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A defence of Kuhn's incommensurability thesis Kosub, Timothy Alexander 1989

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A DEFENCE OF KUHN'S INCOMMENSURABILITY THESIS By TIMOTHY ALEXANDER KOSUB B . S c , The U n i v e r s i t y of B r i t i s h Columbia, 1980 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n THE FACULTY OF GRADUATE STUDIES (Philosophy) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October 1989 © Timothy Alexander Kosub, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date 1 DE-6 (2/88) u A b s t r a c t Kuhn's incommensurability t h e s i s i s the c l a i m that s ucces-s i v e s c i e n t i f i c t h e o r i e s o f t e n c o n f l i c t not o n l y l o g i c a l l y but a l s o n o r m a t i v e l y : i . e . they d i f f e r both about nature and a l s o about the use of common apparatus, concepts and experimental r e -s u l t s , and what are proper s c i e n t i f i c g o a l s and methods. C r i t i c s commonly o b j e c t t h a t Kuhn's t h e s i s a t t a c k s such t r a d i t i o n a l s c i -e n t i f i c v a l u e s as o b j e c t i v i t y and r a t i o n a l i t y . But t h e i r s t r o n g -e s t response can be expressed as a dilemma: e i t h e r , i f taken l i t -e r a l l y , the incommensurability t h e s i s i s s e l f - c o n t r a d i c t o r y ; o r , i f t h a t l i t e r a l r e a d i n g i s r e j e c t e d , t h i s t h e s i s has no p h i l o s o -p h i c a l Import. Kuhn cl a i m s h i s c r i t i c s have m i s i n t e r p r e t e d h i s t h e s i s and he maintains both i t s i n t e l l i g i b i l i t y and r e l e v a n c e . The problem i s whether h i s p o s i t i o n can be s u s t a i n e d . In support of Kuhn, I argue t h a t h i s c r i t i c s ' r e a d i n g of h i s t h e s i s Is based on a mistaken I d e n t i f i c a t i o n of l o g i c with formal l o g i c and, more g e n e r a l l y , of c o m p a r a b i l i t y with commensurabil-l t y . I argue t h a t l o g i c a l comparison of t h e o r i e s t h a t lack com-mon concepts i s p o s s i b l e i f one can compare t h e o r i e s d i r e c t l y , as whole to whole, and t h a t such d i r e c t l o g i c a l comparison i s a c t u -a l l y commonplace i n n a t u r a l languages. I a l s o argue more gener-a l l y t h a t Kuhn's c r i t i c s ' i d e n t i f i c a t i o n of comparison with com-mensuration l e a d s to a v i c i o u s r e g r e s s . My attempt a t r e s o l v i n g the d i s p u t e between Kuhn and h i s c r i t i c s i s informed by a simple "hermeneutic" p r i n c i p l e : I f one view seems e i t h e r u n i n t e l l i g i b l e or i r r e l e v a n t to the other, then both s i d e s p r o b a b l y d i s a g r e e on the I n t e r p r e t a t i o n of shared con-c e p t s . Once the focus of the d i s p u t e i s l o c a t e d , arguments can o f t e n be g i v e n f o r p r e f e r r i n g one i n t e r p r e t a t i o n over another. Thus i f I am r i g h t t h a t Kuhn's c r i t i c s ' view wrongly equates c o m p a r a b i l i t y with commensurability and l o g i c with formal l o g i c , t h a t view c l e a r l y must be r e p l a c e d by one t h a t d i s t i n g u i s h e s them. I argue t h a t i f those d i s t i n c t i o n s are made, incommensur-a b i l i t y can be seen t o r e p r e s e n t no e s s e n t i a l t h r e a t t o s c i e n t i f -i c r a t i o n a l i t y and o b j e c t i v i t y . In t h i s l i g h t , I suggest Kuhn's major a n a l y t i c concepts be viewed as Improvements on more t r a d i -t i o n a l n o t i o n s drawn from formal l o g i c . I a l s o use a h i s t o r i c a l case study of the o r i g i n a l d i s c o v e r y of g e o m e t r i c a l incommensur-a b i l i t y to i l l u s t r a t e f u r t h e r Kuhn's concepts and t o develop a more g e n e r a l n o t i o n of a proof of incommensurability t h a t i s a p p l i c a b l e t o s c i e n t i f i c t h e o r i e s . Contents iv ABSTRACT i i TABLE OF CONTENTS i v FIGURES V i Chapter 1: Commensurability, C o m p a r a b i l i t y , and L o g i c 1 1.0 I n t r o d u c t i o n 1 1.1 The Incommensurability T h e s i s 3 1.1.1 H i s t o r y and The Incommensurability T h e s i s 7 1.2 The C r i t i c a l Response: L o g i c and Formal L o g i c s 14 1.3 Commensurability and C o m p a r a b i l i t y 18 1.4 I n t e r i m Summary 26 1.5 How to D i s t i n g u i s h C o m p a r a b i l i t y from Commensurability 27 1.5.1 Kuhn and D i r e c t Comparison 32 1.6 L o g i c and Formal L o g i c s 33 1.6.1 Argument 1: Formal Comparison of T h e o r i e s 36 1.6.2 Argument 2: Formal A n a l y s i s of T h e o r i e s 36 1.6.3 A G e s t a l t - S w i t c h Model of T h e o r e t i c a l I n c o m p a t i b i l i t y 40 1.7 D i r e c t Comparison: A Thought Experiment 44 1.8 Summary, S y n t h e s i s and C o n c l u s i o n 47 Notes to Chapter 1 50 Chapter 2: Kuhn's A l t e r n a t i v e A n a l y t i c Concepts 60 2.0 I n t r o d u c t i o n 60 2.1 Paradigms and U n i v e r s a l G e n e r a l i z a t i o n s 61 2.1.1 Normal Science as a Process of Paradigm A r t i c u l a t i o n 63 2.1.2 Paradigms, Commensurability and D i r e c t C o m p a r a b i l i t y 65 2.1.3 D i r e c t Comparison of Competing Paradigms 70 2.2 F e l t Anomaly 73 2.3 C r i t i c a l Anomaly and C r i s i s 75 2.4 Thought Experiment: I s o l a t i n g the Root of C r i s i s 77 2.4.1 A P i a g e t i a n Thought Experiment: Volume Conservation 80 2.4.2 G a l i l e o ' s Thought Experiment 83 2.5 Morals 87 Notes to Chapter 2 90 Chapter 3: P r o v i n g Kuhnian Incommensurability: A Case Study 95 3.0 I n t r o d u c t i o n 95 3.1 The Pythagorean Paradigms 96 3.2 Dot-Diagramme A r t i c u l a t i o n s 100 3.3 Rules f o r F i n d i n g Pythagorean T r i p l e s 104 3.4 A r i t h m e t i c & Geometric V e r s i o n s of Pythagorus Theorem 105 3.5 Thought Experiment: Proof of Incommensurability 107 3.6 C o n c l u s i o n : Kuhnian and Mathematical Incommensurability 108 Notes to Chapter 3 111 B i b l i o g r a p h y 113 vi L i s t of F i g u r e s 1. The d u c k - r a b b i t G e s t a l t f i g u r e 41 2. An i n c l i n e d plane 85 3. A pebble r e p r e s e n t a t i o n of twelve 101 4. A pebble r e p r e s e n t a t i o n of t h i r t e e n 101 5. The seventh t r i a n g u l a r number 101 6. The f i f t h square number 102 7. The f o u r t h oblong number 102 8. D i v i d i n g an oblong f i g u r e 103 9. D i v i d i n g a square f i g u r e 103 chapter 1 Commensurability, C o m p a r a b i l i t y and L o g i c 1.0 i n t r o d u c t i o n In The structure of Scientific Revolutions, T. S. Kuhn mar-s h a l s much h i s t o r i c a l , p s y c h o l o g i c a l and p h i l o s o p h i c a l evidence f o r h i s t h e s i s t h a t s u c c e s s i v e s c i e n t i f i c t h e o r i e s are o f t e n both l o g i c a l l y incompatible and incommensurable (Kuhn 1970a, 96, 103). Kuhn contends t h a t the e x i s t e n c e of incommensurable t h e o r i e s i s r u l e d out by t r a d i t i o n a l eplstemology and p h i l o s o p h y of s c i e n c e ; so he concludes t h a t the t r a d i t i o n a l v i e v must be r e p l a c e d by a " v i a b l e a l t e r n a t i v e " t h a t can accommodate incommensurability (Kuhn 1970a, 121). Kuhn's c r i t i c s have r a i s e d many, s e r i o u s o b j e c t i o n s t o the incommensurability t h e s i s . Most are methodological and "moral" ( S c h e f f l e r 1967, 8) o b j e c t i o n s , which c l a i m incommensurability leads t o such bogies as r e l a t i v i s m and i r r a t i o n a l i s m . But Kuhn's opponents' deepest c h a l l e n g e i s t h a t h i s t h e s i s i s e i t h e r s e l f -c o n t r a d i c t o r y , i f taken l i t e r a l l y , or p h i l o s o p h i c a l l y i r r e l e v a n t , i f not. Thus h i s opponents take the standard meaning of "incom-mensurable" to be " a b s o l u t e l y incomparable" and argue t h a t , be-cause l o g i c a l l y incompatible t h e o r i e s are comparable, Kuhn's t h e s i s i s " s e l f - r e f u t i n g " (e.g. Putnam 1981a, 114). And they counter t h a t i f , i n Kuhn's usage, "Incommensurable" does not mean " a b s o l u t e l y Incomparable," then the evidence he musters f o r h i s 2 t h e s i s can have o n l y p s y c h o l o g i c a l or pragmatic s i g n i f i c a n c e . N e v e r t h e l e s s , Kuhn r e j e c t s h i s c r i t i c s ' a n a l y s i s and continues to maintain the p h i l o s o p h i c a l importance of h i s t h e s i s . In t h i s c hapter, I defend Kuhn's r e b u t t a l by shoving t h a t h i s c r i t i c s ' arguments depend on a mistaken i d e n t i f i c a t i o n of l o g i c v i t h formal l o g i c and, more g e n e r a l l y , of comparison v i t h comraensuratlon. I argue t h a t Kuhn's t h e s i s Is p e r f e c t l y i n t e l l i -g i b l e i f those n o t i o n s are d i s t i n g u i s h e d . In a d d i t i o n , I provide t e x t u a l evidence t h a t Kuhn h i m s e l f d i s t i n g u i s h e s those concepts i n h i s vork. My arguments cannot, by t h e i r n ature, provide p o s i -t i v e evidence f o r the a c t u a l e x i s t e n c e of incommensurable theo-r i e s . P r o v i d i n g such p o s i t i v e evidence i s not my aim here s i n c e Kuhn has a l r e a d y s a t i s f i e d t h i s demand, as shovn by the v i g o r o u s c r i t i c a l r e a c t i o n t o h i s t h e s i s . 1 Instead of a r g u i n g f o r a c t u a l Incommensurability, I seek to remove an important o b s t a c l e t o a f a i r e v a l u a t i o n of Kuhn's e v i -dence. But I b e l i e v e my arguments perform more than t h a t nega-t i v e f u n c t i o n . I f I am r i g h t t h a t the t r a d i t i o n a l p h i l o s o p h y of Kuhn's c r i t i c s p r e c l u d e s i n c o m m e n s u r a b i l i t y because i t i s founded on a mistaken i d e n t i f i c a t i o n of d i s t i n c t concepts, my arguments a l s o p r o v i d e p o s i t i v e support f o r Kuhn's c l a i m t h a t ve need a nev " e p i s t e m o l o g l c a l paradigm" (Kuhn 1970a, 121): one t h a t embodies those c o n c e p t u a l d i s t i n c t i o n s , and thus a l l o w s f o r incommensura-b i l i t y . I t t u r n s out t h a t , c o n t r a r y t o vhat some of Kuhn's c r i t -i c s have claimed, t h i s nev epistemology need not eschev t r a d i -t i o n a l s c i e n t i f i c v a l u e s such as o b j e c t i v i t y and r a t i o n a l i t y , 3 1.1 The Incommensurability T h e s i s On the t r a d i t i o n a l view, s c i e n c e Is seen ( i d e a l l y ) t o pro-gress through t h e o r y change by accumulating f a c t s whose char a c -t e r i z a t i o n i s t h e o r y - n e u t r a l and by an I n c r e a s i n g v e r i s i m i l i t u d e t o a theory-independent r e a l i t y . The r e j e c t i o n of a c u r r e n t s c i -e n t i f i c t h e o r y i n favour of a new one i s deemed r a t i o n a l on t h i s view i f I t can be shown neutrally—i.e. on grounds common to both t h e o r i e s — t h a t the new th e o r y i n c o r p o r a t e s the p o s i t i v e r e s u l t s of i t s predecessor while a v o i d i n g i t s e r r o r s and t h a t the new th e o r y has g r e a t e r e x p l a n a t o r y and p r e d i c t i v e scope ( C o l l i e r 1984, 9 ) . A premise of such n e u t r a l comparison, and t h e r e f o r e , of t r a d i t i o n a l p h i l o s o p h y of s c i e n c e , i s t h a t s u c c e s s i v e s c i e n t i f i c t h e o r i e s have the same b a s i c l o g i c a l s t r u c t u r e , a t l e a s t f o r those domains i n which they compete. Thus t o compare s u c c e s s i v e t h e o r i e s n e u t r a l l y f o r e m p i r i c a l adequacy i n those domains, the t r a d i t i o n a l view r e q u i r e s t h a t the meanings of each theory's r e -f e r r i n g e x p r e s s i o n s and other n o n - l o g i c a l concepts do not change (Newton-Smith 1981, 10). But i t i s p r e c i s e l y t h i s b a s i c premise of c o n c e p t u a l s t a b i l i t y between s u c c e s s i v e t h e o r i e s t h a t Kuhn cl a i m s h i s h i s t o r i c a l s t u d i e s show to be mistaken. In f a c t , Kuhn's n o t o r i o u s Incommensurability thesis i s the c l a i m t h a t the-ory change i n s c i e n c e i s o f t e n , i f not always, so r e v o l u t i o n a r y t h a t s u c c e s s i v e t h e o r i e s l a c k the common concepts and r e f e r r i n g 4 e x p r e s s i o n s r e q u i r e d f o r n e u t r a l l o g i c a l comparison (Kuhn 1970a, 102).* H i s t o r i c a l l y , Kuhn argues, the cause of an e a r l i e r t h e o r y ' s 9 f a i l u r e t o cope with experience o f t e n l i e s not j u s t with t h a t theory's e m p i r i c a l content but a l s o with i t s concepts, which are i t s means of e x p r e s s i n g t h a t content (Kuhn 1970a, 53). That i s , o f t e n the d i s c o v e r y of an "unexpected n o v e l t y " ( I b i d . , 35) i s not o n l y e m p i r i c a l l y , but a l s o c o n c e p t u a l l y , anomalous because t h a t novel phenomenon appears to f a l l v i t h i n the t h e o r y ' s scope, but a c t u a l l y c o n f l i c t s v i t h assumptions i m p l i c i t i n the theory's c a t e g o r i e s ( i b i d . , 63-4). The nev t h e o r y , Kuhn c l a i m s , has "to change the meaning of (such p r o b l e m a t i c ] c o n c e p t s " ( i b i d . , 102) i f i t i s t o a v o i d both of i t s predecessor's d i f f i c u l t i e s ( i b i d . , 56, 64). 4 Moreover, Kuhn a l s o argues t h a t a s c i e n t i f i c theory's con-cepts form an " i n t e g r a t e d " vhole ( i b i d . , 129) or "conceptual veb" ( i b i d . , 149); so the r e q u i r e d change of meaning of the p a r t i c u -l a r l y p r o b l e m a t i c concepts a l s o changes the meaning of a l l the other concepts the nev t h e o r y r e t a i n s from i t s predecessor. Such a r e v o l u t i o n a r y conceptual change betveen s u c c e s s i v e t h e o r i e s , he says, makes them "fundamentally i n c o m p a t i b l e " (Kuhn 1970a, 98): i t s i m u l t a n e o u s l y r e s u l t s i n both " s u b s t a n t i v e " ( i . e . l o g i c a l ) and "normative" ( i . e . c o n c e p t u a l , m e t h o d o l o g i c a l , o b s e r v a t i o n a l ) i n c o m p a t i b i l i t y ( I b i d . , 103, 109). ...paradigms [or, t h e o r i e s ; see n3] d i f f e r i n more than substance, f o r they are d i r e c t e d not o n l y to nature but a l s o 5 back upon the s c i e n c e t h a t produced them. They are the source of the methods, p r o b l e m - f i e l d , and standards of s o l u -t i o n accepted by any mature s c i e n t i f i c community a t any given time. As a r e s u l t , the r e c e p t i o n of a new paradigm o f t e n n e c e s s i t a t e s a r e d e f i n i t i o n of the c o r r e s p o n d i n g s c i -ence ( i b i d . , 103. I t a l i c s mine). That r e d e f i n i t i o n of s c i e n c e , Kuhn c l a i m s , "changes...the standards governing p e r m i s s i b l e problems, concepts, and explana-t i o n s " (Kuhn 1970a, 106. I t a l i c s mine). As a r e s u l t , not o n l y does the t r u t h of one view imply the other's f a l s i t y ( i b i d . , 9 8 ) 3 but o f t e n the "problems, c o n c e p t s , and e x p l a n a t i o n s " of one view appear e i t h e r i ncoherent or s c i e n t i f i c a l l y i r r e l e v a n t by the standards of the other (Kuhn 1970a, 103-110). In b r i e f : "The n o r m a l - s c i e n t i f i c t r a d i t i o n t h a t emerges from a s c i e n t i f i c r e v o -l u t i o n i s not o n l y [ l o g i c a l l y ] incompatible but o f t e n a c t u a l l y incommensurable with t h a t which has gone b e f o r e " (Kuhn 1970a, 103. I t a l i c s mine). Kuhn draws the f o l l o w i n g g e n e r a l l o g i c a l and e p i s t e m o l o g i c a l c o n c l u s i o n s from the incommensurability t h e s i s : (1) because there are no t h e o r y - n e u t r a l grounds f o r determining which of two incom-mensurable views i s t r u e (or f a l s e ) , the concept of t r u t h has an "unproblematic" a p p l i c a t i o n o n l y l n t r a t h e o r e t i c a l l y (Kuhn 1970b, 264);' hence, (2) the t r a d i t i o n a l i d e a l of s c i e n c e p r o g r e s s i n g toward an a n t e c e d e n t l y - u n d e r s t o o d aim of t r u t h i s a chimaera. S t i l l , Kuhn b e l i e v e s t h a t , d e s p i t e the absence of a n e u t r a l s t a n d p o i n t , i t i s p o s s i b l e t o compare c u r r e n t t h e o r y with p r e v i -ous ones (Kuhn 1970b, 264). And such backward-looking comparison o f t e n shows l a t e r t h e o r y t o be an improvement on e a r l i e r ones 6 (Kuhn 1970b, 264). That suggests, Kuhn says, t h a t we should " s u b s t i t u t e evolution-from-what-we-do-know f o r the ( t r a d i t i o n a l ] e v o l u t i o n - t o v a r d - v h a t - v e - v i s h - t o - k n o v " (Kuhn 1970a, 171) and come to see pro g r e s s i n s c i e n c e as "a process of e v o l u t i o n from p r i m i -t i v e b e g i n n i n g s " ( i b i d . , 170. I t a l i c s i n o r i g i n a l ) . P h i l o s o p h e r s have r a i s e d v a r i o u s s e r i o u s o b j e c t i o n s t o the incommensurability t h e s i s : they have claimed t h a t i t Is simply f a l s e ; t h a t i t i s based on a vague t h e o r y of meaning; t h a t i t makes r a t i o n a l t h e o r y c h o i c e i m p o s s i b l e ; t h a t I t e l i m i n a t e s the c r i t i c a l Impact of experiment on theory; t h a t i t l e a d s t o i d e a l -ism and r e l a t i v i s m ; and t h a t i t i s otherwise " m e t h o d o l o g i c a l l y and e p i s t e m o l o g i c a l l y u n d e s i r a b l e " (Lodynski 1982, 91). But by f a r the s t r o n g e s t c r i t i c i s m p h i l o s o p h e r s have made of Kuhn's t h e s i s , and the most r e v e a l i n g of t h e i r b a s i c p h i l o s o p h i c a l com-mitments, i s t h a t i t i s e i t h e r " u n i n t e l l i g i b l e " because " s e l f - r e -f u t i n g " (e.g., Newton-Smith 1981, 148-9; Putnam 1981a, 126; Lo-dyn s k i 1982, 91); or p h i l o s o p h i c a l l y - i r r e l e v a n t because d e s c r i b -i n g p u r e l y "humdrum," p s y c h o l o g i c a l f a c t s about s c i e n t i s t s (Mus-grave 1980, 51) or e n t i r e l y "modest" c o n c e p t u a l d i s p a r i t i e s be-tween s c i e n t i f i c t h e o r i e s (Davidson 1984, 184). Ne v e r t h e l e s s , Kuhn c o m p l e t e l y r e j e c t s both a l t e r n a t i v e s and the i n t e r p r e t a t i o n s of h i s t h e s i s t h e y are based on. 7 Kuhn's c r i t i c s ' p o r t r a y a l of the incommensurability t h e s i s , as e i t h e r s e l f - r e f u t i n g or p h i l o s o p h i c a l l y - i r r e l e v a n t , s t r i k i n g l y resembles those he says g e n e r a l l y " c h a r a c t e r i z e d i s c o u r s e be-tween p a r t i c i p a n t s i n incommensurable p o i n t s of view" 7 (Kuhn 1970b, 230; c . f . p5 above). Thus the f a i l u r e of h i s c r i t -i c s t o f i n d an i n t e r p r e t a t i o n of h i s t h e s i s t h a t makes i t both i n t e l l i g i b l e and p h i l o s o p h i c a l l y - r e l e v a n t suggests t h e i r views may simply be incommensurable wi t h h i s ( i b i d . , 230). In other words, Kuhn's view may not be u n i n t e l l i g i b l e or i r r e l e v a n t sim-p l i c i t e r . J u s t so relative t o h i s c r i t i c s ' p e r s p e c t i v e . And s i n c e , by d e f i n i t i o n , incommensurable p o i n t s of view i n t e r p r e t common concepts d i f f e r e n t l y (see pp3-4 above), we should expect t h a t Kuhn's p o s i t i o n , i f i t i s i n t e l l i g i b l e or philosophically-relevant, must Involve a d i f f e r e n t and coherent understanding of those concepts t h a t l e a d t o t h e i r argumentative impasse." Before defending the i n t e l l i g i b i l i t y and r e l e v a n c e of Kuhn's p o s i t i o n , however, I s h a l l f i r s t i l l u s t r a t e the h i s t o r i c a l argument f o r the incommensurability t h e s i s w i t h Kuhn's d i s c u s s i o n of the P r o u s t - B e r t h o l l e t c o n t r o v e r s y . T h i s d i s c u s s i o n w i l l h o p e f u l l y p r o v i d e both a c l e a r e r p i c t u r e of the n o t i o n of loglco-normatlve i n c o m p a t i b i l i t y and an i n d i c a t i o n of one source of the Impasse between Kuhn and h i s c r i t i c s : the nature of l o g i c . 1.1.1 H i s t o r y and the Incommensurability T h e s i s One h i s t o r i c a l example Kuhn c i t e s of a p a i r of "fundamental-l y i n c o m p a t i b l e " t h e o r i e s are those of the French chemists J . L. Proust and C. L. B e r t h o l l e t . " P r o u s t h e l d t h a t a l l chemical r e -a c t i o n s take p l a c e i n f i x e d , i n t e g r a l r a t i o s but B e r t h o l l e t In-s i s t e d t h a t most occur i n c o n t i n u o u s l y - v a r y i n g p r o p o r t i o n s . On 8 the face of I t , these p r o p o s i t i o n s are examples of the more f a -m i l i a r , l e s s g l o b a l , n o t i o n of i n c o m p a t i b i l i t y s i n c e both seem t o be e x p r e s s i b l e as formal c o n t r a d i c t o r i e s u s i n g common ( l o g i c a l and n o n - l o g i c a l ) c o n c e p t s . 1 0 But Kuhn p o i n t s out t h a t not o n l y d i d P roust d i s p u t e B e r t h o l l e t ' s view of c h e m i c a l processes, he a l s o d i f f e r e d v i t h him about the meaning of the concept, chemical r e a c t i o n , used i n e x p r e s s i n g t h e i r a p p a r e n t l y Incompatible c l a i m s . Hence he a l s o d i f f e r e d v i t h him about the meanings of the r e l a t e d concepts of chemi c a l compound and p h y s i c a l m i x t u re. 1 1 Thus though both chemists allowed t h a t , "when mixing (of chemicals] produced heat, l i g h t , e f f e r v e s c e n c e or something e l s e of the s o r t , " a chemical r e a c t i o n had o c c u r r e d , o n l y B e r t h o l l e t took the behaviour of " s a l t i n v a t e r , a l l o y s , g l a s s , oxygen i n the atmosphere, and so on" as a l s o evidence of a chemical r e a c -t i o n (Kuhn 1970a, 131). Because t h i s l a t t e r c l a s s of substances can form combinations of smoothly-varying p r o p o r t i o n s , i n r a t i o s from zero t o a f i x e d l i m i t , B e r t h o l l e t argued t h a t they provided c l e a r evidence f o r h i s theory* And s i n c e such compounds are more commonly found than those combining i n f i x e d p r o p o r t i o n s , he con-cluded t h a t v a r i a b l e compounds vere the norm, f i x e d ones the ex-c e p t i o n (Meldrum 1910, 5). By c o n t r a s t , because he r e j e c t e d the idea t h a t s o l u t i o n s vere chemical compounds, P r o u s t denied t h a t B e r t h o l l e t had proved h i s p o i n t . And he i n s i s t e d t h a t o n l y those cases of mixing t h a t produced compounds of f i x e d p r o p o r t i o n s vere chemical r e a c t i o n s . 9 Kuhn here argues t h a t we should not construe t h i s d i f f e r e n c e of concepts as a t r i v i a l d e f i n i t i o n a l e q u i v o c a t i o n (Kuhn 1970a, 131). F o r , he c l a i m s , P r o u s t ' s and B e r t h o l l e t ' s r e s p e c t i v e c h o i -ces of d e f i n i t i o n of chemical reaction and compound had substan-t i v e , incompatible consequences f o r the ways i n which they ex-p l a i n e d t h e i r common environments. Thus, because he b e l i e v e d t h a t compounds of v a r i a b l e composition were the norm, B e r t h o l l e t sought t o f i n d i n s t a n c e s of them produced by chemical r e a c t i o n s meeting the c r i t e r i a t h a t P r o u s t a l s o accepted ( g i v i n g o f f heat, l i g h t , gas, e t c . ) . And such experimental evidence was not hard to f i n d . Thus B e r t h o l l e t was a b l e to show t h a t "metals such as copper, t i n and l e a d , on h e a t i n g i n a i r can take up oxygen con-tinuously i n p r o p o r t i o n s I n c r e a s i n g to a f i x e d l i m i t , g i v i n g a continuous s e r i e s of o x i d e s , as shown i n some cases (e.g. lead) by v a r y i n g c o l o u r changes" ( P a r t i n g t o n 1957, 156. I t a l i c s i n o r i g i n a l ) . On the other hand, because of h i s commitment to h i s " d e f i n i -t i o n , " Proust repeated B e r t h o l l e t ' s experiments and claimed t h a t the same o p e r a t i o n s showed " t h a t these oxides were mixtures of two, or a s m a l l number, of d e f i n i t e o x i d e s , and he c a r e f u l l y d i s t i n g u i s h e d between mixtures and s o l u t i o n s . . . a n d chemical com-pounds" ( i b i d . , 156). N e v e r t h e l e s s , the experimental evidence was c o m p l e t e l y e q u i v o c a l ; "the two men n e c e s s a r i l y t a l k e d through each other and t h e i r debate was e n t i r e l y I n c o n c l u s i v e " (Kuhn 1970a, 132). Thus i t seems t h a t Proust's and B e r t h o l l e t ' s con-f l i c t i n g d e f i n i t i o n s of concepts a l s o made t h e i r t h e o r e t i c a l con-10 e l u s i o n s incompatible. And, e q u a l l y , as ve s h a l l see, t h e i r c h o i c e of incompatible t h e o r i e s a l s o seemed t o f o r c e those s c i e n -t i s t s t o i n t e r p r e t shared concepts i n c o n f l i c t i n g vays. B e r t h o l l e t ' s c l a i m t h a t s a l i n e i s a chemical compound, and not a p h y s i c a l mixture, vas a n a t u r a l consequence of h i s nev t h e -ory of a f f i n i t y , v h i c h he c o n s t r u c t e d to overcome the f l a v s of the o l d e r v e r s i o n . A f f i n i t y t h e o r y took chemical compounds to r e s u l t from a mutual " e l e c t i v e " a f f i n i t y betveen t h e i r c o n s t i t u -ent elements. Thus lumps of elemental substances l i k e s i l v e r h e l d together because of the a f f i n i t y of t h e i r c o r p u s c l e s f o r each o t h e r ; mixed compounds formed because the c o r p u s c l e s of the d i f f e r e n t elements had a g r e a t e r mutual a f f i n i t y f o r each other than f o r those of t h e i r ovn s o r t ; and s o l u t i o n s formed because the s o l u t e ' s p a r t i c l e s had a g r e a t e r a f f i n i t y f o r those of the s o l v e n t than f o r each o t h e r . "In the e i g h t e e n t h c e n t u r y the t h e o r y of e l e c t i v e a f f i n i t y vas an admirable chemical paradigm, v i d e l y and sometimes f r u i t f u l l y deployed i n the d e s i g n and a n a l y -s i s of chemical e x p e r i m e n t a t i o n " (Kuhn 1970a, 130). P r i o r t o B e r t h o l l e t , chemists had combined a b e l i e f i n a f -f i n i t y t h e o r y v i t h an acceptance of the g e n e r a l i t y of R i c h t e r ' s l a v of c o n s t a n t p r o p o r t i o n s (Meldrum 1910, 2 ) . But the o l d a f -f i n i t y t h e o r y vas plagued by anomalies: f o r example, i t c o u l d n ' t e x p l a i n the phenomenon of "mass a c t i o n . " The chemist Bergman had shovn t h a t , f o r r e a c t i o n s of the form: "Ad + c = Ac + d, where Ac i s p r e c i p i t a t e d " (Meldrum 1910, 4 ) , 1 2 depending on the p a r t i c u l a r A, d and c i n v o l v e d , from t v o to s i x times the mass of c was needed to s a t u r a t e A when combined with d, than when A was uncom-bined. T h i s phenomenon was c l e a r l y incompatible with the o l d e r a f f i n i t y t h e o r y's acceptance of the d o c t r i n e of f i x e d composi-t i o n ; f o r under d i f f e r e n t circumstances f o r c e s of a f f i n i t y seemed to permit the formation of compounds of w i d e l y - v a r y i n g propor-t i o n s . B e r t h o l l e t chose t o r e s o l v e the o l d e r theory's anomaly by keeping the proven n o t i o n of chemical a f f i n i t y but " o b l i t e r a t -[ ing] the ( t r a d i t i o n a l ] d i s t i n c t i o n between chemical and p h y s i c a l f o r c e s " (Meldrum 1910, 4 ) . Thus he took " a l l f o r c e s of a f f i n i t y [to be] m o d i f i e d g r a v i t a t i o n a l a t t r a c t i o n " ( P a r t i n g t o n 1957, 157n*). That c h o i c e allowed B e r t h o l l e t t o keep i n t a c t much of the t h e o r e t i c a l apparatus of the o l d e r t h e o r y of a f f i n i t y but made i t impossible to d i f f e r e n t i a t e any longer between chemical compounds and s o l u t i o n s . Mere p h y s i c a l mixtures were now j u s t those combinations of substances t h a t f a i l e d t o show c o h e s i o n : f o r example, those whose c o n s t i t u e n t s "could be d i s t i n g u i s h e d by eye or m e c h a n i c a l l y s e p a r a t e d " (Kuhn 1970a, 131). And B e r t h o l l e t claimed the r e l a t i v e l y few compounds of f i x e d p r o p o r t i o n s oc-c u r r e d because of the ways i n which t h e i r s p e c i f i c s t r u c t u r e s I n t e r a c t e d with c e r t a i n environments. F i x e d compounds, he s a i d , "were formed as a r e s u l t of the i n t e r f e r e n c e of extraneous phys-i c a l f o r c e s . For example, c e r t a i n p r o p o r t i o n s of the elements c o u l d produce a compound which was l e a s t s o l u b l e or of g r e a t e s t d e n s i t y ( i n f l u e n c e of c o h e s i o n ) , or the most v o l a t i l e ( i n f l u e n c e 12 of e l a s t i c i t y ) of a l l the p o s s i b l e compounds, and hence t h i s compound was formed i n p r e f e r e n c e " ( P a r t i n g t o n 1957, 157). Instead of choosing to keep the t i m e - t e s t e d concept of a f -f i n i t y from the o l d e r theory, however, Pr o u s t opted to r e t a i n t h a t t h e ory's law of c o n s t a n t p r o p o r t i o n s and i t s d i s t i n c t i o n between s o l u t i o n (qua p h y s i c a l mixture) and chemical compound. However, t h a t l e f t him with no a l t e r n a t i v e p o s i t i v e account of what f o r c e s kept substances l i k e s a l t and sugar, as mere p h y s i c a l m i xtures, In s o l u t i o n ; he c o u l d o n l y r e p l y t h a t such s o l u t i o n s c l e a r l y d i f f e r e d i n nature from standard chemical compounds l i k e s a l t and sugar themselves. So the f o r c e s t h a t bound s o l v e n t and s o l u t e together were conceivably v e r y d i f f e r e n t from chemical f o r c e s . Hence, he argued, the behaviour of s o l u t i o n s should not be used as counterexamples to h i s t h e o r y ( P a r t i n g t o n 1957, 157). That r e b u t t a l d i d not i n i t i a l l y prove v e r y c o n v i n c i n g to chemists because B e r t h o l l e t ' s new a f f i n i t y t h e o r y was the o n l y method then a v a i l a b l e f o r e x p l a i n i n g "mass a c t i o n " (now under-stood as "chemical r e a c t i o n r a t e s " (Day, General Chemistry, 1974). Proust's s i d e had to wait f o r Dalton's atomic theory and the work of B e r z e l i u s before they c o u l d begin to match B e r t h o l -l e t ' s account i n t h i s area (Meldrum 1910, 8, 12). Dalton's new t h e o r y , based on a law of m u l t i p l e (as opposed to e i t h e r f i x e d or c o n t i n u o u s l y - v a r y i n g ) p r o p o r t i o n s had much g r e a t e r apparent scope and i m p l i c a t i o n s than e i t h e r B e r t h o l l e t ' s or P r o u s t ' s t h e o r i e s . I t made sense of a c l e a r demarcation of s o l u t i o n s from compounds; handled the formation of metal oxides and s a l t s i n a l e s s equivo-13 c a l manner than e i t h e r B e r t h o l l e t or Proust; accounted f o r Gay-Lussac's l a v of combining volumes of gases; and vas a l s o a b l e encompass the phenomenon of mass a c t i o n (Meldrum 1910, 12-14). Dalton's "nev vay of p r a c t i c i n g chemistry...proved so r a p i d l y f r u i t f u l t h a t o n l y a f e v of the o l d e r chemists i n France and B r i -t a i n vere a b l e t o r e s i s t i t " (Kuhn 1970a, 134). S t i l l , i t " r e -s o l v e d " ( f o r most chemists, but not B e r t h o l l e t ) the d i s p u t e by i n t r o d u c i n g a nev t h e o r y and s e t of concepts t h a t vere incompat-i b l e v i t h both those of B e r t h o l l e t and P r o u s t . And i t a l s o l e f t unsolved a problem B e r t h o l l e t ' s t h e o r y c o u l d handle: the beha-v i o u r of s o l u t i o n s ( i b i d . , 131). In summary, i t seems, a t l e a s t I n t u i t i v e l y , t h a t B e r t h o l -l e t ' s and P r o u s t ' s (and Dalton's) competing t h e o r i e s of chemical r e a c t i o n s and compounds a l s o i n v o l v e d competing concepts f o r p i c k i n g out i n s t a n c e s of those r e a c t i o n s and compounds. Kuhn p o i n t s out a s t i l l f u r t h e r aspect of t h i s p u t a t i v e logico-norma-t i v e i n c o m p a t i b i l i t y . Based on c o n c l u s i o n s he dravs from G e s t a l t psychology and the work of the p h i l o s o p h e r N. R. Hanson, he a r -gues t h a t the v i e v s of Proust and B e r t h o l l e t vere not o n l y l o g i -c a l l y and c o n c e p t u a l l y , but a l s o perceptually or observatlonally, incompatible (see Kuhn 1970a, 112-114). Thus he c l a i m s t h a t not o n l y vas i t i m p o s s i b l e f o r e i t h e r s c i e n t i s t t o understand or f o r -mulate the o t h e r ' s t h e o r y u s i n g h i s ovn concepts; i t vas a l s o im-p o s s i b l e f o r e i t h e r s c i e n t i s t t o p e r c e i v e the v o r l d i n the same vay as h i s opponent. Thus "vhere B e r t h o l l e t sav a compound t h a t c o u l d v a r y i n p r o p o r t i o n , P r o u s t sav o n l y a p h y s i c a l mixture" 14 ( i b i d . 