zeno's second argument against plurality

5/26/2018 Zeno's Second Argument against Plurality

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Zeno's Second Argument against Plurality

Peterson, Sandra Lynne.

Journal of the History of Philosophy, Volume 16, Number 3, July

1978, pp. 261-270 (Article)

Published by The Johns Hopkins University Press

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Z eno s Second A rgum entagainst PluralityS N D R P E T E R S O N

1. I N TR O D U C TI O N P r o c l u s r e p o r t s t h a t Z e n o g a v e f o r t y a r g u m e n t s a g a i n s t p l u -r a l i t y . ' P l a t o r e p o r t s t h a t t h e s e w e r e i n s u p p o r t o f th e th e s is o f Z e n o ' s c o l l e a g u e P a r -m e n i d e s t h a t t h e r e i s e x a c t l y o n e b e i n g . 2 I t a k e i t t h a t w e a r e t o u n d e r s t a n d t h e P a r -m e n i d e a n t h e s is to b e i n c o m p a t i b l e w i t h Z e n o ' s b e i n g d i s ti n c t f r o m P a r m e n i d e s a n dZ e n o ' s g i v i n g f o r t y a r g u m e n t s a g a i n s t p l u r a l i t y . 32 . A N E W R E C O N S TR U C T IO N Z e n o ' s s e c o n d a r g u m e n t a g a i n s t p l u r a l i t y a t t e m p t st o r e d u c e t h e a s s u m p t i o n t h a t t h e re a r e m a n y t o th e a b s u r d i t y t h a t t h e r e w o u l d b e b o t hf i n i t e l y m a n y a n d i n f i n i t e l y m a n y . I o f f e r a r e c o n s t r u c t i o n o f th e p a r t o f t h e a r g u m e n tf o r t h e c o n s e q u e n c e t h a t t h e r e w o u l d b e f i n i t e l y m a n y . T h e t e x t is :I f th e r e a r e m a n y , i t i s n e c e s s a r y th a t t h e y b e a s m a n y a s t h e y a r e , n e i t h e r m o r e n o r f e w e r . B u t i ft h e y a r e a s m a n y a s t h e y a r e , t h e y m u s t b e f i n it e ly m a n y . 4

I f t h e r e a r e m a n y i s t h e a s s u m p t i o n f o r r e d u c t i o n t o a b s u r d i t y . T h e i t is n e c e s -s a r y i n d i c a t e s a n i m p l i c a t i o n o f t h e r e b e i n g m a n y : w h a t e v e r n u m b e r x i s , i f x i sg r e a t e r t h a n 1 , i t i s e x a c t l y w h a t i t i s , t h a t i s , x = x . N e i t h e r m o r e n o r f e w e r I t a k et o b e a n e x p l i c a t i o n o f x ' s b e i n g a s m a n y a s i t i s: f o r a n y n u m b e r n , g r e a t e r t h a n 0 ,x :~ x + n a n d x ~ x - n . F o r t h e p u r p o s e o f t h e r e c o n s t r u c t i o n w e C a n u s e a s i n g l e i n -s t a n c e o f t h e g e n e r a l c l a i m f o r t h e r e s u l t o f a d d i t i o n : x : ~ x + 1 . T h e y m u s t b e f i n i te -l y m a n y i s t h e c o n c l u s i o n t h a t t h e n u m b e r o f t h i n g s t h e r e a r e is f i n it e .

M y r e - e x p r e s s i o n o f Z e n o ' s a r g u m e n t h a s m a d e u s e o f t h e w o r d n u m b e r ; t h eG r e e k 6 p t 0 p o q d o e s n o t o c c u r i n t h e te x t o f t h e s e c o n d a r g u m e n t a g a i n s t p l u r a l i t y , n o ra n y w h e r e e l se i n t h e F r a g m e n t e f r o m Z e n o . U s e o f n u m b e r a n d o f t h e v a r i a b l e x ,

The label seco nd argum ent against p lu ra l i ty i s f rom E n c y c l o p e d i a o f P h i l o s o p h y , Edwards ed. , s .v .Z en o of El ea (wri tten by G. Vlastos).I H . D ie ls and W . K ranz , D i e F r a g m e n t e d e r V o r s o k r a ti k e r , 12th ed. (Dublin, 1966), 29a15.2 Prin. 128c-d.3 G . E . L . O w en ( Z en o an d t h e M a t h em a t i c i an s , P r o c e e d i n g s o f t h e A r i s t o te l i a n S o c i e t y 58 [195 8] : 199)says tha t Ze no cer ta in ly he ld . . . tha t there i s on ly one th ing in ex is tence . H . Chern iss (Ar i s to t l e sC r i ti c is m o f P r e s o c ra t ic P h i l o s o p h y [New York , 1971] , p . 145), com me nt ing on one o f Z eno ' s a rguments ,says , Th is a rgum ent i s a re fu ta t ion o f the doct r ine tha t there can be m ore than a s ing le Rea l Being ; it i s inaccord wi th the purpose o f Zeno ' s d ia lec t ic as g iven by Pla to .' V las tos ' s t rans la t ion ( Ze no , p . 371) o f D ie ls and Kranz , 29b3 , and G . S . K i rk and J . E . Raven , T h ePresocra t i c Ph i losophers (Cam bridge , 1964) , no . 366 . F in i te ly m an y t rans la tes ~e~spao p~ vct. A s m a n yas there are t rans la tes x o t ~ t O x a . . . 6o~t ~,ax't.

