Abstract
The text aims to reconstruct the genesis of a core diagram in Antoine Culioli's Theory of Predicative and Enunciative Operations: the so-called “cam structure”. For the purpose of a clear and rigorous exposition, the formal aspects of this diagram are examined in relation to some key problems of Culiolian theorization: the generative role of the Lexis with respect to the plane of assertion and, more generally, with respect to the work of enunciation. In the last part of the paper, an attempt is made to connect the main results of Culiolian thinking with some of the most stimulating theoretical works on the epistemological value of the diagram: Gilles Deleuze's heterogenesis and Gilles Châtelet's philosophy of mathematics.
Why speak of the cam? Because it shifts from one plane to another, only to return to the starting plane at one point or another
Antoine Culioli
Just like the Chinese painter painting a reed becomes a reed so as to achieve the right gesture, I had to turn myself into language, to shape myself into a text so that I might be able to capture something that otherwise eluded me.
Antoine Culioli
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Notes
- 1.
Culioli [1].
- 2.
Ivi.
- 3.
Jean-Pierre Desclés, sharing the same opinion, observed that “Culioli s’est montré très léger avec l’introduction de la came”. He also added that “Il faut savoir distinguer d’une part, la notion de catégorie (utilisée par la linguistique à différents niveaux: morphologique, classes syntaxiques, grammatical, lexical) qui peut être formellement approchée et éclaircie par la structure mathématique des types fonctionnels de Church et par les Grammaires Catégorielles, et d’autre part, le concept de catégorie caractérisé par des objets, des morphismes algébriques, des diagrammes commutatifs, des foncteurs entre categories”. Desclés, J.-P. (2019-12-25). Letter to Francesco La Mantia [personal communication]. For my part, however great my appreciation of Mr. Desclés’s arguments, I will not engage here in an examination of the differences he may have noted between the two usages of the word “category”. However, and thanking him for his comment, I propose to return to the issue in a future article.
- 4.
Culioli [1].
- 5.
Desclés, J.-P. (2019-04-09). Letter to Francesco La Mantia [personal communication].
- 6.
Culioli and Normand [4: 73–74].
- 7.
Culioli [1].
- 8.
Lawvere and Schanuel [5: 263].
- 9.
Hazewinkel [3: 308].
- 10.
In what follows, I will not distinguish between “diagrammatisation interne” [internal diagrammatisation] and “diagrammatisation externe” [external diagrammatisation], this distinction pertaining to category theory and exceeding the scope of this work. This being said, however, with regard to the formal distinction between the two types of diagrammatisation, the reader is encouraged to refer to Lawvere and Schanuel [5: 13–60]. A very fine philosophical analysis of the theme of the “diagrammatique” in category theory can be found in Alunni [6: 83–94]. See also Alunni ([7]: 315–393).
- 11.
Victorri, B. (2019-12-20). Letter to Francesco La Mantia [personal communication].
- 12.
I would like thank Jean-Pierre Desclés for kindly donating this image.
- 13.
Coulardeau [8: 9].
- 14.
Ballier [9: 43–72].
- 15.
Ibid., p. 51.
- 16.
Ivi.
- 17.
The cam is not a “3D representation”, but what we could call, echoing Ballier, a “2D-3D representation”. Besides, this can be seen when tracing along its path mentally.
- 18.
Ballier [9: 51].
- 19.
Ivi.
- 20.
Ballier [9: 55]: “The paper tries to account for the fact that French has some uses of il which can be translated as he/it but also has a whole range of possible interpretations. (…) The figure is meant to formalise the ambivalence of the marker il (he/it) and to account for two sets of linguistic observations. (A) In certain cases, il is replaced by ça, especially in cases where ça resumes generic subjects (les chats, ça griffe, cats will scratch). (B) The semantic reference of il range from definite (noted as il1) to non specific (noted as il2). Culioli makes the point in a footnote that il is morphologically masculine but semantically neither masculine nor feminine. The schema circumscribes the range of possible uses of il and ce, showing how some references can be definite (il1) or beyond any specific or generic uses (il2). In Il2 est arrivé trois personnes, the pronoun corresponds to a plural interpretation, for il pleut (it’s raining) the metereological it does not have any specific reference. On top of the illustration, in the inside of the circle, il1 has a unique single reference (not unlike he). Moving to the downside part of the figure comes the interpretation the one that (right), opposite to this interpretation is left hand side this or that. On top of initial il1 comes the so-called impersonal il2 (which in French corresponds to uses of meteorological of it and potential uses where il stands for a formal subject which can be followed by a plural (Il2 est arrivé trois personnes)”. For the impersonal mode in Romanian, cf. Vilkou-Pustovaja [38: 287].
