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The Reason for Structural Causation

Denn das Selbstgedachte versteht man viel grundlicher, als das Erlernte, und erhalt, wenn man es nachmals bei jenen Friihern fmdet, unverhofft eine stark fur die Wahr- heit desselben zeugende Bestatigung, durch fremde, aner- kannte Auktoritat, wo durch man sodann Zuversicht und Standhaftigkeit gewinnt, es gegen jeden Widerspruch zu verfechten.

Arthur Schopenhauer

"Ueber Philosophie und ihre Methode"

1 The Problem

S u p p o s e y o u have a n u r n c o n t a i n i n g fifty w h i t e b a l l s a n d fifty b l a c k b a l l s . Y o u p e r f o r m a series of a h u n d r e d d r a w i n g s w i t h replacements, a n d take d o w n the results. A s it h a p p e n s , there are 52 w h i t e b a l l s i n the series a n d 48 b l a c k ones.

T h i s is a n o r m a l result under the c i r c u m s t a n c e s d e s c r i b e d .1 In fact, we are i n c l i n e d t o say t h a t the colour r a t i o o f b a l l s i n the u r n i n some way explains or causes the colour r a t i o a m o n g the b a l l s a c t u a l l y d r a w n i n our e x p e r i m e n t . W e r e we to p e r f o r m a great number o f such d r a w i n g e x p e r i m e n t s o f l e n g t h 100, we w o u l d f i n d t h a t i n the vast m a j o r i t y of cases, the n u m b e r o f w h i t e b a l l s d r a w n is a p p r o x i m a t e l y the same as the n u m b e r o f black balls. In other words, the r a t i o o f w h i t e to b l a c k balls r e m a i n s a p p r o x i m a t e l y constant i n our repeated sequences o f d r a w i n g s . L a p l a c e ([19], p . x l v i i i ) c a l l e d the colour r a t i o o f balls i n the u r n the constant cause of the c o l o u r r a t i o a m o n g the o u t c o m e s . A c c o r d i n g to L a p l a c e , it is due to J a k o b B e r n o u l l i ' s t h e o r e m (a s p e c i a l case o f the weak l a w o f large n u m b e r s ) t h a t the constant cause w i l l i n the l o n g r u n get the better of the so-called variable causes: causes w h i c h are effective i n an u n c o n t r o l l e d way i n every single d r a w i n g , but the effects o f w h i c h f i n a l l y "average o u t " .

1 We may also say that this is a lucky result which allows for the kind of explanation to be discussed in this paper. We might as well end up with a series of 97 white and only 3 black balls. It is this realization of the improbable that made Stegmuller ([50], pp. 281-285, 313-314; [51]) argue against the very feasibility of statistical explanations.

See Humphreys ([13], pp. 117-118) for a rejoinder.

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O n the other h a n d , there is the mechanistic p i c t u r e of e x p l a n a t i o n . A c c o r - d i n g to t h i s v i e w , an e x p l a n a t i o n consists i n a d e t a i l e d account o f how an i n i t i a l state o f affairs develops a n d l a w f u l l y produces the e x p l a n a n d u m . In the case at h a n d , the d e s c r i p t i o n of the i n i t i a l c o n d i t i o n s consists i n a perfect s p e c i f i c a t i o n of the p o s i t i o n s a n d m o m e n t a of a l l b a l l s , o f the f o r m of their surface as well as the surface of the u r n , a n d a precise d e s c r i p t i o n of the d r a w i n g p r o c e d u r e . If we h a d a l l this i n f o r m a t i o n , t h e n i t c o u l d — i n p r i n c i p l e — be c a l c u l a t e d e x a c t l y w h i c h of the b a l l s were b o u n d to be d r a w n . A f t e r a d d i n g i n f o r m a t i o n a b o u t the colours of the balls, one gets the c o l o u r r a t i o as a b y p r o d u c t o f the m o r e i n f o r m a t i v e e x p l a n a t i o n . T h e e x p l a n a t i o n of the colour r a t i o is as it were o n l y p a r a s i t i c o n the genuine e x p l a n a t i o n of the a c t u a l order of d r a w i n g s o f exactly these balls ( w i t h " i n t e r n a l n u m b e r s " 56, 02, 88, 39, . . . , s a y ) .2 A n advocate o f this account of e x p l a n a t i o n w i l l say t h a t he does not u n d e r s t a n d w h a t justifies us to endow the s t a t e m e n t of so-called constant causes w i t h the h o n o u r a b l e t e r m ' e x p l a n a t i o n ' . I n fact, L a p l a c e h i m s e l f s h o u l d have fallen v i c t i m to t h i s a r g u m e n t . H i s o w n creature, the famous d e m o n , w i l l challenge his d o c t r i n c e o f constant causes. B e i n g i n a n i d e a l e p i s t e m i c s i t u a t i o n ,3 the o m n i s c i e n t i n t e l l i - gence s i m p l y w o u l d not u n d e r s t a n d the recourse to constant causes. H e does not need t h e m , he has no place for t h e m .4

Is, t h e n , L a p l a c e i n the s i t u a t i o n of the p r o v e r b i a l sorcerer's apprentice w h o does n o t come to t e r m s w i t h the d e m o n he has c o n j u r e d u p ? It is the a i m o f this paper to help L a p l a c e convince the d e m o n t h a t his — L a p l a c e ' s — d o c t r i n e makes sense after a l l a n d c a n i n fact be r e c o n c i l e d w i t h the d e m o n ' s m e c h a n i s t i c account of the w o r l d .

T h i s l a t t e r account is t h o r o u g h l y deterministic. It is of c r u c i a l i m p o r t a n c e , however, not to m i x up t w o different senses of ' d e t e r m i n i s m ' . W e s h a l l use the t e r m to denote the thesis t h a t whatever h a p p e n s i n the w o r l d is d e t e r m i n e d by antecedent c o n d i t i o n s ; t h a t there are no i r r e d u c i b l e chance p h e n o m e n a w h i c h cannot be e x p l a i n e d away by a c o n s i d e r a t i o n of a d d i t i o n a l (sometimes h i d d e n ) causal f a c t o r s .5 W e do not use the t e r m ' d e t e r m i n i s m ' to signify the thesis t h a t

2 Against a suggestion of Max Urchs, I am inclined to group the following condition as a paradigmatic non-theorem rather than a paradigmatic theorem of the logic of causation or explanation: If A causes/explains B and B logically entails C , then A causes/explains C. Parasitic explanations are no genuine explanations.

3 Naively, this is just the situation we attempt to reach in the course of the scienctific enterprise; but cf. footnote 2.

4 W.V.O. Quine ([41], §§ 46-47) appears to be a disciple of the Laplacian intelligence when he insinuates that dispositions and causal statements are to be reduced to the existence of certain "connecting" or "definite mechanisms'' ([41], pp. 223, 225). In [42], quite the same explanatory role is played by the notion of natural kind, or — what "varies together"

(p. 121) with the notion of kind — the notion of similarity. He holds, however, that "we can take it as a very special mark of the maturity of a branch of science that it no longer needs an irreducible notion of similarity and kind." (p. 138)

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e v e r y t h i n g is p r e d i c t a b l e by h u m a n s c i e n t i s t s .6

W e k n o w t h a t there are f u n d a m e n t a l l i m i t s to our capacities of d e t e r m i - n i n g e x a c t l y the antecedent c o n d i t i o n s , as w e l l as to our capacities o f s o l v i n g c o m p l e x e q u a t i o n systems. Hence there are f u n d a m e n t a l l i m i t s to our c a p a c i - ties o f p r e d i c t i n g future events. It is m i s l e a d i n g to identify d e t e r m i n i s m w i t h p r e d i c t a b i l i t y - i n - p r i n c i p l e , as l o n g as i t is not p o i n t e d out t h a t w h a t is m e a n t is p r e d i c t a b i l i t y for the Laplacian demon r a t h e r t h a n p r e d i c t a b i l i t y for humans.

In the f o l l o w i n g we s h a l l assume, for the sake of a r g u m e n t , t h a t the w o r l d is de- t e r m i n i s t i c , even i f t h a t does not m a t c h the current state of scientific t h e o r i z i n g . T h e reason for this a s s u m p t i o n is, first, t h a t we w o u l d be h a p p y to get clear a b o u t the ( s u p p o s e d l y ) s i m p l e r case o f a d e t e r m i n i s t i c w o r l d , a n d , secondly, t h a t i t is the L a p l a c i a n d e m o n w h o m we are g o i n g to argue w i t h anyway.