132; c . f . i b i d . , 198). In other words, even t h e i r "obser-v a t i o n languages" were " t h e o r y - l a d e n , " and thus i n c o m p a t i b l e . Because of t h i s p u t a t i v e t o t a l i n c o m p a t i b i l i t y o f t h e i r p o i n t s of view, Kuhn concludes t h a t Proust and B e r t h o l l e t c o u l d not f i n d any common ground t o s e t t l e t h e i r d i s p u t e "and t h e i r debate was e n t i r e l y i n c o n c l u s i v e " (Kuhn 1970a, 132). 1.2 The C r i t i c a l Response: L o g i c and Formal L o g i c s I f Kuhn i s r i g h t t h a t there are t h e o r i e s , l i k e P r o u s t ' s and B e r t h o l l e t ' s , t h a t are s i m u l t a n e o u s l y l o g i c a l l y , c o n c e p t u a l l y and p e r c e p t u a l l y i n c o m p a t i b l e , i t seems c l e a r t h a t the l o g i c a l incom-p a t i b i l i t y of those t h e o r i e s c o u l d not be formally demonstrated. T h i s i s because: (1) t o show on the b a s i s of " l o g i c a l form" t h a t two e n t i t i e s (e.g. sentences, t h e o r i e s ) are i n c o m p a t i b l e , formal l o g i c s r e q u i r e t h a t those e n t i t i e s have common concepts;*" and (2) the f a c t t h a t the concepts of two p u t a t i v e l y incommensurable t h e o r i e s form d i s t i n c t , "conceptual webs" means t h a t one cannot c o n s t r u c t commonly-acceptable concepts from the two t h e o r i e s from s p e c i f i c , l o c a l i n s t a n c e s of l o g i c a l c o n f l i c t or congruence be-tween them. That i s , because the meanings of the ot h e r , incom-p a t i b l e t h e o r y ' s concepts are giv e n by t h e i r e n t i r e r o l e i n t h a t theory, t h a t theory's content cannot be d e s c r i b e d except by con-cepts which presuppose t h a t content's t r u t h . 1 4 I t seems c l e a r , t h e r e f o r e , t h a t i f there are t h e o r i e s t h a t are both l o g i c a l l y and c o n c e p t u a l l y i n c o m p a t i b l e , s i n c e formal 15 l o g i c s a p p a r e n t l y cannot demonstrate t h a t i n c o m p a t i b i l i t y , the gen e r a l concept of l o g i c a l i n c o m p a t i b i l i t y must have a l a r g e r scope than the s p e c i f i c formal one. On the other hand, however, the f a c t t h a t Kuhn's c r i t i c s do not g e n e r a l l y r e c o g n i z e t h i s op-t i o n , suggests s t r o n g l y t h a t they b e l i e v e t h a t l o g i c a l incompati-b i l i t y i s a p u r e l y formal n o t i o n . 1 8 Thus a standard o b j e c t i o n to Kuhn's analyses i s t h a t , be-cause debates l i k e those of P r o u s t and B e r t h o l l e t seem s t a l l e d l a r g e l y by t h e i r e q u i v o c a l use of common e x p r e s s i o n s , Kuhn has not " j u s t i f i e d " the c l a i m t h a t such views are actually incompat-i b l e (Newton-Smith 1981, 149). Instead i t i s claimed t h a t I f they were t r u l y Incompatible, i t should be p o s s i b l e t o analyze f u r t h e r those t h e o r i e s ' concepts and express t h e i r c o n f l i c t as a formal c o n t r a d i c t i o n u s i n g common concepts. For example, i t might be the case t h a t Proust and B e r t h o l l e t shared t a c i t , common c r i t e r i a f o r d i s t i n g u i s h i n g c h e m i c a l compounds from p h y s i c a l mix-t u r e s , which might be made e x p l i c i t through formal a n a l y s i s ; i n which case, t h e i r views would d i f f e r o n l y l o g i c a l l y , not concep-t u a l l y . On the other hand, i f Pr o u s t ' s and B e r t h o l l e t ' s concepts were d i f f e r e n t , Kuhn's c r i t i c s i n f e r t h a t , t h e r e f o r e , appearances and i n t u i t i o n s t o the c o n t r a r y , t h e i r t h e o r i e s were r e a l l y com-p a t i b l e ( c . f . Newton-Smith 1981, 150). Kuhn's r e p l y to the f i r s t a l t e r n a t i v e i s to note t h a t s c i e n -t i s t s seldom r e c o g n i z e the r e s u l t s of such formal " a n a l y s e s " as e q u i v a l e n t with t h e i r own usage (Kuhn 1970a, 47; Kuhn 1977, 305). Moreover, he a l s o notes t h a t when a h i s t o r i a n l i k e h i m s e l f t r i e s 16 to analyse the l o g i c a l bases of o l d e r t h e o r i e s , he I n v a r i a b l y f i n d s t h a t "phrased i n j u s t t h a t way, or i n any other way he can imagine, (those l o g i c a l r e c o n s t r u c t i o n s ] would almost c e r t a i n l y have been r e j e c t e d by some members of the group he s t u d i e s " (Kuhn 1970a, 44). In other words, Kuhn i s s a y i n g t h a t , i n h i s e x p e r i -ence, formal r e c o n s t r u c t i o n s of h i s t o r i c a l t h e o r i e s always misre-present t h e i r content. And he appeals t o W i t t g e n s t e i n ' s work on " f a m i l y resemblance" concepts t o argue t h a t t h e r e need be no t a -c i t , formal s t r u c t u r e s u n d e r l y i n g s c i e n t i s t s ' r e a s o n i n g ( i b i d . , 4 7 ) . " Of course, one might o b j e c t t h a t s c i e n t i s t s ' d e n i a l s of the Congruence of t h e i r concepts with the r e s u l t s of a n a l y s i s are beside the p o i n t s i n c e the i s s u e s i n v o l v e d i n those r e c o n s t r u c -t i o n s l i e o u t s i d e t h e i r e x p e r t i s e ( C o l l i e r 1984, 96). To see tha t t h i s o b j e c t i o n f a i l s t o address Kuhn's p o i n t one must keep separate the f o l l o w i n g i s s u e s : (1) whether or not s c i e n t i s t s are n e c e s s a r i l y q u a l i f i e d t o judge, say, the r e l a t i o n s between t h e i r own and h i s t o r i c a l t h e o r i e s or the p h i l o s o p h i c a l i m p l i c a t i o n s of t h e i r work; and (2) whether or not s c i e n t i s t s are e x c e p t i o n a l l y q u a l i f i e d t o determine when formal r e c o n s t r u c t i o n s change the meanings of t h e i r concepts i n t h e i r s c i e n t i f i c context. I t seems c l e a r t h a t the answer t o the f i r s t q u e s t i o n i s t h a t s c i e n t i s t s are not s p e c i a l l y q u a l i f i e d by t h e i r s c i e n t i f i c work i n e i t h e r h i s t o r i c a l r e c o n s t r u c t i o n or p h i l o s o p h i c a l a n a l y s i s . But i t seems e q u a l l y c l e a r t h a t the answer to the second q u e s t i o n i s t h a t s c i e n t i s t s are s u r e l y the most q u a l i f i e d to recognize the noncongruence of t h e i r ovn usage v i t h formal r e c o n s t r u c t i o n s (supposing t h a t they understand the formal language). And Kuhn o b v i o u s l y has the second q u e s t i o n i n mind (along v i t h i t s suppo-s i t i o n ) i n the above c l a i m (pl5-16; see a l s o Kuhn 1977c, 303nl3). Thus i f a " r e l n t e r p r e t a t i o n i n the broader c o n t e x t " ( C o l l i e r 1984, 96) seems to a s c i e n t i s t t o change h i s concepts, t h a t be-l i e f deserves s e r i o u s c o n s i d e r a t i o n . More important, s i n c e the p o i n t of such r e i n t e r p r e t a t i o n s i s t o provide a n e u t r a l framework f o r s c i e n t i f i c debate, the f a c t t h a t s c i e n t i s t s from both camps vou l d l i k e l y f i n d those r e i n t e r p r e t a t i o n s i n a c c u r a t e v o u l d o n l y d e f e a t the purpose of t h a t supposedly n e u t r a l r e l n t e r p r e t a t i o n . Kuhn's counter to the second c l a i m t h a t t h e o r i e s t h a t cannot be r e c o n s t r u c t e d as f o r m a l l y incompatible are simply d i f f e r e n t and compatible has been, as ve have seen, to t r y t o make the i n -c o m p a t i b i l i t y of such t h e o r i e s seem i n t u i t i v e l y obvious. And, though i n t u i t i o n s n o t o r i o u s l y d i f f e r , the t h e o r i e s he d e s c r i b e s as incompatible are u s u a l l y a l s o taken as such by h i s c r i t i c s (e.g., Nevton-Smith 1981, 158). Thus, again , Kuhn's c r i t i c s ' i n -s i s t e n c e t h a t t h e o r i e s t h a t seem i n t u i t i v e l y Incompatible are e i -ther r e c o n s t r u c t i b l e as f o r m a l l y incompatible or not r e a l l y i n -compatible suggests that the r e a l i s s u e betveen them and Kuhn i s vhether or not the canons of formal l o g i c s f u l l y capture the no-t i o n of l o g i c a l i n c o m p a t i b i l i t y . In the f o l l o w i n g s e c t i o n s , I use comments and arguments from Kuhn's c r i t i c s t o show t h a t , to preserve the i n t e l l i g i b i l i t y and r e l e v a n c e of h i s t h e s i s , he must d i f f e r with them not o n l y on the 18 nature of l o g i c but a l s o on the more g e n e r a l n o t i o n of compari-son. And I g i v e t e x t u a l evidence t o imply t h a t Kuhn would a c t u -a l l y want t o d i f f e r from h i s c r i t i c s i n the ways I suggest. I then l i n k t h a t d i s p u t e about comparison and l o g i c t o the a n c i e n t problem of u n i v e r s a l s and argue t h a t the p o s i t i o n of Kuhn's c r i t -i c s leads t o an i n f i n i t e r e g r e s s . 1.3 The C r i t i c a l Response: C o m p a r a b i l i t y and Commensurabillty As we have j u s t seen, one primary source of Kuhn's opponents problems with the incommensurability t h e s i s i s t h e i r i d e n t i f i c a -t i o n of l o g i c with formal l o g i c . But there i s a s t i l l more b a s i c source of h i s c r i t i c s ' d i f f i c u l t i e s : t h e i r e q u a t i o n of "incommen-s u r a b l e " with e i t h e r " a b s o l u t e l y incomparable" or "not i n t e r -t r a n s l a t a b l e . " As noted above (p6), o f t e n Kuhn's opponents' s t r a t e g y i s to c o n f r o n t him with a dilemma. With r e s p e c t t o the concept of incommensurability i t s e l f , they commonly argue t h a t e i t h e r (1) i f "incommensurable" i s read ' l i t e r a l l y ' — a s meaning " a b s o l u t e l y i n -comparable"—Kuhn's t h e s i s t u r n s out to be the s e l f - c o n t r a d i c t o r y c l a i m t h a t c e r t a i n t h e o r i e s are both ( a b s o l u t e l y ) incomparable and comparable;* 7 or (2) i f t h a t prima f a c i e r e a d i n g i s r e j e c t e d , h i s t h e s i s i s not s e l f - c o n t r a d i c t o r y , but "incommensurability" turns out to be o n l y h i s m i s l e a d i n g e x p r e s s i o n f o r what are mere-l y p s y c h o l o g i c a l or pragmatic problems i n comprehending, compar-ing or t r a n s l a t i n g a l i e n v iews. 1 8 19 I t i s easy t o see why Kuhn's opponents' I n t e r p r e t a t i o n of "incommensurable," as meaning l i t e r a l l y e i t h e r " a b s o l u t e l y incom-p a r a b l e " or "not i n t e r t r a n s l a t a b l e , " makes h i s t h e s i s seem u n i n -t e l l i g i b l e . F o r , given t h e i r Interpretation, s i n c e i t i s t r i v i -a l l y obvious t h a t to c a l l t vo t h e o r i e s incompatible i s both to compare and c o n t r a s t them,'9 i t f o l l o w s t h a t Kuhn c o u l d not pos-sibly f i n d h i s t o r i c a l examples of s u c c e s s i v e t h e o r i e s t h a t are "not o n l y i n c o m p a t i b l e but o f t e n a c t u a l l y incommensurable" (Kuhn 1970a, 103). F u r t h e r , on this Interpretation, s i n c e i n h i s v r i t -ings Kuhn r e p e a t e d l y t r a n s l a t e s , i n a common ( n a t u r a l ) language (contemporary E n g l i s h ) , t h e o r i e s he c l a i m s are incommensurable v i t h contemporary ones, i t f o l l o v s t h a t he c o u l d not p o s s i b l y use t h i s vork to prove those t h e o r i e s vere t r u l y incommensurable. Donald Davidson, vho takes "incommensurable" to mean "not i n t e r t r a n s l a t a b l e " (Davidson 1984b, 190), makes e x a c t l y t h i s l a s t p o i n t vhen he remarks i r o n i c a l l y : "Kuhn i s b r i l l i a n t a t s a y i n g vhat t h i n g s vere l i k e before the r e v o l u t i o n u s i n g — v h a t e l s e ? — o u r p o s t - r e v o l u t i o n a r y idiom" ( i b i d . 184). Even a p h i l o s o p h e r as sympathetic t o Kuhn's o v e r a l l v i e v of s c i e n c e as H i l a r y Putnam ( c . f . Putnam 1981b, esp. 69-78) f i n d s h i s t h e s i s to be " s e l f -r e f u t i n g " (Putnam 1981a, 114), and f o r e x a c t l y the same reasons as Davidson ( I b i d . , 116). Thus Putnam observes t h a t , s i n c e " i n -commensurable" means "not I n t e r t r a n s l a t a b l e " ( i b i d . , 114), f o r Kuhn (or P a u l Feyerabend) t o " t e l l us t h a t G a l i l e o had 'incommen-s u r a b l e * n o t i o n s and then to go on to d e s c r i b e them a t l e n g t h i s Incoherent" (Putnam 1981a, 115. I t a l i c s i n o r i g i n a l ) . Kuhn's c r i t i c s ' i d e n t i f i c a t i o n o f "in-commensurable" with " a b s o l u t e l y in-comparable" (and " n o t - i n t e r t r a n s l a t a b l e " ) pro-v i d e s a c l u e t o the deep source of t h e i r d i s p u t e with him: i t shows t h a t they take "comparable" (and " i n t e r t r a n s l a t a b l e " ) t o mean "commensurable/" which i t s e l f means l i t e r a l l y "measurable [ b r o a d l y : comparable] by a common s t a n d a r d " (Concise Oxford En-glish Dictionary, s.v. "commensurable"). But "comparable" and "comparable by a common st a n d a r d " are not s e l f - e v i d e n t l y synony-mous e x p r e s s i o n s . F u r t h e r , i t a l s o seems p o s s i b l e t o d i s t i n g u i s h the l o c u t i o n s "comparable i n a common natural language" ( i n t e r -t r a n s l a t a b l e ) and "comparable i n a common system of concepts" (commensurable) ( c . f . n8, n l 4 ) ; f o r natural languages may not be f u l l y uniform conceptual systems. So whether "commensurable" should be c o n f l a t e d with, or d i s t i n g u i s h e d from, e i t h e r "compar-a b l e " or " i n t e r t r a n s l a t a b l e " seems t o be a s i g n i f i c a n t p h i l o s o -p h i c a l i s s u e . And Kuhn has gone on r e c o r d as denying t h a t " i n -commensurable" means e i t h e r "incomparable" or "not i n t e r t r a n s -l a t a b l e " (e.g., Kuhn 1970b, 267; Kuhn 1976, 191; Kuhn 1970a, 201-4) . On the other hand, however, i f one took c o m p a r a b i l i t y to r e q u i r e commensurablllty, because l o g i c a l I n c o m p a t i b i l i t y c l e a r l y i n v o l v e s ( l o g i c a l ) c o m p a r a b i l i t y , he would n a t u r a l l y a l s o con-clude t h a t l o g i c a l l y incompatible t h e o r i e s cannot be incommensur-a b l e . Thus we f i n d t h a t one of Kuhn's main c r i t i c s , Dudley Sha-pere, who e x p l i c i t l y equates "Incommensurable" with " a b s o l u t e l y incomparable" (Shapere 1981, 55), a l s o says t h a t " i n order f o r 21 Itwo t h e o r i e s ] t o be i n c o n s i s t e n t with one another, they must be formulated, or a t l e a s t f o r m u l a b l e , i n a common language." ( i b i d . 44nl0. I t a l i c s mine). Shapere concludes t h a t " i t i s d i f f i c u l t t o see how one c o u l d c o n s t r u c t a t h e o r y which, while d i f f e r i n g i n the meanings of a l l i t s terms from another t h e o r y ( i . e . , i s i n -commensurable w i t h i t ] , can n e v e r t h e l e s s be I n c o n s i s t e n t w i t h t h a t other t h e o r y " ( i b i d . , 44). As the f o l l o w i n g passage r e v e a l s , another prominent opponent of Kuhn's views, I s r a e l S c h e f f l e r , a l s o i d e n t i f i e s "comparable" with "commensurable"; thus he wonders how paradigms t h a t are ...based i n d i f f e r e n t worlds, and address themselves to d i f -f e r e n t problems with the h e l p of d i f f e r e n t standards . . . l e a n be] s a i d t o be i n c o m p e t i t i o n ? To d e c l a r e them i n competi-t i o n i s , a f t e r a l l , t o p l a c e them w i t h i n some common frame-work, to view them w i t h i n some shared p e r s p e c t i v e s u p p l y i n g , i n p r i n c i p l e a t l e a s t , comparative and e v a l u a t i v e c o n s i d e r a -t i o n s a p p l i c a b l e t o both. I t i s i n f a c t to c o n s i d e r them as o r i e n t e d i n d i f f e r e n t ways toward the same purposes, as making r i v a l appeals from the s t a n d p o i n t of s c i e n t i f i c g o a l s taken t o be o v e r r i d i n g and with r e s p e c t to a common s i t u a -t i o n taken as a p o i n t of r e f e r e n c e . ( S c h e f f l e r 1967, 82. I t a l i c s mine). Thus both Shapere and S c h e f f l e r imply t h a t Kuhn's a s s e r t i o n t h a t incommensurable t h e o r i e s can be i n c o m p e t i t i o n i s s e l f - c o n -t r a d i c t o r y f o r t h e y c l a i m t h a t the v e r y n o t i o n of i n c o m p a t i b i l i t y presupposes t h a t of commensurability. Importantly, however, n e i -t h er Shapere nor S c h e f f l e r g i v e an argument f o r t h e i r c l a i m ; i n -stead they s i m p l y a s s e r t i t as though i t s t r u t h were s e l f - e v i -d e n t . 8 0 A c c o r d i n g to Ian Hacking, the deepest c r i t i c i s m of incommen-s u r a b i l i t y i s t h a t of Davidson (Hacking 1983, 73-4; Davidson 1984b, 183-198). Davidson a s s o c i a t e s the idea of incommensur-a b i l i t y w i t h the more commonly accepted view t h a t t h e r e can be more than one "conceptual scheme" ( i b i d . , 185, 190) but d enies t h a t t h i s i d e a makes sense a t all: t h a t i s , even when a p p l i e d t o t h e o r i e s or schemes t h a t a r e n ' t competing ( i b i d . , 198). L i k e Shapere and S c h e f f l e r , he understands "incommensurable" to mean " a b s o l u t e l y incomparable" ( i b i d . , 184) a t and, as we have a l s o seen, "not i n t e r t r a n s l a t a b l e " ( i b i d . 190). Davidson g i v e s a s u b t l e argument, based on h i s p h i l o s o p h i c a l s t u d i e s , t o t r y to show t h a t we cannot s e n s i b l y a t t r i b u t e any t h e o r i e s , b e l i e f s or concepts to o t h e r s unless we can f i r s t ( e x a c t l y ) t r a n s l a t e what they are s a y i n g " i n t o a f a m i l i a r tongue" such as E n g l i s h ( i b i d . 186). But Davidson's argument i s "deeper" j u s t because i t i s more e x p l i c i t l y committed than those of Kuhn's other c r i t i c s to an e q u a t i o n of c o m p a r a b i l i t y with commensurability. Thus, while he acknowledges t h a t the examples Kuhn (and o t h e r s ) p o i n t to of con-c e p t u a l change and d i s p a r i t y are " [ o c c a s i o n a l l y ] i m p r e s s i v e " (Da-v i d s o n 1984b, 184), he i n s i s t s t h a t they "are not so extreme but t h a t the changes and the c o n t r a s t s can be e x p l a i n e d and d e s c r i b e d u s i n g the equipment of a single language" (Davidson 1984b, 184. I t a l i c s mine). And t h i s c l a i m shows t h a t Davidson i m p l i c i t l y equates i n t e r t r a n s l a t l o n ( s u c c e s s f u l e x p l a n a t i o n and d e s c r i p t i o n of a l i e n views i n a s i n g l e , n a t u r a l language) with commensuration ( t h e i r s u c c e s s f u l e x p l a n a t i o n and d e s c r i p t i o n i n a n e u t r a l system of concepts: "the equipment of a s i n g l e language.") 8 8 However, as I have argued above (p20), i t Is j u s t t h i s p u t a t i v e I d e n t i t y of l n t e r t r a n s l a t a b i l i t y and commensurability t h a t Kuhn d e n i e s . That Kuhn's c r i t i c s i m p l i c i t l y i d e n t i f y "comparable" (and " I n t e r t r a n s l a t a b l e " ) v i t h "commensurable" i s a l s o made c l e a r by the o n l y a l t e r n a t i v e s they seem v i l l l n g t o c o n s i d e r i f t h e i r f i r s t c o n s t r u a l of " i n c o m m e n s u r a b i l i t y " i s not vhat he has i n mind. Thus they commonly counter t h a t i f , c o n t r a r y to f i r s t ap-pearances, h i s n o t i o n i s n ' t intended t o imply a b s o l u t e incompar-a b i l i t y or u n t r a n s l a t a b i l i t y , then the t h e o r i e s he terms "incom-mensurable" must, In the l i t e r a l sense, be commensurable. But, i n t h a t case, they say, Kuhn's t h e s i s cannot have any p h i l o s o p h i -c a l import; a t most i t can have o n l y e i t h e r p s y c h o l o g i c a l or pragmatic s i g n i f i c a n c e . For example, Alan Musgrave says t h a t i f comparison or t r a n s -l a t i o n of 'incommensurable' t h e o r i e s i s p o s s i b l e , "incommensura-b i l i t y has ceased t o be a l o g i c a l a f f a i r and [ i t s import! has be-come a p u r e l y p s y c h o l o g i c a l . . . m a t t e r " (Musgrave 1982, 50): i t merely r e f e r s t o the commonplace empirical f a c t t h a t o l d s c i e n -t i s t s o f t e n have c o n s i d e r a b l e d i f f i c u l t y i n a c c e p t i n g and even understanding nev t h e o r i e s . 2 3 S i m i l a r l y , Davidson argues t h a t examples Kuhn and others ad-duce to show t h a t , "what comes e a s i l y i n one language may come hard i n another," can o n l y r e v e a l "modest," pragmatic d i f f i c u l -t i e s i n s e c u r i n g adequate t r a n s l a t i o n s ; they cannot provide e v i -dence f o r c o n c e p t u a l d i s c o n t i n u i t i e s (Davidson 1984b, 184). I t f o l l o w s t h a t Kuhn's term "incommensurable'' i s e i t h e r a m i s l e a d i n g e x p r e s s i o n f o r a n o n - l o g i c a l issue ( i f i t does not mean "not i n t e r t r a n s l a t a b l e " ) or embodies a c o n f u s i o n ( i f I t d o e s ) : " i f t r a n s l a t i o n [of another's c o n c e p t u a l scheme] succeeds, we have shown t h e r e Is no need to speak of two conceptual schemes, while i f t r a n s l a t i o n f a i l s there i s no ground f o r speaking of two" (Davidson 1980b, 243). 2 4 As a l r e a d y noted (p20), Kuhn denies h i s opponents' charge t h a t by "incommensurable" he ever meant " a b s o l u t e l y incomparable" or "not i n t e r t r a n s l a t a b l e " ; moreover, he c o n t i n u e s t o i n s i s t t h a t h i s t h e s i s has l o g i c a l and e p l s t e m o l o g i c a l r e p e r c u s s i o n s , not merely p s y c h o l o g i c a l or pragmatic ones (e.g. Kuhn 1970b, 232-3). But p h i l o s o p h e r s are o f t e n r e l u c t a n t to l e t go of even a straw man-opponent. Thus Kuhn's c r i t i c s have taken h i s d e n i a l s t o show simply t h a t he i s "but a pale r e f l e c t i o n of the o l d , r e v o l u t i o n -a r y Kuhn"; t h a t , i n the face of t h e i r cogent c r i t i c i s m , he beat a f u l l - s c a l e r e t r e a t from h i s o r i g i n a l , " c h a l l e n g i n g " ideas towards the t r a d i t i o n a l canons of r a t i o n a l i t y t h a t he o r i g i n a l l y s l u r r e d (e.g. Musgrave 1981, 51). 2 8 Kuhn's r e t o r t i s t h a t the debate be-tween him and h i s c r i t i c s has the c h a r a c t e r i s t i c " t a l k l n g -through-each-other" f e a t u r e of arguments between p r a c t i t i o n e r s of incommensurable paradigms l i k e Proust and B e r t h o l l e t (Kuhn 1970b, 231-2). Other p h i l o s o p h e r s are more sympathetic to Kuhn's view than Davidson and Musgrave but a l s o imply, l i k e Davidson, t h a t the d i f f i c u l t i e s he terms " i n c o m m e n s u r a b i l i t i e s " are not l o g i c a l and e p i s t e m o l o g i c a l but a r i s e o n l y from temporary pragmatic problems In p r e c i s e l y f o r m u l a t i n g the r e l a t i o n s betveen s c i e n t i f i c t h e o r -i e s . Incommensurability v o u l d be s o l e l y a pragmatic, and thus temporary, i s s u e i f , f o r example, the lack of common concepts vas o n l y apparent and removable by a n a l y s i s ; or i f "metascience" vere a b l e t o f i n d other common elements than concepts or phenomena f o r n e u t r a l t h e o r y comparison. 8' As noted above (ppl6-17), John C o l l i e r takes the f i r s t op-t i o n : he suggests t h a t ve can make a r t i c u l a t e the " i m p l i c i t p a r t of a t h e o r y " through "the c r e a t i o n of a broader, more i n c l u s i v e language capable of e x p r e s s i n g the content of both t h e o r i e s " ( C o l l i e r 1984, 8-9). Dale Moberg takes the second approach: he argues t h a t incommensurability because of " r a d i c a l meaning v a r i -ance" may simply p o i n t t o a need t o develop a nev "theory of t h e -ory comparison" (Moberg 1979, 261). Though there are no c o n c l u -s i v e a priori reasons vhy such p r o j e c t s must f a i l , ve have seen t h a t Kuhn argues i n e f f e c t t h a t there are no a p r i o r i reasons vhy they must succeed e i t h e r ; and th e r e are p l e n t y of e m p i r i c a l r e a -sons to b e l i e v e t h a t they v i l l not ( c . f . ppl6-17 above). S t i l l , i t i s m i s l e a d i n g t o see the d i s p u t e betveen Kuhn and h i s opponents as p u r e l y e m p i r i c a l . As ve have seen (ppl9-20), h i s c r i t i c s ' p h i l o s o p h y makes incommensurability a l o g i c a l impos-s i b i l i t y ; so t h a t p h i l o s o p h y can have no c r i t e r i a f o r d e c i d i n g vhether a g i v e n p a i r of t h e o r i e s vere or vere not commensurable. Kuhn's opponents' o f t e n unspoken, but c o n s i s t e n t , i d e n t i f i c a t i o n s of c o m p a r a b i l i t y with commensurablllty; and l o g i c a l i n c o m p a t i b i l -i t y with formal l o g i c a l i n c o m p a t i b i l i t y show t h a t those i d e n t i -f i c a t i o n s are not a r b i t r a r y , but b u i l t i n t o t h e i r whole p h i l o s o -p h i c a l approach. Davidson makes t h i s normally t a c i t commitment e x p l i c i t when he says a t the b e g i n n i n g of h i s c r i t i q u e of Kuhn: " D i f f e r e n t p o i n t s of view make sense, but only i f t h e r e Is a com-mon co-ordinate system on vhich to plot them} y e t the e x i s t e n c e of a common system [standard] b e l i e s the c l a i m of dramatic incom-p a r a b i l i t y " (Davidson 1984, 184. I t a l i c s mine). And Davidson, l i k e Shapere and S c h e f f l e r , never argues f o r t h i s a s s e r t i o n ; and he t a c i t l y assumes t h a t p h i l o s o p h e r s l i k e Kuhn a l s o accept i t . 