l s t r u c t u r e t h a t one of its object m i g h t have. T h e i n i t i a l s t r u c -
20
STRUCTURALISM AND STRUCTURES
t u r e c o u l d , however, be extended so as to i n c o r p o r a t e some o f the i n t e r n a l s t r u c t u r e of its objects. T h e case o f relations is analogous t o t h a t o f objects, a l t h o u g h the s i t u a t i o n for t h e m is somewhat more c o m p l e x . In a given s y s t e m , a r e l a t i o n m a y possess properties not recognized b y a specified s t r u c t u r e w i t h i n the syst e m . A s i n the case of o b j e c t s , however, such properties m a y be recovered by respecifying the s t r u c t u r e . T h e a p p r o x i m a t e nature of the d e f i n i t i o n of s t r u c t u r e resides to a large extent i n the i m p r e c i s i o n of the n o t i o n of a r e l a t i o n , a n d the m a i n p r o b l e m s encountered i n the analysis of a s y s t e m u s u a l l y involve the relations. F u r t h e r m o r e , even i n o r d i n a r y systems, relations are often c o m p l e x a n d difficult t o describe. M e t h o d s of d e a l i n g w i t h these p r o b l e m s i n a n u m b e r of special s i t u a t i o n s w i l l be discussed i n later sections. T h e d e f i n i t i o n of s t r u c t u r e , t h o u g h s i m p l e i n f o r m a n d very general, serves the purpose o f p o i n t i n g us i n the desired d i r e c t i o n . It also has the v i r t u e o f not e x c l u d i n g a n y t h i n g t h a t m i g h t conceivably be regarded as a s t r u c t u r e , a fact t h a t is i m p o r t a n t i n our general a p p r o a c h . Its p r i n c i p a l role, however, is t o p r o v i d e a focus for o u r efforts to expose a n d to f o r m u late some of the i m p o r t a n t general characteristics of structures. Therefore, m u c h o f the discussion here and i n the succeeding sections is more or less s y s t e m a t i c u n f o l d i n g of the d e f i n i t i o n . A l t h o u g h a general definition is essential to any f o r m a l t r e a t m e n t of s t r u c t u r e s , it m a y fail to convey the whole p i c t u r e i n some cases. T h e reason is t h a t a p a r t i c u l a r s t r u c t u r e is u s u a l l y not presented i n i s o l a t i o n b u t as a s u b s t r u c t u r e o f a larger "universe" s t r u c t u r e . T h e l a t t e r , w h i c h m a y also c o n t a i n n u m e r o u s other s t r u c t u r e s relevant t o the subject b e i n g s t u d i e d , is often not recognized e x p l i c i t l y when a t t e n t i o n is fixed on a p a r t i c u l a r s u b s t r u c t u r e . For e x a m p l e , m a n y structures, such as those associated w i t h the b u i l d i n g f r a m e w o r k , appear as substructures of p h y s i c a l or (the more a b s t r a c t ) E u c l i d e a n space. T h i s is an i m p o r t a n t and generally u n a v o i d a b l e p r o b l e m w h i c h w i l l be considered i n some d e t a i l later. T h e r e is one m o r e p o i n t concerning the a p p l i c a b i l i t y of a general theory of s t r u c t u r e s t h a t must be m e n t i o n e d . I n any g i v e n s u b j e c t , s t r u c t u r e s are n a t u r a l l y dealt w i t h f r o m the p o i n t of v i e w a n d i n the a p p r o p r i a t e language of t h a t s u b j e c t , a fact already noted i n c o n n e c t i o n w i t h the L e v i Strauss and P i a g e t definitions. A l t h o u g h t h i s practice tends t o obscure the i n d e p e n d e n t l y i m p o r t a n t u n i v e r s a l role of s t r u c t u r e s , it suggests t h a t the general view m a y be p r i m a r i l y of t h e o r e t i c a l , rather t h a n p r a c t i c a l , significance i n c e r t a i n subjects. A t the same t i m e , a general theory of s t r u c t u r e s can p r o v i d e special insights i n t o v i r t u a l l y any subject a n d its connections w i t h other subjects.
II. G E N E R A L
8. I s o m o r p h i s m s
STRUCTURE CONCEPTS
21
of Structures
W e have already encountered at an i n t u i t i v e level the i d e a t h a t two concrete s t r u c t u r e s , such as a b u i l d i n g f r a m e w o r k a n d a m o d e l of i t , m a y "have the same s t r u c t u r e " . T h a n k s t o the f o r m a l d e f i n i t i o n of " s t r u c t u r e " , i t is n o w possible to give a precise m e a n i n g to this idea as well as t h a t of an a b s t r a c t s t r u c t u r e . It is based on the concept of an " i s o m o r p h i s m " of s t r u c t u r e s , a concept t h a t is i n t i m a t e l y b o u n d up w i t h the idea of s t r u c t u r e itself, a n d is essential t o the precise f o r m u l a t i o n of c e r t a i n basic properties of s t r u c t u r e s . T h e d e f i n i t i o n is i n s p i r e d b y s i m i l a r ideas f r o m m a t h e m a t i c s . A n i s o m o r p h i s m between two structures consists of a one-to-one correspondence between the collections of objects of the two s t r u c tures, such t h a t a, p o s s i b l y ordered, set o f objects f r o m one s t r u c t u r e w i l l be related if, a n d only if, the corresponding o b j e c t s o f the other s t r u c t u r e are also r e l a t e d . I n this case, the t w o structures are said to be i s o m o r p h i c . A n i s o m o r p h i s m between one s t r u c t u r e a n d a subs t r u c t u r e of another is called an e m b e d d i n g of the first w i t h i n the second. A "one-to-one correspondence" between the elements of two sets (or c o l lections) is s i m p l y an a s s o c i a t i o n , or "correspondence", of a l l elements f r o m one set w i t h the elements of the other i n such a way t h a t each element of the second is associated w i t h one, and only, element of the first. T h i s last cond i t i o n is the "one-to-one" requirement. T h e d e f i n i t i o n of an i s o m o r p h i s m m a y be a p p l i e d to either concrete or abstract s t r u c t u r e s . If two structures are i s o m o r p h i c , they are s a i d to have the "same s t r u c t u r e " . T h i s is s o m e t h i n g c o m m o n t o any collection of m u t u a l l y i s o m o r p h i c s t r u c t u r e s , a n d is precisely w h a t we w i l l m e a n by an " a b s t r a c t s t r u c t u r e " . It is o b v i o u s l y preserved by i s o m o r p h i s m s , a n d is assumed t o exist i n its o w n r i g h t . (Some of the p h i l o s o p h i c a l p r o b l e m s raised by t h i s p o i n t of view w i l l be discussed briefly i n Section 14.) A n abstract s t r u c t u r e is regarded as i s o m o r p h i c t o the associated concrete structures a n d is s a i d to be represented b y the l a t t e r . Conversely, d e p e n d i n g o n the p o i n t of v i e w , an a b s t r a c t s t r u c t u r e m a y also be s a i d to represent a concrete s t r u c t u r e . G e n e r a l l y s p e a k i n g , a representation could be any s y s t e m t h a t contains a s t r u c t u r e i s o m o r p h i c to the given one. S u c h a s y s t e m w i l l n o r m a l l y i n v o l v e m u c h irrelevant i n f o r m a t i o n w h i c h might therefore be changed more or less a r b i t r a r i l y w i t h o u t d e s t r o y i n g the representation. A s far as an abstract s t r u c t u r e is concerned, relations are c o m p l e t e l y d e t e r m i n e d b y the collection o f (possibly ordered!) sets of objects t h a t are connected b y t h e m . T h e reason for this is t h a t a general i s o m o r p h i s m preserves o n l y the s i m p l e fact t h a t objects are r e l a t e d . T h e r e f o r e , the assoc i a t e d collection of sets m a y even be taken as the d e f i n i t i o n of the r e l a t i o n .
12
STRUCTURALISM AND
STRUCTURES
W e h a d a g l i m p s e of this i n our brief look at the order structure of the real n u m b e r s i n Section 7. It is w o r t h n o t i n g here t h a t the elements of a set of objects connected by a r e l a t i o n need not be d i s t i n c t . In other words, a p a r t i c u l a r o b j e c t may appear i n m o r e t h a n one way i n a g i v e n a p p l i c a t i o n of the r e l a t i o n . T h e above definition o f i s o m o r p h i s m of s t r u c t u r e s ignores a l l of the extraneous i n f o r m a t i o n u s u a l l y c o n t a i n e d i n the various realizations of the u n d e r l y i n g abstract s t r u c t u r e . T h i s includes, for e x a m p l e , a n y t h i n g associated w i t h a larger s t r u c t u r e t h a t m i g h t c o n t a i n the representing s t r u c t u r e as a s u b s t r u c t u r e . T h e r e are i m p o r t a n t cases, however, such as the b u i l d i n g structures e m b e d d e d i n E u c l i d e a n space, i n w h i c h i t is necessary t o preserve some of the e x t r a i n f o r m a t i o n . T h e p r o b l e m m a y sometimes be avoided by a more careful specification of the s t r u c t u r e (so t h a t an isomorp h i s m w i l l carry more i n f o r m a t i o n ) , or by r e s t r i c t i n g the t y p e of r e a l i z a t i o n p e r m i t t e d (say, to substructures of E u c l i d e a n space). T h e r e are also i n stances i n w h i c h it is n a t u r a l to formulate a more restrictive definition of an i s o m o r p h i s m . T h i s i d e a is touched u p o n i n Section 10 a n d is i m p l i c i t i n the definition of " e x t e r n a l " properties g i v e n below. It w i l l be t a k e n up s y s t e m a t i c a l l y for a special case i n C h a p t e r I X . U n t i l then, the u n r e s t r i c t e d definition w i l l serve o u r purposes. N e x t , we d i s t i n g u i s h t w o k i n d s of properties t h a t m a y be associated w i t h an abstract s t r u c t u r e . T h e first concerns o n l y the s t r u c t u r e , w h i l e the second involves e m b e d d i n g s of the given s t r u c t u r e i n larger s t r u c t u r e s . A p r o p e r t y of a s t r u c t u r e is s a i d to be i n t e r n a l i f it depends o n l y o n relations w i t h i n the s t r u c t u r e itself. It is said to be e x t e r n a l if it is not i n t e r n a l a n d depends on relations t h a t involve objects of the s t r u c t u r e w h e n i t is realized as a s u b s t r u c t u r e of some larger s t r u c t u r e . E a c h e x t e r n a l p r o p e r t y is always associated w i t h a specific e m b e d d i n g o f the given s t r u c t u r e i n a larger one. T h i s concept is p a r t i c u l a r l y relevant t o biological s t r u c t u r e s , w h i c h are considered i n C h a p t e r V I I I . In the case of an i s o m o r p h i s m of concrete structures, i t m a y be i m p o r t a n t t o consider w h a t effect the i s o m o r p h i s m has o n some of those special properties of objects and relations t h a t are not d i r e c t l y recognized by the i n volved structures a n d therefore need not be preserved by the i s o m o r p h i s m . O n the other h a n d , because a u x i l i a r y properties can depend to some degree on the given structures, there m a y be some r e g u l a r i t y i n the way they are t r a n s f o r m e d . Such p h e n o m e n a are i m p l i c i t , for e x a m p l e , i n L e v i - S t r a u s s ' c o m p a r i s o n o f m y t h s a n d k i n s h i p structures w i t h i n different cultures [L5]. H e e v i d e n t l y also h a d t h e m i n m i n d i n f o r m u l a t i n g the definition of s t r u c ture quoted i n Section 5. Dependencies of this k i n d are also covered by the concept of " s t r u c t u r a l d e t e r m i n i s m " discussed i n Sections 26 and 57. It is necessary i n some s i t u a t i o n s to consider s t r u c t u r e t r a n s f o r m a t i o n s
II. G E N E R A L
STRUCTURE CONCEPTS
23
m o r e general t h a n i s o m o r p h i s m s . O n e i m p o r t a n t instance concerns the way in w h i c h m e n t a l images (structures) are recorded i n the b r a i n , a process t h a t c l e a r l y m u s t i n v o l v e m o r e t h a n a s i m p l e i s o m o r p h i s m . T h e r e are also m a n y e x a m p l e s i n m a t h e m a t i c s , one of w h i c h is t h e F o u r i e r t r a n s f o r m . A l t h o u g h most of the m a t h e m a t i c a l e x a m p l e s are m u c h too t e c h n i c a l t o be dealt w i t h here, i t is p e r h a p s w o r t h w h i l e t o l o o k at one very s i m p l e case i n v o l v i n g the p a i r of p o i n t - l i n e structures i l l u s t r a t e d i n the n e x t figure. T h e p o i n t s (objects) i n s t r u c t u r e (1), l a b e l e d A , B , C, D are supposed to represent the vertices of a t e t r a h e d r o n i n space. T h e l i n e (relation) determ i n e d b y t w o p o i n t s , say A a n d B , is denoted b y the p a i r A B . S t r u c t u r e (2) is o b t a i n e d b y t a k i n g the lines i n (1) as the objects a n d the p o i n t s where t h e y intersect as the r e l a t i o n s . T h u s , we have s i x (line) o b j e c t s a n d four (point) relations f r o m (1), represented i n (2) as s i x p o i n t s a n d f o u r lines respectively. T h e t r a n s f o r m a t i o n f r o m (1) to (2) o b t a i n e d i n this w a y is o b v i o u s l y not a s t r u c t u r e i s o m o r p h i s m . It is a s p e c i a l case of w h a t is c a l l e d a "duality".