[ 2 6 1 ]


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262 HISTORY OF PHILOSOPH Yrepresenting particular numbers o f collections, shortens cons iderably the pres entationof the argument. It is in no other way crucial to my reconstruction. The reconstruct ioncould be rephrased without using nu mb er . For example, instead of saying fo r anynumb er x, x ~: x + 1, one could say, f or an y collection C, that collection C doesnot have just as m an y items in it as the collection consisting of C tog ether with somenew item i not in C. I stipulate that here the latter is all I mean by the former. The fol-lowing is my reconstruction:

1. Suppose there are k, where k is more than 1. [Ass umpti on for reduction toabsurdity]

2. For any x, if x is greater th an 1, x -- x. [Premise ]3. For any x, if x = x, then x ~e x + 1. [Premise]4. For an y x, if x r x + 1, then x is finite. [Premise]5. k is finite. [Conc lusion]

The assumption might also be expressed: the number of things there are is k. Of thethree premises, 2 and 3 are directly from the text. Premise 4 has no counterpart in thetext: I have added it as, for me, the most natural means to bridge the inferential gapbetween 1, 2, 3, and 5. The conjectured premise 4 has been expressed, in parallel withpremise 3, by use of the variable x, to be thought of as representing a number of a col-lection.3 RECONSTRUCTING For a stated premise of an ancient argument we may sepa-rate two exegetical questions. First, a question of use: Did the aut hor use this premise?Second, a question of justificat ion: W hat reasons did he or might he give for it? Wemay separa te indicates that we can have evidence for an answer to the first questionand no evidence for an answer to the second. However, the two questions may belinked when, to fill a gap in an argument as in the present case, we have conjecturedthat a certain premise was used. Considera tion o f the question of justification mayhelp us to determine whether it is likely that an author would have used a conjecturedpremise if he had seen its relevance. A second kind of linking occurs when we have evi-dence fo r the use of a premise tha t seems obvious ly false. Since we are disposed not toattribute to anyone the use of an obviously false premise, we may need to convinceourselves that our evidence for the use of the premise is not suspect, so that we mayinquire what reasons the author may have had for the premise.

Obvi ousl y false, of course, means obvio usly false to som eon e. It is possible todispute what would or would not have been obviously false to an ancient author; it iseven possible to dispute what is or is not obviously false to oneself.

Where an author is arguing directly for a view of his own, the exegetical question ofuse amounts to, Did he hold the claim? The question of justification amount s to, Forwhat reasons? In the case of reductio ad absurdum arguments like Zen o's the situationis different. We ma y not be able to tell if a premise used is one the author holds or onethat he denies but considers relatively indispensable to those whose view he is attempt-ing to reduce to absurd ity. By relative ly indispensable I mean less dispensable thanthe assumption for the reductio those who hold a relatively indispensable premise willnot give it up in preference to giving up the assumption. If the author of a reductiouses a premise he considers false but relatively indispensable to his opponent s, the two


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ZENO ON PLUR ALITY 263questions amount to, Did he think his opponents ought to accept i t? What beliefs oftheirs did he l ikely think should have led them to accept i t?

For stated premises 2 and 3, the exegetical questions are separable. The qu estion ofhow Zeno would have justified them is not, so far, o f urgency for the reconst ructio n.Premise 2 looks obviously true, and thoug h premise 3 is false because the infinit e car-dinals are coun ter exa mpl es to it, s it is less than o bvi ousl y false to me, and, I suppose,to Ze no. That is, I am not unwi ll ing to suppose Z eno would use 3 in this r e d u c t i oarg umen t unless I also have evidence that Z eno k new that his oppo nen ts had an argu-me nt e stabl ishing that 3 is false.

For the conjectured premise 4, which was supplied solely for the reason--a strongon e- -t ha t i t fil ls very naturally an argu mentativ e hiatus, the exegetical questions arenot separable. Since 4 is, though true, not obviousl y true, one cann ot confide ntlyattr ibute 4 to Zeno. It is then app ropr iate to raise the second exegetical questio n, Wh atwould Zeno's reasons have been for 4?

In what follows I have not tackled this questi on but have reflected on a related spec-ulative question: Ho w c o u l d Zeno have argued for 4? The force of co ul d will be,How could Zeno, without using any beliefs that he and his oppone nts l ikely did nothave, have argued for 4? I offer fou r answers to the speculative questi on, fou r candi-dates for Z eno' s justification for 4. They demonst rate that rather l it t le suffices tomake 4 plausible.4. SPECULATIONS ON HOW ZENO MIGHT HAVE JUSTIFIED USING PREMISE 4: FIRSTC NDID TE JUSTIFIC TION Premi se 4 is the universa l gener aliza tion of the condi-tion al, if x ~ x + 1, then x is finite. A way to esta blish this co ndi ti on al wou ld be toestabli sh its contra posi tive , if x is no t finit e, then x = x + 1.6 A vague im ag ina bl edefense of this contrap ositive comes quickly to mind: take a collection of infinitelymany items; the result of addition of one more item to them will still be a collectionwith infinitely ma ny items, tha t is, just as man y items as there were before the addi-tion. (Accor ding to the rem arks in Section 2, above, the latter is all I mea n if I say thatthe numb er of the collection is the same as that nu mbe r plus 1.) Altho ugh the tha tis (unwa rrant ed because of the existence of different kinds of infinity, some largerthan others) 7 makes the vague justification inadequate, this--les s than ob vi ou s- -inadequ acy does not in turn immedia tely make the vague justification an impr obabl ecandidate for Zeno 's justification for 4. The vagueness, however, ma kes the justifica-t ion a rather unsatisfying speculation.

For a discussion of how they are counterexamples, see e.g., B. Russell, In t roduct ion to Ma themat ica lP h i l o s o p h y (London, 1963), chap. 8. Something like premise 3 turns up in Aristotle, Metaphys ics 1043b37:When one of the parts of which a number consists has been taken away from or added to the number, it isno longer the same number, but a different one, even if it is the very smallest part that has been taken awayor added (Ross trans.). But number in this citation amounts to finite number. See n. 16 below. In asimilar vein is Plato's Cratylus 432a7-9, pointed out to me by my colleague Vicki Harper: Ten, for in-stance, or any number you like, if you add or subtract anything is immediately another number (Fowlertrans.).Something close to this feature of a collection x is sometimes chosen as one way of defining infinite--so-called reflexive--sets. See e.g., A. A. Fraenkel, Abs tract Se t Theory 3rd ed. (Amsterdam, 1966), p. 29Definition VII: A set R is called inf ini te and more strictly,reflexive if R has a proper subset that is equiv-alent to R.7For discussionestablishing hat some infinite sets are larger than others, see Fraenkel, chaps. 1, 2 (especi-ally Sec. 5, theorem 2, Cantor's Theorem).