W h y is t h i s a n i n t e r e s t i n g p r o b l e m ? I have two answers. F i r s t a n d foremost, the p r o b l e m s w i t h the u r n m o d e l are s y m p t o m a t i c of a p r o b l e m c o n c e r n i n g e x p l a n a t i o n i n the s o c i a l sciences. T h e r e s t a t i s t i c a l tables are s u p p o s e d to be of e x p l a n a t o r y value. T h e y e x h i b i t the same k i n d of c o n s t a n c y as repeated e x p e r i m e n t s w i t h sequences o f d r a w i n g s . N u m b e r s w h i c h are constant over a s u b s t a n t i a l t i m e i n t e r v a l are t a k e n to be caused by s o m e t h i n g — by some "con- s t a n t causes." M o r e specifically, the p a r a l l e l m a y be d r a w n a l o n g the f o l l o w i n g lines:

• D u e to the colour r a t i o of the b a l l s i n the u r n , we m a y say t h a t each i n d i v i d u a l d r a w i n g has a p r o p e n s i t y o f 0.5 of s h o w i n g a black b a l l . C h a n g i n g the r a t i o i n the u r n w i l l result i n a l o w e r i n g or i n c r e a s i n g of this p r o p e n s i t y o f each i n d i v i d u a l d r a w i n g . T h o u g h this does n o t give a n account o f the i n d i v i d u a l

" k i n e t i c histories" o f a c t u a l d r a w i n g s o f b l a c k b a l l s , we m a y s t i l l assume t h a t i n our d e t e r m i n i s t i c but " c h a o t i c " u r n m o d e l the i n i t i a l c o n d i t i o n s are d i s t r i b u t e d i n such a way t h a t i n a c o r r e s p o n d i n g p r o p o r t i o n of cases, b l a c k d r a w i n g s w i l l be prevented or p r o m o t e d .

• D u e to the s o c i a l a n d legislative c o n d i t i o n s i n a society, each i n d i v i d u a l has a p r o p e n s i t y o f 0.0027 of c o m m i t t i n g a c e r t a i n c r i m e i n G e r m a n y i n 1993, say. C h a n g i n g the s o c i a l b o u n d a r y c o n d i t i o n s w i l l result i n a l o w e r i n g or i n c r e a s i n g of t h i s p r o p e n s i t y o f each i n d i v i d u a l . T h o u g h t h i s does not give a n account of the i n d i v i d u a l " m o r a l histories" o f a c t u a l c r i m e s , we m a y s t i l l

5 This is roughly what Anscombe ([1], p. 63) calls the assumption of relevant differences:

"Kan effect occurs in one case and a similar effect does not occur in an apparently similar case, there must be a relevant further difference."

6 Even if they are equipped with the best measuring devices and computers one can think of. This kind of explication of determinism is aimed at by Popper ([38], pp. 1-2): "what I call 'scientific' determinism the doctrine that the structure of the world is such that any event can be rationally predicted, with, any desired degree of precision, if we are given a sufficiently precise description of past events, together with all the laws of nature." (Popper's italics)

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assume t h a t i n our d e t e r m i n i s t i c b u t chaotic w o r l d the i n i t i a l c o n d i t i o n s are d i s t r i b u t e d i n such a way t h a t i n a c o r r e s p o n d i n g p r o p o r t i o n of cases, crimes w i l l be prevented or p r o m o t e d .

A c o m m o n cause p h i l o s o p h y was a d v o c a t e d i n the s o c i a l sciences by the B e l g i a n s t a t i s t i c i a n A d o l p h e Q u e t e l e t i n the 1830s a n d '40s.7 A s W e y m a L i i b b e ([23]) p o i n t s out, b o t h i n the u r n e x a m p l e and i n s o c i a l s t a t i s t i c s , we assume t h a t the i n d i v i d u a l events ( d r a w i n g s a n d crimes) are i n d e p e n d e n t of each other, a n d yet the collective results show a s u r p r i s i n g constancy. T h i s has led to the p o s t u l a t i o n of a constant cause for the constant m a c r o s c o p i c effects. W h i l e it is p r e t t y clear w h a t is the n a t u r a l c a n d i d a t e for b e i n g the constant cause i n the u r n m o d e l — v i z . , the c o l o u r r a t i o amongst the b a l l s i n the u r n — we are at a loss w h a t to say i n the case of the s o c i a l sciences. T h e m a i n p r o b l e m i n contrast to the u r n m o d e l is t h a t i t is at least d o u b t f u l whether we c a n l e g i t i m a t e l y s t i p u l a t e the existence of such a t h i n g as a "constant cause" of the c r i m e rate i n G e r m a n y i n 1993. T h i s is not the o n l y p r o b l e m w i t h the p a r a l l e l between u r n e x p e r i m e n t s a n d the tables o f s o c i a l s t a t i s t i c s . It m u s t be a d m i t t e d t h a t considerable w o r k needs to be done i n order to reveal t h a t the a n a l o g y is indeed i n s t r u c t i v e .

M y second reason for addressing u r n e x p e r i m e n t s i n this paper is s i m p l e r . T h e r e is e n o u g h discussion i n the l i t e r a t u r e t h a t w a r r a n t s a new a t t e m p t at c l a r i f y i n g the concept of c a u s a l i t y i n u r n m o d e l s . I n his paper o n " T h e Slow R i s e of P r o b a b i l i s m " , L o r e n z K r i i g e r identifies a lack o f d i s c e r n m e n t of the differences between the t w o a b o v e - m e n t i o n e d accounts of c a u s a t i o n or e x p l a n a t i o n d u r i n g the greater p a r t of the 19th c e n t u r y .8

T h e purpose of the present paper is a n a t t e m p t to reconcile t w o very different approaches to the concept o f c a u s a t i o n . I n the o r i g i n a l f o r m , it is the o p p o s i - t i o n f o u n d i n L a p l a c e between his d o c t r i n e of constant a n d variable causes o n the one h a n d 9 a n d his mechanistic determinism o n the o t h e r .1 0 T h e t e n s i o n between the t w o was f o r m u l a t e d i n a l l c l a r i t y o n l y by M a x w e l l w h o r e p e a t e d l y stressed the contrast between the statistical a n d the dynamical m e t h o d ( w h i c h latter he also called the historical or strictly kinetic m e t h o d ) . T h i s p r o b l e m has a g a i n been brought to the fore a n d identified as a reason for the slow rise of d e t e r m i n i s m by K r i i g e r w h o used the terms structural a n d dynamical causes

7 See e.g. Quetelet ([40], Vol. I, pp. 3-16).

8 Like Kriiger and pace Stegmuller ([52], pp. 9, 633-634), I shall not distinguish between the problem of causation and the problem of explanation. Causation and explanation are closely tied in the work of Wesley Salmon.

9 This doctrine of Laplace's is no well-defined theory. It is given in the section "Des lois de la Probability qui resultent de la multiplication indefmie des evenements." ([19], pp. xlvii- lv) He does not even address the question of what "causes regulieres et constantes"

(p. xlviii) are. Presumably they are causes which are supposed to be constantly effective over a substantial period of time.

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for the d i s t i n c t i o n i n q u e s t i o n .1 1 In the recent A m e r i c a n d i s c u s s i o n , , a s i m i l a r d i c h o t o m y surfaces i n the work of W e s l e y S a l m o n ([46], p . 99) w h o distinguishes statistical f r o m aleatory12 c a u s a t i o n . A t any rate, we have prima facie con- f l i c t i n g i n t u i t i o n s a b o u t c a u s a l i t y a n d e x p l a n a t i o n .1 3 I s h a l l argue t h a t as far as games of chance are concerned, to assign p r o b a b i l i s t i c laws a f u n d a m e n t a l role i n scientific e x p l a n a t i o n does not conflict w i t h the a s s u m p t i o n o f L a p l a c i a n d e t e r m i n i s m . M y vague hope is t h a t i n the l o n g r u n a proper u n d e r s t a n d i n g of games of chance w i l l also shed some l i g h t o n the role o f p r o b a b i l i s t i c laws as e m p l o y e d i n the s o c i a l sciences.

