kant on the method of mathematics

Kant on the Method of Mathematics

Carson, Emily.

Journal of the History of Philosophy, Volume 37, Number 4, October1999, pp. 629-652 (Article)

Published by The Johns Hopkins University PressDOI: 10.1353/hph.2008.0905

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Kant on the M e t h o d

o f M a t h e m a t i c s

E M I L Y C A R S O N

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

THIS PAPER WILL TOUCH on three very genera l bu t closely re la ted quest ions abou t Kant ' s phi losophy. First, on the role of ma themat i c s as a p a r a d i g m of knowledge in the d e v e l o p m e n t of Kant ' s Critical ph i losophy; second, on the na ture o f Kant ' s oppos i t ion to his Le ibn izean predecessors and its role in the d e v e l o p m e n t of the Critical phi losophy; and finally, on the specific role o f intui t ion in Kant ' s ph i losophy o f ma themat i c s . O n e o f the points tha t I wan t to make is tha t recogniz ing the i m p o r t a n c e of these first two issues is essential to giving an adequa te accoun t of the third. T h a t is, only by apprec ia t ing how Kant uses the exam pl e of ma thema t i ca l knowledge, and in par t icu lar the objections he had to prev ious accounts o f such knowledge, can we u n d e r - s tand the phi losophica l role that he ascribes to in tui t ion in his own accoun t of mathemat ics .

I d o n ' t p r o p o s e to expla in the details o f Kant ' s ph i losophy of m a t h e m a t - ics, bu t r a the r to c o m p a r e his views on m a t h e m a t i c s in the pre-Cri t ical Prize Essay of 1764 to those of the Critique of Pure Reason, with the a im of showing how Kant ' s doct r ine of const ruct ion in p u r e in tui t ion arises out o f a combina - tion of two factors: first, his a t t e m p t to g r o u n d the dist inction be tween the respect ive m e t h o d s o f ma thema t i c s and phi losophy, and second, his conce rn to de f end the reality of ma thema t i ca l knowledge against the views of those he called ' the metaphysic ians . ' T h e first f ac to r d e m a n d s an account of the certainty of mathemat ics , the second requi res an accoun t o f its content. I shall a rgue tha t there is a possible conflict be tween Kant ' s accounts of these features in the Prize Essay, a possibility which is finally ru led ou t only by m e a n s of the Critical doctr ine o f p u r e intuit ion. So I h o p e to show that the re is a significant phi losophical role for in tui t ion in Kant ' s accoun t of ma thema t i ca l knowledge (over and above the m o r e logical role a t t r ibuted to it on read ings

[6~91

630 JOURNAL OF THE HISTORY OF PHILOSOPHY 3 7 : 4 OCTOBER 1 9 9 9

such as t h o s e o f F r i e d m a n , Be th , a n d H i n t i k k a l) in r e c o n c i l i n g the c o n t e n t o f m a t h e m a t i c s a n d its c e r t a i n ty .

2. THE MATHEMATICIANS VS. THE METAPHYSICIANS

T h r o u g h o u t his e a r l y w r i t i n g s , K a n t a p p e a l e d to t h e e v i d e n c e a n d c e r t a i n t y o f m a t h e m a t i c s ; h is a t t i t u d e t o w a r d s m e t a p h y s i c s , h o w e v e r , u n d e r w e n t a d r a - m a t i c shif t . I n t he Physical Monadology o f 1756, K a n t a s k e d h o w m e t a p h y s i c s a n d g e o m e t r y can b e u n i t e d ,

If] or the former pe remptor i ly denies that space is infinitely divisible, while the latter, with its usual certainty, asserts that it is infinitely divisible. Geomet ry contends that empty space is necessary for free motion, while metaphysics hisses the idea off the stage. Geomet ry holds universal at traction or gravitation to be hardly explicable by mechanical causes but shows that it derives from the forces which are inheren t in bodies at rest and which act at a distance, whereas metaphysics dismisses the notion as an empty delusion of the imaginat ion [1:475-6]. 2

H e a t t e m p t e d in th is w o r k to r e c o n c i l e t h e r e s p e c t i v e p o s i t i o n s o f m e t a p h y s i c s a n d g e o m e t r y r e g a r d i n g in f in i t e d iv is ib i l i ty , t h e r e b y d e m o n s t r a t i n g t h a t "it is n e i t h e r t he case t ha t t he g e o m e t e r is m i s t a k e n n o r t h a t t h e o p i n i o n to be f o u n d a m o n g m e t a p h y s i c i a n s d e v i a t e s f r o m t h e t r u t h " [1 :48o] . B u t in a se r ies o f w o r k s f r o m 1 7 6 2 - 3 , K a n t r evea l s a m a r k e d l y d i f f e r e n t a t t i t u d e t o w a r d s m e t a - phys ics . F o r e x a m p l e , in " T h e o n l y p o s s i b l e a r g u m e n t in s u p p o r t o f a d e m o n - s t r a t i o n o f t he e x i s t e n c e o f G o d , " K a n t says:

the mania for me thod and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the sl ippery g round of metaphysics. These mishaps are constantly before one 's eyes, but there is little hope that people will be warned by them, or that they will learn to be more circumspect as a result [2:7a].

I n c o n t r a s t , K a n t d e s c r i b e s t he g e o m e t e r as u n c o v e r i n g "with t h e g r e a t e s t c e r t a i n t y " t h e m o s t s e c r e t p r o p e r t i e s o f t h a t w h i c h is e x t e n d e d [2:7o]. B u t t he less c o n c i l i a t o r y a t t i t u d e to m e t a p h y s i c s c o m e s o u t m o s t c l ea r ly in K a n t ' s ac tua l p r o c e d u r e in " T h e o n l y p o s s i b l e a r g u m e n t . " I n t h e S e v e n t h Re f l ec t ion , he p r e s e n t s a n a t t e m p t to e x p l a i n the o r i g i n o f t he sys tem o f t he u n i v e r s e in t e r m s o f t h e g e n e r a l laws o f m e c h a n i c s . I n d o i n g so, h e p r e s u p p o s e s t he u n i v e r s a l

See for example, Michael Friedman, Kant and theExact Sciences (Cambridge: Harvard Univer- sity Press, 1992);Jaakko Hintikka, "Kant's 'New Method of Thought' and His Theory of Mathe- matics," in Knowledge and the Known (Boston: D. Reidel Publishing Company, 1974), 126-134; Evert W. Beth, "0ber Lockes 'allgememes Dreieck,' " in Kant-Studien 48 0956-7): 361-38o.

~References to the pre-Critical writings are to the translations in the Cambridge Edition of TheoretzcalPhdosophy z755-177 o, translated and edited by David Walford, (Cambridge: Cambridge University Press, 1992); page references are to the Akademie edition. References to the Cntzque of Pure Reason are to Kemp Smith's translation, except where indicated.

K A N T ON T H E M E T H O D O F M A T H E M A T I C S 6 3 1

g r a v i t a t i o n o f m a t t e r as f o r m u l a t e d by N e w t o n a n d his fo l lowers . T o fo r e s t a l l s p u r i o u s o b j e c t i o n s , h e w a r n s :

I f there are any who think that, by employing a definit ion drawn from metaphysics and formula ted according to their own taste, they can demolish the conclusions established by men of perspicacity on the basis of empir ical observation and by means of mathematical i n f e r e n c e - - i f there are such persons, they may ignore what follows as some- thing which has only a remote bear ing on the main purpose of the book [~: 139].

1n the case o f t he conf l i c t ove r t h e h y p o t h e s i s o f a c t i o n a t a d i s t ance , K a n t n o l o n g e r s e e m s p r e p a r e d , as he was in t he Physical Monadology, to t ake s e r i o u s l y t h e o b j e c t i o n s o f t he m e t a p h y s i c i a n s . H e d e s c r i b e s t h e N e w t o n i a n s as p r o c e e d - i n g o n t h e basis o f e m p i r i c a l o b s e r v a t i o n a n d m a t h e m a t i c a l i n f e r e n c e to " i n d u - b i t a b l y c o r r e c t " a n d " se l f - ev iden t " conc lus ions ,3 w h i l e t he m e t a p h y s i c i a n s , h e says, b e g i n wi th s u r r e p t i t i o u s d e f i n i t i o n s , a n d t h e r e f o r e "have n o s o u n d r e a - son to o b j e c t to t he i d e a o f i m m e d i a t e a t t r a c t i o n at a d i s t a n c e " [5: 288] .

K a n t ' s o b j e c t i o n s to the m e t a p h y s i c i a n s a r e n o t c o n f i n e d to p a r t i c u l a r a r g u - m e n t s l ike t he o n e j u s t e x a m i n e d , h o w e v e r . H e e x p l a i n s in t h e e s say o n n e g a t i v e m a g n i t u d e s t h a t h e t h i n k s t h a t th is is j u s t o n e i n s t a n c e o f a g e n e r a l p r o b l e m in m e t a p h y s i c s : n o t h i n g , K a n t says, ha s b e e n m o r e d a m a g i n g to p h i l o s o p h y t h a n its a t t e m p t to i m i t a t e t h e m e t h o d o f m a t h e m a t i c s . T h e application o f t he m a t h e - m a t i c a l m e t h o d in p h i l o s o p h y , h o w e v e r , ha s a l l o w e d the l a t t e r to a t t a i n to h e i g h t s to w h i c h i t o t h e r w i s e c o u l d n o t h a v e a s p i r e d [2:167] . B u t i n s t e a d o f t h u s t u r n i n g the i n s igh t s o f m a t h e m a t i c s to its own a d v a n t a g e , m e t a p h y s i c s has " a r m e d i t s e l f a g a i n s t them'J :

where it might, perhaps , have been able to gain secure foundat ions on which to base its reflections, it is to be seen trying to turn mathematical concepts into subtle fictions, which have little t ruth to them outside the field of mathematics.

