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Journal of the History of Philosophy, Volume 30, Number 4, October

1992, pp. 497-522 (Article)

DOI: 10.1353/hph.1992.0082 

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M a t h e m a t i c a l C o n s t r u c t i o n

S y m b o lic o g n i ti o n a n d t h e

I n f i n i t e I n t e l l e c t : R e f l e c t i o n s o n

M a i m o n a n d M a i m o n i d e s

D A V I D R . L A C H T E R M A N t

S AL O MO N M A IM O N W A S , p e r h a p s , t h e f i rs t , a l t h o u g h c e r t a i n l y n o t t h e l a st , J e w -

i sh th i n k e r t o u n d e r t a k e a t h o r o u g h c o l la t io n o f m o d e r n p h i l o s o p h y i n i ts

m o s t c r it ic a l a r t ic u l a t io n , b y K a n t , a n d M a i m o n i d e a n i s m , t h e m o s t r ad i c al l y

p h i l o s o p h ic a l e x p r e s s i o n o f t h e p r e - m o d e r n J e w i s h t r a d i t i o n ., U n l i k e S p i-

n o z a , w h o s e r e l a ti o n t o M a i m o n i d e s is , o n t h e s u r f a c e , u n r e l e n t i n g l y a n t a g o n i s -

t i c, a n d u n l i k e H e r m a n n C o h e n , f o r w h o m o n l y t h e e t h i c a l t e a c h i n g s o f t h e

p r o p h e t s a n d t h e s a ge s a r e t o b e b r o u g h t i n l in e w i t h K a n t i a n m o r a l t h e o r y , ,

M a i m o n s o u g h t t o h a r m o n i z e h i s r e v i s e d K a n t i a n i s m w i t h t h e n o e ti c s a n d

e p i s t e m o l o g y o f M a i m o n i d e s . H i s a d o p t e d n or a d e p l u m e s e e m s t o b e a p l e d g e

o f d u r a b l e , i f a l w a y s a m b i v a l e n t , l o y a l t y t o R a m b a m . s

tDav id R. Lach te rm an d ied on M ay zo , 1991.

' See the studies collected in

Wolftnibfa~ltr S tud itn zur AtgOd~rung

(WolfenbOttei, 1977), espe-

c ia l ly F r i ed r i ch Niew6hner , " ' P r ima t de r Eth ik ' ode r ' e rkennm is theo re ti sche B egr~nd ung d e r

Eth ik '? Thesen z u r K an t -Rezep tion in de r j0d i schen Ph ilosoph ic," 119-61 ; and the "Nachw or t "

to Solomon M aimons Lebtnsgeschichu, neu hrsg. yon Zwi Batscha (Fra nkfurt a .M.: Suhrka mp , 1984),

329-9 z . A m ore gene ra l conspec tus i s o f fe red by Na than Ro tens tr ei ch , J tws and German Philoso-

phy: The Polemics of Em ancipation (New York: Schocken, 1984) .

9 For Spinoza and Maimonides , see the c lass ic s ta tement by Leo Strauss , spinoza 's Crit ique of

Religion (N.Y.: Schocken, 1965). Strauss's view o f the conflict between Maim onides and Sp inoz a

has been mit igated or re jected by Shlom o Pines , "Spinoza 's Traaa~us

Thtologico-Pol;aicm,

Maimoni-

des and Kant ," Scripta Hitroso ymilana ao (1968) : $ -54 and by W. Zev Harvey , "A Por t r a i t o f

Sp inoza a s a Ma im on idean ," Jou rna /o f the Histo ~ ofPM losophy 19 0981 ) : 15 t -T a . F o r Cohen , s ee

h is conc ise s t a t emen t in " Inn e re Bez iehungen de r Kan ti schen Ph i lo soph ie zum Juden tum ,"

Jiidische Schrifttn (Berlin , 19~4) Bd. 1 a84 -3o 5; cf . Sylvain Zac, Laphidosophit re lig ituse de Hermann

Cohen, Av ant-prop os de Paul Ricoeu r (Paris : Vrin , 1986) , esp. t81 --zo L

) On h i s change o f name f rom Sh lomo ben Yehoshua to Sa lomon M aimon , see Samue l H ugo

Be r g m a n ,

The Philosophy of Solomon Maimo n,

t rans . No ah Jacobs (Jerusalem : M agnes Press , 1967

[Hebrew original, 193z]) , t - $ and n . z .

[ 4 9 7 ]

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4 9 8 J O U R N L O F T H E H I S T O R Y O F P H I L O S O P H Y 3 0 : 4 O C T O B E R X 9 9 2

T o i l l u st ra t e h o w t h is l o y a lt y is m o r e t h a n m e r e l y p r o g r a m m a t i c

a n d

t o

b r i n g i n t o r e l i e f s o m e o f it s a m b i v a l e n t c o n t o u r s , I s h al l e x a m i n e c e r t a i n

f e a t u re s o f M a i m o n ' s p h i l o s o p h y o f m a t h e m a t ic s .

T h e c e n t r a l p la c e o f m a t h e m a t i c s i n M a i m o n ' s e l a b o r a t io n o f K a n t ' s t h e o r y

o f k n o w l e d g e h a s, o f c o u r s e , b e e n n o t e d a n d a n a l y z e d , s o m e t i m e s i n m a r k e d l y

d i f f e r e n t w a ys , b y a ll m a j o r s t u d e n t s o f h is w o r k , f r o m K u n t z e , G u e r o u l t , a n d

B e r g m a n t o A t la s a n d , m o s t r e c e n t l y , Za c.4 I n t h e m a i n t h e s e s c h o l a r s h a v e

c o n c e n t r a t e d t h e i r a t te n t i o n o n t h e p a r t a s s ig n e d t o

d i f f e r e n t i a l s

a s t h e b r i d g e

b e tw e e n n o u m e n a a n d p h e n o m e n a a n d o n M a i m o n 's a n t ic i pa t io n s o f l at e

n i n e t e e n t h - a n d e a r l y t w e n t i e t h - c e n t u r y c o n c e p t i o n s o f t h e p u r e l y f o r m a l ,

a r b i t r a ry , o r s y s t e m - r e l a t iv e c h a r a c t e r o f m a t h e m a t i c a l a x i o m s .5

M y a d d e n d a t o th e i r w o r k a r e f o c u s s e d o n t h e r o l es p l a y e d b y c o n s t r u c t i o n

( S e c ti o n i ) a n d b y s y m b o l i z a t i o n ( S e c ti o n ~) i n M a i m o n ' s t e a c h i n g s a b o u t

m a t h e m a t i c a l k n o w l e d g e . O n c e t h e s e ro l es h av e b e e n e x p o u n d e d , I sh a ll t u r n

t o M a i m o n ' s a n d M a i m o n i d e s ' n o e t ic , in t h e h o p e o f e s t ab l i sh n i n g b o t h r e s e m -

b l a n c e s a n d d i s c r e p a n c i e s ( S e c t io n 3 ) .

1 . C O N S T R U C T I O N

M a i m o n ' s c o n c e p t o f m a t h e m a t i c a l c o n s t r u c ti o n r e m a i n s t ie d , in s o m e r e -

s p e c t s, to it s K a n t i a n s o u r c e i n t h e

C r i t iq u e o f P u r e R e a s o n

e v e n w h i l e b e i n g

s i g ni f ic a n t ly l i b e r a t e d f r o m i t i n o t h e r r e s p e c t s.

L e t m e b e g i n , t h e n , b y c i ti n g M a i m o n ' s m o s t v iv id f o r m u l a t i o n o f t h e

p o w e r o f m a t h e m a t i c a l c o n s t r u c t i o n , f r o m h is e n t r y in t h e R o y a l A c a d e m y

( B e r li n ) c o m p e t i t i o n o f i 7 9 ~ , O b e r d i e P r o g r e s s e n d e r P h i l o s o p h i e 0 7 9 3 ) :

G o d , a s an i n f i n i te p o w e r o f r e p r e s e n t a t i o n

[Vorstellungsvermtgen]

f r om a l l e t e r n i t y ,

t h i nks h i m se l f a s a ll pos s i b l e e s sences

[Wesen]

t ha t i s , he t h i nks h i mse l f a s

restricted in

every possible way. H e d o e s n o t t h i n k a s w e d o , [ n am e l y ] , discursively; r a t h e r , h i s t h o u g h t s

4A full bibliography o f secondary studies of Maimon (to 1966 appears in Noah Jacobs,

Schrifttum fiber Salomo n Maim on, in

Wolfenbfitteler Studien

353-9 5 (translated b y G. Leiser-

sohn from Kiryat Sepher 41 [1966]: ~45-62). T he most important boo k-length studies to appea r

after Jacob's

terminus ante quem

are F. Moist, La

filosofia di Salomone Maimon

(Milan: Feltrinelli,

1972), Sylvain Zac , Salomon Maimon. Critique de Kant (Paris: Editions du Cerf, 1988) and Ja n

Bransen,

The Antinomy of Thought: Maim onian Skepticism and the Relation between Thought and Objects

(Ph.D. Dissertation, Rijksuniversiteit Utrecht, 1989).

5This emphasis is at least partly d ue to the Net-Kantian rediscovery of M aimon, as described

in Friedrich Kuntze, Die P hilosophie Salomon M aimons (Heidelberg: C. W inter, 1 912 . On his

rendition of differentials, see 33 a if.; Er nst Cassirer,

Das Erkenntnisproblem in der Philosophie und

Wissenschaft der neueren Z eit

Dritter Band (reprint ed., Darmstadt: W issenschaft liche Buchgesell-

schaft, 1917), 97-1 o4; Samuel H . Bergman, The Philosophy of Solomon Maimon 59-68; Martial

Gueroult,

La philosophie transcendentale de Salomon Maimon

(Paris: Alcan, 19~9), 54-6 8; and Zac,

Salomon Maimon 155--71. For his treatm ent of the relat ivity of axioms, see Sam uel Adas, From

Critical to Speculative Idealism: The Philosophy of Solomon Maimon

(The Hague: M. Nijhoff, 1964),

z19-48.

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R E F L E C T IO N S O N M A IM O N A N D M A IM O N ID E S 99

a r e a t o n e a n d s a m e t i m e

presentat ions~complete exhibi t ions [Darstel lungen].

I f s o m e o n e

o b j e c ts t h a t w e h a v e n o c o n c e p t o f s u c h a s t y le o f t h i nk i n g , m y a n s w e r i s: W e d o i n f a c t

h av e a co n cep t o f i t , s i n ce w e p a r t l y h av e t h i s s t y l e i n o u r p o sses s io n . A l l m a th em at /~ a /

concep ts a re

t h o u g h t

b y u s a n d a t t h e s a m e t i m e

exhibited

as

rea l objects [ree l le O bjekte]

t h r o u g h c o n s t r u c t i o n a p r i o r i . T h u s , w e a r e i n t h i s . r e s p e c t

s imi la r t o God .

( IV, 2o ;

M a i m o n ' s e m p h a s e s ) 6

M a i m o n ' s f o r m u l a t i o n is , o n f i rs t h e a r i n g , a n e x t r e m e v e r s i o n o f K a n t ' s

q u i t e C a r t e s i a n a p p r a i s a l i n t h e f i rs t C r /t /q ue , vi z. , t h a t t h e m e t h o d o f c o n s t r u c t -

i n g c o n c e p t s i n p u r e i n t ui t io n a p r i o r i b e c o m e s , s o t o s p e a k , t he m a s t e r o f

n a t u r e ( A 7 2 5 / B 7 53 ) . ( O n cl o s e r h e a r i n g t h e r e s o n a n c e s o f M a i m o n ' s s i m i l ar -

i ty t h e s i s e c h o n o t o n l y D e s c a r t e s a n d K a n t , b u t h i s sp e c i a l r e a d i n g o f

M a i m o n i d e s a s w e l l. )

T w o a x e s o f c o n s t r u c t i o n n e e d t o b e d i s t in g u i s h e d , e v e n t h o u g h t h e y

m a y b e s ai d t o m e e t a t th e p o i n t o f o ri g i n . I sh a l l c a ll t h e s e t h e e v i d e n t i a r y

a n d t h e o p e r a t i o n a l a x e s.