I s t a r t e d to t h i n k a b o u t s t r u c t u r a l c a u s a t i o n a n d the r e l a t i o n between deter- m i n i s m a n d p r o b a b i l i s m i n response to the s t i m u l a t i n g paper of L o r e n z K r i i g e r a n d the e q u a l l y s t i m u l a t i n g discussions w i t h m y colleague W e y m a L i i b b e . I t h o u g h t t h a t some basic insights of m o d e r n chaos theory s h o u l d be i n v o k e d i n order to overcome a p u r e l y s u b j e c t i v e v i e w of p r o b a b i l i t y a n d c o n v i n c e the L a p l a c i a n intelligence t h a t i t does m a k e sense to use p r o b a b i l i s t i c p a t t e r n s i n c a u s a l e x p l a n a t i o n s . O n l y then d i d I discover t h a t b a s i c a l l y the same i d e a h a d been p u t to g o o d use by H e n r i P o i n c a r e as e a r l y as 1 8 9 6 .1 4 H e p u s h e d the i d e a m u c h further by offering a m a t h e m a t i c a l t h e o r e m o n w h i c h p h i l o s o p h i c a l i n - t e r p r e t a t i o n s can t u r n . F o r some reason, P o i n c a r e ' s c o n t r i b u t i o n tends to be neglected i n the current l i t e r a t u r e o n p r o b a b i l i s t i c causality. O n e n o t a b l e ex- c e p t i o n is J a n v o n P l a t o ([31] - [34]) w h o i n a series of excellent papers traces the fate of P o i n c a r e ' s method of arbitrary functions up to the m o d e r n t h e o r y of e r g o d i c systems. I have benefitted a great deal f r o m his w o r k .

2 Four Concepts of Causation

W e n o w present a very r o u g h scheme o f four different concepts o f c a u s a t i o n to w h i c h we w i l l refer i n our subsequent d i s c u s s i o n .

1. Causation of the Laplacian intelligence: Covering law, deductive-nomolo-

10 Which found its immortal expression in the metaphor of Laplace's ([19], pp. vi-vii) demon:

"Nous devons done envisager l'etat present de I'univers comme 1'efTet de son etat anterieur et comme la cause de celui qui va suivre. Une intelligence qui, pour uii instant doime, connaitrait toutes les forces dont la nature est animee et la situation respective des etres qui la composent, si d'ailleurs elle etait assez vaste pour soumettre ces donnees a 1'Analyse, embrasserait dans la meme formule les mouvements des plus grands corps de I'univers et ceux du plus leger atome: rien ne serait incertain pour elle, et l'avenir, comme le passe, serait present a ses yeux."

11 "The ratio of the two effects 'white ball drawn' and 'black ball drawn', however, cannot be attributed to any dynamical cause operative in the process. The constant "cause"

that brings about this ratio is, of course, the ratio of white to black balls in the urn. This is clearly a real condition of the dynamical process . . . , but we may call it a "structural"

condition, since it is not a dynamical part of the history of the system." ([18], p. 65)

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gical, deterministic causation. C o n s i d e r a complete d e s c r i p t i o n of the state of the w o r l d at a g i v e n t i m e , take a l l laws of n a t u r e a n d derive f r o m these t w o c o m p o n e n t s the state of the w o r l d a t any other t i m e , a g a i n described i n full completeness. It makes sense t o s t i p u l a t e t h a t a cause m u s t t e m p o r a l l y precede its effects; moreover, a n element of the d e s c r i p t i o n of the " i n i t i a l state"

qualifies as a ("determining") cause for an effect under c o n s i d e r a t i o n o n l y i f it a c t u a l l y enters i n t o the d e r i v a t i o n of the effect. W e have no o c c a s i o n to consider c o u n t e r f a c t u a l s i t u a t i o n s .

2. Laplacian causation: The doctrine of constant and variable causes. In order to ascertain whether s o m e t h i n g (a "factor") is a cause o f the effect under dis- cussion, one has to consider a l l / a large n u m b e r o f / a select set of s i t u a t i o n s i n w h i c h this factor is c o n s t a n t l y present, b u t i n w h i c h a l l other factors are changed a r b i t r a r i l y / w i t h sufficient v a r i a b i l i t y / a c c o r d i n g to c e r t a i n c r i t e r i a of choice. I f the effect is o b t a i n e d u n i f o r m l y or w i t h sufficient frequency a n d a p p r o x i m a t i o n , t h e n the factor under c o n s i d e r a t i o n is i n fact to be regarded as a ( "constant")

cause. Here we c o m p a r e a real s i t u a t i o n w i t h — a c t u a l or c o u n t e r f a c t u a l — s i t u a t i o n s where everything except the putative cause is varied.

(In n u m e r o u s s p e c i a l cases the constant causes of L a p l a c e m a y be i d e n t i f i e d w i t h the statistical or structural causes a l r e a d y m e n t i o n e d . )

S. Counterfactual causation, forward direction: sine q u a n o n causation. I n order to ascertain whether s o m e t h i n g (a "factor") is a cause o f the effect under discussion, one has to consider a l l / a large n u m b e r o f / a select set of s i t u a t i o n s i n w h i c h a l l other factors are c o n s t a n t l y present, b u t i n w h i c h t h i s factor is changed a r b i t r a r i l y / w i t h sufficient v a r i a b i l i t y / a c c o r d i n g t o c e r t a i n c r i t e r i a o f choice. If the effect is lost u n i f o r m l y or w i t h sufficient frequency as a result of the change i n the p u t a t i v e causes, t h e n the factor under c o n s i d e r a t i o n is i n fact to be regarded as a ("counterfactual") cause. Here we c o m p a r e a real s i t u a t i o n w i t h — a c t u a l or c o u n t e r f a c t u a l — s i t u a t i o n s where everything except the putative cause is kept fixed.

4. Counterfactual causation, backward direction: Why ask, "Why?"? ( S a l m o n [45], H u m p h r e y s [13]) W h y - q u e s t i o n s arise when we are s u r p r i s e d by an effect,

12 T h i s t e r m is due to H u m p h r e y s .

13 I s h a l l not p u r s u e here the i n t e r e s t i n g q u e s t i o n w h e t h e r the two t y p e s o f c a u s a t i o n are i n some essential way r e l a t e d t o the d i s t i n c t i o n b e t w e e n a discrete a n d a continuous c o n c e p t u a l i z a t i o n of the w o r l d . — F o r m o r e c o n f l i c t i n g i n t u i t i o n s , see S t e g m u l l e r ([50], p p . 311-317), S u p p e s ([53]) a n d S p o h n ([48]).

14

It m u s t b e p o i n t e d out t h a t P o i n c a r e was n o i s o l a t e d figure. T h e gist of his a r g u m e n t c a n b e f o u n d i n r e m a r k a b l e d e t a i l a l r e a d y i n v o n K r i e s ([17], C h a p t e r III, S e c t i o n 2), a n d a p h i l o s o p h i c a l l y m o r e p e n e t r a t i n g assessment of the p r o b l e m is p r o v i d e d i n the b e a u t i f u l p a p e r of v o n S m o l u c h o w s k i ([47]).

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w h e n we have been e x p e c t i n g , at least w i t h considerable p r o b a b i l i t y , a diffe- rent course o f events. In order to a s c e r t a i n w h e t h e r s o m e t h i n g ( a "factor") is a cause o f t h e effect under d i s c u s s i o n , one has t o consider a l l / a large n u m b e r o f / a select set o f s i t u a t i o n s i n w h i c h the effect is changed a r b i t r a r i l y / w i t h sufficient v a r i a b i l i t y / a c c o r d i n g to c e r t a i n c r i t e r i a o f choice. If the factor under conside- r a t i o n m u s t have been absent u n i f o r m l y or w i t h sufficient frequency i n order to a c c c o m o d a t e t h e change i n the effects, t h e n i t is i n fact t o be regarded as a ("counterfactual") cause. Here we c o m p a r e a r e a l s i t u a t i o n w i t h — a c t u a l or c o u n t e r f a c t u a l — s i t u a t i o n s where everything except the putative effect is kept fixed.15

(End of list)

S o m e readers w i l l m i s s a reference t o t h e p o s i t i v e relevance theories as discussed i n c o n t e m p o r a r y approaches to " p r o b a b i l i s t i c c a u s a l i t y . " I n fact there w i l l be none. I have f o u n d t h a t the recent discussions i n this area are m o r e or less i r r e l e v a n t t o the t o p i c s treated i n t h e present p a p e r . T h e account defended below is m o r e l i k e a "causal t h e o r y o f p r o b a b i l i t y " ([29], p . 59) t h a n a " p r o b a b i l i s t i c t h e o r y o f c a u s a t i o n " .

T h e r e is a very s t r o n g i n t u i t i o n t h a t the " d e m o n i c " account o f c a u s a t i o n is the best one. T h e L a p l a c i a n d e m o n is i n a n i d e a l e p i s t e m i c s i t u a t i o n w h i c h we can o n l y e n v y .1 6 W h a t m o r e c o u l d one ask f r o m a n e x p l a n a t i o n t h a n t h e k i n d of i n f o r m a t i o n g i v e n b y the o m n i s c i e n t d e m o n ?

