Diagrams and Gestures: Mathematics, Philosophy, and Linguistics (Lecture Notes in Morphogenesis) 3031291107, 9783031291104

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Lecture Notes in Morphogenesis Series Editor: Alessandro Sarti

Francesco La Mantia Charles Alunni Fernando Zalamea   Editors

Diagrams and Gestures Mathematics, Philosophy, and Linguistics

Lecture Notes in Morphogenesis Series Editor Alessandro Sarti, CAMS Center for Mathematics, CNRS-EHESS, Paris, France Advisory Editors Henri Berestycki, Ecole des hautes etudes en sciences sociales, Paris, France Paul Bourgine, Ecole Polytechnique, Palaiseau, France Giovanna Citti, Dipartimento di Matematica, Università di Bologna, Bologna, Bologna, Italy Paolo Fabbri, Universitá di Venezia, Venzia, Italy Vincenzo Fano, Department of Basic Sciences and Foundations, Urbino University, Urbino, Italy Sara Franceschelli, ENS, Lyon, France Maurizio Gribaudi, EHESS, Paris, France Annick Lesne, LPTMC UMR 7600, CNRS UMR 7600 case courrier 121, Paris cedex 05, France Giuseppe Longo, Centre Cavaillès, Ecole Normale Supérieure, Paris, France Thomas Lorenz, Efiport GmbH, Frankfurt School Verlag, Frankfurt am Main, Hessen, Germany Jean-Pierre Nadal, Service Facturier, EHESS, Paris, France Nadine Peyriéras, Institut de Neurobiologie Alfred Fessard, CNRS-N&D, Gif sur Yvette, France Jean Petitot, EHESS, Paris, France Jan-Philip Schmidt, University of Heidelberg, Heidelberg, Germany Carlos Sonnenschein, Department of Integrated Physio and Pathobiology, Tufts University School of Medicine, Boston, MA, USA Ana Soto, Department of IPP, Tufts University, Boston, MA, USA Federico Vercellone, University of Turin, Turin, Italy

Lecture Notes in Morphogenesis is an interdisciplinary book series which aims to face the questions of emergence, individuation and becoming of forms from several different points of view: those of pure and applied mathematics, of computational algorithms, of biology, of neurophysiology, of cognitive and social structures. The set of questions above concerns all the manifestations of Being, all the manifestations of Life. At the heart of contemporary embryogenesis lies an essential question: How can form emerge from the constant, chaotic flow? How can a sequence of purely informational elements — an a-signifying combination of chemical substances organized in the DNA molecule — evolve into the highly complex and structured forms of the living organism? A similar question can be asked when we deal with the morphogenesis of vision in neural systems and with the creation of evolving synthetic images, since digital technology makes possible the simulation of emergent processes both of living bodies and of visual forms. Finally the very idea that abstract structures of meaning could be captured in terms of morphodynamic evolution opens the door to new models of semiolinguistics, semiotic morphodynamics, and cognitive grammars. An entire heritage of ideas and concepts has to be reconsidered in order to face new and challenging problems: the theoretical framework opened by Goethe with the introduction of the word “Morphogenesis” is developed by D’Arcy Thompson in “On Growth and Form”, it is reorganized with new theoretical insights by the classical structuralism of Levi-Strauss and formalized by the dynamical structuralism of René Thom. The introduction of the post-structuralists ideas of individuation (in Gilbert Simondon and Gilles Deleuze) and plasticity of structures builds a bridge to contemporary problems of morphogenesis at a physical, biological, social and transindividual level. The objective of this book series is to provide suitable theoretical and practical tools for describing evolutionary phenomena at the level of Free boundary problems in Mathematics, Embryogenesis, Image Evolution in Visual Perception, Visual Models of Morphogenesis, Neuromathematics, Autonomy and Self-Organization, Morphogenetic Emergence and Individuation, Theoretical Biology, Cognitive Morphodynamics, Cities Evolution, Semiotics, Subjectivation processes, Social movements as well as new frontiers of Aesthetics. To submit a proposal or request further information, please use the PDF Proposal Form or contact directly: Dr. Thomas Ditzinger ([email protected])

Francesco La Mantia · Charles Alunni · Fernando Zalamea Editors

Diagrams and Gestures Mathematics, Philosophy, and Linguistics

Editors Francesco La Mantia Department of Humanities Università di Palermo Palermo, Italy

Charles Alunni Centre d’Archives en Philosophie Histoire et Édition des Sciences Ecole Normale Supérieure Paris, France

Fernando Zalamea Departamento de Matemáticas Universidad Nacional de Colombia Bogotá, Colombia

ISSN 2195-1934 ISSN 2195-1942 (electronic) Lecture Notes in Morphogenesis ISBN 978-3-031-29110-4 ISBN 978-3-031-29111-1 (eBook) https://doi.org/10.1007/978-3-031-29111-1 © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Vilém Flusser (1920–1991) To Per Aage Brandt (1944–2021) In memoriam

Foreword

Francesco La Mantia begins his comprehensive, theoretically sophisticated editorial introduction to the Diagram-Gesture pair by way of an exposition of the two concepts as separate entities: gestures as certain movements of bodies and diagrams as oscillations between a graphic and symbolic object. He then explores, one might say urges, the idea of their interconnection, hypothesizing how they might be ultimately inseparable, polar instances of a single concept he dubs “diagrammatic gesture”, the justification for which he locates in the influential theory of Gilles Châtelet on gesture’s role in the creation of mathematical operations. According to Châtelet, whose writings can be considered an enabling background or proof of concept of the very idea of pairing the two concepts, diagrams are not independent of gestures, but make their appearance as schematic objects that “freeze” gestures mid-flight and “cut out” new ones; creating from their gestural inputs something new by mobilizing the gestures that exist in an implicit or virtual state “inside” them. A similar, but informal and spontaneous, creation of novelty, an “archeological version of it” (13), that appears to support the phenomenon of a diagram cutting out a new gesture, occurs when children, engaged in drawing diagrams, react to their own constructions by recognizing unintended shapes that surprise them. La Mantia characterizes a diagram as a dynamic object, an oscillation between two poles: a graphic “figure”, a configuration of lines and shapes inscribed on a surface, and a symbolic expression or “formula”, which operates through an “interactive relationship which allows to transit from the figure to the formula, and vice-versa” (6). Charles Alunni in his gnomically entitled “Diagram: Demon of Proof”, approaching the diagram from an “Imaginative” (156) rather than functional/ technical perspective, and prefacing his essay with a display of schemas of diagrammatic mathematical proofs, asks What role does the diagram play in a diagrammatic demonstration, one that proceeds by evidencing or showing that is constructed from a series of operations on the subparts of the diagram? After establishing various features of diagrams which include writing—the gram—precedes the diagrammatic act; diagrams act like writing machines; diagrams manifest the dialectical structure of the gram, he concludes that the diagram in a demonstrative proof acts to “orient” the evidence, guiding it—as if by a demon—toward a proof, a demonstration. vii

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One can also view diagrams through a semiotic lens. According to Charles Sanders Peirce, whose phenomenological categories of (firstness/secondness/thirdness) and classification of signs are taken as theoretical givens by a number of contributors here, a diagram is an epistemically useful sign, one whose sign-body has an aspect that bears a similarity to its object: a diagram “represents parts of a thing by analogous relations to its own parts” (reference A), and he understood diagrams as icons, placing them between pictures and metaphors. The two definitions are complimentary but not equivalent: the first phenomenological and symmetric foregrounding dynamic effect as a “fluctuation between polarities” (8), and the second pragmatic and directional from a sign’s body to its object, foregrounding its use. Aage Brandt adds a cognitive dimension to Peirce’s definition. “Diagrams represent the way our inner eye apperceives thoughts” (271); their graphic components of arrows, strings, boxes, and so on—linked to mental operations of movement, connection, and the like—making them appropriate instruments for figuring “how things work” (273). A gesture can be in analogical relation to its object; for example, a back and forth movement of the hand signifying an oscillation. Such iconic gestures can be seen as virtual diagrams, yet to be made actual through the construction of a mapping of them onto a two-dimensional surface. The classical mathematical diagrams of a point, a line, a tally mark, and others would appear to have originated as actualizations from gestural predecessors in this way. Of course, not all mathematical diagrams are related to gestures iconic or otherwise. They can be a gestural in the sense that their parts are abstract, making no reference to gestures. Such is the case for the two contrasting classes of diagrams—continuous (catastrophe theory) and discrete (cellular automata)—exposited by Wolfgang Wildgen who in his semiotic analysis chooses to focus on the “rational aspects of diagram construction … ignoring bodily rooted and manifested diagrams” (299). An assemblage of iconic gestures, 54 in all, is at the heart of Fernando Zalamea’s profoundly simple diagrammatic answer to the question: What is mathematics? The gestures, hand-finger configurations, have been crystalized from 9 body movements, each capturing some aspect of a mathematical activity involving a particular dialectic, such as discrete/continuous, and so on, spanning mathematics and organized in 3 triads in accordance with Peirce’s phenomenological categories. Further, each is extended in 6 different mathematically significant ways making 54 iconic gestures of mathematics. The result is a virtual, yet to be actualized, diagram of mathematics, offering a corporeal purchase on mathematical thought, creating a “gestural blowup … in which the body is expanded through gestures in order to enact conceptual thinking” (149) that reveals the essential carnality of mathematical thought. In contrast to simple iconic gestures capturing mathematics, there are mathematical diagrams capturing simple gestures. For Guerino Mazzola, the gestures are musical, and accepting that in music “time is essentially the result of the composers’ and musicians’ constructive activity” (60), he asks how assemblages of them combine to create the experience of musical time. In particular how “vertical” time, the temporal musical experience of an extended “now”, might be created. He proposes representing musical gestures as arrows, morphisms in a category, so that assemblages of them might be captured by diagrams from the rich diagrammatic resources