T h e E v / d e nt / a ry A x/s . A l t h o u g h M a i m o n i s o f t e n t h o u g h t o f a s b a s i c a ll y

m o r e L e i b n i z ia n t h a n K a n t i a n , t h e e v i d e n t i a r y w e i g h t h e g iv e s t o c o n s tr u c -

t i o n s ti lt s t h e b a l a n c e o f h is t h o u g h t t o w a r d s a s t r o n g l y K a n t i a n p o s i t i o n .

N e i t h e r t h e p r i n c i p l e o f i d e n t i t y ( a = a ), n o r i t s n e g a t i v e c o u n t e r p a r t , t h e

p r i n c i p l e o f n o n c o n t r a d i c t i o n , n o r , f i na ll y , M a i m o n ' s o w n m u c h - v a u n t e d

p r i n c i p l e o f d e t e r m i n a b i l i t y s u f f ic e s , i n i ts o w n r i g h t , o r in c o m p a n y w i t h

a n y o f t h e o t h e r s , t o s e c u re

a n

o b j e c t iv e t r u t h f o r m a t h e m a t i c a l p r o p o s i t i o n s

( I I I , 1 6 o) ; t h a t i s, n o n e o f t h e m a l l o w u s , a s fi n i te m i n d s , t o c o n f e r o n t h e

r e l e v a n t r e f e r e n t s o f s u c h p r o p o s i t i o n s ( r i g h t - a n g l e d t r i a n g l e, f o r e x a m p l e )

t h e s t a t u s o f r e a l o b j e c t s ( I V , 3 5 ). 7

T h a t s o m e t h i n g w o r l d l y a n s w e r s to a n o n c o n t r a d i c t o r y c o n c e p t c a n n o t b e

a s c e r t a i n e d b y a n a ly s i s o f t h e c o n c e p t o n i ts o w n . B u t n e i t h e r c a n t h e p r i n c i p l e

o f d e t e r m i n a b i l i t y , w h i c h p e r m i t s u s to e s ta b l i s h p o s s i b le , n o n a r b i t r a r y l i a i s o ns

b e t w e e n s u b j ec t c o n c e p t s - - t h e d e t e r m i n a b l e s ( e.g ., t r i a n g l e ) - - a n d p r e d i c a te

c o n c e p t s - - t h e d e t e r m i n a t i o n s ( e .g ., is o sc e le s , s c a l e ne a n d e q u i l a t e r a l ) - - r e v e a l

6 Unless otherwise noted, all citations fr om Maimon r efer to Salomon M aimon, Gesammel te

Werke, hrsg. Valerio V erra (Hildesheim: O lms, 1965-1976), Bde. l - 7 . Volumes are indicated by

Rom an numerals; th e pa ge numb ers are those o f the original editions (printed in Verra), since

several othe r re prints (especially in the Aetas Kantiana-Series) are also currently available. All

translations are m y own.

7On the shadowy or vagrant distinctions between reel and wirklich in Maimon, see M.

Gueroult, La

philosophie transcendentale,

47-4 9- A comprehensive study of the uses and vagaries of

Sein, RealiSt, Wirklichkeit, Existenz, Objekt, Gegenstand , and Ding in Ka nt and

the post-K antian t radi tion is still outstanding. Essential to such a study would be Anneliese Maier,

Kants Qualit~tskategorien, in her

Zw ei Untersu chung en zur nachscholastischen Philosophie

(Rome,

1968), 7 l-14 7. He idegg er's remark s in

Kants These f iber das Sein

(Frankfurt a.M.: Kiostermann,

1967), 273-3o7, w ould also be of fund am ental relevance.

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5 0 0 J O U R N L O F T H E H I S T O R Y O F P H I L O S O P H Y 3 0 : 4 O C TO B ER 1 9 9 9

w h e t h e r t h e s e p o s si b l e l ia i so n s a r e r e a l i z a b le o r w o r l d l y . 's C o n s i d e r t h e

e x a m p l e M a i m o n c i te s b o t h i n / b e r d i e P r o g r e s s e n d e r P h i l o s o p h i e a n d , a y e a r l a t e r ,

i n D i e K a t e g o r i e n d e s A r i s t o t e le s 1 7 9 4 ) , i .e ., th e r e g u l a r d o d e c a h e d r o n ( IV , 3 5 ; V I ,

l o l - x o 2 ) : I c a n t h i n k o f a r e g u l a r s o li d w h i c h i s e n c l o s e d b y te n e q u a l s id e s ,

j u s t a s w e l l a s I c a n t h i n k o f a r e g u l a r s o l id e n c l o s e d b y s ix e q u a l s i d e s ( V I , i o x -

1o ~t).9 T h e s e t w o o b j e c t s o f t h o u g h t a r e a l i k e i n o b e y i n g

b o t h

t h e p r i n c i p l e o f

n o n c o n t r a d i c t i o n ( u n l i k e n o n c i r c u l a r c i rc l e ) a n d t h e p r in c i p l e o f d e t e r m i n -

a b i li ty (u n l i k e b l a c k ci r cl e ) ; t h e d i f f e r e n c e b e t w e e n t h e m is t h a t o n l y t h e

s e c o n d y i e ld s , vi a c o n s t r u c t i o n , c o g n i t i o n o f a n a c t u a l o b j e c t, t h e c u b e , w h i l e

t h e r e a r e n o r e a l o r o b j e c ti v e r e g u l a r d o d e c a h e d r a . H e n c e , t h e t w o a n a ly t i-

c a l p r i n c i p l e s a r e n e c e s s a r y , b u t n o t e v e n j o i n t l y s u f fi c i en t , t o g r o u n d t h e o b j e c -

t iv e r e a l it y o r o b j e c t i v e t r u t h o f m a t h e m a t i c a l c o n c e p t s o r j u d g m e n t s . ' ~ ( I

p o s t p o n e d i s c u s s i o n o f w h a t i t m e a n s t h a t a n i n f i n i te in t e l le c t k n o w s m a t h e m a t -

i cs a n a l y t i c a ll y , n o t s y n t h e t i c a l l y . )

T h e m o d e r n n o t i o n o f c o n s t r u c t i o n a s t h e k e y t o m a t h e m a t i c a l t r u t h a n d

e x i s t e n c e h a s a r i c h h i s t o r y . S u f f i c e i t t o s ay h e r e t h a t a d i r e c t l i n e l e a d s f r o m

D e s c a r te s ' s c o n s t r u c t i o n o f a p r o b l e m i n L a G ~ o m ~ t r ie , t h r o u g h t h e c o n s t r u c -

t io n o f a n e q u a t i o n i n L e i b n i z a n d W o l f f , t o K a n t 's c o n s t r u c t i o n o f a c o n -

c e p t . A l l t h e s e i n s ta n c e s i n v o l v e a c o m p l e x i n t e r p l a y b e t w e e n i n t e r i o r m i n d

( D e s c a r t e s ' s m e n s

p u r a )

a n d i ts e x t e r i o r o t h e r , w h e t h e r t h i s i s e x t e n s i o n , o r

b o d y , o r t h e g i v e n , o r s e n s i b l e i n tu i t i o n s . C o n s t r u c t i o n i s t h e m i n d ' s e x h i b i t i o n -

i sm , it s w a y o f m a s t e r i n g i ts o t h e r b y i m p o s i n g u p o n , o r i n s e r t in g i n t o it a s

i n to a m e d i u m , t h e c o u n t e r p a r t s o f it s p u r e l y i n te l le c t ua l o r d i a n o e ti c f o r m a -

t io n s . A n d y e t , t h is a l ie n m e d i u m is n o t h o s p i t a b l e t o a ll m i n d - o r i g i n a t e d

s t r a n g e r s ; i ts p a r t i a l x e n o p h o b i a i s b r o u g h t h o m e b y th e f a c t t h a t n o t a l l

( C a rt e si a n) e q u a t io n s h a v e r e a l (a s o p p o s e d t o i m a g i n a r y ) r o o t s o r b y t h e

i m p o ss ib i li ty o f f a s h i o n i n g a r e g u l a r d o d e c a h e d r o n .

M a i m o n s e t s o u t , n o t t o e l i m i n a t e o r t o r e s t r i c t t h is x e n o p h o b i c r e s i s t a n c e

t o th e m i n d , b u t t o r e c o n c e i v e a n d r e p o s i t io n t h e v e r y m e d i u m o f c o n s t r u c t i o n

i n s u c h a w a y t h a t t h e m i n d ' s d e p e n d e n c e o n g i v e n n e s s o r e x t e r i o r i t y is s e -

sSee the valuable study by Sylvain Nacht-Elade, Aristotle's Doctrine o f the Differentia Spe-

cifica and Maimon's Law of Determinabili ty, Scripta Hierosolyraitana6 096o ) : 292-48.

9 nterest ingly, Kant gives the same example of the regular hexagon in Der einzig mOgliche

Beweisgrund zu einer D emonstration des D aseins Gottes Theorie-Werkausgabe,hrsg. W. Weichs-

edel , Bd. 2, p. 639). Leibniz discusses the inconstructible do deca hedro n in Nouveaux Essais (Livre

III, ch. iii).

~oFor M aimon 's way(s) of d rawing the an alytic/synthetic distinction, see his Versuch einer neuen

Logik oder Theorie des Den kens

(1794) (V, 28-30; 193-24) and the comments of Martial Gueroult ,

La philosophie transcendemale, 91-99 and Jan Bransen, Antinomy of Thought, 69-83. For the mat rix

and wider setting o f the question of an alyticity/syntheticity, see the ex cellent study b y JoOlle

Proust,

Questions de forme: Logique et propo sition analytique de K ant g~ Carn ap

(Paris: Fayard, 1986).

~ See my

The Ethics o f GeometTy

(New York: Roufledge, t989), 9o 6-909 .

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R E F L E C T I O N S O N M A I M O N A N D M A I M O N I D E S 5 1

v e r e l y l i m i t e d . T h i s b r i n g s m e t o t h e s e c o n d a x i s : H o w d o e s c o n s t r u c t i o n

o p e r a t e , f o r M a i m o n ?

The O p e ra t io na l Ax i s

A g a i n w e m u s t s e t o u t f r o m t h e K a n t ia n f r a m e w o r k .

F o r K a n t c o n s t r u c t i o n a p r io r i i s a t r a n s a ct i o n a m o n g ( 1 ) p u r e c o n c e p t s o f

t h e u n d e r s t a n d i n g , ( 2 ) tr a n s c e n d e n t a l i m a g i n a t i o n , a n d ( 3 ) t h e a p ri o r i f o r m s

o f i n t u i t i o n o r s e n s i b il i ty , n a m e l y , s p a c e a n d t im e . ~ , A c o n s t r u c t e d c o n c e p t i s

a c o r r e s p o n d i n g s t r u c t u r e m a d e p r e s e n t b y im a g i n a t i o n i n t e m p o r a l i n t u it i o n

( i .e . , a r i t h m e t i c a l n u m b e r ) o r i n s p a t ia l i n t u i t i o n ( i . e ., g e o m e t r i c a l f i g u re ) .~ s

S a l i e n t h e r e i s t h e a l t e r i t y o f i n t u i t i o n v i s -a - v i s u n d e r s t a n d i n g , a n a l t e r i ty

m a n i f e s t i n a t l e a s t t w o r e s p e c t s : f i r s t , t h e c o n c e p t i s u n i v e r s a l , w h i l e t h e

c o n s t r u c t e d i n t u i t i o n i s, p e r f o r c e , in d i v i d u a l, a n d s e c o n d , t h e c o n c e p t h a s a n

i n te r i o r m u l t i p l i c it y ( t h e p l u r a l i t y o f i ts m a r k s ) , w h i l e t h e i n t u i t i o n i s c h a r a c -

t e r i z e d b y a n e x t e r i o r o r e x t r i n s i c d i v e r s i t y o f it s c o m p o n e n t s ( i . e. , d i s t i n c t

t e m p o r a l u n i t s o r s p a t i a l s e g m e n t s ) .