P r i m a facie, constant causes are very suspect entities for t h e a n a l y t i c p h i - losopher. H e or she w o u l d be a f r a i d o f i l l e g i t i m a t e l y h y p o s t a t i z i n g s o m e t h i n g w h i c h does n o t " r e a l l y e x i s t " , or does n o t have a n y s y s t e m a t i c or scientific s i - gnificance. B u t s t i l l i t is i n t u i t i v e l y very c o m p e l l i n g to t h i n k t h a t there m u s t be a cause (or p e r h a p s : a reason) for t h e r e g u l a r i t i e s i n observed r a t i o s . A f t e r a l l , this inference f r o m regularities i n observed frequencies to u n d e r l y i n g c a u s a l processes is a c e n t r a l m o t i v e for e m b r a c i n g the s t a t i s t i c a l m e t h o d .

However, I s h a l l n o t a d v o c a t e one p a r t i c u l a r concept o f c a u s a t i o n against another one as the true concept. A c t u a l l y I t h i n k t h a t this w o u l d at least i n p a r t be a q u a r r e l a b o u t t e r m i n o l o g i c a l questions d e v o i d o f any genuine p h i l o s o p h i c a l interest. R a t h e r , i t w i l l emerge i n t h e course of our a r g u m e n t t h a t c o u n t e r f a c t u a l c a u s a t i o n c a n help us o n o u r w a y f r o m L a p l a c i a n c a u s a t i o n t o the concept o f c a u s a t i o n w h i c h fits the L a p l a c i a n d e m o n . I n t h i s w a y we hope to be able to

15 In the analysis of conditional and causal connectives, the the contrast between the forward and the backward directions may be traced back to papers of Nelson Goodman ([10]) and Gilbert Ryle ([44]). It is discussed in Rott ([43]) where it is argued that the backward direction is preferable. A similar conclusion is reached by McCall ([28], p. 315). Also compare Gardenfors ([8], Chapters 8-9) where the anlysis of causality employs a con- traction with respect to causes, while explanations involve a contraction with respect to effects.

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help L a p l a c e out of the p r e d i c a m e n t o f the sorcerer's apprentice w h o c a n n o t take a firm h o l d of the ghost he has conjured u p .

3 How to Convince the Laplacian Intelligence of the Existence of Structural Causes

Counterfactual Reasoning

W e ask a b o u t a cause or a b o u t an e x p l a n a t i o n of a n event w h e n we d i d not expect i t to take place. T h i s is not the same as to say t h a t we e x p e c t e d t h a t the e x p l a n a n d u m event w o u l d not h a p p e n . W e merely t h i n k t h a t things might have t u r n e d out otherwise, t h a t is, t h a t the e x p l a n a n d u m event was not necessary.

If s o m e t h i n g has been necessary a l l the t i m e , we do not ask for its c a u s e .1 7 T h i s line o f reasoning s h o u l d be p l a u s i b l e even for the L a p l a c i a n i n t e l l i g e n c e . T r u e , i n the first instance the d e m o n does not p a r t i c u l a r l y concern i t s e l f a b o u t a l t e r n a t i v e w o r l d s or a l t e r n a t i v e developments o f our real w o r l d . B u t g r a n t e d the a b o v e - m e n t i o n e d p r a g m a t i c p r e s u p p o s i t i o n o f our search for causes, t h e intelligence can of course satisfy our c u r i o s i t y . If y o u ask, " W h y d i d this m i x t u r e of c h e m i c a l substances e x p l o d e ? " y o u i n d i c a t e t h a t y o u rather e x p e c t e d the w h o l e t h i n g to r e m a i n i n a stable c o n d i t i o n . T h e L a p l a c i a n intelligence c a n t e l l y o u w h a t sort of i n i t i a l c o n d i t i o n s w o u l d have to be g i v e n , i f the m i x t u r e were t o r e m a i n stable.

F o r the L a p l a c i a n intelligence, c o u n t e r f a c t u a l r e a s o n i n g i n the b a c k w a r d d i - r e c t i o n is not essentially different f r o m c o u n t e r f a c t u a l r e a s o n i n g i n the f o r w a r d d i r e c t i o n . R e m e m b e r t h a t by h y p o t h e s i s , i t is c a p a b l e of perfect d e t e r m i n i s t i c p r e d i c t i o n as w e l l as perfect d e t e r m i n i s t i c r e t r o d i c t i o n . T h i s applies t o i n i t i a l c o n d i t i o n s g i v e n i n the m a x i m a l specific d e s c r i p t i o n o f a m i c r o s t a t e as w e l l as to i n i t i a l c o n d i t i o n s given i n the f o r m o f macrostates. In the first case, the d e m o n has to c o m p u t e o n l y one trajectory, i n the second case, i t has to p e r f o r m the c o m p u t a t i o n o f a whole — a n d i n general, a very large — set o f trajectories.

T h i s is s t i l l M a x w e l l ' s ([27], p . 439) o p i n i o n . B u t there are differing views. P o p p e r ([38],

§§ 10-11) gives the d e m o n a r a t h e r h u m a n face a n d thus misrepresents, I t h i n k , L a p l a c e ' s intentions. S i m i l a r l y , P r i g o g i n e a n d Stengers ([39], p . 271) m a i n t a i n t h a t "[w]hen f a c e d w i t h . . . u n s t a b l e s y s t e m s , L a p l a c e ' s d e m o n is j u s t as powerless as we". S a l m o n ([45], p . 701) h o l d s t h a t the L a p l a c i a n intelligence m a y lack "knowledge of the m e c h a n i s m s of p r o d u c t i o n a n d p r o p a g a t i o n of s t r u c t u r e i n the w o r l d " , a n d S p o h n ([49], p . 186) c o n t e n d s t h a t i t lacks a n "inductive scheme" a n d h e n c e "would n o t k n o w w h a t to b e l i e v e , if i t were to discover t h a t i t is w r o n g " . W h i l e P o p p e r a n d P r i g o g i n e a n d Stengers d o u b t the d e m o n ' s p r e d i c t i v e c o m p e t e n c e , S a l m o n a n d S p o h n d e n y t h a t the intelligence's p r e d i c t i v e c o m p e t e n c e suffices for its e x p l a n a t o r y c o m p e t e n c e .

17 So far we are i n perfect a g r e e m e n t w i t h K a n t [15].

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A s the b a c k w a r d a n d f o r w a r d d i r e c t i o n s o f c o u n t e r f a c t u a l r e a s o n i n g are per- fectly s y m m e t r i c for the L a p l a c i a n intelligence, we m a y not o n l y ask " W h a t w o u l d have been different, i f the a c t u a l effect h a d not s h o w n u p ? " W e m a y as w e l l ask the converse question " W h a t w o u l d have been different, i f the a c t u a l i n i t i a l c o n d i t i o n s h a d been s l i g h t l y different?" F o r the L a p l a c i a n intelligence, these questions are not f u n d a m e n t a l l y different. F o r finite creatures, however, w h o l i k e us assume t h a t the causal d i r e c t i o n follows the arrow o f t i m e , it seems m o r e n a t u r a l to pose the second q u e s t i o n . W e c o u l d have changed the effect by m a n i p u l a t i n g the cause, but not vice versa. T h i s a c t i o n - o r i e n t e d i n t u i t i o n of c o n t r o l m a y be regarded as a g o o d reason for preferring c o u n t e r f a c t u a l reasoning i n the f o r w a r d d i r e c t i o n t h a n i n the b a c k w a r d d i r e c t i o n .1 8

T h e essential i d e a to be a p p l i e d i n our u r n m o d e l , however, is n e u t r a l w i t h respect t o the t e m p o r a l d i r e c t i o n of c o u n t e r f a c t u a l reasoning. F i x any set o f i n i t i a l c o n d i t i o n s . F o r the L a p l a c i a n intelligence, the whole sequence of d r a w i n g s (or, for t h a t m a t t e r , a sequence o f sequences of drawings) is j u s t a t i n y p a r t o f the course of the w h o l e d e t e r m i n i s t i c w o r l d . So e v e r y t h i n g , i n c l u d i n g the acts o f d r a w i n g , is settled f r o m the s t a r t .1 9 T h e n , of course, the L a p l a c i a n intelligence c a n p r e d i c t the results o f our sequence (of sequences) of d r a w i n g s . C l e a r l y , we c a n say " I f these-or-those i n i t i a l c o n d i t i o n s h a d o b t a i n e d , the result w o u l d have been s u c h - a n d - s u c h . " T h e p r o b l e m is w h i c h i n i t i a l c o n d i t i o n s to consider i n this d e t e r m i n i s t i c t h o u g h t e x p e r i m e n t .