Fig. 9. A n a l o g i e s arid
8.1
Isomorphisms
In Section 1, the e x a m p l e of analogies was g i v e n as a n i l l u s t r a t i o n of the fact t h a t the n o t i o n of s t r u c t u r e is i m p l i c i t i n m a n y everyday experiences. T h e p o i n t was t h a t the i m p l i e d s i m i l a r i t y between a g i v e n s t r u c t u r e a n d an analogous s t r u c t u r e a c t u a l l y m e a n s t h a t the two "possess some c o m m o n s t r u c t u r e " . I n other words, t h e t w o s t r u c t u r e s c o n t a i n s u b s t r u c t u r e s t h a t a r e i s o m o r p h i c . T h e purpose of a n a n a l o g y is t o c a l l a t t e n t i o n t o , or to emphasize, some aspect of the g i v e n s t r u c t u r e (as represented b y one of its substructures). It is i n s t r u c t i v e to l o o k m o r e closely at a p a r t i c u l a r analogy t h a t most people w i l l have l i t t l e difficulty u n d e r s t a n d i n g . W e choose as a n e x a m p l e a news s t o r y t h a t a p p e a r e d i n the N e w H a v e n R e g i s t e r j u s t before the second debate between George B u s h a n d M i c h a e l D u k a k i s d u r i n g the 1988 p r e s i d e n t i a l c a m p a i g n . It b o r e the h e a d l i n e , " D u k a k i s needs to score k n o c k o u t
24
STRUCTURALISM
AND STRUCTURES
i n debate t o n i g h t " , and the l e a d i n g sentence r e a d , " M i c h a e l D u k a k i s needs t o h i t a home r u n i n t o n i g h t ' s debate, w h i l e G e o r g e B u s h can lose it a n d s t i l l w i n the W h i t e House — as long as he doesn't strike o u t " . T h e reference, of course, is to the relative s t a n d i n g s of the two candidates going i n t o the debate. W e w i l l ignore the prize fight r e m a r k a n d concentrate o n the baseball reference. In this case the debate s t r u c t u r e is the g i v e n , a n d the baseball s t r u c t u r e is the analogy. T h e purpose o f the analogy was t o emphasize the effect of the debate o u t c o m e o n the c a n d i d a t e s ' relative s t a n d i n g i n the c a m p a i g n . O u r o b j e c t i v e then is to make e x p l i c i t the i m p l i e d s t r u c t u r e i s o m o r p h i s m between the baseball a n d debate contexts. A s it t u r n s o u t , the a n a l y s i s is s o m e w h a t more complex t h a n m i g h t be expected f r o m the obviousness of the e x a m p l e . It must also be understood t h a t the details, w h i c h are rather tedious, do not represent the a c t u a l t h o u g h t process experienced b y anyone w h o u n d e r s t a n d s the analogy. O n the other h a n d , they do make e x p l i c i t the s t r u c t u r a l content o f the e x a m p l e and at the same t i m e serve t o b r i n g out some very i m p o r t a n t features o f general structures. W e w i l l not a t t e m p t t o give a d e s c r i p t i o n o f either o f the f u l l structures, b u t w i l l concentrate o n the p o r t i o n s of these s t r u c t u r e s i n v o l v e d i n the analogy. T h e basic picture, o n w h i c h e v e r y t h i n g depends, consists of the debate setting w i t h B u s h l e a d i n g D u k a k i s i n the polls a n d slated t o w i n the election, plus the i m a g i n a r y baseball s e t t i n g , w h i c h m i g h t be a best player c o m p e t i t i o n i n w h i c h B u s h and D u k a k i s are l e a d i n g candidates, w i t h B u s h presently o n t o p . In the latter case, we m a y t h i n k of the c o m p e t i t i o n as consisting of a one t i m e at b a t for each. A t this p r e l i m i n a r y stage, the s t r u c t u r e s are t r i v i a l , each consisting of o n l y t w o objects ( B u s h a n d D u k a k i s ) , and one r e l a t i o n (that of one person b e i n g ahead of the o t h e r ) . T h e basic p i c t u r e is clear enough, b u t some of the i m p l i e d properties of the two s i t u a t i o n s need to be m a d e e x p l i c i t . C o n s i d e r first the baseball sett i n g . A c c o r d i n g t o our a n a l y s i s , w h i c h is by no means unique, the s t r u c t u r e m u s t c o n t a i n five objects and one rather c o m p l e x r e l a t i o n i n order t o represent the desired i n f o r m a t i o n . W e denote these i t e m s by suggestive s y m b o l s whose " m e a n i n g s " wilt be specified below. T h e objects w i l l be denoted b y B , D , H , N , S, a n d the relation(s) by B u -r D v
=> x
> y,
where x , y , u , t> are variables whose values are objects. Observe t h a t denoti n g (or n a m i n g ) objects and relations b y s y m b o l s need not be a p a r t of the p e r c e p t i o n o f the s t r u c t u r e , b u t o n l y serves to facilitate the d e s c r i p t i o n (or c o m m u n i c a t i o n ) of i t . In s i m p l e cases such as this, the s t r u c t u r e w o u l d u s u a l l y be perceived more or less directly as a " p i c t u r e " . T h i s is an i m p o r t a n t p o i n t , w h i c h is e l a b o r a t e d i n C h a p t e r V .
II. G E N E R A L S T R U C T U R E C O N C E P T S
25
B a s e b a l l meanings must now be assigned t o b o t h objects a n d relations, a n d the values of the r e l a t i o n a l variables must be restricted to fit the i m a g i n a r y baseball s e t t i n g : B a n d D s t a n d for B u s h a n d D u k a k i s . H a n d S s t a n d for " H o m e r u n " a n d " S t r i k e o u t " , w h i l e N s t a n d s for a performance different f r o m either of these. T h u s , H is a better performance t h a n either JV or S, w h i l e S is worse t h a n either H or N . T h e variables x a n d y m a y take either S o r D a s values, w h i l e u and v take the performance values H , N , or S. In the r e l a t i o n , B u + D v stands for the performances of B a n d D . For e x a m p l e , B N + D H means t h a t B neither h i t a h o m e r u n nor s t r u c k out, while D hit a home run. x > y means t h a t x r a n k s above y, so is restricted t o the two cases B > D and D > B . B u + D v => x > y means t h a t the i n d i c a t e d performances i m p l y (or w i l l result in) the i n d i c a t e d r a n k i n g . T h e values of the variables i n the r e l a t i o n are restricted as follows: B H
+ Dv
=>
B N B S
+ Dv + D S + Dv + D H
=>
B B
=> =*• =>
B D D
B S B N
> D , for v = H , N , or S. > D , for v = N or S > D . > B , for v = H or N . > B .
T h e reasons for these restrictions are o b v i o u s f r o m the prescribed m e a n i n g s . T h e first three express the fact t h a t B u s h w i l l r e t a i n the higher r a n k i n g p r o v i d e d he t u r n s i n a performance at least as good as t h a t of D u k a k i s . T h e f o u r t h says t h a t i f B u s h strikes out then D u k a k i s w i l l g a i n the l e a d , p r o v i d e d , of course, t h a t he does not also strike out. T h e last one says t h a t a h o m e r u n w i l l give D u k a k i s the lead unless B u s h also h i t s a h o m e r u n . T h i s is a c o m p l e t e d e s c r i p t i o n of the s t r u c t u r e for the baseball s e t t i n g . It consists of five objects plus nine d i s t i n c t relations a m o n g t h e m i m p l i e d by the five r e s t r i c t i o n statements. ( N o t e t h a t the first statement accounts for three r e l a t i o n s , one for each value of the variable v , w h i l e the second a n d f o u r t h each accounts for two.) In s p e c i f y i n g objects a n d relations for the debate s e t t i n g , we choose n o t a t i o n s t h a t w i l l suggest i m m e d i a t e l y the i s o m o r p h i s m t h a t i m p l e m e n t s the a n a l o g y : T h e objects are B , D , E , M , P , a n d the relations are i d e n t i c a l w i t h those i n the baseball case, except H , N , a n d S are replaced respectively by E , M , a n d P . B a n d D s t a n d , as before, for B u s h a n d D u k a k i s , w h i l e E , M , P are debate performances, s t a n d i n g for E x c e l l e n t , M e d i o c r e , a n d P o o r , respectively. B y v i r t u e of the i s o m o r p h i s m , the baseball analogy serves t o emphasize
26
STRUCTURALISM
AND
STRUCTURES
the fact t h a t B u s h ' s o u t r a n k i n g of D u k a k i s w i l l be changed b y the debate o n l y i f D u k a k i s ' performance is excellent w h i l e B u s h ' s is n o t , or B u s h ' s performance is p o o r w h i l e D u k a k i s ' is n o t . Let us consider now the a b s t r a c t s t r u c t u r e t h a t the debate and b a s e b a l l settings have i n c o m m o n . If the suggested m e a n i n g s are i g n o r e d , either of the s y m b o l i c representations of the t w o structures g i v e n above m a y be t h o u g h t o f as a representation o f the abstract s t r u c t u r e . Because the m e a n ings are irrelevant as far as the abstract s t r u c t u r e is concerned, we i n t r o d u c e new n o t a t i o n s t h a t are not associated i n any way w i t h the e x a m p l e s . D e n o t e the five objects and the v a r i a b l e r e l a t i o n respectively b y the (neutral) symbols, I , J , K , L , M ,
and
(u,v;x,y),
where the letters i n the r e l a t i o n are variables whose values (as before) are objects yet to be d e t e r m i n e d . In other words, the d o m a i n of the r e l a t i o n r e m a i n s to be denned. T h e d o m a i n c o u l d t h e o r e t i c a l l y be p r e s c r i b e d i n a completely a r b i t r a r y m a n n e r , y i e l d i n g a different s t r u c t u r e for each choice. B u t because we are interested i n the special structures i n v o l v e d i n the a n a l ogy, i t m u s t be specified so t h a t the abstract s t r u c t u r e is i s o m o r p h i c w i t h each o f the concrete s t r u c t u r e s . C o n s i d e r , for e x a m p l e , the correspondence t h a t associates the g i v e n abstract objects 7, J , K , L , M respectively w i t h the b a s e b a l l objects B , D , H , N , S ; and the abstract r e l a t i o n ( u , v ; x , y ) w i t h the baseball r e l a t i o n , B u -+ D v =3- x > y . T h e n u a n d v w i l l t a k e o n the values K , L , M w h i l e x and y take values 7, J . In order for the correspondence to determine a s t r u c t u r e i s o m o r p h i s m , the following restrictions o n the v a r i ables i n the abstract r e l a t i o n are also needed: ( K , v\ I , J ) , where the value of v is K , L , or M . ( L , v ; I , J ) , where the value of v is L or M . ( M , M ; I , J ) . (A7, v ; J , I ) , where the value of v is K or L . (LJ
STRUCTURALISM
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derivable f r o m t h e m . T h e d e d u c t i o n process, w h i c h is analogous t o the e l a b o r a t i o n discussion for an a p p r o x i m a t e d e f i n i t i o n (Section 4), is o b v i o u s l y a process of successive a p p r o x i m a t i o n . T h e s y s t e m o f a x i o m s defines d i r e c t l y a basic s t r u c t u r e , w h i c h is a s u b s t r u c t u r e o f the desired one a n d a first a p p r o x i m a t i o n to i t . T h e basic s t r u c t u r e is then developed i n t o successively larger substructures. T h e l a t t e r grow t h r o u g h discovery or c o n s t r u c t i o n of new objects a n d relations i m p l i e d b y the a x i o m s a n d their consequences. T h e r o u g h idea here is t h a t the larger the s u b s t r u c t u r e the m o r e i n f o r m a t i o n i t w i l l i n c l u d e , a n d the more i n f o r m a t i o n t h a t a s u b s t r u c t u r e contains the better it w i l l a p p r o x i m a t e the f u l l s t r u c t u r e . In a c t u a l practice t h i s assertion needs to be qualified b y some m e t h o d of w e i g h t i n g the i n f o r m a t i o n , because some i t e m s of i n f o r m a t i o n w i l l generally be more i m p o r t a n t t h a n others. F o r e x a m p l e , properties are not derived at r a n d o m f r o m a set of a x i o m s b u t are n o r m a l l y a i m e d t o w a r d some g o a l and s u b j e c t to c e r t a i n s t a n d a r d s of q u a l i t y . T h e l a t t e r m a y involve c r i t e r i a such as "usefulness", or "relevance t o the b o d y of e x i s t i n g k n o w l e d g e " , or s i m p l y "elegance". It is no d o u b t possible t o f o r m u l a t e a s y s t e m o f a x i o m s t h a t m i g h t capt u r e , at least for a very wide class of abstract s t r u c t u r e s , a general n o t i o n of s t r u c t u r e consistent w i t h t h a t b e i n g developed here. O u r d e f i n i t i o n of s t r u c t u r e , a l o n g w i t h certain f o r m a l properties such as the ones already discussed a n d others t h a t w i l l c o m e u p l a t e r , are a step i n t h a t d i r e c t i o n . A n a t u r a l a p p r o a c h m i g h t be t h r o u g h category theory, w h i c h s h o u l d be general enough to a c c o m m o d a t e the desired result. O n the other h a n d , f r o m o u r current p o i n t o f v i e w , a completely f o r m a l t r e a t m e n t s t i l l seems s o m e w h a t p r e m a t u r e at t h i s stage o f development. T h e r e r e m a i n s t o o m u c h yet to be exposed a b o u t general s t r u c t u r e s . 26.