Foreword

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of category theory; a theory which internalizes the figure/formula oscillation of the diagram through its built-in equivalence of equations and commuting squares. Jean-Claude Dumoncel proposes to illustrate the pair (gesture, diagram) in Proust’s Remembrance of Time Passed through an interplay of two “cycles” of gestures: the diagram of Gilberte’s moves along the paths of Combray; and the history of Albertine. While the first is quickly narrated using some simple diagrams, the second involves Dumoncel in a complex engagement with modal logic, “a system which pervades the total extension of the novel and of its Universe” (113); an engagement that embraces Leibniz’s concept of a best possible world, Proust’s invention, before its mid-century formulation, of “hybrid logic”, a conjuration of Albertine as a Boolean triple of sensory predicates, Deleuze’s paradoxical suggestion that Albertine “is” a possible world, and much else. Peirce’s interest in the power of diagrams as an epistemic aid to thought led him to invent a species of diagram, “existential graphs”, that could function as the means of ratiocination, specifically as a gestural form of symbolic logical reasoning. Arnold Oostra, for whom the graphs are an “outstanding use of diagrams in mathematics” (3), illuminates how Peirce’s graphs translate the rules of inference of first-order and modal logic from their standard formulation—an abstract manipulation of symbols— to a gestural handling of diagrammatic figures. Giovanni Maddalena, likewise appreciating the power of Peirce’s graphs, extracts their vital principle, namely reasoning by doing; and his project is to show how the principle can be extended from demonstrating purely logical reasoning to demonstrating synthetic truths of the form A = B. This requires him to construct a post-Kantian notion of “synthetic”, as well as to elaborate a concept of continuity through change, enabling him to construct a theory of reasoning by gesture “that respects all the characters we find in Peirce’s logic of existential graphs” (176). In his phenomenological studies, Villem Flusser defined gesture as “a movement of the body or a tool attached to the body for which there is no satisfactory causal explanation” (reference B). The editors have included an essay of his devoted to alphabetic writing, the tool in question being the stylus. He patiently elaborates how phenomenologically onerous as well as neurologically demanding it is to carry out the extended one-dimensional sequences of gestures necessary to “materialize one’s thoughts by scratching lines on surfaces” (214) and convert our “whispered” thoughts into the written word-sentence diagrams of alphabetic inscription, and he concludes that the process (little changed from its cuneiform forebear) which has served as the organizational medium of western culture, confronted with the mediational demands of contemporary (circa 1990) visual and algorithmic technologies, is about to fall into desuetude. A diagram can use the analogical or oscillatory capacity of another diagram as part of itself. A strikingly intricate example emerges from Rocco Gangle’s account of the gestures of philosophical friendship in Plato’s Meno, which recounts Socrates demonstrating to Meno’s slave a geometrical fact involving diagrams while at the same time demonstrating a metaphysical truth about the nature of the soul to Meno. By augmenting Peirce’s triadic definition of a sign with a fourth figure of motivation, Gangle is able to represent the “self-participating multi-scale nature of diagrammatic

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reasoning and its role in … exemplifying the relation of friendship (philia) in the Socratic image of Plato” (198) as a complex diagram, one that encloses a diagram of Socrates’ lesson concerning a geometrical diagram. Three contributors to the collection pursue the pairing of gestures and diagrams at play within Culioli’s linguistic theory of predicative enunciation. Introduced here by Dominique Ducard, the theory understands language “as the product of symbolic activity through gestures, according to a process that transmutes an interiorized sensorimotricity into mental representations constituting diagrams of the movement of thought” (223) that are “traceable in linguistic forms” (225). In relation to such mental movement, Lionel Dufaye offers a meditation on the puzzle of how such static, fixed-on-paper, diagrams are able to capture language’s dynamic character, its “diachronic evolution and semantic plasticity and impermanence”? (233). For Francesco La Mantia, the nature of the diagrams themselves needs scrutiny. In particular, he asks, what does Culioli intend by his famous cam diagram? What is its mode d’emploi? What are the rules relating it to the form of mental movement it diagrammatizes? After suggesting many possible answers to his question, which he offers as notes to a diagrammatology of the entire field, he concludes “without an adequate explanation of the formal rules characterizing the cam, Culioli’s curved morphologies remain but a ‘toy’” (265). The editors have put together a rich, unfailingly interesting, and provocative collection of essays involving pairings of a wide range of gestures and diagrams that organize, illuminate, represent, or utilize them within mathematical, philosophical, linguistic, and semiotic projects. Whether, to return to the introduction, they offer any support to a Châtelet-inspired gesture-diagram fusion is for the reader to decide. What is not in doubt is a certain affinity, a kind of operational linkage between the two concepts, that is evidently productive of swathe of scholarly activity. Brian Rotman Emeritus Professor at the Department of Comparative Studies of the Ohio State University Columbus, USA

Acknowledgements The editors of this volume would like to warmly thank Maria Rita Abramo, Antonino Bondì, Jean-Pierre Desclés, Sophie Fisher, Maria Giulia Dondero, Franco Lo Piparo, Rosa Maria Lupo, Claudio Paolucci, David Piotrowski, Cecilia Rofena, Brian Rotman, Salvatore Tedesco, Sebastiano Vecchio, Bernard Victorri, and Yves-Marie Visetti for reading and commenting on earlier versions of this work.

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References The Collected Papers of Charles Sanders Peirce, Harvard U. Press, CP2, Elements of Logic, p. 274 Vilem Flusser, “Gestures”, translated Nancy Ann Roth, U. of Minnesota Press, (2014)

Contents

1

Almost an Introduction. From the Basilar Notions to the Legacy of Gilles Châtelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Francesco La Mantia

Part I

1

Diagrams and Gestures: Mathematics

2

The Gestural Construction of Musical Time . . . . . . . . . . . . . . . . . . . . . Guerino Mazzola

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3

Gilberte’s Gesture and the Commutative Combray . . . . . . . . . . . . . . . Jean-Claude Dumoncel

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4

Existential Graphs as an Outstanding Case of the Use of Diagrams in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Arnold Oostra

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54 Gestures on Higher Mathematics, and Their Use for a Diagrammatic Approach to the Question “What Is Mathematics” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Fernando Zalamea

Part II

Diagrams and Gestures: Philosophy

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The Diagram: Demon of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Charles Alunni

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Gesture, a New Tool for a Different Vision of Synthetic Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Giovanni Maddalena

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Diagrammatic Gestures of Friendship in Plato’s Meno . . . . . . . . . . . . 207 Rocco Gangle

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The Act of Writing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Vilém Flusser and Charles Alunni xiii

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10 The Diagram on Stage: Movement, Gesture and Writing . . . . . . . . . . 243 Catherine Paoletti Part III Diagrams and Gestures: Linguistics and Semiotics 11 Meta-Morphosis: Kinesis and Semiosis in Language Concerning a Theory of Enunciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Dominique Ducard 12 Fluid Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Lionel Dufaye 13 But What About the Cam Structure? Notes for an Enunciative Diagrammatology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Francesco La Mantia 14 Diagrams, Gestures, and Meaning. A Cognitive-Semiotic View . . . . 319 Per Aage Brandt 15 Continuous, Discrete Diagrams and Transitions. Applications in the Study of Language and Other Symbolic Forms . . . . . . . . . . . . . 331 Wolfgang Wildgen 16 Conclusions: As a Kleiner Narr in Trance Towards a Diagrammatic Model of Enunciation . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Francesco La Mantia Appendix A: Diagrammatic Eidos and Dynamical Platonism . . . . . . . . . . 407 Appendix B: The Illusions of the Speaking Subject. A Note on the Lapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Chapter 1

Almost an Introduction. From the Basilar Notions to the Legacy of Gilles Châtelet Francesco La Mantia

Come on, my reader, and let us construct a diagram to illustrate the general course of thought. (C. S. Peirce) Wie unterscheidet sich eine Geste von irgend einer anderen Bewegung? Dadurch daß sie etw. ausdrückt? (L. Wittgenstein)

1.1 Diagrams and Gestures: Preliminary Remarks In recent years, the theoretical investigation about the notions of “diagram” and “gesture” has become increasingly popular and widespread.1 ,2 Fields like mathematics, semiotics and linguistics have been particularly attentive to the ways in which such notions are used and codified.3 Philosophers and scholars of various tendencies have devoted many efforts to investigating the variety of meanings, theoretical implications and practical results 1

Peirce [1, p. 80]. “How does a gesture differ from some other movement? In that it expresses something?–” 3 As to Mathematics, see at least Clemence [2, pp. 409–415]; as to Semiotics, see at least Everhart 2018; as to Linguistics, see at least [3, p. 184], as to Psychoanalysis, see [4, pp. 133–200]. 2

Although the empirical author of the essay is Francesco La Mantia, the contents exposed are the result of shared and fruitful conversations between all three editors. This is why the writer wishes to express his gratitude to Charles Alunni, who has discussed and examined in detail all the paragraphs of the introduction, and to Fernando Zalamea, who has carefully re-read the text. Special thanks go to Maria Giulia Dondero who has been an attentive and valuable reader. Of course, La Mantia is solely responsible for any error. F. La Mantia (B) Dipartimento Di Scienze Umanistiche, Università Degli Studi Di Palermo, Palermo, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_1

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F. La Mantia

underlying it.4 In any way, we owe one of the most important analysis of such a conceptual pair to French mathematician and philosopher of mathematics Gilles Châtelet.5 In his theoretical work, the two notions reach such a level of reciprocal interpenetration that it is impossible to uncouple them (cf. [8]: 62). In this sense, we find the locution of “diagrammatic gesture” to be particularly appropriate. Indeed, such an expression6 perfectly captures the degree of coalescence between the notions. Before investigating this theoretical nexus in more details, we prefer to examine each of these notions individually. For sake of clarity and simplicity, the etymology of “diagram” and “gesture” will play a relevant role in the four next paragraphs (Sects. 1.2, 1.2.1, 1.2.2 and 1.3).

