Structuralism and structures : a mathematical perspective

Structuralism and Structures

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2. Structuralism.

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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

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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

STRUCTURALISM

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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.

^ 1

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i

TV

i

T\

1

1 ' p

1

Fig.

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

STRUCTURALISM

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STRUCTURES

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

.

•

Fig.

•

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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

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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 .

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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

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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<;J,I). T h i s set of nine ordered q u a d r u p l e s of objects is called the d o m a i n of the abstract r e l a t i o n and m a y be t a k e n as a d e f i n i t i o n of t h a t r e l a t i o n . A l t h o u g h abstract structures are t h e o r e t i c a l l y q u i t e independent of concrete representations, they do not j u s t appear out o f nowhere, b u t are u s u a l l y a b s t r a c t e d f r o m concrete settings, as i n the above e x a m p l e . O u r a t t e m p t here t o focus o n an abstract s t r u c t u r e i l l u s t r a t e s some of the d i f ficulties i n discussing abstract structures a p a r t f r o m concrete settings. I n fact, i t m a y be v i r t u a l l y i m p o s s i b l e to consider an abstract s t r u c t u r e i n itself. P e r h a p s the best t h a t can be done is to p r o d u c e , as we d i d here, a s y m b o l i c representation for w h i c h the s y m b o l s have no m e a n i n g a p a r t f r o m the representation itself. Some such representation is o b v i o u s l y es-

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sential for discussing or c o m m u n i c a t i n g a n y t h i n g a b o u t the s t r u c t u r e . A t the same t i m e , c e r t a i n f o r m a l l y presented (abstract) structures, such as m a t h e m a t i c a l s t r u c t u r e s , can possess a k i n d o f i n t e g r i t y t h a t enables one u l t i m a t e l y t o f o r m m e n t a l representations of t h e m i n d e p e n d e n t l y of their i n i t i a l p r e s e n t a t i o n . Such representations, t h o u g h t e c h n i c a l l y concrete, are often regarded as a b s t r a c t , p a r t l y because they appear t o depend o n l y on the given a b s t r a c t s t r u c t u r e . S o m e p h i l o s o p h i c a l aspects of the p r o b l e m s concerning abstract structures discussed here are considered i n S e c t i o n 14. A f t e r a l l of t h i s , the fact r e m a i n s t h a t v i r t u a l l y everyone w h o sees the analogy w i l l u n d e r s t a n d it i m m e d i a t e l y w i t h o u t benefit of any e x p l a n a t i o n whatsoever. In other words, the m i n d deals w i t h the p r o b l e m a u t o m a t i c a l l y a n d unconsciously a n d w i t h no apparent effort. T h e c o m p l e x i t y of o u r a n a l y s i s o n l y accentuates the mystery as to how the m i n d accomplishes feats of this k i n d . A t the same t i m e , i n t r o s p e c t i o n suggests t h a t the a c t u a l process depends i n one way or another on v i s u a l or geometric representat i o n s of the relevant s t r u c t u r e s , thus p r o v i d i n g the advantage of p i c t u r e s over s y m b o l s . U n f o r t u n a t e l y , i t is v i r t u a l l y impossible to c a p t u r e such p i c tures except t h r o u g h i n t r o s p e c t i o n , a l t h o u g h their existence w o u l d seem t o be m o r e or less essential to e x p l a i n the relevant p h e n o m e n a . A s far as the p a r t i c u l a r e x a m p l e is concerned, the above a n a l y s i s is o b v i o u s l y not w o r t h a l l the t r o u b l e , especially since i t c o n t r i b u t e s l i t t l e or n o t h i n g to an u n d e r s t a n d i n g of the analogy itself. O n the other h a n d , it e x h i b i t s the fact t h a t an analogy does indeed d e p e n d o n a s t r u c t u r e isom o r p h i s m . It also brings out some i m p o r t a n t features of general structures, i n c l u d i n g , for e x a m p l e , the fact t h a t a f o r m a l i d e n t i f i c a t i o n of s t r u c t u r e i n some relatively s i m p l e cases m a y be rather difficult. T h e role played b y structures i n an analogy also occurs i n the more i n c l u sive case of " a s s o c i a t i o n s " , as when one object or s i t u a t i o n suggests another. T h e c o n n e c t i o n m a y be quite superficial w i t h no apparent s t r u c t u r a l content, as i n a s i m p l e coincidence, b u t is often made t h r o u g h a n o n t r i v i a l s t r u c t u r e i s o m o r p h i s m , t h o u g h the latter m a y not be as easy t o i d e n t i f y as i n an analogy. These c o m m o n occurrences of s t r u c t u r e , especially the r o u t i n e a p p e a l to analogies and the ease w i t h w h i c h everyone u n d e r s t a n d s t h e m , gives s t r o n g s u p p o r t to the c l a i m t h a t m u c h , i f not a l l , of o u r m e n t a l a c t i v i t y u l t i m a t e l y consists of the processing of structures. S u c h a v i e w , w h i c h is not new a n d s o m e w h a t c o n t r o v e r s i a l , o b v i o u s l y raises an i m p o r t a n t a n d f u n d a m e n t a l question concerning m e n t a l a c t i v i t y i n general a n d c o g n i t i o n i n p a r t i c u l a r . W e w i l l r e t u r n to the subject i n C h a p t e r V I . 10.

A n A n a l y s i s of P o i n t - L i n e S t r u c t u r e s

Let us look a g a i n , i n the l i g h t of our d e f i n i t i o n , at two o f the s i m p l e s t r u c t u r e s associated w i t h a b u i l d i n g f r a m e w o r k , the s t r u c t u r e represented

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b y the three d i m e n s i o n a l p o i n t - l i n e m o d e l a n d the t w o d i m e n s i o n a l p o i n t line s t r u c t u r e o b t a i n e d by i g n o r i n g a l l properties o f the former except the p r o p e r t y t h a t c e r t a i n sets of p o i n t s lie o n a c o m m o n l i n e segment. A s a l r e a d y suggested, the objects i n these structures m a y he t a k e n to consist o n l y of the p o i n t s , w i t h t w o or more p o i n t s related p r o v i d e d they lie o n a c o m m o n line segment. It is o b v i o u s t h a t the two s t r u c t u r e s , w h e n described i n this way, are i s o m o r p h i c . T h i s means t h a t differences, such as the fact t h a t one is t h r e e - d i m e n s i o n a l w h i l e the other is o n l y t w o - d i m e n s i o n a l , or t h a t the relative distances between points m a y be different, are p r o p e r ties of the concrete representations rather t h a n of the associated a b s t r a c t structures. In order to i n c l u d e t h r e e - d i m e n s i o n a l i t y i n the d e s c r i p t i o n of the b u i l d i n g s t r u c t u r e , i t is necessary t o specify the relative p o s i t i o n s of the p o i n t s i n space, as w e l l as w h i c h p o i n t s are j o i n e d b y lines. O n e m e t h o d of d o i n g t h i s m i g h t be i n t e r m s of a c o o r d i n a t e s y s t e m . W h a t e v e r m e t h o d is used, the result is a r e a l i z a t i o n of the b u i l d i n g s t r u c t u r e as a s u b s t r u c t u r e of E u c l i d e a n three-space. T h e l a t t e r is regarded as a s t r u c t u r e whose objects (infinite i n n u m b e r ) consist of p o i n t s , lines, and planes, and whose relations are p r e s c r i b e d b y the E u c l i d e a n space a x i o m s . A s these r e m a r k s suggest, an assertion t h a t a s t r u c t u r e is " t h r e e - d i m e n s i o n a l " is equivalent to s a y i n g t h a t i t is a s u b s t r u c t u r e of E u c l i d e a n three-space (and is not c o n t a i n e d i n a p l a n e ) . T h e y also suggest t h a t , i n order for an i s o m o r p h i s m to preserve the t h r e e - d i m e n s i o n a l character of a s t r u c t u r e , as a n t i c i p a t e d i n Section 8, it m u s t be restricted i n some way or other. T h i s p r o b l e m , w h i c h involves s o m e m a t h e m a t i c s t h a t is a b i t o n the technical side and a d d i t i o n a l properties of s t r u c t u r e s , w i l l be dealt w i t h i n C h a p t e r I X . Because of the a d d i t i o n a l i n f o r m a t i o n contained i n the t h r e e - d i m e n s i o n a l representation of the b u i l d i n g s t r u c t u r e , several l i n e segments m a y c o m b i n e to f o r m a larger segment (for e x a m p l e , a representative of a full girder or p i l l a r ) , so a d d i t i o n a l relations a m o n g the points m a y be i n t r o d u c e d b y defining sets o f p o i n t s to be related i f they lie o n one o f the extended segments. T h e s u b s t r u c t u r e s d e t e r m i n e d by these sets of p o i n t s m i g h t also be regarded as new objects c o r r e s p o n d i n g t o girders a n d p i l l a r s . T h e new objects a n d relations are " i m p l i e d " , so to speak, by the " e x t r a " i n f o r m a t i o n c o n t a i n e d i n the s t r u c t u r e as presented. N e i t h e r of the two representations of the b u i l d i n g s t r u c t u r e recognizes the l i n e segments, except as i n d i c a t o r s of r e l a t i o n s . S t r u c t u r e s w h i c h do recognize the line segments are o b t a i n e d by defining the objects t o consist o f b o t h the p o i n t s a n d the line segments, i n w h i c h case the relations are between lines and p o i n t s , w i t h a l i n e b e i n g related to a p o i n t i f it meets the p o i n t i n the a p p r o p r i a t e m a n n e r . O n e c o u l d also consider a s t r u c t u r e i n w h i c h o n l y the l i n e segments are objects. I n t h i s case the relations are

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represented by the p o i n t s , w i t h a set o f lines b e i n g related by a p o i n t i f t h e y intersect i n t h a t p o i n t . (See F i g u r e 8.1 i n Section 8.) It w i l l be recalled t h a t we presented F i g u r e 6.1 as an i l l u s t r a t i o n of a b u i l d i n g s t r u c t u r e . It is, i n fact, o n l y a t w o - d i m e n s i o n a l figure so does not c o n t a i n e x p l i c i t l y the t h r e e - d i m e n s i o n a l i n f o r m a t i o n . Few people w i l l have any difficulty, however, i n p e r c e i v i n g it as " r e p r e s e n t i n g " a threed i m e n s i o n a l o b j e c t . T h e p o i n t here is t h a t , t h r o u g h the i n t e r v e n t i o n of the m i n d , a t w o - d i m e n s i o n a l figure c a n a c t u a l l y represent a t h r e e - d i m e n s i o n a l s t r u c t u r e . T h i s p h e n o m e n o n is so c o m m o n p l a c e t h a t our emphasis on i t may seem unnecessary. B u t the way i t enters i n t o the A l b e r s e x a m p l e discussed i n Section 17 suggests t h a t the whole t h i n g m a y not be q u i t e as s i m p l e as m i g h t first appear. F u r t h e r m o r e , the a b i l i t y of the m i n d t o read i n t o c e r t a i n s t r u c t u r e s significant i n f o r m a t i o n not o b v i o u s l y c o n t a i n e d i n those s t r u c t u r e s is expressed i n m a n y s i t u a t i o n s considerably more subtle t h a n the one considered here. T h e s t r u c t u r e i n F i g u r e 6.1 could be a n a l y z e d as a s u b s t r u c t u r e of the E u c l i d e a n p l a n e so t h a t its i m p l i c i t t h r e e - d i m e n s i o n a l content w o u l d be i n c l u d e d . T h i s m i g h t be done, for e x a m p l e , by u s i n g m e t h o d s of descriptive g e o m e t r y to locate precisely the p o i n t s i n the p l a n e , say by s p e c i f y i n g their coordinates. T h e weakest p o i n t - l i n e s t r u c t u r e represented by F i g u r e 6.1 (that is, the one c a r r y i n g the least i n f o r m a t i o n ) is the s t r u c t u r e o b t a i n e d by i g n o r i n g e v e r y t h i n g a b o u t the figure except t h a t there is a finite set of p o i n t s , some of w h i c h are connected by line-segments. In this case, F i g u r e 6.1 c o u l d be greatly d i s t o r t e d a n d s t i l l represent the associated abstract s t r u c t u r e . It is n o t e w o r t h y t h a t figures as s i m p l e as those considered here already e x h i b i t i m p o r t a n t general properties of structures, such as a d m i t t i n g a v a riety of i n t e r p r e t a t i o n s as structures, i n v o l v i n g s t r u c t u r e d objects, and i m p l y i n g a d d i t i o n a l relations a m o n g objects. T h e y also i l l u s t r a t e the fact t h a t any e x p l i c i t representation o f an abstract s t r u c t u r e w i l l i n e v i t a b l y c o n t a i n e x t r a i n f o r m a t i o n irrelevant to the latter. T h e various s t r u c t u r a l i n t e r p r e t a t i o n s of a given s y s t e m represent different p o r t i o n s or different aspects o f the i n f o r m a t i o n i n v o l v e d i n the s y s t e m . T h e s t r u c t u r a l i s t ' s p r o b l e m is to discover the most significant of the v a r i ous possible s t r u c t u r a l interpretations. W h a t is "most s i g n i f i c a n t " m a y , of course, depend o n the observer as well as o n the general state of knowledge a n d u n d e r s t a n d i n g of the subject. Therefore, a serious s t r u c t u r a l a n a l y s i s is n o r m a l l y a difficult process r e q u i r i n g m u c h knowledge a n d experience w i t h the field i n question. 11. S p e c i a l K i n d s o f R e l a t i o n s A s already r e m a r k e d , the relations i n a given s t r u c t u r e m a y be quite

30

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c o m p l i c a t e d . For e x a m p l e , a r e l a t i o n may i n v o l v e m a n y o b j e c t s , or the same objects m a y be i n v o l v e d i n a v a r i e t y o f different r e l a t i o n s . I n our general t h i n k i n g a n d discussion of s t r u c t u r e s , we t r y to a v o i d as far as possible any r e s t r i c t i o n s o n the r e l a t i o n s , so as not to exclude i n advance u n a n t i c i p a t e d structures t h a t m i g h t be i m p o r t a n t . T h e s t u d y of c e r t a i n properties of s t r u c t u r e w i l l be easier, however, i f we do r e s t r i c t a t t e n t i o n to a s p e c i a l class of relations. R e c a l l t h a t , i n the case of an abstract s t r u c t u r e , a r e l a t i o n is d e t e r m i n e d , or defined, as soon as the sets of objects w h i c h i t involves are specified. T h e collection of these sets is the d o m a i n of d e f i n i t i o n of the r e l a t i o n . A r e l a t i o n is finite p r o v i d e d each of the sets i n its d o m a i n is finite. I n most of our r e m a r k s a b o u t r e l a t i o n s , it w o u l d be sufficient t o consider o n l y finite relations. If a r e l a t i o n is independent of o r d e r i n g (that is, it relates the objects i n a g i v e n set regardless of how they are a r r a n g e d ) , t h e n the r e l a t i o n is s a i d to be s y m m e t r i c . A finite r e l a t i o n is called a b i n a r y r e l a t i o n i f i t involves o n l y (ordered) pairs of the objects. N o t e t h a t because the pairs are o r d e r e d , an o b j e c t x m a y be related to an o b j e c t y w h i l e y m a y not he related to x , j u s t as for the "less t h a n " r e l a t i o n a m o n g n u m b e r s . If the d o m a i n of the r e l a t i o n contains ( y , x ) whenever i t contains ( x , y ) (i.e., b o t h y is related t o x a n d x is related to y), then i t is s y m m e t r i c , and an e x a m p l e is the r e l a t i o n of congruence a m o n g geometric figures. R e c a l l t h a t the (less t h a n ) order r e l a t i o n discussed i n Section 7 was a f i i i s e m m e t r i c . A r e l a t i o n t h a t o n l y involves ordered t r i p l e s of the objects is c a l l e d a t e r n a r y r e l a t i o n , a n d , i n general, a r e l a t i o n t h a t involves ordered n-tuples is a n n - a r y r e l a t i o n . R e l a t i o n s are often subjected to a d d i t i o n a l c o n d i t i o n s , w h i c h m a y be expressed as (structure) c o n d i t i o n s on the defining collections of ordered sets. E x a m p l e s of this w i l l be seen below. A given s t r u c t u r e m i g h t , of course, have a m o n g its relations several of these s p e c i a l types. W i t h a few exceptions, we w i l l have occasion to deal e x p l i c i t l y o n l y w i t h b i n a r y a n d t e r n a r y relations. A s t r u c t u r e is s a i d to be e l e m e n t a r y i f its o n l y r e l a t i o n is a single b i n a r y r e l a t i o n . A n elementary s t r u c t u r e w i t h a finite n u m b e r of o b j e c t s m a y a l ways be represented by a p o i n t - l i n e s t r u c t u r e i n a plane, where o b j e c t s are represented b y points a n d relations b y " d i r e c t e d " line segments (arrows) goi n g f r o m one o b j e c t to a related one. Conversely, such p o i n t - l i n e structures are o b v i o u s l y elementary. 12.

Structural Stability If an a b s t r a c t s t r u c t u r e changes i n any way, then the new s t r u c t u r e o b v i -

ously m u s t be n o n i s o m o r p h i c to the o r i g i n a l . T h i s means t h a t an abstract

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s t r u c t u r e is b y d e f i n i t i o n s t a b l e , i n the sense t h a t i t cannot change i n t o another s t r u c t u r e w i t h o u t the a d d i t i o n or s u b t r a c t i o n of objects a n d / o r rel a t i o n s . O n the other h a n d , a s y s t e m t h a t represents a s t r u c t u r e m a y change a great deal i n i t s other structures w i t h o u t ceasing to be a representation o f the first. A s a representation, however, i t cannot change " s m o o t h l y " i n t o a representation o f a different n o n i s o m o r p h i c s t r u c t u r e . S u c h a change w o u l d require a r e s t r u c t u r i n g of the representing s y s t e m i n v o l v i n g a n a b r u p t redefinition of objects a n d / o r relations. T h e suggestion is t h a t two s t r u c t u r e representations cannot be very " n e a r " t o one another w i t h o u t b e i n g i s o m o r p h i c . T h i s is the general idea i n 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 " w h i c h we w i l l r e t u r n t o i n C h a p t e r I X , where the i d e a is e x p l o r e d i n a n e l e m e n t a r y m a t h e m a t i c a l s e t t i n g . I n a d d i t i o n , C h a p t e r I X contains a n elem e n t a r y account of R e n e T h o r n ' s C a t a s t r o p h e T h e o r y , w h i c h also involves stability phenomena. A s a s i m p l e i l l u s t r a t i o n of s t a b i l i t y i n v o l v i n g p e r c e p t i o n (or "object r e c o g n i t i o n " , ) consider a d r a w i n g of a circle. T h e o b j e c t o f interest here is a n a b s t r a c t c i r c l e , w h i c h is a s u b s t r u c t u r e of (abstract) E u c l i d e a n space. Therefore, no chalk or p e n c i l d r a w i n g , or any other concrete p i c t u r e , c a n p o s s i b l y he a n e x a c t representation of i t . O n the other h a n d , such drawings m a y be very crude a n d yet, i n a n a p p r o p r i a t e context, say i n a g e o m e t r y s e t t i n g , w i l l c a l l u p for a viewer the precise n o t i o n of a circle. M o r e o v e r , i f a c e r t a i n d r a w i n g is perceived as representing a c i r c l e , t h e n d r a w i n g s t h a t do n o t deviate g r e a t l y f r o m the g i v e n one w i l l also be perceived as representing a c i r c l e . T h e r e are m a n y other e x a m p l e s of " p e r c e p t u a l s t a b i l i t y " , some of w h i c h w i l l come u p l a t e r . A n o t h e r e x a m p l e o f s t r u c t u r a l s t a b i l i t y , w h i c h is a n obvious case b u t difficult t o a n a l y z e a n d e x p l a i n , is c o n t a i n e d i n the t w o s i g n a t u r e samples i n F i g u r e 12.1. T h e first s a m p l e , w h i c h goes back to 1935, is a X e r o x c o p y f r o m one of the a u t h o r ' s college t e x t b o o k s , w h i l e the second, fifty years l a t e r , was w r i t t e n (not copied!) q u i t e i n d e p e n d e n t l y o f the first. A l s o , use of the a b b r e v i a t e d first n a m e , w h i c h appears i n b o t h s a m p l e s , was a b a n d o n e d over t h i r t y years ago.

1935

1985 Fig.

12.1

32

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AND STRUCTURES

T h e m a i n p r o b l e m w i t h the above n o t i o n o f s t a b i l i t y is t h a t we do not have a general definition of "nearness" t h a t m i g h t a p p l y to a r b i t r a r y s t r u c t u r e representations. It may even be i m p r a c t i c a l to t r y t o f o r m u l a t e such a d e f i n i t i o n , since the w o r d m a y be interpreted i n a variety of ways. T h e r e are, however, m a t h e m a t i c a l examples for w h i c h a satisfactory s o l u t i o n of the p r o b l e m exists. (See C h a p t e r I X . ) Despite the lack of a d e f i n i t i o n of nearness for a r b i t r a r y s t r u c t u r e s , there are i m p o r t a n t n o n m a t h e m a t i c a l settings, as i n the above e x a m p l e s , i n w h i c h a general s t a b i l i t y p r i n c i p l e definitely seems to operate. A d d i t i o n a l e x a m ples o c c u r i n gestalt p h e n o m e n a , the c o m m u n i c a t i o n of ideas, a n d the p r o cess of u n d e r s t a n d i n g . F o r n o w , we can only conjecture, t h r o u g h a n a l o g y w i t h e x a m p l e s s i m p l e enough to be a n a l y z e d completely, w h a t is a c t u a l l y h a p p e n i n g i n such cases. In most of these " n o n a n a l y z a b l e " e x a m p l e s , the nature o f the i s o m o r p h i s m , a l o n g w i t h the nearness c r i t e r i o n , m a y also be unclear a n d perhaps needs to be replaced by some n o t i o n of an " a p p r o x i m a t e i s o m o r p h i s m " i n v o l v i n g ideas s i m i l a r to those associated w i t h a p p r o x i m a t i o n of s t r u c t u r e s discussed later i n Section 24. T h e s t a b i l i t y p h e n o m e n a suggested by the preceding r e m a r k s , b e i n g more or less direct consequences of the definition of s t r u c t u r e , are of a rather f o r m a l n a t u r e . In the case of concrete s t r u c t u r e s , however, we m a y also have s t r u c t u r a l s t a b i l i t y associated w i t h p h y s i c a l p h e n o m e n a . F o r exa m p l e , a s t r u c t u r e w h i c h depends o n p h y s i c a l forces, m a y be stable because it embodies a m i n i m a l energy state. C h a n g e s t h a t d o not increase the energy of such a s y s t e m by too m u c h w i l l not alter the u n d e r l y i n g s t r u c t u r e . T h e r e are m a n y instances of this k i n d of s t a b i l i t y , i n c l u d i n g such t h i n g s as a t o m i c and m o l e c u l a r structures, c r y s t a l s , phase lock p h e n o m e n a i n elect r o n i c s , c h e m i c a l processes, fluid flow, and so f o r t h . A t a n o t h e r level of c o m p l e x i t y , we have the b i o l o g i c a l s t r u c t u r e s , w h i c h e x h i b i t a h i g h degree o f " d y n a m i c " s t a b i l i t y w i t h the e n v i r o n m e n t . T h i s k i n d o f s t a b i l i t y is t h e o r e t i c a l l y reducible t o p h y s i c a l a n d c h e m i c a l t e r m s , a l t h o u g h i t is often difficult t o see j u s t how the r e d u c t i o n m i g h t be made. Some o f these questions are t a k e n u p i n C h a p t e r V I I I . A s y s t e m a t i c discussion of the role of p h y s i c a l forces i n the s h a p i n g of c e r t a i n b i o l o g i c a l s t r u c t u r e s w i l l be f o u n d i n the classic work by D ' A r c y T h o m p s o n , " O n G r o w t h a n d F o r m " [T2]. T h e remarkable fact is t h a t a relatively s m a l l n u m b e r of basic f o r m s o c c u r i n a wide v a r i e t y of o r g a n i s m s . T h e s e p a r a t i o n between abstract and concrete s t a b i l i t y p h e n o m e n a m a y not be quite as great as m i g h t first a p p e a r , since some of the l a t t e r are a m e n a b l e to a purely m a t h e m a t i c a l t r e a t m e n t . I n other words, the s t r u c tures i n v o l v e d m a y be identified w i t h m a t h e m a t i c a l structures, b r i n g i n g us back t o a n essentially f o r m a l t r e a t m e n t . In fact, it is not unreasonable t o expect t h a t a theory of general structures w i l l eventually be developed

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33

( t h o u g h perhaps not i n a s t r i c t l y m a t h e m a t i c a l f o r m ) by w h i c h m a n y more of these s t r u c t u r a l p h e n o m e n a may be dealt w i t h i n a s y s t e m a t i c way. A n analogous conjecture r e g a r d i n g C a t a s t r o p h e T h e o r y has been proposed by R e n e T h o r n . (See his statement quoted i n Section 65.) 13.

Structural

Information

In this section we consider the concept of " i n f o r m a t i o n " c o n t a i n e d i n a s t r u c t u r e . In an a b s t r a c t s t r u c t u r e , an i t e m of i n f o r m a t i o n is equivalent to a p r o p e r t y of the s t r u c t u r e , and is represented by one of its s u b s t r u c t u r e s . T h i s applies to b o t h i n t e r n a l and e x t e r n a l properties, t h o u g h , i n the first case, specification of the s u b s t r u c t u r e occurs, so to speak, w i t h i n the g i v e n s t r u c t u r e , w h i l e i n the second it depends o n e x t e r n a l considerations. W e sometimes refer t o the external properties as "higher level p r o p e r t i e s " of the g i v e n s t r u c t u r e . For reasons t h a t w i l l become clearer i n S e c t i o n 35 of C h a p t e r V I , t h i s t e r m i n o l o g y is suggested b y the n o t i o n o f "higher level mental phenomena". T h e r e m a i n d e r of this section is devoted t o the question o f " c o m p a r a b i l i t y " of structures w i t h respect to the " a m o u n t " of (internal) s t r u c t u r a l i n f o r m a t i o n t h a t they c o n t a i n . T h e n o t i o n of c o m p a r a b i l i t y also has a b e a r i n g o n the question of " c o m p l e x i t y " o f structures, w h i c h is discussed i n Section 54 i n connection w i t h biological structures. T h e o b j e c t i v e is to define w h a t it s h o u l d m e a n for one s t r u c t u r e to c o n t a i n more s t r u c t u r a l i n f o r m a t i o n t h a n another. A s m i g h t be expected the d e f i n i t i o n depends o n the n o t i o n of an i s o m o r p h i s m . O n e s t r u c t u r e is s a i d to contain more (internal) i n f o r m a t i o n t h a n another i f there exists an i s o m o r p h i s m of the second s t r u c t u r e w i t h a s u b s t r u c t u r e of the first. T h e d e f i n i t i o n asserts, i n effect, t h a t the first s t r u c t u r e contains a l l of the (internal) s t r u c t u r a l i n f o r m a t i o n carried by the second. ( T h e r e s t r i c t i o n to i n t e r n a l properties is necessary because the second s t r u c t u r e may a d m i t embeddings i n larger structures not c o m p a t i b l e w i t h embeddings of the first.) T h e second s t r u c t u r e is also s a i d to c o n t a i n l e s s i n f o r m a t i o n t h a n the first. A n y two structures t h a t s t a n d i n this r e l a t i o n t o one another are said to be " c o m p a r a b l e " ( w i t h respect t o i n f o r m a t i o n ) . It w i l l be convenient to express the fact t h a t "the s t r u c t u r e B contains more i n f o r m a t i o n t h a n the s t r u c t u r e A " by the n o t a t i o n , " A < B " . A l t h o u g h the s y m b o l " < " used here suggests the o r d i n a r y "less t h a n " s y m b o l for n u m b e r s , it is not to be construed as suggesting t h a t one m i g h t assign a n u m e r i c a l value t o the " a m o u n t " of i n f o r m a t i o n i n a given s t r u c t u r e . O n the other h a n d , " < " is t r a n s i t i v e , w h i c h means t h a t A < B and B < C i m p l y A <

C.

34

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STRUCTURES

T h i s is one of the requirements for a n order r e l a t i o n a n d is a n easy consequence o f the d e f i n i t i o n of a n i s o m o r p h i s m . A t the s a m e t i m e , " < " is (at best) a " p a r t i a l " o r d e r i n g (Section 48), because not a l l structures are c o m p a r a b l e . I n other words, there exist p a i r s of s i m p l e structures n e i t h e r of w h i c h is i s o m o r p h i c to a s u b s t r u c t u r e of the other! F o r e x a m p l e , consider the structures A a n d B i l l u s t r a t e d i n F i g u r e 13.1. c

a

c

b

a Fig.

b

13.1

T h e s t r u c t u r e A consists of three objects, a , b , c, a n d an ordered b i n a r y r e l a t i o n : ( a , b ) , ( b , c), (c, a ) . S i m i l a r l y , B consists of three o b j e c t s , also denoted b y a , b , c , a n d a n ordered b i n a r y r e l a t i o n : (a, &), ( b , c ) , ( a , c). T h e key i d e a i n s h o w i n g t h a t A a n d B are not c o m p a r a b l e is t h a t , since A a n d B have the same finite n u m b e r of objects, a n y i s o m o r p h i s m of e i t h e r o n e w i t h a s u b s t r u c t u r e of t h e o t h e r w o u l d r e q u i r e a o n e - t o - o n e c o r r e s p o n d e n c e between t h e s e t s of a l l o b j e c t s i n t h e t w o s t r u c t u r e s . F u r t h e r m o r e , ( i n t h i s e x a m p l e ) one need o n l y consider the correspondence t h a t associates elements w i t h the s a m e l a b e l , a n d observe t h a t ( a , c) is a r e l a t i o n i n A b u t n o t i n B , w h i l e (c, a ) is a r e l a t i o n i n B b u t not i n A , so the s t r u c t u r e s are not c o m p a r a b l e . In a d d i t i o n to t r a n s i t i v i t y , any order r e l a t i o n is also r e q u i r e d to be " a n t i s y m m e t r i c " . T h i s means t h a t , i f two objects are d i s t i n c t , t h e n o n l y one of t h e m c a n be r e l a t e d to the other. It t u r n s out t h a t the r e l a t i o n " < " f a i l s to be " a n t i s y m m e t r i c " , so it not a n order r e l a t i o n for s t r u c t u r e s . I n other words, there exist structures A a n d B such t h a t b o t h A < B a n d B < A , b u t A a n d B are d i s t i n c t i n the sense t h a t t h e y are not i s o m o r p h i c . A l t h o u g h t h i s is not a c r u c i a l m a t t e r for w h a t follows, we sketch a n e x a m p l e s i m p l y because i t t h r o w s a d d i t i o n a l l i g h t o n the general s t r u c t u r e concept. In t h i s case, the structures m u s t be i n f i n i t e . T h e e x a m p l e is based o n the s t r u c t u r e N of a l l p o s i t i v e integers u n d e r t h e i r n a t u r a l "less t h a n " o r d e r i n g " < " . O n e s t r u c t u r e A is t a k e n t o be i d e n t i c a l w i t h N , a n d the second s t r u c t u r e B is also i d e n t i c a l w i t h N except t h a t a l l r e l a t i o n s i n N t h a t i n v o l v e the integer 1 are suppressed. I n other words, 1 is, so to speak, " i s o l a t e d " i n B . T h i s setup suggests the f o l l o w i n g

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35

diagrams: A : B :

1, (1),

2, 2,

3 3

4,

where the r e l a t i o n is u n d e r s t o o d to be " < " i n b o t h cases, except the n o t a t i o n (1) is supposed to suggest the i s o l a t i o n of 1 i n B . It is obvious t h a t B < A because B is a s u b s t r u c t u r e of A o b t a i n e d by s i m p l y d r o p p i n g the relations i n A t h a t involve 1. It is also easy to see t h a t A is i s o m o r p h i c to the s u b s t r u c t u r e o f B d e t e r m i n e d b y a l l o f the integers except 1. F i n a l l y , the fact t h a t 1 is i s o l a t e d i n B , w h i l e no o b j e c t i n A is i s o l a t e d , is a n o b s t r u c t i o n t o the existence of a n i s o m o r p h i s m between A and B . 14.

O n Abstract

Structures

T h i s section is devoted t o a few r e m a r k s c o n c e r n i n g p h i l o s o p h i c a l quest i o n s raised by the assumed existence of abstract s t r u c t u r e s . T h e m a i n e m p h a s i s is o n a t t e m p t s t o deal w i t h such questions f r o m the p o i n t o f v i e w of s t r i c t m a t e r i a l i s m . A l t h o u g h m a t t e r s of this k i n d m i g h t be ignored as far as our m a i n objectives are concerned, the concept of an a b s t r a c t s t r u c t u r e is so i m p o r t a n t to o u r general a p p r o a c h , t h a t i t is desirable to pay a t t e n t i o n to some o f the questions i t generates. Needless t o say, there are m a n y persons, especially i n the n a t u r a l s c i ences, w h o do not accept the suggestion of p h i l o s o p h i c a l i d e a l i s m i m p l i c i t i n the a s s u m p t i o n t h a t abstract structures do exist. I n c l u d e d are m a n y m a t h e m a t i c i a n s , w h o , p r o b a b l y because of the l o n g association o f m a t h e m a t i c s w i t h physics, w o u l d , i f pressed, classify themselves as m a t e r i a l i s t s . A t the same t i m e , m a t h e m a t i c i a n s r o u t i n e l y treat m a t h e m a t i c a l s t r u c t u r e s as entities t h a t exist i n d e p e n d e n t l y of m a t e r i a l t h i n g s , t h a t is, as abstract s t r u c t u r e s . T h i s p r a c t i c e , whether or not i t derives f r o m a conscious p h i l o s o p h i c a l p o s i t i o n , is consistent w i t h general m a t h e m a t i c a l experience and w o u l d be a w k w a r d to a v o i d . I n any case, p h i l o s o p h i c a l views s e l d o m enter i n t o the way m a t h e m a t i c i a n s t h i n k a b o u t m a t h e m a t i c s , a n d r e l a t i v e l y few are either interested i n or sensitive enough t o p h i l o s o p h i c a l m a t t e r s to w o r r y seriously a b o u t the p r o b l e m . T h e y are a c c o r d i n g l y able to shift easily f r o m one p o i n t of v i e w t o the other. F o r a s t r i c t m a t e r i a l i s t , however, abstract structures do not e x i s t . E v e r y s t r u c t u r e is assumed to be concrete and t o exist i n the m a t e r i a l w o r l d . A l t h o u g h a concrete s t r u c t u r e obviously m u s t consist of m a t e r i a l objects, the s t a t u s of the relations is not so clear. A r e l a t i o n is o b v i o u s l y not i n the same category as a m a t e r i a l o b j e c t . It m u s t , of course, relate s o m e t h i n g , t h o u g h t h a t s o m e t h i n g m a y be different o n different occasions. C o n s e q u e n t l y , it is not u n c o m m o n , regardless of p h i l o s o p h i c a l c o n v i c t i o n s , for i n d i v i d u a l s t o t h i n k and speak o f a r e l a t i o n as t h o u g h i t were s o m e t h i n g a p a r t f r o m the

36

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AND STRUCTURES

objects t h a t it relates. T h i s practice, of course, m a y be regarded as an a r t i f a c t o f language, a l t h o u g h i t suggests a step i n the d i r e c t i o n of a d m i t t i n g the existence o f abstract relations. O n e m e t h o d of a v o i d i n g the a d m i s s i o n of abstract r e l a t i o n s , at least i n some cases, is t o assume t h a t the relations i n a concrete s t r u c t u r e are u l t i m a t e l y d e t e r m i n e d by p h y s i c a l forces analogous t o the forces t h a t determine the s t r u c t u r e o f atoms a n d molecules. It is also possible to declare a r e l a t i o n t o be concrete s i m p l y because it involves o n l y m a t e r i a l objects. A n o t h e r m e t h o d , w h i c h was o u t l i n e d i n Section 8, consists i n r e d u c i n g the r e l a t i o n to a specification of (possibly ordered) subsets of objects, so c e r t a i n objects are related p r o v i d e d they constitute one of the specified subsets. A g a i n , i n order t o a v o i d the u s u a l practice of r e g a r d i n g sets t o be a b s t r a c t , i t is necessary to declare a set to be concrete p r o v i d e d i t consists of m a t e r i a l objects. T h i s is o b v i o u s l y a special case of the s i m i l a r convention for r e l a t i o n s . W e come now to the m a i n question of how one m i g h t deal w i t h the n o t i o n of an abstract s t r u c t u r e f r o m a m a t e r i a l i s t p o i n t o f v i e w . R e c a l l first t h a t i n Section 8 we defined an abstract s t r u c t u r e to be the " t h i n g " c o m m o n to a collection of i s o m o r p h i c (concrete) structures. Hence, i n any concrete s t r u c t u r e , there is i m p l i c i t an abstract s t r u c t u r e t h a t represents those features t h a t d i s t i n g u i s h the former as a s t r u c t u r e . F u r t h e r m o r e , we regard an a b s t r a c t s t r u c t u r e as e x i s t i n g i n d e p e n d e n t l y of any concrete s t r u c t u r e i n w h i c h it m i g h t be perceived. It w i l l consist not only of abstract relations b u t also o f abstract objects. For an abstract concept t h a t depends o n the i d e n t i f i c a t i o n of properties c o m m o n t o a collection of concrete entities (as do s t r u c t u r e s ) , one of the most c o m m o n m e t h o d s of a v o i d i n g the a s s u m p t i o n t h a t a c o r r e s p o n d i n g abstract o b j e c t a c t u a l l y exists, is to identify the concept w i t h the class of a l l concrete t h i n g s t h a t e x h i b i t the property. B y this m e t h o d , w h i c h is sometimes used to deal w i t h m a t h e m a t i c a l s t r u c t u r e s (e.g., the concept of a g r o u p ) , an " a b s t r a c t " structure w o u l d be identified w i t h a n " i s o m o r p h i s m class" of concrete s t r u c t u r e s , i.e., the collection of a l l concrete structures i s o m o r p h i c to a given one. A l t h o u g h t h i s has a n appearance of concreteness, i t depends o n the general n o t i o n of i s o m o r p h i s m , w h i c h has an abstract character s i m i l a r to t h a t of a r e l a t i o n , as m e n t i o n e d above. F u r t h e r m o r e , the n o t i o n of the class of a l l concrete representations o f a g i v e n s t r u c t u r e is also o p e n to challenge, because i t is u s u a l l y i m p o s s i b l e t o k n o w or specify a l l members of the class. T h e a p p r o a c h , nevertheless, has a p r a c t i c a l a p p e a l , because it is i m p o s sible to c o m m u n i c a t e any i n f o r m a t i o n a b o u t a s t r u c t u r e w i t h o u t e m p l o y i n g i n one way or another a concrete representation (say b y the use of language, d i a g r a m s , etc.). I n the same way, one cannot even t h i n k o f a s t r u c t u r e w i t h out f o r m i n g a m e n t a l representation, w h i c h also m i g h t be identified w i t h a

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37

(concrete) b r a i n s t r u c t u r e . These facts also m i g h t be used to argue t h a t , for a l l p r a c t i c a l purposes, an abstract s t r u c t u r e cannot exist [ C 2 , p. 221]. A n o t h e r way of dealing w i t h some of the p r o b l e m s is s i m p l y to a v o i d any reference to "objects" i n the d e f i n i t i o n of s t r u c t u r e . A s t r u c t u r e then consists o n l y of a collection of relations, w i t h no direct m e n t i o n o f the t h i n g s t h a t are r e l a t e d . A s noted i n S e c t i o n 5, Peter C a w s gives such a d e f i n i t i o n , a n d i t is i m p l i c i t i n P i a g e t ' s d e f i n i t i o n . T h e a p p r o a c h o b v i o u s l y ignores the p r o b l e m s already m e n t i o n e d c o n c e r n i n g the n a t u r e of relations, t h o u g h an idealist w o u l d no doubt take i t for granted t h a t relations enjoy an independent existence, a n d so w o u l d have no trouble w i t h the d e f i n i t i o n . O u r final c o m m e n t s o n the p r o b l e m of abstract structures concern the fact, m e n t i o n e d at the end of Section 9, t h a t they are sometimes identified w i t h m e n t a l s t r u c t u r e s . F r o m the idealist p o i n t o f v i e w , m e n t a l p h e n o m e n a o b v i o u s l y occur i n the " m i n d " , so m e n t a l structures are a f o r t i o r i ideal objects, b u t f r o m the m a t e r i a l i s t p o i n t of v i e w , they m u s t be concrete brain structures. E v e n when m e n t a l structures are identified w i t h concrete b r a i n / n e r v e s t r u c t u r e s , they nevertheless have a c e r t a i n s p e c i a l character t h a t sets t h e m a p a r t . T h i s stems p a r t l y f r o m the fact t h a t so l i t t l e is k n o w n of how m e n t a l s t r u c t u r e s are a c t u a l l y f o r m e d , a n d also f r o m the fact t h a t they are i n v o l v e d in consciousness, an even greater m y s t e r y . W h a t e v e r the reason, they t e n d t o be treated as t h o u g h they were different f r o m t y p i c a l concrete s t r u c t u r e s . T h e u n i q u e character of m e n t a l s t r u c t u r e s is a basis for the i d e a t h a t they are neither i d e a l nor m a t e r i a l objects, b u t lie somewhere between these extremes. S u c h ideas appear i n discussions t h a t a t t e m p t to a v o i d s o m e o f the familiar problems in both materialism and idealism. A n o t h e r compromise accepts m e n t a l structures as ideal i n character but rejects the e x t r e m e idealist concept of a u n i v e r s a l " M i n d " . T h i s is a p p a r e n t l y the p o s i t i o n of Peter C a w s , w h o describes h i m s e l f i n his b o o k on S t r u c t u r a l i s m [p. 234] as " a m a t e r i a l i s t i n the r e a l m of the n a t u r a l sciences a n d an idealist i n the r e a l m of the social sciences". I find myself i n an analogously a m b i g u o u s p o s i t i o n , b e i n g a t least a m a t h e m a t i c a l i d e a l i s t i c but u n a b l e to accept a c o m p l e t e l y idealist p o i n t of v i e w . M e n t a l structures are discussed more t h o r o u g h l y i n C h a p t e r V I . T h e y are of s p e c i a l interest to us because of the m y s t e r y as to how the m i n d processes so easily enormously c o m p l e x s t r u c t u r e s .

CHAPTER

SOME

15.

EXAMPLES

III

OF

STRUCTURES

Introduction

In order to clarify further the concept of s t r u c t u r e a n d t o give a better i d e a of w h a t it involves, a n u m b e r of special examples of s t r u c t u r e s w i l l now be described a n d at least p a r t i a l l y a n a l y z e d i n the l i g h t of the d e f i n i t i o n . T h e examples also serve t o b r i n g out some o f the less obvious characteristics of s t r u c t u r e s . I n c l u d e d are some i l l u s t r a t i o n s of ways t h a t s t r u c t u r e s m a y change or evolve. W e already k n o w (Section 12) t h a t a concrete representation o f an a b stract s t r u c t u r e m a y change a great deal w i t h o u t losing the p r o p e r t y of b e i n g a representation, b u t , a s a s t r u c t u r e , it cannot shift " s m o o t h l y " i n t o a representation of a different s t r u c t u r e . T h i s is a c o r o l l a r y o f the fact t h a t , for a b s t r a c t s t r u c t u r e s , "different" m e a n s " n o n i s o m o r p h i c " , so a shift f r o m one s t r u c t u r e t o a n o n i s o m o r p h i c one must involve a c t u a l a d d i t i o n s or deletions of some of the objects a n d / o r relations i n the i n i t i a l s t r u c t u r e . In spite of the inherent s t a b i l i t y suggested by the above r e m a r k , s t r u c tures (or their representations) can a n d do change i n t o n o n i s o m o r p h i c s t r u c tures. A n d the p o t e n t i a l for change often occurs i n the most i m p o r t a n t a n d u s u a l l y more complex structures. I n c l u d e d , for e x a m p l e , is a t e n d e n c y for c e r t a i n k i n d s of structures t o evolve and to grow i n c o m p l e x i t y u n d e r a p p r o p r i a t e circumstances. These properties m a y be observed i n concrete s t r u c t u r e s , as i n c r y s t a l g r o w t h or the development of a l i v i n g o r g a n i s m , and i n m e n t a l structures, as i n the development of concepts i n the process of u n d e r s t a n d i n g some subject of s t u d y . T h e y are of great i m p o r t a n c e a n d w i l l t u r n u p frequently i n m u c h of w h a t we have to say a b o u t s t r u c t u r e s . As is often the case, e x a m p l e s s i m p l e enough t o be described i n reasonable d e t a i l can o n l y suggest the full significance of such p h e n o m e n a . A m a j o r i t y of the following examples are m a t h e m a t i c a l i n character. Nevertheless, except possibly for the somewhat more t e c h n i c a l group s t r u c tures, a l l of the examples are relatively s i m p l e and easy to u n d e r s t a n d w i t h l i t t l e or no m a t h e m a t i c a l b a c k g r o u n d . In presenting the m a t h e m a t i c a l exa m p l e s , we have relied wherever possible on geometric " p i c t u r e s " , o m i t t e d all proofs, a n d have avoided v i r t u a l l y a l l of the technical m a t h e m a t i c a l det a i l s . T h e n o n m a t h e m a t i c a l e x a m p l e s , w h i c h e x h i b i t various " d y n a m i c " 39

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AND STRUCTURES

properties of s t r u c t u r e s , include the classical B o h r m o d e l o f the a t o m , a t y p i c a l m a c h i n e , a n d line drawings by Josef A l b e r s . A l b e r s ' s w o r k , w h i c h brings out a r e m a r k a b l e q u a l i t y o f the h u m a n m i n d w i t h respect to the way i t deals w i t h certain s t r u c t u r e s , is a n a l y z e d i n m o r e d e t a i l elsewhere [R3]. 16.

Atoms and Machines C o n s i d e r first the B o h r a t o m , i l l u s t r a t e d i n F i g u r e 16.1 o n the f o l l o w i n g page. T h e a t o m consists of a nucleus s u r r o u n d e d by o r b i t i n g electrons. Despite the s p i n n i n g electrons, the a t o m is m o r e often t h a n not t h o u g h t o f as a " f i x e d " object, i n w h i c h the m o v i n g electrons are represented by their o r b i t s . T h i s e x a m p l e illustrates one way of representing a c h a n g i n g s t r u c t u r e as a single " t i m e - i n d e p e n d e n t " s t r u c t u r e . N o t e t h a t i n e l e m e n t a r y c h e m i s t r y a molecule is also a d y n a m i c space s t r u c t u r e w h i c h is regarded as essentially independent of t i m e a n d whose objects are a t o m s . T h e e l e m e n t a r y c h e m i s t r y p i c t u r e of m o l e c u l a r s t r u c t u r e w i l l be e x a m i n e d later (Section 27) i n a very different context. C o n s i d e r next a t y p i c a l m a c h i n e such as an a u t o m o b i l e engine. T h i s is a r e l a t i v e l y c o m p l e x m e c h a n i s m consisting of various p a r t s i n c l u d i n g an engine b l o c k , c y l i n d e r h e a d , pistons, c o n n e c t i n g rods, c r a n k shaft, gears, p u l l e y s , b e l t s , and so f o r t h . A s s e m b l e d i n their proper r e l a t i o n s h i p s , these objects o b v i o u s l y c o n s t i t u t e a p h y s i c a l s t r u c t u r e , each of whose p a r t s (objects) m a y also be seen as a s t r u c t u r e . A s the engine r u n s , its p a r t s change their r e l a t i v e p o s i t i o n s , r e t u r n i n g u l t i m a t e l y t o their s t a r t i n g p o s i t i o n s , so the m o t i o n is c y c l i c , a property u s u a l l y expected i n a m a c h i n e . It is the nature of a m a c h i n e of this k i n d t h a t its essential i d e n t i t y is preserved t h r o u g h o u t the f u n c t i o n i n g cycle. In t e r m s o f s t r u c t u r e s , this means t h a t the concrete structures at two different t i m e s are i s o m o r p h i c . In other words, there is associated w i t h the machine a fixed abstract s t r u c ture w h i c h is represented at each p o i n t of t i m e b y the concrete s t r u c t u r e . It w o u l d be possible t o a n a l y z e a sufficiently s i m p l e m a c h i n e - s t r u c t u r e i n t e r m s o f objects a n d r e l a t i o n s , using some of the ideas already e m p l o y e d i n a n a l y z i n g the b u i l d i n g s t r u c t u r e s . B u t because an a n a l y s i s w o u l d be q u i t e tedious a n d not so very i n s t r u c t i v e , it is o m i t t e d . A l t h o u g h a machine exists i n p h y s i c a l t h r e e - d i m e n s i o n a l space, a r u n n i n g m a c h i n e m a y be regarded as a concrete s t r u c t u r e e x i s t i n g i n p h y s i c a l / o w r - d i m e n s i o n a l space, where three of the d i m e n s i o n s represent o r d i n a r y space a n d the f o u r t h represents t i m e . A p h y s i c a l space "cross s e c t i o n " at a p a r t i c u l a r t i m e w i l l give the engine i n its state at t h a t t i m e . T h e f o u r - d i m e n s i o n a l concrete s t r u c t u r e is i s o m o r p h i c w i t h a s u b s t r u c t u r e of m a t h e m a t i c a l space-time ( w h i c h is an abstract f o u r - d i m e n s i o n a l E u c l i d e a n space). T h i s s t r u c t u r e has cross sections " p e r p e n d i c u l a r " to the t i m e axis w h i c h are i s o m o r p h i c to the fixed abstract machine s t r u c t u r e described above.

III.

SOME EXAMPLES OF

STRUCTURES

B o h r ' s R a d i u m A t o m [B2] F i g . 16.1

42

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AND

STRUCTURES

17. L i n e D r a w i n g s b y J o s e f A l b e r s T h e f o l l o w i n g p a i r of d r a w i n g s is t a k e n f r o m a d e l i g h t f u l b o o k , " D e s p i t e S t r a i g h t L i n e s " , b y the l a t e a r t i s t , Josef A l b e r s [ A l , p . 52]. It is the second o f four p a i r s w h i c h bear the t i t l e , "4 P a i r s of S t r u c t u r a l C o n s t e l l a t i o n s " . A n y one of the r e m a i n i n g p a i r s , as well as m a n y other e x a m p l e s f r o m the A l b e r s b o o k , c o u l d have been used as a n i l l u s t r a t i o n i n place of the one we have chosen.

Fig.

17.1

A l b e r s ' s p o e t i c b u t revealing c o m m e n t s a b o u t these figures are quoted below. " W i t h i n a f o r m a l l i m i t a t i o n of e q u a l contours as m u t u a l s i l h o u ette, these p a i r s show different but related p l a s t i c m o v e m e n t s of lines, planes, volumes. T h u s , they change i n m o t i o n : f r o m c o m i n g to g o i n g , i n e x t e n s i o n : f r o m i n w a r d to o u t w a r d , i n g r o u p i n g : f r o m together t o separated, i n v o l u m e : f r o m f u l l to e m p t y , or reversed. A n d a l l t h i s , i n order t o show extended f l e x i b i l i t y " . A s suggested b y his c o m m e n t s , one of A l b e r s ' s objectives i n these d r a w ings is t o create a n i l l u s i o n o f m o t i o n for the observer. H e accomplishes this t h r o u g h a r e m a r k a b l y clever use of l i n e arrangements. N o t e t h a t the d r a w ings themselves are t w o d i m e n s i o n a l structures c o n s i s t i n g s i m p l y of several s t r a i g h t lines l y i n g i n a p l a n e . T h e a r r a n g e m e n t o f lines is s u c h , however, t h a t m o s t i n d i v i d u a l s w i l l first perceive the d r a w i n g s as representations of three d i m e n i s o n a l objects i n space. O n the other h a n d , the viewer w i l l q u i c k l y realize t h a t no such objects c a n p o s s i b l y exist i n space. A l t h o u g h different p a r t s o f a d r a w i n g m a y each be given two or m o r e s p a t i a l i n t e r -

III.

SOME EXAMPLES OF STRUCTURES

43

p r e t a t i o n s , these are o b v i o u s l y inconsistent w i t h one another. A p a r t i c u l a r i n t e r p r e t a t i o n i n one p a r t cannot be extended t o the whole d r a w i n g . O n e m i g h t t h i n k , w h e n presented w i t h such an o b v i o u s c o n t r a d i c t i o n , a r a t i o n a l m i n d w o u l d a b a n d o n any a t t e m p t at a s p a c i a l i n t e r p r e t a t i o n , a n d s i m p l y r e t u r n to a two d i m e n s i o n a l reading. Instead, the m i n d insists o n r e s o l v i n g the c o n t r a d i c t i o n . T h i s is done by i n t r o d u c i n g m o t i o n i n t o the i n t e r p r e t a t i o n . So the perceived three d i m e n s i o n a l s t r u c t u r e changes its shape as one's a t t e n t i o n moves f r o m one p o r t i o n of the d r a w i n g to another. T h e s a m e p h e n o m e n o n also occurs w i t h the f a m i l i a r g e o m e t r i c a l " o p t i c a l i l l u s i o n s " . T h e s e , however, tend s i m p l y t o " f l i p - f l o p " , w h i l e the m o r e s u b t l e A l b e r s constructions, i f observed carefully, e x h i b i t a more or less continuous motion. A l b e r s o b t a i n s s i m i l a r effects i n some of his paintings by e x p l o i t i n g the fact t h a t we tend t o see c e r t a i n colors as either foreground or as b a c k g r o u n d i n the presence of c e r t a i n other colors. I n these p a i n t i n g s , w h i c h are even more s u b t l e t h a n the l i n e d r a w i n g s , the c o n t r a d i c t i o n u s u a l l y involves the relative depths of certain parts of a p a i n t i n g , as suggested by their c o l o r i n g , as opposed to the relative depths suggested by the arrangement of parts i n the p a i n t i n g itself. T h e most subtle use of this technique is f o u n d i n A l b e r s ' s m a n y " H o m a g e to the S q u a r e " p a i n t i n g s . A p e n e t r a t i n g a n a l y s i s of how the v i s u a l s y s t e m a p p a r e n t l y organizes ambiguous i n f o r m a t i o n , backed up by some serious m a t h e m a t i c s , w i l l be found i n a n article by D o n a l d D . H o f f m a n [H5], (See also [M3].) T h e m a i n purpose o f the above e x a m p l e is to show t h a t the m i n d is able to f o r m d y n a m i c s t r u c t u r e s t h a t , as far as the " r e a l " w o r l d is concerned, i n c o r p o r a t e ostensibly c o n t r a d i c t o r y sets of i n f o r m a t i o n . T h i s is a r e m a r k able f a c i l i t y w h i c h suggests a general p r i n c i p l e , t h a t "the m i n d a b h o r s a c o n t r a d i c t i o n " a n d w i l l a t t e m p t i n one way or another to resolve an a p p a r ent c o n t r a d i c t i o n i n t o s o m e t h i n g m e a n i n g f u l . D e s p i t e the " u n r e a l " results in some cases, it is perhaps not too s u r p r i s i n g t h a t such a f a c i l i t y m i g h t evolve i n response to the p r o b l e m of s u r v i v a l i n a c o m p l e x e n v i r o n m e n t . A n o b j e c t - r e l a t i o n a n a l y s i s of the s t r u c t u r e represented by the A l b e r s d r a w i n g s w o u l d require m e t h o d s s i m i l a r to those o u t l i n e d for the t w o d i m e n s i o n a l representation of the three d i m e n s i o n a l b u i l d i n g s t r u c t u r e , and the r e l a t i v e p o s i t i o n s of the points a n d even the relative w i d t h s o f the lines i n v o l v e d must be carefully specified i n order to p r o d u c e the desired effect. A n a n a l y s i s of the color constructions w o u l d he considerably more c o m p l e x . W e find i n physics a different t y p e of c o n t r a d i c t i o n r e s o l u t i o n d e m a n d e d by the various elementary particles. A n electron, for e x a m p l e , n o r m a l l y behaves like a p a r t i c l e but may also be diffracted as though i t were a wave p h e n o m e n o n . It is j u s t as difficult to resolve this p a r a d o x i n t r a d i t i o n a l p h y s i c a l terms as it is to v i s u a l i z e the o b j e c t suggested by an A l b e r s d r a w i n g

44

STRUCTURALISM

AND STRUCTURES

i n o r d i n a r y space. A d d i t i o n a l examples c o u l d be c i t e d f r o m m a t h e m a t i c s as w e l l as a l m o s t any other field of i n t e l l e c t u a l endeavor. T h e s e p h e n o m e n a are, i n fact, so c o m m o n t h a t they have been regarded as c h a r a c t e r i s t i c , not o n l y of m u c h m e n t a l a c t i v i t y b u t of s t r u c t u r a l e v o l u t i o n i n general. A l t h o u g h the great b u l k of e x a m p l e s m a y be far m o r e c o m p l e x a n d less t r a n s p a r e n t t h a n the A l b e r s d r a w i n g s , i t is p l a u s i b l e to conjecture t h a t a n a p p r o p r i a t e a n a l y s i s w o u l d reveal i n most cases t h a t the m i n d ' s r e s o l u t i o n (or synthesis) of the perceived c o n t r a d i c t i o n is at least analogous t o the i n t r o d u c t i o n of the d y n a m i c s t r u c t u r e i n the A l b e r s e x a m p l e s . 18.

Configurations

A s i m p l e c o n f i g u r a t i o n is a m a t h e m a t i c a l o b j e c t w h i c h consists of m p o i n t s a n d n lines such t h a t e x a c t l y j p o i n t s lie o n each l i n e a n d e x a c t l y k l i n e s lie o n each p o i n t , where m , n , j , a n d k are p o s i t i v e integers. W e c a l l the t w o pairs of integers, [ m , n ; j , k ] , the t y p e o f the c o n f i g u r a t i o n . A c o n f i g u r a t i o n is t h u s a s p e c i a l k i n d of p o i n t - l i n e s t r u c t u r e . A s i m p l e e x a m p l e of a c o n f i g u r a t i o n consists o f a finite n u m b e r of p o i n t s , no three of w h i c h lie o n a l i n e , p l u s a l l of the lines d e t e r m i n e d b y the p o i n t s . T h i s is the " c o m p l e t e " c o n f i g u r a t i o n d e t e r m i n e d b y the p o i n t s . I f there are m p o i n t s , such a c o n f i g u r a t i o n w i l l be o f t y p e [m, m ( m — l ) / 2 ; 2, m — 1]. C o n s i d e r , for e x a m p l e , the c o m p l e t e h e x a g o n i l l u s t r a t e d i n the n e x t figure. T h i s is a c o n f i g u r a t i o n d e t e r m i n e d b y s i x p o i n t s , n o three o f w h i c h l i e o n a l i n e , so is of t y p e [ 6 , 1 5 ; 2,5]. T h e r e are 60 different s i m p l e hexagons t h a t m a y b e f o r m e d b y t a k i n g the s i x p o i n t s i n v a r i o u s orders as vertices of a h e x a g o n . T h e i r (extended) sides account for the 15 lines i n the c o n f i g u r a tion.

Fig.

18.1

45

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

T h e simplest o b j e c t - r e l a t i o n d e s c r i p t i o n of a n a r b i t r a r y c o n f i g u r a t i o n of type [ m , n ; j , k] is to take the m points as objects and the n lines as r e l a t i o n s ; t h a t is, any j - t u p l e of p o i n t s t h a t lie on one of the lines is a related set. In this case, we have a single ( s y m m e t r i c ) j - n a r y r e l a t i o n defined i n the s t r u c t u r e . T h e p r o p e r t y of being a c o n f i g u r a t i o n of the i n d i c a t e d t y p e requires the a d d i t i o n a l c o n d i t i o n t h a t each o b j e c t be involved i n e x a c t l y k relations. T h e r e is also a " d u a l " d e s c r i p t i o n i n w h i c h the lines are objects and the points are relations, as w e l l as one i n w h i c h b o t h points and lines are objects. T h e i n f o r m a t i o n c o n t a i n e d i n a given c o n f i g u r a t i o n m a y be recorded b y an a l i g n m e n t t a b l e i n w h i c h b o t h p o i n t s a n d lines are listed a l o n g w i t h an i n d i c a t i o n o f w h i c h points lie o n each line. T h i s is a k i n d of " s y m b o l i c " representation of the p o i n t - l i n e s t r u c t u r e version of the configuration. S u c h f o r m a l s t r u c t u r e representations come i n m a n y different forms and are i m p o r t a n t i n a l l contexts, especially i n m a t h e m a t i c s . T h e f o l l o w i n g table is for the complete hexagon i l l u s t r a t e d i n F i g . 18.1. T h e points are numbered f r o m 1 to 6 and each line is i n d i c a t e d by the p a i r of points t h a t determine i t . For e x a m p l e , the line d e t e r m i n e d b y the points 1 a n d 2 is denoted by 12. A l i g n m e n t T a b l e for a C o m p l e t e H e x a g o n ( p 0 i n t s

12

13

14

15

16

1

X

X

X

X

X

2 3 4 5 6

X X

i 24

n

e

23

25

26

X

X

X

X

X X

X X

35

36

X

X

X

X

X X

a 34

X X

45

46

X

X

X X

56

X X

X

T h e collection of lines i n a c o n f i g u r a t i o n w i l l , i n general, determine a n u m b e r of a d d i t i o n a l points not i n c l u d e d i n the configuration. S i m i l a r l y , the new points may d e t e r m i n e a d d i t i o n a l lines. T h e s e e x t r a points a n d lines m a y i n t u r n determine a d d i t i o n a l lines and p o i n t s , and the process m a y continue indefinitely. A l t h o u g h a r b i t r a r y e x a m p l e s of this k i n d are u s u a l l y not very i n t e r e s t i n g , the i n d i c a t e d process suggests one type of s t r u c t u r e e v o l u t i o n . M o r e interesting examples m a y be o b t a i n e d by a p p r o p r i a t e l y r e s t r i c t i n g the given p o i n t s , and being s o m e w h a t more selective i n the choice of the lines a n d points to be added to the s t r u c t u r e . T h i s is shown b y the next e x a m p l e .

46

STRUCTURALISM AND

STRUCTURES

19. T h e P a s c a l C o n f i g u r a t i o n The

classical c o n f i g u r a t i o n , w h i c h is described below, is c a l l e d the P a s c a l

C o n f i g u r a t i o n because i t rests o n the f o l l o w i n g t h e o r e m due t o P a s c a l : If a s i m p l e h e x a g o n is i n s c r i b e d i n a conic, t h e n the three p o i n t s d e t e r m i n e d b y p a i r s of opposite sides are c o l l i n e a r . T h e three p o i n t s are k n o w n as the P a s c a l p o i n t s a n d the l i n e c o n t a i n i n g t h e m as the P a s c a l L i n e of t h e h e x a g o n . T h e conic m a y b e either a p a r a b o l a , ellipse ( i n c l u d i n g a circle), or h y p e r b o l a . T w o e x a m p l e s of the t h e o r e m are i l l u s t r a t e d i n the next figure, where s i x p o i n t s o n a n ellipse are t a k e n i n two different ways as the vertices of a s i m p l e h e x a g o n . T h e t w o hexagons have vertices 123456 a n d a b c d e f respectively, where the vertices o f the second h e x a g o n coincide w i t h those of the first b u t i n the order 132654. T h e P a s c a l p o i n t s for the first h e x a g o n are P \ , P 2 , P 3 , a n d its P a s c a l l i n e is L . T h e first p o i n t , for e x a m p l e , is d e t e r m i n e d b y sides 12 a n d 4 5 . S i m i l a r l y , the P a s c a l p o i n t s of the second h e x a g o n are Q \ , Q i , Q z , a n d i t s P a s c a l l i n e is the d o t t e d l i n e M . Pascal's T h e o r e m

Fig.

19.1

For the sake of accuracy, i t m u s t be added here t h a t the a p p r o p r i a t e s e t t i n g for the P a s c a l t h e o r e m , a n d hence the c o n f i g u r a t i o n defined b e l o w , is a p r o j e c t i v e p l a n e , r a t h e r t h a n a E u c l i d e a n p l a n e . I n fact, the P a s c a l p o i n t s need not exist i n the E u c l i d e a n plane. T h i s is s h o w n b y the e x a m p l e of a regular h e x a g o n i n s c r i b e d i n a circle, where opposite sides of the h e x a g o n are p a r a l l e l so do not intersect. O n the other h a n d , a projective p l a n e

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

47

m a y be represented as an extension of the E u c l i d e a n plane o b t a i n e d b y the a d d i t i o n of a " l i n e at i n f i n i t y " consisting o f " p o i n t s at i n f i n i t y " . W i t h this extension, every pair of lines intersect a n d , i n p a r t i c u l a r , p a r a l l e l lines intersect i n a p o i n t at infinity. T h u s , for the regular h e x a g o n , the P a s c a l line coincides w i t h the l i n e at infinity. It w i l l not be necessary for us to become more deeply i n v o l v e d w i t h details c o n c e r n i n g projective planes at this time. Six points o n a conic m a y be regarded i n 60 different ways as vertices of a s i m p l e h e x a g o n , so there are 60 d i s t i n c t P a s c a l lines. T h e r e are n o t , h o w ever, 180 P a s c a l p o i n t s , since some of t h e m necessarily coincide, r e d u c i n g the n u m b e r to at most o n l y 45. In special cases, there m a y be a d d i t i o n a l coincidences, but these o c c u r only by threes a n d there can be at most four such t r i p l e p o i n t s . T h e m a x i m u m n u m b e r occurs for a regular hexagon i n s c r i b e d i n a circle, three o f the t r i p l e p o i n t s b e i n g at i n f i n i t y a n d one at the center of the circle. It t u r n s out t h a t , i n general, t h e P a s c a l p o i n t s l i e by t h r e e s o n t h e P a s c a l l i n e s a n d t h e P a s c a l l i n e s i n t e r s e c t by f o u r s o n t h e P a s c a l p o i n t s . B y d e f i n i t i o n , the P a s c a l c o n f i g u r a t i o n consists of the 60 P a s c a l lines together w i t h the 45 P a s c a l p o i n t s ( b a r r i n g coincidences), so is of type [45,60; 3,4]. N o t e t h a t it does not contain any of the vertices or sides of the 60 hexagons. T h e c o n f i g u r a t i o n is o b v i o u s l y too c o m p l e x t o be i l l u s t r a t e d here i n its entirety. T h e s t r u c t u r e represented by the P a s c a l c o n f i g u r a t i o n extends i n a n i n teresting a n d n o n t r i v i a l m a n n e r t h r o u g h the a d d i t i o n of c e r t a i n new points and lines d e t e r m i n e d b y the given points and lines of the c o n f i g u r a t i o n . W e list below some o f the p r i n c i p a l results o b t a i n e d by m a t h e m a t i c i a n s w h o have s t u d i e d the c o n f i g u r a t i o n . T h e s p e c i a l points a n d lines are n a m e d for the m a t h e m a t i c i a n s w h o discovered t h e m . T h e 60 P a s c a l lines intersect by threes i n 20 Steiner p o i n t s . T h e 20 Steiner points lie by fours o n 15 Steiner- PIiicker lines. T h e 60 P a s c a l lines also intersect by threes i n 60 K i r k m a n p o i n t s . T h e r e are 20 C a y l e y - S a l m o n lines each of w h i c h contains one Steiner p o i n t and three K i r k m a n p o i n t s . T h e 20 C a y l e y - S a l m o n lines intersect by fours o n 15 S a l m o n p o i n t s . It is not p r a c t i c a l to t r y to i n d i c a t e how these properties m i g h t be obt a i n e d , but some i d e a of w h a t is going o n is suggested b y the fact t h a t a K i r k m a n p o i n t is d e t e r m i n e d by the three P a s c a l lines associated w i t h the three hexagons t h a t can be formed w i t h o u t using any o f the sides of one specified hexagon. A s suggested by the respective n u m b e r s of the s p e c i a l p o i n t s a n d lines (20 Steiner points a n d 20 C a l e y - S a l m o n lines, 15 S a l m o n points a n d 15 S t e i n e r - P l i i c k e r lines, 60 K i r k m a n points a n d 60 P a s c a l lines), there exists a p o i n t - l i n e d u a l i t y given by relevant one-to-one correspondences between

48

STRUCTURALISM AND STRUCTURES

t h e associated classes of p o i n t s a n d lines. T h e s e correspondences m a y he described i n r e l a t i v e l y s i m p l e t e r m s , b u t we w i l l n o t pursue the m a t t e r . A few m o r e details, a l o n g w i t h references a n d s o m e results for the configur a t i o n i n a p r o j e c t i v e p l a n e h a v i n g o n l y a f i n i t e n u m b e r of p o i n t s , w i l l be f o u n d i n [R2]. A l t h o u g h the s t u d y o f configurations is n o longer f a s h i o n a b l e , as i t was s i x t y t o one h u n d r e d years ago, the above properties o f the P a s c a l configur a t i o n a n d its extensions are nonetheless genuine m a t h e m a t i c a l results w i t h n o n t r i v i a l proofs. T h e s y s t e m a t i c discovery of s p e c i a l structures w i t h i n the m y r i a d of p o i n t s a n d lines t h a t c a n be generated f r o m the basic h e x a g o n , is another e x a m p l e o f s t r u c t u r a l i s m i n a c t i o n . T h e m a i n p o i n t o f interest i n t h i s case is the w a y the s t r u c t u r e evolves i n t o m o r e a n d m o r e c o m p l e x structures u n d e r the d r i v i n g force o f i n q u i r i n g m i n d s . T h i s p o t e n t i a l for g r o w t h is n o t restricted t o g e o m e t r i c a l s t r u c t u r e s , n o r even t o a r b i t r a r y m a t h e m a t i c a l s t r u c t u r e s , b u t is a general p r o p e r t y o f a l l s t r u c t u r e s , except perhaps the most t r i v i a l a n d u n i n t e r e s t i n g ones. 20. T h e T r i a n g l e G r o u p T h i s s i m p l e e x a m p l e is based o n a n e q u i l a t e r i a l t r i a n g l e whose vertices are l a b e l e d w i t h the n u m b e r s 1 , 2 , 3 (as i n F i g u r e 20.1(a)). A l t h o u g h the t r i a n g l e is another c o n f i g u r a t i o n , we are not p r i m a r i l y interested i n i t as such, b u t r a t h e r i n the set of r o t a t i o n s t h a t t r a n s f o r m the t r i a n g l e i n t o itself. T w o r o t a t i o n s are defined t o be e q u i v a l e n t ( o r e q u a l ) i f each takes the t r i a n g l e i n t o e x a c t l y the s a m e p o s i t i o n . I n other words, the end p o s i t i o n s of the l a b e l e d vertices are the s a m e i n b o t h cases. T h e r e are two k i n d s of r o t a t i o n s corresponding respectively t o the t w o m e t h o d s o f choosing a n a x i s o f r o t a t i o n . O n e choice is the l i n e p e r p e n d i c u l a r t o the t r i a n g l e at its center, a n d the other is a l i n e t h r o u g h one v e r t e x p e r p e n d i c u l a r t o the opposite side. T h e t w o k i n d s of r o t a t i o n s are i l l u s t r a t e d i n F i g u r e 20.1 (b) a n d (c). i

i i Fig.