Structural Determinism and Reductionism
T h e topics considered i n this section, t h o u g h not d i r e c t l y concerned w i t h the m a i n t h e m e o f the chapter, are i m p l i c i t i n the a p p r o x i m a t i o n m e t h o d s discussed i n the preceding two sections. In a d d i t i o n , the n o t i o n of s t r u c t u r a l d e t e r m i n i s m is of f u n d a m e n t a l i m p o r t a n c e a n d w i l l a p p e a r later i n a variety of different contexts. Before b e g i n n i n g a f o r m a l discussion o f d e t e r m i n i s m , we describe a very s i m p l e e x a m p l e t o i l l u s t r a t e i n concrete t e r m s w h a t is i n v o l v e d . T h e e x a m ple, w h i c h is d r a w n f r o m personal experiences, w i l l no d o u b t suggest m a n y s i m i l a r e x a m p l e s to the reader. O n e day years ago w h e n m y eldest son was q u i t e y o u n g , a neighbor gave h i m a toy g u n as a present. T h e g u n made a l o u d p o p p i n g noise when fired, but by the t i m e I a r r i v e d h o m e t h a t evening it h a d ceased to o p e r a t e . In an a t t e m p t to relieve the crisis, I i m m e d i a t e l y took the g u n a p a r t h o p i n g to
IV. M A N A G E M E N T O F C O M P L E X S T R U C T U R E S
61
fix i t . It was easy to see how the g u n was supposed t o w o r k , a n d also to see w h y i t h a d f a i l e d — there was a p a r t m i s s i n g . T h e r e u p o n , I asked m y wife i f she h a d seen a s m a l l piece o f m e t a l " a b o u t so b i g " , and I drew a sketch o f i t . It so h a p p e n e d t h a t she d i d remember p i c k i n g up a m e t a l o b j e c t f r o m the floor, w o n d e r i n g at the t i m e w h a t it was. She was also p u z z l e d t h a t I c o u l d k n o w a b o u t the existence a n d more or less exact appearance o f s o m e t h i n g t h a t I h a d never seen. T h e m y s t e r y was not very deep, of course, since the gun s t r u c t u r e , m i n u s the m i s s i n g piece, a c t u a l l y d e t e r m i n e d i n a n o b v i o u s way the m i s s i n g p a r t . D e s p i t e the t r i v i a l i t y , " M r . F i x i t " was credited w i t h another success a n d everyone was h a p p y . N o w let us t r y t o define m o r e precisely a general n o t i o n of s t r u c t u r a l det e r m i n i s m . It w i l l be useful t o consider a setup considerably more i n c l u s i v e t h a n t h a t suggested by the e x a m p l e . C o n s i d e r a s t r u c t u r e S and t w o of its s u b s t r u c t u r e s , S' a n d S". If it is possible t o construct ( w i t h i n S ) the s t r u c t u r e S" f r o m S', then we say t h a t S" d e t e r m i n e s S" w i t h i n S. If S" contains S', i n p a r t i c u l a r i f S" = S, then S' determines S" i n t e r n a l l y . If S' a n d S" are d i s j o i n t , then S' determines S" e x t e r n a l l y w i t h i n S. A n y s u b s t r u c t u r e contains a p o r t i o n of the i n f o r m a t i o n i n c o r p o r a t e d i n its parent s t r u c t u r e . A l s o , a s t r u c t u r e w h i c h determines another contains i m p l i c i t l y a l l of the i n f o r m a t i o n possessed b y the l a t t e r . I n c i d e n t a l l y , the toy g u n e x a m p l e , as described above, is a case of e x t e r n a l d e t e r m i n i s m , because the m i s s i n g p a r t (substructure) was d e t e r m i n e d by the s u b s t r u c t u r e consisting of the g u n m i n u s the p a r t . A t the same t i m e , i t c o u l d be regarded as i n t e r n a l , because the full s t r u c t u r e was d e t e r m i n e d by a s u b s t r u c t u r e . Because the m e a n i n g of the w o r d " c o n s t r u c t " is not e n t i r e l y clear, the above d e f i n i t i o n is more or less a m b i g u o u s , so is not a c t u a l l y c o m p l e t e . A s i m i l a r p r o b l e m is also present i n the n o t i o n o f " e v o l u t i o n " , or " e x t e n s i o n " , of a s t r u c t u r e , s i m p l y because the a c t u a l m e t h o d o f g r o w t h is often not specified. These details, t h o u g h often not c r u c i a l i n p a r t i c u l a r cases, are sometimes rather tedious to s u p p l y . T h e general i d e a is we 11-illustrated, however, by a n a x i o m s y s t e m . T h e s t r u c t u r e d i r e c t l y associated w i t h the set of a x i o m s d e t e r m i n e s the full s t r u c t u r e , i n the sense t h a t the l a t t e r is p o t e n t i a l l y c o n s t r u c t i b l e f r o m the former t h r o u g h a general process of d e d u c t i o n r e s u l t i n g i n the discovery or creation o f new o b j e c t s a n d relations i m p l i e d b y the a x i o m s a n d their consequences. T h i s is also an e x a m p l e of internal determinism. A n o t h e r e x a m p l e of external d e t e r m i n i s m is p r o v i d e d by the P a s c a l configuration (Section 19), w h i c h is d e t e r m i n e d by a complete hexagon i n s c r i b e d i n a conic. R e c a l l t h a t the 45 p o i n t s a n d 60 lines of the P a s c a l c o n f i g u r a t i o n are disjoint f r o m the 6 points and 15 lines of the complete
62
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h e x a g o n . T h e p o i n t - l i n e s t r u c t u r e c o n s i s t i n g o f the u n i o n of these t w o conf i g u r a t i o n s m a y be t a k e n as the parent s t r u c t u r e S, so the P a s c a l configur a t i o n is d e t e r m i n e d w i t h i n S b y the complete h e x a g o n . I n t h i s e x a m p l e , the m e t h o d of c o n s t r u c t i o n is essentially geometric. M a n y a d d i t i o n a l exa m p l e s exist a m o n g p h y s i c a l s t r u c t u r e s , a n o b v i o u s one b e i n g the g r o w t h of a c r y s t a l . S o m e b i o l o g i c a l e x a m p l e s w i l l be discussed i n C h a p t e r V I I I . A s already observed i n the preceding section, a n a t u r a l a n d m o r e or less a u t o m a t i c a p p r o a c h to u n d e r s t a n d i n g a c o m p l e x s t r u c t u r e is t h r o u g h its substructures. I n a given case, the effectiveness of the a p p r o a c h w i l l d e p e n d u p o n the degree t o w h i c h the chosen s u b s t r u c t u r e determines the f u l l s t r u c t u r e . A n especially desirable case, f o u n d p r i m a r i l y i n the p h y s i c a l sciences, is a d e t e r m i n i n g s u b s t r u c t u r e w h i c h a d m i t s a m a t h e m a t i c a l repres e n t a t i o n , a n d m a y a c c o r d i n g l y be developed m a t h e m a t i c a l l y t o give precise i n f o r m a t i o n c o n c e r n i n g the parent s t r u c t u r e . T h e concept o f s t r u c t u r a l det e r m i n i s m also casts some l i g h t o n p r o b l e m s associated w i t h " r e d u c t i o n i s m " as an a p p r o a c h t o u n d e r s t a n d i n g c o m p l e x systems. R o u g h l y s p e a k i n g , red u c t i o n i s m is a n a t t e m p t t o u n d e r s t a n d a s y s t e m b y r e d u c i n g i t t o c e r t a i n basic p r i n c i p l e s w h i c h are a l r e a d y u n d e r s t o o d . T h e a p p r o a c h w i l l o b v i o u s l y be effective i n s i t u a t i o n s , such as those described above, i n w h i c h a r e l a t i v e l y s i m p l e s u b s t r u c t u r e determines the whole s t r u c t u r e . P h y s i c a l science serves, d i r e c t l y or i n d i r e c t l y , as the p r i n c i p l e m o d e l for r e d u c t i o n i s m of this kind. D e s p i t e the u n i v e r s a l success of the a p p r o a c h i n science a n d technology, the w o r d " r e d u c t i o n i s m " carries a negative c o n n o t a t i o n . T h e m e t h o d is frequently c r i t i c i z e d i n other contexts, because i t ignores the p r i n c i p l e t h a t "the w h o l e is greater t h a n the s u m of i t s p a r t s " . It m a y also be regarded, often w i t h g o o d reason, as d o i n g violence t o a subject b y either i g n o r i n g or d i s t o r t i n g the very t h i n g s t h a t need to be u n d e r s t o o d . F r o m the s t r u c t u r e p o i n t of v i e w , the difficulties i n these cases result i n one w a y or another f r o m r e d u c t i o n s to s u b s t r u c t u r e s t h a t are not d e t e r m i n i n g . T h o u g h a n o n d e t e r m i n i n g s u b s t r u c t u r e m a y be i n t e r e s t i n g i n its o w n r i g h t , p o s s i b l y i m p o r t a n t i n f o r m a t i o n carried b y the f u l l s t r u c t u r e m a y be inaccessible f r o m i t , so w i l l be i r r e t r i e v a b l y lost i n the r e d u c t i o n . I n other words, the general o b j e c t i o n t o such r e d u c t i o n s is n o t s i m p l y a loss of i n f o r m a t i o n , w h i c h m a y be m o r e or less i n e v i t a b l e , b u t r a t h e r the i r r e t r i e v a b l e loss of essentia? i n f o r m a t i o n . Defects of t h i s k i n d are often present i n efforts t o describe p h e n o m e n a outside of the p h y s i c a l sciences i n p u r e l y p h y s i c a l t e r m s . W e m e n t i o n , i n p a s s i n g , another t y p e of r e d u c t i o n i s m w h i c h is a s p e c i a l -case of the c o n t r a c t i o n process discussed i n the next section. It is i l l u s t r a t e d b y the e x a m p l e of h u m a n society a n d based o n the fact t h a t the l a t t e r is c o m p o s e d of very c o m p l e x i n d i v i d u a l s . T h e i d e a is t h a t i n d i v i d u a l h u m a n beings have c o m p l e x i n t e r n a l structures t h a t o b v i o u s l y p l a y a n essential
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role i n m a n y aspects of the society to w h i c h they b e l o n g . F u r t h e r m o r e , the i n t e r n a l m a k e u p of the i n d i v i d u a l s cannot be deduced s t r i c t l y f r o m the o v e r a l l s o c i a l s t r u c t u r e . A t the same t i m e , sociologists, and also a n t h r o p o l ogists (e.g., R a d c l i f f e - B r o w n ) , regard h u m a n society as a s t r u c t u r e whose o b j e c t s are i n d i v i d u a l h u m a n beings, a r e d u c t i o n t h a t a c c o r d i n g l y excludes e x p l i c i t c o n s i d e r a t i o n of the i n t e r n a l structures of the i n d i v i d u a l members of society. T h e result is therefore an irretrievable loss of i n f o r m a t i o n essent i a l for the u n d e r s t a n d i n g of a variety of h u m a n social p r o b l e m s . T h i s does not m e a n , o f course, t h a t such r e d u c t i o n s are necessarily w i t h o u t value. Some of these p r o b l e m s w i t h social structures are discussed by Peter C a w s [C2, Sec. 40]. M a n y examples of r e d u c t i o n i s t failures result f r o m i g n o r i n g a n i m p o r t a n t feature of c e r t a i n s t r u c t u r e representations. A n extreme e x a m p l e of w h a t we have i n m i n d is p r o v i d e d by the c o m m o n practice of i l l u s t r a t i n g properties of a geometric figure by d r a w i n g s on a piece of paper. I n t h i s case, no one i n t h e i r r i g h t m i n d w o u l d try t o deduce those properties f r o m the physi c a l properties of the paper. Y e t , it is easy to f a l l i n t o e x a c t l y this t y p e of error i n more subtle examples. T h e difficulty lies i n the fact t h a t , a l t h o u g h one s t r u c t u r e m a y be representable as a s u b s t r u c t u r e of another, one c a n not expect to be able to describe, or recover, the s u b s t r u c t u r e s t r i c t l y i n t e r m s of the second s t r u c t u r e . T h e point is t h a t specification of the s u b s t r u c t u r e requires i n f o r m a t i o n external to the representing s t r u c t u r e . T h i s is a s p e c i a l case of the following more general p h e n o m e n o n . G i v e n any n o n t r i v i a l s t r u c t u r e , it is always possible t o b u i l d on it other s t r u c t u r e s whose objects a n d relations may be formed more or less a r b i t r a r i l y out of the objects, substructures, and relations w i t h i n the given s t r u c t u r e . T h i s process m a y be repeated as often as desired, y i e l d i n g an h i erarchical s t r u c t u r e t h a t m a y be q u a l i t a t i v e l y very different f r o m the i n i t i a l s t r u c t u r e . Observe t h a t the c o n s t r u c t i o n w i l l generally fail to be d e t e r m i n e d b y the o r i g i n a l s t r u c t u r e , s i m p l y because the choices at each stage c a n be quite independent of the l a t t e r . In other words, i n d e t e r m i n a t e e x t e r n a l factors m a y enter i n t o the c o n s t r u c t i o n . B i o l o g i c a l s y s t e m s o b v i o u s l y i n v o l v e c o m p l e x s t r u c t u r e s b u i l t o n u n d e r l y i n g c h e m i c a l - p h y s i c a l s t r u c t u r e s i n this m a n n e r , the d r i v i n g force b e i n g the process of e v o l u t i o n . A n o t h e r e x a m p l e is the higher m e n t a l p h e n o m e n a associated w i t h b r a i n s t r u c t u r e , also discussed i n Section 36. Ideas s i m i l a r t o some of the above w i l l be found i n the first a r t i c l e by F o d o r a n d P y l y s h y n i n the b o o k , C o n n e c t i o n s a n d S y m b o l s , edited by P i n k e r a n d M e h l e r [P5, p. 63], If a c o n s t r u c t i o n involves i n d e t e r m i n a t e e x t e r n a l factors, t h e n a reductionist a t t e m p t t o derive i t f r o m the u n d e r l y i n g s t r u c t u r e is l i k e l y to f a i l . O n the other h a n d , it is possible t h a t a r e d u c t i o n f r o m one hierarchy i n the c o n s t r u c t i o n to a lower one w i l l be successful. In fact, this is a c o m m o n