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2 6 4 H I S T O R Y O F P H I L O S O P H YT h e v a g u e d e f e n s e o f 4 is t h e f i r s t c a n d i d a t e f o r Z e n o ' s j u s t if i c a t io n f o r 4. I n o w

t u r n t o s o m e a d d i t i o n a l c a n d i d a t e s t h a t i n c lu d e f u r t h e r s p e c u l a t io n o n w a y s Z e n oc o u l d h a v e t ri e d t o j u s t i fy t h e t r a n s i t i o n f r o m i n f i n i te l y m a n y p l u s o n e t o j u s t asm a n y a s b e f o r e .5 SECOND CANDIDATE T h e s e c o n d c a n d i d a t e m a k e s c r u c i a l u s e o f tw o p r e m -i se s. ( i) A w a y t o e s t a b l i sh t h a t t h e r e a r e j u s t a s m a n y i t em s i n o n e c o l l e c t i o n a s i na n o t h e r is t o s et u p a o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n t h e c o ll e ct io n s , p a i r in g e a c hi t em f r o m o n e c o l l e c t i o n w i t h e x a c tl y o n e i te m f r o m t h e s e c o n d c o l l e c t i o n , s o t h a t n oi t e m i n t h e s e c o n d i s l e f t w i t h o u t a p a r t n e r . ( ii ) A n y c o l l e c t i o n t h a t i s n o t f i n i t e , t h a t is ,is i n f in i t e , c a n b e r e p r e s e n t e d b y a n ( i m a g i n e d ) u n e n d i n g l is t; f o r e x a m p l e ,

a t a 2 a 3 a 4

I t s e e m s t o m e n o t u n r e a s o n a b l e t o a t t r i b u t e a b e l i e f i n s o m e v e r s i o n o f ( i) , i n s o m ew e a k w a y , t o ju s t a b o u t a n y o n e , i n c l u d i n g Z e n o . P r e m i s e ( ii) is l es s a c c e p t a b l e e v e n asa s p e c u l a t i o n a b o u t w h a t Z e n o b e l i e v e d . I w i ll g o in t o t h e r e a s o n s w h y l a te r . F o r n o w ,le t u s c a r r y o u t a n a r g u m e n t t h a t Z e n o c o u l d h a v e g iv e n i f h e h a d u s e d ( i i ) .

A n i n f i n i t e c o l l e c t i o n o f i t e m s , r e p r e s e n t e d , b y ( ii ), b y t h e li st a t a 2 a 3 . . . . c a n b ep u t i n to o n e - t o - o n e c o r r e s p o n d e n c e w i t h i ts e lf i n th e w a y s u g g e s te d b y t h es e t w op a i r e d l i s t s .

at a2 as 9 9 9

at az a3 . . .

B y ( i) , th i s c o r r e s p o n d e n c e s h o w s t h a t t h e c o l l e c ti o n h a s a s m a n y i t e m s a s it s el f ( i. e. ,s h o w s t h a t t h e n u m b e r o f t h e c o l le c t io n i s e q u a l t o t h e n u m b e r o f t h e c o ll e ct io n ) . F u r -t h e r, t h e c o l le c t io n c a n b e a u g m e n t e d b y o n e m o r e i t e m w i t h o u t s p o i l in g t h e p o ss i-b i l it y o f a o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n i ts e lf a n d i ts el f a u g m e n t e d . T h e n e wc o r r e s p o n d e n c e m a y b e i n d i ca t e d b y t h e se p a i r e d l is ts :

at az a3 . . .b a ~ a x

B y ( ii ), t h e f a c t t h a t t h e r e a r e , b y ( i) , j u s t a s m a n y i t e m s o n t h e o l d l is t a s o n t h e n e w , i sg r a p h i c c o n f i r m a t i o n t h a t i f x is n o t f i n it e , x = x + 1 .6 . D I S P E N S IN G W I T H ( ii ) P r e m i s e ( ii ) is f a l se , 8 t h o u g h n o t o b v i o u s l y . I t is o fs o m e i n te r es t t o c o n s i d e r o t h e r a r g u m e n t s Z e n o c o u l d h a v e g i v e n th a t d o n o t u s e (ii) .

T h e r e i s s o f ar as I k n o w n o r e a s o n t o s u p p o s e t h a t Z e n o o r a n y c o n t e m p o r a r y o fh is k n e w t h e d i s t i n c t io n b e t w e e n d e n u m e r a b l e a n d n o n d e n u m e r a b l e i n fi n it ie s th a tm a k e s ( ii) fa ls e , a l t h o u g h Z e n o s e e m s t o h a v e e x p l o i t e d t h e in f i n i t y o f c e r t a in n o n -d e n u m e r a b l e c o l le c t io n s in s o m e o f h is a r g u m e n t s . F o r e x a m p l e , i n th e o t h e r h a l f o ft h e s e c o n d a r g u m e n t a g a i n s t p l u r al i t y , t h e w a y Z e n o e s ta b l is h e s t h a t i f t h e r e a re m a n y ,t h e r e a r e i n f i n i t e l y m a n y w o u l d b e a p p r o p r i a t e t o e s t a b l i s h i n g t h a t t h e r e a r e i n f i n i t e l ym a n y p o i n t s b e t w e e n a n y t w o p o i n t s o n a l i ne . T h e r e is a n o n d e n u m e r a b l e i n fi n it y o ft h es e , b u t th e m e t h o d Z e n o g i ve s f o r f i n d i n g p o in t s ( o r w h a t e v e r h e h a d i n m i n d ) - - f o r

8 See Fraenkel, chap. 1, for the distinction, w hich shows (ii) false , between d e n u m e r a b l e infinite sets,who se mem bers can be indicated by an infinite list, and infinite sets that are no t so listable.