1.2 Diagrams: Etymology and First Conceptual Analyses The word “diagram” comes from the Latin diagramma, and this, in turn, from the Greek διαγ ´ ραμμα. According to what was observed by Noëlle Batt, other two Greek words are involved in the etymological profile of this term: the verb διαγ ´ ραϕειν (“to ´ ραμμα is indeed inscribe”) and the substantive γ ρ αμμα ´ (“line” or “letter”).7 Διαγ ´ you can see—is entirely contained in a deverbal of διαγ ´ ραϕειν 8 and γ ρ αμμα—as διαγ ´ ραμμα. From this point view, “diagram” refers to a « figure dessinée9 ». A more detailed analysis allows the enlargement of the semantic potential of such a word under at least two aspects. First, let’s consider the high polysemy of διαγ ´ ραϕειν: this verb doesn’t only mean “to inscribe” but also “to describe”, “to record”, to “attribute” and even “to delete”.10 Second, let’s remember that two Indo-European roots are at the origin of γ ρ αμμα ´ and διαγ ´ ραϕειν: (1) grbh-mn, grbhmn—“to trace”, “to sketch”, “to write”11 and (2) mn, which generate words like “image”, “letter”, “text”, and so on.12 All of this implies, on the one hand, that “diagram” refers also to “un registre et un décret”13 ; on the other hand, it implies that the same word means the “Inscription […] qui peut se faire lettre ou image, lettre et image”.14 These contents are sufficient to give a first approximative portrait of the semantic richness of the term “diagram”.

4

See among others [5, pp. 50–110], [6, pp. 87–137]. See at least [7]. 6 For which see at least [9, p. 147], [10], [11]. 7 Cf. [12], p. 6]. See also Lo Piparo [13]. 8 Cf. [14, p. 85]. 9 [15, p. 46]. 10 Ibid. 11 Cf. [12, p. 6]. 12 Ibid. 13 [15, p. 46]. 14 [12, p. 6]. 5

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1.2.1 Diagrams as Analytical Potentialities We can reach a more complete knowledge of this simply by considering the Greek preposition δια. ´ It means “through”15 and deploys a variety of technical usages identifying some peculiar features of diagrams. The first one concerns their analytical potentialities: from δια´ one may trace back to the dividing activity of the intellectual discernment. Indeed, literally taken, this Greek term also means “en divisant”.16 Thus, to the extent that they are “outils de la réflexion”,17 diagrams are able to operate the most subtle differences (δια) ´ in the most disparate theoretical frameworks.

1.2.2 Diagrams as Interplay Between Figures and Formulas As for the second feature, it relates to what is perhaps the most significant aspect of the diagrams. We refer to their mobile character: from δια´ (“through”18 ), you can also trace back to a sophisticated interplay between polarities on which diagrammatic operativity rests. This is the case of the concepts of figure and formula.19 Insofar as the δια´ of διαγ ´ ραμμα also means “between”20 and “reciprocally”,21 figures and formulas are what through which (δiα) ´ a diagrammatic praxis is formed and constituted. You could say, then, that “diagram” is not simply a synonym of “figure” (or “graph”). It refers rather to the interactive relationship which allows the transition from the figure to the formula,22 and vice versa. As will later become clearer, this specialized use of “diagram” does not exhaust the semantic potential of the word. But, with reference to this particular case, it is sufficient to consider the reciprocal relationships that are established between strings of symbols (or formulas) and pictures (or figures) in any branch of mathematical reflection.23

1.2.3 Diagrammatic Interplay: A Peircean Point of View It was the American mathematician and semiotician Charles Sanders Peirce who clearly grasped the form of the aforementioned relationships (Sect. 1.2.2). According 15

[15, p. 46]. Ibid. 17 [16], cit. in [17, p. 17]. 18 [18, p. 479], see also [15, p. 46]. 19 Ibid. 20 Ibid. 21 Ibid. 22 Ibid. 23 Ibid. 16

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to him, mathematics can be conceived as “the study of the substance of hypotheses”.24 This definition preludes a diagrammatic interplay (Sect. 1.2.2) insofar for Peirce, hypotheses “must be […] embodied in […] dots, […] lines, […] letters, and the like”.25 Such “immaterial”26 objects, in fact, are parts of an interplay because they constitute the formal poles within which “[t]he mathematician often passes from one mode of embodiment to another”27 . As it has been said, since the diagram is neither the figure nor the formula, but, rather, the interplay between the one and the other, the transitions performed by the working mathematician are exemplar cases of diagram in the sense of the previous section. The entire philosophical project of Descartes’ analytical geometry is thus diagrammatic par excellence: the foundational act of such geometry—i.e. the biunivocal correspondence of planar points with ordered pairs of real numbers and even the earlier, the “one-to-one correspondence between the points of a straight line and the real numbers”28 —is what allows the transition from the geometrical universe of figures to the algebraic one of numbers and formulas, and vice versa. Therefore, a same “system of relations”29 —or a same “hypothesis”,30 in Peirce’s lexicon—can be formally embodied using qualitatively different “objects”, such as geometric sets and equations. And this is because “[s]uch a change is no change in the hypothesis but only in the […] embodiment of the hypothesis”.31 In this sense “[a] simple example is a drawing of a circle of radius 1 with center in the origin of a coordinate system which can be used as a […] [formal embodiment] of the same relational structure of the algebraic equation x 2 + y2 = 1”.32 (see Fig. 1.1). 24

Peirce [19] in Moore 2010 p. 3. Peirce [20] in Moore 2010 p. 46. 26 Ibid. 27 Ibid. 28 Korn and [21, p. 32]. NB: Although Analytic Geometry has been called upon here as a paradigmatic example of diagrammatic thinking, there is at least one aspect of the intellectual enterprise of “the stupendous Descartes” (cf. [22] in Moore 2010 p. 182) that certainly could not meet the favor’s Peirce. We refer to one of the most important implicit consequences of the foundational act of analytic geometry, that is, the identification of “arithmetic line (cf. [23, p. 4]) and of the “geometric line” (cf. [23, p. 4]). Such a theoretical result, above all in its Cantorian construal, could only cause “Peirce’s protest” (cf. [23, p. 4]). Whereas for Cantor, in fact, “the real numbers [i.e. the arithmetic continuum] corresponded to the [geometric] continuum” (cf. [23, p. 4]), Peirce rejected this correspondence denouncing, if anything, its risks of reductionism. It would be interesting to discuss the reasons behind this critique, to show its strengths as well as its weaknesses, and to indicate, finally, the eventual contact points with other non-Cantorian conceptions of the mathematical continuum. However, this is far beyond the scope of our essay. It will, therefore, suffice to mention some of the more theoretically elaborated approaches to the study of the Peircean theorization of the continuum. Among these, from a mathematical perspective, we recall the fundamental works of Fernando Zalamea [24–26], while, from a semiotic perspective, we recall those of Claudio Paolucci [27–29], that of Emanuele Fadda [30] and that of Sebastiano Vecchio [31]. 29 Peirce [20] in Moore 2010 p. 46. 30 Ibid. 31 Ibid. The italics are ours. 32 Steensen and Johansen [32, p. 185]. 25

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Fig. 1.1 “the circle of radius 1”. The picture is taken from Wikipedia

The interplay taking place between the formula (“x 2 + y2 = 1”) and the figure (“the circle of radius 1 with center in the origin of a coordinate system”) is then the dynamical form of the diagram as a fluctuation33 between polarities that are formally unfolded by matching immaterial qualitatively different objects. And yet for Peirce “diagram” is a term which primarily refers to the aforementioned immaterial objects. It is not by chance that he asserts that “[t]he diagrams in which the hypotheses are embodied are of two kinds”,34 that is to say, “[g]eometrical figures”35 and “algebraic formulae”36 . Thus, while it is true that it is to Peirce that we owe a deep insight of the diagrammatic interplay, the usage of “diagram” that the author promotes in his mathematical writings, or at least in those we consulted, is qualitatively different from the one we have here tried to formulate. When the American scholar presents geometrical figures and algebraic formulae such as kinds of diagram, the focus of attention is centered more on the polarities than on the fluctuations. On the contrary, when we have described diagrams in terms of transitions from figures to formulae and vice versa (Sect. 1.2.2), the focus of attention was centered more on the fluctuations than on the polarities. Though subtle, it is an important difference, and means that the notion of interplay provides an excellent key for grasping the dynamical nature of diagrams. As we hope, the reader will be able to evaluate all the heuristic potentials of such a fluctuational approach to diagrams in the sections devoted to practice of mathematical education (Sects. 1.4 and 1.5). For the moment, however, we prefer to postpone the analysis of this aspect, focusing instead on the notion of gesture. It is the level of generality of these notes that allows us to do so, but, above all, it is the theoretical path that we have set ourselves that requires it.

33

The choice of this term is freely inspired from [26 p. 195]. Peirce [19] in Moore 2010 p. 46. 35 Ibid. p. 47. 36 Ibid. 34

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1.3 Gestures: Etymology and First Conceptual Analyses The word “gesture” comes from “the Latin for action, for carrying out an activity, and for performing”.37 More precisely, it derives from two Latin roots: gestus and gerere.38 Gestus is the “neutre substantivé”39 of the past participle of gerere, which means the fact of accomplishing something.40 Put in other terms, the Latin verb would refer to the capacity of executing a task.41 It would therefore seem to fall under the domain of management (gestion42 ) or praxis.43 A closer look at the Latin origins of “gesture” reminds us of the locution res gestae.44 The latter was referred to the heroic deeds of exceptional men.45 Moreover, in the period between the end of the seventeenth century and the entirety of the twentieth century, intentionality46 and spontaneity,47 but also meaningfulness48 and corporeality,49 were among the notions that are increasingly intertwined in the definition of “gesture”.