20.1

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

49

Since the t r i a n g l e is e q u i l a t e r a l , every r o t a t i o n o f the first k i n d is e q u i v alent to either a clockwise or counter clockwise r o t a t i o n t h r o u g h a n angle of 120°. T h e counter clockwise r o t a t i o n ( u s u a l l y regarded as positive) is denoted by the s y m b o l ft a n d the clockwise by R - . T h e results o f t h e i r a p p l i c a t i o n t o t r i a n g l e 20.1(a) are i n d i c a t e d i n F i g u r e 20.2 (a) a n d (b). +

A r o t a t i o n a x i s o f the second k i n d is d e t e r m i n e d by the v e r t e x t h a t contains i t , so there are three of t h e m . A r o t a t i o n a b o u t one of these axes w i l l at m o s t interchange the endpoints of the side opposite the d e t e r m i n i n g vertex a n d is therefore equivalent to a r o t a t i o n of 180° ( i n either d i r e c t i o n ) . T h e s e w i l l be d e n o t e d respectively by S , S t , a n d S , a c c o r d i n g as the vertex w h i c h determines the a x i s o f r o t a t i o n is at the t o p , left, or r i g h t o n the r o t a t e d t r i a n g l e . T h e result o f a p p l y i n g S to t r i a n g l e 20.1(a) is i n d i c a t e d i n F i g u r e 20.2(c). Observe t h a t each o f these r o t a t i o n s acts i n the same w a y regardless of the l a b e l i n g of the vertices. F o r e x a m p l e , i f the t o p vertex has the l a b e l 3, a n a p p l i c a t i o n of S w i l l r o t a t e the t r i a n g l e a b o u t the line t h r o u g h vertex 3 p e r p e n d i c u l a r to the side d e t e r m i n e d by vertices 1 a n d 2. t

T

t

t

3

2

R.

R .

Fig.

1

S

t

20.2

For reasons t h a t w i l l become clear i n a m o m e n t , we i n c l u d e as a r o t a t i o n the t r i v i a l t r a n s f o r m a t i o n t h a t does not change the t r i a n g l e at a l l . It is called t h e i d e n t i t y r o t a t i o n a n d is d e n o t e d by the s y m b o l / . W i t h the i d e n t i t y , we have identified s i x n o n e q u i v a l e n t r o t a t i o n s . It t u r n s out t h a t every r o t a t i o n of the t r i a n g l e i n t o itself is equivalent to one o f the s i x . R o t a t i o n s are i n t e r r e l a t e d t h r o u g h the n o t i o n o f a " p r o d u c t " , w h i c h we n o w define. T h e p r o d u c t ( o r c o m p o s i t i o n ) of t w o r o t a t i o n s is defined as the result of a p p l y i n g first one t h e n the other. T h e process is also called " m u l t i p l i c a t i o n " . T h e p r o d u c t is always another r o t a t i o n , so is equivalent to one of the above s i x . I n v i e w of t h i s fact, the s i x r o t a t i o n s are s a i d t o be "closed under the o p e r a t i o n of f o r m i n g p r o d u c t s " . F o r e x a m p l e , the p r o d u c t of

STRUCTURALISM

50

AND STRUCTURES

R a n d S t , w h i c h is w r i t t e n as R + S t a n d means " f i r s t a p p l y R and then a p p l y S t " , is a r o t a t i o n equivalent to S t , as is i n d i c a t e d i n F i g u r e 20.3. +

+

1

3

3

Fig. 20.3 T h e v a r i o u s p r o d u c t s of the s i x basic r o t a t i o n s are recorded i n the f o l l o w ing m u l t i p l i c a t i o n t a b l e , where the first factor i n a p r o d u c t is t a k e n f r o m the left h a n d c o l u m n , the second f r o m the top r o w , a n d the p r o d u c t is e q u i v alent to the c o r r e s p o n d i n g t a b l e entry. F o r e x a m p l e , the p r o d u c t is equivalent to S t , the entry i n the R + row a n d the S t c o l u m n . Multiplication Table I R R -

R RI

R1

St Si Sr

Sr s

Si

t

Sr

s

s,

+

+

t

R

+

s

s

St Sr I

S

R-

R-

R

t

S

t

r

T

s

t

s,

Si R+

1

RI

+

T h e m a i n reason for c o n s i d e r i n g t h i s e x a m p l e is t h a t i t is one of m a n y classic e x a m p l e s of a " g r o u p " i n m a t h e m a t i c s . A l t h o u g h there is m o r e t o be s a i d a b o u t i t , we need the general d e f i n i t i o n o f a g r o u p a n d the associated n o t i o n of a "group s t r u c t u r e " , before c o n t i n u i n g w i t h the s p e c i a l case. 21. G r o u p S t r u c t u r e s T h e general concept o f a group is one o f the most i m p o r t a n t a n d most s t u d i e d concepts i n m a t h e m a t i c s . It covers a w i d e v a r i e t y o f e x a m p l e s t a k e n f r o m v i r t u a l l y every b r a n c h of m a t h e m a t i c s a n d also f r o m other fields t h a t depend o n m a t h e m a t i c s , such as physics. T h e c u s t o m a r y d e f i n i t i o n of a group is i n t e r m s of a s y s t e m of a x i o m s .

III. S O M E E X A M P L E S

OF

STRUCTURES

51

Group Axioms A n a r b i t r a r y set G of elements (denoted by letters g , h , k . . . ) is called a g r o u p i f it satisfies the f o l l o w i n g a x i o m s : (1) T h e r e is defined for each p a i r of elements g a n d h , a u n i q u e element g h , called the p r o d u c t of g a n d h . (2) T h e p r o d u c t satisfies an associative p r o p e r t y g ( h k ) = ( g h ) k for all g , h , a n d fc, where the p r o d u c t s w i t h i n the parentheses are t a k e n first. (3) T h e r e exists an element e such t h a t e g — g e = g for a l l g . It is c a l l e d an i d e n t i t y e l e m e n t . (4) F o r each g , there exists an element g " It is called an i n v e r s e of g .

99~

y

= 9 ~ 9 1

1

such t h a t

= e.

It r e a d i l y follows f r o m the a x i o m s t h a t the i d e n t i t y element and inverses are u n i q u e l y d e t e r m i n e d . P a s s i n g f r o m a p a i r of elements t o their p r o d u c t is c a l l e d a g r o u p o p e r a t i o n a n d is s a i d to be c o m m u t a t i v e i f g h = h g for a l l g a n d h . In s p e c i a l cases, the g r o u p o p e r a t i o n m a y be denoted b y a s p e c i a l s y m b o l (such as " + " for the o p e r a t i o n of a d d i t i o n ) . T w o groups are s a i d to be i s o m o r p h i c i f there exists a one-to-one correspondence between their elements t h a t preserves the g r o u p p r o d u c t s . T h e n u m b e r of elements i n a g r o u p , w h i c h may be either finite or i n f i n i t e , is c a l l e d the o r d e r of the g r o u p . A subset of a g r o u p is called a s u b g r o u p i f i t is closed under p r o d u c t s ( t h a t is, contains a l l p r o d u c t s i n v o l v i n g its elements) a n d also contains the inverses of its elements. A s u b g r o u p is o b v i o u s l y a g r o u p i n its o w n r i g h t . T h e set c o n s i s t i n g of only the i d e n t i t y element is t r i v i a l l y a s u b g r o u p . S u b g r o u p s different f r o m the f u l l group a n d the i d e n t i t y element are s a i d to be p r o p e r . A subset of a g r o u p is s a i d to g e n e r a t e the g r o u p i f the smallest s u b g r o u p t h a t c o n t a i n s the subset is the g r o u p itself. In t h i s case, each element o f the g r o u p m a y be o b t a i n e d by a finite succession of p r o d u c t s i n v o l v i n g elements of the subset and their inverses. T h e subset is c a l l e d a s y s t e m of g e n e r a t o r s for the g r o u p . T w o of the m o s t f a m i l i a r e x a m p l e s of a g r o u p are the real n u m b e r s under a d d i t i o n , and the nonzero real numbers u n d e r m u l t i p l i c a t i o n . A s already n o t e d , the o p e r a t i o n of a d d i t i o n is denoted by " + " , t h e i d e n t i t y element is 0, a n d the inverse of a n u m b e r x is its negative — x . F o r m u l t i p l i c a t i o n , the p r o d u c t is u s u a l l y w r i t t e n as x x y (or x y ) , the i d e n t i t y element is 1, a n d the inverse of a nonzero n u m b e r x is its r e c i p r o c a l l / x , w h i c h m a y also be w r i t t e n as x ~ . B o t h groups are c o m m u t a t i v e . l

52

STRUCTURALISM

AND

STRUCTURES

W e r e t u r n n o w to the t r i a n g l e g r o u p . It m a y be verified f r o m the m u l t i p l i c a t i o n table t h a t the rotations of an e q u i l a t e r a l t r i a n g l e , w i t h the p r o d u c t w h i c h was defined for t h e m , constitute a finite ( n o n c o m m u t a t i v e ) g r o u p of order 6. T h e existence of a n i d e n t i t y and inverses are o b v i o u s f r o m the t a b l e . F o r e x a m p l e , the inverse of R + is R _ , a n d the inverse of S is itself. T h a t the associative p r o p e r t y is satisfied is m o r e tedious b u t not difficult t o verify. A n e q u i l a t e r a l t r i a n g l e is the simplest of the n o n t r i v i a l regular geometric figures, a l l of w h i c h have interesting groups of r o t a t i o n s . t

F u r t h e r e x a m i n a t i o n of its m u l t i p l i c a t i o n table w i l l show t h a t the t r i a n g l e g r o u p also has the f o l l o w i n g s p e c i a l g r o u p p r o p e r t i e s . T h e r o t a t i o n s of the first k i n d (/, R , a n d R - ) c o n s t i t u t e a s u b g r o u p of order three, w h i l e r o t a t i o n s a b o u t a fixed a x i s o f the second k i n d (for e x a m p l e , I a n d S ) constitute a s u b g r o u p of order t w o , of w h i c h there are three. T h e s e four are a l l o f the proper subgroups of the full g r o u p . ( N o t e t h a t the set of a l l r o t a t i o n s of the second k i n d is not a subgroup.) F u r t h e r m o r e , any two elements w h i c h do not b o t h b e l o n g to one of the four subgroups a c t u a l l y generate the f u l l g r o u p . These properties are only a few of the m a n y t h a t are r o u t i n e l y s t u d i e d i n g r o u p theory. +

t

E l e m e n t s of m a n y g r o u p s , such as the t r i a n g l e g r o u p , are n a t u r a l l y presented as one-to-one m a p p i n g s of a g i v e n fixed set onto itself. S u c h a m a p p i n g is called a t r a n s f o r m a t i o n , a n d the g r o u p a t r a n s f o r m a t i o n g r o u p . T h e successive a p p l i c a t i o n of any t w o t r a n s f o r m a t i o n s is also a t r a n s f o r m a t i o n a n d is defined as the p r o d u c t of the two. It is easy to verify t h a t the collection o f all t r a n s f o r m a t i o n s of the given set o n t o itself constitutes a g r o u p under the p r o d u c t as defined. T h e i d e n t i t y is the m a p p i n g t h a t takes each p o i n t onto itself, a n d the inverse of a t r a n s f o r m a t i o n is s i m p l y the reverse one-to-one m a p p i n g . T h e t r a n s f o r m a t i o n g r o u p on a finite set w i t h n elements has order n f a c t o r i a l . T h e t r i a n g l e g r o u p a c t u a l l y coincides w i t h the f u l l t r a n s f o r m a t i o n g r o u p o n the three vertices o f the t r i a n g l e . T h i s is not generally the case for the r o t a t i o n groups, but h a p p e n s to be t r u e here because the t r a n s f o r m e d set contains o n l y three elements. A n a r b i t r a r y g r o u p G , regardless of i t s n a t u r a l p r e s e n t a t i o n , is i s o m o r p h i c t o a s u b g r o u p of the t r a n s f o r m a t i o n g r o u p o n the set G itself. T h e i s o m o r p h i s m associates each group element g w i t h the t r a n s f o r m a t i o n T t h a t m a p s a n a r b i t r a r y element x of G to the element g x . If G c o n t a i n s more t h a n t w o elements, t h i s representation o f G is always a proper s u b g r o u p o f the f u l l t r a n s f o r m a t i o n group o n G , b u t is often not a " n a t u r a l " representation of G . s

A n y set w h i c h satisfies the group a x i o m s is said t o have g r o u p s t r u c t u r e . It r e m a i n s to give an o b j e c t - r e l a t i o n analysis of a n a r b i t r a r y g r o u p s t r u c ture. T h e objects m a y o b v i o u s l y be t a k e n to be the g r o u p elements, a n d we need o n l y one r e l a t i o n to complete the s t r u c t u r e . It is a t e r n a r y r e l a t i o n

HI.

SOME EXAMPLES

OF

STRUCTURES

S3

a n d is defined i n t e r m s of the g r o u p p r o d u c t as follows: T h e group elements i n an ordered t r i p l e , [ g , h , k ) , are r e l a t e d i f g h = k . T h i s is, of course, not an a r b i t r a r y t e r n a r y r e l a t i o n , but m u s t be res t r i c t e d so t h a t the g r o u p a x i o m s are satisfied. In other words, the d o m a i n of d e f i n i t i o n o f the r e l a t i o n m u s t also satisfy " a x i o m " c o n d i t i o n s , one for each of the four g r o u p a x i o m s . F o r e x a m p l e , the c o n d i t i o n c o r r e s p o n d i n g t o the first g r o u p a x i o m requires t h a t there exist for every p a i r o f elements g , h i n the g r o u p a unique element k such t h a t { g , k , k ) is i n the d o m a i n of the r e l a t i o n . C o n d i t i o n s c o r r e s p o n d i n g t o the other a x i o m s are also easy to f o r m u l a t e . Conversely, any set of ordered t r i p l e s t h a t satisfies these a x i o m c o n d i t i o n s is associated w i t h a g r o u p whose p r o d u c t o p e r a t i o n " o " is defined b y the e q u a t i o n g o h = t , where ( g , h , k ) is the t r i p l e required by the first c o n d i t i o n . P r o o f s of the above assertions are s t r a i g h t f o r w a r d enough t h a t the interested reader m a y w i s h to s u p p l y the d e t a i l s . It is also not difficult to prove t h a t two g r o u p s t r u c t u r e s w i l l be isom o r p h i c as s t r u c t u r e s if, and o n l y if, they are i s o m o r p h i c as groups. I n other words, an abstract g r o u p is d e t e r m i n e d by its associated a b s t r a c t g r o u p s t r u c t u r e . F o r more or less o b v i o u s reasons, g r o u p s t r u c t u r e s are n o r m a l l y considered i n t e r m s of the group o p e r a t i o n r a t h e r t h a n the assoc i a t e d t e r n a r y r e l a t i o n . T h e o n l y purpose i n defining the r e l a t i o n here is s i m p l y to show t h a t g r o u p structures are covered by o u r general d e f i n i t i o n o f s t r u c t u r e s . T h e g r o u p example also illustrates the fact t h a t m a n y s t r u c tures are associated w i t h special definitions t h a t are " n a t u r a l " for t h e m . A s i n the case o f the s i m p l e structures considered previously, a g r o u p s t r u c t u r e m a y be looked at, or represented, i n a variety of ways. A l t h o u g h m a n y g r o u p s , such as the t r i a n g l e g r o u p , m a y be defined geom e t r i c a l l y , thus p r o v i d i n g a k i n d of geometric representation o f their g r o u p s t r u c t u r e , groups i n general m a y not a d m i t a convenient geometric type representation. O n the other h a n d , there is associated w i t h each finite g r o u p its m u l t i p l i c a t i o n table w h i c h records s y s t e m a t i c a l l y each of the ordered triples involved i n the r e l a t i o n a n d a c c o r d i n g l y provides, at least t h e o r e t i cally, a representation of the given group s t r u c t u r e . W e have already seen how this works i n the case of the t r i a n g l e g r o u p . If the order of a g r o u p is very large or infinite, then it w i l l be i m p o s s i b l e to write d o w n its full t a b l e , so the s t r u c t u r e i n f o r m a t i o n m u s t be recorded i n some other m a n n e r d e p e n d i n g u p o n the p a r t i c u l a r g r o u p i n v o l v e d . F o r e x a m p l e , the g r o u p of all r o t a t i o n s of a sphere i n t o itself is an infinite g r o u p . In order to s t u d y it (that is, discover its various s t r u c t u r a l properties) one m u s t either work directly w i t h the geometry or o b t a i n formulas for a r b i t r a r y r o t a t i o n s and a p p l y general algebraic m e t h o d s . A l s o , some infinite groups are generated b y a finite n u m b e r of elements, w h i c h therefore d e t e r m i n e

54

STRUCTURALISM AND

STRUCTURES

the g r o u p . T h e generators thus m a y provide a " m a n a g e a b l e " base f r o m w h i c h to s t u d y the f u l l g r o u p . It is also possible i n s o m e cases to s t u d y a given g r o u p t h r o u g h its subgroups. T h e v a r i o u s m e t h o d s for s t u d y i n g large groups i l l u s t r a t e some o f the techniques for o b t a i n i n g properties of structures too large or too c o m p l i c a t e d to be dealt w i t h i n their entirety. O u r discussion o f groups is perhaps longer and c o n t a i n s more t e c h n i c a l d e t a i l t h a n w o u l d n o r m a l l y be desirable i n a w o r k o f t h i s k i n d . G r o u p s , however, are of such great i m p o r t a n c e a n d e x h i b i t so m a n y o f the characteristics, as well as the p r o b l e m s , associated w i t h general s t r u c t u r e s , t h a t t i m e is well spent i n t r y i n g to o b t a i n some u n d e r s t a n d i n g o f t h e m . A l t h o u g h the triangle g r o u p served well as a vehicle for b r i n g i n g out i m p o r t a n t g r o u p properties, there are m a n y other examples t h a t c o u l d have served the same purpose. T h e i d e n t i f i c a t i o n and study of the g r o u p structures i n these n a t u r a l l y o c c u r r i n g objects is an excellent e x a m p l e of the s t r u c t u r a l i s t a p p r o a c h . 22.

The Real Number System

O u r final i l l u s t r a t i v e e x a m p l e of a s t r u c t u r e (or system) is the set of a l l r e a l n u m b e r s . It is a n infinite set w h i c h contains a l l of the o r d i n a r y n u m b e r s t h a t one takes for granted i n elementary m a t h e m a t i c s courses. T h i s set, however, is not s i m p l y a collection of " t h i n g s " , b u t , as the w o r d " s y s t e m " suggests, i t also has s t r u c t u r e . T h i s is by v i r t u e of the a r i t h m e t i c o p e r a t i o n s ( a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n ) a n d the order r e l a t i o n "less t h a n " (or, e q u i v a l e n t l y , "greater t h a n " ) . T h e s t r u c t u r e associated w i t h the order r e l a t i o n was o u t l i n e d i n Section 7, a n d , as was p o i n t e d out i n Section 2 1 , there are two group structures associated respectively w i t h the o p e r a t i o n s of a d d i t i o n and ( w i t h 0 excluded) m u l t i p l i c a t i o n . T h e real n u m b e r s y s t e m contains as subsystems the s y s t e m o f r a t i o n a l n u m b e r s and the s y s t e m of integers. A s u b s y s t e m of the real n u m b e r s is u s u a l l y u n d e r s t o o d to be a subset w h i c h is closed under 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 . It is c u s t o m a r i l y r e q u i r e d to be a g r o u p under a d d i t i o n b u t not necessarily under m u l t i p l i c a t i o n (even w i t h o u t 0). F o r e x a m p l e , the r a t i o n a l numbers f o r m a g r o u p under b o t h 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 ( w i t h o u t 0), but the integers f o r m a g r o u p o n l y u n d e r a d d i t i o n . T h e real n u m b e r s y s t e m itself is a s u b s y s t e m o f a larger n u m b e r s y s t e m consisting of the c o m p l e x n u m b e r s . T h e base objects i n the real n u m b e r s y s t e m are, of course, the i n d i v i d u a l n u m b e r s , so constitute an infinite set. T h e r e l a t i o n s i n c l u d e the t e r n a r y g r o u p r e l a t i o n s , given by the a d d i t i v e a n d m u l t i p l i c a t i v e g r o u p s , p l u s the b i n a r y order r e l a t i o n a n d the various r e l a t i o n s t h a t l i n k the above. O n e s i m p l i f y i n g feature is the fact that the groups i n v o l v e d are c o m m u t a t i v e . T h e f a m i l i a r number line provides a satisfactory geometric representation of the order s t r u c t u r e of the real numbers. T h e a d d i t i v e g r o u p m a y also be

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so represented i n a fairly s t r a i g h t f o r w a r d way, b u t the m u l t i p l i c a t i v e group is not so clearly represented. T h e r e are a great m a n y s t r u c t u r e s associated i n one way or another w i t h the real n u m b e r s y s t e m a n d c e r t a i n s p e c i a l ones do a d m i t nice geometric representations, t h r o u g h a n a l y t i c geometry, for e x a m p l e . T h e most precise m e t h o d of defining the f u n d a m e n t a l s t r u c t u r e of the real numbers is by means of a s y s t e m of a x i o m s . T h e a x i o m s , a m o n g other things, specify the basic properties of the operations a n d the order r e l a t i o n . In p a r t i c u l a r , the a x i o m s assert t h a t the real n u m b e r s c o n s t i t u t e a group under a d d i t i o n a n d , i f 0 is o m i t t e d , a g r o u p under m u l t i p l i c a t i o n . T h e y specify t h a t b o t h groups are c o m m u t a t i v e , a n d i n c l u d e a n a x i o m of d i s t r i b u t i v i t y w h i c h connects 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 . T h e r e are also a x i o m s t h a t characterize the order r e l a t i o n a n d connect i t to the g r o u p o p erations, p l u s a s p e c i a l "completeness" a x i o m , w h i c h ensures the existence of certain n u m b e r s , such as the square root of 2, for e x a m p l e . T h e f u n d a m e n t a l s t r u c t u r e and the m a n y associated s t r u c t u r e s m a y then be b u i l t u p o n the a x i o m base t h r o u g h a process i n v o l v i n g definitions and c h a i n s of logical deductions. A s y s t e m of a x i o m s m a y be regarded as a ( p a r t i a l ) representation of the extended s t r u c t u r e t h a t it defines. A l s o , the development o f an a x i o m a t i c a l l y defined s t r u c t u r e f r o m the a x i o m s is another i l l u s t r a t i o n o f the e v o l u t i o n of structures m e n t i o n e d i n connection w i t h the P a s c a l C o n f i g u r a t i o n i n Section 19. W e w i l l have m u c h more t o say i n S e c t i o n 25 c o n c e r n i n g the a x i o m a t i c definition of m a t h e m a t i c a l s t r u c t u r e s a n d the general p r i n c i p l e s t h a t guide the b u i l d i n g of such a s t r u c t u r e u p o n its a x i o m a t i c base. A l t h o u g h the s y s t e m of real n u m b e r s m a y be p a r t i a l l y represented by the f a m i l i a r n u m b e r l i n e , its s t r u c t u r e is u l t i m a t e l y " a n a l y t i c " , as o p p o s e d to g e o m e t r i c , i n character. In other words, its properties are u s u a l l y expressed in the f o r m a l s y m b o l i c language c o m m o n l y associated w i t h m a t h e m a t i c s . T h e same r e m a r k is also true for m a n y groups. These structures are a c c o r d i n g l y more difficult to v i s u a l i z e t h a n are the other structures t h a t we have e x a m i n e d . O n the other h a n d , m o s t people have a fairly accurate i n t u i t i v e u n d e r s t a n d i n g o f the basic s t r u c t u r e of the real n u m b e r s y s t e m f r o m their experiences w i t h o r d i n a r y a r i t h m e t i c a n d elementary a l g e b r a , a l t h o u g h they perhaps have not thought of numbers as c o n s t i t u t i n g a s t r u c t u r e , or s y s t e m . In any case, there is no such t h i n g as a c o m p l e t e u n d e r s t a n d i n g of the f u l l s t r u c t u r e since i t is p o t e n t i a l l y infinite i n extent a n d s u p p o r t s e n o r m o u s l y c o m p l e x structures i m p o r t a n t i n their o w n r i g h t . These "super s t r u c t u r e s " encompass a very large p a r t of m a t h e m a t i c s a n d are c o n s t a n t l y b e i n g developed and extended t h r o u g h m a t h e m a t i c a l research.

CHAPTER

MANAGEMENT

IV

OF COMPLEX STRUCTURES

23.

T h e Analysis of Structures M a n y of the structures t h a t are e x p l i c i t l y dealt w i t h i n m a t h e m a t i c s and the sciences are far more c o m p l e x t h a n the examples of s t r u c t u r e s t h a t we have e x a m i n e d so far. T h e same is true o f structures i m p l i c i t i n m a n y other fields a n d those r o u t i n e l y a n d a u t o m a t i c a l l y processed by the m i n d . D e spite t h e i r relative s i m p l i c i t y , however, o u r e x a m p l e s i l l u s t r a t e i m p o r t a n t properties of the o b j e c t s a n d relations i n v o l v e d i n any s t r u c t u r e . T h e y also u n d e r l i n e the fact t h a t structures are u s u a l l y not presented i n a n a n a l y z e d f o r m , a n d t h a t except for very special or s i m p l e cases, the a n a l y s i s of cert a i n structures i n t o objects a n d relations m a y be q u i t e difficult. F i n a l l y , the examples show t h a t a given s y s t e m m a y be a n a l y z a b l e i n more t h a n one way, d e p e n d i n g u p o n w h i c h p o r t i o n s or aspects of the i n f o r m a t i o n cont a i n e d i n the s y s t e m one wishes to emphasize. It is fortunate t h a t i n a general discussion of structures as presented here, the a n a l y s i s is p r i m a r i l y of t h e o r e t i c a l i m p o r t a n c e . It is u t i l i z e d m a i n l y t o i l l u s t r a t e p r i n c i p l e s a n d t o suggest ways o f t h i n k i n g a b o u t very c o m p l e x structures for w h i c h detailed analyses m a y not be p r a c t i c a l . In a c t u a l practice, we tend to deal w i t h most s t r u c t u r e s o n an i n t u i t i v e level w i t h o u t need for a detailed analysis. T h i s is a very i m p o r t a n t a b i l i t y t h a t o b v i o u s l y suggests a g a i n t h a t the m i n d (or b r a i n ) is s p e c i a l l y o r g a nized for d e a l i n g d i r e c t l y w i t h s t r u c t u r e s . Nevertheless, an a n a l y s i s i n t o objects a n d r e l a t i o n s , or s o m e t h i n g analogous t o i t , is s o m e t i m e s required for a deeper a n d more s u b t l e u n d e r s t a n d i n g of a subject. F o r e x a m p l e , such analyses, t h o u g h often disguised i n one way or another, are c o m m o n t o m a t h e m a t i c s . T h e same is true o f any serious, n o n s u p e r f i c i a l a p p l i c a t i o n of s t r u c t u r a l i s m to other fields. T h i s is the technical side o f the s u b j e c t , a n d its appearance w i l l vary greatly w i t h the field. A l t h o u g h such a n a l y ses are n a t u r a l i n m a t h e m a t i c s a n d closely related subjects, the p r o b l e m m a y be more difficult i n certain other areas, because the s t r u c t u r e s m a y be concealed w i t h i n m a t e r i a l t h a t contains large a m o u n t s o f s t r u c t u r a l l y irrelevant i n f o r m a t i o n .

57

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24.

STRUCTURALISM

A p p r o x i m a t i o n of

AND

STRUCTURES

Structures

O n e o f the most difficult t h i n g s t o u n d e r s t a n d a b o u t m e n t a l a c t i v i t y is how the m i n d c a n manage, b o t h a u t o m a t i c a l l y a n d efficiently, so m a n y ext r e m e l y c o m p l e x s t r u c t u r e s . F u r t h e r m o r e , t h i s is a f a c u l t y e x h i b i t e d by a l l h u m a n s ( a n d perhaps also some a n i m a l s ) i n their everyday encounters w i t h things a r o u n d t h e m . A t y p i c a l e x a m p l e is the a b i l i t y to recognize almost i n s t a n t l y a f a m i l i a r face, even when i t has not been observed recently and has perhaps changed s u b s t a n t i a l l y i n the i n t e r i m . It is a deep mystery as to how the mass of i n f o r m a t i o n necessarily i n v o l v e d i n such cases is recorded and processed, t h o u g h structures m u s t c e r t a i n l y p l a y a f u n d a m e n t a l role t h r o u g h o u t . It appears v i r t u a l l y hopeless to do more t h a n suggest w h a t m i g h t conceivably take place, by e x a m i n i n g c e r t a i n r e l a t i v e l y s i m p l e but general m e t h o d s of d e a l i n g w i t h c o m p l e x s t r u c t u r e s . T h e m e t h o d s are m a i n l y ways of " a p p r o x i m a t i n g " or " r e d u c i n g " s t r u c tures, the general i d e a b e i n g to derive f r o m a given s t r u c t u r e a s i m p l e r a n d m o r e manageable one t h a t contains a significant p o r t i o n of the c r u c i a l i n f o r m a t i o n carried by the o r i g i n a l . T h e y are i n s p i r e d by techniques f r o m m a t h e m a t i c s and other more or less f o r m a l l y organized subjects. A l t h o u g h such techniques are c e r t a i n l y not e x p l i c i t i n everyday experience, it is nevertheless p l a u s i b l e t o assume t h a t s i m i l a r or analogous a p p r o x i m a t i o n s are p r o d u c e d a u t o m a t i c a l l y i n the m e n t a l processing of quite a r b i t r a r y s t r u c tures. T h e u n d e r l y i n g i d e a is t h a t the m i n d , when confronted w i t h the p r o b l e m of c o m p r e h e n d i n g a very c o m p l e x s t r u c t u r e , w i l l a u t o m a t i c a l l y replace t h a t structure by certain s i m p l e r a p p r o x i m a t i n g s u b s t r u c t u r e s . O n e possible choice is a "skeleton 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 b t a i n e d by o m i t t i n g the "fine" (or " l o c a l " ) details of the o r i g i n a l . A n e x a m p l e w o u l d be the s t r u c t u r e o b t a i n e d by o m i t t i n g a l l but the t r u n k and m a i n branches of a tree. A skeleton is p r e s u m e d to be s i m p l e r t h a n the given s t r u c t u r e , a n d t o share some of its characteristic " g l o b a l " properties. In this sense, it is a k i n d o f inner a p p r o x i m a t i o n to the o r i g i n a l . O n c e a s u b s t r u c t u r e is identified, by whatever m e t h o d , one is d r i v e n to " i m p r o v e " the a p p r o x i m a t i o n by e x t e n d i n g it t o i n c o r p o r a t e more of the i n f o r m a t i o n contained i n the o r i g i n a l . T h i s process m a y be m o r e or less a u t o m a t i c , as i n c o m m o n experience, or m a y be h i g h l y d i s c i p l i n e d , as i n the case o f scientists a n d scholars w o r k i n g o n problems i n their respective fields. T h e extension process may involve more or less i n f o r m a l g r o w t h processes, analogous to those observed i n s o m e of our examples, or m a y he q u i t e f o r m a l , as i n the a x i o m a t i c process discussed i n the next section. 25. A x i o m a t i c s a n d

Approximation

T h e a x i o m a t i c m e t h o d , also called the m e t h o d of " i m p l i c i t d e f i n i t i o n " ,

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is n o r m a l l y associated w i t h m a t h e m a t i c s a n d is o b v i o u s l y not a n everyday technique. It nevertheless has a n u m b e r of features t h a t are a l m o s t c e r t a i n l y present i n m o r e i n f o r m a l a n d i n t u i t i v e m e t h o d s o f d e a l i n g w i t h c o m p l e x s t r u c t u r e s . T h e o b j e c t i v e here is to give a d e s c r i p t i o n o f the m e t h o d f r o m the p o i n t of v i e w of a p p r o x i m a t i o n of structures. A very different m e t h o d of a p p r o x i m a t i o n , w h i c h we c a l l a " c o n t r a c t i o n " , w i l l be i n t r o d u c e d i n S e c t i o n 27. T h e s t a r t i n g p o i n t for an a x i o m a t i c setup is a more or less i n f o r m a l l y u n d e r s t o o d s y s t e m , a n d the goal is to define or describe the l a t t e r as precisely as possible. T h e idea is t o f o r m u l a t e a x i o m s t h a t express c e r t a i n basic properties of the s y s t e m a n d w h i c h " c h a r a c t e r i z e " i t , i n the sense t h a t " a l l " of its "expected" properties are derivable f r o m the basic ones t h r o u g h l o g i c a l d e d u c t i o n . A g o o d a x i o m s y s t e m thus s h o u l d define a basic system w h i c h p o t e n t i a l l y determines the full s y s t e m t h r o u g h successive extensions of the basic one. T h e requirement t h a t the a x i o m s d e t e r m i n e the f u l l s y s t e m m u s t , of course, be q u a l i f i e d , because the o r i g i n a l s y s t e m m a y not be s h a r p l y defined a n d the extension process m a y continue indefinitely. In other words, i t m a y not be m e a n i n g f u l or possible t o describe the f u l l s y s t e m i n this f o r m a l sense. A n e x a m p l e of the a x i o m a t i c m e t h o d , t h a t is f a m i l i a r to everyone, is given b y the geometry o f o r d i n a r y " p h y s i c a l s p a c e " , w h i c h we " k n o w " f r o m direct experience and w h i c h is described t h r o u g h E u c l i d ' s a x i o m s . A n o t h e r e x a m p l e is the real n u m b e r s y s t e m (as described i n S e c t i o n 22), w h i c h we " k n o w " f r o m e l e m e n t a r y m a t h e m a t i c s p r i o r to the s t a n d a r d a x i o m a t i c t r e a t m e n t . T h e reader w i l l note t h a t these classical e x a m p l e s , i n w h i c h an a p p r o p r i a t e s y s t e m of a x i o m s is f o r m u l a t e d t o describe an e x i s t i n g inform a l l y u n d e r s t o o d s y s t e m , are also excellent i l l u s t r a t i o n s o f the s t r u c t u r a l i s t approach. W h e t h e r or not a given set of a x i o m s does indeed characterize a p a r t i c u l a r , i n t u i t i v e l y u n d e r s t o o d s y s t e m , is o b v i o u s l y s o m e t h i n g t h a t is also i n t u i t i v e a n d m u s t be agreed u p o n by everyone concerned. T h e a x i o m s c o u l d t u r n out to be i n a d e q u a t e at any t i m e , a n d w o u l d a c c o r d i n g l y have t o be a u g m e n t e d or replaced. O n the other h a n d , i f they p r o d u c e enough of the i m p o r t a n t expected properties, they w i l l be t a k e n , at least t e n t a t i v e l y , as a w o r k i n g d e f i n i t i o n for the given s y s t e m . If no " u n a c c e p t a b l e " properties are deduced, the a x i o m s w i l l e v e n t u a l l y become the preferred d e f i n i t i o n of the s y s t e m . A l t h o u g h a x i o m s can p r o v i d e a p r e c i s e d e f i n i t i o n , they u s u a l l y w i l l not c o n s t i t u t e a c o m p l e t e d e f i n i t i o n , i n the sense a l l u d e d t o i n Section 4 i n c o n n e c t i o n w i t h the definition o f s t r u c t u r e . T h e y nevertheless do c o n t a i n i m p l i c i t l y a l l properties of the s y s t e m d e t e r m i n e d by t h e m , because these properties are p r e s u m e d t o be either already c o n t a i n e d i n the a x i o m s or

<>(•>

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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

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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

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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.

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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

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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-

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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)

< <

(c,d)0 (o,e)

l e d )

| Fig.

27.3

C o n s i d e r the r e a l c o o r d i n a t e p l a n e represented i n F i g u r e 27.3. It has a n a t u r a l l e x i c o g r a p h i c order s t r u c t u r e derived f r o m the "less t h a n " order r e l a t i o n " < " o n the r e a l n u m b e r s . M o r e precisely, let ( a , b) a n d (c,d") be the coordinates of any t w o d i s t i n c t p o i n t s i n the p l a n e , a n d define (a,b)<(c,d)

70

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if a < c, or , i n case a = c, i f 6 < d . T h e v e r t i c a l lines i n the p l a n e constitute a d e c o m p o s i t i o n of the latter i n t o disjoint s u b s t r u c t u r e s . It is easy t o see t h a t the c o n t r a c t i o n s t r u c t u r e , associated w i t h this d e c o m p o s i t i o n , is i s o m o r p h i c w i t h the " l e s s - t h a n " s t r u c t u r e o n the x - a x i s , where the i s o m o r p h i s m is g i v e n b y the correspondence t h a t associates a v e r t i c a l l i n e (substructure) w i t h the p o i n t where the l i n e intersects the x - a x i s . In our general d e s c r i p t i o n , a c o n t r a c t i o n is u s u a l l y presented as the e n d r e s u l t of a process, w h i l e i n m a n y cases, such as i n the previous three examples, the a c t u a l process goes i n the o p p o s i t e d i r e c t i o n ! T h e c o n t r a c t i o n s t r u c t u r e is perceived first, w h i l e the i n t e r n a l s t r u c t u r e of its o b j e c t s must be exposed later t h r o u g h further i n v e s t i g a t i o n . T h i s reverse process, c o m b i n e d w i t h the extension processes o u t l i n e d i n Section 25, provides a fair d e s c r i p t i o n of how a scientific field is r o u t i n e l y developed. B e y o n d this, of course, i t is possible for an advance i n knowledge or u n d e r s t a n d i n g to force a n o v e r a l l r e s t r u c t u r i n g of a s u b s t a n t i a l p o r t i o n o f the whole field. T h e result is a "scientific r e v o l u t i o n " and also an " i n f o r m a l " e x a m p l e of a "catastrophe" (Chapter I X ) . Despite the prevalence of processes opposite t o c o n t r a c t i o n s , our general d e s c r i p t i o n is a p p r o p r i a t e for m a n y m a t h e m a t i c a l e x a m p l e s , a n d also serves to b r i n g out the c o m p l e t e p i c t u r e i n every case. F u r t h e r m o r e , cont r a c t i o n m a p p i n g s are i n c l u d e d i n the s o m e w h a t m o r e general n o t i o n of a " h o m o m o r p h i s m " of s t r u c t u r e s , w h i c h we now define. H o m o m o r p h i s m s include the i s o m o r p h i s m s defined i n S e c t i o n 8, a n d dep e n d o n the general i d e a of a " m a p p i n g " . T h e l a t t e r is s i m p l y an association of each element of a g i v e n set w i t h an element of a second set. Since several elements o f the i n i t i a l set m a y go i n t o the same element o f the second set, m a p p i n g s are generally m a n y - t o - o n e . T h e y also need not involve every e l ement o f the target set. C o n t r a c t i o n m a p p i n g s are o b v i o u s l y special cases of general m a p p i n g s . A h o m o m o r p h i s m o f one s t r u c t u r e i n t o another is a m a p p i n g o f the o b j e c t s of the first i n t o those of the second so t h a t relations are preserved. T h e requirement t h a t the m a p p i n g preserve relations means s i m p l y t h a t related o b j e c t s of the first s t r u c t u r e m a p i n t o related objects of the seco n d . Observe t h a t the definition of relations a m o n g s u b s t r u c t u r e s is precisely w h a t is needed for the c o n t r a c t i o n m a p p i n g to preserve r e l a t i o n s . Therefore, c o n t r a c t i o n m a p p i n g s are s t r u c t u r e h o m o m o r p h i s m s . A l s o , a h o m o m o r p h i s m , whose m a p p i n g is one-to-one, is an i s o m o r p h i s m . N o w let h denote a h o m o m o r p h i s m of one s t r u c t u r e S\ i n t o a second s t r u c t u r e S%. T h e n the set of "values" i n S i of the m a p p i n g h is o b v i o u s l y

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71

a s u b s t r u c t u r e of 52 w i t h respect to those r e l a t i o n s i n S r e q u i r e d by the h o m o m o r p h i s m . It is c a l l e d a " h o m o m o r p h i c i m a g e " of S i . A c o n t r a c t i o n of a s t r u c t u r e S is thus a h o m o m o r p h i c image of S . C o n v e r s e l y , any hom o m o r p h i c image of S is i s o m o r p h i c t o a c o n t r a c t i o n o f S. I n the l a t t e r case, the c o n t r a c t i o n is w i t h respect to those s u b s t r u c t u r e s o f S d e t e r m i n e d by sets of its objects t h a t are m a p p e d by the h o m o m o r p h i s m o n t o a single o b j e c t of the image. 2

28. C o n t r a c t i o n o f G r o u p S t r u c t u r e s A n i n t e r e s t i n g a n d i m p o r t a n t m a t h e m a t i c a l e x a m p l e of a c o n t r a c t i o n is given by c e r t a i n g r o u p structures. C o n s i d e r first an a r b i t r a r y ( m u l t i p l i c a tive) g r o u p G , a n d let S denote a p a r t i c u l a r s u b g r o u p of G . T h e n the g r o u p s t r u c t u r e o f S is a s u b s t r u c t u r e of the group s t r u c t u r e of G . T h e i d e a is to decompose the G s t r u c t u r e i n a " n a t u r a l " way w i t h respect to the s u b g r o u p S a n d to f o r m the associated c o n t r a c t i o n . O n e such d e c o m p o s i t i o n consists of sets of the f o r m g S , c o n s i s t i n g of a l l p r o d u c t s g s where g is a fixed element of G a n d s ranges over S. A set g S is called a left c o s e t of S, and it follows f r o m the g r o u p a x i o m s t h a t two such cosets either coincide or are d i s j o i n t . Since the i d e n t i t y element e o f G belongs t o the s u b g r o u p S , g o b v i o u s l y belongs to g S , so the left cosets c o n s t i t u t e a d e c o m p o s i t i o n of G i n t o d i s j o i n t subsets. F u r t h e r m o r e , S is one o f these sets because S = t S . A s i m i l a r d e c o m p o s i t i o n o f G is given by the right cosets o f S , namely, sets of the f o r m S g . The G s t r u c t u r e m a y be c o n t r a c t e d w i t h respect t o either of the coset dec o m p o s i t i o n s . A l t h o u g h these contractions are s o m e t i m e s needed i n m a t h e m a t i c s , they are generally not g r o u p s t r u c t u r e s , so are not very i n t e r e s t i n g in the present c o n t e x t . In order to o b t a i n a g r o u p , it is necessary to restrict S. T h e r e s t r i c t i o n is a u t o m a t i c a l l y satisfied i f G is c o m m u t a t i v e . A s u b g r o u p S is called a n o r m a l ( o r i n v a r i a n t ) s u b g r o u p of G i f gSg-

1

= S

for each element g o f the g r o u p G . The g r o u p S w i l l be n o r m a l p r o v i d e d t h a t g S = S g for each g . In other words, the right a n d left cosets associated w i t h each element g are e q u a l . In p a r t i c u l a r , the t w o coset d e c o m p o s i t i o n s of G d e t e r m i n e d b y S are i d e n t i c a l , so there is a unique coset c o n t r a c t i o n o f the G s t r u c t u r e w i t h respect to S. N o t i c e t h a t i n the t r i a n g l e g r o u p (Section 20), r o t a t i o n s of the first k i n d ( i n c l u d i n g /) is a n o r m a l subgroup of order three. A l s o , each of the subgroups o f order t w o , c o n s i s t i n g of r o t a t i o n s o f the second k i n d a b o u t a fixed a x i s , is n o r m a l .

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It m a y now be proved t h a t the coset c o n t r a c t i o n d e t e r m i n e d by a n o r m a l subgroup is indeed a group s t r u c t u r e . In fact, consider two a r b i t r a r y elem e n t s g a n d g ' of G . T h e n the ordered t r i p l e [ g , g ' , g g ' ) is i n the d o m a i n of the ternary r e l a t i o n for the G s t r u c t u r e . Hence, by d e f i n i t i o n , the ordered t r i p l e ( S g , S g ' , S g g ' ) is i n the d o m a i n of the r e l a t i o n for the c o n t r a c t i o n . Therefore we m a y define the coset p r o d u c t , (Sg)(Sg') =

Sgg'.

T h e p r o o f t h a t this p r o d u c t is well-defined (i.e., i f S g = S h a n d S g ' = S h ' , t h e n S g g ' = S h h ' ) a n d t h a t it satisfies the group a x i o m s , t h o u g h not difficult, w i l l be o m i t t e d . O b s e r v e , for e x a m p l e , t h a t S itself, w h i c h is e q u a l to S e , is an i d e n t i t y element, a n d S g ' is an inverse for S g . T h e coset group o b t a i n e d i n the above c o n t r a c t i o n is called the q u o t i e n t of the given group by the n o r m a l s u b g r o u p . 1

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29. T h e R o l e o f L a n g u a g e A s we have repeatedly m a i n t a i n e d , higher m e n t a l a c t i v i t y m u s t consist p r i m a r i l y of the c o n s t r u c t i o n a n d m a n i p u l a t i o n o f s t r u c t u r e s . T h i s process, w h i c h m a y be either conscious or unconscious, often involves the use of language, at least for the conscious p o r t i o n . M o r e o v e r , i t is o b v i o u s t h a t language, i n some f o r m or other, is generally required for the c o m m u n i c a t i o n o f ideas to others, a n d also m a y enter i n t o the s t r i c t l y i n t e r n a l m e n t a l a c t i v i t y i n v o l v e d i n t h i n k i n g . In fact, language is so prevalent i n o u r m e n t a l processes t h a t it has been c l a i m e d t h a t a l l t h o u g h t is dependent o n language. T h e idea is t h a t a t h i n k i n g i n d i v i d u a l is s i m p l y c o m m u n i c a t i n g w i t h himself, and t h a t the associated awareness a n d use of ideas a l w a y s i n volves words, a l o n g w i t h their interrelations based o n m e a n i n g and language structure. W h e t h e r or not such a c l a i m is i n any sense l i t e r a l l y true m a y d e p e n d u l t i m a t e l y o n the definitions of b o t h t h i n k i n g a n d language. A t any r a t e , the m a t t e r is d e b a t a b l e , because the m i n d is c l e a r l y able t o deal d i r e c t l y w i t h m a n y s t r u c t u r e s a p p a r e n t l y w i t h o u t the i n t e r v e n t i o n of language. F o r e x a m p l e , i t is difficult t o see h o w o r d i n a r y language can be seriously i n volved i n the sudden deep insights a c c o m p a n y i n g creative experiences, nor i n the very s i m i l a r "face r e c o g n i t i o n " experience f a m i l i a r to everyone. T h e r e are also m a n y less s p e c t a c u l a r cases, such as the use of analogies, i n w h i c h u n d e r s t a n d i n g of r e l a t i v e l y c o m p l e x p h e n o m e n a seems to precede v e r b a l i z a t i o n . It r e m a i n s t r u e , of course, t h a t i n c o m m u n i c a t i n g or r e c a l l i n g these experiences a person w i l l n o r m a l l y use some f o r m of language. F u r t h e r more, language is a very special powerful t o o l for d e a l i n g w i t h s t r u c t u r e s , whatever i t s degree o f involvement i n m e n t a l a c t i v i t y c o n c e r n i n g t h e m . O u r u l t i m a t e objective is t o t r y to u n d e r s t a n d , i n s t r u c t u r a l t e r m s , s o m e t h i n g of how language enters i n t o the management of s t r u c t u r e s . L a n g u a g e i t s e l f is a r e l a t i v e l y c o m p l e x s t r u c t u r e a n d m a y be a n a l y z e d at different levels, represented, for e x a m p l e , b y the v a r i o u s l i n g u i s t i c a n a l yses at one e x t r e m e a n d the f a m i l i a r sentence s t r u c t u r e t y p e of a n a l y s i s at the other. Its effectiveness as a t o o l for m a n a g i n g s t r u c t u r e s is o b v i o u s l y dependent o n its o w n s t r u c t u r e , b u t e x a c t l y how e v e r y t h i n g w o r k s is not 73

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at a l l clear. A careful analysis of the c o m m u n i c a t i o n process i n t e r m s of the m a n n e r i n w h i c h structures are broken d o w n , represented piecewise i n the language s t r u c t u r e , a n d then c o m m u n i c a t e d i n a l i n e a r sequential fashion w o u l d be difficult b u t revealing for b o t h language a n d s t r u c t u r e s . I n any case, the general p r o b l e m of language is o b v i o u s l y very c o m p l e x , a n d involves m a n y t e c h n i c a l aspects of b o t h language and c o g n i t i o n . A l t h o u g h some of these technicalities w i l l be dealt w i t h l a t e r , this section is l i m i t e d to general c o m m e n t s o n a few rather obvious ways t h a t language enters i n t o the m a n a g e m e n t of s t r u c t u r e s . In its simplest f o r m , a language t r e a t m e n t of a s t r u c t u r e m i g h t b e g i n w i t h n a m i n g of the various objects a n d relations, w h i c h then m a y be represented b y t h e i r names. I n some cases, however, relations need not be recognized e x p l i c i t l y but m a y be d e t e r m i n e d b y k n o w n properties of the objects i n a concrete representation or carried by the b u i l t i n s t r u c t u r e o f the language itself. F u r t h e r m o r e , these methods of s p e c i f y i n g relations m a y overlap a n d v a r y c o n s i d e r a b l y w i t h i n a g i v e n t r e a t m e n t of a s t r u c t u r e . C o m p l i c a t i o n s of t h i s k i n d , a l o n g w i t h the fact t h a t a large part of the process is unconscious, c o n t r i b u t e t o the difficulty of d e t e r m i n i n g e x a c t l y how language is i n v o l v e d in d e a l i n g w i t h s t r u c t u r e s . A n o t h e r o b v i o u s , but f u n d a m e n t a l , use o f language is i n the n a m i n g of s t r u c t u r e s themselves. T h i s is a f o r m a l recognition of a s t r u c t u r e as an o b j e c t . T h e s t r u c t u r e m a y thereafter be represented by its n a m e i n the language t r e a t m e n t of larger structures w h i c h involve the first as an o b j e c t . T h e p o t e n t i a l i m p o r t a n c e of language i n the c o n t r a c t i o n process is t h u s clear. F i x i n g the perception of the substructures as o b j e c t s , a n d u n d e r s t a n d i n g the c o n t r a c t i o n itself, are g r e a t l y f a c i l i t a t e d by the n a m i n g process a n d the subsequent language representation of the c o n t r a c t i o n . T h e p e r c e p t i o n a n d n a m i n g of a s t r u c t u r e as an o b j e c t b r i n g s up a n i n t e r esting p o i n t c o n c e r n i n g a possible loss of i n f o r m a t i o n t h r o u g h the process. T h i s i d e a , or rather its reverse, is b e a u t i f u l l y expressed i n the c o m m e n t , " S e e i n g is f o r g e t t i n g the n a m e o f the t h i n g one sees", It appears as the t i t l e of a b i o g r a p h y of the a r t i s t , R o b e r t I r w i n , w r i t t e n by L a w r e n c e Weschler [W3]. A s i m i l a r o b s e r v a t i o n c o n c e r n i n g the use of words is expressed b y L e v i - S t r a u s s i n " T h e E l e m e n t a r y S t r u c t u r e s of K i n s h i p " [ L 5 , p. 496]; B u t to the extent t h a t words have become c o m m o n p r o p e r t y , a n d their s i g n i f y i n g f u n c t i o n has s u p p l a n t e d their character as values, language, a l o n g w i t h scientific c i v i l i z a t i o n , has helped to i m p o v e r i s h p e r c e p t i o n and t o s t r i p it of its affective, aesthetic a n d m a g i c a l i m p l i c a t i o n s , as well as t o schematize t h o u g h t . T h i s is a curious c o m m e n t f r o m one w h o has e m p h a s i z e d the f u n d a m e n t a l role of language t h r o u g h o u t m u c h of his w r i t i n g s , a n d w h o regards

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s t r u c t u r a l l i n g u i s t i c s as the m o d e l for s t r u c t u r a l a n a l y s i s i n a l l o f the soc i a l sciences. H i s views o n the subject are i l l u s t r a t e d i n s t a t e m e n t s q u o t e d in Section 32. A l t h o u g h these views are not inconsistent w i t h the above c o m m e n t , they do focus a t t e n t i o n o n i t a n d c a l l for some c l a r i f i c a t i o n . A possible, t h o u g h a b s t r a c t , s t r u c t u r a l e x p l a n a t i o n o f the m a i n i d e a c o n t a i n e d i n the statement is suggested by the p r e c e d i n g d i s c u s s i o n . In the first place, as soon as a s t r u c t u r e is perceived as a n object, its i n t e r n a l properties tend to be pushed i n t o the b a c k g r o u n d . A l t h o u g h an alert m i n d does not suppress these properties c o m p l e t e l y , there is nevertheless s o m e loss of p e r c e p t i o n . B e y o n d this, n a m i n g the s t r u c t u r e tends to b r i n g about a d d i t i o n a l loss. In extreme cases, the n a m e m a y reduce to l i t t l e more t h a n an e m p t y s y m b o l , no longer able t o c a l l up the o r i g i n a l s t r u c t u r e i n its entirety. A more c o m p l e x version of the s a m e t h i n g is represented by a cliche, an expression whose o r i g i n a l content has been lost or d i s t o r t e d by thoughtless r e p e t i t i o n so t h a t it is no longer m e a n i n g f u l . It is a fact t h a t m u c h of e v e r y d a y conversation consists of cliches, a l o n g w i t h m a n y words a n d expressions l a c k i n g i n any genuine content. T h i s does not m e a n , however, t h a t such exchanges are necessarily devoid of content, b u t o n l y t h a t whatever i n f o r m a t i o n is exchanged i n the process is not o n the o b v i o u s v e r b a l level. A more extensive c o m m e n t a r y o n l i n g u i s t i c s t r u c t u r e , a l o n g w i t h its i n v o l v e m e n t w i t h general s t r u c t u r e s a n d w i t h the s t r u c t u r a l i s t m o v e m e n t , is c o n t a i n e d i n Sections 31 a n d 32. 30.

Simple Communication

C o m m u n i c a t i o n is a general social p h e n o m e n o n i n v o l v i n g the transfer of m e n t a l s t r u c t u r e s (i.e., concepts or ideas), u s u a l l y f r o m one person to another. A l t h o u g h the m o s t c o m m o n a n d generally most accurate f o r m of c o m m u n i c a t i o n between h u m a n s is t h r o u g h a spoken or w r i t t e n language, pictures a n d even b o d y language are also used, a l o n g w i t h various c o m b i n a t i o n s of these forms. T h e p i c t u r e m e t h o d , where the word " p i c t u r e " is u n d e r s t o o d t o m e a n a representation of a whole s t r u c t u r e , reduces u l t i m a t e l y to the transference of the m e n t a l s t r u c t u r e by means of such a r e p r e s e n t a t i o n . It i n c l u d e s , for e x a m p l e , the use of analogy, w h i c h norm a l l y involves c a l l i n g a t t e n t i o n to a s i m i l a r i t y between a f a m i l i a r s t r u c t u r e a n d another less f a m i l i a r one. In the case of b o d y language, certain f a c i a l expressions and b o d y postures or movements have come t o be associated w i t h special a t t i t u d e s , so m a y serve (often unconsciously) to represent the latter to a n observer. O r d i n a r y language constitutes a s p e c i a l k i n d of s y s t e m w i t h i n w h i c h a g i v e n m e n t a l s t r u c t u r e (idea) m a y be represented i n a f o r m t h a t m a y be t r a n s m i t t e d f r o m one i n d i v i d u a l to another. L a n g u a g e c o m m u n i c a t i o n

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differs f r o m the p i c t u r e m e t h o d i n t h a t the s t r u c t u r e is not u s u a l l y t r a n s ferred as a whole, b u t is broken d o w n by the sender i n t o s i m p l e p a r t s , each o f w h i c h may be m o d e l e d i n the language s t r u c t u r e a n d passed on to be reconstructed b y the recipient. A n i m p o r t a n t feature of language c o m m u n i c a t i o n is t h a t the m a t e r i a l is s t r u n g out l i n e a r l y i n t i m e . T h e recipient m u s t f o r m his copy of the s t r u c ture, piece-by-piece as i t is presented, w i t h o u t the p r i o r benefit of a t o t a l v i e w of the m a t e r i a l . It follows f r o m the d e v e l o p m e n t a l n a t u r e of the p r o cess, t h a t an effective s t r u c t u r e c o m m u n i c a t i o n m u s t be c o o r d i n a t e d w i t h s o m e of the n a t u r a l or p o t e n t i a l g r o w t h properties of s t r u c t u r e s discussed i n previous sections. T h i s t y p e of c o o r d i n a t i o n , w h i c h is o b v i o u s l y a necessary feature o f any good p r e s e n t a t i o n , depends o n a reasonably accurate n o t i o n of the s t a t u s of the g r o w i n g s t r u c t u r e at each stage of the process. W h e n it is i g n o r e d , the result is a c e r t a i n a m o u n t of confusion and a possible b r e a k d o w n of c o m m u n i c a t i o n . T o w h a t degree the received s t r u c t u r e is i s o m o r p h i c to the o r i g i n a l w i l l d e p e n d o n the receiver's knowledge and u n d e r s t a n d i n g as w e l l as the accuracy o f the language representation. F o r e x a m p l e , i f one were to relate s o m e t h i n g a b o u t the work of a c e r t a i n f e m a l e m a t h e m a t i c i a n , w i t h o u t m e n t i o n i n g the sex, t h e n another ( m a n or woman!) m i g h t very w e l l w i n d up w i t h the i m a g e of a m a l e m a t h e m a t i c i a n . In order for a t r a n s a c t i o n of this k i n d to be successful, the recipient m u s t possess knowledge w h i c h either already c o n t a i n s , or is capable of c o n t a i n i n g , the i d e a . A t the same t i m e , i n order t o f o r m u l a t e an a p p r o p r i a t e language s t r u c t u r e representation o f the i d e a , a sender m u s t be t o some degree aware of the status of t h a t p o r t i o n of a recipient's knowledge where the concept is supposed to find its place. (See Section 38.) T h e c o m m u n i c a t i o n process becomes m u c h more interesting w h e n the s t r u c t u r e s i n question are more c o m p l e x , and represent ideas t h a t are u n f a m i l i a r to the recipient. I n this case, the c o m m u n i c a t i o n m a y take the f o r m of a discussion i n v o l v i n g definitions, examples, e x p l a n a t i o n s , and so f o r t h . F o r a successful c o m m u n i c a t i o n , a recipient m u s t be more or less f a m i l i a r w i t h the objects a n d k i n d s of relations i n v o l v e d i n the given s t r u c t u r e s . For e x a m p l e , the objects m u s t have names w i t h the same meanings for b o t h , i n the sense t h a t a n a m e m u s t call u p p r e s u m a b l y i s o m o r p h i c o b j e c t s t r u c t u r e s i n the m i n d s of b o t h . G i v e n these c o n d i t i o n s , the sender can t h e n n a m e the objects and describe t h e i r m u t u a l r e l a t i o n s , e n a b l i n g the recipient to f o r m the desired m e n t a l s t r u c t u r e . F u r t h e r e m b e l l i s h m e n t s o n this process are easy to i m a g i n e . In the process described above, language takes on the character of a "dev i c e " by w h i c h one person o b t a i n s p a r t i a l control of the m e n t a l e q u i p m e n t of a n o t h e r , for the purpose of b u i l d i n g a certain s t r u c t u r e w i t h i n the l a t t e r ' s

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m i n d . T h i s is not q u i t e as insidious as i t sounds, because the recipient w i l l s e l d o m p l a y a role q u i t e as passive as suggested b y the u n q u a l i f i e d statem e n t , a n d the a c t u a l process u s u a l l y involves m u t u a l consent, often w i t h the roles o f sender a n d receiver p e r i o d i c a l l y reversed. It is o b v i o u s t h a t these language features are i m p o r t a n t i n " t e a c h i n g a n d l e a r n i n g " , the t o p i c of Section 39 i n the next chapter. In a c t u a l p r a c t i c e , it w o u l d be s u r p r i s i n g i f words h a d e x a c t l y the same m e a n i n g t o different persons, so there is always a p o s s i b i l i t y t h a t a c o m m u n i c a t e d s t r u c t u r e w i l l not be an accurate c o p y of the o r i g i n a l . O n the other h a n d , as we have already observed (Section 12), structures t e n d to be s t a ble, so t h a t i f the two s t r u c t u r e s are sufficiently " n e a r " to one a n o t h e r , t h e n t h e y w i l l be i s o m o r p h i c . It is therefore possible t h a t less t h a n perfect c o m m u n i c a t i o n m a y s t i l l result i n essentially i s o m o r p h i c m e n t a l s t r u c t u r e s , so an imprecise c o m m u n i c a t i o n m a y nevertheless y i e l d a precise result. T h i s p h e n o m e n o n , easily identified i n m a t h e m a t i c a l c o m m u n i c a t i o n , no d o u b t p l a y s a role i n most exchanges between i n d i v i d u a l s . A l t h o u g h every concrete s t r u c t u r e is a representation of a n a b s t r a c t s t r u c ture, one is s e l d o m aware of the l a t t e r under o r d i n a r y c i r c u m s t a n c e s , because of the c h a r a c t e r i s t i c "noise" i n concrete s t r u c t u r e s . T h i s is even true i n the case of analogies t h a t involve c o m p a r i s o n s of w i d e l y different concrete objects, so the connection is genuinely a b s t r a c t . S i m i l a r e x a m p l e s are p r o v i d e d b y c e r t a i n types of poetry. Despite a n a b u n d a n c e of e x a m p l e s of this k i n d , it is very difficult for m a n y people t o conceive of a n abstract s t r u c t u r e . T h i s difficulty accounts for some of the p r o b l e m s t h a t the average person has w i t h m a t h e m a t i c s , where the a c t u a l content of the subject consists u l t i m a t e l y of a b s t r a c t structures. A concrete s t r u c t u r e m a y be " a l m o s t " a b s t r a c t , i n the sense t h a t i t c o n t a i n s a m i n i m u m of i n f o r m a t i o n a p a r t f r o m the abstract s t r u c t u r e t h a t it represents. It m a y a c c o r d i n g l y convey a q u i t e accurate n o t i o n of the a b stract s t r u c t u r e . E x a m p l e s of this k i n d include such t h i n g s as p o i n t - l i n e structures a n d carefully d r a w n geometric figures. W i t h respect to c o m m u n i c a t i o n , they are i n the p i c t u r e category. I n m o r e c o m p l e x cases, c o m p a r a b l y precise c o m m u n i c a t i o n m u s t depend a g a i n o n language. Because o r d i n a r y language, by its n a t u r e , u s u a l l y contains a great deal of " n o i s e " , precise c o m m u n i c a t i o n of abstract structures often requires the use of a very f o r m a l language, w h i c h is v i r t u a l l y free of extraneous i n f o r m a t i o n . T h e p r i m e e x a m p l e here is m a t h e m a t i c a l language, w h i c h w i l l be treated i n d e t a i l i n C h a p t e r V I I . M o r e generally, a precise l o g i c a l t r e a t m e n t of any subject serves to expose the abstract logical s t r u c t u r e of t h a t subject. 31. S t r u c t u r a l L i n g u i s t i c s In a d d i t i o n t o the s i m p l e c o m m u n i c a t i o n process, discussed i n the pre-

T8

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v i o u s s e c t i o n , language is i n v o l v e d w i t h s t r u c t u r e s and s t r u c t u r a l i s m at another level, t h r o u g h " s t r u c t u r a l l i n g u i s t i c s " . T h e l a t t e r is a s t u d y of s t r u c t u r a l properties more or less c o m m o n to a l l languages, as o p p o s e d to a t r a d i t i o n a l s t u d y of a p a r t i c u l a r l a n g u a g e . T h i s is o b v i o u s l y not the place for us t o a t t e m p t an a n a l y s i s o f this c o m p l e x a n d technical s u b j e c t , so we w i l l r e s t r i c t a t t e n t i o n here a n d i n the next section t o a few s p e c i a l topics t h a t b r i n g out s t r u c t u r a l notions relevant t o our general p r o g r a m . M o s t of the discussion i n these sections w i l l be devoted t o basic concepts due to F e r d i n a n d de Saussure, the Swiss l i n g u i s t (1857-1913), w h o is credited w i t h l a u n c h i n g m o d e r n s t r u c t u r a l l i n g u i s t i c s . Saussure's ideas, t h o u g h form u l a t e d m a n y years ago, cover very well the m a i n p o i n t s t h a t we w i s h to m a k e . M o r e recent l i n g u i s t i c developments c o u l d no d o u b t a d d to the discussion b u t w o u l d be a digression f r o m our i m m e d i a t e objectives. F i n a l l y , S e c t i o n 33 c o n t a i n s a few r e m a r k s o n the general n a t u r e a n d possible o r i g i n of l a n g u a g e . Saussure h a d a m a j o r influence o n the s t r u c t u r a l i s t m o v e m e n t , not o n l y i n language a n d l i t e r a t u r e , b u t also i n other areas a w e l l . H i s m a i n ideas are o u t l i n e d i n a b o o k w i t h the t i t l e , " C o u r s e i n G e n e r a l L i n g u i s t i c s " , and based o n notes t a k e n by his students i n lectures given at G e n e v a f r o m 1906 to 1911. T h e f o l l o w i n g excerpts f r o m the E n g l i s h t r a n s l a t i o n [SI] b r i n g out those ideas t h a t concern us. B u t w h a t is language? It is not to be confused w i t h h u m a n speech, of w h i c h i t is o n l y a definite p a r t , t h o u g h c e r t a i n l y a n essential one. It is b o t h a s o c i a l p r o d u c t o f the f a c u l t y o f speech a n d a collection of necessary conventions t h a t have been a d o p t e d by a s o c i a l b o d y t o p e r m i t i n d i v i d u a l s t o exercise t h a t f a c u l t y . T a k e n as a whole, speech is m a n y - s i d e d a n d heterogeneous; s t r a d d l i n g several areas s i m u l t a n e ously — p h y s i c a l , p h y s i o l o g i c a l , a n d p s y c h o l o g i c a l — it belongs b o t h t o the i n d i v i d u a l and t o society; we c a n n o t put it i n t o any category of h u m a n facts, for we c a n n o t discover its u n i t y , [p.9] L a n g u a g e is a well-defined object i n the heterogeneous mass of speech facts ... i t is the social side o f speech, outside the i n d i v i d u a l w h o can never create nor m o d i f y it by himself; it exists o n l y b y v i r t u e of a sort of contract signed b y the members of a c o m m u n i t y . M o r e over, the i n d i v i d u a l m u s t always serve an a p p r e n t i c e s h i p i n order to l e a r n the f u n c t i o n i n g o f language; a c h i l d assimilates it o n l y g r a d u a l l y . [p.14] ... w h a t is n a t u r a l t o m a n k i n d is not o r a l speech b u t the f a c u l t y o f c o n s t r u c t i n g a language, i.e., a s y s t e m o f d i s t i n c t signs c o r r e s p o n d i n g t o d i s t i n c t ideas, [p.10] ... b e y o n d the f u n c t i o n i n g of the various organs, there exists a m o r e general f a c u l t y w h i c h governs signs a n d w h i c h w o u l d be the l i n g u i s t i c

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f a c u l t y proper, [p.11] W e s h o u l d also a d d the associative a n d c o o r d i n a t i n g f a c u l t y t h a t we f i n d as soon as we leave i s o l a t e d signs; t h i s f a c u l t y p l a y s the d o m i n a n t role i n the o r g a n i z a t i o n of language as a s y s t e m , [p.13] F r o m our p o i n t of v i e w , the last three s t a t e m e n t s i n the q u o t a t i o n s h o u l d be b r o a d e n e d to assert t h a t w h a t is n a t u r a l t o m a n k i n d (or perhaps t o any a n i m a l w i t h a sufficiently c o m p l e x b r a i n ) is the f a c u l t y of c o n s t r u c t i n g a n d m a n i p u l a t i n g ( m e n t a l ) structures. T h i s faculty, at least i n m a n , includes the p o t e n t i a l for c o n s t r u c t i n g a n d o r g a n i z i n g language as a s y s t e m . Saussure s y m b o l i z e s , by the d i a g r a m i n F i g u r e 31.1 [ S I , p.12], the process of c o m m u n i c a t i o n between t w o i n d i v i d u a l s . I n the figure, b o t h "concept" a n d " s o u n d - i m a g e " are u n d e r s t o o d t o be s t r i c t l y m e n t a l (or psychological) constructs. T h e general i d e a is t h a t the p h y s i c a l s o u n d p r o d u c e d b y speech s t i m u l a t e s the a u d i t o r y organs, t h u s g e n e r a t i n g a " s o u n d - i m a g e " (s) t h a t evokes a concept (c) b y a s s o c i a t i o n . Conversely, a concept m a y c a l l up a s o u n d - i m a g e t h a t activates the v o c a l a p p a r a t u s p r o d u c i n g speech, etc. Hence, the t r a n s f o r m a t i o n "s —* c" is " p a s s i v e " , w h i l e "c —* s" is " a c t i v e " . Audition

Phonatica

Phonation

Audition

F i g . 31.1 It is clear t h a t o r d i n a r y language is based u l t i m a t e l y u p o n speech. T h e s o u n d - i m a g e , b y d e f i n i t i o n , is p r o d u c e d b y a spoken w o r d (or m o r p h e m e ) . O n the other h a n d , a s o u n d - i m a g e , once established, m a y also be e l i c i t e d b y a w r i t t e n w o r d , or b y the m e n t a l p i c t u r e o f a w r i t t e n w o r d , or s i m p l y b y a n act of m e m o r y . It is t h r o u g h this last p o s s i b i l i t y t h a t language m a y enter i n t o t h i n k i n g , a s i t u a t i o n i n w h i c h a n i n d i v i d u a l is l i t e r a l l y " c o m m u n i c a t i n g w i t h h i m s e l f " . W h a t h a p p e n s i n " s e l f - c o m m u n i c a t i o n " is t h a t one uses language as a t o o l t o help i n the f o r m a t i o n of m o r e c o m p l e x m e n t a l s t r u c tures out of s i m p l e r ones. T h e result m a y t h e n be coded b y the language a n d stored i n m e m o r y for easy r e t r i e v a l . A n o t h e r c o m m o n i n t e r n a l use o f language is i l l u s t r a t e d b y the e x a m p l e of a person c a r r y i n g o n m e n t a l l y a n i m a g i n a r y conversation w i t h someone w h o is not present. T h i s is a s i t u a -

