Descartes's Mathesis Universalid - F Van de Pitte

7/29/2019 Descartes's Mathesis Universalid - F Van de Pitte

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Descartes' Mathesis Universalisb y F r e d e r i c k P . V a n d e P i t t c ( E d m o n t o n )

I t is a c u r i ou s m a t t e r t h a t an es se ntia l aspect of Descartes' m e t h o dshould he the subjec t of (or play a significant role in) a n u m b e r ofi m p o r t a n t commentar ie s 1 w i t h o u t ever being fully de f ined , or evenproper ly character ized. Indeed, wi th respect to Descartes' mathesisuniversalis, even such a s imple open ing s t a t emen t will b e subject tocr i t ic ism. For on the one h a n d i t has been ma in t a ined t ha t in the strictsense the t e rm ^me thod ' is reserved b y Descartes to refer precisely tot h i s mathesis universalis2 and, thus , th e lat ter cannot in justice b ecalled merely "an essential aspect" of his method . On the other h a n d ,i t h as also been argued tha t the MU was merely a passing phase inDescartes ' search for a method, later replaced b y a more t ruly universalprocedure.3Disagreements am ong com menta tors are p r imar i ly of tw o k inds : The f ir st concernsw h e t h e r t he MU is simply a universa l math emat ics ; or w h e t h e r it is to b e understood asa still broader method, embracing all aspects of know ledge. The second involves whetherth e MU is a method wh ich Descartes actually possessed; or s imply a dream tha t w asnever fulfi lled. Liard considers t he MU to be essentially identical with th e algebra ofsymbols w h i c h Descartes applied so frui tfully to geometry, and t hus a universal mathe-matics ready at hand (p. 591).4 Boutroux sees the MU as an a t t empt at a universalmethod such as the mind might employ if it were dissociated from th e senses andimagina t ion . But t h i s is a mere dream of y o u t h , h e concludes (p . 42), and since such a

1 A complete list of such works would make up a very extensive bibl iography, but thefollowing t it les provide a representative sample: Louis Liard, "La methode et lam a t h e m a t i q u e universelle de Descartes," Revue philosophique de la France et del'etranger, X (1880), pp. 596600. Pierre Boutroux, L'imagination et les mathema-tiques selon Descartes (Paris: Alcan, 1900). Ernst Cassirer, Descartes: Lehre, Persn-lichkeit, Wirkung (Stockholm: Bermann-Fischer, 1939). L. J. Beck: The Method ofDescartes (Oxford: The Clarendon Press, 1952). Jean-Paul Weber, La Constitutiondu texte des Regulae (Paris: Societe d'edition d'enseignement superieur, 1964). Jean-Luc Marion, Sur l'ontologie grise de Descartes (Paris: J. Vrin , 1975).2 Albert G. A. Balz, Descartes and the Modern Mind (Hamden Conn.: Archon Books,1967), p. 338. Also, p. 340: "Mathesis universalis would be the whole of Reason'sscience." Hereafter, except in quota t ions , 'mathesis universalis' will be simply 'MU.'3 Jean-Paul Weber, "La Methode de Descartes d'apres les Regulae," Archives dePhilosophie, XXXV (1972), pp. 53-54.4 Page numbers in this paragraph refer to the works listed in note 1, above.0003-9101/79/0612-0003$2.00Copyright b y Walter de Gruyter & Co. Brought to you by | George Mason UniAuthenticated | 129.174.21.5Download Date | 4/24/13 12:33 A


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Descartes' Mathesis Universalis 155science is impossible, Descartes settled for a make-shift method in which the imaginationplays only a minor role (pp. 34-36). Cassirer has written of the MU in several works,but his essential position is that it is a universal science which only breaks down whenthe attempt ismade to apply it to metaphysics (pp. 39-68). Beck tries to show the MU asa universal science extending beyond mathematics (p. 229), but unfortunately hecon-tinues to equate the term w i t h 'universal mathematics' (the same passage from theRegulae is rendered both ways, pp. 194 and 199),and to insist that Descartes' unif iedscience is dependent upon an extension of the "mathematical method" (pp. 230 and237). Weber concludes that the MU is a universal mathematics which represents anearly stage in Descartes' development, later replaced by a more general method (p. 11) .Marion sees the MU as a universal method which involves an additional level of ab-straction beyond mathematics, thus attaining the "mathematicite non-mathematique desmathematiques" (pp. 62-64). But the method fails because there is "no return" fromt h i s level of abstraction, and the objects of the method (order and measure) belong to them i n d alone (p.69).In part, at least, this disagreement over th e role of the M U inDescartes' work must reduce to a disagreement concerning the properunderstanding of Rule IV in the Regulae ad directionem ingenii, sincethat is the only context in which th e term is used b y Descartes. This isone of those fortunate occasions, therefore, when a careful analysis ofa single passage can potentially provide a defini t ive resolution ofdifferences which persist in Cartesian scholarship. Th e following is anattempt to provide just such an analysis.Th e most careful structural analysis of the Regulae to date h as beendone b y Weber in La Constitution du texte des Regulae. A great deal ofinteresting material wasbrought to light in this work, and the volume hasbeen accepted as an important contribution to our under s tand ing of theRegulae.5 But, unfortunately, Weber's analysis of Rule IV is incorrect,and i t provides a formidable barrier to any ultimate resolution of th econtroversy surrounding Descartes' MU. It will be necessary, there-fore, to point out exactly h ow this interpretation is incorrect, andprovide th e necessary revisions. Th e intended result of th is revisedinterpretation will be to put an end to the long tradition of miscon-ceptions, and consequent inaccurate translations, of Rule IV .Th e essential problem with Weber's interpretation lies in h is attempt to provide def ini t ive statement of Rule IV by splitting what he refers to as t h e traditional versionof the text into two parts: IV-A (AT,X, pp. 368-374, l ine 15); and IV (. \.

5 Weber's structural revisions were incorporated into t h e Philosophische Hibhtnhckedition b y Heinrich Springmeyer: Regular ad directionem mgenii Kegeln rw r t u *richtung der Krkcnntniskrafl, cd. & t r an s . , He in r i c h S p r i n p m cv cr , I inlet (ialn\ .nulHans G nter Zckl (Hamburg: Felix Meiner, 1 9 7 ^ )