1.3.1 Gestures: Meaningfulness and Corporeality For example, the second pair of terms already appears in a text dating back to the 1694. We refer to the first edition of the Dictionnaire of the French Academy: under the heading “gesture” one could read about a “[…] mouvement du corps, du bras, de la tête, de la main”,50 with particular reference to the actions “qui accompagnent le discours”.51 In the same spirit, the Dictionnaire Universel of Antoine Furetière remarks that a gesture is a “mouvement du corps qui se fait […] pour signifier quelque chose”.52 This semiological dimension of gesture will predominate in many contemporary works of linguistics and cognitive science. According to authors such as Adam Kendom53 37

[33 p. 5]. Cf. [34, p. 18]. 39 [35, p. 152]. 40 Cf. [36, p. 27]. 41 Cf. Ibid. 42 Ibid. 43 Cf. Ibid. 44 Cf. Ibid. 45 Cf. Ibid. 46 Ibid. p. 28. 47 Ibid. p. 29. 48 Ibid. p. 28. 49 Ibid. 50 Ibid. 51 Ibid. 52 Ibid. The italics are ours. 53 Cf. [37]. 38

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or Geneviève Calbris,54 “gesture” exclusively denotes any neuromuscular action (or visible movement of body) involving the execution of a communicative act.55 It is in these kinds of theoretical conceptions that intentionality appears as a constitutive element of gestures.

1.3.2 Directionality of Gestures: From Corporeality to Inter-Corporeality Indeed, performing a communicative act implies intentionality in at least two senses of the word. The first one is “directionality”: gestures are intentional insofar as they are addressed to someone—besides, as the etymology attests, “intendere est in aliud tendere”.56 As to the second sense, it is “deliberation”: gestures are intentional insofar as they are consciously and/or voluntarily performed for the aforementioned communicative purposes. From a certain point of view, it could be argued that discussing about gestures in terms of directionality is advantageous. This aspect tells us in fact something very profound about their corporeal nature: insofar as every gesture is supposed to be addressed to someone, it is not simply a bodily movement, but rather “a bodily movement toward another body”.57 So, the intentional parameter would reveal that the corporeality of gesture is more properly an “inter-corporeality”.58 However, this parameter is also strictly connected to the deliberative dimension to human actions. But this last aspect, although fundamental, does not cover the whole sphere of the gestural.

1.3.3 An Incorrect Reduction If gestures are reduced to particular kinds of communicative acts, and if these are presented as addressing someone, it is certain that consciousness and will (or voluntariness) can only be constitutive ingredients. A more profound view regarding the praxeology of gesture, however, reveals the incorrect character of this reduction.

54

Cf. [38]. But consider also [39, p. 35]: “Chiamiamo “gesto” ogni azione con un inizio e una fine che porta un significato”. 56 Tommaso D’Aquino I-II. 12–15. 57 [40, p. 87]. 58 Regarding this concept, one may refer to [41], but also to Meyer et al. [42], as well as to [43]. 55

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Two Reasons

Firstly, because the sphere of gesture does not exactly cover the sphere of communicative acts; secondly, because consciousness and voluntariness are only one aspect of the gestural.

Dissimilation Between Gestures and Acts As regards the first point, it is sufficient to remember a theoretical distinction that has been introduced in some domains of contemporary semiotic debate. Indeed, according to the research program known as the Semiotic Ecology of Culture (henceforth, SEC),59 notions such as those of act and gesture differ from one another in virtue of their referring to distinct but related agencies. More precisely, they have to be differentiated insofar as every act—variously qualified as communicative,60 enunciative61 or linguistic62 —corresponds to forms of linguistic praxis presupposing for their success a whole set of stabilized regularities (conventions, shared values, univocal plans of relevance) to which gestures adhere only partially. This adhesion is partial in the sense that it concerns only a local aspect of gestures: the one that makes them available to be “absorbed” within an “espace codé”63 i.e. within a preorganized semiosis identifying them as being acts of a certain kind. On the other hand, however, always according to the SEC, every gesture is endowed with a potential for agentivity (or puissance d’agir 64 ) which puts it in the conditions for exceeding such a preorganized semiosis—or “for tearing it apart”65 —in the form of “clumps of meaning” (coagulations de sens66 ) preluding new praxeological frameworks. Let’s take for example the case of a conductor.67 The movements the latter performs with his or her baton are not made at random: they respond, if anything, to the execution of a precise musical score. It is precisely for this reason that they are readable (cf. Fossali [44], p. 86), that is to say, recognizable as the gestures of a conductor. So, from this point of view, they adhere to a codified space of action which express them as being parts of a well-determined sequence of acts. Nevertheless, “[c]haque performance musicale échappe quelque peu à l’exécution prévue de la partition”.68 And this is because the fidelity to the musical score, on which the semiotic readability of conductor’s gestures depends, does not exhaust the praxeological 59

Cf. the monumental book by Basso Fossali [44]. Cf. Huang 2012. 61 Cf. [45]. 62 Cf. [46]. 63 Basso Fossali [44, p. 87]. 64 [36, p. 27]. 65 Basso Fossali [44, p. 87]. 66 Ibid. p. 87. 67 Ibid. p. 86. 68 Ibid. p. 85. 60

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potential of movements mobilized by the performer. We refer in particular to all those possible variations that emerge all while taking on the most disparate forms: stylistic virtuosities, specific interpretations (or dramatizations), which, on the one hand, are the outcome of the “étude de la partition et (des) heures consacrées aux épreuves”,69 on the other hand, manifest the gaps always ready to infiltrate between “le texte de l’auteur que l’on se doit de posséder au mieux”70 and “l’effet produit sur l’orchestre et l’auditoire par l’interprétation proposée”.71 Thus, to put it another way, gestures differ from acts to the extent that the former constitute everything in the latter that exceeds the boundaries of codified frames of action or scènes actantielles.72 This praxeological surplus also includes all those aspects of the gestural which transcend the limits of consciousness and will.

Gestures as What Betrays Us Consciousness and will are only one aspect of the gestural: its expressiveness (expressivité) often “nous traverse sans que nous pouissions l[a] controler”.73 And this is because gestures are “never just the enactment of an intention”.74 Of course it is a double sense of “intention” that is questioned by this kind of remarks. As the reader will remember from Sect. 1.3.2, the intentional features of gestures have been described in terms of two aspects: (a) directionality (gestures are intentional insofar as they are addressed to someone; (b) deliberation (gestures are intentional to the extent that they are voluntarily performed in order to accomplish communicative tasks). If we envisage the very real possibility that gestures can betray us, and, in this sense, escape to our control by revealing “ce que (l’on aurait) préféré cacher”,75 then (a) and (b) can never exhaust the praxeological potential of gestural. Precisely because some bodily movements escape us before we can repress them76 (e.g. “croiser les jambes, allumer une cigarette, se passer la main dans les cheveux”77 ), directionality and deliberation cease to be the main features of gesture. Certainly it could be argued in this respect that what is beyond our control is anyway performed in front of someone else. Moreover, in accordance with scholars like Kendom and Calbris (Sect. 1.3.1), one could classify as gestures only those bodily movements that are 69

Ibid. Delcroix 2006 p. 101, cit. in Basso Fossali [44, p. 86]. 71 Ibid. 72 Cf. Basso Fossali [44, p. 86]: “Le geste […] participe d’une scène actantielle [and in this sense it is an act – editor’s note], mais laisse entrevoir toute l’épaisseur potentielle de l’acteur et tout le possible qui se situe au-delà des cadres systématisés des actes codés”. 73 [36, p. 31]. 74 [8, p. 64]. 75 [36, p. 42]. 76 Cf. Ibid. 77 Ibid. 70

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intentional in the double sense explained here. Nevertheless, these countermoves do not debunk the effective scope of criticisms addressed to the idea of the gesture as a communicative act.

Inter-Corporeality Without Intentionality Indeed, although what escapes us, and therefore betrays us, is presumably grasped or induced by someone else, the gesture eluding our control is not at all meant… for someone else. To be so, it should fulfil the main function of any communicative act: to convey information you actually want to get across to someone.78 It happens, however, that “what escapes us”—the gesture that betrays us, in fact—is the furthest away from the sphere of our conscious control.79 Therefore, it simply lies beyond what one may or may not want to convey to someone. Hence, from this point of view, local conditions would be given for detaching the inter-corporeality of gestures (Sect. 1.3.2) from their presumed intentionality. As we have already said, insofar as the gesture that betrays us is presumably grasped or induced by someone else, without however actually being meant for that someone, it is inter-corporeal but not intentional. In other words, it exists as such for another body—and in this sense it is inter-corporeal—but it isn’t deliberatively addressed to the latter. As regards the intentional conceptions of gesture (henceforth, ICGs ) the theoretical move which aims at classifying all those not-voluntary bodily movements as “not-gestural” can be downgraded in terms of a fundamental remarks. It concerns the theories of gesture currently available in the field of philosophical-linguistic ideas: whatever one thinks of them, there are some that are completely neutral with respect to the voluntary/non-voluntary opposition. This is the case, for example, with Marcel Jousse’s theoretical approach. According to the French anthropologist, gestures are in fact all “mouvements […] Visibles ou invisibles, conscients ou inconscients, volontaires ou involontaires […] qui s’exécutent dans le composé humain”.80 Furthermore, although the distinction between “voluntary” and “involuntary” is in principle quite clear, gestures are actually a changing mixture of the two. And this is because “il(s) se situe(nt) à distance variable entre ces deux pôles que sont «l’exécution d’une action», qui est généralement intentionnelle, et «l’expression (d’un état d’âme)», qui m’échappe toujours en partie”.81 So, in this respect, we can 78

Cf. Ibid. p. 29. Cf. Ibid. p. 42. 80 [47, p. 686]. See also Cadoz and Wanderley (2000), p. 73, cit. in Arias-Valero and LluisPuebla [48] p. 88: “We consider that the word gesture (or the French equivalent geste) necessarily makes reference to a human being and to its body behavior—whether they be useful or not, significant or meaningless, expressive or inexpressive, conscious or not, intentional or automatic/ reflex, completely controlled or not, applied or not to a physical object, effective or ineffective, or suggested”. 81 [36, p. 42]. 79

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conclude that “[l]e geste découle d’une va-et-vient entre action délibérée et inertie du corps, entre visée et gestion par défaut, entre concentration et laissez-faire”.82 But there is more. ICGs have been also efficaciously criticized in the field of the “mathematics education research program”.83