W e y m a L i i b b e ([23]) considers v a r i a t i o n s of the colour r a t i o of the balls i n the u r n i n order to find out a b o u t its c a u s a l relevance. I do not t h i n k , however, t h a t t h i s is a very n a t u r a l line o f reasoning, or i n the s p i r i t of the L a p l a c i a n intelligence. A s I s a i d above, the c o l o u r r a t i o i n our series o f d r a w i n g s is e x p l a i n e d o n l y i n a p a r a s i t i c m a n n e r , v i z . , by first e x p l a i n i n g the fact t h a t these very balls were d r a w n a n d t h e n l o o k i n g at the colour of the b a l l s . T h e c o l o u r s do not figure i n the c a l c u l a t i o n o f the intelligence, so they are no genuine causes for i t . In our i n t u i t i v e d e l i b e r a t i o n s , t o o , we are not g o i n g to v a r y some s p e c i a l c a n d i d a t e cause (like the c o l o u r r a t i o ) , but we vary something. W e p r o d u c e as slight d e v i a t i o n s f r o m the a c t u a l i n i t i a l c o n d i t i o n s as possible. W e c o u n t e r f a c t u a l l y shift b a l l N o 38 h a l f a m i l l i m e t e r to the r i g h t , give N o 82 a l i t t l e p u s h , remove a t i n y unevenness i n the u r n ' s w a l l , or p e r h a p s we exchange the p o s i t i o n s of balls N o 97 a n d N o 98. T h e full d e s c r i p t i o n o f the w o r l d , as far

18 Contra Ryle, McCall and Rott. — Why do people on trains prefer seats facing the engine?

19 The idea of interpreting repeated experiments as a part of one large experiment has also been used by von Plato ([31], p. 65, [33], p. 45) in his attempt to reconcile the initial- probability account (focusing on abstract "ensembles") with the time-average account suggested by ergodic theory. Von Plato claims that "objective" distributions of initial conditions can be derived with the help of the new physico-mathematical theories of ergodicity. If this were true, the explication of chance would stay completely in the realm of the objective.

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as i t enters i n t o the d e m o n ' s c a l c u l a t i o n s , s h o u l d be counted as the cause of the effect. E v e r y conceivable d e v i a t i o n f r o m the i n i t i a l state s h o u l d be t a k e n i n t o c o n s i d e r a t i o n . S l i g h t d e v i a t i o n s s h o u l d receive more serious c o n s i d e r a t i o n t h a n d r a s t i c a n d p h a n t a s t i c ones. V a r i a t o n o f the colour r a t i o w i l l not be the obvious i d e a .2 0

M y i n t u i t i o n s , t h o u g h not p a r t i c u l a r l y f i r m , suggest t h a t c o u n t e r f a c t u a l re- asoning i n o u r case does not a c c o r d w e l l w i t h the s i m p l e schemes prepared i n the p r e v i o u s section. It seems to me t h a t we reflect o n h y p o t h e t i c a l changes of any arbitrary kind r e l a t i n g to the " i n i t i a l t i m e " when the e x p l a i n i n g event is s u p p p o s e d to have o c c u r r e d . W e do not consider changes w i t h a s p e c i a l v i e w to some preconceived c a n d i d a t e cause, b u t w i d e s p r e a d p r o b a b i l i t y d i s t r i b u t i o n s over a great v a r i e t y of possible d e v i a t i o n s f r o m the a c t u a l course of events.

Subjectivity

N o E q u a l i t y

W e want to k n o w the p r o b a b i l i t i e s for the outcomes o f our d r a w i n g s . T o c o m p u t e t h e m , we need to s u p p l y the L a p l a c i a n intelligence w i t h a p r i o r d i s t r i b u t i o n for i n i t i a l states. O n the basis of t h i s i n f o r m a t i o n , a n d o n l y o n the basis of t h i s i n f o r m a t i o n , c a n a l l subsequent d e t e r m i n i s t i c c a l c u l a t i o n s be based.

T h e first a n d most o b v i o u s i d e a is to say t h a t a l l possible i n i t i a l states s h o u l d be assigned a n equal p r o b a b i l i t y . T h i s is the answer L a p l a c e w o u l d have g i v e n . B u t u n f o r t u n a t e l y , the answer is not well-defined. A s a l r e a d y p o i n t e d out by v o n K r i e s ([17], C h a p t e r I, S e c t i o n 4), i t is not clear, even w h e n we are d e a l i n g w i t h finite spaces of p o s s i b i l i t y , which elements to count as e q u i p r o b a b l e . T h e p r o b l e m raises its head w i t h m u c h m o r e severity w h e n infinite p o s s i b i l i t y spaces are to be considered. T h i s is borne out by the famous p a r a d o x of B e r t r a n d . I n his i n f l u e n t i a l Calcul des Probabilites o f 1899, he showed t h a t there are three e q u a l l y p l a u s i b l e ways of e s t i m a t i n g the p r o b a b l i t y t h a t the l e n g t h of a c h o r d i n a circle is greater t h a n the l e n g t h o f the sides of an i n s c r i b e d e q u i l a t e r a l t r i a n g l e

— b u t t h a t the three m e t h o d s y i e l d three different p r o b a b i l i t y values.

N o w , pure geometry has c e r t a i n l y never been the i n d e n d e d d o m a i n o f a p p l i - c a t i o n o f p r o b a b i l i t y theory, a n d i n p r a c t i c a l p r o b l e m s of a s i m i l a r s t r u c t u r e we m a y expect h i n t s f r o m the e m p i r i c a l l y g i v e n Versuchsanordnung w h a t the r i g h t f o r m a l representation of e q u i p o s s i b i l i t y w i l l be. B u t the p r o b l e m nevertheless

20 What would have been the result of changing the colour ratio in the urn of our intro- ductory example by substituting a few white balls for black ones (or vice versa)? It may well be that we finally happen to come up with exactly the same colour ratio, 52:48, in our resulting sequence of drawings — although the a priori probability of doing so has increased or decreased. Is, then, the colour ratio in the urn a cause for the colour ratio in that particular sequence?

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r e m a i n s acute. W h a t is the j u s t i f i c a t i o n for c o u n t i n g c e r t a i n i n i t i a l c o n d i t i o n s as m o r e or less p r o b a b l e ? I take i t t h a t B e r t r a n d m a d e clear once a n d for a l l t h a t there is n o absolute sense o f e q u i p r o b a b i l i t y ( e q u i p o s s i b i l i t y ) , n o o b j e c t i v e f o o t h o l d for d e c i d i n g w h i c h frame o f reference t o use for t h e fixation o f e q u i p r o - b a b i l i t y . It is n o t at a l l o b v i o u s how t o choose the d i s t r i b u t i o n over the i n i t i a l states. It seems, therefore, t h a t t h e p r o b l e m necessarily involves subjective or

conventionalist ( P o i n c a r e ) elements.

S i m i l a r i t y

T h i s is n o t as b a d as i t m a y a p p e a r . I n considerations o f c a u s a l i t y , we do n o t e n t e r t a i n t h e a s s u m p t i o n t h a t t h e w o r l d m i g h t have been totally different f r o m w h a t i t is l i k e a c t u a l l y . W e do n o t r e c k o n w i t h d r a m a t i c d e v i a t i o n s f r o m the a c t u a l state o f affairs, or w i t h m a r k e d l y " e x o t i c " i n i t i a l c o n d i t i o n s . I n o u r u r n e x a m p l e we do n o t consider i t a serious p o s s i b i l i t y t h a t the u r n ' s o p e n i n g is closed, t h a t someone reshapes t h e b a l l s i n t o cubes, t h a t there are o n l y three i n - stead o f 100 b a l l s i n the u r n , t h a t t h e b a l l s are green a n d r e d r a t h e r t h a n b l a c k a n d w h i t e , t h a t some h i d d e n m a g n e t i c c o n t r i v a n c e has been i n s t a l l e d i n order to i n t r o d u c e a bias, etc. W e j u s t t h i n k t h a t t h i n g s m i g h t have been somewhat different. T h a t is, i n m o s t cases ( t h o u g h p e r h a p s less c h a r a c t e r i s t i c a l l y i n g a - mes o f chance) we are i n c l i n e d t o w e i g h i n i t i a l c o n d i t i o n s a c c o r d i n g t o h o w close they come t o the a c t u a l i n i t i a l c o n d i t i o n s , or t o some s t a n d a r d o f " n o r m a l i t y " . W h e n l o o k i n g for causes, we t r y to let i n t a c t a large n u m b e r o f ceteris paribus c o n d i t i o n s w h i c h o n l y m a k e the causes effective at a l l . I n the set o f a l l concei- vable i n i t i a l c o n d i t i o n s , s i n g u l a r i t i e s i n p h y s i c a l state spaces w i l l be a t t r i b u t e d a p r o b a b i l i t y measure zero.