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Z E N O O N P L U R A L I T Y 265a n y tw o , p i c k o n e b e t w e e n t h e m - - a l l o w s o n e to p ic k o u t o n l y d e n u m e r a b l y m a n yp o i n t s . 9

H o w e v e r , m u c h l es s i n f o r m a t i o n t h a n t h e d is t in c t io n b e t w e e n t h e d e n u m e r a b l e a n dt h e n o n d e n u m e r a b l e m i g h t m a k e o n e h e s i ta n t a b o u t u s i n g (ii). Z e n o m a y h a v e k n o w nt h a t a m e t h o d s u c h a s ( a ) t a k i n g e v e r y p o i n t m a r k i n g o f f m / n t h s o f t h e l i n e s e g m e n tf o r e v er y i nt eg er rn a n d n - - t h e r a t i o n a l p o i n t s - - w a s n o t a n a d e q u a t e m e t h o d o fc o v e r i n g al l t h e p o i n t s o f t h e li n e. T h a t i s , h e m a y h a v e k n o w n t h a t ( b ) f o r e v e ry p o i n tm a r k i n g o f f m / n t h s o f t h e li ne t h er e w o u l d b e a n e w p o i n t n o t a m o n g t h e r a t i o n a lp o i n t s m a r k i n g o f f t h e l e n g th o f a d i a g o n a l o f a s q u a r e w i t h s id e m / n , ~ / 2 ( r e ~ n) 2.A l t h o u g h t h e p o i n t s t a k e n i n (a ) a n d ( b ) t o g e t h e r a r e s ti ll o n l y a d e n u m e r a b l e , l i st a b lei n f i n i t y , k n o w l e d g e o f th e p o i n t s d e s c r i b e d i n ( b) i s s u c h a s t o m a k e o n e h e s i t a t e t oa s s u m e t h a t a p u r p o r t e d l is t o f p o i n t s is a d e q u a t e . O n e m i g h t f e a r t h a t o t h e r n e wp o in t s, m a r k i n g o f f f u r t h er i n c o m m e n s u r a b l e s - - l e n g t h s i n c o m m e n s u r a b l e w i t h b o t h( a ) a n d ( b ) - - w e r e y e t t o b e d i s c o v e r e d .

R e l e v a n t q u e s t i o n s h e r e a re : H o w m a n y o f t h e p o i n t s o f a l in e d i d Z e n o k n o w o f ?D i d h e k n o w t h e i n c o m m e n s u r a b i l i t y o f t h e d i a g o n a l , o r o f a n y t h i n g e l se ? I f s o , d i dh is k n o w l e d g e t h a t t h e r e w e r e s o m e le n g th s i n c o m m e n s u r a b l e w i t h o t h er s m a k e h i ms u s pe c t th a t t h e r e m i g h t b e p o i n t s m a r k i n g o f f y e t u n d i s c o v e r e d i n c o m m e n s u r a b l el e n g t h s? I d o n o t k n o w t h e a n s w e r t o a n y o f th e s e q u e s t i o n s . B u t i f t h e a n s w e r s t o t h es e c o nd t w o s h o u l d b e y es , th e s e c on d c a n d i d a t e a r g u m e n t w o u l d n o t h a v e b e e n Z e n o ' sr e a s o n f o r p r e m i s e 4 .

7 . T H I R D C A ND ID A TE T h e t h i r d c a n d i d a t e a r g u m e n t i s o n e Z e n o c o u l d h a v eg i v e n h a d h e b e e n , f o r t h e r e a s o n s i n d i c a t e d a b o v e , t o o s c r u p u l o u s t o u s e (i i) . T h et h i r d c a n d i d a t e u s e s :

( i ) a g a in( i i ' ) P a r t o f a n y i n f i n i te c o l l e c ti o n c a n b e r e p r e s e n t e d b y a n i m a g i n e d u n e n d i n g

lis t .( ii i) T h e w h o l e i s g r e a t e r t h a n a n y p a r t .

H e r e ( i i ' ) a m o u n t s t o : p a r t o f , a s u b s e t le s s t h a n t h e w h o l e o f , a n y i n f i n i t e c o l l e c t i o n isd e n u m e r a b l y i n f i n i t e .T o r e t u r n t o t h e e x a m p l e o f a n i n f in i t e c o l le c t i o n th a t w o u l d h a v e b e e n a t t h e b a c ko f Z e n o ' s m i n d , t h e re a r e a n u m b e r o f w a y s o f s el ec ti ng d e n u m e r a b l y i n fi n it e s u b se t so f t h e s e t o f p o i n t s o n a li n e; f o r e x a m p l e , c h o o s e m i d p o i n t , m a r k o f f t h ir d s , f o u r t h s ,f i f th s , a n d s o o n . Z e n o c o u l d h a v e p r o d u c e d s o m e s u c h li s ti n g a n d t h e n h a v e a r g u e df r o m ( i ), ( i i ' ) , a n d (iii) t h u s : s i n c e a d d i n g o n e m o r e t o p a r t o f t h e c o l le c t io n w o u l d n o ti n c r e a s e i t , a d d i n g o n e m o r e t o t h e w h o l e , w h i c h i s b y (i ii ) l a r g e r s t il l t h a n t h e p a r t ,w o u l d n o t i n c r e a s e it . B u t ( iii) is a m b i g u o u s . I t m a y m e a n t h a t t h e r e is s o m e i t e m i n t h ew h o l e t h a t is n o t a ls o in t h e p a r t . O r i t m a y b e e q u i v a l e n t t o t h e f a l s e h o o d t h a t n op r o p e r p a r t o f a c o l le c t i o n c a n b e m a d e t o c o r r e s p o n d o n e - t o - o n e t o t h e c o l l e ct i o n .

9 Vlastos's comm ent (p. 371) that this part of the argum ent enunciates the densenessof a continuum(my emphasis) does not, of course, imp ly that Zeno recognized hereby the distinction between a c ontinuumand a denum erable infinity.


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2 6 6 H I S T O R Y O F P H I L O S O P H YT h e r e i s e v i d e n c e t h a t s o m e a n c i e n t a u t h o r s a s s e r t e d ( i i i ) . ' ~ Z e n o c o u l d r e a s o n a b l yh a v e u s e d t h e t h i r d c a n d i d a t e a r g u m e n t a g a i n s t s o m e o n e w h o d i d h o l d ( ii i) .8 FOURTH CANDIDATE A f o u r t h c a n d i d a t e a v a i l a b l e t o Z e n o u s e s s e v e r al r e -s o u r c e s , w h i c h I h a v e i n d i c a t e d i n b r a c k e t s b e s i d e t h e r e l e v a n t s t e p s in t h e j u s t i f i c a -t i on . I t s e e m s t o m e r e a s o n a b l e t o a t t r i b u t e t o Z e n o ( o r j u s t a b o u t a n y o n e ) a re a d i n es st o u s e , t h o u g h n o t a n e n u n c i a t i o n o f , t h e s e p r i n c i p l e s o f o f r e a s o n i n g . F o r s p e c ia l a t -t e n t i o n a r e ( i ) a n d ( i i ' ) f r o m a b o v e .