M a i m o n d e n i e s t h e r a d i c a l o r a b s o l u te d i s t in c t n e ss o f u n d e r s t a n d i n g a n d

s e n s i b i l i t y ( I I , i 8 2 ; c f . I I , 6 3 ) . I n t h e p r e s e n t c o n t e x t , h i s d e n i a l h a s t h i s

f u n d a m e n t a l c o n s e q u e n c e : t i m e a n d s p a c e t h e m s e l v e s b e l o n g , i n t h e f i r s t

i n s t a n c e , t o t h e c o n c e p t u a l o r d e r ,

no t

t o t h e i n t u i t i v e . I n a f f i r m i n g t h i s ,

M a i m o n , i n S a m u e l B e r g m a n ' s w o r d s , i n di c a te s hi s o p p o s i t i o n t o t h e K a n t i a n

N e e d l e s s t o s a y , t h i s i s f a r f r o m b e i n g t h e w h o l e s t o r y , e v e n i n s k e l e t a l f o r m . A p i v o t a l

c h a p t e r w o u l d b e l o n g t o K a n t ' s o f t e n - i g n o r e d d i s t i n c t io n (KdrV B 1 61 A n m . ) b e t w e e n s p a c e ( a n d

t i m e ) a s f o r m o f in t u i t i o n a n d s p a c e (a n d ti m e ) a s f o r m a l i n t u it i o n . T o d a t e , th e m o s t i l l u m i n a t -

i n g o b s e r v a t i o n s o n t h i s d i s t i n c t io n c a n b e f o u n d i n H a n s G r a u b n e r ,

Form and Wesen: Ein Bei trag

zur Deutu ng des Formbegriffs in Kants Kritik der reinen V ernu nfi (= K ant-Studien,

E r g ~ n z u n g s h e f t , B d .

1 o4 ) ( B o n n : B o u v i e r , x 9 7 :t ), 1 3 8 - 6 8 a n d G e r d B u c h d a h l ,

Metaphysics and the Philosophy of Science

( C a m b r i d g e , M a s s . : M . I . T . P r e s s , 1 9 6 9 ), 6 0 6 - 1 5 . P r e s u m a b l y , t h e c o n s t r u c t e d i n t u i t i o n s in m a t h e -

m a t i c s a r e f a s h i o n i n g s o r s e q u e n t i a l a r ra y s o f s p a c e o r t i m e a s f o r m a l i n t u i t i o n s .

,s I t i s, I t h i n k , i m p o s s i b l e t o d i s c e r n a n d t o f o l lo w t h e t h r e a d s w o v e n t o g e t h e r i n t o K a n t ' s

t h e o r y o f m a t h e m a t i c a l c o n s t r u c t i o n i f o n e f o l l o w s J a a k k o H i n t i k k a ' s r e c o n s t r u c t i o n o f t h i s a s a

l o gi c al t h e o r y o f t h e i n t r o d u c t i o n o f s i n g u l a r t e r m s i n t o p r oo f s . S e e, a m o n g H i n t i k k a ' s m a n y

f o r a ys i n t o t h i s d o m a i n , K a n t ' s T h e o r y o f M a t h e m a t i c s R e v i si t ed , i n

Essays on Kant s Critique of

Pure Reason,

e d . J . w . M o h a n t y a n d R . W . S h a h a n ( N o r m a n , O k l a .: U n i v . o f O k l a h o m a P r e s s ,

1 9 82 ) , ~ o l - x 5 . I s e t o u t m y c r it i c is m s o f H i n t i k k a ' s i n t e r p r e t a t i o n i n The So vereignty of Construction

( N e w Y o r k : R o u t l e d g e , f o r t h c o m i n g ) , C h . 3 . O t h e r i n s i g h t f u l re n d i t i o n s o f K a n t i a n c o n -

s t r u c ti v i s m m a y b e f o u n d i n R o b e r t E . B u t t s ,

Kant and the Double-Government Methodology

( D o r d r e c h t : R e i d e l , 1 98 4 ), 1 4 6 - ~ o l a n d G r e g o r B f i c h e l, Geom etrie un d Philosophie. Zura Verhiiltnis

beider Vernu~t~senschaf ten im For tgang yon der Kr i t ik der reinen Vernunf t zum Opus pos tumum

( =

Kant-Studien,

E r g ~ n z u n g s h e f t e , B d . 1~ l ) ( B e r l i n : d e G r u y t e r , 1 9 8 7) , 3 7 - 1 3 1 .

F i c h t e , i n t h e w a k e o f M a i m o n , a d h e r e s t o a s t r o n g c o n s t r u c t i v i s t t h e s i s. S o , f o r i n s t a n c e , i n

a p o s t h u m o u s l y p u b l i s h e d t e x t f r o m 1 81 e h e w r i te s : E v i d e n c e i s j u s t t h e a b s o l u t e in s i g h t w h i c h

m a k e s i t s e n t r a n c e w h e n w e c o n s t r u c t , fo r e v i d e n c e e x t e n d s o n l y t o t h e s p h e r e w h i c h i s t o b e

p r o d u c e d b y c o n s t r u c t i o n . . . . T h i s i s p r e c is e l y a s it is i n m a t h e m a t i c s , w h i c h is c o m m o n l y a d -

d u c e d a s a n e x a m p l e o f c o n s t r u c t i o n

(Nachgelassene Werke,

h r sg . I . H . F i c h t e , B d . I X , p . 1 4 9 ) .

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5 0 2 J O U R N A L O F T H E H I S T O R Y O F P H I L O S O P H Y 3 : 4 O C T O B E R 1 9 9 ~

b e l i e f t h a t i n t u i t i o n c o n t a i n s a n e l e m e n t t h a t i s b e y o n d t h e c o n c e p t b u t w h i c h ,

n e v e r t h e l e s s , d e t e r m i n e s t h e p h e n o m e n a . ~ 4

M a i m o n r a t i o n a l iz e s t h e i n t u i t iv e a n d , t h e r e b y , e l i m i n a te s a n y f u n d a m e n -

t al h e t e r o g e n e i t y b e t w e e n c o n c e p t a n d i n t u i ti o n b y e x t ra c t i n g s p ac e a n d t i m e

f r o m t h e d i v e r s i t y o r d i s t i n c tn e s s

V e r s c h i e d e n h e i t )

i n h e r e n t i n t h e c o n c e p t u a l

o r d e r i t se l f. A b i t m o r e p r e c is e l y : I t b e l o n g s t o t h e v e r y n a t u r e o f o u r t h i n k i n g

o r r e p r e s e n t i n g t o r e f e r i n t e n t i o n a l l y t o m a n y d i s t in c t o b j e ct s , w h e t h e r o b -

j e c t h e r e i s t a k e n t o m e a n a s ta t ic s e g m e n t o f t h e m a n i f o l d o f e x t e r n a l s e n s a -

t io n o r a n e p i s o d i c in c i d e n t i n t h e m a n i f o l d o f i n n e r e x p e r i e n c e . T h i s m a n y -

n e s s o r d i v e r s it y i n h e r e n t i n t h e i n t e n t i o n a l s t r u c t u r e o f a t h o u g h t o r a c o n c e p t

is p r i o r t o a n y q u a l i t a t i v e sp e c i f ic a t i o n o f t h e d i f f e r e n c e s i n q u e s t i o n . F o r t h e

o b j ec ts m u s t f i r st b e t h o u g h t a s d i v e r s e

i ~ b e r h a u p t

b e f o r e t h e d i v e r s i t y c a n b e

d e t e r m i n e d i n a d e f i n i t e f a s h i o n ( I I I , 1 6 ). I n tr i n s i c d i v e r s it y s h o w s u p , a g a i n

a t t h e c o n c e p t u a l l e v e l , i n t w o f o r m s : t o p i c a l o r p o s i t i o n a l d i s c r e t e n e s s

A u s e i n a n d e r s e i n )

a n d s e q u e n t i a l d i s c r e te n e s s

F o l g e )

( I I , x 7 ) , i n o t h e r w o r d s , a s

s p a ti a li t y ( b u t n o t e m p i r i c a l s p a c e) a n d t e m p o r a l i t y ( b u t n o t e m p i r i c a l t im e ) .

A t t h i s p u r e l y f o r m a l o r c o n c e p t u a l l e v e l, sp a t ia l it y a n d t e m p o r a l i t y a r e d e v o i d

o f a n y m a t e r i a l c h a r a c t e ri s t ic s , e. g ., t h r e e - o r n - d i m e n s i o n a l i t y ( V I I , 8 o ),

i n f i n i t e d i v i si b i li t y , i s o t r o p h y , e q u a b l e s u c c e s s io n , a n d t h e l i ke .~ s S u c h m a t e -

r i a l c h a r a c t e r i s t i c s b e l o n g , n o t t o s p a c e a n d t i m e a s a p r i o r i c o n c e p t s ( s p ec i fi c a -

t io n s o f p u r e d i v e r si t y ), b u t t o t h e i r

s c h e m a t a

o r i m a g e s p r o d u c e d b y t h e i m a g i -

n a t i o n a n d w e d d e d t o t h e p a r t i c u l a r s e ts o f e m p i r i c a l o b j ec t s a p p e a r i n g i n , o r

t o , s e n s ib l e i n t u i t i o n . 6

T h i s d i s t i n c t io n b e t w e e n s p a c e a n d t i m e a s a p r i o r i c o n c e p t s a n d t h e i r

i m a g i n a t i v e s c h e m a t a o p e n s t h e g a t e s t o a fl o o d o f p h e n o m c n o l o g i c a l a n d

s y s te m a t i c p r o b l e m s I m u s t c i r c u m n a v i g a t e h e r e . H o w e v e r , i t is c r u c ia l t o

e x p l o r e t h e i m p a c t o f t h is d i s t in c t i o n o n M a i m o n ' s c o n c e p t i o n o f t h e o b j e ct s o f

m a t h e m a t i c s a n d , c o r r e l a ti v e l y , o n h i s u n d e r s t a n d i n g o f h o w i m a g i n a t i v e c o n -

s t r u c t i o n r e n d e r s t h o s e o b j e c t s e p i s t e m i c a l l y a c c es s ib l e to u s .

,4 Bergm an,

Phi losophy of Solomon M aimon,

38. Cs also Charlotte Katzoff, Solomo n Maim on's

Interpretation of Kant's Copernican Revolution,

Kant-Studien

66 (1975): 342-56. For a recent,

and deep ly intriguing, atte mp t to mediate between sensibility and u nderstanding, see Werner

Flach,

Zur Prinzipienlehre der Anschauung,

Bd.

1: Das spekulat ive Grundproblem der Vereinzelung

(Ham burg: M einer, a9.63). (Flach is a close student of the Neo-K antians.)

'sCf. Atlas,

From Cri t ical to Speculat ive Ideal ism, 172.

'r II, 133

Schema) ,

347

Schema) ,

427 (B//d); III,

15 ein sinnl iches Bi ld) .

See, too, the

extended statement: Die VorsteUungen yon Zeit un d Raum als Objekte der Anschauung an sich,

sind so lche durch eine T ~iuschung entstandene figtirlicbe Vorstellungsarten ( IV , 260-66).

Maimo n's doctrine o f schemata is complicated by his (and K ant's) distinction between the empiri-

car' and the transcendental schema and even more so by his claim: Un d eben dieses

ranscenden-

tale Schema

ist de r C, genstand der

Elementargeometrie,

so wie das dadurch Vorgestellte Gegen stand

der hthern

Geometric

ist

Kri t ische Untersuchungen i iber den menschl ichen Geist ,

VII, 188).

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R E F L E C T I O N S O N M I M O N N D M I M O N I D E S 5 0 3

I f s p a c e a n d t i m e a r e i n th e f i r st p la c e c o n c e p t s a n d " a f t e r w a r d s " i n t u i t i o n s

o n l y b y w a y o f t h e i m a g i n a t i o n ' s l ic it f ic t io n s ,' 7 w h a t k i n d o f m a t h e m a t a a n d

w h a t m a n n e r o f c o n s tr u c t i o n w o u l d s u it c o n c e p t u a l s p a c e a n d t i m e ? T o a n -

s w e r t h e s e t w i n q u e s ti o n s w e m u s t b e w i l li ng t o e x t r a p o l a t e f r o m M a i m o n ' s

e x p r e s s r e m a r k s , s i n c e t h e s e d o n o t a l w a y s e l a b o r a t e c l e a r ly a n d d i s ti n c t ly t h e

d i f f e r e n t s t r a t a o f " m a t h e m a t i c a l o b j e c t " a n d " c o n s t r u c t io n " r e q u i r e d t o sa t-

i s f y t h e t h e s e s h e a f f i r m s .