I n c a u s a l r e a s o n i n g , t h e n , we invoke a n o t i o n o f s i m i l a r i t y i n the style o f D a v i d L e w i s ([21], [22]). T h e r e are three p o i n t s o n w h i c h o u r suggestion dif- fers f r o m L e w i s , however. F i r s t , L e w i s ' s s i m i l a r i t y takes the f o r m o f a t e r n a r y r e l a t i o n o f c o m p a r a t i v e s i m i l a r i t y between " w o r l d s " or states. It is o n l y the worlds w h i c h are closest to the a c t u a l one t h a t m a t t e r i n L e w i s ' s semantics, a l l t h e other w o r l d s do n o t have a n y relevance. T h i s a l l - o r - n o t h i n g p r i n c i p l e for c o u n t i n g w o r l d s alias states seems t o o r e s t r i c t i v e . W e w o u l d r a t h e r like t o a t t r i b u t e different weights t o different states, so as to m a k e m o r e r e m o t e possi- b i l i t i e s count less t h a n close p o s s i b i l i t i e s , b u t s t i l l have some non-zero influence.

W e propose t o measure the relevance o f states w i t h the help o f a s m o o t h pro- b a b i l i t y d i s t r i b u t i o n over the set o f a l l possible i n i t i a l states. S e c o n d , i t seems t h a t distance f r o m t h e a c t u a l w o r l d is n o t the o n l y t h i n g t h a t m a t t e r s . I f the real w o r l d h a p p e n s t o e x h i b i t e x o t i c features i n some s p e c i a l s i t u a t i o n , t h e n we are i n c l i n e d t o give m o r e weight t o the p o s s i b i l i t y t h a t t h i n g s m i g h t have been more o r d i n a r y t h a n t o the p o s s i b i l i t y t h a t t h i n g s m i g h t have been s t i l l m o r e e x o t i c . W h a t counts, then, is some c o m p o u n d o f closeness t o t h e a c t u a l w o r l d a n d closeness t o a s t a n d a r d o f n o r m a l i t y .2 1 T h i r d , I do n o t see a n y o b j e c t i v e

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basis for s i m i l a r i t i e s between possible w o r l d s , or even possible courses of the a c t u a l w o r l d . S i m i l a r i t y is always s i m i l a r i t y - i n - a - c e r t a i n - r e s p e c t , a n d there is a huge n u m b e r o f possible respects to p a y a t t e n t i o n to, b u t there is no u n i v e r s a l rule how to c o m b i n e s i m i l a r i t i e s a n d d i s s i m i l a r i t i e s i n different respects i n t o a p l a u s i b l e o v e r a l l s i m i l a r i t y . It seems o b v i o u s , t h e n , t h a t s i m i l a r i t y is a hope- lessly subjective n o t i o n . B u t to say t h i s is to aggravate the p r o b l e m t h a t has a l r e a d y been l i n g e r i n g since our d i s c u s s i o n of B e r t r a n d ' s p a r a d o x . W e cannot hope to convince the L a p l a c i a n intelligence of our o w n s u b j e c t i v e s t a n d a r d s o f s i m i l a r i t y a n d n o r m a l i t y . T h i s w o u l d definitely o v e r s t r a i n its readiness to en- gage i n a n t h r o p o m o r p h i c p a t t e r n s o f r e a s o n i n g . W e s o m e h o w have to t r a n s c e n d the r e a l m o f the p u r e l y s u b j e c t i v e .

A s s i g n i n g weights to i n i t i a l c o n d i t i o n s i n the f o r m of p r o b a b i l i t i e s (density functions) m i g h t t e m p t one to t h i n k t h a t we are d e a l i n g here w i t h a f o r m o f u n c e r t a i n t y . Indeed the p r o b l e m o f u n c e r t a i n t y is stressed i n m u c h o f the rele- vant l i t e r a t u r e . B u t i t is essential to u n d e r s t a n d t h a t i t is not o u r concern here t h a t we are never able to measure the a c t u a l i n i t i a l c o n d i t i o n s w i t h absolute p r e c i s i o n . T r u e as this is, i t is not at a l l w h a t we are interested i n . W e do not even a i m at a precise d e t e r m i n a t i o n of the e m p i r i c a l d a t a . I n the c o u n t e r f a c t u a l d e l i b e r a t i o n s i n v o l v e d i n c a u s a l r e a s o n i n g , we are g o i n g to v a r y the i n i t i a l c o n d i - t i o n s a n y w a y — a n d we do so i n accordance w i t h some i r r e m e d i a b l y s u b j e c t i v e s t a n d a r d s o f s i m i l a r i t y a n d n o r m a l i t y .

Objectivity Regained

C o n s i d e r a single draw of a b a l l f r o m the u r n . A s s u m e t h a t the i n i t i a l c o n d i t i o n s are fixed b u t u n k n o w n . T h e L a p l a c i a n intelligence c a n specify the r e g i o n i n the state space o f i n i t i a l c o n d i t i o n s w h i c h leads to the result t h a t b a l l N o 01 is d r a w n ; the r e g i o n t h a t leads to the d r a w i n g of b a l l N o 02; the r e g i o n t h a t leads to the d r a w i n g of b a l l N o 03; etc. etc. C a l l the set o f a l l p o i n t s i n the state space of a " c h a n c e - p r o d u c i n g " d e t e r m i n i s t i c s y s t e m w h i c h are b o u n d to p r o d u c e one a n d the same effect (e.g., " N o 01 is d r a w n " ) an equivalence region. M y c l a i m now is t h a t the equivalence regions f o r m fractal structures i n the space o f a l l possible i n i t i a l c o n d i t i o n s . B y a p p r o a c h i n g the a c t u a l i n i t i a l state ( a c c o r d i n g to some s t a n d a r d of closeness) we do not i n the same way a p p r o a c h the a c t u a l result of the d r a w i n g . W e c a n n o t resolve the e x t r e m e l y i n v o l v e d s t r u c t u r e o f the equivalence regions by u s i n g a huge m a g n i f y i n g glass. T h e c o m p l e x i t y of the regions repeats itself at a l l scales of m a g n i f i c a t i o n . T h i s is a s p e c u l a t i o n so far, but i t s h o u l d present o n l y t e c h n i c a l p r o b l e m s to e s t a b l i s h the p o i n t r a t h e r m o r e r i g o r o u s l y .

21 For a discussion of these two kinds of closeness, viewed as two out of five "faces of minimahty", see [25].

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T h e essential p a r t o f the a r g u m e n t consists o f the f o l l o w i n g i d e a :

For any (subjective) standard of closeness and for every initial state close to the actual one, there is a huge number of initial states still closer to the actual one which lead to entirely different results (i.e., different from the actual result, different from the result effected by the initial state

mentioned first, and different from each other).

T h e e x t r e m e s e n s i t i v i t y o f the results t o m i n u t e v a r i a t i o n s o f the i n i t i a l c o n d i t i o n s has a d o u b l e effect: T h e equivalence regions are, first, i n t r i n s i c a l l y c o m p l e x a n d , second, o f quite the same s i z e .2 2 T h u s we arrive at the f o l l o w i n g t r a n s f o r m a t i o n o f the above idea:

Any probability distribution over initial states mirroring our subjective standards of closeness and normality passes smoothly over the equiva- lence regions which are of bizarre shape but of equal size. Because these regions form an extremely — even infinitely — complex, criss-crossing fi- ligree, every such probability distribution will in the end lead to the same probability distribution over the space of possible effects.

