Khalfa - Jean Cavaillès on the Effectiveness of Symbolic Thought

8/9/2019 Khalfa - Jean Cavaillès on the Effectiveness of Symbolic Thought

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Jean Cavaillès on the Effectiveness of Symbolic Thought

JEAN KHALFA

Abstract:The philosopher of Mathematics Jean Cavaillès (1903–1944) plays an importantrole in Claude Imbert’s thought. His published work had a signicant impactafter the war. It is largely a reection on debates on the foundation of mathematics and on two opposed models of axiomatics, foundationalist andconstructionist. The philosophy he announced (cut short by his death duringWorld War II) was to be a study of the generativity of conceptual structures, asopposed to a phenomenology of knowledge. He derived from his reection oninvention in mathematics a great scepticism on the ideas of the separateness andunity of consciousness and a criticism of the teleologies inherent in philosophiesof consciousness. In that, his work, according to Claude Imbert, made possiblethe reections on structures and symbolisms which were to dominate theFrench context in the following decades.

Keywords: philosophy of mathematics, formal systems, epistemology,teleology, Jean Cavaillès

In Claude Imbert’s genealogy of the abandonment of Kantian and neo-Kantian theories of knowledge, logic and symbolism, Jean Cavaillèsholds a prime position. He represents a moment in the late thirtiesin France when, for her, a number of antinomies and false problemswere abandoned, in particular the oppositions of form and intuition,theory and object, concept and experience, and the obsession withabsolute foundations. Because Cavaillès’s life was tragically cut short,in 1944, at the age of forty one, when he was murdered by the Nazis(he was one of the founders and heroes of the French Resistance),1 heleft a work that only announced a transformation in the philosophicalunderstanding of the operations of thought, but one which he did not

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have the time to carry through to a full philosophical theory or a fullyarticulated movement beyond philosophy as did Claude Lévi-Strauss.His publications are mainly a reection on the development of modernmathematics and of the logical systems designed to explain andsupposedly ground their procedures. Nevertheless his work on thephilosophy of mathematics played a signicant role in anticipating thepost-World War II replacement of a consciousness-based philosophicalanalysis of truth and intelligibility with a process-based one, where thestudy of the autonomous generativity of systems of thought, structuresand symbolisms, what he called ‘effective thought’ ( pensée effective )2replaced that of the intractable relationships between consciousness and

its objects.Within the pre-war French tradition where, with the exception of Léon Brunschvicg (1869–1944), questions regarding the foundation of mathematics and mathematical logic were often considered marginalby mathematicians and philosophers alike, Cavaillès was an originalthinker. His work was essentially concerned with the concreteprocedures of the construction of mathematical theories, of whichrecent developments showed the irreducible diversity. It was now clear that their warranty could only come from their own development,which is why he declared himself a Spinozist.3 This was the conclusionof his work on the history of Set Theory and of his edition andcommentary of the correspondence between Cantor and Dedekind.4For Imbert what is interesting here is Cavaillès’s differentiationbetween two types of axiomatic theories, those of Hilbert andDedekind. The rst one is closed and still aims to be compatible withKant’s transcendentalism, while what the correspondence betweenCantor and Dedekind shows is an open system, a thought by naturein movement, (more than simply a historical record), constantlymodifying its axioms, inventing operations and new types of proof,and a new language, an extensional symbolic production, partiallyborrowed from algebra and suited to these new types of axiomatic.Here is of course where Frege’s syntactic invention was important.5

From this perspective, neither the conceptions of knowledge of the philosophies of consciousness (from Kant to Husserl), nor thoseof formalism (Hilbert) and of logicism (from Russell to Carnap)were satisfactory. Philosophies of consciousness tend to reducethe complexities and unpredictability of the inner development of demonstrations to descriptions of forms of judgment and consequentlyoften rely on logical theories that are simplistic when comparedwith what is at stake. The rst and last sections of Cavaillès best