S e e n f r o m t h e p e r s p e c t i v e o f u n d e r s t a n d i n g , a l l c o n c e p t s a r e r e l a t i o n s ,

p u r e l y a b s t r a c t s tr u c t u r e s , a n d a r e l o g ic a ll y i n d e p e n d e n t o f p a r t i c u l a r r e l a t a o r

s u b s t r a t a t h a t " f u l f il l " t h e m , a s so m a n y s u b s t i t u t i o n i n s t a n c e s f o r t h e v a r i a b l e s

' x ' a n d ' y ' i n t h e f o r m a t x R y .

S o , w i t h r e g a r d t o t h e c o n c e p t t i m e , M a i m o n c a n s a y t h a t " th e f o r m o f

n u m b e r i s p u r e t i m e a s c o n c e p t " ( I I , 2 ~ : c f . i b i d . $ 6 , 6 9) . H o w i s t h i s t o b e

i n t e r p r e t e d ? F o r M a i m o n , a n u m b e r i s a p u r e r a t i o ( o f i n t e g e r s ), i .e ., t h e

" n u m b e r 2 " is t h e r a t i o ~ : 1 o r , t o g e n e r a l i z e , a n y p o s i t iv e i n t e g e r n i s s i m p l y

t h e r a t i o n : 1. A p a r t i c u l a r n u m b e r i s w h o l l y d e t e r m i n e d b y it s p o s i t i o n i n t h e

s e q u e n c e o f r a t i o s u : l , 2 + 1 : 1 . . . 9 + k : 1 ( w h e r e k > 1). I n b r i e f , M a i m o n

f o l l o w s t h e m o d e r n , C a r t e s i a n , t r a d i t i o n i n r e p l a c i n g & ~ t O ~6 g w i t h ~ .d ry og o r

r a ti o . *s T i m e a s t h e c o n c e p t o f o r d e r e d s e q u e n t i a l i t y i s, t h e n , t h e f o r m o f

n u m b e r a s su c h. *9 E n u m e r a t e d s e ts o f ( se n s ib l e ) o b j e c t s a r e u n n e c e s s a r y h e r e :

t h e p u r e r e l a t i o n n : 1 s u f f ic e s t o d e f i n e a n d , i n d e e d , t o g e n e r a t e a n y n u m b e r

(w h e re n > 1 ).

W h e r e , t h e n , d o i m a g i n a t i o n a n d , w i t h i t, c o n s t r u c t i o n e n t e r t h e p i c t u r e ?

A t tw o p o i n t s , w h i c h n e e d t o b e c a r e f u l l y d i s c r i m i n a t e d . F i rs t , p u r e u n d e r -

s t a n d i n g i s i t s e lf a l r e a d y " i m a g i n a t i v e , " t h a t i s, p r o d u c t i v e o f t h e o p e r a t i o n s i t

e n a c ts i n t h i n k i n g t h r o u g h t h e n u m e r i c a l r e l a t i o n o f n : l , o r , a s i n D e s c a r t e s 's

R e g u l a e , t h e c o n t i n u o u s p r o p o r t i o n ~ : m :: m : n . *~ W i t h t h e s e o p e r a t i o n s i t

17See II, 133; III, 15 and 44, an d com pare Zac, Saloraon Maimon, 173-98.

,s See Jacob Klein,

Greek Mathematical Thought and the O rigin of Algebra,

trans. E. Brann (Cam-

bridge, Mass. : M.1.T . Press, 1968), 46 -6 o and cf. John Mayberry, "A New Begriffsschrift 0), "

British Jour nal for the Philosophy of Science 3 t (198o): ~31-54 for the fu rthe r repercussions of the

mo dern aband onm ent of the Greek notion of dtQt~l~6g as an assembled account o f a collection of

enum erable items.

,91 bypass here the question of the Status of the "unit" when infinitely divisible time is

cons trued as the fo rm of number. In an addendu m to h is Versuc h fiber die T ranscendentalphilosophie

Maimon writes: "Die absolute Einheit (wie sie in der reinen Arithmetik betrachtet wird) ist eine

Idee, die niemals in der Anschauung (deren Form en Zeit und Raum sind, welche ins unendliche

theilbar sind) dargestellt werden kann" (II, 35o).

9 Regulae ad directionem ingenii, Rule 6 and the com men tary of J .-L. Marion and P.

Costa bel ad. loc., Regles utiles et clairespour la direction de l esprit en la recherchede la v~rit~(The Hague :

Nijhoff, 1977).

As always in such matters one is tempted to discover "anticipations" of m ode rn conceptuality

in prem odern thought. One sympto m in late medieval and Renaissance arithmetical o r logistical

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5 0 4 J O U R N A L O F T H E H I S T O R Y O F P H I L O S O P H Y 3 0 " 4 O C T O B E R 1 9 9 2

c r e a t e s it s o w n m a t t e r o r s t u f f , i n t h e s h a p e o f th e a b s t r a c t m e d i u m o f r e la -

t i ona l i t y

i iberhaupt,

t o b e f i l le d i n b y p a r t i c u l a r r e l a t io n s .

I n a k e y p a s s a g e f r o m h i s

Ve r s uch e iner ne ue n Lo g ik

( 1 7 9 4 ) M a i m o n w r i te s :

E v e r y

object

o f p u r e m a t h e m a t i c s c o n s i st s o f th e

ge ne r a l s tu f f ,

s p a c e o r t i m e ,

a n d a p a r t /c u / a r f o r m ; b o t h a r e a p r /o r /, o n l y w i t h th e d i f f e r e n c e t h a t t h e s t u f f is

n o t p r o d u c e d [hervorgebracht] b y t h e c o g n i t i v e p o w e r , b u t i s g i v e n a p r i o r i t o it ,

w h i l e t h e f o r m i s

p r o d u c e d

b y t h e c o g n i ti v e p o w e r i n a c c o r d w i th t h e p r i n c i p l e

o f d e t e r m i n a b i l i t y ( V , 1 2 5 ).

T h e a p p a r e n t i n c o n si s te n c y b e t w e e n o n e p o r t i o n o f th is p a s s a g e - - t h e

s t u f f is n o t p r o d u c e d b y th e c o g n i ti v e p o w e r - - a n d m y p r e v i o u s c l ai m c a n , I

t h i n k , b e e l i m i n a t e d , o r a t l e a s t m i t i g a t e d , a s fo l lo w s . T h e i m a g i n a t i o n ,

s e m u

stricto, t o w h i c h M a i m o n c o n s i s t e n tl y a s s ig n s m a t h e m a t i c a l c o n s t r u c t i o n a n d

w h i c h h e c a ll s t h e f a c u l t y o f f ic t i o n s

Er d ic h tungs v e r mtge n) ,

i s i t se l f pu t t o

w o r k i n t w o d i s t i n g u i s h a b l e r e g i s t e r s , t h e s y m b o l i c a n d t h e o s t e n s iv e . T h e

m o d el f o r t h e f i rs t r e g i s t e r i s a l g e b r a ; t h e m o d e/s f o r t h e s e c o n d a r e a r i t h m e t i c

a n d g e o m e t r y a s o r d i n a r i l y o r e l e m e n t a r i l y u n d e r s t o o d . '~

2 . T H E L G E B R I C M O D E L

I n t h e t o o o f t e n i g n o r e d A p p e n d i x t o h i s f ir st e r m a n b o o k , V e r su c h t ib er / e

T r a n s c e n d e n m I p h il o s o p h i e 1 7 9

I), M a i m o n d i sc u s s e s a t l e n g t h S y m b o l i c C o g n i -

t i on a n d P h i l o s o p hi c a l L a n g u a g e ( II , s 6 5 - 3 3 2 ) . O f c e nt r a l c o n c e r n t o m e

h e r e is h i s d e f i ni t i o n o f a n o b j e c t o f s y m b o l i c c o g n i t i on : , It is a m a n n e r o r

m o d e [ Ar t] f t h i n k i n g a n o b j e ct o f i n t ui t io n c o n s i d e r e d a s it se lf n o b j e ct ( b u t

n o t o f i n tu i ti o n ( ib id ., 7 2 ) . S y m b o l i c o b j e ct s c o r r e s p o n d t o w h a t t h e S c h ol a s -

t ic s c a l l e d s e c o n d i n t e n t i o n s o r ~ t u s s i g n a t us , t h a t is, t h e y o b j e c t i f y t h e

m a n n e r o f o u r t h i n k i n g f ir st -i n te nt io n al b j e c t s ( as i n a ct u s e x e rc / t ~) . S y m -

p r a c t ic e f o r e s h a d o w i n g C a r t e s i a n a n d p o s t - C a r t e si a n d e v e l o p m e n t s is t h e e m p h a s i s o n n u m b e r s

a s s i g ns t o b e m a n i p u l a t e d i n c a l c u la t i o n ; s e e t h e i m p o r t a n t s t u d y b y H a n s G e o r g K n a p p , " Z a h l ai s

Z e i c h en . Z u r ' T e c h n i s i e r u n g ' d e r A r i t h m e t i k i n M i t t e l a lt e r ," Historia Mathematica 1 5 (1 9 8 8 ) : 1 1 4 -

3 4 . O n t h e a d v e n t a n d e a r l y " p r a c t i c a l" a p p li c a t io n s o f A r a b i c a l g e b r a i n E u r o p e , s e e F r a n k J .

Swe t z , Capital ism and Ar i thmetic: The New Ma th of the x Sth Centu~ (La Sa i | e , I l l .: Op e n C o u r t , 1 9 87 ) .

O n e m i g h t c o n j e c t u r e t h a t D e s c a r t e s ( a n d h i s fo l lo w e r s ) " th e o r i z e s " m e d i e v a l a n d R e n a i s s a n ce

" T w o g o o d " f i r s t a p p r o x i m a t i o n s " t o th e s t a tu s a n d r o l e o f a l g e b r a i n K a n t ' s t h o u g h t m a y b e

f o u n d i n A . T . W i n t e r b o u r n e , " C o n s t r u c t i o n a n d t h e R o l e o f S c h e m a t i s m i n K a n t ' s P h i l o s o p h y o f

M a t h e ma t i c s , " Studies in History and Philosophy of Science 12 ( 1 9 8 1 ) : 3 3 - 4 6 a n d C h a r l e s P a r s o n s ,

" A r i t h m e t i c a n d t h e C a t e g o r i e s , " Topoi 3 0 9 8 4 ) : 1 ~ ( A k e y K a n t i a n t e x t o c c u r s i n h i s

" R e f l e x i o n e n z u r M a t h e m a t i k " [ A k . - A u s g . , B d . X I V , p p . 5 4 - 5 5 ] . )

9 Th e two pai rs o f d is t in c t ion s (v iz . , f i rs t / se cond in ten t ions, actu,~ exerc /tu.#actu ,~ s /gnatu .s) do no t

c o i n c id e , b u t t h e y d o o v e r l a p , i n t h e M e d i e v a l tr a d i ti o n . S e e L . A . H i c k m a n , Logical Second

Intentions: Late Scholastic Theories of Higher-Level Predicates (Ph .D. Di s s . , Un i v e rs i t y o f Te x a s , Au s t i n ,

1 97 1 ) a n d G a b r i e l N u c h e l m a n s , " T h e D i s t in c t io n Actuz Exercitus/Actus Significatm i n M e d i e v a l

Se ma n t i c s , " i n Meaning and Inference in Medieval Phi losophy, e d . N . K r e t z m a n n ( D o r d r e c h t :

K l u w e r , 1 9 8 8) , 5 8 - 9 o . I t i s o n e o f t h e m a n y m e r i t s o f J a c o b K l e i n 's w o r k , Greek Mathematical

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R E F L E C T I O N S O N M I M O N N D M I M O N I D E S

5 0 5

b o li c c o g n i t io n , f u r t h e r m o r e , r e a c h e s f a r b e y o n d t h e a m b i t o f o u r i n tu i ti v e

t h i n k i n g ( ib i d. , 27 9 ) . M a i m o n t a k e s a s e x a m p l e s t h e n u m b e r l o o o , D e s c a r te s ' s

c h i l i g o n a n d E u c l i d ' s p r o o f ( E / e m e n t s I I , 3 8 ) t h a t t r i a n g l e s o n e q u a l b a s e s a n d

b e t w e e n t w o p a r a l l e l l i n e s a r e e q u a l t o o n e a n o t h e r , i n o r d e r t o s e c u r e t h i s

f u n d a m e n t a l p o i n t : W e g r a s p th e i r m o d e o f o r ig i n at io n [ E n t s t ehungs ar t ] w i t h -

o u t i n t u it i n g t h e m a s a l r e a d y o r i g i n a t e d ( ib i d. , 2 7 3 ). T h e E n t s t e h u n g s a r t o f

a n y s u c h u n i n t u i t a b l e m a g n i t u d e , a s w e ll a s o f g e o m e t r i c a l f i g u r e s t a k e n i n

t h e i r a b s o l u t e e x a c t i t u d e , i s a l s o c a l l e d i t s s c h e m a o r f o r m ( i b i d . , 2 7 4 - 7 5 ) . A n d

t h e n M a i m o n d r a w s th e p e r h a p s u n e x p e c t e d c o n c lu s i o n : a ll e m p i r i c a l c o n -

c e p t s a n d p r o p o s i t i o n s , i n d e e d a l l a p r i o r i c o n c e p t s , i n s o f a r a s t h e y a r e n o t

m e r e l y f o r m s , b u t o b j e c t s o f i n t u i ti o n , a s a ls o a ll a x io m s o f m a t h e m a t i c s , a r e

e x c l u d e d f r o m s y m b o l i c c o g n i t io n ; o n l y f o r m s , t h e r e f o r e , t h a t is , r u l e s o f t h e

m o d e o f o r i g i n a ti o n o f o b j e c ts , b e l o n g t o s y m b o li c c o g n i t io n . T h e c a t e g o r ie s ,

as wel l as a l g e b r a ic fo r m u l a e , a r e o f t h is s o r t . . . ( ib id ., 2 7 7 ; m y e m p h a s is ) .