C o n s i d e r t h e in f i n i t e c o l l e c t i o n W . ( a ) I t h a s a d e n u m e r a b l y i n f i n i t e su b s e t D . [ B y( i i ' ) . ] (~ ) W i s t h e s a m e a s ( W m i n u s D ) p l u s D . C a l l W m i n u s D R , f o r r e m a i n -d e r . W i s t h e n t h e s a m e a s R p l u s D . [ B y a p r i n c i p l e t h a t c o l l e c t i o n s a r e th e s a m e i ft h e y h a v e e x a c t l y t h e s a m e i t e m s i n t h e m . ] (3 ') W h a s j u s t a s m a n y i t e m s a s R p l u s D .[ B y (i ) a n d ( /3 ).] (6 ) D h a s j u s t a s m a n y i t e m s a s t h e c o l l e c t i o n c o n s i s t i n g o f D p l u ss o m e n e w i t e m u . [B y ( i) .] (~ ) S o , W h a s j u s t a s m a n y i t e m s a s R p l u s ( D p l u s u ). [ B y ap r i n c i p l e t h a t i f a c o l l e c t i o n C , h a s t h e s a m e n u m b e r o f i t e m s i n i t a s a c o l l e c ti o n C 2,a n d a c o l l e c t i o n (73 t h e s a m e n u m b e r o f i t e m s a s a c o l l e c t i o n C 4, t h e n t h e c o l l e c t i o nc o n s i s t i n g o f t h e i t e m s i n C , t o g e t h e r w i t h t h o s e i n C3 h a s t h e s a m e n u m b e r o f i t e m s a st h e c o l l e c t i o n c o n s i s t i n g o f t h e i t e m s i n C2 p l u s t h e i t e m s i n (74. I f Z e n o h a d e n u n c i a t e dt h e p r i n c i p l e , h e m i g h t h a v e s a i d , I f e q u a l s a r e a d d e d t o e q u a l s , e q u a l s r e s u l t . ] ( ~')( R p l u s D ) p l u s u i s t h e s a m e c o l l e c t i o n a s R p l u s ( D p l u s ~ ,). [ B y th e p r i n c i p l e t h a t c o l -l e c t i o n s a r e t h e s a m e i f t h e y h a v e t h e s a m e i t e m s in t h e m . ] (r/) S o , W h a s j u s t a s m a n yi t e m s a s ( R p l u s D ) p l u s u . [B y ( ~') , (E ), a n d t h e p r i n c i p l e t h a t i f t w o i t e m s a r e t h e s a m e ,w h a t e v e r h o l d s o f t h e o n e h o l d s o f t h e o t h e r . ] (O ) S o , W h a s e x a c t ly a s m a n y i t e m s a sW p l u s v . [ B y 0 /) , t h e p r i n c i p l e c i t e d a t 0 /) , a n d 0 3 ) . ]

9 . H I S T O R I C A L I N TE R E S T O F T H E SPECULATIONS I f Z e n o h a d g i v e n o n e o f t h el a s t th r e e o f t h e c a n d i d a t e a r g u m e n t s f o r p r e m i s e 4, h e w o u l d h a v e b e e n n o t i c in g th eo d d i t y t h a t a d e n u m e r a b l y i n f i n i te se t c a n b e m a d e t o c o r r e s p o n d o n e - t o - o n e t o ap r o p e r p a r t o f i t se l f . '2 T h e f o u r t h c a n d i d a t e a r g u m e n t i s a s k e tc h o f a v a l id a r g u m e n tt h a t w o u l d h a v e e s t a b l i s h e d p r e m i s e 4 . T h e f o u r t h c a n d i d a t e s h o w s t h a t Z e n o c o u l dh a v e e s t a b l i s h e d t h e c o n j e c t u r e d p r e m i s e 4 , i n t h e s e n s e o f c o u l d i n d i c a t e d i n S e c -t i o n 2 a b o v e ( a l t h o u g h 4 is a t r u t h a b o u t t h e d e n u m e r a b l e a n d t h e n o n d e n u m e r a b l e ) ,w i t h o u t h a v i n g a p r o o f t h a t t h e re a r e n o n d e n u m e r a b l e i n f i n it ie s . H e m i g h t s i m p l yh a v e d o u b t e d t h a t s o m e o f t h e i n f i n i t e c o l l e c t i o n s h e k n e w o f w e r e l i s t a b le , t h a t i s,d e n u m e r a b l e . ,3

,o I t appears as a Euclidean comm on notion, although T . L. H eath ( E u c l i d s E l e m e n t s [New York, 1956],1:232 thinks it is not genuine. ,~. Szab6 raises the interesting possibility that it had to b e included in Euclidbecause someone, i.e., Zeno, had denied it ( T he Transformation of Mathematics into Deductive Scienceand the Beginning of I ts Foundations on Definit ions and Axioms, S c r i p t a M a t h e m a t i c a 27 [1964]:131-136). It does not see m to me, how ever, tha t S zab 6's discussion raises this interesting possibility to thelevel of m uch probabili ty .1 am indebted to my colleague John Wallace for helping me to shorten this justif icatio n.,2 S. C . K leene evidently credits G alileo with the discovery of this odd ity ( I n t r o d uc t i o n t o M e t a m a t h e -m a t i c s [Am sterdam, 1962], p. 3).'~ Szab6 (pp. 134-135) suggests that the Zenonian parado x reported by Aristotle as the half of the t imeequals i ts dou ble (Aristotle, P h . 239b33ff., discussed by Vlastos under the head T he M oving Blocks ) ismeant to convey that part of an infinite set may be made to correspond one-to-one to the whole set. Herealso it seems to me th at Sz ab6 's thought-provoking article has raised a very interesting possibility withoutmaking i t very probab le. The relevant point here is that in S zab6 's reconstruction of the parad ox the sets in


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Z E N O O N P L U R A L I T Y 2 6710. VLASTOS'S RECONSTRUCTION Th e re co ns tr uc ti on give n in Sec ti on 2 dif fer sf ro m th e re c o n s t ru c t io n o f V la s to s in Ze n o (p . 3 71 ) . I wo u ld se t o u t V la s to s ' sr e c o n s t r u c t i o n t h u s:

v l . S u p p o s e t h e r e a r e m a n y . [ A s s u m p t i o n ]v 2 . I f th e re a re ma n y , th e re a re a s ma n y a s th e re a re . [P re mis e ]v 3. I f th e re a re a s ma n y a s th e re are , th e re i s s o me d e f in i t e n u mb e r o f th e m .