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known philosophical work, Sur la logique et la théorie de la science , arean analysis of the naivety of Kant’s claim to repudiate psychologywhile deriving his philosophy of science from an inherited logic of judgement, and a criticism of the early Husserl’s attempt at groundinglogic in a philosophy of consciousness. He famously concluded:‘A doctrine of science can only be given by a philosophy of theconcept and not a philosophy of consciousness. The generativenecessity [of scientic demonstrations] is not that of an activity butof a dialectics.’6 The aim of a philosophy of the concept is thus tostudy the mechanisms of this generativity, which operates by splittingand overcoming, and this initially by analysing the development of

specic mathematical and scientic theories. On the other hand, apurely formal logical syntax is unable to draw a priori the outlines of real mathematical theories because it misses an important dimension of formal productivity: ‘its progress is necessary and always conditionedby effectivity’. The proper understanding of the notion of a formalsystem, for Cavaillès, implies studying a generation by ‘bursts andsuccessive overcomings’ as opposed to juxtaposing abstract systemsconstructed without relationship to effective processes:7

These historical reversals where the result destroys the method and the wholesystem it derives from, have often been described: the processes that are necessaryto the solution of a problem cause, in the very actualisation that gives thema meaning, a change in outlook such that it is already necessary to abandonthe notions constitutive of their structure. But intellectual linkages go beyondempirical history: their dialectical development causes both their movement andthe permanence of their validity.8

Finally, when formalism is linked to empiricism, as in Logicism, oneis unable to explain the inner relationship of mathematics to physics,or theory to object, other than simply as a choice between abstractsystems based on experience, a contingent choice which recreatesthe problem of the heterogeneity of the object to the theory, whichis precisely what this approach was supposed to overcome (Cavaillès1962, 42; Cavaillès 1981, 169–83).

In his study of the history of Set Theory Cavaillès stresses the factthat metatheories use procedures that do not belong to the theoriesstudied and yet are now of a fully mathematical nature. So, as ClaudeImbert noted:

instead of looking yet again for a fundamental, originary, elementary theory, andan axiomatic categoricity, Cavaillès suggests associating constructivism with its


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metamathematical control and thus reveal the solidarity of all parts of mathematics,so that it can never be said : “here is the simple”. All projects of inventory, of denition, origin, architectonics, but also of the intentional determination of ascience by its object were suddenly abolished. (Imbert 2003b, 7–8)9

The purpose of a philosophy of mathematics was to reect on theprocesses of thought themselves from the inside, in their singularitiesand multiplicity and not to t them into a logic of scientic thoughtaimed at reducing the alleged difference between subject and objectthat had been instituted in the rst place. Thus he wrote aboutIntuitionism, the theory that however complex, mathematical objects

are mental constructs, as opposed to autonomous entities:The necessity of the generation of an object can only ever be grasped throughthe recognition of a success; existence within the thematic eld has meaningonly as a correlate of an effective act. (. . .) The question of the meaning of anoperation as posed by the Intuitionists derives from the prejudice— of a non-critical ontology — that the object must be dened prior to the operation, whileit is inseparable from it. (Cavaillès 1981, 177–8)10

The privilege of philosophy as a foundational theory was thusirremediably abolished, but this could be seen positively rather thanas a loss. Cavaillès had attended the famous Davos disputation betweenCassirer and Heidegger, in 1929, where both speakers started fromthe recognition that Kant’s formulation of transcendentalism couldnot serve as a foundation to contemporary science. Seen from theperspective of this debate his reections on mathematics take on a moregeneral philosophical meaning. Imbert notes that Cavaillès did not takesides but was already struck there by the impasse of both perspectives:Cassirer’s residual neo-Kantism and the transcendentalism still at theheart of his philosophy of symbolic forms as stages in the linear cultural development of awareness, as well as Heidegger’s hermeneuticphenomenology, his ‘ontology of nitude’ and of the situatedness of existence, designed to replace Husserl’s conception of transcendentalphenomenology as rigorous science. Though transcendentalism wasan impasse as far as grounding modern science was concerned,Cavaillès was not drawn to the mystical celebration of nitude andthe ‘existentialism’ of Heidegger’s students. He had studied youthmovements during a prior stay in Germany and looked with ironyas well as worry on their obsession with Kierkegaardian or Pascaliananxiety, on their questioning of the meaningfulness of reason andculture, seen as instruments to disguise existence’s emptiness. Cavaillès