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t i o n i n w h i c h the person creates a m e n t a l c o n s t r u c t o f another i n sufficient d e t a i l to produce a sense of a c t u a l presence. A c t i v i t i e s of this k i n d o b v i ously represent very c o m p l e x psychological p h e n o m e n a w i t h i m p l i c a t i o n s e x t e n d i n g far b e y o n d the mere use of language. A l t h o u g h speech is an i n d i v i d u a l a c t i v i t y , language is a s o c i a l p r o d u c t , shared b y members of a c o m m u n i t y . T h e ( c , s ) associations, p o s t u l a t e d in Saussure's d e s c r i p t i o n of c o m m u n i c a t i o n , are more or less c o m m o n to those i n d i v i d u a l s w h o are able to c o m m u n i c a t e w i t h one a n o t h e r . E a c h w i l l reproduce, a p p r o x i m a t e l y i f not e x a c t l y , the same associations between concepts a n d sound-images. T h r o u g h the f u n c t i o n i n g of the receptive a n d c o o r d i n a t i n g f a c u l ties, impressions t h a t are perceptively the same for a l l are m a d e o n the m i n d s o f speakers....If we c o u l d embrace the s u m of word images stored i n the m i n d s of all i n d i v i d u a l s , we c o u l d i d e n t i f y the s o c i a l b o n d t h a t constitutes language. It is a storehouse filled by the m e m b e r s of a given c o m m u n i t y t h r o u g h t h e i r active use of s p e a k i n g , a g r a m m a t i c a l s y s t e m t h a t has a p o t e n t i a l existence i n each b r a i n , or, m o r e specifi c a l l y , i n the brains of a group o f i n d i v i d u a l s . F o r language is not complete i n any speaker; i t exists perfectly o n l y w i t h i n a c o l l e c t i v i t y . [ S I , p p . 13, 14] T h e " l i n g u i s t i c u n i t " , w h i c h occurs i n the s i m p l e c o m m u n i c a t i o n process, consists of an association of a concept w i t h a s o u n d - i m a g e , a n d is c a l l e d a s i g n by Saussure. T h e two elements of the s i g n , b o t h of w h i c h are psychol o g i c a l , are i n t i m a t e l y u n i t e d so t h a t each recalls the other. Since there is a c e r t a i n a m o u n t o f a m b i g u i t y i n the o r d i n a r y usage of these t e r m s , Saussure proposes t h a t the concept and associated s o u n d - i m a g e , i n v o l v e d i n a s i g n , be c a l l e d the s i g n i f i e d and s i g n i f i e r respectively. In everyday terminology, the signifier is the language element, a n d the signified is its m e a n i n g . M o r e o v e r , the b a s i c signifiers are the m i n i m a l m e a n i n g f u l elements of the language, v i z . , the m o r p h e m e s . Since the most c o m m o n m o r p h e m e s are associated w i t h i n d i v i d u a l words, s i m p l e sound-images are often called " w o r d - i m a g e s " rather t h a n the more accurate " m o r p h e m e images" . A c c o r d i n g to Saussure, the l i n g u i s t i c sign has t w o " p r i m o r d i a l " characteristics: (1) T h e A r b i t r a r y N a t u r e of the S i g n , a n d (2) T h e L i n e a r N a t u r e of the Signifier [ S I , pp. 69, 70]. P r o p e r t y (2) is s i m p l y the recognition t h a t a signifier, b e i n g an a u d i t o r y p h e n o m e n o n , occupies an i n t e r v a l of t i m e , so the signifiers occur i n a linear succession of intervals s t r u n g o u t a l o n g the t i m e a x i s . T h i s is i n contrast w i t h pictures ( v i s u a l signifiers) w h i c h can exist i n several dimensions. W e have already seen the i m p o r t a n c e of the l i n e a r character o f language i n connection w i t h the c o m m u n i c a t i o n o f

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structures discussed i n S e c t i o n 3 0 , a n d i t w i l l not be necessary t o a d d to those r e m a r k s here. P r o p e r t y (1), however, calls for some c o m m e n t s . T h e m e a n i n g o f (1) is t h a t the a s s o c i a t i o n b e t w e e n s i g n i f i e d a n d s i g n i f i e r i s q u i t e a r b i t r a r y . T h i s does not m e a n , h o w e v e r , " t h a t the choice of the signifier is left entirely to the speaker",since "the i n d i v i d u a l does not have the power t o change a sign i n a n y w a y once i t has become established i n the l i n g u i s t i c c o m m u n i t y " . T h e p o i n t is t h a t the signifier need not have any n a t u r a l connection w i t h the signified. T h e p r i n c i p l e is q u i t e g e n e r a l , a n d is not c o n t r a d i c t e d by the occasional instances i n w h i c h there appears to be a connection, as for e x a m p l e when the s o u n d of a w o r d suggests the associated concept. T h o u g h it is conceivable t h a t such connections m i g h t have p l a y e d a role i n the b e g i n n i n g , they are not essential to the associations themselves, a n d tend to become quite irrelevant once the signs are i n c o r p o r a t e d i n t o the language t h r o u g h the s o c i a l process. M u c h of the c o m p l e x i t y a n d f l e x i b i l i t y i n use of language depends u p o n the a r b i t r a r i n e s s of its signs. In s t r u c t u r a l t e r m i n o l o g y , the sign is a s i m p l e s t r u c t u r e , "s ^ c", cons i s t i n g of t w o o b j e c t s , the s o u n d - i m a g e s a n d the concept c, c o u p l e d by t w o (ordered) b i n a r y r e l a t i o n s , "s - * c" and "s « - c", associated w i t h the p s y c h o l o g i c a l steps i n the s i m p l e c o m m u n i c a t i o n process. Because these objects w i l l generally have some s t r u c t u r e o f t h e i r o w n , we may interpret the a r b i t r a r y character of the sign as an expression of the fact t h a t its s t r u c ture does not involve i n any way the object s t r u c t u r e s . O n the other h a n d , cases w h i c h appear to violate the a r b i t r a r i n e s s , are precisely those cases i n w h i c h the t w o o b j e c t structures are s i m i l a r i n some way or o t h e r , r e s u l t i n g i n the p o s s i b i l i t y of a more c o m p l e x s i g n - s t r u c t u r e i n w h i c h these s t r u c t u r a l s i m i l a r i t i e s are recognized. T h e fact is, however, t h a t t h i s a d d i t i o n a l s t r u c ture does not n o r m a l l y enter i n t o the role of the sign as a l a n g u a g e - e l e m e n t . In other words, as the sign becomes i n c o r p o r a t e d i n t o the language, such e x t r a s t r u c t u r e w i l l tend t o be e l i m i n a t e d . A m o r e general version of this p h e n o m e n o n w i l l be considered i n the next section. W e t u r n now t o the question of j u s t w h a t constitutes a "language s t r u c t u r e " . T h e answer is suggested b y the preceding discussion a n d , i n t e r estingly e n o u g h , is stated precisely as we w o u l d have i t i n the f o l l o w i n g q u o t a t i o n f r o m the " C o u r s e " . T h e first e m p h a s i s is ours. T h e signs t h a t make up language are not a b s t r a c t i o n s but real objects; s i g n s a n d t h e i r r e l a t i o n s are w h a t l i n g u i s t i c s studies; they are the c o n c r e t e e n t i t i e s of o u r science, [p. 102] Saussure goes o n t o e m p h a s i z e t h a t the l i n g u i s t i c e n t i t y exists i n neither the signifier nor the signified p o r t i o n s o f a sign but o n l y i n their a s s o c i a t i o n . T h u s , " a succession of sounds is l i n g u i s t i c o n l y i f i t s u p p o r t s an i d e a " . S i m i l a r l y , concepts "become l i n g u i s t i c entities o n l y when associated w i t h

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sound-images". T h e o b j e c t s w i t h i n a language s t r u c t u r e are the signs, a n d the relations consist of (or at least i n c l u d e ) those specified by the g r a m m a r a n d s y n t a x . A s i n the case of most systems, there are possibly other m e t h o d s b y w h i c h s t r u c t u r e c o u l d be reasonably identified w i t h i n the mass o f language d a t a , but the one suggested here is most n a t u r a l for our purposes. T h e r e are, of course, m a n y d i s t i n g u i s h e d substructures i n any language s t r u c t u r e t h a t m a y be s i n g l e d out and treated as objects i n their o w n r i g h t . I n other words, these substructures become objects i n new s t r u c t u r e s t h a t c o n t a i n significant i n f o r m a t i o n a b o u t the language. T h i s is one of the features t h a t make languages so very c o m p l e x . L i n g u i s t i c s (that is, s t r u c t u r a l linguistics) is therefore a s t u d y o f the general characteristics or properties c o m m o n t o these s t r u c t u r e s . 32. Semiotics In this s e c t i o n , we consider another Saussure i d e a w h i c h , t h o u g h i n s p i r e d b y language, has a m u c h broader connection w i t h o u r s t u d y of general s t r u c tures t h a n do the s p e c i a l features of a language. S t r u c t u r a l l y s p e a k i n g , it a m o u n t s to the suggestion t h a t v a r i o u s other social systems e x h i b i t s t r u c t u r e analogous to t h a t of a language. T a k i n g language as his m o d e l , Saussure proposed a new scientific field of s t u d y , a " s t u d y of s i g n s " , for w h i c h he suggested the n a m e " s e m i o l o g y " . It w o u l d n a t u r a l l y i n c l u d e the study o f language as its most i m p o r t a n t subfield. T h e idea has h a d a great deal of influence i n c e r t a i n areas (for e x a m p l e , l i t e r a t u r e , a n t h r o p o l o g y , and psychoanalysis) a n d has undergone considerable development since i t was first l a u n c h e d . W e w i l l not a t t e m p t , however, t o explore these developments i n any d e t a i l , because o u r m a i n purpose is o n l y to b r i n g out some o f the basic connections w i t h genera! s t r u c t u r e s . W e b e g i n w i t h two c o m m e n t s b y Saussure c o n c e r n i n g the p r o posal: L a n g u a g e is a s y s t e m of signs t h a t express ideas, a n d is therefore c o m p a r a b l e to a s y s t e m o f w r i t i n g , the a l p h a b e t o f deaf mutes, s y m b o l i c rites, p o l i t e f o r m u l a s , m i l i t a r y signals, etc. B u t i t is the m o s t i m p o r t a n t o f a l l these systems. A s c i e n c e t h a t s t u d i e s t h e life of s i g n s w i t h i n s o c i e t y is conceivable; it w o u l d be a part o f social psychology and consequently of general psychology; I s h a l l c a l l it s e m i o l o g y (from G r e e k s e m e i o n ' s i g n ' ) . Semiology w o u l d show w h a t constitutes signs, w h a t laws govern t h e m . [ S I , p. 16]. Before c o m m e n t i n g u p o n a "general theory of s i g n s " , we i n c l u d e two i m p o r t a n t q u o t a t i o n s f r o m L e v i - S t r a u s s , taken f r o m his b o o k c m " S t r u c t u r a l A n t h r o p o l o g y " [L6]. T h e y b r i n g out the idea suggested i n the second S a u s -

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sure q u o t a t i o n a n d i n d i c a t e the basis for L e v i - S t r a u s s ' s t h i n k i n g c o n c e r n i n g the r e l a t i o n s h i p o f language to s o c i a l p h e n o m e n a i n general. A m o n g a l l social p h e n o m e n a , language alone has t h u s far been s t u d i e d i n a m a n n e r w h i c h p e r m i t s it to serve as the o b j e c t o f t r u l y scientific a n a l y s i s , a l l o w i n g us to u n d e r s t a n d its f o r m a t i v e process a n d to predict its m o d e of change. T h i s results f r o m m o d e r n researches i n t o the p r o b l e m o f p h o n e m i c s , w h i c h have reached beyond the s u perficial conscious a n d h i s t o r i c a l expression of l i n g u i s t i c p h e n o m e n a to a t t a i n f u n d a m e n t a l a n d objective realities consisting of systems of relations w h i c h are the p r o d u c t s of unconscious t h o u g h t processes. T h e q u e s t i o n w h i c h now arises is this: Is it possible to effect a s i m i l a r r e d u c t i o n i n the a n a l y s i s of other forms of social p h e n o m e n a ? If so, w o u l d t h i s a n a l y s i s lead to the s a m e result? A n d i f the answer to this last question is i n the affirmative, can we conclude t h a t a l l forms of s o c i a l life are s u b s t a n t i a l l y of the same n a t u r e — t h a t is, do they consist o f systems o f behavior t h a t represent the p r o j e c t i o n , o n the level of conscious and socialized t h o u g h t , of u n i v e r s a l laws w h i c h regulate the unconscious a c t i v i t i e s of the m i n d ? [pp. 58, 59]. ...the question m a y be raised whether the different aspects of social life ( i n c l u d i n g even art a n d religion) cannot o n l y be s t u d i e d by the m e t h o d s of, a n d w i t h the help of concepts s i m i l a r to those e m p l o y e d i n l i n g u i s t i c s , but also w h e t h e r they do not c o n s t i t u t e p h e n o m e n a whose i n m o s t nature is the same as t h a t of language, [p. 62] L e v i - S t r a u s s thus suggests the p o s s i b i l i t y t h a t language, i n w h i c h the use of scientific m e t h o d s is well-established, m i g h t serve as a m o d e l for the i n t r o d u c t i o n o f these m e t h o d s i n the study of a l l social p h e n o m e n a . O b s e r v e t h a t the idea goes m u c h deeper t h a n j u s t the o b s e r v a t i o n of s i m i l a r i t i e s between language and c e r t a i n aspects of s o c i a l p h e n o m e n a . It suggests t h a t these p h e n o m e n a , along w i t h language, have a c o m m o n o r i g i n i n the " u n i v e r s a l l a w s " w h i c h govern unconscious a c t i v i t y of the m i n d . T h i s is an i m p o r t a n t p o i n t , and o b v i o u s l y relates to the suggestion t h a t the m i n d has a b u i l t - i n a u t o m a t i c a n d unconscious a b i l i t y to c o n s t r u c t and m a n a g e structures. W e w i l l r e t u r n t o the subject i n the next section. W e now have a clear statement of the c l a i m t h a t social p h e n o m e n a are s t r u c t u r e d l i k e a language, a n d a suggestion o f w h y i t m i g h t be t r u e . T h e idea has h a d a deep influence u p o n L e v i - S t r a u s s ' s w o r k , b u t , as he indicates himself, the a c t u a l extent to w h i c h the general p r o p o s i t i o n is true r e m a i n s t o be settled t h r o u g h further c o l l a b o r a t i v e research between a n t h r o p o l o g i s t s and l i n g u i s t s . H o w m u c h of this c o l l a b o r a t i o n has a c t u a l l y t a k e n place is not clear, b u t i t is p l a u s i b l e t h a t the process m i g h t be f a c i l i t a t e d by an a p p l i c a t i o n of s o m e of the theory of general s t r u c t u r e s . A n a p p r o a c h to

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the p r o b l e m f r o m the p o i n t of v i e w of l i n g u i s t i c s , based o n the thesis t h a t "language is a variety of b e h a v i o r " , has been s y s t e m a t i c a l l y developed by K e n n e t h L . P i k e . H i s b o o k , " L a n g u a g e i n R e l a t i o n t o a U n i f i e d T h e o r y of t h e S t r u c t u r e of H u m a n B e h a v i o r " [P4], also contains 51 pages of references covering a w i d e range of source m a t e r i a l r e l a t i n g i n one way or another to the subject. T h e p r o b l e m now is to t r y to u n d e r s t a n d better the sense i n w h i c h various k i n d s of social p h e n o m e n a have language-type structures. W h a t characteristics d i s t i n g u i s h a language-type s t r u c t u r e f r o m any other s t r u c t u r e ? A s already suggested i n the Saussure q u o t a t i o n s , the answer to this question w i l l i n v o l v e the general n o t i o n of a " s i g n " . T h e r e already exist w i t h i n a language a n u m b e r of psychological objects more c o m p l e x t h a n the basic sound-images (or signifiers) identified w i t h spoken (or w r i t t e n ) language u n i t s . T h e y are p r o d u c e d by various m u l t i p l e w o r d c o n s t r u c t i o n s , such as phrases, sayings, cliches, a n d so f o r t h , w h i c h m a y p l a y the role of signifiers. A s suggested by the first Saussure statement quoted above, a general theory of signs w o u l d allow a w i d e variety of s i g n i fiers i n a d d i t i o n to a l l of those associated w i t h speech. I n order t o discuss these more complex objects, we w i l l replace the t e r m , " s o u n d - i m a g e " , b y the more i n c l u s i v e t e r m , " s t i m u l u s - i m a g e " , d e n o t i n g the " m e n t a l p r o d u c t " elicited by any one of a variety of social s t i m u l i not restricted to those associated w i t h language. T h e n , as i n the case of language, a general sign w o u l d consist o f a s t i m u l u s - i m a g e signifier coupled w i t h a signified concept. Necessary req u i r e m e n t s for a general sign are t h a t the s t i m u l u s - i m a g e be evoked by a c o m m o n social act, and t h a t the c o u p l i n g be essentially the same for a l l members of the c o m m u n i t y . A l t h o u g h i n d i v i d u a l s are u s u a l l y not aware of any s t r u c t u r e possessed b y s i m p l e sound-images, a s t i m u l u s - i m a g e a n d its associated concept m a y be complex enough t h a t their structures cannot be easily i g n o r e d . So, i n case the s t r u c t u r e s were s i m i l a r , the i n i t i a l association between s t i m u l u s - i m a g e a n d concept m i g h t be expected to recognize t h a t s i m i l a r i t y . T h i s means t h a t the c o u p l i n g c o u l d involve a ( p a r t i a l ) s t r u c t u r e i s o m o r p h i s m , so m a y not be completely a r b i t r a r y . E v e n i n the general case, however, the social process w h i c h converts a s t i m u l u s - i m a g e a n d concept c o m b i n a t i o n i n t o a sign w i l l tend to e l i m i n a t e differences i n the way i n d i v i d u a l s see the r e l a t i o n s h i p between the t w o o b jects. Therefore, unless there is a clear a n d c o m m o n l y perceived s t r u c t u r a l connection, the s o c i a l process w i l l , as i n the language case, t e n d t o p r o d u c e a sign i n w h i c h the c o u p l i n g is relatively t r i v i a l . In other words, u n d e r these circumstances the p r i n c i p l e of a r b i t r a r i n e s s w i l l a p p l y . A t the same t i m e , the p o s s i b i l i t y r e m a i n s t h a t the c o u p l i n g i n an established sign could

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retain i n d e f i n i t e l y a significant s t r u c t u r a l c o m p o n e n t . T h e r e is also another feature t h a t sets general signs a p a r t f r o m the s i m p l e language signs when the s t i m u l u s - i m a g e a n d its concept possess sufficient s t r u c t u r e . I n such cases, each i n d i v i d u a l w i l l tend t o s u p p l y a s t r u c t u r a l c o n n e c t i o n consistent w i t h his o w n u n d e r s t a n d i n g of the s i t u a t i o n , quite a p a r t f r o m the s o c i a l l y d e t e r m i n e d c o u p l i n g w i t h i n the s i g n . In v i e w o f the spontaneous m a n n e r i n w h i c h the h u m a n m i n d deals w i t h s t r u c t u r e s , it w o u l d be s u r p r i s i n g indeed i f such connections were not m a d e a u t o m a t i c a l l y whenever possible, regardless of whether or not the c o u p l i n g i n the sign itself involves any s t r u c t u r e features. A s an e x a m p l e of the last p o s s i b i l i t y , consider the s a y i n g : " N e v e r look a gift horse i n the m o u t h " . T h i s expression e l i c i t s a s t i m u l u s - i m a g e w h i c h is c o u p l e d i n a purely f o r m a l way w i t h the concept t h a t " i t is generally i n a d visable for one t o e x a m i n e a gift t o o c r i t i c a l l y " . T h e s a y i n g is based o n the fact t h a t a horse's age m a y be e s t i m a t e d by o b s e r v i n g how m u c h its g u m s have receded, so the p o i n t o f an e x a m i n a t i o n w o u l d be to discover whether o r not the horse is u n d e s i r a b l y o l d . T h i s c o u l d result i n e m b a r r a s s m e n t to the giver, a n d , i n any case, w o u l d i n d i c a t e an in sensitiveness to the s p i r i t of g i v i n g . T h e p o i n t is t h a t a c r i t i c a l e x a m i n a t i o n o f any gift w o u l d tend to produce a s i m i l a r undesirable result. M o s t people w i l l no d o u b t u n d e r s t a n d the a c t u a l sign content more or less as described, but m a n y w i l l not k n o w a b o u t the m e t h o d of e s t i m a t i n g a horse's age. Nevertheless, everyone c o u l d p r o b a b l y come up w i t h a p l a u s i b l e " e x p l a n a t i o n " of the s t i m u l u s - i m a g e t h a t w o u l d serve to connect it to the concept. Such e x p l a n a t i o n s c o u l d v a r y g r e a t l y w i t h o u t v i o l a t i n g the sign content. F o r e x a m p l e , it might be argued t h a t checking the horses m o u t h , for whatever reason, is not a good idea because the horse m a y bite. Such b e h a v i o r , j u s t as b e i n g over age, w o u l d constitute an undesirable q u a l i t y i n a horse. T h e fact t h a t i n d i v i d u a l s m i g h t interpret the c o u p l i n g for a general sign in n o n t r i v i a l ways m a y be irrelevant as far as the sign i t s e l f is concerned. Unless, of course, everyone w i n d s up d o i n g i t the same way. I n w h i c h case, as already n o t e d , the c o m m o n c o u p l i n g w o u l d a u t o m a t i c a l l y become a p a r t of the s i g n . S o m e t h i n g like this is suggested by an e x a m p l e f r o m a crossword puzzle, t h a t appeared i n T h e N e w H a v e n R e g i s t e r o n O c t o b e r 30, 1987. A clue i n the p u z z l e was the single word " V i o l e n t l y " , a n d the expected s o l u t i o n was the phrase, " H a m m e r a n d t o n g s " . T h e l a t t e r o b v i o u s l y refers t o c o m m o n expressions such as, " H e attacked the p r o b l e m w i t h h a m m e r a n d t o n g s " . O n the other h a n d , i n T h e O x f o r d U n i v e r s a l D i c t i o n a r y under the w o r d " H a m m e r " , we find the phrase, " H a m m e r a n d t o n g s (colloq.): w i t h m i g h t a n d m a i n (like b l a c k s m i t h s m i t i n g the i r o n taken w i t h the tongs f r o m the forge-fire)".

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Observe t h a t the b l a c k s m i t h context suggests a v i g o r o u s a n d c o n t r o l l e d , t h o u g h perhaps very p h y s i c a l , attack o n the p r o b l e m , but there is no s u g gestion at a l l o f an u n r e s t r a i n e d a p p l i c a t i o n of force as i m p l i e d by the w o r d " v i o l e n t l y " . T h e puzzle clue t h u s appears to be at best m i s l e a d i n g , a c o n c l u sion t h a t agrees completely w i t h m y o w n c h i l d h o o d m e m o r i e s of w a t c h i n g a r u r a l K a n s a s b l a c k s m i t h repair f a r m m a c h i n e r y . I was a c c o r d i n g l y s u r p r i s e d to f i n d the f o l l o w i n g entry i n W e b s t e r ' s S e v e n t h N e w C o l l e g i a t e D i c t i o n a r y : " h a m m e r a n d t o n g s (adv.): w i t h great force a n d v i o l e n c e " . It is u n d e r s t a n d a b l e , o f course, how an i n d i v i d u a l i g n o r a n t of the art of b l a c k s m i t h i n g m i g h t confuse the s m i t h y ' s v i g o r w i t h violence. O n the other h a n d , despite the d e a r t h of b l a c k s m i t h s i n o u r m o d e r n society, it is distressing to discover i n the W e b s t e r d e f i n i t i o n the fact t h a t this false i m a g e of a venerable profession has already become fixed i n our language. W e r e t u r n now to the question of w h a t i t s h o u l d m e a n for a social s y s t e m t o e x h i b i t s t r u c t u r e analogous to t h a t of a language. O b v i o u s l y the first r e q u i r e m e n t is t h a t i t consist of a s y s t e m o f signs, where the t e r m " s i g n " refers to a coupled s t i m u l u s - i m a g e and concept as described above. A c t u ally, m u c h of the s t r u c t u r a l i s t l i t e r a t u r e is devoted, d i r e c t l y or i n d i r e c t l y , to the i d e n t i f i c a t i o n a n d d e s c r i p t i o n o f signs a n d the way they are i n v o l v e d in the i n t e r a c t i o n s of i n d i v i d u a l s w i t h i n the given society. O n c e signs have been identified, the next requirement for a language-type s t r u c t u r e m u s t be t h a t relations analogous to g r a m m a r and s y n t a x exist a m o n g those signs. B u t the description of g r a m m a r a n d s y n t a x is a l r e a d y a n o n t r i v i a l task for o r d i n a r y language, so i t is not clear to w h a t extent the existence of analogous relations has been d e m o n s t r a t e d i n more general systems. P e r h a p s systems of t h i s type, because of the i n t r i n s i c character of signs, m a y a u t o m a t i c a l l y e x h i b i t the expected r e l a t i o n a l s t r u c t u r e of a language. A t any rate, s o m e t h i n g s i m i l a r t o the p i c t u r e we have sketched appears t o be w h a t those w h o suggest t h a t a s o c i a l s y s t e m resembles a language have i n m i n d . A n o t h e r field i n w h i c h s t r u c t u r a l l i n g u i s t i c s has had a significant i n f l u ence is p s y c h o a n a l y s i s , the strongest proponent being the F r e n c h p s y c h o a n a l y s t , Jaques L a c a n . T h e s p e c i a l role of language i n p s y c h o a n a l y s i s was already u n d e r s t o o d by F r e u d , as i n d i c a t e d by his emphasis o n the i n f o r m a t i o n unconsciously revealed i n language t h r o u g h d r e a m s , slips of the tongue, p u n s , a n d so f o r t h . L a c a n , however, w h o is a follower o f the master, goes farther b y suggesting t h a t the F r e u d i a n unconscious is i t s e l f s t r u c t u r e d like a language. [ L I , p . 20; L3] T h e " F r e u d i a n unconscious" is t h a t p o r t i o n of the "general unconscious" (or "subconscious") of p r i m a r y concern i n p s y c h o a n a l y s i s , a n d is a very special p a r t of the whole. T h e general unconscious contains m u c h m a t e r i a l

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t h a t is easily r e c a l l e d , a n d accounts for the great b u l k of a l l m e n t a l a c t i v i t y . B y c o m p a r i s o n , the conscious appears t o be h a r d l y m o r e t h a n a r i p p l e on the surface, p l a y i n g m a i n l y a role i n the various modes o f c o m m u n i c a t i o n . T h e unconscious a c c o m m o d a t e s most of the r o u t i n e m e n t a l f u n c t i o n s , such as the processing a n d s t o r i n g of i n f o r m a t i o n , as well as m a j o r p o r t i o n s of the highest m e n t a l functions. T h e " F r e u d i a n u n c o n s c i o u s " , o n the other h a n d , is h i g h l y r e s t r i c t e d . Peter G a y i n his recent b i o g r a p h y of F r e u d [G3, p. 128], describes it as follows: M o s t of the unconscious consists o f repressed m a t e r i a l s . T h i s u n conscious, as Freud c o n c e p t u a l i z e d i t , is not the segment o f m i n d h a r b o r i n g t h o u g h t s t e m p o r a r i l y out of sight a n d easily r e c a l l e d ; t h a t is w h a t he called the preconscious. R a t h e r , the unconscious proper resembles a m a x i m u m security p r i s o n h o l d i n g a n t i s o c i a l i n m a t e s ... forever a t t e m p t i n g to escape. A t the same t i m e , the F r e u d i a n unconscious is s o m e t h i n g more or less c o m m o n t o a l l i n d i v i d u a l s , at least w i t h i n a g i v e n c u l t u r a l g r o u p . It is also a p r o d u c t of s o c i a l i n t e r a c t i o n , and m a y be t h o u g h t o f as a s u b s t r u c t u r e of the m i n d , " p r o g r a m m e d " by c e r t a i n c o m m o n experiences of the g r o u p but u s u a l l y inaccessible t h r o u g h n o r m a l channels. In order for this special s u b s t r u c t u r e of the m i n d to be l i k e a l a n g u a g e , it m u s t consist o f a s y s t e m of general signs. A s i n a l l c o m p l e x signs, the coup l i n g of s t i m u l u s - i m a g e to concept m a y range f r o m a r b i t r a r y associations t o connections t h a t involve i n t e r n a l s t r u c t u r e o f the two. T h e r e is a m p l e evidence of the different k i n d s of c o u p l i n g i n p s y c h o a n a l y t i c p h e n o m e n a . A m o n g the various functions expected of a process s i m i l a r to a language, are the i n t e r n a l exchanges analogous t o those associated w i t h o r d i n a r y selfc o m m u n i c a t i o n . In this case, however, these exchanges cannot n o r m a l l y be m o n i t o r e d a n d c o n t r o l l e d d i r e c t l y by the conscious m i n d . I n fact, the v i r t u a l e x c l u s i o n o f conscious i n t e r v e n t i o n i n the business of the F r e u d i a n unconscious, t h o u g h it is s t r u c t u r e d like a language, s h a r p l y distinguishes it f r o m o r d i n a r y language. It is clear t h a t this unconscious is not t o t a l l y isolated f r o m either the rest of the m i n d or the e x t e r n a l w o r l d . It receives c e r t a i n outside inform a t i o n a n d gives up pieces of i n f o r m a t i o n i n one f o r m or another, as, for e x a m p l e , i n dreams and other b e h a v i o r of w h i c h the i n d i v i d u a l m a y not always be aware. T h e u t i l i z a t i o n of outside m a t e r i a l is u n p r e d i c t a b l e , and any m a t e r i a l f o r m e d i n the unconscious c a n , as a rule, enter consciousness only i n d i r e c t l y , i f at a l l . Because these exchanges are not under c o n t r o l of the conscious m i n d , they often appear o n the surface to be t o t a l l y i r r a t i o n a l and to occur s t r i c t l y by chance. T h e a c t u a l i n f o r m a t i o n t h a t they c o n t a i n is s e l d o m evident to the casual observer u n f a m i l i a r w i t h the " l a n g u a g e " ,

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and the m e a n i n g of a g i v e n signifier as i t emerges f r o m the unconscious m a y be q u i t e obscure. In other words, it m a y be very difficult t o s u p p l y the second h a l f of the sign i n v o l v e d i n an exchange a n d to determine its r e l a t i o n s h i p to the rest of the message. D e c i p h e r i n g the messages p r o d u c e d by the unconscious, d i s c o v e r i n g the " r u l e s " w h i c h govern s t r u c t u r e f o r m a t i o n w i t h i n the unconscious, e x p l a i n ing the ways i n w h i c h the unconscious interacts w i t h the conscious, a n d e x p o s i n g the general significance of the unconscious a c t i v i t y to the i n d i v i d ual are m a j o r p r o b l e m s i n the field of p s y c h o a n a l y s i s . A serious t r e a t m e n t of these topics f r o m the p o i n t o f v i e w of structures w o u l d surely b r i n g out m a n y interesting s t r u c t u r a l p h e n o m e n a , but is beyond b o t h the scope of this w o r k a n d the a u t h o r ' s knowledge of the subject. 33.

The Language Faculty

T h e a b i l i t y to use a language of great c o m p l e x i t y a n d flexibility is one of the most obvious q u a l i t i e s t h a t set h u m a n beings a p a r t f r o m a l l other a n i m a l s . F u r t h e r m o r e , the c o m p a r a t i v e ease and s p o n t a n e i t y w i t h w h i c h a child learns a language suggests t h a t h u m a n s are equipped w i t h a n i n n a t e language faculty of some k i n d or other. A t the same t i m e , it is not clear j u s t how s p e c i a l such a language f a c u l t y m i g h t be. Is it unique to h u m a n beings, or do certain other a n i m a l s also possess a language p o t e n t i a l ? These are controversial questions w h i c h o b v i o u s l y cannot be settled by purely theoretical a r g u m e n t s . T h e r e are, however, s t r u c t u r a l considerations t h a t m a y t h r o w some l i g h t o n the s i t u a t i o n . L i n g u i s t N a o m C h o m s k y is an o u t s t a n d i n g a d v o c a t e of the v i e w t h a t h u m a n s are equipped w i t h a special i n n a t e f a c u l t y for l e a r n i n g a language. T h e following q u o t a t i o n s , w h i c h o u t l i n e C h o m s k y ' s ideas c o n c e r n i n g the significance of such a faculty to the s t u d y of l i n g u i s t i c s , are t a k e n f r o m his b o o k , " L a n g u a g e a n d P r o b l e m s of K n o w l e d g e " [C4]. .... we a t t e m p t to construct a g r a m m a r , a theory of u n i v e r s a l g r a m mar, a theory of the fixed a n d i n v a r i a n t p r i n c i p l e s t h a t c o n s t i t u t e the h u m a n language f a c u l t y a n d the parameters of v a r i a t i o n associated w i t h t h e m . W e can t h e n , i n effect, deduce p a r t i c u l a r languages b y s e t t i n g the parameters i n one or another way.... L a n g u a g e l e a r n i n g , t h e n , is the process of d e t e r m i n i n g the values of the parameters left unspecified b y u n i v e r s a l g r a m m a r , of s e t t i n g the switches t h a t m a k e the network f u n c t i o n . . . . (It) is not r e a l l y somet h i n g t h a t the c h i l d does; it is s o m e t h i n g t h a t happens t o the c h i l d placed i n an a p p r o p r i a t e e n v i r o n m e n t , m u c h as the c h i l d ' s b o d y grows and matures i n a predetermined way when p r o v i d e d w i t h a p p r o p r i a t e n u t r i t i o n and e n v i r o n m e n t a l s t i m u l a t i o n ....environment determines the way the parameters of universal g r a m m a r are set, y i e l d i n g differ-

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ent languages, [pp. 133, 134] A n u m b e r of e x p e r i m e n t s t h a t tend t o s u p p o r t the existence of a n i n b o r n language f a c u l t y are described by Peter D . E i m a s i n a survey a r t i c l e , " T h e P e r c e p t i o n of S p e e c h i n E a r l y I n f a n c y " [ E l ] . A f t e r o u t l i n i n g briefly the a p p r o a c h a n d s o m e of the results of these e x p e r i m e n t s , w h i c h are concerned w i t h speech p e r c e p t i o n at the phoneme level, we w i l l m a k e a few c o m m e n t s concerning t h e m . T h e basic acoustic u n i t s , or segments, i n speech vary w i t h respect t o a n u m b e r of acoustic p a r a m e t e r s , w h i c h c a r r y the i n f o r m a t i o n necessary for the p e r c e p t i o n of phonemes. T h e r e l a t i o n s h i p between acoustic u n i t s a n d phonemes is n o t , however, a s i m p l e one. T h o u g h a s m a l l change i n a single p a r a m e t e r m a y s i g n a l a change i n the perceived phoneme, it appears t o be more c o m m o n t h a t p a r a m e t e r s c a n vary w i d e l y w i t h o u t a c o r r e s p o n d i n g p h o n e m e change. (Note the suggestion of " s t r u c t u r a l s t a b i l i t y " i n this c o m m e n t . ) T h e a u t h o r cites e x p e r i m e n t a l results c o n f i r m i n g " t h a t i n the p e r c e p t i o n of speech we are o r d i n a r i l y aware o f discrete p h o n e m i c categories rather t h a n o f the continuous v a r i a t i o n i n each acoustic p a r a m e t e r : we perceive speech c a t e g o r i c a l l y " . I n other words, a listener is able to e x t r a c t f r o m a v a r i e t y of speech segments a c o m m o n s t r u c t u r e t h a t identifies a p a r t i c u l a r phoneme. T h u s , as far as perception is concerned, t h i n g s are already q u i t e c o m p l e x even at the lowest level of language s t r u c t u r e . A l t h o u g h m a n y of the experiments o n phoneme p e r c e p t i o n have been carried out o n a d u l t s or c h i l d r e n , E i m a s , a m o n g others, s t u d i e d i n f a n t s . Because i n f a n t s cannot give v e r b a l reports, experimenters use s u c k i n g rate or heart rate as i n d i c a t o r s . F o r e x a m p l e , w h e n an i n f a n t is first presented w i t h a s o u n d , representing, say, a p a r t i c u l a r consonant, its s u c k i n g rate w i l l first increase a n d t h e n decrease w i t h f a m i l i a r i t y . U p o n presentation of a new s t i m u l u s , t h e s u c k i n g rate w i l l increase s h a r p l y , i n d i c a t i n g p e r c e p t i o n o f the change. U s i n g techniques of this k i n d , various aspects o f phoneme perception i n i n f a n t s have been s t u d i e d , l e a d i n g to the conclusion t h a t i n f a n t s also "perceive speech c a t e g o r i c a l l y " . S i m i l a r results have been o b t a i n e d for infants f r o m a v a r i e t y o f language b a c k g r o u n d s . A l t h o u g h the above i t e m s cover m o s t of the results t h a t bear o n our discussion, the e x p e r i m e n t s act u a l l y i n v o l v e d a more d e t a i l e d i n v e s t i g a t i o n of the p h e n o m e n a t h a n c a n be covered i n a b r i e f sketch. T h e results o f these experiments s u p p o r t , or at least do not c o n t r a d i c t , the existence o f an i n n a t e language f a c u l t y . P e r c e i v i n g phonemes is o b v i ously a necessary c o n d i t i o n for l e a r n i n g a spoken l a n g u a g e , b u t i t is not clear t h a t i t is necessarily peculiar to language c a p a b i l i t y . F o r e x a m p l e , there are e x p e r i m e n t s w h i c h i n d i c a t e t h a t Japanese q u a i l also are able to d i s t i n g u i s h p h o n e m i c categories [ K l ] , an a b i l i t y t h a t no one w o u l d c l a i m indicates a language p o t e n t i a l , at least i n the sense u n d e r s t o o d here. In

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fact, some p h o n e m i c categories may s i m p l y represent a general c o n t r a c t i o n process (as described i n Section 27) t h a t facilitates d e a l i n g w i t h a mass of sound data. A n o t h e r i t e m relevant t o the n a t u r e of a language f a c u l t y is the fact t h a t i n d i v i d u a l s w h o have been deaf f r o m b i r t h are able to l e a r n a sign language. A l t h o u g h the s t r u c t u r e of a s t a n d a r d sign language closely resembles t h a t of an o r d i n a r y language (perhaps because of an influence of o r d i n a r y language o n the i n i t i a l c o n s t r u c t i o n of the f o r m e r ) , the p o i n t here is t h a t l e a r n i n g the sign language is a very different process f r o m l e a r n i n g a spoken language. So the language faculty, whatever its n a t u r e , cannot be as s p e c i a l i z e d as one m i g h t first expect. R e c a l l t h a t b o t h Saussure a n d L e v i - S t r a u s s c l a i m e d t h a t s o c i a l p h e n o m e n a were s t r u c t u r e d l i k e a language. A t t h i s p o i n t , the c l a i m appears quite reasonable, since any social p h e n o m e n a o b v i o u s l y m u s t i n v o l v e c o m m u n i c a t i o n i n s o m e f o r m or another. F u r t h e r m o r e , L e v i - S t r a u s s also conjectured t h a t the language-type s t r u c t u r e is d e t e r m i n e d b y " u n i v e r s a l laws w h i c h regulate the unconscious a c t i v i t i e s of the m i n d " . F r o m our p o i n t o f view, t h i s w o u l d m e a n t h a t the u n d e r l y i n g s t r u c t u r e of each type of c o m m u n i c a t i o n is d e t e r m i n e d b y the general laws t h a t regulate the f o r m a t i o n a n d processing of s t r u c t u r e s . T h e r e exists the p o s s i b i l i t y t h a t a language type s t r u c t u r e for any c o m m u n i c a t i o n process m i g h t eventually be deduced f r o m general properties of the s t r u c t u r i n g process itself. T h i s w o u l d require, of course, the cons t r u c t i o n of an a p p r o p r i a t e a x i o m a t i z a t i o n of the s t r u c t u r i n g process, f r o m w h i c h the desired result m i g h t be derived. A l t h o u g h it is c o m m o n to view the language f a c u l t y as a p r o d u c t of the e v o l u t i o n of language, the suggestion here is t h a t language is an a d a p t a t i o n o f a more f u n d a m e n t a l p r o d u c t of e v o l u t i o n , the general s t r u c t u r i n g a b i l i t y of the m i n d . G i v e n the l a t t e r , language proper c o u l d evolve f r o m the s i m p l e c o m m u n i c a t i o n signals t h a t m a n y a n i m a l s use for w a r n i n g and i d e n t i f i c a t i o n . T h i s d e s c r i p t i o n of the h u m a n language f a c u l t y does not d i r e c t l y c o n t r a d i c t either the innateness hypothesis or the other ideas expressed b y C h o m s k y . It o n l y suggests t h a t the language f a c u l t y is j u s t one m a n i f e s t a t i o n o f a very general c a p a b i l i t y of the m i n d . In other words, it is b o t h i n n a t e a n d s p e c i a l , t h o u g h special i n a sense rather different f r o m t h a t i m p l i e d by the usual hypothesis. T h e p o s s i b i l i t y t h a t language type structures are as u n i v e r s a l as c l a i m e d , has i m p l i c a t i o n s for a n i m a l s other t h a n h u m a n s . A f t e r a l l , any l i v i n g org a n i s m , i n order t o adjust t o its e n v i r o n m e n t , m u s t be e q u i p p e d i n some degree t o s t r u c t u r e the flood of s t i m u l i t h a t impinges o n i t . T h e r e f o r e , given a sufficient level of c o m p l e x i t y a n d the a b i l i t y t o adjust to s o c i a l s i t u a t i o n s , an a n i m a l m i g h t be expected to e x h i b i t , i n some f o r m or another, a

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l i m i t e d language a b i l i t y . Therefore, some o f the c l a i m s for language a b i l i t y in a n i m a l s become more p l a u s i b l e . I n fact, when one considers the extent a n d i n t e n s i t y o f the experience to w h i c h a c h i l d is n o r m a l l y subjected w h i l e l e a r n i n g the language, it seems q u i t e r e m a r k a b l e t h a t a c h i m p a n z e e , for exa m p l e , i n an a r t i f i c i a l e n v i r o n m e n t i n v o l v i n g a few h u m a n s , a n d presented w i t h the task o f l e a r n i n g a r e l a t i v e l y u n n a t u r a l s i g n language, is able to a c c o m p l i s h as m u c h as i t a p p a r e n t l y does.

CHAPTER

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34.

IN

VI

MENTAL

PHENOMENA

Introduction

T h i s chapter is devoted t o several rather different t h o u g h not u n r e l a t e d topics p e r t a i n i n g m o s t l y to the role of structures i n higher m e n t a l p h e n o m e n a . It depends heavily, b o t h d i r e c t l y a n d i n d i r e c t l y , o n the s t r u c t u r e ideas developed i n the preceding chapters, and the topics raise f a m i l i a r p h i l o s o p h i c a l questions t h a t have been s t u d i e d i n one f o r m or another since ancient times. A l t h o u g h some of these questions were touched u p o n i n Section 14 and others w i l l be considered i n Section 37, our m a i n o b j e c t i v e is rather different and a great deal m o r e l i m i t e d i n scope, ft is s i m p l y t o s k e t c h , f r o m the p o i n t of view of s t r u c t u r e s , a plausible description of how p e r c e i v i n g a n d u n d e r s t a n d i n g m i g h t conceivably take place. T h e scenarios are t o a large extent deduced f r o m general properties of s t r u c t u r e s a n d are not offered as an accurate account o f w h a t a c t u a l l y occurs. T h e y are nevertheless valuable for o r g a n i z i n g ones t h i n k i n g o n a subject, a n d p r o v i d e a h e l p f u l a p p r o a c h i n Section 39 to certain p r o b l e m s of t e a c h i n g a n d l e a r n i n g . For o u r i m m e d i a t e purposes, it is i m m a t e r i a l whether the m i n d is regarded as an e n t i t y e x i s t i n g apart f r o m the b r a i n , or as a s t r i c t l y secondary p h e n o m e n o n associated w i t h the p h y s i c a l f u n c t i o n i n g of the b r a i n . A l s o , e x a c t l y how m e n t a l structures are represented i n the b r a i n is a m y s t e r y t h a t we w i l l not a t t e m p t to resolve, a l t h o u g h the representations o b v i o u s l y m u s t involve the n e u r a l s t r u c t u r e i n one way or another. In some contexts i t is helpful to t h i n k of the n e u r a l representations as analogous to o r d i n a r y e l e c t r i c a l networks. F o r e x a m p l e , t h i s analogy is used i n Section 46 of the next chapter to s u p p o r t a s t r u c t u r a l d e s c r i p t i o n of the creative process as it appears t o o c c u r i n m a t h e m a t i c s . A t the same t i m e , the electrical network m o d e l , t h o u g h sometimes useful, is m u c h too s i m p l e to represent accurately the s t r u c t u r e a n d f u n c t i o n of neural networks, w h i c h involve c o m p l e x synapses a n d d e p e n d not o n l y o n e l e c t r i c a l b u t o n c h e m i c a l processes as w e l l . In S e c t i o n 35 some of the details associated w i t h the role o f 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 are discussed, a n d S e c t i o n 36 consists of some observations o n the nature o f t h i s involvement w i t h s t r u c t u r e s . Section 37 is devoted to some p h i l o s o p h i c a l r e m a r k s c o n c e r n i n g the r e l a t i o n s h i p between 93

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m e n t a l s t r u c t u r e s a n d the things they represent, and the s i m i l a r i t y of m e n t a l experiences of different i n d i v i d u a l s . It also contains the suggestion of a s t r u c t u r a l s e t t i n g i n w h i c h consciousness m i g h t c o n c e i v a b l y occur. S e c t i o n 38 contains a d e s c r i p t i o n o f a person's general b a c k g r o u n d s t r u c t u r e , where each i t e m o f ones knowledge is recorded as a s u b s t r u c t u r e , a l o n g w i t h a s t r u c t u r a l d e s c r i p t i o n of the process of u n d e r s t a n d i n g . S e c t i o n 39 deals w i t h some s t r u c t u r a l aspects of teaching a n d l e a r n i n g i n the l i g h t of ideas developed i n earlier sections. 35.

T h e C e n t r a l R o l e of Structures

O u r general p o i n t of v i e w , t h a t most m e n t a l a c t i v i t y is based o n the direct processing of structures, is at least i m p l i c i t i n m u c h o f the precedi n g m a t e r i a l a n d underlies a l l t h a t we w i l l have t o say concerning m e n t a l p h e n o m e n a . W e also assume t h a t the m i n d is able t o deal d i r e c t l y w i t h n o n t r i v i a l w h o l e s t r u c t u r e s , b a s i c a l l y w i t h o u t i n t e r v e n t i o n of s y m b o l i c or other i n t e r m e d i a t e devices for representing the s t r u c t u r e s . T h e s e are l i k e c o m plex p i c t u r e s t h a t are v i s u a l i z e d a l l at once. T h e r e are m a n y c o m m o n p l a c e experiences (such as the use of analogies discussed i n S e c t i o n 9), as well as m o r e s o p h i s t i c a t e d p h e n o m e n a , t h a t are most conveniently e x p l a i n e d by this assumed a b i l i t y t o process s t r u c t u r e s . A n a l o g o u s a s s u m p t i o n s also u n d e r l i e m u c h of cognitive psychology, w i t h v a r y i n g e m p h a s i s o n structures a n d how they are processed. F o r e x a m p l e , i n a b o o k by H o w a r d M a r g o l i s , " P a t t e r n s , T h i n k i n g , a n d C o g n i t i o n " [M2] , the general s t r u c t u r a l a p p r o a c h is quite e x p l i c i t . T h e a u t h o r develops the thesis t h a t p a t t e r n (i.e. s t r u c t u r e ) recognition is f u n d a m e n t a l to t h i n k i n g a n d j u d g e m e n t , b u t rejects the more c o m m o n i d e a t h a t these processes d e p e n d p r i m a r i l y o n rules and logic. I n other words, the b r a i n is " a - r a t i o n a l " . M a r g o l i s [ C h a p t e r 4] also offers some controversial suggestions [ D l ] as to how the b r a i n a c t u a l l y deals w i t h p a t t e r n s . In a b o o k o n , M e n t a l I m a g e s a n d T h e i r T r a n s f o r m a t i o n s [S4], p . 119, Roger N . S h e p a r d a n d L y n n A . C o o p e r also give evidence for the direct processing of m e n t a l structures i n a series o f s i m p l e e x p e r i m e n t s , m a n y of t h e m i n v o l v i n g r o t a t i o n s a n d other s p a c i a l t r a n s f o r m a t i o n s o f various m e n t a l objects. It seems t h a t most cognitive psychologists adopt either a " c o n n e c t i o n i s t " or " t r a d i t i o n a l i s t " p o i n t o f v i e w . C o n n e c t i o n i s t s a p p r o a c h the subject t h r o u g h " n e u r a l n e t w o r k s " , and are m u c h influenced b y c o m p u t e r m o d els, w h i l e t r a d i t i o n a l i s t s emphasize the m a n i p u l a t i o n o f " s t r u c t u r e d s y m b o l i c e x p r e s s i o n s " . W h a t e v e r the differences, a l l such approaches a m o u n t u l t i m a t e l y to m e t h o d s of representing a n d processing s t r u c t u r e s . S t r u c t u r e processing o b v i o u s l y must o c c u r t o some degree i n a l l a n i m a l s w i t h a b r a i n , the m a i n differences b e i n g i n c o m p l e x i t y a n d s o p h i s t i c a t i o n . T h u s , the i d e n t i f i c a t i o n of basic m e n t a l a c t i v i t y w i t h pure s t r u c t u r e p r o -

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cessing provides a sort, of bridge between h u m a n s a n d other a n i m a l s . T h e r e r e m a i n s , however, a w i d e g a p , s i m p l y because the h u m a n process is so enorm o u s l y c o m p l e x , a c c o m m o d a t i n g , a m o n g other secondary features, a h i g h l y developed language facility. D e s p i t e the fact t h a t so l i t t l e is k n o w n c o n c e r n i n g the a c t u a l process of f o r m i n g m e n t a l s t r u c t u r e s (i.e., images), there are a few general c o m m e n t s on the subject t h a t are perhaps w o r t h m a k i n g . In the first place, the f o r m a t i o n o f a m e n t a l s t r u c t u r e c o r r e s p o n d i n g t o i n c o m i n g i n f o r m a t i o n is never a s t r i c t l y passive process. For e x a m p l e i n v i s i o n , the r e t i n a l i m a g e , t h o u g h t w o - d i m e n s i o n a l , is perceived whenever possible as representing a threed i m e n s i o n a l o b j e c t . A l t h o u g h this is to be expected because we evolved i n a t h r e e - d i m e n s i o n a l e n v i r o n m e n t , i t does i l l u s t r a t e how the m i n d s y s t e m a t i c a l l y adds to the r a w d a t a . A n even more i n t e r e s t i n g e x a m p l e a l o n g these same lines is the u s u a l response t o the A l b e r s c o n s t r u c t i o n s discussed i n Section 17. R e c a l l t h a t the c o n s t r u c t i o n s are plane figures t h a t a p p e a r to represent t h r e e - d i m e n s i o n a l objects. T h e full representation, however, c a n not occur because the figures c o n t a i n c o n t r a d i c t o r y i n f o r m a t i o n . Despite this " i m p o s s i b l e " s i t u a t i o n , the m i n d is able to resolve the c o n t r a d i c t i o n by i n t r o d u c i n g m o t i o n i n t o the picture. A n o t h e r significant aspect of m e n t a l i m a g e r y arises f r o m the fact t h a t images m a y c o r r e s p o n d t o d a t a f r o m one (or m o r e ) of the senses other t h a n sight. F o r e x a m p l e , a perception of a s o l i d o b j e c t i n space m a y be derived f r o m the sense of touch as well as sight, no doubt w i t h different results i n the t w o cases. In t h i s s i t u a t i o n , a person b l i n d f r o m b i r t h is w h o l l y dependent on the sense of touch and could conceivably w i n d up w i t h a more or less nonperspective m e n t a l image of the o b j e c t . T h e r e are i n d i c a t i o n s t h a t even sighted persons m a y have s i m i l a r experiences when the sense of touch is c o m b i n e d w i t h t h a t of sight. T h i s is suggested by i n t r o s p e c t i o n as w e l l as some c h i l d r e n ' s d r a w i n g s t h a t show s i m u l t a n e o u s l y the f r o n t , back, a n d interior of a house. A t the same t i m e , because sight is so i m p o r t a n t to an a n i m a l ' s s u r v i v a l , i t is l i k e l y t h a t e v o l u t i o n e a r l y o n b u i l t i n t o a l l o f us some of the space i n t u i t i o n derived f r o m the v i s u a l experience. Therefore, a b l i n d person m a y possess some v i s u a l space i n t u i t i o n w i t h o u t ever h a v i n g experienced v i s i o n . F o r this reason, it is also conceivable t h a t a b l i n d person's m e n t a l image of a solid object, t h o u g h derived s t r i c t l y f r o m the sense of t o u c h , m a y s t i l l involve some i n t u i t i v e l y added p e r s p e c t i v i t y . It is obvious t h a t the b r a i n already e x h i b i t s at b i r t h a great deal of " h a r d w i r e d " s t r u c t u r e , w h i c h provides not o n l y c o n t r o l of o r d i n a r y b o d i l y f u n c t i o n s b u t also some n o n t r i v i a l m e n t a l functions as w e l l . In other words, h u m a n beings, a n d no d o u b t also m a n y other a n i m a l s , are b o r n w i t h a w e l l developed c a p a c i t y for d e a l i n g w i t h s t r u c t u r e s , a l o n g w i t h the p o t e n t i a l for f o r m i n g a n d m a n i p u l a t i n g new s t r u c t u r e s . T h e fact t h a t i n d i v i d u a l s

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are able r o u t i n e l y to c o m m u n i c a t e relatively c o m p l e x i n f o r m a t i o n a m o n g themselves also suggests t h a t the s t r u c t u r i n g process does not differ greatly f r o m one person t o another. T h i s is not to say, of course, t h a t everyone processes i n f o r m a t i o n i n e x a c t l y the same way, but o n l y t o c l a i m t h a t the u n d e r l y i n g m e c h a n i s m m u s t be essentially the same. T h o u g h the basic s t r u c t u r i n g process m a y i n v o l v e w h o l e s t r u c t u r e s , it is obvious t h a t s o m e structures w i l l be t o o complex or extensive t o be processed as wholes a n d w i l l require b r e a k i n g d o w n i n t o s m a l l e r structures. A s we have a l r e a d y seen i n Sections 29 a n d 30, such d e c o m p o s i t i o n s are n o r m a l l y i n v o l v e d i n language c o m m u n i c a t i o n of a s t r u c t u r e . In t h i s case, the language s t r u c t u r e provides a m e t h o d of representation t h a t enables piecewise c o m m u n i c a t i o n of the s t r u c t u r e . L a n g u a g e , however, is not l i k e l y to be the o n l y s t r u c t u r a l process by w h i c h s t r u c t u r e s m i g h t be dealt w i t h in a piecewise f a s h i o n . T h e m i n d can no d o u b t decompose and reassemble structures d i r e c t l y a n d q u i t e i n d e p e n d e n t l y of language. A l s o , the need to decompose structures for purposes of c o m m u n i c a t i o n is the p o i n t at w h i c h s y m b o l s , codes, etc., as well as language, m a y enter i n t o the p i c t u r e . Because o f our ignorance concerning the m a n n e r i n w h i c h s t r u c t u r e s are recorded and processed i n the b r a i n , we m a y as well assume t h a t the processing, i n whatever way i t is a c c o m p l i s h e d , proceeds more or less i n accordance w i t h the properties of general structures. T h i s a p p r o a c h avoids any need to speculate a b o u t a c t u a l p h y s i o l o g i c a l mechanisms t h a t m i g h t be capable of d e a l i n g a p p r o p r i a t e l y w i t h structures, and serves to focus even more a t t e n t i o n o n the structures themselves. It r e m a i n s t r u e , of course, t h a t any knowledge at the p h y s i o l o g i c a l level concerning these m a t t e r s w o u l d be b o t h i n t e r e s t i n g a n d i m p o r t a n t , t h o u g h it c o u l d not by itself reveal the essential n a t u r e o f higher m e n t a l processes. L o w level b r a i n a c t i v i t y m a y be largely devoted t o more or less routine control functions o f a type, for e x a m p l e , t h a t m i g h t be easily s i m u l a t e d o n a d i g i t a l c o m p u t e r . B y contrast, higher m e n t a l processes have a very different character, due t o the variety a n d flexibility of their involvement w i t h s t r u c t u r e s . I n whatever m a n n e r structures m i g h t be represented a n d m a n i p u l a t e d at the p h y s i o l o g i c a l level, the p h e n o m e n a t h a t are o f concern t o us must o c c u r at one of the higher levels of structure o r g a n i z a t i o n . A t this p o i n t , it is desirable to say m o r e precisely w h a t is m e a n t by expressions such as "higher level structure o r g a n i z a t i o n " . F i r s t , however, we need t o r e m i n d the reader of c e r t a i n general facts a b o u t s t r u c t u r e s . R e c a l l t h a t a s t r u c t u r e is by d e f i n i t i o n d e t e r m i n e d as soon as its objects and relations are specified. T h i s statement, t h o u g h t r u e , u n f o r t u n a t e l y tends t o encourage an oversimplified view of the s t r u c t u r e concept, p a r t l y because i t does not focus o n the p o t e n t i a l c o m p l e x i t y of the r e l a t i o n s . In p a r t i c u l a r , it is easy t o f a l l i n t o the t r a p o f v i s u a l i z i n g a general s t r u c t u r e

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more or less as a s i m p l e p o i n t - l i n e s t r u c t u r e . A l t h o u g h the p o i n t - l i n e i n t e r p r e t a t i o n serves the desired purpose i n m a n y cases, it is o b v i o u s t h a t there can be m u c h more to a s t r u c t u r e t h a n is suggested by these s i m p l e pictures. F o r e x a m p l e , as we have seen earlier, a s t r u c t u r e m a y i n v o l v e certain d y n a m i c features. It is also necessary sometimes to a p p r o a c h s t r u c t u r e s i n ways t h a t m a y even d e p e n d o n considerations t h a t extend b e y o n d the basic o b j e c t - r e l a t i o n specifications. In t h i s d i r e c t i o n , a g i v e n s t r u c t u r e m i g h t be considered f r o m one of the f o l l o w i n g v i e w p o i n t s : (1) as a s u b s t r u c t u r e of a larger s t r u c ture (not d e t e r m i n e d by i t ) , (2) i n t e r m s of one o f its c o n t r a c t i o n s , (3) t h r o u g h images under h o m o m o r p h i s m s or other more c o m p l e x t r a n s f o r m a t i o n s , (4) i n t e r m s of s p e c i a l concrete representations. N o t e t h a t each of these depends on e x t e r n a l i n f o r m a t i o n not carried by the s t r u c t u r e itself. F o r e x a m p l e , a c o n t r a c t i o n of the s t r u c t u r e (Section 27) is d e t e r m i n e d by a d i s j o i n t collection of s u b s t r u c t u r e s , possibly chosen q u i t e i n d e p e n d e n t l y of a n y o b v i o u s features o f the given s t r u c t u r e . It is these e x t e r n a l l y det e r m i n e d properties of s t r u c t u r e , as opposed t o a s i m p l e o b j e c t - r e l a t i o n properties, t h a t are referred t o i n expressions such as " h i g h e r level s t r u c t u r e o r g a n i z a t i o n " or " h i g h e r m e n t a l processes". T h e p o i n t is t h a t one cannot derive, s t r i c t l y f r o m the s t r u c t u r e s t h e m selves, higher level s t r u c t u r a l p h e n o m e n a t h a t depend o n independent external connections. E x t e r n a l factors m a y be l i m i t e d by, but generally not d e t e r m i n e d by, the u n d e r l y i n g s t r u c t u r e s . I n p a r t i c u l a r , a l t h o u g h higher m e n t a l p h e n o m e n a are somehow associated w i t h s u b s t r u c t u r e s of the general b r a i n s t r u c t u r e , they cannot be u n d e r s t o o d i n s t r i c t l y p h y s i o l o g i c a l terms. A n y effort to do so is analogous to an a t t e m p t t o u n d e r s t a n d c o m puter software o n l y i n t e r m s of the h a r d w a r e . 36.

T h e D r i v e for I n t e l l i g i b i l i t y

T h e basis for most of the discussion i n this section is the fact t h a t the h u m a n m i n d manages t o perceive s t r u c t u r e w i t h i n v i r t u a l l y any s y s t e m presented to i t . O n e t h i n g t h a t seems to be clear is t h a t the process is a u t o m a t i c a n d d r i v e n by the necessity t o m a k e sense of the given s y s t e m . In other words, we have a b u i l t - i n d r i v e for i n t e l l i g i b i l i t y . A n obvious expression of t h i s d r i v e has already been noted i n the way most people respond to an A l b e r s d r a w i n g . These ideas are also i l l u s t r a t e d b y the s i m p l e e x a m p l e presented below i n F i g u r e 36.1 w h i c h was a d a p t e d f r o m a b o o k o n " V i s i o n " by D a v i d M a r r [ M 3 , p. 50]. N o t i c e t h a t the figure consists of a r e p e t i t i o n of several s i m p l e elements t h a t are perceived to group themselves i n a variety of more c o m p l e x a n d c o n s t a n t l y c h a n g i n g patterns t h a t appear a n d disappear i n a more or less r a n d o m f a s h i o n . T h e r e is also a p e r c e p t i o n of m o t i o n , as o c c u r r e d i n the

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A l b e r s e x a m p l e s . T h e p o i n t is t h a t the c o m p l e x p a t t e r n s , a l t h o u g h they are p o t e n t i a l substructures of the complete figure, i n a sense do not r e a l l y exist u n t i l the m i n d a c t u a l l y "creates" (or discovers) t h e m i n i t s search for m e a n i n g . T h e created s t r u c t u r e s , t h o u g h realized i n the g i v e n s y s t e m , are i n s p i r e d b y the observer's previous knowledge a n d experience. T h u s , the p a t t e r n f o r m a t i o n i n the e x a m p l e does not differ i n p r i n c i p l e f r o m the conjecture f o r m a t i o n t h a t takes place i n the m o r e c o m p l e x s i t u a t i o n s w h i c h interest us. B o t h depend o n the observer for t h e i r " a c t u a l " , as opposed t o " p o t e n t i a l " , existence. A l s o , as i n the p r e v i o u s cases, one c o u l d go o n t o a n a l y z e the g i v e n s y s t e m i n order t o discover how the p a t t e r n s ( a n d their apparent m o t i o n s ) relate to the basic s t r u c t u r e .

Fig.

36.1

F r o m J . L . M a r r o q u i n M a s t e r ' s Thesis M I T Electrical Engineering l i Computer

1976 Science

A s we have a l r e a d y m e n t i o n e d , a n i d e n t i f i c a t i o n o f " S t r u c t u r a l i s m " w i t h " T h e A r t of I n t e l l i g i b i l i t y " was m a d e b y Peter C a w s i n his b o o k o n S t r u c t u r a l i s m [C2]. A t the s a m e t i m e , he makes the very curious p o i n t of e x c l u d i n g the n a t u r a l sciences f r o m "the area of s t r u c t u r a l i s t concern, because

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their structures are obliged to c o n f o r m t o those of n a t u r e ; they therefore become as c o n c e p t u a l l y a n d m a t h e m a t i c a l l y c o m p l e x as t h a t o b l i g a t i o n d i c tates a n d t h r o w n o l i g h t o n the s t r u c t u r i n g tendencies o f h u m a n t h o u g h t " [pp. 146, 147]. Because so m u c h of the s t r u c t u r a l i s t l i t e r a t u r e is associated w i t h fields outside the n a t u r a l sciences, the exclusion does not d e t r a c t greatly f r o m C a w s ' i n f o r m a t i v e discussion of the s u b j e c t . O n the other h a n d , the reason given for the e x c l u s i o n reveals the same confusion concerning the practice of science t h a t we discussed earlier i n c o n n e c t i o n w i t h the " t w o c u l t u r e " p r o b l e m (Section 26), a n d leads C a w s to conclude t h a t scientific thought Is so d o m i n a t e d by nature t h a t i t lacks the s t r u c t u r i n g freedom f o u n d i n other areas. T h a t the conclusion is u n w a r r a n t e d , can be seen f r o m a n e x a m i n a t i o n of the c r e a t i v i t y i n v o l v e d i n any s u b s t a n t i a l piece of scientific work. T h e r e are t w o i m p o r t a n t issues raised b y the C a w s s t a t e m e n t : (1) the special character o f n a t u r a l science s t r u c t u r e s , a n d (2) the w a y i n w h i c h the " s t r u c t u r i n g tendencies of h u m a n t h o u g h t " are expressed. A b r i e f d i s cussion o f these issues f r o m an e x p l i c i t s t r u c t u r a l p o i n t o f v i e w w i l l remove some o f the confusion s u r r o u n d i n g the subject a n d t h r o w a d d i t i o n a l l i g h t o n general s t r u c t u r e theory itself. It is t r u e , o f course, t h a t n a t u r a l science s t r u c t u r e s are u l t i m a t e l y det e r m i n e d by n a t u r e . A t the same t i m e , the d e s c r i p t i o n and a n a l y s i s of the s t r u c t u r e s , w h i c h constitutes the t h e o r e t i c a l side of the science, is a h u m a n p r o d u c t , as is the design a n d e x e c u t i o n of c r u c i a l e x p e r i m e n t s suggested by the theory. It is also t r u e , at least i n p r i n c i p l e , t h a t the t h e o r e t i c a l structures are i n some sense derivable f r o m a relatively s m a l l set of b a sic p r i n c i p l e s , a n d t h a t the d e r i v a t i o n is often ( t h o u g h not e x c l u s i v e l y ) m a t h e m a t i c a l i n character. In the t e r m i n o l o g y of S e c t i o n 26 (on s t r u c t u r a l d e t e r m i n i s m ) , t h i s means t h a t the t h e o r e t i c a l s t r u c t u r e is " d e t e r m i n e d " by the r e l a t i v e l y s m a l l s u b s t r u c t u r e defined b y the basic p r i n c i p l e s . It is here t h a t a nonscientist m a y get the i d e a t h a t the p r o d u c t i o n of science is a more or less m e c h a n i c a l process. A n d t h a t scientists are l i k e robots w h o b e g i n w i t h the basic p r i n c i p l e s , p l u g i n t o f o r m u l a s , a n d t u r n m a t h e m a t i c a l c r a n k s u n t i l they o b t a i n scientific results, a p i c t u r e t h a t resembles the c a r i c a t u r e o f a m o n k e y p e c k i n g away at a t y p e w r i t e r e v e n t u a l l y t o reproduce one of Shakespeare's p l a y s . T h e p i c t u r e , w h i c h has l i t t l e to do w i t h r e a l i t y , ignores the i m p o r t a n t fact t h a t a scientific result of any significance o r i g i n a t e s , not f r o m m e c h a n i c a l c a l c u l a t i o n , b u t i n a conjecture, or h y p o t h e s i s , f o r m u l a t e d by a scientist after m u c h s t u d y and t h o u g h t . Once the conjecture is f o r m u l a t e d , there follows an a t t e m p t to verify the result by e x p e r i m e n t (or observation) a n d t o derive it f r o m basic principles or p r e v i o u s l y established results. T h e

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latter m a t e r i a l is often the o n l y part of the entire process t h a t is recorded i n the l i t e r a t u r e . W h a t is s e l d o m recorded is the c o m p l e x , often i n t u i t i v e , process by w h i c h the conjecture is produced. T h i s can be the m o s t creative p a r t o f the w o r k , t h o u g h the final verification m a y also involve significant creative i n s i g h t s and is s e l d o m of a p u r e l y m e c h a n i c a l n a t u r e . It is i n these creative p o r t i o n s of the o v e r a l l process where " t h e s t r u c t u r i n g tendencies o f h u m a n t h o u g h t " a b o u n d . It is very d o u b t f u l t h a t a c c o m p l i s h m e n t s exist i n s o c i a l science t h a t c o m p a r e i n t h i s respect to the f o r m u l a t i o n , v e r i f i c a t i o n , a n d d e r i v a t i o n o f the m o s t significant results i n n a t u r a l science. T h e r e l a t i v e lack of precision i n social science structures a n d the fact t h a t they o r i g i n a t e i n o r g a n i z a t i o n s o f h u m a n beings give an i m p r e s s i o n o f greater h u m a n i n v o l v e m e n t t h a n exists i n n a t u r a l science. B u t the h u m a n o r g a n i z a t i o n s s t u d i e d by the social scientist are j u s t as o b j e c t i v e l y " r e a l " a n d are as independent o f the s o c i a l scientist as n a t u r a l p h e n o m e n a are of the n a t u r a l scientist. T h i s is t r u e despite the fact t h a t the s o c i a l scientist is a h u m a n b e i n g s t u d y i n g h u m a n o r g a n i z a t i o n s . In each case, the o b j e c t i v e is t o discover a n d describe the relevant s t r u c t u r e s . F u r t h e r m o r e , i t is o u r c l a i m t h a t the w a y i n w h i c h the s t r u c t u r e s are dealt w i t h is u l t i m a t e l y d e t e r m i n e d by general properties o f s t r u c t u r e s , so, except for d e t a i l s , is b a s i c a l l y the s a m e i n a l l areas of study. T h e r e may be more significant differences i n the " v e r i f i c a t i o n " process. In the s o c i a l sciences, the verification m a y t a k e the f o r m o f r e l a t i n g the new result to previous knowledge a n d checking for it i n other societies, for e x a m p l e , w h i l e i n the n a t u r a l sciences, verification m a y involve d e r i v i n g the new result f r o m e x i s t i n g theory or p e r f o r m i n g e x p e r i m e n t s to test i t . 37.