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m e t h o d of s t u d y i n g such c o n s t r u c t i o n s . F o r e x a m p l e , t h o u g h it m a y be i m possible to make a satisfactory analysis of higher m e n t a l functions s t r i c t l y in t e r m s of b r a i n physiology, i t is often possible to f o r m u l a t e a m e a n i n g f u l analysis at a psychological level. A general m i s u n d e r s t a n d i n g of the successes of r e d u c t i o n i s m i n science (and technology) versus its failures i n the h u m a n i t i e s is an i m p o r t a n t factor i n the " T w o C u l t u r e s " gap described by C P . S n o w [S6]. T h i s is a serious s p l i t , not w e l l - u n d e r s t o o d by m a n y on either side. M a n y scientists look w i t h s u s p i c i o n o n any subject t h a t is inaccessible to a precise s t r a i g h t f o r w a r d t r e a t m e n t , a n d w i l l regard it as not w o r t h their serious a t t e n t i o n . T h e y also have l i t t l e patience w i t h the w o r d y discussion style t h a t is so t y p i c a l of the h u m a n i t i e s . T h e i r p r o b l e m is clearly an i n a b i l i t y to see any c o n n e c t i o n between scientific m e t h o d a n d the necessarily different approaches i n the h u m a n i t i e s . M a n y h u m a n i s t s , o n the other h a n d , regard the scientific m e t h o d as c r u d e l y m e c h a n i c a l , a n d , despite the p r o f o u n d i m p a c t t h a t t e c h n o l o g i c a l developments have h a d o n m o d e r n society, u n w o r t h y o f the h u m a n i n t e l lect. A s c o m p a r e d to t r a d i t i o n a l scholars, scientists are often regarded as r e s e m b l i n g robots. T h e p r o b l e m i n this case seems to be s i m p l e ignorance o f the true n a t u r e o f science as a p r o f o u n d l y creative endeavor. A n extreme version of t h i s h u m a n i s t v i e w of science is i l l u s t r a t e d by the f o l l o w i n g r e m a r k made by h i s t o r i a n , S i r I s a i a h B e r l i n [ B l ] , a n d q u o t e d i n an a r t i c l e b y P . J . D a v i s [D2]: " A person w h o lacks c o m m o n intelligence c a n be a physicist of genius, but not even a mediocre h i s t o r i a n " . T h i s s t a t e m e n t , w h i c h is based on a m u c h d i s t o r t e d view of physics, is perhaps not representative o f the m a j o r i t y of h u m a n i s t s , t h o u g h m i l d e r versions are c e r t a i n l y not u n c o m m o n . Despite the a b s u r d i t y of the l i t e r a l s t a t e m e n t , it contains a g e r m of t r u t h , w h i c h is expressed more cogently, interestingly enough, b y a p h y s i c i s t , E . D . C . C o h e n . T h e C o h e n r e m a r k , w h i c h follows, was m a d e i n reference to the c a n d i d a c y of D a v i d B a l t i m o r e , a N o b e l laureate i n biology, for the presidency o f Rockefeller U n i v e r s i t y [C5]: " W h a t Rockefeller needs is a president w h o is wise i n the b i b l i c a l sense. T o w i n a N o b e l P r i z e doesn't m e a n t h a t y o u are wise even t h o u g h y o u are s m a r t and clever. W e w i l l see how wise D a v i d B a l t i m o r e i s " . T h e images of a dedicated scientist projected b y the two s t a t e m e n t s have a c o m m o n element, t h o u g h the second contains far m o r e w i s d o m t h a t the first. A l t h o u g h the c u l t u r e gap is very real a n d the extreme views o n b o t h sides are u n d e r s t a n d a b l e , I believe t h a t a serious s t r u c t u r a l analysis of c e r t a i n p o r t i o n s of the o p p o s i n g d i s c i p l i n e s , a l o n g w i t h an i n d i c a t i o n of how workers deal w i t h the s t r u c t u r e s , w o u l d reveal m u c h t h a t they have i n c o m m o n . T h e r e is n o question t h a t the role of structures is more difficult t o d o c u -
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ment i n the h u m a n i t i e s t h a n i n science and technology, a n d t h a t the s t r u c tures a p p e a r i n g i n the latter are s p e c i a l , often b e i n g o f m a t h e m a t i c a l t y p e . It is also true t h a t these special "scientific" s t r u c t u r e s have m a n y p r o p e r ties t h a t one cannot hope to f i n d elsewhere, and a t t e m p t s to a p p l y t h e m to nonscientific subjects are the source of m a n y r e d u c t i o n i s t f a i l u r e s . Nevertheless, s t r u c t u r e s must be dealt w i t h consciously or u n c o n s c i o u s l y in a l l areas, a n d an awareness of some o f their u n i v e r s a l properties w o u l d d o m u c h t o b r i n g out s i m i l a r i t i e s as opposed to differences between fields. A s y s t e m a t i c exposure o f these s t r u c t u r a l s i m i l a r i t i e s w o u l d do m u c h t o bridge the w i d e n i n g c u l t u r e g a p , a n d m i g h t also help t o reduce the widespread scientific i l l i t e r a c y t h a t plagues our society. 27.
Contractions
T h e a p p r o x i m a t i o n process discussed i n Section 25 m a y be t h o u g h t o f as a n a p p r o a c h t o s t r u c t u r e s " f r o m b e l o w " , or " f r o m w i t h i n " , because it begins w i t h a relatively " s m a l l " part and proceeds to increasingly larger p o r t i o n s of the given s t r u c t u r e . It m a y a p p l y t o structures t h a t are inaccessible as a whole, p o s s i b l y because of their i n f i n i t e extent. A t the other e x t r e m e , there are structures t h a t are l o c a l l y rather t h a n g l o b a l l y inaccessible, perhaps because of u n c e r t a i n or c o m p l e x l o c a l s t r u c t u r e . In such cases, it m a y be possible to d i s t i n g u i s h an o v e r a l l s t r u c t u r e t h a t effectively ignores the l o c a l p r o b l e m s . T h e basic i d e a is t h a t a subs t r u c t u r e ( c o n t a i n i n g , say, the troublesome local i n f o r m a t i o n ) m a y , because of "wholeness", be regarded as an o b j e c t a p a r t f r o m its i n t e r n a l s t r u c t u r e . F u r t h e r m o r e , a given s t r u c t u r e w h i c h is decomposed i n t o (disjoint) s u b structures, w i l l d e t e r m i n e , as we s h a l l see, a second s t r u c t u r e h a v i n g the s u b s t r u c t u r e s as objects. T h e second is a k i n d of a p p r o x i m a t i o n " f r o m a b o v e " , w h i c h ignores the l o c a l i n f o r m a t i o n contained i n the s u b s t r u c t u r e s . T h i s is a very i m p o r t a n t concept w h i c h we c a l l a c o n t r a c t i o n because of the way i t is c o n s t r u c t e d . Its relevance to social s t r u c t u r e s m e n t i o n e d i n the preceding section w i l l become apparent. C o n s i d e r any d e c o m p o s i t i o n o f the objects of the g i v e n s t r u c t u r e i n t o disjoint s u b s t r u c t u r e s . A s far as theory is concerned, such a d e c o m p o s i t i o n c o u l d be d e t e r m i n e d b y a q u i t e a r b i t r a r y d e c o m p o s i t i o n o f the set of objects. T h i s , however, w o u l d generally result i n s o m e t h i n g more or less irrelevant a n d u n i n t e r e s t i n g , so i n a c t u a l p r a c t i c e the d e c o m p o s i t i o n w o u l d n o r m a l l y recognize some key properties of the i n i t i a l s t r u c t u r e . I n any case, (Ae s e t of d i s j o i n t s u b s t r u c t u r e s b e c o m e s t h e s e t of o b j e c t s i n t h e c o n t r a c t i o n , so it o n l y r e m a i n s to give an a p p r o p r i a t e d e f i n i t i o n of the relations i n t e r m s of those i n the g i v e n s t r u c t u r e . F i r s t , we define a c o n t r a c t i o n m a p p i n g f r o m the g i v e n s t r u c t u r e to the c o n t r a c t i o n , b y associating w i t h each object of the g i v e n s t r u c t u r e the s u b -
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s t r u c t u r e t h a t contains i t . E x c e p t i n the case o f a t r i v i a l d e c o m p o s i t i o n , this m a p p i n g w i l l be m a n y - t o - o n e , because at least some o f the substructures w i l l c o n t a i n m o r e t h a n one object. N o w , v i a the c o n t r a c t i o n m a p p i n g , we transfer relations f r o m t h e given s t r u c t u r e t o t h e c o n t r a c t i o n , thus o b t a i n i n g t h e f o l l o w i n g d e f i n i t i o n o f rel a t i o n s for substructures: A collection o f substructures is defined t o be related p r o v i d e d i t is the i m a g e , u n d e r t h e c o n t r a c t i o n m a p p i n g , o f a set o f related o b j e c t s in the given structure. W i t h t h i s d e f i n i t i o n , t h e c o n t r a c t i o n f i n a l l y becomes a b o n a fide s t r u c t u r e . I t w i l l b e c a l l e d t h e c o n t r a c t i o n of t h e g i v e n s t r u c t u r e w i t h r e s p e c t t o t h e p r e s c r i b e d d e c o m p o s i t i o n i n t o s u b s t r u c t u r e s . In passing to a c o n t r a c t i o n , relations t e n d to s i m p l i f y , o r lose some o f t h e i r properties, o r even d i s a p p e a r . T h i s results f r o m t h e i d e n t i f i c a t i o n o f o b j e c t s w i t h i n t h e substructures. W e give n e x t a brief d e s c r i p t i o n o f five s i m p l e b u t very i n s t r u c t i v e exa m p l e s . T h e y w i l l show clearly w h a t is going on a n d suggest t h e w i d e a p p l i c a b i l i t y o f the c o n t r a c t i o n n o t i o n . A m o r e f o r m a l e x a m p l e f r o m group theory, w h i c h i l l u s t r a t e s t h e m a t h e m a t i c a l m o t i v a t i o n for t h e d e f i n i t i o n o f a c o n t r a c t i o n , i s discussed i n the n e x t section. A Geometric Example T h i s e x a m p l e is i l l u s t r a t e d i n F i g u r e 27.1 (a,b,c). T h o u g h i t is n o t i m p o r t a n t i n itself, i t does p r o v i d e a very s i m p l e i l l u s t r a t i o n o f h o w t h e c o n t r a c t i o n process w o r k s . T h e i n i t i a l s t r u c t u r e (a) is a complete h e x a g o n (See F i g . 18.1.), i n w h i c h each i n f i n i t e l i n e is replaced b y a l i n e segment d e t e r m i n e d b y t w o vertices. It i s a p o i n t - l i n e s t r u c t u r e w i t h t h e s i x vertices as objects a n d l i n e segments as r e l a t i o n s .
to]
lb) Fig.
(c)
27.1
A s suggested b y t h e d o t t e d contours, t h e s t r u c t u r e is d e c o m p o s e d i n t o three disjoint substructures, P , L , a n d T (for " p o i n t " , " l i n e " , a n d " t r i a n g l e " ) , d e t e r m i n e d respectively b y t h e three sets o f vertices, {5}, { 1 , 6 } , a n d { 2 , 3 , 4 } w i t h i n the contours.