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5 > I ' l e d e r i c k . V a n de P i t t eP M4. l ine 1 (t .WO).'1 He is q u i t e cor rec t , of course, in f i nd ing f au l t w i th t h e t r a d i t io na lrsion H u t . u n f o r t u n a t e l y , in d o i n g so h is cri t ic ism foeuses upon t h e wrong i s sue ,r i m . m l v . h e m a i n t a i n s t h a t in I V - w e a re presented w i t h me re l y a universal m a t h e -a t ies . an d t h a t t h i s t ex t must the re fore represen t an earl ier stage in Descartes' develop-e n t a s t age w h i c h W e b e r d i s t i n g u i s h e s from w h a t h e cal ls th e "u n iv e r sa l method"IV .' W h a t W e h e r has no t recognized is t h a t a correct u n d e r s t a n d i n g o f R u l e IVml even o l hi s own d iv i s io n of the t e x t ) is e n t i r e l y de p e nde n t u p o n a proper in te r -e t a t i o n of the t e rm 'muthesis universalis.' In br i e f , th e crucial flaw in h is in te rpre ta t ionis t h a t in cri t ic izing th e t rad i t iona l vers ion of the tex t , he ha s neglected to criticize th et r a d i t i o n a l conception of M U as universal mathemat ics . A care fu l examinat ion of thet ex t reveals several misconceptions in t h i s in t e rp re t a t io n .Almost every s tandard t ransla t ion of the Regulae, whether in toDutch , French , German or English, h as employed th e equivalent of'ma themat ic s ' for the term 'mathesis.' Even worse perhaps, the fewremain ing t ranslations employ 'mathesis' in some passages, and 'ma the-matics ' in others , wi thou t adequate notes to permit th e reader torecognize th e equivocation. The equivalence of these tw o terms istechnically defensible in some cases. But w h en these terms are treatedas equivalent t h r o u gho u t th e entire text of Rule IV, the resultingconfusion is not defensib le. M ore to the point, th e fact tha t Descartesemploys three separate terms in Rule IV (^mathematica(e)? ^mathesis,'and 'mathesis universalis') would much more naturelly lead one tosuppose that he intends to convey three dist inct meanings. An im-portant part of the following analysis will be the at tempt to show thatthese terms are essentially distinct, and that these distinctions must bepreserved if Rule IV is to retain its significance.Th e pr imary th ing to keep in mind is tha t Rule IV is above all adiscussion of method.8 Th erefore, th e significance th at these termsconvey must provide a contribution to the question of method in someform. In addition, since Descartes has already mentioned in Rule IIIt ha t he feels free to employ words in h is own way by referring themback to their original Latin significance, it would surely be wise toconsider the origin of these terms as well .6 All references to the text of Rule IV will be to the "traditional version": (Euvres deDescartes (AT), ed. C. Adam and P. Tannery (Paris: Cerf, 1897-1913), X, pp. 371-379. This set is currently being revised. However, for the sake of uni formi ty ,references to Descartes' wo rks will be to the original edition, since none of thechanges proposed in the new edition will effect the s tudy at hand. For Weber'srecommendations on splitting the text, see Constitution, p. 3.7 Constitution, pp. 7 1 1 .8 This is made clear by its title: "Necessaria est Methodus ad rerum veritatem in-

vestigandam" (AT,X, p. 371).Brought to you by | George Mason UnivAuthenticated | 129.174.21.5Download Date | 4/24/13 12:33 A


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Descartes' Malhcsis Universal is 157The term 'maihesis' is, of course, orig inal ly a Greek word (), referring to theact or process of learning; the acquisition of knowledge. As the word was taken over intoLatin, however, 'maihesis' began to be used more in the sense of '': somethinglearned (i.e., the object of learning), or science. In Descartes' time, the term wascommonly used to mean "mathematical science." This much tends to just ify the standard

translation.But this is only one part of the tradition. Since the excellent studyof Giovanni Crapulli, Mathesis Universalis, genesi di una idea nel XVIsecolo,9 it has become necessary to recognize other aspects of thet radi t ion as well. Of primary importance, th e term 'maihesis' h ad beenacquir ing a technical use in this period that must also b e t aken intoaccount. Tracing the development of the notion of a universal orgeneralized mathematics,10 Crapulli shows that several authors of the16th Century had written works presenting mathesis as just such adiscipline. In at least tw o cases, th e term 'mathesis universalis' h adbeen employed,11 implying the extension of this science to encompassall aspects of mathematical knowledge. At the same time, however,th e conception w as growing that this discipline could b e someth ingmore: a prima mathesis, which (like prima philosophia in relation toother areas of philosophy) could serve as an ult imate foundat ion forall mathematical sciences.12 In other words, rather than simply a9 Roma: Edizioni dell'Ateneo, 1969.10 The roots of this science have been discerned in the works of Aristotle and Eudoxus.See Beck, The Method of Descartes, pp. 200-201; and the Philosophische Bibliothekedition of the Regulae (note 5, above), p. 207, n. 6.11 Crapulli (Mathesis Universalis, p. 209) mentions only Van Roomen ( Adr i an usRomanus, 1561 1615), Apologia pro Archimede (1597). Jean Laporte creditsJ.-P. Weber with the discovery of this source: Le Rationalisme de Descartes (Paris:PUF, 1945), p. 9, n. 7. But Van Roomen used the term at least as early as 1593 inIdeae mathematicae pars prima, sive methodus polygonorum ( A n t w e r p i a e : ApudJoannem Keerbergium). The original Privilegio is dated 1590. In UCEuvre dc

Descartes (Paris: Vrin, 1971), Mme G. Rodis-Lewis also refers (Vol. II, p.501.n. 52) to the work of J. H. Alsted (1588-1638), Methodus admirandorum mathe-maticorum complectens novem libros matheseos universae (Herborn, 1613). Webergives an account of Van Roomen's sources: Constitution, Appendix A, pp. 247-249.Fina l l y , th e long list of distinguished mathematicians an d their contributions, w h i c his provided b y V a n Roomen h imse l f , makes it clear t h a t he is not the or ig ina tor ofthe concept 'mathesis universalis,' but rather one who has coordinated and s y s t e m a -tized a previous t r ad i t ion . See Ideae mathematicae. Dedication to Oavius (first u n -numbered leaf fo l lowing th e t i t le page) an d fol lowing pages.12 Th e Apologia pro Archimede of Van Roomen is not read i ly ava i lab le , b ut Oapi i l l iquotes a passage from this work in w h i c h t h e t e rm 'prima maihcsis' is employed"Jnscribemus au tcm s c i en t i am h a n c nomine primac mathcmatuac, sou / M U M Cmatheseos, ad s i rn i l i t ud inem primae philosophiac" (Maihcsis ( / / / . 2\*

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15S F r e d e r i c k I . V a n clc P i t t cg e n e r a l set of ru les i i n c l procedures , it could become a set of generalpr inc iples in v i r tu e of w h i c h t h e m a t h e m a t i c a l sc iences achieve the i rv a l i d i t y .H o w e v e r , t h i s t r a d i t i o n s t i l l accepted prima mathfsis as a divis ion within t h e scienceof m a t h e m a t i c s V an R o o m e n div ides m a t h e m a t i c s first in to "pura & impura sive mixta."' iv m a t h e m a t i c s is t h e n b r o k e n d o w n in to ''universalis < f c specialty." A n d f inal ly ,u n i v e r s a l m a t h e m a t i c s is div ided i n t o " Logist ic e & prima Mathesis."*3 This es tabl ishestw o t h i n g s c lear ly : First , tha t prima mathesis i s the essen t ia l core of basic principles (andt h u s 'nmthesis' m e a n s h e r e not "mathemat ic s , " b u t " the principles of mathematics");and s econd ly , t h a t t hese principles are f i rmly f ixed within th e discipline of m a t h e m a t i c si tself .Descar tes unders tands th i s t r ad i t ion ve ry wel l ,1 4 since h e defines'mathesis' ( the sc ience , ra ther than its his tory) as :th e ab i l i t y to resolve all prob lems , and moreover to discover b y one's ow n indus t rye v e r y t h i n g t h a t can be discovered b y t h e h u m a n m i n d in t h i s science.15Th us , mathesis is a command of the basic principles which govern theprocess of discovery (and therefore the process of learning) w i th in the