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Questions

I n this section we consider briefly some o f the p h i l o s o p h i c a l p r o b l e m s associated w i t h the general s t r u c t u r a l a p p r o a c h t o m e n t a l p h e n o m e n a . T h e s e are not new questions, as they have been discussed at l e n g t h by m a n y t h i n k e r s . W h a t e v e r novelty there m i g h t be i n o u r t r e a t m e n t of the s u b j e c t stems f r o m a consistent a n d e x p l i c i t s t r u c t u r a l p o i n t of v i e w . T h e m e n t a l p h e n o m e n a o f interest depend u l t i m a t e l y on the p e r c e p t i o n o f o b j e c t s i n the e x t e r n a l w o r l d . A n d perceptions m a y be t h o u g h t o f as m e n t a l s t r u c t u r e s t h a t are i s o m o r p h i c w i t h characteristic s t r u c t u r a l representations of the o b j e c t s . B u i l d i n g these m e n t a l structures involves not o n l y sense d a t a f r o m the objects b u t also relevant p r e v i o u s l y a c q u i r e d i n f o r m a t i o n a b o u t the e x t e r n a l w o r l d . V e r y l i t t l e is k n o w n , of course, c o n c e r n i n g the a c t u a l f o r m a t i o n of m e n t a l s t r u c t u r e s a n d how b r a i n p h y s i o l o g y is i n v o l v e d i n the process. However, some of the p r o b l e m s are d o c u m e n t e d i n a recent a r t i c l e by S e m i r Z e k i [Z2] o n " T h e V i s u a l Image i n M i n d and B r a i n " . T h e a r t i c l e appeared i n a special issue o f Scientific A m e r i c a n devoted to a

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variety o f " M i n d and B r a i n " questions. F o r our purposes, i t is possible t o ignore the m a n y t e c h n i c a l p r o b l e m s associated w i t h p e r c e p t i o n , a n d s i m p l y assume the existence of a process t h a t produces m e n t a l s t r u c t u r e s t h a t not o n l y represent e x t e r n a l o b j e c t s t r u c t u r e s b u t are also p r o p e r l y placed ( s t r u c t u r a l l y ) w i t h respect t o p r e v i o u s l y established knowledge s t r u c t u r e s . Despite the c o m p l e x i t y a n d lack of precise t e c h n i c a l d e t a i l s , a s t r u c t u r a l p o i n t of view based o n general p r o p erties of s t r u c t u r e s c a n nevertheless y i e l d useful i n s i g h t s i n t o the subject. S o m e of the questions t h a t interest us here concern the a c t u a l r e l a t i o n s h i p between e x t e r n a l objects a n d t h e i r m e n t a l representations. N o t i c e first t h a t the representations o f e x t e r n a l objects w i l l necessarily be more or less i n c o m p l e t e . In other words, they c a n n o t l i t e r a l l y i n c l u d e a l l o f the i n f o r m a t i o n c o n t a i n e d i n the objects. F o r e x a m p l e , consider the p e r c e p t i o n o f an o r d i n a r y c h a i r . T h e c h a i r is a concrete o b j e c t t h a t m a y be regarded as a s t r u c t u r e at v a r i o u s levels. T h e n a t u r a l l e v e l , a n d the one n o r m a l l y recorded i n p e r c e p t i o n , is as a s t r u c t u r e m a d e up o f pieces of w o o d ( m e t a l , or plastic) of various shapes (i.e., s u b s t r u c t u r e s ) . It m a y be a n a l y z e d f u r t h e r , however, as a s t r u c t u r e m a d e up of w o o d fibers, for e x a m p l e , or molecules, or a t o m s , or elementary p a r t i c l e s . T o i n c o r p o r a t e the b u l k of such i n f o r m a t i o n i n the perception o f a chair is o b v i o u s l y i m p r a c t i c a l . T h i s o b s e r v a t i o n already raises questions a b o u t the u l t i m a t e m e a n i n g a n d r e l i a b i l i t y o f our knowledge of concrete o b j e c t s . I n fact, we seem to need an a s s u m p t i o n to the effect t h a t an o b j e c t ' s very existence is somehow e m b o d i e d i n , or d e t e r m i n e d by, its various s t r u c t u r a l characteristics. A n e x t r e m e version w o u l d be t h a t a concrete o b j e c t a c t u a l l y consists of a confluence o f abstract s t r u c t u r e s . W e consider next the case o f " t h o u g h t e x p e r i m e n t s " i n physics. A famous e x a m p l e is E i n s t e i n ' s t r a i n e x p e r i m e n t w h i c h he used to conclude t h a t n e i ther absolute t i m e nor absolute space exists. A t h o u g h t e x p e r i m e n t u s u a l l y consists of an i m a g i n e d e x t r e m e s i t u a t i o n , more often t h a n not i m p o s s i b l e to realize, for w h i c h a certain o u t c o m e is more or less o b v i o u s . T h e q u e s t i o n of interest here concerns whether or not the i n f o r m a t i o n c o n t r i b u t e d b y the e x p e r i m e n t is a c t u a l l y new. S o m e of the m a n y t h o u g h t e x p e r i m e n t s (not a l l of t h e m successful!) t h a t have been proposed i n physics are discussed i n a n article b y R o y Sorensen t h a t a p p e a r e d recently i n A m e r i c a n S c i e n t i s t [S7]. M u c h of the a r t i c l e is devoted t o E r n s t M a c h ' s theory o f w h y t h o u g h t e x p e r i m e n t s m a y be expected to give v a l i d results. M a c h ' s ideas o n the s u b j e c t are i n t e r e s t i n g because he was a s t r i c t e m p i r i c i s t , w h i l e there a p p e a r to be s t r o n g r a t i o n a l i s t i m p l i c a t i o n s i n a t h o u g h t e x p e r i m e n t . H i s theory is also i n t e r e s t i n g because it depends on an a p p e a l t o e v o l u t i o n . T h e f o l l o w i n g o u t l i n e of M a c h ' s reasoning is f r o m the Sorensen a r t i c l e :

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M a c h argues t h a t t h o u g h t e x p e r i m e n t s w o r k because biology forces t h o u g h t t o c o n f o r m t o the e n v i r o n m e n t , e n s u r i n g a s i m i l a r i t y between our inner p r i v a t e w o r l d a n d the outer p u b l i c one. A m i n d t h a t wanders t o o far f r o m the t r u t h is destroyed b y the selective forces described by D a r w i n , a n d the m i n d s of alt a n i m a l s m u s t m i m i c the p a t t e r n s of n a t u r e . L e a r n - o r - d i e e v o l u t i o n a r y pressure o n consciousness endows us w i t h a treasure-store of a c c u m u l a t e d experiences t h a t are "ever close at h a n d a n d o f w h i c h o n l y the smallest p o r t i o n is e m b o d i e d i n clear a r t i c u l a t e t h o u g h t " . M a c h c a l l e d t h i s treasure-store " i n s t i n c t i v e k n o w l e d g e " , [p. 253] B u t M a c h insists t h a t whatever i n s i g h t we o b t a i n f r o m t h o u g h t e x p e r i m e n t s is o n l y o l d i n f o r m a t i o n reorganized [p.255]. T h e last statement represents a b l a n k e t rejection of any r a t i o n a l i s t c l a i m s t h a t one m i g h t read i n t o a successful t h o u g h t e x p e r i m e n t . It also a m o u n t s to a negative answer t o the question t h a t interests us, a l t h o u g h the w o r d , " r e o r g a n i z e d " , m i g h t be general enough to a d m i t c e r t a i n i n f o r m a t i o n t h a t c o u l d p r o p e r l y be c a l l e d " n e w " . A t the same t i m e , there are cases t o w h i c h an a n a l y s i s i d e n t i c a l to t h a t used for a t h o u g h t e x p e r i m e n t w o u l d seem to a p p l y , but w h i c h give rise to new i n f o r m a t i o n t h a t is not s i m p l y a r e o r g a n i z a t i o n of o l d i n f o r m a t i o n . C o n s i d e r , for e x a m p l e , a successful p r e d i c t i o n i n a science such as p h y s i c s , as discussed i n S e c t i o n 47 i n the next c h a p t e r . In this case, a f o r m a l , often m a t h e m a t i c a l , extension o f a k n o w n theory predicts genuinely new facts before t h e y are a c t u a l l y observed. T h e theory extension is made q u i t e i n d e p e n d e n t l y o f the concrete setting o f the theory a n d is analogous t o the e x t e n s i o n i n a t h o u g h t e x p e r i m e n t . E x c e p t i n t r i v i a l cases, no one c o u l d seriously c l a i m t h a t such predictions are merely r e o r g a n i z a t i o n s o f o l d i n formation. These are s p e c i a l cases of a general q u e s t i o n : " U n d e r w h a t c i r c u m s t a n c e s w i l l an extension o f a representation of a given concrete o b j e c t y i e l d new i n f o r m a t i o n a b o u t the l a t t e r ? " T h e answer t h a t we propose is t h a t " T h e given representation m u s t d e t e r m i n e the e x t e n s i o n " . T h e w o r d " d e t e r m i n e " here refers to 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 " discussed i n S e c t i o n 26. T h e idea is t h a t , u n d e r this c o n d i t i o n , the represented concrete s t r u c t u r e , b e i n g i s o m o r p h i c t o the representing s t r u c t u r e , w o u l d also d e t e r m i n e a c o n c r e t e e x t e n s i o n t h a t is i s o m o r p h i c w i t h the e x t e n s i o n o f the o r i g i nal representation. T h e r e f o r e , the a d d i t i o n a l i n f o r m a t i o n c o n t a i n e d i n the concrete extension w o u l d be new i n f o r m a t i o n " p r e d i c t e d " b y the extended representation. T h e a c t u a l existence o f the p r e d i c t e d new concrete i n f o r m a t i o n m u s t , of course, be verified e v e n t u a l l y b y o b s e r v a t i o n . A v e r i f i c a t i o n f a i l u r e is u s u a l l y t a k e n t o m e a n t h a t the o r i g i n a l representation was t o o s u p e r f i c i a l

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or was flawed i n s o m e other m a n n e r . T h e e v o l u t i o n a r y a r g u m e n t , when it applies, o n l y serves to suggest t h a t the g i v e n representing s t r u c t u r e is a g o o d one a n d w h y an e x t e n s i o n of it m i g h t also be expected t o represent the e x t e r n a l w o r l d , t h u s y i e l d i n g new i n f o r m a t i o n a b o u t the l a t t e r . T h e p h i l o s o p h y u n d e r l y i n g a r g u m e n t s o f the above t y p e a d m i t t e d l y constitutes a break w i t h M a c h ' s s t r i c t e m p i r i c i s m i n favor o f a r a t i o n a l i s t p o s i t i o n . In the case of o r d i n a r y experiences, i t is u s u a l l y t a k e n for g r a n t e d t h a t different i n d i v i d u a l s w i l l n o r m a l l y react to a given aspect of the e n v i r o n m e n t i n s i m i l a r ways. F o r e x a m p l e , i n Section 35 i t was suggested t h a t the a b i l i t y t o construct a n d process ( m e n t a l ) structures does not differ g r e a t l y f r o m one person t o another. I n p a r t i c u l a r , i t is a s s u m e d t h a t perceptions o f a g i v e n concrete s i t u a t i o n w i l l n o r m a l l y possess b a s i c a l l y the s a m e c h a r a c t e r for everyone. A l t h o u g h we t e n d a u t o m a t i c a l l y t o m a k e such a s s u m p t i o n s , i t is reasonable t o ask how they m i g h t be j u s t i f i e d . T h e simplest j u s t i f i c a t i o n is again p r o v i d e d b y an a p p e a l t o the theory o f e v o l u t i o n , a l o n g the same lines as E r n s t M a c h ' s . N o t e t h a t a n e v o l u t i o n a r y a r g u m e n t w i l l be m o r e or less relevant i n any s i t u a t i o n c o n c e r n i n g the r e l a t i o n s h i p o f i n d i v i d u a l s either t o one another or t o the e n v i r o n m e n t . T h e basic i d e a is t h a t o r g a n i s m s , closely related i n the sense of e v o l u t i o n , w i l l e x h i b i t s i m i l a r s t r u c t u r a l a d j u s t m e n t s t o the e n v i r o n m e n t . T h e r e f o r e , different h u m a n s m a y be expected t o e x h i b i t considerable s i m i l a r i t y (if not i d e n t i t y ) i n t h e i r basic responses to e x t e r n a l s t i m u l i . In p a r t i c u l a r , m e n t a l representations o f the same e n v i r o n m e n t a l m a t e r i a l c a n be e x p e c t e d t o be more or less i s o m o r p h i c . Despite the basic s i m i l a r i t y of h u m a n s a n d the c o r o l l a r y t h a t their m i n d s w o r k i n essentially the same way, there r e m a i n difficult questions c o n c e r n i n g their i n t e r a c t i o n s w i t h one another. F o r e x a m p l e , we take it for g r a n t e d t h a t one can k n o w i n m a n y cases w h a t another person is t h i n k i n g , t h o u g h there appears to be no direct access to such i n f o r m a t i o n . It is therefore reasonable t o a s k , " I s it possible for one ever to k n o w the a c t u a l content o f a n o t h e r person's m i n d ? " T h e belief t h a t such knowledge i s possible is no d o u b t based o n k n o w l edge o f ones o w n experiences plus the a s s u m p t i o n t h a t others resemble us. O n the other h a n d , the o n l y d i r e c t evidence t h a t one c a n have c o n c e r n i n g a m e n t a l experience o f another, is the s u b j e c t ' s overt b e h a v i o r ( i n c l u d i n g , e.g., use of l a n g u a g e ) , p r e s u m a b l y i n response t o t h a t experience. A l t h o u g h such p r o b l e m s have received considerable a t t e n t i o n (see, e.g., [C3]), a gene r a l l y s a t i s f a c t o r y s o l u t i o n appears t o be as elusive as ever. F r o m o u r p o i n t o f v i e w , the p r o b l e m involves two s t r u c t u r e s , representing respectively the m e n t a l experience a n d the associated b e h a v i o r . I n t h i s s e t t i n g , the general p r o b l e m takes the f o r m o f another q u e s t i o n : " T o w h a t

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extent d o the t w o s t r u c t u r e s d e t e r m i n e one a n o t h e r ? " T h e d e t e r m i n i s m u n d e r s t o o d here is a g a i n the " s t r u c t u r a l d e t e r m i n i s m " discussed i n S e c t i o n 26. T h i s f o r m u l a t i o n suggests t h a t i f the second s t r u c t u r e determines the first then one c o u l d t h e o r e t i c a l l y discern the content of another person's m i n d . T h e confidence o f o r d i n a r y h u m a n s i n their a b i l i t y to k n o w the m i n d s o f others p r o b a b l y stems f r o m t h e i r o w n i n t u i t i v e experience w i t h two such structures w i t h i n themselves. It w o u l d be n a i v e t o expect a s i m p l e s t r u c t u r a l s o l u t i o n to a p r o b l e m t h a t has received so m u c h a t t e n t i o n f r o m philosophers over the years. I n f a c t , there are f o r m i d a b l e t e c h n i c a l difficulties t h a t m u s t be overcome i n order to c o n s t r u c t precise t h e o r e t i c a l descriptions of the relevant s t r u c t u r e s , a n d also to e s t a b l i s h the desired connections between those s t r u c t u r e s . Nevertheless, the s t r u c t u r a l f o r m u l a t i o n constitutes a general a p p r o a c h a n d , despite the t e c h n i c a l difficulties, promises to be far m o r e s y s t e m a t i c a n d manageable t h a n the t r a d i t i o n a l a t t e m p t s to deal w i t h the p r o b l e m . Regardless o f whether such a n a p p r o a c h w i l l y i e l d a satisfactory answer t o the m a i n q u e s t i o n , i t s h o u l d at least give some idea of w h a t k i n d o f an answer, i f any, one m i g h t reasonably expect to o b t a i n . It is w o r t h n o t i n g t h a t a s t a n d a r d a t t a c k on questions o f this k i n d c o n centrates o n language b e h a v i o r . T h e m e t h o d is to a n a l y z e the language expressions c o m m o n l y used to describe p a r t i c u l a r instances of the p h e n o m e n a i n q u e s t i o n . A t first sight, an a t t e m p t to get at the essence o f such an experience by e x p l o r i n g the ways we t a l k a b o u t it w o u l d a p p e a r t o be f u t i l e . T h e a t t e m p t appears m o r e reasonable, however, i f we recall f r o m the p r e c e d i n g chapter t h a t the role o f language is to p r o v i d e a c o m m u n i c a b l e s t r u c t u r e representation o f the subject of interest, a n d also recognize t h a t the g o a l here is to i d e n t i f y t h a t s t r u c t u r e . W e end t h i s section w i t h a b r i e f observation o n consciousness f r o m the p o i n t of view of s t r u c t u r e s . T h o u g h there is clearly no h o p e of o b t a i n i n g a very precise d e s c r i p t i o n of s o m e t h i n g as s u b j e c t i v e as consciousness, i t is nevertheless possible t o i m a g i n e a s t r u c t u r a l setup t h a t m i g h t a c c o m m o date the p h e n o m e n o n . Suppose, for e x a m p l e , t h a t the general s t r u c t u r i n g process were t o i n c l u d e a n a u t o m a t i c self-checking feature i n v o l v i n g a p a r allel b a c k u p copy of s t r u c t u r e s c u r r e n t l y b e i n g processed. T h e p r a c t i c a l i m p o r t a n c e o f such a f a c i l i t y is o b v i o u s , so i t is not unreasonable t o expect t h a t it m i g h t be produced i n the n o r m a l course o f e v o l u t i o n a r y developm e n t . In any case, g i v e n the f a c i l i t y , it is p l a u s i b l e t o conjecture t h a t the experience o f consciousness is p r o d u c e d b y an i n t e r a c t i o n between the m a i n flow o f m e n t a l s t r u c t u r e s a n d the p a r a l l e l b a c k u p copy i n the self-checking process. T h i s could u n d e r l i e the sense of awareness o f those s t r u c t u r e s a n d the t h i n g s they represent. If an e x p l a n a t i o n a l o n g these lines is v a l i d , t h e n one m i g h t also expect c e r t a i n a n i m a l s other t h a n h u m a n s t o experience at

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least a r u d i m e n t a r y f o r m of consciousness. 38.

T h e Background Structure and Understanding

B e g i n n i n g w i t h the " c o n g e n i t a l " s t r u c t u r e , the t o t a l m e n t a l s t r u c t u r e a c q u i r e d or developed by a n i n d i v i d u a l u p t o the present, is c a l l e d the b a c k g r o u n d s t r u c t u r e . It is c o n t i n u a l l y b e i n g extended by a d d i t i o n of m a t e r i a l f r o m the e x t e r n a l w o r l d , first i n the f o r m of s t r u c t u r e s d e r i v e d s t r i c t l y f r o m sense d a t a , a n d l a t e r also i n the f o r m o f ideas f r o m others t h r o u g h language c o m m u n i c a t i o n . It also grows t h r o u g h i n t e r n a l processes such as those i n v o l v e d i n t h i n k i n g . Because i t contains a complete u p - t o - d a t e record of ones knowledge a n d experience, the b a c k g r o u n d s t r u c t u r e is not o n l y e x t r e m e l y c o m p l e x b u t is also i n a constant state of f l u x , a n d w i l l v a r y g r e a t l y f r o m one i n d i v i d u a l to another. A l l o f t h i s is based o n our general a s s u m p t i o n t h a t the b r a i n , i n response to e x t e r n a l s t i m u l i , is able t o f o r m m e n t a l s t r u c t u r e s (concepts) t h a t represent, at least a p p r o x i m a t e l y , structures inherent i n the presented d a t a . S o m e of the new s t r u c t u r e s m a y be identified d i r e c t l y w i t h e x i s t i n g s t r u c tures w i t h i n the b a c k g r o u n d ( r e c o g n i t i o n ) , w h i l e others may require an a p p r o p r i a t e extension of the b a c k g r o u n d before they can be a c c o m m o d a t e d . T h e a c t u a l end result w i l l n a t u r a l l y d e p e n d , not o n l y o n the g i v e n d a t a , b u t also on the extent a n d degree o f development o f the i n d i v i d u a l ' s backg r o u n d s t r u c t u r e . N o t e t h a t , a l t h o u g h a p o r t i o n of the b a c k g r o u n d m a y at t i m e s be an o b j e c t o f a t t e n t i o n , one is generally more or less u n a w a r e of b o t h its existence and f u n c t i o n i n g . T h e b o d y o f i n f o r m a t i o n , w h i c h is c o m m o n to the members o f a g i v e n i n t e r a c t i v e g r o u p , constitutes a k i n d o f " m u t u a l " b a c k g r o u n d s t r u c t u r e a n d provides the basis for c o m m u n i c a t i o n w i t h i n the g r o u p . A t the other ext r e m e , we have a " g l o b a l " s t r u c t u r e consisting o f the collective k n o w l e d g e a n d experience o f the entire g r o u p . A l t h o u g h the average m e m b e r o f the group m a y possess 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 collective s t r u c t u r e , a n " e x p e r t " w i l l have i n c o r p o r a t e d a c o m p a r a t i v e l y large p o r t i o n of i t i n t o his background. W e t u r n n o w t o the question o f w h a t i t m e a n s to " u n d e r s t a n d " somet h i n g , say a concept or a b o d y o f i n f o r m a t i o n . A s is c u s t o m a r y , we use the w o r d "concept" t o refer t o either a m e n t a l s t r u c t u r e or the abstract s t r u c t u r e t h a t i t represents. U n d e r s t a n d i n g is the end result of a subjective process, a n d therefore can o n l y be s t u d i e d either d i r e c t l y t h r o u g h i n t r o s p e c t i o n , or i n d i r e c t l y t h r o u g h observations of b e h a v i o r supposedly i n d i c a t i v e o f u n d e r s t a n d i n g . T h e p r o cess is not o n l y very c o m p l e x , but is also very fast and proceeds more or less a u t o m a t i c a l l y . T h e r e f o r e , one m u s t resort a g a i n to an over s i m p l i f i e d a n a l y sis t h a t c a n o n l y suggest w h a t m i g h t a c t u a l l y occur. A s u s u a l , a s t r u c t u r a l

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a p p r o a c h makes the task m u c h easier. It is rather obvious t h a t " u n d e r s t a n d i n g " m u s t depend first of a l l o n the c o n s t r u c t i o n o f a m e n t a l s t r u c t u r e t h a t represents the m a t e r i a l o f interest a n d is p r o p e r l y e m b e d d e d i n the person's b a c k g r o u n d . T h e result w i l l also d e p e n d o n b o t h i n t e r n a l a n d e x t e r n a l properties o f the representing s t r u c t u r e (Section 7). G i v e n these a s s u m p t i o n s , it is n a t u r a l t o m a k e the following "definition": T o u n d e r s t a n d s o m e t h i n g is to be a w a r e of t w o t h i n g s : (1) the i n t e r n a l properties o f the representing s t r u c t u r e a n d (2) the various e x t e r n a l properties o f the representation, w h i c h d e p e n d o n its i n c l u sion i n the b a c k g r o u n d s t r u c t u r e . T h e second i t e m refers to relations of the given piece o f i n f o r m a t i o n to e x i s t i n g i n f o r m a t i o n already represented i n the b a c k g r o u n d s t r u c t u r e . T h e representation of i n f o r m a t i o n w i t h i n ones b a c k g r o u n d a n d the i d e n t i f i c a t i o n of its various properties u s u a l l y cannot be separated i n a c t u a l p r a c t i c e , it is also necessary t o a l l o w for "degrees" o f u n d e r s t a n d i n g , d e p e n d i n g o n the a m o u n t of relevant knowledge one m i g h t possess a n d the q u a l i t y o f the r e p r e s e n t a t i o n . These factors w i l l o b v i o u s l y v a r y greatly f r o m one person to a n o t h e r . W h e t h e r or not t w o i n d i v i d u a l s have the " s a m e " u n d e r s t a n d i n g of s o m e t h i n g c a n o n l y be conjectured i n d i r e c t l y f r o m t h e i r actions w i t h respect t o the m a t t e r i n q u e s t i o n . I n fact, one person's " u n d e r s t a n d i n g " m a y be a " m i s u n d e r s t a n d i n g " as far as others are concerned. O n the other h a n d , since people often do agree t h a t they have a c o m m o n u n d e r s t a n d i n g , the m a t t e r o f u n d e r s t a n d i n g o b v i o u s l y cannot be as v a r i a b l e as m i g h t be i n d i c a t e d b y these r e m a r k s . Indeed, t h a n k s a g a i n to s t r u c t u r a l s t a b i l i t y , i t is q u i t e possible t h a t t w o i n d i v i d u a l s m a y conclude w i t h some confidence t h a t t h e y a c t u a l l y do have a c o m m o n u n d e r s t a n d i n g . S i n c e u n d e r s t a n d i n g is c l e a r l y a relative m a t t e r , it is n a t u r a l to ask w h a t it means for one's u n d e r s t a n d i n g of s o m e t h i n g to be "correct" or " c o m p l e t e " . E v i d e n t l y , the best t h a t can be done here is t o measure a n i n d i v i d u a l ' s u n d e r s t a n d i n g w i t h respect to the collective b a c k g r o u n d . T h u s , a definition o f "correctness" m i g h t require a n awareness of the properties o f a s t r u c t u r e , p r o p e r l y represented w i t h i n the c o l l e c t i v e b a c k g r o u n d s t r u c t u r e . A correct u n d e r s t a n d i n g w o u l d therefore d e p e n d i n part o n whether or not one's b a c k g r o u n d s t r u c t u r e incorporates enough o f the relevant p o r t i o n of the collective b a c k g r o u n d , i n other words, o n how " e x p e r t " one h a p p e n s t o be o n the s u b j e c t i n question. A l s o i n v o l v e d , of course, is the adequacy of the r e p r e s e n t a t i o n w i t h i n the b a c k g r o u n d . F o r t h i s reason, e x p e r t n e s s m a y be n e c e s s a r y b u t i s o b v i o u s l y n o t s u f f i c i e n t f o r a c o r r e c t u n d e r s t a n d i n g . A l t h o u g h correctness of u n d e r s t a n d i n g is a r e l a t i v e m a t t e r t h a t c a n o n l y

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be d e t e r m i n e d i n d i r e c t l y , s t r u c t u r a l s t a b i l i t y , as i n the case of m u t u a l u n d e r s t a n d i n g , suggests the p o s s i b i l i t y of m a k i n g f a i r l y precise j u d g e m e n t s as to w h e t h e r or not a g i v e n person's u n d e r s t a n d i n g o f s o m e t h i n g is correct. N o t e t h a t , correct or n o t , a person w i l l generally s t r u c t u r e presented m a t e r i a l i n one way or another. S u c h representations, t h o u g h perhaps not s t r i c t l y correct, m a y s t i l l serve the useful purpose of o r g a n i z i n g one's b e h a v i o r w i t h respect to the given s i t u a t i o n . P h e n o m e n a o f t h i s k i n d are analogous to the g r o w t h o f a c r y s t a l . L a r g e c r y s t a l s are s e l d o m perfect since they u s u a l l y c o n t a i n displacements or other defects caused, for e x a m p l e , b y i m p u r i t i e s . T h e y nevertheless grow i n t o r e l a t i v e l y stable s t r u c t u r e s w h i c h i n c o r p o r a t e the defects ( t h o u g h perhaps at some cost o r other) a n d m a y f u n c t i o n i n m a n y contexts as w o u l d perfect c r y s t a l s . T h e defects w i l l show up o n l y w h e n more precise d e m a n d s are placed o n the s t r u c t u r e . T h e s a m e is t r u e of defects i n u n d e r s t a n d i n g . If they are not r e a d i l y altered w h e n i n conflict w i t h r e a l i t y , t h e y w i l l r e m a i n i n d e f i n i t e l y as a block t o f u r t h e r understanding. T h e above r e m a r k s suggest t h a t u n d e r s t a n d i n g is a m o r e or less s t r a i g h t f o r w a r d process, w h i c h takes place completely a u t o m a t i c a l l y . A l t h o u g h t h i s is no d o u b t correct i n m a n y everyday s i t u a t i o n s , i t o b v i o u s l y c a n n o t be true i n a l l cases. A given t o p i c m a y be entirely new t o a p e r s o n , w h o cannot p o s s i b l y u n d e r s t a n d it w i t h o u t first a c q u i r i n g m o r e i n f o r m a t i o n o n the s u b j e c t . C o n s i d e r a b l e effort m a y therefore be r e q u i r e d , not o n l y t o c o n s t r u c t a b a c k g r o u n d representation of a new s t r u c t u r e , but also t o i d e n t i f y the properties, b o t h i n t e r n a l a n d e x t e r n a l , t h a t are needed for an u n d e r s t a n d ing. T h e p o i n t t h a t we w i s h t o m a k e here is t h a t the c o n s t r u c t i o n o f a repr e s e n t a t i o n , a l o n g w i t h the r e c o g n i t i o n o f its properties, is a process of discovery not essentially different f r o m any other discovery process. T h e r e fore, regardless of how well a subject is u n d e r s t o o d by others, an i n d i v i d u a l w h o meets i t for the first t i m e m u s t rediscover i t for himself. S i n c e the p r e s e n t a t i o n of a well u n d e r s t o o d subject w i l l n o r m a l l y i n c l u d e m a n y clues t h a t t e n d t o help one a v o i d difficulties, a rediscovery m a y not i n v o l v e as m a n y u n c e r t a i n t i e s a n d false starts as an o r i g i n a l discovery. Nevertheless, the rediscovery m u s t s t i l l i n v o l v e m a n y o f the features c h a r a c t e r i s t i c of a discovery. I n p a r t i c u l a r i t w i l l depend o n the same k i n d o f creative i n s i g h t s experienced i n an o r i g i n a l discovery. T h e beauty i n t h i s is t h a t the process o f u n d e r s t a n d i n g (if it is not obs t r u c t e d by b a c k g r o u n d defects or a defective presentation) c a n p r o v i d e a creative experience for anyone w h o pays a t t e n t i o n , regardless o f how p r o f o u n d or c o m m o n p l a c e the subject happens to be. It offers a n u n e n d i n g source o f pleasure for a l l w h o wish t o p a r t a k e . T h i s fact is o f great i m p o r tance i n e d u c a t i o n , a n d s h o u l d d o m i n a t e the presentation of every s u b j e c t

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o f s t u d y . T h e s p e c i a l role of s t r u c t u r e i n the e d u c a t i o n a l process is discussed i n the n e x t section. 39.

Teaching and Learning

T h e process o f u n d e r s t a n d i n g m u s t begin w i t h a p r e s e n t a t i o n o f a s u b j e c t , a b o d y of i n f o r m a t i o n , to the person w h o is supposed t o acquire a n u n d e r s t a n d i n g of i t . A l t h o u g h the i n f o r m a t i o n m a y consist o f raw sense d a t a , a more c o m m o n s i t u a t i o n is t h a t i t w i l l be c o m m u n i c a t e d either verb a l l y or by means of w r i t t e n m a t e r i a l . T h e most f o r m a l s e t t i n g for the l a t t e r is i n the c l a s s r o o m , where teachers a t t e m p t t o convey t h e i r o w n u n d e r s t a n d i n g o f a subject to the students, u s u a l l y w i t h the a i d o f a t e x t b o o k . W h a t a c t u a l l y takes place i n a c l a s s r o o m is a c o m p l e x a n d l i t t l e u n d e r s t o o d process, consisting of far more t h a n f o r m a l exchanges between teacher a n d students. A l s o i n v o l v e d are such things as the m a n y a n d varied i n t e r a c tions a m o n g the students themselves, events d u r i n g previous class meetings, an u p c o m i n g test, the weather outside, a n d so o n a n d o n . A l o n g w i t h v e r b a l exchanges a m o n g the p a r t i c i p a n t s , m u c h i n f o r m a t i o n is also c o m m u n i c a t e d b y nuances i n voice and b o d y language. F o r e x a m p l e , a sensitive p e r son (student or teacher) can often t e l l whether another t r u l y u n d e r s t a n d s s o m e t h i n g b y the way the l a t t e r e x p l a i n s i t , regardless o f whether or not the e x p l a n a t i o n i t s e l f is " c o r r e c t " . A n y a t t e m p t t o make a d e t a i l e d a n a l y sis of a given c l a s s r o o m t e a c h i n g - l e a r n i n g experience, reveals very q u i c k l y j u s t h o w s u b t l e the whole process can be. T h i s fact b e c a m e very clear to s o m e o f us d u r i n g the 60's w h e n we a t t e m p t e d to record the m a t e r i a l for a c a l c u l u s course o n film. M o r e recently, the advent of the c o m p u t e r has p r o m i s e d new approaches to the p r o b l e m o f i m p r o v i n g t e a c h i n g and l e a r n i n g i n the schools. T h e r e are, o f course, excellent possibilities for d o i n g some t h i n g s better w i t h the c o m p u t e r . These i n c l u d e the type of l e a r n i n g based o n r o u t i n e d r i l l , such as t h a t i n v o l v e d i n m a s t e r i n g a language or the e l e m e n t a r y techniques of m a t h e m a t i c s . T h e r e are also the recent developments i n a r t i f i c i a l i n t e l l i gence, w h i c h offer the prospects for c o m p u t e r a i d i n even m o r e s o p h i s t i c a t e d kinds of learning. D e s p i t e the obvious p o t e n t i a l of c o m p u t e r s i n e d u c a t i o n , at least i n very special a n d c o m p l e t e l y u n d e r s t o o d s i t u a t i o n s , i t is difficult to believe t h a t they c a n ever d u p l i c a t e the more s u b t l e aspects of a c t u a l i i v e i n t e r a c t i o n s of h u m a n m i n d s i n a c l a s s r o o m . Regardless o f advances i n s o p h i s t i c a t i o n a n d " i n t e l l i g e n c e " , it is d o u b t f u l t h a t a c o m p u t e r w i l l ever be able t o " r e a d " the m i n d s of students w h o cannot express i n words w h a t it is t h a t t r o u bles t h e m , s o m e t h i n g t h a t g o o d teachers do r o u t i n e l y b y o b s e r v i n g f a c i a l expressions as w e l l as l i s t e n i n g carefully t o confused r e m a r k s a n d " w r o n g " answers t o posed questions. Such things involve a large c o m p o n e n t of i n -

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t u i t i o n a n d are possible because of the teacher's k n o w l e d g e , derived f r o m previous contact, of h o w the s t u d e n t ' s m i n d " w o r k s " . N o r c a n the c o m p u t e r m i m i c a n experienced teacher's a b i l i t y to adjust a discussion t o the d e v e l o p i n g s i t u a t i o n , b y t a k i n g i n t o account the constant a n d often u n v e r b a l i z e d reactions of students t o the whole process. T h e r e is also n o s u b s t i t u t e for the i n s p i r a t i o n t h a t m a n y students w i l l derive f r o m the messages of " p l e a sure i n u n d e r s t a n d i n g " unconsciously p r o j e c t e d b y teachers w h o love a n d appreciate t h e i r subject. These are some of the m o r e s u b t l e , a n d perhaps i n the l o n g r u n the m o s t i m p o r t a n t , aspects of the t e a c h i n g - l e a r n i n g process. If such factors are neglected i n a n e d u c a t i o n a l p r o g r a m , the p r o d u c t m a y be i n d i v i d u a l s w i t h o u t a s e n s i t i v i t y a n d love for l e a r n i n g , w h o m a y possess c e r t a i n skills b u t lack the deeper u n d e r s t a n d i n g a n d m o t i v a t i o n necessary for d e a l i n g i n t e l l i g e n t l y w i t h new p r o b l e m s as t h e y arise. O n the basis of the preceding discussion, it is possible to o u t l i n e some o f the m o r e t e c h n i c a l s t r u c t u r a l requirements for the t e a c h i n g - l e a r n i n g process t o result i n the desired g o a l of knowledge a n d u n d e r s t a n d i n g . T h e c o n c l u sions, t h o u g h not l i m i t e d s t r i c t l y t o a m a t h e m a t i c a l s e t t i n g , are s t r o n g l y influenced b y the a u t h o r ' s m a n y years of experience i n t e a c h i n g m a t h e m a t ics. A l t h o u g h i t is easier to o u t l i n e the role of s t r u c t u r e i n the t e a c h i n g o f m a t h e m a t i c s t h a n i n m o s t other fields, the results s h o u l d , o n the whole, a p p l y to the t e a c h i n g of any subject at any l e v e l . T h e r e is one very f u n d a m e n t a l a n d c r u c i a l p o i n t w h i c h is often i g n o r e d or overlooked i n discussions of l e a r n i n g . It concerns the f a c t , a l r e a d y m e n t i o n e d several t i m e s before, t h a t the h u m a n m i n d , w h e n confronted w i t h m a t e r i a l of a l m o s t a n y k i n d , w i l l t e n d t o f o r m m e n t a l structures c o n s t i t u t i n g perceptions of t h a t m a t e r i a l . T h i s process, w h i c h is largely spontaneous a n d no d o u b t a p r o d u c t of e v o l u t i o n a r y a d a p t a t i o n , is d r i v e n b y a need to e x t r a c t m e a n i n g f r o m the barrage of i n f o r m a t i o n t o w h i c h we are c o n s t a n t l y b e i n g subjected, i t is n o t a b l y s t r o n g i n y o u n g c h i l d r e n , b u t m a y be d i s t o r t e d i n older c h i l d r e n b y considerations o f " w h a t the teacher or parent e x p e c t s " . A n i l l u m i n a t i n g p e r s o n a l experience, i n v o l v i n g one of the a u t h o r ' s c h i l d r e n at a b o u t the age of five, w i l l serve to i l l u s t r a t e t h i s i m p o r t a n t p o i n t . O n a f a m i l y o u t i n g one nice s p r i n g day, we c a m e u p o n a grove o f trees w h i c h were b e a u t i f u l l y green, w i t h the e x c e p t i o n of one large dead tree. O u r eldest s o n , w h o was i n t h a t w o n d e r f u l stage w h e n his interest i n t h i n g s a n d his s u p p l y of questions a b o u t t h e m seemed t o be i n e x h a u s t i b l e , n o t i c e d the grove of trees a n d asked, " D a d , w h y is t h a t tree dead?" M y rather evasive reply was to the effect t h a t there are a great m a n y t h i n g s t h a t m i g h t have k i l l e d i t , a n d we h a d no way of k n o w i n g w h a t a c t u a l l y h a p p e n e d . H i s i m m e d i a t e response was, " B u t D a d , w h y m a y b e is the tree dead?" M e m o r y fails t o s u p p l y the rest o f the conversation, b u t the desired p o i n t is adequately m a d e w i t h o u t i t .

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I f anyone bothers t o pay a t t e n t i o n t o the way they deal w i t h the constant f l o o d of i n f o r m a t i o n t h a t comes t h e i r way, they w i l l q u i c k l y realize t h a t most o f their " e x p l a n a t i o n s " o f t h a t i n f o r m a t i o n w i l l f a l l i n t o the " m a y b e " category. S u c h e x p l a n a t i o n s are i n e v i t a b l e , especially w h e n i t comes to m o r e t e c h n i c a l m a t e r i a l . W h e t h e r the e x p l a n a t i o n s are g o o d or b a d w i l l depend t o a large extent o n the q u a l i t y of the b a c k g r o u n d k n o w l e d g e o n w h i c h they are based. It is therefore i m p o r t a n t t h a t such knowledge, t h o u g h perhaps meager, be as accurate a n d to the p o i n t as possible. T o p r o v i d e b a c k g r o u n d knowledge over w i d e areas of l e a r n i n g is the goal of a l i b e r a l a r t s e d u c a t i o n , the u n d e r l y i n g p h i l o s o p h y b e i n g t h a t the v a r i o u s fields of knowledge are i n t e r r e l a t e d a n d t h a t students m u s t not o n l y l e a r n a b o u t each one b u t s h o u l d also u n d e r s t a n d some o f the connections between t h e m . I n other w o r d s , the idea is to p r o v i d e a n e d u c a t i o n t h a t w i l l enable i n d i v i d u a l s t o organize the i n f o r m a t i o n t o w h i c h they have been exposed and t o relate the s a m e to new i n f o r m a t i o n as i t arises. Because o f the c o m m o n h u m a n experience, the goal is not so difficult t o a t t a i n w i t h i n the h u m a n i t i e s . O n the other h a n d , analogous connections w i t h i n the sciences a n d between the h u m a n i t i e s a n d sciences are more difficult t o e s t a b l i s h , because o f the everpresent t e c h n i c a l barriers i n science a n d the differences i n a p p r o a c h between the sciences a n d the h u m a n i t i e s . O n e way t o a t t a c k the p r o b l e m , as already suggested i n S e c t i o n 2 6 , m i g h t be t o p r o v i d e a basis for r e l a t i n g subjects by e x p o s i n g general s t r u c t u r a l s i m i l a r i t i e s between t h e m . T h i s w i l l not be easy t o do, because i t d e m a n d s careful a t t e n t i o n t o aspects o f the subject not o r d i n a r i l y e m p h a s i z e d w i t h i n i n d i v i d u a l d i s c i p l i n e s . N o t e t h a t we are a s k i n g for more t h a n is u s u a l l y i n v o l v e d i n a t r a d i t i o n a l s t r u c t u r a l i s t a p p r o a c h , where general n o t i o n s o f s t r u c t u r e are often o n l y i m p l i c i t i n the m e t h o d of a n a l y s i s . I n order to reveal s t r u c t u r a l s i m i l a r i t i e s between subjects, i t is necessary to focus o n s t r u c t u r e s themselves so as t o b r i n g out t h e i r u l t i m a t e independence of special subject m a t t e r . P e r h a p s a goal o f this k i n d requires, i n a d d i t i o n to i n t e r n a l a d j u s t m e n t s w i t h i n disciplines, a special i n t e r d i s c i p l i n a r y p r o g r a m to d o c u m e n t s o m e o f the s t r u c t u r a l s i m i l a r i t i e s between t h e m . I n d e p e n d e n t l y of the goal o f r e l a t i n g fields o f s t u d y , an e m p h a s i s o n s t r u c t u r e s h o u l d i m p r o v e the teaching of m a n y subjects. C e r t a i n l y , i n the case of science a n d m a t h e m a t i c s , greater a t t e n t i o n t o basic structures w o u l d have a l a s t i n g effect, s i m p l y because unused techniques a n d bare facts are soon f o r g o t t e n , w h i l e basic ideas (structures, p r i n c i p l e s ) t e n d to be r e t a i n e d . F u r t h e r m o r e , some u n d e r s t a n d i n g of the m a i n s t r u c t u r e s i n science is exa c t l y w h a t is needed i n order to deal i n t e l l i g e n t l y w i t h the mass of t e c h n i c a l i n f o r m a t i o n t h a t permeates m o d e r n society. M a t h e m a t i c s , w h i c h is o b v i o u s l y w o r t h y of b e i n g s t u d i e d i n its o w n r i g h t as a field of k n o w l e d g e , is often proposed as the key to b r i d g i n g the gap

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between science a n d the h u m a n i t i e s . Reasons g i v e n are u s u a l l y t h a t i t is the " l a n g u a g e " of the p h y s i c a l sciences, or t h a t the s t u d y o f m a t h e m a t i c s helps students t o " t h i n k s y s t e m a t i c a l l y " . A l t h o u g h there m a y be a n e l ement of t r u t h i n these suggestions, the average l i b e r a l a r t s s t u d e n t w i l l s e l d o m master enough m a t h e m a t i c s t o deal adequately w i t h the t e c h n i c a l language o f the sciences, a n d the c l a i m for s y s t e m a t i c t h i n k i n g is difficult t o d o c u m e n t . O n the other h a n d , a s t u d y o f m a t h e m a t i c s w o u l d o b v i o u s l y s u p p o r t a s t r u c t u r a l a p p r o a c h . A l m o s t any decent m a t h e m a t i c s course (say, at the level of a n a l y t i c geometry a n d above) offers a u n i q u e o p p o r t u n i t y for students to l e a r n a b o u t general s t r u c t u r e s , especially i f the s t r u c t u r a l c o n tent is made e x p l i c i t . Needless t o say, it is necessary for students t o become s o m e w h a t f a m i l i a r w i t h the language (i.e., the techniques!) o f m a t h e m a t i c s i n order t o g a i n access t o the ideas. A l t h o u g h everyone tends a u t o m a t i c a l l y t o s t r u c t u r e whatever i n f o r m a t i o n comes their way, i t is a n u n f o r t u n a t e fact t h a t the result m a y be h i g h l y defective. F o r e x a m p l e , the b a c k g r o u n d s t r u c t u r e m a y be i n a d e q u a t e or defective, so t h a t i t c a n n o t i n c o r p o r a t e a proper representation of the presented m a t e r i a l . In this case the i n d i v i d u a l w i l l be i n c a p a b l e of p e r c e i v i n g the r e q u i r e d s t r u c t u r e , so w i l l necessarily f a l l back o n some s u p e r f i c i a l or perhaps irrelevant aspect of i t . T h e same t h i n g w i l l also h a p p e n , even w h e n the b a c k g r o u n d is adequate, i f the person is u n a b l e for s o m e other reason t o perceive the essential s t r u c t u r e i n the presented m a t e r i a l . T h u s , regardless of the desired result, false s t r u c t u r e s are often f o r m e d a n d w i l l y - n i l l y take their p e r m a n e n t place w i t h i n the i n d i v i d u a l ' s b a c k g r o u n d s t r u c t u r e . S o m e of these can be m o s t b i z a r r e , as the f o l l o w i n g e x a m p l e suggests. M a n y years ago, we owned an o l d t w o - f a m i l y house j o i n t l y w i t h some friends. It was e v e n t u a l l y sold t o a s a l e s m a n w h o m o v e d i n t o the p o r t i o n v a c a t e d b y our friends, w h i l e we s t a y e d on for a t i m e as tenants. T h e new owner was very naive (to say the least!) a b o u t t a k i n g care of a house, so I made a s p e c i a l effort t o p o i n t out t o h i m some o f the t h i n g s t h a t m i g h t be helpful for h i m t o k n o w . O n one o c c a s i o n , w h e n t r y i n g t o show h i m how the furnace w o r k e d , I opened the d o o r t o the firebox w h i l e the furnace was r u n n i n g . W h e r e u p o n , he j u m p e d back i n surprise w i t h the e x c l a m a t i o n , " T h e r e ' s fire i n there!" I never d i d f i n d out w h a t he expected t o see i n a n o i l fired furnace, b u t whatever it was h a d served h i m c o m f o r t a b l y for a g o o d fifty years. T h e false " k n o w l e d g e " c o n t a i n e d i n such incorrect representations is a c o m m o n p r o b l e m w i t h m a n y students, l e a d i n g to m u c h c o n f u s i o n , a n d s t a n d i n g as an o b s t r u c t i o n t o a c q u i r i n g further knowledge. T h e p r o b l e m is especially prevalent a m o n g m a t h e m a t i c s s t u d e n t s , w h o often are b u r dened w i t h several layers o f confusion, the result of a t t e m p t i n g t o b u i l d new concepts u p o n previous misconceptions. I n the case of m a t h e m a t i c s ,

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the first defect m a y occur e a r l y i n the s t u d y of a r i t h m e t i c . B u t for college students w h o t a k e m a t h e m a t i c s , the t r o u b l e seems t o o r i g i n a t e most often i n e l e m e n t a r y algebra. A m i n i m u m requirement for l e a r n i n g ( u n d e r s t a n d i n g ) to t a k e place is t h a t the s t u d e n t ' s b a c k g r o u n d knowledge be sufficiently developed to accept the new i n f o r m a t i o n . W i t h o u t this, l e a r n i n g w i l l be severely l i m i t e d , or even i m p o s s i b l e . T h e level o f d e v e l o p m e n t , w h i c h m a y v a r y g r e a t l y f r o m one s t u d e n t t o the next, also determines the degree or d e p t h o f u n d e r s t a n d i n g t h a t c a n take place. W h e n b a c k g r o u n d s are i n a d e q u a t e , the o b v i o u s s o l u t i o n is t o r e p a i r the deficiencies w i t h r e m e d i a l work o f some k i n d or other. O t h e r w i s e the s t u dent w i l l either w i t h d r a w completely or a u t o m a t i c a l l y develop a defective " u n d e r s t a n d i n g " of the new m a t e r i a l . U n f o r t u n a t e l y , at least i n the case of m a t h e m a t i c s , teachers often a i d a n d abet the second a l t e r n a t i v e i n order t o a v o i d the first. M a t e r i a l m a y be presented i n a f o r m irrelevant to its ess e n t i a l s t r u c t u r e i n order t o m a k e i t "easier" or more " i n t e r e s t i n g " , so t h a t m a n y students w i n d up w i t h a shared, but i n c o r r e c t , p i c t u r e o f the subject. " C o o k b o o k " m a t h e m a t i c s often falls i n t o t h i s category. E v e n s t u d e n t s who m i g h t otherwise be able to u n d e r s t a n d the m a t e r i a l can easily be m i s l e d by such ill-conceived pedagogical tactics. T o c o m p o u n d the e r r o r , an incorrect p i c t u r e m a y be certified as a t r u e u n d e r s t a n d i n g o f the m a t e r i a l by the use of " p h o n y " t e s t i n g , either t h r o u g h rigged tests or by d r i l l i n g s t u d e n t s i n advance o n the " c o r r e c t " answers to test questions. R e m e d i a l courses often consist o f o n l y a r e p e t i t i o n o f m a t e r i a l c o n t a i n e d i n an earlier course. T h e y a c c o r d i n g l y tend t o be b o r i n g a n d m a y even reinforce e x i s t i n g p r o b l e m s , so are s e l d o m very successful. F o r m a n y s t u dents, the result is no better t h a n it was the first t i m e a r o u n d a n d m a y even be worse. O n e c o m m o n difficulty is t h a t a s t u d e n t often does not have adequate b a c k g r o u n d even for the r e m e d i a l m a t e r i a l . F o r e x a m p l e , a p r o b l e m w i t h a l g e b r a m a y i n fact be a p r o b l e m w i t h a r i t h m e t i c . A n a d d i t i o n a l difficulty is t h a t , perhaps because of previous u n p l e a s a n t experiences, a s t u d e n t m a y have a psychological block w h i c h essentially precludes any f u r t h e r progress i n a subject. A l l of these p r o b l e m s are c o m m o n p l a c e i n m a t h e m a t i c s , b u t no d o u b t occur t o some degree i n other areas as w e l l , where they m a y be less easily identified. In n o case can any of t h e m be resolved unless the u n d e r l y i n g defects i n u n d e r s t a n d i n g are exposed a n d corrected. S t u d e n t s are s e l d o m able to do this for themselves, a n d it is often difficult for a teacher t o d o the j o b . Because each s t u d e n t ' s troubles tend to be s p e c i a l a n d t o lie well below the surface, u n c o v e r i n g t h e m requires i n d i v i d u a l a t t e n t i o n and extensive p r o b i n g . U n c o r r e c t e d deficiencies cause m i s c o n c e p t i o n s t h a t i n t u r n generate m o r e misconceptions, l e a d i n g e v e n t u a l l y t o a complete b l o c k i n g of

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the l e a r n i n g process. T h e i m p o r t a n c e of d e a l i n g w i t h the p r o b l e m as e a r l y as possible therefore cannot be overemphasized. A s s u m i n g t h a t students possess backgrounds adequate t o deal w i t h a g i v e n s u b j e c t , the p r o b l e m t h e n is t o present new m a t e r i a l , b o t h i n class a n d i n the t e x t b o o k , i n a f o r m t h a t w i l l enable students t o s t r u c t u r e the m a t e r i a l for themselves. Ideally, the new structures s h o u l d develop d i r e c t l y out o f the s t r u c t u r e s p r e v i o u s l y u n d e r s t o o d , a n d i n a m a n n e r consistent w i t h the n a t u r a l g r o w t h properties o f s t r u c t u r e s . T h i s is a delicate m a t t e r for b o t h teacher a n d a u t h o r , r e q u i r i n g a t h o r o u g h u n d e r s t a n d i n g o f the subject b y b o t h a n d a careful o r g a n i z a t i o n o f the t e x t b o o k m a t e r i a l b y the a u t h o r . If the teacher's view o f the subject or the t e x t b o o k ' s o r g a n i z a t i o n o f the m a t e r i a l is s t r u c t u r a l l y defective i n a n y way, t h e n the defects w i l l t e n d t o be passed o n t o the students. S i m i l a r l y , any pedagogical device or s h o r t c u t , w h i c h c o n t r a d i c t s or distorts the essential s t r u c t u r e o f the new m a t e r i a l , w i l l have the s a m e result. O n l y a gifted student a l r e a d y blessed w i t h a reliable i n t u i t i o n w i l l be able t o overcome these v i o l a t i o n s o f s u b j e c t integrity. T h e e m p h a s i s o n a s t r u c t u r a l presentation of a subject does not m e a n t h a t one s h o u l d always t r y t o teach the relevant s t r u c t u r e itself. T h i s c a n be a serious m i s t a k e , especially i n the case o f y o u n g students for w h o m the necessarily f o r m a l presentation m a y interfere w i t h the s p o n t a n e i t y o f u n d e r s t a n d i n g . V i o l a t i o n of t h i s p r i n c i p l e tended t o weaken some aspects o f a n otherwise s o u n d c u r r i c u l u m r e f o r m effort d u r i n g the 60's, m i s l e a d i n g l y labeled the " N e w M a t h e m a t i c s " . W h e n a subject is presented s t r i c t l y i n accordance w i t h its essential s t r u c t u r e , t h a t s t r u c t u r e w i l l be present, at least i m p l i c i t l y , i n the s t u d e n t ' s a c q u i r e d knowledge o f the m a t e r i a l a n d m a y be m a d e e x p l i c i t l a t e r i f desired. In the case o f m a t u r e students, w h o are more conscious of t h e i r m e n t a l processes, e x p o s i n g the s t r u c t u r e where convenient m a y f a c i l i t a t e and e n r i c h the process of u n d e r s t a n d i n g . A n o t h e r aspect o f a s t r u c t u r a l a p p r o a c h t o a subject is the level at w h i c h s t r u c t u r e s are perceived. T h i s p o i n t was i l l u s t r a t e d i n o u r discussion of c o n t r a c t i o n s (Section 27) b y the black b o x e x a m p l e c o n c e r n i n g the f u n c t i o n i n g of an a u t o m o b i l e . T h e level of u n d e r s t a n d i n g there r a n g e d f r o m t h a t needed j u s t t o d r i v e a n d m a i n t a i n the m a c h i n e , t h r o u g h the k n o w - h o w required b y a m e c h a n i c , to a f u l l a p p r e c i a t i o n of the u n d e r l y i n g p r i n c i p l e s of mechanics, physics, a n d c h e m i s t r y i n v o l v e d i n its o p e r a t i o n . A s i m i l a r a n a l y s i s m a y be a p p l i e d t o any sufficiently c o m p l e x s t r u c t u r e , and the level at w h i c h i t is perceived w i l l o b v i o u s l y depend o n the s o p h i s t i c a t i o n o f the person's b a c k g r o u n d s t r u c t u r e relative to the given s t r u c t u r e . T h e teachi n g of any s u b j e c t is a l m o s t certain t o be unsuccessful unless the s t r u c t u r a l level a t w h i c h the basic m a t e r i a l of the s u b j e c t is presented is m o r e or

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less c o m p a t i b l e w i t h most student b a c k g r o u n d s . F i n a l l y , the goal i n every course s h o u l d c l e a r l y be one o f knowledge a n d u n d e r s t a n d i n g . Nevertheless, there is a g r o w i n g tendency ( a m o n g b o t h students a n d teachers) t o equate h i g h test scores w i t h p r o o f t h a t the goal has been a t t a i n e d . T h i s a t t i t u d e applies especially t o s t a n d a r d i z e d tests, but is not l i m i t e d t o t h e m . T h e a b s u r d i t y o f the whole t h i n g is c o m p o u n d e d , for e x a m p l e , w h e n s t u d e n t s are o n l y required to p a r r o t back p r e v i o u s l y m e m o r i z e d m a t e r i a l , o r have been "coached" for the test. T h e result is t h a t m a n y o t h e r w i s e c a p a b l e college students have respectable grade records b u t are seriously deficient in understanding. T h e r e is a c t u a l l y n o t h i n g i n t r i n s i c a l l y w r o n g w i t h s t a n d a r d i z e d tests. W h a t is w r o n g , however, is the way the e d u c a t i o n a l c o m m u n i t y ( i n c l u d i n g s t u d e n t s , teachers, professors, a n d a d m i n i s t r a t o r s alike) has responded t o t h e m . It is a p p a l l i n g to realize t h a t performance o n such tests has c o m e t o be regarded, not as for knowledge a n d u n d e r s t a n d i n g , b u t as t h e i r

definition.

evidence

A l l tests s h o u l d c o n t a i n some questions t h a t challenge a s t u d e n t ' s u n d e r s t a n d i n g . I n m a t h e m a t i c s , for e x a m p l e , this means p r o b l e m s (necessarily very s i m p l e ! ) t h a t are new to the students b u t involve i m p o r t a n t concepts already covered i n the course. It is o n l y t h r o u g h challenges of t h i s k i n d t h a t students w i l l l e a r n w h a t it means, and also how it a c t u a l l y to understand something. A l t h o u g h such goals are c e r t a i n l y desirable, they are not always easy to i m p l e m e n t i n the current e d u c a t i o n a l e n v i r o n m e n t . T h e p r o b l e m is i l l u s t r a t e d by an experience I h a d i n teaching e l e m e n t a r y C a l c u l u s . T h e students c o m p l a i n e d a b o u t questions o f the type described a b o v e , so I t r i e d t o e x p l a i n t o t h e m m y reasons for i n c l u d i n g such questions i n their tests. A l l of t h e m r e a d i l y agreed w i t h the objective a n d appeared to l i k e the i d e a , w h e r e u p o n several o f the students suggested t h a t I give t h e m a s u p p l y of " s a m p l e " questions of the i n d i c a t e d t y p e covering the test m a t e r i a l , so t h a t they c o u l d better prepare themselves for the test! In other words, the i d e a of using an a c q u i r e d u n d e r s t a n d i n g of c e r t a i n concepts to deal w i t h a novel s i t u a t i o n i n v o l v i n g those concepts was completely missed. E x p e r i e n c e s of t h i s k i n d convince m e t h a t current m e t h o d s o f t e a c h i n g , d o m i n a t e d b y d r i l l i n g a n d c o a c h i n g students for s t a n d a r d i z e d tests, or tests t h a t i m i t a t e t h e m , is an i n s u l t t o the h u m a n m i n d , a n d is t u r n i n g out generations of m e n t a l robots, w h o neither k n o w nor care w h a t i t means to u n d e r s t a n d something.

feels,

CHAPTER

MATHEMATICAL

40.

VII

STRUCTURES

Introduction

In the preceding sections, we have discussed i n considerable d e t a i l several e x a m p l e s of m a t h e m a t i c a l s t r u c t u r e s , so the reader s h o u l d already have some i d e a of w h a t such structures are like. O u r purpose now is t o i n d i c a t e more c l e a r l y ( i n terms!) w h a t i t is t h a t makes t h e m s p e c i a l . U n f o r t u n a t e l y , the class of a l l m a t h e m a t i c a l structures is s t i l l t o o b r o a d t o a d m i t a precise d e f i n i t i o n . F u r t h e r m o r e , the n o t i o n is not fixed b u t tends to change g r a d u a l l y as m a t h e m a t i c s develops, so c e r t a i n s t r u c t u r e s t h a t are recognized as m a t h e m a t i c a l at the present t i m e perhaps w o u l d have been rejected at a n earlier p e r i o d . Nevertheless, i t is possible to give a k i n d o f " o p e r a t i o n a l " d e f i n i t i o n i n t e r m s of w h i c h anyone w i t h sufficient m a t h e m a t i c a l knowledge c o u l d decide whether a given s t r u c t u r e s h o u l d be classified as m a t h e m a t i c a l or n o t . M a t h e m a t i c a l s t r u c t u r e s are t o v a r y i n g degrees i n t e r r e l a t e d , a n d together c o n s t i t u t e one a l l - i n c l u s i v e s y s t e m , the The latter is not a fixed object, since it is i n a constant state of g r o w t h t h r o u g h the creation (or discovery?) of new m a t h e m a t i c s . It also has a lesser tendency to c o n t r a c t , as certain p o r t i o n s o f o l d m a t h e m a t i c s become obsolete or irrelevant a n d are " f o r g o t t e n " . W i t h i n the whole of m a t h e m a t i c s , we have w h a t is generally called the " m a i n of m a t h e m a t i c s . T h i s is the m a t h e m a t i c s t h a t has passed the test o f t i m e a n d continues to be i m p o r t a n t b o t h i n a p p l i c a t i o n s a n d as an i n s p i r a t i o n , as well as a base, for the c r e a t i o n of new m a t h e m a t i c s . T h e m a i n b o d y also changes, but very g r a d u a l l y as a r u l e , a n d m u c h slower t h a n the whole. M a t h e m a t i c s consists o f a n u m b e r of subfields (such as, for e x a m p l e , a l g e b r a , a n a l y s i s , a n d geometry), each w i t h its d i s t i n c t i v e characteristics. D e s p i t e t h e i r differences, the various subfields overlap c o n s i d e r a b l y a n d are i n a m o r e or less constant state of i n t e r a c t i o n . These i n t e r a c t i o n s , a l o n g w i t h o c c a s i o n a l contacts w i t h disciplines outside of m a t h e m a t i c s , c o n s t i t u t e an i m p o r t a n t d r i v i n g force for the development of the subject.

nontechnical

"body of mathematics".

body"

115

116

41.

STRUCTURALISM AND STRUCTURES

Mathematical

Language

For a l i n g u i s t , the t e r m " l a n g u a g e " u s u a l l y refers to one of the " n a t u r a l " languages, spoken ( a n d perhaps w r i t t e n ) by a more or less well-defined c u l t u r a l group of h u m a n s . C o m m o n usage of the t e r m , however, is m u c h more i n c l u s i v e , referring s i m p l y t o any s y s t e m for c o m m u n i c a t i n g or r e c o r d i n g i n f o r m a t i o n . T h e l a t t e r is o b v i o u s l y the way the t e r m is used i n the expression " m a t h e m a t i c a l l a n g u a g e " . A m a t h e m a t i c a l language is a l m o s t always w r i t t e n , a n d is s e l d o m " s p o k e n " except i n fragments a n d rather i n f o r m a l l y . A s we have already noted, a conspicuous feature of m a t h e m a t i c s is its h i g h l y f o r m a l language. A l t h o u g h this u s u a l l y refers o n l y t o the collection of s y m b o l s a n d equations, such as those i n o r d i n a r y a l g e b r a , a m a t h e m a t i cal language w i l l also involve whatever other m a t e r i a l is needed for precise c o m m u n i c a t i o n of the ideas. I n c l u d e d are graphs, figures, a n d tables, a l o n g w i t h c e r t a i n p o r t i o n s of o r d i n a r y language needed to express such things as theorems a n d their proofs. I n a d d i t i o n , c o m m o n words f r o m o r d i n a r y language are frequently assigned s p e c i a l m a t h e m a t i c a l m e a n i n g s different f r o m , but perhaps suggested by, t h e i r u s u a l m e a n i n g s or c o n n o t a t i o n s . D e s p i t e the appearance of c o m p l e x i t y a n d p o t e n t i a l confusion, m a t h e m a t i c a l languages are very precise a n d r e m a r k a b l y free o f a m b i g u i t y for the c o m m u n i c a t i o n of m a t h e m a t i c a l ideas. T h i s is possible largely because o f the nature o f the ideas themselves. T h e s y m b o l s i n the more f o r m a l p a r t of a m a t h e m a t i c a l language are u s u a l l y regarded as corresponding to words i n a n a t u r a l language. In c e r t a i n cases t h i s is correct, as for e x a m p l e w i t h the s y m b o l s for special n u m b e r s (such as the integers and p i ) and the s y m b o l s for the a r i t h m e t i c o p e r a t i o n s ( + , x , —,/,=}. I n m a n y other cases, however, a n isolated s y m b o l has no m e a n i n g whatsoever, a n d w i l l acquire m e a n i n g o n l y as a p a r t of a f o r m u l a . S y m b o l s o f t h i s k i n d are not like words, since they do not correspond to the signifier i n any l i n g u i s t i c sign a n d can take o n such a role o n l y as part of a m a t h e m a t i c a l expression. T h i s is analogous i n a n a t u r a l language t o a c o n t e x t u a l d e t e r m i n a t i o n of the m e a n i n g of a w o r d . T h e r e is a difference, however, because i n the l a t t e r case the context t y p i c a l l y fixes o n one of several possible m e a n i n g s o f the word (that is, selects one sign of the seve r a l i n w h i c h the w o r d is a signifier), w h i l e i n the case of the m a t h e m a t i c a l s y m b o l there is s i m p l y no m e a n i n g a p a r t f r o m the context. A n o t h e r p r a c tice, rare i n o r d i n a r y usage but c o m m o n i n m a t h e m a t i c s , is the t e m p o r a r y assignment o f a special m e a n i n g to a p a r t i c u l a r s y m b o l b y an i n d i v i d u a l m a t h e m a t i c i a n . N o t e t h a t this c o u p l i n g is not the s a m e as i n a l i n g u i s tic s i g n , because i t is t e m p o r a r y a n d not o r d i n a r i l y s a n c t i o n e d by general usage. A s the above r e m a r k s suggest, a large p a r t of m a t h e m a t i c a l i n f o r m a t i o n is carried p r i m a r i l y by language s t r u c t u r e . T h e s i t u a t i o n is different for a

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STRUCTURES

117

n a t u r a l language, where m u c h of the i n f o r m a t i o n is c a r r i e d by w o r d m e a n ings. In the l a t t e r case, however, some rather s u b t l e i n f o r m a t i o n m a y be c a r r i e d by s t r u c t u r e , especially when the language is s p o k e n . It is a p p r o p r i a t e t o recall here B e r t r a n d R u s s e l l ' s m u c h quoted d e f i n i t i o n of p u r e m a t h e m a t i c s [R5], as "the subject i n w h i c h we never k n o w w h a t we are t a l k i n g a b o u t , nor whether w h a t we are s a y i n g is t r u e " . T h i s catchy a n d clever, but m i s l e a d i n g , " d e f i n i t i o n " m a y be reassuring to l a y m e n (as suggested b y R u s s e l l ) a n d u n d o u b t e d l y u n d e r s t o o d b y m a t h e m a t i c i a n s , b u t it obscures the very i m p o r t a n t fact t h a t most m a t h e m a t i c i a n s d o indeed k n o w w h a t they are t a l k i n g a b o u t . W e refer, o f course, to F u r t h e r m o r e , t r u t h i n m a t h e m a t i c s is t r u t h , w h i c h is independent of the t r u t h or falsity of an assertion t h a t m i g h t arise f r o m an assignment o f e m p i r i c a l meanings to the s y m b o l s .

structures.

empirical

logical

mathematical

S t r u c t u r e and f o r m a l i s m tend to be l i n k e d , a n d an awareness of s t r u c t u r e leads n a t u r a l l y t o the c o n s t r u c t i o n of a f o r m a l language t o describe i t . T h i s is a consequence o f the u l t i m a t e abstract nature of s t r u c t u r e s . E s s e n t i a l l y the same idea was recognized by P i a g e t w h o expressed it as follows: T h e discovery o f s t r u c t u r e m a y , i m m e d i a t e l y or at a m u c h later stage, give rise to f o r m a l i z a t i o n . S u c h f o r m a l i z a t i o n is, however, a l ways the creature of the t h e o r e t i c i a n , whereas s t r u c t u r e i t s e l f exists a p a r t f r o m h i m . ... the m o d e of existence of the s t r u c t u r e he earlier discovered m u s t be d e t e r m i n e d separately f r o m each p a r t i c u l a r area of i n v e s t i g a t i o n . [ P 3 , p . 5] T h e last statement refers, i n our t e r m i n o l o g y , to p a r t i c u l a r representations of the abstract s t r u c t u r e . T h e use of a m o r e or less f o r m a l language is b y no means u n i q u e to m a t h e m a t i c s . E a c h field, at least i f it is at a l l technical i n n a t u r e , requires an a p p r o p r i a t e m e t h o d for the u n a m b i g u o u s c o m m u n i c a t i o n of its special subject m a t t e r . Some fields, such as theoretical physics, are even able to use the language of m a t h e m a t i c s for this purpose. T h a t such usage is possible i n these cases is an interesting fact w h i c h w i l l be discussed i n Section 46 w h e n we consider, f r o m the p o i n t of view o f s t r u c t u r e s , w h a t is i n v o l v e d i n an a p p l i c a t i o n o f m a t h e m a t i c s . A n o t h e r rather different k i n d of e x a m p l e is the professional j a r g o n associated w i t h certain s e m i t e c h n i c a l subjects. A l t h o u g h j a r g o n is sometimes used (or misused) t o exclude or m i s l e a d outsiders, it m a y also serve the l e g i t i m a t e role of i n c r e a s i n g p r e c i s i o n of c o m m u n i c a t i o n . A n interesting p o i n t , w h i c h is often confusing t o outsiders, is t h a t m a t h e m a t i c i a n s when c o m m u n i c a t i n g w i t h one another a l m o s t never use the language of m a t h e m a t i c s i n its f u l l f o r m a l i t y . I n fact, such usage w o u l d n o r m a l l y be a h i n d r a n c e to c o m m u n i c a t i o n , since it w o u l d force a t t e n t i o n

118

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STRUCTURES

o n m a n y already u n d e r s t o o d details and tend to d i s t r a c t a t t e n t i o n f r o m the " l a r g e r " i d e a w h i c h is u s u a l l y the object of interest. O n the other h a n d , i f there is a m i s u n d e r s t a n d i n g or disagreement c o n c e r n i n g some p o i n t , t h e n the f o r m a l language w i l l be invoked to whatever degree is needed t o clear up the p r o b l e m . It is o b v i o u s t h a t t h i s i n f o r m a l i t y i n language use is possible o n l y when the p a r t i c i p a n t s are q u i t e f a m i l i a r w i t h one anothers k n o w l edge a n d u n d e r s t a n d i n g of the s u b j e c t under discussion. Therefore, the r e p o r t i n g of m a t h e m a t i c s i n a research j o u r n a l , for e x a m p l e , w i l l u s u a l l y be considerably more f o r m a l , since the i n d i v i d u a l reader is u n k n o w n . E v e n i n this case, however, a n a u t h o r u s u a l l y has i n m i n d a f a i r l y definite audience whose p r e s u m e d knowledge o f the subject w i l l d e t e r m i n e the liberties t h a t c a n be t a k e n w i t h the language. A more or less f u l l use of the t e c h n i c a l m a t h e m a t i c a l language is u s u a l l y called a " treatment.

rigorous"

Because o f the e x t r e m e f o r m a l i t y of m u c h of the language of m a t h e m a t ics, the i n f o r m a l i t y w i t h w h i c h m a t h e m a t i c i a n s use i t is very conspicuous. A t the same t i m e , s i m i l a r practices o b v i o u s l y m u s t o c c u r i n one f o r m or another i n a l l language c o m m u n i c a t i o n . These are i m p o r t a n t , s t r u c t u r a l l y s i g n i f i c a n t , v a r i a t i o n s i n the use of a language. L a r g e p o r t i o n s o f the m a t e r i a l (structures) f a m i l i a r t o the p a r t i c i p a n t s m a y be c o m m u n i c a t e d b y the use of t e m p o r a r y a d hoc labels, a n d c o m p l i c a t e d logical a r g u m e n t s m a y be s p a n n e d b y i n t u i t i v e leaps quite i m p o s s i b l e for outsiders t o follow. A l t h o u g h errors m a y be made by such practices, there is always the p o s s i b i l i t y , a n d also the u l t i m a t e necessity, for a more careful a n d detailed t r e a t m e n t . These t h i n g s are interesting enough, but they are not i n the m a i n l i n e of our d i s c u s s i o n , so w i l l not be p u r s u e d any farther at t h i s t i m e . W e close t h i s section w i t h an extension of the r e m a r k s i n S e c t i o n 29 c o n c e r n i n g the tendency for language representations t o degrade the perception of a s t r u c t u r e . T h i s tendency is even stronger i n the case of a f o r m a l language t h a n it is w i t h o r d i n a r y language. In fact, any f o r m a l i s m tends t o be d i a m e t r i c a l l y opposed to a direct p e r c e p t i o n of the s t r u c t u r e to w h i c h it applies, so the two approaches are essentially c o n t r a d i c t o r y . A n e x t r e m e e x a m p l e o f d e g r a d a t i o n i n m a t h e m a t i c s is the response o f m a n y students to the s t u d y of a l g e b r a . A t an e l e m e n t a r y l e v e l , a l g e b r a is a form a l i s m for d e a l i n g w i t h a n u m b e r s y s t e m . A l t h o u g h most students have at least a r u d i m e n t a r y i n t u i t i o n a b o u t o r d i n a r y n u m b e r s , they w i l l often disassociate a l g e b r a m o r e or less completely f r o m n u m b e r s a n d t r e a t i t as a collection o f rules to be mastered for their o w n sake. S o m e m a y even be rather adept at algebraic m a n i p u l a t i o n s b u t have no i d e a w h a t is b e h i n d t h e m . A c o m p l e t e d is association of t h i s k i n d is i n t e l l e c t u a l l y possible o n l y because the f o r m a l i s m has, as does o r d i n a r y language, an i n t r i n s i c s t r u c t u r e o f its o w n . A l t h o u g h an expert at t i m e s m a y also do algebraic c a l c u l a t i o n s quite m e c h a n i c a l l y , he s i m u l t a n e o u s l y m a i n t a i n s a " l a t e n t awareness" of

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the rich s t r u c t u r e of the n u m b e r system (or other s y s t e m t o w h i c h a l g e b r a m i g h t a p p l y ) w h i c h m a y be called up as needed. T h i s is a t y p i c a l resol u t i o n of the basic c o n t r a d i c t i o n between f o r m a l i s m a n d p e r c e p t i o n t h a t occurs t h r o u g h o u t m a t h e m a t i c s as well as other 42.