8 7 1.4 I n t e r i m Summary We have seen t h a t the Incommensurability t h e s i s i s the c l a i m t h a t s u c c e s s i v e s c i e n t i f i c t h e o r i e s are o f t e n both l o g i c a l l y and n o r m a t i v e l y ( i . e . c o n c e p t u a l l y , o p e r a t i o n a l l y , o b s e r v a t i o n a l l y ) i n c o m p a t i b l e . We have a l s o seen t h a t Kuhn's c r i t i c s f i n d h i s t h e s i s e i t h e r u n i n t e l l i g i b l e or p h i l o s o p h i c a l l y i r r e l e v a n t be-cause they equate l o g i c with formal l o g i c s and c o m p a r a b i l i t y and i n t e r t r a n s l a t a b i l l t y with commensurablllty. T h i s leaves us with two ( m u t u a l l y - e x c l u s i v e ) o p t i o n s : (1) e i t h e r those n o t i o n s cannot be m e a n i n g f u l l y d i s t i n g u i s h e d ; i n which case, Kuhn's t h e s i s i s "impaled on the dilemma"; or (2) i t i s p o s s i b l e to d i s t i n g u i s h those n o t i o n s s i g n i f i c a n t l y ; i n which case, Kuhn "escapes between 27 the horns" but the t r a d i t i o n a l " e p l s t e m o l o g l c a l paradigm" l o s e s i t s p h i l o s o p h i c a l impetus. 2' In the f o l l o w i n g s e c t i o n , I show how to make a p h i l o s o p h i -c a l l y - s i g n i f i c a n t d i s t i n c t i o n between comparison and commensura-t i o n ; and, as p roof my i n t e r p r e t a t i o n i s on the r i g h t t r a c k , I g i v e t e x t u a l evidence to i n d i c a t e t h a t Kuhn a l s o makes such a d i s t i n c t i o n i n h i s p h i l o s o p h y . I then g i v e a more d e t a i l e d a n a l -y s i s of the p r e s u p p o s i t i o n s of s y s t e m a t i c , formal l o g i c s , l i k e the p r o p o s i t l o n a l and p r e d i c a t e c a l c u l i , t o h e l p p i n p o i n t those p r e s u p p o s i t i o n s which s p e c i f i c a l l y embody a commitment to commen-s u r a b i l i t y . That a n a l y s i s w i l l h o p e f u l l y y i e l d a c l e a r e r p i c t u r e of what a n o t i o n of l o g i c t h a t i s compatible with the Incommen-s u r a b i l i t y t h e s i s would be l i k e . 2 9 1.5 How to D i s t i n g u i s h C o m p a r a b i l i t y from Commensurability As a l r e a d y mentioned (p20), the most obvious p o t e n t i a l d i f -ference between "comparable" and "commensurable" i s t h a t the l a t -t e r e x p r e s s i o n i n v o l v e s an a d v e r b i a l q u a l i f i c a t i o n of the f i r s t : t h a t i s , "commensurable" means "comparable by a common standard." And i f the a d v e r b i a l c l a u s e "by a common standard" i s not redun-dant, then "incommensurable" means "not comparable by a common standard," not s i m p l y "not comparable a t a l l " (nor: "absolutely incomparable"). Hence, i f c e r t a i n t h i n g s were comparable, but not commensurable, they would be comparable, though not by a com-mon s t a n d a r d . 28 To get a more s p e c i f i c idea of what comparison without com-mon standards would be l i k e , f i r s t c o n s i d e r the n o t i o n of a standard of comparison. T r i v i a l l y , a standard of comparison i s a norm, paradigm, pattern, or r u l e f o r comparing t h i n g s (The Con-cise Oxford Dictionary). And two o b j e c t s are comparable Just i n case those o b j e c t s are r e l a t e d i n a p a r t i c u l a r way; t h a t i s , j u s t i n case they are both i n s t a n c e s of a particular b i n a r y r e l a t i o n : e.g. one of s i m i l a r i t y , d i f f e r e n c e , i n c o m p a t i b i l i t y , relevance, entailment, length, intensity, volume, e t c . Two o b j e c t s w i l l be commensurable, t h e r e f o r e , j u s t In case they are ins t a n c e s of a p a r t i c u l a r r e l a t i o n t h at i s determinable by a common standard. And, by c o n t r a s t , two o b j e c t s w i l l be comparable and incommen-s u r a b l e j u s t i n case they have such r e l a t i o n s , but those r e l a -t i o n s a re not determinable by a common standard. There are (at l e a s t ) two d i s t i n g u i s h a b l e cases of commensur-a b i l i t y : (1) a s i n g l e standard i s ab l e t o compare a given p a i r of o b j e c t s because each o b j e c t i s a simple, hence i r r e d u c i b l e or d i -r e c t , i n s t a n c e of i t s c a t e g o r i e s ; and (2) a s i n g l e standard i s ab l e t o compare a given p a i r of o b j e c t s because each o b j e c t i s a complex, hence " d e f i n l t i o n a l l y - e l i m i n a b l e " or i n d i r e c t , instance of more p r i m i t i v e c a t e g o r i e s embodied In t h a t standard. In both cases, however, the comparison of one o b j e c t with the other by the standard i s i n d i r e c t : each o b j e c t i s s e p a r a t e l y compared with the standard t o determine how they compare with each other. In other words, commensuration Is comparison of two t h i n g s by t h e i r comparison with a t h i r d : t h e i r common standard. Thus, s i n c e i n -29 commensurability means t h a t t h e r e are no common standards of comparison, a f u r t h e r c o n d i t i o n on our n o t i o n of comparable and incommensurable o b j e c t s i s t h a t each such o b j e c t must be compared d i r e c t l y with the other t o determine t h e i r i n t e r r e l a t i o n s . However, d e f i n i t i o n (1) does not, as i t stands, g i v e us the resources f o r making a u s e f u l d i s t i n c t i o n between commensurabil-i t y and c o m p a r a b i l i t y ; f o r , s i n c e , by h y p o t h e s i s , the o b j e c t s that we are c o n s i d e r i n g a re comparable, they c o u l d a l s o be made "commensurable" by a t r i v i a l technique compatible with t h a t d e f -i n i t i o n . Thus even though t h e y shared no common s t r u c t u r a l e l e -ments, and so were incommensurable by our e a r l i e r d e f i n i t i o n (p4 above), those o b j e c t s would be, i n a weaker sense, comparable by a common st a n d a r d : one c o n s t r u c t e d from t h e i r mere u n i o n . 8 0 D e f i n i t i o n (2) i s more h e l p f u l . Thus, on the b a s i s of (2) and our e a r l i e r d e f i n i t i o n of "Incommensurability," we may say: D e f i n i t i o n 3: Tvo structurally-complex objects are compar-able and incommensurable vhen they have relations that are not determinable on the basis of more p r i m i t i v e elements common to both.31 And s i n c e Incommensurable o b j e c t s ' r e l a t i o n s cannot be de-termined Indirectly as a f u n c t i o n of common p a r t s , t h e i r r e l a -t i o n s can o n l y be determined d i r e c t l y . That i s , the r e l a t i o n s between such o b j e c t s — w h e t h e r of p a r t to p a r t ; part/whole t o part/whole; whole t o whole; e t c . — c a n be determined and determin-able o n l y by the simple j u x t a p o s i t i o n of one o b j e c t or i t s a s -pects with the other. And ( i f such d i r e c t comparison makes 30 sense) t h a t s p a t i a l , temporal, e t c . , p a i r i n g w i l l be enough t o show t h a t those o b j e c t s c o n s t i t u t e an i r r e d u c i b l e i n s t a n c e of a p a r t i c u l a r comparative r e l a t i o n . " In summary, then, the i n t e l l i g i b i l i t y of the n o t i o n of com-para b l e but Incommensurable o b j e c t s depends on the i n t e l l i g i b i l -i t y of the n o t i o n of d i r e c t comparison. T h i s attempt to b i f u r -c a t e the n o t i o n of c o m p a r a b i l i t y i n t o d i r e c t c o m p a r a b i l i t y and commensurablllty has p o t e n t i a l p h i l o s o p h i c a l s i g n i f i c a n c e : i t suggests t h a t comparison of s t r u c t u r a l l y complex e n t i t l e s need not r e l y on the presence of elements common to both; i t need not r e l y on common standards of comparison. To make t h i s r a t h e r a b s t r a c t d i s c u s s i o n more a c c e s s i b l e , c o n s i d e r the a p p l i c a t i o n of my f o r m u l a t i o n t o commensurablllty and i n c o m m e n s u r a b i l i t y i n geometry, where those n o t i o n s were f i r s t i n t r o d u c e d . In geometry, two lengths are s a i d to be com-mensurable (by length) i f one can c o n s t r u c t a common u n i t l e n g t h t h a t w i l l d i v i d e both e x a c t l y . Thus a s t i c k t h a t i s ten c e n t i -metres long i s commensurable with one seven c e n t i m e t r e s long be-cause, f o r example, a one cent i m e t r e l e n g t h w i l l d i v i d e both without remainder. And both lengths are, i n a sense, ' d e f i n i -t l o n a l l y - e l l m l n a b l e ' s i n c e each l e n g t h can be reduced to m u l t i -p l e s of ( " l o g i c a l c o n s t r u c t i o n s on") t h a t common u n i t (see p28 above). Thus the r a t i o of the l e n g t h of the f i r s t to the second s t i c k can be broken down f u r t h e r t o the r a t i o 10 : 7. On the other hand, as Is w e l l known, the hypotenuse and s i d e of a r i g h t angle i s o c e l e s t r i a n g l e are incommensurable: i t i s 31 impossible to c o n s t r u c t a common u n i t l e n g t h t h a t w i l l d i v i d e both hypotenuse and s i d e e x a c t l y . N e v e r t h e l e s s , those l e n g t h s are d i r e c t l y comparable, s i n c e by mere I n s p e c t i o n of the t r i a n g l e one can see immediately t h a t the hypotenuse i s longer than the s i d e ; and from s i m i l a r t r i a n g l e s one can a l s o see d i r e c t l y t h a t those lengths form a consta n t , and i r r e d u c i b l e , ratio ( c . f . Von F r i t z 1945, 261). The s i d e and hypotenuse are thus both incom-mensurable and d i r e c t l y comparable as i n my d e f i n i t i o n . More-over, i t i s a l s o p o s s i b l e t o compare d i r e c t l y the p a r t s of those l e n g t h s : e.g., every p a r t of the hypotenuse i s (approximately) 14,142/10,000 times the l e n g t h of a p a r t of the s i d e , i f both those p a r t s are i n t h e i r o r i g i n a l r a t i o s . 9 3 Moreover, i n l i n e with the comments I made on d e f i n i t i o n (1) (p29 above), there i s a l s o a simple way i n which those l e n g t h s can be made "commensur-a b l e . " That i s , there i s a "common standa r d " f o r comparing the lengths of the hypotenuse and s i d e : the r e a l number "system"; and ac c o r d i n g to t h a t common system, the r a t i o s of those lengths i s ^ 2 : 1 . But s i n c e t h a t r e a l number system i s simply the union of two d i s j o i n t s e t s : the r a t i o n a l (e.g. which c o n t a i n s 1) and i r r a t i o n a l numbers (e.g. which c o n t a i n s v 2 ) , t h e i r "commensurabil-i t y " i n that system does not e n t a i l t h e i r c o m p a r a b i l i t y by more b a s i c elements common to bo t h . 9 4 Before I extend my model t o the problem of l o g i c a l l y incom-p a t i b l e and incommensurable s c i e n t i f i c t h e o r i e s , I s h a l l f i r s t p resent evidence t h a t Kuhn a l s o makes use of a d i s t i n c t i o n be-32 tween (what I have c a l l e d ) d i r e c t and i n d i r e c t comparison i n h i s p h i l o s o p h y . 1.5.1 Kuhn and D i r e c t Comparison The most e x p l i c i t example of a d i s t i n c t i o n between d i r e c t and I n d i r e c t comparison, and thus c o m p a r a b i l i t y and commensura-b l l l t y , i n Kuhn's w r i t i n g s i s i n h i s n o t i o n of a s i m i l a r i t y s e t (see Kuhn 1970a, 192, 200; Kuhn 1977c, 305-18). However, t h a t n o t i o n i s I n t i m a t e l y r e l a t e d t o h i s a n a l y t i c concept of a s c i e n -t i f i c paradigm, which he i n t r o d u c e s to r e p l a c e the t r a d i t i o n a l , formal n o t i o n of a u n i v e r s a l g e n e r a l i z a t i o n . And s i n c e I s h a l l d i s c u s s Kuhn's a l t e r n a t i v e t o the t r a d i t i o n a l formal schema i n chapter two, I s h a l l here use more su g g e s t i v e evidence of the above d i s t i n c t i o n . As the f o l l o w i n g passage shows, Kuhn a p p a r e n t l y b e l i e v e s t h a t r e c o g n i t i o n of r e l a t i o n s between d i f f e r e n t p o i n t s of view does not r e q u i r e a common, n e u t r a l b a s i s of comparison. What occurs d u r i n g a s c i e n t i f i c r e v o l u t i o n i s not f u l l y r e d u c i b l e t o a r e i n t e r p r e t a t i o n of i n d i v i d u a l and s t a b l e ( i . e . , t h e o r y - n e u t r a l ] d a t a . In the f i r s t p l a c e , the data are not u n e q u i v o c a l l y s t a b l e . [A B e r t h o l l e t i a n compound Is not a P r o u s t l a n p h y s i c a l m i x t u r e ] , nor i s oxygen d e p h l o g l s -t i c a t e d a i r . . . . [ T h u s ] r a t h e r than being an i n t e r p r e t e r [of " f i x e d " d a t a ] , the s c i e n t i s t who embraces a new paradigm i s l i k e [a] man [who undergoes a g e s t a l t s w i t c h ] . C o n f r o n t i n g the same c o n s t e l l a t i o n of o b j e c t s as before and knowing that he does so, he n e v e r t h e l e s s f i n d s them transformed through and through In many of t h e i r d e t a i l s (Kuhn 1970a, 121-2). I t a l i c s mine). 33 1.6 Commensurability and Formal L o g i c s Thus f a r I have claimed, without much e l a b o r a t i o n , t h a t f o r -mal l o g i c s r e q u i r e t h a t the l i n g u i s t i c e n t i t i e s they compare or analyse be e x p r e s s i b l e w i t h i n a common v o c a b u l a r y (e.g. p l 4 ) : i n other words, t h a t those e n t i t i e s be commensurable. I now want t o suggest t h a t formal l o g i c s be taken q u i t e s t r i c t l y as l o g i c a l "measures." I f I am r i g h t , t h i s means e n t i t i e s t h a t can be f o r -m a l l y compared are logically commensurable because they are com-par a b l e by a common l o g i c a l measure. By c o n t r a s t , i n my use of "formal comparison," those e n t i t i e s t h a t cannot be f o r m a l l y com-pared are l o g i c a l l y incommensurable. 8 3 Of course, i f one i s to take the m e t r i c a l analogy s e r i o u s l y , formal l o g i c s have t o have " u n i t s " of l o g i c a l measure: i . e . they have t o have s p e c i f i c means f o r d e t e r m i n i n g l o g i c a l r e l a t i o n s h i p s . In what f o l l o w s , I sug-gest t h a t those u n i t s are the s o - c a l l e d logical concepts. And I argue t h a t the view t h a t language d i v i d e s i n t o s t r i c t l y l o g i c a l and n o n - l o g i c a l concepts, which i s e s s e n t i a l to formal systems, i s incompatible with the t r u t h of the incommensurability t h e s i s . There are two b a s i c uses made of formal l o g i c s l i k e the p r e d i c a t e c a l c u l u s i n t r a d i t i o n a l p h i l o s o p h y of s c i e n c e : (1) they are used to d e s c r i b e the i n t e r n a l l o g i c a l s t r u c t u r e of a theory; and (2) they are used to formulate the l o g i c a l r e l a t i o n s between s c i e n t i f i c t h e o r i e s (e.g. Popper 1968, 32 e t . seq.). The second use c l e a r l y depends on the f i r s t , s i n c e i n d i v i d u a l t h e o r i e s have f i r s t t o be f o r m a l i z e d before t h e i r l o g i c a l i n t e r r e l a t i o n s with others can be determined. Now, i n v a r i a b l y , formal l o g i c s d i v i d e t h e i r o b j e c t languages i n t o l o g i c a l and non- or e x t r a - l o g i c a l concepts (e.g. Suppes 1957, 3, 43, 48, 68; Mates 1972, 16, 45; Boolos and J e f f r e y 1980, 97). For example, i n the p r e d i c a t e c a l c u l u s the standard l o g i c a l concepts (words, terms, symbols) a r e : "not, "and," " o r , " " i f . . . , t hen," " i f and o n l y i f , " " a l l , " and "some," and t h e i r synonyms. And p h i l o s o p h e r s who a p p l y formal l o g i c s t o s c i e n t i f i c languages a l s o commonly speak of the l o g i c a l and n o n - l o g i c a l v o c a b u l a r i e s of those languages (e.g. Suppes 1957, 68)." 8 The i m p l i c a t i o n of t h i s d i v i s i o n of language i s t h a t a l l (or a t l e a s t the c l e a r e s t , c . f . n l 3 ) l o g i c a l work i n t h a t language i s done by j u s t these concepts. Such a d i v i s i o n of language seems to be a p r e - c o n d i t i o n f o r any f o r m a l , ( n o n - t r i v l a l l y ) s y s t e m a t i c l o g i c ( c . f . Mates 1972, p l 6 ) . Unless l o g i c a l r e l a t i o n s between, f o r example, p r o p o s i -t i o n s and t h e o r i e s were based o n l y on the meanings of a r e l a t i v e -l y s m a l l number of concepts, i t would be g e n e r a l l y Impossible to determine those r e l a t i o n s f o r m a l l y . For, i n t h a t case, the l o g i -c a l r e l a t i o n s between many p a i r s of e n t i t i e s would given by the meanings of those e n t i t l e s a l o n e ; so there would o f t e n be no com-mon f e a t u r e l i n k i n g v a r i o u s i n s t a n c e s of a p a r t i c u l a r l o g i c a l r e l a t i o n l i k e i n c o m p a t i b i l i t y . N e v e r t h e l e s s , I s h a l l now argue t h a t the incommensurability t h e s i s presupposes t h a t l o g i c a l r e l a -t i o n s are i n f a c t l a r g e l y non-systematic and non-systematizable. 35 I t must be emphasized, however, t h a t t h i s p r e s u p p o s i t i o n does not imply the p a t e n t l y absurd c l a i m t h a t there are no s y s t e m a t i c l o g -i c a l r e l a t i o n s nor t h a t such l o g i c s are not Important and power-f u l tools of s c i e n c e ( c . f . Kuhn 1977, 313). Rather, i t s i m p l y means that such s y s t e m a t i c r e l a t i o n s o n l y c o n s t i t u t e a s m a l l p r o -p o r t i o n of l o g i c a l r e l a t i o n s proper. I argue f o r two c l a i m s : (1) t h a t the r e l a t i o n of l o g i c a l i n c o m p a t i b i l i t y between incommensurable t h e o r i e s cannot be cap-tu r e d by formal systems; and (2) t h a t , i f there were incommen-su r a b l e t h e o r i e s , I t c o u l d not g e n e r a l l y be t r u e t h a t e v e r y I n d i -v i d u a l s c i e n t i f i c t heory c o u l d be analysed by formal l o g i c s . Both arguments work by r e d u c t i o ad absurdurn. I have t r i e d t o make the arguments completely g e n e r a l and not dependent on the f e a t u r e s of p a r t i c u l a r formal systems l i k e the p r e d i c a t e c a l c u -l u s . One should keep i n mind t h a t , by d e f i n i t i o n (see pp4, 29), incommensurable t h e o r i e s , or incommensurable p a r t s of t h e o r i e s , have no common concepts. And because ve are here c o n s i d e r i n g the hypothesis t h a t formal l o g i c s can e i t h e r compare or analyse i n -commensurable t h e o r i e s , i t i s a l s o t h e r e f o r e assumed t h a t incom-mensurable t h e o r i e s share the same l o g i c a l concepts but have no common non-logical concepts. 1.6.1 Argument 1: L o g i c a l R e l a t i o n s Between T h e o r i e s Here we c o n s i d e r the hyp o t h e s i s t h a t f o r m a l l y comparable 36 t h e o r i e s can a l s o be incommensurable. By the h y p othesis of f o r -mal c o m p a r a b i l i t y , the l o g i c a l r e l a t i o n s between any p a i r of e l e -ments, e.g. ( t l j ; t 2 k ) , of any p a i r of such t h e o r i e s , e.g. ( T l ; T 2 ) , must depend only on the meanings of the l o g i c a l con-c e p t s . T h i s means th a t one such element i s e q u i v a l e n t to a l o g i -c a l f u n c t i o n , f , of the o t h e r : e.g. t l j = / ( t 2 k ) ; and t h a t means, s i n c e the t h e o r i e s share the l o g i c a l concepts, the t h e o r y ( T l ) t h a t c o n t a i n s the f i r s t element ( t l j ) a l s o c o n t a i n s the second element ( t 2 k ) . Which c o n t r a d i c t s the h y p o t h e s i s of incommensur-a b i l i t y . 1.6.2 Argument 2: Formal A n a l y s i s of I n d i v i d u a l T h e o r i e s Here we c o n s i d e r the h y p othesis t h a t any i n d i v i d u a l s c i e n t i -f i c t h e o r y can be f o r m a l l y analysed even though there are p a i r s of t h e o r i e s t h a t cannot be f o r m a l l y compared because of t h e i r i n c o m m e n s u r a b i l i t y . Consider a p a i r of incommensurable t h e o r i e s : s i n c e , by h y p o t h e s i s , each i s f u l l y a n a l y s a b l e f o r m a l l y , each w i l l share the same l o g i c a l c oncepts; s i n c e those t h e o r i e s are incommensurable, they w i l l share no n o n - l o g i c a l concepts. Now, by h y p o t h e s i s , those t h e o r i e s are l o g i c a l l y r e l a t e d . And, c l e a r -l y , those l o g i c a l r e l a t i o n s must be due e i t h e r to the meanings of the t h e o r i e s ' l o g i c a l or n o n - l o g i c a l concepts. By argument 1, those r e l a t i o n s can't be due to the l o g i c a l concepts, s i n c e t h a t would imply, c o n t r a r y to the h y p o t h e s i s of incommensurability, t h a t the t h e o r i e s share common n o n - l o g i c a l concepts. On the 37 , other hand, i f those l o g i c a l r e l a t i o n s f o l l o w from the meanings of the n o n - l o g i c a l concepts, then the n o n - l o g i c a l concepts a r e n ' t (at l e a s t a b s o l u t e l y ) n o n - l o g i c a l . Which i s a a l s o a c o n t r a d i c -t i o n . Hence i f there are incommensurable p a i r s of t h e o r i e s , the languages of s c i e n t i f i c t h e o r i e s do not d i v i d e i n t o s t r i c t l y l o g i c a l and n o n - l o g i c a l concepts. In other words, i f there are incommensurable t h e o r i e s , l o g i c i s not i d e n t i c a l with formal l o g i c . For each of the above arguments i t i s p o s s i b l e t o r e t o r t that l o g i c j u s t i s formal l o g i c and t h a t , t h e r e f o r e , the incom-m e n s u r a b i l i t y t h e s i s i s f a l s e . Or i n s t e a d I t might be claimed that i n d i v i d u a l t h e o r i e s form d i f f e r e n t l o g i c a l systems; and each such system does d i v i d e f u r t h e r i n t o d i f f e r e n t p u r e l y l o g i c a l and n o n - l o g i c a l concepts (e.g. Goodman 1984, 44-5; but c . f . ibid.f 94-5); hence, formal l o g i c s don't preclude incommensurability. But, with r e s p e c t to t h i s second o p t i o n , I have not here claimed t h a t Kuhn's t h e s i s i m p l i e s t h a t there are no r e l a t i o n s based on l o g i c a l form or even t h a t no t h e o r i e s may be f o r m a l l y analysed. I have o n l y s a i d t h a t the incommensurability t h e s i s means that f o r -mal l o g i c s do not exhaust l o g i c proper. In any case, even i f each of an Incommensurable p a i r of t h e o r i e s c o n s t i t u t e d a d i f f e r -ent formal systems vould s t i l l leave u n e x p l a i n e d — i . e . unformal-i z e d — t h e l o g i c a l r e l a t i o n s between those d i s c r e t e systems. With r e s p e c t to the f i r s t r e t o r t , i t should be noted t h a t there are i n s t a n c e s of statements i n o r d i n a r y language whose l o g i c a l r e l a t i o n s seem to a r i s e from the meanings of t h e i r "non-38 l o g i c a l " c oncepts. For example, the c l a i m , "George i s a bachelor and George i s a married man," i s not c o n t r a d i c t o r y because of i t s form but because of the meanings of bachelor and married man. Moreover, t o t u r n t h i s c l a i m i n t o a formal c o n t r a d i c t i o n by f i r s t d e f i n i n g bachelor as unmarried man, presupposes t h a t one a l r e a d y knows those concepts t o be synonymous, and so begs the q u e s t i o n of the source of the i n c o m p a t i b i l i t y of bachelor and married man. Examples of such meaning-dependent l o g i c a l r e l a t i o n s are not mere c u r l o u s i t i e s . The s o - c a l l e d n a t u r a l l o g i c i a n , Steven Thom-as, c l a i m s t h a t "samplings i n d i c a t e I that]...more than 90% of d e d u c t i v e l y v a l i d i n f e r e n c e s i n n a t u r a l language...depend on the semantics, r a t h e r than on the l o g i c a l form or syntax, of the premise(s) and c o n c l u s i o n " (Thomas 1986, 448. I t a l i c s i n o r i g i -n a l ) . " 7 Moreover, Davidson h i m s e l f i s a l s o aware of deep prob-lems formal approaches have with the most common f e a t u r e s o£ n a t u r a l languages ( l i k e t h a t of s c i e n c e ) , though with the oppo-s i t e c o n c l u s i o n t o Thomas. Since I t h i n k there is no alternative, I have taken an o p t i -m i s t i c and programmatic view of the p o s s i b i l i t i e s f o r a formal c h a r a c t e r i z a t i o n of a t r u t h p r e d i c a t e f o r a n a t u r a l language. But i t must be allowed t h a t a s t a g g e r i n g l i s t of d i f f i c u l t i e s and conundrums remains (Davidson 1984, 35. I t a l i c s mine). What my arguments, Thomas's s t a t i s t i c s and Davidson's "con-undrums" suggest i s th a t the l o g i c a l r e l a t i o n s between incommen-su r a b l e t h e o r i e s must l i k e w i s e depend on the semantics of t h e i r " n o n l o g i c a l " c oncepts. That should not be s u r p r i s i n g i f we r e -c a l l Kuhn's c h a r a c t e r i z a t i o n of incommensurable t h e o r i e s as "fun-39 d a m e n t a l l y i n c o m p a t i b l e " (see p 4 ) ; f o r t h a t l o g i c o - n o r m a t i v e h y -b r i d n o t i o n i s , i n r e t r o s p e c t , c l e a r l y a t odds wi th any r i g o r o u s d i v i s i o n of the l o g i c a l from the n o n - l o g i c a l . And i f we a l s o r e -member t h a t Kuhn takes s c i e n t i f i c t h e o r i e s to be h o l i s t i c , "con-c e p t u a l webs," we s h o u l d a l s o expec t t h a t the l o g i c a l r e l a t i o n s between those "webs" w i l l have to be a consequence of the seman-t i c s of those systems as c o n c e p t u a l who le s . I t i s t r u e t h a t sentences a r e n o r m a l l y h e l d to be the u n i t o f l o g i c a l r e l a t i o n s such as i n c o m p a t i b i l i t y ; so i t w i l l undoubt -e d l y appear p e c u l i a r to speak of w h o l e s a l e t h e o r e t i c a l incompat -i b i l i t y . But i f we s t a t e the n o t i o n o f l o g i c a l I n c o m p a t i b i l i t y q u i t e b r o a d l y a s : " i f this were t r u e , then t h a t c o u l d not a l s o be t r u e , " then the idea of d i r e c t t h e o r e t i c a l i n c o m p a t i b i l i t y may not seem so odd . Thus one c o u l d s a y t h a t two h o l i s t i c t h e o r e t i -c a l systems are ( d i r e c t l y ) l o g i c a l l y i n c o m p a t i b l e j u s t i n c a s e , i f the whole wor ld were as i n one t h e o r y , i t c o u l d not a l s o be as i n the o ther t h e o r y . S t i l l , g i v e n the obv ious l i m i t s of the human mind , s i n c e t h e o r i e s are e n t i t l e s w i t h v e r y complex i n t e r n a l s t r u c t u r e s , i t might seem i m p o s s i b l e t h a t t h e i r i n c o m p a t i b i l i t i e s c o u l d be d e -t ermined d i r e c t l y . To h e l p h i s r e a d e r s get an i n t u i t i v e u n d e r -s t a n d i n g of the u n f a m i l i a r i d e a of p r i m i t i v e t h e o r e t i c a l incom-p a t i b i l i t y , Kuhn employs the i d e a o f a "swi tch i n v i s u a l g e s t a l t " as I t s "e lementary p r o t o t y p e " (Kuhn 1970a, 1 1 1 ) . 9 0 1.6 .3 A " G e s t a l t - S w i t c h " Model of T h e o r e t i c a l I n c o m p a t i b i l i t y 40 A c l a s s i c example of a v i s u a l g e s t a l t s w i t c h i s the response to the s o - c a l l e d duck-rabbit f i g u r e . S u r p r i s i n g l y , t h i s s t r u c -t u r a l l y - c o m p l e x f i g u r e i s seen a l t e r n a t e l y as wholly e i t h e r a duck or a r a b b i t , though never both a t once. Hence, s i n c e i t i s c l e a r l y impossible to see the f i g u r e s i m u l t a n e o u s l y as both a duck and a r a b b i t — a duck i s not a r a b b i t and v i c e v e r s a — t h e s e seem to be d i r e c t l y incompatible c o n s t r u a l s of the same domain. The d u c k - r a b b i t f i g u r e thus appears to provide a v e r y good model f o r e x p l i c a t i n g how the d i r e c t l o g i c a l comparison of ( p u t a t i v e ) incommensurables can take p l a c e without implying t h e i r commensur-a b l l l t y . 41 The Duck-Rabbit F i g u r e 1. Thus, having experienced both g e s t a l t s , we can d i r e c t l y com-pare and c o n t r a s t the duck and r a b b i t views as f o l l o w s . The up-per b i l l of the duck p e r s p e c t i v e " i s " the l e f t ear of the r a b b i t p e r s p e c t i v e ; the l e f t duck eye of the duck p e r s p e c t i v e " i s " the r i g h t r a b b i t eye of the r a b b i t p e r s p e c t i v e ; the t h r o a t of the duck p e r s p e c t i v e " i s " the nape of the r a b b i t p e r s p e c t i v e ; the notch on the back of the head of the duck p e r s p e c t i v e " i s " the mouth on the r a b b i t p e r s p e c t i v e ; and so on. 3 9 I have used shudder-quotes around the word " I s " In each case to emphasize t h a t , while the r e s p e c t i v e p a r t s are i n a sense co-e x t e n s i v e , I t Is n e v e r t h e l e s s impossible t h a t they both o b t a i n (from e i t h e r p e r s p e c t i v e ) . For ducks have no f l o p p y ears and r a b b i t s no r i g i d beaks, and so on. N o t i c e f u r t h e r t h a t while both the duck and r a b b i t are i n a sense composed of t h e i r c o n s t i -tuent b i l l s , e a r s , e t c . , these p a r t s are what they are because they are p a r t s of the whole o b j e c t . Thus i t i s more acc u r a t e to say i n s t e a d t h a t the duck and r a b b i t d i v i d e into t h e i r c o n s t i t u -ent p a r t s . 