2 1 4 ) . See a lso Van Roomen 's work , Universae mathesis idea, qua mathematicaeuniversim sumptae natura, praestantia, usus et distributio brevissime proponuntur(Herb ipol i : Apud Georgium Fleischmann, 1602). Chapter IV is ent i t led "De pr imamathesi" (p . 20) . Under th is heading: "Principia habe t t an tum propria. Locum inM a t h e s i ob t ine t p r i m u m . Eadem rat ione qua Prima Philosophia inter rel iquas philo-sophicas scientias prima est" (lines 69). Crapulli shows clearly that while there doesnot seem to be a long t rad i t ion involving th e term 'prima mathesis', there w asnonetheless a significant his tory of the concept involved. See Chapter IV: "La 'scientiam a t h e m a t i c a communis ' in analogia alia 'prima philosophia' secondo B. Pereira"(Benito Pereira, 1535-1610), Mathesis Universalis, pp. 93-99.13 Universae mathesis idea, p. 14. In accordance w ith t rad it ion , Logistice ispresented asth e organum scientiae; prima Ma thesis as scientia itself.14 It may be noted th a t wh i le " the t rad it ion" has been me nt ioned several t imes, only VanRoomen's posit ion has actually been presented. The following excerpt fro m the workof Alsted will conf i rm tha t V an Roomen's position was reflected elsewhere: "Caput I.Hexilogia Matheseo>s: Mathesis est pars encyclopaediae philosophicae, tractans dequantitate communiter. . . . Mathesis est generalis, vel specialis. Ilia etiam diciturc o mmu ni s & universal is , haec propria. Ma thesis generalis proponit praecognita mathe-matica, quae praecipiunt de disciplinarum mathematicarum natur & studio . . .."Methodus admirandorum mathematicorum, novem libris e xhibens universam mathesin(Herbornae: Nassaviorum, 1623), pp. 56. All italics are in the original. This is the"secunda editio passim castigata & ornatius elaborata," and the t i t le varies from theedition of 1613 referred to by Mme Rodis-Lewis.15 "In eumdem enim fere sensum duo soleo in Mathesi distinguere: historiam scilicet &scientiam. . . . Per Scientiam ver, perit iam quaestiones omnes resolvendi, atque adeoinveniend i propri industri illud omne quod ab hu ma no ingen io in e scientia potestinvenir i . . .. Letter to Hogelande, 8 February 1640 (AT. XII, Supplement [Indexvolume], p. 2, lines 1016).

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Descartes' Ma thesis Universalis 159m ath em atical disciplines: precisely th e meaning which 'prima mathesis'had come to convey. But , as usual, Descartes did not merely acceptt radi t ional teaching as he found it; he added his own contribution, aswell .

In this case, w h a t Descartes added was the recognition that theaccepted classification of principles within mathematics would stillpermit of (in fact demand) a distinct set of principles which falls out-s ide mathematics , and is logically prior. Tha t is, if prima mathesis isunderstood to be the most basic principles concerning quantity,16 it isstill necessary to consider those principles which alone account forhow i t is that questions involving quanti ty arise in the first place.Descartes' essential divergence from the tradit ion, therefore, concernsthe source of the most basic principles of mathematics . He recognizest h a t if one must account for principles outside mathematics, as wel l asthose internal to the discipline, then the tradition is faced with adilemma: Either prima mathesis is a division of mathemat ics inwhich case it is not self-contained (principia habet tantum propria,note 12, above), since it must relate to prior principles as wel l ; or it isself-contained ("prima" mathesis in the strict sense), and thus outs idemathematics , since it is logically prior to all questions of quanti f ica t ion.This insight prompted Descartes to be critical of the mathemat i c s ofhis own period.17 He felt t ha t i t had not yet at ta ined an awareness ofth e fundamental principles which were required to ground it as ascience. By t ak ing th e concept ''prima mathesis' seriously, and recog-nizing t h a t it must be logically prior to (i.e., outside) mathematics , hew as enabled to do two things: First of all, he was able to provide themissing foundat ion, and give mathematics the s ta tus which i t hadalway s claimed. This in itself ma y be seen as no mean ach ievement , b uth e saw the second as his more s ignificant contrib ut ion to science.Descartes determined th e prior principles required by mathematics to be those oforder and measure, i.e., th e primary relations throug h which any mater ial is brought toth e state of organization and systematizationwhich is essential to science. What is mostimportant here is t ha t the basic elemen ts should display the logically necessary relation-ships which alone can provide the basis for ab solutely certain inferent ial relations as th estructure of the science develops. But although thes e principles provide the essential16 Universae malhesis idea: "Mathemat ics princeps est quac soli qu a n t i t a t u m specu-lationi i n t e n ta est" (p . 14); "prima Mathes is cst quae vcrsatur circa quan t i ta lcmabsolute sumptam" (p . 20).17 Th e precise criticism in Rule IV is t h a t th e mathemat ical disciplines f a i l to demon-strate w h y t he i r propositions arc true (/. c., h ow they arc related to first principles)and how fu r ther t r u th s may be discovered. AT, X, p 375, lines 7 >



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160 iT cderick P. V a n de P i t t cf o u n d a t i o n w h i c h p e r m i t s m a t h e m a t i c s to claim t h e s t a t u s o f science, they a rc none-t h e l e s s not nnitftu to m a t hematics. Rather, as the basis of inference i tself , they arecommon to al l di sc ip l ines . U y focus ing on these principles, therefore, Descartes is ableto t a k e the c r u c i a l step f rom the restrictions of a purely mathematical discipline to ap o t e n t i a l l y u n l i m i t e d f ield of i n q u i r y , i.e., to a universal science as such.1HFrom t h i s perspect ive, it seems very l ikely t h a t Descartes w asor ig ina l ly a t t e m p t i n g to es tab l ish th e m a t h e m a t i c s of h i s period as ascience w h e n h e discovered th e principles of a universal method. W ek n o w t h a t he had a universal mathemat ica l procedure as early as1619. | 4> Me m ay very wel l have been t r y ing to provide th is method withp rope r founda t iona l principles. In any case, h e discovered th e essentialaspects of order: relat ions, proportions, etc., which make it possible toes tablish a discipline as a science, i.e., as certain and indub i tab lek nowl e d g e . 2 0As we have already noted, Descartes determined that in order to become a science,any subject matter under consideration must be able to achieve a certain kind oforganization and systematization. That is, unless we are able to detect patterns, and giveorder and structureto the material being dealt with, it could not be a proper object ofscience. In addition, w h e n this organization can be provided, its elements must be sobasic, and related so rigorously, that one can readily grasp both the truth involved andit s inherent necessity.21 Thus the principles in terms of which a discipline may beconstituted (and recognized as) a science are not merely the formal principles whichprovide validi ty on the basis of definitions, axioms, etc. They are, at the same time, theprinciples which render the subject intelligible. When these principles are once grasped,it is possible both to recognize all the truths of this science that have already beendiscovered (its "history"), and to extend the science through further discoveries.2218 Thus emerges the essential role of mathesis in Rule IV: It is the "bridge" by means ofw h i c h Descartes is able to move outside the confines of mathematics. It should benoted that, because prima mathesis is considered to be outside mathematics, andtherefore unrestricted, Descartes sees it as equivalent to mathesis universalis, and theformer term is never used.19 This is the "scientia penitus nova" of which Descartes writes to Beeckman (26 March