T h i s p a s s a g e , t o g e t h e r w i th o t h e r t e x t s r e c o l le c t e d b e f o r e , w a r r a n t s a n u m -

b e r o f s p e c u l a t io n s .

( t ) W h e n t h e p u r e u n d e r s t a n d i n g is a t w o r k in m a t h e m a t ic s , f o r m a n d

m a t t e r a r e o n l y m o d a l l y d i s t i n g u i s h a b l e , s i n c e t h e o p e r a t i o n s p e r f o r m e d a r e

i d e n ti c a l w i th t h e o p e r a t i o n s s y m b o l ic a l ly r e n d e r e d o r o b j e c t i fi e d .

( 2 ) T h e exp l i c i t s y m b o l s o f t h e s e o b j e c ti f ie d o p e r a t i o n s a r e t h e d e e d s o f t h e

i m a g i n a t io n , a n d t h is i n a t w o f o l d m a n n e r :

(a ) T h e s y m b o l s a r e t h e m s e l v e s p r o d u c e d t o s ta n d i n f o r t h e o p e r a t i o n s

t h e y s y m b o l i z e a n d , i n f ac t , b e c o m e i n t u r n t h e m a t t e r f o r f u r t h e r o p e r a t i o n s .

( b) T h e i m a g i n a t i o n , a p i n g t h e u n d e r s t a n d i n g , a s M a i m o n s ay s , p r o -

c e e d s in a c c o r d a n c e w i t h a c o n c e p t o f th e u n d e r s t a n d i n g a s i ts r u l e o f c o n d u c t

( R i c h t s c h n u r ;

i b i d ., 1 3 3 ). H e n c e , t h e d i s c r e t e e l e m e n t s o f a c o g n i t i v e o p e r a t i o n ,

a, 'b, ' c , f o r e x a m p l e , a r e a r r a y e d b y t h e i m a g i n a t io n i n a sp a t ia l fi e ld a n d ,

v ia t h e la t te r , i n a t e m p o r a l l y i n d e x e d s e q u e n c e , a , , b , , ca . T h i s s p a t i o -

t e m p o r a l a r r a y / s i n t u i t a b le o r s e n s u o u s , a n d t h e s p a c e a n d t i m e i n g r e d i e n t s i n

i t a r e s c h e m a t a o r B i l d e o f th e p u r e c o n c e p t s o f A u s e i n a n d e r s e in a n d F o l g e .

( 3) C r u c ia l t o t h is e n t i r e p r o c e s s o f s y m b o l i c c o n s t r u c t i o n b y t h e p u r e

i m a g i n a t i o n i s t h e r o l e o f t h e s c h e m a a s a g u i d e t o t h e m o d e o f o r ig i n a t i o n o f

t h e r e l e v a n t m a t h e m a t i c a l o b j e c t o r c o m p l e x o f o b je c t s. T h e i m a g i n a t i o n f o l-

l ow s a r u l e o r a l g o r i t h m

( E n t s t e h u n g s r e g e l ;

c f . I I , 3 3 a n d 3 5 ) f o r t h e l o g i c a l

g e n e s i s o f th e c o n c e p t u a l r e l a ti o n s. T h e p a r a d i g m f o r s u c h a ru l e , o n

M a i m o n ' s a c c o u n t , i s a n a l g e b r a i c f o r m u l a ( ib i d ., 2 7 7 ), s in c e t h e s e q u e n t i a l

Thought, tha t he calls attention to the importance of this distinction for the ge nesis of early

mod ern m athematics and physics (esp. p. 3o6 , n. 324); one c ould also note th at Leibniz still

employs the notion in a ctu s ignato in his early Dissertatio de arte c0ndnnator/a 0666), wh en spea king

of quality, quantity, and relation (see D/e philosophische Sch riften, ed. Gerhardt [reprint, Hildes-

heim: Olms, 1965], IV, p. 35).

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~ o 6

J O U R N A L O F T H E H I S T O R Y O F P H I L O S O P H Y 3 : 4 O C T O B E R 1 9 9 2

d i s p la y o f t h e t e r m s o f s o m e s u c h f o r m u l a is , i n p r i n c ip l e , a g u i d e t o t h e

a r i t h m e t i c a l o r g e o m e t r i c a l o b j e c t s s a t i s f y i n g i t.

( 4) W h e n M a i m o n t u r n s e x p r e s s l y t o t h e s c i e n ce o f a l g e b r a ( o r t o it s c o g -

n a t es ) t h e p i c t u r e I h a v e s k e t c h e d t h u s f a r c o m e s i n t o s h a r p e r f o c u s ; in p a r t i c u -

l a r t h e n e c e s s it y o f o s te n s iv e c o n s t r u c t i o n a s a c o m p l e m e n t t o s y m b o l i c c o n -

s t r u c t io n i s b r o u g h t i n t o r e l i ef . T w o p a i r s o f o v e r l a p p i n g d i s t in c t i o n s a r e

p e r t i n e n t h e r e : f ir s t, th a t b e t w e e n

d i e a l lg e m e i n e G r 6 s s e n l e h r e

a n d

d i e r e i n e ( n i c h t

a l l g e m e i n e ) G r 6 s s e n l e h r e

( V , 2) a n d , s e c o n d , t h a t b e t w e e n

d i e B u c h s t a b e n r e c h n u n g

a n d

d i e e i g e n t l i c h e A l g e b r a

( V I I , 2 5 f f. ). *s A b r i e f i n s p e c t i o n o f t h e s e t w o d i s t i n c -

t i o n s w il l h e l p u s s e e h o w m u c h f o r M a i m o n , a s f o r K a n t , o r d i n a r y ( o r t r a d i -

t io n a l) a r i th m e t i c a n d g e o m e t r y b e l o n g t o t h e e so t e r ic o r a c r o a m a t i c p l a n e o f

e x p o s i ti o n , w h i l e m o r e s o p h i s t i c a t e d m a t h e m a t i c s f u r n i s h e s t h e m a i n p h i l o -

s o p h i c a l l e a d s a n d t h e m o s t e x o t i c d i l em m a s . * 4

I n h i s

V e r s u c h e i ne r n e u e n L o g i k

( 1 79 4 ) M a i m o n c o m p a r e s t h e r e l a t i o n b e -

t w e e n f o r m a l l o gi c a n d t r a n s c e n d e n t a l l o g i c t o th e r e l a t i o n b e t w e e n " t h e u n i -

v e r s a l t h e o r y o f

m a g n i t u d e s - - a r i t h m e t i c a u n i v e r s a l i s

a n d " t h e p u r e , b u t n o t

u n i ve r sa l , t h e o r y o f m a g n i t u d e s . "

T h e f i rs t i t e m i n t h i s s e c o n d p a i r " a b s t r a c ts n o t o n l y f r o m a ll e m p i r i c a l

o b j ec t s t o w h i c h i t is a p p l i e d , b u t e v e n f r o m a ll a p r i o r i d e t e r m i n a t e o b j e ct s o f

p u r e m a t h e m a t i c s , a n d c o n s i d e r s o n l y a ll p o s si b le f o rm s o r

r e l a t i o n s

i n w h i c h

m a g n i t u d e s g e n e r a l l y c a n b e t h o u g h t , " w h i l e t h e s e c o n d d i s c i p l in e " h a s f o r i ts

t h e m e

[ G e g e n s t a n d ]

r e a l o b j e c t s d e t e r m i n a b l e t h r o u g h c o n s t r u c t i o n

a p r i o r i

(V,

2 ). T h e p u r e , b u t n o t u n i v e r s a l , t h e o r y o f m a g n i t u d e s " m a k e s u s e o f t h e

u n i v e r sa l t h e o r y o f m a g n i t u d e s o n l y u n d e r t h e c o n d i t i o n s o f a p o s si b le c o n -

s t r u c t i o n " ( i b i d . ) .

A r i t h m e t i c a u n i v e r s a l i s

i s N e w t o n ' s p h r a s e f o r w h a t D e s c a r t e s a n d W a l li s

c a l l e d

m a t h e s i s u n i v e r s a l is . * 5

M a i m o n u n d e r s t a n d s

a r i t h m e t i c a u n i v e r s a l i s

a s u n i -

v e r s a l

a l g e b r a ,

i n t h e se n s e t h a t t h e r o o t s o f e q u a t i o n s c a n b e " i n d e t e r m i n a b l e

( i n f in i t e ) " o r " i m p o s s i b l e " ( i .e ., i m a g i n a r y ) . T h a t i s , i n s u c h c a s e s n o l o c u s a n d

n o m a g n i t u d e s a t i s fy i n g t h e v a l u e s o f t h e e q u a t io n ( s ) c a n b e e x h i b i t e d i n p u r e

,s A good sense of how

die al lgemeine Gr6ssenlehre

was construed towards the end of the

eighteenth century m ay be gleaned from Friedrich M urhard,

System der Elemente der al lgememen

Gr6ssenlehre nach ihrem Zustand am nebst Li teratur und Geschichte

(Lemgo, 1798). Murhard's book,

and the relation between

die allgemeine Gr6ssenlehre

and sev enteenth-century algebra and analysis,

deserve closer study.

9 This should be o bvious from Kant's reliance on the ca lculus in his

Metaphysische Anfangs-

gri inde der N aturwissenschaf ten (cf. Biichel, Geometrie und Phi losophie, 221-99) and from Maimon's

references to Cavalieri's

Methodus indivisibi lium

(II, 174; V, 2o5) and to

analysis situs

(II, 69).

,s On the pre-Cartesian history of the phrase see the indispensable monograph by Giovanni

Crapulli,

Mathesis U niversalis . Genesi di u na idea nel XV1 secolo

(Rom e: Ateneo, a969). Joh n WaUis's

treatise bears the tide

Mathesis universal is seu O pus ari thmet ieum

(London, 1657 and is, in part, a

critique of Descartes's geometrical perspective.