[P re mis e s u p p l i e d b y Vla s to s ]v 4. A d e f in i t e n u mb e r i s a f in i t e n u mb e r . [P re m is e s u p p l i e d b y Vla s to s ]v 5 . Th e re a re f in i t e ly ma n y . [Co n c lu s io n ]

Vla s to s t e rms th e in fe r e n c e f ro m a d e f i n i t e t o t a l i ty ( ' s o ma n y a s th e re a re ' ) to a f i n i t eo n e , ''~4 a n d h e re ma r k s th a t i t i s h a rd to s ee h o w a n y o n e c o u ld h a v e b ro k e n [ th ea r g u m e n t ] b e f o r e t h e d e m o n s t r a t i o n o f t r a n s f i n i t e c a r d i n a l s a n d s u p e r d e n u m e r a b l e

question are no ndenumerably infinite. The way of generating the correspondence is not, of course, via themethod of listing. Even if one were to grant that Zeno had in mind Szabo's method of correspondence,one would not thereby be supposing that Ze no recognized nondenumerability.,4 p. 371. In introducing the notion o f definite number (which I comment on later) to make what is evi-dently felt to be an otherwise lacking connection in the argument, Vlastos's reconstruction is similar to thatof H. D. P. Lee ( Zen o o f E l ea [Cambridge, 1936], p. 180): [Zeno] . . . says that any plurality of thingsmust consist of a definite number of things and so be a finite num ber . Likewise, F. M. Cornford (Platoand Parmenides [New York, 1957], p. 180): Ze no . . . regards the many as a plurality . . . which mustamount to some definite number.A number of authors cite the second argument against plurality without expanding at all its first half.Their nonexpansion may indicate that they find it more or less persuasive as it stands or tha t they do not findit profit able to conjecture what is missing: John Burnet, Ear ly Greek Phi losophy (Oxford, 1930), p. 316;E. Zeller, A His tory o f Greek Phi losophy, trans. S. F. Alleyne (Lo ndon, 1881), p. 617; W. D. Ross, Aris-to t le s Phys ics (Oxford, 1936), p. 479; Kirk and Raven, pp. 288-289. Owen (p. 210) rephrases it: It mustcontain just the number that it does, whatever that number is.Anothe r reconstruction of the argument that, like mine and the Vlastos-Lee-Cornford reconstruction,adds some material to Ze no's premises is that of Hermann Frtinkel ( Zen o of Elea's Attacks on Plurality,A m er i ca n Jo u r n a l o f P h i l o l o g y 63 [1942] : 1-25, 193-206). Frtinkel expounds the first half of the secondargument against plurality as follows: If we admit pl ur al it y. ., and divisibility, all the parts together willmake up the whole of the universe. [F] In order to be complete, their number, whatever it is, must be finit e.He cites Parmenides 144e-145a in connection with (F). I take it he means tha t Pla to is there giving the sameargument as Zeno's second argument against plurality. Plato argues, in Cornfor d's translation, Further,since its parts are par ts o f a whole [6kou], the One, in respect of its wholeness, will be limited [nene-pao~t~vov]. For the parts are contained by the whole; and a container must be a limi t. Plato seems to beusing the premise that anything that is a whole will be a container, i.e., a limit, to what it contains; so itscontained parts will have a limit, i.e., be finite.Frtinkel's reconstruction introduces the notion o f being complete, which does not appear in Zeno 's frag-ment 3. Because of the citation from Plato one expects that Fr~.nkel thinks the notion of being completeenters via the notion of being whole, so that perhaps Frtinkel' s full reconstruction would be: in order to be awhole, i.e., complete, the number of the many must be finite. Plato makes the connection between whole-ness and completeness at 137c7-8: a whole means that f rom which no part is miss ing. Frtinkel 's recon-struction takes no notice of the phrase neith er more nor fewer. He evidently takes as many as they areto be closely connected with comple te. There seems no pressing reason to suppose that the particularParmenides passage Frtinkel cites is an echo of the first half of Ze no' s second argument against plurality.The reconstruction of Paul Tannery ( Le Concept scientifique du continu Z6non d'El6e et Georg Can-tor, Rev uephi los ophiqu e de la France e t de l ~ t ranger 20 [1885]) is: To say that a body is a sum of pointsis to admit implicitly that the number of those points is limited (p. 392). Tannery explains sum of poin tsas a totality of juxtaposed poin ts (p. 392). His reconstruction is influenced by his view of the special targetof Zeno's arguments.Over the reconstructions that add to Zeno's words, mine has the advantage that it takes account of thepresence of the phrase neither more nor fewer.


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2 68 HI ST OR Y OF PHI L O SOPHYsets by Georg Ca ntor . ' ' '~ Vlastos has in mind an a rgum ent wi th a premise , v4, towhich the in f ini t e ca rd ina l s a re coun te rexamples . The phrase ne i t he r more norfew er , which adds premise 3 to my reconst ruc t ion and thereby grea t ly inf luences i t,plays no role in Vlastos's. If we accept his reconstruction, we must suppose the phrasee i ther pointless or mere ly repet it ive of as many as they ar e . Vlastos takes th is la t terphrase as a nonre la t ional predica te br inging in the not ion of def ini te number . Thereconst ruc t ion in Sect ion 2 regards the phrase as re la t ional , br inging in the not ion ofthe self-identi ty of a numb er of a collect ion (or, the noti on that a co llect ion has just asmany i tems as i tself) .