4 0 For Kuhn, then, on analogy with t h i s "elementary p r o t o t y p e , " the comparison or c o n t r a s t of incommensurable t h e o r i e s must u l t i -mately i n v o l v e t h e i r d i r e c t comparison as whole to whole. That those t h e o r i e s are w h o l l y incompatible, however, does not mean t h a t t h e i r p a r t s cannot be compared or even found to be s i m i l a r . I t j u s t means t h a t any comparison of those p a r t s w i l l i n v o l v e the j u x t a p o s i t i o n , not of theory-independent terms, concepts and sen-tences, but of those e n t i t i e s as theory-dependent s u b d i v i s i o n s . Thus each element i n the comparison or c o n t r a s t p a i r s w i l l be there as a r e p r e s e n t a t i v e of the whole t h e o r y from which they are a b s t r a c t e d . For example, j u s t as the b i l l gua b i l l of the duck can be seen d i r e c t l y to be c o - e x t e n s i v e with the ears qua ears of the rabbit, so a l s o a chemical compound gua Bertholletian sub-stance of g r e a t l y variable proportions can be seen d i r e c t l y to be (often) c o - e x t e n s i v e with a p h y s i c a l mixture gua Proustian solu-tion. Two o b j e c t i o n s have been r a i s e d a g a i n s t the use of the ges-t a l t model f o r t h e o r y change ( C o l l i e r 1984, 109). The f i r s t ob-j e c t i o n — t h a t the d u c k - r a b b i t example i n v o l v e s r e p r e s e n t a t i o n s , not r e a l o b j e c t s - - i s , I t h i n k , r a t h e r e a s i l y d isposed o f . For, mutatis mutandis, t h a t example works j u s t as w e l l If we think of the o b j e c t as e i t h e r a p icture-duck or a p i c t u r e - r a b b i t , which we do i n any case ( c . f . a l s o Feyerabend 1975, 258 & n98). The second o b j e c t i o n i s more s e r i o u s . Thus i t i s c l a i m e d , with some (I t h i n k mistaken) encouragement from Kuhn (Kuhn 1970a, 114), t h a t i n the d u c k - r a b b i t case, there i s a n e u t r a l base of comparison: "the l i n e s which make up the f i g u r e " ( C o l l i e r 1984, 109). I f so, the duck and r a b b i t views would be a l t e r n a t i v e , but compatible c o n s t r u a l s of a common domain. But j u s t as i t i s im-p o s s i b l e to see the f i g u r e s i m u l t a n e o u s l y as both a duck and a r a b b i t , i t i s a l s o impossible to see the f i g u r e s i m u l t a n e o u s l y as both mere l i n e s and as e i t h e r a duck or a r a b b i t . Thus, while the drawing i s , i n a sense, composed of the l i n e s on paper, and while one may draw the f i g u r e o n e s e l f , s e e i n g those l i n e s as e i -ther a duck or a r a b b i t , or v i c e v e r s a , must a l s o take p l a c e "as a r e l a t i v e l y sudden and u n s t r u c t u r e d event" (Kuhn 1970a, 122). T h i s second o b j e c t i o n i s a more s p e c i f i c v e r s i o n of such c l a i m s as: "what the d u c k - r a b b i t shows i s that r e a l i t y can have a l t e r n a t i v e or complementary r e p r e s e n t a t i o n s , each of which grasps some a s p e c t , but which are not o n l y not i n c o n f l i c t but are each needed i f one i s to get 'the whole p i c t u r e ' (so to speak)" ( S a v l t t 1989, 2; see a l s o Hacking 1981, 4). But t h i s more g e n e r a l c l a i m I t s e l f c o n t a i n s an i m p l i c i t demand f o r commen-s u r a b l l l t y In making judgements about c o n f l i c t . Thus i t suggests t h a t unless there are common standards f o r showing views to be i n c o n f l i c t , those views are compatible by d e f a u l t . On the other hand, as we have seen, there are two b a s i c ways i n v h i c h one might judge v i e v s to be e i t h e r c o n f l i c t i n g or com-plementary: by i n d i r e c t or d i r e c t comparison. But i n cases l i k e the d u c k - r a b b i t f i g u r e , and incommensurable t h e o r i e s , I have argued, there i s no n e u t r a l p e r s p e c t i v e f o r making an i n d i r e c t judgement. T h e r e f o r e , i n such cases, the judgement must be made by d i r e c t comparison; and such comparison shovs those perspec-t i v e s to be c l e a r l y incompatible. That i s , i t i s im p o s s i b l e , from one p o i n t o f view, t o have the other p o i n t of v i e v ; and there i s no t h i r d s t andpoint from v h i c h one might have both. Hence t o c a l l incommensurable v i e v s compatible by d e f a u l t i s t o imply t h a t comparison i s commensuration. But as I s h a l l nov argue, such an i d e n t i f i c a t i o n leads t o a b s u r d i t i e s . 1.7 D i r e c t Comparison: A Thought Experiment From the f a c t t h a t commensuration i n v o l v e s the comparison of tvo t h i n g s by t h e i r comparison with a t h i r d t h i n g (see SI.5, p28-29), i t should be c l e a r t h a t t o d e f i n e "comparison" as "commen-s u r a t i o n " vould y i e l d a v i c i o u s r e g r e s s . 4 ' In vhat f o l l o v s , I argue t h a t t o demand that to e s t a b l i s h s i m i l a r i t y and d i f f e r e n c e of r e f e r e n c e ; to determine s i m i l a r i t y and d i f f e r e n c e of meaning; and t o r e c o g n i z e l o g i c a l i n c o m p a t i b i l i t y always r e q u i r e s common elements leads i n each case to a v i c i o u s r e g r e s s . On the other hand, I argue t h a t i f Kuhn's c r i t i c s were t o agree t h a t such com-p a r i s o n s w i l l a t some p o i n t have to be d i r e c t , but hold t h a t t h i s p o i n t i s o s t e n s i v e d e f i n i t i o n , a "common c o - o r d i n a t e system" 45 (Davidson 1984b, 184), or formal l o g i c they must o f f e r an i n d e -pendent argument t o prove t h a t t h i s i s so. And, t h e r e f o r e , i f the prima facie evidence i s t h a t no such system does or c o u l d e x i s t , t h a t evidence w i l l have t o be taken a t face v a l u e . F i r s t , i f the c o - r e f e r e n c e of two terms always depended on each one's being c o - r e f e r e n t i a l with some common t h i r d term, then we should a l s o r e q u i r e a s t i l l f u r t h e r term t o mediate c o - r e f e r -ence between each of the f i r s t two terms and the t h i r d ; and so on. On the other hand, i f the c o - r e f e r e n c e of the f i r s t two terms w i t h the t h i r d can be determined d i r e c t l y , we have t o be give n a reason why t h i s c o u l d not a l s o be the case f o r the f i r s t two. For example, c o n s i d e r the f o l l o w i n g c o - r e f e r e n t i a l expres-s i o n s , " B r i a n Mulroney" and "the prime m i n i s t e r of Canada." I f we hold t h a t t o e s t a b l i s h t h e i r c o - r e f e r e n c e we f i r s t have t o show t h a t each i s c o - r e f e r e n t i a l with a "more p r i m i t i v e " expres-s i o n l i k e a demonstrative pronoun (e.g. t h i s ) , we s h a l l a l s o need a s t i l l "more b a s i c " e x p r e s s i o n to mediate the c o - r e f e r e n c e of both " B r i a n Mulroney" and "the prime m i n i s t e r of Canada" with t h a t demonstrative pronoun. 4 2 On the other hand, i f these l a s t r e l a t i o n s h i p s can be determined d i r e c t l y , 4 3 we need a reason why t h i s c o u l d not, In p r i n c i p l e , a l s o be done with " B r i a n Mulroney" and "the prime m i n i s t e r of Canada" alon e . Second, i f the s i m i l a r i t i e s or d i f f e r e n c e s of meanings of two concepts always r e q u i r e d a "common co - o r d i n a t e system" (Da-vidso n 1984, 184) t o determine those r e l a t i o n s , then we should a l s o r e q u i r e a s t i l l f u r t h e r c o - o r d i n a t e system t o show t h a t those concepts vere i n s t a n c e s of t h a t f i r s t system; and so on. Again, i f i t Is admitted t h a t the r e l a t i o n s the tvo concepts have v i t h the c o - o r d i n a t e system must be determined d i r e c t l y , ve need a f u r t h e r reason vhy t h i s c o u l d not a l s o be the case f o r the r e l a t i o n s betveen the concepts themselves. F i n a l l y , I f the i n c o m p a t i b i l i t y of tvo concepts, sentences, t h e o r i e s , e t c . a l v a y s had to be determined by a standard l i k e l o g i c a l form, then ve should a l s o need s t i l l f u r t h e r standards to shov t h a t one such e n t i t y Is i d e n t i c a l i n meaning v i t h the nega-t i o n of the o t h e r . And, on the other hand, i f i t i s countered t h a t the r e c o g n i t i o n t h a t one such e n t i t y i s synonymous v i t h the negation of the other i t s e l f needs no standard, ve need an inde-pendent reason vhy t h e i r i n c o m p a t i b i l i t y c o u l d not be judged d i r e c t l y . The f o r e g o i n g e x e r c i s e i s not meant to e s t a b l i s h t h a t a l l uses of terms, concepts, sentences, and t h e o r i e s i n v o l v e no com-mon s t a n d a r d s . That i s c l e a r l y f a l s e . Appeals to standards, pa-radigms, r u l e s , t r a d i t i o n , e t c . o f t e n p l a y an extremely important r o l e i n s c i e n c e , law, l i t e r a t u r e , a r t , e t c . 4 4 My argument o n l y makes the obvious p o i n t that such appeals must end somewhere; and so even those p r a c t i c e s t h a t Involve appeals to standards must at some p o i n t i n v o l v e d i r e c t comparison. T h e r e f o r e , when h i s t o r i -c a l , a n t h r o p o l o g i c a l , s o c i o l o g i c a l and conceptual evidence shows th a t c e r t a i n communities and t r a d i t i o n s use a p p a r e n t l y q u i t e d i f f e r e n t standards o f , say, t r u t h , meaning, or j u s t i c e , and determined attempts to a t t r i b u t e more b a s i c , common standards to 47 those groups are always " r e j e c t e d by...members of the group" (Kuhn 1970a, 44), t h a t evidence must be taken a t face v a l u e . The a b i l i t y to compare t h i n g s d i r e c t l y , however, should not be thought of as a "mysterious" (Shapere 1980, 32) c a p a c i t y to i n t u i t t h e i r r e l a t i o n s (see a l s o n28). That would be tantamount to i n t r o d u c i n g i n t u i t i o n as a commensurating f a c u l t y ; but as we have j u s t seen, the demand t h a t any such standards are necessary leads to a b s u r d i t i e s . W i t t g e n s t e i n puts t h i s l a s t p o i n t as f o l -lows: " I f i n t u i t i o n i s an inner v o i c e — h o w do I know how I am t o obey i t ? And how do I know t h a t i t doesn't mislead me? For i f i t can guide me r i g h t , i t can a l s o guide me wrong. ( ( I n t u i t i o n an unnecessary s h u f f l e . ) ) " ( W i t t g e n s t e i n 1953, 8213). i n the f i n a l a n a l y s i s , then, a l l t h a t can be s a i d to e x p l a i n d i r e c t comparison i s simply to remark t h a t i t i s a b a s i c f a c t of human " n a t u r a l h i s t o r y " ( i b i d . , §25) t h a t we group c e r t a i n t h i n g s as s i m i l a r , i d e n t i c a l , d i f f e r e n t , i n c o m p a t i b l e , and so on. No more can be s a i d without redundancy. 1.8 summary, Sy n t h e s i s and C o n c l u s i o n In t h i s chapter I have t r i e d t o show t h a t the argumentative impasse between Kuhn and h i s c r i t i c s on incommensurability a r i s e s from a fundamental d i f f e r e n c e about the I n t e r p r e t a t i o n of two concepts: l o g i c and comparison. Thus i t turns out t h a t Kuhn's c r i t i c s i m p l i c i t l y I d e n t i f y l o g i c with formal l o g i c s ; but h i s ho-l i s t i c n o t i o n of t h e o r e t i c o - c o n c e p t u a l i n c o m p a t i b i l i t y c o n f l i c t s 48 with the i d e a , b a s i c t o formal l o g i c s , t h a t there i s a completely r i g o r o u s d i s t i n c t i o n between the l o g i c a l and n o n - l o g i c a l . More d e e p l y s t i l l , I t tu r n s out t h a t Kuhn's c r i t i c s I d e n t i f y l o g i c w ith formal l o g i c s because they i d e n t i f y comparison with commen-s u r a t i o n . But I showed, by p o i n t i n g out t h a t commensuration i s comparison of two t h i n g s by t h e i r comparison with a t h i r d , t h a t such an i d e n t i f i c a t i o n would make commensuration i t s e l f impos-s i b l e (see pp28-9, 44-6). One proof t h a t the d i s t i n c t i o n between ( d i r e c t ) comparison and commensuration i s on the r i g h t t r a c k i s i t s a b i l i t y t o r e -s o l v e a r e s i d u a l p u z z l e Kuhn f e e l s about the c h a r a c t e r i s t i c s of the new " e p l s t e m o l o g l c a l paradigm" (see 1.0 above). Kuhn says t h a t "I am convinced t h a t we must l e a r n t o make sense of [seem-i n g l y p a r a d o x i c a l ] statements t h a t a t l e a s t resemble t h e s e . . . though the world does not change with a change of paradigm the s c i e n t i s t a f t e r w a r d works i n a d i f f e r e n t world" (Kuhn 1970a, 121). These a p p a r e n t l y c o n t r a d i c t o r y c l a i m s can be made sense of as f o l l o w s . On the one hand, j u s t as d i r e c t comparison of the duck and r a b b i t views shows them t o be coe x t e n s i v e (p41), so too d i r e c t comparison of incommensurable t h e o r i e s shows t h a t t h e i r domains are o f t e n l a r g e l y o v e r l a p p i n g ; so i n one sense those t h e o r i e s d e s c r i b e the same world. On the other hand, j u s t as d i r e c t comparison of the duck and r a b b i t views does not r e v e a l a f u r t h e r , common d u c k - r a b b i t view, one can a l s o say th a t Incom-mensurable t h e o r i e s d e s c r i b e d i f f e r e n t worlds. Kuhn's r e j e c t i o n of a primary r o l e f o r (formal) l o g i c i n 49 a n a l y s i n g s c i e n c e has been construed by many c r i t i c s to i n v o l v e "denying o b j e c t i v i t y to the processes by which s c i e n t i f i c t h e o r -i e s are c r i t i c a l l y e v a l u a t e d " ( S c h e f f l e r 1967, 88). By c o n t r a s t , I suggested e a r l y i n t h i s chapter t h a t Kuhn should be understood i n s t e a d as o f f e r i n g an alternative c o n s t r u a l of s c i e n t i f i c o b j ec-t i v i t y and r a t i o n a l i t y t h a t i s not based on formal l o g i c , and thus commensurability. In t h i s r e g a r d , S c h e f f l e r a l s o notes, though with some p e r p l e x i t y , t h a t " P r o f e s s o r Kuhn himself con-c l u d e s by r e i n t r o d u c i n g the v e r y [ e v a l u a t i v e ] ideas he has been a t pains to deny i n the main tendency of h i s work...[for example] he opposes r e c e i v e d n o t i o n s of f a l s i f i c a t i o n , but h i m s e l f i n t r o -duces the concepts of anomaly and c r i s i s , which have a parallel f u n c t i o n i n h i s account" ( i b i d . , 89. I t a l i c s mine). The main reason Kuhn r e j e c t s a primary a n a l y t i c r o l e f o r formal l o g i c i n s c i e n t i f i c epistemology i s , as we have seen, t h a t he b e l i e v e s t h e o r i e s are o f t e n incommensurable. In the next c h a p t e r , I s h a l l b r i n g out s t i l l other reasons Kuhn has f o r the demotion of formal l o g i c . Thus i n chapter two I d i s c u s s Kuhn's n o t i o n s of paradigm, normal s c i e n c e , anomaly, c r i s i s , thought ex-periment and s c i e n t i f i c r e v o l u t i o n and show how they p a r a l l e l , and d i v e r g e from, more t r a d i t i o n a l c a t e g o r i e s drawn from formal l o g i c . I s h a l l a l s o show th e r e how Kuhn embodies a d i s t i n c t i o n between commensuration and what I have c a l l e d d i r e c t comparison i n those a l t e r n a t i v e a n a l y t i c n o t i o n s . Notes 50 1. I f Kuhn had not made a p e r s u a s i v e case f o r incommensurability, i t i s d o u b t f u l t h a t opposing p h i l o s o p h e r s would have made such co n c e r t e d e f f o r t s to respond t o h i s work. I s r a e l S c h e f f l e r , a noted c r i t i c of Kuhn, admits t h a t , a t f i r s t g l a n c e , Kuhn's r e -s u l t s seem undeniable: "For the c l a i m s i n q u e s t i o n are supported by d e t a i l e d c o n s i d e r a t i o n s t o which we o u r s e l v e s are i n c l i n e d to ass e n t , a t l e a s t i n i t i a l l y " ( S c h e f f l e r 1967, 22). 2. In t h i s paper, I s h a l l i n c l u d e under the d e s c r i p t i o n s " t r a d i -t i o n a l image," " r e c e i v e d view," e t c . not o n l y the c l a s s i c s t a t e -ments of the L o g i c a l E m p i r i c i s t s and K a r l Popper but a l s o more r e c e n t v a r i a t i o n s formulated i n response to Kuhn's (and other like-minded p h i l o s o p h e r s ' ) c r i t i c i s m s . Thus the p h i l o s o p h i e s of Lakatos, Laudan and Newton-Smith, f o r example, may be seen as " t r a d i t i o n a l " f o r my purposes s i n c e they a l s o seek what Kuhn argues does not and need not e x i s t : t h e o r y - n e u t r a l c r i t e r i a f o r e v a l u a t i n g progress i n s c i e n c e . Of course, such a schematic a c -count of 'the' t r a d i t i o n a l view i s bound to be o v e r l y s i m p l i s t i c and lump together incompatible p h i l o s o p h i e s , s t i l l i t i s s a f e to conclude t h a t , whatever the d i f f e r e n c e s of those p h i l o s o p h i e s , they are u n i t e d i n being attempts to f i n d t h e o r y - n e u t r a l compara-t i v e c r i t e r i a , and thus i n t h e i r r e j e c t i o n of incommensurability. 3. I s h a l l u s u a l l y use the term "theory" and Kuhn's p r e f e r r e d ex-p r e s s i o n "paradigm" as though they were synonymous. T h i s i s i n keeping with Kuhn's usual p r a c t i c e , though, given what he means by "paradigm," i t does not agree with h i s opponents' understand-ing of "theory." Thus t h e o r i e s t r a d i t i o n a l l y are thought to be u n i v e r s a l g e n e r a l i z a t i o n s ( c . f . G i e r e 1983, 271); but Kuhn holds t h a t h i s concept of a paradigm, or "concrete s c i e n t i f i c a c h i e v e -ment" (Kuhn 1970a, 11) i s more b a s i c t o s c i e n c e than i s t h a t t r a -d i t i o n a l c o n c e p t i o n ( I b i d . , 182). I s h a l l d i s c u s s paradigms i n chapter two. 4. By c o n t r a s t , i f the e m p i r i c a l anomaly does not i m p l i c a t e the theory's c a t e g o r i e s , theory change need not i n v o l v e c o n c e p t u a l change, and so a l s o need not i n v o l v e incommensurability. For example, the theory's concepts may a l l o w f o r the f o r m u l a t i o n of d i f f e r e n t e m p i r i c a l laws, so the replacement of the f a l s e law with a more adequate one can leave the b a s i c theory unchanged. On the t r a d i t i o n a l view, however, s i n c e t h e o r i e s are taken to be g e n e r a l i z a t i o n s , such a change of l a v s would a l s o be taken as a change of theory. But, as I poin t e d out i n note 3, Kuhn takes paradigms to be the b a s i c u n i t of s c i e n c e . And he suggests t h a t 51 change of t h a t p u t a t i v e l y more b a s i c u n i t almost always produces the conceptual and e v a l u a t i v e change he c a l l s incommensurability (Kuhn 1970a, 103). 5. I t i s o c c a s i o n a l l y claimed t h a t Kuhn cannot r e a l l y mean t h a t incommensurable t h e o r i e s are logically incompatible; so he must have some other n o t i o n of i n c o m p a t i b i l i t y In mind. Thus Newton-Smith says t h a t " K u h n . . . i n c o n s i s t e n t l y e x p l i c a t e s the n o t i o n of c o m p e t i t i o n (between incommensurable t h e o r i e s ] i n terms of the n o t i o n of l o g i c a l i n c o m p a t i b i l i t y " (Newton-Smith 1981, 159). Newton-Smith k i n d l y o f f e r s what he b e l i e v e s to be a more s u i t a b l e n o t i o n of i n c o m p a t i b i l i t y to get Kuhn's i n t e n t i o n s a c r o s s : "prag-matic t e n s i o n " ( i b i d . , 159). But, pace Newton-Smith, Kuhn e x p l i -c i t l y s t a t e s t h a t the t h e o r i e s he deems incommensurable—e.g. Newtonian and E i n s t e i n i a n d y n a m i c s — a r e a l s o logically incompati-b l e : " E i n s t e i n ' s t h e o r y (Kuhn argues] can be accepted o n l y with the r e c o g n i t i o n t h a t Newton's was wrong" (Kuhn 1970a, 98; see a l s o ibid., 97). In any case, i n t h i s paper I s h a l l take Kuhn a t h i s word and t r y to g i v e an i n t e r p r e t a t i o n of the n o t i o n of " f u n -damentally i n c o m p a t i b l e " t h e o r i e s t h a t coheres both with Kuhn's e x p l i c i t l y - s t a t e d I n t e n t i o n s and with good sense. 6. That i s , though Kuhn a l l o w s t h a t i n comparing two incommen-s u r a b l e t h e o r i e s one can see t h a t , from the vantage of one theo-ry's standards, t h a t t h e o r y i s t r u e while the other i s f a l s e , he c l a i m s there i s no common, theory-independent perspective f o r s e t t l i n g which of them are i n f a c t true or f a l s e . 7. Kuhn a l s o denies most of h i s c r i t i c s ' other c l a i m s : v i z . , t h a t h i s p o s i t i o n means t h a t experiment p l a y s no c r i t i c a l r o l e i n s c i -ence (e.g. Kuhn 1970a, 132); t h a t i t leads to i d e a l i s m (Kuhn 1970b, 2); (strong) r e l a t i v i s m ( i b i d . , 264-5); l r r a t l o n a l l s m (ibid., 263-4); and he a l s o b e l i e v e s t h a t theories of meaning p l a y no e s s e n t i a l r o l e In these Issues ( i b i d . , 266n2). In t h i s paper, however, I s h a l l not d i s c u s s these more s p e c i f i c q u e s t i o n s i n any d e t a i l s i n c e I b e l i e v e t h a t they are a r e f l e c t i o n of the more g e n e r a l d i f f e r e n c e s Kuhn has with h i s c r i t i c s over l o g i c and comparison. That Is, I hope I t w i l l become c l e a r t h a t Kuhn does not r e j e c t the fundamental concepts of e m p i r i c a l s c i e n c e , o n l y p a r t i c u l a r c o n s t r u a l s of them. 8. Kuhn suggests t h a t when seek i n g the source of incommensurabil-i t y between two views t h a t one "can f i r s t attempt to d i s c o v e r the terms and l o c u t i o n s t h a t , used u n p r o b l e m a t i c a l l y w i t h i n each com-munity, are n e v e r t h e l e s s f o c i of t r o u b l e f o r i n t e r - g r o u p d i s c u s -s i o n s " (Kuhn 1970a, 202). Kuhn i s here o f f e r i n g a hermeneutic p r i n c i p l e t h at i s s i m i l a r to t h a t of Quine: "The maxim of t r a n s -l a t i o n u n d e r l y i n g a l l t h i s Is t h a t a s s e r t i o n s s t a r t l l n g l y f a l s e 52 on the face of them are l i k e l y t o t u r n on hidden d i f f e r e n c e s of language" (Quine I960, 59). Kuhn's p r i n c i p l e d i f f e r s from Quine's, as we s h a l l see, i n not r e q u i r i n g t hat the t r a n s l a t e d c l aims make sense from the s t a n d p o i n t of the home language's c u r -r e n t concepts ( c . f . Feyerabend 1987, 77 and note 14). 9. Kuhn d e s c r i b e s t h e i r views as being "fundamentally a t c r o s s -purposes" (Kuhn 1970a, 132). R e c a l l t h a t "fundamentally incom-p a t i b l e " means "both l o g i c a l l y i n c o m p a t i b l e and incommensurable." A l s o note t h a t , i n t h i s essay, I s h a l l o n l y be concerned with those p u t a t i v e cases of p h i l o s o p h i c a l l y - I n t e r e s t i n g incommensura-b i l i t y : those which a l s o i n v o l v e l o g i c a l i n c o m p a t i b i l i t y ; thus t h e o r i e s here d e s c r i b e d as Incommensurable w i l l a l s o be assumed to be l o g i c a l l y i n c o m p a t i b l e . 10. For example, i f we l e t R and c stand f o r , r e s p e c t i v e l y , the 1-place p r e d i c a t e s " i s a c h e m i c a l r e a c t i o n " and "takes p l a c e i n constant ( i n t e g r a l ) p r o p o r t i o n s , " and i f we take the p r e d i c a t e "takes p l a c e i n c o n t i n u o u s l y - v a r y i n g p r o p o r t i o n s " to be l o g i c a l l y e q u i v a l e n t to ->Cf then i t seems t h a t we can r e p r e s e n t Proust's and B e r t h o l l e t ' s p o s i t i o n s as (formal) l o g i c a l c o n t r a d i c t o r i e s u s i n g the same concepts: Proust: (x)(Rx -• Cx) B e r t h o l l e t : -"(x)(Rx - Cx) = (3x)(Rx & ->Cx) 11. A l t e r n a t i v e l y — i n the " m a t e r i a l mode"—Proust d i s a g r e e d with B e r t h o l l e t over which pr o c e s s e s were chemical r e a c t i o n s and over which t h i n g s were chemical compounds as opposed to p h y s i c a l mix-t u r e s . The t h r e e concepts, chemical reaction, chemical compound and physical mixture are l o g i c a l l y - i n t e r r e l a t e d as f o l l o w s : a chemical compound i s formed when two or more chemical substances are mixed together i f and o n l y i f t h e y undergo a chemical reac-tion} otherwise they c o n s t i t u t e a mere physical mixture. 1 2 , A, d, and c are c h e m i c a l s u b s t a n c e s ; Ad and Ac are compounds of those substances. 1 3 . T h i s has been noted by most of Kuhn's c r i t i c s , though t h e i r arguments are u s u a l l y expressed as the c o n t r a p o s i t l v e of mine. That i s , Kuhn's c r i t i c s g e n e r a l l y i n f e r t h a t , s i n c e formal l o g i c s would show no l o g i c a l i n c o m p a t i b i l i t y between p u t a t i v e l y - i n c o r a -mensurable t h e o r i e s , those t h e o r i e s c o u l d not be l o g i c a l l y incom-p a t i b l e (e.g. Shapere 1 9 8 1 , 4 3 - 4 ; Newton-Smith 1 9 8 1 , 1 4 9 , 1 5 9 ; S c h e f f l e r 1 9 6 7 , 8 2 ; Davidson 1 9 8 4 , 1 8 4 ) . However, g r a n t i n g t h a t formal l o g i c s would not show Incommensurable t h e o r i e s to be i n -53 compatible, one may argue i n s t e a d t h a t those l o g i c s are simply inadequate t o d e s c r i b i n g l o g i c a l r e l a t i o n s between incommensur-abl e t h e o r i e s . I s h a l l p r ovide a more complete a n a l y s i s of the source of c o n f l i c t between Kuhn's t h e s i s and assumptions of f o r -mal l o g i c s i n s e c t i o n s 1.6 e t . seg., where I s h a l l a l s o argue th a t the e x i s t e n c e of incommensurable t h e o r i e s means t h a t formal l o g i c s c o u l d not, as a general r u l e , g i v e the l o g i c a l s t r u c t u r e s of a l l i n d i v i d u a l t h e o r i e s . 14. The problem (2) of f i n d i n g common concepts f o r incommensur-able t h e o r i e s i s s i m i l a r to the problem of t r a n s l a t i n g between c e r t a i n n a t u r a l languages. Thus t r a n s l a t o r s o f t e n f i n d t h a t f o r many words of one language there e x i s t s no s i n g l e word, or w e l l -d e f i n e d s e t of words, t h a t matches the meaning of the other i n a l l c ircumstances of i t s a p p l i c a t i o n ; to t r a n s l a t e they must a d j u s t t h e i r c h o i c e of word from s i t u a t i o n to s i t u a t i o n . But s i n c e those s i t u a t i o n - v a r i a n t c h o i c e s o f t e n proceed a c c o r d i n g to an a l i e n p a t t e r n of thought, such t r a n s l a t i o n s o f t e n r e s u l t In cla i m s t h a t seem b i z a r r e to the speakers of the home ( t r a n s l a t -ing) language ( c . f . Quine 1960, 57-9; Nida 1964, 92; Hacking 1983, 69-71). And i f , pace Quine (Quine 1960, 58), such t r a n s l a -t i o n s are c o r r e c t , but seem b i z a r r e o n l y because they r e p r e s e n t non-standard usage of e x i s t i n g concepts, t h i s l a s t f a c t suggests t h a t t r a n s l a t i o n may a l t e r the home language by i n t r o d u c i n g i n t o i t concepts t h a t are incommensurable with c u r r e n t ones (see a l s o Feyerabend 1977 and note 8 ) . 15. T h i s i s not s t r i c t l y t r u e . Thus, to borrow a l o c u t i o n of Kuhn's, those c r i t i c s o c c a s i o n a l l y "dimly r e c o g n i z e " t h a t there are phenomena t h a t seem incompatible with t h e i r views. For ex-ample, the l o g i c i a n s P a t r i c k Suppes and Benson Mates acknowledge as a " s u b t l e and complicated matter" the f a c t t h a t some l o g i c a l r e l a t i o n s seem not to be p u r e l y formal, but to d e r i v e from the meanings of ' n o n - l o g i c a l * e x p r e s s i o n s (Suppes 1957, 68; Mates 1972, 14-5, 80). And Shapere wonders b r i e f l y I f "Feyerabend [and Kuhn have] i n mind some s p e c i a l sense of ' i n c o n s i s t e n t ' (though [they] c l a i m not to be abandoning the law of n o n c o n t r a d i c t i o n ) or 'meaning'" (Shapere 1981, 44). But such b r i e f acknowledgements of problems are i n v a r i a b l y taken back; f o r such p h i l o s o p h e r s im-mediately go on to disparage u n f o r m a l i z a b l e l o g i c a l i n t u i t i o n s as i r r e l e v a n t to l o g i c (Mates 1972, 9-10); or irredeemably vague (Suppes 1957, 4); or f a r too "broad" (Shapere 1981, 54). 16. Because i t o f t e n seems Impossible to analyse f o r m a l l y a c t u a l s c i e n t i f i c t h e o r i e s , Kuhn o f f e r s h i s own a l t e r n a t i v e which he b e l i e v e s i s more a c c u r a t e . I s h a l l d i s c u s s Kuhn's a l t e r n a t i v e a n a l y t i c schema i n chapter two. 54 17. C o r r e l a t i v e l y , Kuhn's c r i t i c s a l s o b e l i e v e t h a t , read a t face v a l u e , "incommensurable" means "not i n t e r t r a n s l a t a b l e , " and t h a t on t h i s c o n s t r u a l , t h e r e f o r e , h i s t h e s i s amounts to the s e l f - c o n -t r a d i c t o r y c l a i m t h a t c e r t a i n t h e o r i e s are both not i n t e r - t r a n s -l a t a b l e and I n t e r t r a n s l a t a b l e . 18. F o r m a l l y , Kuhn's c r i t i c s ' argument i s a "complex c o n s t r u c t i v e dilemma" (Encyclopedia of P h i l o s o p h y , s.v. " T r a d i t i o n a l L o g i c , " and " G l o s s a r y of L o g i c a l Terms"). Thus, on the f o l l o w i n g symbol-i z a t i o n , ct: prima facie c o n s t r u a l of the Incommensurability thesis': " i n -commensurable" means e i t h e r " ( a b s o l u t e l y ) incomparable" or "not i n t e r t r a n s l a t a b l e . " Ci\ a l t e r n a t i v e c o n s t r u a l : t h e o r i e s can be both incommensurable, comparable and i n t e r t r a n s l a t a b l e . I: the incommensurability t h e s i s i s i n t e l l i g i b l e ; i n p a r t i c u l a r , not s e l f - r e f u t i n g . R: the incommensurability t h e s i s i s p h i l o s o p h i c a l l y - r e l e v a n t ; i n p a r t i c u l a r , i t r e q u i r e s changes i n the r e c e i v e d , p h i l o s o p h i c a l view of s c i e n c e . the argument of Kuhn's c r i t i c s I s : 1. Ci - ->l 2. Ci - ->R 3. c, v c* C. T v "»R As I argue l a t e r ( s e c t i o n s 1.4, 1.5 and 1.5.1), Kuhn's r e -sponse to h i s c r i t i c s amounts to an attempt to "escape between the horns of the dilemma": he e f f e c t i v e l y denies t h a t e i t h e r c o n s t r u a l c a p t u r e s h i s i n t e n t i o n s . 