- 21.
Culioli [1].
- 22.
Culioli [11]. In italics in the original.
- 23.
According to Sarah De Vogüé, “le début de la came est là où est a, donc là où est l’affirmation dans le dernier schéma. Lexis (et ā −1) est la fin (ou fin-début, parce qu’elle se projette sur a), en surplomb, intégrant tout le mouvement de a à ā”. De Vogüé, S. (2019-12-11). Letter to Francesco La Mantia [personal communication]. I must confess that I had not considered this aspect of the cam. Indeed, it is assertion, “la forme affirmative du dire” [assertive speech], which inaugurates enunciation, any negative form presupposing affirmative predication. However, if we consider the “potentiels de préparation” [potentials of preparation] of enunciation, that is, all which precedes the act of saying, it is indeed the lexis which is the cam’s starting point. A problem then arises: how to hold these two different readings of Culioli’s diagram together? Since I do not have an immediate answer, I will limit myself to raising the question. In any event, I would like to thank Sarah De Vogüé for her stimulating remarks. This being said, however, I remain persuaded that in what concerns the cam, it is the letter a which represents the germinal activity of the lexis.
- 24.
Culioli [1].
- 25.
Ibid., p. 29.
- 26.
Cf. Freud [12].
- 27.
- 28.
Gray [15: 49].
- 29.
In fact, in a first version of my text, I said that “la came culiolienne a la propriété singulière d’être un cycle” [Culioli's cam has the peculiar property of being a cycle]. If I realised that this was not quite right, it is thanks to Jean-Pierre Desclés. Indeed, according to the Parisian mathematician, “si les extrémités sont identiques, on a affaire à une structure cyclique”, whereas “la structure en came vise à sortir du simple cycle”. (Desclés, J.-P. (2019-12-25). Letter to Francesco La Mantia [personal communication]). Hence the choice of a much more prudent terminology. I also know that this does not suffice. And this, as Desclés remarked in his letter, inasmuch as the cam is not reducible to a spiral “où l’extrémité (…) est complètement différente du début” nor to a cycle as such “où elle se projette sur le début, car on insiste sur l’analogie entre début et extrémité”. And yet, this being said, it remains an unshakable fact that ā −1 is projected onto ā. It is then for this reason that, in spite of Mr. Desclés’s invaluable remarks, I have not cast aside the topological image of the cycle, the latter having been suggested to me by Bernard Victorri. However, not being a mathematician, I am unable to assess whether this is the right choice. Nevertheless, I remain convinced that this is what makes it possible to best appreciate the differences between Gosselin’s cam and Culioli’s. In any case, I would like to thank both Jean-Pierre Desclés and Bernard Victorri: as divergent as their theoretical positions may be, they have helped me understand all the difficulties linked to the complexity of Culioli’s cam.
- 30.
Desclés, J.-P. (2019-12-25). Letter to Francesco La Mantia [personal communication].
- 31.
Desclés, J.-P. (2019-12-25). Letter to Francesco La Mantia [personal communication]. In fact, it is a much more complex remark which I reproduce here in full: “On voit bien que la structure de la came est liée d’un côté, à la relation de repérage par rupture # entre ā −1 et a: [ā −1 # a] et à la relation de repérage par différenciation entre a et ā: [ ā ≠ a] et entre ā −1 et ā: [ ā −1 ≠ ā]. Cette structuration indique le lien entre la structuration en came et la structuration en spirale où ã−1 est en rupture (#) avec a tout en introduisant éventuellement une projection sur a avec [ā −1 = a] en réduisant alors la spirale à n’être qu’une structure cyclique ou à maintenir la rupture d’où la continuation des changements dans la nouvelle boucle de la spirale.” To better understand what is at issue, the reader is asked to not forget the change that I proposed in Paragraph 5 for the position of the letters and which was well received by Desclés.
- 32.
Culioli [1].
- 33.
Deleuze and Guattari [16: 134].
- 34.