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R E F L E C T I O N S O N M I M O N N D M I M O N I D E S 5 0 7

o r s e n s u o u s i n t u it i o n . T h e

p u r e

t h e o r y o f m a g n i t u d e s is r e s t r i c t e d t o t h o s e

a l g e b r a i c a l l y s y m b o l i z a b l e f o r m u l a e w h o s e v a l u e s c a n b e c o n s t r u c t e d : l o c i w i t h

r e a l ( n o t c o m p l e x ) r o o t s a n d , a t l e a s t f o r M a i m o n , p o s i ti v e r a t i o n a l n u m b e r s . '8

T h e s e c o n d d i s t i n c ti o n n o t e d a b o v e h a s t h e s a m e t e n o r . T h e " l it e ra l c a lc u -

l u s " d i e B u c h s t a b e n r e c h n u n g ) a b s t r a c ts f r o m t h e i ss u e o f t h e c o n s t r u c t ib i l i ty o f

t h e r e l a t i o n s it y i e l d s b y p u r e l y c o m b i n a t o r i a l m e a n s . S o , f o r a p r i m e i n s t a n c e ,

x = ~ / - a , w h e r e " a " is a p o s i ti v e i n t e g e r , s a y 2 , d o e s n o t l e a d to a n y " d e t e r m i n -

a b le m a g n i t u d e " ( V I I , 2 5) . " G e n u i n e a l g e b ra , " o n th e c o n t r a r y - - a n d n o t e

h o w f a i t h f u l l y M a i m o n e c h o e s D e s c a r t e s i n c a l l i n g t h i s a l s o " t h e s o l u t i o n o f

p r o b l e m s " ( i b id . ) - -i s c o n c e r n e d o n l y w i th c o n s t r u c ti b l e m a g n i t u d e s .

T h u s , w h i l e th e " u n i v e r s a l t h e o r y o f m a g n i t u d e s " o r t h e " l i te r a l c a l c u lu s "

s e e m a t f ir s t s i g h t t o e n j o y l i m i tl e ss c o m b i n a t o r i a l f e r ti l it y , m a n y o f t h e i r

o f f s p r i n g p r o v e t o b e " w i n d - e g g s , " i n a s m u c h a s t h e y a r e r e s is t a n t to c o n s t r u c -

t io n . O n l y " g e n u i n e a l g e b r a " o r " t h e p u r e t h e o r y o f m a g n i t u d e s " s e rv e s as a

b r i d g e b e t w e e n t h e p u r e t h i n k i n g o f r e l a t i o n s a n d " r e a l (i .e ., a c t u a l) o b j e c ts . "

W h a t M a i m o n t o u c h e s u p o n h e r e a n d e l s ew h e r e is t h e g r o u n d a n d t r e a c h-

e r o u s e n i g m a o f s y m b o l iz a t io n p e r s e . O n t h e o n e h a n d , i f s y m b o l s tr a n s c r ib e

t h e o p e r a t i o n s o f p u r e t h o u g h t t h e r e a r e n o i n t ri n si c li m it s o n t h e c o m b i n a t o -

r ia l po s s ib i li ti e s o f p u r e t h o u g h t i t s e lf ; t h e r e a r e , h o w e v e r , n o g u a r a n t e e s t h a t

a n y a r b i t r a r y s y m b o l i c f o r m a t i o n ( a r r a y o f si g ns ) wi ll c o r r e s p o n d t o a " r e a l "

o b j e c t o r s t a te o f a f fa i r s (s e e, a g a i n , t h e c a s e o f ~ / - 2 ) . O n t h e o t h e r h a n d ,

t h e r e i s n o m e c h a n i c a l m e t h o d f o r t h e d i s c o v e r y o f j u s t t h o s e s y m b o l ic e q u a -

t i o n s w h o s e s o l u t i o n s a r e c o n s t r u c t i b l e , t h a t i s , d o y i e l d " r e a l " o b j e c t s s u c h a s

l oc i o r r a t i o n a l n u m b e r s .

A s M a i m o n r e m a r k s , a f t e r q u o t i n g a t le n g t h a p as s a g e f r o m N e w t o n ' s

A r i t h m e t i c a U n i v e r s a l i s

o n t h e r u l e s f o r a m a t h e m a t i c a l h e u r i s ti c : " N e w t o n ' s

p r e s c r i p t i o n s f o r d i s c o v e r y in m a t h e m a t i c s s t r i k e m e t h e w a y K l o p s t o c k 's p r e -

s c r ip t io n s f o r m o r e e l e v a te d p o e t r y d o . L u c r e t i u s w a s n o t a l t o g e t h e r w r o n g

w h e n h e c o m p a r e d d i s c o v e r e rs to b l o o d h o u n d s " ( II , 3 7 o ). '7

A t a l l e v e n t s , t h e e v i d e n c e a s s e m b l e d s o f a r p o i n t s u n i v o c a l ly t o th e c o n c l u -

s i o n t h a t c o n s t r u c t i o n , f a r f r o m b e i n g m e r e l y a n " e x t r in s i c c r i t e r i o n , ' 's is f o r

M a i m o n t h e h e a r t o f t h e m a t t e r . A s f ir s t s y m b o l i c , a n d t h e n o s t e n s iv e , i m a g i n a -

t iv e c o n s t r u c t i o n a l lo w s u s t o tr a f f ic b e t w e e n t h e r e g i o n o f p u r e t h o u g h t a n d

t h e d o m a i n o f t h e i n t u it i v e .

~ Se e Versuch fiber die Transcendentalphilosophie

(II, 58, 77, lOO, 2e7-29).

9 Leibniz, too, was intent o n finding a systematic me thod for the discovery of algori thms; see

Paul Schreker, "Lcibniz and the A rt of Inventing Algorithms,"

Journal of the History of Ideas 7

(1947): lo7-1 6. Lack of success in this domain explains why V ico refers to algebra as a divinatory

art Operefilosofiche, ed. P. C ristofolini [Florence: Sansoni, a97 1 , 353)-

9 M. Guerou lt La philosophic transcendentale, 5o. Cf. Valerio Verra's remarks on construction

in Maimon in his "Nachwort" to Salomon Maimon,

Gesammelte Werke,

Bd. VII, p. 712.

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5 0 8 J O U R N A L O F T H E H I S TO R Y O F P H I L O SO P H Y 3 0 : 4 O C T O B E R 1 9 9 2

I t d o e s s o, as w e h a v e a l r e a d y b e g u n t o s ee , b e c a u s e i m a g i n a t i o n , o b e y i n g

t h e d i r e c t io n s o f p u r e u n d e r s t a n d i n g , e x h i b i ts t h e r u l e s o f o r i g i n a t io n i n h e r -

e n t i n t h e o p e r a t i o n s o f t h e u n d e r s t a n d i n g i ts el f. C o n s t r u c t i o n a n d o r i K i n a t i o n

w o r k i n t a n d e m , s i n c e t h e m i n d , f o r M a i m o n , a s w e ll a s f o r D e s ca r te s , H o b b e s ,

S p i n o z a, a n d e v e n K a n t , i s t h e m a t r i x o f g e n e r a t i o n s .

T h i s c h a r a c t e r i z a t i o n i s n o t t r a n s p a r e n t a n d , i n d e e d , i ts o p a q u e n e s s w i ll

l e a d u s s h o r t l y t o M a i m o n ' s n o t i o n o f t h e i n f in i t e o r d i v i n e i n t el le c t. F o r t h e

m o m e n t , t w o f e a t u r e s n e e d e m p h a s i s a l o n g t h e r o u t e t o t h a t c l im a t ic n o t io n .

F i r st , a s t h e m a t r i x o f g e n e r a t i o n o r l o g i c a l g e n e s i s , t h e m i n d i s n o t i t s e l f o n

t h e m o v e . B o r r o w i n g , a s d o e s M a i m o n , f r o m t h e i d i o m o f t h e c a l cu l u s o f

f l u x io n s , G u e r o u l t m a k e s t h e s a m e p o i n t a s f o ll o w s : " T h e m o d e o f p r o d u c -

t i o n , o r , i f o n e l i ke s, th e r u l e a c c o r d i n g t o w h i c h t h e u n d e r s t a n d i n g t h i n k s t h e

o b j e c t, is c l e a r ly n o t i t s e l f t h o u g h t o f a s i n f l u x ( fl ue n te ), b u t t h e d e t e r m i n a t i o n

o f t h e o b j e c t f o l l o w i n g th i s r u l e c a n o n l y b e c o n c e i v e d a s i n f l u x . ". 9

S e c o n d , t h e m o t i o n s o r f l u x i o n s b e l o n g i n g t o m a t h e m a t i c s i n i ts es s e n ti a ll y

c o n s t r u c ti v i s t m o d e i n t r o d u c e i n e x o r a b l y a h i a tu s b e t w e e n t h e f i n it e e x p r e s s-

i b le r u l e o r , a s w e w o u l d n o w a d a y s s ay , r e c u r s iv e a l g o r i t h m , a n d i ts n e c e s s a r i l y

i n f in i t e w o r k i n g - o u t i n i n t u i t i o n . T h i s is th e p e n u l t i m a t e l in k i n t h e c h a i n

l e a d i n g t o t h e t h es i s t h a t " o u r u n d e r s t a n d i n g i s a s c h e m a f o r t h e i d e a o f a n

i n f in i t e u n d e r s t a n d i n g " ( I I, 3 6 5 ).

T h e p r o b l e m o f t h e i n c o m p l e t ab l e c h a r a c t er o f ce r t ai n m a t h e m a t i c a l

n o t i o n s - - t h e l i m i t- v a l u e o f a n i n f i n it e c o n v e r g i n g s e ri e s o f r at i o n al s , o r t h e

q u a n t i t y o f a r a t i o o f d i f f e r e n t i a l s d y / d x - - i s f a m i l i a r to M a i m o n ' s r e a d e r s ( cf .

I I , 35 1 ; 3 5 8 ). L e s s fa m i l i a r, b u t e v e n m o r e t h o u g h t - p r o v o k i n g , i s M a i m o n ' s

d i s c u ss i o n o f t h e in c o m p l e t a b l e c h a r a c t e r o f m o r e e l e m e n t a r y m a t h e m a t i c a l

t a sk s , s u c h a s t h e c o n s t r u c t i o n o f a c ir c le . I n t h e a r t i c l e o n " T r u t h " i n h i s

P h i l o s o p h i s c h e s W O r t e r b u c h

M a i m o n c o m m e n t s i n d e t a i l o n D e s c a r t e s ' s f a m o u s

c r i t i q u e ( L a

G ~ o m ~ t r i e ,

B k . I I ) o f t h e G r e e k g e o m e t e r s ' d i v i si o n o f c o n s t r u c t i o n

i n t o " g e o m e t r i c a l " a n d " m e c h a n i c a l " ( I I I , 1 7 o - 7 2 ) . A t i ss u e is w h e t h e r t h e

g r a p h c o n s t r u c t e d i n a c c o r d w i t h t h e a l g e b r a i c e q u a t i o n o f a c ir c le ( o r o f o t h e r

c u r v e s ) c a n c a p t u r e , s o t o sp e a k ,

a l l

t h e p o i n t s b e l o n g i n g t o t h e l oc u s. M a i m o n

O p p o se s D e s c a r te s i n d e n y i n g t h a t i t c a n . H e w r i te s : " H e r e r e a s o n r e q u i r e s

t h a t w e c o n t i n u e t o m u l t i p l y th e p o i n ts t h o u g h t o f a d

i n f i n i t u m ,

w h e r e b y t h i s

c o n s t r u c t i o n a p p r o x i m a t e s i ts c o n c e p t ; o n l y t h r o u g h t h e c o m p l e t e a c hi e v e -

m e n t o f t h i s c o n c e p t d o w e a c q u i r e a ' r e a l ' o b j e c t a p r i o r i , a n o b j e c t w h i c h i s

n o t o t h e r w i s e p o s s i b l e " ( V , 1 7~ t). T h e a l t e r n a t i v e , t o c o n s t r u c t t h e c i r c le s i m p l y

9 For Newton's distinction between

luentes

(the variables in an equation) and their

f luxions

("the speeds with which they each flow and are increased by their generating motion"), see A

Treatise of the Method o f Flu.x6ons and Infinite Series ~

737), in

The Mathemat ical Papers o f l sao.c Newton,

ed. D. T. W hiteside (Cam bridge: Cambridge U.P., 1967), 1: z7.

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R E F L E C T I O N S O N M A I M O N A N D M A I M O N I D E S 5 0 9

by mechanical or kinematic means--a compass, say--yields only an a poste-

riori object.so A

progressus in inf ini tum

Maimon asserts, is the only way we

have of constructing a concept in a totally a priori fashion (ibid).

Every finitely expressse d symbolic equatio n (e.g., x* + y' = k) is, then, the

format of an infinite task. Hence, even the capacity of our understanding to

generat e a priori its own objects is held in bounds by the gap separati ng the

finite form of its rules of origination from the infinitude of the execution of

those rules in each particular case. We are incapable of passing,

uno iactu

from

the finite rule to its infinite realization.