19. T r i v i a l l y , two views are incompatible I f they make opposing c l a i m s about the same s i t u a t i o n . Kuhn a l s o a c c e p t s t h i s d e f i n i -t i o n ( c . f . Kuhn 1970a, 122). 20. S c h e f f l e r acknowledges t h a t Kuhn may be r i g h t about incommen-s u r a b i l i t y between the standards of s c i e n t i f i c t h e o r i e s a t a " f i r s t - o r d e r " ( i . e . i n t r a t h e o r e t i c a l ) l e v e l ; but he hypothesizes t h a t e v i d e n t c o n f l i c t between two t h e o r i e s presupposes a "second-order r e f l e c t i v e and c r i t i c a l l e v e l of d i s c o u r s e , " which "presup-poses a c e r t a i n s h a r i n g of standards a t the second-order ( i n t e r -t h e o r e t i c a l ] l e v e l " ( S c h e f f l e r 1967, 83). 21. Davidson's a c t u a l e x p r e s s i o n i s " d r a m a t i c a l l y incomparable." 55 22. What Davidson Is r e f e r r i n g to with the l o c u t i o n "the equip-ment of a s i n g l e language" are a language's p u t a t i v e " r e f e r e n t i a l a p p a r a t u s . . . p r e d i c a t e s , q u a n t i f i e r s , v a r i a b l e s , and s i n g u l a r terms," and i t s "equipment" f o r d e a l i n g v i t h the t r u t h of "whole sentences," as d i s p l a y e d i n T a r s k i ' s Convention T ( I b i d . , 193-4). In other words, Davidson b e l i e v e s there i s a s t r o n g resem-blance betveen n a t u r a l and formal languages ( c . f . Davidson 1984a, 29-30). In f a c t , much of Davidson's p h i l o s o p h y of language i s based on a b e l i e f t h a t n a t u r a l languages must have an u n d e r l y i n g s t r u c t u r e v e r y much l i k e t h a t of formal ones ( c . f . i b i d . , 3, 17, 30, 55). I t i s t r u e t h a t , i f n a t u r a l languages had the s t r u c t u r e formal languages are supposed to have, they vould form a s i n g l e system of concepts, and t r a n s l a t i o n vould imply commensuration (See s e c t i o n 1.6 e t . seg.). Hovever, the Chomskyan argument Da-v i d s o n o f f e r s f o r t h i s a priori n e c e s s i t y ( i b i d . , 3 ) — t h a t l a n -guages vould not be l e a r n a b l e v i t h o u t formal r e c u r s i v e r u l e s — i s a v e r s i o n of the demand f o r commensurability, and so begs the q u e s t i o n . 23. I t i s I r o n i c t h a t , though s c i e n t i s t s o f t e n disparage the con-cerns of t h e i r opponents as "mere metaphysical s p e c u l a t i o n " (Kuhn 1970a, 103), Kuhn's p h i l o s o p h i c a l c r i t i c s o f t e n s l a n d e r h i s vork as "mere e m p i r i c a l s p e c u l a t i o n . " 24. R e c a l l t h a t Davidson l i n k s the ideas of d i s t i n c t c o n c e p t u a l schemes v i t h t h a t of Incommensurable s c i e n t i f i c t h e o r i e s . 25. Musgrave somevhat d i s a p p o i n t e d l y concedes t h a t "Kuhn's 'I never s a i d i t ' p l o y o f t e n c o n v i n c e s . " I b i d . p51. For s i m i l a r v i e v s on Kuhn's ' r e t r a c t i o n s ' see, Putnam (1981a) pl26 and W.H. Nevton-Smith (1981) p l 0 3 . 26. One should keep i n mind here t h a t , though Kuhn b e l i e v e s i n -commensurable t h e o r i e s address a common v o r l d , he does not h o l d that t h i s commonality can be expressed i n terms ac c e p t a b l e to both t h e o r i e s . That means, i n p a r t i c u l a r , t h a t even s o - c a l l e d " o s t e n s i v e d e f i n i t i o n " i s not n e u t r a l between t h e o r i e s . See s e c t i o n 1.7. 27. T h i s should a l r e a d y be c l e a r from h i s equation of Kuhn's no-t i o n "incommensurable" v i t h " d r a m a t i c a l l y incomparable" and "not i n t e r t r a n s l a t a b l e . " In f a c t , the s t r a t e g y of Davidson's much di s c u s s e d paper, "On the Very Idea of a Conceptual Scheme" (Da-v i d s o n 1984b, 183-198), i s not t o show t h a t comparison and t r a n s -l a t i o n e n t a i l common st a n d a r d s , and thus meet Kuhn head on; i n -stead, he argues t h a t other standards f o r i n d i v i d u a t i n g concep-t u a l s c h e m e s , p u t a t i v e l y independent of t r a n s l a t i o n , t u r n out to 56 presuppose the a b i l i t y to l n t e r t r a n s l a t e those schemes: "Studying the c r i t e r i a of t r a n s l a t i o n i s . . . a way of f o c u s i n g on criteria of i d e n t i t y f o r c o n c e p t u a l schemes ( i b i d . , 184)....Can we then say t h a t two people have d i f f e r e n t c o n c e p t u a l schemes i f they speak languages t h a t f a i l of i n t e r t r a n s l a t a b i l i t y ? . . . M y s t r a t e g y w i l l be t o argue t h a t we cannot make sense of t o t a l f a i l u r e , and then to examine more b r i e f l y cases of p a r t i a l f a i l u r e " ( i b i d . , 185). 28. The t r a d i t i o n a l e p i s t e m o l o g l c a l paradigm would l o s e i t s p o i n t because there would then be no reason to s t r u g g l e to f i n d e l u -s i v e , common bases f o r e v a l u a t i n g s c i e n t i f i c t h e o r i e s . For i f I am r i g h t , r a t i o n a l theory c h o i c e would not need those common bases. 29. I should c a u t i o n t h a t the f o l l o w i n g d i s c u s s i o n , while i n t e n -ded to be i l l u m i n a t i n g and c o m p e l l i n g , must n e v e r t h e l e s s be con-s i d e r e d as programmatic. I t would, i n any case, be f o o l h a r d y to expect t h a t p r o p o s a l s f o r modifying c u r r e n t l y - a c c e p t e d concepts w i l l not themselves have to be r e f i n e d c o n t i n u a l l y t o f a c i l i t a t e what Nelson Goodman c a l l s "the t r a n s i t i o n from s t a t i c a b s o l u t i s m to dynamic r e l a t i v i s m i n epistemology" (Goodman 1984, 19. I t a l i c s mine) . 30. T h i s relativity of commensurablllty i s not, however, a com-p l e t e l y u n d e s i r a b l e r e s u l t . Thus Kuhn notes t h a t " s c i e n t i f i c r e v o l u t i o n s . . . n e e d seem r e v o l u t i o n a r y o n l y to those whose para-digms are a f f e c t e d by them. To o u t s i d e r s they may, l i k e the B a l -kan r e v o l u t i o n s of the e a r l y t w e n t i e t h c e n t u r y , seem normal p a r t s of the developmental process" (Kuhn 1970a, 92-3). And Kuhn a l s o suggests t h a t "we may come to see ['the t r a n s i t i o n from Newtonian to E l n s t e l n i a n mechanics'] as a prototype f o r r e v o l u t i o n a r y r e -o r i e n t a t i o n s i n s c i e n c e " ( I b i d . , 102. I t a l i c s mine). Thus we may come to c o n s t r u c t prototypes or standards f o r comparing t h e o r i e s by u s i n g t h e o r y p a i r s l i k e t h a t of Newton and E i n s t e i n as i r r e d u -c i b l e Instances of the r e l a t i o n o f, f o r example, fundamental the-o r e t i c a l I n c o m p a t i b i l i t y . 31. The e x p r e s s i o n " s t r u c t u r a l l y - c o m p l e x " i s i n c l u d e d i n the d e f -i n i t i o n to exclude from the c l a s s of Incommensurable o b j e c t s t h i n g s l i k e phenomenal c o l o u r s . Thus, without t h a t q u a l i f i c a -t i o n , i t might be argued t h a t , s i n c e one cannot determine the r e l a t i o n s between, say, blue and y e l l o w on the b a s i s of t h e i r more p r i m i t i v e p a r t s — t h e y have none--blue and y e l l o w are incom-mensurable. But, at l e a s t i n t h i s paper, speaking of c o l o u r s as incommensurables would introduce unnecessary c o m p l e x i t i e s . On the other hand, s i n c e incommensurability i s c l e a r l y r e l a t e d to i r r e d u c i b i l i t y , i t might be p o s s i b l e to i n c l u d e such I r r e d u c i b l e t h i n g s as phenomenal c o l o u r s as l i m i t i n g cases of incommensurable 57 p a i r s of o b j e c t s ( c . f . note28). 32. I t should be noted t h a t d i r e c t c o m p a r a b i l i t y does not mean untutored c o m p a r a b i l i t y ; t h a t i s , I am not implying t h a t people who know nothing about, say, p a r t i c u l a r s c i e n t i f i c t h e o r i e s c o u l d j u s t look a t them and judge t h e i r r e l a t i o n s . In f a c t , I am im-p l y i n g j u s t the o p p o s i t e . To be a b l e to compare d i r e c t l y two t h e o r i e s one c l e a r l y must f i r s t have an independent, sympathetic, f u l l understanding (verstehen) of each. (I owe a r e c o g n i t i o n of t h i s problem to Rudy Vogt.) 33. T h i s l a s t remark may be more f a m i l i a r as the c l a i m t h a t the r a t i o of the hypotenuse to the s i d e of a r i g h t i s o c e l e s t r i a n g l e i s approximately 1.4142 : 1. I n c i d e n t a l l y , i t should not be thought t h a t , because t h e r e i s a r u l e f o r approximating v2 with r a t i o n a l numbers, t h i s r u l e makes them commensurable. For the s e r i e s of approximations always remains d i f f e r e n t from ^2; and the o p e r a t i o n ( m u l t i p l i c a t i o n ) by v h i c h the approximation i s compared v i t h v2 i s a l s o d i s j o i n t i n i t s separate a p p l i c a t i o n s . Thus, f o r example, one compares <r2 v i t h i t s approximation by s q u a r i n g both numbers. But v h i l e the m u l t i p l i c a t i o n ( l ) of v2 by I t s e l f i s simply d e f i n e d to be 2; the m u l t i p l i c a t i o n ( 2 ) of any r a t i o n a l number ( r a t i o of I n t e g e r s ) by i t s e l f can be broken dovn f u r t h e r as the r a t i o of tvo numbers, each produced by successive a d d i t i o n . That i s , though m u l t i p l i c a t i o n of i r r a t i o n a l and r a -t i o n a l numbers are f o r m a l l y similar} the f i r s t i s a p r i m i t i v e o p e r a t i o n , vhereas the second i s r e a l l y s u c c e s s i v e a d d i t i o n . 34. I am here i g n o r i n g the c o n s t r u a l of r e a l numbers as e i t h e r "the t o t a l i t y of i n f i n i t e d e c i m a l s , " as " [ i n f i n i t e ] sequences of nested I n t e r v a l s " or as "Dedeklnd c u t s " (see Courant and Robbins 1958, 68-75) These c o n s t r u a l s r e p r e s e n t attempts to analyse the notions of r a t i o n a l and i r r a t i o n a l number i n terms of p u t a t i v e l y -more b a s i c n o t i o n s ; and i f they vere s u c c e s s f u l , they vould show th a t r a t i o n a l and i r r a t i o n a l numbers (or p a i r s of l e n g t h s ) are commensurable ( i n the sense of " r e d u c i b l e " ) . However, a d i s c u s -s i o n of those attempts i s both I r r e l e v a n t to my present concerns and beyond my present e x p e r t i s e . Let me j u s t remark t h a t they i n v o l v e the ( f o r me and many p h i l o s o p h e r s ) suspect n o t i o n of ac-t u a l i n f i n i t y ; and i t i s a l s o d o u b t f u l t h a t the "analyzed" no-t i o n s of r a t i o n a l and i r r a t i o n a l number are the same as the o r i -g i n a l ones. 35. In the v e r y g e n e r a l sense of "formal comparison" I u s e — t w o t h i n g s are f o r m a l l y comparable j u s t i n case they are comparable by some common f e a t u r e - - e n t i t i e s t h a t aren't f o r m a l l y comparable are incommensurable. 58 36. Suppes t h e r e s a y s : " F o r t u n a t e l y the s y s t e m a t i c d e d u c t i v e de-velopment of mathematics or t h e o r e t i c a l s c i e n c e can proceed w i t h -out e x p l i c i t r e c o u r s e to sentences vhose t r u t h f o l l o w s s i m p l y from the meanings of the predicates used" ( I t a l i c s mine). But see (Kuhn 1977, 303 & nl3) f o r an a l t e r n a t i v e view of the a c c u r -acy and r e a l s i g n i f i c a n c e of such f o r m a l i z a t i o n s and a s p e c i f i c r e f e r e n c e t o Suppes. 37. T h i s c l a i m can c l e a r l y be extended t o cover other l o g i c a l r e -l a t i o n s and o b j e c t s than v a l i d arguments, premises and c o n c l u -s i o n s . I t s h o u l d a l s o be c l e a r here t h a t Thomas does not mean by "semantics" e i t h e r "formal semantics" or "model t h e o r y " ( i b i d . 449-52). 38. I n c i d e n t a l l y , the s e e m i n g l y - i r r e f u t a b l e p s y c h o l o g i c a l e v i -dence t h a t people do reason h o l i s t i c a l l y has l e d some b r a i n t h e -o r i s t s (e.g. M i c h a e l A r b i b ) and p h i l o s o p h e r s (e.g. Mary Hesse) t o develop non-computational models of b r a i n f u n c t i o n t h a t p a r a l l e l such semantic behaviour. Thus A r b i b and Hesse t r y to c o r r e l a t e what they c a l l semantic nets or schemas with what they term neu-ral nets or columns ( A r b i b & Hesse 1986, 34-41, 69-72). 39. I take t h i s l a s t p a r a l l e l from C r a i g D i l w o r t h ' s S c i e n t i f i c Progress ( D i l w o r t h 1986, 73). D i l w o r t h develops what he c a l l s a " p e r s p e c t i v i s t " view of incommensurability between s c i e n t i f i c t h e o r i e s by e x t e n s i v e use of the d u c k - r a b b i t f i g u r e , and he shows how (what I c a l l " d i r e c t comparison") can be used to favour one aspect over another. Thus he notes t h a t one can argue f o r the g r e a t e r a c c u r a c y of the r a b b i t p e r s p e c t i v e by n o t i n g t h a t an " i n -d e n t a t i o n " on the duck's head i s somewhat anomalous from the duck p o i n t of view while the rabbit-mouth i n t e r p r e t a t i o n i s to be ex-pected on the r a b b i t view ( i b i d . , 73). 40. The Idea of d e f i n i n g a p a r t by a b s t r a c t i o n from a whole should not seem strange t o those f a m i l i a r w i t h Frege's work. Thus Frege d e r i v e d the c o n s t i t u e n t p r e d i c a t e s (concepts) of a sentences by a b s t r a c t i n g away those sentences' s i n g u l a r terms. For example, the p a r t i c u l a r sentence, "John l o v e s Mary," d i v i d e s i n t o the one-place p r e d i c a t e s , "x l o v e s Mary" and "John l o v e s y," and the two-place p r e d i c a t e , "x loves y." 41. I h a v e i n m i n d h e r e t h e " t h i r d - m a n " a r g u m e n t a g a i n s t P l a t o ' s t h e o r y o f I d e a l f o r m s and W i t t g e n s t e i n ' s more g e n e r a l r u l e - f o l -l o w i n g c o n s i d e r a t i o n s w h i c h a t t a c k t h e i d e a t h a t u n i v e r s a l s must be d e t e r m i n e d b y f o r m a l d e f i n i t i o n s : e . g . b y n e c e s s a r y a n d s u f f i -c i e n t c o n d i t i o n s . T h a t W i t t g e n s t e i n ' s work on r u l e s i s r e l e v a n t h e r e s h o u l d n o t be s u r p r i s i n g s i n c e Kuhn's b a s i c i d e a o f t h e p a r -59 adigm, which presumes d i r e c t comparison (see chapter two), i s a s p e c i f i c a d a p t a t i o n of W i t t g e n s t e i n ' s n o t i o n of f a m i l y resem-bl a n c e . But W i t t g e n s t e i n ' s d i s c u s s i o n of f a m i l y resemblance con-cepts i n the Investigations j u s t precedes an e x t e n s i v e d i s c u s s i o n of r u l e - f o l l o w i n g which s e r v e s as a propaedeutic to h i s s t i l l more e x t e n s i v e d i s c u s s i o n i n §§143-242. 42. Here i t may be u s e f u l to keep i n mind W i t t g e n s t e i n ' s c r i t i q u e of the p r i m i t i v e nature of o s t e n s i v e d e f i n i t i o n ( W i t t g e n s t e i n 1968, 8S28-30). There he p o i n t s out t h a t o s t e n s i v e d e f i n i t i o n s are not e s s e n t i a l l y c l e a r e r i n meaning than other r e f e r e n t i a l e x p r e s s i o n s . For example, i f someone p o i n t s t o a c e r t a i n t a l l animal and say s : "I'm r e f e r r i n g t o t h i s , " he may be r e f e r r i n g to the animal, i t s shape, c o l o u r , age, e t c . The g e n e r a l p o i n t about dete r m i n i n g c o r e f e r e n c e i s s i m p l y t h a t a t some p o i n t one must j u s t see, without any f u r t h e r grounds, i n c l u d i n g o s t e n s i o n , t h a t two terms are c o r e f e r e n t i a l . 43. "Determined d i r e c t l y " does not mean here: "determined by ' d i -r e c t r e f e r e n c e ' t o a common o b j e c t . " That i s , I am not c l a i m i n g t h e r e must be some s o r t of unmediated p e r c e p t i o n of o b j e c t s a l a Kripke with which c o r e f e r e n c e can be determined. Rather, I am sim p l y a r g u i n g t h a t the r e c o g n i t i o n t h a t two d i f f e r e n t ways of d e s i g n a t i n g some o b j e c t are c o r e f e r e n t i a l must, a t some p o i n t , not r e q u i r e any f u r t h e r means of de t e r m i n i n g r e f e r e n c e . Thus I am a c t u a l l y denying t h a t there need be any p r i v i l e g e d means, such as o s t e n s i o n , f o r deter m i n i n g c o r e f e r e n c e . See a l s o note 42. 44. For example, Kuhn notes t h a t the s y s t e m a t i c , f o c u s i n g c h a r a c -t e r of standards (paradigms) i n s c i e n c e permits " s c i e n t i s t s to i n v e s t i g a t e some p a r t of nature In a d e t a i l and depth t h a t would otherwise be unimaginable" (Kuhn 1970a, 24). Chapter 2 Kuhn's A l t e r n a t i v e A n a l y t i c Concepts 2.0 I n t r o d u c t i o n In t h i s c hapter, I show how Kuhn's concepts f o r a n a l y s i n g s c i e n c e perform many of the same e v a l u a t i v e f u n c t i o n s as t r a d i -t i o n a l n o t i o n s drawn from formal l o g i c , but without p r e c l u d i n g e i t h e r incommensurability o r , as we s h a l l see, the h i s t o r i c a l c h a r a c t e r of s c i e n t i f i c t h e o r i e s . I s h a l l devote p a r t i c u l a r a t -t e n t i o n t o Kuhn's a n a l y s i s of the " d i a l e c t i c a l " r o l e of thought experiments i n s c i e n c e because t h a t a n a l y s i s i l l u m i n a t e s h i s l o g -ico-normative c o n c e p t i o n of t h e o r i e s and pr o v i d e s the b a s i s f o r d e v e l o p i n g i n the next chapter a g e n e r a l i z e d proof of Incommen-s u r a b i l i t y . In what f o l l o w s , I d e s c r i b e Kuhn's a l t e r n a t i v e t o the t r a d i -t i o n a l p i c t u r e of s c i e n t i f i c t h e o r i e s as u n i v e r s a l g e n e r a l i z a -t i o n s . I th e r e show t h a t Kuhn b e l i e v e s h i s c o u n t e r p a r t n o t i o n , the paradigm, Is the more b a s i c both because paradigms g i v e s c i -e n t i f i c g e n e r a l i z a t i o n s t h e i r e m p i r i c a l meaning and because t h i s l a s t process must be repeated anew f o r novel s i t u a t i o n s . I t thus turns out t h a t by making the n o t i o n of a paradigm b a s i c t o s c i -ence, Kuhn a l s o makes t h e o r i e s i n t o h i s t o r i c a l l y - e v o l v i n g t r a d i -t i o n s , not t i m e l e s s , formal s t r u c t u r e s as on the r e c e i v e d view. Paradigms a l s o e x p l a i n t h e o r e t i c a l h o l i s m and make room f o r i n -commensurability . 61 2.1 Paradigms and U n i v e r s a l G e n e r a l i z a t i o n s Ronald G i e r e notes t h a t " s i n c e A r i s t o t l e I t has been assumed th a t the o v e r a l l g o a l of s c i e n c e i s the d i s c o v e r y of t r u e u n i v e r -s a l generalizations of the form: A l l A's are B" (Giere 1983, 271. I t a l i c s In o r i g i n a l ) . However, the n o t i o n of a u n i v e r s a l gener-a l i z a t i o n i s an elementary c a t e g o r y of formal ( s y l l o g i s t i c and p r e d i c a t e ) l o g i c . Hence, to make such a n o t i o n , and thus the l o g i c a l system of which i t i s a p a r t , the b a s i c a n a l y t i c u n i t f o r s c i e n c e would a l s o , as we saw In chapter one, make incommensura-b i l i t y i m p o s s i b l e (see S1.6). 1 But h i s t o r i c a l evidence of actual incommensurability and co n c e p t u a l evidence of i t s p o s s i b i l i t y , suggests t h a t u n i v e r s a l g e n e r a l i z a t i o n s cannot p l a y such a funda-mental p a r t i n s c i e n c e . There i s more s p e c i f i c evidence f o r t h a t f a c t : the e m p i r i c a l and formal indeterminacy of s c i e n t i f i c con-cepts . Science commonly c o n t a i n s many g e n e r a l i z a t i o n s . For ex-ample, Newton's f i r s t law of motion begins with a c l a i m about the behaviour of " e v e r y body" (Enc. of Phil., s.v. "Newtonian Mechan-i c s and Mechanical E x p l a n a t i o n . " I t a l i c s mine). And such c l a i m s are e a s i l y rendered f o r m a l l y as, e.g., (x ) ( F x -• Gx). But f o r gen-e r a l i z a t i o n s not to be p u r e l y f o r m a l , 2 but a l s o t o have e m p i r i c a l i m p l i c a t i o n s , t h e i r c o n s t i t u e n t concepts (here, F and G) must have determinate e m p i r i c a l meaning p r i o r to t h e i r .comparison with nature. For u n l e s s i t were c l e a r what a g e n e r a l i z a t i o n s a i d i n a given i n s t a n c e , no c o n c l u s i o n s c o u l d there be drawn about i t s e m p i r i c a l t r u t h or f a l s i t y (see a l s o Kuhn 1977d, 284). I t f o l -lows t h a t i f e m p i r i c a l indeterminacy i n new cases were the r u l e , n o rmally the " o v e r a l l goal of s c i e n c e " would be t o remove t h a t Indeterminacy by d i s c o v e r i n g how to extend the scope of concepts; i t c o u l d not, i n any n o n - t r i v i a l sense, be "the d i s c o v e r y of tr u e g e n e r a l i z a t i o n s . " F or, a t any g i v e n time, s c i e n t i f i c g e n e r a l i z a -t i o n s would be t r u e o n l y of those s i t u a t i o n s t o which t h e i r con-c e p t s have a l r e a d y been extended and indeterminate f o r those t o which they have n o t . 9 A s u r p r i s i n g example of e m p i r i c a l l y i ndeterminate g e n e r a l i -z a t i o n s are Newton's f o r m u l a t i o n s i n the P r i n c i p i a of the laws of motion and g r a v i t a t i o n . Even though these laws are o f t e n c i t e d as the s c i e n t i f i c model of both p r e c i s i o n and g e n e r a l i t y , 4 they f r e q u e n t l y proved noncommital or vague i n t h e i r import f o r p r e v i -o usly-unexplored s i t u a t i o n s (Kuhn 1970a, 3 2 ) . T h e r e f o r e , s i n c e i t was o f t e n impossible to p r e d i c t the i m p l i c a t i o n s of Newton's laws f o r r e l e v a n t s i t u a t i o n s , i t s concepts were not e m p i r i c a l l y determinate. Hence, i t s l a t e r a p p l i c a t i o n s were not s t r i c t l y c o n t a i n e d i n Newton's o r i g i n a l f o r m u l a t i o n . 8 "These l i m i t a t i o n s of agreement l e f t many f a s c i n a t i n g t h e o r e t i c a l problems f o r New-ton's s u c c e s s o r s . . . ( F o r example,] E u l e r , Lagrange, Laplace and Gauss a l l d i d some of t h e i r most b r i l l i a n t work on problems aimed to improve the match between Newton's paradigm and o b s e r v a t i o n s of the heavens" (Kuhn 1970a, 3 2 ) . Kuhn says t h a t he " i n t r o d u c e d the term 'paradigm' to under-score the dependence of s c i e n t i f i c r e s e a r c h on concrete examples t h a t b r i d g e what would otherwise be gaps i n the s p e c i f i c a t i o n of the content and a p p l i c a t i o n of s c i e n t i f i c t h e o r i e s " (Kuhn 1970d, 284). T h e r e f o r e , s i n c e paradigms s p e c i f y "the content and ap-p l i c a t i o n of s c i e n t i f i c t h e o r i e s , " a paradigm i s c o n c e p t u a l l y " p r i o r t o the v a r i o u s concepts, laws, ( s u b ] t h e o r i e s , and p o i n t s of view t h a t may be a b s t r a c t e d from i t " (Kuhn 1970a, 11). From the p r e c e d i n g d i s c u s s i o n of e m p i r i c a l indeterminacy, t h e r e f o r e , one would a l s o expect t h a t , while use of paradigms might h e l p to s p e c i f y the content of t h e o r i e s by s e t t i n g broad parameters, i t c o u l d not r i g o r o u s l y determine t h a t content. Kuhn c l a i m s t h a t a n a l o g i c a l r e a s o n i n g based on paradigms, or "concrete s c i e n t i f i c achievements" ( i b i d . , 11), helps t o s p e c i f y the content of t heo-r i e s i n j u s t t h i s rough way. Thus i f he i s r i g h t , t h i s a n a l o g i -c a l r e a s o n i n g i s more b a s i c t o s c i e n c e than the t r a d i t i o n a l l y -p r e f e r r e d formal d e d u c t i o n from g e n e r a l i z a t i o n s , a u x i l i a r y hypo-theses and i n i t i a l c o n d i t i o n s . 2.1.1 Normal s c i e n c e as a Process of Paradigm A r t i c u l a t i o n Since f o r Kuhn, a s c i e n t i f i c theory's g e n e r a l i z a t i o n s get t h e i r content from "concrete examples," h i s p h i l o s o p h y i n v e r t s the t r a d i t i o n a l view of the r e l a t i o n of u n i v e r s a l to p a r t i c u l a r . Instead of s t a r t i n g out as completed g e n e r a l i z a t i o n s c o n t a i n i n g a l l t h e i r I m p l i c a t i o n s , Kuhn says t h a t t h e o r i e s begin as " s e l e c -ted and s t i l l incomplete examples" (Kuhn 1970a, 23), or p a r a -digms, which are c o n t i n u a l l y made more general by refinement and 64 by d e v e l o p i n g s t i l l f u r t h e r i n s t a n c e s of s u c c e s s f u l a p p l i c a t i o n s (Kuhn 1970a, 30). I t i s j u s t these g i v e n p a r t i c u l a r a p p l i c a t i o n s t h a t show how the theory's concepts, laws, and g e n e r a l i z a t i o n s can be a p p l i e d t o "some [new] co n c r e t e range of n a t u r a l phenom-ena" ( i b i d . , 46). E x p l o i t a t i o n of these accepted a p p l i c a t i o n s p r o v i d e s s c i e n t i s t s v i t h i n t e r p r e t a t i o n s f o r the t h e o r y . Where Kuhn's account d i f f e r s from more t r a d i t i o n a l m o d e l - t h e o r e t i c n o t i o n s of i n t e r p r e t a t i o n i s t h a t , f o r him, a t no p o i n t i s the theory's i n t e r p r e t a t i o n f i x e d f o r a l l r e l e v a n t c a s e s . Instead of being actual g e n e r a l i z a t i o n s , t h e r e f o r e , i t i s more a c c u r a t e to say t h a t , f o r Kuhn, t h e o r i e s a re programmatic g e n e r a l i z a t i o n s . For he c l a i m s the c r e a t i v e input of s c i e n t i s t s doesn't end v i t h the i n i t i a l e x p r e s s i o n of a t h e o r y or a t any stage vhen i t i s s t i l l c o n s i d e r e d v i a b l e . Rather, s c i e n t i s t s a re f o r c e d t o adapt c o n t i n u a l l y t h e i r c u r r e n t t h e o r e t i c a l r e s o u r c e s t o handle nev problems. T h i s ongoing, a m p l i a t i v e process of r e a l i z i n g a t h e -ory's promise by constant e x p l o r a t i o n and refinement, Kuhn c a l l s paradigm articulation or normal science. Normal science c o n s i s t s i n the a c t u a l i z a t i o n of t h a t prom-i s e , an a c t u a l i z a t i o n achieved by extendi n g the knovledge of those f a c t s t h a t the paradigm d i s p l a y s as p a r t i c u l a r l y r e -v e a l i n g , by Increasing the extent of the match betveen those f a c t s and the paradigm's p r e d i c t i o n s , and by f u r t h e r a r t i c u -l a t i o n of the paradigm i t s e l f ( i b i d . , 23-4. I t a l i c s mine). Kuhn c l a i m s h i s s t u d i e s show s c i e n t i s t s u s u a l l y do not ab-s t r a c t from e x i s t i n g a p p l i c a t i o n s e x p l i c i t r u l e s f o r determining f u r t h e r a p p l i c a t i o n s ( i b i d . , 46). Instead they use previous 65 achievements as s u g g e s t i v e models of how to approach new prob-lems. In so doin g , they i n t e n t i o n a l l y r e p l i c a t e the e n t i r e con-c e p t u a l s t r u c t u r e of t h e i r t h e o r y i n the new s i t u a t i o n (Kuhn 1982, 566). Thus a n a l o g i c a l r e a s o n i n g based on paradigms natur-a l l y leads to h o l i s m . " S c i e n t i s t s . . . never l e a r n concepts, laws, and t h e o r i e s i n the a b s t r a c t and by themselves. Instead, these I n t e l l e c t u a l t o o l s are from the s t a r t encountered i n a h i s t o r i -c a l l y and p e d a g o g i c a l l y p r i o r u n i t t h a t d i s p l a y s them with and through t h e i r a p p l i c a t i o n s " ( I b i d . , 46). I t i s i n h i s d e t a i l e d account of how s c i e n t i s t s "extend," " i n c r e a s e " and thus " a r t i c u l a t e " t h e i r paradigms without e x p l i c i t r u l e s f o r guidance t h a t Kuhn e v i n c e s most c l e a r l y a d i s t i n c t i o n between the concepts of ( d i r e c t ) c o m p a r a b i l i t y and commensura-b i l i t y , and thus a l s o c o m p a r a b i l i t y and incommensurability. He gets t h i s account by drawing a c l o s e analogy between h i s idea of s c i e n t i f i c t h e o r i e s as paradigms and W i t t g e n s t e i n ' s n o t i o n of the f a m i l y resemblance concept. 