67
IV. M A N A G E M E N T O F C O M P L E X S T R U C T U R E S
T h e c o n t r a c t i o n w i t h respect t o these substructures is represented i n (c), w h i l e (b) represents a n i n t e r m e d i a t e stage. T h e r e l a t i o n i n the c o n t r a c t i o n is also a ( s y m m e t r i c ) b i n a r y r e l a t i o n , whose d o m a i n of d e f i n i t i o n consists of a l l the d i s t i n c t p a i r s of the substructures P , L , a n d T, (P,L),(P,T),(P,T), p l u s ( L , L ) a n d ( T , T ) , w h i c h are images of p a i r s of objects c o n t a i n e d i n L a n d T, respectively. T h e l a t t e r two do not c a r r y any essential structure i n f o r m a t i o n , so m i g h t as w e l l be o m i t t e d . T h e i n t e r m e d i a t e figure (b) suggests h o w several relations i n the i n i t i a l s t r u c t u r e m a y collapse i n t o a single one i n the c o n t r a c t i o n . Block Diagrams T h e next e x a m p l e , w h i c h is t o t a l l y n o n m a t h e m a t i c a l i n character, m a k e s use o f a t y p i c a l "block d i a g r a m " . N o t e t h a t block d i a g r a m s are h i g h l y s i m plified versions of r e l a t i v e l y c o m p l e x structures, a n d are often used, for e x a m p l e , to present s c h e m a t i c versions o f such t h i n g s as e l e c t r i c a l c i r c u i t s a n d flow chart representations of c o m p l e x c o m p u t e r p r o g r a m s . A n y block d i a g r a m is essentially a d i a g r a m o f a c o n t r a c t i o n , a n d , conversely, m a n y contractions m i g h t be conveniently represented as block d i a g r a m s . T h e exa m p l e , i l l u s t r a t e d i n F i g u r e 27.2, represents a possible c o m p u t e r s y s t e m for a n office c o m p l e x . T h e blocks represent, o f course, the c o n t r a c t e d s u b s t r u c tures of a n i n i t i a l s t r u c t u r e .
Typical office. microcomputer local printer
N.
remote micro. interface
communications controller remote Terminal
terminal
central disc
storage
central computer facilities
central printer backup storage
F i g . 27.2 A p l a n , such as the one suggested, m a y be f o r m u l a t e d w i t h p r a c t i c a l l y no t e c h n i c a l knowledge as to h o w i t m i g h t be i m p l e m e n t e d . I n other words, the detailed i n f o r m a t i o n i m p l i c i t i n the i n d i v i d u a l b l o c k s does not enter
(58
STRUCTURALISM
AND STRUCTURES
d i r e c t l y i n t o the o v e r a l l p l a n . T h e a c t u a l w o r k i n g out of the i n t e r n a l s t r u c tures of the blocks, a l o n g w i t h the details of the relations a m o n g t h e m (that i s , p r o d u c i n g the s t r u c t u r e of w h i c h t h i s is a c o n t r a c t i o n ) , w o u l d require considerable knowledge of c o m p u t e r technology. E x a m p l e s of this k i n d , o f w h i c h there are m a n y , i l l u s t r a t e the general fact t h a t , an u n d e r s t a n d ing o f a c o n t r a c t i o n m a y be r e l a t i v e l y u n s o p h i s t i c a t e d as c o m p a r e d t o a n u n d e r s t a n d i n g of the i n i t i a l s t r u c t u r e . Black Boxes A n o t h e r rather different k i n d of c o n t r a c t i o n is i l l u s t r a t e d by w h a t m i g h t be called the "black box" a p p r o a c h t o a complex m a c h i n e . F r o m t h i s p o i n t of v i e w , the m a c h i n e is regarded as consisting of a c o l l e c t i o n of parts (the " b l a c k b o x e s " ) each of w h i c h performs a p a r t i c u l a r f u n c t i o n i n the o v e r a l l o p e r a t i o n of the m a c h i n e . K n o w l e d g e of the various p a r t s , a l o n g w i t h their special functions and their f u n c t i o n a l r e l a t i o n s h i p s , w o u l d c o n s t i t u t e one level of u n d e r s t a n d i n g of the m a c h i n e a n d its f u n c t i o n . S u c h u n d e r s t a n d i n g need not involve any knowledge of the i n t e r n a l structure of the p a r t s , hence the t e r m "black b o x e s " . O n e m a y be t o t a l l y i g n o r a n t of how the a c t i o n of each p a r t is p r o d u c e d a n d yet u n d e r s t a n d i n a very p r a c t i c a l sense how the whole m a c h i n e works. T h i s is the k i n d of u n d e r s t a n d i n g w h i c h the great m a j o r i t y of us depend o n i n d e a l i n g w i t h the m a n y machines t h a t are taken for g r a n t e d i n our m o d e r n society. C o n s i d e r , for e x a m p l e , the level of u n d e r s t a n d i n g t h a t a n average person must possess i n order t o operate a n a u t o m o b i l e a n d keep it i n reasonable r u n n i n g c o n d i t i o n . A possible collection o f (black box) a u t o m o b i l e parts m i g h t consist o f the engine, fuel t a n k , b a t t e r y , gear shift, c l u t c h , accelerator, steering m e c h a n i s m , wheels, brakes, etc. E a c h person w i l l have some i d e a , perhaps rather vague, o f the functions of the various parts a n d how they interact to produce a w o r k i n g a u t o m o b i l e , but m a y not u n d e r s t a n d a n y t h i n g concerning the parts themselves. T h e black box a p p r o a c h m a y , of course, be rather s o p h i s t i c a t e d . For e x a m p l e , an expert a u t o m o b i l e m e c h a n i c w o u l d have some u n d e r s t a n d i n g of the i n t e r n a l w o r k i n g s of each p a r t a n d precisely how the various parts i n t e r a c t , a n d his o v e r a l l u n d e r s t a n d i n g w o u l d be such t h a t he c o u l d trace a m a l f u n c t i o n to a defective p a r t and replace it w i t h a g o o d one. A t the same t i m e , he m i g h t have l i t t l e or no knowledge of the p r i n c i p l e s of m e chanics, physics, and c h e m i s t r y u p o n w h i c h the o p e r a t i o n of an a u t o m o b i l e u l t i m a t e l y depends. Elementary Chemistry O u r f o u r t h e x a m p l e o f a c o n t r a c t i o n , w h i c h is a b i t more t e c h n i c a l t h a n the others, is a m u c h s i m p l i f i e d account of the way t h a t a t o m s and molecules enter i n t o the s u b j e c t of elementary chemistry. A s b a c k g r o u n d , i t is interest-
IV. M A N A G E M E N T O F C O M P L E X
STRUCTURES
69
i n g f r o m the p o i n t of v i e w of s t r u c t u r a l i s m t o k n o w t h a t , u n t i l the l a t t e r h a l f of the 19th century, the t e r m " a t o m " was c o m m o n l y used t o i n c l u d e b o t h a t o m s a n d molecules. F u r t h e r m o r e , even as l a t e as the 1890's, there were disagreements a m o n g chemists concerning the a c t u a l existence o f a t o m s . S o m e believed t h a t they were n o t h i n g m o r e t h a n convenient artifacts of the theory, a m o u n t i n g o n l y t o an efficient m e t h o d o f o r g a n i z i n g c h e m i c a l k n o w l e d g e . F o r a b r i e f account of this controversy, see the b i o g r a p h y of E i n s t e i n b y A b r a h a m P a i s [ P I , C h a p t e r 5]. G e n e r a l l y s p e a k i n g , e l e m e n t a r y c h e m i s t r y is concerned w i t h t w o k i n d s of structures: (1) m o l e c u l a r s t r u c t u r e s , i n w h i c h the objects are a t o m s a n d the relations are d e t e r m i n e d b y the forces between a t o m s , a n d (2) the structures represented b y c h e m i c a l substances, i n w h i c h the objects are molecules a n d the r e l a t i o n s are d e t e r m i n e d b y the forces t h a t b i n d molecules together. B e y o n d these are the structures o f a t o m s themselves, the s t u d y o f w h i c h lies i n the p r o v i n c e of a t o m i c physics. A t o m s appear as substructures of the general p h y s i c a l s t r u c t u r e w h i c h underlies a t o m i c theory. A c o n t r a c t i o n of the l a t t e r therefore produces the basic c h e m i c a l s t r u c t u r e w i t h a t o m s as objects. It m a y be t h o u g h t of as c o n s i s t i n g of a l l the a t o m s i n the universe a n d described w i t h o u t reference t o the i n t e r n a l s t r u c t u r e o f the a t o m s . M o l e c u l e s a p p e a r as substructures of the basic c h e m i c a l s t r u c t u r e , so another c o n t r a c t i o n produces the s t r u c t u r e w i t h molecules as o b j e c t s , t h a t underlies m o l e c u l a r chemistry. It m a y be described w i t h o u t reference t o the i n t e r n a l a t o m i c s t r u c t u r e of molecules. C h e m i c a l substances a p p e a r as substructures of the m o l e c u l e s t r u c t u r e , so a f i n a l c o n t r a c t i o n produces a s t r u c t u r e w i t h substances as objects. A C o n t r a c t i o n of the Plane V
[,«>.
Each point (x,y) is mapped into the point x on the x-axis.
to, M. (o,6) (o,6)
<
For a general s y s t e m of t h i s k i n d , a p o i n t P of E is defined to be regular if
i t a d m i t s a n e i g h b o r h o o d such t h a t the structures c o r r e s p o n d i n g to p o i n t s of the n e i g h b o r h o o d are i s o m o r p h i c . O t h e r w i s e , is s a i d t o be T h e regular p o i n t s (if any exist) constitute the a n d the r e m a i n i n g p o i n t s , a l l of w h i c h are s i n g u l a r , c o n s t i t u t e the It w o u l d be unreasonable t o expect regular p o i n t s t o exist w i t h o u t c o n d i t i o n s o n / t h a t somehow recognize s p e c i a l properties of the t w o spaces. R a t h e r t h a n p u r s u e these general questions, we go d i r e c t l y to our e x a m p l e , w h i c h , t h o u g h very s p e c i a l , e x h i b i t s some o f the i n t e r e s t i n g p h e n o m e n a associated w i t h such systems.
P singular. region of stability, singular set.
T h e space E i n the e x a m p l e w i l l be a E u c l i d e a n p l a n e , w i t h coordinates a n d the structures w i l l be conic sections represented i n a second Euclidean plane w i t h coordinates T h e conic i n the i y - p l a n e associated w i t h the p o i n t (s, f) is g i v e n b y the s p e c i a l e q u a t i o n ,
(s,t),
E'
(x,y).
sx2 + y2 - 2tx - 2ty +12 C(s,t).
= 0,
C(s,t)
a n d denoted b y The manner i n which t h r o u g h o u t the s t - p l a n e is suggested b y F i g u r e 6 4 . 1 .
depends o n
and (
1
I! H hyperbolas
s
E
P
parabolas
ellipses circles
D" lines H hyperbolas
D' points P
parabolas
HI
£ ellipses IV
Fig.
64.1
-> s
IX. S P A C E S T R U C T U R E S A N D
STABILITY
187
O b s e r v e t h a t the s t - p l a n e is d i v i d e d i n t o five d i s j o i n t subsets denoted by ( D ' , D", P, E, H) a n d consisting respectively of those p o i n t s (s, t) for w h i c h is a p o i n t (the o r i g i n ) , a l i n e , a p a r a b o l a , an ellipse ( i n c l u d i n g the circle), or a h y p e r b o l a . D' is the r i g h t h a l f and D" is the left h a l f of the _-axis, w i t h the o r i g i n assigned to D". P is the (-axis m i n u s the o r i g i n . T h e set E consists of q u a d r a n t s I and I V , w h i l e H consists of q u a d r a n t s II a n d III, w i t h the axes o m i t t e d i n each case. N o t e t h a t E and H are open sets. T h e s i t u a t i o n m a y be described precisely as follows:
C(s,t)
t=
C(s,t)
If 0, then the conic is degenerate, r e d u c i n g t o a single p o i n t w h e n s > 0, two d i s t i n c t lines when s < 0, a n d two copies of the x - a x i s w h e n s = 0. If ( / 0, then C ( s , ( ) is always a nondegenerate conic tangent to the y - a x i s at the p o i n t (0, f), a vertex of the o c n i c . T h e type of conic depends o n the value of s: If 8 = 0, then C ( 0 , () is a p a r a b o l a w i t h vertex (0,i).
s
C(s,t)
If > 0, then is an ellipse w i t h center C ( l , t ) is a circle w i t h center ((,() and r a d i u s (.
(t/s,t).
If s < 0, then C ( s , f ) is a h y p e r b o l a w i t h center
In p a r t i c u l a r ,
(t/s,t).
If we now define s t r u c t u r e i s o m o r p h i s m s i n t e r m s o f the affine t r a n s f o r m a t i o n s of the p l a n e , then the sets E and H c o n s t i t u t e the regular p o i n t s . T h i s means t h a t each p o i n t of E (or of H) a d m i t s a n e i g h b o r h o o d in w h i c h is an ellipse (or a h y p e r b o l a ) . A l l other points of the s f - p l a n e are s i n g u l a r . A l t h o u g h every n e i g h b o r h o o d of a s i n g u l a r p o i n t contains points whose associated s t r u c t u r e s are not i s o m o r p h i c , the b e h a v i o r f r o m one s i n gular p o i n t to another may vary a great deal. T h u s C ( s , t ) degenerates to a p o i n t (the o r i g i n ) on D' and to a pair of lines o n D", but is a p a r a b o l a on
C(s,t)
P.