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admired Pascal, but Imbert rightly insists on the epistemological rather than existential dimension of this interest.11 At least, he said, Pascalwas a mathematician. Indeed, he was one who constantly stressed theirreducibility of scientic theories to each other, in opposition to theCartesian paradigm of a Mathesis Universalis.12 So, in Davos, Cavaillèscould see the obsolescence of both paradigms (and incidentally of thesubsequent opposition usually inferred from the disputation, between a‘continental’ and an ‘analytic’ approach to philosophy) and the generalneed to rethink the forms of the production of intelligibility, which heset out to do rst through its most eminent example, contemporarymathematics. Philosophy had to be reinvented by reecting on the

new sources of rationality.13

The urgent task was to grasp processes of intelligence that it does not produce, and to penetrate at the pointof appearance of ‘that upon which it does not have competencebut which it cannot ignore’ (Imbert 2003b, 22). From then on ‘onecould consider the whole work of Cavaillès as the decisive explorationof a new space of thought, revealing a diverse cognitive capacity,one that is never denitively stabilised, as well as calling for ananthropological history’.14 As Imbert wrote of Lévi-Strauss elsewhere,Cavaillès was a representative of a generation ‘conscious that it hadinherited exhausted intellectual procedures it now had to reshape’(Imbert 2004b, 24).15

If it is impossible to predict what Cavaillès’ subsequent philosophicalthoughts might have been, there are nevertheless a few indicationsin his works and echoes with his contemporaries that are worthmentioning. For instance, a constant questioning of the idea of a separate and unied consciousness. Cavaillès replaced it withthe notion of ‘thematic eld’ [champ thématique ],16 permanentlytransformed by the procedures of abstraction that are constantly appliedto each successive object in it:But the sensible, the immediate concrete consciousness, is not abandoned:acting upon it is not departing from it (any abstract object, produced for instance by thematisation, is a gesture on a gesture, (. . .) on a gesture upon theprimitive sensible). The thematic eld is not situated beyond the world, rather it is a transformation of it: the effective thought (requiring a more completeconsciousness) of things is the thought of its own objects (the adequate thoughtof a plurality is the thought of its number). (Cavaillès 1981, 179–80)17

A second constant is the criticism of the teleology seeminglyinherent to the philosophies of consciousness.18 There certainly is aconsciousness of progress but no progress of consciousness: ‘The term


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‘consciousness has no univocity of application— no more than thething has a unity that could be isolated. There is no such thing asa consciousness that generates its products, or is simply immanentto them. It is each time in the immediacy of the idea, lost in itand losing itself in it and only linking with other consciousnesses(which one is tempted to call other moments of consciousness)through the internal links of the ideas to which they belong.’19Claude Imbert has noted that at the time when Cavaillès produced,on the basis of a philosophy of Mathematics, a non-teleogical theoryof consciousness and the generativity of symbolic systems, MarcelMauss did the same from an anthropological point of view, ‘replacing

the process of a philosophical epic of consciousness with an openended anthropological differentiation’ (Imbert 2003b, 24). This in turnopened up the eld to Lévi-Strauss’s study of ‘symbolic productionsand objectied mental maps’ (Imbert 2003b, 24).

She also notes that in 1938, Cavaillès renewed the philosophicalgenre of the essay in the list he created with Raymond Aron andAlbert Lautman for Éditions Hermann, of which the purpose was toproduce ‘essays of a thought inected by its encounter with new formsof knowledge’ (Imbert 2003b, 21).20 Sartre’s Esquisse d’une théorie desémotions was one of the titles published in an endeavour the war wassoon to interrupt.21 This is also when Merleau-Ponty, who was closeto Cavaillès, wrote The Structure of Behaviour , again an essay of a thoughtreecting on and inected by the new sciences.22

Cavaillès was not given the chance to take part in the newphilosophy that was to develop after the war, exploring the notionof structure and transforming our understanding of cognition in itsrelationship to symbolism. Nevertheless, if, as Imbert writes, ‘in these years that immediately preceded the war, philosophical activity hadentered its experimental phase, Cavaillès had initiated its possibility’(Imbert 2003b, 25).23