2.1.2 Paradigms, Commensurablllty and D i r e c t C o m p a r a b i l i t y Famously, W i t t g e n s t e i n has p o i n t e d out t h a t I t i s o f t e n impossible to f i n d a s e t of common c h a r a c t e r i s t i c s informing many of our o r d i n a r y concepts. For example, when we search f o r a determinate s e t of f e a t u r e s common to a l l and onl y games, upon c a r e f u l s c r u t i n y we f i n d Instead "a complicated network of s i m i -l a r i t i e s o v e r l a p p i n g and c r i s s - c r o s s i n g : sometimes o v e r a l l s i m i -l a r i t i e s , sometimes s i m i l a r i t i e s of d e t a i l " ( W i t t g e n s t e i n 1968, S66). Thus we d i s c o v e r t h a t some games are merely amusing, while others are s e r i o u s l y c o m p e t i t i v e ; some i n v o l v e s k i l l , but ot h e r s depend o n l y on l u c k ; some i n v o l v e v e r y s t r i c t r u l e s , though others are f l e x i b l e and spontaneous; and s t i l l o t h e r s c o n t a i n v a r i o u s combinations of these f e a t u r e s ; and so on. Because t h e r e i s no f i x e d s e t of s i m i l a r i t i e s common to a l l games, but o n l y a growing ( i b i d . , 868) f a m i l y of resemblances c o n n e c t i n g p a r t i c u l a r i n s t a n c e s , i n l e a r n i n g the concept of game, examples (paradigms) must p l a y an e s s e n t i a l , not simply i l l u s t r a t i v e r o l e . "Here g i v -i n g examples i s not an indirect means of e x p l a i n i n g — i n d e f a u l t of a b e t t e r [e.g. l i k e an e x p l i c i t d e f i n i t i o n ] " ( i b i d . , 871. I t a l i c s i n o r i g i n a l ) . I t f o l l o w s t h a t the e x t e n s i o n of the con-cept of game i s , a t any given time, " u n c i r c u m s c r i b e d " ( I b i d . , 870), s i n c e i t i s determined by nothing more b a s i c than the par-t i c u l a r games themselves. Kuhn puts W i t t g e n s t e i n ' s n o t i o n of f a m i l y resemblance t o work i n the f o l l o w i n g way. He says t h a t s c i e n t i s t s o b t a i n new a p p l i c a t i o n s "by the d i r e c t i n s p e c t i o n of paradigms," (Kuhn 1970a, 44. I t a l i c s mine) and t h a t these new a p p l i c a t i o n s a re l i n k e d with each other and the o r i g i n a l paradigms "by [ f a m i l y ] resemblance and [ d i r e c t ] modeling to one or another p a r t of the s c i e n t i f i c corpus" ( i b i d . , 45). Because a s c i e n t i f i c paradigm r e p r e s e n t s a more complex conceptual s t r u c t u r e than the si m p l e r n o t i o n of a game, such paradigms might i n s t e a d be c a l l e d : family resemblance conceptual systems. E t y r a o l o g i c a l l y , the term "paradigm" n a t u r a l l y lends i t s e l f t o a d i s c u s s i o n of incommensurability, s i n c e i t means "standard of comparison." Thus one would expect t h a t the concepts and laws of incommensurable t h e o r i e s would get t h e i r r e s p e c t i v e meanings from incompatible "manipulative procedures and paradigm a p p l i c a -t i o n s " (Kuhn 1970a, 142-3). But Kuhn's account of paradigms a l s o has deep p a r a l l e l s with my d i s c u s s i o n of c o m p a r a b i l i t y and com-m e n s u r a b l l l t y i n chapter one. Thus because each new a p p l i c a t i o n employs the same b a s i c s e t of common s t r u c t u r a l elements ( i . e . concepts) as i t s paradigms, each i s i n e f f e c t commensurable with any other such a p p l i c a t i o n . Kuhn a l s o r e c o g n i z e s t h a t commensur-a t i o n i t s e l f r e s t s on d i r e c t comparison. For each a p p l i c a t i o n i s o b t a i n e d by a d i r e c t comparison with e i t h e r the paradigms them-s e l v e s or one of t h e i r c u r r e n t a p p l i c a t i o n s , which l i n k them to those paradigms by a "network of o v e r l a p p i n g and c r i s s c r o s s r e -semblances" ( i b i d . , 45). L i k e W i t t g e n s t e i n , Kuhn argues t h a t the s i m i l a r i t i e s l i n k i n g the v a r i o u s a p p l i c a t i o n s themselves form no f i x e d , standard s e t ; so those s i m i l a r i t i e s cannot p r o v i d e an i n t e r p r e t i v e standard more b a s i c than the paradigms (examples) themselves. Though a d i s c u s s i o n of some of the a t t r i b u t e s shared by a number of [exemplary s o l u t i o n s t o problems] o f t e n helps us [as a temporary h e u r i s t i c ] l e a r n how to employ [the t h e o r y ] , there i s no s e t of c h a r a c t e r i s t i c s t h a t i s s i m u l t a n e o u s l y a p p l i c a b l e to a l l members of the c l a s s [of s o l u t i o n s ] and to them a l o n e . ( i b i d . , 45) 68 In other words, because i t cannot be known i n advance which s i m i l a r i t i e s w i l l prove f r u i t f u l i n extending a paradigm, no r u l e can be a b s t r a c t e d from e x i s t i n g s i m i l a r i t i e s t h a t w i l l determine l a t e r e x t e n s i o n s . So such s i m i l a r i t i e s cannot determine s c i e n -t i s t s i n f i n d i n g new applications} r a t h e r , s c i e n t i s t s must o b t a i n new a p p l i c a t i o n s by f i n d i n g f r u i t f u l similarities. Because the network of s i m i l a r i t i e s cannot p l a y the r o l e of an i n t e r p r e t i v e standard, Kuhn c a l l s the resemblances betveen paradigms and ap-p l i c a t i o n s " d i r e c t . " Paradigms and a p p l i c a t i o n s , t h e r e f o r e , con-s t i t u t e a p r i m i t i v e similarity set, s i n c e they are a l l s i m i l a r t o each other, s i m p l i c i t e r . As j u s t noted, Kuhn c l a i m s t h a t such unmediated p a t t e r n i n g of nev problems on o l d s o l u t i o n s i s b a s i c to the s c i e n t i f i c e n t e r p r i s e . The p r a c t i c e of normal s c i e n c e depends on the a b i l i t y , a c -q u i r e d from exemplars [paradigms], to group o b j e c t s and s i t u a t i o n s i n t o s i m i l a r i t y s e t s v h i c h are p r i m i t i v e i n the sense t h a t the grouping i s done v i t h o u t an ansver to the q u e s t i o n , " S i m i l a r v i t h r e s p e c t to vhat?" (Kuhn 1970a, 200) Kuhn p r o v i d e s s e v e r a l h i s t o r i c a l examples of vhat he means by " d i r e c t modeling" ( i b i d . , 47). For i n s t a n c e , G a l i l e o l e a r n e d to see a ( p o i n t ) pendulum's c y c l e as i m p o r t a n t l y s i m i l a r to the seemingly v e r y d i f f e r e n t s i t u a t i o n of a b a l l r o l l i n g down one i n -c l i n e and up another ( i b i d . , 190). G a l i l e o shoved hov, i n both cases, the moving o b j e c t s c o u l d ideally be seen as s t a r t i n g from a c e r t a i n h e i g h t , p a s s i n g through a lowest p o i n t , and r e t u r n i n g to t h e i r o r i g i n a l e l e v a t i o n . L a t e r , Huyghen's found an important 69 analogy between G a l i l e o ' s treatment of (Imaginary) p o i n t pendula and the behaviour of ( r e a l ) extended ones. By c o n s t r u i n g a r e a l pendulum as a c l u s t e r of p o i n t pendula t h a t c o u l d each swing i n -dependently upon r e l e a s e , Huyghens was a b l e t o s o l v e the problem of f i n d i n g a r e a l pendulum's c e n t r e of o s c i l l a t i o n (ibid., 190). D a n i e l B e r n o u l l i l a t e r adapted Huyghen's model %o t r e a t "the flow of water from an o r i f i c e " ( i b i d . , 190)." The o n l y g e n e r a l i z a t i o n common t o each of these s o l u t i o n s i s the " P r i n c i p l e of v i s v i v a . . . A c t u a l descent equals p o t e n t i a l descent" ( i b i d . , 191). But each s i t u a t i o n — b a l l s r o l l i n g down i n c l i n e s , p o i n t and r e a l pen-d u l a , water f l o w i n g from an o r i f i c e — d o e s not a t f i r s t glance seem r e l e v a n t l y s i m i l a r to any o f t h e o t h e r s . And u n t i l G a l i l e o , Huyghens and B e r n o u l l i found s p e c i f i c ways to see these new c i r -cumstances as i n s t a n c e s of " v i s v i v a " — u n t i l they a r t i c u l a t e d t h e i r c o n c e p t s — t h e y c o u l d not a p p l y the formal techniques of t h e i r e a r l i e r a p p l i c a t i o n s t o them ( c . f . Kuhn 1977, 305). Perhaps the most s t r i k i n g i l l u s t r a t i o n Kuhn g i v e s of d i r e c t m o d e l l i n g and resemblance i s the s e t of a r t i c u l a t i o n s developed from a formal statement of Newton's second law. These extensions make h i s p o i n t i n a p a r t i c u l a r l y dramatic way s i n c e they show t h a t even formal g e n e r a l i z a t i o n s , whose use supposedly i n v o l v e s merely mechanical m a n i p u l a t i o n s , o f t e n r e q u i r e c r e a t i v e a r t i c u -l a t i o n b efore they can be used f o r p h y s i c a l problems. Moreover, the p u r e l y formal resemblances such a r t i c u l a t i o n s have with each other and the o r i g i n a l e q uation i s o f t e n hard to d i s c e r n . For example, the standard ( s i n c e Gauss) e x p r e s s i o n of the second law 70 i s £ = ma. But that law can a l s o be a p p l i e d to such seemingly d i s t i n c t s i t u a t i o n s as: f r e e l y - f a l l i n g b o d i e s , where i t takes the form mg = m d a s / d t a ; simple pendula, where i t becomes mgsin 8= mld ae/dt a; i n t e r a c t i n g harmonic o s c i l l a t o r s , where I t i s expressed with two equations, one of which i s m » d 2 S i / d t a + k i S i = ka(sa - Si + d ) ; "and f o r more complex s i t u a t i o n s , such as the gyroscope, i t takes s t i l l other forms, the f a m i l y resemblance of which t o f=ma i s s t i l l harder t o d i s c o v e r " ( i b i d . , 189). 2.1.3 D i r e c t Comparison of Competing Paradigms For Kuhn, then, each s c i e n t i f i c t h e o r y c o n s i s t s of a s m a l l s e t of paradigms and an expanding s e t of a p p l i c a t i o n s d e r i v e d from them by " d i r e c t modeling." 7 These a d d i t i o n a l a p p l i c a t i o n s are a l l i n d i r e c t l y r e l a t e d to each other but are directly r e l a t e d to the s e t of paradigms: t h e i r common standar d s . And because t h a t l a s t r e l a t i o n i s d i r e c t , there are no f u r t h e r standards more b a s i c t o a s c i e n t i f i c t h e o r y than i t s paradigms and a r t i c u l a t i o n s ( c . f . Kuhn 1970a, 11). 8 I t f o l l o w s , t h e r e f o r e , on t h i s model, t h a t s c i e n t i f i c t h e o r i e s t h a t employ d i f f e r e n t paradigms won't share some common standards, and so w i l l be (at l e a s t p a r t l y ) incommensurable. 9 In chapter one, I claimed t h a t i n c o m p a t i b l e , yet incommen-s u r a b l e t h e o r i e s were c o n c e i v a b l e i f i t were p o s s i b l e to re c o g -n i z e those t h e o r i e s ' i n c o m p a t i b i l i t y by the d i r e c t comparison of one t h e o r y with another. Such a d i r e c t r e c o g n i t i o n of incompati-b i l i t y would depend on being a b l e to see, by d i r e c t l y comparing how each whole t h e o r y used i t s s e t of r e f e r r i n g terms, concepts and laws, that those terms were o f t e n c o - e x t e n s i v e and t h a t t h e i r r e s p e c t i v e t h e o r i e s s a i d incompatible t h i n g s about t h e i r common domain. We have j u s t seen how d i r e c t comparison works i n p r a c -t i c e w i t h i n the scope of a paradigm. S t i l l , though d i r e c t compa-r i s o n between and w i t h i n paradigms are very much a l i k e , they are a l s o i m p o r t a n t l y d i f f e r e n t . On Kuhn's " g e s t a l t " model, incommensurable t h e o r i e s can be d i r e c t l y compared by f i n d i n g p a i r s of s i m i l a r , but d i s t i n c t con-cepts (terms, laws, e t c . ) from each t h e o r y and d i s p l a y i n g t h e i r s i m i l a r i t i e s and d i f f e r e n c e s by si m p l y j u x t a p o s i n g them i n a " d i s c i p l i n e d " (Kuhn 1977d, 267) manner. For example, one c o u l d d i s p l a y the s i m i l a r i t i e s and d i f f e r e n c e s between the Newtonian concept of mass and i t s E i n s t e l n i a n c o u n t e r p a r t i n the f o l l o w i n g way. 1 0 Newtonian mass equals the r a t i o of f o r c e to a c c e l e r a t i o n , and i s constant; (at low r e l a t i v e v e l o c i t i e s ) Einstelnian mass a l s o (roughly) equals the r a t i o of f o r c e to a c c e l e r a t i o n , and i s a l s o (roughly) c o n s t a n t . Newtonian mass i s a p r o p e r t y of bodies t h a t i s a measure of t h e i r i n e r t i a ; Einsteinian mass i s a l s o a p r o p e r t y of bodies t h a t i s a measure of t h e i r i n e r t i a . By con-t r a s t : "Newtonian mass i s conserved; E i n s t e i n i a n i s c o n v e r t i b l e with energy. Only a t low r e l a t i v e v e l o c i t i e s may the two be mea-sured i n the same way, and even then they must not be conceived to be the same" (Kuhn 1970a, 102). 1 1 C l e a r l y , these comparative j u x t a p o s i t i o n s of the concepts of incommensurable t h e o r i e s bear a s t r o n g ' f a m i l y resemblance' to Kuhn's idea of a r r a n g i n g paradigms and a p p l i c a t i o n s i n a s i m i l a -r i t y s e t . N e v e r t h e l e s s , there i s an important d i f f e r e n c e between the e x t e r n a l and i n t e r n a l s o r t s of comparisons. For while s c i e n -t i s t s o b t a i n s o l u t i o n s to t h e i r problems by d i r e c t m o d e l l i n g on t h e i r paradigms, comparing incommensurable t h e o r i e s c o u l d not serve t h a t same f u n c t i o n . T h i s i s because n o t i n g the s i m i l a r i -t i e s and d i f f e r e n c e s i n the ways d i s t i n c t paradigms determine what i s t r u e , i s not i n i t s e l f a way of determining what i s true (see i b i d . , 122-3); f o r no paradigms mediate that process. Nev-e r t h e l e s s , t h a t e x t e r n a l comparative process can a i d i n d e t e r m i -ning which ( i f any) of two competing t h e o r i e s i s more ac c u r a t e or promising by d i s p l a y i n g each one's r e l a t i v e m e r i t s i n i t s own terms. "That e x h i b i t can be Immensely p e r s u a s i v e , even compel-l i n g l y so" (Kuhn 1970a, 94). In summary, Kuhn's no t i o n of a s c i e n t i f i c paradigm performs many of the same f u n c t i o n s as the more t r a d i t i o n a l n o t i o n of a u n i v e r s a l g e n e r a l i z a t i o n . Thus, j u s t as on the t r a d i t i o n a l model where s c i e n t i s t s deduce p a r t i c u l a r e m p i r i c a l consequences from t h e i r t h e o r i e s ( g e n e r a l i z a t i o n s ) , so too on Kuhn's approach s c i -e n t i s t s o b t a i n "concrete achievements" by " d i r e c t modeling" on t h e i r e x i s t i n g t h e o r i e s (paradigms). Where Kuhn's approach d i f -f e r s from the more t r a d i t i o n a l one, however, i s by making the d e t e r m i n a t i o n of t h e o r e t i c a l content an h i s t o r i c a l process and by p e r m i t t i n g incommensurability. His account a l s o d i f f e r s i n s i g -n l f i c a n t l y r e d u c i n g the s e n s i t i v i t y of t h e o r y to experiment. S t i l l , as we s h a l l see, because i t does not completely e l i m i n a t e experimental c o r r i g i b i l i t y , i n the end Kuhn's view a l s o makes ex-perience the f i n a l c o u r t of a p p e a l f o r t h e o r i e s . 2.2 F e l t Anomaly As noted above, Kuhn's account of how a theory's concepts and g e n e r a l i z a t i o n s get t h e i r c o n t e n t from exemplary achievements (paradigms) by a r t i c u l a t i o n i s based on h i s t o r i c a l evidence of the e m p i r i c a l and formal indeterminacy of s c i e n t i f i c concepts. Since a r t i c u l a t i o n i s (roughly) Kuhn's c o u n t e r p a r t f o r the t r a d i -t i o n a l n o t i o n of l o g i c a l d e r i v a t i o n , and s i n c e the usual concept of theory f a l s i f i c a t i o n depends on t h a t of l o g i c a l d e r i v a t i o n , one may expect t h a t Kuhn w i l l a l s o employ i n h i s approach an a l t e r n a t i v e n o t i o n to t h a t of f a l s i f i c a t i o n . That a l t e r n a t i v e n o t i o n i s anomaly or anomalous exper i e n c e (Kuhn 1970a, 77, 80, 146). Anomalous exp e r i e n c e s are s i m i l a r t o and d i f f e r e n t from f a l -s i f y i n g e x periences i n the f o l l o w i n g ways. F i r s t , j u s t as on the t r a d i t i o n a l l o g i c a l model where a (genuinely) f a l s i f y i n g e x p e r i -ence r e s u l t s i n the r e j e c t i o n of a t h e o r y , anomalous experiences a l s o (often) r e s u l t i n t h e o r y r e j e c t i o n ( I b i d . , 146). 1 2 More-over, j u s t as f a l s i f y i n g e x p e r i e n c e s are of s i t u a t i o n s t h a t are d i f f e r e n t from t h a t i m p l i e d by a t h e o r y , so too anomalous e x p e r i -ences are of circumstances which f a i l t o f i t "paradigm-induced e x p e c t a t i o n s " ( i b i d . , 52-3). By c o n t r a s t , however, while (genu-i n e l y ) f a l s i f y i n g experiences always demand the r e j e c t i o n of the t h e o r y which e n t a i l s them, t h i s i s seldom the case f o r anomalous e x p e r i e n c e s . An anomaly i s , f o r Kuhn, o n l y a f a i l u r e of nature to f i t a t h e o r y i n i t s c u r r e n t s t a t e of articulation} i n other words, i t i s a f a i l u r e pro tem. In support of h i s own concept, Kuhn p o i n t s out t h a t h i s t o r i c a l and contemporary evidence show t h a t r i g h t from t h e i r v e r y i n c e p t i o n , a l l t h e o r i e s f a i l t o f i t nature i n many a r e a s , and even i n those p l a c e s where they are n o t a b l y s u c c e s s f u l , the theory-nature match i s o f t e n f a r from p e r f e c t . Hence, i f one accepted the t r a d i t i o n a l view t h a t "any and every f a i l u r e t o f i t were ground f o r t h e o r y r e j e c t i o n , [ t h i s would r e s u l t i n the p a r a d o x i c a l view t h a t ) a l l t h e o r i e s ought to be r e j e c t e d a t a l l times" ( i b i d . , 146). 4 3 U s u a l l y , even the "most stubborn" anomalies a t h e o r y i s con-f r o n t e d w i t h , prove to be r e s o l v a b l e by paradigm a r t i c u l a t i o n , the method of normal s c i e n c e (Kuhn 1970a, 81; see a l s o Kuhn 1977b, 262). While such circumstances are r e c o g n i z a b l y i n c o n g r u -ous from the b e g i n n i n g , because t h e i r import f o r the paradigm cannot y e t be known with any p r e c i s i o n , Kuhn sometimes d e s c r i b e s them as " f e l t anomalies" (Kuhn 1977, 264). And j u s t because of t h a t c u r r e n t u n c l a r i t y of import, s c i e n t i s t s o f t e n s h e l v e such problems to be readdressed by "a f u t u r e g e n e r a t i o n with more developed t o o l s " (Kuhn 1970a, 84). Some h i s t o r i c a l examples of f e l t anomalies t h a t l a t e r proved to be r e s o l v a b l e by normal s c i -ence were the i n i t i a l f a i l u r e s of Newtonian mechanics to get a good value f o r the speed of sound In a i r ; to g i v e the a c t u a l o r b i t of the moon; and to f i n d a law f o r the f o r c e s between e l e c -t r i c a l charges (Kuhn 1970a, 33, 81; Kuhn 1977, 196-200). The f i r s t problem was l a r g e l y s o l v e d by Laplace (about 1815), the second by C l a l r a u l t (1750), and the t h i r d by Coulomb (1785). 2.3 C r i t i c a l Anomaly and C r i s i s While most ( f e l t ) anomalies prove to be r e s o l v a b l e by the paradigm's (growing) r e s o u r c e s , o c c a s i o n a l l y some prove to r e s i s t the g r e a t e s t e f f o r t s a t a r t i c u l a t i o n by t h a t paradigm's best p r a c t i t i o n e r s (Kuhn 1970a, 5). When t h a t happens, Kuhn says, ( i n the area of the stubborn anomaly) the p r a c t i c e of normal s c i e n c e becomes impossible and the " e x t r a o r d i n a r y s c i e n c e " which takes i t s p l a c e , i n d i c a t e s a " c r i s i s s t a t e . " Because i t i s the s t u b -born anomaly's r e s i s t a n c e to being " a s s i m i l a b l e by paradigm a r t i -c u l a t i o n " ( I b i d . , 35) t h a t provokes the c r i s i s , I s h a l l c a l l such anomalies " c r i t i c a l . " H i s t o r i c a l l y , such " t r a n s l t l o n l s ) from normal to e x t r a o r d i n a r y r e s e a r c h " share the f o l l o w i n g c h a r a c t e r -i s t i c s : "the p r o l i f e r a t i o n of competing a r t i c u l a t i o n s , the w i l -l i n g n e s s to t r y a n y t h i n g , the e x p r e s s i o n of e x p l i c i t d i s c o n t e n t , the recourse to p h i l o s o p h y and the debate over fundamentals..." (Kuhn 1970a, 91. I t a l i c s mine). 1" The process of normal s c i e n c e o f t e n generates "competing a r -t i c u l a t i o n s " — t h a t i s to say, a l t e r n a t i v e ways of extending e s t a -b l i s h e d precedent. For example, Kuhn p o i n t s out that the exten-76 s l o n of the c a l o r i c t h e o r y of heat, from s i t u a t i o n s i n v o l v i n g mixing and changes of s t a t e , to those i n v o l v i n g chemical r e a c -t i o n s , f r i c t i o n and gas compression and a b s o r p t i o n c o u l d have taken p l a c e i n d i f f e r e n t vays. And "many experiments were under-taken to e l a b o r a t e these v a r i o u s p o s s i b i l i t i e s and to d i s t i n g u i s h between them" ( i b i d . , 29).*° In a s t a t e of c r i s i s , however, the number of competing a r t i c u l a t i o n s " p r o l i f e r a t e " because no one of them proves s u f f i c i e n t l y s u c c e s s f u l to provide the new b a s i s of r e s e a r c h i n the problem a r e a . S c i e n t i s t s become " w i l l i n g t o t r y a n y t h i n g " because t h e i r p r e v i o u s exemplars no longer serve t o guide r e s e a r c h . That new openness t o a l t e r n a t i v e procedures even turns s c i -e n t i s t s "to p h i l o s o p h i c a l a n a l y s i s as a d e v i c e f o r u n l o c k i n g the r i d d l e s of t h e i r f i e l d " ( I b i d . , 88). P h i l o s o p h i c a l a n a l y s i s can serve t h i s purpose, Kuhn says, f o r two reasons: (1) the very pro-cess of s e a r c h i n g f o r assumptions can weaken the b e l i e f t h a t o n l y those procedures s a n c t i o n e d by the c u r r e n t paradigm(s) can r e -s o l v e the i s s u e s ; and, more deeply, (2) i t can "expose the para-digm to e x i s t i n g knowledge i n ways t h a t i s o l a t e the r o o t of c r i -s i s with a c l a r i t y u n a t t a i n a b l e i n the l a b o r a t o r y " ( i b i d . , 88). S i n c e , f o r Kuhn, " e x i s t i n g knowledge" i s embodied i n a l l the a r -t i c u l a t i o n s a paradigm has undergone s i n c e i t s i n c e p t i o n , what p h i l o s o p h i c a l a n a l y s i s does on h i s model i s to uncover which a r -t i c u l a t i o n s u l t i m a t e l y come i n t o c o n f l i c t with t h e i r paradigm. A n a l y s i s a c h i e v e s t h a t r e s u l t , Kuhn says, "by t r a n s f o r m i n g f e l t anomaly i n t o c o n c r e t e c o n t r a d i c t i o n " (Kuhn 1977, 264). As exam-p i e s of j u s t t h i s n o t i o n of c r i s i s - i n s p i r e d p h i l o s o p h i c a l a n a l y -s i s , he c i t e s the c l a s s i c a l s c i e n t i f i c "thought experiments" o f -G a l i l e o , E i n s t e i n and Bohr, among o t h e r s . As we s h a l l see, one i n t r i g u i n g outcome of Kuhn's approach i s th a t thought experiments are not p u r e l y c o n c e p t u a l i n import, but a l s o have e m p i r i c a l con-sequences. But t h a t should not be s u r p r i s i n g i f one r e c a l l s Kuhn's arguments, d i s c u s s e d i n chapter one, th a t c h o i c e of con-cepts i s not t h e o r e t i c a l l y , and thus a l s o not e m p i r i c a l l y , neu-t r a l . 2.4 Thought Experiment: I s o l a t i n g the Root of C r i s i s Kuhn says t h a t , on the t r a d i t i o n a l p h i l o s o p h i c a l understand-ing of the s c i e n t i f i c r o l e of thought experiments, t h a t "func-t i o n . . . i s to a s s i s t i n the e l i m i n a t i o n of p r i o r c o n f u s i o n by f o r -c i n g the s c i e n t i s t to re c o g n i z e c o n t r a d i c t i o n s t h a t had been i n -herent i n h i s way of t h i n k i n g from the s t a r t " (Kuhn 1977, 242. I t a l i c s mine). In other words, on t h a t t r a d i t i o n a l understand-in g , s c i e n t i s t s who o r i g i n a l l y formulate concepts t h a t are l a t e r shown t o be p r o b l e m a t i c , i n i t i a l l y make conceptual e r r o r s t h a t were, even a t t h e i r f o r m u l a t i o n , eminently a v o i d a b l e . The t r a d i -t i o n a l view l e a d s , Kuhn argues, to such i m p l a u s i b l e suggestions as t h a t A r i s t o t l e , one of the most b r i l l i a n t l o g i c i a n s , made extremely elementary conceptual e r r o r s ( i b i d . , 253). The d i s c u s s i o n i n s e c t i o n 2.1 of s c i e n t i f i c g e n e r a l i z a t i o n s suggests t h a t one reason the t r a d i t i o n a l p h i l o s o p h i c a l paradigm 78 i s d r i v e n to such i m p l a u s i b l e c l a i m s i s t h a t i t presupposes t h a t concepts have f u l l y determinate meanings a t i n c e p t i o n . But on Kuhn's view s c i e n t i f i c concepts a c q u i r e meaning over time by a r -t i c u l a t i o n t o n o v e l , but r e l e v a n t c i r c u m s t a n c e s . Thus depending on how the world i s , t h a t a r t i c u l a t i o n c o u l d introduce a p p l i c a -t i o n s t h a t l a t e r come to c o n f l i c t with e a r l i e r ones. The f a l s i -f i c a t i o n of the e m p i r i c a l content embodied i n the e a r l i e r usage would then a l s o r e v e a l a conceptual c o n f l i c t between the e a r l i e r and the l a t e r a r t i c u l a t i o n s . But i f the world to which the s c i -e n t i f i c concepts were a r t i c u l a t e d had turned out to be d i f f e r e n t , then the e m p i r i c a l commitments they came to embody might not have l e d t o c o n f l i c t . So f o r Kuhn the p r o b l e m a t i c concepts have no " i n t r i n s i c d e f e c t " ; they j u s t f a i l " t o f i t the f u l l f i n e s t r u c -t u r e of the [ a c t u a l ] world" (Kuhn 1977b, 258). Another, more s u b t l e source of the t r a d i t i o n a l view's con-s t r u a l of thought experiments i s i t s presumption of commensura-b i l i t y : i t s b e l i e f that a l l p o s s i b l e e m p i r i c a l t h e o r i e s are f o r -mulable w i t h i n a common system of s t r i c t l y l o g i c a l and n o n - l o g i -c a l c o n cepts. Hence, on t h a t r e c e i v e d paradigm, the l o g i c a l con-t r a d i c t i o n s r e v e a l e d by thought experiments w i l l be s o l e l y a t t r i -b u t a b l e to a m i s a p p l i c a t i o n of the common l o g i c a l apparatus a t the time when the now problematic concepts were f i r s t formulated. Which means, of course, t h a t no e m p i r i c a l c o n c l u s i o n s can be drawn from the r e s o l u t i o n of those c o n t r a d i c t i o n s . N e v e r t h e l e s s , i n o p p o s i t i o n to the t r a d i t i o n a l view, much p h i l o s o p h i c a l , h i s t o r i c a l , and p s y c h o l o g i c a l evidence suggests both t h a t concepts are seldom, i f ever, completely determinate, and t h a t s c i e n t i f i c t h e o r i e s are o f t e n incommensurable. Such indeterminacy and incommensurability imply t h e r e f o r e t h a t the c o n t r a d i c t i o n s which o f t e n s u r f a c e i n p e r i o d s of c r i s i s need not a l s o have been present a t the paradigm's i n c e p t i o n . They a l s o suggest t h a t those a r t i c u l a t i o n - g e n e r a t e d c o n t r a d i c t i o n s can r e v e a l the ways i n which the system of concepts i s (now) mal-formed, and thus p r o v i d e a h i n t t o the r e s o l u t i o n of those con-t r a d i c t i o n s ( c . f . Kuhn 1977b, 264). That r e s o l u t i o n w i l l c l e a r l y i n v o l v e a r e o r i e n t a t i o n of the previous paradigm's c o n c e p t u a l r e l a t i o n s , and so w i l l c r e a t e a completely new, hence incommen-s u r a b l e , c o n c e p t u a l system whose a r t i c u l a t i o n w i l l embody p r e v i -o u s l y i n a c c e s s i b l e e m p i r i c a l knowledge. Kuhn's c o n c e p t i o n of thought experiments has a d i s t i n c t l y " d i a l e c t i c a l " c h a r a c t e r . Marx Wartofsky d e f i n e s d i a l e c t i c as "the d e s t r u c t i o n of an argument [ t h e s i s ] , or b e t t e r y e t , a u n i -v e r s a l p r o p o s i t i o n by an argument—by the d i s c o v e r y of a c o n t r a -d i c t i o n [ a n t i t h e s i s ] i n the views of those who propound i t " (War-t o f s k y 1977, 13). That d e s t r u c t i o n however o f t e n has a "con-s t r u c t i v e consequence" or " s y n t h e s i s " ; i t o f t e n p i n p o i n t s the problematic commitments of the t h e s i s thus showing how to con-s t r u c t " a l t e r n a t i v e s to the f a i l e d premises" ( i b i d . , 14). S i m i -l a r l y , Kuhn says t h a t the famous ( d i a l e c t i c a l ) thought e x p e r i -ments of " G a l i l e o , E i n s t e i n , Bohr and o t h e r s " (Kuhn 1970a, 88) not o n l y "transformed f e l t anomaly i n t o concrete c o n t r a d i c t i o n " ; they s i m u l t a n e o u s l y "provided the c l u e s to s e t the s i t u a t i o n 80 r i g h t " (Kuhn 1977b, 264). As we s h a l l see In chapter t h r e e , thought experiments a l s o prove t h a t the new " s y n t h e s i s " must be incommensurable with the previous " t h e s i s . " In order both to i l l u s t r a t e the concept of a thought e x p e r i -ment and to e x p l i c a t e Kuhn's account of them, I s h a l l now g i v e a b r i e f a n a l y s i s of two t y p i c a l members of t h i s c l a s s : P i a g e t ' s vo-lume c o n s e r v a t i o n and G a l i l e o ' s v e l o c i t y thought e x p e r i m e n t s . " 2.4.1 A P i a g e t i a n Thought Experiment: Volume c o n s e r v a t i o n In the P i a g e t i a n experiment, t y p i c a l l y a young c h i l d (four to f i v e y e a rs o l d ) 1 7 w i l l c l a i m t h a t there i s more water i n a t a l l , narrow g l a s s than i n a s h o r t , wide g l a s s — e v e n though the c h i l d knows t h a t both g l a s s e s have been f i l l e d with water poured from two, i d e n t i c a l r e c e p t a c l e s with equal l e v e l s of water i n them, and whose volumes he o r i g i n a l l y pronounced e q u a l . What i s even more s u r p r i s i n g than the c h i l d ' s i n i t i a l response i s t h a t a t t h i s stage i t i s normally impossible to convince him t h a t he has made a mistake. Thus, when one pours the water back i n t o the o r i g i n a l v e s s e l s , where the water l e v e l s are once a g a i n apparent-l y e q u a l , the c h i l d then r e a d i l y agrees t h a t both v e s s e l s now have the same volume. C l e a r l y , a t t h i s stage, the c h i l d ' s c r i -t e r i o n of comparative volume f o r l i q u i d s i s something l i k e r e l a -t i v e h e i g h t . 