1619): "qua generaliter solvi possint quaestiones omnes, quae in quolibet generequantitatis, tarn continuae quam discretae, possunt proponi." AT, X, pp. 156157.20 Order provides the clarity required of intuition; inferential relations provide the basisfor deduction. In Rule III we are told that intuition and deduction are the most certainavenues to science, and the only two w h i c h ought to be employed. AT, X, p. 370,lines 16-17.21 These criteria will immediately be recognized as referring to the simple natures andnecessary relations of the Regulae.22 Just as in his definition of 'mathesis,' Descartes maintainsin the Regulae that it is themark of having mastered a science to be able to solve any of its problems, and to beable to make firm judgments on the matters to which it pertains. Otherwise we havelearned only its history. See Rule III (AT, X, p. 367, lines 16-13).

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Descartes ' Ma thesis Universal is 161W h a t Descartes learned with respect to mathematics, therefore, heconsidered a truth applicable to all other sciences as well : That th eprinciples of validity of any science are at the same t ime th e principlesof discovery or learning wi th in the science. And since order and

relations provide th e common basis for validity in all th e sciences, t heymu s t also provide th e basis for a common method.23 This is thefoundat ion that our argument requires.. I t is difficult to say whether Descartes began wi th th e originalm e a n i ng of the term '' and gradually worked Out h is ins ightconcerning th e essential nature of scientia, or whetherh e followed th ereverse procedure. In any case, it is clear that h e employs th e word'mathesis' primarily in its Greek sense. H e recognized t h a t mathematicsis (science) or (the "learnable") only in vir tueof th e fact that it is subject to , th e process of learning.24 Sincethis same process is common to all the sciences, it is perfectly na tu ra lt h a t h e should employ th e term 'mathesis universalis' to mean th euniversal principles of learning, or universal method. And the term'mathesis' (when used alone) m ay mean th e under ly ing principlesw h ic h render mathematics a science, or simply th e principles oflearning (i.e., method) or indeed both, since Descartes understandsthese to be the same principles.25W i t h all of this in mind, we can now turn to Rule IV in order to determine its properinterpretation. Since Descartes is clearly aware of the intellectual movements of histime, it would be quite wrong to assume that his proposal in this rule ismerely to provideanother version of generalized mathematics. Throughout the rule (in both I V A andI V B ) , the emphasis is on something new or more precisely, on something as old aslearning itself, but just now set in clear focus and declared to be a project capable offu l f i l lmen t . Because Descartes mentions the MU only in Rule IV, where he iscon-centrating on the essential problem of finding an appropriate method, the real importof his statements should be fairly clear. He is attempting to fulfill the various aspectsof23 It should be noted that the identification of the principles of val id i ty w i th t h eprinciples of in te l l ig ib i l i ty (learning) is the underlying significance of Descartes"assertion that the principles of knowledge may be called first philosophy or meta-physics. See note 53, below.24 Heidegger has recognized this distinction in the context of his work on K a n t , but h asnot given the point itsappropriate application to the work of Descartes. M . Heidegger.

Die Frage nach dem Ding (T bingen: Niemeyer, 1962), . 54. This distinction isu n d o u b t e d l y also what Marion is attempting to express in the phrase "la tnaihr-maticile non-mathematiquc des mathematiques" referred to above.25 That Descartes employs the word 'mathesis' in the sense of "method" becomes moteclear when w e recognize that in R u l e IV , w h i c h is t o t a l l y dedicated to a discussion olmethod, the word 'method' docs not occur at all on pages 374- 379 (i. e., in IV It)In these same pages, 'mathesis' and its v a r i a n t s occur eleven l i m e s



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162 Frederick P. Van de Pitte( M e t r a d i t i o n w h i c h w e have o u t l i ne d . Th e u n iv e r s a l wisdom (sapientia) envisioned inK u l e I is to include all the sciences, and is to he attained b y means of th e universalm e t h o d ( M i l ) ol R u l e IV . 2"But t h e conflict among commentators remains: It is generallya g re e d , on the one h a n d , t h a t th e in ten t ion of Descartes in Rule IV isto show how i t is possible to work toward th e universal wisdom ofRul e I b y m e a n s of a single, universa l method. But t h i s would b eimpossible i f the MU were simply a universal mathematics. Yet, insurpr i s ing contrast, t he p redominan t we igh t o f commentary continuesto main ta in that the MU is precisely that a science of quantification.U n d o u b t e d l y th e primary reason for th is problem i s the fact that th et r ad i t iona l readingof I V B is both incorrect and seriously misleading.W h e n th e text is properly understood, there is no need to see Descartesas offer ing anything less than a truly universal method.The tendency for commentators and translators to equate 'mathesis'and ' ma thema t i c s ' in Rule IV leads them to present us with severalvery strange constructions. For example, when Descartes tells us that,in h is search for a method, he was led from th e consideration ofpar t icular mathematical disciplines (arithmetic and geometry) to ageneral study of mathesis, the latter is interpreted to mean simplymathemat i c s . Because there isa clear distinction between the particularmathemat ica l disciplines and the general field of mathematics, thisin terpre ta t ion has a pr ima facie cogency. But when, in the nextsentence , the terms 'mathesis' and 'mathematicae' are both employed,a similar reduction of both terms to 'mathematics' makes the distortionmore serious. It is normally translated:

I first sought to determine precisely what everyone understands by this term["mathematics"], and why not only the sciences already mentioned [arithmetic andgeometry], but also Astronomy, Music, Optics, Mechanics, and several others are calledparts of mathematics.27Once the two terms 'mathesis' and 'mathematics' are equated, thisis a reasonable translation of the passage; but it completely obliterates26 Paolo Rossi tells us that during this period the ideal of a pansophia dominated allaspects of culture. Clavis Universalis, arti mnemoniche e logica combinatoria da Lulloa Leibniz (MilanoNapol i : R. Ricciardi Editore, 1960), p. 53. In a section onDescartes, Rossi makes it clear that Descartes was caught up in the movement towarda universal science. (Ibid., pp. 153 161).27 ". . . Quaesivi imprimis q u i d n a m practise per illud nomen {Mathesis} omnes in-telligant, & quare non modo jam dictae, sed Astronomia etiam, Musica, Optica,Mechanica, aliaeque complures, Mathematicae partes dicantur" (AT, X, p. 377,