T o i l l u s t r a t e this , K a n t r e t u r n s to t h e d i s p u t e o v e r t h e m o n a d s . I n its i n q u i r y in to t h e n a t u r e o f space , n o t h i n g w o u l d be o f m o r e u se to m e t a p h y s i c s t h a n " the r e l i a b l y e s t a b l i s h e d d a t a " o f g e o m e t r y : fo r e x a m p l e , t ha t s p a c e d o e s n o t cons i s t o f s i m p l e pa r t s . B u t

these data are ignored and one relies s imply on one 's ambiguous consciousness of the concept, which is thought in entirely abstract fashion. I f it should then happen that speculation, conducted in accordance with this p rocedure , should fail to agree with the proposi t ions of mathematics, then art a t t empt is made to save the artificially contr ived concept by raising a specious objection against this science, and claiming that its funda- mental concepts have not been der ived f rom the true nature of space at all, bu t arbitrarily invented.

3Kant actually reserves judgement with regard to the hypothesis of action at a distance, though he does say that it "has much to be said for it" [2:288].

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In the face of the conflict between the doctr ine of simple substances and the infinite divisibility o f geometr ical space, the metaphysicians re ta ined the fo rmer by treating the objects of geomet ry as fictions, creatures of the imagina- tion by means o f which we a t tempt to order our confused sensible representa- tions of the monad ic realm, a realm which is known clearly only by the intel- lect.4 Whereas in the Physical Monadology, Kant sought to show that both the opinions o f the metaphysicians and those of the mathematic ians were true, he now dismisses the fo rmer as "obscure abstractions which are difficult to test," while elevating the latter to "a science which admits only intelligible and obvious insights" [2: 168]. Indeed , far f rom turn ing mathematics into a body of 'subtle fictions' when it conflicts with the proposi t ions of metaphysics, metaphysicians should look to mathematics as a source o f basic data. Mathematics is thus the pa rad igm of certain cognition.

3. THE METHOD OF MATHEMATICS IN THE PRIZE ESSAY

3. Z. In the Prize Essay of 1764, Kant under takes to answer the Prize Ques- tion set by the Berlin Royal Academy concern ing the certainty o f metaphysical cognit ion by compar ing it with the certainty of mathematical cognition, in order to de te rmine whe ther metaphysics is capable of the same degree of certainty as mathematics , and the path by which that certainty may be attained. Once again, mathemat ics serves as a model of certain cognition. Kant claims that his treatise "contains no th ing but empirical proposit ions which are certain" [2:275], so it seems that he takes himself to be offering a philosophically neutral description of mathematical and metaphysical methods respectively.

The compar ison Kant presents is twofold. In the First Reflection, he compares the "manner in which certainty is at tained" in mathemat ical and philosophical cognition. In the Th i rd Reflection, he compares the nature of philosophical and mathematical certainty. I will consider these in turn.

3" 2. Comparison of the methods of mathematics and metaphysics (i) Definitions The first difference in m e t h o d which Kant considers is the

role of definitions. This difference is summarized in the following passage:

In mathematics I begin with the definition of my object, for example, of a triangle or a circle, or whatever. In metaphysics I may never begin with a definition. Far from being

4Wolff's expositor, the Marquise de Ch~itelet, m her Instltutzons physiques of 1742, puts the metaphysicians' position particularly bluntly: "So the infinite dlvislbihty of extension is at the same time a geometrical truth and a physical error" (in Christian Wolff" Gesammelte Werke, Materialen, und Dokumente, Band 28 [Hildesheim: Georg Olms Verlag, 1988]): whence Kant's charge that they have turned mathematical concepts into fictions with little truth to them outside mathematics. For more on this dispute, see Friedman's Kant and the Exact Sciences, 4, and the references cited there.

K A N T O N T H E M E T H O D O F M A T H E M A T I C S 633

the first thing I know about the object, the definition is nearly always the last thing I come to know. In mathematics, namely, I have no concept of my object at all until it is furnished by the definition. In metaphysics I have a concept which is already given to me although it is a confused one. My task is to search for the distinct, complete and determinate concept [9: 283].

This difference in m e t h o d is made possible by the fact that "mathematics arrives at all its definitions synthetically, whereas ph i losophy arrives at its definitions analytically" (2:276). A synthetic definition, according to Kant, is arrived at by "the arbitrary combinat ion o f concepts." The concept thus de- fined is no t given pr ior to the definition, but ra ther "comes into existence" as a result of the definition. For example,

[w]hatever the concept of cone may ordinarily signify, in mathematics the concept is the product of the arbitrary representation of a right-angled triangle which is rotated on one of its sides [3: 376].

Similarly, the concept of a square is the arbitrary combinat ion o f the concepts four-sided, equilateral, and rectangle.5 This is not the result of an analysis of some concept given in ano ther w a y - - i t is not, for example, abstracted f rom our exper ience of squares in nature; the concept is, as Kant says, first given by the definit ion itself.

In philosophy, on the o ther hand, the concepts are always given in some way, but "confusedly or in an insufficiently de te rmina te fashion." The task of the ph i losopher is then to discover by means o f analysis the characteristic marks in the confused concept in o rder to arrive at a complete and determinate concept , that is, a definition. T h u s Kant says for example, "everyone has the concept of time." This idea that everyone has must be examined in all kinds o f relations, and once the characteristic marks have been made distinct, and then combined together, the result ing concept has again to be c o m p a r e d with the concep t of time which was originally given in o rder to de te rmine whe ther or not it is adequate , whe the r it has cap tu red the original idea. If, by contrast, we tried to arrive at a definit ion of time synthetically, by arbitrarily combin ing concepts, it would have been a "happy coincidence" if the result- ing concept had been exactly the same as the idea of time which is given to us

[~:277]. Kant attributes much mistaken phi losophy to the failure to recognize this

fundamen ta l methodologica l difference between ph i losophy and ma themat - ics. Indeed , it underl ies his diagnosis o f the main p rob lem of phi losophy: "noth ing has been more damaging to phi losophy," he says, than the imitation of the me thod of mathematics "in contexts where it cannot possibly be era-

5 Dohna-Wundlacken logic, 757, translated in the Cambridge Edition of Kant's Lectures on Logzc byJ. Michael Young, (Cambridge: Cambridge University Press, a999).

634 J O U R N A L OF T H E H I S T O R Y OF P H I L O S O P H Y 3 7 : 4 O C T O B E R 1 9 9 9

ployed" [2: 283]. For example , a ph i lo sophe r could offer a synthet ic definit ion by "arbitrari ly th ink ing of a subs tance endow ed with a facul ty o f reason and calling it a spirit ." However , this would no t be a phi losophical definit ion, bu t r a t he r a "grammat ica l" one, a m e r e linguistic st ipulation, and "no ph i losophy is n e e d e d to say wha t n a m e is to be a t tached to an arb i t ra ry concep t" [2: 277]. I ndeed , Kan t accuses Leibniz o f having m a d e this mistake in imagin ing "a s imple substance which had no t h i ng bu t obscure represen ta t ions" and calling it a ' s lumber ing m o n a d . ' H e did not the reby define the m o n a d , "he merely inven ted it, for the concep t o f a m o n a d was not given to h im bu t created by h im."

Similarly, a m a t h e m a t i c i a n could offer an analytic definit ion, and indeed K a n t der ides Wol f f for a t t e m p t i n g to subsume the geometr ica l concep t o f similarity u n d e r the genera l concep t o f similarity which, however , is "of no concern wha teve r to the g e o m e t e r . " For tuna te ly for ma themat ics , though ,

in the end, nothing is actually inferred from such definitions, or, at any rate, the immediate inferences which [the geometer] draws ultimately constitute the mathematical definition itself. Otherwise this science would be liable to exactly the same wretched discord as philosophy itself [2: 277].

Kan t e laborates on his theory of defini t ions in his lectures on logic. He a t t r ibutes to the t e rm 'definit ion' a very strict sense. He defines it as a distinct, comple t e and precise concep t o f a thing [24: 263]. A concep t is distinct"insofar as there is clarity o f marks" in it [24: x 2o], that is, insofar as one is conscious of the marks con ta ined in the concept . Bu t in addi t ion those marks mus t be "clear ground[s] o f cogni t ion of the th ing" [24: 264]: a tautology, for example , is not a definit ion, for the marks in the p u r p o r t e d defini t ion are not distinct f r o m those o f the def in i tum, and, as Kan t puts it, "what me re ly says the same thing names no g r o u n d " [24: 265]. A concep t is complete (or comple te ly distinct) when the marks are sufficient to cognise, first, the d i f fe rence of the definitum f rom all o the r things, and second, the ident i ty of it with o the r things. Finally, a definit ion is precise when n o n e of the marks in the defini t ion is a l ready con ta ined in an- o ther : for example , "a body is ex t ended divisible ma t t e r " is not precise because the m a r k of divisibility lies in the mark of mat te r , and thus is r e d u n d a n t .

Kan t claims that ma themat i ca l definitions, because they are definit ions of a rb i t ra ry concepts , are defini t ions in this sense. In fact, ma thema t i ca l concepts are the only ones that admi t o f definit ion. First o f all, Kan t says that all fabr ica ted concep ts are "p roduced s imul taneous ly with thei r distinctness": I am conscious of each of the marks inc luded in the concep t because I pu t them there in def ining the concept , and "one can mos t easily be conscious of that which one has onese l f inven ted" [24: 153]. Similarly, the defini t ion is complete because the ma thema t i c i an


thinks everything that suffices to distinguish the thing from all others, for [it] is not a thing outside him, which he has cognised in part according to certain determinations, but rather a thing in his pure reason, which he thinks of arbitrarily and in conformity with which he attaches certain determinations, whereby he intends that the thing should be capable of being differentiated from all other things [24:125].