1 8 A s h o r t time (about a year) l a t e r , however, while our t y p i -c a l c h i l d w i l l , when confronted by the same s i t u a t i o n , once again 81 c l a i m that the t a l l , narrow g l a s s has more water i n i t , he now d i m l y recognizes t h a t h i s r e p l y i s anomalous. And when he i s s t i l l o l d e r , he immediately c l a i m s t h a t both the t a l l , narrow and the s h o r t , wide g l a s s e s have equal volumes of water i n them. Ob-v i o u s l y , something has happened t o the c h i l d i n the i n t e r i m be-tween the f i r s t and l a t e r t r i a l s t o change h i s p e r c e p t i o n of the experimental s i t u a t i o n . That i s , i t i s c l e a r t h a t somewhere alon g the way, the c h i l d has a c q u i r e d the concept t h a t l i q u i d volumes do not change merely through t r a n s f e r e n c e from one v e s s e l to another. P l a g e t ' s thought-experimental s i t u a t i o n i s p e r f e c t l y designed to f o r c e a c o n f l i c t between the c h i l d ' s o r i g i n a l n o t i o n and t h a t l a t e r development. Here i s one p l a u s i b l e Kuhnian r e c o n -s t r u c t i o n of the c h i l d ' s p r o g r e s s . I n i t i a l l y , our t y p i c a l c h i l d p robably learned the concept of r e l a t i v e volume i n simple s i t u a t i o n s where the v e s s e l s were of q u i t e s i m i l a r s i z e and shape. In other words, the c h i l d ' s o r i g i -n a l "paradigms" of r e l a t i v e volume l i k e l y o n l y took i n t o account circumstances i n which v e s s e l shape and s i z e were roughly I d e n t i -c a l . I t i s q u i t e easy to see how those paradigms co u l d then have been " a r t i c u l a t e d " to develop the r e l a t e d concept of volume con-s e r v a t i o n through l i q u i d t r a n s f e r e n c e . For the c h i l d would a l -most c e r t a i n l y a l s o have a c q u i r e d t h a t e x t e n s i o n of h i s o r i g i n a l paradigm In f a m i l i a r s i t u a t i o n s where the v e s s e l s were of q u i t e s i m i l a r s i z e and shape. N o t i c e t h a t , on Kuhn's view, the a r t i c u -l a t e d paradigm w i l l not yet embody a c o n t r a d i c t i o n ; f o r the a r t i -c u l a t i o n has not yet i t s e l f been extended to s i t u a t i o n s where 82 volume t r a n s f e r e n c e takes p l a c e between v e s s e l s of d i f f e r e n t s i -zes and shapes. Moreover, Kuhn wants to argue t h a t even a f t e r such a f u r t h e r e x t e n s i o n of the o r i g i n a l paradigm, the c h i l d ' s "conceptual apparatus" s t i l l does not y e t c o n t a i n a c o n t r a d i c -t i o n . That i s because l i f e need never present a c h i l d with the more t r i c k y s i t u a t i o n c o n t r i v e d by P i a g e t , and because i t a l s o seems unreasonable to demand t h a t when h i s concepts were formula-ted he must have had such s i t u a t i o n s i n mind ( c . f . Kuhn 1977b, 254). In other words, i t seems u t t e r l y " P i c k w i c k i a n " (Kuhn 1977d, 288) t o imagine t h a t , when forming concepts, they be de-s i g n e d t o "be a p p l i c a b l e t o any and every s i t u a t i o n t h a t might c o n c e i v a b l y a r i s e i n any p o s s i b l e world" (Kuhn 1977b, 254). N e v e r t h e l e s s , t h i s world a p p a r e n t l y does o f t e n present c h i l -dren with P i a g e t ' s s o r t s of anomalous circumstances, f o r they i n v a r i a b l y go on to form the more s o p h i s t i c a t e d a d u l t n o t i o n of r e l a t i v e volume. In f a c t , a t a c e r t a i n age the P i a g e t i a n e x p e r i -mental s i t u a t i o n can i t s e l f be used to achieve t h a t end." Once the c h i l d r e c o g n i z e s t h a t t r a n s f e r r i n g l i q u i d s i n such circum-stances doesn't change t h e i r volume, but t h a t h i s paradigm of r e l a t i v e h e i g h t says t h a t the comparative volumes have changed, he i s f o r c e d t o r e - e v a l u a t e the r e l a t i v e importance of h i s o r i g i -n a l paradigm and i t s l a t e r a r t i c u l a t i o n s . Often now the c o n s e r -v a t i o n c r i t e r i o n takes p r i d e of p l a c e , and t h a t of r e l a t i v e h e ight becomes s u b o r d i n a t e . 2 0 T h i s conceptual r e o r g a n i z a t i o n i s t h e r e f o r e a r e v o l u t i o n ; f o r even those s i t u a t i o n s adequately handled by the p r e v i o u s paradigm are now construed q u i t e d i f f e r -83 e n t l y . Thus because the thought experiment transforms the ways i n which " e x i s t i n g [ e m p i r i c a l ] knowledge" about such s i t u a t i o n s i s seen, i t t h e r e f o r e a l s o g i v e s new e m p i r i c a l i n f o r m a t i o n . 2.4.2 G a l i l e o ' s Thought Experiment: T o t a l Motion and A c c e l e r a t i o n In the f o l l o w i n g thought experimental s i t u a t i o n , G a l i l e o shows t h a t an a r t i c u l a t e d form of A r i s t o t l e ' s concept of v e l o c i t y l e a d s t o c o n t r a d i c t i o n s . A r i s t o t l e t r e a t e d the n o t i o n of change of p l a c e , t o which the term "motion" i s now u s u a l l y r e s t r i c t e d , on analogy with q u a l i t a t i v e changes of s t a t e l i k e h e a l i n g or burning (Physics, Bk. V ) : thus he says t h a t "every change i s from something to s o m e t h i n g — a s the word i t s e l f metabole i n d i c a t e s " (Physics, Bk V, 224b35-225al; Kuhn 1977b, 246). That choice of d e f i n i t i o n leads A r i s t o t l e to make an e m p i r i c a l commitment: he makes e s s e n t i a l use of the b e g i n n i n g and t e r m i n a l p o i n t s of a change of p l a c e to determine the q u a n t i t y and speed of that mo-t i o n . For example, he c l a i m s t h a t "The q u i c k e r of two t h i n g s t r a v e r s e s a g r e a t e r magnitude i n an equal time, an equal magni-tude i n l e s s time, and a g r e a t e r magnitude i n l e s s time" ( i b i d . , 232a25-7). A r i s t o t l e ' s n o t i o n of v e l o c i t y i s thus v e r y much l i k e the modern concept of average v e l o c i t y ; however, i t i s not i d e n t i c a l with the modern concept because i t embodies no d i s t i n c t i o n be-tween average and instantaneous v e l o c i t y . The l a t t e r concept, s i n c e G a l i l e o , has been seen to be necessary to d e a l with c o n t i -84 n u o u s l y - v a r y i n g motions or a c c e l e r a t i o n s . In f a c t , because he doesn't r e c o g n i z e the modern d i s t i n c t i o n s , A r i s t o t l e formulates p h y s i c a l laws l i k e the f o l l o w i n g : " I f , then, A the movent have moved B [the t h i n g moved] a [ t o t a l ] d i s t a n c e C i n a [ t o t a l ] time D, then i n the same time the same f o r c e A w i l l move 1/2 B twice the d i s t a n c e C, and i n 1/2 D i t w i l l move 1/2 B the whole d i s -tance C: f o r thus the r u l e s of p r o p o r t i o n w i l l be p r e s e r v e d " ( i b i d . , 249b30-250a4; Kuhn 1977, 256). Such a law, while i t seems i n i t i a l l y p e c u l i a r from the s t a n d p o i n t of Newtonian phy-s i c s , i s i n f a c t p e r f e c t l y adequate f o r those s i t u a t i o n s t h a t do not i n v o l v e a c c e l e r a t i o n . 2 1 While A r i s t o t l e was largely unaware of s i t u a t i o n s i n v o l v i n g a c c e l e r a t i o n s , he o c c a s i o n a l l y d i m l y recognized the presence of changing motions. For example, when he d i s t i n g u i s h e s " n a t u r a l " or unforced from " v i o l e n t " or f o r c e d motions, A r i s t o t l e makes no mention of t h e i r endpoints: "But whereas the v e l o c i t y of t h a t which comes to a s t a n d s t i l l seems always to i n c r e a s e , the v e l o c i -t y of t h a t which i s c a r r i e d v i o l e n t l y seems always to decrease" (Physics, 230b23-5; Kuhn 1977, 247). In other words, A r i s t o t l e seems abl e to compare v e l o c i t i e s without h i s standard c r i t e r i a . N e v e r t h e l e s s , as the need f o r the G a l i l e a n thought experiment l a t e r showed, he probably presumed t h a t e v e n t u a l l y such motions cou l d be a s s i m i l a t e d by h i s motion-as-change-of-state paradigm. Before G a l i l e o c o u l d c o n v i c t the A r i s t o t e l i a n concept of i n -adequacy, however, t h a t concept f i r s t had to be f u r t h e r a r t i c u l a -ted so t h a t i t c o u l d begin to formulate more p r e c i s e l y s i t u a t i o n s 85 i n v o l v i n g c o n t i n u o u s l y - c h a n g i n g motions. That a r t i c u l a t i o n was achieved by the impetus t h e o r i s t s of the middle ages through t h e i r n o t i o n of l a t i t u d e of forms. That n o t i o n allowed a d i s -t i n c t i o n between the " t o t a l " v e l o c i t y of a motion and the " i n t e n -s i t y " of a v e l o c i t y a t any p o i n t d u r i n g t h a t motion (Kuhn 1977, 247). While t h i s n e o - A r i s t o t e l i a n d i s t i n c t i o n i s c l e a r l y a l o t c l o s e r t o the p o s t - G a l i l e a n one, as Kuhn p o i n t s out, i t s t i l l r e q u i r e s t h a t the d i s t a n c e s and times ("extensions") of the com-pared motions are e q u a l . I t i s j u s t t h i s a r t i c u l a t i o n of the o r i g i n a l A r i s t o t e l i a n paradigm t h a t G a l i l e o shows leads t o con-t r a d i c t i o n s i n the thought-experimental s i t u a t i o n he c o n t r i v e s i n h i s Dialogue Concerning the Tvo chief World Systems. In t h a t f i c t i t i o u s d i a l o g u e , G a l i l e o asks two A r i s t o t e l i a n s to imagine a s i t u a t i o n i n which two bodies s l i d e without f r i c t i o n down two planes (CB and CA) from a common p o i n t (C). (Note t h a t CA i s s i g n i f i c a n t l y longer than CA so t h a t the extensions of the two motions are not equal.) C 86 G a l i l e o f i r s t asks h i s A r i s t o t e l i a n opponents t o agree t h a t vhen the o b j e c t s s l i d e down CB and CA and reach t h e i r r e s p e c t i v e t e r m i n i , they w i l l have the same impetus or I n t e n s i t y of motion: i n other words, they w i l l have a c q u i r e d s u f f i c i e n t v e l o c i t y t o r e t u r n them to t h e i r common s t a r t i n g p o i n t , C. He then asks them which of the two bodies i s the f a s t e r . T h e i r f i r s t ( n a t u r a l ) r e -sponse i s to say t h a t the o b j e c t s l i d i n g down CB i s f a s t e r : i t gets t o i t s terminus i n l e s s time and has g r e a t e r apparent v e l o c -i t y a l o n g the way. N e v e r t h e l e s s , t h a t response seems Incompati-b l e w i t h the a r t i c u l a t e d c r i t e r i o n of the impetus t h e o r i s t s ; f o r t h a t c r i t e r i o n says t h a t s i n c e both bodies s t a r t e d v i t h the same I n i t i a l v e l o c i t y (at r e s t ) and ended v i t h the same f i n a l v e l o c i -t y , t h e i r mean speed must be the same. G a l i l e o then suggests t h a t the problem a r i s e s from t r y i n g t o compare tvo motions v i t h d i f f e r e n t s p a t i a l e x t e n s i o n s . So the s o l u t i o n t o the A r i s t o t e l i a n dilemma might be to choose equal e x t e n s i o n s , say, by o n l y r e q u i r i n g the body on the i n c l i n e d plane to t r a v e l a d i s t a n c e CB. U n f o r t u n a t e l y , t h i s o n l y vorsens the problem. Thus, i t turns out t h a t , depending on where one l a y s out the l e n g t h CB on the plane CA, the A r i s t o t e l i a n c r i t e r i o n permits the v e l o c i t y on the i n c l i n e t o be a t once slower, f a s t e r , and the same. For example, i f the l e n g t h CB i s measured on CA from C, the v e r t i c a l l y moving body v i l l reach i t s terminus i n l e s s time; i f t h a t l e n g t h i s measured up from the p o i n t A, then the body on CA w i l l reach i t s terminus f i r s t ; and i f that l e n g t h begins a t some well-chosen p o i n t a l o n g CA between C and A, both 87 bodies w i l l reach t h e i r t e r m i n i a t the same time. Thus the ap-p l i c a t i o n of the a r t i c u l a t e d A r i s t o t e l i a n concept t o a comparison of the v e l o c i t y of a c c e l e r a t i n g bodies r e s u l t s i n c o n t r a d i c t i o n . G a l i l e o used t h i s thought experiment t o b i f u r c a t e the o l d e r A r i s t o t e l i a n n o t i o n i n t o the modern kinematic concepts of i n s t a n -taneous and average v e l o c i t y . In G a l i l e o ' s r e f o r m u l a t i o n , the concept of instantaneous v e l o c i t y i s the more b a s i c s i n c e i t can compare both constant and changing speeds. Thus a f t e r G a l i l e o ' s c o n c e p t u a l r e v o l u t i o n , the concept of instantaneous v e l o c i t y becomes the paradigm, and A r i s t o t l e ' s " q u a l i t a t i v e change" n o t i o n i s transformed i n t o the modern concept of average v e l o c i t y . 2.5 Morals In t h i s chapter I have t r i e d t o show t h a t Kuhn's account does no gre a t v i o l e n c e to t r a d i t i o n a l s c i e n t i f i c v a l u e s . Thus h i s v e r s i o n of how paradigms g i v e meaning t o s c i e n t i f i c concepts does not make t h e o r i e s c o m p l e t e l y Immune to e m p i r i c a l evidence; nor, t h e r e f o r e , does i t make them r a t i o n a l l y incomparable. For when an experimental anomaly becomes c r i t i c a l , the new the o r y which "renders the anomaly l a w l i k e " (Kuhn 1970a, 97) by concep-t u a l r e v o l u t i o n can be seen to be a b e t t e r t h e o r y than the one I t supplants--even though the two t h e o r i e s employ d i f f e r e n t con-c e p t s . For example, even though Dalton's new chemical t h e o r y em-ployed d i f f e r e n t concepts from B e r t h o l l e t ' s a f f i n i t y theory, no common concepts were r e q u i r e d to show t h a t Dalton's view was 88 b e t t e r . Chemists were a b l e to see directly t h a t Dalton's con-cepts d e a l t w i t h chemical phenomena i n a b e t t e r manner than Ber-t h o l l e t ' s because, among other t h i n g s , those concepts avoided B e r t h o l l e t ' s anomalous c o n f l a t i o n of s e e m i n g l y - d i s t i n c t substan-ces . I t i s o f t e n d i f f i c u l t t o convince those who s t r o n g l y a s s o c i -ate p a r t i c u l a r f o r m u l a t i o n s of some s u b j e c t with t h a t s u b j e c t i t -s e l f t h a t when one c h a l l e n g e s those f o r m u l a t i o n s , he i s not thereby n e c e s s a r i l y c h a l l e n g i n g the b a s i c i n t e g r i t y of t h a t sub-j e c t . For example, because e t h i c s i s , f o r many, i n e x t r i c a b l y a s s o c i a t e d with the idea of God, d e n i a l s of God's e x i s t e n c e o f t e n lead those people t o e x c l a i m t h a t " i f God i s dead, e v e r y t h i n g i s p e r m i t t e d " (Dostoyevsky, The Brothers Karamazov). S i m i l a r l y , i n the p h i l o s o p h y of s c i e n c e , some p h i l o s o p h e r s conclude, whether i n alarm (e.g. S c h e f f l e r ) or g l e e (e.g. Feyerabend), t h a t I f t r a d i -t i o n a l , formal canons of r a t i o n a l i t y don't a p p l y t o s c i e n c e , then "anything goes" (Feyerabend 1975, 23-8). But t h a t c o n c l u s i o n doesn't f o l l o w anymore than the p r e v i o u s one. Thus j u s t as So-c r a t e s showed ( i n the Euthyphro) t h a t the concepts of e t h i c s and God must be d i s s o c i a t e d to a v o i d c o n t r a d i c t i o n , so too, I have argued i n t h i s c h a p t e r , must the concepts of e m p i r i c a l r a t i o n a l -i t y be defeased from formal accounts of them to a v o i d making out as i r r a t i o n a l our epitome of r a t i o n a l i t y , s c i e n c e . F i n a l l y , i n c o m m e n s u r a b i l i t y should not be seen as "methodo-l o g i c a l l y u n d e s i r a b l e " but as a f a c t of l i f e . Anyone who has taken p a r t i n " p o i n t l e s s " p o l i t i c a l , s c i e n t i f i c or r e l i g i o u s 89 debates knows t h a t incommensurability i s not a problem to be wished away but one needing s e r i o u s p h i l o s o p h i c a l a t t e n t i o n . But any methodology t h a t suggests such a n t a g o n i s t s r e a l l y share a common viewpoint cannot h e l p but make an opponent who di s a g r e e s about e v e r y t h i n g seem j u s t p l a i n s t u p i d . I t f o l l o w s t h a t t r a d i -t i o n a l eplstemology i s inadequate t o the problem of incommensura-b i l i t y ; f o r i t n a t u r a l l y c o n f l a t e s (Carnap: " e x t e r n a l " ) questions about which standards should govern a s u b j e c t with ( " i n t e r n a l " ) q u e s t i o n s about t h a t s u b j e c t under a p a r t i c u l a r standard. T h i s f a i l i n g suggests t h a t i f i n s t e a d " p a r t i c i p a n t s i n Incommensurable p o i n t s of view" are a l e r t to the "fundamental" nature of t h e i r debate, t h a t both comprehension and r a t i o n a l r e s o l u t i o n of t h e i r c o n f l i c t may be p o s s i b l e . 90 Notes 1. T h i s does not mean no p a i r of t h e o r i e s c o u l d be f o r m a l i z e d — i . e . w r i t t e n out i n l o g i c a l n o t a t i o n , but o n l y t h a t such f o r m a l i -z a t i o n , i f a c h i e v e d , would be m i s l e a d i n g because imp l y i n g t h a t c e r t a i n of the t h e o r i e s ' concepts were a b s o l u t e l y n o n l o g i c a l . 2. A g e n e r a l i z a t i o n v o u l d be ( l a r g e l y ) formal i n c o n t e n t , f o r ex-ample, i f i t f u n c t i o n e d (more) as a definition or r u l e d e t e r m i n -in g r e l a t i o n s among i t s c o n s t i t u e n t concepts than as an e m p i r i -c a l l y - t e s t a b l e l a v . (I have p l a c e d " l a r g e l y " and "more" i n p a r -entheses to accommodate Kuhn's b e l i e f , d i s c u s s e d i n chapter one (see §1.1.1), t h a t even d e f i n i t i o n s are not f u l l y e i t h e r l o g i c a l -l y or e m p i r i c a l l y n e u t r a l . ) 3. As ve s h a l l see In s e c t i o n 2.2, t h a t s c i e n t i s t s s t r i v e t o make t h e i r g e n e r a l i z a t i o n s come out t r u e does not make t h e i r t h e o r i e s e m p i r i c a l l y i n c o r r i g i b l e or t a u t o l o g i c a l l y t r u e (see Lodynski 1982, 98). For o c c a s i o n a l l y "a normal p r o b l e m . . . r e s i s t s the r e i t e r a t e d onslaught of the a b l e s t members of the group v i t h i n vhose competence i t f a l l s " (Kuhn 1970a, 5). Such f a i l u r e s of e x p e c t a t i o n then " l e a d the p r o f e s s i o n a t l a s t to a nev s e t of commitments" ( i b i d . , 6 ) . 4. For example, tvo hundred years a f t e r Nevton p u b l i s h e d the P r i n c i p i a , the p h y s i c i s t E r n s t Mach v o i c e d a common ( i f h i s t o -r i c a l l y i n a c c u r a t e ) sentiment vhen he claimed t h a t Nevton "com-p l e t e d the formal e n u n c i a t i o n of the mechanical p r i n c i p l e s nov g e n e r a l l y accepted. Since h i s time no e s s e n t i a l l y nev p r i n c i p l e has been s t a t e d " ( E n c y c l o p e d i a of P h i l . s.v. "Nevtonlan Mechanics and Mechanical E x p l a n a t i o n " ) . 5. Kuhn does not b e l i e v e , hovever, t h a t because Nevton's P r i n -c i p i a d i d not i n f a c t a l v a y s g i v e a c l e a r p i c t u r e of every s i t u a -t i o n t h a t l a t e r seemed r e l e v a n t , t h a t the P r i n c i p i a vas p o o r l y thought out or t h a t i t s concepts were, i n any r e a l i s t i c sense, impr e c i s e . J u s t the o p p o s i t e . He says t h a t "no other vork knovn to the h i s t o r y of s c i e n c e has s i m u l t a n e o u s l y permitted so l a r g e an i n c r e a s e i n both the scope and p r e c i s i o n of r e s e a r c h " (Kuhn 1970a, 30). 6. In another paper, "Second Thoughts on Paradigms" (Kuhn 1977c), Kuhn d e s c r i b e s B e r n o u l l i ' s achievement i n more d e t a i l . \ 91 Determine the descent of the c e n t e r of g r a v i t y of the water i n tank and j e t d u r i n g an i n f i n i t e s i m a l i n t e r v a l of time. Next imagine that each p a r t i c l e of water afterwards moves s e p a r a t e l y upward to the maximum he i g h t o b t a i n a b l e with the v e l o c i t y i t possessed a t the end of the i n t e r v a l of descent. The ascent of the c e n t e r of g r a v i t y of the separate p a r t i -c l e s must then equal the descent of the c e n t e r of g r a v i t y of the water i n tank and j e t . From t h a t view of the problem, the long-sought speed of e f f l u x f o l l o w e d a t once. (Kuhn 1977a, 306). While the resemblance of B e r n o u l l i ' s a p p l i c a t i o n to Huy-ghen's can be immediately r e c o g n i z e d — o n c e Bernoulli has drawn our attention to them as a possible construal—there seem no ob-v i o u s "more b a s i c " r u l e s he c o u l d have used to get h i s r e s u l t ; moreover, any of the other attempts he made or c o u l d have made befor e t h i s one i n f a c t succeeded, c o u l d a l s o have been compati-b l e with Huyghen's model and c o u l d a l s o have proved s u c c e s s f u l . 7. T h i s i s not Kuhn's complete view. Thus he c o n s i d e r s paradigms to be o n l y a p a r t of what he c a l l s the t o t a l " d i s c i p l i n a r y ma-t r i x " which i n c l u d e s such other elements a s : symbolic g e n e r a l i z a -t i o n s , b e l i e f s i n p a r t i c u l a r models, and shared values (Kuhn 1970a, 181-7). N e v e r t h e l e s s , Kuhn c l a i m s t h a t the idea of a c h i e v i n g concrete problem solutions by d i r e c t resemblance i s "the most n o v e l " and " p h i l o s o p h i c a l l y . . . d e e p e s t " aspect of h i s view ( i b i d . , 175, 187). 8. Kuhn says here t h a t "...the c o n c r e t e s c i e n t i f i c achievement [ i s ] a fundamental u n i t . . . a u n i t t h a t cannot be f u l l y reduced to l o g i c a l l y atomic components which might f u n c t i o n i n i t s s t e a d . " 9. A l s o , s i n c e c e r t a i n " p r o f e s s i o n a l s u b s p e c i a l t i e s " make d i f -f e r e n t uses of the same p a r a d i g m s — l i k e p h y s i c i s t s and chemists i n quantum m e c h a n i c s — t h e paradigm " i s not the same paradigm f o r them a l l " (Kuhn 1970a, 50). Hence, at the p o i n t where t h e i r ap-p l i c a t i o n s d i v e r g e , those s u b s p e c i a l t i e s c o u l d a l s o be s a i d to be incommensurable, though not u s u a l l y a l s o i n c o m p a t i b l e . 10. R e c a l l t h a t Kuhn says: "From the viewpoint of t h i s essay [New-t o n i a n and E i n s t e i n i a n dynamics] are fundamentally incompatible ( i . e . l o g i c a l l y incompatible and Incommensurable) i n the sense i l l u s t r a t e d by the r e l a t i o n of [ B e r t h o l l e t i a n to P r o u s t i a n chem-i s t r y ] : E i n s t e i n ' s theory can be accepted o n l y with the r e c o g n i -t i o n t h a t Newton's was wrong" (Kuhn 1970a, 98). 92 11. Kuhn's c r i t i c s o f t e n complain t h a t h i s examples don't show t h a t E i n s t e i n and Newton had d i f f e r e n t concepts of mass; i n s t e a d those examples show merely t h a t they had d i f f e r e n t t h e o r i e s of mass (e.g. Musgr-ave 1978, p339). As we saw i n chapter one, how-ever, Kuhn b e l i e v e s t h a t f o r such cases one cannot draw a sharp d i s t i n c t i o n between t h e o r e t i c a l and conceptual d i f f e r e n c e s , here would be a p u r e l y t h e o r e t i c a l d i f f e r e n c e o n l y i f both theo-r i e s were: agreed on v h i c h t h i n g s had mass and how to measure that mass; but d i s a g r e e d on p r o p e r t i e s of t h a t mass t h a t both recognized as c o n c e p t u a l l y p o s s i b l e . 12. I i n c l u d e " g e n u i n e l y " i n parentheses to exclude the idea of f a l s i f i c a t i o n of " a u x i l l i a r y hypotheses." T h i s l a t t e r i d e a , motivated by the work of P i e r r e Duhera, i s s i m i l a r to Kuhn's no-t i o n s of anomaly i n a r t i c u l a t i o n . I t d i f f e r s from (my recon-s t r u c t i o n of) Kuhn's p o s i t i o n i n implying t h a t a theory's a u x i l -l i a r y hypotheses are independent of the theory. But f o r Kuhn, choosing a theory's a r t i c u l a t i o n s help to i n t e r p r e t i t . 13. I t would be mistaken here t o conclude t h a t Kuhn's account confuses the mere f a c t t h a t s c i e n t i f i c t h e o r i e s are o f t e n v e r y r e s i s t a n t t o f a l s i f i c a t i o n v i t h normative, l o g i c a l c r i t e r i a t h a t those t h e o r i e s should attempt to meet (e.g. Lakatos 1970, 177; c . f . Kuhn, 1970b, 233, 235-7). For example, i t vould be a misun-d e r s t a n d i n g of Kuhn's p o s i t i o n to suggest t h a t s c i e n t i s t s vho behave i n the vay he d e s c r i b e s , dishonestly endulge i n "conven-t i o n a l i s t strategems" (Popper 1968, 37) or "ad hoc r e s c u e s " (Giere 1984, 163-4) to p r o t e c t t h e i r t h e o r i e s from t h e i r r e a l consequences. Those r e j o i n d e r s to Kuhn vould be moot o n l y i f s c i e n t i f i c concepts were (or c o u l d be) a c t u a l l y determinate p r i o r to t h e i r a p p l i c a t i o n to u nforseen, but r e l e v a n t , circumstances. For only thus c o u l d l o g i c a l c r i t e r i a be i d e n t i f i e d with norms of s c i e n t i f i c r e a s o n i n g . S i n c e , as we have seen, Kuhn argues t h a t s c i e n t i f i c concepts are seldom, i f ever, determinate i n advance, he r e j e c t s the view t h a t l o g i c a l c r i t e r i a can serve t h a t norma-t i v e r o l e . Hence, f o r Kuhn, the response of s c i e n t i s t s to anoma-lous s i t u a t i o n s o f t e n shows t h a t i n such cases the a p p l i c a t i o n of t h e i r paradigms are not y e t determinate} i t does not imply t h a t t h e i r a p p l i c a t i o n i s a l l too c l e a r , and t h a t , t h e r e f o r e , the s c i e n t i s t s i n q u e s t i o n are l a c k i n g i n i n t e l l e c t u a l courage or honesty. 14. I n c i d e n t a l l y , j u s t because, as i s obvious, the s t a t e s of c r i -s i s Kuhn d e s c r i b e s have an i r r e d e e m a b l y - p s y c h o l o g i c a l aspect to them, i t should not be i n f e r r e d t h a t the concept of crisis can have no e p i s t e m o l o g i c a l import. R e c a l l t h a t f o r Kuhn the process by which paradigms are a r t i c u l a t e d and, t h e r e f o r e , by which know-ledge i s a c q u i r e d , i s u l t i m a t e l y based on the unmediated judge-ments of those paradigm's p r a c t i t i o n e r s . Hence, a breakdown i n 93 t h a t p r o c e s s , Is a breakdown i n the normal e p l s t e m o l o g l c a l pro-c e s s : hence, such c r i s e s are eplstemologlcal c r i s e s . 15. Another p o s s i b l e example of the need to choose among compet-ing a r t i c u l a t i o n s of the same paradigm i s Newton's p a r t i c u l a t e and Huyghen's wave t h e o r i e s of l i g h t . Thus while the l i g h t - p a r -t i c l e v e r s i o n of the o r i g i n a l P r i n c i p i a n - m e c h a n i c a l paradigm to the behaviour of l i g h t was perhaps more n a t u r a l , the more accu-r a t e wave t h e o r y was a l s o seen as an a r t i c u l a t i o n of the P r i n c i p -i a . Thomas Young's t w i n - s i l t experiments were Instrumental i n choosing the wave theory over the p a r t i c l e model (Mason 1962, 468-70). Of course, as Kuhn notes, from the more r e s t r i c t e d p o i n t of view of the predecessor p a r t i c l e - o p t i c s (sub) paradigm, wave t h e o r y was q u i t e r e v o l u t i o n a r y (Kuhn 1970a, 86). 16. By " P i a g e t , " I am r e f e r r i n g to the famous Swiss c h i l d psycho-l o g i s t and " g e n e t i c e p l s t e m o l o g l s t , " Jean P i a g e t . I came a c r o s s t h i s P i a g e t i a n thought experiment through my wife and have t r i e d i t out s e v e r a l times with young c h i l d r e n , always with P i a g e t ' s r e s u l t s . My d i s c u s s i o n of the G a l i l e a n thought experiment i s based on t h a t i n Kuhn's paper "A F u n c t i o n f o r Thought E x p e r i -ments" (Kuhn 1977b, 240-65). There Kuhn c a u t i o n s t h a t not a l l those t h i n g s termed "thought experiments" In the h i s t o r y of s c i -ence have the same f e a t u r e s as the ones to be d i s c u s s e d ; s t i l l , they are " t y p i c a l of an important c l a s s " ( i b i d . , 241). 17. P i a g e t ' s stages of conceptual development are most commonly— though not n e c e s s a r i l y — a s s o c i a t e d with c e r t a i n age groups. 18. T h i s f o r m u l a t i o n of the c h i l d ' s c r i t e r i o n i s perhaps mi s l e a d -ing because I t Implies a d i s t i n c t i o n between volume and height t h a t the c h i l d h i m s e l f a p p a r e n t l y cannot yet r e c o g n i z e . 19. When my n i e c e was about four years o l d , she d i d not yet even dimly r e c o g n i z e an anomaly when my wife c o n f r o n t e d her with the P i a g e t i a n s i t u a t i o n . My wife then made s e v e r a l I n t e r e s t i n g per-mutations on P i a g e t ' s o r i g i n a l format: she switched the water f o r cream soda, put s i g n i f i c a n t l y more cream soda i n the f i r s t of the two s i m i l a r v e s s e l s , and then poured them i n t o two other v e s s e l s t h a t had g r e a t l y d i f f e r e n t diameters. In t h a t s i t u a t i o n , my puz-z l e d n i e c e claimed both t h a t the narrower v e s s e l had more cream soda i n i t than the wider one, and t h a t n e v e r t h e l e s s she would r a t h e r have the cream soda i n the wider v e s s e l . 