It is in this mathematical setting that Maimon's exotic, but not unparalle d,

conception of an infinite intellect might be re-inspected. Recall the claim

about construction fr om which I began: We are, in this respect, similar to

God. Similarity is not, of course, the same as identity. How, then, does the

constructive or generati ve activity of the infinite intellect differ from our own ?

What light do Maimon's interpretations

ofMoreh Nebuchim

I, 68 throw on this

question?

Permit me to state my speculations in a very summa ry fashion:

(i) Maimon ofte n contrasts finite with infinite intellect by saying that the first

is symbolic or discursive, while the second is intuitive (see, e.g., II, ~oo). In

keeping with what we have learned up to now about his theory of mathematics,

we can take these cont rasts to mean tha t a finite intellect is dianoetic; an infinite,

noetic. That is, intuitive in this context has to do with the immediat e grasp or

appreh ension of some whole in its complete integrity; its objects are

omnimodo

determinatum. Symbolic knowl edge, as we saw, conveys formulaically he struc-

ture of a whole (an infinite series, fo r example) but must always be expo und ed

stepwise, indeed by a discursive

progressus in infinitum.

Does intuitive have any other connotations when applied to infinite intel-

lect? Does the latter require, for example, any analogue to the matter of our

sensible intuition? In other terms, is anyt hing given to it? Maimon addresses

this question in a formid ably obscure passage of his

Versuch fibe r die Transcen-

dentalphilosophie

(II, 250- 5 l) and concludes that what is given to the infinite

understanding and (thereby) intuited by it is either an

objectum reale

which is

not thought by it or a simple idea of the relation of a co nc ep t. ., to some-

thing external to it. It cannot be said with any confidence that Maimon's

solutio n is clear. On the secon d optio n, the given as a relatio n, is the re lation in

some sense given to the infinite understanding by itself?. Is theoretical intel-

so T h e q u e s t i o n w h e t h e r C a r t e s i a n a l g e b r a i c /m e c h a n i c a l lo c u s - c o n s t ru c t i o n s e x h a u s t t h e

i n f in i t y o f p o i n t s o n a n e v e r y w h e r e - d e n s e g e o m e t r i c a l c o n t i n u u m i s p o s e d b y L e i b n i z a n d t a k e n

u p , f o r e x a m p l e , b y V i c o 's m a t h e m a t i c a l m e n t o r P a o l o M a t t i a D o r ia . S e e m y e s sa y V i c o , D o r i a e

l a g e o m e t r i a s i n t e t i c a , BoUtttino de centro di studi Vichiani lo ( 1 9 8 o ) : l o - 3 5 .

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5 1 0 J O U R N A L O F T H E H I S T O R Y O F P H I L O S O P H Y 3 0 : 4 O C T O B E R 1 9 9 2

l e c t i o n a n a f f a i r o f p o s i t i n g i t s o w n a b s t r a c t , i m p e l l i n g , o r s h o c k i n g o c c a s i o n s

(AnstOsse)? (F ich te , in sp i red by M aim on , wi ll exp lo r e th i s poss ib i l i ty .)

( ii ) Still a n o t h e r d i s t in c t i o n i s m o b i li z e d i n o r d e r t o l o c a t e t h e d i f f e r e n c e

b e t w e e n f i n i t e a n d i n f i n i t e i n t e l l e c t .

W e

c a n a c h i e v e o n l y

Vors t eUungen ,

r e p r e -

s e n t a t i o n s , n o t

D a r s t e l l u n g e n ,

p re s e n t a t i o n s o r t o t a l e x h i b i t i o n s , w h i l e i n t h e

c a se o f t h e d i v i n e m i n d t h e r e p r e s e n t a t i o n s . . , a r e a t t h e s a m e ti m e p re s e n t a -

t io n s, t h a t i s, b y t h e v e r y f a c t t h a t t h e y a r e r e p r e s e n t a t i o n s t h e y

[ s c . t h e

p r e s e n -

t a t i o n s ] m a i n t a i n o b j e c t i v e r e a l i t y o u t s i d e t h e p o w e r o f i n f i n i t e r e p r e s e n t a -

t ion ( IV, 62 ) . s,

V o r s t e l l u n g

is M a i m o n ' s t e r m f o r t h e n e c e s s a r i l y pa r t ia l a n d

i n c o m p l e t a b l e s y n t h e s is o u r f a c u l t y o f c o g n i t i o n c a n e f f e c t ( I I , 3 4 9) ; a c o m -

p l e t e sy n t h e s i s w o u l d b e t h e p e r f e c t c o n s c i o u s n e s s n o t o n l y o f al l t h e i n f i n i t e

p a r t s o f a n o b j e c t , b u t a l s o o f t h e i r r e l a t i o n s t o t h e s y n t h e s i s , t h a t is , t o t h e

s u b je c t o f j u d g m e n t i ts e lf .s , ( T h e i n t e g r a t i o n o f d i f f e r e n t i a l s w o u l d s e r v e

M a i m o n a s a g o o d e x a m p l e h e r e . ) T h u s , d i v i n e a c ts o f t h i n k i n g a r e p r o d u c t i v e

e x h i b i t i o n s o f r e a l o b j e c t s i n t h e i r i n d i v i d u a l t o ta l i t y a n d m u t u a l i n t e g ra t i o n .

(Le i b n iz ' s i n f i n i t e a n a l y s i s i s, t h e r e f o re , f o r M a i m o n , t h e in i t i a t i o n o f a c o m -

p l e t e s y n t h e s i s . ) I n d e e d , t h e s e a r e a c t s , a s w e s h a l l s e e , o f d i v i n e s e l f -

exh ib i t ion .

( ii i) O n e f i n a l t r a i t s e p a ra t e s f i n i t e f ro m i n f i n i t e i n t e l le c t . O u r f a c u l t y o f

c o g n i t io n a n d o u r ( t r a n s c e n d e n t a l ) i m a g i n a t i o n w o r k i n a c c o r d a n c e w i t h ru l e s

regelmi iss ig) ,

b u t

n o t

i n v i rt u e o f a n u n d e r s t a n d i n g o f t h o se r u l e s

r e g e l v e r s ~ n d i g )

( cf . I I , 2 o) . C o n s e q u e n d y , t h e r u l e s f ir s t a c q u i r e t h e i r r e a l i ty t h r o u g h t h e

p r e s e n c e

[Gegenwar t ]

o f t h e o b j e c ts p r o d u c e d , i. e. , c o n s t r u c t e d i n a c c o r d a n c e

w i t h t h e m . B e fo r e t h e o b j e c t s e x i s t i n t h e m i n d [G e m/ /t he ] w e c o u l d n o t k n o w

u n d e r w h a t r u l e s t h e y m u s t b e s u b s u m e d a f t e r t h e y h a v e b e e n o r i g in a t e d . S o,

f o r i n s ta n c e , t h e u n d e r s t a n d i n g p r e s c r i b e s a r u le to th e p r o d u c t i v e i m a g i n a -

t io n , n a m e l y , t o p r o d u c e a s p a c e e n c l o s e d b y t h r e e l i ne s . T h e i m a g i n a t i o n o b e y s

a n d c o n s t r u c t s t h e t r i l a t e r a l f i g u r e ; b u t , l o o k , a t t h e s a m e t i m e t h r e e a n g l e s

o b t r u d e t h e m s e lv e s , w h ic h th e u n d e r s t a n d i n g h a d n o t a s k e d f o r . . . . T h e u n -

s , On DarsteUung as d 6 v o i l e m e n t o r e x h i b i t i o n t o t a l e s e e S . Z a c , L ' I d 6 e d e c h o s e e n s o l

d a n s l a p h i l o s o p h i e d e S o l o m o n M a i m o n , Archives de Philosophic 51

1988):

~ 2 9 . I t i s o n e o f t h e

l a c u n a e in J a n B r a n s e n ' s t e x t , The Ant inoray of ThougM, t h a t n o h e e d i s p a i d t o M a i m o n ' s s y s t em -

a t i c d i s t i n c t i o n b e t w e e n VorsteUunga n d DarsteUung.

s , I n a v e r y d i f f i c u l t n o t e i n h i s Versuch iiber die Transcendentalphilosophie ( l I , 3 4 9 ) M a i m o n

w r i t e s : S o is t a u c h d a s v o l i s t , 'i n d i g e B e w u s s t s e i n a l l e r T h e i l e d e r S y n t h e s i s u n d f o l g l ic h a u c h d e r

S y n t h es i s s e ib s t, k e i n e V o r s t e l l u n g s o n d e r n , e i n e D a r s t e l l u n g d e s ( V e r s t a n d e s ) D i n g s s e lb s t . T h e

d i f f e r e n c e b e t w e e n

Vorstel lung

a n d

DarsteUung

t h u s s e e m s t o b e a n a l o g o u s t o t h e K a n t i a n d i s ti n c -

t i o n b e tw e e n t h e a n a l y ti c a l a n d t h e s y n t h e t i c u n i ty o f a p p e r c e p t i o n ; m o r e o v e r , t h e c u r i o u s f i n a l

p h r a s e e i n e D a r s t e t l u n g d e s ( V e r s t a n d e s ) D i n g s s e l b st m i g h t h i n t a t M a i m o n ' s r e s u s c it a t io n o f

t h e K a n t i a n D i n g a n s i c h. T h i s m a y c a s t d o u b t o n S . H . B e r g m a n ' s a s s e r t io n . . . f o r t h e

r e m o v a l o f t h e t h i n g - i n - i t s e lf s i g ni f ie s n o t h i n g m o r e t h a n h i s [ M a i m o n ' s ] r e t u r n t o t h e s t a n d p o i n t

o f A r is t o d e a n d M a i m o n i d e s

Phi losophy of Solomon M aimon,

35) .

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R E F L E C T I O N S O N M A I M O N A N D M A I M O N I D E S 5 1 1

d e r s t a n d i n g p u t s o n a n i m p e r i o u s f a c e a n d s a ys : ' A t r il a te r a l f i g u r e m u s t h a v e

t h r e e a n g l e s ' , a s i f i t w e r e i t s e l f t h e l e g i s l a t o r i n t h i s a f f a i r , a l t h o u g h i n f a c t it

m u s t o b e y a l e g i s la t o r c o m p l e t e l y u n k n o w n t o i t ( I I I , 1 7 4 - 7 5 ) -

I ci te t h is p a s s a g e a t s o m e l e n g t h s i nc e it b r i n g s o u t m o r e c l e a r ly t h a n a n y

o t h e r i n M a i m o n ' s w o r k t h e l o g ic a l, i f n o t a l so te m p o r a l , g a p b e t w e e n t h e r u l e -

g o v e r n e d p r o d u c t i o n o f a n o b j e c t o r s c h e m a ( a n

Ents tehungsregel ;

I I , 3 3 ) a n d

w h a t f lo w s o f a p p a r e n t n e c e s s i ty f r o m t h e e n a c t m e n t o f t h a t r u le . T h e l eg i sl a -

t iv e , o r c o n s t r u c t i o n a l , a c t i v it y o f t h e f i n it e m i n d i s s e l f - le g i s l a ti o n o n l y i n a n

a t t e n u a t e d s e n s e , b e c a u s e t h e f u ll m e a n i n g o f t h e r u l e i s a l w a y s o p a q u e t o it .

C o n v e r s e l y , f o r a n i n f i n i t e i n t e l l e c t , s i m i l a r l y f a c i l e i n t h e p r o d u c t i o n o f

o b j e c ts b y v i r tu e o f a r u l e o f o r ig i n a t i o n , t h e r e c a n b e n o s u c h g a p . J u s t a s in

t h e c a s e o f

D a r s t e l l u n g

vis-a-vis

Vors teUung,

h e r e , t o o , c o g n i t i o n o f t h e r u l e

e n t ai ls s i m u l t a n e o u s r e c o g n i t i o n o f a ll t h e i n fi n it e c o n s e q u e n c e s o f t h e r u l e .