C(s,t)
If for each p o i n t (s, f) we replace by the abstract s t r u c t u r e defined b y the collection of conies affinely isomorphic to C ( x , f ) , the result is a s t r u c t u r e - v a l u e d f u n c t i o n C ( s , ( ) w h i c h is constant i n each of the sets E and _ / , but is not constant i n any neighborhood of a s i n g u l a r p o i n t . In other words, the s t r u c t u r e values of the function undergo "changes of f o r m " near s i n g u l a r p o i n t s . F u r t h e r m o r e , as we have already noted (Section 12), such changes must i n one way or another take place a b r u p t l y . It is therefore suggestive to describe t h i s phenomenon by s a y i n g t h a t a s t r u c t u r e - v a l u e d f u n c t i o n is " d i s c o n t i n u o u s " at its singular p o i n t s . However, as the above example already shows, this k i n d of d i s c o n t i n u i t y is generally more c o m p l e x t h a n m i g h t be s u r m i s e d f r o m the f a m i l i a r p i c t u r e of a " j u m p " d i s c o n t i n u i t y i n the g r a p h of a s i m p l e f u n c t i o n . Because a l l o f the change takes place near the s i n g u l a r p o i n t s , it is obvious t h a t properties of the s i n g u l a r set are c e n t r a l to a s t u d y of systems of this k i n d . A
18S
65.
STRUCTURALISM
Catastrophe
AND
STRUCTURES
Theory
T h e m a t h e m a t i c s u n d e r l y i n g catastrophe theory, w h i c h includes m a n y i m p o r t a n t c o n t r i b u t i o n s to pure a n d a p p l i e d m a t h e m a t i c s , goes back to work by P o i n c a r e , and is c o m m o n l y k n o w n under the less d r a m a t i c labels of " s i n g u l a r i t y " or " b i f u r c a t i o n " theory. T h e s y s t e m a t i c development of the subject i n its present f o r m , however, is due to Rene T h o r n [ T l ] , w h o suggested m a n y of the p o t e n t i a l a p p l i c a t i o n s to other fields r a n g i n g f r o m physics t h r o u g h cosmology and b i o l o g y to language and t h o u g h t . H e also i n t r o d u c e d the t e r m "catastrophe" to describe the relevant p h e n o m e n a perceived i n each of these areas. Because of the s t r o n g analogy between catastrophe theory and p h e n o m e n a i n so m a n y other fields, few other m a t h e m a t i c a l subjects have a t t r a c t e d as m u c h general a t t e n t i o n . It has been covered b y stories i n m a j o r newspapers and magazines as one of the great i n t e l l e c t u a l movements of the century, a m o u n t i n g to a m a t h e m a t i c a l r e v o l u t i o n c o m p a r a b l e to t h a t brought o n by N e w t o n ' s i n v e n t i o n of the C a l c u l u s ! A l t h o u g h m u c h of the reason for this a t t e n t i o n is due to the broad (and often m i s u n d e r s t o o d ) claims made for the subject by T h o r n himself, perhaps even more is due to its vigorous p r o m o t i o n by E . C . Z e e m a n , who has o u t l i n e d i n some d e t a i l a p p l i c a t i o n s to a wide variety of p r o b l e m s . These i n c l u d e , a l o n g w i t h t r a d i t i o n a l topics f r o m physics and engineering, developmental biology, conflicting j u d g e ments caused by stress, the stock exchange, and p r i s o n disturbances [ Z l ] . A fact w h i c h accounts for m u c h of the general interest a n d e n t h u s i a s m for catastrophe theory is t h a t the m a t h e m a t i c a l p h e n o m e n a i n the elementary theory are easy to visualize and are s t r i k i n g l y s i m i l a r to p h e n o m e n a equally easy to observe i n m a n y other fields. S i m i l a r i t i e s as s t r o n g as these cert a i n l y suggest a c o m m o n u n d e r l y i n g s t r u c t u r e of some k i n d . O n the other h a n d , it is a very large step to presume t h a t the m a t h e m a t i c a l s t r u c t u r e s , or even future generalizations of t h e m , w i l l be adequate to treat a l l such p h e n o m e n a . Y e t s o m e t h i n g close to this is suggested i n some discussions of the subject. S k e p t i c i s m at such c l a i m s need not extend to the possible value of using the m a t h e m a t i c a l s t r u c t u r e as a descriptive or m e t a p h o r i c a l device to "exp l a i n " p h e n o m e n a not yet susceptible to m a t h e m a t i c a l t r e a t m e n t . S u c h use m a y suggest a theoretical treatment a p p r o p r i a t e to the field i n question, and m i g h t even have some predictive value, quite apart f r o m the p o s s i b i l i t y or not of c o n s t r u c t i n g a q u a n t i t a t i v e m o d e l . W i t h respect to this p o i n t , i t is b o t h relevant and i l l u m i n a t i n g to see w h a t T h o r n has to say on the question of q u a n t i t a t i v e m o d e l i n g i n the s o c i a l sciences [ Z l , p. 637], H i s r e m a r k s , w h i c h refer i n d i r e c t l y to a classification of s t r u c t u r e s , are o b v i ously a p p l i c a b l e to a m u c h wider class of structures t h a n those involved i n catastrophe theory.
IX. S P A C E S T R U C T U R E S A N D S T A B I L I T Y
189
In s o c i a l sciences, s t i l l more t h a n i n exact sciences, the hope of findi n g q u a n t i t a t i v e m o d e l l i n g of catastrophes is very s l i g h t . G r a n t e d t h a t C T leads t o b a s i c a l l y q u a l i t a t i v e m o d e l l i n g , w h a t m a y be the i n t e r est of such m o d e l s ? C e r t a i n l y not e x p e r i m e n t a l c o n f i r m a t i o n , w h i c h w o u l d not be at a l l s u r p r i s i n g , since the m o d e l is c o n s t r u c t e d precisely to generate the given morphology. A first answer, I t h i n k , is as f o l l o w s : C T is ( q u i t e l i k e l y ) the first coherent a t t e m p t (since A r i s t o t e l i a n L o g i c ) to give a theory o n W h e n n a r r o w - m i n d e d scientists o b j e c t to C T t h a t it gives no more t h a n analogies, o r m e t a p h o r s , they do not realize t h a t they are s t a t i n g the proper a i m of C T , w h i c h is to classify a l l possible types of analogous s i t u a t i o n s ... N o w the posi t i v i s t o b j e c t i o n may be rephrased as follows: W h e r e a s q u a n t i t a t i v e m o d e l l i n g allows us t o use c o m p u t a t i o n , a n d therefore is more powerful t h a n c o m m o n sense i n t u i t i o n , how c o u l d q u a l i t a t i v e m o d e l l i n g be stronger t h a n u s u a l , o r d i n a r y language d e d u c t i o n ? H o w can a q u a l i t a t i v e m o d e l be s o m e t h i n g more t h a n an i d l e , superfluous geometric p i c t u r e of c o m m o n sense i n t u i t i o n ? T h i s o b j e c t i o n , I believe, has some v a l i d i t y . B u t i t w i l l lose its s t r e n g t h , precisely i n so far as a complete C T w i l l be c o n s t r u c t e d , w h i c h w i l l a l l o w f o r m a l d e d u c t i o n , and c o m b i n a t o r i a l generation of new forms f r o m a set of g i v e n forms. In as m u c h as C T develops i n t o a f o r m a l s y n t a x of ( p l u r i d i m e n s i o n a l ) c a t sastrophes, we w i l l be able t o go f r o m a purely v e r b a l d e s c r i p t i o n to an a b s t r a c t , t o p o l o g i c a l m o r p h o l o g y w h i c h we w i l l be able t o h a n d l e w i t h p u r e l y f o r m a l , algebraic tools. Hence we m i g h t p u t i n t o c o n n e c t i o n a p p a r e n t l y disjoint facts, predict unexpected s i t u a t i o n s , or, at least, reduce the a r b i t r a r i n e s s o f the d e s c r i p t i o n . A s I s a i d earlier, reducing the a r b i t r a r i n e s s of the d e s c r i p t i o n r e a l l y is the proper d e f i n i t i o n of scientific e x p l a n a t i o n . . . .
analogy.
T h e t e r m " c a t a s t r o p h e " is an extreme e x a m p l e of the use o f a word f r o m o r d i n a r y language to designate a pure m a t h e m a t i c a l concept. T h e u s u a l m o t i v a t i o n i n such usage is t h a t some aspect of the m e a n i n g or c o n n o t a t i o n of the chosen word suggests i n some way the associated m a t h e m a t i c a l concept. ( N o t i c e t h a t this practice is analogous to the m e t a p h o r i a l use of m a t h e m a t i c a l language i n n o n m a t h e m a t i c a l fields, a use sometimes c r i t i cized because it gives a false impression of scientific content!) A l t h o u g h the practice is c o m m o n t h r o u g h o u t m a t h e m a t i c s , it is sometimes confusing to outsiders, w h o often t r y to discover a t e r m ' s m a t h e m a t i c a l m e a n i n g by s t u d y i n g its etymology, or t o connect some of its irrelevant m e a n i n g s or c o n n o t a t i o n s to the m a t h e m a t i c s . F r o m the s t a n d p o i n t of s t r u c t u r e s , this approach makes a certain a m o u n t of sense, but the choice of such m a t h e m a t i c a l t e r m i n o l o g y is often so superficial t h a t n o t h i n g of m u c h significance can result. A t any r a t e , the w o r d " c a t a s t r o p h e " is c e r t a i n l y suggestive, i f
190
STRUCTURALISM
AND
STRUCTURES
s o m e w h a t overly d r a m a t i c , for some of the m a t h e m a t i c a l p h e n o m e n a o b served i n s i n g u l a r i t y theory. O n the other h a n d , i t is also different i n t h a t i t offers u n u s u a l l y s t r o n g encouragement t o the i l l u s i o n t h a t the m a t h e m a t ics i n q u e s t i o n m i g h t carry over to areas where even the p o s s i b i l i t y o f any r i g o r o u s m a t h e m a t i c a l t r e a t m e n t is d e b a t a b l e . 66.
Zeeman's Catastrophe M a c h i n e
T h e i n g e n i o u s l y s i m p l e " m a c h i n e " described here is due to E . C . Z e e m a n [ Z l ] . Since i t m a y be easily c o n s t r u c t e d b y anyone w h o is interested, it offers a n o p p o r t u n i t y t o observe the catastrophe p h e n o m e n o n first h a n d . T h e m a c h i n e , w h i c h is i l l u s t r a t e d i n F i g u r e 6 6 . 1 , consists of a r e c t a n g u l a r b o a r d , a ( r i g i d ) c i r c u l a r disc, a n d two r u b b e r b a n d s . T h e d i a m e t e r o f the disc s h o u l d be a b i t larger t h a n the n a t u r a l l e n g t h of a r u b b e r b a n d . A s i n d i c a t e d i n the figure, the center 0 of the disc is fastened t o the b o a r d so t h a t the disc m a y r o t a t e freely. T h e r u b b e r b a n d s are fastened to the disc at a p o i n t Q near its p e r i m e t e r , a n d the other end o f one b a n d is fastened t o a p o i n t A of the b o a r d , so t h a t distance AO is e q u a l t o a p p r o x i m a t e l y twice the n a t u r a l l e n g t h o f the b a n d . T h e free end C of the other b a n d m a y be m o v e d freely over the surface o f the b o a r d . T h e l a t t e r is called the " c o n t r o l space" a n d the p o i n t C is c a l l e d the " c o n t r o l p o i n t " . T h e value z of the angle between the line a n d the r a d i u s of the disc is c a l l e d the " s t a t e " of the s y s t e m .
/.AOQ,
AO
Fig.
OQ
66.1
W i t h each choice of the p o i n t C , the s y s t e m w i l l assume b y disc r o t a t i o n a state za for w h i c h the p o t e n t i a l energy c o n t a i n e d i n the stretched bands is a l o c a l m i n i m u m (i.e., a m i n i m u m a m o n g those energy values of states z near „ _ ) If the p o i n t C is m o v e d c o n t i n u o u s l y over the surface o f the b o a r d , the angle z w i l l also change continuously, except for c e r t a i n l o c a t i o n s at w h i c h a s m a l l change of p o s i t i o n m a y cause a n a b r u p t change o f state. B y experi m e n t a t i o n , one m a y locate enough of these e x c e p t i o n a l p o i n t s t o suggest t h a t they lie o n a s y m m e t r i c , d i a m o n d - s h a p e d curve w i t h four cusps, as
IX. S P A C E S T R U C T U R E S A N D
STABILITY
191
i n d i c a t e d i n the figure. W h e n C is m o v e d u p w a r d s along a v e r t i c a l line w h i c h intersects the r i g h t h a l f of the d i a m o n d , the angle z w i l l change c o n t i n u o u s l y u n t i l C reaches a p o i n t D of the u p p e r b o u n d a r y of the d i a m o n d , w h e r e u p o n the disc w i l l s u d d e n l y t u r n counter-clockwise, s h i f t i n g the p o i n t Q f r o m below to a b o v e the center line. If C is m o v e d back d o w n the v e r t i c a l line, the s u d d e n shift back w i l l take place as C crosses the lower b o u n d a r y of the d i a m o n d . If the v e r t i c a l line intersects the left h a l f of the d i a m o n d , t h e n the same t h i n g w i l l h a p p e n , except t h a t the i n i t i a l a b r u p t disc r o t a t i o n w i l l be c l o c k w i s e . T h e p o i n t C m a y be m o v e d c o n t i n u o u s l y f r o m any p o i n t outside the d i a m o n d to any other p o i n t of the b o a r d w i t h o u t sudden changes of state, p r o v i d e d a p a t h is chosen t h a t does not enter the d i a m o n d f r o m below or above a n d leave f r o m the o p p o s i t e (upper or lower) b o u n d a r y . T h e s a m e is true w h e n the s t a r t i n g p o i n t is inside the d i a m o n d , except t h a t i f the p a t h e x i t s the d i a m o n d , then i t m u s t do so by crossing the " c o r r e c t " side, d e p e n d i n g o n the i n i t i a l state o f the s y s t e m . G i v e n a fixed control p o i n t C , it is possible t o w r i t e d o w n a f o r m u l a for the p o t e n t i a l energy E i n t e r m s of the state v a r i a b l e z a n d coordinates of C , a n d t h e n to o b t a i n an e q u a t i o n i n v o l v i n g coordinates of C a n d z w h i c h m u s t be satisfied b y z i n order for the energy to be a m i n i m u m . Despite the s i m p l i c i t y of the e x a m p l e , the equations involve features t h a t do not concern us here, so we t u r n to a s i m p l e r m a t h e m a t i c a l e x a m p l e c o n s t r u c t e d expressly t o b r i n g out the desired ideas. A n e q u a l l y s i m p l e ( n o n m a t h e m a t i c a l ) e x a m p l e f r o m biology w i l l be described i n the n e x t section. 67.