In o the r words, if the th ing de f ined is first given by the defini t ion, then the defini t ion is o f course comple te . Any t h i ng that satisfies the def ini t ion o f a t r iangle is a triangle. Precision is the only respec t in which the m a t h e m a t i c i a n may go wrong , bu t according to Kant , "this is no t an e r r o r bu t mere ly a mis take" or an imper fec t ion : it concerns the "neatness" of the definit ion, r a the r than its essence [94: 969].

Th is is in sharp contras t to empir ica l concepts which, Kan t says, are capable only o f descript ion, no t o f definit ion. Since in that case the concep t is given, in o rde r to make it distinct I mus t " e n u m e r a t e all the marks tha t I th ink in connec t ion with the express ion of the definitum." But one can neve r know that the marks that one has e n u m e r a t e d at any po in t are "sufficient to distinguish the th ing f r o m all r e m a i n i n g things" [94:194]. T h e mos t we can h o p e for is compara t ive comple teness , "when the marks o f a th ing suffice to dist inguish it f r o m every th ing that we have cognised in exper ience until now." This com- parat ive comple teness is character is t ic o f m e r e descr ip t ion [94:968]. Since phi losophical concepts , like empir ica l ones, are also given, "the ph i l o sophe r canno t so easily be certain that he has t ouched on all the marks tha t be long to a thing, and that he has insight into these comple te ly perfect ly"; consequent ly , m a n y marks "may still be long to the th ing of which he knows no th ing" [94:153]. This suggests that phi losophica l concepts , like empir ica l ones, do not in the end admi t o f def ini t ion ei ther. At best, any p u r p o r t e d def ini t ion will be uncer ta in .

In addi t ion to the dist inction be tween analytic and synthetic definit ions, Kan t dist inguishes nom i na l f r o m real definitions. A def ini t ion is nominalwhen its marks are "adequa te to the whole concep t that we think with the express ion o f the definitum"; a real defini t ion is one "whose marks const i tu te the whole possible concep t o f the th ing" [24:139]. Alternat ively, nomina l defini t ions "contain every th ing that is equal to the whole concep t that we make for ourselves of the thing," whereas real defini t ions "contain every th ing tha t be longs to the thing in itself." This distinction becomes clearer w h e n we see how it is app l ied to d i f fe ren t kinds of concepts . In par t icular , Kan t says that all def ini t ions of a rb i t ra ry concepts that are made , as o p p o s e d to given, are real definit ions. Why?

Just because it lies solely with me to make up the concept and to establish it as it pleases me, and the whole concept thus has no other reality than merely what my fabrication wants; consequently I can always put all the parts that I name into a thing, and these


must then constitute the complete, possible concept of the thing, for the whole thing is actual only by means of my will [24: 268].

Empir ical concepts , on the o ther hand, are capable o f at bes t nomina l definition since "I do no t def ine the object but instead only the concep t that one thinks in the case of the th ing" [~4:271]. T h e d i f ference seems to be that in the case of a rb i t ra ry concepts , the marks of the "whole possible concep t o f the t h i n g ' j u s t are the marks o f the concep t that we think in the case of the thing: there is no th ing m o r e to the object itself than what we at t r ibute to it. T h e i m p o r t a n t po in t for now, in any case, is that because ma themat i ca l defini t ions are of a rb i t ra ry concepts , they are also, by Kant ' s lights, real definitions.

Oi) Signs T h e second p a r t o f the First Reflect ion compa re s the examina t ion of the universal u n d e r signs in concreto in ma themat ic s with the examina t ion of the universal by means o f signs in abstracto in phi losophy. Still p r o c e e d i n g solely on the basis o f "conclusions der ived immedia te ly f r o m o u r exper ience ," Kant singles ou t two features o f ma thema t i ca l p roofs and inferences [~: 278]. T h e first is the use of signs as i l lustrated by the case o f ar i thmetic . Ins tead of opera t - ing with "the things themselves ," one ope ra t e s with signs accord ing to "easy and cer tain rules, by means of substi tut ion, combina t ion , and m a n y kinds of trans- fo rma t ion . " T h e second fea tu re is that in g e o m e t r y universal conclusions are drawn f r o m par t icular examples . For example , to discover the p rope r t i e s of all circles, we draw one circle; instead of d rawing all possible lines which could intersect each o the r within tha t circle, we draw two lines. In this way, we con- s ider the universal rule gove rn ing the re la t ions ho ld ing be tween intersect ing lines in all circles in these two lines in concreto.

In ph i losophy , on the contrary , the only signs used are words. Words as signs, accord ing to Kant , lack two crucial fea tures which signs in ma themat i c s have: first, they do not "show in their compos i t ion the cons t i tuent concepts of which the whole idea indicated by the word consists"; and second, they are "not capable of indicat ing in their combina t ions the relat ions o f the phi losophical t hough t s to each o ther" [~: 279]. These fea tures are i l lustrated by a par t icularly salient example , the geomet r ica l demons t r a t i on that space is infinitely divisible. 6 T h e g e o m e t e r takes a s t ra ight line s tanding vertically be tween two paral lel lines, and f r o m a po in t on one of these draws lines to intersect the o the r two. "By means of this symbol," Kant claims, the g e o m e t e r "recognises with the grea tes t cer ta inty that the division can be carr ied on ad infinitum."

Since signs in ph i l o sophy do not have the fea tures of signs in mathemat ics , however , the ph i lo sophe r mus t deal directly with the "universal concepts of the things themselves" in abstracto. T o illustrate the ph i lo sophe r ' s p rocedure ,

61 think it is important to note that at this point Kant is contrasting the procedure of philosophy with that of geometry specifically, and not mathematics m general.


Kan t again takes an exam pl e f r o m the Physical Monadology, the demons t r a t i on that all bodies consist o f s imple substances. T h e p h i l o s o p h e r first "assures himself" that bodies in genera l are wholes c o m p o s e d o f substances, and that composi t ion is an accidental state of substances. H e then infers that substances could con t inue to exist if we imagine away all compos i t ion in a body. Since what r ema ins of a c o m p o u n d when all composi t ion has been cancel led is simple, he concludes that bodies mus t consist o f s imple substances.

T h e di f ference seems to be that the geomet r ica l cons t ruc t ion comple te ly represen t s the re levant geomet r i ca l facts, tha t is, it "shows in its compos i t ion the cons t i tuen t concepts" which we are in te res ted i n - - s t r a i g h t lines, for e x a m p l e - - a n d indicates in their combina t ions the relat ions o f the geomet r ica l thoughts to each o t h e r - - e . g . , that they intersect. T h e ph i l o sophe r has no such visual aids, bu t relies only on abs t rac t reflection. Because the symbolic na tu re of ma themat ic s is l inked to the na tu re o f ma thema t i ca l certainty, we'll r e t u rn to this topic below in cons ider ing the T h i r d Reflection.

(iii) Indemonstrable propositions and unanalyzable concepts A l though Kan t claims in the third section of the First Reflect ion to focus on the d i f fe rence in n u m b e r of unanalyzable concepts and indemons t r ab l e p ropos i t ions in the two disciplines, he in fact goes on to describe the d i f fe ren t roles they play. T h e unanalyzable concepts which m a t h e m a t i c s begins with are "the concepts of m a g n i t u d e in genera l , o f unity, o f plurali ty, o f space, and so on." In his lectures on logic, he also lists "a few given concepts" which the m a t h e m a t i c i a n canno t define, such as "a place, a direct ion, a s t ra ight line, etc." But because ma thema t i c s employs only the synthet ic me thod , the def ini t ion of these concepts does not be long to that science; while they m a y admi t o f def ini t ion e l s e w h e r e - - f o r example , in p h i l o s o p h y - - t h e def ini t ions have no m a t h e m a t i - cal consequences at all.

I t is the business of phi losophy, on the o the r hand , to analyze concepts . "Every analysis which can occur is actual ly necessary": bo th "the dist inctness of the cogni t ion and the possibility of valid in fe rences d e p e n d u p o n such analy- ses" [e: e8o]. Th is process inevitably leads to unana lyzable concepts , indeed, to " u n c o m m o n l y many" unana lyzable concepts , since the vastness and complex - ity of universal cogni t ion could not be reducible to jus t a few f u n d a m e n t a l concepts .

So in the case of ma themat ics , unanalyzable concepts are given, they are what we begin ou r inquiry with. In the case of ph i losophy , unana lyzab le concepts are sought, the goal o f ou r inquiry is to r educe the concep ts that are given to these unanalyzable concepts . T h e given concep ts of ma thema t i c s are thus "clear and certain," whereas the concepts o f p h i l o s o p h y are "given in a confused fash ion" [2:278 ]. As m e n t i o n e d earlier, though , some ma thema t i ca l concepts are capable of phi losophica l defini t ion, as for example , "the concep t o f

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space in genera l . " But the m a t hem a t i c i an "accepts such a concept given in accordance with his clear and ord inary r ep resen ta t ion . " I t would seem then that such a concep t can be given in d i f fe ren t ways, one way which is clear and certain, and a n o t h e r way which is confused.