Vlastos supplies two conjec tural premises, v3 and v4, to fil l in the inferential gap;my reconstruction supplies only one. For his conjectured premise v4 there is an answerto the second exegetical quest ion, wheth er Zen o or his oppo nent s h ad r eason to be-l ieve v4: the connect ion betw een defini te num ber and fini te nu mbe r was very close inantiquity, so that v4 would have been accepted by anyone. '611. A DIFFICULTY FOR VLASTOS S RECONSTRUCTION The conjec tured premisev3 gives rise to the difficulty that v3 and v4 lead immedi ately to the result: If there areas many as there are, there is some fini te number of them. It seems to me that v3, inthe absence of some ancil lary just if i cat ion for i t , would, accordingly, have been unac-ceptable to any of Zeno 's opponents who wished to make as many as comparisonsregarding the infini te . No one who wished to say, for example, that there are as manypoints of division on a l ine as there are, or that there are as many poi nts on one side ofan equilateral t r iangle as on another, or that there are as many integers as there are,would have agreed to v3 without seeing an argument for i t . Because of the extremenaturalness to us of as man y as comp aris ons regarding the infini te , I suppose i tl ike ly , in the absence of counterevidence , tha t some of Z eno ' s audience o f opponentswoul d hav e wished to make such compa rison s. ,7

Needed to comple te Vlastos ' s reconst ruc t ion i s an answer to the quest ion of howZeno could have made v3 plausible to his opponents . Without some accompanyingsketch of how i t might have been just if ied v3 makes an implausible intrusion into theargument . One doubts tha t Zeno would have used i t .

,s p. 371. If Zeno had given as justification for the conjectured premise 4 one of the speculative justi-fications sketched here, he could have broken the argument. He could have employed closely related linesof reasoning to show that premise 3 was false, i f there were any infinite numbers.That there is a close relation between the speculative justification for 4 and an argument against 3 leads toa difficulty for my reconstruction. The difficulty is discussed in Sec. 13.16 1 owe to Vlastos this information, as well as a reference to Aristotle, Ph. 204b7-10, as an example ofancient usage. To paraphrase Aristotle: a number of things is something you can count, i.e., can f inishcounting, can get through. It should be noted that even one not willing to say that the number of numbers isinfinite (on the ground that numbers must be finite) could say that number is infinite, i.e., that there are in-finitely many numbers (as Aristotle says, 203b22, apparently citing it as a widely held, respectable, belief).At 206a8 Aristotle notes that one of the impossible consequences of the supposition that the infinite does notexist in any way at all is that number will not be infinite.~7 The presumption that some of Zeno's contemporaries held it appropriate to say There are as manynumbers as there are, or some instance of There are as many of the infinitely many F' s as there are of theinfinitely many G's , is the kind of thing for which there could be direct evidence. I have not found any pas-sages in Zeno's contemporaries that use the locution, but there is a passage in the Parmenides that is strongevidence that Plato would at least have found the locution appropriate. The usage of Zeno's interested suc-cessors is some evidence for the usage of his contemporaries; usage in the Parmenides may be particularlygood evidence because of the claim of the second half of the Parmenides to be an exercise in the mannertropos) of Zeno (135d7-8).


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Z E N O O N P L U R A L I T Y 2 6 9a s s k e t c h e d u p t o S e c t i o n 3 a r e t h e f o l l o w i n g . F i r s t , t h e c o n j e c t u r e d p r e m i s e o f t h er e c o n t r u c t i o n i s n o l e ss a n a t u r a l w a y o f f i l l i n g t h e i n f e r e n t i a l g a p i n t h e a r g u m e n t t h a nt h e o t h e r w a y s t h a t h a v e b e e n p r o p o s e d . S e c o n d , t h e r e c o n s t r u c t i o n a d d s n o n e wn o t i o n s t o t h o s e i n t r o d u c e d b y Z e n o ' s a c t u a l w o r d s - - o f w h i c h w e h a v e , u n f o r t u -n a t e l y , s o f e w . T h e o n l y n o t i o n u s e d i n m y r e c o n s t r u c t i o n t h a t is n o t i n t r o d u c e d b yZ e n o ' s w o r d s i s t h e n o t i o n o f n u m b e r , w h i c h c a n b e e l i m i n a t e d f r o m t h e r e c o n s t r u c -t i o n . ~s V l a s t o s ' s r e c o n s t r u c t i o n i n t r o d u c e s t h e n o t i o n o f d e f i n i t e n u m b e r . I h a v e e x -p e r i m e n t e d w i t h r e m o v i n g i t i n n o t e 1 8 , b e l o w . T h i r d , t h e r e c o n s t r u c t i o n i n S e c t i o n 2 ,a n d n o o t h e r I k n o w o f , g i v e s s o m e ro l e t o Z e n o ' s p h r a s e n e i t h e r m o r e n o r f e w e r . ' ' a91 3 . Q U E S T IO N S F O R T H E N E W R E C O N ST R U C T IO N T h e p r e m i s e s o f t h e a r g u m e n ta s r e c o n s t r u c t e d i n S e c t i o n 2 y i e ld s o m e s t r i k i n g c o n s e q u e n c e s . F o r e x a m p l e , f r o m 3a n d 4 it f o l l o w s t h a t i f x ( s o m e n u m b e r o f a c o l l e c t io n ) i s n o t f i n i t e , t h e n x b o t h i s a n di s n o t d i s t i n c t f r o m x + 1 . S u c h a c o n s e q u e n c e w o u l d h a v e b e e n o b j e c t i o n a b l e t o t h o s et h e o r i s t s o f p l u r a l i t y w h o h e l d t h a t t h e r e w e r e i n f i n i t e l y m a n y t h i n g s ; i t w o u l d h a v eg i v e n t h e m w a r n i n g - - i n a d v a n c e o f Z e n o ' s c o n c l u s i o n t h a t t h e r e w e r e f i n i t e l y m a n yt h i n g s - - t h a t t h e y c o u l d n o t h o l d s o m e o f t h e p r e m i s e s Z e n o w a s u s i n g . S i n c e t h e o b -j e c t i o n a b l e c o n s e q u e n c e s a r e v e r y e a s i l y o b t a i n e d , i t i s l ik e l y , f i r s t , t h a t s u c h t h e o r i s t so f p l u r a l i t y w o u l d h a v e s e e n t h e m , a n d i t m a y s e e m l i k e ly , s e c o n d , t h a t t h e y w o u l dt h e r e f o r e h a v e r e j e c t e d t h e f a l s e p r e m i s e 3 . T h e s e c o n d l i k e l i h o o d w o u l d m e a n t h a t