1.4 A Mathematical Game: Sets of Possibilities and Virtuality Let’s start with an example. It is an educational experience reported to us by Wolff-Michael Roth and Jean-François Maheux.84 The two scholars, who have been working for years on the epistemology of mathematical learning, tell of “secondgrade children […] invited to build […] tangram shapes and trace these and their parts on sheets of paper”.85 The outcomes acquired during the analysis of such conducts contribute to the reassessment of ICGs (Sect. 1.3.3.1.3): they highlight how the “shapes result[ing] from the children’s manipulations”86 are often “unexpected”,87 that is, irreducible to an “object of intention”.88 In this perspective, Nelly’s hand movements give a very interesting insight about the point at issue. By means of them, she triggers a complex morphogenetic work from which emerges a whole series of shapes (or “configurations”89 ) surprising their performer and attesting to the praxeological potential (Sect. 1.3.1.1.1) of mobilized conducts. In particular, it is once again the duality of gestures and acts (Sect. 1.3.1.1.1) that come to our attention. When we discussed that in the SCE’s framework (Sect. 1.3.1.1.1), we used the expression “praxeological potential” (or one of “agentiveness potential”) to refer a specific endowment of gestures as opposed to acts. Thanks to this endowment, gestures are distinguished from acts because they are always in the conditions to exceeds the codified regularities of a given semiosis. In other words, while acts would be resolved within the boundaries provided by these regularities, gestures would be always capable of violating (or “tearing”) them in the form of “clumps of meaning” that prelude new and unforeseen frames of action. Now, it is precisely such clumps that surface in Nelly’s movements. Indeed, according to Roth and Maheux, the work they constitute is exposed to sudden changes that their performer seems to squander (i.e. Nelly). In the first part of mathematical game, the child moves and brings together 82

Basso Fossali [44, p. 96], [8, p. 62]. 84 Cf. [49, pp. 221–238]. 85 Ibid. p. 223. 86 Ibid. 87 Ibid. 88 Ibid. p. 221. 89 Ibid. p. 224. 83

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a whole series of triangles while “saying ‘That makes a boat’90 or “‘And that goes into the boat’”.91 It would seem, therefore, that Nelly’s work stops permanently at the configuration named “boat”. Nevertheless, the mathematical game continues: after a short pause of “about 4s”,92 Nelly’s morphogenetic movements resume their course. Yet, they are already heading for another configuration. Although there is not apparent indication of this (the new movements are “associated with talk about a boat”93 ), something completely different takes shape in the terminal phases of game. Then, from the boat, a new configuration emerges: a square. In fact, “When the movement has ended, she steps back and says, with apparent surprise, ‘I got a square’”.94 Hence, the apparition of “clumps of meaning” deploying a new “set of possibilities”,95 that is, a “[v]irtuality”96 : it “becomes a new whole configuration around which subsequent actions can take place97 ” by exceeding the codified regularities of previous actions. We thus find the “gestures/ acts” duality alluded above: emerging as a new configuration, and germinating on a previous form playing the attractor’s role with respect to a series of bodily movements, the square, or rather, the morphogenetic work underlying its “revelation”98 constitutes a sequence of gestures in the sense of SEC, namely, a series of actions which, in virtue of their praxeological potential, alter the codified regularities (i.e. the attractor) of previous stabilized chains of action (or acts). Of course, defined in this way, “acts” and “gestures” are nothing more than transitory forms of the praxis: “guises” that human action takes on time from time depending on whether it conforms to some codified regularities or radically subverts them. From this point of view, diagrams are (also) the “territory”99 in which the tensions between such transitory forms emerge by deploying different potentialities, or, in Guattari’s terms, “divers registres d’altérité”.100

1.5 Towards the Diagrammatic Gestures of Gilles Châtelet In Sect. 2.2 we have presented diagrams in terms of an interplay between figures and formulas. Although the previous example (Sect. 1.4) is a case of “informal mathematics”, we believe that the morphogenetic movements of Nelly fit neatly into 90

Ibid. p. 223. Ibid. 92 Ibid. 93 Ibid. p. 25. 94 Ibid. 95 Ibid. p. 26. 96 Ibid. 97 Ibid. 98 Ibid. 99 For this term referred to the notion of diagram see [50, p. 198]. 100 [51, p. 78]. But see also Guattari [51, p. 59]. 91

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such kind of process. More precisely, we think that they fit into an archeological version of it, that is to say, within an interactive framework in which the symbolic polarity of the formulas is absent and yet enacted by the verbal materiality of oral discourse. In this perspective the manipulative activity described in Sect. 1.4 is an “embryo” of diagrammatic practice. Indeed, to the extent that the Nelly’s work concern not only the figural level of triangles she moves and brings together but also the symbolic level of the words she pronounces in commentary of such movements, the configurations emerging during the mathematical game are the outcome of a dynamical interplay having in itself all the germs of a diagram in the sense of Sect. 1.2.2. Indeed, as De Freitas and Sinclair remark, “the concept itself is entailed in the hands that gesture, the mouth that speaks, and the affect that circulates across an interaction”.101 It is however true that only the usage of an explicit algebraic scripture sets the conditions for a real diagrammatic inter-play (Sect. 1.2.2). Why then is the informal Nelly’s manipulative work so important to the issues we are addressing in this Introduction? For essentially two reasons. First, because it enacts one of the main features of the gesture in the Châtelet’s sense. Second, because it reveals how the virtualities deployed by the mathematical game (Sect. 1.4) fit into a dynamical aspect of diagrams that the French mathematician discusses at length in his work. Let’s start from the last point.

1.5.1 Germs of Deformation: “Becoming-Other” and “Pressures of the Virtuality” In the Introduction to Les Enjeux du mobile (henceforth, EM) one reads that “[d]iagrams […] blossom with dotted lines in order to engulf images that that were previously figured in thick lines. […] The dotted line refers neither to the point and its discrete destination, nor to the line and its continuous trace, but to the pressure of the virtuality […] that worries the already available image in order to create space for a new dimension: the diagram’s mode of existence is such that its genesis is comprised in its being”.102 Moreover, it is asserted elsewhere that “le diagramme saute par-dessus les figures et contraint à créer des nouveaux individus”.103 Seemingly unrelated to the spirit of these remarks, Nelly’s manipulations are actually an effective intuitive exemplification of what is described in the execerpts you have just read from Châtelet’s texts. In essence, these aim to emphasize that any diagrammatic practice worthy of the name inscribes a germ of deformation in the figures being worked on. Indeed, it is to this germ that phrases such as “the pressure of the virtuality” or “le diagramme saute par-dessus les figures” refer. And it is similar germs that can be traced in the morphogenetic work of the child: to 101

[8, p. 66]. [52, p. 10]. 103 [53, pp. 111–112], cit. in [15, p. 41]. 102

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the extent that, as we have seen (Sect. 1.4), from the “boat” configuration emerges an unexpected “square” configuration, such a work is diagrammatic by definition. In this context, in fact, the square is the very deformation of the boat, that is, the outcome of an inter-play between figural and symbolic planes whose “trademark” is their “becoming-other”.

1.5.2 Diagrammatic Alterations: Gestures and Algorithmic Legalities This “becoming-other” is furthermore an aspect reverberating on Nelly’s movements: precisely because they transit from the genesis of a “boat” configuration to that of a “square” configuration, “the pressure of the virtuality” inoculates germs of deformation—and thus of “otherness”—in the morpho-genetic work performed by the child’s hands. It is in this sense that such a work exemplifies one of the main features of the gesture in Châtelet’s sense (Sect. 1.5). According to the French mathematician, the gesture—far from being caught by an “algorithm [that] controls its staging”104 —is characterized among others things by its capacity of “awaken[ing] in us other gestures.105 ” And to say “other gestures” is, for Châtelet, to point out all the difference that separates a diagrammatic practice as an activity “haunted” by germs of otherness, from an algorithm, as a finite set of “mere instructions”.106 If we insist on this aspect, however, it is not because we aim to belittle the importance of the algorithmic dimension of mathematical activity, but rather because we mean to deepen the duality between acts and gestures that diagrammatic practices deploy in their own making. As we have seen in the previous sections, it has been possible to appreciate the distinctive features of the forms of both praxes in the context of Nelly’s movements (Sect. 1.4), and even earlier in the one of a conductor’s movements (Sect. 1.3.3.1.1). Nevertheless, it is again the words of Gilles Châtelet, and precisely those we have quoted here, which provide a new conceptual parameter for evaluating the differences in question. The critique of algorithmic reason is one of the main values of this parameter to the extent that the germs of otherness of the gesture are emphasized by the author against the mechanical execution of the algorithm. That is to say, more precisely, against what such execution guarantees, namely “a protocol for decomposing the action into endlessly repeatable acts”.107 The Châteletian duality gesture/act fully lies therein. Indeed, if one takes care to observe an association implicitly suggested in the text, the polarities in question will clearly emerge. For Châtelet, an act is a “endlessly repeatable” segment of action, or at least “act” seems to be the name chosen by the author to designate those minimal units of action—those segments, precisely—on which the instauration of an algorithmic 104

[52, p. 9]. Ibid. 106 Ibid. 107 Ibid. 105

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legality rests. On the other hand, in opposition to such a name and to what it seems to entails in terms of “endlessly repeatability”, the word “gesture” is used just few lines earlier to designate everything that perturbs the equilibria presupposed by such a legality.108 We find thus again a tension that should be familiar to us at this point: the one between what conforms to a given system of codified regularities (the act as an “endlessly repeatable” segment of action) and what is instead constituted as a tearing of previous regularities (the gesture as a source of “other gestures”, or, put in other words, the gesture as a vector of diagrammatic alterations).