T h e r e is a similar d i cho tomy in the case o f i ndemons t r ab l e proposi t ions , o f which there are only a few in ma themat i c s : "even if they admi t o f p r o o f e lsewhere, they are none the less r ega rded as immedia te ly cer ta in in this science" [2: 281]. But wha t does it m e a n to regard a p ropos i t ion as immedia te ly certain? Is this d i f ferent f r o m recogniz ing it with i m m e d i a t e certainty? Kant gives as examples the propos i t ions that the whole is equal to all its par ts taken toge ther , and that there can only be one s t ra ight line be tween two points "and so forth."7 These pr inciples are set u p at the beg inn ing of a mathemat ica l inquiry "so tha t it is clear that these are the only obvious p ropos i t ions which are immedia te ly p r e s u p p o s e d as true, and tha t all o the r propos i t ions are sub- j ec t to strict proof ." T o say tha t they are " immedia te ly p r e s u p p o s e d as t rue" seems to leave o p e n the possibility that they need no t in fact be true; at the very least, it does not imply that they are recognized with immed ia t e certainty. Moreover , as we'll see below, where Kan t does discuss the cer ta inty of ma the - matics, he does no t explain the certainty o f its i ndemons t r ab l e proposi t ions .

In teres t ingly , Kan t descr ibes the mos t i m p o r t a n t business of ph i losophy as seeking ou t the " indemons t rab le f u n d a m e n t a l truths," which "consti tute a scheme o f immeasu rab le scope" [2:281, my emphasis] . For any object, the characterist ic marks which the u n d e r s t a n d i n g "initially and immedia t e ly per - ceives in the object" const i tute the data for the same n u m b e r of i ndemon- strable proposi t ions . Defini t ions are then d rawn up on the basis o f these p r o p o - sitions. Kan t considers as an example the concep t o f space. H e starts by seeking out those characterist ic marks "initially and immedia te ly t h o u g h t in that concept . " H e notices that there is a mani fo ld in space, that this mani fo ld is no t cons t i tu ted by substances, and that space can only have three dimensions. T h e r e is, Kan t says, no basis on which these propos i t ions could be proved , for they const i tu te "the first and simplest though t s I can have of my object, when I first call it to mind . " Wher ea s in mathemat ics , the defini t ions are the first though t s one can en te r ta in o f one ' s object, in phi losophy, the first thoughts of the object serve to "generate this distinct cogni t ion and to produce the definit ion sought" [2:281-~] .

So we see that a l though Kan t focuses on the d i f ference in n u m b e r of such concepts and proposi t ions in the two disciplines, he also shows that they serve

7 The choice of examples is interesting since they represent the distinction between Euclidean axioms and postulates, and since Kant later describes the first as analytic and the second as synthetic. Thus they represent two different kinds of indemonstrable propositions.


quite d i f ferent funct ions. Mathematics combines the unanalyzable given concepts by means of synthesis into definitions, and then draws inferences f rom these definit ions by means of the indemons t rab le proposi t ions. Mathematical knowledge thus requires unanalyzable concepts, definitions, and indemonstrable proposi t ions at the outset. In the case of phi losophical knowledge, however, the indemons t rab le proposi t ions are ob ta ined immedia te ly f rom the given (analyzable) concepts, and serve as prerequisi tes for the definitions, which are the goal of analysis. T h e unanalyzable concepts are arrived at by means o f analysis, whereas they are presupposed by synthesis.

I focused above on the d i f ferent ways Kant seems to speak o f the claims to t ru th o f the indemons t rab le proposi t ions of mathemat ics and of philosophy respectively. T h a t is, he says of the f o r m e r that they are " regarded as immedia te ly certain"; "presupposed as t rue"; the latter, he describes as "fundamenta l indemonst rab le t ruths." I don ' t want to place too mu ch weight on this difference, but there is a similar no tewor thy d i f fe rence in his t r ea tmen t of the epistemology of the indemons t rab le proposi t ions, about how they are known. Kant actually says very little about this, bu t what he does say suggests that they are known in d i f fe ren t ways. First Of all, in the third section of the T h i r d Reflection, Kant addresses the quest ion o f "the true charac ter o f the first f undamen ta l t ruths of metaphysics," which, he says, is "not of a kind d i f fe ren t f rom that of any o ther rat ional cognit ion, apart from mathematics" [2: 293-4 , my emphasis]. O f the fundamen ta l t ruths of phi losophy, he says for example that "the unde r s t and ing immedia te ly perceives" the characteristic marks in the object, they are "initially and immedia te ly t h o u g h t in the concept ." Later, he describes them as sought out by means o f "certain inner exper ience ," "an immedia te and self-evident inner consciousness" [9:986]. T h e indemonst rab le mathemat ical proposi t ions cannot , like their phi losophical counterpar ts , be perce ived in the object because that would no t take us at all beyond the def ini t ion (by means of which the object is first given), and presumably, taking us beyond the defini t ions is exactly the task of the indemons t rab le proposi t ions. I suggested above that Kant attr ibutes no role to symbols in provid ing us with knowledge of the indemons t rab le proposit ions. So while Kant at least gestures towards an ep is temology for the i n d emo n - strable proposi t ions o f phi losophy, he does no t seem to address the quest ion of h o w - - o r even w h e t h e r - - w e know those of mathematics .

(iv) Objects In the four th section, Kant asserts that "the object of mathe- ma t i c s" - -which is m a g n i t u d e - - " i s easy and simple, whereas tha t of philosophy is difficult and involved" [9: 989]. It is, Kant says, easier to unde r s t and "an ari thmetical object which contains an immense multiplicity" than it is to grasp a philosophical idea like f r e e d o m in terms of its e lements . T r u e to the empirical na tu re of Kant 's inquiry, the p r o o f he re is in the pudding : "the ou tcome of

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the two inquir ies shows the d i f fe rence be tween t h e m . . . Claims to phi losophical cogni t ion vanish, but m a t h e m a t i c s endures . "

This takes us to Kant ' s discussion in the T h i r d Reflection of the nature of metaphys ica l and ma thema t i ca l certainty, where one migh t expec t to find an- swers to some o f the quest ions raised by the discussion of the First Reflection.

3.3. Mathematical and metaphysical certainty

Kant begins the T h i r d Reflection with the assert ion that "one is cer ta in if one knows that it is impossible that a cogni t ion should be false" [2: ~9o]. Mathemat i - cal cer ta in ty differs f r o m phi losophical cer ta in ty in two ways, bo th objectively and subjectively. T h e objective deg ree of cer ta inty "depends on the sufficiency in the characterist ic marks of the necessi ty o f a t ru th" ; the subjective degree of cer ta in ty "increases with the degree of intui t ion to be f o u n d in the cognit ion of this necessity."

Mathemat ics is g rea te r in objective certainty because it arr ives at its concepts synthetically: "it can say with cer ta in ty that wha t it did not in tend to represen t in the object by m e a n s of the def ini t ion is not con ta ined in that object" since the concep t only "comes into exis tence" by means o f the de f in i t i on - - i t has "no o the r significance." T h u s one is g u a r a n t e e d of the sufficiency and necessity of the characterist ic marks. In phi losophy, on the contrary , one m i g h t j u d g e that a characterist ic m a r k does not be long to a concep t s imply because one fails to notice that mark . Consequent ly , ph i losophy and metaphysics are m o r e uncer- tain "in their defini t ions." T h e g rea te r degree of objective cer ta inty in ma the - matics again follows f r o m its p r o c e e d i n g synthetically.

Mathemat ics is also g rea te r in subjective cer ta inty because "in its inferences and proofs , [it] regards its universal knowledge u n d e r signs in concreto, whereas ph i lo sophy always regards its universal knowledge in abstracto, as existing along- side signs." This, Kan t says, const i tutes a substantial d i f ference in the way the two inquir ies at tain to certainty. Because the signs in ma thema t i c s are "sensible means to cogni t ion" [2:291], we can know that no concep t has been over- looked and tha t the rules have been observed "with the deg ree of assurance characteris t ic o f seeing some th ing with one ' s own eyes." T h e signs in philosophy, words, only remind us of the universal concepts they signify. I t requires cons tant a t ten t ion to be aware of their significance, for if a characterist ic mark escapes ou r a t tent ion, " there is no th ing sensible which can reveal to us the fact that the characterist ic m a r k has been over looked": the result is e r ror .

So the fact that ma themat ic s p roceeds synthetically cont r ibutes to its certainty because the concepts def ined have no significance outside the definition, and we the re fo re canno t mis takenly a t t r ibute marks to it which do not be long to it. T h e fact that it p roceeds by means of sensible signs cont r ibutes to the cer- ta inty of its inferences and p roofs since we can assure ourselves that we have

KANT ON THE METHOD OF MATHEMATICS 64a

followed the rules correctly. Not ice however that no th ing has been said yet abou t the i ndemons t r ab l e proposi t ions . I t is at this po in t that one migh t have expec ted Kan t to suggest an epis temological l ink be tween the role of symbols and the i ndemons t r ab l e propos i t ions of geomet ry . Bu t the only cont r ibut ion which the use of symbols makes to the cer ta in ty o f ma themat i c s , according to Kant , is to ensure tha t the p roofs and inferences p r o c e e d correctly. Th is leaves open the in te rp re ta t ion that, a l though the i n d e m o n s t r a b l e p ropos i t ions are r ega rded as true, they need not in fact be true. H e does say that the cer ta inty o f g e o m e t r y is even g rea te r because "the signs are s imilar to the things signified," but again, does no t link this to the i ndemons t r ab l e proposi t ions .