H o w to R e c o g n i z e a M a t h e m a t i c a l

subjects.

Structure

W e are now ready to tackle the p r o b l e m of " d e f i n i n g " w h a t we m e a n by a m a t h e m a t i c a l s t r u c t u r e . O t h e r w i s e s t a t e d , the p r o b l e m is to i n d i c a t e , i f possible, j u s t w h a t q u a l i t i e s a g i v e n s t r u c t u r e must possess i n order t o be classified as a m a t h e m a t i c a l s t r u c t u r e . W e suggest t h a t it must satisfy the f o l l o w i n g two c o n d i t i o n s : (1)

T h e s t r u c t u r e must be susceptible t o f o r m a l a n a l y s i s .

I n other words, i t must be possible to use an e x i s t i n g p o r t i o n , or to create a new p o r t i o n , of m a t h e m a t i c a l language i n t e r m s of w h i c h the s t r u c t u r e m a y be described w i t h the precision d e m a n d e d i n m a t h e m a t i c s . O n e m e t h o d of a c c o m p l i s h i n g this, for e x a m p l e , is t h r o u g h an a x i o m a t i c t r e a t m e n t . (2) T h e s t r u c t u r e must have s u b s t a n t i a l connections w i t h the b o d y of mathematics. T h i s is a c r u c i a l c o n d i t i o n because it is possible t o have a quite f o r m a l t r e a t m e n t o f a s t r u c t u r e w h i c h has no c o n n e c t i o n at a l l to any e x i s t i n g m a t h e m a t i c s . In other words, it is possible t o have the f o r m w i t h o u t the content of m a t h e m a t i c s . S o m e of the more s u p e r f i c i a l , or forced, a t t e m p t s to a p p l y m a t h e m a t i c s produce results of this type. A g o o d e x a m p l e is S p i n o z a ' s " g e o m e t r i c " t r e a t m e n t of ethics. C r i t e r i a for e v a l u a t i n g connections t o e x i s t i n g m a t h e m a t i c s m a y v a r y considerably f r o m one m a t h e m a t i c i a n to another a n d f r o m one t i m e p e r i o d to another, so there m a y exist a m a r g i n a l gray area where differences of o p i n i o n as to whether or not the results s h o u l d be regarded as genuine m a t h e m a t i c s c o u l d occur. A l t h o u g h this is generally not a serious m a t t e r , m a t h e m a t i c a l developments i n connection w i t h c e r t a i n a p p l i c a t i o n s may o c c a s i o n a l l y f a l l w i t h i n the gray area. B e y o n d the question of whether a s t r u c t u r e is m a t h e m a t i c a l or not, is the question o f its i m p o r t a n c e . It is obvious t h a t not a l l of the l e g i t i m a t e m a t h e m a t i c a l results are of equal i m p o r t a n c e . S o m e m a y be t r a n s i t o r y , to be replaced l a t e r by new and better results, and some m a y be r e m e m b e r e d o n l y for h i s t o r i c a l reasons. A few c o n t r i b u t i o n s w i l l be recognized i m m e d i ately as of l a s t i n g i m p o r t a n c e . These w o u l d i n c l u d e , for e x a m p l e , s o l u t i o n s of certain f u n d a m e n t a l o u t s t a n d i n g problems t h a t have challenged m a t h e m a t i c i a n s for a long t i m e . A l s o i n c l u d e d w o u l d be m a j o r " b r e a k t h r o u g h s " t h a t clarify a n d u n i t e s u b s t a n t i a l p o r t i o n s o f m a t h e m a t i c s , perhaps r e q u i r i n g extensive r e s t r u c t u r i n g of the m a t e r i a l . O t h e r results m a y require a test

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of t i m e . H o w l o n g w i l l they s u r v i v e as recognized p a r t s of m a t h e m a t i c s ? W i l l they eventually be i n c o r p o r a t e d i n t o the m a i n b o d y of m a t h e m a t i c s ? T h e o u t c o m e i n these cases m a y depend i n p a r t o n developing fashions or the interests of a few recognized leaders. T h e l a t t e r , t h r o u g h t h e i r i n fluence o n other m a t h e m a t i c i a n s , c a n d e t e r m i n e , at least t e m p o r a r i l y , the p r i n c i p l e directions i n w h i c h the subject w i l l develop T h e m a t t e r is o b v i ously very c o m p l e x and u n d e r s t a n d a b l y involves m a n y subjective factors. Nevertheless there is u s u a l l y r e m a r k a b l e agreement a m o n g m a t h e m a t i c i a n s c o n c e r n i n g the e v a l u a t i o n of c o n t r i b u t i o n s to their subject. 43.

Research and Development

of

Mathematics

M o s t serious i n t e l l e c t u a l a c t i v i t i e s are i n one way or another creative. In t h i s a n d the next two sections, we w i l l t r y t o b r i n g out some of the special features of the c r e a t i v i t y i n v o l v e d i n m a t h e m a t i c a l research. Research i n m a t h e m a t i c s has always m e a n t a c t i v i t y t h a t leads to new m a t h e m a t i c s , t h a t is, new m a t h e m a t i c a l s t r u c t u r e s . A research m a t h e m a t i cian n a t u r a l l y has m u c h i n c o m m o n w i t h theoretical scientists i n a l l fields, but i n some ways is more like a composer or an a r t i s t . A t the same t i m e , despite a few m a t h e m a t i c i a n s w h o are expert expositors or h i s t o r i a n s , and a n u m b e r of first rate m a t h e m a t i c i a n s w h o have an e x c e p t i o n a l l y b r o a d knowledge of m a t h e m a t i c s , the m a t h e m a t i c a l analogue of a t y p i c a l a c a d e m i c scholar, say i n the h u m a n i t i e s , is conspicuously rare. For the l a t t e r , research m a y m e a n intensive work i n the l i b r a r y to discover and c o o r d i n a t e m a t e r i a l i n the l i t e r a t u r e , p r e v i o u s l y neglected or not fully u n d e r s t o o d . A l t h o u g h the m a t h e m a t i c a l researcher n a t u r a l l y m u s t do a c e r t a i n a m o u n t of t r a d i t i o n a l l i t e r a t u r e (re)search i n order t o acquire necessary knowledge of w h a t has already been done on a p r o b l e m , the m a i n objective is new m a t h e m a t i c s , a n d the end p r o d u c t is j u d g e d a l m o s t entirely u p o n whether or not it is a significant o r i g i n a l c o n t r i b u t i o n to the e x i s t i n g b o d y of m a t h e m a t i c s . T h e b o d y o f m a t h e m a t i c s , w h i c h consists of the current collective k n o w l edge of m a t h e m a t i c s , is recorded m a i n l y i n books a n d articles a n d also i n the m i n d s of p r a c t i c i n g m a t h e m a t i c i a n s . A l t h o u g h it is v i r t u a l l y i m p o s s i ble n o w for a single i n d i v i d u a l to k n o w well a l l of even the most i m p o r t a n t p a r t s of m a t h e m a t i c s , each active m a t h e m a t i c i a n does k n o w a s u b s t a n t i a l p o r t i o n of at least his o w n s p e c i a l field. T h i s means t h a t he has i n his m e m ory a representation of a significant p o r t i o n of the structures c o n t a i n e d i n his field o f interest. M a t h e m a t i c i a n s w i l l sometimes speak of their " m a t h e m a t i c a l i n t u i t i o n " i n reference to those m a t h e m a t i c a l structures w i t h w h i c h they are especially f a m i l i a r . In this sense, i n t u i t i o n is a m e n t a l p h e n o m e n o n associated w i t h a deep u n d e r s t a n d i n g of some p o r t i o n of a subject. T h e t e r m means m o r e , however, t h a n j u s t " k n o w l e d g e " of subject m a t t e r . It is analogous to the

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p e r c e p t i o n of a picture as a whole as opposed to a n a r r a t i v e d e s c r i p t i o n o f the p i c t u r e . In fact, the t e r m suggests a degree of i n t i m a c y w i t h the relevant s t r u c t u r e s t h a t enables one to "see" and u n d e r s t a n d t h e m w i t h o u t h a v i n g t o resort t o a f o r m a l a n a l y s i s . For e x a m p l e , an e x p e r t m a t h e m a t i c i a n m i g h t possess a deep u n d e r s t a n d i n g of a certain t h e o r e m and its connections to other m a t h e m a t i c s , but s t i l l require considerable effort to p r o d u c e a f o r m a l proof. A t the same t i m e , as every b e g i n n i n g g r a d u a t e student k n o w s , it is possible to u n d e r s t a n d and produce on d e m a n d the i n d i v i d u a l steps of a proof, w i t h o u t b e i n g able to appreciate fully the theorem p r o v e d .

research

Mathematical consists of e x t e n d i n g i n one way or another these personal s t r u c t u r e s a n d thus tends t o be a p r i m a r i l y i n d i v i d u a l a c t i v i t y . O n the other h a n d , it is not n o r m a l l y done i n i s o l a t i o n , but is c a r r i e d o n , d i r e c t l y or i n d i r e c t l y , i n concert w i t h workers i n the same or related fields. T h e results of these research efforts, plus their subsequent r e c o r d i n g i n the l i t e r a t u r e , c o n t r i b u t e to the general g r o w t h and development o f m a t h e m a t ics. S t r u c t u r a l extensions i n m a t h e m a t i c a l research tend to f a l l a n y w h e r e between two e x t r e m e types. T h e first is analogous to filling i n a m i s s i n g p o r t i o n of a f a b r i c h a v i n g an i n t r i c a t e p a t t e r n . For e x a m p l e , one m i g h t est a b l i s h a p r e v i o u s l y u n k n o w n connection between k n o w n results or p r o d u c e a new proof for a k n o w n theorem. H o w problems of this k i n d arise varies greatly, f r o m m o r e or less obvious gaps i n the theory, to conjectures. T h e latter may represent i n t u i t i v e insights by experts i n the field, or m a y be suggested b y analogies w i t h other p o r t i o n s of m a t h e m a t i c s or w i t h other fields such as physics. A l l of these a d d i t i o n s are " i n t e r n a l " i n the sense t h a t they shed l i g h t o n but do not change s u b s t a n t i a l l y the o r i g i n a l s t r u c t u r e . Nevertheless, some of the most i m p o r t a n t a n d s a t i s f y i n g c o n t r i b u t i o n s to m a t h e m a t i c s are o f this k i n d , because they result i n a s i g n i f i c a n t l y deeper u n d e r s t a n d i n g of the subject. T h e second type o f extension is more " e x t e r n a l " i n n a t u r e , u s u a l l y i n v o l v i n g new concepts (objects) a n d relations t h a t lie more or less outside of the given s t r u c t u r e , thus a l t e r i n g to some degree its overall character. T h i s k i n d of m a t h e m a t i c s is presently very fashionable a n d accounts for m u c h of the current research. A l t h o u g h i t includes m a n y u n i m p o r t a n t results t h a t w i l l soon be forgotten, it also includes some of the t r u l y great c o n t r i b u t i o n s , since i t can lead to vigorous new fields of research and a m u c h deeper u n d e r s t a n d i n g of broad areas of e x i s t i n g m a t h e m a t i c s . T h i s b r i n g s us finally to the m a i n questions: " H o w a n d i n w h a t f o r m do these extensions come a b o u t , a n d how are they finally i n c o r p o r a t e d i n the b o d y of m a t h e m a t i c s ? " A l t h o u g h it w o u l d be p r e s u m p t u o u s for us t o c l a i m final answers to these l o n g - s t a n d i n g questions concerning m a t h e m a t ical c r e a t i o n , a fresh look at t h e m f r o m the p o i n t of view of structures is

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nonetheless i n s t r u c t i v e . Before d e a l i n g w i t h the m o r e subtle aspects of the above q u e s t i o n , we m e n t i o n briefly a n obvious m e t h o d b y w h i c h a m a t h e m a t i c a l s t r u c t u r e is sometimes developed. T h e s p e c i a l i m p o r t a n c e of a m a t h e m a t i c a l language stems f r o m the fact t h a t it is a p o w e r f u l t o o l for the s y s t e m a t i c m a n a g e m e n t of c o m p l e x m a t h e m a t i c a l s t r u c t u r e s , offering the p o s s i b i l i t y of d e v e l o p i n g certain o f t h e m , at least t o a l i m i t e d extent, t h r o u g h f o r m a l language m a n i p u l a t i o n . I n other words, the language s t r u c t u r e is sufficiently "close" t o the associated m a t h e m a t i c a l s t r u c t u r e t h a t a development of the former translates i n t o a development of the l a t t e r . O n the other h a n d , a " b l i n d " m a n i p u l a t i o n of s y m b o l s cannot be expected t o produce a n y t h i n g very rem a r k a b l e . T h e process m u s t a c c o r d i n g l y be directed t o w a r d some goal. T h i s u s u a l l y involves a guess or conjecture, suggested, for e x a m p l e , b y a n analogy. A goal m i g h t also arise f r o m an i n t u i t i v e i n s i g h t i n t o the s u b j e c t o f interest. S u c h i n s i g h t s , w h i c h are an i m p o r t a n t element i n creative a c t i v i t y of a l l k i n d s , are discussed i n the next section.

44. T h e R o l e of Insight i n Research T h e theme of this section is perhaps best expressed b y an often q u o t e d statement o f A l b e r t E i n s t e i n ' s t h a t " I n v e n t i o n is not the p r o d u c t of l o g i c a l t h o u g h t , even t h o u g h the final p r o d u c t is t i e d to a l o g i c a l s t r u c t u r e " . T h e i d e a i m p l i c i t i n t h i s o b s e r v a t i o n is i l l u s t r a t e d by the sudden insights t h a t seem to be an i n e v i t a b l e feature of the creative process. T h e p r o d u c t i o n of new m a t h e m a t i c s is a s p e c i a l case of a general creative process, w h i c h is expressed i n m u c h the same f o r m i n a l l fields, t h o u g h each n a t u r a l l y has i t s o w n s p e c i a l features. T h e u n i v e r s a l character o f c r e a t i v i t y is b r o u g h t out i n the p e r s o n a l accounts o f creative experiences by workers i n different fields. S o m e examples f r o m a wide variety of i n d i v i d u a l s are c o n t a i n e d i n a b o o k edited by B r e w s t e r G h i s e l i n and e n t i t l e d , " [G4]. F o r discussions of m a t h e m a t i c a l c r e a t i v i t y (or discovery), we note the famous essay b y H e n r i P o i n c a r e o n [P6, p p . 383-394], w h i c h is quoted below, and a b o o k by Jacques H a d a m a r d o n f i ' e / d " [HI]. T h e r e are also the b o o k s , and by George P o l y a [P7], [P8]. T h e P o l y a books are h i g h l y regarded b y m a t h e m a t i c i a n s , b u t are o n the t e c h n i c a l side. T h a t is, i n stead o f discussions " a b o u t " m a t h e m a t i c s , they deal d i r e c t l y w i t h specific m a t h e m a t i c a l topics, t h o u g h the first involves o n l y e l e m e n t a r y m a t e r i a l . B y c o m p a r i s o n , the following r e m a r k s are entirely n o n t e c h n i c a l a n d s t e m f r o m a very different p o i n t of v i e w . T h e y are perhaps m o r e i n the s p i r i t of P a u l H a l m o s ' article, [H2], w h i c h is also rather t h a n mathematics.

The Creative

Process"

"Mathematical Creation"

"The Psychology of Invention in the Mathematical "Mathematics and Plausible Reasoning" ical Discovery",

about,

of,

"Mathematics as a Creative Art'

"Mathemat-

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M a n y d e s c r i p t i o n s of the creative process place so m u c h e m p h a s i s o n great discoveries t h a t they w i n d u p g i v i n g an i m p r e s s i o n t h a t the e x p e r i ence is more or less l i m i t e d to "great m i n d s " . B y contrast, the p o i n t t h a t we w i s h t o m a k e is t h a t c r e a t i v i t y , far f r o m b e i n g confined t o genius types, is a rather c o m m o n occurrence t h a t m a y be i d e n t i f i e d , b u t is often overl o o k e d , i n everyday experiences of o r d i n a r y people. It seems i n fact t o be an i m p o r t a n t aspect of a great deal o f m e n t a l a c t i v i t y at a l l levels, t h o u g h this o b s e r v a t i o n does not make the a n a l y s i s any easier. Despite its u n i v e r sal character, the creative experience, for reasons e x p l a i n e d b e l o w , is easier t o observe i n m a t h e m a t i c s t h a n i n a l m o s t any other s e t t i n g . A t the same t i m e , the final p r o d u c t of m a t h e m a t i c a l research, u s u a l l y a p u b l i s h e d paper, w i l l s e l d o m c o n t a i n any evidence of the creative experience t h a t b r o u g h t it i n t o existence. M o r e o v e r , the experience is often so intensely p e r s o n a l t h a t m a n y m a t h e m a t i c i a n s t e n d to be r e l u c t a n t , a n d perhaps even s o m e w h a t e m b a r r a s s e d , to discuss i t . O u r u l t i m a t e objective is to consider, i n rather general t e r m s , m a t h e m a t i c a l creation 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 d e v e l o p m e n t . I n other words, we w i s h to give some idea of how m a t h e m a t i c a l structures evolve t h r o u g h research. A l t h o u g h the research process takes place i n the m i n d s of m a t h e m a t i c i a n s , the n a t u r e of m a t h e m a t i c s is such t h a t i t is possible to give at least some idea of w h a t is g o i n g o n . It is obvious t h a t any discussion of a t o p i c of t h i s k i n d w i l l be s t r o n g l y colored by an a u t h o r ' s o w n personal experiences a n d biases. T h e present one is no e x c e p t i o n . O n e o f the difficulties i n s t u d y i n g m e n t a l processes of the k i n d we have here is t h a t m u c h of the a c t i v i t y occurs i n the unconscious. T h e results of unconscious m e n t a l a c t i v i t y are injected, often s u d d e n l y a n d unexpectedly, i n t o consciousness i n the f o r m of S u d d e n insights m a y of course occur d u r i n g conscious m e n t a l a c t i v i t y , t h o u g h the unconscious is a l m o s t c e r t a i n l y i n v o l v e d i n these events as w e l l . In any case, it w i l l help to t r y t o u n d e r s t a n d s o m e t h i n g of the "insight p h e n o m e n o n " before we t a c k l e the s t r u c t u r e questions i n the next section.

"insights".

A l t h o u g h i n s i g h t is c e r t a i n l y a u n i v e r s a l p h e n o m e n o n , i t is not genera l l y recognized i n everyday experiences, such as "face r e c o g n i t i o n " . T h i s is especially true of the unconscious a c t i v i t y . O n the other h a n d , the u n conscious processing o f m a t h e m a t i c s is often experienced a n d reported b y m a t h e m a t i c i a n s . T h a t m a t h e m a t i c s is different i n t h i s respect, is no d o u b t e x p l a i n e d by its abstractness a n d the relative absence o f irrelevant d i s t r a c t i o n s , so c o m m o n i n o r d i n a r y experiences of a l l k i n d s . In other words, the p u r i t y a n d relative i s o l a t i o n of the m a t h e m a t i c a l experience makes i t easily recognized a n d recalled. F o r our purposes, i t w o u l d not be necessary to d i s t i n g u i s h between conscious and unconscious i n s i g h t . I n fact, events of this k i n d occur w i t h such speed t h a t it is often difficult to d r a w a sharp

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line between the two. Because the l a t t e r is so interesting i n itself, we w i l l l i m i t a t t e n t i o n to i t . A t y p i c a l " i n s i g h t s c e n a r i o " , associated, for e x a m p l e , w i t h an a t t e m p t to solve a n elusive p r o b l e m , consists o f four stages. T h e s e have been described i n one f o r m or another by other m a t h e m a t i c i a n s , i n c l u d i n g H e n r i P o i n c a r e w h o is quoted below. T h e process begins w i t h a p e r i o d of intensive work d u r i n g w h i c h the researcher becomes very f a m i l i a r w i t h the p r o b l e m , but despite persistent effort is unable to p r o d u c e a s o l u t i o n . T h i s is followed by a p e r i o d o f r e l a x a t i o n or p r e o c c u p a t i o n w i t h s o m e t h i n g t o t a l l y u n r e l a t e d to the i n t r a c t a b l e p r o b l e m . N e x t , quite u n p r e d i c t a b l y a n d often i n another context, the o u t l i n e o f a s o l u t i o n occurs to the researcher w i t h o u t any conscious effort on his p a r t . A l t h o u g h m a n y details m a y be m i s s i n g , there is u s u a l l y a feeling o f a l m o s t c o m p l e t e certainty t h a t the s o l u t i o n w i l l work out. T h e f i n a l stage consists of an a t t e m p t t o provide a f o r m a l v e r i f i c a t i o n of the s o l u t i o n . Successful verification w i l l prove the insight to be correct, an o u t c o m e w h i c h m a y be rare e a r l y o n i n research, b u t occurs s u r p r i s i n g l y often i n the later stages. O n the other h a n d , the verification a t t e m p t m a y reveal t h a t the insight is after a l l incorrect. T h i s e v e n t u a l i t y , t h o u g h negative i n character a n d i n i t i a l l y depressing, m a y a c t u a l l y result i n a s u b s t a n t i a l increase i n u n d e r s t a n d i n g of the p r o b l e m . F i n a l l y , the verification a t t e m p t may s i m p l y f a i l , w h i c h leaves one w i t h a question of whether or not the insight is correct. T h e s e are the insights t h a t m a y lead to new p r o b l e m s a n d conjectures. Insights may o c c u r w i t h varied i n t e n s i t y at a l l levels of p r o b l e m s o l v i n g . T h e experience is a m e m o r a b l e one, enjoyed frequently by m o s t m a t h e m a t i cians i n the course of their research. T h e r e are numerous v a r i a t i o n s o n the above scenario, b u t , as far as structures are concerned, a l l such insights, conscious or unconscious, m a y be " e x p l a i n e d " i n essentially the same way. T h i s depends a g a i n o n our a s s u m p t i o n t h a t the m i n d possesses the c a p a b i l ity of d e a l i n g actively a n d s y s t e m a t i c a l l y w i t h s t r u c t u r e s . R e c a l l also t h a t m o s t of the m i n d ' s processing of structures is a u t o m a t i c , w h i c h means t h a t it does not involve conscious i n t e r v e n t i o n . H e n r i P o i n c a r e (1854-1912), the author o f the f a m o u s essay m e n t i o n e d above, was one of the very great m a t h e m a t i c i a n s of o u r t i m e . T h e essay, entitled Creation",contains i n some d e t a i l an account of one of his research experiences t h a t i n v o l v e d several events of the k i n d described above. P o i n c a r e then speculates o n how the unconscious (the " s u b l i m i n a l s e l f " ) manages t o come up w i t h possible s o l u t i o n s to p r o b l e m s . H e t h i n k s of the s o l u t i o n as a " g o o d c o m b i n a t i o n " of m a t h e m a t i c a l entities already k n o w n . T h e question is how the unconscious is able t o select such a c o m b i n a t i o n f r o m the e n o r m o u s n u m b e r of possibilities. P o i n c a r e suggests t h a t

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one factor is t h a t the " g o o d " c o m b i n a t i o n s have an aesthetic value t h a t b r i n g s t h e m i n t o consciousness. N o t e the s t r u c t u r a l concepts i m p l i c i t i n the f o l l o w i n g q u o t a t i o n f r o m the essay [P6, p p . 3 9 1 , 392]: N o w , w h a t are the m a t h e m a t i c s entities to w h i c h we a t t r i b u t e this character of b e a u t y a n d elegance, a n d w h i c h are capable of developing i n us a sort o f aesthetic e m o t i o n ? T h e y are those whose elements are h a r m o n i o u s l y disposed so t h a t the m i n d w i t h o u t effort c a n embrace their t o t a l i t y w h i l e r e a l i z i n g the d e t a i l s . T h i s h a r m o n y is at once a satisfaction o f o u r aesthetic needs a n d an a i d t o the m i n d , s u s t a i n i n g a n d g u i d i n g . A n d at the same t i m e , i n p u t t i n g under our eyes a well-ordered whole, i t makes us foresee a m a t h e m a t i c a l l a w . N o w , as we have s a i d above, the o n l y m a t h e m a t i c a l facts w o r t h y o f f i x i n g o u r a t t e n t i o n a n d capable of b e i n g useful are those w h i c h c a n teach us a m a t h e m a t i c a l l a w . So t h a t we reach the f o l l o w i n g c o n c l u s i o n : T h e useful c o m b i n a t i o n s are precisely the m o s t b e a u t i f u l , I m e a n those best able t o c h a r m t h i s special s e n s i b i l i t y t h a t a l l m a t h e m a t i c i a n s k n o w , b u t of w h i c h the profane are so ignorant as often t o be t e m p t e d t o s m i l e at i t . W h a t h a p p e n s t h e n ? A m o n g the great n u m b e r s of c o m b i n a t i o n s b l i n d l y f o r m e d by the s u b l i m i n a l self, almost a l l are w i t h o u t interest a n d w i t h o u t u t i l i t y ; b u t j u s t for t h a t reason they are also w i t h o u t effect u p o n the aesthetic sensibility. Consciousness w i l l never k n o w t h e m ; only c e r t a i n ones are h a r m o n i o u s a n d , consequently, at once useful a n d b e a u t i f u l . T h e y w i l l be capable of t o u c h i n g t h i s s p e c i a l s e n s i b i l i t y o f the geometer of w h i c h I have j u s t s p o k e n , and w h i c h , once aroused, w i l l c a l l our a t t e n t i o n to t h e m , a n d t h u s give t h e m occasion to become conscious. T h e p r o b l e m r e m a i n s t h a t the unconscious is a l m o s t c e r t a i n l y u n a b l e i n most cases to consider the t o t a l i t y of possible c o m b i n a t i o n s i n the process of f i n d i n g a good one. P o i n c a r e , i n the f o l l o w i n g lengthy q u o t a t i o n [ P 6 , p . 393], suggests a n answer to t h i s p r o b l e m i n the f o r m o f an analogy. It is v i v i d enough, but less s t r u c t u r a l i n character t h a n the above. W e w i l l offer a q u i t e different a n a l o g y i n the next section. Perhaps we ought t o seek the e x p l a n a t i o n i n t h a t p r e l i m i n a r y per i o d o f conscious work w h i c h always precedes a l l f r u i t f u l unconscious l a b o r . P e r m i t me a rough c o m p a r i s o n . F i g u r e the future elements of our c o m b i n a t i o n s as s o m e t h i n g like the h o o k e d a t o m s of E p i c u r u s . D u r i n g the c o m p l e t e repose of the m i n d , these a t o m s are motionless, they are, so t o speak, hooked to the w a l l ; so this complete rest m a y be indefinitely p r o l o n g e d w i t h o u t the atoms m e e t i n g , and consequently w i t h o u t any c o m b i n a t i o n between t h e m .

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O n the other h a n d , d u r i n g a p e r i o d o f apparent rest a n d u n c o n scious w o r k , certain of t h e m are detached f r o m the w a l l a n d put i n m o t i o n . T h e y flash i n every d i r e c t i o n t h r o u g h the space (I was a b o u t t o say the r o o m ) where they are enclosed, as w o u l d , for e x a m p l e , a s w a r m o f gnats or, i f y o u prefer a more learned c o m p a r i s o n , l i k e the molecules o f gas i n the k i n e m a t i c theory of gases. T h e n their m u t u a l i m p a c t s may p r o d u c e new c o m b i n a t i o n s . W h a t is the role o f the p r e l i m i n a r y conscious w o r k ? It is e v i d e n t l y to m o b i l i z e c e r t a i n of these a t o m s , t o u n h o o k t h e m f r o m the w a l l a n d p u t t h e m i n s w i n g . W e t h i n k we have done no g o o d , because we have m o v e d these elements a t h o u s a n d different ways i n seeking to assemble t h e m , a n d have f o u n d no satisfactory aggregate. B u t , after t h i s s h a k i n g u p i m p o s e d u p o n t h e m by our w i l l , these a t o m s do not r e t u r n t o their p r i m i t i v e rest. T h e y freely continue t h e i r dance. N o w , our w i l l d i d not choose t h e m at r a n d o m ; i t p u r s u e d a perfectly d e t e r m i n e d a i m . T h e m o b i l i z e d atoms are therefore not any atoms whatsoever; they are those f r o m w h i c h we m i g h t reasonably expect the desired s o l u t i o n . T h e n the m o b i l i z e d a t o m s undergo i m p a c t s w h i c h m a k e t h e m enter i n t o c o m b i n a t i o n s a m o n g themselves or w i t h other atoms at rest w h i c h they struck against i n their course. A g a i n I beg p a r d o n , m y c o m p a r i s o n is very r o u g h , b u t I scarcely k n o w how otherwise to make m y t h o u g h t u n d e r s t o o d . T h e r e are m a n y other interesting r e m a r k s i n the P o i n c a r e essay, b u t the above q u o t a t i o n s w i l l be sufficient for our purposes. In order t o r o u n d out the p i c t u r e , however, we report t w o rather more c o m m o n p l a c e p e r s o n a l m a t h e m a t i c a l experiences w h i c h have a s o m e w h a t different twist t o t h e m . S o m e years ago I received the p r o o f sheets for a paper t h a t was to he p u b l i s h e d i n a s t a n d a r d research j o u r n a l . Since o n l y t r i v i a l corrections were required, the n o r m a l procedure w o u l d have been to r e t u r n t h e m p r o m p t l y t o the j o u r n a l . In t h i s case, however, for no conscious reason, I delayed s e n d i n g t h e m back for a p e r i o d of a couple of weeks. In the m e a n t i m e , I g r a d u a l l y became aware of repeatedly r e v i e w i n g i n m y m i n d the p r o o f of a c e r t a i n l e m m a of the paper. T h i s w o u l d occur at o d d times a n d w i t h o u t previous t h o u g h t a b o u t the paper. T h e experience i t s e l f was not so s u r p r i s i n g , because the l e m m a was a n i m p o r t a n t one, a n d I was quite pleased w i t h its proof. A s a m a t t e r of fact, i t is not u n u s u a l for one to " r e p l a y " i n t h i s way the proofs of especially s a t i s f y i n g results, for the sheer pleasure of e x p e r i e n c i n g t h e m a g a i n . T h e practice is no different f r o m r e m e m b e r i n g at o d d t i m e s the m e l o d y i n a favorite piece o f m u s i c . O n the other h a n d , there was an element of compulsiveness a b o u t these events, i n t h a t they occurred a b i t t o o often a n d at inconvenient times. A s a result, they event u a l l y aroused m y suspicions, l e a d i n g m e to sit d o w n a n d e x a m i n e carefully

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the p r o o f of the l e m m a . A s the reader has no d o u b t already guessed, there was a n error i n the p r o o f ! It was f o r t u n a t e i n this instance t h a t the error was not a deep one so was rather easy t o correct, whereupon I h a p p i l y a n d p r o m p t l y r e t u r n e d the p r o o f sheets. T h e second experience i n v o l v e d a former colleague, a n d concerned a certain w e l l - k n o w n conjecture i n our c o m m o n field o f interest. W e h a d t r i e d off a n d o n to settle the conjecture, w i t h o u t success u n t i l one day m y colleague a n n o u n c e d t h a t he h a d a s u r p r i s i n g l y s i m p l e proof. In fact, it required o n l y a few m i n u t e s for h i m to o u t l i n e the proof, w h i c h was rather elegant a n d settled the question i n a very satisfactory way. Since we were i n v o l v e d w i t h e x a m i n a t i o n s at the t i m e , it was not convenient for us to pursue the m a t t e r i m m e d i a t e l y . A couple of weeks later, a n d again for no obvious reason, I awoke i n the m i d d l e of the n i g h t and found m y s e l f r e v i e w i n g m y colleague's proof. I c o u l d r e m e m b e r it u p to a certain p o i n t where I w o u l d lose the t h r e a d of a r g u m e n t a n d have to r e t u r n to the b e g i n n i n g . T h i s occurred repeatedly d u r i n g the course of perhaps an h o u r . M y e x p l a n a t i o n o f the difficulty at the t i m e was t h a t I c o u l d not arouse myself sufficiently to s u p p l y the c r u c i a l step o f the a r g u m e n t . E v e n t u a l l y I went back t o sleep a n d p u t the whole t h i n g out of m y m i n d u n t i l a few days later when I encountered m y colleague a n d t o l d h i m t h a t I h a d been u n a b l e to recall his proof. H e t h e r e u p o n proceeded to refresh m y m e m o r y , t r a c i n g the same steps t h a t I h a d followed and b e c o m i n g s t u c k , o f course, at e x a c t l y the same p o i n t where I was s t o p p e d . T h e p r o o f was defective! In fact, several years l a t e r a clever y o u n g m a t h e m a t i c i a n c o n s t r u c t e d a c o u n t e r e x a m p l e to the conjecture, t h a t i t was a c t u a l l y false.

showing

T h e above are o n l y t w o out of a n u m b e r o f s i m i l a r experiences, each of w h i c h i n v o l v e d a c o m p u l s i v e r e v i e w i n g of some p o r t i o n of a f a m i l i a r piece of m a t h e m a t i c s . A s a result, when I catch myself i n v o l u n t a r i l y " r e p l a y i n g " any piece of m a t h e m a t i c s , I have learned to " l i s t e n " very carefully so as to be sure t h a t i t is s t r i c t l y for pleasure. E x a m p l e s such as the above suggest t h a t the unconscious is not o n l y a s o m e w h a t u n i n h i b i t e d creator but appears t o be a rather s u b t l e c r i t i c as w e l l . Nevertheless, as P o i n c a r e also points o u t , the unconscious, r e m a r k able as i t is, seems never to present one w i t h a l l the details of a s o l u t i o n . F i l l i n g i n details a p p a r e n t l y requires the d i s c i p l i n e of the conscious, p r o b a b l y because it involves (self-) c o m m u n i c a t i o n . T h e message i n each of the examples was s u b t l e a n d i n d i r e c t . T h e r e was not even a suggestion of error, o n l y a n enforced " r e p l a y i n g " of the proof. W e w i l l t r y to " e x p l a i n " i n the f o l l o w i n g section w h y messages f r o m the unconscious seem t o take this form.

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A Structural Interpretation of Creativity

T h e p r o b l e m at this p o i n t is to t r y to give some idea of how insights m i g h t o c c u r when the m i n d deals w i t h a s t r u c t u r e . T h i s w i l l require a closer look at h o w m e n t a l structures m i g h t be represented i n the b r a i n . W e have been able to gloss over the question u n t i l now by r e g a r d i n g m e n t a l structures as more or less independent of these representations. Despite the d e a r t h of i n f o r m a t i o n on the s u b j e c t , a m o d e l o f how m e n t a l s t r u c t u r e s m i g h t be formed is s t i l l needed, t h o u g h the chances are rather s l i m t h a t it can be more t h a n a very r o u g h analogy to w h a t a c t u a l l y o c c u r s . T h e point is t h a t t h i n k i n g about a n y t h i n g , i n c l u d i n g t h i n k i n g itself, requires some k i n d of " v i s u a l i z a t i o n " of the o b j e c t . It is h o p e d t h a t the rather fuzzy and h y p o t h e t i c a l p i c t u r e of a m e n t a l s t r u c t u r e sketched here w i l l c o n t a i n e n o u g h grains of t r u t h to be of some help i n this c o n n e c t i o n . T h e h u m a n b r a i n is c o m p o s e d o f an e n o r m o u s n u m b e r o f i n d i v i d u a l nerve cells, w h i c h , t h r o u g h the synapses (100 t r i l l i o n or so i n n u m b e r ) , a d m i t the p o s s i b i l i t y of a l m o s t u n l i m i t e d interconnections. A s a l r e a d y suggested i n Section 34, we w i l l t h i n k of t h i s e x t r e m e l y c o m p l e x s t r u c t u r e as analogous t o a massive electrical network, w h i c h contains our m e n t a l structures as subnetworks. Despite its o v e r s i m p l i f i c a t i o n , the analogy provides a useful means of p i c t u r i n g the illusive m e n t a l structures. In order for a p o t e n t i a l interconnection w i t h i n the b r a i n network to be effective, it must be a c t i v a t e d i n some way or other, a c o n d i t i o n t h a t m a y or m a y not be p e r m a n e n t . T h e a c t i v a t i o n of a g r o u p of interconnections produces a s t r u c t u r e , w h i c h exists, so to speak, w i t h i n the mass of nerve cells. It is h e l p f u l to t h i n k of the a c t i v a t i o n of a s t r u c t u r e w i t h i n the b r a i n as a k i n d of " h i g h l i g h t i n g " process t h a t accentuates the s t r u c t u r e against an u n a c t i v a t e d a n d undifferentiated b a c k g r o u n d i n the nerve mass. M e n t a l structures m a y now be identified w i t h these a c t i v a t e d nerve structures. N o t e t h a t a s t r u c t u r e of t h i s k i n d is a p o t e n t i a l s u b s t r u c t u r e o f m a n y larger nerve s t r u c t u r e s t h a t are capable of being activated to c o n t a i n i t . A l t h o u g h we w i l l not t r y to guess j u s t how m e n t a l structures are formed i n the first place (say, i n response t o a set of e x t e r n a l s t i m u l i ) , it w i l l f a c i l i t a t e the discussion w h i c h follows to consider briefly the extension of a m e n t a l s t r u c t u r e t h r o u g h a c t i v a t i o n of a larger nerve s t r u c t u r e . R e c a l l t h a t we defined (in S e c t i o n 7) i n t e r n a l a n d e x t e r n a l properties o f a s t r u c t u r e . T h e first are concerned w i t h the objects and relations w i t h i n the s t r u c t u r e a n d the seco n d w i t h the various relations t h a t involve the s t r u c t u r e when it appears as an o b j e c t or a s u b s t r u c t u r e of another s t r u c t u r e , as for e x a m p l e i n a c o n t r a c t i o n (Section 27). E x t e r n a l properties, a p a r t f r o m the c o n t a i n i n g s t r u c t u r e s t h a t define t h e m , enjoy o n l y a p o t e n t i a l existence. O n the other h a d , a p o t e n t i a l e x t e r n a l p r o p e r t y s h o u l d somehow be " a n t i c i p a t e d " i n the g i v e n s t r u c t u r e

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itself, at least i f the associated extension is a significant one i n the sense t h a t it does not ignore completely the i n t e r n a l s t r u c t u r e o f the f o r m e r . W e m a y t h i n k of these " p o t e n t i a l " properties as represented b y special "connector p o i n t s " o n the " p e r i p h e r y " of the o b j e c t (or " w h o l e " ) associated w i t h the given s t r u c t u r e . I n the e l e c t r i c a l network m o d e l , they m a y be t h o u g h t of as c a r r y i n g e l e c t r i c a l charges t h a t can activate possible connections near t h e m . T h e y m a y also be t h o u g h t of as analogous to b u d s o n a tree b r a n c h , w h i c h are p o t e n t i a l new branches t h a t m a y be realized as the tree s t r u c t u r e develops. W i t h the above p i c t u r e i n m i n d , let us r e t u r n t o the case of a m a t h e m a t i cian w h o is t r y i n g to extend a m a t h e m a t i c a l s t r u c t u r e already recorded i n his b r a i n as an a c t i v a t e d nerve s t r u c t u r e . In a n a c t u a l case, there w i l l be m a n y other a c t i v a t e d structures i n the " n e i g h b o r h o o d " o f the given one. These represent possibly relevant knowledge, a n d m a y or m a y not have direct connections to the structure of interest. T h e o b j e c t i v e then is to activate a larger s t r u c t u r e w h i c h contains as a s u b s t r u c t u r e the given one, perhaps a l o n g w i t h c e r t a i n n e i g h b o r i n g s t r u c t u r e s . It is reasonable t o assume t h a t an extension w i l l not be a c t i v a t e d unless triggered by one or more connector p o i n t s , t h u s e x c l u d i n g at the outset m a n y irrelevant extensions. O n e obvious m e t h o d o f extension is to t r y t o activate a series of connections t o one of the n e i g h b o r i n g structures. O t h e r extensions arise more u n p r e d i c t a b l y a n d spontaneously. T h e y are a result o f the tendency of the connector points t o activate connections t h a t grow o u t , so to speak, i n t o the nerve mass. S o m e m a y eventually coalesce i n t o the desired extension, b u t most w i l l p r o b a b l y be rejected, because they f a i l to be as m e a s u r e d b y a variety of c r i t e r i a , not the least of w h i c h is the aesthetic one described by P o i n c a r e . T r a n s i e n t extensions of this k i n d are not merely r a n d o m growths, because they involve i n an essential way the connector p o i n t s . Moreover, they tend to arise spontaneously w i t h l i t t l e or no conscious i n t e r v e n t i o n , even for the rejections.

significant,

A t this point we recall the " i n s i g h t s c e n a r i o " , w h i c h was o u t l i n e d i n the previous section, to see how it fits i n t o the above p i c t u r e . R e c a l l t h a t the first stage consisted of an intensive conscious s t u d y of the g i v e n s t r u c t u r e i n order to u n d e r s t a n d it t h o r o u g h l y a n d to extend it i f possible. T h e g o a l of the extension m i g h t be to include one of the n e i g h b o r i n g s t r u c t u r e s , as suggested b y the above r e m a r k s . T h i s m a y be successful, i n w h i c h case the i m m e d i a t e p r o b l e m is solved. If the a t t e m p t is t o t a l l y unsuccessful, t h a t is, the f a i l u r e is s t r u c t u r a l i n character, then perhaps the o n l y g a i n is the knowledge t h a t c e r t a i n k i n d s of extensions are e v i d e n t l y o f no use. In other words, c e r t a i n connector p o i n t s , or some of the connections that they a c t i v a t e , may now be " b l o c k e d " , so t h a t they w i l l not arise i n future a t t e m p t s . A n intermediate p o s s i b i l i t y m i g h t be a failure w h i c h is due t o a

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collapse o f the extension process, perhaps because i t b e c a m e too c o m p l e x for the conscious to m a n a g e , or s i m p l y because i t encountered t o o m a n y d i s t r a c t i o n s . In such a case, there r e m a i n s a p o s i t i v e residue c o n s i s t i n g of a p o t e n t i a l route t o an e x t e n s i o n . T h e i d e a is t h a t , i f t h i s first stage has been of sufficient i n t e n s i t y , t h e n the extension process m a y a n d w i l l c o n t i n u e at the unconscious l e v e l . A n u l t i m a t e advantage w i l l be a r e d u c t i o n , d u r i n g the first stage, of the n u m b e r of p o t e n t i a l extensions t h a t need be t r i e d . U n c o n s c i o u s processes also have the advantage o f b e i n g able to proceed w i t h o u t the m a n y e x t e r n a l d i s t r a c tions t h a t c o n s t a n t l y i n t e r r u p t and i n h i b i t conscious m e n t a l a c t i v i t y . T h e unconscious c a n therefore deal w i t h s p e c t a c u l a r e x t e n s i o n forays t h a t m a y be too c o m p l e x or u n s t a b l e to be sustained i n the noisy conscious. These excursions c a n either involve an e x i s t i n g n e i g h b o r i n g s t r u c t u r e , or activate a new one, w h i c h m a y then enter as an e n t i t y i n t o consciousness. S u c h a n event constitutes one f o r m o f sudden i n s i g h t . A l t h o u g h f u l l details of the connections between the o r i g i n a l s t r u c t u r e and a new or n e i g h b o r i n g one may not enter i n t o consciousness, the fact t h a t connections were a c t u a l l y established gives rise to the s t r o n g i n t u i t i v e feeling t h a t a c o n n e c t i o n does e x i s t . T h u s , we see (1) how an i n s i g h t m a y arise, (2) the o r i g i n of the feeling t h a t i t "solves" the p r o b l e m , a n d (3) the reason t h a t feeling is so often correct. It remains for the researcher t o verify, if possible, the correctness by f i l l i n g i n step b y step the m i s s i n g connections, a n d record the result i n accepted m a t h e m a t i c a l f o r m . These final steps, w h i c h depend more or less c o m p l e t e l y o n the conscious, are n o r m a l l y the o n l y elements of the whole process t h a t are (or can be) revealed t o others. In the t w o examples i n v o l v i n g the incorrect proofs, we began w i t h w h a t was s u p p o s e d l y a correct proof, w h i c h was r o u t i n e l y s t o r e d i n m e m o r y as an i t e m of m a t h e m a t i c a l knowledge. B u t since the p r o o f was incorrect, the c o r r e s p o n d i n g s t r u c t u r e h a d to be flawed i n some way, perhaps w i t h m i s s i n g or i m p o s s i b l e connections. Because of some s t i m u l u s or o t h e r , such as c o r r e c t i n g p r o o f sheets, the unconscious is p r o m p t e d t o reactivate the s t r u c t u r e a n d is, of course, b l o c k e d b y the flaw. T h i s is a n event w h i c h m a y be t r a u m a t i c enough to force the m a t t e r i n t o consciousness, where the error m a y e v e n t u a l l y be discovered. O n e m i g h t t h e n ask w h y the event d i d not occur before, when the error was first m a d e . P e r h a p s it d i d occur, b u t the process was sidetracked b y some d i s t r a c t i o n or o t h e r , such as p r e o c c u p a t i o n w i t h the m a i n results of the paper (in the case of the l e m m a ) , or s i m p l y wishful thinking. A s s o m e o f the above r e m a r k s suggest, the s t r u c t u r e developments i n volved i n i n s i g h t p h e n o m e n a are not r e s t r i c t e d to the unconscious, a l t h o u g h unconscious processing m a y always p l a y at least an i n d i r e c t role. In any case, i n s i g h t s t h a t occur d u r i n g conscious work o n a p r o b l e m m a y be a n a -

VII. M A T H E M A T I C A L S T R U C T U R E S

131

l y z e d i n a m a n n e r s i m i l a r to the above. 46. 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 m a t e r i a l , or d a t a , i n any subject constitutes a " s y s t e m " , as defined in Section 7, consisting o f c e r t a i n d i s t i n g u i s h e d objects o f s t u d y a l o n g w i t h a mass of p r e v i o u s l y established i n f o r m a t i o n c o n c e r n i n g t h e m . A n i n v e s t i g a t i o n n o r m a l l y leads to the c o n s t r u c t i o n of a " t h e o r y " , designed to " o r g a n i z e " a n d perhaps " e x p l a i n " t h a t p o r t i o n o f the s y s t e m b e i n g s t u d i e d . T h e t e r m " o r g a n i z e " o b v i o u s l y means to identify s t r u c t u r e , and " e x p l a i n " means t o connect t h a t s t r u c t u r e t o other m o r e f a m i l i a r ones, or t o derive a l l or p a r t of it f r o m a relatively s i m p l e s u b s t r u c t u r e . A g o o d theory is therefore the result of a successful s t r u c t u r a l a n a l y s i s . It is a c o n c e p t u a l m o d e l a n d is tested, at least i n the sciences, by c o m p a r i s o n w i t h k n o w n facts, such as e x p e r i m e n t a l results. W e s h a l l c a l l any s t r u c t u r e w i t h i n a subject s y s t e m of this k i n d a " d a t a s t r u c t u r e " . A p p l i c a t i o n s of m a t h e m a t i c s concern theories t h a t involve s t r u c t u r e s of m a t h e m a t i c a l t y p e . In these cases, the f o r m a l statement of the theory consists of a m a t h e m a t i c a l d e s c r i p t i o n of a d a t a s t r u c t u r e . Ideally, this a m o u n t s to setting up an i s o m o r p h i s m between the d a t a s t r u c t u r e a n d a m a t h e m a t i c a l s t r u c t u r e . In a c t u a l practice, however, it is u s u a l l y necessary t o settle for an a p p r o x i m a t e i s o m o r p h i s m , one t h a t may not q u i t e fit the d a t a s t r u c t u r e or incorporates only p o r t i o n s of the two s t r u c t u r e s . F o r this reason, the general significance of an a p p l i c a t i o n w i l l depend b o t h on the i m p o r t a n c e o f the two structures a n d also o n the degree of the a p p r o x i m a t i o n . T h e l a t t e r m a y vary greatly f r o m one a p p l i c a t i o n to another, r a n g i n g f r o m near i s o m o r p h i s m d o w n to l i t t l e more t h a n a m e t a p h o r i c a l use of m a t h e m a t i c a l language. A m i n i m a l role for any theory is to describe the m a t e r i a l to w h i c h it applies. T h e r e f o r e , a first test of its effectiveness w i l l n a t u r a l l y concern the accuracy a n d completeness of the d e s c r i p t i o n . O n the other h a n d , a purely descriptive theory is o f l i m i t e d value. In fact, one u s u a l l y expects a theory w o r t h y of the n a m e t o be able, i n one way or another, t o cover m a t e r i a l outside t h a t for w h i c h it was constructed. T h i s c o u l d arise by an e x t e n s i o n of either the i n i t i a l d a t a s t r u c t u r e or the theory s t r u c t u r e t h a t represents it. In the first case, an extension amounts t o a discovery, t h r o u g h either e x p e r i m e n t or other research, of properties of the s y s t e m outside the dom a i n of the theory. T h e question then is whether or not the theory, or some extension of i t , w i l l cover the new m a t e r i a l . O n the other h a n d , an extension of the theory s t r u c t u r e (perhaps v i a an extension of an associa t e d m a t h e m a t i c a l s t r u c t u r e ) w i l l predict the existence of properties of the s y s t e m corresponding to the extension, and the question is whether or not

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i t is possible {perhaps b y e x p e r i m e n t ) t o verify the p r e d i c t i o n . I n either case, a negative answer t o the q u e s t i o n casts doubt o n the theory w h i l e a p o s i t i v e one is evidence of its v a l i d i t y , t h o u g h the a c t u a l force of these conclusions w i l l depend o n how n e a r l y i s o m o r p h i c are the i n i t i a l s t r u c t u r e s , a n d t o w h a t extent the extensions are d e t e r m i n e d b y t h e m . T h e v a r i o u s p o s s i b i l i t i e s o u t l i n e d a b o v e are i l l u s t r a t e d i n F i g u r e 4 6 . 1 . D, and respectively represent and structures, w h i l e D', T', a n d M ' represent possible extensions of t h e m . D o u b l e ended arrows suggest associated i s o m o r p h i s m s , a n d dashed lines i n d i c a t e assumed or conjectured i s o m o r p h i s m s , w h i c h m a y or m a y not exist.

T,

M

data, theory

Data

Theory

Fig.

mathematical

Mathematical

4e.l

P e r h a p s the m o s t i m p o r t a n t advantage of m a t h e m a t i c a l a p p l i c a t i o n s is the a v a i l a b i l i t y of m a t h e m a t i c a l techniques for m a n i p u l a t i n g a n d e x t e n d i n g structures. Because of the n a t u r e of m a t h e m a t i c s , m a t h e m a t i c a l p r e d i c t i o n s m a y also e x h i b i t a precision a n d extent s e l d o m f o u n d i n n o n m a t h e m a t i c a l theories. T h e a b i l i t y t o encompass new m a t e r i a l , a n d especially t o p r e d i c t , as described above, is w i d e l y regarded i n science as a n essential r e q u i r e m e n t for any theory. W h e n a theory fails t h i s test, i t is reduced t o at most a d e s c r i p t i o n of a l i m i t e d p o r t i o n of the subject, a n d its scientific s t a t u s is open to challenge. Such failures are u s u a l l y i n t e r p r e t e d t o m e a n t h a t the theory either does n o t represent essential features of the subject or t h a t the representation is flawed. A l t h o u g h t h i s c o n c l u s i o n is p r o b a b l y v a l i d i n most contexts, i t m a y be a b i t t o o severe i n some instances, such as w h e n a p o t e n t i a l c r u c i a l e x p e r i m e n t requires techniques n o t yet a v a i l a b l e . T h e r e are also the cases of e x t r e m e l y c o m p l e x systems or systems t h a t a d m i t " c h a o t i c " b e h a v i o r , for w h i c h i t is v i r t u a l l y i m p o s s i b l e to m a k e r e l i a b l e predictions. T h e m o s t successful a p p l i c a t i o n s of m a t h e m a t i c s are n a t u r a l l y of greatest

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interest t o us, because they e x h i b i t m o r e significant s t r u c t u r a l p h e n o m e n a . Therefore, the e x a m p l e of physics m o t i v a t e s d i r e c t l y or i n d i r e c t l y m u c h of our t h i n k i n g o n the subject. T h e f o l l o w i n g section is devoted to the u n i q u e r e l a t i o n s h i p t h a t exists between m a t h e m a t i c s a n d physics. A p p l i c a t i o n s to n o n p h y s i c a l subjects u s u a l l y involve features q u i t e different f r o m the case of physics, and s e l d o m e x h i b i t the precision f o u n d i n the l a t t e r . S o m e of these differences a n d the special p r o b l e m s t h a t a c c o m p a n y t h e m are discussed i n Section 48. A l o n g w i t h the general p r o b l e m of a p p l y i n g m a t h e m a t i c s to other fields, is the fact t h a t m a t h e m a t i c s , because o f its h i g h l y abstract subject m a t t e r and f o r m a l language, is i n t r i n s i c a l l y different f r o m most other d i s c i p l i n e s . O n e consequence is t h a t m a t h e m a t i c i a n s tend t o a p p r o a c h another field rather differently f r o m workers i n t h a t field. T h i s difference, w h i c h stems f r o m a difference i n i n t u i t i o n , is discussed b y B a r r y C i p r a i n a magazine report o n recent m a t h e m a t i c a l approaches t o D N A [C5], H e a t t r i b u t e s to G o e t h e the r e m a r k t h a t " M a t h e m a t i c i a n s are l i k e the F r e n c h . T h e y take whatever y o u tell t h e m a n d translate i t i n t o their o w n language — a n d f r o m then o n it is s o m e t h i n g e n t i r e l y different". A l s o , i n reference to the difference between m a t h e m a t i c i a n s a n d biologists, he quotes S y l v i a Spengler, a b i o p h y s i c i s t , as s a y i n g (about m a t h e m a t i c i a n s ) t h a t " I t ' s not j u s t t h a t they are not s p e a k i n g the same language, i t ' s t h a t they are not t h i n k i n g the same w a y " . C i p r a has the final w o r d w i t h the suggestion t h a t G o e t h e ' s m a t h e m a t i c i a n m i g h t have a d d e d : " V i v e l a difference".

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T h e differences a l l u d e d to i n the above r e m a r k s are not necessarily l i m i t e d to r e l a t i v e l y n o n m a t h e m a t i c a l subjects. E v e n physicists a n d m a t h e m a t i c i a n s m a y approach the same p r o b l e m i n very different ways. A p h y s i cist m a y be able t o make an easy i n t u i t i v e j u m p to a f o r m a l m a t h e m a t i c a l result t h a t m i g h t be difficult for a m a t h e m a t i c i a n t o j u s t i f y . T h i s is not an u n u s u a l p h e n o m e n o n , two classical examples b e i n g the H e a v y s i d e C a l culus a n d the D i r a c d e l t a f u n c t i o n , b o t h o f w h i c h were, o n the surface, m a t h e m a t i c a l l y unjustified a n d only later placed o n a s o u n d m a t h e m a t i c a l basis. A n o t h e r f a m i l i a r e x a m p l e is the physicist's casual use o f divergent series a n d integrals as t h o u g h they were convergent. M o s t of these practices can e v e n t u a l l y be recast i n rigorous m a t h e m a t i c a l terms, t h o u g h sometimes o n l y after considerable effort. In such cases, as we have discussed a b o v e , a p h y s i c i s t m a y be able t o establish a s t r u c t u r a l extension by p u r e l y p h y s i cal a r g u m e n t s , w h i l e the corresponding m a t h e m a t i c a l extension m a y even require the creation of new m a t h e m a t i c s . 47. T h e Effectiveness o f M a t h e m a t i c s i n P h y s i c s H i s t o r i c a l l y , m a t h e m a t i c s and physics have always h a d a very close rel a t i o n s h i p based on the numerous 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 t o physics

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and the role of physics as a source of new m a t h e m a t i c a l ideas. A s u b s t a n t i a l p a r t o f classical m a t h e m a t i c s was i n i t i a l l y i n s p i r e d by physics, a n d the influence of physics o n the development of m a t h e m a t i c s continues, t h o u g h g r e a t l y d i m i n i s h e d by c o m p a r i s o n . In fact, despite the o n g o i n g i n j e c t i o n of ideas f r o m physics a n d other fields, the s p e c t a c u l a r g r o w t h of m a t h e m a t i c s i n m o d e r n times has been d r i v e n largely by forces i n t e r n a l t o m a t h e m a t i c s . A t the s a m e t i m e , s o m e of the newest a n d s o p h i s t i c a t e d m a t h e m a t i c a l crea t i o n s have found a p p l i c a t i o n not o n l y i n physics b u t i n c e r t a i n other areas as w e l l . T h e case o f physics, however, r e m a i n s u n i q u e . E v e n i n m o d e r n a p p l i c a t i o n s of m a t h e m a t i c s t o physics the c o n n e c t i o n between m a t h e m a t i c a l a n d p h y s i c a l structures continues to be e x t r e m e l y close. T h i s is t r u e despite the m a n y developments t h a t have o c c u r r e d i n m a t h e m a t i c s q u i t e independently of physics. T h e c o n t i n u e d existence o f such a n i n t i m a t e r e l a t i o n s h i p between t w o u l t i m a t e l y very different subjects is s o m e w h a t of a n e n i g m a . T h e m a t t e r has been discussed b y a w e l l - k n o w n p h y s i c i s t , Eugene P . W i g n e r , i n a lecture o n [W5]. (See also a lecture b y c o m p u t e r scientist, R . W . H a m m i n g [H3].) T h e W i g n e r lecture is devoted m a i n l y to a discussion of examples t h a t illustrate how very effective m a t h e m a t i c s is i n physics. A l t h o u g h he does not deal i n d e p t h w i t h the question of j u s t w h y it is so effective, some of his r e m a r k s i n the lecture are very instructive.

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"

It is true, of course, t h a t physics chooses certain m a t h e m a t i c a l concepts for the f o r m u l a t i o n of the laws of n a t u r e , a n d surely o n l y a fract i o n of a l l m a t h e m a t i c a l concepts is used i n physics. It is t r u e also t h a t the concepts w h i c h were chosen were not selected a r b i t r a r i l y f r o m a l i s t i n g of m a t h e m a t i c a l t e r m s b u t were developed, i n m a n y i f not most cases, i n d e p e n d e n t l y b y the physicist a n d recognized t h e n as h a v i n g been conceived before by the m a t h e m a t i c i a n . It is not true, however, as is so often s t a t e d , t h a t this h a d t o h a p p e n because m a t h e m a t i c s uses the s i m p l e s t possible concepts and these were b o u n d t o occur i n any f o r m a l i s m . A s we saw before, the concepts of m a t h e m a t i c s are not chosen for their conceptual s i m p l i c i t y ... b u t for their a m e n a b i l i t y t o clever m a n i p u l a t i o n s a n d to s t r i k i n g , b r i l l i a n t a r g u m e n t s . ... It is difficult t o a v o i d the impression t h a t a m i r a c l e confronts us here, q u i t e c o m p a r a b l e i n its s t r i k i n g n a t u r e t o the m i r a c l e t h a t the h u m a n m i n d c a n s t r i n g a t h o u s a n d arguments together w i t h o u t g e t t i n g itself i n t o c o n t r a d i c t i o n s or t o the t w o m i r a c l e s of the existence of laws of n a t u r e a n d o f the h u m a n m i n d ' s c a p a c i t y to d i v i n e t h e m . T h e o b s e r v a t i o n w h i c h comes closest to an e x p l a n a t i o n for the m a t h e m a t i c a l concepts' c r o p p i n g up i n physics w h i c h I k n o w is E i n stein's statement t h a t the o n l y p h y s i c a l theories w h i c h we are w i l l i n g

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to accept are the b e a u t i f u l ones. It stands to argue t h a t the concepts of m a t h e m a t i c s , w h i c h i n v i t e the exercise of so m u c h w i t , have the q u a l i t y of beauty, [p.7] A possible e x p l a n a t i o n of the physicists's use o f m a t h e m a t i c s to f o r m u l a t e his laws of nature is t h a t when he finds a c o n n e c t i o n between two q u a n t i t i e s w h i c h resembles a connection w e l l - k n o w n f r o m m a t h e m a t i c s , he w i l l j u m p at the conclusion t h a t the c o n n e c t i o n is t h a t discussed i n m a t h e m a t i c s s i m p l y because he does not k n o w of any other s i m i l a r connection. ... However, i t is i m p o r t a n t to p o i n t out t h a t the m a t h e m a t i c a l f o r m u l a t i o n of the p h y s i c i s t ' s often crude experience leads i n an u n c a n n y n u m b e r of cases t o an a m a z i n g l y acc u r a t e d e s c r i p t i o n of a large class of p h e n o m e n a . T h i s shows t h a t the m a t h e m a t i c a l language has more to c o m m e n d i t t h a n b e i n g the o n l y language w h i c h we can speak; i t shows t h a t it is, i n a very real sense, the correct language, [p. 8] W i g n e r goes o n t o discuss a d d i t i o n a l e x a m p l e s and some p h i l o s o p h i c a l questions i n physics. A l t h o u g h we cannot pursue the m a t t e r , it is w o r t h w h i l e t o m e n t i o n one of the questions because it is relevant to the p r o b l e m t h a t interests us here. T h e question is purely p h i l o s o p h i c a l a n d asks w h y , i n the presence of the o v e r w h e l m i n g c o m p l e x i t y of the w o r l d , there exists the r e m a r k a b l e r e g u l a r i t y expressed so efficiently i n the laws of n a t u r e ? W e w i l l not a t t e m p t t o deal w i t h the general question here, t h o u g h s t r u c t u r a l i s m w o u l d o b v i o u s l y have some b e a r i n g o n i t , b u t w i l l t r y i n s t e a d to give a s t r u c t u r a l e x p l a n a t i o n of the narrower p r o b l e m as t o w h y m a t h e m a t i c a l m e t h o d s are so u n i v e r s a l l y effective i n physics. Despite s o m e over s i m p l i f i c a t i o n , the a p p r o a c h s h o u l d shed l i g h t o n the p r o b l e m a n d remove some of the m y s t e r y . A s we see i t , the p r o b l e m may be expressed i n the f o r m of a q u e s t i o n : " W h y is it so often possible for the physicist t o find e x a c t l y the m a t h e m a t ical tools he needs a m o n g the m a n y c o n t r i b u t i o n s to m a t h e m a t i c s , when the c r e a t i o n of those tools was m o t i v a t e d b y forces t h a t h a d n o t h i n g to do with physics?" T h e general p o i n t raised by the question, v i z . , the seeming irrelevance of a b s t r a c t m a t h e m a t i c s t o the p h y s i c a l w o r l d , is already covered by the a s s u m p t i o n , stated i n Section 6, t h a t a s t r u c t u r e is u l t i m a t e l y " a n abstract entity t h a t exists q u i t e independently o f any concrete s i t u a t i o n i n w h i c h i t m i g h t be p e r c e i v e d " . T h i s ( P l a t o n i c ) a s s u m p t i o n applies t o s t r u c t u r e s f r o m b o t h physics a n d m a t h e m a t i c s , thereby r e m o v i n g any to t h e i r c o m p a t i b i l i t y . E v e n w i t h c o m p a t i b i l i t y , however, the m a i n p o i n t of the question r e m a i n s , because o f the independent o r i g i n s of p h y s i c a l a n d m a t h e m a t i c a l s t r u c t u r e s . In other words, " W h y do p h y s i c a l s t r u c t u r e s t u r n out so often t o be i s o m o r p h i c t o m a t h e m a t i c a l s t r u c t u r e s ? " A possible

jection

philosophical ob-

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answer t o the question depends o n two observations. R e c a l l first t h a t the contact between physics (or n a t u r a l p h i l o s o p h y ) a n d m a t h e m a t i c s a c t u a l l y b e g a n w i t h the representation o f p h y s i c a l space b y E u c l i d e a n geometry, a l o n g w i t h the u l t i m a t e dependence o f m e a s u r e m e n t o n n u m e r i c a l s t r u c t u r e . T h i s m e a n s t h a t the two fields were e x t r e m e l y close at the very beginnings of t h e i r existence. I n a d d i t i o n , as already m e n t i o n e d , physics has always depended h e a v i l y o n m a t h e m a t i c s a n d m u c h o f the e a r l y development of m a t h e m a t i c s was i n s p i r e d b y i t s contacts w i t h physics. Observe next t h a t a characteristic p r o p e r t y o f the s y s t e m o f m a t h e m a t i c a l structures is t h a t i t is " s t r u c t u r a l l y d e t e r m i n i s t i c " , i n the sense t h a t s u b s t a n t i a l p o r t i o n s of the s y s t e m are derivable f r o m r e l a t i v e l y s m a l l subsystems. P h y s i c s is s i m i l a r l y d e t e r m i n i s t i c w i t h s u b s t a n t i a l p o r t i o n s derivable f r o m a few basic laws or p r i n c i p l e s . F u r t h e r m o r e , the l a t t e r is not s i m p l y a reflection o f the extensive use of m a t h e m a t i c s i n physics, b u t r a t h e r is w h a t m a k e s t h a t use possible. O n the basis of these observations, a n answer t o the question m a y n o w be f o r m u l a t e d . T h e i d e a is t h a t the b u l k of m a t h e m a t i c a l s t r u c t u r e s , t h o u g h i n d e p e n d e n t l y c o n s t r u c t e d , are u l t i m a t e l y d e t e r m i n e d d i r e c t l y or i n d i r e c t l y b y m a t h e m a t i c s already associated w i t h physics. Because o f p a r a l l e l determ i n a t i o n s i n physics, i t is reasonable t o conjecture t h a t the derived m a t h e m a t i c a l structures m i g h t also a d m i t connections t o (possibly yet u n d i s covered) p h y s i c a l structures. S i m i l a r l y , a g i v e n (new) p h y s i c a l s t r u c t u r e is l i k e l y to be d e t e r m i n e d b y physics already associated w i t h m a t h e m a t i c s , so m i g h t be expected i t s e l f t o be connected t o a m a t h e m a t i c a l s t r u c t u r e (also perhaps not yet discovered) d e t e r m i n e d b y the associated m a t h e m a t i c s .

given

isomorphism physical structures

determines

determines

desired isomorphism

Fig.