1.5.3 Diagrammatic Gestures as Heterogenetic Work Diagrammatic gestures in Châtelet’s sense can be defined in terms of the aforementioned vector (Sect. 1.5.2). It must be said, however, that “diagrammatic gesture” is not an expression belonging to the technical lexicon of the French mathematician. Such a phrase does not occur in either EM or in L’enchantement du virtuel (henceforth, EV). Nevertheless, notions such as those of “diagram” and “gesture” are so intertwined in Châteletian thinking that—as has already been noted in Sect. 1.1—we found “diagrammatic gesture” to be a more than appropriate locution for capturing such a connection. On the basis of the findings in Sects. 1.5.1 and 1.5.2, one of the most significant points of this interweaving is given by the fact that, for Châtelet, diagrams and gestures are both dynamical units of change: “The two participate in each other’s provisional ontology”.109 Some of the most important textual traces of such an ontology—that we called the “becoming-other” (Sects. 1.5.1 and 1.5.2)— can be found in aforementioned pages concerning respectively the “dotted lines” of diagrams and the gestures capacity for “awaken(ing) other gestures”. Undoubtedly, it is Jean-Blaise Alcantara who clearly grasped this double aspect. In his Un schématisme de forces: la fulguration du diagramme entre Deleuze et Châtelet, he reflects upon the “becoming-other” of the gesture which produces diagram and on that of the diagram which “transfix(es) a gesture”,110 by observing that “[l]e geste proprement diagrammatique tend […] à libérer la ‘Figure’”111 and it (i.e. “[l]e geste diagrammatisé”112 ) “appartient à l’ordre du virtuel qui […] appelle son prolongement”.113 Indeed, “figure” is a term that, already for Deleuze of Francis Bacon.

108

From this point of view, Châtelet’s remarks devoted to Penrose’s book Shadows of the Mind constitute an excellent starting point for developing and deepening the diagrammatic potential of such perturbations. In this regard, see [54, pp. 245–252]. 109 [8, p. 64]. 110 [52, p. 10]. 111 [55, p. 151]. 112 Ibid, p. 156]. 113 Ibid.

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Logique de la sensation,114 designates the outcome of an heterogenetic work.115 The diagrammatic gesture “liberates figures” in the sense that, from a given form (e.g. a “boat configuration”), it generates “une forme d’une tout autre nature”116 (e.g. a “square configuration”). And it “appelle son prolongement” in the sense of propelling “new plastic unities”.117 Hence, precisely, “becoming-other” appears as shared formal feature of diagrams and gestures as complementary polarities of the heterogenetic work we call “diagrammatic gesture”. But what, exactly, is meant by “heterogenetic”?

1.5.4 Heterogenesis: A Key for Diagrammatic Gestures? The adjective “heterogenetic” is a paronym of the substantive “heterogenesis”. This term literally means “genesis (or deployment) of alterity”. As is well known, it belongs to the technical lexicon of Deleuzian-Guattarian philosophy. In such a theoretical framework, the word “heterogenesis”, while remaining whitin the limits of etymology, acquires a series of specialized meanings which greatly enrich its semantic potential. Guattari, for instance, introduced the expression “puissance d’hétérogenèse”118 in order to denote an “ouverture sur des processus irréversibles de différenciation […] singularisants”.119 Deleuze, for his part, connects “heterogenesis” to a production of radical novelty: “a system […] must be a hetero-genesis: something which […] has never been attempted”.120 Moreover, Deleuze and Guattari employ the adjective “heterogeneous” as a quasi-synonym of “in continuous variation”121 : “Hétérogène, en variation continue”.122 The list of meanings of “heterogenesis” could be extended further.123 Nevertheless, for the level of generality of these notes, the meanings of the term explained here are more than sufficient to provide a complete overview of the main conceptual implications underlying “heterogenesis”. On the basis of such an overview, heterogenesis in Deleuzian-Guattarian sense of the term can easily be assimilated to the “becoming-other”—or “pure becoming124 ”—of Châteletian diagrammatic gestures (Sect. 1.5.3). We are not sure

114

Ibid, p. 150. For more details, see [56, pp. 49–51]. 116 [57, p. 95], cit. in [55, p. 150]. 117 [52, p. 20]. 118 Guattari [51, p. 82]. 119 Ibid. See also [58, p. 77]. 120 Deleuze [57, p. 10], cit. in [59, p. 239]. 121 Cf. [59, p. 240]. 122 Deleuze and Guattari [60, p. 656]. 123 For more details, see [59, pp. 239–241]. 124 [61, p. 19]. 115

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whether such a becoming is in continuity with Deleuzian-Guattarian heterogenesis. Two points, however, seem, if not to prove, at least to encourage this line of interpretation: (1) The presence of explicit references to the work of Gilles Deleuze in some theoretically crucial passages of EM125 (e.g. the reference in a footnote of EM to Différence et Répétition as a philosophical work within which the theme of the “Differential” is explored126 ); (2) The remarks that the last chapter of EV develops on the subject of “becoming” starting from a heartfelt remembrance of Deleuze.127 The second point, in particular, provide an excellent basis for developing the supposed presence of a heterogenetic theme in Châtelet’s thinking.

1.5.5 Châtelet and “Les Embryons Larvaires” A clear proof that Châtelet was particularly attentive to Deleuzian theme of heterogenesis lies, in our opinion, in the lines reserved by the French mathematician to the love of “Gilles (for) les ‘embryons larvaires’”.128 The figure of the embryo is, in fact, evocated by Châtelet as a site of heterogenesis par excellence. Indeed, the development potentials of the embryo are presented in terms of their plasticity (leur plasticités129 ) and of their capacity for propagating through the environment (pousser par le milieu130 ). These two dynamical features prelude heterogenesis insofar as the environment or milieu is in turn presented as a catalysator (catalysateur 131 ), that is to say, as a something which triggers transformations (déclenche la transformation132 ). All this, as a first approximation, would be sufficient to hypothesize a special attention of Châtelet to Deleuzian theme of heterogenesis. Nevertheless, even assuming that this is the case, a clarification of terminology is necessary.

125

As well as, among the acknowledgements, Châtelet’s words of gratitude to Guattari for having “reawakened [him] to philosophy” [52, p. xxvii]. 126 Ibid, p. 37 n. 17. 127 Châtelet (2010), pp. 262–266. 128 Châtelet (2010), p. 264. 129 Ibid. 130 Ibid. 131 Ibid. 132 Ibid.

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Heterogenesis is not a Transformation

Although this is precisely the word used by the author, we don’t believe that “transformation” is a term that can properly refer to heterogenesis. In a Deleuzian perspective, the becoming-other of heterogenesis is “pure” in the sense of being free from any pre-fixed constraint which controls in a way or another its morphogenetic potential. It is, in other words, a “smooth space”,133 isnamuch as smooth “is amorphous and not homogeneous”.134 On the other hand, a transformation in the usual sense of the term is a becoming-other that is subject to preset constraints (or rules). Let’s take for instance the case of transformations in some areas of mathematical research. Generally, they are bijective maps from a space to itself,135 and they work “selon une règle précise qui définit la transformation”.136 Now, it is from the binding power of such rules that Deleuzian heterogenesis escapes. While in a homothety or in an isometry, the parts of space (or figures) change with respect to a pre-given and immutable structure (e.g. the group of transformation identifying once and for all what varies and what does not vary in the figures involved), a heterogenetic change is irreducible to the constraints of any structure. Of course, in saying “irreducible” we don’t aim to suggest that heterogenesis is an absolute chaos: if it were, its morphogenetic potential would be null (any form, however provisional, emerges always thanks to a system of constraints). Hence, what we want to suggest is very different: heterogenesis is perfectly equidistant from immutable structures and wild chaos. It is a becoming-other which gives itself laws that it however modifies “en cours de route de manière spatiale et temporelle”.137 Therefore, the difference between transformations and heterogenesis can be summarized as follows: the former are structure-dependent dynamics; the latter are structure-free dynamics or, more accurately, “dynamics of […] fluxes in continuous recombination”.138 So, from this point of view, it would be preferable to not assimilate heterogenesis to a type of transformation.

1.5.6 Several Uses of “Transformation” It is nonetheless true that a mathematician of Deleuzian inspiration such as Alessandro Sarti, to whom we owe, among others things, the first attempt to formally define heterogenesis,139 did not hesitate to informally present the dynamics of

133

Deleuze and Guattari (1987), p. 476, cit. in [59, p. 240]. Ibid. 135 Cf. [62, p. 6]. 136 [63, p. 120]. 137 Sarti in [64, p. 5]. 138 [65, p. 5]. 139 See the fundamental [66]. 134

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heterogenesis in terms of transformations. For example, by discussing some theoretical assumptions of Deleuze and Guattari, he asserted that heterogenesis is “to be conceived as a transformation from one multiplicity to another multiplicity”.140 If, however, the theoretical framework in which the previous definition has been enucleated is relatively unambiguous, many other are not at all. Put more simply, there are varying uses of “transformation”. For the previous reasons (Sect. 1.5.5.1), the use of “transformation” emerging in some areas of mathematical research is very problematic. The one emerging from Sarti’s words is instead unproblematic because of the terms it involves. In a Deleuzian-Guattarian perspective, a multiplicity is already in itself a structure-free dynamics: as you can read in Thousands Plateaus “every multiplicity is already composed by heterogeneous terms in symbiosis.”141 By defining then heterogenesis in terms of transformation, Sarti—but so also Deleuze and Guattari—in no way reduce the germs of morphogenesis to predetermined spaces of possibilities.

1.5.7 “Promesses De Papillons” Having clarified this point, it must be said that Châtelet’s usage of “transformation” fits perfectly into this way of thinking. After pointing out the love of “Gilles [for] les embryons larvaires”, he remarks that the Deleuzian attention to the development potentials concerns “champs de forces (ou virtualités)”142 transcending the limits of preestablished possibilities. Within such limits larvae are only germs of expected forms or “promesses de papillons”.143 The vivid Châteletian locution must be taken seriously: the transformation of a larva into a butterfly (or an ant) is a “promise” in the sense of being a developmental trajectory that may or may not be realized within the preset constraints of a precise evolutive history. It is then a becoming that undoubtedly has margins of contingency and that however is “extracted” from a range of trajectories which a priori excludes many other development possibilities: for instance, the possibility for a larva to become a wolf. In this sense, the becoming-other of heterogenesis is a completely different process. From a structure-free dynamic point of view, a larva is not the promise of anything in particular. And this is because—far from being subject to the aforementioned constraints—it can become everything: be it a wolf or a sheep, a mountain or an eagle, and, ultimately, even a butterfly or an ant. So, if it is a germ, heterogenesis makes it to be a generator of radically unexpected (and in a certain sense impossible) forms.