A Mathematical Example T h e f o l l o w i n g basic e x a m p l e , w h i c h is i l l u s t r a t e d b y the g r a p h s i n F i g u r e 67.1 b e l o w , i n one f o r m or another is a s t a n d a r d i t e m i n most e l e m e n t a r y t r e a t m e n t s o f catastrophe. It is s t r i c t l y m a t h e m a t i c a l i n content, b u t m a y be t h o u g h t of as a r i s i n g f r o m a h y p o t h e t i c a l m e c h a n i c a l s y s t e m . S u p p o s e , as i n Z e e m a n ' s e x a m p l e , t h a t a c e r t a i n m e c h a n i c a l s y s t e m depends o n (or is " c o n t r o l l e d b y " ) two p a r a m e t e r s x a n d y. A choice o f p a r a m e t e r values m a y be represented by a p o i n t (x, y) i n a c o o r d i n a t e p l a n e . T h e collection o f a l l such p o i n t s is called the " c o n t r o l space" o f the s y s t e m . A l s o , let z be a p a r a m e t e r whose values d e t e r m i n e the " s t a t e " of the syst e m . N o w assume t h a t the " p o t e n t i a l energy" E of the s y s t e m , for a fixed p o i n t (*, If) of the control space a n d state z, is given b y the f o r m u l a ,
E= zA -2yz2
-4.xz.
T h e v a r i a b l e s x, y, z are independent of one another, a n d it is assumed t h a t the s y s t e m is free to move f r o m one state to another.
19_
STRUCTURALISM AND
Fig.
67.1
STRUCTURES
IX. S P A C E S T R U C T U R E S A N D
STABILITY
19.'1
For each p o i n t o f the c o n t r o l space, the s y s t e m w i l l a u t o m a t i c a l l y assume a state such t h a t the value of the p o t e n t i a l energy is a l o c a l m i n i m u m . F o r a g i v e n c o n t r o l p o i n t (x, y), the value (or values) of z for w h i c h E is a l o c a l e x t r e m e ( m a x i m u m or m i n i m u m ) must satisfy the f o l l o w i n g e q u a t i o n (in three v a r i a b l e s ) ,
z3 = yz + x. dE/dz
It is o b t a i n e d by s e t t i n g the p a r t i a l d e r i v a t i v e e q u a l t o zero. A p o r t i o n of the g r a p h of this e q u a t i o n , w h i c h is an infinitely extended surface in three-space, is represented below i n F i g u r e 67.1 (a). T h e front edge of the p o r t i o n of the surface represented i n the figure is the curve traced o n the surface b y a v e r t i c a l plane p e r p e n d i c u l a r to the y - a x i s at the p o i n t y = 3. Its e q u a t i o n is given by z
3
= 3z-r~,
a n d its g r a p h i n the x z - p l a n e is represented by F i g u r e 67.1 (b). N o t i c e h o w the surface folds back on itself and t h a t , to each p o i n t ( x , y) i n the x y - p l a n e , there corresponds one, two, or three points on the surface. T h e fold-back p o i n t s , such as, for e x a m p l e , the points (—2,3,1) and ( 2 , 3 , - 1 ) , are those points o n the surface where the tangent plane to the surface is v e r t i c a l . These points f o r m a curve o n the surface, a n d project onto a curve i n the x y - p l a n e w i t h the e q u a t i o n , 4y
3
= 27x . 2
Its g r a p h is i n d i c a t e d i n F i g u r e 67.1 (c), a n d , for reasons t o appear later, it is called the "catastrophe s e t " . For each c o n t r o l p o i n t ( x , y ) outside of the catastrophe set, the z - c o o r d i n a t e of the c o r r e s p o n d i n g p o i n t ( x , y , z) o n the surface determines a state for w h i c h E has a l o c a l extreme value. In order to observe the catastrophe p h e n o m e n o n i n this e x a m p l e , we note the b e h a v i o r of the point P o n the surface as its c o n t r o l p o i n t C i n the x y - p l a n e moves f r o m far left to far right a l o n g line y = 3. A s C traces the l i n e , P moves s m o o t h l y a l o n g the surface (i.e., a l o n g the curve represented i n (b)) u n t i l C reaches the p o i n t ( 2 , 3 ) , w h e r e u p o n P m u s t j u m p s u d d e n l y to the u p p e r p o r t i o n of the curve. T h i s is the catastrophe o c c u r r i n g at the p o i n t near ( 2 , 3 ) . In the same way, i f C moves f r o m far right t o far left a l o n g y = 3, a catastrophe occurs when C reaches the p o i n t (—2,3). T h e p o i n t s (2,3) a n d ( - 2 , 3 ) o b v i o u s l y belong to the catastrophe set c o n s i s t i n g of points of the curve (c). P h e n o m e n a s i m i l a r to the above w i l l o c c u r i f the c o n t r o l p o i n t C moves c o n t i n u o u s l y a l o n g any s m o o t h curve i n the x y - p l a n e t h a t crosses the " g r a y "
STRUCTURALISM AND
194
STRUCTURES
region enclosed b y the curve (c), e n t e r i n g the region f r o m one side {left or r i g h t ) a n d l e a v i n g f r o m the o t h e r , w h i l e a v o i d i n g the cusp p o i n t 0 at the o r i g i n . T h e catastrophe occurs o n l y w h e n C leaves the r e g i o n . T h e cusp p o i n t 0 is s p e c i a l , because C m a y leave t h e region at 0 w i t h o u t a d i s c o n t i n u o u s change of state. T h e effect o f e n t e r i n g at 0 is a m b i g u o u s since a curve m a y be extended c o n t i n u o u s l y f r o m the cusp p o i n t o n the surface a l o n g any one of t h e surface levels. N o t i c e t h a t , except for the m i d d l e p o r t i o n o f the f o l d i n the surface, the p o i n t P m a y be m o v e d c o n t i n u o u s l y to any p o i n t o n the surface b y m o v i n g t h e c o n t r o l p o i n t C a l o n g a s m o o t h curve t h a t p r o p e r l y avoids the catastrophe set. F o r e x a m p l e , the p o i n t B o n the u p p e r level of the surface ( F i g u r e 67.1 (a)) m a y be reached f r o m the p o i n t A o n the lower l e v e l a l o n g a curve L w h i c h projects onto the c o n t r o l space i n a curve V t h a t passes above the cusp a n d approaches the p r o j e c t i o n A' o f A f r o m the r i g h t . C r o s s i n g the catastrophe set f r o m the r i g h t i n t h i s w a y w i l l not produce a d i s c o n t i n u o u s change o f state. T h e p o i n t s o n the m i d d l e o f t h e f o l d c o r r e s p o n d t o states for w h i c h the energy f u n c t i o n has a m a x i m u m . T h e y a c c o r d i n g l y represent u n s t a b l e states a n d are inaccessible t h r o u g h stable states, except p o s s i b l y t h r o u g h the cusp. O n e of the results f r o m s i n g u l a r i t y theory is t h a t the s i n g u l a r i t y set associated w i t h a n a r b i t r a r y ( s m o o t h ) surface i n three space consists at m o s t of f o l d p o i n t s a n d cusps. T h e theory also extends t o control spaces of d i m e n s i o n s different f r o m t w o . T h u s , i f the c o n t r o l space is one d i m e n s i o n a l , t h e n the o n l y possible s i n g u l a r i t i e s are f o l d p o i n t s . A n e x a m p l e is o b t a i n e d f r o m the above e x a m p l e b y r e s t r i c t i n g e v e r y t h i n g t o the l i n e y = 3. T h e s i t u a t i o n becomes progress!vewly m o r e c o m p l e x as the d i m e n s i o n increases, b u t there are s t i l l o n l y a finite n u m b e r o f possible t y p e s o f s i n g u l a r i t i e s , l a b e l e d b y such descriptive t e r m s as " l i p s " , " b e a k s " , a n d " s w a l l o w t a i l s " , t h u s p r o v i d i n g m o r e examples of " c o l o r f u l " m a t h e m a t i c a l t e r m i n o l o g y . For another v i e w of w h a t is h a p p e n i n g here, we present i n F i g u r e 67.2 3 graphs of the energy f u n c t i o n , = z —
E
E
(-3,3)
E
(-2,31
4
2yz — 4xz.
E
[-1,31 Fig.
E
E
E
(1,3)
(2,3)
13,3)
67.2
IX. S P A C E S T R U C T U R E S A N D
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195
T h e g r a p h s , w h i c h correspond to a few s p e c i a l c o n t r o l p o i n t s a l o n g the line y = 3, are not d r a w n t o scale since the result w o u l d be c u m b e r s o m e a n d i t is o n l y necessary t o see clearly the n u m b e r a n d t y p e of c r i t i c a l p o i n t s i n each case. T h e i d e a is t o suggest how the graphs l o c a l m i n i m a , change as the control p o i n t moves f r o m left to r i g h t along the l i n e . T h e l o c a l m i n i m u m in each case is i n d i c a t e d by a heavy " d o t " o n the curve. I n c o o r d i n a t i n g the following observations w i t h the preceding r e m a r k s , it is i m p o r t a n t to a v o i d a confusion of values of z w i t h the c o r r e s p o n d i n g values of E. I n other words, at a l o c a l m i n i m u m of the energy f u n c t i o n , i t is the value o f E t h a t is a m i n i m u m a n d not the value o f z. N o t i c e t h a t i t is the value o f z, rather t h a n the value o f E, t h a t determines a p o i n t o n the surface i n F i g u r e 67.1 (a). T h e energy f u n c t i o n has f r o m one t o three c r i t i c a l p o i n t s . W h e n the c o n t r o l p o i n t C lies i n the " g r a y " region (See F i g u r e 67.1 (c)), there are three p o i n t s , one m a x i m u m a n d t w o m i n i m a . W h e n C is i n the catastrophe set (except for the c u s p ) , there are two p o i n t s , one m i n i m u m a n d one inflection p o i n t . T h e l a t t e r is neither a m a x i m u m nor a m i n i m u m . F o r all other choices of C , there is o n l y one c r i t i c a l p o i n t , a m i n i m u m . A s C moves f r o m left to r i g h t a l o n g the l i n e y = 3, the r - c o o r d i n a t e of the m i n i m u m p o i n t changes c o n t i n u o u s l y u n t i l we reach the p o i n t ( 2 , 3 ) , where the m i n i m u m p o i n t i n question becomes an inflection p o i n t . T h i s is an u n s t a b l e s t a t e , a n d any further change i n C w i l l force z to shift a b r u p t l y f r o m the value -1 to 2 because the l a t t e r value gives the other m i n i m u m point. It is i n s t r u c t i v e t o consider the above e x a m p l e f r o m the p o i n t of v i e w of structures as discussed i n Section 64. T h e idea is to look at the g r a p h of the energy f u n c t i o n (a differentiable f u n c t i o n of z ) as a s t r u c t u r e associated w i t h each p o i n t o f the x y - p l a n e . T h e g r a p h , for a fixed control p o i n t , m a y be a n a l y z e d i n different ways as a s t r u c t u r e , d e p e n d i n g o n w h i c h p r o p e r t i e s one wishes t o e m p h a s i z e . I n the present case, it is the " c r i t i c a l p o i n t s t r u c t u r e " t h a t is i m p o r t a n t . A " c r i t i c a l p o i n t " is a p o i n t o f the g r a p h at w h i c h the tangent is h o r i z o n t a l ( t h a t is, where the d e r i v a t i v e is zero), a n d is either a l o c a l e x t r e m e p o i n t or a n inflection p o i n t . T h e " c r i t i c a l p o i n t s t r u c t u r e " is the ordered sequence of c r i t i c a l p o i n t s o f the g r a p h , a l o n g w i t h the i n f o r m a t i o n as t o whether the p o i n t is a l o c a l m a x i m u m , a l o c a l m i n i m u m , or a n inflection p o i n t . T h e o r d e r i n g of the sequence is w i t h respect to the n a t u r a l o r d e r i n g o n the z-axis.
dE/dz
If we denote a m i n i m u m p o i n t by m (or in'), a m a x i m u m p o i n t by M, a n d an i n f l e c t i o n p o i n t b y I, t h e n the six c r i t i c a l p o i n t s t r u c t u r e s of the graphs i l l u s t r a t e d i n F i g u r e 67.2 may be s y m b o l i z e d as follows: m,
m < I,
m < M < m',
m<M
< in',
I < m,
m,
196
STRUCTURALISM
where " < " means "lies to the left o f .