3.4. So the following p ic ture of the m e t h o d of m a t h e m a t i c s emerges f r o m the Prize Essay. I t begins with a few given concepts , which ma thema t i c i ans canno t and mus t no t define, such as m a g n i t u d e in genera l , unity, plurali ty, and space, and a small n u m b e r of i n d e m o n s t r a b l e p ropos i t ions which are " regarded as" immedia te ly certain, "p r e supposed as t rue," such as the p ropos i t ions that the whole is equal to all its par ts taken together , and that there can only be one s t raight line be tween two points. F u r t he r concepts are bui l t u p out o f these given ones by a rb i t ra ry c o m b i n a t i o n - - b y synthesis. T h e m a t h e m a t i c i a n then derives f u r t he r p ropos i t ions f r o m these complex concepts toge ther with the f u n d a m e n t a l proposi t ions . In the proofs and inferences , however , the m a t h e - matician does no t consider the objects themselves or thei r universal concepts , bu t considers r a the r thei r signs. T h e g rea te r deg ree of objective cer ta in ty in ma thema t i c s derives f r o m its use of the synthet ic m e t h o d ("it can say with cer ta inty that wha t it did no t in tend to r e p r e s e n t in the object by m e a n s of the defini t ion is not con ta ined in that object") and the g rea te r deg ree of subjective cer ta inty arises f r o m the role of sensible signs (by m e a n s of which "things can be known with the degree o f assurance character is t ic o f seeing some th ing with one 's own eyes").

T h e p r o b l e m ! wish to raise is tha t Kant ' s accoun t of the cer ta inty of mathemat ics , wi thout ph i losophica l supp lemen ta t ion , risks be ing subject to his own object ions to the metaphysic ians ' t r e a t m e n t of mathemat ics . As we 've seen, Kant ' s object ions to this view go back at least as far as the Physical Monadology, where he attacks the metaphys ic ians for th ink ing that " they had to main ta in that the p roper t i e s of geomet r ica l space are as good as imagi- nary." H e p u r p o r t s to show that "ne i ther is g e o m e t r y deceived no r does the t hough t of the me taphys ic i an deviate f r o m the t ru th ." So clearly Kan t takes the claim that geomet r ica l space is imag ina ry or ideal to u n d e r m i n e the claim to t ru th of geomet ry . T h u s they tu rn ma thema t i ca l concepts into "subtle fictions, which have little t ru th to t hem outs ide the field o f ma themat i c s . " In an a t t e m p t to save the "artificially contr ived" concep t s o f metaphysics , they


raise spec ious ob j ec t i ons aga ins t m a t h e m a t i c s , " c l a iming tha t its f u n d a m e n t a l concep t s have n o t b e e n de r i ved f r o m the t rue n a t u r e of space at all, b u t a rb i t r a r i l y i n v e n t e d . "

B u t we have j u s t seen tha t for K a n t h imse l f , n o n - p r i m i t i v e m a t h e m a t i c a l concep t s are a rb i t r a ry concep t s : they are "concep ts tha t are m a de , are such as are c r e a t e d by us arb i t rar i ly , or f ab r i ca ted , w i t h o u t p rev ious ly h a v i n g b e e n g iven" [24:133]. H e even goes so far as to call t h e m "ficti t ious concep ts . " I n d e e d , K a n t ' s a c c o u n t o f the object ive ce r t a in ty of m a t h e m a t i c s rests on the a rb i t r a r ine s s o f its de f in i t ions , o n its p r o c e e d i n g by the synthe t ic m e t h o d . T h e q u e s t i o n n a t u r a l l y arises as to how he d i f f e r en t i a t e s his view tha t m a t h e m a t i c a l concep t s are a rb i t r a r i ly c rea ted , f ab r i ca ted , f r o m the view which he at tacks so v e h e m e n t l y tha t t hey are ' subt le f ict ions, ' ' a rb i t ra r i ly i n v e n t e d , ' a n d ' f i gmen t s of the i m a g i n a t i o n ' ? B o t h views m i g h t s e e m to be k inds o f f o r m a l i s m a b o u t m a t h e m a t i c s , a c c o r d i n g to which m a t h e m a t i c a l concep t s are c o n v e n i e n t ideal- iza t ions , p e r h a p s i n d i s p e n s a b l e tools for physics, b u t wi th n o c la im to t r u t h o n the i r own.

A c c o r d i n g to Beth , K a n t ho lds j u s t such a view in the Pr ize Essay:

We see that with regard to the methodology of mathematics Kant defends views which essentially agree with Leibniz's and Nieuwentyt 's conceptions, and which come quite near to formalist and logicist doctrines in contemporary research on the foundations of mathematics, s

U n l i k e Beth , I d o n ' t w a n t to say tha t K a n t does ho ld such a view in the Prize Essay, j u s t t ha t m o r e m u s t be said to d i f f e r en t i a t e his view f r o m the ones he objects to. P e r h a p s the c la im tha t m a t h e m a t i c a l de f i n i t i ons are also real

de f in i t i ons is s u p p o s e d to deal wi th this k i n d o f worry , b u t it 's n o t c lear tha t it does so, for in w h a t sense is the t h i n g thus d e f i n e d 'ac tual '? I n a n y case, a n o t h e r way to p u t the q u e s t i o n t h e n is to ask how is it tha t m a t h e m a t i c s admi t s o f real de f in i t ions . H o w is it tha t m y fab r i ca t ing a g e o m e t r i c a l c o n c e p t

s Evert W. Beth, The Foundations of Mathematzcs (Amsterdam: North Holland Publishing Co., 1965), 45. More recently, it has been argued by Brigina-Sophie yon Wolff-Metternich that in the Prize Essay, Kant held the inconsistent view that mathematics proceeds synthetically in the forma- tion of its concepts, but that it proceeds analytically in its judgements, in the same way that philosophy proceeds analytically from given concepts. In other words, mathematical judgements are simply logical consequences of arbitrary definitions, and are thus, in Kant's later terminology, analytic. But this fails to take into account the role that Kant sets out for indemonstrable propositions at the beginning of the mathematical enterprise. See D~e Ubenaindung des mathemat~schen Erkenntnisideals (Berlin: Walter DeGruyter, 1995), 36. Von Wolff-Metternich gives an account in Chapter 1 of the development of Kant's account of mathematics from the Prize Essay to the Cntzque similar to that presented here, but she argues that Kant appeals to intuition in the later work in order to capture the syntheticity of mathematics. I shall argue rather that it is the truth and certainty of mathematical knowledge which require the doctrine of mtumon; syntheticity then follows.

K A N T O N T H E M E T H O D O F M A T H E M A T I C S 643 confers 'actuality' on it, where my fabr ica t ing a concep t o f a s lumber ing m o n a d does not?

These quest ions po in t to the d e m a n d for an exp lana t ion of the re levant d i f ference be tween ma themat i ca l concep ts and metaphys ica l ones, a difference which accounts for the admissibili ty of a rb i t r a ry concepts in the one bu t not in the other : why is invent ion permiss ible , even requi red , in mathemat ics , bu t no t in phi losophy? W h y is the synthet ic defini t ion o f a t r apez ium legiti- mate , and Leibniz 's invent ion o f the s lumber ing m o n a d not? Af te r all, bo th involve the f o r m a t i o n of complex concepts f r o m given pr imi t ive ones. More pointedly, why does the arbi t rary r e p r e s e n t a t i o n of a r ight -angled t r iangle which is ro ta ted on one of its sides issue in a legi t imate ma thema t i ca l concept , while the r ep r e s en t a t i on o f a f igure enclosed by two s t ra ight lines does not?

T h e obvious answer to this lat ter ques t ion is that the f igure descr ibed canno t be def ined in accordance with the i ndemons t r ab l e proposi t ions , for it contradicts the p ropos i t ion that be tween two points only one s t ra ight line m a y be drawn. But this then s imply pushes the ques t ion on to the i ndemons t r ab l e propos i t ions which const ra in the arbi t rar iness o f definit ions. W h a t is their sta- tus? Kan t says that they are taken to be immedia t e ly certain. I indicated above that Kan t gives no epis temological accoun t o f these concep t s and proposi t ions , and it is at this point , I think, that his fai lure to do so becomes impor tan t . Wi thou t f u r t h e r a r g u m e n t , the re ject ion o f s lumber ing m o n a d s on these g rounds seems as dogmat ic as thei r aff i rmat ion. T h e p r o b l e m is that Kant ' s descr ip t ion of the ma themat i ca l m e t h o d seems to c o r r e s p o n d to that a p p r o - pr ia te to a f o rm a l ax iomat ic system; unless some exp lana t ion is given of the con ten t o f those pr imi t ive concepts and p ropos i t ions and, especially, the g r o u n d of the i r certainty, this accoun t s imply collapses into the fo rmal i sm tha t Kan t so obviously opposes . In addit ion, this th rea t o f fo rma l i sm u n d e r m i n e s his a t t e m p t to dist inguish the m e t h o d s a p p r o p r i a t e to ma thema t i c s and me ta - physics. I f the g e o m e t e r is s imply deduc ing p rope r t i e s and rela t ions of imagina ry or ideal objects g iven by a rb i t ra ry definit ions, wha t is to s top the me ta - physician f r o m deve lop ing an axiomat ic system for s l umber ing m o n a d s in a similar way? In what sense can we say tha t m a t h e m a t i c s is a body of truths, a n d the theory o f s l umber ing m o n a d s is not? More impor tan t ly , given Kant ' s concern with the relative cer ta in ty of m a t h e m a t i c s and metaphysics , how can we say that we know these t ruths with certainty?