Culioli [17: 122]. In this respect, Jean Petitot pointed out to me “que la structure en came ressemble à la monodromie dans les espaces fibrés”. (Petitot, J. (2019-11-05). Letter to Francesco La Mantia [personal communication]). Being unable to develop the formal aspects of this precious remark here, I will limit myself to a few brief reflections. In particular, I will try to explain the monodromic nature of the cam from an intuitive point of view. In short, as written by Pierre Lochak in his Mathématiques et finitude, “l’application de monodromie a à voir avec le transport d’une propriété interne le long d’un lacet [ou d’un cycle, ndr]” (Lochak [18]: 381), the gesture by which the cam is traced is monodromic par excellence! Indeed, the cam diagrammatises the morphogenetic activity of a lexis as a trajectory along a closed path. Now, this representation is close to monodromy inasmuch as the trajectory of the cam corresponds to a particular type of transportation. Of course, I am referring to the circulation of notional virtualities which the shaping of “enunciables” triggers during enunciation, and, more specifically, to the fact that this circulation instantiates an arrow directed from one extremity of the cam to the other. But the affinities with monodromy do not end there. Lochak reminds that the monodromic transportation of properties also supports that they change over the course of the loop (cf. Lochak [18]: 384). Well, this is precisely what Culioli’s diagram represents. Indeed, the local steps of the cam’s trajectory each correspond to a modification of notional virtualities, this modification itself consisting in the transition from enunciables to enunciations, and vice versa. This is why, with each complete cycle across the cam, the trajectory’s starting and arrival points, all the while being identifiable with one another, are nonetheless different. And this, inasmuch as the lexis at the beginning of the enunciation (the “point de départ” [start point]) is retroactively modified by effect of the act of enunciation itself (“point d’arrivée” [end point]). And this, insofar as everything that is said in the course of enunciation operates in turn upon the notional materials of the lexis by modifying them for future enunciative actions. Hence the monodromic character of the cam, or, to quote Petitot once again, “si l’on a (…) des déterminations supplémentaires, alors ā −1 est bien “au dessus” de a mais différent de a (monodromie)”. This being said, I remain convinced that the correct collocation of the letters is as reported in Paragraph § 5.
- 35.
Desclés [19: 14].
- 36.
Le Goffic [20: 86].
- 37.
Ibid., pp. 84–85.
- 38.
On this matter, Sarah De Vogüé seems to be in agreement with me: “Oui, c’est mieux que potentialités, ça marque comment l’énonciation est tiraillée, prise dans le tumulte de la came”. However, she also points out that “pour que ce soit vraiment un tumulte, de l’instabilité, il faut qu’il y ait des positions sur cette came (ce que marquera effectivement la bifurcation, celle-ci ayant perdu cependant d’abord deux puis un point, qu’elle remplace par des chemins)”. Furthermore, she adds that “on n’a pas un continuum, et on se trouve pris soit dans une valeur soit dans l’autre, qui sont comme des trous dans lesquels on tombe: tu es dans l’affirmation, le doute, le rejet, la came t’interdit l’indécision. Le dire est instable, mais il est aussi impliquant”. This last remark deserves attention: contrary to what I have argued, it seems to exclude that the cam be of the order of the continuous. Well, from a certain point of view, this is precisely the case: Culioli’s device diagrammatises enunciative positions which, once constituted, “interdi(sent) l’indécision” [preclude(s) indecision] and therefore the gradualities typical of continuity. But, if this is true, it is also true that an analysis of the processes of constitution of each of them can only be based on a model that is of the order of continuity. And this, for at least two reasons: firstly, because the transitions from one position to another are made through a whole series of notional (or enunciable) virtualities which constitute a field of intermediate values, this field instantiating in turn a continuous space which is by definition “insécable” [unbreakable] [21]; secondly, because the constitution of each position can easily be described as a qualitative singularity (or discontinuity) that unfolds from this notional space.
- 39.
A very fine and stimulating analysis of these states of suspension is provided in Authier-Revuz ([22]: 149–160).
- 40.
Cf. for example Ducard [23], Ducard [24]. In this respect, Jean-Pierre Desclés also insisted upon the necessity of distinguishing between “(1) schème (au sens de Kant = intermédiaire entre le sensible et le concept); (2) représentation sémantico-cognitive SSC; (3) représentation iconique ou figurale en général associé à un SSC; (4) diagramme d’enchaînement d’opérations (comme dans la théorie des catégories de Eilenberg, Mac Lane, Lawvere…) (5) schéma (comme schéma syntaxique = schème qui en représente l’organisation sémantique associée). Les graphes doivent être pris au sens de la “théorie des graphes”. (C. Berge et autres): un graphe est un ensemble d’objets X et un ensemble de flèches F, accompagnés par deux applications l’application I de F dans X (qui détermine l’objet initial de la flèche ou arète) et l’application T de F dans X (qui détermine l’objet terminal de la flèche ou arc orienté). Il y a des rapports entre la théorie de graphes et la “théorie des catégories” de Eilenberg et autres… Les graphes servent à représenter de nombreuses situations où les objets et flèches ont des statuts homogènes. Ils ne doivent pas être confondus avec les diagrammes”. Desclés, J.-P. (2019-12-25). Letter to Francesco La Mantia [personal communication]. I find these kinds of distinctions, like the ones Ducard discusses in his texts, to be quite valuable. Nevertheless, because of the level of generality at which my text is situated, I prefer to analyse them in a future article.