- t —>

math, structures

M

47.1

A s i m p l e version of the r e l a t i o n s h i p between physics a n d m a t h e m a t i c s o u t l i n e d above is represented s c h e m a t i c a l l y i n F i g u r e 4 7 . 1 , i n w h i c h the

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d o t t e d lines i n d i c a t e conjectured objects. P is a p h y s i c a l s t r u c t u r e repres e n t i n g a new p h y s i c a l result t h a t may have been suggested by an a c t u a l e x p e r i m e n t or a t h o u g h t e x p e r i m e n t or a n i n t u i t i v e i n s i g h t , b u t has yet t o be g i v e n a m a t h e m a t i c a l t r e a t m e n t . P' is a conjectured p h y s i c a l s t r u c ture assumed t o determine P ( w i t h i n physics) a n d already k n o w n to be i s o m o r p h i c w i t h a m a t h e m a t i c a l s t r u c t u r e M'. T h e i d e a then is t h a t , u n der these circumstances, M' s h o u l d determine a m a t h e m a t i c a l s t r u c t u r e M i s o m o r p h i c w i t h P. A s W i g n e r p o i n t s o u t , one does not find a s t r u c t u r e M by m a k i n g a r b i t r a r y choices f r o m a l i s t i n g of m a t h e m a t i c a l structures. I n fact, the required s t r u c t u r e m a y not exist, so our physicist m a y be forced t o p l a y the role of a m a t h e m a t i c i a n and a t t e m p t t o create a new piece o f m a t h e m a t i c s . If the s t r u c t u r e P' does not determine P , t h e n the o u t c o m e w i l l be less clear, a n d a s o l u t i o n , i f it exists, may be considerably m o r e difficult t o c o n s t r u c t . F o r e x a m p l e , it may be necessary t o t r y other choices for P'. W h a t e v e r the s i t u a t i o n m i g h t be, the g o a l , despite the u n c e r t a i n t y , is at least as well-defined as is frequently the case i n routine m a t h e m a t i c a l research. W e finish off this section w i t h a f e w a d d i t i o n a l c o m m e n t s suggested the special r e l a t i o n s h i p t h a t exists between physics and m a t h e m a t i c s . -

by

O n e consequence of the u n d e r l y i n g s t r u c t u r a l s i m i l a r i t y between the t w o fields is i m p l i c i t i n the r e m a r k at the end of the preceding section. It is the fact t h a t , up to a p o i n t , p h y s i c a l i n t u i t i o n m a y be quite adequate to s u p p o r t the f o r m a l i s m of the c o r r e s p o n d i n g m a t h e m a t i c s . A s i m p l e e x a m p l e is the practice i n elementary calculus of i n t e r p r e t i n g the d e r i v a t i v e i n t e r m s of velocity T h i s was a useful pedagogical device w h e n students b r o u g h t to C a l c u l u s a previous experience w i t h elementary physics. E v e n so, the der i v a t i v e concept needed eventually to be separated f r o m v e l o c i t y because derivatives arise i n so m a n y other contexts. B e y o n d this, i n d i c a t i o n s are t h a t s u b s t a n t i a l p o r t i o n s of some p h y s i c i s t s ' u n d e r s t a n d i n g of m a t h e m a t ics m a y a c t u a l l y be supported by p h y s i c a l concepts rather t h a n the u s u a l m a t h e m a t i c a l structures. Therefore, the presumed s e p a r a t i o n o f physics and m a t h e m a t i c s m a y not always be as complete as we sometimes take for granted. T h e above r e m a r k s also suggest another "measure" of the significance of a n a p p l i c a t i o n of m a t h e m a t i c s , n a m e l y , the degree to w h i c h the relevant m a t h e m a t i c a l f o r m a l i s m is supported by the i n t u i t i o n of an e x p e r t i n the field of a p p l i c a t i o n . A discussion o f the problems raised i n this section w o u l d not be c o m p l e t e w i t h o u t at least a b r i e f r e m a r k o n E i n s t e i n ' s s t a t e m e n t , recalled b y W i g n e r , t h a t o n l y b e a u t i f u l theories are acceptable i n physics, a n d its i m p l i c a t i o n t h a t those concepts f r o m m a t h e m a t i c s chosen by physics for the f o r m u l a t i o n

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of the l a w s of nature m u s t be b e a u t i f u l . M a t h e m a t i c i a n s w i l l often refer to c e r t a i n p u r e l y m a t h e m a t i c a l results as " b e a u t i f u l " a n d describe certain proofs a s " e l e g a n t " . O n the other h a n d , a l t h o u g h there is u s u a l l y b r o a d agreement a m o n g m a t h e m a t i c i a n s a b o u t such j u d g e m e n t s , there is very l i t t l e i n d i c a t i o n or a n a l y s i s of precisely w h a t such t e r m s m i g h t m e a n as a p p l i e d t o m a t h e m a t i c s . W h a t is there a b o u t a g i v e n m a t h e m a t i c a l s t r u c t u r e , or its e x t e n s i o n , t h a t sets i t a p a r t as b e a u t i f u l or elegant? W i g n e r concludes t h a t the m a t h e m a t i c a l concepts are chosen i n physics, not for their s i m p l i c i t y , as is often c l a i m e d , b u t "for their a m e n a b i l i t y to clever m a n i p u l a t i o n s and t o s t r i k i n g , b r i l l i a n t a r g u m e n t s " , a r g u i n g " t h a t concepts, w h i c h i n v i t e the exercise o f so m u c h w i t , have the q u a l i t y of b e a u t y " . T h i s suggests t h a t the beauty i n m a t h e m a t i c a l concepts depends on the i n t e n s i t y of the i n t e l l e c t u a l (or creative) experience i n d u c e d b y t h e m . If we a d d a requirement t h a t the concepts be significant (that is, have a s u b s t a n t i a l c o n n e c t i o n to the m a i n b o d y of m a t h e m a t i c s ) , the result is a reasonable d e f i n i t i o n of w h a t constitutes beauty i n m a t h e m a t i c s . A s u b jective element r e m a i n s , of course, i n j u d g i n g the i n t e l l e c t u a l experience, b u t this is more or less i n e v i t a b l e i n any d e f i n i t i o n of this t y p e . T h e close s i m i l a r i t y between the first p a r t of the " d e f i n i t i o n " a n d the p e r c e p t u a l experience associated w i t h the A l b e r s c o n s t r u c t i o n s discussed i n Section 17 is o b v i o u s l y no accident. 48.

O t h e r Applications of

Mathematics

A t one t i m e , the serious a p p l i c a t i o n s of m a t h e m a t i c s were l i m i t e d a l m o s t entirely to the p h y s i c a l sciences a n d engineering. N o w , however, useful a p p l i c a t i o n s are f o u n d i n m a n y other areas as w e l l . T h e m a i n purpose of this section is to o u t l i n e some of the differences a n d p r o b l e m s associated w i t h these n o n p h y s i c a l a p p l i c a t i o n s . It is an i m p o r t a n t fact t h a t physics is i n m a n y respects a m o d e l b y w h i c h a l l other sciences are measured. T h e r e f o r e , because of the close r e l a t i o n s h i p between physics and m a t h e m a t i c s , a p p l i c a t i o n s of m a t h e m a t i c s are generally looked u p o n as evidence of scientific content. T h i s p o i n t of v i e w fuels a d r i v e , a p p r o p r i a t e l y called "physics e n v y " , to i n t r o d u c e m a t h e m a t i c a l techniques i n t o any subject w h i c h c l a i m s or aspires t o be scientific. A l t h o u g h m a t h e m a t i c s is u n q u e s t i o n a b l y an i m p o r t a n t t o o l w h e n a p p l i c a b l e , i t s i n v o l v e m e n t is neither necessary nor sufficient for a subject to be scientific. 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 r o u t i n e i n the v a r i o u s p h y s i c a l sciences ( i n c l u d i n g engineering), m a i n l y because these subjects depend u l t i m a t e l y u p o n physics. T h e s i t u a t i o n is very different, however, as soon as we pass to other subjects, such as the s o c i a l and b e h a v i o r a l sciences or biology. M o s t

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of their development has t a k e n place quite i n d e p e n d e n t l y of m a t h e m a t i c s , so there is no general b a c k l o g of m a t h e m a t i c a l connections o n w h i c h new connections m i g h t be b u i l t . A l t h o u g h none o f this rules out the p o s s i b i l i t y of a p p l i c a t i o n s of m a t h e m a t i c s , i t reduces considerably the o p p o r t u n i t i e s for m a k i n g t h e m . For e x a m p l e , i n fields other t h a n the p h y s i c a l sciences, the a p p l i c a t i o n s t e n d to be restricted to r e l a t i v e l y s m a l l p o r t i o n s of the target m a t e r i a l w i t h l i t t l e or no p o s s i b i l i t y of extension to larger p o r t i o n s . A l t h o u g h isolated instances o f this k i n d m a y also occur i n physics, the t y p i c a l s i t u a t i o n is u s u a l l y quite different, i n t h a t an a p p r o p r i a t e m a t h e m a t i c a l s y s t e m w i l l often represent s u b s t a n t i a l p o r t i o n s o f the f i e l d . O n the other h a n d , t h a n k s to the wide v a r i e t y a n d abstractness of m a t h e m a t i c a l s t r u c t u r e s , o p p o r t u n i t i e s do exist for a p p l i c a t i o n s to areas h a v i n g no a p r i o r i connection w i t h either physics or m a t h e m a t i c s . It is not feasible for us to a t t e m p t a detailed account of specific a p p l i c a tions of m a t h e m a t i c s t o any o f the n o n p h y s i c a l sciences. T h i s w o u l d require discussions of t e c h n i c a l m a t e r i a l i n b o t h m a t h e m a t i c s a n d the target fields a n d w o u l d not c o n t r i b u t e s i g n i f i c a n t l y t o o u r m a i n objective, a deeper u n d e r s t a n d i n g of s t r u c t u r e s . Instead, therefore, our a t t e n t i o n w i l l be directed t o some of the general p r o b l e m s i n v o l v e d i n a p p l y i n g m a t h e m a t i c s to these fields as c o m p a r e d to physics. W e are interested, i n c i d e n t a l l y , o n l y i n those a p p l i c a t i o n s t h a t i n v o l v e proper use of g o o d m a t h e m a t i c s . T h i s excludes those p u r p o r t e d a p p l i c a tions i n w h i c h the m a t h e m a t i c s is either t r i v i a l or used i n c o r r e c t l y . T h e latter are i l l u s t r a t e d i n m a n y of the a t t e m p t s to use m a t h e m a t i c a l f o r m u l a s a n d s y m b o l s t o describe m a t e r i a l t h a t does not a c t u a l l y possess the necessary m a t h e m a t i c a l s t r u c t u r e . In some cases, t h i s practice m i g h t f a l l under the m e t a p h o r l a b e l , b u t the practice is often m i s l e a d i n g because i t conveys to m a n y persons a false i m p r e s s i o n of m a t h e m a t i c a l precision. It also suggests a l e g i t i m a t e scientific a p p r o a c h , w h i c h m a y or m a y not be correct. A l t h o u g h such usage is often easy t o identify, most m a t h e m a t i c i a n s , as a r u l e , have neither sufficient interest nor knowledge c o n c e r n i n g the subject m a t e r i a l to pursue the m a t t e r when it arises. T h e r e are, however, exceptions t o the r u l e . T w o of these, whose observations are relevant to the above r e m a r k s , are N e a l K o b l i t z [K2] a n d Serge L a n g [L2]. B o t h m a t h e m a t i c i a n s target a r e l a t i v e l y s m a l l p o r t i o n o f w o r k b y one p o l i t i c a l scientist, S a m u e l H u n t i n g t o n . T h e i r c r i t i c i s m s , w h i c h have i m p l i c a t i o n s t h a t e x t e n d w e l l bey o n d this p a r t i c u l a r case, have aroused m u c h controversy. T h e p r i n c i p a l d e t a i l s of the whole affair w i l l be found i n an even more c o n t r o v e r s i a l work b y C h a r l e s J . Sykes [S10], C h a p t e r 12. L a n g ' s m a n u s c r i p t u n f o r t u n a t e l y r e m a i n s u n p u b l i s h e d , but he has given i t w i d e c i r c u l a t i o n v i a an extensive m a i l i n g list. A s we k n o w , genuine a p p l i c a t i o n s depend first of a l l o n the i d e n t i f i c a -

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t i o n of m a t h e m a t i c a l type structures w i t h i n the subject field. E v e n when they exist, these structures are s e l d o m as clear-cut here as they are i n the p h y s i c a l sciences, a n d their m a t h e m a t i c a l representations u s u a l l y cannot be expected to a p p r o a c h the precision f o u n d i n the l a t t e r . T h i s is the o r i g i n of most of the p r o b l e m s encountered i n a p p l y i n g m a t h e m a t i c s o u t s i d e the p h y s i c a l sciences. In order to construct a representation, i t is often necessary to m a k e compromises. In e x t r e m e cases, this c a n result i n a p p l i c a t i o n s t h a t are forced a n d l a c k i n g i n c r e d i b i l i t y . C o m p r o m i s e s m a y take the f o r m of unverified a s s u m p t i o n s concerning the given s t r u c t u r e , or the disregard of t e c h n i c a l requirements essential for the m a t h e m a t i c s t o m a k e any sense. S i m i l a r compromises m a y also occur i n a p p l i c a t i o n s to the p h y s i c a l sciences, b u t , t h a n k s t o the m a n y connections w i t h m a t h e m a t i c s t h a t already exist, the chances are far better t h a t they w i l l eventually be j u s t i f i e d . Because m a n y of the most effective m a t h e m a t i c a l tools i n physics i n v o l v e techniques f r o m the field of a n a l y s i s , the l a t t e r is seen as a p r i m e source of m a t e r i a l for a p p l i c a t i o n s to other fields. T h e i d e a i n a n a p p l i c a t i o n of a n a l ysis to any subject is to i d e n t i f y the m a t h e m a t i c a l variables w i t h subject p a r a m e t e r s , so the given f u n c t i o n relations a m o n g the variables correspond t o m e a n i n g f u l relations a m o n g the parameters, thus e s t a b l i s h i n g an isom o r p h i s m between a m a t h e m a t i c a l s t r u c t u r e and a c o r r e s p o n d i n g subject s t r u c t u r e . In a d d i t i o n , values of the variables are u s u a l l y real n u m b e r s to w h i c h there s h o u l d correspond m e a n i n g f u l measurements of the associated parameters, it is t h i s requirement t h a t accounts i n p a r t for the pressure to i n t r o d u c e q u a n t i t a t i v e methods i n t o a subject. A n i m p o r t a n t p r o p e r t y of the real numbers is t h a t they are " s i m p l y o r d e r e d " w i t h respect to the u s u a l n o t i o n of "less t h a n " ( S e c t i o n 22). T h i s is a s p e c i a l case o f the f o l l o w i n g more inclusive n o t i o n of " p a r t i a l " order: A n y set o f objects is s a i d to be p a r t i a l l y ordered w i t h respect to an order r e l a t i o n " < " (read, " i s less t h a n " ) p r o v i d e d i t satisfies the f o l l o w i n g two c o n d i t i o n s : (1) T h e r e exist pairs of ( d i s t i n c t ) o b j e c t s , say x a n d j / , such t h a t x < y or y < x, but not b o t h . ( T h i s is called (2) If three d i s t i n c t o b j e c t s , say and z, are such t h a t x < y and y < z, then also x < z. ( T h i s is c a l l e d

"antisymmetry".)

x,y,

"transitivity".)

T h e t e r m " p a r t i a l " refers to the fact t h a t not a l l pairs are required to be c o m p a r a b l e w i t h respect to " < " . I n case a l l pairs are c o m p a r a b l e , " < " is c a l l e d a N u m e r o u s examples of p a r t i a l and s i m p l e orderings occur i n m a t h e m a t i c s , often subject t o a variety of a d d i t i o n a l c o n d i t i o n s , as i n the case o f the real numbers.

"simple ordering".

In m a n y contexts, there are properties t h a t m a y v a r y f r o m one object to another, or f r o m one t i m e to another, so t h a t it is m e a n i n g f u l to c o m -

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pare the different occurrences w i t h respect to some n o t i o n of " m a g n i t u d e " . F o r e x a m p l e , one m a n u f a c t u r e d i t e m m a y be of higher " q u a l i t y " or m o r e " d e s i r a b l e " t h a n another, so one occurrence m a y be described as "greater t h a n " another w i t h respect t o the d i s t i n g u i s h e d property. I n d e a l i n g w i t h properties of t h i s type, under pressure to i n t r o d u c e measurement wherever possible, the first tendency is to identify the comparisons w i t h an o r d e r i n g of real n u m b e r s . In other words, the i d e a is t o associate w i t h each o c c u r rence of the p r o p e r t y a real n u m b e r , i n d i c a t i n g " m a g n i t u d e " , so t h a t the n a t u r a l o r d e r i n g of the real n u m b e r s expresses c o m p a r a b i l i t y . If successful, the result is a s i m p l e e x a m p l e of a m a t h e m a t i c a l representation ( i n v o l v i n g the order s t r u c t u r e of the real n u m b e r s ) , a n d is one m e t h o d of i n t r o d u c i n g numerical parameters. A p r o b l e m , w h i c h is always present i n a t t e m p t s t o i n t r o d u c e real n u m bers i n this way, is to ensure t h a t the required association is a c t u a l l y a m e a n i n g f u l one. F o r e x a m p l e , it is not always clear t h a t it is even possible to i d e n t i f y the desired c o m p a r a b i l i t y w i t h a s i m p l e o r d e r i n g . In such cases, some type of p a r t i a l o r d e r i n g m i g h t be more m e a n i n g f u l . T h e l a t t e r , of course, m a y not p r o v i d e the desired q u a n t i f i c a t i o n . F u r t h e r m o r e , even when an i d e n t i f i c a t i o n is successful, there is no assurance t h a t it involves any m o r e t h a n j u s t the order s t r u c t u r e of the n u m b e r s . In other words, one c a n not a u t o m a t i c a l l y assume, j u s t because the order s t r u c t u r e applies, t h a t i t is legitimate t o e x p l o i t the r e m a i n i n g structures of the real n u m b e r s y s t e m . A n y appeal to these s t r u c t u r e s m u s t be preceded by their proper i d e n t i f i c a t i o n w i t h i n the subject s t r u c t u r e . T h e p r o b l e m is c o m p o u n d e d when several n u m e r i c a l p a r a m e t e r s are i n v o l v e d , a n d i t is desired, for exa m p l e , to a d d , m u l t i p l y , or d i v i d e parameters. Neglect of t h i s s t r u c t u r a l l y obvious rule, is the source of m a n y f o o l i s h " 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 . A s s u m e now t h a t we have a subject s t r u c t u r e consisting of several p a r a m eters a l o n g w i t h some relations a m o n g t h e m . F o r e x a m p l e , t w o p a r a m e t e r s m a y be related so t h a t one of t h e m decreases w h e n the other increases. T h e goal t h e n is to represent t h i s s t r u c t u r e m a t h e m a t i c a l l y b y a s s o c i a t i n g w i t h each p a r a m e t e r a real variable and w i t h each p a r a m e t e r r e l a t i o n a f u n c t i o n r e l a t i o n i n v o l v i n g the c o r r e s p o n d i n g variables. A t t e m p t s are o c c a s i o n a l l y m a d e to carry out the above p r o g r a m w i t h out first e s t a b l i s h i n g t h a t the parameters a c t u a l l y do take values i n the real n u m b e r s . In such a case, an a p p e a l to the m a t h e m a t i c a l s t r u c t u r e c o u l d be q u i t e meaningless. E v e n w i t h real parameters, there m a y s t i l l be p r o b lems because of restrictions on the m a t h e m a t i c s . F o r e x a m p l e , a c o m m o n r e q u i r e m e n t i n a n a l y s i s is t h a t the range of values of a v a r i a b l e be an ino f real numbers, w h i l e the associated p a r a m e t e r m i g h t take values i n a discrete set, say, the integers. A l s o , functions are often required to be continuous or even differentiable, c o n d i t i o n s t h a t m a y not c o r r e s p o n d

terval

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STRUCTURES

t o a n y t h i n g m e a n i n g f u l i n the subject s t r u c t u r e . W h e n there are discrepancies o f this k i n d , i t is obvious t h a t o n l y a p o r t i o n of the m a t h e m a t i c s s t r u c t u r e m a y a c t u a l l y enter i n t o the p i c t u r e . In the most e x t r e m e cases, the correspondence between s t r u c t u r e s m a y be reduced t o l i t t l e m o r e t h a n a weak analogy. N o t e t h a t discreteness, w h i c h m a y cause p r o b l e m s i n some cases, also comes up i n physics, for e x a m p l e , t h r o u g h the a t o m i c s t r u c t u r e o f m a t t e r a n d q u a n t u m effects. In classical a p p l i c a t i o n s , however, t h i s discreteness is u s u a l l y not observable under o r d i n a r y c i r c u m s t a n c e s , a l l o w i n g the i n t r o d u c t i o n of parameters h a v i n g continuous ranges. A g o o d e x a m p l e is p r o v i d e d by the gas l a w , PV = cT (c a constant), w h i c h relates the pressure, v o l u m e , a n d t e m p e r a t u r e of an " i d e a l " gas. It ignores the (discrete) m o l e c u l a r c o m p o s i t i o n o f an a c t u a l gas, but a p p r o x i m a t e s closely its g l o b a l b e h a v i o r . A s t u d y of gases w h i c h does take i n t o account t h e i r m o l e c u l a r c o m p o s i t i o n requires the use of " s t a t i s t i c a l m e c h a n i c s " . M o s t o f the above r e m a r k s are suggested b y a p p l i c a t i o n s of a n a l y s i s i n one f o r m or another. T h e s i t u a t i o n is somewhat different i n a p p l i c a t i o n s of a l g e b r a , for e x a m p l e , w h i c h does not make the same d e m a n d s for q u a n t i f i c a t i o n a n d encourages greater e m p h a s i s on general s t r u c t u r a l properties. A l t h o u g h a p p l i c a t i o n s of a l g e b r a may be less impressive t h a n some a p p l i c a tions of a n a l y s i s , perhaps because they m a y f a i l to p r o v i d e the d e m a n d for q u a n t i f i c a t i o n , the a p p r o a c h can be less forced a n d more n a t u r a l f r o m the s t r u c t u r a l p o i n t of v i e w . N o t e t h a t P i a g e t ' s s t r u c t u r a l a p p r o a c h to p s y c h o l o g y is entirely t h r o u g h algebra. A l s o , anthropologists have used algebraic s t r u c t u r e to represent k i n s h i p r e l a t i o n s , a n d economists a p p e a l r e g u l a r l y t o a l g e b r a , convexity theory, game theory, a n d even t o p o l o g y (as w e l l as analysis). D e s p i t e the difficulties associated w i t h a p p l i c a t i o n s outside the p h y s i c a l sciences, some very sophisticated m a t h e m a t i c s has been a p p l i e d t o subjects t h a t a p p a r e n t l y do not satisfy a l l o f the c o n d i t i o n s required by the m a t h ematics. Y e t , the m a t h e m a t i c a l f o r m u l a t i o n s often represent, or at least s t r o n g l y suggest, the p h e n o m e n a of interest s u r p r i s i n g l y w e l l . Nevertheless, w h e n the deeper s t r u c t u r a l connections are a c t u a l l y m i s s i n g , the represent a t i o n tends t o be reduced t o mere d e s c r i p t i o n of a very l i m i t e d fragment of the s u b j e c t . P e r h a p s because of this fact, s o m e a p p l i c a t i o n s appear t o have d e s c r i p t i o n as the m a i n g o a l , a n d regard m a t h e m a t i c a l s o p h i s t i c a t i o n as a measure o f their "significance". T h e success of such representations p r o b a b l y depends more o n the variety a n d f l e x i b i l i t y o f 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 n o n a serious c o n n e c t i o n w i t h m a t h e m a t i c s . T h o u g h certain of these results m a y be useful, i t is fair to ask i n some cases i f they c o n t r i b u t e more to a n u n d e r s t a n d i n g and development of a subject t h a n w o u l d a careful e x p o s i t i o n by an expert.

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U n d e r l y i n g any concrete a p p l i c a t i o n of m a t h e m a t i c s is the q u e s t i o n of how a c c u r a t e l y the a p p l i c a t i o n reflects the " r e a l w o r l d " w h i c h it p u r p o r t s t o represent. A s we already k n o w , one test of this is p r e d i c t a b i l i t y . T h e r e m a r k s i n the preceding p a r a g r a p h s suggest a couple o f reasons w h y an a p p l i c a t i o n m a y give false p r e d i c t i o n s . In the first place, i f o n l y a p o r t i o n o f the m a t h e m a t i c a l s t r u c t u r e is i n v o l v e d i n a representation, t h e n it is p l a u s i ble t h a t some results deduced f r o m the m a t h e m a t i c s m a y f a i l to c o r r e s p o n d to any subject properties. A t the other e x t r e m e , a representation m a y be r e l a t i v e l y complete as far as the m a t h e m a t i c a l s t r u c t u r e is concerned, but f a i l t o involve a significant p o r t i o n of the subject s t r u c t u r e . I n fact, a s u b ject m a y be so c o m p l e x a n d involve so m a n y independent p a r a m e t e r s t h a t it is p r a c t i c a l l y i m p o s s i b l e t o represent a significant p o r t i o n of its s t r u c t u r e b y a reasonable m a t h e m a t i c a l m o d e l . Here again i t is easy to u n d e r s t a n d w h y predictions m a y f a i l . I n this c o n n e c t i o n , the reader is referred to a r e p o r t by G i n a K o l a t a p u b l i s h e d i n Science [ K 3 ] , which contains some i n t e r e s t i n g observations f r o m several economists. P e r h a p s the most reveali n g , because it suggests the k i n d of p r o b l e m s i n v o l v e d , is the r e m a r k t h a t " T h e r e are t w o t h i n g s y o u are better off not seeing i n the m a k i n g - sausages a n d econometric e s t i m a t e s " .

"Asking Impossible Questions About the Economy and Getting Impossible Answers",

A l t h o u g h representation deficiencies of the k i n d o u t l i n e d above w i l l account for m a n y of the p r e d i c t i o n p r o b l e m s associated w i t h r o u t i n e a p p l i cations of m a t h e m a t i c s , they do not cover a l l p o s s i b i l i t i e s . P r e d i c t i o n m a y also f a i l for a s y s t e m w h i c h a d m i t s " c h a o t i c " b e h a v i o r . T h e l a t t e r is a p h e n o m e n o n associated i n its purest f o r m w i t h c e r t a i n d e t e r m i n i s t i c systems w h i c h a d m i t precise m a t h e m a t i c a l f o r m u l a t i o n s a n d e x h i b i t i n i t i a l l y well-defined regular b e h a v i o r . E v e n t u a l l y , however, the r e g u l a r i t y r a p i d l y breaks d o w n a n d the s y s t e m falls i n t o a state of u n p r e d i c t a b l e chaotic beh a v i o r . T h e p h e n o m e n o n , w h i c h can occur i n r e l a t i v e l y s i m p l e systems, is not due t o incompleteness of the m a t h e m a t i c a l representation. It occurs because the s y s t e m is e x t r e m e l y sensitive t o m i n u t e changes i n values of the s y s t e m parameters. A change (or " e r r o r " ) grows e x p o n e n t i a l l y a n d results e v e n t u a l l y i n a t o t a l loss of s t a b i l i t y a n d regularity. C h a o s is by n o means a new t o p i c of s t u d y . It was considered i n connect i o n w i t h d y n a m i c a l systems by P o i n c a r e , and has been discussed r e g u l a r l y since then by m a n y other m a t h e m a t i c i a n s . R e c e n t l y , however, there has occurred a m a j o r upsurge of interest i n the subject, p a r t l y because of the general a v a i l a b i l i t y of h i g h speed computers equipped w i t h large memories, w h i c h make it possible to m o d e l the relevant systems o n a c o m p u t e r a n d to e x p e r i m e n t w i t h t h e m . T h e r e is also an increasing awareness of the w i d e occurrence of chaotic behavior. It may be observed i n v i r t u a l l y a l l fields, r a n g i n g f r o m m a t h e m a t i c s t h r o u g h the p h y s i c a l a n d b i o l o g i c a l sciences to

14-1

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the s o c i a l sciences. F o r e x a m p l e , i t e x p l a i n s i n p a r t the u n r e l i a b i l i t y of l o n g range weather forecasts as w e l l as e c o n o m i c forecasts. B o t h of these cases, however, are also very c o m p l e x and depend o n large n u m b e r s of p a r a m e ters, a m a j o r p r o b l e m i n itself. A n excellent account of chaotic p h e n o m e n a w i l l be found i n a b o o k by J a m e s G l e i c k [G5]. It bears the t i t l e , a n d is directed to the general reader.

making a new science",

"Chaos:

T h e r e is one r e m a i n i n g point t h a t we w i s h to m a k e c o n c e r n i n g a p p l i c a tions o f m a t h e m a t i c s . T h i s is the fact t h a t such a p p l i c a t i o n s m a y shape a n d even define the content of a subject. P e r h a p s the most e x t r e m e exa m p l e is m a t h e m a t i c a l economics, w h i c h is so d o m i n a t e d b y m a t h e m a t i c a l techniques t h a t it often appears as m u c h a b r a n c h of m a t h e m a t i c s as of economics, t h o u g h i t is s e l d o m evaluated as s u c h . T h i s tends to separate it f r o m economics proper, a n d perhaps e x p l a i n s some o f the " i m p o s s i b l e answers" o b t a i n e d i n a t t e m p t s t o a p p l y the subject t o everyday e c o n o m ics. T h e m a t t e r has also d r a w n c r i t i c i s m f r o m some t r a d i t i o n a l economists, i n c l u d i n g J o h n K e n n e t h G a l b r a i t h [ G l ] a n d W a s s i l y L e o n t i e f [L4]. It is i n s t r u c t i v e t o c o m p a r e the case of economics w i t h t h a t of physics. M a t h e m a t i c a l physics is also d o m i n a t e d b y m a t hematics, and some of the " f a r - o u t " theories i n m o d e r n physics are v i r t u a l l y i n d i s t i n g u i s h a b l e f r o m m a t h e m a t i c s . B e y o n d t h i s , one can even argue t h a t a l l of physics is u l t i m a t e l y d e t e r m i n e d by its a m e n a b i l i t y to m a t h e m a t i c a l techniques. T h e r e is, however, a c r u c i a l difference i n this case. T h e entire content of physics depends i n the end o n observations of the m a t e r i a l w o r l d . A n d , a l t h o u g h it is true t h a t the content m a y be d e t e r m i n e d by m a t h e m a t i c s , the d e t e r m i n i n g process is one of selection rather t h a n d e f i n i t i o n . A n y p h y s i c a l theory, regardless of its m a t h e m a t i c a l beauty or s o p h i s t i c a t i o n , w i l l u l t i m a t e l y s t a n d or f a l l d e p e n d i n g o n its success i n e x p l a i n i n g a n d p r e d i c t i n g a c t u a l p h y s i c a l observations. N o p h y s i c a l theory w o u l d l o n g s u r v i v e i f i t were c o n t r a d i c t e d by a n e x p e r i m e n t or gave impossible answers to real w o r l d questions.

CHAPTER

BIOLOGICAL

49.

VIII

STRUCTURES

Introduction

In biology, as i n a l l scientific fields, notions of s t r u c t u r e are everywhere present. A l t h o u g h structures are often not the d i r e c t objects of a t t e n t i o n i n science, they t e n d to lie r e l a t i v e l y near the surface. T h i s is especially t r u e i n biology, where s t r u c t u r e stands out i n m u c h of the m a t e r i a l a n d is often dealt w i t h e x p l i c i t l y . S t r u c t u r e s are evident at a l l levels, r a n g i n g f r o m m o l e c u l a r biology, where they are perhaps m o s t e x p l i c i t a n d merge into purely chemical and physical structures, through traditional biology to p o p u l a t i o n a n d sociobiology, where they blend i n t o the general s t r u c t u r e s associated w i t h s o c i a l p h e n o m e n a . C e n t r a l i n a l l of this is the h u m a n b r a i n . It is perhaps the most c o m p l e x o f a l l the structures t h a t we ever encounter, a n d is c a p a b l e , i n a sense, of m o d e l i n g a l l of the others. A comprehensive s t r u c t u r a l analysis of any p a r t of biology w o u l d o b v i ously be interesting s t r i c t l y f r o m the p o i n t of v i e w of s t r u c t u r e s , b u t w o u l d be a difficult task even for a biologist, a n d p r o b a b l y not of great i m p o r tance to biology proper. Needless to say, o u r i m m e d i a t e o b j e c t i v e , as i n other areas t h a t we have considered, is not to a t t e m p t a serious s t r u c t u r a l analysis o f any p o r t i o n of the subject, but r a t h e r t o t r y to l e a r n more a b o u t general structures by e x a m i n i n g some o f the special ways t h a t they enter i n t o biology. A s t r u c t u r a l approach of this k i n d , t h o u g h it m a y not be r e v o l u t i o n a r y i n effect, does throw a different l i g h t o n some of the topics to w h i c h it is a p p l i e d . It is also obvious t h a t i t w i l l influence the choice of m a t e r i a l to be e x a m i n e d . In the case of biology, the result is a p r e p o n derance o f topics f r o m the theory o f e v o l u t i o n . T h e subject of e v o l u t i o n is p a r t i c u l a r l y a p p r o p r i a t e here since m u c h of its m a t e r i a l is f a i r l y accessible a n d also d i r e c t l y concerned w i t h s t r u c t u r e . A n excellent source of m a t e r i a l is the series o f elegant essays by S t e p h e n J a y G o u l d t h a t have appeared r e g u l a r l y for years i n under the general t i t l e , M a n y have been collected a n d p u b l i s h e d (often w i t h different titles) i n b o o k f o r m [G6],[G7],[G8]. M o s t o f the essays are devoted t o i n t r i g u i n g a n d easily u n d e r s t o o d questions conc e r n i n g e v o l u t i o n , and c o n t a i n n u m e r o u s i l l u s t r a t i o n s of b i o l o g i c a l s t r u c tures. T h e y have p r o v i d e d the i n s p i r a t i o n for m u c h of w h a t follows. A l s o ,

zine

"This View of Life".

145

Natural History Maga-

146

STRUCTURALISM

AND STRUCTURES

since e v o l u t i o n i s t s u s u a l l y assume a more or less e x p l i c i t s t r u c t u r a l p o i n t of view i n d e a l i n g w i t h their s u b j e c t , m a n y o f the s t r u c t u r a l ideas t h a t are e x t r a c t e d f r o m t h i s m a t e r i a l , t h o u g h f o r m u l a t e d i n general t e r m s , w i l l be f a m i l i a r i n one f o r m or another t o t h e m . 50.

Classification of

Organisms

S y s t e m a t i c s , the classification of b i o l o g i c a l o r g a n i s m s a c c o r d i n g to t h e i r s i m i l a r i t i e s , was at one t i m e more or less coincident w i t h the field of general biology, a n d continues to be a n i m p o r t a n t p a r t of the s u b j e c t . A l t h o u g h systematics tends t o be more or less descriptive i n n a t u r e , i t has t a k e n o n a m o r e theoretical character under the influence of the theory of evol u t i o n . A t the same t i m e , those features of o r g a n i s m s t h a t are considered i n j u d g i n g s i m i l a r i t y are c o n s t a n t l y b e i n g e x p a n d e d , a n d m a y range f r o m r e l a t i v e l y superficial appearances, t h r o u g h more s u b t l e a n a t o m i c a l a n d beh a v i o r a l characteristics, to the c h e m i s t r y of c e r t a i n b i o l o g i c a l processes a n d the s t r u c t u r e of the genome. It is obvious t h a t the classification p r o b l e m is not a s i m p l e one. F r o m an abstract p o i n t of v i e w , a s y s t e m of classification m i g h t be based o n any agreed-upon sets of s i m i l a r i t y c r i t e r i a , i n t e r m s of w h i c h each o r g a n i s m c o u l d , i n p r i n c i p l e , be assigned to one group or a n other. I n a c t u a l p r a c t i c e , of course, no s y s t e m of classification is ever this arbitrary. A n y set o f s i m i l a r i t y features a n d t h e i r r e l a t i o n s h i p s w i l l c o n s t i t u t e a s t r u c t u r e a c c o r d i n g to our general d e f i n i t i o n . A l s o , any o r g a n i s m t h a t exh i b i t s these features " c o n t a i n s " , b y d e f i n i t i o n , a representation o f the s t r u c t u r e . T h e r e f o r e , the class associated w i t h a set of s i m i l a r i t i e s consists precisely o f a l l those o r g a n i s m s t h a t c o n t a i n representations of the s t r u c t u r e . N o t i c e , however, t h a t various representations of a s t r u c t u r e m a y a p p e a r o n the surface t o be q u i t e different. F o r e x a m p l e , an appendage, such as a leg, m a y v a r y g r e a t l y a m o n g a group of a n i m a l s , either i n size, relative d i m e n s i o n s , or even f u n c t i o n , a n d yet be identifiable as a representation of a specific s t r u c t u r e . In c e r t a i n cases, of course, it m a y be necessary t o recognize s o m e of these v a r i a t i o n s . F o r e x a m p l e , a leg a n d a w i n g at one level are s t r u c t u r a l l y i s o m o r p h i c b u t o b v i o u s l y need t o be d i s t i n g u i s h e d i n some contexts. M a n y such e x a m p l e s e x i s t , v a r y i n g w i d e l y i n subtlety a n d c o m p l e x i t y . T h e i r t r e a t m e n t is not a r b i t r a r y nor is i t d e t e r m i n e d It w i l l d e p e n d o n the context as w e l l as the general state o f knowledge a n d u n d e r s t a n d i n g at a g i v e n t i m e , a n d m a y change s i g n i f i c a n t l y w i t h new developments. In all cases, however, any properties t h a t d i s t i n g u i s h one representation of a s t r u c t u r e f r o m another could be expressed i n t e r m s of a n a p p r o p r i a t e extension of the represented s t r u c t u r e , thus s h a r p e n i n g the classification a n d r e d u c i n g the n u m b e r of representatives.

a priori.

147

VIII. B I O L O G I C A L S T R U C T U R E S

T h e theory of e v o l u t i o n is based o n the p r i n c i p l e t h a t a l l l i v i n g o r g a n i s m s have evolved f r o m e a r l i e r , u l t i m a t e l y s i m p l e r o r g a n i s m s , m a i n l y t h r o u g h r a n d o m v a r i a t i o n and a selection process d r i v e n b y e n v i r o n m e n t a l pressures. H e n c e , the question of how ( a n d perhaps w h y ) the v a r i o u s life f o r m s evolved or differentiated f r o m earlier forms becomes of p r i m a r y interest. T h e m a i n objectives t h e n are t o reconstruct the f a m i l y tree of lifeforms, t h r o u g h a s t u d y of e x i s t i n g o r g a n i s m s plus fossils of e x t i n c t o r g a n i s m s , and t o u n d e r s t a n d , i f possible, h o w the e v o l u t i o n a r y process w o r k s . Therefore, f r o m the s t a n d p o i n t of e v o l u t i o n , a classification s y s t e m m u s t involve more t h a n the m o r p h o l o g y ( f o r m and s t r u c t u r e ) of o r g a n i s m s . It m u s t also recognize phylogeny, their e v o l u t i o n a r y history. A s i m i l a r i t y o f o r g a n i s m s w i l l be significant o n l y i f the shared features were i n h e r i t e d f r o m a c o m m o n e v o l u t i o n a r y ancestor. I n h e r i t e d characteristics of t h i s k i n d are k n o w n as m a n y of w h i c h are i m p l i c i t i n the s t a n d a r d descriptions of the various k i n d s of o r g a n i s m s . A n e x a m p l e of a s i m i l a r i t y t h a t is not a h o m o l o g y is the a b i l i t y t o fly. it is shared by some e x t i n c t reptiles, most b i r d s , bats, a n d m a n y insects, b u t was not i n herited f r o m a c o m m o n ancestor to these a n i m a l s . S i m i l a r i t i e s o f this k i n d are a p r o d u c t of convergent a n d are called T h e y are d e t e r m i n e d by s p e c i a l e n v i r o n m e n t a l c o n d i t i o n s , a n d therefore, as far as e v o l u t i o n is concerned, secondary i n i m p o r t a n c e to homologies i n the basic classification process. O n the other h a n d , as w i l l be seen i n Section 58, analogies are of s p e c i a l interest f r o m the p o i n t o f view of s t r u c t u r e s . A classification begins w i t h a set of s i m i l a r i t i e s observed i n some group of i n d i v i d u a l o r g a n i s m s . T h e collection of s i m i l a r i t i e s , or features, i n t u r n distinguishes a generally larger group consisting of a l l those o r g a n i s m s t h a t share the observed features. If the s i m i l a r i t i e s consist o f homologies, then they m a y c o n s t i t u t e a basis for classification t h a t is significant f r o m the p o i n t o f view of e v o l u t i o n . In a c t u a l practice, the question of whether a p a r t i c u l a r s i m i l a r i t y is a n analogy or a h o m o l o g y may be very difficult to settle a n d is often a m a t t e r of controversy. T h e role of homologies as opposed t o analogies, plus the use o f shared c o m p l e x i t y as a guide t o h o m o l o g y a n d some of the p r o b l e m s encountered, are clearly o u t l i n e d i n the f o l l o w i n g q u o t a t i o n f r o m one of the G o u l d essays [ G 9 , p. 14].

homologies,

evolution

analogies.

T h e d e c o d i n g o f phylogeny requires no more t h a n a m e t h o d for r e c o g n i z i n g h o m o l o g y a n d e l i m i n a t i n g analogy. B i o l o g i s t s have l o n g realized t h a t c o m p l e x i t y offers the best guide to homology. W h e n s i m ilarities between t w o species are sufficiently widespread a n d i n t r i c a t e - t h a t is, c o m p o s e d of m a n y h u n d r e d u t t e r l y u n r e l a t e d , c o m p l e x parts - they cannot arise by independent e v o l u t i o n ; they m u s t record the passive i n h e r i t a n c e of shared descent. A n a l o g o u s s i m i l a r i t y produces

148

STRUCTURALISM AND

STRUCTURES

u n c a n n y likeness - j u s t consider the m a r s u p i a l moles, dogs, a n d mice of A u s t r a l i a - but its results c a n e x t e n d o n l y so far. N a t u r a l selection m a y converge f r o m different directions u p o n the m e c h a n i c a l o p t i m a of c e r t a i n life styles (thus p r o d u c i n g a s t r i k i n g o u t w a r d s i m i l a r i t y i n s w i m m i n g machines, whether they be fish, s q u i d , or m a m m a l ) , but it cannot u n d o the h o m o l o g o u s h a n d i w o r k of ages a n d so restructure a n o r g a n i s m t h a t its ties to h i s t o r y are forever lost i n current a d a p t a t i o n s . If we can look at enough independent details, we w i l l always find h o m o l o g y i n t a c t . M o r p h o l o g y m a y be our first (and u s u a l l y adequate) g u i d e , but it j u s t doesn't p r o v i d e enough independent d e t a i l for s o r t i n g h o m o l ogy f r o m analogy i n difficult cases. M o r p h o l o g y l o o k s c o m p l e x and m u l t i f a c e t e d , b u t p a r t s do not always m a i n t a i n their apparent i n d e pendence. G r o w t h a n d development l i n k the features o f o r g a n i s m s i n t o c o m p l e x webs of c o r r e l a t i o n ; s m a l l changes m a y trigger c a s c a d i n g effects t h r o u g h o u t the b o d y . S i m i l a r i t i e s t h a t look too c o m p l e x for analogy m a y a c t u a l l y arise as consequences of single changes i n these triggers. M o r p h o l o g y also acts as the p r i m a r y b r e e d i n g g r o u n d of analogy, as n a t u r a l selection guides different lineages t o the s a m e o p t i m a for s i m i l a r roles i n c o m m o n e n v i r o n m e n t s . Because of possible confusion to those of us w h o are not experts, a b r i e f r e m a r k is i n order c o n c e r n i n g the use of the t e r m " o r g a n i s m " w i t h reference to one lifeform or another. It u s u a l l y refers to a type of b e i n g rather t h a n to a n i n d i v i d u a l , a n d , even w h e n a p p l i e d t o an i n d i v i d u a l , the reference is to the i n d i v i d u a l as an example of a certain t y p e . In other words, a single i n d i v i d u a l is a c t u a l l y a concrete (biological) as defined i n Section 7, regarded as a p a r t i c u l a r representation o f a c e r t a i n b i o l o g i c a l N o t e t h a t the b i o l o g i c a l s t r u c t u r e m a y v a r y w i t h the p o i n t of v i e w i n a p a r t i c u l a r case. T h u s , one m i g h t see a given i n d i v i d u a l as an e x a m p l e of an a n i m a l , a vertebrate, a m a m m a l , or s i m p l y a d o g , d e p e n d i n g o n the circumstances. A l t h o u g h a m b i g u i t i e s of t h i s k i n d are c o m m o n p l a c e a n d a u t o m a t i c a l l y dealt w i t h i n everyday perception of objects of a l l k i n d s , it is well t o keep t h e m i n m i n d i n references to ancestors and the i n h e r i t a n c e of characteristics. F o r e x a m p l e , an ancestor of a p a r t i c u l a r type of o r g a n i s m is generally not an i n d i v i d u a l but another t y p e o f o r g a n i s m t h a t precedes it i n a sequence of e v o l u t i o n a r y events.

system,

structure.

51. T h e Genetic Structure T h e most c o m m o n structures associated w i t h b i o l o g i c a l o r g a n i s m s , at least i n the context of e v o l u t i o n , are those complexes of features by w h i c h o r g a n i s m s are o r d i n a r i l y described a n d classified. In a d d i t i o n t o these " g l o b a l " s t r u c t u r e s , however, there are the less conspicuous cell a n d bio-

VIII. B I O L O G I C A L S T R U C T U R E S

149

c h e m i c a l s t r u c t u r e s . A m o n g these m i c r o s t r u c t u r e s , the m o s t i m p o r t a n t one for our purposes is the o r g a n i s m ' s genetic s t r u c t u r e , or a copy of w h i c h resides i n each of the o r g a n i s m ' s l i v i n g cells. T h e genetic s t r u c t u r e has become i n c r e a s i n g l y i m p o r t a n t i n the s t u d y of b i o l o g y d u r i n g the last f o r t y or fifty years, a n d m o r e recently is p l a y i n g a role i n d e t e r m i n i n g the i n t e r r e l a t i o n s of c e r t a i n o r g a n i s m s i n e v o l u t i o n a r y h i s t o r y .

genome,

T h e basic substance of the genome is DNA (deoxyribonucleic a c i d ) . DNA is a n e x a m p l e of a " m a c r o m o l e c u l e " , a " l i n e a r p o l y m e r " c o n s i s t i n g of a large n u m b e r of s i m p l e s u b u n i t s s t r u n g together i n l o n g s t r a n d s . O r d i n a r y DNA is m a d e up o f two s u c h s t r a n d s t h a t w i n d a r o u n d one another to f o r m the f a m o u s double h e l i x , the d e s c r i p t i o n of w h i c h earned J a m e s W a t s o n a n d F r a n c i s C r i c k a N o b e l p r i z e i n 1953. T h e s u b u n i t s i n t h i s case are c a l l e d nucleotides, a n d each consists of b o n d e d sugar a n d p h o s p h a t e groups p l u s one of the four bases: adenine (A), cytosine ( C ) , g u a n i n e ( G ) or t h y m i n e (T). I n each s t r a n d , successive nucleotides are connected t h r o u g h a b o n d between the sugar g r o u p of one w i t h the phosphorus g r o u p of the other. E a c h s t r a n d has a b u i l t - i n o r i e n t a t i o n , g o i n g , say, f r o m the p h o s p h o r u s of one nucleotide t o the sugar of the next. T h e c h a i n of a l t e r n a t i n g sugar a n d phosphorus (SP) groups is the " b a c k b o n e " of the s t r a n d . T w o s t r a n d s differ b y the order i n w h i c h the four bases are d i s t r i b u t e d a l o n g the s t r a n d . T h i s o b v i o u s l y a d m i t s the p o s s i b i l i t y of a v i r t u a l l y u n l i m i t e d n u m b e r of d i s t i n c t s t r a n d s . These ordered sequences o f the f o u r bases a r r a n g e d a l o n g an SP b a c k b o n e are the f u n d a m e n t a l s t r u c t u r a l u n i t s of the genome. A n u n t w i s t e d s a m p l e of DNA is s y m b o l i z e d i n F i g u r e 51.1.

nucleotide

—f-SP ^— S P \ A J G v

f

T PS

I

C PS

SP C

SP T

SP G

SP A

SP C

G PS

A PS

C PS

T PS

G PS

1

I

I

I

I

5=sugar group, F=phosphorus group. ^4=adenine, G = c y t o s i n e , G = g u a n i n e , T " = t h y m i n e . F i g . 51.1. T h e DNA is f o r m e d f r o m its t w o s t r a n d s b y j o i n i n g each base f r o m one s t r a n d to a base of the o t h e r . It t u r n s out t h a t b o n d i n g o n l y occurs between adenine a n d t h y m i n e a n d between cytosine a n d g u a n i n e . T h e r e f o r e , the two s t r a n d s i n a s a m p l e of DNA m u s t be c o m p l e m e n t a r y , w i t h each base i n one s t r a n d c o r r e s p o n d i n g to a base i n the other w i t h w h i c h i t c a n b o n d . DNA s t r u c t u r e is the r e p o s i t o r y of i n f o r m a t i o n for the synthesis of p r o teins, the w o r k i n g substances of l i v i n g o r g a n i s m s . T h e i n f o r m a t i o n needed for a p a r t i c u l a r p r o t e i n is c o n t a i n e d i n one or m o r e sections of DNA that together constitute a gene. T h e sequence of bases i n the gene determines

1.50

STRUCTURALISM

AND STRUCTURES

the order i n w h i c h the different a m i n o acids must be assembled i n the p r o tein c h a i n . T h e synthesis involves, i n a d d i t i o n t o DNA, several k i n d s of RNA (ribonucleic acid). T h e latter are also macromolecules resembling single strands of DNA. A general ( m u c h oversimplified) o u t l i n e of the p r o cesses is t h a t the relevant p o r t i o n of a s t r a n d of DNA is copied as a s t r a n d of RNA. T h e RNA i n t u r n directs the synthesis of the desired p r o t e i n . F o r more details of the complex processes by w h i c h protein synthesis is a c t u a l l y c a r r i e d o u t , we refer to an issue of devoted to the general subject of m o l e c u l a r biology. In a d d i t i o n to a general overview article on by A . W e i n b e r g [W2], it contains excellent articles on b o t h DNA a n d RNA w r i t t e n by experts i n the field. A l o n g w i t h clear general e x p l a n a t i o n s , these articles also o u t l i n e some o f the ingenious l a b o r a t o r y techniques, such as c l o n i n g and r e c o m b i n a n t DNA, by w h i c h some of the knowledge a b o u t DNA a n d its functions has been o b t a i n e d . A l t h o u g h there r e m a i n m a n y difficult questions, the a c c o m p l i s h ments so far are impressive w i t h no end of progress i n sight.

Scientific American

"The Molecules of Life",

R.

M o s t o f the necessary i n f o r m a t i o n for the development a n d f u n c t i o n i n g of the o r g a n i s m is encoded i n its DNA, w h i c h is therefore a very c o m p l e x d e t e r m i n i n g s t r u c t u r e for the o r g a n i s m . A t the same t i m e , this f u n d a m e n t a l s t r u c t u r e contains a great deal of r e d u n d a n c y a n d m a n y sections t h a t a p p a r e n t l y either have no f u n c t i o n at a l l or are not d i r e c t l y i n v o l v e d i n p r o t e i n synthesis. In a d d i t i o n t o the basic order s t r u c t u r e concerned w i t h protein s y n t h e sis, some of the more " g l o b a l " s t r u c t u r e s o f D A ' A are also of b i o l o g i c a l significance. A s i m p l e e x a m p l e is the f a m i l i a r o r g a n i z a t i o n of genes o n the chromosomes. M o r e interesting, however, are certain g l o b a l s t r u c t u r e s i n v o l v i n g the overall shape of the double h e l i x itself. F o r e x a m p l e , a s t r i n g m a y fold back u p o n itself i n c o m p l e x ways, thus f o r m i n g a v a r i e t y of closed loops, c h a i n s , and k n o t s . T h e precise d e f i n i t i o n and classification of such configurations are m a t h e m a t i c a l topics long s t u d i e d i n the field of topology. T h e classification of k n o t s , i n p a r t i c u l a r , has received a lot of a t t e n t i o n . F o r the double h e l i x , the general idea is t h a t b i o l o g i c a l significance of a c o n f i g u r a t i o n is associated w i t h its " t y p e " . T h e l a t t e r , at least for strings w i t h o u t loose ends, m a y be defined i n terms of properties t h a t are u n changed (stable) under any space deformation of the configuration t h a t can be made w i t h o u t b r e a k i n g the s t r i n g . T h e s t u d y of the b i o l o g i c a l s i g n i f i cance of such type-forms is a p p r o p r i a t e l y called " b i o l o g i c a l t o p o l o g y " [ W l ] . A s m i g h t be expected, the f u n c t i o n of some of these g l o b a l s t r u c t u r e s is more s u b t l e t h a n the role of base orderings for the synthesis of proteins. M u c h progress has been made i n recent years i n i d e n t i f y i n g the roles of a n u m b e r of genes i n the p r o d u c t i o n o f certain proteins a n d , t h r o u g h t h e m , the i n i t i a t i o n of p a r t i c u l a r chemical reactions, but there r e m a i n s m u c h to

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be learned as to how the process a c t u a l l y works i n its entirety. T h e whole t h i n g is c o m p l i c a t e d by the fact t h a t c h e m i c a l processes, once i n i t i a t e d , proceed a c c o r d i n g to their o w n laws, so the final result m a y be far removed f r o m the gene itself. It is also unclear e x a c t l y h o w genes are t u r n e d o n a n d off at the a p p r o p r i a t e times. T h i s e v i d e n t l y involves s p e c i a l " r e g u l a t o r y " elements, but the m e t h o d of control a n d how the necessary p r e c i s i o n of the a c t i o n is m a i n t a i n e d , are difficult t o d e t e r m i n e . A l l of these p r o b l e m s i n d i c a t e t h a t the genetic structures are not only very c o m p l e x b u t have a d y n a m i c character as w e l l . A s m i g h t be expected f r o m the key role of DNA i n heredity a n d i n the development of a n o r g a n i s m , c e r t a i n DNA segments have the p o t e n t i a l of s e r v i n g as very s p e c i a l test structures for the i d e n t i f i c a t i o n o f homology. A s w i t h other s t r u c t u r e s , however, a segment m a y be useful or not as a test o f h o m o l o g y d e p e n d i n g o n whether a case c a n be made t h a t its s i m i l a r i t y i n two o r g a n i s m s a c t u a l l y i m p l i e s relatedness. A t the s a m e t i m e , because o f the e n o r m o u s c o m p l e x i t y of DNA, the p r o b l e m o f m a k i n g such d e t e r m i n a t i o n s m a y be very difficult. In most c o m p l e x s t r u c t u r e s , one can expect t o find t i g h t l y o r g a n i z e d s u b s t r u c t u r e s . These are d i s t i n c t i v e structures t h a t are l o c a l l y d e t e r m i n e d i n the sense t h a t a s m a l l l o c a l change m a y force changes t h r o u g h o u t the s t r u c t u r e . DNA no doubt contains m a n y such components. A t the same t i m e , the DNA m a y v a r y g r e a t l y f r o m one type of o r g a n i s m t o another, w i t h some components v a r y i n g more or less independently of one another. T h i s p r o p e r t y o f independence, a l o n g w i t h c o m p l e x i t y , is the basis for a r g u i n g t h a t DNA is a g o o d c a n d i d a t e for testing homology. G o u l d [ G 9 , p. 16] expresses i t at follows: Such a l o n g s t r i n g of independent i t e m s cannot evolve detailed s i m i l a r i t y . A h i g h percentage o f i t e m - b y - i t e m m a t c h i n the DNA of two species m u s t represent h o m o l o g y and shared descent. T h i s q u o t a t i o n , as w e l l as the l o n g one i n the previous section, was t a k e n f r o m the last of a series of three of G o u l d ' s essays d e a l i n g w i t h a n o l d p r o b l e m of how t o classify the flamingo. T h e question was w h e t h e r the f l a m i n g o s h o u l d be i n c l u d e d w i t h geese or s t o r k s . S t r o n g a r g u m e n t s based o n c o m p a r i s o n of p h y s i c a l characteristics ( m o r p h o l o g y ) existed for either decision. T h e d i l e m m a , i t seems, was f i n a l l y resolved i n favor of the storks t h r o u g h a c o m p a r i s o n o f

"View of Life"

DNA's.

T h e DNA i n f o r m a t i o n t h a t is c l a i m e d to settle the f l a m i n g o question is f r o m a larger project, by C h a r l e s S i b l e y a n d J o n A h l q u i s t [S5], to cons t r u c t an e v o l u t i o n a r y tree for b i r d s . T h e i r methods, w h i c h a p p e a r t o be s t r a i g h t f o r w a r d b u t t e c h n i c a l l y difficult, are o u t l i n e d b y G o u l d i n the above essay. It w i l l be sufficient here to note t h a t the s i m i l a r i t i e s are m e a s u r e d

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i n d i r e c t l y i n t e r m s of the degree of affinity between single strands o f genes f r o m the t w o b e i n g c o m p a r e d . T h e affinity is i n t u r n m e a s u r e d b y the t e m p e r a t u r e at w h i c h d i s s o c i a t i o n occurs, w i t h low t e m p e r a t u r e i n d i c a t i n g low affinity, or less s i m i l a r i t y , a n d hence a m o r e distant r e l a t i o n s h i p . T h e m e t h o d t h u s reduces a p r o b l e m of d e t e r m i n i n g the degree o f s i m i l a r i t y o f t w o very c o m p l e x structures to the measurement of a single p a r a m e t e r . A t the s a m e t i m e , the m e t h o d , w h i c h measures degree of r e l a t i o n s h i p , m i g h t also serve as an e v o l u t i o n a r y clock. It is w e l l - k n o w n t h a t genes m a y evolve at different rates, b u t i n d i c a t i o n s are t h a t the o v e r a l l changes, as measured here, occur at a nearly constant r a t e . If the rate is indeed constant, a l l that r e m a i n s is to c a l i b r a t e the clock, using independent t i m e checks o b t a i n e d f r o m other sources. A l t h o u g h not a l l experts agree o n the general r e l i a b i l i t y of these m e t h o d s and have seriously c r i t i c i z e d s o m e of the work [L8], the a p p r o a c h is very i n t e r e s t i n g f r o m a s t r u c t u r a l p o i n t of v i e w .

DNA's

In the next sections, we w i l l t r y t o isolate a few of the s p e c i a l s t r u c t u r a l characteristics p e c u l i a r t o l i v i n g o r g a n i s m s , a n d to a n a l y z e t h e m f r o m the p o i n t of view of general s t r u c t u r e s . U n d e r l y i n g e v e r y t h i n g t h a t we have to say is the fact t h a t b i o l o g i c a l structures are not o n l y very c o m p l e x but also have a s p e c i a l d y n a m i c character. A s s u c h , they are q u i t e different f r o m machines (Section 16), since they involve c o m p l e x i n t e r a c t i o n s w i t h the e n v i r o n m e n t i n a d d i t i o n t o their i n t e r n a l processes. T h e y also t e n d to be stable w i t h i n r e l a t i v e l y wide v a r i a t i o n s i n the e n v i r o n m e n t , one aspect of w h i c h is an a b i l i t y , w i t h i n l i m i t s , to repair themselves. A n even more s u b t l e characteristic is the p o t e n t i a l to evolve i n response to e n v i r o n m e n t a l influences. A l l of these processes depend u l t i m a t e l y o n exchanges of m a t t e r a n d / o r energy w i t h the e n v i r o n m e n t . 52. T h e E n v i r o n m e n t o f a S t r u c t u r e M o s t of the t i m e we have tended to regard a s t r u c t u r e more or less as an "object i n i t s e l f " , a p a r t f r o m possibly larger structures t h a t m i g h t c o n t a i n it. It is, of course, the wholeness of a s t r u c t u r e t h a t enables us t o regard it t h u s as an independent o b j e c t . P a r t i a l exceptions t o t h i s p o i n t o f v i e w occur i n the consideration o f s u b s t r u c t u r e s and i n the d e f i n i t i o n of e x t e r n a l , as opposed to i n t e r n a l , properties o f any s t r u c t u r e . It is obvious t h a t the general g r o w t h or e v o l u t i o n of s t r u c t u r e s must also take place w i t h i n larger s t r u c t u r e s , a l t h o u g h i n o u r earlier discussion of these processes i t was not necessary t o emphasize the p o i n t . In d e a l i n g w i t h a s u b s t r u c t u r e , there are s i t u a t i o n s i n w h i c h the ambient s t r u c t u r e cannot be ignored even w h e n we are interested i n the s u b s t r u c t u r e as an independent o b j e c t . T h i s is o b v i o u s l y the case w i t h b i o l o g i c a l s t r u c tures, w h i c h are p r a c t i c a l l y impossible to isolate because of their almost c o n t i n u o u s i n t e r a c t i o n w i t h the e n v i r o n m e n t .

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A s far as an abstract s t r u c t u r e is concerned, there is n o a p r i o n reason for preferring one s u b s t r u c t u r e representation over another. O n the other h a n d , for a concrete s t r u c t u r e , there exist " n a t u r a l " representations w i t h i n those p o r t i o n s of the real w o r l d t h a t contain i t , the u l t i m a t e one b e i n g the entire universe. V a r i o u s of these c o n t a i n i n g s t r u c t u r e s , d e p e n d i n g o n the p o i n t of v i e w , m a y be regarded as of the given s t r u c t u r e . In this d e f i n i t i o n , a g i v e n o r g a n i s m is a part of its e n v i r o n m e n t , w h i l e i n c o m m o n usage an e n v i r o n m e n t is u s u a l l y thought of as e x t e r n a l t o the o r g a n i s m . T h e l a t t e r m i g h t be called an a n d defined as the s u b s t r u c t u r e c o m p l e m e n t a r y t o the given s t r u c t u r e w i t h i n an e n v i r o n m e n t .

environments

external environment

E a c h concrete s t r u c t u r e w i l l n o r m a l l y be perceived i n a s e t t i n g t h a t is o b v i o u s l y a p p r o p r i a t e to i t u n d e r the given circumstances. T h a t is, it w i l l be m o r e or less a u t o m a t i c a l l y associated w i t h a p a r t i c u l a r e n v i r o n m e n t a l s t r u c t u r e t h a t we w i l l c a l l its W e also define its t o be the s u b s t r u c t u r e (of the n a t u r a l e n v i r o n m e n t ) d e t e r m i n e d by the objects "nearest" to the given s t r u c t u r e , the nearest o b jects b e i n g the ones t h a t appear i n relations i n v o l v i n g at least one o b j e c t of the s t r u c t u r e i n question. T h e i m m e d i a t e e n v i r o n m e n t clearly determines the most relevant e x t e r n a l properties of the g i v e n s t r u c t u r e . N o t e t h a t the i m m e d i a t e e n v i r o n m e n t of a b i o l o g i c a l o r g a n i s m is essentially the ecological niche t h a t i t occupies.

mediate environment

natural environment.

im-

T h e fact t h a t a s t r u c t u r e is embedded i n an e n v i r o n m e n t means t h a t it m u s t t o some degree c o n f o r m to the l a t t e r . I n the t e r m i n o l o g y of S e c t i o n 26, this means t h a t certain features o f the given s t r u c t u r e m u s t be e x t e r n a l l y d e t e r m i n e d w i t h i n its e n v i r o n m e n t . T h i s is an i m p o r t a n t p o i n t t h a t w i l l come up a g a i n i n Section 58. 53. T h e E v o l u t i o n a r y Process C o n s i d e r a b i o l o g i c a l o r g a n i s m as i t interacts w i t h its e n v i r o n m e n t . If the t w o r e m a i n e d constant w i t h i n l i m i t s , they m i g h t c o n c e i v a b l y continue indefinitely i n e q u i l i b r i u m . A static c o n d i t i o n w i l l s e l d o m p r e v a i l , however, since there is a tendency for b o t h the o r g a n i s m a n d its e n v i r o n m e n t to change. V a r i a t i o n s i n the o r g a n i s m are believed to be p r i m a r i l y a result of more or less r a n d o m changes i n the genetic m a t e r i a l . T h e s e m a y be caused, for e x a m p l e , by r a d i a t i o n , injection of foreign m a t e r i a l , such as a c h e m i c a l or a v i r u s , a n d errors i n r o u t i n e t r a n s f o r m a t i o n s o f the D N A . W h e t h e r or not the o r g a n i s m s t h a t result f r o m genetic changes c a n survive w i l l depend o n their a b i l i t y to a d a p t t o the (possibly changed) e n v i r o n m e n t . Hence, there is a p o s s i b i l i t y t h a t an o r g a n i s m w i l l be destroyed by either a genetic or a n e n v i r o n m e n t a l change. A t the same t i m e , an a d a p t a b l e o r g a n i s m is not u n i q u e l y d e t e r m i n e d , so there m i g h t be several s u r v i v i n g v a r i a n t s , each of w h i c h c o u l d t h e o r e t i c a l l y b e g i n a new l i n e . T h i s

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process, c o n s i s t i n g of genetic v a r i a t i o n s plus n a t u r a l selection of the more a d a p t a b l e v a r i a n t s , a n d u s u a l l y c o n t i n u i n g t h r o u g h m a n y generations, constitutes the s t a n d a r d m o d e r n account of how a n o r g a n i s m m a y evolve i n t o a new f o r m u n d e r the pressure or d i r e c t i o n o f the e n v i r o n m e n t . It is k n o w n as the or the a n d is based o n the classical D a r w i n i a n n o t i o n of n a t u r a l selection plus a n i d e n t i f i c a t i o n of gene m u t a tions as the source o f the variants u p o n w h i c h n a t u r a l selection acts. I n a d d i t i o n , the theory also recognizes the i m p o r t a n c e of p o p u l a t i o n s t r u c t u r e i n the d e v e l o p m e n t o f new species. M o r e recently, the s y n t h e t i c theory has been subject t o some c r i t i c i s m s t h a t w i l l be considered i n Section 56.

synthetic theory,

modem synthesis,

A f u n d a m e n t a l , and no longer seriously challenged, p r i n c i p l e i n m o d e r n e v o l u t i o n a r y theory is t h a t the process is not i n any sense g o a l - d i r e c t e d . In other words, the process is not g u i d e d b y a p l a n or p a t t e r n (derived f r o m a " h i g h e r " p r i n c i p l e , n a t u r a l or otherwise) b y w h i c h a p r e d e t e r m i n e d o r g a n i s m w i l l result. M o s t c o n t r a r y proposals are based o n unscientific a s s u m p t i o n s t h a t are u l t i m a t e l y more c o m p l e x a n d often more difficult to e x p l a i n t h a n the p r o b l e m itself. A t the s a m e t i m e , the v a r i e t y of possible s t r u c t u r e s t h a t can occur m a y be s i g n i f i c a n t l y l i m i t e d b y c h e m i c a l or p h y s i c a l c o n d i t i o n s , e n v i r o n m e n t a l restrictions, c e r t a i n r e g u l a r i t y requirements (such as s y m m e t r y ) , associated for e x a m p l e w i t h f u n c t i o n , a n d the necessity of b u i l d i n g o n e x i s t i n g s t r u c t u r e s . Despite the absence of goals, e v o l u t i o n is s t i l l a creative force i n the very real sense t h a t the process brings i n t o existence u n p r e d i c t a b l y new types o f o r g a n i s m s . T h e basic feature of the e v o l u t i o n a r y process is t h a t i t produces s t r u c t u r a l changes i n the o r g a n i s m s o n w h i c h it acts. W h e t h e r or not the changes result i n a different k i n d of o r g a n i s m , or s i m p l y a v a r i a n t o f the o r i g i n a l , w i l l d e p e n d o n whether or not they h a p p e n to involve the defining s t r u c ture of the o r i g i n a l . T h e perceived result w i l l therefore depend o n how the classification is made. F u r t h e r m o r e , v a r i a t i o n s i n the o r g a n i s m s t h a t represent a p a r t i c u l a r defining s t r u c t u r e may be s u b s t a n t i a l , as i l l u s t r a t e d by the v a r i e t y o f a n i m a l s t h a t share a m a m m a l i a n s t r u c t u r e . R e c a l l t h a t any s t r u c t u r a l change m u s t consist of a d d i t i o n s or deletions of objects a n d / o r relations. I n p a r t i c u l a r , the objects may r e m a i n constant w h i l e relations a m o n g t h e m change. F o r e x a m p l e , a n o r g a n i s m m a y undergo o n l y t r i v i a l changes i n i t s p h y s i c a l components w h i l e e x p e r i e n c i n g significant r e l a t i o n a l changes i n i t s i n t e r n a l s t r u c t u r e or i n its r e l a t i o n s h i p to the e n v i r o n m e n t . A s we c o n t i n u e , i t is w e l l t o remember t h a t the e v o l u t i o n a r y process m a y b r i n g a b o u t any c o m b i n a t i o n of the possible k i n d s of s t r u c t u r a l change. T h e r e are m a n y different ways t h a t o r g a n i s m s m a y develop as they a d a p t t o the e n v i r o n m e n t . A t one e x t r e m e the process m a y result i n a " s p e c i a l i s t " , finely a d a p t e d to a r e l a t i v e l y stable a n d perhaps narrow e n v i r o n m e n t a l niche, a n d at the other to a " g e n e r a l i s t " , capable o f s u r v i v i n g i n a w i d e l y

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c h a n g i n g e n v i r o n m e n t . In b o t h cases, the o r g a n i s m s are dependent o n a stable e n v i r o n m e n t a l s u b s t r u c t u r e . I n the first case, the s u b s t r u c t u r e (the niche) is special b u t relatively stable w i t h respect to s h o r t t e r m e n v i r o n m e n t a l change. T h e longer a n a n i m a l occupies such a niche, the more specialized a n d the more v u l n e r a b l e it becomes t o disturbances of its niche. Here, for e x a m p l e , are m a n y endangered species c u r r e n t l y threatened w i t h e x t i n c t i o n by d i s r u p t i o n of their h a b i t a t s . I n the second case the s u b s t r u c ture m a y not be special (in the sense of a niche), but lies " d e e p l y " enough t o be essentially unaffected, not o n l y by " n o r m a l " e n v i r o n m e n t a l shifts, but more e x t r e m e changes t h a t m i g h t otherwise be f a t a l . T h e generalists (a f a m i l i a r e x a m p l e of w h i c h is the o r d i n a r y o p o s s u m ) m a y therefore surv i v e w i t h very l i t t l e e v o l u t i o n a r y change over e x t r a o r d i n a r i l y long periods of t i m e a n d r e l a t i v e l y extreme e n v i r o n m e n t a l changes. L a r g e l y because of t h e i r e a r l y appearance o n the "tree of l i f e " , generalists have been described as " p r i m i t i v e " or " s i m p l e " . A t the same t i m e , the specialists, because they t e n d to a p p e a r l a t e , have been described as " a d v a n c e d " or " c o m p l e x " . A s w i l l be seen i n Section 54, such labels c a n be m i s l e a d i n g . It is e v i d e n t l y the view of most e v o l u t i o n i s t s , i n c l u d i n g D a r w i n , t h a t the e v o l u t i o n a r y process must be a g r a d u a l one. T h i s stems f r o m the reasonable a s s u m p t i o n t h a t a m a j o r , genuinely v a r i a t i o n w o u l d involve extensive changes quite independent o f the e n v i r o n m e n t a n d therefore be e x p e c t e d t o result i n an o r g a n i s m w i t h l i t t l e chance of s u r v i v a l . A t the same t i m e , i t is p l a u s i b l e t h a t sufficiently s m a l l changes c o u l d p r o d u c e variants w i t h an a d a p t a b i l i t y range consistent w i t h the e x i s t i n g e n v i r o n m e n t .

random,

G r a d u a l i s m i s , of course, a relative m a t t e r , b u t appears to be an essent i a l aspect of the e v o l u t i o n a r y process i n some f o r m or another. O n the other h a n d , i t is very difficult t o e x p l a i n how a s t r i c t l y g r a d u a l process c o u l d p r o d u c e m a n y of the r e m a r k a b l e structures t h a t e x i s t . These are c o m p l e x s t r u c t u r e s t h a t are h i g h l y specialized for some p a r t i c u l a r f u n c t i o n . A s t a n d a r d e x a m p l e is the eye, w h i c h functions like a very s o p h i s t i c a t e d c a m e r a . It is not so difficult to imagine how a crude eye m i g h t be g r a d u a l l y refined b y the e v o l u t i o n a r y process i n t o the r e m a r k a b l e i n s t r u m e n t t h a t we take for g r a n t e d , t h o u g h the details o f the process m i g h t r e m a i n rather vague. W h a t is more difficult t o i m a g i n e is how even a crude s t r u c t u r e w i t h a c a m e r a - l i k e f u n c t i o n c o u l d be the result of a g r a d u a l process b e g i n n i n g w i t h s o m e t h i n g h a v i n g no resemblance to a c a m e r a . S i m i l a r questions also arise w i t h respect t o the emergence of new species (Section

56).