140

[59, p. 240]. Italics are ours. Deleuze and Guattari (1987), p. 249, cit. in [59, p. 240]. 142 Châtelet (2010), p. 264. 143 Ibid. 141

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Thus, the production of radical nolvelty posits heterogenesis within new contingency margins. While ones featuring the transformation of a larva into a butterfly (or an ant) are inscribed in a predetermined “champ de forces” which make it a “promise” in Châtelet’s sense, the ones underlying the transformation of a larva into a wolf (or into a sheep) are inscribed in a “champs de forces” which is anything but preestablished. Therefore, the metaphor of the “promise”, from which Châtelet seeks to emancipate himself, clearly shows how “transformation” is a word employed by the author in a sense that is different from the usual one. In this perspective, “deformation” could be a more appropriate term.

1.6 Deformation and Becoming-Other: General Remarks We already employed this word when analyzing Nelly’s movements (Sects. 1.5.1 and 1.5.2). For instance, the morphogenetic transition from the boat to the square has been presented in terms of deformation (Sect. 1.5.1). Moreover, in the same observational framework, “deformation” and “becoming-other” have been assimilated to one another (Sect. 1.5.2). One could say, then, that such a theoretical context ensures the provision of all the conditions for appropriately replacing “transformation” (in Châtelet’s sense) with “deformation”. One point, however, requires more precision: the assimilation between “deformation” and “becoming-other”. The latter has been introduced in an operative setting in which the usual notion of transformation had not yet been clearly distinguished from the Deleuzian becomingother. To understand this, it is sufficient to return to Sects. 1.5.1 and 1.5.2. There is nothing in those pages that suggests the distinction at play here: the germs of otherness operating in the aforementioned transition have been described in terms of deformation. Nonetheless, due to their level of generality and, above all, for the absence of successively introduced distinctions, the previous descriptions might have been described in terms of transformation. Of course the supposed unpredictability of the forms emerging from the child’s manipulations (Sect. 1.5.2) would already seem to prelude one of the main features of Deleuzian becoming-other: i.e. its being a structure-free process (Sect. 1.5.5.1). Once again, however, the lack of specific information in this regard, and, on the other hand, the necessity for more detailed analysis induce us to go back on the assimilation between deformation and becoming-other. That is why we will seek to deepen some of its more important aspects. In particular, we will attempt to justify the proposed assimilation by developing a two-step strategy. Firstly, we shall identify some theoretical contexts within which the “transformation/deformation” distinction has been explicitly highlighted. Secondly, we shall explain some mathematical-formal aspects of diagrammatic gestures.

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Fig. 1.2 “Painting” (Francis Bacon). The picture is taken from [67]

1.6.1 Spaces of Latency One can find a theoretical locus in which transformation and deformation have been clearly distinguished in some remarks made by the French comparatist Noëlle Batt. In L’expérience diagrammatique, she discusses the Deleuzian reading of works by Irish painter Francis Bacon. In particular, by focusing on Deleuze’s commentaries about Painting, a 1946 work by Bacon (Fig. 1.2), the author presents transformation and deformation as changes differing, among others things, by the presence or absence of intentionality.144 We find the choice of this differential parameter to be coherent with some of our basic assumptions about the praxeology of gesture. The reader will remember that we tried to disentangle gestures and intentionality by means of two theoretical steps: (1) the dissimilation of gestures and acts (Sect. 1.3.3.1.1); (2) the reconceptualization of the gesture as a potential for action transcending the limit of a pure conscious voluntariness (Sect. 1.3.3.1.2). It is the second step that encounters Batt’s perspective. If understood correctly, a deformation is a change involving a radical difference between the actual form and the intentional project 145 underlying it. This is the case of Painting insofar as what is shown on the canvas is radically dissimilar from the conscious idea of the painter. As we know from Bacon himself, he “was attempting to make a bird alighting on a field […] but suddenly the lines that [he]’d drawn suggested something totally different”.146 That is to say, “l’homme au parapluie devenant l’animal de boucherie écartelé”.147 If it was instead a transformation, the radical dissimilarity of deforming gesture would have given away a 144

Cf. [12, p. 14], see also [56, pp. 50–51]. Cf. [56, p. 50]. 146 Bacon in Sylvester (1975), p. 11, cit. in [56, p. 48]. 147 [12, p. 14]. 145

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series of structural affinities, or, more properly, of form similarities (rassemblances de forme148 ) reducing the qualitative difference between the image on the canvas and the preceding conscious idea. Put in these terms, the difference between deformation and transformation seems to be clearly articulated in its main polarities. Above all, it seems efficaciously highlight all aspects of the diagrammatic gesture that we called “heterogenetic”. The gap between the bird in the painter’s head and the “whole image”149 on the canvas is a space of latency teeming with germs of otherness (Sect. 1.5.2) and on which the painter’s gesture can unfold the potential for action…unbeknownst to the painter himself. Here, in our opinion, is the main feature of heterogenesis: one of being an unforeseeable becoming-other. As we saw, many scholars have dwelt on this theme (Sects. 1.5.1 and 1.5.2). In any case, there are still thinkers such as Batt or Maria Giulia Dondero who drew from such a subject particularly relevant outcomes with respect to the distinction we are discussing here. The former, for instance, made unpredictability to be a distinctive feature of deformation by observing it involves a zone of indetermination150 which works as a generator of markings and lines “qui [font] encore partie du non-su, du non-encore pensé”.151 As for the latter, instead, her remarks about chance152 and the gestural manipulation of chance153 allow to deepen the unpredictability of deformation under the profile of the figurative links it interrupts or shatters.154 These are, of course, remarks that are primarily inscribed within the framework of art theory and visual semiotics. Therefore, despite their undoubted merits, they do not have sufficient scope to establish general criteria for distinguishing between transformation and deformation. On the other hand, however, thanks to a deeply philosophically inspired vision, both authors have been able to place such observations within broader and more articulate theoretical frameworks. Hence, the possibility of deriving from them useful insights into the distinction here at play: these need only be specified in order to confirm it.

1.6.1.1

Deformation and the Gesture’s Flight

Let us return to Bacon’s painting. As mentioned above, Batt makes it to be a sort of implicit “test-bed” on which to identify the differential properties of deformation and transformation. The first one would work as a generator of “random markings and lines” (marques et lignes au hasard 155 ) insofar as the painter’s gestures escape 148

Ibid. Bacon in Sylvester (1975), p. 2021 p. 48. 150 Cf. [12, pp. 14–15] and [56, p. 48]. 151 Cf. [12, p. 21] and [56, p. 51]. 152 Cf. [68, pp. 85–86]. 153 Ibid. 154 Ibid. 155 [12, p. 21]. 149

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the conscious intentions of the painter himself (Sect. 1.6.1). In this sense, therefore, Painting would be the visual outcome of a gesture’s flight from the constraints that the conscious mind tries to impose upon the genesis of form. More recent studies in the field of the so-called “improvisation esthetics”156 could, however, considerably moderate the oppositional terms of this tension. Nonetheless, entrusting the discussion of such musings in other research papers, it is undoubtably that the flight from which Painting emerges is one of the main distinctive keys to Bacon’s esthetics. Indeed, if all of the Irish painter’s work, not only Painting, is read by Batt and, before Batt, by Deleuze in terms of deformation, it is because Bacon systematically returns on the heterogenetic role by which “involuntary marks”,157 “accident(s)”158 and “not conscious doing”159 are put into play “on the canvas”.160 Thus, it is the set of these vectors of heterogenesis—or “graph”, in Bacon’s words161 —that makes the deformation operate as a “creation de rapports originaux substitués à la forme”.162 Deformation, in short, is not the replacement of a previous form with a different but related one. Rather it is the “liberation” of figures (Sect. 1.5.3), i.e. thr genesis of forms, if you prefer, which inaugurates radically new relationships and plans of composition. When we read in L’expérience diagrammatique about “redistributions des rapports”,163 it is precisely to the liberation of figures that Batt is referring. Nevertheless, she does take an additional step, that of explicating the heuristic value of such “redistribution” by connecting it to a specific function of Châteletian diagrams. After having remembered that random markings and lines are able for Bacon “to suggest much deeper ways by which you can trap the fact you are obsessed by”,164 she remarked that verbs such as “to suggest” and “to trap” are not only suited to Bacon’s “marques”,165 but also diagrams in Châtelet’s sense as sources of thought experiments (or “allusive stratagems”166 ) which disorient and successively reorient the “working mathematician”.167 Dondero’s valuable remarks about change and the gestural manipulation of chance (Sect. 1.6.1), though they do not mention Châteletian diagrams, fit perfectly into this order of ideas. Before discussing them, however, a terminological clarification is necessary.