NOTES

1 See Georges Canguilhem, Vie et mort de Jean Cavaillès (Paris: Allia, 1996);Alya Aglan and Jean-Pierre Azema, Jean Cavaillès, Résistant ou la pensée en actes(Paris: Flammarion, 2002). Cavaillès created several resistance networks, wasarrested by the Vichy police and escaped twice. He wrote Sur la logique et lathéorie de la science in the prison camp of Eyjeaux, near Montpellier, betweenSeptember and December 1942, escaped to London where he met De Gaullein London in 1943. He was nally arrested in August 1943 and shot in


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February 1944. One of the main characters in Jean-Pierre Melville’s 1969lm on the French Resistance, L’Armée des ombres, was inspired by Cavaillès.

2 See for example Jean Cavaillès, Méthode axiomatique et formalisme. Essai sur le problème du fondement des mathématiques [1938] (Paris: Hermann, 1981), 91–2.

3 For the early Spinoza, the idea that before our knowledge can be deemedtrue it must rst demonstrate the certainty of its method, leads to an inniteregression since one would need a method for proving the certainty of the rst method, etc. Stating that working iron requires the possession of hammers is tantamount to saying that men are unable to work iron sincehammers must be made out of iron. In fact better and better tools havebeen developed on the go, starting with whatever material was available innature. ‘So, in like manner, the intellect, by its native strength, makes for itself intellectual instruments, whereby it acquires strength for performing other intellectual operations, and from these operations gets again fresh instruments,or the power of pushing its investigations further, and thus gradually proceedstill it reaches the summit of wisdom [Sic etiam intellectus vi sua nativa facit sibi instrumenta intellectualia, quibus alias vires acquirit ad alia opera intellectualia, et exiis operibus alia instrumenta seu potestatem ulterius investigandi; et sic gradatim pergit,donec sapientiae culmen attingat ]. Spinoza, Tractatus de Intellectus Emendatione , VI,31, in Opera, edited by von C. Gebhardt, vol. II, (Heidelberg: C. WintersUniversitätsverlag, 1972), 14.

This being said, in terms of epistemology, Imbert stresses Pascal’s inuencerather, not just because of his questioning of the classical notions of axiom anddenition, but also because of the irreducibility of his own scientic theoriesto a system of science.

4 Jean Cavaillès, Méthode axiomatique et formalisme. Essai sur le problème du fondement des mathématiques [1937] (Paris: Hermann, 1981). Hereafter (Cavaillès 1981); Philosophie mathématique [1938] (Paris: Hermann, 1962).Hereafter (Cavaillès 1962). Cavaillès’s works have been collected in Œuvrescomplètes de philosophie des sciences, (Paris: Hermann, 1994).

5 Personal communication with Claude Imbert, 10 August 2010.6 ‘Ce n’est pas une philosophie de la conscience mais une philosophie du

concept qui peut donner une doctrine de la science. La nécessité génératricen’est pas celle d’une activité, mais d’une dialectique’ (Jean Cavaillès, Sur la logique et la théorie de la science (Paris: Presses Universitaires de France,

1947), 78). Hereafter (Cavaillès 1947). When Cavaillès refers to dialecticshe has in mind generative processes such as generalisation, formalisation or thematisation, where for instance certain operations are transformed intoelements within a higher-level eld of operations. See (Cavaillès 1981, 177).(Translations are mine unless otherwise indicated.)

7 ‘Son progrès nécessaire est chaque fois conditionné par l’effectif.’ ‘Unegénération par éclatements et dépassements successifs’ (Cavaillès 1947, 35).


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8 ‘La description est bien connue de ces renversements historiques où lerésultat fait éclater la méthode et le système tout entier dont il est issu: lesprocédés exigés par la solution d’un problème provoquent, dans l’actualisationmême qui leur donne un sens, un tel changement d’éclairage qu’il fautdéjà abandonner les notions qui forment leur structure. Mais les liaisonsintellectuelles dépassent l’histoire empirique: c’est leur développementdialectique qui assure à la fois le mouvement de celles-ci et par elles-mêmesla permanence de leur validité’ (Cavaillès 1962, 274).