E x a m p l e s o f this type a p p e a r o n the surface t o involve an u n a v o i d a b l e d i l e m m a : E i t h e r the end result was p r o d u c e d by some k i n d of "emergent" process, c o n t r a d i c t i n g the p r i n c i p l e of g r a d u a l i s m , or the development was g r a d u a l , b u t s o m e h o w g u i d e d to the end result, c o n t r a d i c t i n g the p r i n c i p l e of no g o a l - d i r e c t i o n . Interestingly enough, the first c o n t r a d i c t i o n is o n l y

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a p p a r e n t , as w i l l be e x p l a i n e d i n Sections 55 a n d 56. A n even more f u n d a m e n t a l e x a m p l e is the genetic m a t e r i a l itself. T h i s c o m p l e x , h i g h l y s t r u c t u r e d system is u n i v e r s a l a m o n g a l l life f o r m s a n d a l m o s t i d e n t i c a l i n general f o r m a n d f u n c t i o n i n otherwise r a d i c a l l y d i f ferent o r g a n i s m s . A n o r g a n i c chemist, A . G . C a i r n s - S m i t h , i n a article e n t i t l e d [ C I ] , makes the following c o m m e n t s o n this r e m a r k a b l e " u n i t y o f b i o c h e m i s t r y " .

American

"The First Organisms"

Scientific

Surely the proper conclusions to be d r a w n f r o m such a detailed l o o k at the u n i t y of b i o c h e m i s t r y are t h a t (1) a l l life now o n the e a r t h is descended f r o m a c o m m o n ancestor, (2) this ancestor was q u i t e h i g h up the e v o l u t i o n a r y tree a n d (3) the central b i o c h e m i c a l s y s t e m was already fixed by t h a t t i m e . T h a t i t s h o u l d have r e m a i n e d fixed for so l o n g is surely because of its curious interdependent k i n d o f c o m p l e x i t y . T h i s is the c o m p l e x i t y of " h i g h t e c h " engineering where m a n y w e l l chosen c o m p o n e n t s depend so m u c h on each other t h a t they c a n n o t , any of t h e m be changed. T h a t k i n d of cleverness c o u l d o n l y have been a p r o d u c t of e v o l u t i o n . It is at least o n the cards t h a t the choice of the c o m p o n e n t s t h a t b e c a m e fixed was also a p r o d u c t of e v o l u t i o n . T o conclude, the u n i t y of b i o c h e m i s t r y does not refer to the s t a r t of e v o l u t i o n but to a m u c h later stage. T h e p r o b l e m is to e x p l a i n how such a delicately o r g a n i z e d s y s t e m c o u l d have evolved f r o m s i m p l e r s t r u c t u r e s . O t h e r s have proposed s o l u t i o n s to the p r o b l e m , but C a i r n s - S m i t h offers a p a r t i c u l a r l y i n t e r e s t i n g one f r o m the p o i n t of v i e w o f s t r u c t u r e . W e w i l l r e t u r n to this p r o b l e m i n the next section. O n e c a n not help m a r v e l i n g at the e x q u i s i t e l y elaborate d e t a i l t h a t e v o l u t i o n effects i n so m a n y o f its creations. T o e x p l a i n how the basic e v o l u t i o n ary process develops these r e m a r k a b l e creatures f r o m i n i t i a l l y very s i m p l e s t r u c t u r e s , is a f u n d a m e n t a l ( a n d difficult!) p r o b l e m i n the theory of evol u t i o n . Because the process cannot a c t u a l l y be observed i n m o s t instances, an e x p l a n a t i o n w i l l u s u a l l y take the f o r m of a p l a u s i b l e d e s c r i p t i o n of how the development m i g h t have o c c u r r e d . O b j e c t i v e evidence for the correctness of such an e x p l a n a t i o n w o u l d consist of examples of other o r g a n i s m s , either a m o n g the l i v i n g or i n the fossil record, t h a t a c t u a l l y e x h i b i t the c r u c i a l i n t e r m e d i a t e structures presumed t o have o c c u r r e d . A c c u m u l a t i o n of evidence of t h i s k i n d m a y eventually lead to general acceptance of the e x p l a n a t i o n as the "correct" one. U n f o r t u n a t e l y , even when the p i c t u r e is f a i r l y clear, it is a l m o s t i m p o s s i b l e t o c o m p r e h e n d i n any d e t a i l j u s t how the process a c t u a l l y w o r k s . T h e difficulties here are r e m i n i s c e n t o f those encountered i n the a t t e m p t to c o m p r e h e n d c o s m o l o g i c a ! p h e n o m e n a . T h e t i m e scale, plus the n u m b e r and variety of events, is so great as to rule out

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an i n t u i t i v e a p p r e c i a t i o n based on o r d i n a r y experiences, so the process of u n d e r s t a n d i n g tends to reduce to an i n t e l l e c t u a l exercise. 54.

Complexity in Evolution

T h e r e are m a n y casual references t o c o m p l e x i t y i n the present chapter as well as t h r o u g h o u t the rest o f this b o o k . T h e subject of e v o l u t i o n , however, raises some questions c o n c e r n i n g c o m p l e x i t y t h a t are not adequately covered b y the u s u a l i n f o r m a l t r e a t m e n t . T h e r e f o r e , it is desirable to consider the n o t i o n somewhat more carefully before we continue. It is clear t h a t " c o m p l e x i t y " is one of those c o m m o n t e r m s t h a t everyb o d y uses r o u t i n e l y w i t h o u t q u e s t i o n i n g e x a c t l y w h a t they m e a n . I n fact, c o m p l e x i t y is a h i g h l y i n t u i t i v e n o t i o n , a n d it is difficult, i f not i m p o s s i b l e , t o f o r m u l a t e a precise definition consistent w i t h i t . Nevertheless, even i n its most i n t u i t i v e versions, c o m p l e x i t y is u l t i m a t e l y concerned w i t h s t r u c ture. F o r e x a m p l e , v i r t u a l l y everyone w o u l d perceive c o m p l e x i t y s o m e h o w in t e r m s of the interconnected p a r t s of the t h i n g i n question. I n other words, the t h i n g is viewed as a F r o m the most naive p o i n t of v i e w , the i d e a is t h a t c o m p l e x i t y of a s t r u c t u r e depends u p o n the n u m b e r of its objects and relations. T h o u g h t h i s m a y at first seem f a i r l y n a t u r a l , the mere n u m b e r of objects a n d relations i n a s t r u c t u r e carries very l i t t l e a c t u a l s t r u c t u r a l i n f o r m a t i o n a b o u t t h a t s t r u c t u r e . T h e r e f o r e , any useful n o t i o n of c o m p l e x i t y , however i n t u i t i v e it m i g h t be, s h o u l d involve more s u b t l e s t r u c t u r e properties t h a n can be specified s i m p l y b y a few n u m b e r s . T h e fact t h a t we find i t so easy t o f a l l back i n t h i s way o n i n a d e q u a t e descriptions of structures arises f r o m a tendency to forget j u s t w h a t constitutes a structure.

structure.

D e s p i t e the f u n d a m e n t a l p r o b l e m o f f o r m u l a t i n g a d e f i n i t i o n o f c o m p l e x i t y , we discover o n closer e x a m i n a t i o n t h a t the most c r i t i c a l appeals to c o m p l e x i t y a c t u a l l y involve of c o m p l e x i t y rather t h a n c o m p l e x i t y itself. A n d for this, we o n l y need to be able to say precisely w h a t i t means for one s t r u c t u r e to be more t h a n another. I n other words, it is possible i n most cases t o bypass the p r o b l e m of c o n c o c t i n g a d e f i n i t i o n of c o m p l e x i t y a n d concentrate instead o n c o m p l e x i t y , for w h i c h a precise d e f i n i t i o n is rather easy to f o r m u l a t e .

comparisons

complex

relative

T h e s o l u t i o n is t o identify c o m p l e x i t y of a s t r u c t u r e w i t h the " a m o u n t " of s t r u c t u r a l i n f o r m a t i o n contained i n the s t r u c t u r e . ( I n c i d e n t a l l y , S. L l o y d [L9] suggests a s i m i l a r a p p r o a c h to c o m p l e x i t y . ) T h i s i d e a seems reasonable enough, but it does not help i n " c o m p l e x i t y " , because a precise defi n i t i o n o f the a m o u n t of s t r u c t u r a l i n f o r m a t i o n is j u s t as difficult t o devise as a d e f i n i t i o n of c o m p l e x i t y . O n the other h a n d , we m a y use the d e f i n i t i o n of c o m p a r a b i l i t y o f s t r u c t u r e s w i t h respect to s t r u c t u r a l i n f o r m a t i o n , given in Section 13, to o b t a i n a definition of relative c o m p l e x i t y .

defining

158

STRUCTURALISM

B

AND STRUCTURES

If A a n d B are structures a n d A is i s o m o r p h i c to a s u b s t r u c t u r e of (i.e., then is s a i d t o be more than

A < B),

B

complex

A.

A s u s u a l i n definitions o f t h i s k i n d , we m a y have t o a d m i t a p p r o x i m a t e i s o m o r p h i s m s . T h e d e f i n i t i o n t h e n w i l l serve o u r purposes a n d is consis tent w i t h the i n t u i t i v e n o t i o n of r e l a t i v e c o m p l e x i t y . O n the other h a n d , there m a y be structures t h a t w o u l d be regarded as c o m p a r a b l e a c c o r d i n g to i n f o r m a l usage, b u t are not covered b y the d e f i n i t i o n . A s i n the case of c o m p l e x i t y itself, however, such c o m p a r i s o n s are l i k e l y t o i n v o l v e n o n s t r u c t u r a l considerations, a n d w o u l d a c c o r d i n g l y be m i s l e a d i n g . W e are p r e p a r e d n o w t o resume the discussion o f e v o l u t i o n . A s b a c k g r o u n d , two s c h e m a t i c versions o f the e v o l u t i o n a r y "tree of l i f e " (for fauna!) are depicted i n the next figure. T h e y are f r o m a recent b o o k b y G o u l d [ G i l , p . 46], w i t h the t i t l e , and subtitle,

"Wonderful Life", Burgess Shale and the Nature of History".

T h e Cone of Increasing Diversity

Fig.

"The

Decimation and Diversification

54.1

T h e first v e r s i o n , " T h e C o n e o f i n c r e a s i n g D i v e r s i t y " , is c o n v e n t i o n a l , w h i l e the second, " D e c i m a t i o n a n d D i v e r s i f i c a t i o n " , is suggested b y the m o d e r n r e c o n s t r u c t i o n o f the Burgess Shale Fossils. T h e w o r d " d i v e r s i t y " , as n o r m a l l y used b y b i o l o g i s t s , m a y refer either t o n u m b e r s o f species or to differences i n b o d y p l a n s . G o u l d a n d colleagues suggest t h a t " d i v e r s i t y " s h o u l d be reserved for the first, a n d t h a t " d i s p a r i t y " be used for the second. I n t h i s t e r m i n o l o g y , it is " a c e n t r a l a n d s u r p r i s i n g fact o f life's h i s t o r y m a r k e d decrease i n d i s p a r i t y followed b y a n o u t s t a n d i n g increase i n d i v e r s i t y w i t h i n the few s u r v i v i n g designs" [p. 49]. T h e t e r m " d e c i m a t i o n " i n the t i t l e o f the second " t r e e " , refers t o a d r a m a t i c r e d a c t i o n i u d i s p a r i t y t h a t followed the e a r l y emergence of a p l e t h o r a o f b o d y p l a n s , as evidenced b y the B u r g e s s fossils. T h i s r e m a r k a b l e c o l l e c t i o n of fossils was discovered i n 1909 b y the A m e r i -

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c a n P a l e o n t o l o g i s t , C D . W a l c o t t . H e also, as G o u l d points out [p. 24], c o n sistently " m i s i n t e r p r e t e d " t h e m , " v i e w i n g the f a u n a collectively as a set of p r i m i t i v e or ancestral versions of later, i m p r o v e d f o r m s " . H i s ( m i s ) i n t e r p r e t a t i o n s t o o d u n t i l 1971 when it was challenged by H . W . W h i t t i n g t o n of C a m b r i d g e U n i v e r s i t y , w h o showed t h a t the Burgess a n i m a l s , far f r o m bei n g s i m p l y p r i m i t i v e versions of m o d e r n forms, e x h i b i t a range of a n a t o m i c a l types m u c h greater t h a n those t h a t have s u r v i v e d . " T h e sweep o f a n a t o m i c a l variety reached a m a x i m u m right after the i n i t i a l diversification of m u l t i c e l l u l a r a n i m a l s . T h e later h i s t o r y o f life proceeded by e l i m i n a t i o n , not e x p a n s i o n . T h e current e a r t h may h o l d more species t h a n ever before, but most are i t e r a t i o n s u p o n a few basic a n a t o m i c a l d e s i g n s " . It is t h i s p i c t u r e o u t l i n e d by G o u l d [p. 47], w h i c h suggests t h a t the second version of the "tree" is nearer the t r u t h t h a n the c o n v e n t i o n a l one. I n one of his c r i t i c i s m s of the conventional (inverted cone) version of the tree of life, G o u l d [p. 45] has this to say about the difficulty t h a t h u m a n beings have i n a d j u s t i n g t o the knowledge t h a t a l l life t h a t came before t h e m was not j u s t an elaborate p r e p a r a t i o n for their u l t i m a t e place i n the scheme of things: T h e o l d chain of b e i n g w o u l d provide the greatest c o m f o r t , but we now k n o w t h a t the vast m a j o r i t y of " s i m p l e r " creatures are not h u m a n ancestors or even prototypes, but o n l y c o l l a t e r a l branches o n life's tree. T h e cone of increasing progress a n d diversity therefore becomes our i c o n o g r a p h y o f choice. T h e cone implies predictable development f r o m s i m p l e to c o m p l e x , f r o m less to more. may form o n l y a t w i g , b u t i f life moves, even fitfully, t o w a r d greater c o m p l e x i t y a n d higher m e n t a l powers, then the eventual o r i g i n of self-conscious intelligence m a y be i m p l i c i t i n a l l t h a t came before. In short, I cannot u n d e r s t a n d our continued allegiance to the m a n i f e s t l y false i c o n o g r a phies of l a d d e r a n d cone except as a desperate finger i n the dike o f cosmically j u s t i f i e d hope and arrogance.

Homo sapiens

It is clear f r o m a l l of this t h a t h u m a n beings are prone to assume (perhaps unconsciously) t h a t t h e i r species represents the "highest" f o r m of life a n d , moreover, t h a t i t was a n i n e v i t a b l e o u t c o m e of the process of e v o l u t i o n . I n other words, i f the process c o u l d be s t a r t e d a g a i n , the f i n a l o u t c o m e , as well as m a n y o f the i n t e r m e d i a t e details, w o u l d be m u c h the same. T h e i d e a is t h a t e v e r y t h i n g is d e t e r m i n e d , i f not by a supreme b e i n g , t h e n a c c o r d i n g to some general law or p r i n c i p l e . A related i d e a is to equate m o v e m e n t up the tree of life w i t h progress (toward the u l t i m a t e h u m a n i d e a l ) . A n o t h e r is t o j u d g e an o r g a n i s m at a low p o s i t i o n as " p r i m i t i v e " and one at a h i g h p o s i t i o n as " a d v a n c e d " . T e r m s t h a t are related to " p r i m i t i v e " a n d " a d v a n c e d " , b u t perhaps more precise, are "generalist" a n d " s p e c i a l i s t " .

160

STRUCTURALISM AND

STRUCTURES

W e w i l l have more t o say about t h e m below. F i n a l l y , there is the i d e a t h a t p o s i t i o n o n the tree correlates w i t h c o m p l e x i t y , the higher the p o s i t i o n the greater the c o m p l e x i t y , and conversely. T h e r e are examples t h a t agree superficially w i t h each of the above ideas, b u t , as general " l a w s " , none are f u l l y j u s t i f i e d b y theory, nor s u p p o r t e d by o b s e r v a t i o n . E a c h is based to some degree o n n o n e v o l u t i o n a r y p r i n c i ples. A l s o , except p o s s i b l y for the allusions to c o m p l e x i t y , none contains significant s t r u c t u r a l content, the o n l y f o r m i n w h i c h the a c t u a l d a t a of evol u t i o n c a n reasonably be described. Because references t o c o m p l e x i t y may, however, have some s t r u c t u r a l content, they deserve a closer e x a m i n a t i o n . In a d d i t i o n t o the a t t e m p t to correlate c o m p l e x i t y w i t h p o s i t i o n o n the " t r e e " , the " i n e v i t a b i l i t y " thesis is also sometimes f o r m u l a t e d i n t e r m s of c o m p l e x i t y , w i t h a s i m u l t a n e o u s suppression of the g o a l directed a s s u m p tions u s u a l l y d e m a n d e d b y h u m a n prejudice. In this f o r m , the thesis reduces t o two basic ideas m o r e or less a m e n a b l e to s t r u c t u r a l a n a l y s i s :

tends

(1) T h e process o f e v o l u t i o n to increase c o m p l e x i t y . (2) T h e r e fore, i t m i g h t be expected eventually t o produce an o r g a n i s m of sufficient c o m p l e x i t y to e x h i b i t some f o r m of intelligence. In order to a v o i d the u s u a l h u m a n biases, it is necessary to insist t h a t "sufficiently c o m p l e x " does not necessarily m e a n " h u m a n l i k e " a n d t h a t " i n telligence" does not refer s t r i c t l y t o the h u m a n variety. O n the other h a n d , j u s t w h a t "intelligence" m i g h t mean i n this general context is not o b v i o u s , t h o u g h it m i g h t be identified w i t h c e r t a i n h i g h level s t r u c t u r a l p h e n o m e n a . For n o w , we w i l l restrict a t t e n t i o n to (1) a n d r e t u r n briefly to the more difficult question at the end of the next section. Q u i t e a p a r t f r o m the m i s d i r e c t e d a t t e m p t s to correlate degree of c o m p l e x i t y w i t h p o s i t i o n on the tree of life, the fact r e m a i n s t h a t e v o l u t i o n has m a n y t i m e s p r o d u c e d e n o r m o u s l y c o m p l e x beings f r o m e x t r e m e l y s i m ple ones. A s a m a t t e r of fact, a l l life is presumed to have evolved f r o m single-celled o r g a n i s m s . In other words, a c c o r d i n g to any reasonable n o t i o n of c o m p l e x i t y , e v o l u t i o n does indeed " t e n d t o increase c o m p l e x i t y " . I n c i d e n t a l l y , it is not c l a i m e d t h a t an increase i n c o m p l e x i t y of a g i v e n org a n i s m is a consequence o f the e v o l u t i o n a r y process, o n l y t h a t an increase m i g h t be f r o m the general nature of the process. O n the other h a n d , " s i m p l i f i c a t i o n " {a decrease i n c o m p l e x i t y ) , t h o u g h conceivable, is a p p a r e n t l y not a c o m m o n occurrence i n e v o l u t i o n even when i t w o u l d appear to be a n advantage.

necessary expected

T h e n o t i o n of increase i n c o m p l e x i t y i m p l i c i t i n these r e m a r k s w i l l o b v i o u s l y have to d e p e n d o n the above c o m p a r a b i l i t y d e f i n i t i o n o f relative c o m p l e x i t y . I n p a r t i c u l a r , each o r g a n i s m m u s t be identified as a s t r u c t u r e , a n d a p r o d u c t of e v o l u t i o n w i l l represent an increase i n c o m p l e x i t y i f it contains an i s o m o r p h i c copy of an i m m e d i a t e ancestor as a s u b s t r u c t u r e .

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T h e r e are some u n u s u a l aspects of s t r u c t u r e s t h a t enter i n t o this p i c ture t h a t need to be e m p h a s i z e d . In the first place, a certain a m o u n t of f l e x i b i l i t y m a y be necessary i n the s t r u c t u r a l d e s c r i p t i o n of the o r g a n i s m s , a n d the presumed i s o m o r p h i s m s may have to be replaced b y a p p r o x i m a t e i s o m o r p h i s m s . I n a d d i t i o n , i t is w e l l t o recall the variety of ways t h a t a s u b s t r u c t u r e m a y be identified w i t h i n a given s t r u c t u r e , some of w h i c h were p o i n t e d out i n Section 52. In general, the s u b s t r u c t u r e m a y involve either some or a l l o f the objects a n d / o r relations i n the given s t r u c t u r e . A n d the relations may be either i n t e r n a l or e x t e r n a l ( e n v i r o n m e n t a l ) , i n c l u d i n g any c o m b i n a t i o n of these. I n p a r t i c u l a r , a more t h a n c u s t o m a r y recognition of the e n v i r o n m e n t a l s t r u c t u r e is r e q u i r e d . T h e u s u a l o v e r s i m p l i f i e d p i c t u r e o f a m o r e or less isolated s t r u c t u r e , t h a t we tend to fall back o n i n m u c h of o u r v i s u a l i z a t i o n of s t r u c t u r a l p h e n o m e n a , m a y not be adequate i n the present case. Let us r e t u r n now to the o r i g i n a l p r o b l e m of w h y e v o l u t i o n m i g h t be expected to increase c o m p l e x i t y . W e note first t h a t the predecessor of a p r o d u c t of e v o l u t i o n d i d exist and reproduce, so its basic s t r u c t u r e w i l l tend to be stable i n the face of r o u t i n e f l u c t u a t i o n s i n its e n v i r o n m e n t . T h i s means t h a t it already possesses an advantage for s u r v i v a l , so w i l l tend t o be preserved i n some f o r m or degree under the n o r m a l a c t i o n of the evol u t i o n a r y process. It is more likely t h a t a d j u s t m e n t to an e n v i r o n m e n t a l change w o u l d be m a d e t h r o u g h e x p l o i t a t i o n of an e x i s t i n g r e l a t i v e l y s t a ble s t r u c t u r e rather t h a n b y r a d i c a l development of a new s t r u c t u r e , even t h o u g h the l a t t e r s o l u t i o n m i g h t be t h e o r e t i c a l l y more efficient. S u c h a process, i n w h i c h most of the o r i g i n a l s t r u c t u r e persists as a more or less stable core, w i l l o b v i o u s l y increase c o m p l e x i t y a c c o r d i n g to our d e f i n i t i o n . It is clear f r o m this p o i n t of view why the specialists described i n Section 53 are regarded as ( r e l a t i v e l y ) c o m p l e x . P r o d u c e d by an i n c r e a s i n g l y det a i l e d a d a p t a t i o n to a stable e n v i r o n m e n t a l niche, they are the result of progressive refinement of s t r u c t u r e . T h e above arguments a p p l y d i r e c t l y to g r a d u a l changes, b u t also cover changes t h a t seem to c o n t r a d i c t g r a d u a l i s m . T h e l a t t e r w i l l be e x p l a i n e d i n the next two sections. F u r t h e r m o r e , an increase i n c o m p l e x i t y as conceived a b o v e , because it may take a variety of forms, serves to e x p l a i n several e v o l u t i o n a r y p h e n o m e n a t h a t , o n the surface, seem to be u n r e l a t e d .

"Rube Goldberg effect".

O n e e x a m p l e is the so-called T h e reference is to a f a m o u s c a r t o o n i s t , R u b e G o l d b e r g , whose cartoons were a regular i t e m i n m a n y newspapers some years ago. G o l d b e r g presented e l a b o r a t e constructions i n v o l v i n g r i d i c u l o u s c o m b i n a t i o n s of m a c h i n e r y , a n i m a l s , and so f o r t h , t h a t were supposed to interact i n a rather u n l i k e l y m a n n e r t o produce a s i m p l e effect f r o m an e q u a l l y s i m p l e cause. A l t h o u g h m a n y e v o l u t i o n a r y a d a p t a t i o n s are elegant to say the least, others are very c u m b e r s o m e and

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overly e l a b o r a t e , m u c h like the R u b e G o l d b e r g c o n s t r u c t i o n s . A t the same t i m e , i n d i v i d u a l p a r t s often e x h i b i t the perfection of d e t a i l t h a t one comes to expect f r o m the process. A n e x a m p l e o f the R u b e G o l d b e r g effect is p r o v i d e d b y the s u b t e r r a n e a n t e r m i t e , whose i m m a t u r e workers, because o f t h e i r lack o f a protective outer cover, m u s t b u i l d elaborate m u d t u n nels i n order t o reach f o o d . It appears o n the surface t h a t selection for a protective cover w o u l d precede the development of a c o m p l i c a t e d b e h a v i o r such as t u n n e l b u i l d i n g , especially since the analogous f o r m s of m a n y other Insects, as w e l l as the m a t u r e r e p r o d u c t i v e f o r m s o f the t e r m i t e itself, are so protected. T h e r e are m a n y other e x a m p l e s of this k i n d , i n w h i c h the end p r o d u c t appears to be overly elaborate a n d o c c a s i o n a l l y even a n inefficient solutions to a p r o b l e m o f a d a p t a t i o n . T h e p r o b l e m is to e x p l a i n h o w such seemingly a w k w a r d and inefficient a d a p t a t i o n s arise. Since previous structures tend t o be preserved, an e n d s o l u t i o n m a y carry s t r u c t u r a l features t h a t are no longer needed a n d c o u l d t h e o r e t i c a l l y be e l i m i n a t e d . F o r e x a m p l e , i f the f i n a l p r o d u c t is a r r i v e d at b y a r o u n d a b o u t route, i t m a y record the solutions to different a n d perhaps u n r e l a t e d p r o b l e m s t h a t h a d to be dealt w i t h along the way. T h e elaborate m u d t u n nels constructed by the s u b t e r r a n e a n t e r m i t e s no d o u b t represent a c a r r y over f r o m a n earlier, a n d perhaps efficient, s o l u t i o n to a n e n v i r o n m e n t a l problem. A given s t r u c t u r e m a y c o n t a i n p o r t i o n s t h a t are not relevant t o the s u r v i v a l of the o r g a n i s m , either because they have ceased t o be f u n c t i o n a l or were never f u n c t i o n a l , perhaps a result of benign m u t a t i o n s . F u r t h e r m o r e , j u s t because of their irrelevance, there may be no significant advantage for t h e m t o be e l i m i n a t e d b y n a t u r a l selection, so t h e y m a y t e n d t o a c c u m u l a t e . F o r e x a m p l e , i t appears t h a t s u b s t a n t i a l p o r t i o n s o f the genetic s t r u c t u r e m a y f a l l i n t o t h i s category.

tendency

These r e m a r k s have e m p h a s i z e d the n a t u r a l for the p r e s e r v a t i o n of structures i n the e v o l u t i o n a r y process, often even b e y o n d t h e i r usefulness. O n the other h a n d , there is no guarantee t h a t a g i v e n s t r u c t u r e w i l l be preserved i n d e f i n i t e l y , since i t could at some p o i n t b e c o m e t o t a l l y irrelevant a n d a great enough b u r d e n to the o r g a n i s m (say, i n energy cost) to be a c a n d i d a t e for e l i m i n a t i o n by n a t u r a l selection. A possible e x a m p l e o f this is contained i n the C a i r n - S m i t h account o f the o r i g i n o f the genetic s t r u c t u r e [ C I ] o u t l i n e d d i r e c t l y below. M o s t theories c o n c e r n i n g the o r i g i n of life take the f o r m of proposals as t o h o w some f o r m of the genetic m a t e r i a l m i g h t have arisen. G i v e n t h i s , one m a y t h e n speculate o n how a l l l i v i n g o r g a n i s m s m i g h t have e v o l v e d . S u c h theories are s u p p o r t e d i n part by the o b s e r v a t i o n t h a t c e r t a i n a m i n o acids (the f u n d a m e n t a l b u i l d i n g blocks of a l l p r o t e i n s ) , as w e l l as short s t r i n g s of R N A , m a y be produced by purely p h y s i c a l processes, such as a t m o s p h e r i c

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electrical discharges a c t i n g o n a " p r i m o r d i a l s o u p " presumed to have existed p r i o r t o the appearance of life o n the e a r t h . T h i s fact also s u p p o r t s a suggestion t h a t R N A s t r u c t u r e preceded D N A i n the e v o l u t i o n a r y sequence. C a i r n s - S m i t h , o n the other h a n d , argues t h a t the genetic s t r u c t u r e as a whole does not represent an i n i t i a l stage o f e v o l u t i o n o f life forms b u t a m u c h later stage of the process. (See the q u o t a t i o n i n the preceding section.) He insists t h a t the genetic s t r u c t u r e c o u l d not have arisen g r a d u a l l y by an a s s e m b l i n g of a m i n o acids, a n d t h a t i t o b v i o u s l y represents, even i n its simplest f o r m , a " h i g h t e c h " s y s t e m t h a t m u s t have been preceded by a "low t e c h " s y s t e m i n v o l v i n g q u i t e different basic u n i t s . He proposes t h a t the p r i m i t i v e structures c o u l d easily have been formed i n c e r t a i n clays, a n d t h a t they a c t u a l l y c o n s t i t u t e d a s u p p o r t i n g m a t r i x u p o n w h i c h the genetic s t r u c t u r e itself e v o l v e d . T h e more s o p h i s t i c a t e d s t r u c t u r e s , once created, continued t o develop i n d e p e n d e n t l y o f the m a t r i x , a n d also assumed the essential functions of the l a t t e r . Subsequently, the now irrelevant p r i m i t i v e s t r u c t u r e s were, for one reason or a n o t h e r , e l i m i n a t e d . T h e theory t h u s suggests one way i n w h i c h a h i g h l y c o m p l e x specialized ( " h i g h tech") s t r u c t u r e m i g h t have arisen t h r o u g h a g r a d u a l process. A l t h o u g h a scenario o f this k i n d is not u n i v e r s a l l y accepted, i t is a p p a r e n t l y a t t r a c t i n g considerable s u p p o r t a m o n g e v o l u t i o n i s t s a n d is also very interesting f r o m the p o i n t of view of structures. O u r e x a m p l e of the P a s c a l configuration i n S e c t i o n 19 offers a p a r t i a l analogy t o the C a i r n s - S m i t h p r o p o s a l . T h e former is d e t e r m i n e d i n a s t r a i g h t f o r w a r d way b y a complete hexagon i n s c r i b e d i n a conic. O n the other h a n d , the result does not involve any of the lines or p o i n t s of the hexagon and its conic. Therefore, i f the latter were deleted f r o m the p i c ture, the P a s c a l C o n f i g u r a t i o n itself w o u l d r e m a i n i n t a c t , but w o u l d be very difficult to " e x p l a i n " w i t h o u t knowledge of the i n i t i a l s t r u c t u r e o n w h i c h it was c o n s t r u c t e d . 55. M u l t i p l e F u n c t i o n W e have e m p h a s i z e d the d y n a m i c character of b i o l o g i c a l o r g a n i s m s , espec i a l l y w i t h respect t o their i n t e r a c t i o n w i t h the e n v i r o n m e n t . F u r t h e r m o r e , each p a r t of an o r g a n i s m w i l l p l a y its o w n special role i n this o v e r a l l i n t e r a c t i o n . S u c h a role w i l l be c a l l e d a of the s t r u c t u r e represented b y the p a r t i n question. Defined i n this way, a f u n c t i o n w i l l i n v o l v e b o t h the concrete representing s t r u c t u r e a n d its e n v i r o n m e n t . A l t h o u g h i t w o u l d be possible t o e x t e n d the d e f i n i t i o n o f the s t r u c t u r e i n question so as to i n clude w i t h i n i t a p a r t i c u l a r f u n c t i o n , this w o u l d obscure the m a i n reason for i n t r o d u c i n g the n o t i o n .

"function"

It is possible for a g i v e n s t r u c t u r e to be associated w i t h m a n y different f u n c t i o n s . A s a m a t t e r o f fact, even a p a r t i c u l a r concrete representation

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of a s t r u c t u r e m a y interact w i t h its e n v i r o n m e n t i n more t h a n one way. I n other words,

function.

the representation may be able to support more than one "multiple function"

T h i s is the n o t i o n of referred t o i n the section h e a d i n g . F r o m the p o i n t of v i e w o f a r b i t r a r y s t r u c t u r e s , the p o s s i b i l i t y of m u l t i ple functions is not s u r p r i s i n g , because a s u b s t r u c t u r e of a larger s t r u c t u r e generally m a y be perceived i n m o r e t h a n one way i n its r e l a t i o n to the l a t t e r , d e p e n d i n g o n w h i c h of the associated e x t e r n a l properties are e m p h a s i z e d . T h e o b j e c t i v e here is to t r y t o use these ideas to e x p l a i n how c e r t a i n o r g a n i s m s t r u c t u r e s , w h i c h a p p e a r to violate the p r i n c i p l e of g r a d u a l i s m , m i g h t have e v o l v e d . W e have already seen i n the preceding section one approach to this p r o b l e m for the s p e c i a l case of the genetic s t r u c t u r e , v i z . , the p r o p o s a l by C a i r n s - S m i t h t h a t the genetic s y s t e m developed i n the context of a n earlier p r i m i t i v e s y s t e m w h i c h later was e l i m i n a t e d . T h e a p p r o a c h t h r o u g h m u l t i p l e f u n c t i o n is more t r a d i t i o n a l . Before o u t l i n i n g i n very general t e r m s the ideas i n v o l v e d , we i n c l u d e some r e m a r k s by G o u l d f r o m one of his essays [ G 7 , p . 50], i n w h i c h he discusses the " h y p e r s e l e c t i o n i s t " v i e w t h a t e v o l u t i o n proceeds t h r o u g h n a t u r a l selection l e a d i n g to an always " b e t t e r " o r g a n i s m , a v i e w t h a t was e x p l i c i t l y r e p u d i a t e d b y D a r w i n himself:

only

D a r w i n , on the other h a n d , was a consistent p l u r a l i s t g a z i n g u p o n a messier universe. H e saw m u c h fit and h a r m o n y , for he believed t h a t n a t u r a l selection holds pride of place a m o n g e v o l u t i o n a r y forces. B u t other processes w o r k as w e l l , a n d o r g a n i s m s d i s p l a y an a r r a y of features t h a t are not a d a p t a t i o n s a n d do not p r o m o t e s u r v i v a l d i rectly. D a r w i n t o o k p a r t i c u l a r interest i n t w o p r i n c i p l e s l e a d i n g to n o n a d a p t i v e change: (1) O r g a n i s m s are integrated systems a n d a d a p t i v e change i n one part can lead t o n o n a d a p t i v e m o d i f i c a t i o n s of other features (" correlations of g r o w t h " i n D a r w i n ' s phrase); (2) A n o r g a n i s m b u i l t under the influence of selection for a specific role m a y be a b l e , as a consequence o f its s t r u c t u r e , t o p e r f o r m m a n y other unselected functions as w e l l . I t e m (1) i n the q u o t a t i o n brings up an i m p o r t a n t p o i n t c o n c e r n i n g s t r u c t u r a l d e t e r m i n i s m , w h i c h we w i l l r e t u r n to i n the next section. I t e m (2) expresses e x a c t l y the n o t i o n of m u l t i p l e f u n c t i o n . It is e l a b o r a t e d later o n i n the essay i n a discussion o f A l f r e d Russel W a l l a c e ' s i n a b i l i t y to accept the h u m a n intellect as a p r o d u c t of e v o l u t i o n : B u t h y p e r s e l e c t i o n i s m is not v a l i d . It is a c a r i c a t u r e of D a r w i n ' s subtler v i e w , and it b o t h ignores a n d m i s u n d e r s t a n d s the n a t u r e of organic f o r m and f u n c t i o n . N a t u r a l selection m a y b u i l d an o r g a n " f o r " a specific function or group o f functions. B u t this purpose need not f u l l y specify the c a p a c i t y o f a s t r u c t u r e . O b j e c t s designed for definite

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purposes c a n , as a result of their s t r u c t u r a l c o m p l e x i t y , p e r f o r m m a n y other tasks as w e l l . A factory m a y i n s t a l l a c o m p u t e r o n l y t o issue the m o n t h l y pay checks, b u t such a machine can analyze the election r e t u r n s or w h i p anyone (or at least p e r p e t u a l l y tie t h e m ) i n t i c k - t a c k toe. O u r large brains m a y have o r i g i n a t e d for some set of necessary s k i l l s i n g a t h e r i n g f o o d , s o c i a l i z i n g , or whatever; b u t these s k i l l s do not exhaust the l i m i t s of w h a t such a complex m a c h i n e can do A s the above q u o t a t i o n s show, the n o t i o n of m u l t i p l e f u n c t i o n is an o l d a n d f a m i l i a r one i n the theory of e v o l u t i o n . Moreover, i t has been offered as an e x p l a n a t i o n for the sudden emergence o f c e r t a i n organisms or s t r u c t u r e s t h a t appear to c o n t r a d i c t the p r i n c i p l e of g r a d u a l i s m . Nevertheless, it is i l l u m i n a t i n g to o u t l i n e an e x p l i c i t s t r u c t u r a l account of these apparent c o n t r a d i c t i o n s . It t u r n s o u t , i n fact, t h a t such p h e n o m e n a m a y be regarded as genuine s t r u c t u r a l i n the development of the o r g a n i s m , w h i c h , at the same t i m e ,

discontinuities do not violate the principle of gradualism.

In order t o u n d e r s t a n d t h i s seeming p a r a d o x , let us sketch briefly the process as it is p r e s u m e d to take place. C o n s i d e r a p r i m i t i v e s t r u c t u r e w h i c h s u p p o r t s some f u n c t i o n i m p o r t a n t to the s u r v i v a l of the associated o r g a n i s m . T h e e v o l u t i o n a r y process a c t i n g i n the s t a n d a r d m a n n e r w i l l t e n d to develop and " i m p r o v e " the p r i m i t i v e s t r u c t u r e w i t h respect to its o r i g i n a l f u n c t i o n . T h e l i k e l y result w i l l be structures of increasing s o p h i s t i c a t i o n a n d c o m p l e x i t y . A t the same t i m e , the p o t e n t i a l of the structures for functions other t h a n the o r i g i n a l one w i l l also increase. A l t h o u g h some of the new functions c o u l d be very different f r o m the o r i g i n a l , they are nevertheless associated w i t h structures s h a p e d b y the e n v i r o n m e n t , a n d are therefore already more or less c o m p a t i b l e w i t h the e n v i r o n m e n t . M o r e o v e r , it is possible for a new f u n c t i o n t o be m o r e f a vorable to s u r v i v a l of the o r g a n i s m t h a n the o r i g i n a l , so a r e l a t i v e l y s m a l l change i n the basic structure or the e n v i r o n m e n t , quite consistent w i t h the p r i n c i p l e of g r a d u a l i s m , c o u l d trigger a shift to a new f u n c t i o n . F u r t h e r e more, the change i n f u n c t i o n c o u l d be d r a m a t i c a l l y a b r u p t and appear as a true d i s c o n t i n u i t y i n the process o f development. O n the other h a n d , as far as s t r u c t u r a l changes are concerned, the d i s c o n t i n u i t y does not involve the s t r u c t u r e , but rather the s t r u c t u r e c o n s i s t i n g of the p h y s i c a l s t r u c t u r e plus its f u n c t i o n . Therefore, because the e v o l u t i o n a r y process acts d i r e c t l y o n the p h y s i c a l s t r u c t u r e , so is o n l y i n d i r e c t l y i n v o l v e d w i t h f u n c t i o n , the p r i n c i p l e of g r a d u a l i s m r e m a i n s i n t a c t . T h e e v o l u t i o n a r y p r o cess m a y , of course, continue the g r a d u a l development of the basic s t r u c t u r e w i t h respect t o the new f u n c t i o n , thus a c c e n t u a t i n g the " d i s c o n t i n u i t y " . A s is p o i n t e d out i n the next section, a change of this k i n d is also a n e x a m p l e of a " c a t a s t r o p h i c " change f r o m one stable state to another.

physical

T h e above r e m a r k s b r i n g out the fact t h a t the appearance of a c o n t r a -

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d i c t i o n i n s i t u a t i o n s of t h i s k i n d stems f r o m an unconscious replacement of one s t r u c t u r e by another, n a m e l y , the basic p h y s i c a l s t r u c t u r e b y the basic structure T h e p r i n c i p l e of g r a d u a l i s m applies s t r i c t l y to the first, w h i l e the p h e n o m e n o n o f interest involves the second.

plus function.

In the case of the eye, i t is reasonable t o conjecture t h a t the s t a r t i n g p o i n t i n v o l v e d the general s e n s i t i v i t y of l i v i n g cells t o l i g h t . O n t h i s basis, the g r a d u a l e v o l u t i o n of a s i m p l e light-sensitive o r g a n , w h i c h w o u l d be an advantage for s u r v i v a l of the parent o r g a n i s m , m i g h t be expected. Such an o r g a n m i g h t continue to " i m p r o v e " g r a d u a l l y , developing, for e x a m p l e , a very s i m p l e means for c o n c e n t r a t i n g the l i g h t to increase s e n s i t i v i t y , u n t i l its s t r u c t u r e b e c a m e c o m p l e x enough t o s u p p o r t a p r i m i t i v e c a m e r a - l i k e f u n c t i o n , a n d the obvious s u r v i v a l advantage of the latter w o u l d favor a shift o f f u n c t i o n . O n c e a camera-like f u n c t i o n takes over, the p o s s i b i l i t y of d e v e l o p i n g a genuine eye becomes p l a u s i b l e , t h o u g h a d d i t i o n a l f u n c t i o n changes w o u l d , no d o u b t , be necessary a l o n g the way. A l t h o u g h a scenario a l o n g these lines is reasonable enough, the m a n y u n k n o w n s i n the process make i t difficult to conjecture i n any d e t a i l w h a t m i g h t a c t u a l l y have o c c u r r e d , especially for s o m e t h i n g as c o m p l e x as the eye. A n o t h e r p h e n o m e n o n , w h i c h depends o n m u l t i p l e f u n c t i o n for its e x p l a n a t i o n , concerns evolved structures t h a t , at first sight, appear t o be q u i t e m a l a d a p t i v e . A n e x a m p l e , discussed b y G o u l d i n one of his essays [ G 6 , p. 79], is the Irish E l k ( a c t u a l l y a deer). A m o n g these a n i m a l s , now e x t i n c t , the males developed enormous antlers t h a t were replaced each year, o b v i ously at a h i g h cost i n energy. T h e antlers must also have been i n m a n y other respects a severe h a n d i c a p , s i m p l y because o f their great size a n d weight, a n d even have been cited as the reason the a n i m a l b e c a m e e x t i n c t ! A n e x p l a n a t i o n of their development as a shift o f f u n c t i o n is also g i v e n by G o u l d i n his essay [G6, p p . 79-90]. I n the first place, it is obvius t h a t antlers were developed i n i t i a l l y as weapons o f attack or defense, a n d , as such, p r o v i d e d the a n i m a l w i t h a greater o r less of an advantage i n c o m p e t i t i o n w i t h other a n i m a l s , p r o b a b l y for the privilege of m a t i n g . U p t o a p o i n t , the size of antlers w o u l d correlate w i t h their effectiveness as weapons. B u t eventually, as i n so m a n y cases of this k i n d , the d e t e r m i n a t i o n of s u p e r i o r i t y m i g h t be decided by a d i s p l a y or other r i t u a l e x h i b i t i n g a n t l e r size w i t h o u t a n a c t u a l test of s t r e n g t h . T h u s , we have a shift i n f u n c t i o n f r o m " a n t l e r as a w e a p o n " t o "size of antler as a s y m b o l o f s t r e n g t h " . O n c e t h i s shift is made, the e v o l u t i o n a r y process, no d o u b t t h r o u g h sexual selection, c o u l d continue the development of antlers p r i m a r i l y w i t h respect t o size, r e s u l t i n g e v e n t u a l l y i n the seemingly a b s u r d monster antlers. Y e t a n o t h e r example of m u l t i p l e f u n c t i o n , already a l l u d e d to b y G o u l d

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i n the above q u o t a t i o n s , is p r o v i d e d by the h u m a n b r a i n . T h e challenge i n t h i s case is to e x p l a i n how the e v o l u t i o n a r y process c o u l d produce such a s t r u c t u r e . M o r e precisely, w h a t c o m b i n a t i o n of p r i m i t i v e e n v i r o n m e n t a l pressures c o u l d possibly result i n a device capable o f c o m p o s i n g and a p p r e c i a t i n g m u s i c , creating a n d u n d e r s t a n d i n g p u r e m a t h e m a t i c s , c o n t e m p l a t i n g the n a t u r e of the universe, a n d so on? F u n c t i o n s of t h i s k i n d , few o f w h i c h c o u l d conceivably have any s u r v i v a l value for p r e h u m a n s i n their struggle w i t h the e n v i r o n m e n t , can o n l y be e x p l a i n e d i n t e r m s of m u l t i p l e function. T h e b r a i n is already a n exceedingly c o m p l e x o r g a n , even i n a n i m a l s m u c h less developed t h a n h u m a n s , as far as b r a i n f u n c t i o n is concerned. I n fact, any a n i m a l t h a t is c o m p a r a b l e t o m a n o n a s t r i c t l y b i o l o g i c a l level requires a h i g h l y developed b r a i n j u s t for c o n t r o l of its p h y s i c a l s y s t e m a n d r o u t i n e i n t e r a c t i o n s w i t h the e n v i r o n m e n t . A necessary feature o f this e q u i p m e n t is a n a b i l i t y to m a n a g e those structures w h i c h c a r r y the e n v i r o n m e n t a l i n f o r m a t i o n necessary for s u r v i v a l . T h u s , we have the beginnings of a general f a c i l i t y for the m a n a g e m e n t of s t r u c t u r e s , a n d a context for the developm e n t of higher m e n t a l processes (Section 35). These observations suggest t h a t e v o l u t i o n set the stage q u i t e e a r l y i n the g a m e for the d e v e l o p m e n t o f h u m a n l i k e m e n t a l q u a l i t i e s , i n c l u d i n g a language p o t e n t i a l (Section 33). It is also reasonable t o conjecture t h a t these qualities are already adequate t o s u p p o r t the m o r e esoteric m e n t a l p h e n o m e n a m e n t i o n e d above. A l t h o u g h the c a p a c i t y for higher m e n t a l a c t i v i t y is no d o u b t an instance of m u l t i p l e f u n c t i o n p r o d u c e d b y r o u t i n e e v o l u t i o n a r y development of the b r a i n , the new functions are f u n d a m e n t a l l y different f r o m the u s u a l i n t e r actions of an o r g a n i s m w i t h the e n v i r o n m e n t . T h e y not o n l y do not replace or interfere w i t h any o f the o r i g i n a l more r o u t i n e b i o l o g i c a l f u n c t i o n s b u t also involve a v a r i e t y o f essentially " a b s t r a c t " s t r u c t u r a l objects. F u r t h e r more, i n d i c a t i o n s are t h a t the b r a i n of p r i m i t i v e m a n p r o b a b l y d i d not differ very m u c h f r o m t h a t of m o d e r n m a n , so the impressive i n t e l l e c t u a l development e x h i b i t e d b y m o d e r n h u m a n s is to a large extent the result of c u l t u r a l rather t h a n p h y s i c a l e v o l u t i o n . A t some p o i n t i n its h i s t o r y , the b r a i n became subject t o a n entirely different type of e v o l u t i o n a r y developm e n t , one t h a t e x p l o i t e d the e n o r m o u s capacity o f a very c o m p l e x s y s t e m to f u n c t i o n i n ways t h a t had l i t t l e to do w i t h the o r i g i n a l purpose for w h i c h it was designed. T h e e x a m p l e o f the b r a i n , raises a question as to whether or not s i m i l a r p h e n o m e n a occur i n other systems. A r e there b i o l o g i c a l systems t h a t have evolved a b s t r a c t f u n c t i o n s analogous to those e x h i b i t e d by the b r a i n ? Is it conceivable t h a t such functions m i g h t a c t u a l l y appear a u t o m a t i c a l l y i n certain cases of extreme c o m p l e x i t y ? N o t e t h a t the last question is p u r e l y s t r u c t u r a l i n content. It is not intended t o i m p l y (or expect) t h a t these

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" h i g h e r " functions w o u l d necessarily have a n y t h i n g t o do w i t h creatures r e s e m b l i n g h u m a n beings. O n the other h a n d , some of the functions m i g h t suggest intelligence, s i m p l y because intelligence is so deeply i n v o l v e d w i t h the processing of structures. It is o n l y i n this sense t h a t one c a n l e g i t i m a t e l y c l a i m t h a t increasing c o m p l e x i t y m i g h t be expected to produce intelligence. T h e various e x p l a n a t i o n s of p h e n o m e n a such as those o u t l i n e d i n this a n d the preceding t w o sections, are not o n l y based o n s t r u c t u r e concepts, but involve some o f the most interesting aspects of s t r u c t u r e s , such as complexity, stability, determinism, and multiple function. 56.

Biological

Catastrophes

R o u g h l y s p e a k i n g , the i d e a b e h i n d catastrophe theory Is t h a t c e r t a i n systems, under " c o n t i n u o u s " change of c o n d i t i o n s , t e n d t o persist i n a given state (i.e., r e m a i n stable) up to a c r i t i c a l p o i n t , at w h i c h an a d d i t i o n a l s m a l l change causes the s y s t e m to shift a b r u p t l y ( c a t a s t r o p h i c a l l y ! ) t o a different (stable) state. It is f u n d a m e n t a l l y a m a t h e m a t i c a l subject a n d w i l l be discussed f r o m t h a t p o i n t of view i n Sections 67-70 i n the next chapter. O n the other h a n d , the m a t h e m a t i c a l subject is h i g h l y suggestive o f m u c h more general p h e n o m e n a , w h i c h m a y be observed i n m a n y different contexts, b u t often w i t h o u t any obvious m a t h e m a t i c a l content. S o m e o f the examples have generated considerable controversy, not a b o u t the p h e n o m e n a themselves, a l l of w h i c h are p l a u s i b l e a n d i n t e r e s t i n g , but c o n c e r n i n g only the question o f whether or not a genuine m a t h e m a t i c a l t r e a t m e n t m i g h t be possible. W e have already considered one type o f catastrophe f r o m biology, the a b r u p t change due to a shift i n f u n c t i o n discussed i n the previous section. T o see t h a t t h i s is an example of a c a t a s t r o p h i c change, the o n l y t h i n g we need t o a d d to the previous discussion is the o b s e r v a t i o n t h a t the very existence o f a b i o l o g i c a l s t r u c t u r e is evidence of a state of e q u i l i b r i u m w i t h the e n v i r o n m e n t . Therefore, a g r a d u a l change l e a d i n g to a significant shift of r e l a t i o n s h i p w i t h the e n v i r o n m e n t , as i n a n a b r u p t change of f u n c t i o n , represents a t r a n s i t i o n f r o m one e q u i l i b r i u m state to another, so is a catastrophe. T h e r e are also catastrophe p h e n o m e n a possible i n biology t h a t are more general, i n the sense t h a t they involve shifts o f s t r u c t u r e rather t h a n j u s t f u n c t i o n . T h e r e m a i n d e r o f this section is devoted to t h i s t o p i c , w h i c h b r i n g s out some interesting aspects of structures. T h e m o d e l e x a m p l e is the concept of " p u n c t u a t e d e q u i l i b r i a " , used to e x p l a i n species f o r m a t i o n . A n o t h e r b i o l o g i c a l e x a m p l e is discussed i n Section 70 i n C h a p t e r I X . I n c i d e n t a l l y , M . M . D o d s o n [D4], [D5] has also a p p l i e d catastrophe theory to certain evolutionary phenomena. A s r e m a r k e d i n Section 53, the t r a d i t i o n a l s y n t h e t i c theory o f e v o l u -

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t i o n a r y development has i n recent years c o m e under a t t a c k . T h e f o l l o w i n g q u o t a t i o n f r o m an article o n the subject b y G . L . S t e b b e n s a n d F . J . A y a l a [S8j o u t l i n e s the issues. T h e molecular studies have ... led to two direct challenges t o the s y n t h e t i c theory. O n e is a p r o p o s a l t h a t a k i n d of m o l e c u l a r determ i n i s m , r a t h e r t h a n pure chance, i m p e l s the development o f v a r i a t i o n s in D N A . T h e other is a c o n t r a s t i n g c l a i m , k n o w n as the n e u t r a l t h e ory, t h a t chance governs not o n l y the i n i t i a l appearance o f genetic v a r i a t i o n s but also their subsequent e s t a b l i s h m e n t i n a p o p u l a t i o n . A different k i n d of challenge, based o n new i n t e r p r e t a t i o n s of the fossil r e c o r d , has emerged f r o m paleontology. K n o w n as p u n c t u a t e d e q u i l i b r i u m , i t holds t h a t e v o l u t i o n proceeds not at a steady pace b u t i r r e g u l a r l y , i n fits a n d s t a r t s . A l t h o u g h the molecular challenges suggest some interest i n s t r u c t u r a l questions, we w i l l restrict a t t e n t i o n to the case of p u n c t u a t e d e q u i l i b r i a , w h i c h was i n t r o d u c e d b y E l d r i d g e a n d G o u l d [E2]. It m a i n t a i n s t h a t "est a b l i s h e d species do not change s u b s t a n t i a l l y i n p h e n o t y p e over a l i f e t i m e t h a t m a y encompass m a n y m i l l i o n s of years (stasis), and t h a t m o s t evol u t i o n a r y change is concentrated i n geologically instantaneous events of b r a n c h i n g s p e d at i o n " . Since m o s t evolutionists t r a d i t i o n a l l y have been c o m m i t t e d i n one way or other to a g r a d u a l i s t p o i n t of view w i t h respect t o species d e v e l o p m e n t , the p r o p o s a l by E l d r i d g e a n d G o u l d has generated considerable controversy. U n f o r t u n a t e l y , the fossil r e c o r d , w h i c h is the l o g i c a l place to check such differences, generally involves a t i m e scale too coarse t o record details of changes as r a p i d as those presumed i n p u n c t u a t e d e q u i l i b r i a . A t the same t i m e , there have been m a n y examples e x t r a c t e d f r o m the record t o s u p p o r t a g r a d u a l i s t account of speciation over p u n c t u a t i o n . M o s t of these, however, have been challenged by G o u l d and E l d r i d g e [G12] as either u n s u b s t a n t i a t e d or a c t u a l l y consistent w i t h p u n c t u a t e d e q u i l i b r i a . A l t h o u g h disagreements c o n c e r n i n g i n t e r p r e t a t i o n s o f the fossil record are not likely t o be settled o n p u r e l y t h e o r e t i c a l g r o u n d s , any r e s o l u t i o n is certain t o involve s t r u c t u r e considerations i n some way or o t h e r . G o u l d , for e x a m p l e , i n a recent a r t i c l e o n [G10], where he responds t o c r i t i c i s m s o f p u n c t u a t e d e q u i l i b r i a , makes an e x p l i c i t appeal t o the idea of s t r u c t u r a l s t a b i l i t y a n d the fact t h a t b r e a k i n g the s t a b i l i t y w i l l n o r m a l l y result i n a r a p i d change to another stable state.

lutionary Theory"

"Darwinism and the Expansion of Evo-

In the largest sense, this debate is b u t one s m a l l aspect of a broader discussion a b o u t the nature o f change: Is our w o r l d (to c o n s t r u c t

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a r i d i c u l o u s l y o v e r s i m p l i f i e d d i c h o t o m y ) p r i m a r i l y one of constant change ( w i t h s t r u c t u r e as a mere i n c a r n a t i o n of the m o m e n t ) , or is s t r u c t u r e p r i m a r y a n d c o n s t r a i n i n g , w i t h change as a difficult p h e n o m enon, u s u a l l y a c c o m p l i s h e d r a p i d l y w h e n a stable s t r u c t u r e is stressed b e y o n d its buffering capacity t o resist a n d absorb? It w o u l d be h a r d t o deny t h a t the D a r w i n i a n t r a d i t i o n , i n c l u d i n g the m o d e r n s y n t h e sis, f a v o r e d the first v i e w w h i l e " p u n c t u a t i o n a l i s t " t h o u g h t i n general, i n c l u d i n g such aspects of classical m o r p h o l o g y as D ' A r c y T h o m p s o n ' s theory of f o r m [T2], prefers the second, [p. 383] T h e E l d r i d g e - G o u l d theory o f p u n c t u a t e d e q u i l i b r i a clearly suggests t h a t s p e c i a t i o n is a c a t a s t r o p h i c event. A c t u a l l y , o n the basis o f the general p r i n c i p l e o f s t r u c t u r a l s t a b i l i t y , i t m i g h t be expected t h a t the e v o l u t i o n of one b i o l o g i c a l s y s t e m i n t o a n o t h e r w o u l d n o r m a l l y i n v o l v e d i s c o n t i n u ities. I n p a r t i c u l a r , a sufficiently fine analysis o f an ostensibly c o n t i n u ous development w o u l d p r o b a b l y show i t t o consist o f a succession o f very s m a l l d i s c o n t i n u i t i e s . S u c h a n e x a m p l e , however, w o u l d not be regarded as a c o n t r a d i c t i o n of g r a d u a l i s m . O n the other h a n d , t h e d i s c o n t i n u i t i e s i n p u n c t u a t e d e q u i l i b r i a are large, a n d so appear a g a i n t o be inconsistent with gradualism. T h e r e are m a n y special features i n s p e c i a t i o n , w h i c h we w i l l not a t t e m p t to deal w i t h . F u r t h e r m o r e , as we have already seen, the p r o b l e m o f a b r u p t , or d i s c o n t i n u o u s , e v o l u t i o n a r y development is not p e c u l i a r to s p e c i a t i o n , b u t is i n v o l v e d i n the development of m a n y b i o l o g i c a l systems. A l t h o u g h s o m e d i s c o n t i n u i t i e s o f t h i s k i n d m a y b e a n a l y z e d , for e x a m p l e , as instances of m u t i p l e f u n c t i o n , the catastrophe a p p r o a c h , o u t i n e d b e l o w , w i l l always a p p l y . T h e g o a l is to see how i t t o o is consistent w i t h the p r i n c i p l e of gradualism. _____

Fig.

56.1

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F i g u r e 56.1 is a n a d a p t a t i o n of the e q u i l i b r i u m surface sketched i n F i g u r e 69.1 (a) of C h a p t e r I X , where i t is a p a r t of the m a t h e m a t i c a l t r e a t m e n t of catastrophe. T h e figure is used here i n a p u r e l y s y m b o l i c way, t o help us v i s u a l i z e the r e l a t i o n s h i p between g r a d u a l i s m a n d p u n c t u a t i o n . P o i n t s of the surface are supposed to be associated w i t h possible s t r u c t u r e s , where t h e structures at n e i g h b o r i n g p o i n t s are not very different. T h e t w o p a t h s , i n d i c a t e d b y arrows, are supposed t o suggest possible routes b y w h i c h one s t r u c t u r e S i m i g h t evolve i n t o a second one S2- T h e p u n c t u a t e d case i n volves a sudden change (catastrophe) w h i l e the other avoids the a b r u p t change b y s k i r t i n g the f o l d i n the surface. A s is clear i n accounts of p u n c t u a t e d e q u i l i b r i a , stasis ( s t a b i l i t y ) is as i m p o r t a n t t o the t h e o r y as are the s u d d e n changes. I n fact, because of the n a t u r e of s t a b i l i t y p h e n o m e n a , one m i g h t expect stasis rather t h a n continuous change t o be the r u l e . W e w i l l concentrate a t t e n t i o n o n p u n c t u a t i o n because of its i n t e r e s t i n g s t r u c t u r a l implications. Needless to say, most o f the f o l l o w i n g r e m a r k s are s p e c u l a t i v e a n d a d m i t t e d l y m a y not agree i n d e t a i l w i t h w h a t a c t u a l l y h a p p e n s . T h e y nevertheless show t h a t a p l a u s i b l e general e x p l a n a t i o n of the p h e n o m e n a c a n be f o r m u l a t e d i n f a i r l y precise s t r u c t u r a l t e r m s , a g a i n w i t h o u t v i o l a t i o n of the p r i n c i p l e of g r a d u a l i s m . T h e various p o s s i b i l i t i e s are i l l u s t r a t e d i n F i g u r e 56.2, suggesting three different p a t h s of d e v e l o p m e n t .

Fig.

56.2

P a t h (a) i n the figure i l l u s t r a t e s the t r a d i t i o n a l g r a d u a l process a n d needs no f u r t h e r discussion. P a t h (b) represents a p u n c t u a t i o n w h i c h is a n a r t i fact o f a coarse t i m e scale. I n other words, a sufficiently fine a n a l y s i s w o u l d reveal a g r a d u a l process, a l b e i t a very r a p i d one as m e a s u r e d i n geological t i m e . T h i s is p l a u s i b l e enough o n p u r e l y p h y s i c a l g r o u n d s , since the d i s r u p -

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t i o n o f a stable e q u i l i b r i u m ( t h r o u g h o r d i n a r y e v o l u t i o n a r y changes), i f it does not destroy the s t r u c t u r e , w o u l d be expected to result i n r a p i d changes of state u n t i l another r e l a t i v e l y stable e q u i l i b r i u m is e s t a b l i s h e d . T h i s poss i b i l i t y is e x p l i c i t l y i n c l u d e d b y G o u l d and E l d r i d g e [G12] as a l e g i t i m a t e e x a m p l e o f p u n c t u a t i o n . P a t h (c) represents true c a t a s t r o p h i c p u n c t u a t i o n , w h i c h is r e l a t i v e l y independent of the t i m e scale, a n d hence not reducible b y a finer analysis t o a g r a u d a l process. Case (c) is the i n t e r e s t i n g one for us, a n d o b v i o u s l y requires some a d d i t i o n a l discussion. A s u s u a l , there are two p r o b l e m s : to e x p l a i n how extreme changes of this k i n d can be consistent w i t h g r a d u a l i s m , a n d t o suggest how they m i g h t y i e l d s t r u c t u r e s t h a t s u r v i v e . T h e s o l u t i o n is a g e n e r a l i z a t i o n of t h a t i n the case of m u l t i p l e f u n c t i o n . It w i l l help to review first the n o t i o n o f a " s y s t e m " as opposed to a " s t r u c t u r e " . I n S e c t i o n 7, a s y s t e m was defined t o be "any c o l l e c t i o n o f 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 a y be identified w i t h i n i t " . F o r our purposes, the most obvious e x a m p l e o f a b i o l o g i c a l s y s t e m is a " w h o l e o r g a n i s m " . A b i o l o g i c a l s t r u c t u r e , o n the other h a n d , is u s u a l l y a s t r u c t u r e involved i n a b i o l o g i c a l s y s t e m , and defined i n t e r m s of a c e r t a i n c o m p l e x of properties of the l a t t e r . It may also be realized i n a variety of different b i o l o g i c a l systems. T h u s i n p a r t i c u l a r , a given i n d i v i d u a l w i l l be a representative of a certain type of o r g a n i s m i f it embodies the defining t r a i t s of t h a t o r g a n i s m . F i n a l l y , a is a b i o l o g i c a l s t r u c t u r e represented by a system consisting of r e p r o d u c t i v e l y c o m p a t i b l e i n d i v i d u a l s . It is evident t h a t the d e f i n i t i o n of an e x i s t i n g b i o l o g i c a l s t r u c t u r e m u s t i n c o r p o r a t e (perhaps i m p l i c i t l y ) the fact t h a t the s t r u c t u r e is a d a p t e d to its e n v i r o n m e n t .

species

T h e general p o s s i b i l i t y for c a t a s t r o p h i c changes of a b i o l o g i c a l s y s t e m , is based o n the fact t h a t the entire s y s t e m is i n e q u i l i b r i u m w i t h the env i r o n m e n t a n d w i l l u s u a l l y a d m i t p o t e n t i a l s t r u c t u r e s different f r o m the s t r u c t u r e w i t h w h i c h i t is n o r m a l l y associated. T h e r e f o r e , a l t h o u g h the l a t t e r s t r u c t u r e m a y be d o m i n a n t and tend to be preserved because o f its greater s t a b i l i t y , a secondary s t r u c t u r e could evolve g r a d u a l l y a n d m o r e or less i n d e p e n d e n t l y t o a p o i n t where it offers s t a b i l i t y c o m p a r a b l e t o the o r i g i n a l . T h i s a g a i n w o u l d set the stage for a s m a l l change, i n v o l v i n g e i ther the o r g a n i s m or its e n v i r o n m e n t , to p r e c i p i t a t e an a b r u p t shift to the e q u i l i b r i u m state p r o v i d e d by the secondary s t r u c t u r e . A l t h o u g h a change of t h i s k i n d m a y be thought of as a shift i n f u n c t i o n of a s y s t e m , it a c t u a l l y involves a shift f r o m one defining s t r u c t u r e to another. It is o b v i o u s , of course, t h a t speculations such as these, despite their reasonableness f r o m the p o i n t of view o f s t r u c t u r e s , can be s u b s t a n t i a t e d o n l y by a c t u a l e x a m ples f r o m the record.

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57. D e t e r m i n i n g Structures In t h i s section, we consider yet another possible way t h a t p h e n o m e n a w h i c h a p p e a r t o c o n t r a d i c t the p r i n c i p l e of g r a d u a l i s m m i g h t arise. R e c a l l t h a t the p r i n c i p l e is based on the a s s u m p t i o n t h a t e v o l u t i o n a r y change depends o n r a n d o m v a r i a t i o n s i n the genetic m a t e r i a l of an o r g a n i s m a n d t h a t the v a r i a t i o n s m u s t be s m a l l i n order to produce a n o r g a n i s m t h a t m i g h t be c o m p a t i b l e w i t h the e n v i r o n m e n t and t h u s have a chance of s u r v i v a l . A t the s a m e t i m e , a very large r a n d o m v a r i a t i o n has a v a n i s h i n g l y s m a l l chance o f b e i n g v i a b l e . Therefore, eschewing m i r a c l e s , one cannot e x p l a i n a p p a r e n t l y n o n g r a d u a l p h e n o m e n a i n t e r m s o f such v a r i a t i o n s . Observe, however, t h a t there is a difference between j u s t a large v a r i a t i o n a n d a large v a r i a t i o n . It is possible for a s m a l l r a n d o m change i n the genetic m a t e r i a l to trigger a large s t r u c t u r a l change i n the o r g a n i s m . T h i s c o u l d occur, for e x a m p l e , i f the s m a l l change i n v o l v e d a s t r u c t u r e t h a t d e t e r m i n e d a m u c h larger one. Such changes, w h i c h were already recognized by D a r w i n (see the first G o u l d q u o t a t i o n i n S e c t i o n 55), need not be a d a p t i v e . In other words, a s m a l l (local) change c o u l d be a d a p t i v e but m i g h t engender other changes t h a t were not a d a p t i v e . O n the other h a n d , a p l a u s i b l e s t r u c t u r a l analysis suggests t h a t this need not always be the case. A large v a r i a t i o n s t r u c t u r a l l y d e t e r m i n e d b y a s m a l l Tandom v a r i a t i o n m i g h t very w e l l be a d a p t i v e . T h e basic reason is t h a t we begin w i t h an o r g a n i s m t h a t is c o m p a t i b l e w i t h the e n v i r o n m e n t . I n p a r t i c u l a r , the f u l l s t r u c t u r e d e t e r m i n e d b y the s m a l l genetic s t r u c t u r e is e n v i r o n m e n t a l l y c o m p a t i b l e , so the d e t e r m i n i n g process m u s t also involve e n v i r o n m e n t a l connections. T h e i d e a is t h a t , because of these connections, the r e s u l t i n g s t r u c t u r a l change w i l l not be independent of the e n v i r o n m e n t (as w o u l d be expected i n the case of a large r a n d o m change), so could be c o m p a t i b l e w i t h the e n v i r o n m e n t . F o r e x a m p l e , a large s t r u c t u r a l change i n d u c e d by a s m a l l genetic change c o u l d appear as a collection o f s m a l l l o c a l changes, each of w h i c h , as far as the e n v i r o n m e n t is concerned, m i g h t lie w i t h i n the a d a p t a b i l i t y range o f the o r g a n i s m . T h i s suggests t h a t a large s t r u c t u r e change of this k i n d m i g h t very well have s u r v i v a l value. It also suggests a possible c a t a s t r o p h i c evol u t i o n a r y change analogous to a shift i n f u n c t i o n . A l t h o u g h the s t r u c t u r a l relationships conjectured to exist i n the above scenario are a d m i t t e d l y rather vague, they are not unreasonable. O n the other h a n d , whether or how instances of this k i n d a c t u a l l y occur is another m a t t e r . O n e p o s s i b i l i t y is by a m u t a t i o n of regulatory genes. T h i s c o u l d trigger m a j o r changes i n the o r g a n i s m ' s ontogeny t h a t m i g h t , for the above reasons, be c o m p a t i b l e w i t h the e n v i r o n m e n t .

random

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58.