- 41.
Batt [25: 6].
- 42.
Ivi.
- 43.
Ivi. In italics in the original.
- 44.
Châtelet [26].
- 45.
Ivi.
- 46.
- 47.
- 48.
This being said, it is, however, necessary to remark that the gesture by which the cam is traced, as a diagrammatical gesture, “n’est pas un simple déplacement spatial”, see Châtelet [29: 32]. It is, rather, a movement of the body in which one can recognise: (1) a whole historical sedimentation of knowledge, that is, in this particular case, a whole sedimentation of mathematical knowledge, and, more precisely, of categorical knowledge; (2) the regulated exercise of a discipline of the body, this discipline consisting, among other things, in conforming to the actions entailed by the mastery of the knowledge mentioned; (3) the activation of virtualities, these virtualities being unforeseen future movements (e.g., the gesture that traces the bifurcation), which differ from the first gesture. It would be very interesting to see how these three characteristics, which do not exhaust the diagrammatic complexity of the gesture by which the cam is traced, can be generalised, according to Châtelet, to the practice of the gesture per se. And yet, insofar as all this goes beyond the aims of this introduction, I can only refer here to Châtelet [29: 32–40].
- 49.
On this matter, see also the excellent publication by De Vogüé (forthcoming).
- 50.
Ducard [23: 556].
- 51.
Ivi.
- 52.
On this matter, one may refer to the invaluable remarks by Batt [25: 8–24].
- 53.
Piotrowski and Visetti [30: 163].
- 54.
Ivi.
- 55.
Ibid., p. 164.
- 56.
The others being, according to these authors, Gilles Deleuze and Maurice Merleau-Ponty.
- 57.
Piotrowski and Visetti [30: 164].
- 58.
Batt [25: 21].
- 59.
Ivi.
- 60.
On this matter, which I can only briefly mention here, cf. the pioneering work by Dondero and Fontanille [31].
- 61.
Culioli and Normand [4: 74–75].
- 62.
Ivi. The italics are in the original.
- 63.
Jean-Pierre Desclés shared an anecdote with me that seems to have some affinities with what I have just said. I reproduce it here in full: “Après un de ses exposés, où il semblait, selon moi, passer, sans prévenir, de l’extension à l’intension, je lui faisais remarquer cette ambiguïté dans son discours théorique, il m’a répondu “Vous avez raison, mais vous êtes mathématicien et moi je raisonne en linguiste, je dois expliquer l’ambiguïté”. I responded “Justement, pour expliquer l’ambiguïté il faut expliquer ce qui est exprimé en extension et ce qui est exprimé en intension”. He answered: “Là encore, vous avez raison, vous êtes mathématicien mais moi, en tant que linguiste, pour expliquer l’ambiguïté du langage, je dois construire une théorie ambiguë”. Desclés, J.-P. (2019-12-25). Letter to Francesco La Mantia [personal communication].
- 64.
Culioli [32: 367].
- 65.
Culioli [17: 89].
- 66.
To give just a few examples, I will limit myself to recalling that Culioli’s cam structure has explicitly been the subject of reflections in the field of ethno-anthropology (cf. Scubla [33: 58–59]), semio-linguistics (cf. Fogsgaard [34]), and cognitive semiotics (cf. the fundamental/pioneering text by Brandt [35]: 102–119). Furthermore, despite the name of Culioli having never been mentioned, the notion of cam structure often appears in psychoanalytical texts (cf. Portes [36: 103–104]) as well as in mathematical texts (cf. Thom [37: 68–69]). Some semioticians have finally likened this topological object to Thom’s diagram of the cusp (cf. Cadiot and Visetti [38: 217 n. 28]).
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La Mantia, F. (2023). But What About the Cam Structure? Notes for an Enunciative Diagrammatology. In: La Mantia, F., Alunni, C., Zalamea, F. (eds) Diagrams and Gestures. Lecture Notes in Morphogenesis. Springer, Cham. https://doi.org/10.1007/978-3-031-29111-1_13
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