T h e p a s s a g e i n Parmenides con ta in ing t he r e l evan t use o f xoact6xct c a occurs i n a l ong s t r e t ch o f a rgu-m en t f rom 144a4-144d5 . T he fo l l owing c l a im s appea r i n t he d i scuss ion . ( a ) I f nu m ber i s , t he re is an un l im i t edp lu ra l i t y [nk l j 0oq f ine tpov] o f be ings (144a60 . (b ) I f eve ry num ber p a r t akes o f be ing , e ach pa r t o f num berpar ta kes of be in g (144a7-9). (c ) Th ere are unl im i tedly m an y pa r ts o f be ing [~t~pTI dtn~pctvxa xfiq ofio~ctg](144b6-7 ) . (d ) T he re fo re , t he re a re t he m os t [n3.~axa] pa r t s o f i t (144c l -2 ). ( e ) Wh a t i s d iv ided i n to pa r t s i s a sm a n y a s i t s p a r ts [xoact~xt~ 6octTtep ~t~pT1] (144d4-5). ( t ) N oth ing t ha t i s lacks u ni ty a nd no thin g th a t i s onel acks be ing (144e l -2 ). (g ) So t he p rev ious c l a im - -a t 144c l -2 - - was w rong ; be ing does no t have t he m o s tpa r t s , s i nce it is no t m o re wide ly [o66~ f6p n~(c o] d i s t r i bu t ed t han u n i ty , bu t t hey a re equa l [ [ ac t] (144d5-8 ).T h i s pa ssage i nd i ca t e s t ha t P l a to he ld i t appr opr i a t e t o say t ha t t he i n f in i t e ly m a ny pa r t s o f wh a t i s one a re a sm any a s t he i n f in i t e ly m any pa r t s o f wha t i s .,8 S i n c e t h e fr a g m e n t s f r o m Z e n o d o n o t u s e t h e w o r d n u m b e r , i t is a p p r o p r i a t e t o s et o u t t h e r e c o n -s t r u c t io n o f t h e a r g u m e n t w i t h o u t i t. M y r a t h e r a w k w a r d r e c o n s t r u c ti o n , w i t h t h e v a r i a b le s t h a t r a n g e o v e rnum bers r em oved , goes a s fo l l ows:

1 ' . S u p p o s e t h e r e a r e m a n y .2 ' . I f t h e r e a r e m a n y , t h e r e a re as m a n y a s t h e r e a r e .3 ' . I f t h e re a re a s m a n y a s t h e r e a r e , th e n a s m a n y a s t h e r e a r e is n o t th e s a m e a s o n e m o r e t h a n t h e r eare .4 ' . B u t i f o n e m o r e t h a n t h e r e a r e is n o t a s m a n y a s th e r e a r e, t h e r e a r e f i ni te l y m a n y .5 . T here a re f i n i t e ly m any .

As no t ed ea r l i e r i n Sec . 2 , t he re a re no im por t an t d i f f e rences be tween wha t i s t o be sa id abou t t h i s r e con-s t r u c ti o n a n d w h a t w a s s a i d a b o u t t h e r e c o n s t r u c ti o n i n t h e b o d y o f t h e p a p e r . H e r e 4 ' i s t h e s u p p l i edprem ise . T he aux i l i a ry j us t i f i c a t i ons fo r 4 ' wou ld be c lose t o t he j us t i f i c a t i ons specu l a t ed fo r 4 ( a s shou ld bee v i d e n t f r o m m y p a r e n t h e t ic a l r e m a r k s t h r o u g h o u t t h e p a p e r ) .T h e r es u l t o f ta k i n g n u m b e r o u t o f V l a s t o s ' s r e c o n s t r u c t i o n s e em s t o b e:v l ' . S u p p o s e t h e r e a r e m a n y .v 2 ' . I f t h e r e a r e m a n y , t h e r e a r e a s m a n y a s t h e r e a r e .v (3& 4) ' . I f t he re a re a s m a ny a s t he re a re , t he re a re f i n it e ly m an y .v 5 ' . T h e r e a r e f i n it e ly m a n y .

H e r e v3 a n d v 4 , w h i c h c o n t a i n e d n u m b e r , d r o p o u t a n d a r e r e p la c e d b y a p r e m i s e t h a t s ee m s t o n e e d a d -d i t i ona l j u s t i f i c a t i on , fo r t he r ea sons v3 seem ed to need i t i f Z eno ' s a rgum ent was t o be pe r suas ive .,9 T an ne ry (p . 392) says o f t he second a rgum ent aga ins t p lu ra l it y t ha t i t s b rev i t y i s ve ry susp ec t . Bu t h i sr e c o n s t r u c ti o n i n a w a y m a k e s t h e a r g u m e n t e v e n b r i e f e r b y l ea v i n g o u t c o n s i d e r a t i o n o f th e p h r a s e n e i t h e rm o r e n o r f e w e r .


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270 HISTORY OF PHI LOS OPHYZen o's argumen t was obviously unpersuasive; obv ious unpersuasiveness is evidence ofinaccurate reconstruction.

Against the appare nt second likelihood I can urge that 3 cannot have been highly re-jectable by or dispensable to Zeno's opponents, for Zeno actually uses it. This de-fense, of course, is only as good as my understanding o f neither more nor fewe r. Itis, however, a defense not possible for Vlastos's supplied v3. A lesser defense of 3 isthat not only is it not obv ious ly false, 2~ it is similar to a true principle for finite num-bers that was enuncia ted by both P lato and Aristotl e. 2'

Consideration of the ancillary speculative arguments, however, brings up anotherdifficulty: the lines of reasoning sketched in the speculative argument for premise 4can easily be applied in constructing an argument that if there are infinitely many o fanything, premise 3 is false. So, if Zeno had offered any of the speculative ancillaryjustifications he would thereby have supplied his opponents with the material for anequally good argument against premise 3.

This difficulty is a damaging--though in the circumstances, not crushing--pointagainst my speculations from Section 4 on.University of Minnesota

~o Vicki Harper has suggested a comparison with A thing is what it is and not another thing.2, See n. 5 above.22 Here I am indebted to Vicki Harper.

zeno's second argument against plurality

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