STRUCTURALISM AND

STRUCTURES

Convergent Evolution

Convergent e v o l u t i o n is the independent development of nearly i d e n t i c a l features i n t w o different a n d essentially unrelated o r g a n i s m s . Such features are not i n h e r i t e d f r o m a c o m m o n ancestor, so are instances of the analogies (as opposed t o homologies) discussed i n Section 50. T h e r e are m a n y e x a m ples o f convergence, such as the c o m m o n f o r m e x h i b i t e d b y a n i m a l s t h a t s w i m or by a n i m a l s t h a t fly. B u t some of the most s t r i k i n g l i v i n g examples are f o u n d a m o n g the m a r s u p i a l s of A u s t r a l i a . ( T h e r e are also m a n y e x t i n c t examples f o u n d i n S o u t h A m e r i c a . ) These are the m a r s u p i a l analogues of p l a c e n t a l m a m m a l s . It is r e m a r k a b l e , to say the least, t h a t t w o v i r t u a l l y u n r e l a t e d a n i m a l s c a n , even o n the surface, be so m u c h a l i k e . It is generally believed t h a t analogies are s o m e h o w a result of e v o l u t i o n ary development w i t h i n nearly i d e n t i c a l e n v i r o n m e n t s , a l t h o u g h i t is rather difficult to u n d e r s t a n d at first how the results c a n be so precise. T h e p r o cess tends to a p p e a r a b i t m o r e p l a u s i b l e f r o m a general s t r u c t u r a l p o i n t of v i e w , w h i c h we w i l l now a t t e m p t to sketch. T h i s , as i n the case o f most of our r e m a r k s i n this c h a p t e r , is not so m u c h an e x p l a n a t i o n of convergence, b u t m e r e l y a way of l o o k i n g at the p h e n o m e n o n . C o n s i d e r , for e x a m p l e , the t h y l a c i n e , a c a r n i v o r o u s wolflike m a r s u p i a l of T a s m a n i a . E x c e p t p o s s i b l y for the black stripes across its back, a casual observer m i g h t have difficulty i n d i s t i n g u i s h i n g this a n i m a l f r o m an o r d i n a r y wolf. T h e resemblances, however, are m a i n l y e x t e r n a l . B e l o w the surface, the t w o are different i n m a n y details easily discerned by an expert. B u t this fact a c t u a l l y makes the w h o l e t h i n g even more m y s t e r i o u s . If the a n i m a l s were more a l i k e i n t e r n a l l y , then it w o u l d be easier t o u n d e r s t a n d how s i m i l a r e n v i r o n m e n t s m i g h t produce the external s i m i l a r i t i e s . It is, of course, these basic differences w h i c h u l t i m a t e l y "prove" t h a t the s i m i l a r i t i e s are indeed analogies rather t h a n homologies. It is clear t h a t a c o m m o n ancestor of m a r s u p i a l s a n d p l a c e n t a l m a m m a l s was a n e a r l y generalized m a m m a l w h i c h possessed the basic characteristics of a t y p i c a l four-legged a n i m a l . Therefore, the p o t e n t i a l to develop i n t o a " d o g l i k e " a n i m a l was present even before the d i v i s i o n . N o w suppose t h a t a l i n e of m a r s u p i a l s and a line o f p l a c e n t a l m a m m a l s undergo an e v o l u t i o n a r y development under essentially i d e n t i c a l e n v i r o n m e n t a l c o n d i t i o n s . T h u s , we have two different b i o l o g i c a l systems w h i c h have a d a p t e d t o the same e x t e r n a l e n v i r o n m e n t despite differences i n t h e i r i n t e r n a l s t r u c t u r e s . F i n a l l y , suppose t h a t a s u b s t r u c t u r e of the i m m e d i a t e e n v i r o n m e n t is d e t e r m i n i n g (Section 26). I n the present case the s u b s t r u c ture w o u l d be associated w i t h existence as a l o n g - b o d i e d , r u n n i n g p r e d a t o r . T h e n it w i l l follow t h a t the e x t e r n a l characteristics, d e t e r m i n e d i n the a n i m a l s b y t h a t s u b s t r u c t u r e , m u s t be essentially i d e n t i c a l . In other words, convergence w o u l d result as a more or less a u t o m a t i c consequence o f the

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supposed s t r u c t u r a l d e t e r m i n a t i o n . W h a t is needed for a f u l l u n d e r s t a n d i n g of convergence is a precise exp l a n a t i o n i n p h y s i c a l t e r m s of how the e n v i r o n m e n t a l s t r u c t u r e a c t u a l l y determines the e x t e r n a l properties of a g i v e n i n t e r a c t i n g o r g a n i s m . I n some cases, such as the s w i m m i n g or flying a n i m a l s , the relative s i m p l i c i t y o f the p h y s i c a l c o n d i t i o n s enables one to conjecture w i t h some confidence how the process m i g h t w o r k . I n other cases, however, the p r o b l e m is far more difficult, a n d the conjectured e x p l a n a t i o n s are c o r r e s p o n d i n g l y less precise. 59.

Anthropomorphism

A n t h r o p o m o r p h i s m , the practice of a s c r i b i n g h u m a n q u a l i t i e s t o beings or t h i n g s n o n h u m a n , has always been a factor, either consciously or u n c o n sciously, i n m a n ' s v i e w o f the w o r l d a r o u n d h i m . In m o d e r n times, h o w ever,the practice has been v i g o r o u s l y condemned as t o t a l l y unscientific. A t the same t i m e , such c o n d e m n a t i o n has p r o b a b l y h a d l i t t l e influence o n the t h i n k i n g o f o r d i n a r y people, or perhaps even o n m o s t scientists i n their everyday dealings w i t h the r e a l w o r l d . M o s t of us are prone to observe, for e x a m p l e , t h a t a pet parakeet does one of its t r i c k s because " i t wants our a t t e n t i o n " or t h a t the c o m p u t e r " t h o u g h t t h a t we d i d n ' t w a n t t o save the i t e m because we neglected to tell i t o t h e r w i s e " . T h e m a i n effect of the rejection of a n t h r o p o m o r p h i s m as unscientific has been t o suppress m o s t o f the i n f o r m a l expressions of i t i n the l i t e r a t u r e . Therefore, the appearance i n recent years o f suggestions by a few scientists t h a t i n some c i r c u m s t a n c e s the practice m a y be reasonable, comes as a definite s u r p r i s e . It w i l l be sufficient for our purposes t o l i m i t a t t e n t i o n to a single e x a m p l e , J o h n M c C a r t h y , one of the pioneers i n the field of a r t i f i c i a l intelligence. H i s views o n the subject are o u t l i n e d i n an a r t i c l e , w h i c h appeared i n

"The Little Thoughts of Thinking Machines", Today[Ml].

Psychology

M c C a r t h y begins by r e m a r k i n g o n w h a t we have called the " b l a c k b o x " a p p r o a c h to c o m p l e x systems (Section 27), by w h i c h one may be able to deal w i t h the e x t e r n a l f u n c t i o n i n g of systems, such as electric lights a n d telephones, w i t h o u t u n d e r s t a n d i n g their i n t e r n a l s t r u c t u r e s . H e then goes o n t o m a k e the f o l l o w i n g c o m m e n t s concerning future p r o b l e m s associated w i t h very c o m p l e x computer-based systems: In the next century, w e ' l l be increasingly faced w i t h m u c h more c o m p l e x c o m p u t e r - b a s e d systems. It w o n ' t be necessary for m o s t people to k n o w very m u c h about how they work i n t e r n a l l y , but w h a t we w i l l have to k n o w a b o u t t h e m i n order t o use t h e m is more c o m p l e x t h a n w h a t we need to k n o w a b o u t electric lights a n d telephones. A s o u r d a i l y lives involve more sophisticated c o m p u t e r s , we w i l l find t h a t a s c r i b i n g l i t t l e t h o u g h t s t o machines w i l l be i n c r e a s i n g l y useful i n

176

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AND STRUCTURES

u n d e r s t a n d i n g how to get the most g o o d out of t h e m . M u c h t h a t we w i l l need to know concerns the i n f o r m a t i o n stored i n c o m p u t e r s , w h i c h is w h y we find ourselves using words l i k e " k n o w s " , " t h i n k s " a n d " w a n t s " i n referring to machines, even t h o u g h these machines are very different f r o m h u m a n s and these words arose f r o m the h u m a n need to t a l k to other h u m a n s , [p. 45] In his l i m i t e d advocacy, M c C a r t h y distinguishes between " g o o d " a n d " b a d " a n t h r o p o m o r p h i s m . It is g o o d i f " i t says s o m e t h i n g t h a t cannot as conveniently be s a i d some other w a y " , b u t is b a d i f i t ascribes e m o t i o n s or personalities to machines. W o r k e r s i n artificial intelligence, whose objective i t is t o p r o g r a m s o m e t h i n g r e s e m b l i n g intelligence i n t o c o m p u t i n g machines, believe t h a t m u c h b e h a v i o r can be u n d e r s t o o d u s i n g the " p r i n ciple of r a t i o n a l i t y " . Beliefs a n d goals are ascribed i n accordance w i t h this p r i n c i p l e , w h i c h says r o u g h l y t h a t certain behavior m a y be accounted for b y o b s e r v i n g t h a t a " m a c h i n e or person or a n i m a l does w h a t i t t h i n k s w i l l achieve its g o a l s " . M c C a r t h y discusses i n some d e t a i l the p r o b l e m s of a p p l y i n g the p r i n c i p l e i n a r t i f i c i a l intelligence a n d w i n d s up w i t h the c a u t i o n t h a t "we m u s t be careful not to ascribe properties t o a machine t h a t the p a r t i c u l a r machine doesn't have. W e h u m a n s can easily fool o u r selves when there is s o m e t h i n g we w a n t to believe". T h i s is, o f course, the s t a n d a r d c r i t i c i s m of a n t h r o p o m o r p h i s m i n general. D e s p i t e the p i t f a l l s , there seems to be no doubt t h a t a l i m i t e d practice of a n t h r o p o m o r p h i s m c a n indeed be h e l p f u l , over a n d above the superficial convenience of d e a l i n g i n f o r m a l l y w i t h n o n h u m a n systems. It is therefore n a t u r a l t o ask how any practice w h i c h appears at first t o be so c o m p l e t e l y unjustified c a n p l a y such a role. Is there more to i t t h a n j u s t a way of f o r m u l a t i n g ideas otherwise difficult to express? A s a m a t t e r of fact, there does seem to be a m u c h deeper and also more i n t e r e s t i n g e x p l a n a t i o n . A given b e h a v i o r , w h i c h is a c a n d i d a t e for a n t h r o p o m o r p h i c i n t e r p r e t a t i o n , m a y be thought o f as a s t r u c t u r e w h i c h develops f r o m a n i n i t i a l d e t e r m i n i n g s t r u c t u r e . In other words, when the i n i t i a l s t r u c t u r e is a c t i v a t e d , the b e h a v i o r w i l l follow, p r o d u c i n g a "desired" goal. Such a s t r u c t u r e m a y a d m i t a variety of representations, i n v o l v i n g machines, a n i m a l s , or h u m a n beings. T h e r e is one t h i n g , however, w h i c h sets the last representation a p a r t f r o m the others. It is the fact t h a t h u m a n beings m a y at the outset be more or less aware of the f u l l s t r u c t u r e . In other words, they m a y have a m e n t a l p i c t u r e of w h a t w i l l h a p p e n when the i n i t i a l s t r u c t u r e is a c t i v a t e d . It is because of this awareness t h a t h u m a n s c a n l e g i t i m a t e l y say t h a t they a c t i v a t e d the i n i t i a l s t r u c t u r e i n order to produce the desired g o a l . O b s e r v e , however, t h a t awareness a c t u a l l y has n o t h i n g t o do w i t h the s t r u c t u r e i t self. A s l o n g as the i n i t i a l s t r u c t u r e is a c t i v a t e d , whether b y pressing keys on a c o m p u t e r i n p u t , by a s p e c i a l s t i m u l u s or i n s t i n c t i n an a n i m a l , or by

VIII. B I O L O G I C A L S T R U C T U R E S

177

r a t i o n a l choice i n a h u m a n , the result w i l l be the same. T h i s a n a l y s i s indicates clearly when an a n t h r o p o m o r p h i c i n t e r p r e t a t i o n may help one to v i s u a l i z e more clearly the process i n question and hence to deal w i t h i t m o r e effectively. O n the other h a n d , t r o u b l e w i l l arise i f the assumed n o n h u m a n s t r u c t u r e representations do not e x i s t . It is also conceivable t h a t an i n i t i a l s t r u c t u r e m i g h t be perceived b y h u m a n s to develop t o w a r d a c e r t a i n goal i n a q u i t e different and perhaps more c o m p l e x m a n n e r t h a n is a c t u a l l y the case. I n other words, despite the c o m m o n g o a l , the t w o representations m i g h t a c t u a l l y involve different structures. I n this case, an a n t h r o p o m o r p h i c i n t e r p r e t a t i o n might read far more i n t o the n o n h u m a n b e h a v i o r t h a n a c t u a l l y exists. C o n s i d e r finally the case of w h i c h is the s t u d y of, or belief i n , evidence for design a n d purpose i n n a t u r e . It m a y also be r a t i o n a l i z e d as an a i d to c o m p r e h e n d i n g certain complex p h e n o m e n a , q u i t e a p a r t f r o m the u s u a l a p p e a l to unscientific p h i l o s o p h i c a l or religious a s s u m p t i o n s . F o r exa m p l e , teleological expressions sometimes appear even i n t e c h n i c a l scientific e x p o s i t i o n . T h e p o i n t here is t h a t teleology is u l t i m a t e l y a n t h r o p o m o r p h i c i n character a n d is s i m i l a r l y based o n a h u m a n o v e r a l l awareness of t i m e dependent p h e n o m e n a . A l t h o u g h an u n f o l d i n g of the p h e n o m e n o n is quite independent of awareness, the n a t u r a l a n t h r o p o m o r p h i c tendency to read "design a n d purpose" i n t o it m a y s t i l l help one t o v i s u a l i z e the process. A s i n the case of a n t h r o p o m o r p h i s m proper, the a p p r o a c h w i l l be nonscientific o n l y i f the design a n d purpose are assumed to be s o m e t h i n g more t h a n an expression of h u m a n awareness.

teleology,

CHAPTER

SPACE

IX

STRUCTURES AND

60.

STABILITY

Introduction M a t h e m a t i c a l ideas have p l a y e d an i m p o r t a n t role i n the preceding chapters, t h o u g h i n m o s t cases t h a t role has been i n d i r e c t a n d sometimes not at a l l obvious t o anyone other t h a n the a u t h o r . T h i s chapter is different i n t h a t the m a t h e m a i c s is m u c h m o r e d i r e c t l y involved and m u c h of the m a t e r i a l c o u l d be given 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 . W i t h the exc e p t i o n o f the last two sections, however, we resist the t e m p t a t i o n to offer such a t r e a t m e n t a n d t r y instead to b r i n g out the u n d e r l y i n g ideas w i t h o u t the technicalities. A t the same t i m e , a serious a t t e m p t to e x p l a i n a c t u a l m a t h e m a t i c a l concepts w i l l i n e v i t a b l y involve the use o f some m a t h e m a t i c a l formalities. T h e chapter is concerned w i t h some s p e c i a l cases of t w o very general s t r u c t u r e topics: (1) certain relationships between structures a n d s u b s t r u c tures, a n d (2) s t a b i l i t y properties o f s t r u c t u r e s . T h e first concerns the fact t h a t a given s t r u c t u r e is s e l d o m isolated, b u t w i l l t y p i c a l l y exist as a s u b structure of a larger one. For e x a m p l e , a s t r u c t u r e representation of an object i n space w i l l n o r m a l l y appear as a s u b s t r u c t u r e of t h r e e - d i m e n s i o n a l E u c l i d e a n space. F u r t h e r m o r e , an i s o m o r p h i s m of the representation m u s t somehow involve the E u c l i d e a n space i f the t h r e e - d i m e n s i o n a l i t y is to be preserved. A s i m i l a r o b s e r v a t i o n applies to the more general case of any s t r u c t u r e and its e x t e r n a l properties {Section 8). In order for a n i s o m o r p h i s m of the s t r u c t u r e t o preserve a given e x t e r n a l property, it m u s t recognize the e m b e d d i n g associated w i t h the d e f i n i t i o n of t h a t property. O n e feature of the second t o p i c , o n s t a b i l i t y properties, was t o u c h e d u p o n i n S e c t i o n 12, where we called it a " p r i n c i p l e o f s t r u c t u r a l s t a b i l i t y " . T h e p r i n c i p l e asserts t h a t , i f two s t r u c t u r e representations are "sufficiently near or s i m i l a r " to one another, then they w i l l be i s o m o r p h i c . It was i l l u s t r a t e d i n Section 12 w i t h a s i m p l e e x a m p l e c o n c e r n i n g the p e r c e p t i o n o f a circle f r o m a d r a w i n g . T h e r e are m a n y other more c o m p l e x a n d subtle examples (such as precision of c o m m u n i c a t i o n m e n t i o n e d i n Section 30) w h i c h suggest a very b r o a d 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 . A l t h o u g h it is not at a l l clear e x a c t l y w h a t is h a p p e n i n g i n most of these cases, i t is obvious t h a t s o m e such p r i n c i p l e m u s t a p p l y despite the u n i v e r s a l difficulty of s p e l l i n g out the 179

STRUCTURALISM AND

180

STRUCTURES

d e t a i l s . A c o m m o n p r o b l e m is the f o r m u l a t i o n o f an 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 structures i n v o l v e d . In order to suggest the n a t u r e o f the p r o b l e m , we offer a precise m a t h e m t i c a l t r e a t m e n t for s i m p l e p o i n t - l i n e structures i n Sections 69 a n d 70, the last two sections of the chapter. I n a d d i t i o n to the above "nearness" e x a m p l e s , there is a variety of other m o r e s u b t l e e x a m p l e s t h a t also e x h i b i t i n one way or another s t a b i l i t y p h e n o m e n a i n v o l v i n g s t r u c t u r e s . O n e of these is " c a t a s t r o p h e t h e o r y " . R o u g h l y s p e a k i n g , the idea b e h i n d the l a t t e r is t h a t c e r t a i n systems, under a c o n t i n u o u s change o f c o n d i t i o n s , tend to persist i n a given state (i.e., r e m a i n stable) up t o a c r i t i c a l p o i n t at w h i c h an a d d i t i o n a l s m a l l change causes the s y s t e m t o shift a b r u p t l y (i.e., c a t a s t r o p h i c a l l y ! ) t o a different (stable) state. A l t h o u g h the subject has its o r i g i n s i n m a t h e m a t i c s , where the p h e n o m e n a i n question m a y be described very precisely, there are m a n y examples of analogous p h e n o m e n a for systems t h a t are not m a t h e m a t i c a l in character. S o m e of the l a t t e r f r o m biology were discussed i n S e c t i o n 56. E x a m p l e s of b o t h k i n d s are i n c l u d e d i n Sections 65-68, w h i c h are devoted to a v e r y b r i e f account of the theory. T h e next section c o n t a i n s an o u t l i n e of elementary m a t e r i a l concerni n g E u c l i d e a n spaces. T h e s e spaces provide the s e t t i n g for m a n y of the e x a m p l e s discussed below. 61. E u c l i d e a n Spaces T h r e e - d i m e n s i o n a l E u c l i d e a n space is the f o r m a l m a t h e m a t i c a l represent a t i o n of the p h y s i c a l space i n w h i c h we live. It is t r a d i t i o n a l l y defined i n t e r m s of a s y s t e m o f a x i o m s not essentially different f r o m those o r i g i n a l l y l a i d d o w n by E u c l i d a n d s t u d i e d to this day i n elementary geometry. A l l of the a x i o m s , except p o s s i b l y the parallel a x i o m , represent o b v i o u s properties o f p h y s i c a l space. In other words, they are consistent w i t h the n a t u r a l space i n t u i t i o n possessed by v i r t u a l l y everyone as a result of ( b o t h i n d i v i d u a l a n d e v o l u t i o n a r y ) experiences w i t h the e n v i r o n m e n t . U n d e f i n e d t e r m s m e n t i o n e d i n the a x i o m s are the " p o i n t s " , " l i n e s " , a n d " p l a n e s " , where the lines a n d planes are s p e c i a l sets of p o i n t s . E u c l i d e a n space is a s t r u c t u r e , i n the sense of our d e f i n i t i o n , i n w h i c h p o i n t s , lines, a n d planes are the objects a n d relations are specified b y the a x i o m s . T h e s u b s t r u c t u r e o b t a i n e d b y r e s t r i c t i o n to a single plane i n the space, is called a " E u c l i d e a n p l a n e " or a " t w o - d i m e n s i o n a l E u c l i d e a n space". O n e o f the basic concepts i n o r d i n a r y E u c l i d e a n space is the n o t i o n of p o i n t s (also called a distance denoted by for any two points P a n d Q of the space. It is p o s i t i v e or zero, zero o n l y if a n d is s y m m e t r i c i n and (i.e., | P Q | = It also satisfies the i n e q u a l i t y , \PR\ < \PQ\ + \QR\, w h i c h h o l d s for any three points a n d asserts t h a t the length of one side of a t r i a n g l e is never

distance between X = Y,

P.Q.R,

function),

P

Q

\PQ\

\QP\).

IX. S P A C E S T R U C T U R E S A N D

181

STABILITY

greater t h a n the s u m o f the lengths of the other two sides. T h e e q u a l i t y holds o n l y i f the points are collinear a n d hence d e t e r m i n e a "degenerate" triangle. A E u c l i d e a n space m a y also be represented " a n a l y t i c a l l y " b y use o f a (usually C a r t e s i a n ) c o o r d i n a t e s y s t e m . T h i s leads to a r e p r e s e n t a t i o n of p o i n t s by ordered pairs (x,y) o f real n u m b e r s , i n two d i m e n s i o n s , a n d by ordered triples i n three d i m e n s i o n s . T h e real n u m b e r s are the coordinates o f the p o i n t s . L i n e s a n d planes are represented (or defined) by l i n e a r equations i n v o l v i n g the c o o r d i n a t e variables. T h e v a r i o u s relations a m o n g these objects specified by the a x i o m s (such as the intersection p r o p erties) m a y be w o r k e d out using elementary algebra. T h e distance \AA'\ between t w o p o i n t s (a,fc,c) a n d i n three d i m e n s i o n s , is given b y the f o r m u l a ,

(x,y,z),

x,y,z

A =

\AA'\

=

{{a - a')2

A' = (a',b',^),

+ (b - b')2

+ (c-

c') ] ' . 2

1

2

A s i m i l a r t w o - t e r m f o r m u l a holds i n two d i m e n s i o n s . If P is a g i v e n p o i n t of the space a n d e is an a r b i t r a r y positive n u m b e r , then the set of a l l points such t h a t < e is c a l l e d a of A subset of the space is s a i d t o be i f each of its points a d m i t s a n e i g h b o r h o o d contained w i t h i n the set.

Q

\PQ\ open

neighborhood

P.

analytic geometry,

A l l of t h i s is p a r t of the subject of a n d it m a y be verified t h a t we indeed have a representation of a E u c l i d e a n space. N o t e t h a t the i n t r o d u c t i o n o f a c o o r d i n a t e s y s t e m a m o u n t s to a representation of the geometry system i n t e r m s of the real n u m b e r s y s t e m . 62. S u b s t r u c t u r e s of E u c l i d e a n Space A given s y s t e m is u s u a l l y regarded as characterized b y c e r t a i n o f its s t r u c t u r e properties. F r o m t h i s p o i n t of v i e w , the general i s o m o r p h i s m p r o b l e m is to construct t r a n s f o r m a t i o n s of the s y s t e m t h a t preserve these characteristic properties. It is not difficult to see t h a t the c o l l e c t i o n of a l l t r a n s f o r m a t i o n s of a s y s t e m to itself, t h a t preserve specified s t r u c t u r a l properties, c o n s t i t u t e a group under the t r a n s f o r m a t i o n p r o d u c t defined i n Section 2 1 . In this a n d the next section, we describe s i m p l e examples w h i c h b r i n g out v a r i o u s aspects of the above p r o b l e m i n a E u c l i d e a n space, a n d at the same t i m e set the stage for a special e x a m p l e o f s t r u c t u r a l s t a b i l i t y i n Section 64. W e k n o w f r o m elementary geometry t h a t the m a p p i n g s , or t r a n s f o r m a tions, of a E u c l i d e a n space t h a t preserve the E u c l i d e a n s t r u c t u r e are the r i g i d m o t i o n s . T h e y preserve distances, as w e l l as lines, planes, a n d angles, a n d c o n s t i t u t e a g r o u p , the E u c l i d e a n g r o u p , w h i c h is generated ( i n the sense o f Section 21) by t r a n s l a t i o n s , r o t a t i o s , a n d reflections. These are

182

STRUCTURALISM

AND STRUCTURES

the t r a n s f o r m a t i o n s t h a t i m p l e m e n t the congruences i n E u c l i d e a n geometry. F o r e x a m p l e , a t r i a n g l e is a s p e c i a l s u b s t r u c t u r e of E u c l i d e a n space and is transformed by each element of the group i t o a congruent t r i a n g l e . A l t h o u g h the r i g i d t r a n s f o r m a t i o n s are e x a c t l y those t h a t preserve the s t r u c t u r e of E u c l i d e a n space, we concluded i n Section 6 t h a t the "size" of a b u i l d i n g s t r u c t u r e was irrelevant to its a c t u a l s t r u c t u r e . In other words, it is the " s h a p e " o f a figure t h a t is often i m p o r t a n t to s t r u c t u r e , not the d i m e n s i o n s . T h i s a m o u n t s to a l l o w i n g t r a n s f o r m a t i o n s t h a t o n l y preserve relative distances as opposed to a c t u a l distances. These are t r a n s f o r m a t i o n s such t h a t , i f d i s t i n c t p o i n t s are t a k e n i n t o p o i n t s then m u s t be e q u a l t o |AB|/|C_>|. S u c h t r a n s f o r m a t i o n s c o n s t i t u t e a g r o u p , called the g r o u p . It o b v i o u s l y contains the r i g i d t r a n s f o r m a t i o n s as a s u b g r o u p . Its elements, w h i c h s t i l l preserve lines, planes, and angles, are called s i m i l a r i t y t r a n s f o r m a t i o n s , a n d m a y take a t r i a n g l e , for e x a m p l e , i n t o a s i m i l a r rather t h a n a congruent t r i a n g l e . In a l l o w i n g i s o m o r p h i s m s o f this k i n d , we o b v i o u s l y a b a n d o n those s t r u c t u r e properties of E u c l i d e a n space t h a t are scale-dependent.

A,B,C,D

\A'B'\l\C'Ly\

A',B',C',D',

similarity

F i n a l l y , i f the t r a n s f o r m a t i o n s are o n l y required t o preserve lines ( a n d hence planes), the result is a larger g r o u p , the group. Afhne transform a t i o n s need not preserve either angles or ratios of distances, but o b v i o u s l y w i l l t r a n s f o r m triangles i n t o triangles. In fact, a l l (nondegenerate) t r i a n gles are equivalent under the affine g r o u p . T h i s means t h a t , g i v e n any two t r i a n g l e s , there always exists an affine t r a n s f o r m a t i o n of the space t h a t takes one i n t o the other. W i t h this step, the E u c l i d e a n space is i n effect replaced b y an whose properties depend on p o i n t s , lines, a n d their intersection properties.

affine

affine space

T h e three groups described above give progressively weaker notions of s t r u c t u r e i s o m o r p h i s m . In other words, the larger the g r o u p , the easier it is for t w o structures to be i s o m o r p h i c a l l y equivalent. These r e m a r k s w i l l be i l l u s t r a t e d by some examples f r o m a n a l y t i c geometry i n the next section. 63. T h e C o n i c Sections T h e conic sections are substructures o f a E u c l i d e a n plane a n d consist of the f a m i l i a r circles, ellipses, p a r a b o l a s , a n d h y p e r b o l a s . A l t h o u g h the conies m a y be o b t a i n e d as plane sections of a r i g h t c i r c u l a r cone, it is g e o m e t r i c a l l y more convenient t o describe t h e m i n t e r m s o f the f o c u s - d i r e c t r i x property. T h i s is a characteristic geometric p r o p e r t y and involves a fixed p o i n t F (the a fixed line (the assumed not be c o n t a i n a n d an a r b i t r a r y fixed p o s i t i v e real n u m b e r (the

focus),

L

directrix) e

eccentricity).

F,

In the plane determined by F a n d L, denote by P a n a r b i t r a r y p o i n t and by D the p e r p e n d i c u l a r p r o j e c t i o n of P onto L. I n t h i s setup, the locus of points P such t h a t \FP\ = e\DP\ is a conic w i t h d i r e c t r i x L, focus F, and

IX. S P A C E S T R U C T U R E S A N D

STABILITY

183

eccentricity e. T h e result is a n ellipse, p a r a b o l a , or h y p e r b o l a a c c o r d i n g as e < 1, e = 1, or e > 1. N o t i c e t h a t the d e f i n i t i o n suggests i m m e d i a t e l y a s t r u c t u r a l d e s c r i p t i o n of a conic. It is a s u b s t r u c t u r e of a E u c l i d e a n p l a n e whose objects consist of the focus F a n d d i r e c t r i x L p l u s the set of a l l p o i n t s P such t h a t the t e r n a r y r e l a t i o n \FP\ = e\DP\ is satisfied. T h e one defect of the d e f i n i t i o n is t h a t i t gives no circles, except p e r h a p s the degenerate p o i n t - c i r c l e "F" o b t a i n e d b y t a k i n g e = 0. Nevertheless, a r b i t r a r y circles are often i n c l u d e d a m o n g the ellipses a n d regarded as h a v i n g zero eccentricity. T h e three conies covered b y the d e f i n i t i o n are i l l u s t r a t e d i n F i g u r e 63.1.

Parabola: e = 1 .

Hyperbola: Fig.

e>1.

63.1

Because s i m i l a r i t y t r a n s f o r m a t i o n s of the p l a n e preserve lines a n d r a t i o s of distances, i t is evident t h a t t h e y w i l l t r a n s f o r m each conic i n t o a s i m i l a r one w i t h the same eccentricity. It is also t r u e t h a t , i f t w o conies possess the s a m e eccentricity, t h e n there exists a s i m i l a r i t y t h a t takes one i n t o the other. I n p a r t i c u l a r , a l l p a r a b o l a s {and also a l l circles) are s i m i l a r , b u t t w o ellipses or two h y p e r b o l a s are not s i m i l a r unless their eccentricities are equal. A l t h o u g h one c a n perceive the difference i n shape between t w o ellipses w i t h different eccentricities, we also recognize t h a t t h e y have i n c o m m o n the p r o p e r t y of b e i n g a n ellipse. I n ther words, there is a s t r u c t u r e c o m m o n

184

STRUCTURALISM AND

STRUCTURES

to a l l ellipses independent o f their eccentricities. T h e same is also true of h y p e r b o l a s a n d , o f course, p a r a b o l a s . F u r t h e r m o r e , a l t h o u g h it is not q u i t e as o b v i o u s , a s i m i l a r r e m a r k holds for all o f the conies. It t u r n s out t h a t the affine group i n the p l a n e is the group t h a t preseves the structures c o m m o n t o each o f the three classes of conies, the p a r a b o l a s , ellipses ( i n c l u d i n g circles), a n d hyperbolas. T h i s means t h a t affine t r a n s f o r m a t i o n s m a p each class i n t o itself, a n d , given any two conies i n the same class, there w i l l exist an affine t r a n s f o r m a t i o n w h i c h t r a n s f o r m s one i n t o the o t h e r . T h i s is the f o r m a l i z a t i o n of a p r o p e r t y t h a t we already expected to be t r u e a b o u t conies. For the sake o f completeness, it is desirable to give a very brief e x p l a n a t i o n of the above less obvious r e m a r k a b o u t s t r u c t u r e c o m m o n to all of the conies. T h e t r u t h of the r e m a r k m i g h t already be expected f r o m the fact, m e n t i o n e d o n the next page, t h a t each conic m a y be represented i n a n a l y t i c geometry as the g r a p h of a q u a d r a t i c e q u a t i o n . In order t o m a k e t h i n g s precise i n t e r m s o f t r a n s f o r m a t i o n s , however, we need to replace the E u c l i d e a n plane by a projective p l a n e . T h e l a t t e r m a y be defined a x i o m a t i c a l l y , t h u s p r o v i d i n g a basis for T h e a x i o m s are s i m i l a r t o the E u c l i d e a n a x i o m s (but w i t h o u t the p a r a l l e l a x i o m or a n o t i o n of distance), a n d the f u n d a m e n t a l objects are s t i l l p o i n t s and lines. T r a n s f o r m a t i o n s w h i c h m a p a projective plane pointwise onto itself a n d preserve the p r o jective geometry c o n s t i t u t e a g r o u p , the and its elements are c a l l e d In a d d i t i o n t o a p r o j e c t i v e p l a n e , there are higher d i m e n s i o n a l projective spaces.

projective geometry.

projeciivities.

projective group,

A s was r e m a r k e d i n Section 19, a projective plane m a y be represented b y a E u c l i d e a n plane plus a line at infinity consisting of points at infinity. N o t e , however, t h a t the points a n d l i n e at infinity are no different f r o m any other points and lines i n the projective plane. T h e y are d i s t i n g u i s h e d o n l y t h r o u g h their r e l a t i o n s h i p to the E u c l i d e a n p l a n e regarded as embedded i n the p r o j e c t i v e p l a n e . I n this s e t t i n g , because the graphs o f p a r a b o l a s and h y p e r b o l a s "go off t o i n f i n i t y " , i t is n a t u r a l t o e x t e n d t h e m i n the project i v e p l a n e . W h e n t h i s is p r o p e r l y done, a p a r a b o l a w i l l c o n t a i n one p o i n t at i n f i n i t y a n d a h y p e r b o l a w i l l c o n t a i n two. A n ellipse is c o n t a i n e d entirely w i t h i n the E u c l i d e a n p o r t i o n so does not involve any infinite p o i n t s . Because the designated l i n e at infinity is not generally preserved by a p r o j e c t i v i t y , the above setup m a y also f a i l to be preserved. I n p a r t i c u l a r , an ellipse m a y not be t r a n s f o r m e d i n t o another ellipse. It t u r n s out t h a t p r o j e c t i v i t i e s do m a p conies i n t o conies, a n d , given any p a i r of (nondegenerate) conies, there exists a p r o j e c t i v i t y t h a t m a p s one i n t o the o t h e r . A n y t w o conies are a c c o r d i n g l y s a i d t o be " p r o j e c t i v e l y e q u i v a l e n t " . T h u s , we o b t a i n the desired f o r m a l i z a t i o n o f the n o t i o n t h a t the conies possess a c o m m o n s t r u c t u r e . F i n a l l y , i f a p r o j e c t i v i t y leaves a

IX. S P A C E S T R U C T U R E S A N D

STABILITY

!85

" l i n e at i n f i n i t y " i n v a r i a n t , then i t induces an affine t r a n s f o r m a t i o n i n the associated E u c l i d e a n plane. Conversely, every affine t r a n s f o r m a t i o n of the given E u c l i d e a n plane is the r e s t r i c t i o n of a p r o j e c t i v i t y . In this sense the affine group is a subgroup of the projective g r o u p . N o t i c e t h a t , i n the above discussion of the conies, we have regarded t h e m as substructures of successively weaker geometric s t r u c t u r e s . F i r s t they were defined i n t e r m s of E u c l i d e a n geometry, w h i c h was i m m e d i a t e l y a b a n d o n e d for the geometry o f shapes. N e x t i n order were the affine a p r o j e c t i v e geometries of the p l a n e . A s s o c i a t e d w i t h each geometry was its c h a r a c t e r i s t i c group o f s t r u c t u r e - p r e s e r v i n g i s o m o r p h i s m s : the s i m i l a r i t y g r o u p , the affine g r o u p , a n d the projective g r o u p respectively. I n each case, the abstract structures d e t e r m i n e d by the conies were defined b y a n o t i o n of i s o m o r p h i s m p r o v i d e d by the group associated w i t h the ambient geometric s t r u c t u r e . T h u s i n the shape geometry, there is a (conic) s t r u c t u r e for each value of the eccentricity. In the affine geometry, there are three structures associated respectively w i t h the ellipses, p a r a b o l a s , and h y p e r b o l a s . F i n a l l y , i n the projective geometry there is o n l y one s t r u c t u r e c o m m o n t o a l l of the conies. It is possible ( a n d also interesting) to s t u d y conies f r o m a s t r i c t l y geom e t r i c p o i n t o f view, either as conic sections or using the f o c u s - d i r e c t r i x d e f i n i t i o n , b u t the most powerful a n d convenient a p p r o a c h is t h r o u g h an¬ geometry. In the l a t t e r , a coordinate s y s t e m is i n t r o d u c e d i n t o the p l a n e whereby a conic m a y then be represented as the g r a p h of a n equat i o n . F o r e x a m p l e , i n the o r d i n a r y C a r t e s i a n c o o r d i n a t e s y s t e m , each conic is the g r a p h of a q u a d r a t i c equation i n the c o o r d i n a t e variables x a n d y. Conversely, except for some "degenerate" cases, it m a y be proved t h a t each such e q u a t i o n represents a conic. T h e general e q u a t i o n has the f o r m ,

alytic

Ax2 + 2Bxy + Cy2 + Dx + Ey + F = 0, A,B,C,D,E,F

A,B,C

where the coefficients are real constants and are not a l l zero. T h e g r a p h of such a n e q u a t i o n m a y reduce to a single point or f a i l t o e x i s t . T h e e q u a t i o n m a y also be factorable, i n w h i c h case its g r a p h reduces to one or two s t r a i g h t lines. These are the degenerate cases. O t h e r w i s e , the g r a p h w i l l be an ellipse, p a r a b o l a , or a h y p e r b o l a , a c c o r d i n g as the " d i s c r i m i n a n t " B2 —AC of the e q u a t i o n is negative, zero, or p o s i t i v e . C i r c l e s are i n c l u d e d a m o n g the ellipses a n d are o b t a i n e d w h e n B = 0 and

A = C.

T h e r e m a i n i n g sections of this chapter are devoted t o e x a m p l e s t h a t i l l u s t r a t e a v a r i e t y of p h e n o m e n a associated w i t h the general n o t i o n of s t r u c t u r a l s t a b i l i t y . T h e first one involves a f a m i l y o f conies.

186

STRUCTURALISM AND STRUCTURES

64. Stability i n a F a m i l y of Conies T h e e x a m p l e described i n this section i l l u s t r a t e s a n aspect of s t r u c t u r a l s t a b i l i t y q u i t e different f r o m previous e x a m p e s . It is a s p e c i a l case of a general s i t u a t i o n i n w h i c h one is g i v e n a f u n c t i o n (or m a p p i n g ) / denned i n one ( u s u a l l y E u c l i d e a n ) space and taking of a second space E' as values. T h u s , / associates t o each p o i n t P of E, for w h i c h i t is defined, a s u b s t r u c t u r e f(P) of E'. N o w , i f / is denned for points P a n d Q, t h e n the structures f(P) a n d f(Q) m a y be defined t o be near to one another i f P a n d Q are near i n E. T h e s t a b i l i t y q u e s t i o n t h e n takes the form:

E

substructures

"IfQ is sufficiently near to P, then is the structure f(Q) isomorphic to f{P)?> 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

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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

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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

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STABILITY

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 <

m',

w h i c h , f u r t h e r m o r e , exhaust a l l o f the possibilities i n the present e x a m p l e . T h u s , we have a s y s t e m consisting o f e x a c t l y one of these four s t r u c t u r e s associated w i t h each p o i n t of the i j / - p l a n e . R e c a l l t h a t a p o i n t of the x y - p l a n e is called a regular p o i n t of the s y s t e m if it has a n e i g h b o r h o o d i n w h i c h the associated structures are i s o m o r p h i c . O t h e r w i s e the p o i n t is called a s i n g u l a r p o i n t . T h e set of regular p o i n t s is called the a n d the set of s i n g u l a r points is c a l l e d the set. I n the present e x a m p l e , the s i n g u l a r set consists of the p o i n t s o n the g r a p h of the curve i n F i g u r e 67.1 (c), a n d the s t a b i l i t y region consists of a l l other p o i n t s . In the " g r a y " region enclosed by the g r a p h , the s t r u c t u r e s are of the type m < M < m'. O t h e r w i s e they are a l l of t y p e m outside of the g r a p h a n d at the cusp. ( T h e energy f u n c t i o n reduces t o E = z4 a t the c u s p , a n d its g r a p h has a m i n i m u m at z = 0.) O n the left h a n d b r a n c h of the g r a p h they are of type m < I, a n d o n the r i g h t h a n d b r a n c h t h e y are o f t y p e I < m. T h e reader w i l l no d o u b t notice the resemblance o f this e x a m p l e to the one i n Section 63. O n the other h a n d , i n order t o b r i n g out the c a t a s t r o p h y p h e n o m e n o n i n t h i s s e t t i n g , i t is necessary t o look at these s i m p l e s t r u c t u r e s a b i t differently.

stability region,

singular

In the first place, the catastrophe b e h a v i o r concerns o n l y the l o c a l s t r u c ture o f the graphs. In fact, i t only involves a p a r t i c u l a r l o c a l m i n i m u m p o i n t a n d the w a y the l a t t e r varies w i t h the control p o i n t In order to see this, we have o n l y to fix a t t e n t i o n either on a left h a n d m i n i m u m p o i n t or on a r i g h t h a n d m i n i m u m p o i n t , where those g r a p h s w i t h o n l y one m i n i m u m m a y be i n c l u d e d i n either case. T h e c r i t i c a l p o i n t structures are thus reduced t o o n l y t w o , w h i c h m a y be represented b y " m _ " (left h a n d m i n i m u m ) a n d " m + " (right h a n d m i n i m u m ) , respectively. T h e regular set for m _ is the x y - p l a n e m i n u s the r i g h t - h a n d b r a n c h o f the curve i n F i g ure 67.1 (c). S i m i l a r l y , the regular set for m+ is the p l a n e m i n u s the left h a n d b r a n c h of the curve. T h u s , the s i n g u l a r sets are respectively the t w o branches of the curve a n d include the cusp i n b o t h cases. T h e catastrophe p h e n o m e n o n occurs because the r e g u l a r i t y sets overlap.

(x,y).

68.

A t t a c k or R e t r e a t

N o discussion of catastrophe theory w o u l d be c o m p l e t e w i t h o u t a des c r i p t i o n of at least one of its controversial a p p l i c a t i o n s . T h e controversy is a c t u a l l y not a b o u t the p h e n o m e n a themselves, a l l o f w h i c h are p l a u sible a n d most i n t e r e s t i n g , but o n l y concerns the question of whether or

IX. S P A C E S T R U C T U R E S A N D

STABILITY

197

not a genuine m a t h e m a t i c a l t r e a t m e n t is possible. I n c i d e n t a l l y , the s u g gested catastrophes are often of the elementary cusp t y p e i l l u s t r a t e d i n the preceding section. T h e e x a m p l e o u t l i n e d here is t y p i c a l a n d is also due t o Z e e m a n [ Z l ] , w h o discusses it and a v a r i e t y of other s i m i l a r e x a m p l e s i n considerable d e t a i l . It concerns the "fight or f l i g h t " b e h a v i o r of a t e r r i t o r i a l fish t o w a r d other fish t h a t m i g h t enter the t e r r i t o r y s u r r o u n d i n g i t s nest. F o r s i m p l i c i t y , we assume t h a t the b e h a v i o r m a y range c o n t i n u o u s l y f r o m the one e x t r e m e of a n a l l out a t t a c k on a n invader t o the opposite e x t r e m e of a hasty retreat. It is also reasonable to assume t h a t the b e h a v i o r w i l l depend p r i m a r i l y o n t w o v a r i a b l e s , the size s of an invader a n d the distance r of the invader f r o m the nest. T h e general s i t u a t i o n is suggested b y F i g u r e 68.1 a n d e x p l a i n e d below.

r

0 Fig.

68.1

S m a l l values of s a n d r w i l l n o r m a l l y lead to an a t t a c k , w h i l e large values w i l l n o r m a l l y c a l l for a retreat. These conflicting m o d e s p r o v i d e the s e t t i n g for a catastrophe p h e n o m e n o n . T h e control variables are s a n d r w h i c h assume o n l y p o s i t i v e values. A c r u c i a l p o i n t i n the c o n t r o l space w i l l be ( o,rrj), e r e srj is the size o f the defending fish, a n d rn is the distance f r o m the nest to the perceived b o u n d a r y of the t e r r i t o r y . A p o t e n t i a l a c t i o n at a g i v e n p o i n t (s, r ) of the control space is represented i n the figure b y an arrow p a r a l l e l t o the r - a x i s . T h e arrow i n d i c a t e s a n a t t a c k or retreat m o d e a c c o r d i n g as i t p o i n t s i n the d i r e c t i o n of i n c r e a s i n g or decreasing r , a n d its l e n g t h suggests the i n t e n s i t y of t h p o t e n t i a l a c t i o n . It is assumed t h a t the larger of any two fish w i l l tend to a t t a c k the s m a l l e r . T h u s , i f the defender encounters a fish s m a l l e r t h a n itself (i.e. s < so), i t w i l l d r i v e the latter beyond the b o u n d a r y of its t e r r i t o r y , w i t h the i n t e n s i t y of the a t t a c k decreasing t o w a r d zero as r increases. s

w n

198

STRUCTURALISM AND

STRUCTURES

Suppose now t h a t the defender D encounters a n i n v a d e r I of m o d e r a t e size s> 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 : <*l(P. 3). (f*. 4')] = m a x [ d ( p , p ' ) , <%,

_01>

derived f r o m the m e t r i c i n 5 . T h e p r o o f t h a t t h i s is indeed a m e t r i c is s t r a i g h t f o r w a r d a n d w i l l be o m i t t e d . T h e representation of F* i n S x 5 thus reduces the nearness p r o b l e m to defining nearness for T h e i d e a consists i n s h o w i n g t h a t the collection of subsets of any m e t r i c space is i t s e l f a m e t r i c space. Therefore, let X denote a n a r b i t r a r y m e t r i c space w i t h m e t r i x d(x, y), a n d denote by X* the collection o f a l l finite subsets o f X. W e define the distance between two " p o i n t s " and of (i.e., finite subsets o f X) as follows:

finite subsets of a metric space. all finite d(A,B)

A

B

X*

F i r s t , let p be a fixed p o i n t of the space X a n d A any finite subset of X. T h e n define the distance d(p, A) f r o m p to the set A as the value of a l l the distances as ranges over

minimum

d(p, x)

x

A.

IX. S P A C E S T R U C T U R E S A N D

STABILITY

N e x t , for a r b i t r a r y finite subsets A a n d B o f X, be the value of a l l the distances ranges over A and y ranges over B.

maximum

d(x,B)

d{A,B)

201

define d(A, and

d(y, A)

to

B) as

x

X'.

W e prove now t h a t is indeed a m e t r i c for E v e r y t h i n g is m o r e or less obvious f r o m definitions except the t r i a n g l e i n e q u a l i t y , w h i c h asserts t h a t , for a r b i t r a r y finite subsets A, B, a n d C o f X,

d{A,C)
+ d{B,C).

B y the preceding definitions, there exists either a p o i n t a . i n A or a point c in such t h a t or T h e two 0 cases are s i m i l a r , so we w i l l concentrate o n the first.

C

0

d(A,C) = d(ao, C)

d(A,C) = d(c , A).

B y d e f i n i t i o n of _ ( _ , C ) , it is less t h a n or e q u a l to d(on, c) for a r b i t r a r y c i n the set C. H e n c e , by the t r i a n g l e i n e q u a l i t y i n X, we have, 0

d(A,C)
and c i n C.

C h o o s e 60 i n B such t h a t

d(a0,b0) = T h e n , since d(o.o, B)

< d(A,

+ d(b,c), d(a0,B).

we o b t a i n

B),

d(A,C)
+ d(b0,c),

for a r b i t r a r y c i n C . N o w choose CQ i n C such t h a t d(6 ,c ) = 0

Again,

d(bo,C) < d(B,C),

0

d(6o,C).

so it follows t h a t

d(A,C)
+ d(B,C)l

c o m p l e t i n g the proof. 70.

Stability of Point-Line Structures.

W e are now i n a p o s i t i o n to give a precise d e f i n i t i o n of distance between t w o finite p o i n t - l i n e structures F a n d G i n a m e t r i c space S a n d therefore state a n d prove 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 these s t r u c t u r e s . Because F a n d G m a y be represented as finite subsets [ F ] and [ G ] of the C a r t e s i a n p r o d u c t 5 x 5 , they m a y accordingly be regarded as points i n the space ( 5 x S)' and their distance defined i n t e r m s of the ( 5 x 5 ) * metric. A

A

A

A

A

A

STRUCTURALISM AND S T R U C T U R E S

202

objects minimum

F i n a l l y , we w i l l need a measure of the " s p a c i n g " o f b e l o n g i n g to a structure F i n O n e o b v i o u s measure is the distance i n between d i s t i n c t p o i n t s of F. For t e c h n i c a l reasons, however, we define the spacing t o be the of t h i s m i n i m u m d i s t a n c e a n d the n u m b e r 1. T h e n s(F) w i l l be a f u n c t i o n w i t h values always less t h a n or equal t o 1 and p o s i t i v e i f F contains m o r e t h a n one p o i n t .

S.

A

s(F)

S

smaller

A f t e r a l l of these technical p r e l i m i n a r i e s , it is finally possible to state the s t a b i l i t y result t o w a r d w h i c h we have been w o r k i n g . N o t i c e t h a t the c o n d i t i o n r e q u i r i n g the sets [ F ] a n d [GA] i n SxS to have the same n u m b e r of p o i n t s is o b v i o u s l y for F a n d G to be i s o m o r p h i c , so m u s t be satisfied i n one way or another. W e prefer to include it e x p l i c i t l y as p a r t of the nearness c r i t e r i o n . A

necessary

A

A

Let F be a finite point-line structure in S with (nonzero) spacing S{F), and let G 6e any finite point-line structure in S suck that [GA] and [Fn] have the same number of points and A

A

d([F ],[G ])<s(F)/2. A

(*)

A

Then G must be isomorphic with FA. A

T h e p r o o f of this s t a t e m e n t , t h o u g h n o t so very difficult, is a d m i t t e d l y rather tedious, so m a y t r y the patience of m a n y readers. It is i n c l u d e d , h o w ever, as a n accessible piece of m a t h e m a t i c s i n v o l v i n g w o r t h w h i l e s t r u c t u r e ideas. W e begin w i t h a list of the key steps i n the proof:

(1) There exists a special one-to-one mapping M of[Gn] onto [FA] such that, for each (g, g') in [G ], A

d(M{g,g'),(g,g'))<s(F)/2. (2) There exists a one-to-one mapping m of G onto F such that, for each g in G, M(g,g) = (m(g),m(g)). (3) The mapping m of G onto F defines an isomorphism between the structures FA and G " For the p r o o f of (1), let (g,g') be a n a r b i t r a r y p a i r i n [ G ] a n d observe A

t h a t , by definition of the d i s t a n c e f u n c t i o n in (S

x S)'

a n d c o n d i t i o n (*),

d{[FA],(g,g'))
(g,g')

[F ].

A A where is the distance o f t o the set A l s o , by definition of t h i s " p o i n t - t o - s e t " d i s t a n c e , there exists ( / , / ' ) i n [ F ] such that d([FA],(g,9')), A

d((f,f>),(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.

<*(/, e) < d(f, g) + d(e, g) < s(F)/2 + s{F)/2 = s{F), s(F)

e.

M(g,g) {g,g') = (g,g)-

so _ ( / , e ) < a n d hence / = In other words, = (/,/)• Again, since is one-to-one, i t follows t h a t T h i s proves t h a t restricts to a one-to-one m a p p i n g o f the d i a g o n a l elements o f [ G ] o n t o t h e d i a g o n a l elements o f [ F ] . T h e r e f o r e , there exists a one-to-one m a p p i n g m of G o n t o F such t h a t , for each d i a g o n a l element (g,g) of [ G ] ,

M

M

A

A

A

M(g,g) = (m(g),m(g)), proving (2). T h e p r o o f o f ( 3 ) , w h i c h w i l l complete the p r o o f o f t h e s t a b i l i t y result, a m o u n t s to s h o w i n g t h a t the m a p p i n g m o f G o n t o F preserves relations i n G a n d FA. Since the relations are represented by the ( n o n d i a g o n a l ) elements o f [ G ] a n d [ F ] , i t w i l l be sufficient t o prove t h a t A

A

A

m(g,g') = (g,g')

(m(j),

m(g'))

for each p a i r i n [ G ] . A t this p o i n t , we k n o w o n l y t h a t t h i s is t r u e for d i a g o n a l elements. Therefore, let (_,_') be a n a r b i t r a r y pair i n [ G ] a n d set A

A

M

M

=

(/,/')•

Then

d(f,g) < s(F)/2

a n d _(/',
s(F)/2.

We also have

M(g,g) =

(m(.),m( )) and f f

M(9'l9')

=

{m(g'),m{g')),

a n d hence

d(m{g),g) < s(F)/2 and d(m(g'),g') < s(F)/2. Therefore,

d(f, m(g)) < d(f, g) + d(m(g), g) < s(F)/2 + s(F)/2 = s(F),

IX. S P A C E S T R U C T U R E S A N D

d(f,m(g)) < s(F), m(g'),

STABILITY

205

m(g).

so which implies / = T h e same a r g u m e n t also yields / ' = a n d completes the proof t h a t p o i n t - l i n e s t r u c t u r e s i n a m e t r i c space satisfy 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 . T h e above s t a b i l i t y result is c e r t a i n l y not s u r p r i s i n g a n d it is n a t u r a l to ask why the p r o o f seems t o require so m u c h tedious d e t a i l . P a r t o f the e x p l a n a t i o n is t h a t the reader was not assumed t o be f a m i l i a r w i t h most of the m a t h e m a t i c a l concepts, so m a n y o f the details spelled out c o u l d have been o m i t t e d as " o b v i o u s " to an expert. P e r h a p s m o r e relevant is the fact t h a t , even for a t o p i c t h a t is rather easily u n d e r s t o o d i n f o r m a l l y , a more or less c o m p l e t e " r i g o r o u s " t r e a t m e n t c a n t u r n out to be s u r p r i s i n g l y difficult. W e saw this early o n i n Section 9, for e x a m p l e , w h e n we gave a f o r m a l t r e a t m e n t o f analogies. In the above discussion, c o n s t r u c t i o n of an 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 p o i n t - l i n e structures t u r n e d out t o be less r o u t i n e t h a n one m i g h t have guessed i n advance. P a r t o f the p r o b l e m is t h a t one tends to forget t h a t any d e f i n i t i o n o f nearness for a s t r u c t u r e must recognize b o t h objects a n d relations, and the l a t t e r are a l m o s t always less manageable t h a n the former. A n o t h e r u n a n t i c i p a t e d feature of the above proof is the mass of c o m p l i c a t i o n s i n v o l v e d i n the c o n s t r u c t i o n o f a structure i s o m o r p h i s m f r o m an a p p a r e n t l y very n a t u r a l definition o f nearness.

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Caws, P., Structuralism: The A r t of the Intelligible, (Contemporary Studies in Philosophy and the Human Sciences), Humanities International Press, Atlantic Highlands, NJ, 1988. Chappell, V.C. (ed.), The Philosophy of M i n d , Prentice-Hall, Englewood Cliffs, NJ, 1962; Dover Publications, New York, NY, 1981. Chomsky, N., Language and Problems of K n o w l e d g e , The M I T Press, Cambridge,, 1988. Cipra, B., M a t h e m a t i c s Untwists the Double Helix, Research News, Science 247 (1990), 913-915. Culliton, B.J., Rockefeller Braces for B a l t i m o r e , News & Comment, Science 247 (1990), 150-151. Daston, L. and Gigerenzer, G., B o o k R e v i e w s , Science 244 (1989), 1094. Davis, P.J., Applied M a t h e m a t i c s as a Social Contract, Mathematics Magazine 61 (1988), 139-147. DeGeorge, R.T. and F.M. (ed.), The Structuralists: From Strauss, Anchor Books, Doubleday, Garden City, NY, 1972. Dodson, M.M., Quantum Evolution and the Fold Catastrophe, (1975), 107-118. phes,

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and Thorn's in Early

Infancy,

Theory

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Levi-

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Eldridge, N. and Gould, S.J., Models in Paleobiology ( J . M . Schopf, e d . ) , Freeman, Cooper, San Francisco, C A , 1972, 82. Gailbraith, J.K., Gold Card Talks to John K e n n e t h Gaibraith, The Gold Card Update, American Express, April, 1988. Gardner, H., The Quest for M i n d , Second edition, The University of Chicago Press, Chicago, IL, 1981. Gay, P., Freud: A Life f o r O u r T i m e , W.W. Norton & Co., New York, 1988. Ghiselin, B. (ed.), The Creative Process, A Mentor Book, The New American Library, New York, 1952. Gleick, J., Chaos, Penguin Books, New York, 1988. Gould, S.J., E v e r Since D a r w i n , Norton paperback, New York, 1979. , The Panda's Thumb, Norton paperback, New York, 1982. , Hen's Teeth and H o r s e ' s Toes, Norton paperback, New York, 1984. , A Clock of E v o l u t i o n , Natural History Magazine 94 (1985), 12-26. 207

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210 [W3] [W4j [W5] [Zl] [Z2]

STRUCTURALISM AND STRUCTURES Weschler, L., Seeing i i Forgetting the Name o j the Thing One S e e s , (A Life of Contemporary Artist Robert Irwin), Univ. of Calif. Press, 1982. Whitehead, A.N., Matfcemitic*
INDEX

A b s t r a c t s t r u c t u r e s , 3,21, 26, 27, 35-37, 167 almost abstract, 77 Affine space a n d g r o u p , 182 A l b e r s l i n e d r a w i n g s , 42, 43 A n a l o g i e s , 3, 23-27 in e v o l u t i o n , 147 A n a l y s i s o f structures, 57 A n t h r o p o l o g y , 12 A n t h r o p o m o r p h i s m , 175-177 A p p l i c a t i o n s of m a t h e m a t i c s , 130-133 d a t a , theory, a n d m a t h s t r u c t u r e s , 130, 133, 139 effectiveness of m a t h i n physics, 133-138 emphasis o n n u m e r i c a l methods, 4, 15, 140, 141 extension a n d p r e d i c t i o n , 131 "physics e n v y " , 138 problems i n n o n p h y s i c a l a p p l i c a t i o n s , 138-142 significance of a n a p p l i c a t i o n , 131 A p p r o x i m a t i o n s of a s t r u c t u r e , 58, 59 a x i o m a t i c a p p r o x i m a t i o n , 59 A s s o c i a t i o n s , 27 A x i o m a t i c m e t h o d , 58-60 A x i o m a t i c representation, 59 B a c k g r o u n d s t r u c t u r e , 104 B e a u t y and elegance i n m a t h e m a t i c s , 125, 137, 138 B i o l o g i c a l catastrophes, 165, 168 B i o l o g i c a l o r g a n i s m s , 146, 147 classification, 146 analogies a n d homologies, s y s t e m a t i c s , 146 B l a c k boxes, 68, 113

211

147

212

STRUCTURALISM AND STRUCTURES

B l o c k d i a g r a m s , 67 B o d y o f m a t h e m a t i c s , 115 B o h r ' s r a d i u m a t o m , 40, 41 B u i l d i n g f r a m e w o r k , 16 B u s h - D u k a k i s debate, 23-26 C a i r n s - S m i t h : the u n i t y of b i o c h e m i s t r y , 156 C a t a s t r o p h e s , 3 1 , 168, 169, 180, 188-198 m a t h e m a t i c a l e x a m p l e , 191-196 C h a o s , 132, 143 C h o m s k y : u n i v e r s a l g r a m m a r , 88 C o g n i t i o n , 2 7 , 94 C o m m u n i c a t i o n , 74-77, 79 o f structures, 75 Complexity, i n e v o l u t i o n , 156-161 and s t r u c t u r a l i n f o r m a t i o n , 157 of s t r u c t u r e s , 2, 157, 159 C o n f i g u r a t i o n s ( m a t h ) , 44, 45 C o n i c sections,

182

focus-directrix d e f i n i t i o n , 182 graphs of q u a d r a t i c equations, 185 shared s t r u c t u r e s , 184 projective equivalence, Consciousness, 104 Content of mathematics, 6 C o n t r a c t i o n s , 65, 66

184

examples, 66-72 C r e a t i v i t y , 120-130 P o i n c a r e : m a t h e m a t i c a l c r e a t i v i t y , 124-126 s t r u c t u r a l i n t e r p r e t a t i o n , 127-130 C u l t u r e g a p , 64, 65 D a t a structures ( a n d systems), 131 D e f i n i t i o n ( s ) of s t r u c t u r e general, 17 C a w s , 13 L e v i - S t r a u s s , 12 P i a g e t , 13 R a d c l i f f - B r o w n , 12 D e s c r i p t i o n vs p r e d i c t i o n , 142

INDEX

D e t e r m i n i n g s t r u c t u r e s , 6 1 , 62, 136, 172 D o u b l e h e l i x , 149 E i n s t e i n : i n v e n t i o n , 122 E l e m e n t a r y chemistry, 68 E l e m e n t a r y s t r u c t u r e s , 30 E n v i r o n m e n t of a s t r u c t u r e , 152 E u c l i d e a n spaces, 179, 180 congruences, 181 distance f u n c t i o n ( f o r m u l a ) , 180, 181 euclidean g r o u p , 181 neighborhoods, 181 r i g i d m o t i o n s , 181 s i m i l a r i t y g r o u p , 182 s u b s t r u c t u r e s , 181 three d i m e n s i o n a l structures, 28, 179 E v o l u t i o n , 95, 103, 145 analogies, 147 catastrophes, 172 c a t a s t r o p h i c vs stable state change, 165 c o m p l e x i t y , 156-161 convergent, 147, 173, 174 d i s c o n t i n u i t i e s , 165 diversity, 158 emergence, 155 e v o l u t i o n a r y process, 153 g r a d u a l i s m , 155, 164, 16, 170-172 homologies, 147, 151 h y p e r s e l e c t i o n , 164 increasing c o m p l e x i t y , 160 p u n c t u a t e d e q u i l i b r i a , 168-172 R u b e G o l d b e r g effect, 161 s t a b i l i t y , 171 s y n t h e t i c theory (or m o d e r n synthesis), 154 Face r e c o g n i t i o n , 58 Formalists, 6 F o r m a l language, 6, 77, 116, 118 F r e u d i a n unconscious, 86-88 G a y : the F r e u d i a n unconscious, 87 G e n e t i c s t r u c t u r e , 148

213

214

STRUCTURALISM AND STRUCTURES

D N A , R N A , 149, 150 genes, 149 genome, 148 g l o b a l s t r u c t u r e , 150 o r i g i n of, 156, 163 G i f t horse, 85 Gould: h o m o l o g y a n d analogy, 147, 148 h o m o l o g y i n D N A , 151 the c o n v e n t i o n a l version o f the tree of life, 159 hyperselection i s m , 164 s t r u c t u r a l s t a b i l i t y i n e v o l u t i o n , 169 Groups (math affine g r o u p , 182 c o n t r a c t i o n , 71 generators, 51 group a x i o m s , 50, 51 group m u l t i p l i c a t i o n t a b l e , 50 group s t r u c t u r e , 52, 53 projective g r o u p , 184 t r a n s f o r m a t i o n g r o u p , 52 triangle g r o u p , 48-52 H a m m e r a n d tongs, 85 H i g h e r level structure o r g a n i z a t i o n s , 63, 96-98, 167 higher m e n t a l a c t i v i t y , 96, 167 H o m o l o g i e s i n e v o l u t i o n , 147 H o m o m o r p h i s m s o f s t r u c t u r e s , 70 H u m a n awareness, 176 H u m a n b r a i n , 2, 166 H u m a n i t i e s vs n a t u r a l sciences, 98-100 I m p l i c i t d e f i n i t i o n , 58 I n f o r m a t i o n i n structures, 33, 157 Insights, 100, 122 research a n d c r e a t i v i t y , 99, 122 role of the unconscious, 123 I n t e l l i g i b i l i t y , 5, 97, 98 I s o m o r p h i s m s of structures, 20, 23 Irish e l k , 166 Japanese q u a i l , 89

INDEX

J a r g o n , 117 K i n s h i p systems, 14 Language a n d a b s t r a c t i o n , 77 m o r p h e m e s , 80 " B u t w h a t is language?" (Saussure), 78 signs, 80-82, 84-87 s t r u c t u r e , 73, 81 i n t h i n k i n g , 73 L a n g u a g e f a c u l t y , 88-90 L a n g u a g e vs p e r c e p t i o n , 74, 75, 118 Language-type structures social systems, 86 F r e u d i a n unconscious, 86 Levi-Strauss: language a n d social p h e n o m e n a , 83 measure a n d s t r u c t u r e , 15 words a n d p e r c e p t i o n , 74 Linguistics in structuralism, 5 M a c h i n e s , 40 M a r s u p i a l s , 174 M a t e r i a l i s m vs i d e a l i s m , 35-37 " M a t h e m a t i c i a n s are different", 133 M a t h e m a t i c a l i n t u i t i o n , 120 M a t h e m a t i c a l language, 116, 122 Mathematics c r e a t i v i t y , 120-124 definition of a m a t h e m a t i c a l s t r u c t u r e , 119 fields defined by use of, 144 i n s i g h t s , 122, 130 i n l i b e r a l a r t s , 110 research and development, 120, 121 rigorous t r e a t m e n t , 118 R u s s e l l d e f i n i t i o n , 117 special role, 6, 7 a n d the unconscious, 123-125, 127 M a t h e m a t i c s a n d physics abstract m a t h vs real w o r l d , 135 effectiveness of m a t h e m a t i c s , 134, 135

215

216

STRUCTURALISM AND

h i s t o r i c a l connections,

STRUCTURES

135

p h y s i c a l i n t u i t i o n m a y s u p p o r t m a t h e m a t i c s , 137 p h y s i c i s t as m a t h e m a t i c i a n , 137 special r e l a t i o n between, 133 M c C a r t h y : " t h i n k i n g m a c h i n e s " , 175 Measure a n d s t r u c t u r e , 4, 15 M e n t a l a c t i v i t y , 27 M e n t a l structures, 93-95 as electrical networds, 93, 128 extensions, 128, 129 f o r m a t i o n , 95 M e n t a l processing o f structures, 2, 2 7 , 58, 73, 7 9 , 94 M e t r i c spaces, 199 p o i n t - l i n e s t r u c t u r e s , 199 cartesian p r o d u c t ,

199

M u l t i p l e f u n c t i o n , 163, 165 i n biology, 161-168 i n higher m e n t a l a c t i v i t y , 167 N e u r a l networks, 94 " N e w M a t h " , 113 O r d e r relation(s), 18, 33, 54, 140 P a r t i a l order, 3 3 , 140 P a s c a l c o n f i g u r a t i o n , 47, 48, 6 1 , 163 P a s c a l ' s t h e o r e m , 46 P e r c e p t i o n , 101 P h i l o s o p h y and s t r u c t u r a l i s m , 35-37, 100-104, 135 P h y s i c a l i n t u i t i o n i n m a t h e m a t i c s , 137 P i a g e t : f o r m a l i z a t i o n , 117 P l a t o ' s lecture o n the G o o d , 8, 9 P o i n c a r e : m a t h e m a t i c a l creation, 124-126 P o i n t - l i n e structures, 28, 201-205 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 , 3 1 , 7 7 , 106, 170 P r o j e c t i v e (group a n d plane), 184 P r o p e r t i e s of structures i n t e r n a l and e x t e r n a l , 106 P s y c h o a n a l y s i s , 8 2 , 86, 87 P u n c t u a t e d e q u i l i b r i a , 168-172 R a n d o m (genetic) v a r i a t i o n , 173 R e a l n u m b e r s y s t e m , 54, 55

INDEX

R e d u c t i o n i s m , 60-64 Relations a n t i s y m m e t r y , 18, 30 b i n a r y , 18, 30 finite, 30 n - a r y , 30 order, 18, 33, 54 p a r t i a l order, 33, 140 s y m m e t r i c , 30 ternary, 30 R e l a t i v e c o m p l e x i t y , 157 Representations, 21 d e s c r i p t i o n vs p r e d i c t i o n , 142 Research a n d c r e a t i v i t y aesthetics, 134 conjectures, 124 i n s i g h t , 122 role of the unconscious, 123, 125, 127 s t r u c t u r a l a n a l y s i s , 127-130 R u b e G o l d b e r g effect, 161 R u s s e l l d e f i n i t i o n of m a t h e m a t i c s , 117 Saussure: the n a t u r e of language, 78, 79 signs, 81 semiology, 82 Science c r e a t i v i t y , 99 discovery a n d verification, 99 influence o n s t r u c t u r a l i s m , 3 relation to mathematics, 4 theory, 99 Scientific r e v o l u t i o n , 70 S e m i o t i c s , 82 S i g n language, 90 Snow, C . P . (the t w o cultures), 64 Sorensen: M a c h ' s ideas on thought e x p e r i m e n t s , 101, 102 S t a b i l i t y , 30-32, 39, 106 c o m m u n i c a t i o n , 75 e v o l u t i o n , 170

217

218

STRUCTURALISM AND STRUCTURES

a f a m i l y of conies, 186 organisms, 161 p e r c e p t i o n , 31 p o i n t - l i n e structures, 201 p r i n c i p l e , 31 region, 186 signature, 31 Stebbens a n d A y a l a : s y n t h e t i c theory o f e v o l u t i o n , 168, 169 S t r u c t u r a l d e t e r m i n i s m , 60-62, 103 S t r u c t u r a l l i n g u i s t i c s , 77 S t r u c t u r a l i s m , 1, 3 h u m a n i t i e s vs n a t u r a l sciences, 98-100 S t r u c t u r e (s) a p p r o x i m a t i o n of, 58 b i o l o g i c a l , 145 d a t a s t r u c t u r e s , 131 d e t e r m i n i n g , 172 extensions, 58 a n d f o r m a l i s m , 117 a n d f u n c t i o n , 165 m e n t a l p h e n o m e n a , 93 m a t h e m a t i c a l , 115 objects & relations, 17, 20 order (simple a n d p a r t i a l ) , 140 p o i n t - l i n e (analysis of), 28 substructures of, 17, 179 S t r u c t u r e h o m o m o r p h i s m , 70 S t r u c t u r e i n f o r m a t i o n , 33 S t r u c t u r e properties i n t e r n a l a n d e x t e r n a l , 22 S t r u c t u r e representations, 21 S t r u c t u r e t r a n s f o r m a t i o n s , 2 2 , 23 S y s t e m , 19, 172 d a t a (or subject) s y s t e m , 130 d e f i n i t i o n , 19 real n u m b e r s , 54 s u b s y s t e m , 19 S y s t e m a t i c s ( b i o l o g y ) , 146 T e a c h i n g a n d l e a r n i n g , 107

INDEX

computers v s live i n t e r a c t i o n s , 108 defective p e r c e p t i o n f o r m a t i o n , 111, 113 drive for m e a n i n g , 109 knowledge a n d u n d e r s t a n d i n g , 113 l i b e r a l e d u c a t i o n , 110 in m a t h e m a t i c s , 110 nature of a n d p r o b l e m s i n , 108 pedagogical errors, 111-113 role of m a t h e m a t i c s , 110 spontaneous f o r m a t i o n of perceptions, 109 s t a n d a r d i z e d tests, 113 s t r u c t u r a l requirements, 109, 113 use o f c o m p u t e r s , 108 " w h y m a y b e " , 109 Teleology, 177 T e r m i t e s , 1623 T e r r i t o r i a l fish, 197 T h e o r e t i c a l s t r u c t u r e s , 131 T h e o r y systems, 131 T h o r n : q u a n t i t a t i v e m o d e l i n g i n catastrophe theory, 189 T h o u g h t e x p e r i m e n t s , 101 T h e t w o cultures ( C . P . S n o w ) , 64 T r i a n g l e g r o u p , 48-50 T h e unconscious, 123-125, 127, 130 Understanding as a creative experience, 107 b a c k g r o u n d for, 104 correct or complete, 106 definition of, 106 defects i n , 106 degrees of, 106 m u t u a l , 106 W h i t e h e a d : P l a t o ' s lecture o n the G o o d , 8 W h o l e n e s s , 19 W i g n e r : effectiveness of m a t h i n physics, 134, 135 Zeeman's catastrophe m a c h i n e , 190

219

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C

A s s o c i a t e

MATHEMATICS

H s i u n g Editors:

S

S

a n d

C h e r n , M

S

K o b a y a s h i ,

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Y a m a g u t i

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