T h i s is a n invariance argument. O n the basis o f w h a t seems t o be a n i n - nocent m a t h e m a t i c a l a s s u m p t i o n — the a s s u m p t i o n o f a k i n d o f s m o o t h n e s s of the i n i t i a l d i s t r i b u t i o n — i t becomes evident t h a t the final d i s t r i b u t i o n w i l l c o r r e s p o n d e x a c t l y t o the respective n u m b e r s o f b a l l s i n the u r n . T h i s c a n be a t t r i b u t e d t o objective, n o n - p r o b a b i l i s t i c features o f the chance set-up. W e have fifty w h i t e b a l l s a n d fifty black b a l l s , a n d we assume t h a t there is n o b u i l t - i n m e c h a n i s m p r o d u c i n g a n y bias. F o r every i n i t i a l p r o b a b i l i t y d i s t r i b u t i o n m i r r o - r i n g o u r s u b j e c t i v e s t a n d a r d s o f closeness, t h e n , each b a l l ' s p r o b a b i l i t y o f b e i n g d r a w n i n a single d r a w i n g w i l l be 1:100. T h e p r o b a b i l i t y t h a t a w h i t e b a l l is d r a w n i n a single d r a w i n g w i l l be 50:100, a n d so o n .

T h e o b j e c t i v i z a t i o n o f the final p r o b a b i l i t i e s o f o u r d r a w i n g s is due to the fact t h a t a l l k i n d s o f well-behaved subjective d i s t r i b u t i o n s over i n i t i a l c o n d i t i o n s lead — after sufficient m i x i n g — to the s a m e r e s u l t a n t d i s t r i b u t i o n . T h i s a g a i n is due to the chaotic b e h a v i o u r o f the s y s t e m . W h a t we get i n the e n d are single- case p r o b a b i l i t i e s2 3 w h i c h are objective i n the sense o f being i n t e r s u b j e c t i v e b u t not i n the sense t h a t they are due t o i r r e d u c i b l y i n d e t e r m i n i s t i c processes.

4 Maxwell and Poincare: Some Historical Remarks

A s p o i n t e d o u t b y K r i i g e r ([18]), J a m e s C l e r k M a x w e l l was p e r h a p s the first t o perceive very c l e a r l y t h a t the n a t u r a l sciences i n the m i d d l e o f the 19th c e n t u r y

22 Cf. von Smoluchowski's ([47], p. 87) somewhat technical definition of what it means that a "causal relation" y = f(x) has "'oscillating' character".

23 Von Smoluchowski ([47], p. 85) uses the term "Wahrscheirdichkeit schlechthin" ("proba- bility as such").

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were a b o u t to a p p l y two m e t h o d s w h i c h were, o n the face of i t , f u n d a m e n t a l l y different. W e quote this at f u l l l e n g t h :

T h e modern atomists have therefore adopted a method which is, I believe, new in the department of mathematical physics, though it has long been in use in the section of Statistics. W h e n the working members of Section F get hold of a report of the Census, or any other document containing the numerical data of Economic and Social Science, they begin by distributing the whole population into groups, according to age, income-tax, education, religious belief, or criminal convictions. T h e number of individuals is far too great to allow of their tracing the history of each separately, so that, in order to reduce their labour within human limits, they concentrate their attention on a small number of artificial groups. The varying number of individuals in each group, and not the varying state of each individual, is the primary datum from which they work. This, of course, is not the only method of studying human nature. We may observe the conduct of individual men and compare it with that conduct which their previous character and their present circumstances, according to the best existing theory, would lead us to expect. Those who practise this method endeavour to improve their knowledge of the elements of human nature in much the same way as an astronomer corrects the elements of a planet by comparing its actual position with that deduced from the received elements.

T h e study of human nature by parents and schoolmasters, by historians and statesmen, is therefore to be distinguished from that carried on by registrars and tabulators, and by those statesmen who put their faith in figures. T h e one may be called the historical, and the other the statistical method. ([26], p p . 373-374)

I n an u n p u b l i s h e d paper of the s a m e year, M a x w e l l ([27], p . 438) elaborates o n the d i s t i n c t i o n of the t w o m e t h o d s , now called "two k i n d s of k n o w l e d g e ,

. . . the D y n a m i c a l a n d the S t a t i s t i c a l " 2 4

W h a t is interesting is t h a t M a x w e l l establishes a c o n n e c t i o n w i t h the d i - s t i n c t i o n between " h i s t o r i c a l " a n d " p r o p h e t i c a l " i n q u i r y , a n d w i t h the contrast between stable a n d u n s t a b l e systems. H e calls a s y s t e m unstable " w h e n a n i n - f i n i t e l y s m a l l v a r i a t i o n i n the present state m a y b r i n g a b o u t a finite difference i n the state of the s y s t e m i n a finite t i m e " ([27], p . 440); otherwise i t is c a l l e d stable. In a l l p r a c t i c a l a p p l i c a t i o n s o f physics, the metaphysical axiom " t h a t f r o m the same antecedents follow the same consequents" has to be r e p l a c e d by the physical axiom "[t]hat f r o m like antecedents follow like consequents". S i m i - lar d i s t i n c t i o n s were to be d r a w n i n the 2 0 t h century under the names " ' w e a k ' d e t e r m i n a c y " vs. " ' s t r o n g ' d e t e r m i n a c y " by B o r n ([3], p . 80), " m e t a p h y s i c a l d e t e r m i n i s m " vs. " ' s c i e n t i f i c ' d e t e r m i n i s m " by P o p p e r ([38], § 1), a n d "weak

24 The distinction has been a subject of lively discussion ever since. See for instance Planck ([30]) and Levi-Strauss ([20], 528-531).

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p r i n c i p l e of c a u s a l i t y " vs. "strong p r i n c i p l e o f c a u s a l i t y " by D e k e r a n d T h o m a s ([5]).2 5 T h e c e n t r a l role of the s t r o n g a x i o m s i n everyday reasoning as well as science has often been stressed, for e x a m p l e by Q u i n e :

Every reasonable expectation depends on resemblance of circumstances, to- gether with our tendency to expect similar causes to have similar effects. ([42], p . 117)

a n d by G l e i c k :

Scientists marching under Newton's banner actually waved another flag that said something like this: Given an approximate knowledge of a system's initial conditions and an understanding of natural law, one can calculate the approxi- mate behavior of the system. ([9], p . 117)

I n u n s t a b l e s y s t e m s a n d i n systems near some s i n g u l a r i t y , M a x w e l l ' s " p h y s i - c a l " a x i o m is s i m p l y false. M a x w e l l wants to u n d e r m i n e the c o m m o n "prejudice i n favour o f d e t e r m i n i s m " ([27], p . 444), b u t he does not o u t r i g h t reject deter- m i n i s m . It can be conceded t h a t m e t a p h y s i c a l d e t e r m i n i s m is a d o c t r i n e w h i c h is n o t scientifically testable (because we never meet exactly the same i n i t i a l c o n - d i t i o n s twice) a n d perhaps is altogether useless for the a d v a n c e m e n t o f science.

Nevertheless, i t is a s u b s t a n t i a l p h i l o s o p h i c a l d o c t r i n e for the m e r i t s a n d de- m e r i t s of w h i c h i t is i n t e r e s t i n g to argue — o n the basis o f current scientific k n o w l e d g e .

M a x w e l l does not m a k e a s u s t a i n e d effort at e x p l i c a t i n g the characteristics o f the s t a t i s t i c a l m e t h o d a n d its precise r e l a t i o n to the h i s t o r i c a l / d y n a m i c a l m e t h o d , nor does he argue i n s u p p o r t of a l i n k a g e between t h i s f u n d a m e n t a l d i s t i n c t i o n a n d his e m p h a s i s on i n s t a b i l i t i e s .

B u t as already m e n t i o n e d , I t h i n k t h a t P o i n c a r e ' s m e t h o d o f a r b i t r a r y func- t i o n s answers the questions s u r r o u n d i n g M a x w e l l ' s diagnosis very generally a n d w i t h s u r p r i s i n g effectiveness. H i s p a r a d i g m a t i c e x a m p l e is a n i d e a l i z e d roulette, i.e., a game of chance, b u t he suggests t h a t the m e t h o d has a m u c h w i d e r field of a p p l i c a t i o n . I n c o n n e c t i n g d y n a m i c a n d s t a t i s t i c a l reasoning i t is first i m p o r t a n t w h a t to count as chance, or a "fortuitous p h e n o m e n o n " :

A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that that effect is due to chance.

. . . it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. ([37], p p . 6 7 - 6 8 )

T h i s p i c t u r e c l e a r l y reconciles L a p l a c i a n d e t e r m i n i s m w i t h the s e e m i n g l y r a n d o m result o f the d r a w i n g s of our u r n e x p e r i m e n t . It m u s t be stressed t h a t

25 The predicates "weak" and "strong" are slightly misleading, since the strong principle does not logically entail the weak one.

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P o i n c a r e does n o t at a l l renounce d e t e r m i n i s m .2 6 H e r a t h e r suggests a k i n d o f r e d u c i b i l i t y o f chance t o m e c h a n i s t i c p a t t e r n s , or m o r e e x a c t l y , the r e s o l u t i o n of the appearance of chance i n a d e t e r m i n i s t i c w o r l d p i c t u r e .