G o d m a y h a v e n o i m a g i n a t i o n , f o r M a i m o n , b u t h e d o e s k n o w a l l t h e r u le s .s s

3 M A I M O N A N D M A I M O N I D E S

I t is n o w t im e t o c o n s i d e r , h o w e v e r b r i e fl y , t h e s c a le o f M a i m o n ' s i n d e b t e d n e s s

t o h is n a m e s a k e M a i m o n i d e s f o r t h e d o c t r i n e o f t h e i n fi n it e i n te l le c t . A ll o f t h e

m a j o r i n t e rp r e t e r s h a v e b e e n q u i c k t o d r a w a t te n t io n t o M a i m o n ' s c o m m e n -

t a r y o n B o o k O n e o f

The G ui de o f t he P erp l exed ,

e n t i t l e d

G i bea t h H am ore ,

a n d

e s p e c ia l ly to it s e x e g e s i s o f I , c h . 6 8 , w i t h it s o p e n i n g r e p e t i t i o n o f t h e d i c t u m

o f t h e p h i l o s o p h e r s : . . . t h a t H e i s t h e i n t e l l e c t [ a/ -'a q/ ] a s w e l l a s t h e i n t e ll e c -

t u a l l y c o g n i z i n g s u b j e c t

[al- aqfl]

a n d t h e i n t e l l e c t u a l l y c o g n i z e d o b j e c t

[al-

m a 'q fi/]; a n d t h a t t h e s e t h r e e n o t i o n s f o r m i n H i m . . . o n e s i n g l e n o t i o n i n

w h i c h t h e r e i s n o m u l t i p li c i t y ( P i n e s tr a n s . , p . 1 6 3 ) .

T w o q u e s t io n s m u s t b e k e p t a p a r t fo r p r e s e n t p u r p o s e s : H o w d o e s

M a i m o n i d e s u n d e r s t a n d t h is d i c t u m ? a n d H o w d o e s M a i m o n u n d e r s t a n d

a n d a d a p t

t h i s d i c t u m ?

A s f o r th e f i r s t q u e s t i o n , n o a n s w e r , h o w e v e r t e n t a t iv e , c a n b e o f f e r e d

w i t h o u t d e t a i l e d e x p l o r a t i o n o f t h e s o u r c e s o f t h e d i c t u m M a i m o n i d e s c i te s .

P i n e s 's th e s i s , t h a t i t r e p r e s e n t s a s t o c k f o r m u l a u s e d b y p r a c t i c a ll y a ll A r i s t o t e -

l ia n p h i l o s o p h e r s ( p . x c vi ii ) s e e m s t o m e t o n e e d a m e n d m e n t , s i nc e t h e f o r -

m u l a , i n e x a c t l y t h i s w o r d i n g , d o e s

no t

o c c u r e i t h e r i n A r is t o t le o r i n t h e G r e e k

c o m m e n t a t o r s A l e x a n d e r o f A p h r o d i s i a s o r T h e m i s t iu s . H o w e v e r , it d o e s

a p p e a r , w i t h a s l ig h t d i f f e r e n c e i n t h e s e q u e n c e o f t e rm s , i n a l -F a r a b i, a n d

ss M a i m o n ' s a c c e n t u a t i o n o f t h e r o l e o f r u /e s in m a t h e m a t i c s i s n o t e w o r t h y . T h i s a c c e n t r e a p -

p e a r s i n F i c h t e ( s e e n o t e t 3 , supra a n d g a i n s g r e a t e s t v o l u m e i n W i t t g e n s t e i n ' s m e d i t a t i o n s o n

f o l | o w i n g a r o l e i n h i s Remarks on tM Foundations of Mathematics . C f . C r i s p i n W r i g h t , Witzgenstein

on the Foundations of Mathematics ( C a m b r i d g e , M a s s . : H a r v a r d U . P . , 1 9 8 o ), p a s s/ m a n d S t u a r t G .

S h a n k e r , W ittgemw in an d the Turn ing-Po int in the Philosophy of Mathematics ( A l b a n y : S . U . N . Y . P r e s s ,

a 9 8 7 ) , 2 3 , 5 - 5 6 .

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5 1 2 J O U R N A L O F T H E H I S T O R Y O F P H I L O S O P H Y 3 0 : 4 O C T O B E R 1 9 9 2

m o r e t h a n o n c e ; f o r i n s t a nc e , i n a l - M a d i n a a l -f a d il a ( W a l z e r e d i t i o n , p . 7 o ) a n d

i n a l - S iy a s a a l - m a d a n i y y a ( N a j j a r e d i t i o n , p . 35 ).s 4 T h e s a l i e n t l e x i c a l p h e n o m e -

n o n , i n b o t h a l - F a r a b i a n d M a i m o n i d e s , is t h a t al-'6.cfil ( m o s t l i ke l y ) r e n d e r s t h e

G r e e k 6 v o t b v o r ~ 6 v o os 1 62 h e o n e t h i n k i n g o r i n t e l l e c t i n g , n o t v6 T Io tg , w h i c h

w o u l d b e t a'a q q~ /. s5 S o , t h e q u e s t i o n o f s o u r c e s i s a ll t h e m o r e u n a v o i d a b l e .

T h e l i k e l i h o o d t h a t n e o - P l a t o n i c , n o t A r i s t o t e l i a n , t e x t s s t a n d i n t h e b a c k -

g r o u n d o f t h e d i c t u m is v e r y m u c h w o r t h e x p l o r i n g , s i n c e it i s i n , e . g ., P r o c l u s '

E l e m e n t s o f T h e o l o g y ( P a r a . t 6 8 [ D o d d s ] ) t h a t w e f in d al l t h e t e r m s l a t e r r e p r o -

d u ce d in M ai m o n id es ' c i t a t i o n : ~ ttx~ x ~ tQ a ~ t~ tcp0~ o ~ S e [ sc . 6 v o ~ g ] x a [ ~6 v o ~ 6 v

x a ~ 6 ~L v o ~ ~ x ~ v o xc t~ v o ~ a t ~ q~ ' ~ c nr r v o o f n r t o g . I n t e l l e c t , t h e n , k n o w s

b o t h w h a t i s i n t e l l e c t e d a n d t h e f a c t t h a t i t i n t e ll e c t s it a n d t h a t i t [ s c . t h e

i n t e l l e c t ] i s i n t e l l e c t e d b y i t s e l f in t h e a c t o f i n t e l le c t i n g . 3 6

T h e P r o c l e a n e v i d e n c e , t o g e t h e r w i t h i ts a n t e c e d e n t c o u n t e r p a r t s i n

P l o t i n u s ( e . g ., E n n . V . 3 .1 , 2 4 - 2 7 ; V . 3 .5 , 2 8 - 4 8 ) , s u g g e s t s t h a t t he p a r a m o u n t

i s s u e i n t h e s e n e o - P l a t o n i c s o u r c e s i s s e l f - c o n s c i o u s n e s s o r s e l f - r e f l e c t io n ,

i .e ., h o w v o ~ g c a n k n o w t h a t i t is t h e a g e n t o f t h i n k i n g ( 6 v o ~ v / , 6 v o o f rv ) w h e n

i t i s a t t h e s a m e t i m e i d e n t i c a l w i t h t h e o b j e c t o f t h o u g h t , t h e v o q x 6 v . C o n s e -

q u e n t l y , t h e i s s u e i s n o t t h e i d e n t i t y o f t h e a c t u a l i t y o f t h i n k i n g , v 6 t l ot g , w i t h

t h e a c t u a l i n t e l l i g i b le , a s i t i s i n A r i s t o t l e , M e t a p h . 1 2 . 9 .

I s t h e i n te n t o f M a i m o n i d e s ' r e p e t i t io n o f th is d i c t u m o f t h e p h i l o s o p h e r s

t h e s a m e a s, o r c o g n a t e w i t h , t h e i n t e n t o f t h e n e o - P l a t o n i s t s ?

H o w e v e r t h is la s t q u e s t i o n m i g h t e v e n t u a l l y b e a n s w e r e d , M a i m o n r e a d s

t h e o p e n i n g o f M o r e h I , 6 8 i n a q u i t e p a r t i c u l a r w a y , a w a y d e e p l y a n c h o r e d t o

h is th e s e s c o n c e r n i n g m a t h e m a t i c a l c o n s t r u c t i o n a s th is h a s b e e n u n d e r s t o o d ,

a n d e x e m p l i f i e d , b y

m o d e r n

p h i l o s o p h e r s f r o m D e s c a r t e s t o K a n t .

R e c al l o n c e m o r e t h e p a s s a g e f r o m O b e r d i e P r o g r e s s e n d e r P h i l o s o p h i e w i t h

s4 Alexander Airma n, in his invaluable article Maim onides on the Intellect and the Scope of

Metaphysics, Von der m ittelalterlichen zur raodernen Aufldiirung (T~ibingen: J. C . B. Moh r, 1987),

6o-1 29, ca lls attention to these and o ther Farabian passages (74, n . t lo ), while noting that

neither Aristotle no r Alexa nder had sp oken of the triunity of intellectus, intelligensand intellectum

(ibid.). Cf. Edw in B ooth, Aristotelian Aporetic Ontology in Islam ic and Christian Thin kers (Cambridge:

Cam bridge U.P., 1983), 97- 98 and esp. nn. a 8-19, for furthe r references. M ost imp ortant to the

claims I am making here is the inform ation that in the so-called Long Recension of the ps.-

Plotinian Theology of Aris totle the following passage occurs: It is the knowledge and the knower

and the known, as we have said above. (See Paul B. Fenton, Th e Arabic and Hebrew Versions of

the Theology of A ristotle, ~5 ~ and n. 6~ ). This passage seems strongly to suggest an originally

Neoplatonic inspiration for Maim onides' dictum.

ssCf. Philipp W. Rosemann, N6~lotg vofloe0~g und T A'A QQ UL A T-T A'A QQ UL . Das

Aristotelische Problem de r Selbstbeziigiichkeit des Unbewegten Bewegers in der Ko mm entieru ng

Ibn R usds, geitschriftfftr philosophische Forsch ung 4~ (1986): 543-6 1, a nd th e translator 's notes on

Arabic terminology in

Averro~s. G ran d Com mentaire de la M~taphysique d'Aristote.. . Livre Lam-

Lambda, trans. Aub ert Martin (Paris: Les Belles Lettres, 1984), 257-7 0.

~Cf . the rema rks by E . R . Dodds, Proclus: The Elements of Theology (Oxford, 1933), ad loc.

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R E F L E C T I O N S O N M A I M O N A N D M A I M O N I D E S 5 1 3

w h i c h I b e g a n . W e d o h a v e a c o n c e p t o f G o d a s a n i nf i n it e f a c u lt y o f r e p r e s e n -

t a ti o n t h a n k s to o u r e x p e r i e n c e o f c o n s tr u c t i o n , i n w h i c h a ll t h e c o n c e p t s o f

m a t h e m a t i c s a r e t h o u g h t b y u s a n d a t th e s a m e t i m e p r e s e n t e d o r e x h i b i t e d a s

r e a l o b j e c ts ( I V , 20 ) . M a i m o n g o e s o n , a t t h e c o n c lu s i o n o f th e p a s s a g e : G o d ,

t h e r e f o r e , t h i n k s a l l r e a l o b je c t s , n o t m e r e l y in a c c o r d w i t h . . , t h e p r i n c i p le o f

c o n t r a d i c ti o n , b u t i n t h e w a y w e th i n k t h e o b j e c ts o f m a t h e m a t i c s ( a l t h o u g h , t o

b e s u re , i n a m o r e c o m p l e t e f a s h i o n ) , t h a t is , H e p r o d u c e s t h e m a t t h e s a m e

t i m e a s H e t h i n k s t h e m ( i bi d .) .

N o w c o m p a r e p a r t o f M a i m o n ' s H e b r e w c o m m e n t a r y o n M o r e h I , 68:

T h e s e f o r m s o f s i m i la r it y [ a n d d i f f e r e n c e ] d e f i n e t h e u n d e r s t a n d i n g a n d

d i s t i n g u i s h i t f r o m e v e r y t h i n g e l s e . I t f o ll o w s f r o m t h i s t h a t t h e i n t eUig ib i l ia

t h a t is , t h e f o r m s a f o r e m e n t i o n e d , a r e t h e u n d e r s t a n d i n g i ts e lf ; s i m il a rl y , th e

i n t e l l i g e n s

t h a t is , t h e c a u s e w h i c h p r o d u c e s t h e s a id f o r m s , i s t h e u n d e r s t a n d -

i ng , f o r th e