AND STRUCTURES
T h e r e are four n o n i s o m o r p h i c s t r u c -
tures i n t h i s collection: m,
m < I,
I < m,
m< M
Sq a n d at a s m a l l value of r where the i n s t i n c t t o defend the nest is great, so t h a t D w i l l attack I i n order to d r i v e it away (See F i g . 68.1.). Since the i n s t i n c t t o defend decreases as r increases, there w i l l be a d i s t a n c e at w h i c h the fear o f the larger enemy overrides the d r i v e to a t t a c k . A t this p o i n t , D w i l l shift a b r u p t l y f r o m an a t t i t u d e of attack t o one o f r e t r e a t . T h e u p p e r curve i n the figure represents those points (s, r ) at w h i c h these shifts take place. O n the other h a n d , i f the i n i t i a l encounter o c c u r s at a distance r so t h a t the p o i n t (s, r) lies above t h i s curve, D's first response w i l l be to retreat f r o m the larger i n d i v i d u a l , a n d the retreat w i l l continue t o a p o i n t where i n s t i n c t t o defend the nest becomes s t r o n g enough t o overcome the fear. A t this p o i n t , D ' s a t t i t u d e w i l l shift a b r u p t l y f r o m retreat to a t t a c k . These p o i n t s c o n s t i t u t e the lower curve i n the figure. It is also reasonable to assume t h a t , i f s is larger t h a n some value si (greater t h a n so), the i n s t i n c t t o defend w i l l not be s t r o n g enough to overcome the fear, once the l a t t e r has t a k e n over. U n d e r these c o n d i t i o n s , D w i l l no d o u b t a b a n d o n the nest [ Z l , p . 14]. T h e catastrophe-like phenomenon occurs here because D's b e h a v i o r m o d e when s > so, whether attack or retreat, w i l l tend t o persist u n d e r changes of the v a r i a b l e r beyond a p o i n t where t h a t m o d e w o u l d be i n i t i a t e d . T h e r e fore, when s > so, the intervals of r values t h r o u g h w h i c h the i n i t i a l a t t i tudes persist w i l overlap for the two cases discussed above. T h e above d e s c r i p t i o n , t h o u g h o b v i o u s l y an o v e r s i m p l i f i c a t i o n , is a p p a r ently close enough to the a c t u a l b e h a v i o r of t e r r i t o r i a l fish t h a t i t m i g h t serve as a tentative m o d e l of t h a t b e h a v i o r . F u r t h e r m o r e , the a n a l o g y w i t h the m a t h e m a t i c a l e x a m p l e discussed i n Section 67 is so s t r o n g t h a t i t is t e m p t i n g to assume t h a t a m a t h e m a t i c a l m o d e l m i g h t exist for the present case. T h e p r o b l e m i n the c o n s t r u c t i o n of such a m o d e l is, first, the i d e n t i f i c a t i o n o f a b i o l o g i c a l s t r u c t u r e t h a t w i l l account for the b e h a v i o r i n q u e s t i o n . In a d d i t i o n , t h a t s t r u c t u r e must a d m i t a m a t h e m a t i c a l descript i o n e x h i b i t i n g its dependence on the c o n t r o l variables (s, r) a n d o n one or more i n t e r n a l parameters t h a t d e t e r m i n e the state of the s y s t e m . G i v e n a m o d e l s a t i s f y i n g these c o n d i t i o n s , i t m i g h t then be possible to " p r e d i c t " the cusp-type catastrophe suggested i n F i g u r e 6 8 . 1 . T h e p r a c t i c a l i t y , or even the p o s s i b i l i t y , of s a t i s f y i n g the above c o n d i t i o n s for the e x a m p l e of t e r r i t o r i a l fish, or for any o f the m a n y other s i m i l a r examples, is open t o question. O n the other h a n d , as suggested by T h o r n ' s r e m a r k s concerning q u a n t i t a t i v e m o d e l s for social science quoted a b o v e , i t m a y be too m u c h to d e m a n d for these examples a rigorous m a t h e m a t i c a l m o d e l o f the t r a d i t i o n a l k i n d . It m a y nevertheless be possible to construct a n o n m a t h e m a t i c a l m o d e l , w h i c h e x h i b i t s properties analogous t o those of the m a t h e m a t i c a l e x a m p l e , a n d is also precise enough for m a k i n g at least
I X . SPACE STRUCTURES AND
STABILITY
199
qualitative predictions. 69.
M e t r i c Spaces
T h i s a n d the next section are devoted to a precise m a t h e m a t i c a l t r e a t m e n t o f the " p r i n c i p l e of s t r u c t u r a l s t a b i l i t y " for the case of p o i n t - l i n e s t r u c t u r e s . A s has already been p o i n t e d o u t , the difficult p r o b l e m here is t o f o r m u l a t e a n a p p r o p r i a t e d e f i n i t i o n o f "nearness" for the s t r u c t u r e s . T h i s p r o b l e m accounts for most o f the t e c h n i c a l i t i e s t h a t d o m i n a t e the f o l l o w i n g discussion. A l t h o u g h we are p r i m a r i l y interested i n E u c l i d e a n spaces, i t t u r n s out to be n o t a t i o n a l l y easier here to deal w i t h p o i n t - l i n e s t r u c t u r e s i n a general " m e t r i c space". T h i s section a c c o r d i n g l y contains a d e f i n i t i o n a n d some properties o f m e t r i c spaces.
space
S
A metric is s i m p l y an abstract p o i n t set along with a real-valued "distance f u n c t i o n " d(p, q) defined for each p a i r (p, q) o f p o i n t s i n S. d(p, q) is also c a l l e d a " m e t r i c " a n d is subject to the f o l l o w i n g three c o n d i t i o n s suggested b y c o r r e s p o n d i n g properties of distance i n a E u c l i d e a n space: (1)
PosUiviiy. d(p, q) > 0, w i t h d(p, q) = 0 i f a n d o n l y i f p = q. Symmetry: d(p,q) = d(q,p), for a l l points p a n d q. triangle inequality: d(p,r) < d(p,q) + d(q, r ) , for any
(2) (3) T h e p o i n t s p, q, a n d r o f
three
S.
T h e n u m b e r d(p, q) is defined to be the " d i s t a n c e " between the p o i n t s , so m a y be regarded as a measure of how " n e a r " p is t o q i n S. T h e E u c l i d e a n spaces are o b v i o u s l y m e t r i c spaces, b u t there are m a n y e x a m p l e s of the l a t t e r t h a t are not E u c l i d e a n . I n other words, not a l l properties o f a E u c l i d e a n space are d e t e r m i n e d b y its m e t r i c .
neighborhoods
S
Basic of p o i n t s i n a m e t r i c space are defined e x a c t l y as i n the s p e c i a l case of a E u c l i d e a n space. A t y p i c a l such n e i g h b o r h o o d is denoted by where £ is an a r b i t r a r y p o s i t i v e n u m b e r , a n d consists of the set o f a l l p o i n t s q i n S such t h a t d(p, q) < £. N o w consider a finite p o i n t - l i n e s t r u c t u r e c o n t a i n e d i n the m e t r i c space 5". Its objects w i l l c o n s t i t u t e a set of points in a n d the s t r u c t u r e w i l l be denoted b y F . R e c a l l t h a t the s t r u c t u r e r e l a t i o n is b i n a r y a n d therefore m a y be represented by a d i s t i n g u i s h e d collection of ordered pairs of p o i n t s of F. T h u s , an o b j e c t / w i l l be related t o another o b j e c t / ' p r o v i d e d ( / , / ' ) is i n t h a t collection. It w i l l also be convenient to a d o p t the convention t h a t each object of the s t r u c t u r e is related t o itself, so the d i s t i n g u i s h e d c o l l e c t i o n w i l l c o n t a i n a l l pairs of the f o r m ( / , / ) - O b s e r v e t h a t w i t h these conventions the s t r u c t u r e A is represented as a (finite) set of of p o i n t s of T h i s m e a n s t h a t FA is represented as a subset [ F ] of the C a r t e s i a n 5 x 5 o f the space w i t h itself. T h e l a t t e r consists of pairs
N(p,e),
finite
F
S
A
F , including both objects and pairs F.
relations,
A
product
S
all
STRUCTURALISM
200
AND STRUCTURES
(p, q) of p o i n t s p a n d q f r o m S, a n d the subset consisting of a l l p o i n t s (p, q) w i t h p = q is c a l l e d the " d i a g o n a l " of 5" x S . I n the representation [F*] of F i n 5 x 5 , the objects (points of F) c o r r e s p o n d t o d i a g o n a l elements a n d every d i a g o n a l element i n [ F ] represents a n o b j e c t . I n other words, the objects of FA are i n one-to-one correspondence w i t h the d i a g o n a l elements i n [ F ] . Observe t h a t [FA] also has the p r o p e r t y t h a t , i f i t contains a p o i n t (p, q), t h e n p a n d q are p o i n t s of F so it also contains the d i a g o n a l elements (p, p) a n d (a, q). Conversely, i t is easy t o see t h a t any finite subset of S x 5 w i t h t h i s p r o p e r t y a c t u a l l y represents a p o i n t - l i n e s t r u c t u r e i n S. A
A
A
A l t h o u g h i t is i m p o s s i b l e , i n g e n e r a l , t o c o n s t r u c t a t r u e p i c t u r e of 5 x 5 , it m a y be represented s y m b o l i c a l l y i n an o r d i n a r y c o o r d i n a t e p l a n e , as suggested b y F i g u r e 6 9 . 1 , where the space 5 is represented b y the p o s i t i v e h a l f of each n u m b e r axis. The Cartesian Product,
5 x 5
I P . p ' l
p'
P . P i
! —
/
i • f
s
iaqonal
___
ip'.pl
I P'
Fig.
69.1
It t u r n s out t h a t S x S is also a m e t r i c space under the f o l l o w i n g m e t r i c : ),(g,g'))
=
IX.
SPACE STRUCTURES A N D STABILITY
203
a n d hence,
d({f,n(9,9'))<s(F)/2. B y d e f i n i t i o n o f distances i n S x 5", t h e last i n e q u a l i t y is equivalent to the two inequalities,
d(f,g) < *(F)/2 and d(f'g') < s(F)/2, F u r t h e r m o r e , i f (e,e') were a n y other p o i n t of [FA]
such t h a t
d(e,g) < s(F)/2 and d(e',g') < s{F)/2, t h e n , by the t r i a n g l e i n e q u a l i t y ( a n d s y m m e t r y ) , it w o u l d follow t h a t d(e, / ) < d(e,
g) < s ( / ) / 2 + s ( F ) / 2 =
g) + d(f,
s(F).
d(e,f) < s(F),
This implies that which can hold only i f e = / . A similar a r g u m e n t also gives e' = / ' . I n other words, the o b t a i n e d p o i n t ( / , / ' ) i n A T h i s means t h a t [ F ] is u n i q u e l y d e t e r m i n e d b y t h e p o i n t in the c o n d i t i o n
(g,g')
A
[G ].
d((f,n(g,g'))<s(F)/2 determines a m a p p i n g M
of [ G ] i n t o [FA],
M(g,g') = so
with
A
(fJ'),
d{M(g,g'),(gl9'))<S(F)/2.
F u r t h e r m o r e , i f (e, e') is a n y p o i n t o f [FA], t h e n , as i n the p r e c e d i n g a r g u m e n t , there exists (h, h') i n [GA] such t h a t
d((e,e'),(h,h'))<s(F)/2. M(k,h')
-
M
onto [F ].
A Therefore, ( c , c ' ) . T h i s proves t h a t maps [ G ] F u r t h e r m o r e , since [ F J a n d [GA] have the same n u m b e r of p o i n t s , t h e m a p p i n g m u s t be one-to-one, c o m p l e t i n g t h e p r o o f of statement ( 1 ) . F o r t h e p r o o f o f (2), let (g,g) be a n a r b i t r a r y d i a g o n a l element of [GA], a n d set (e, T h e n , b y the t r i a n g l e i n e q u a l i t y ( a n d s y m m e t r y ) , A
A
e') = M(g,g).
d(e, e') < d{e, g) + _(„', g) < s(F)/2 + s(F)/2 = s(F), d(e,e') < ${F),
so a n d i t follows again t h a t e = e'. I n other words, each d i a g o n a l element of [ G ] is m a p p e d b y M to a d i a g o n a l element of [ F ] . A
A
204
STRUCTURALISM
M
AND STRUCTURES
onto [F ],
A there m u s t exist for each d i a g Since m a p s [ G ] one-to-one o n a l element (/, / ) o f [ F ] a n element (g, g') i n [ G ] such t h a t M(g, g') = (/, / ) . A l s o , since M m a p s d i a g o n a l elements o f [ G ] to d i a g o n a l elements o f A say ( e , e ) , a n d [F ], is a d i a g o n a l element o f Therefore, A
A
A
A
A
M(g,g)
[F ],
d(e,g) < s(F)/2.