For similar reasons, an appea l to the role of d e m o n s t r a t i o n will no t he lp ei ther, for this s imply begs the quest ion. Why is ma thema t i ca l d e m o n s t r a t i o n m o r e t rus twor thy than metaphys ica l dem o ns t r a t i on? W h y should we trust the demons t r a t i on of infinite divisibility given above over the d e m o n s t r a t i o n o f the indivisibility of monads? Kan t has exp la ined how the use o f symbols en- sures that our dem ons t r a t i ons are in accordance with the i ndemons t r ab l e


proposit ions, but what is required to establish the truth o f their conclusions is assurance of the t ruth of those proposit ions, and Kant has given no such assurance. Kant 's goal is to contrast the nature o f mathemat ica l and philosophical certainty with an eye towards justifying the privileged position that he accords to the latter as "reliably established data" which can serve as the "foun- dation" o f philosophical reflections, as against the metaphysicians who raise specious objections against mathemat ics in o rder to rescue metaphysics. It is surely no better, however, simply to assert the reliability o f the data supplied by geometry:9 Kant at the very least has to show that objections o f the metaphysicians are indeed specious, and therefore that the fundamenta l concepts of geomet ry have been "derived f rom the true nature of space."

Kant sometimes seems to suggest that the success and fruitfulness of mathematics in natural science warrants these bold claims on its behalf, for example when he claims that "nature herself seems to yield proofs of no little distinctness" showing that the concept of the infinitely small is "very true," in contrast to the "obscure abstractions" of metaphysics "which are difficult to test." But this is no t sufficient to war ran t the claim that the concepts of mathematics are "true," part icularly in light of the oppos ing view that they are useful fictions. Kant himself admits that these concepts are "d i f f i cu l t . . . to penetrate." He then goes on to make the very weak claim that "this difficulty can, at best, only justify the cautiousness with which hesitant conjectures are made; it cannot justify the dogmat ic declarat ions of impossibility" [2: 169]. In order to justify his claims about the p rope r use of mathemat ics in phi losophy and about the contrast between mathematical and philosophical certainty, Kant must do more than show that the primitive concepts of mathematics are not impossible.

To sum up then, Kant 's account of mathematics in the Prize Essay seems to leave open the question of the relation between the m e t h o d of mathematics and its certainty. First o f all, it's not clear how mathematical concepts are anything but arbitrary inventions with no objective content . Second, mathematical proposi t ions then seem to lose their claims to truth as opposed to mere deducibility f rom axioms and definitions. Kant 's concern with these issues is clear f rom the fact that he articulates them frequently and t h roughou t his career as objections to ' the metaphysicians. ' The key to these questions with respect to geomet ry is the epistemological relationship between geomet ry and space: how can he establish this relation in such a way that avoids the problems he associates with the metaphysicians ' view, and yet retains the privileged claims to certainty accorded to mathematics? Kant has to show that the mathematical m e t h o d of attaining certainty is in fact a me thod of attaining certainty.

0As indeed Kant does in the Prize Essay itself where he demonstrates that space does not consist of simple parts by "employing infallible proofs of geometry" [2: 287].

K A N T O N T H E M E T H O D O F M A T H E M A T I C S 645 4 " C O N S T R U C T I O N I N P U R E I N T U I T I O N

It is certainly familiar that Kant saw his doctr ine o f intuition as the key to formula t ing an adequate account of mathemat ical knowledge. A more controversial quest ion is what role intuition plays in that account. Taking seriously Kant 's concern with the questions raised above, I think, will clarify what that role is. This concern is reflected in a change in Kant 's presenta t ion o f the mathematical m e t h o d in his writings between 1764 and the a 78os. Whereas in the earlier work, Kant identifies the main contrast between mathematical and philosophical me thod in the role of definitions, he later emphasizes the role o f construct ion in pure intuition. Here, Kant is no t simply asserting a mark o f the difference between them, but ra ther he is providing an explanat ion: the possibility o f construct ion in pure intuit ion is invoked in order to explain the difference in me thodo logy and certainty. '~

Kant first begins to spell out his doctr ine o f space and time as the forms of intuition in the Inaugura l Dissertation o f 177 o, "On the fo rm and principles o f the sensible and the intelligible world." Having a rgued that space is a pure intuition, he goes on to say that "this pure intuit ion can easily be seen in the axioms of geomet ry and in any mental const ruct ion of postulates, even of problems" [2:4o9 ]. The not ion o f pure intui t ion is thus invoked to give conten t to, and g r o u n d the t ruth of, the axioms of geometry . In addition, having gone on to argue that space is "the fundamen ta l fo rm of all outer intuition," Kant concludes that "nothing at all can be given to the senses unless it conforms with the fundamenta l axioms of space and its corollaries (as g e o m e t r y teaches)" [2:4o2 ]. Accordingly, "nature is complete ly subject to the prescrip- tions of geomet ry . . . . And this is so, not on the basis of an invented hypothesis, but on the basis of one which has been intuitively given, as the subjective condi t ion of all p h e n o m e n a , in virtue of which condi t ion alone na ture can be revealed to the senses." Geomet ry is no t a 'convenient fiction' but is necessarily true of the sensible world.

It remains now to show how this doctr ine fits in with the account o f mathematical m e t h o d f rom the Prize Essay. In "The Discipline of Pure Reason in its Dogmat ic Employment" near the end of the first Critique, Kant once again takes up the question of whether the mathemat ica l m e t h o d of at taining certainty is identical with the m e t h o d of attaining certainty in phi losophy. The answer, again, is negative, bu t this time, he gives no t just a description o f the differences in m e t h o d and certainty, but an explanation of those differences. The essential difference between these two kinds of knowledge is again a

'~ is in sharp contrast to Hintikka's view that the doctrine of pure intuition doesn't change in any way Kant's earher account of method, and is in fact extraneous to it. See, e.g.,'"Kant on the Mathematical Method" in Knowledge and the Known ( Dordrecht: Reidel, 1974).


formal one, that "philosophical knowledge is the knowledge gained by reason f rom concepts" whereas "mathematical knowledge is the knowledge gained by reason f rom the construction of concepts" [A 713/B742]. To construct a concept is, for Kant, to "exhibit apriori the intui t ion which cor responds to the concept ." T h u s for example , the geome te r constructs a triangle "by represent ing the object which cor responds to this concep t e i ther by imaginat ion alone, in pu re intuition, or in accordance therewith also on paper , in empirical i n tu i t i on - - in bo th cases complete ly apriori , wi thout having bor rowed the pat- tern f rom any exper ience ." Al though the intui t ion is a single object, "it ex- presses universal validity for all possible intuit ions which fall u n d e r the same concept" because it is "de te rmined by certain universal condit ions of construc- t ion" [A714/B74~], that is, the condi t ions imposed by the fo rm of intuition. Cont ras t with this a philosophical concept , like that o f cause or reality. "No one can obtain an intui t ion cor responding to the concep t o f reality otherwise than f rom exper ience ; we can never come into possession o f it apriori ou t o f our own resources, and pr ior to the empirical consciousness of reality."

So far, this is again jus t a descr ipt ion o f the difference: objects cor responding to mathematical concepts can be provided a priori , bu t this is not the case in metaphysics. Recogniz ing the need for an explanat ion, though, Kant asks, "what can be the reason of this radical d i f ference in the for tunes o f the philoso- phe r and the mathemat ic ian , bo th of whom practise the art of reason, the one making his way by means of concepts, the o ther by means of intuit ions which he exhibits apriori in accordance with concepts?" Why is it possible for mathematicians to obtain a priori intuitions cor responding to their concepts, but not for phi losophers?

T h e answer, according to Kant, is given by the " fundamenta l t ranscenden- tal doctr ines." According to these doctrines, there are two elements in the field of appearance : the fo rm of intui t ion (space and time), and the mat te r (the physical element) . Whereas the material e l ement canno t be given in determinate fashion o the r than empirically, the formal e lement , Kant says, "can be known and d e t e r m i n e d complete ly apriori ." T h a t is, because space and time, as the fo rm of appearances , are given a priori, "a concep t o f space and time as quanta, can be exhibi ted apriori in intuition, that is, cons t ructed" [A72o/ B748]. So the concepts of mathemat ics "include in themselves" a pu re intuition. T h e concepts o f phi losophy, on the o the r hand, include "nothing but the synthesis of possible intuit ions which are not given apriori ." Since the objects cor responding to these concepts canno t be given a priori , the only a priori phi losophical knowledge we can have is of "the mere rule of the synthesis of that which percep t ion may give us aposter iori ." As a result, "pure phi losophy is all at sea when it seeks th rough aprior i discursive concepts to obtain insight

K A N T O N T H E M E T H O D OF M A T H E M A T I C S 647 in r ega rd to the na tura l world, be ing unab le to intui t apr ior i (and the reby to confirm) their reality" [A725/B753].

So, Kan t concludes, co r r e spond ing to the two e lements in the field o f appea rance , the re is a two-fold e m p l o y m e n t o f reason: the ma thema t i ca l em- p l o y m e n t o f r eason t h r o u g h the cons t ruc t ion of concepts , and the ph i losoph i - cal e m p l o y m e n t o f reason in accordance with concepts. T h e essential difference is that the concepts o f mathematics, because of their re la t ion to the f o r m of intuit ion, car ry with t h e m conf i rmat ion o f thei r reality, whereas e x a m p l e s of the concepts of ph i losophy can only be suppl ied by exper ience , and there is no a pr ior i gua ran t ee that expe r i ence will p rov ide such an example , l~ In this way, the doc t r ine of intui t ion plays an essential role in d is t inguishing be tween ma thema t i c s and ph i losophy with respec t to the objective reality of the i r re- spective concepts .