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Descartes' M a thesis Universal is 163th e dist inct ion, and subsequent re lat ion, which Descartes is hereat tempt ing to establish. The passage is intended to say:

I f i rst s o u g h t to de te rm ine prec ise ly w ha t everyone unde r s t ands b y t h i s t e rm[Maihesis], and why not only t hose a l ready ment ioned [ar i t hmet ic and geometry] ,b ut a l so As t ronomy, Music , Optics, Mechanics , and several other mathematicae arecalled parts.28This keeps the distinction clear: Mathesis is the m ast er discipline, thescience of principles, of which various mathematical disciplines (mathe-maticae) are the parts, or applications. This distinction is required byDescartes in order that he can display th e relationship (the order oftheoretical dependence) which exists b etw een th e principles of a majorbranch of science and the individual subdisciplines in which they areapplied to some particular kind of object.With this in mind, w h e n Descartes goes on to investigate the mat te rfur ther , there is no tendency to become confused. But the traditionaltransJation has already gone astray. The problem is increased in thenext passage, which reads (incorrectly ):Here indeed it is not suff ic ient to e xamin e the derivation of the w or d : for since theterm 'mathematics' me an s the same t h in g as 'disciplina,' all the other disciplinesw ould w i t h no less right than geometry i t se l f be called mathematics. Yet there ishardly anyone w i t h the least education who cannot readily dis t inguish among mat terspresented to h i m , w h i c h pertain to mathemat ic s and w h i c h to other disciplines.29Again the distortion makes it difficult to see the precise point. Thepassage s ho uld read:Here indeed it is not suf f ic ien t to examine the derivation of the word: for since theterm 'mathesis' me an s the same t h i n g as 'disciplina,' all these other disc ipl ines wouldw i th no less r igh t t han geometry itself be called mathematicae [/'. e., parts of mathesis].

28 This reading derives support f rom an excel lent source. The version of Adrien Bailletis the only one which we have t ha t comes directly from Descartes' own manuscr ip t .(See the remarks of M A d a m , AT, X, p. 352) . Although it is not a l i te ral t ranslat ion,it clearly could not be a paraphrase of a passage in w h i c h the comma af t e r 'compluresoccurred: "Les pensees qui lui vinren t sur ce sujet , lu i f i r en t abondonner I ' e tude par t i -culiere de F A r i t h m e t i q u e & de laGeometrie, pour sedonn er tout en t ie r la reche rchede cette Science generate, mais vraye & infa i l l ib le , que les Grecs ont nommeejudic ieusement MATHESIS, & dont toutes les Mathematiques ne sont quc desparties" (AT, X, p. 484). It is cur iously inconsis tent for M A d a m to call t h i s passage"une traduction assez fidcle" (ibid., p. 353), and yet leave the c omma in place. Itshould be omit ted f rom the text .29 "Hie en i m vocis or ig inem spectare non suff icit ; n am cum Mathcseos nomcn idemt a n t u m sonet quod d isc ip l ina , n on min or i jure , q u a m G e omc t r ia ipsa , M a t h e m at i c a evocarentur. Atqui v idcmus neminem fe re cssc, si prima t a n t u m scholarum l imina le tgerit, qui non facile d i s t i n g u a t ex ijs q uac oc c ur r un t , q u i d n a m ad M a t h e s i m pc-rlin e a t , & q u i d ad a l i as disc ip l inas" (AT, X, p. 377, l ines 16-22).



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Descar tes ' Mathesis Universalis 165This d is t inc t ion perm its Descartes to go on to the n ext s tep, which isto s h ow t h a t mathesis, th e science of principles applicable to q u a n t i t y ,is itself (in the order of theoret ical dependence) subordinate to primamathesis or MU. T h us , wh e n th e principles wi th in mathemat i c s are

clearly distinguished from their applications, th e basis is establ ishedfor mathesis to serve as the bridge for the s tep outs ide mathematics .Descartes prepares t he way fo r this t ransi t ion b y complet ing th eexplanation of w hy the mathematicae are called parts of mathesis. Butth e translation continues to mislead:Thus, af ter a more at tentive consideration, I f inally determined tha t all and onlythose th ings are rela ted to mathema t i c s in which are examined a kind of order ormeasure, irrespective of w h e t h e r it is in number , f igure , stars, sounds, or any otherobject that such measure is invest igated.32

W h e n th e word 'mathematics' is replaced with 'mathesis,' as it shouldbe, one can see more c lear ly w ha t Descartes is about . Only a disciplinewhose objects are subject to order and measure can be considered amathematica (as , and t h u s ). But in addi t ion toproviding the subject m atter for the process of learnin g, each disciplineis referred back to mathesis as the science of principles (o r primamathesis) in terms of wh i c h it is grounded and achieves its validity. Forboth of these reasons, the various mathematicae are "parts" of mathesis.Stated more s imply, th e various mathematicae share wi th a r i t h m e t i cand geometry the basic dependence upon order and measure . They arerelated to mathesis^ therefore, since it is the science of order andmeasure insofar as these apply to qua n t i t y .This clarification is necessary in order that Descartes can go on tomake the fur ther , and more important dist inction: that is, be tweenorder and measure as t h e y are related to q u a n t i t y ; and, on the o t h e rh a n d , the more basic principles of order and measure, which arelogically prior and fall outs ide mathemat ic . Th e fo rm e r is mathesis, th ela t ter M U . This, therefore, is the precise point at wh ic h Descartes setsup mathesis as the "bridge." But the s tandard t rans la t ion gives usmere ly :And therefore I d e te rmined tha t t h e r e ough t to be a general science which \vouldb e ab le to deal w i th all the ques t ions t h a t can ar ise con cernin g order and measure .w i t h o u t reference to any special subject matter, and called not by some fore ign term.b ut by the t r ad i t iona l and accepted name ' universal mal hemat ics . ' since u would32 "Quod a t t e n t i s cons ideran t i t a n d e m i n n o tu i t , ilia omnia t an t i i m . in qu i b u s mlo \ e lmensu ra ex a mina t u r , ad M at he s im refcrri, nee Interesse u t rum in n u m e n s . vel figuris.

ve l aslris, v el sonis, aliovc quovis objecto, lalis me n sur a q uac ic mk t s i t " ( A I , Vpp. 377, l ine 22-378, l ine 4).Brought to you by | George Mason UnivAuthenticated | 129.174.21.5Download Date | 4/24/13 12:33 AM