9 ‘Au lieu de chercher encore une théorie socle, originaire, élémentaire, et unecatégoricité axiomatique, Cavaillès propose d’associer le constructivisme à soncontrôle métamathématique et de faire voir la solidarité de toutes les partiesdes mathématiques entre elles, en sorte qu’on ne peut jamais dire « là est lesimple ». Tout projet d’inventaire, de dénition, d’origine, d’architectoniquemais aussi de détermination intentionnelle d’une science par son objet, s’entrouvait soudainement annulé.’

10 ‘La nécessité de l’engendrement d’un objet n’est jamais saisissable qu’à traversla constatation d’une réussite; l’existence dans le champ thématique n’a desens qu’en tant que corrélat d’un acte effectif. (...) la question du sensd’une opération telle que la posent les intuitionnistes émane du préjugé — d’ontologie non critique— que l’objet doit être déni antérieurement àl’opération, alors qu’il en est inséparable.’ A non-critical ontology is oneasserting the ‘duality of a sensible world in itself and of a thought mistakenfor historical manifestations’ [la dualité d’un monde sensible en soi et d’une pensée confondue avec des manifestations historiques] (Cavaillès 1981, 176).

11 Imbert refers to Cavaillès’s correspondence and his project of working onprobability theory (Imbert 2003b, 16). She has often stressed the importanceof the notion of ‘pari’ or ‘wager’ understood as risk taking, to a different formof philosophical thought. See for instance Imbert 1982 and Imbert 2007b.

12 See Jean Khalfa, ‘Pascal’s Theory of Knowledge’, in The Cambridge Companionto Pascal , edited by Nicholas Hammond (Cambridge: Cambridge UniversityPress, 2003), 122–43.

13 On the history of the Davos disputation, see Peter Eli Gordon, ‘Ernst Cassirer and Martin Heidegger at Davos, 1929 — an allegory of intellectual history’,Modern Intellectual History 1: 2 (2004), 219–48.

14 Personal communication with Claude Imbert, 10 August 2010.

15 ‘consciente d’avoir reçu des procédures intellectuelles épuisées qu’il luiincombait de recongurer’.16 A notion which helps understand Deleuze’s later notion of a ‘plane of

immanence’.17 ‘Mais le sensible, conscience concrète immédiate, n’est pas abandonné: ce

n’est pas le quitter que d’agir sur lui (tout objet abstrait, obtenu, par exemple,par thématisation, est un geste sur un geste, (. . .) sur un geste sur le sensible

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primitif). Le champ thématique n’est donc pas situé hors du monde maisest transformation de celui-ci: la pensée effective (exigeant une conscienceplus complète) des choses est pensée de ses objets (la pensée adéquate d’unepluralité est pensée de son nombre).’

18 For instance Husserl’s idea of a teleology immanent to the history of philosophy. (Cavaillès 1947, 77).

19 ‘Le terme de conscience ne comporte pas d’univocité d’application — pasplus que la chose, d’unité isolable. Il n’y a pas une conscience génératricede ses produits, ou simplement immanente à eux, mais elle est chaque foisdans l’immédiat de l’idée, perdue en elle et se perdant avec elle et ne se liantavec d’autres consciences (ce qu’on serait tenté d’appeler d’autres momentsde la conscience) que par des liens internes des idées auxquelles celles-ciappartiennent’ (Cavaillès 1947, 78).

20 ‘Essais d’une pensée adjectivée par l’effet de nouveaux savoirs.’21 Jean-Paul Sartre, Esquisse d’une théorie des émotions (Paris: Hermann, 1938).22 Maurice Merleau-Ponty, La Structure du comportement (Paris: Presses

Universitaires de France, 1942).23 ‘Si dans ces années de l’immédiat avant-guerre, l’activité philosophique était

entrée dans sa phase expérimentale, Cavaillès avait mis en jeu sa possibilité.’

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Khalfa - Jean Cavaillès on the Effectiveness of Symbolic Thought