I turn now to the ques t ion of how this accoun t of ma themat i ca l m e t h o d in the Critique of Pure Reason relates to the earl ier accoun t given in the Prize Essay. Recall that in the Prize Essay, the d i f fe rence be tween the m e t h o d s o f m a t h e - matics and phi losophy, be tween the synthet ic and the analytic m e t h o d , res ted largely on the d i f fe ren t roles of defini t ions in each. Similarly in the Critique, Kant a t t emp t s to show once and for all tha t ma thema t i c s and ph i lo sophy are so d i f ferent that "the p r o c e d u r e o f the one can neve r be imi ta ted by the o ther . " H e does this by once again cons ide r ing the means of achieving cer ta in ty in mathemat ics , in o r d e r to show tha t these m e a n s are no t available to the phi losopher . Those means , he says, are "definit ions, axioms, and demons t r a t i ons" none o f which "in the sense in which they are u n d e r s t o o d by the m a t h e m a t i - cian, can be achieved or imi ta ted by the ph i lo sophe r " [A726/B754]. I ' l l beg in by c o m p a r i n g the accoun t o f def ini t ion in the Critique with tha t in the Prize Essay.

Again, Kan t says that to def ine means to p re sen t the comple te , original, and precise concep t o f a thing. An empir ica l concep t c anno t be de f ined because the limits o f the concep t are neve r assured: for example , new observa-

~ T h i s d i f ference also under l i e s the dis t inct ion be tween the pr inciples o f ma thema t i ca l em- p l o y m e n t and those o f dynamica l e m p l o y m e n t : a priori c o n d m o n s o f i n tmt ion are absolutely necessary condi t ions of any possible exper ience ; those o f the exis tence o f objects o f a possible empirical in tu i t ion are in themselves only accidental : r a ther , they are necessa ry only u n d e r the condi t ion o f empirical t h o u g h t in some expe r i ence [A16o/Bx99]. Because we c a n n o t cons t ruc t exis ten ce, the pr inc ip les of dynamica l e m p l o y m e n t are mere ly regulat ive [A 179/B~ 2 ~ ]. T h e possibility o f cons t ruc t ion also expla ins a n o t h e r d i f fe rence be tween the pr inc ip les o f ma thema t i ca l e m p l o y m e n t a n d those o f dynamica l e m p l o y m e n t : the latter do no t have tha t " immed ia t e evidence" which ts pecul ia r to the fo rmer , which "allow of in tui t ive certainty, alike as r ega rd s thei r ev idenua l force and as regards their apriori appl ica t ion to appea rances" [ A t 6 t - a / B ~ o o - t ] . (See also Metaphyxzcal Foundations of Natural Science, 4 : 469 -70 . )


t ions r e m o v e s o m e p r o p e r t i e s a n d a d d o the r s . C o n c e p t s g iven a p r i o r i (such as s u b s t a n c e , cause o r r igh t ) c a n n o t be d e f i n e d b e c a u s e t h e c o m p l e t e n e s s o f the ana lys i s wil l a lways b e o n l y p r o b a b l y , n e v e r a p o d e i c t a l l y , ce r t a in . T h e on ly c o n c e p t s w h i c h r e m a i n a r e " a r b i t r a r i l y i n v e n t e d c o n c e p t s . " W i t h r e g a r d to these , K a n t says,

a concept which I have invented I can always define; for since it is not given to me either by the nature of unders tanding or by experience, but is such as I have myself deliberately made it to be, ! must know what I have in tended to think in using it [A7~9/ B757].

H o w e v e r , he goes on , "I have [not] t h e r e b y d e f i n e d a t r ue ob jec t . " I n t h e case o f a n e m p i r i c a l c o n c e p t , " this a r b i t r a r y c o n c e p t o f m i n e d o e s n o t a s s u r e m e o f t he e x i s t e n c e o r o f t he poss ib i l i ty o f its ob jec t . " T o b o r r o w K a n t ' s p h r a s e f r o m the P r i ze Essay, i t w o u l d j u s t be a " h a p p y a c c i d e n t " i f t h e r e w e r e an e m p i r i c a l o b j e c t c o r r e s p o n d i n g to m y i n v e n t e d c o n c e p t . B u t m a t h e m a t i c a l c o n c e p t s , as we 've s een , " c o n t a i n an a r b i t r a r y syn thes i s t h a t a d m i t s o f a p r i o r i c o n s t r u c t i o n , " t ha t is, c o n s t r u c t i o n in p u r e i n t u i t i o n . T h e c o n s t r u c f i b i l i t y o f such c o n c e p t s in p u r e i n t u i t i o n does a s s u r e us o f t he p o s s i b i l i t y o f t he c o r r e s p o n d i n g objec t . C o n s e q u e n t l y , o n l y m a t h e m a t i c s has d e f i n i t i o n s p r o p e r "for the o b j e c t w h i c h it th inks , i t exh ib i t s a p r i o r i in i n t u i t i o n , a n d this o b j e c t c e r t a i n l y c a n n o t c o n t a i n e i t h e r m o r e o r less t h a n t h e c o n c e p t , s ince i t is t h r o u g h the d e f i n i t i o n tha t the c o n c e p t o f t he o b j e c t is g i v e n " [A729 /B757] . I~

So syn the t i c d e f i n i t i o n s a r e a d m i s s i b l e in m a t h e m a t i c s a n d n o t in p h i l o s o - p h y b e c a u s e t he a r b i t r a r y syn thes i s o f c o n c e p t s in m a t h e m a t i c s a d m i t s o f a p r i o r i c o n s t r u c t i o n , w h i c h a s su re s us o f t he e x i s t e n c e , o r be t t e r , t he poss ib i l i ty o f t he ob jec t s . I t is in th is sense , t h e n , t ha t m a t h e m a t i c a l d e f i n i t i o n s a r e also real d e f i n i t i o n s : a r ea l d e f i n i t i o n is o n e w h i c h "does n o t m e r e l y s u b s t i t u t e fo r t he n a m e o f a t h i n g o t h e r m o r e i n t e l l i g ib l e w o r d s , b u t c o n t a i n s a c l e a r p r o p - e r t y by w h i c h the d e f i n e d object can a lways be k n o w n wi th c e r t a i n t y " [A~42n] . T h u s , K a n t says e a r l i e r in t he Critique, a rea l d e f i n i t i o n " m a k e s c l ea r n o t on ly the c o n c e p t b u t also its objective reality." B e c a u s e m a t h e m a t i c a l d e f i n i t i o n s p re s - e n t the o b j e c t in i n t u i t i o n , in c o n f o r m i t y wi th t he c o n c e p t w h i c h is o r i g i n a l l y f r a m e d by the m i n d i tself , t h e y a r e r ea l de f in i t i ons . M a t h e m a t i c a l d e f i n i t i o n s are, K a n t says, c o n s t r u c t i o n s o f c o n c e p t s [A73o /B758] . C o n v e r s e l y , K a n t goes o n to e x p l a i n , i t w o u l d be fu t i l e fo r t h e m a t h e m a t i c i a n to p h i l o s o p h i z e u p o n

1 2 Kant's claims here about presenting the objects of the concepts m pure intmtion are difficult, because elsewhere he claims that the only objects for mathematical concepts are given in empmcal intuition [A239/B998], and that their real possibility is established by the argument of the Tran- scendental Deduction. But this is, I think, consistent with the claim that construction m pure intuition establishes the objective reality of mathematical concepts g~ven the results of transcendental philosophy. I have argued for this m more detail in my "Kant on Intuition in Geometry," CanadzanJournal ofPhzlosophy 27 (1997): 489-5 a 2.

K A N T ON T H E M E T H O D OF M A T H E M A T I C S 649 the triangle, to th ink abou t it discursively, for "I shou ld not be able to advance a single step beyond the m e r e defini t ion, which is wha t I had to begin with."

This r e fe rence to a m e r e def ini t ion m a y a p p e a r to conflict with the claim that ma themat i ca l defini t ions are const ruct ions o f concepts . T h e resolu t ion of the a p p a r e n t conflict can be used to i l luminate Kant ' s dist inction be tween real and nomina l definit ions. In o rde r to gain ma thema t i ca l knowledge, Kan t goes on, "I m u s t pass beyond this def ini t ion to p roper t i es which are no t con ta ined in this concep t but yet belong to it." T h e only way to do this is to " d e t e r m i n e my object in accordance with the condi t ions e i ther o f empir ical or o f p u r e intu- i t ion" [A718/B746]. So the p rope r t i e s not con ta ined in the ma thema t i ca l concep t but be long ing to it do so in vi r tue of the concept ' s re la t ion to the condi t ions o f p u r e intuition. T h e real mathematical defini t ion then is the deter - mina t ion of the object in accordance with the condi t ions of p u r e intuit ion. This is no t to say that one canno t give a 'mere ' nomina l def ini t ion in accordance with concepts alone, but such a def ini t ion has no place in mathemat ics . F r o m the logical po in t o f view, a def ini t ion is g iven by a collection o f concepts. F r o m the ma themat i ca l po in t o f view, it is the const ruct ion of a concep t in p u r e intui t ion?3

Let me try to make clear now how the no t ion of cons t ruc t ion in p u r e intui t ion resolves the p r o b l e m s ra ised above r ega rd ing Kant ' s ear ly accoun t of ma themat i ca l definit ion. T h a t Kan t was wor r ied a b o u t the formal i s t possibili ty left o p e n by the Prize Essay comes out clearly in m a n y passages in the Critique. H e says, for example , that the knowledge ob ta ined f r o m the cons t ruc t ion of f igures in space would be no th ing bu t p laying with m e r e ch imeras "were it no t tha t space has to be r e g a r d e d as a condi t ion of the a p p e a r a n c e s which constitute the mater ia l for ou te r expe r i ence" [A157/Ba96], that is, were it no t for "the f u n d a m e n t a l t r anscenden ta l doctr ines ." T h e key idea he re is that the con ten t o f the a rb i t ra ry concep ts o f ma thema t i c s is given a pr ior i , by const ruc- tion in p u r e intuit ion, whereas there is no th ing given a pr ior i which could c o r r e s p o n d to the concep t o f a s l u m b e r i n g m o n a d . Pure intui t ion thus con- strains the arbi t rar iness of the defini t ions, and gives con ten t to the ax ioms and pr imi t ive concepts . T h e f u n d a m e n t a l p ropos i t i ons of g e o m e t r y asser t the "universal condi t ions of const ruct ion" of figures, that is, the condi t ions imposed by the f o r m of intuit ion. Cons t ruc t ion in p u r e intui t ion is then s imply cons t ruc- t ion accord ing to the (Euclidean) postulates . T h e concep t o f a f igure enclosed by two s t ra ight lines is not in accordance with the f u n d a m e n t a l p ropos i t ions (i.e., be tween any two points only one s t ra ight line can be drawn) , and thus is not construct ible in p u r e intuit ion. No similar constra ints can be p resc r ibed in

~31 have a r g u e d e lsewhere for the s igni f icance of this d is t inct ion. See the p a p e r r e f e r r e d to in n. 1~, 5o4 -6 .