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166 l - r e d e r i c k . Van de P i t t ec o n t a i n e v e r y t h i n g in v i r t u e ol w h i c h t h e other sciences a re called parts ofm a t h e m a t i c s . ' *The most i m p o r t a n t prob lem w i t h t h i s read ing is t h a t t h e last linesshould say in s tead :

. 'mathfsix univerxalis,' since it w o u l d c o n t a i n e v e r y t h i n g i n v i r t u e of w h i c h othersciences and mtittiematicae are cal led parts.34From mathesis (as the principles of mathemat ic s ) , there fore , Descarteshas now expanded th e perspective to a universal mathesis whichwould inc lude th e principles of all o t h e r sciences, in addition to thoseof mat h e ma t i c s .3 5Because t h i s issue is so impor tan t , and the passage h as been th esource of confusion for so long, the essential point should perhaps b erepea ted wi th greater precision: W h a t Descartes is presenting here isthe order of theoretical dependence between th e various branches ofm a the m a t i c s , mathesis, a nd M U . The mathematicae deal wi th th ewhol e r ange of physical objects and the i r properties. Mathesis providest h e i r principles. Mathesis deals with th e various aspects of order andmeasure as t hey apply to quan t i t y . M U provides its principles. But the-" " . . Ac proinde generalem quamdam esse debere scientiam, quae id omne explicet,quod circa ordinem & mensuram null i speciali materiae addictam quaeri potest,eamdemque, non ascititio vocabulo, sed jam inveterato atque usu recepto, Mathesim

universalem nominari, quoniam in hac continetur illud omne, propter quod aliaescientiae Mathematicaepartes appellantur" (ibid., p. 378, lines 4 1 1 ) .34 This reading requires only that the ampersand be replaced between the words'scientiae' and 'Mathematicae' to provide: "propter quod aliae scientiae & Mathe-maticae partesappellantur." This ampersand was omitted from the AT edition becauseit was thought to be added erroneously (see note to p. 378, l ine 10). Yet it occurs inboth the Amsterdam edition and the Hanover manuscript. Perhaps even moreimportant, it appears in Baillet's paraphrase: "... puis qu'elle renferme tout ce quipeut faire meriter le nom de Science & de Mathematique particuliere aux autresconnoissances" (AT, X, p. 484). Again, Baillet's is the only version known to havebeen taken from Descartes' original manuscript. The ampersand must be reinstated.35 The only commentator who has explicitly attempted to support this kind of inter-pretation is Jean-Luc Marion, in Sur Vontologie grise de Descartes, pp. 5 5 6 4 . ButMarion has not provided the necessary analysis of mathesis, or of the passage above inwhich Descartes discusses the transition from mathesis to mathesis universalis.Without these essential elements, the interpretation remains inconclusive. After thiswork was completed, Marion's new translation of the Regulae appeared. He still doesnot make several of the distinctions suggested above, but he offers excellentsupporting material against conflating mathesis and mathematics (Annexe II,pp. 302 309), and against any translation at all of the term 'mathesis' (p. 158,line 21-end). See Regies utileset clairespour ladirectionde l'esprit en la recherchede laverite. Traduction selon le lexique cartesien, et annotation conceptuelle par Jean-LucMarion, avec des notes mathematiquesde Pierre Costabel(La Haye: M. Nijhoff , 1977).

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Descartes' Mathesis Universalis 167principles of order and measure which constituteth e M U are the veryprinciples of knowledge and inference as such, and are thus the basisof all possible scientia.

The contrast between this interpretation and the traditional version of Rule IV isquite clear: If mathesis is understood in the narrower, technical sense of "the principlesinternal to mathematics" (or simply "mathematics"), then no universalization of theconcept could extent it beyond the l imit s of a science of quantification. Moreover, sincesuch principles would already be all-pervasive within these l imits , the addition 'univer-s'alis' would be redundant. But if mathesis is understood in its broader sense, "theprinciples of learning," then the addition of the term 'universalis' is not only m e a n ing f u l ,but extremely important. In effect, by restoring the original meaning of the word'mathesis' Descartes is providing a distinctly different, and more firmly groundedsignificance for the established term 'mathesis universalis.' His intention is that the termshould now legitimately denote what it had come by imprecise usage to suggest: theuniversal first principles of knowledge; the key to a universal science.36It must be emphasized once more that Descartes does conceive hismethod to extend beyond mathematics, and to be universal in scope.The confusion which exists on this point isprompted by the fact that hetends to discuss th e method in terms of mathemat ical problems toe nh anc e th e clarity of the procedures of method w h e n they are appliedto utterly simple objects. However, he apologizes for this procedure,stat ing that he is not concerned with ordinary mathematics,b ut qui teanother discipline. Mathematics may be seen as the outer shell of thismethod, but i ts actual content is the primary rudiments of h u m a nreason, and it is intended to extend to any subject matter w h a t e v e r .37In Rule VIII this position is even more explicitly formulated w h e nan example is employed in which it is shown to be wrong to limitoneself to mathematics in attempting to solve problems. The examplegiven is that of t ry ing to f ind the line of refraction in optics (in Dioptricaanaclasticam vacant). On e w ho attempts to achieve this end t h roughmathemat ic s alone, Descartes says, will necessarily fail. For theproblem is not merely a matter of the principles of mathemat ics , b utrather of physics.38 Instead, one must follow Rule I, seekingto discoverth e truth in all matters (de omnibus quae occurrunt veritatem quaercrecupiat).39 Then th e problem, although difficult, can finally be solved.36 It is extremely interesting to realize t h a t in the very passage where Descartes attempting to eliminate an important error of his period ( t h e conflation of mathcsisarid mathematics), commentators have misunderstood h is i n t e n t i o n because t h e y h a v efal len into the same error.37 AT, X, pp. 373, line 30-374, l ine 9. H ibid., pp. 393-394. Ibid., p. 394.



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Descartes' Mathesis Universalis 169This hierarchy is confirmed when Descartes speaks of what the method is toaccomplish. As a procedure,41 it is to extend itself to the eduction of truths wi th respectto any subject matter whatever; and i t is to be the most powerful of h u m a n knowledge,since it is the source of all the others.42 A s a science it is the source of principles; as aprocedure, it is the means by which knowledge is revealed.43 And f inally, Descartestells us that while t h e M U extends to all that w h i c h is dealt wi th b y t h e subordinatesciences, and much more besides, it nonetheless has fewer diff icult ies . The reason isthat, while these other sciences will contain all the difficulties proper to MU, they wil lalso contain th e additional difficult ies consequent upon their ow n particular objectsb ut these latter problems wil l not occur in the MU.44 Clearly, then, the MU is not auniversal science in the sense that it contains the other sciences, since it does not containtheir difficulties.Thus the MU is ent ire ly distinct from Descartes' universal wisdom,and it must b e investigated separately. It is the science which containsth e principles of human knowledge;45 and the procedure b y whichthese principles are applied to the other sciences. This double sensecarried by the term 'MU' helps to explain some of the problems whichcommentators have encountered in attempting to explain this concept.But the concept is neither equivocal, nor ambiguous, and a carefulreading of Rule IV permits us to characterize it very clearly.It is now possible to deal more decisively with the details of Weber'sargument . Th e prima facie cogency which i t offers is immediately lostw h e n w e realize that 'mathesis universalis' does not mean "universalmathematics." The contention that the "universal method"46 of VI-Ais broader in scope than th e mere universal mathematics of IV-becomes totally inappropriate. Obviously, then, there is no longer anyjustif ication for Weber's assertion t ha t th e method of IV-A and the