W h y is t h i s a s o l u t i o n o f t h e p r o b l e m posed by M a x w e l l a n d recently rein- forced b y K r i i g e r ? W h a t has t h i s achievement got t o d o w i t h the question o f whether i t is reasonable t o t a l k o f "constant causes"? T h e l a t t e r t e r m is f r o m the r e a l m o f the s t a t i s t i c a l m e t h o d . P o i n c a r e i n effect showed h o w a "struc- t u r a l " ( p h y s i c a l , abstract, n o n - p r o b a b i l i s t i c ) d e s c r i p t i o n o f the s y s t e m under c o n s i d e r a t i o n ( i n our case, o f the u r n c o n t a i n i n g w h i t e a n d b l a c k b a l l s ) can "ex- p l a i n " or "cause" the expected p r o b a b i l i t y d i s t r i b u t i o n s o f the results o f single, i n d e p e n d e n t d r a w i n g s . A s t h e e x p l a n a t i o n consists i n t h e c a l c u l a t i o n o f trajec- tories i n t h e state space, c a u s a t i o n is t o be u n d e r s t o o d essentially i n terms o f dynamical processes. T h e final step l e a d i n g f r o m p r o b a b i l i t i e s o f single d r a w i n g s to the c o n s t a n c y of s t a t i s t i c a l b e h a v i o u r can then be t a k e n w i t h the h e l p of the m a t h e m a t i c a l p r o b a b i l i t y c a l c u l u s . It w i l l tell us t h a t t h e c o l o u r r a t i o i n the sequence o f d r a w i n g s w i l l very probably be approximately e q u a l t o t h e single case p r o b a b i l i t i e s . N o t i c e t h a t i n a c c e p t i n g this as a causal e x p l a n a t i o n , we t a c i t l y m a k e use o f a fifth concept o f e x p l a n a t i o n based o n t h e high probability of a n imprecisely specified effect. T h i s concludes o u r r a t i o n a l r e c o n s t r u c t i o n o f L a p l a c e ' s d o c t r i n e o f constant causes.

5 Essentials of Chaos Theory

T h e a t t e n t i o n p a i d b y M a x w e l l t o i n s t a b i l i t i e s a n d s i n g u l a r i t i e s is a s y m p t o m o f the increased interest o f p h y s i c i s t s of the t i m e i n c o m p l e x d y n a m i c a l s y s t e m s .2 7 B y t h e e n d o f the 19th century, i t was n o t o n l y P o i n c a r e w h o c o n t r i b u t e d t o i m p o r t a n t progress i n the field. S o f y a K o v a l e v s k a y a ([16]) a n d A l e k s a n d r L y a p u - nov ([24]) w r o t e classic papers o n the concept of s t a b i l i t y i n d y n a m i c a l systems, a n d Jacques H a d a m a r d ([12]) p r o v e d a result w h i c h is m a d e chief witness for the i m p a c t o f u n p r e d i c t a b i l i t y i n c l a s s i c a l physics by D u h e m ([6], C h a p t e r 7) a n d P o p p e r ([38], § 14).

However, I d o n o t t h i n k t h a t the w o r k o f M a x w e l l , P o i n c a r e , a n d others a c t u a l l y c o n t r a d i c t s the L a p l a c i a n c o n c e p t i o n o f the w o r l d . T h e r e is o n l y a change of e m p h a s i s . W h i l e L a p l a c e m a y be u n d e r s t o o d as i m p l y i n g t h a t stable

26 This is very explicit in Poincare ([37], pp. 64-65, 70). A similar view is expressed by von Smohichowski ([47], p. 80).

27 Maxwell may have been influenced by Barre de Saint-Venant (cf. [11], p. 464). Maxwell ([27]) could be read as a proclamation of the programme of chaos theory, were it not for the fact that his principal concerns were the mass phenomena of the statistical theory of gases. However, he does refer to one very simple system exhibiting chaotic behaviour in Maxwell ([27], p. 442): the singularity in the refraction of light in a biaxial crystal.

(17)

a n d p r e d i c t a b l e systems are the r u l e a n d i n s t a b i l i t i e s the e x c e p t i o n , M a x w e l l a n d i n p a r t i c u l a r P o i n c a r e become aware t h a t the t r u t h is rather the other w a y r o u n d . It is the m e r i t of chaos t h e o r y t h a t t h i s i n s i g h t has g a i n e d w i d e s p r e a d r e c o g n i t i o n t o d a y .

M o d e r n chaos theory has a definite s t a r t i n g p o i n t . It began i n 1963 w i t h the p u b l i c a t i o n o f E d w a r d L o r e n z ' s a r t i c l e o n " D e t e r m i n i s t i c n o n p e r i o d i c flow"

i n the Journal of the Atmospheric Sciences 2 6 Since the '80s at least, chaos t h e o r y has been very fashionable a n d a t t r a c t e d the interest of a l m o s t every scientific d i s c i p l i n e . F o r our purposes a k n o w l e d g e of chaos t h e o r y p r o v i d e d by g o o d p o p u l a r presentations such as D e k e r a n d T h o m a s ([5]), C r u t c h f i e l d et a l . ([4]), Jensen ([14]), or G l e i c k ([9]) is sufficient. S o m e of the m o s t essential c h a r a c t e r i s t i c s of chaos theory are the f o l l o w i n g :

• E v e n very s i m p l e d e t e r m i n i s t i c systems c a n e x h i b i t an i n c u r a b l y u n p r e d i c - t a b l e b e h a v i o u r , a n d this not o n l y at a few i s o l a t e d p o i n t s o f s i n g u l a r i t y b u t over a w i d e range o f possible states. It does not help to gather m o r e infor- m a t i o n a b o u t the i n i t i a l c o n d i t i o n s of such " c h a o t i c " s y s t e m s . A perfectly exact s t a t e m e n t of the i n i t i a l c o n d i t i o n s cannot be o b t a i n e d ( u n c e r t a i n t y l i - m i t a t i o n ) , a n d a n a n a l y t i c a l s o l u t i o n of the equations i n v o l v e d is i m p o s s i b l e ( c o m p l e x i t y l i m i t a t i o n ) . E v e n the s m a l l e s t errors i n t r o d u c e d by i n c o m p l e t e k n o w l e d g e or a p p r o x i m a t i o n techniques t e n d to have disastrous consequences for the c a l c u l a t i o n of the further d e v e l o p m e n t o f the s y s t e m .

• V e r y m a n y real-life systems f u n c t i o n as chance devices, by a m p l i f y i n g m i n u t e differences i n t o extensive m a c r o s c o p i c effects. I t e r a t i v e processes, like c o l l i s i - ons i n the m i x i n g o f the balls i n our u r n , l e a d to a n e x p o n e n t i a l a m p l i f i c a t i o n of i n i t i a l differences.

• T h e d e v e l o p m e n t of chaotic systems is e x t r e m e l y sensitive to the s m a l l e s t changes of parameters or i n i t i a l c o n d i t i o n s ; like causes do not have like effects.

W h e n the d e v e l o p m e n t of i n d i v i d u a l cases (as opposed to mass p h e n o m e n a ) is considered, one m u s t not assume t h a t s m a l l p e r t u r b a t i o n s ( " v a r i a b l e causes") cancel out i n the l o n g r u n . O n the c o n t r a r y , they w i l l quite often b u i l d up to c l e a r l y p e r c e p t i b l e effects. M a x w e l l ' s p h y s i c a l a x i o m is v i o l a t e d , a n d a fortiori P o p p e r ' s ([38], § 3) " p r i n c i p l e of a c c o u n t a b i l i t y " becomes a d e m a n d t h a t c a n n o t be m e t .

• M a n y s t r i c t l y d e t e r m i n i s t i c s y s t e m s e x h i b i t w h a t appears to be perfectly r a n d o m b e h a v i o u r .2 9 T h u s d e t e r m i n i s m a n d p r e d i c t a b i l i t y , w h i c h h a d so often been t a k e n to m e a n the same t h i n g , get separated. It is o n l y the l a t t e r

28 In so far as he insists on the unpredictable behaviour of very simple deterministic systems, one could think of calling Max Born a forerunner of modern chaos theory. However, one should pay attention to the fact that while Born dissociates causality from determinism (cf. [1], p. 78) but not predictability from determinism, the upshot of chaos theory goes just the other way round: its concept of causality is reflected in deterministic difference or differential equations, but predictability is emphatically denied.

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