650 J O U R N A L OF T H E H I S T O R Y OF P H I L O S O P H Y 37:4 O C T O B E R a999

advance rega rd ing the existence of the objects co r r e spond ing to phi losophical concepts . In this way, Kan t says in a let ter to Reinhold , "mathemat ics is the mos t excel lent mode l for any synthetic use of reason, jus t because the intu- i t ions with which it gives objective reality to its concepts are neve r lacking."a4

Note , though , tha t this accoun t requires that those constraints be prescr ibed in advance. W h a t we 've got so far is a story abou t how the f u n d a m e n t a l p ropos i t ions and arbi t rary concepts of ma themat i c s have objective content . T h e ma themat i c i an does no t s imply spin out consequences of a rb i t ra ry theories, bu t r a the r spins out consequences of true theories. But it's no t e n o u g h for Kant ' s pu rposes tha t these theor ies s imply be p r e s u p p o s e d as t rue; they mus t be known to be true. Otherwise, the doo r is still left open for the axiomatic theory o f s lumber ing monads . For this reason, K a n t mus t see a role for his doc t r ine of intui t ion in expla ining no t only the content , but ou r knowledge, of the i ndemons t r ab l e propos i t ions of mathemat ics . Only in this way can he achieve a c o h e r e n t account bo th of the con ten t o f mathemat ics and of the cer ta in ty of our knowledge of mathemat ics .

T h a t Kan t sees intui t ion as playing this epis temological role is clear f rom the rest o f the discussion of the m e t h o d of ma themat ic s in the "Discipline of Pure Reason." T h e second essential d i f ference be tween ma thema t i c s and philosophy, the role of axioms, is also exp la ined by the doctr ine of pu re intuition. Mathemat ics , Kan t says, can have axioms "since by means o f the construct ion of concepts in the intui t ion o f the object it can combine the predicates of the object both apr ior i and immedia te ly" [A733/B761]. I t is the immed iacy of the basic pr inciples o f ma themat i c s that is distinctive: a synthetic p r inc ip le derived f r o m concepts a lone, as are f o u n d in phi losophy, can never be immedia te ly certain. Kan t considers the exam pl e o f the pr inciple that every th ing which h a p p e n s has a cause. One canno t obtain knowledge of such a pr inciple "directly and immedia te ly" f r o m the concepts alone, but r a the r mus t "look a r o u n d for a th i rd someth ing , namely , the condi t ion of t ime-de te rmina t ion in an exper ience . " Wher ea s intuitive pr inciples are evident, discursive pr inciples requi re a deduct ion . For this reason, "no pr inciple deserving the n a m e of an ax iom is to be f o u n d " in ph i l o sophy [A73~/B76o]. In the t e rmino logy of the Prize Essay, Kan t has now m a d e it explicit that the use of 'sensible signs' ex tends beyond the proofs and inferences . By means of construct ions, we have intuit ive knowledge of the f u n d a m e n t a l p ropos i t ions of geomet ry .

Similarly with the third and final essential difference. W h a t is distinctive of the demons t r a t i ons found in ma thema t i c s is that they are intuitive. But a priori concepts, as oppos ed to intuitions, c anno t give rise to intuitive certainty: thus,

~4 Immanuel Kant: Briefwechsel, edited by Otto Sch6nd6rffer (Hamburg: Felix Melner Verlag, 1972), 389 .

K A N T ON T H E M E T H O D OF M A T H E M A T I C S 6 5 1

mathemat ic s a lone contains demons t r a t i ons "since it derives its knowledge no t f r o m concepts bu t f r o m the cons t ruc t ion o f them, that is, f r o m intui t ion, which can be given aprior i in accordance with the concepts" [A734/B762].

5" C O N C L U S I O N

I have tr ied to a rgue that Kant ' s accoun t of the ma themat i ca l m e t h o d in the Prize Essay requ i red s u p p l e m e n t a t i o n by the Critical doc t r ine of p u r e intuition. A l though he did recognize a role for sensibility in ma thema t i ca l knowledge in the Prize E s s a y - - h e says the re that ma thema t i c s is d i f fe ren t f r o m ph i losophy also in that it "examines the universal u n d e r signs in concre to" - -

without a deve loped theory of sensibility and its re lat ion to space, this role was not sufficient to deal with the p r o b l e m s m e n t i o n e d above. I ndeed , the role for sensibility as p r e sen t ed in the Prize Essay is consis tent bo th with the me taphys i - cians' accoun t of space and sensibility which Kan t v e h e m e n t l y rejected, and

with an empir ic is t view of ma thema t i c s accord ing to which the p ropos i t ions o f g e o m e t r y are l ea rned by exper ience . Bu t a b a n d o n i n g the apr ior i ty of m a t h e - matics was not, for Kant , obviously be t te r than mak ing it a 'me re play o f imaginat ion. ' So the full accoun t of ma thema t i ca l me thod , one which gives objective con ten t to ma themat i c s and retains the a pr ior i t ru th and i m m e d i a t e evidence of ma themat i ca l proposi t ions , had to wait for Kan t ' s doct r ine of p u r e intuition.

This de fence of ma thema t i ca l knowledge is r e n d e r e d m o r e power fu l if we consider that Kant ' s a r g u m e n t for the claim that space and t ime are f o r m s of intuit ion, which is the claim that enables h im to prov ide con ten t and objectivity for mathemat ics , is i n d e p e n d e n t of these concerns . In the "Metaphysical Expo- sition o f the Concep t o f Space," K a n t provides an a r g u m e n t , i n d e p e n d e n t o f geomet ry , and based ins tead on "the condi t ions o f the possibility o f exper i - ence," for taking the space in which objects are given to be as g e o m e t r y describes it.'5 T h e "Transcenden ta l Expos i t ion" t hen shows that this analysis of ou r r ep resen ta t ion of space also accounts for ou r knowledge of geomet ry . T h e section which we have jus t b e e n looking at f r o m the " T r a n s c e n d e n t a l Doct r ine of M e t h o d " in tu rn shows how the doct r ine of p u r e in tui t ion accounts for the di f ference be tween ma thema t i ca l and phi losophica l m e t h o d , a d i f ference which is at the hea r t o f the ent i re Critical project . T o see this, consider one final passage:

There is no need of a critique of reason in its empirical employment, because in this field its principles are always subject to the test of experience. Nor is it needed in mathematics, where the concepts of reason must be forthwith exhibited m concreto in

15This is, of course, a controversial claim. I have provided some argument for it m the paper cited in n. 12.

652 JOURNAL OF THE HISTORY o r PHILOSOPHY 3 7 : 4 OCTOBER X999

pure intuition, so that everything un founded and arbitrary in them is at once exposed. But where neither empirical nor pure intuit ion keeps reason to a visible track, when, that is to say, reason is being considered in its transcendental employment , in accordance with mere concepts, it stands so greatly in need of a discip]ine, to restrain its tendency towards extension beyond the narrow limits of possible experience and to guard it against extravagance and error, that the whole philosophy of pure reason has no other than this strictly negative utility [A 711/B739].

W i t h o u t the d o c t r i n e o f p u r e i n t u i t i o n in t he Prize Essay, t h e r e was no g u a r a n - tee tha t r e a s o n in its m a t h e m a t i c a l e m p l o y m e n t was b e i n g k e p t to "a visible t r ack . " O n l y this " f u n d a m e n t a l t r a n s c e n d e n t a l d o c t r i n e " a l l owed K a n t to sub- s tan t ia te his d i s t i n c t i o n b e t w e e n the m e t h o d s o f m a t h e m a t i c s a n d p h i l o s o p h y a n d to e x p l a i n b o t h the ob jec t ive c o n t e n t a n d the n a t u r e o f o u r k n o w l e d g e o f m a t h e m a t i c s . Th is , I h o p e , m a k e s it c l ea r t ha t t h e r e is a subs tan t ia l p h i l o s o p h i - cal ro l e fo r K a n t ' s d o c t r i n e o f i n t u i t i o n o v e r a n d above the logica l ro le w h i c h v a r i o u s c o m m e n t a t o r s h a v e e m p h a s i z e d . '6

McGiU Universit~

161 am very grateful to Charles Parsons, Ahson Laywine, Michael Hallett, Jennifer McRobert and Warren Goldfarb for comments on and/or discussion of various versions of this paper; to an anonymous referee for very helpful remarks; and especxally to Robert DiSalle for making me realize the importance of Kant's Prize Essay. I am also grateful to the Social Sciences and Humani- ties Research Council of Canada for its very generous support of my doctoral and post-doctoral studies

kant on the method of mathematics

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