41 By method, Descartes says, he understands "certain and easy rules" to follow (AT. X.p. 371, lines 25-26). The basic sense of method, therefore, is procedure.42 AT, X, p. 374, lines 8-12.43 This dua l role explains why Descartescan say in Rule VI11 that this method resemblest h a t aspect of the mechanical arts by which they are able to do w i t h o u t outsideassistance, and provide their ow n directions fo r m ak i n g th e i n s t ru m en t s t h e y requi reAT, X, p. 397, lines 4-6. This also clarifies why Descartes can define t h e scienceMathesis as the ability to solve all problems and discover everything t h a t can be k n o w nin thisscience. See note 15, above. This double use of the word 'science* troubles somecommentators. See Descartes: (Euvres philosophiques, ed. F. Alquic (Pans: Ciarn ic iFreres, 1967), vol. II, p. 159, note 5.44 AT, X, p. 378, lines 11-16.45 This phrase, 'the principlesof knowledge/ will be just if ied in t h e concluding discussionof Weber's position.46 Th e term 'universal m e t ho d * is not used b y Descartes in I V- . h u t \Vche i is t j nUrcorrect in m ai n t a i n i n g t h a t Descartes clear ly in tends th e met h o d to lump ^ < > "kno wledge of w h i c h w e are capable. Sec AT, X, p. 372 , l ines 4

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170 Frederick P. Van tie I ' i t teM l ) of IV-B a rc ent i re ly d i f f e r en t . 4 7 In fact , as w e have seen, there isevery reason to see t h e m as iden t ica l .I ;or t h i s reason, it is unnecessary to t a k e up a detai led considerationof W e b e r ' s second an d t h i r d poin t s : T h a t th e universal ma thema t i c scould not be merely a par t icular applicat ion of the method; and (hisu l t i m a t e conclusion) t h a t th e universa l mat h e m a t i cs mu s t b e a more orless r e m o t e prepara t ion for the metho d w h i c h later evolved. But ift hese tw o aspects of the problem can be set aside, th e final point raisedby W e b e r requires more serious consideration.

A t the end of Rule IV, Descartes observes that, being conscious of his own limitations,he has determined to follow an order in his search for knowledge that wi l l always requireh im to begin w i t h what is simplest and easiest, and never permit him to go on untilnothing further seems to be required at this more basic level. That iswhy,he says, he hasspent his time studying this MU to the best of his ability. In this way he hopes that, ashe takes up the somewhat more profound sciences in their proper sequence, he w i l l beable to deal with them accurately and not prematurely. But before making this transition,he w i l l attempt to bring together and set in order whatever in the previous studies seemsworthy of attention.48Weber would like to read two things into this passage: First of all,h e u n d e r s t a n d s Descartes to be turning his attention away fromuniversa l mathemat ic s at thispoint, and toward philosophy.49 Secondly,h e in te rpre ts th is to mean that the same method is not considered validfor both mathematics and philosophy, and consequently, thatDescartes has not yet devised a properly universal method.50 Becauseth i s passage is part of I V , Weber believes it to constitute conclusiveevidence that the universal mathematics cannot be identified with themethod in its final (i.e., universal) form, and therefore, that I V Bmust have been written before I V A .

In part, what has already been said will serve to unravel the threads which are tangledhere. Since the MU is not universal mathematics,but a truly universal method, it can beapplied equally well to both mathematics and philosophy. Again, therefore, Weber's47 "Ainsi, point de doute: la Mathematique universelle englobe toutes les sciencesmathematiques, laisse de cote toutes les autres sciences, est tout autre chose qu'uneMethode universelle." Constitution, p. 8.48 AT, X, pp. 378, line 26-379, line 9.49 "Que signifie ce passage? Evidemment que Descartes est sur le point de Veloigner'de la Mathematique universelle, pour se tourner vers des 'sciences un peu pluselevees' . . . qu'il designe lui-meme ail leurs du nom de 'Philosophie/"Constitution,p. 10.50 "La meme Methode ne vaut pas a lafoispour les Mathematiques etpour laPhilosophie,et, par consequent, // n'y a pas encore de Methode proprement universelle." Ibid.,

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Descartes' Mathesis Universalis 171ma in conten t ion ( tha t IV-B is mere ly a remote preparat ion for IV-A) remains un-supported. But it is even more imp o r t an t to recognize t h a t Descartes is not a t this pointt u rn i n g from mathematics to t ake up philosophy. In fact, it is vital to realize tha t he hasnot in any s ign if icant sense dealt with ma them atics at al l in Rule IV. That i s , ma them at icshas not been the object of his considerationsin this rule . He has referred to mathemat ics(in his search for a method) only because i t i s in that context that , in earl ier ages, th eprocess or learning had been most successful, and the re fore one could most clearlyperceive the basic principles of learning. By this means he was able to characterize theM U as a science distinct from mathemat ics - but the M U is not yet fully defined.W h a t Descartes has been doing in Rule IV becomes more clear inRule VIII. There he gives us the analogy of the blacksmith who wouldset up his trade without having any equipment to w ork w i th . He is t h u sforced at first to usenatural objects - rocks and pieces of wood - asanvil, hammer and tongs . And wi th these, he w ould not a t first a t temptto make objects of trade for the use of others. Instead he w ould beginb y fash ioning proper tools for himself, in order th a t h i s w ork migh t b emore skillfully and efficiently performed.By this example, Descartes tells us, we are taught that since ourini t ial efforts have as yet only discovered certain rough precepts, whichseem to belong rather to the innate capacity of the mind than to anydeveloped technique, we should not at once attempt to settle thecontroversies of philosophy, or solve th e difficulties of mathemat ics .Instead w e must first employ them to dilige ntly search out every th ingelse that is required for the investigation of truth.51 By these remarks,Descartes means simply that w e must make a careful examinat ion ofth e powers of the h um a n m ind , and of the objects to w h i ch t h e y are tobe applied. This is explicitly stated both before and after the passagejust cited.

In Rule IV , therefore, Descartes h as merely dis t inguished h is method as a universalprocess of learning, wi thout s tat ing very much about e i ther th e principles or the processwhich it involves. It is not unti l Rule VIII that he indicates how we are to go abouldef ining matters more clearly: b y determining th e essential principles of the mind i tself .For these alone could serve as the principles of the process of l earn ing , and t h u s t h eprinciples of all science. This point receives i ts most explicit statement in the commentswh i ch Descartes h as included in two of his publ ished w orks : th e Discourse on Method,and The Principles of Philosophy.