20. Because now r e s t r i c t e d to s i t u a t i o n s where other v o l u m e t r i c f a c t o r s l i k e c r o s s - s e c t i o n a l area are held f i x e d . 94 21. For example, i f we r e s t r i c t our a t t e n t i o n t o those s i t u a t i o n s i n which the body B (of mass in) undergoes an impulse (a f o r c e having a l i m i t e d d u r a t i o n ) I = F t (= mv(t) - mv©), and we assume t h a t f r i c t i o n p l a y s no e s s e n t i a l r o l e , Newtonian mechanics w i l l y i e l d something l i k e A r i s t o t l e ' s law. That i s , i n such c i r c u m -s t a n c e s , the momentum p = rav w i l l be a constant; so, f o r example, when m = 1/2B, the average v e l o c i t y w i l l double and B w i l l t r a v e l twice the i n i t i a l d i s t a n c e C i n the same time, j u s t as A r i s t o t l e c l a i m s . Chapter 3 P r o v i n g Kuhnian Incommensurability: A Case Study 3.0 I n t r o d u c t i o n Ian Hacking c l a i m s Kuhn's concept of incommensurable s c i e n -t i f i c t h e o r i e s i s n ' t n e a r l y so p r e c i s e as the o r i g i n a l mathemati-c a l n o t i o n of incommensurable g e o m e t r i c a l magnitudes. " [ P h i l o s o -phers l i k e Kuhn] are t h i n k i n g of comparing s c i e n t i f i c t h e o r i e s , " he says, "but of course there c o u l d be no exact measure f o r t h a t purpose" (Hacking 1983, 67. I t a l i c s i n o r i g i n a l ) . N e v e r t h e l e s s , i n t h i s c h a p t e r , I want to use a d i s c u s s i o n of the h i s t o r i c a l d i s c o v e r y of the incommensurability of the s i d e and diameter ( o r : d i a g o n a l ) of the square, based on Kuhn's account of s o - c a l l e d "thought experiments," to o b t a i n a more g e n e r a l n o t i o n of incom-m e n s u r a b i l i t y t h a t i s q u i t e r i g o r o u s l y provable f o r s u c c e s s i v e s c i e n t i f i c t h e o r i e s . While i t t u r n s out t h a t p r o o f s of i ncommensurability between such t h e o r i e s can themselves be formulated q u i t e r i g o r o u s l y , be-cause they u l t i m a t e l y depend upon a c c e p t i n g c e r t a i n plausible a r -t i c u l a t i o n s as d e f i n i t i v e of the o l d paradigm, con c e p t u a l inde-terminacy means t h a t the exact import of those p r o o f s f o r t h a t paradigm o f t e n t u r n s out to be p a r t l y ambiguous. However, s i n c e i t a l s o t u r n s out t h a t the p r e c i s e s i g n i f i c a n c e of the proof of mathematical incommensurability was s i m i l a r l y open to i n t e r p r e t a -t i o n , the v e r s i o n f o r t h e o r i e s need not s u f f e r by comparison. My s t r a t e g y i n t h i s chapter i s to use the a n a l y t i c concepts d i s c u s s e d i n chapter two t o p o r t r a y the Pythagorean d i s c o v e r y of incommensurability as a Kuhnian process of normal and r e v o l u t i o n -a r y s c i e n c e . In p a r t i c u l a r , I d e s c r i b e the o r i g i n a l proof of i n -commensurability as a thought experiment t h a t d i r e c t l y compares the Pythagorean n o t i o n of arithmos (roughly: natural number) wit h an a r t i c u l a t e d n o t i o n of g e o m e t r i c a l magnitude. I t t u r n s out th a t the su c c e s s o r E u c l i d e a n mathematical t r a d i t i o n took t h i s concept of g e o m e t r i c a l magnitude t o be the more b a s i c n o t i o n and t r e a t e d t h a t of arithmos as i t s " s p e c i a l case." Hence, because the Pythagorean and E u c l i d e a n t r a d i t i o n s i n e f f e c t employed d i f -f e r e n t concepts as t h e i r standards, i n my a r t i c u l a t i o n of the Kuhnian n o t i o n , those t r a d i t i o n s are incommensurable. 3.2 The Pythagorean Paradigms The exact o r i g i n s of the Pythagorean s c h o o l of mathemati-c i a n s are hard t o determine with any p r e c i s i o n . Pythagoras (ca. 580-500 B.C.), h i m s e l f , i s a somewhat legendary f i g u r e who l i k e l y l e a r n e d h i s f i r s t mathematics i n v i s i t s t o Egypt and Ba b y l o n i a (Boyer 1968, 50). Probably, as i s recorded by l a t e r a uthors, when he r e t u r n e d t o the Greek dominions, Pythagoras e s t a b l i s h e d , In the southeast of I t a l y , the s c h o o l of mysticism and mathemat-i c s named a f t e r him ( i b i d . , 52). Some c e n t r a l b e l i e f s of h i s s c h o o l , which we no longer s t r o n g l y a s s o c i a t e w i t h mathematics, 97 were p o l i t i c a l conservatism; t r a n s m i g r a t i o n of s o u l s ; v e g e t a r i a -nism; and the shunning of beans or l e n t i l s ( i b i d . , 53). Much l a t e r ( c a . 384-322 B.C.), A r i s t o t l e was to express the fundamental t e n e t of the Pythagorean s c h o o l with the g e n e r a l i z a -t i o n , " A l l t h i n g s are (or partake of) number" (Metaphysics 1.5, 985b23ff i n Knorr 1975, 22). Of course, as I argued i n chapter two, such f o r m u l a t i o n s should o f t e n be understood as programmatic paradigms, r a t h e r than as already-completed g e n e r a l i z a t i o n s (sec. 2.1.1). Thus to understand the content of t h a t Pythagorean law, i t i s ne c e s s a r y t o see i n which way i t was o r i g i n a l l y a p p l i e d , and how those paradigms came to be f u r t h e r a r t i c u l a t e d , a process t h a t e v e n t u a l l y culminated i n the thought experiment which p o i n -ted the way to the new, E u c l i d e a n method of doing mathematics. 1 To grasp the f u l l s i g n i f i c a n c e of t h a t thought experiment, however, i t i s important t o r e a l i z e t h a t the Pythagoreans meant e n t i r e l y by "number" (Greek: azithmos) what we should now c o n s i -der to be merely n a t u r a l numbers or p o s i t i v e i n t e g e r s ; i n p a r t i c -u l a r , i t i s c r i t i c a l t o note t h a t the Pythagoreans had no n o t i o n of r a t i o n a l number, o n l y t h a t of r a t i o s or p a i r s of i n t e g e r s . In f a c t , modern d i s t i n c t i o n s l i k e r a t i o n a l , a l g e b r a i c , i r r a t i o n a l and r e a l number o n l y arose as a (much l a t e r ) consequence of the proof of incommensurability (Knorr 1975, 9 ) . 2 In essence, what the Pythagorean paradigm expressed vas the b e l i e f t h a t the t r u e natures of a l l t h i n g s — w h e t h e r p r a c t i c a l , t h e o r e t i c a l or geomet-r i c a l — c o u l d be captured e n t i r e l y i n terms of the p r o p e r t i e s of n a t u r a l numbers or t h e i r r a t i o s (Boyer 1968, 79). I t seems t h a t 98 the Pythagoreans were o r i g i n a l l y g r e a t l y impressed by such f a c t s as t h a t the a p p a r e n t l y g u a l i t i v e musical harmonies can be com-p l e t e l y expressed with sequences of p a i r s of i n t e g e r s ; and t h a t the r e l a t i v e l e n g t h s of the s i d e s of many r i g h t t r i a n g l e s can be captured e n t i r e l y by such t r i p l e s of i n t e g e r s as 3 : 4 : 5 and 5 : 12 : 13 (Popper 1968, 76; see a l s o von F r i t z 1945, 245-6). One can get a perhaps deeper i n s i g h t Into the Pythagoreans' world view by n o t i n g t h a t , f o r them, an even more fundamental catego r y than number was the d i s t i n c t i o n between odd and even. Thus A r i s t o t l e says t h a t the Pythagoreans took the "elements of number to be the elements of a l l things...and the elements of number to be the odd and the even" (Knorr 1975, 134; A r i s t o t l e , Metaphysics 986a 17). In f a c t , one group of Pythagoreans e v i n c e d t h a t b a s i c d i s t i n c t i o n i n the f o l l o w i n g s o - c a l l e d "Table of Con-t r a r i e s " : ONE ODD MALE REST (BEING) SQUARE STRAIGHT RIGHT LIGHT GOOD MANY EVEN FEMALE CHANGE (BECOMING) OBLONG CROOKED LEFT DARKNESS BAD (from Popper 1968, 78) The Pythagorean b e l i e f t h a t number i s the ground of a l l i n -d i v i d u a t i o n r e s u l t e d i n t h e i r d e s c r i b i n g n u m e r i c a l l y even what we should now regard as nonmetrical I n t a n g i b l e s : m a s c u l i n i t y , femi-n i n i t y , marriage, harmony, j u s t i c e , c r e a t i o n , e t c . For example, they claimed t h a t ...the number one i s the generator of numbers and [thus] the number of reason [Greek: ratio]} the number two i s the f i r s t even or female number, the number of o p i n i o n ; three i s the f i r s t t r u e male number, the number of harmony, being com-posed of u n i t y and d i v e r s i t y ; four i s the number of j u s t i c e or r e t r i b u t i o n , i n d i c a t i n g the s q u a r i n g of accounts; f i v e i s the number of marriage, the union of the f i r s t t r u e male and female numbers; and s i x i s the number of c r e a t i o n (Boyer 1968, 57). As I noted i n chapter two, Kuhn c l a i m s t h a t o f t e n a t the be g i n n i n g of a new paradigm programme, i t s adherents are aware of c e r t a i n s i t u a t i o n s t h a t are a p p a r e n t l y anomalous f o r t h a t para-digm. In other words, they d i m l y f e e l t h a t such s i t u a t i o n s do not have q u i t e the c h a r a c t e r i s t i c s t h e i r paradigm would lead them to expect. Many times, i t i s t r u e , i n the course of normal s c i -ence, such anomalies prove t o be r e s o l v a b l e with expanding r e -s o u r c e s . But with the b e n e f i t of h i s t o r i c a l h i n d s i g h t , we can say t h a t , f o r the Pythagoreans, the f e l t anomaly t h a t e v e n t u a l l y proved c r i t i c a l was "the computational anomaly posed by such q u a n t i t i e s as v2" (Knorr 1975, 4). i n f a c t , many previous mathe-m a t i c a l t r a d i t i o n s had a l s o r e c o g n i z e d t h a t such q u a n t i t i e s c o u l d seemingly o n l y be represented as a s u c c e s s i v e s e r i e s of a p p r o x i -mating r a t i o s , and not, f o r example, by a s i n g l e p a i r of i n t e g e r s ( i b i d . , 4 ) . 3 100 3.3 Dot-Diagramme A r t i c u l a t i o n s of the Pythagorean Paradigm While the Pythagoreans b e l i e v e d t h a t the essence of a l l t h i n g s were numbers and t h e i r r a t i o s , t h i s d i d not mean t h a t a l l they s t u d i e d d i r e c t l y was a r i t h m e t i c . In f a c t , they d i d much geometry as w e l l . But always, g e o m e t r i c a l problems were s t u d i e d as " a n a l y t i c i n s t r u m e n t [ s i " f o r t h e i r more fundamental aim of extending the a p p l i c a t i o n of t h e i r numerical paradigm (Knorr 1975, 132). The Pythagoreans were a b l e t o hook up t h e i r number theory with geometry u s i n g a pebble (Greek: psephos) or d o t - d l a -gramme method of r e p r e s e n t a t i o n ( I b i d . , 135ff; Popper 1968, 76-For the Pythagoreans and other a n c i e n t s , the n o t i o n of the u n i t was " p o i n t without p o s i t i o n " (Knorr 1975, 136). I t f o l l o w s on analogy with t h a t p i c t u r e , t h e r e f o r e , t h a t one might in t r o d u c e numbers i n t o f i g u r e s by g i v i n g them a " p o s i t i o n . " Now the Greeks had long used pebbles (psephoi, L a t i n : calculi) f o r such v a r i o u s a c t i v i t i e s as p l a y i n g board games and cou n t i n g v o t e s . In f a c t , P l a t o o f t e n drew a n a l o g i e s between p l a y i n g those games and doing mathematics ( i b i d . , 137; P l a t o Gorgias 450CD; Laws 819D-820D). And the best evidence i s t h a t the Pythagoreans a l s o worked out t h e i r a r i t h m e t i c theorems wi t h the help of pebbles to g i v e t h e i r u n i t s p o s i t i o n . Knorr argues, based on E u c l i d e a n d e f i n i t i o n s , that the Pythagoreans r e p r e s e n t e d even numbers by a row of peb-b l e s separated by a l a r g e r space; and odd numbers as even numbers with a s i n g l e pebble i n t h a t space. 101 For example o — o — o — o — o — o / o — o — o — o — o — o F i g . 3 r e p r e s e n t s twelve (where the V " marks the l a r g e r space between the pebbles, " o " ) . The odd number t h i r t e e n i s t h e r e f o r e r e p r e -sented i n t h i s n o t a t i o n as f o l l o w s (Knorr 1975, 140). Knorr i l l u s t r a t e s how i t i s p o s s i b l e with t h i s p r i m i t i v e n o t a t i o n t o prove many a r i t h m e t i c a l theorems; f o r example, t h a t the sum of any number of even i n t e g e r s i s a l s o even; and t h a t the square of an even Integer i s even ( i b i d . , 140-2). 4 The Pythago-reans a l s o found t h a t they c o u l d r e p r e s e n t g e o m e t r i c a l shapes and f i g u r e s w i t h pebble-numbers. These are the s o - c a l l e d " f i g u r e d numbers," the most b a s i c of which are the t r i a n g u l a r , square, and oblong or hetezomecic numbers. The f o l l o w i n g f i g u r e i l l u s t r a t e s t r i a n g u l a r numbers. o — o — o — o — o — o / o / o — o — o — o — o — o F i g . 4 o F i g . 5 The seventh t r i a n g u l a r number, 28, as r e p r e s e n t e d by the neo-Pytha-gorean, Iamblichus ( i b i d . , 146) 102 In Iambllchus' way of o r d e r i n g t h i s f i g u r e ' s c o n s t i t u e n t numbers, we begin with a s i n g l e stone a t the c e n t r e , and then b u i l d up the f i g u r e by adding around t h a t stone, f i r s t two stones, then t h r e e , then f o u r , and so on u n t i l we reach the sev-enth s t e p , and add seven s t o n e s . Thus the t r i a n g u l a r numbers are 3, 6, 10, 15, 21, 28, and so on. The f i g u r e d number t h a t one must add i n each case t o get the next t r i a n g u l a r number i s c a l l e d i t s gnomon (Knorr 1975, 143)." For the t r i a n g l e , these gnomons are t h e r e f o r e 2, 3, 4, 5, 6, 7, .... The f o l l o w i n g f i g u r e i s a r e p r e s e n t a t i o n of the f i f t h square number, t w e n t y - f i v e . The s e r i e s of square numbers are thus (as we might e x p e c t ) : 4, 9, 16, 25, ... and the gnomons are 3, 5, 7, 9, .... The t h i r d main type of f i g u r e d number i s the oblong or hetezomecic number whose r e p r e s e n t a t i o n i s i n d i c a t e d by the f o l l o w i n g diagrarame of the f o u r t h such number, t h i r t y . F i g . 6 o-o-o-o-o F i g . 7 103 The s e r i e s of oblong numbers are thus: 6, 12, 20, 30, ... and the gnomons are 6, 8, 10, .... By comparing the t r i a n g u l a r with the square and oblong num-bers, one can see ( a l l o w i n g the u n i t t o be a number) t h a t square numbers are the sum of c o n s e c u t i v e t r i a n g u l a r numbers, and t h a t oblong numbers are the double of t r i a n g u l a r numbers. One can a l s o p o r t r a y these r e s u l t s i n pebble n o t a t i o n (Knorr 1975, 150. Note, t h i s i s not Iamblichus' p a t t e r n , but an a l t e r n a t i v e one g i v e n by Nichomachus). There i s a c e r t a i n symmetry to the r e p r e s e n t a t i o n of geome-t r i c a l f i g u r e s with numbers. Thus, i f numbers can be used to n i t u d e s may a l s o be used to r e p r e s e n t numbers. So, f o r example, one can p o s t u l a t e minimal, i n d i v i s i b l e l i n e segments, squares, or cubes t o correspond to the I n t e g r a l u n i t . Now, i f ve look a t f i g u r e 6, ve n o t i c e t h a t the r e c t a n g u l a r f i g u r e d e l i n e a t e d by those dots a l s o d i v i d e s i n t o tvo equal t r i a n g l e s , vhen a diameter l i n e i s dravn. On the other hand, as i s r e v e a l e d by f i g u r e 7, v h i l e the pebble-methodology does show t h a t the square i s com-posed of two t r i a n g l e s , because they are s u c c e s s i v e , t h e i r t r i a n -g u l a r numbers are d i f f e r e n t . By c o n t r a s t , the geometric square d i v i d e s a l o n g the diameter i n t o two egual ( i s o c e l e s ) t r i a n g u l a r F i g . 8 F i g . 9 r e p r e s e n t g e o m e t r i c a l magnitudes, then, s y m m e t r i c a l l y , those mag-104 f i g u r e s . I n t e r e s t i n g l y , with r e s p e c t to the d i s c u s s i o n i n chap-t e r two of a r t i c u l a t i o n and f a m i l y resemblance, the Pythagoreans were aware of t h i s " d i s p a r i t [ y ] , but [sought] i n t h e i r p r e s e n t a -t i o n s to exploit the likenesses, r a t h e r than the differences" (Knorr 1975, 174. I t a l i c s mine). 3.4 Rules f o r F i n d i n g Pythagorean T r i p l e s Knorr a l s o shows how i t Is p o s s i b l e to capture many other aspects of g e o m e t r i c a l f i g u r e s i n t h i s pebble-methodology, i n c l u -d i n g a formula f o r g e n e r a t i n g an i n f i n i t e number of the s o - c a l l e d Pythagorean triples. As mentioned above (p4), the f a c t t h a t t r i -p l e s of i n t e g e r s c o u l d c o m p l e t e l y r e p r e s e n t the r e l a t i v e lengths of the p a r t s of some r i g h t t r i a n g l e s had d e e p l y impressed the e a r l y Pythagoreans. Knorr uses t h e i r pebble-methodology to char-a c t e r i z e the p r o p e r t i e s of such t r i p l e s i n terms of the fundamen-t a l c a t e g o r i e s odd and even ( i b i d . , 158-60). The f o l l o w i n g are the most r e l e v a n t theorems. Note here t h a t "C" names the l a r g e s t number, hence t h a t of the hypotenuse, and "A" and "B" name l e g numbers. I. Given the Pythagorean t r i p l e A, B, C, i f two of the terms are even, the t h i r d i s a l s o even (Knorr 1975, 158). I I . I f i n a g i v e n Pythagorean t r i p l e . . . C i s even, then a l l three terms are even ( i b i d . , 158). I I I . Given a Pythagorean t r i p l e A, B, C, i f one of the terms i s odd, then...C i s odd, and of the numbers A and B, one i s odd and the other even ( i b i d . , 159). 105 As mentioned above (p9), the f a c t t h a t i n t e g r a l numbers and t h e i r r a t i o s c o u l d r e p r e s e n t g e o m e t r i c a l f i g u r e s , meant t h a t those r e p r e s e n t e d f i g u r e s c o u l d a l s o stand f o r numbers. Of course, the Pythagoreans b e l i e v e d t h a t t h i s symmetry was o n l y p a r t i a l because numbers c o u l d a l s o be used to r e p r e s e n t other t h i n g s than g e o m e t r i c a l f i g u r e s . N e v e r t h e l e s s , s i n c e these geo-m e t r i c a l f i g u r e s c o u l d go proxy f o r numbers i n g e o m e t r i c a l prob-lems, cases i n which a g e n e r a l g e o m e t r i c a l p a t t e r n was d i s c o v e r e d c o u l d be used t o s e t a problem f o r the a r i t h m e t i c paradigm. 3.5 An A r i t h m e t i c and Geometric V e r s i o n of Pythagoras' Theorem Knorr shows how an a r t i c u l a t i o n of the pebble-methodology to square and oblong f i g u r e s c o u l d l e a d f i r s t t o a g e n e r a l a r i t h m e t -i c formula r e l a t i n g Pythagorean t r i p l e s of i n t e g e r s , and then to a s t i l l more g e n e r a l geometric formula r e l a t i n g the s i d e s of r i g h t t r i a n g l e s t o t h e i r hypotenuses (Knorr 1975, 175-7). These formulas express, a r i t h m e t i c a l l y and g e o m e t r i c a l l y , the famous Pythagorean theorem on r i g h t t r i a n g l e s . Here i s the a r i t h m e t i c form of the theorem Knorr recon-s t r u c t s . Theorem 1. Given a r i g h t t r i a n g l e of which the s i d e s about the r i g h t angle r e p r e s e n t the numbers A and B, and the hypo-tenuse r e p r e s e n t s the number C, then the square of C equals the sum of the squares of A and of B. That i s , the three numbers, A, B, C form a Pythagorean number t r i p l e . 106 The f o l l o w i n g i s the geometric form of Theorem 1. Theorem [ 2 ] . Given a r i g h t t r i a n g l e , the square on the hypo-tenuse equals the sum of the square on the s i d e s about the r i g h t a n g l e . I t i s j u s t a few s h o r t steps from these theorems, t o g e t h e r with theorems I, II and I I I above on the p r o p e r t i e s of Pythago-rean t r i p l e s ( ) , to the r e d u c t i o proof cum thought experiment c a l l e d the proof of the Incommensurability of the s i d e with the diameter of a square." The p u r e l y g e o m e t r i c a l statement of Pythagoras* Theorem was c l e a r l y o b t a i n e d , on Knorr's r e c o n s t r u c t i o n , by e x t e n s i o n of the numerical p a r a d i g m — i n p a r t i c u l a r , v i a the pebble a r t i c u l a t i o n s of t h a t paradigm. T h e r e f o r e , the symmetry of the s i t u a t i o n would have suggested t o a Pythagorean t h a t i t should be p o s s i b l e to f i n d t r i p l e s of i n t e g e r s f o r any r i g h t t r i a n g l e d e f i n e d by the g e o m e t r i c a l formula. In other words, a Pythagorean's w e l l - f o u n -ded, "paradigm-induced e x p e c t a t i o n s " (Kuhn 1970a, 53) would l e a d him to s e a r c h f o r t r i p l e s f o r such r a t i o s as had p r e v i o u s l y proved e l u s i v e : f o r example, the r a t i o of the diameter to the s i d e of a square (Knorr 1975, 179). That search would q u i c k l y be thwarted as the f o l l o w i n g proof shows. 107 3.6 Thought Experiment: A Proof of Incommensurability Le t us suppose that numbers c h a r a c t e r i z e both the legs ( s i d e s ) , A and B, and the hypotenuse (diameter), C, of the i s o c e l e s r i g h t t r i a n g l e ( s q u a r e ) . O b v i o u s l y , the two legs ( s i d e s ) are equal so A = B; moreover, C i s c l e a r l y g r e a t e r than A and B, but l e s s than t h e i r double. Is C odd or even? ( i ) Suppose t h a t C i s even. By II above ( p l O ) , so are A and B. But i f ve d i v i d e each s i d e i n h a l f v i t h a perpendi-c u l a r l i n e , ve o b t a i n another i s o c e l e s r i g h t t r i a n g l e vhose l e g s and hypotenuse are i n the same ( g e o m e t r i c a l l y - a p p a r e n t ) r a t i o . Hence, the new hypotenuse, C and the nev l e g s , A' and B' v i l l a l s o be even and ve have an i n f i n i t e r e g r e s s . Hovever, s i n c e , by hyp o t h e s i s , C i s i n t e g r a l , i t cannot be d i v i d e d ad I n f i n i t u m . Thus C must be odd. ( i i ) Suppose t h a t c i s odd. By I I I above ( p l O ) , e i t h e r A i s even and B i s odd, or v i c e v e r s a . But by hyp o t h e s i s , A = B; so the l e g numbers must be both odd and even, v h i c h i s a c o n t r a d i c t i o n . On the other hand, i f we c l a i m t h a t A and B are the u n i t , v h i c h i s sometimes thought t o be both odd and even, because C i s c l e a r l y g r e a t e r than e i t h e r A or B, but l e s s than t h e i r double, C i s not i n t e g r a l . In other vords, C i s not a number (adapted from i b i d . , 179-80). What c o n c l u s i o n s can be drawn from t h i s r e d u c t i o - t h o u g h t experiment? The most obvious r e s u l t i s t h a t i t i s impossible to a s s i g n i n t e g e r s s i m u l t a n e o u s l y to the s i d e and diameter of a square (or t o the s i d e and hypotenuse of a r i g h t I s o c e l e s t r i a n -g l e ) . S i n c e , f o r the Pythagoreans, hovever, i n t e g r a l numbers were supposed to be the measure of a l l t h i n g s , vhat the proof shows i s th a t i t i s i n f a c t i m p o s s i b l e to measure c e r t a i n geome-t r i c a l r e l a t i o n s by a r a t i o of i n t e g e r s . But s i n c e each Integer has the u n i t as a common f a c t o r , t h i s a l s o means that the s i d e and diameter of the square have no common (numerical) measure; th a t i s , they are ( n u m e r i c a l l y ) incommensurable. 108 Less o b v i o u s l y , perhaps, what the proof shows i s t h a t we must d i s t i n g u i s h between the concepts of i n t e g r a l number (arith-mos) and g e o m e t r i c a l magnitude ( c . f . Knorr, 24). In f a c t , i n the proof, i t i s the hypothesis t h a t these concepts are c o e x t e n s i v e that leads to the c o n t r a d i c t i o n . More to the p o i n t , the proof of mathematical incommensurability shows t h a t we must b i f u r c a t e the o r i g i n a l concept of arithmos i n t o (atomic) Integer and ( c o n t i n u -ous) g e o m e t r i c a l magnitude. L a t e r Greek mathematicians such as Eudoxus and E u c l i d took the mathematical proof to show t h a t the concept of g e o m e t r i c a l magnitude i s more fundamental than t h a t of arithmos (Knorr 1975, 170-1). 7 3.7 C o n c l u s i o n : Kuhnian and Mathematical Incommensurability I claimed i n the I n t r o d u c t i o n t h a t I t i s p o s s i b l e to draw a c l o s e enough p a r a l l e l between Kuhn's n o t i o n of incommensurability and the mathematical one to o b t a i n a (more) r i g o r o u s method of a p p l y i n g Kuhns's concept. A rough c r i t e r i o n Kuhn o f f e r s f o r t a k -ing two (Kuhnian) incommensurable paradigms to be Incompatible may help i n t h a t t a s k . Kuhn says: "Obviously, then, there must be a c o n f l i c t between the paradigm t h a t d i s c l o s e s anomaly and the one t h a t l a t e r renders the anomaly l a w l i k e " (Kuhn 1970a, 97). Thus, on t h i s p r o f f e r e d c r i t e r i o n , the Pythagorean a r i t h m e t i c a l paradigm which d i s c l o s e d the anomaly of (numerical) incommensur-a b i l i t y Is (Kuhnian) Incommensurable with the E u c l i d e a n geometri-c a l paradigm t h a t "render[ed] the anomaly l a w l i k e . " But the Py-109 thagorean paradigm d i s c l o s e d t h a t anomaly i n the most e x p l i c i t w a y — t h a t i s , transformed i t s " f e l t anomaly i n t o c o n c r e t e c o n t r a -d i c t i o n " — w i t h the thought experiment t h a t was the proof of (nu-m e r i c a l ) incommensurability. Moreover, s i n c e t h a t anomaly was r e a l l y an anomaly relative to the Pythagorean s t a n d a r d of a r i t h -mosf but not f o r the l a t e r E u c l i d e a n s t a n d a r d of g e o m e t r i c a l magnitude, t h a t proof showed, more g e n e r a l l y , t h a t these two paradigms were Kuhnian incommensurable. In summary, t h e r e f o r e , we may say thought experiments which prove t h a t what i s anomalous i n one system of concepts i s l a w l i k e i n another, a l s o prove those c o n c e p t u a l systems to be Kuhnian i n -commensurable. Thus my proof i n chapter one (§1.7) t h a t the con-cept of comparison must be b i f u r c a t e d i n t o two concepts, d i r e c t comparison and commensuration, a l s o shows t h a t the t r a d i t i o n a l view, which i d e n t i f i e s those concepts, i s incommensurable with Kuhn's view, which d i s t i n g u i s h e s them. The concept of a thought experiment or proof of incommensur-a b i l i t y connects up with the n o t i o n s of d i r e c t comparison and c o n t r a s t as f o l l o w s . The thought experiment shows d i r e c t l y t h a t the attempt to express w i t h i n one system of concepts r e l a t i o n s t h a t are e x p r e s s i b l e ( l a w l i k e ) i n another i s c o n c e p t u a l l y anoma-l o u s — r e s u l t s i n concrete c o n t r a d i c t i o n — i n t h a t f i r s t system. For example, the attempt to express as an i n t e g e r r a t i o , the r e l -a t i v e magnitudes of the s i d e and diameter of a s q u a r e — a n o b v i -o u s l y i n t e l l i g i b l e geometric n o t i o n (von F r i t z 1945, 261), and one which seems d i r e c t l y s i m i l a r to a r i t h m e t i c r a t i o s , c o n f l i c t s 110 with c e r t a i n p r o p e r t i e s of the i n t e g e r s . S i m i l a r l y , the attempt t o express w i t h i n the t r a d i t i o n a l view Kuhn's i n t u i t i v e l y i n t e l -l i g i b l e concept of fundamental i n c o m p a t i b i l i t y , a n o t i o n t h a t seems d i r e c t l y s i m i l a r t o formal i n c o m p a t i b i l i t y , c o n f l i c t s with c e r t a i n p r o p e r t i e s of formal l o g i c a l systems. I l l Notes 1. The thought experiment/proof of incommensurability that I shall use in this chapter i s W. B. Knorr's historical reconstruc-tion. Thus Knorr argues that the more familiar proof from Aris-totle (Prior Analytics 1.23, 41a29) and Euclid (X, 117) contains elements that show i t to be a much later development. His recon-struction i s based, in part, on a passage in Plato's Keno (82B-85B) and on the Pythagorean "theory of figured numbers." (Knorr 1975, 22-7). 2. Another point that w i l l prove relevant to the (reconstructed) proof of incommensurability i s that the Pythagoreans often did not regard, or were unsure whether to regard, the unit as i t s e l f a number; and they also often considered i t to be either both odd and even or neither (e.g. Knorr 1975, 146). 3. E.g. 1 : 1; 3 : 2; 7 : 5; and so on. This sequence i s genera-ted by assigning, as a f i r s t approximation, the number one to both the side and diameter (diagonal) of the square. The side number of the next approximation i s obtained from the sum of previous one's side and diameter numbers; while the next diameter number i s the sum of the previous approximation's diameter number and twice i t s side number; and so on (Knorr 1975, 16). While this (inductive) sequence of ratios seems to approach a limi t , i t is also, as i t stands, clearly nonterminating. 4. For instance, one can show that the sum of any number of even integers—any Integer times an even in t e g e r — i s also even with the pebble method as follows. Let A, B, C, D, be even integers portrayed by the a r b i t r a r i l y chosen patterns below: A o — o — o — o o — o — o — o B o—o o—o C o — o — o — o — o o — o — o — o — o D o — o — o o — o — o The sum of these integers, E=A+B+C+D, can now clearly be given by adding the l e f t and right parts of each integer, respec-tively. Since equals were added to equals on each side, the resultant, E, w i l l be representable by a single row of pebbles separated by a space with an equal number of pebbles on each side; that i s , i t w i l l be even integer (based on Knorr 1975, 141). 112 5. Interestingly, the word "gnomon" derives from the Greek word for the part of a sundial that casts a shadow showing the time; thus the pebble-arithmetic gnomon is an "indicator" of the next number in i t s sequence (COED s.v. "gnomon"). 6. 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