In Part II of the Discourse, Descartes says tha t all the rest of th esciences m us t borrow their principles from phi losophy, and t h a t h emus t a t t e m p t , first of all, to es t ab l ish ce r t a in ty t he re (in ph i l o s oph y )AT, X, p. 397, lines 6-23.



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172 I ' l e d e r i c k P . V a n c l c I ' i t t e(h e most i m p o r t a n t t h i n g in th e w o r l d .5 2 But t h e aspect of phi losophyf rom which th e o t h e r sciences must bor row t h e i r principles is firstphilosophy, or metaphys ics . And in the Principles w e are told t h a t firstph i lo sophy , or metaphys ics , "con ta ins th e principles of knowledge."53Th e f i r s t task of any m e t h o d , t he re fo re , must be to establ ish i t s ownf o u n d a t i o n as a science, and ( s imul t aneous ly ) the foundation of al lt h a t it can reveal , b y d e t e r m i n i n g th e principles of knowledge uponw h i c h th e w h o l e project depends.l ;roiii t h i s perspect ive, th e error of Weber's in terpre ta t ion becomesperfec t ly clear. For, r a t h e r t h an t u rn ing from mathemat ics to philosophyat the end of Rule IV , w e see t h a t Descartes is simply employing thataspect of the method which he has already isolated (the analysisof theanc ien t s ) in an effort to disclose th e MU as a science: th e principlesofh u m a n cogni t ion . He is tu rn ing from h is consideration of method as aprocess to a more profound examina t ion of it roots as a science inmetaphys ics .The r ema in ing elementsof Weber's interpretation can be dealt wi thbr ief ly . There would b e little point in entering into a detailed debateconcerning the precise date to be assigned to the portion of Rule IVw h i c h h e calls I V B . 5 4 But it can be readily seen that there is littlebasis for the contention that it is an earlier draf t of I V A . In fact, ifI V A and I V B are really tw o versions of the same basic statementif, t h a t is, w e are forced to choose between them then there can be noquest ion of which is to be retained. I V B is a much more detailedstatement of how Descartes' method was first discovered, and how itis to be understood.In fact , all of Weber's evidence concerning th e separate status ofI V B 5 5 works against the interpretation he offers. No trace remainsof Descartes' original manuscript of the Regulae, and we cannot evenbe certain t h a t the versionswe do have, i. e., the Amsterdam edition of1701, and the Hanover manuscript, were taken directly from the52 AT, VI, pp. 21, line 30-22, line 4.53 Preface to the French edition of the Principles of Philosophy (AT, IX, part 2, p. 14).The point isrepeated again when he speaks of the Principles as being divided into fourparts, "the first of which contains the principles of knowledge, which is what may becalled first philosophy or metaphysics" (ibid., p. 16).54 Weber's argument concludes: " I V B a etc, presque certainement, redige entremi-octobre et debut novembre 1619" (Constitution, p. 17).55 Primarily concerning the fact that in the Hanover manuscript it is relegated to the

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Descartes' Mathcsis Universalis 173original .56 Therefore we do not know for w h a t reason the copyist ofth e Hanover manuscr ip t set IV-Bat the end of his transcription.Perhaps i t w as wri t ten on different paper, or in a different ink . In somem a n n e r it must have been set off as distinct . But while W eber sees thisas evidence for an earlier origin, it seems more likely to serve as proofof a later date. One does not tend to retain a single passage from anearlier draft of a work in progress. Whatever is essential in an earlierdraft is incorporated directly into th e new . Wh en tha t h as been done,the earlier version is no longer worth preserving. Such an isolatedpassage as Weber assumes IV-B to have b een would far more likelybe a revision, an improvement on the earlier work, resulting from aparticular insigh t . Because of the extremely compressed manner inwhich IV-Bis expressed, one might even see it as a set of noteshurriedly jotted down as the basis for a later, more thoroug h presen-tation of the material. But this perspective, too, is simply an opiniondeveloped through long association with th e t ex t ; and like t h a t ofWeber, it is unsupported by any objective evidence. The best inter-pretat ion h ere, sure ly, is to see IV-A and IV-B not as a duplicationof th e same text, but as complem entary aspects of the same material.57

Much work is yet required if we are ever to fully define the precise nature ofDescartes' MU. But surely that task is w o r t hy of separate treatment. In the presentcontext it is sufficient to eliminate those misconceptions which could prevent theacceptance of such clarification w h e n it occurs. On th i s poin t Weber is ent i re ly correct:It would be impossible to properly evaluate and explain the essential nature of Descartes'method if it is merely considered to be a universal mathematics. Such a method could notbegin to perform the task that Descartes has set for himself. But a universal science oflearning, w h e n applied to the various objects of knowledge, could eventual ly providethe universal wisdom of w hi c h he speaks in Rule I.56 M Adam maintains that the Latin manuscript employed for the 1701 edition wasundoubtedly the same used as the basis for the "traduction flamande" offered byGlazemaker in 1684.He also feels that there is sufficient reason to believe the 1701edition a fa i th ful rendering of Descartes' original manuscript. But his position isbasedentirely on circumstantial evidence. See AT, X, pp. 353-356. For an evaluat ion olthe re lat ive merits of these various sources, see H. Springmeyer, "Eine neue kr i t i scheTextausgabe der 'Regulae ad direc t ionem ingeni i ' von Rene Descartes." in /.cii-schrift fr philosophische Forschung, XXIV (1970), pp. 101 - 125. Crapull i providesasomewhat d i f f e r e n t perspective in his In t roduct ion to the L a t i n - D u t c h e d i t i o n :

Regulac ad Directionem Ingenii, texte critique etabli par Giovanni Crapulli. am laversion hollandaise du XVII' siede (La Hayc: M. Nijhof f , 1 9 ( > f > ) .57This is the perspective adopted by Marion (Sur I'oniolngic gnsc dc / V . \ < Y / r f r > ,pp. 55-59).

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174 l r e d c r i c k l . V an de T i t t eHowever , because th e M U is a science of principles, as well as aprocedure for discovery, w e are left w i t h several unanswered quest ions.Precisely w h a t are th e princ iples of cognition t h a t provide an adequatef o u n da t io n for the universa l m e t h od ? A nd jus t how a re we to und e r -

s t and t h e m a n n e r in w h i c h th e procedure is g round e d in them? Surely,from Descartes' conf ident m a n n e r of wri t ing , w e must assume tha t h ebelieves himself to be in possession of this science. But , unfor tuna te ly ,in th e Rcgulae w e find only a presentat ion of some of the proceduresw h i c h these principles entail , rather than a statement of the principlesthemselves . We are left merely w i t h the hope that a more carefulanalysis of the se procedures may yet reveal th eir ult imate foundat ion .5 8A deb t of gra t i tude must b e acknowledged to Prof. Dr. H. Wagner for his commentson an ear l ie r draf t of t h i s work , and part icularly for the phrase 'the theoretical orderofdependence, ' which catches so precisely the essential relationship involved inDescartes ' thought .

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