156

Cf. [69]. Bacon in Sylvester. 158 Ibid. 159 Cf. Bacon in Sylvester. 160 Bacon in Sylvester. 161 Ibid. 162 Deleuze. 163 [12, p. 14]. 164 Ibid. 165 [56, p. 13]. 166 [52, p. 13]. 167 See [70]. 157

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1.6.1.2

F. La Mantia

Free Fluctuations of the Body

Words like “random” or “chance” do not fit into Châtelet’s epistemology. EM presents chance as belonging to a “puerile display”168 of notions, such as those of chaos and fractal,169 all despised by the author. We are unable to justify the reasons for this contempt: these are, in fact, critical remarks having not been sufficiently explained by the French mathematician. In any case, we find it necessary to inscribe Batt’s and Dondero’s approaches within a theoretical framework regarding Châtelet’s view. Of course neither of these would need adjustments: as we’ll see, Batt’s analyses flawlessly connect Deleuzian hetero-genesis to Châteletian diagrammatics. Moreover, precisely because they do not refer to Châtelet’s works, Dondero’s analyses enjoy sufficient margins of autonomy to be understood in their original formulation. And yet, there is a notion that would avoid any reference to chance while fitting perfectly into the two authors’ main assumptions. We refer to Zalamea’s free fluctuations of body.170 According to the Colombian mathematician, they designate the ways by means of which spontaneous and/or wavering gestures of the “working mathematician” enact more or less sizeable fragments of “conceptual thinking”.171 Speaking of “conceptual thinking”, certainly, would seem to loosen more than to strengthen the possibility of a bond between Zalamea’s perspective and those of Batt and Dondero: Bacon’s gestures, which the authors reflect upon, are consciousness-free (Sects. 1.6.1 and 1.6.1.1). One of the painter’s key assertions, not coincidentally, is that “If anything ever does work in my case, it works from the moment when I don’t consciously know what I am doing”.172 Thus, it is difficult to assimilate such “doing” to the free bodily fluctuations of the “working mathematician”. The thinking they enact, as “conceptual”, is anything but not conscious. Nevertheless, deeper analysis comfort our original intuition. Indeed, again according to Zalamea, not even the working mathematician initially knows what he’s doing as he let’s his body fluctuate freely. As you can read in his text, “[t]he process of thinking was first enacted by the body […] in an unclear, vague and free disposition and only afterwards it was drawn by the mind”.173 So, if you take “mind” as a short form of “conscious mind”, you can find a relevant contact point between free fluctuations of the body in Zalamea’s sense and not conscious doing in Bacon’s sense. In both cases, that is, a consciousness-free gesticulation develops from the body by releasing figures (Bacon/Deleuze) or by materializing mathematical concepts (Zalamea). The assimilation of the two gestural constructs holds at least in an other respect, namely that of their allusivity: as they develop, thus expanding174 the body that 168

[52, p. 9]. Cf. Ibid. 170 See the paper of Fernando Zalamea in this volume. 171 Cf. Zalamea. 172 Bacon in Sylvester (1993), p. 53. Cit. in [56, p. 46]. 173 Zalamea in this. The italics are in the text. 174 Cf. Zalamea. 169

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articulates them, they unfold potentialities, they let glimpse, in short, what is not yet (new forms, new ways of conceptualization, etc.) in what already is.175 Hence, a third shared feature: their being exposed to gradual “coming into consciousness”. Insofar as they are allusive, they are also variously accessible to the consciousness of the agents from whom, however, they initially escape in order to constitute themselves as such (Sect. 1.6.1.1). We find this last point to be absolutely fundamental. It is indeed an important dynamical feature which fits perfectly into Batt’s and Dondero’s views of the diagrammatic gesture: when the former refers to the double movement of disorientation and orientation induced by Châteletian diagrams (Sect. 1.6.1.1), and when the latter refers to the tension between chance and the gestural adjustment of chance which occurs in Bacon’s gestures (Sect. 1.6.1.1), it is precisely of the gradual re-appropriation of involuntary gesture by consciousness that they are talking about. This dialectic of conscious and not conscious tendencies is the vital epicenter of the diagrammatic gesture, and, though Zalamea’s paper is not directly concerned with heterogenesis, we consider free fluctuations of body to be an ideal candidate for launching an appropriate description of what we previously called vectors of heterogenesis (Sect. 1.6.1.1). More precisely, we believe there are sufficient margins to derivate from the Zalamean concept what we do not hesitate to call its heterogenetic variant. Indeed, insofar as the body of the working mathematician enacts fragments of conceptual thinking in an unclear, vague and free way, all the conditions are there for locating germs of a not conscious doing in the free fluctuations of the body. In other words, these are “sketches of action” which, being able to diverge from previous conscious intentions, trigger deformations in Bacon’s sense. We thus propose to use the locution “free fluctuations of the body” not only to designate enactions of conceptual thinking, but also to denote praxeological preconditions for the emergence of the deforming gesture. This is why in the following paragraphs we will discuss Batt’s and Dondero’s theses in terms of free fluctuations of the body.

1.6.1.3

Diagrammatic Gestures Aren’t a Dadaist Practice

Let’s start from Dondero’s analyses. As we saw, the author presents deformation in terms of a change that is unforeseeable and “non procédurale[…]”176 (Sect. 1.6.1). The term “not procedural” is fundamental for understanding such a process: it should be understood, in our opinion, as a synonym of “not rule-governed”. Indeed, according to some mathematicians, the absence of rules is the main trademark of deformations. For instance, by discussing some aspects of Gestalt Theory, Swiss mathematician Guerino Mazzola observed that “deformation is not transformation since we do not refer to any transformation rule whatsoever […]”.177 If we neglect the deliberate redundancy of text, and if consider only what is intended to be emphasized 175

Cf. [52, p. 19]: “Potential is what, in motion, allows the knotting together of an ‘already’ and a ‘not yet’”. 176 [68, p. 84]. 177 [71, p. 276].

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in it, it is precisely the absence of rules (“we do not refer to any […] rule”) that is chosen by Mazzola as the “watershed” between transformations and deformations. This parameter, of which we have already occasion to refer (Sects. 1.5.5.1 and 1.5.6), in reality captures only one aspect of differences at play. Anyway, it ties in very well with Italo-Belgian semiotician’s point of view. Although in some areas of mathematics (e.g. topology) the word “deformation” is commonly referred to functions (and thus to rules178 ) of a certain type, so-called “homeomorphisms”,179 the choice of disentangling the concept of deformation and the idea of rule-governed processess allows us to capture the shattering of the figurative links underlying the Baconian gesture. Indeed, while a transformation (e.g. a homothety) T is generally a reversible process preserving a figurative link between the geometric set of departure and the geometric set of destination,180 a deformation in Bacon’s/Deleuze’s sense is a change which is just defined in function of figurative links that it radically breaks. It is such an aspect of heterogenesis that Dondero highlights when she remarks that “Lancées au hasard, ces marques libèrent les liens, les font exploser”.181 If one substitutes “Lancées au hasard” with “produced by freely fluctuant gestures”, one will obtain one of those variants of the deforming gesture which is compatible with Châtelet’s diagrammatic epistemology (Sect. 1.6.1.1). If we have chosen to express ourselves in these terms, it is not simply because we wanted to respect the letter of this epistemology (Sect. 1.6.1.2), but also because we aim at avoiding a conceptual misunderstanding that could be precisely suggested by words such as “random” or “chance”. Such expressions, without appropriate clarifications, risk of reducimg the praxis of the “working mathematician” to a sort of Dada-insignificance. For if “Dada does not mean anything”,182 mathematical praxis does have meaning. We believe we can moreover say the same about Bacon’s pictorial gesture. Albeit deformation is the main device underlying the liberation of the figure, “involuntary 178

NB: a function (or map of sets) is not the rule itself, but “what the rule accomplishes” (cf. [72] p. 23]). Therefore, different rules can accomplish the same function or morphism. Nevertheless, the level of generality of such notes allows us to use “function” and “rule” as interchangeable terms. 179 A homeomorphism (or topological transformation) is a “bijective and bi-continuous map” (cf. [73] p. 100]). It is, in other words, an invertible mapping that preserves continuity (cf. [74, p. 193]). Therefore, according to this informal definition, in a homeomorphic framework, “if p is close to q, fp is close to fq” (cf. [72, p. 100]). Indeed, in a general sense, a continuous map is informally defined as a map which preserves contiguity (cf. [75, p. 6]), namely the relationships of proximity or nearness between points belonging to a given space. 180 For instance, let’s take the case of two homothetic triangles on a plane, namely ABC and A’B’C’. By supposing that ABC is the geometric set of departure of T and A’B’C’ one of destination, you can define a T’ inverse map from A’B’C’ to ABC. That is, if T is a homothety with homothetic ratio r = 3, T’ will be a homothety with homothetic ratio r = 1/3. So, once the correspondences and their respective homothetic ratios are established, it will be possible to move from one geometric set to the other. More precisely, thanks to a whole series of geometrical features remaining invariant under T/T’, you can capture some similarities of form between ABC and A’B’C’. In other terms, you can see A’B’C’ as a magnification of ABC and ABC as a shrinkage of A’B’C’. Therefore a reversible and figurative links-preserving process will be deployed by the couple T,T’. 181 [68, p. 85]. 182 Tzara 1918.

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marks”, “accidents”—and not consciously doing—aren’t enough to bring out images on the canvas. The image’s emergence, as indeed the emergence of meaning, needs much more than this, that is to say, what Dondero, and before Dondero, Deleuze, call “marques manuelles”183 or “geste manipulateur”.184 It is to these “praxeological elements” that we referred when we spoke of the gestural adjustment of chance (Sect. 1.6.1.1). If now we try to reconceptualize even this second phase of heterogenesis in terms of Zalamea (Sect. 1.6.1.2), an appropriate reformulation could be one of “gestural adjustment of free fluctuations of the body”. Thanks to this terminological move, we not only preserve heterogenesis from insignificance, but we also set conditions for a general dialectic of diagrammatic gesture. Such a dialectic is for us a fluid intertwining of free bodily fluctuations and gestural adjustments. Dondero elegantly captures such a dynamical interaction when she summarizes the main phases of Bacon’s gesture as follows: “1. […] perception d’une visualité prétablie; 2. geste lancé au hasard [or “free fluctuation of the body”]; 3. reaction de la toile à ce geste [or gestural adjustment]”.185 It is, however Batt who effectively links the terms of this dialectic back to the lexicon of Châtelet’s analysis.

1.6.2 Disorientation/Orientation: Explaining Châtelet’s Lexicon In particular, we see that Batt modulates the dialectics of diagrammatic gestures in terms of the “disorientation/orientation” couple. As the reader will remember, we referred to such a couple in previous paragraphs (Sects. 1.6.1.1 and 1.6.1.2). If we correctly understood, “disorientation” is the term Batt employs to denote an epistemic effect coinciding with the “subversion of the habits associated with sensible clichés”.186 Conversely,“orientation” is not a term employed by the author. Nonetheless, it belongs to a part of Châtelet’s lexicon187 that she knows well and that designates a sort of internal potentiality of disorientation: a kind of not conscious reservoir of creative resources coming into consciousness and from which novelties emerge. There is no doubt that the main phases of Bacon’s gesture (Sect. 1.6.1.3) fit perfectly into such an interplay of complementary polarities: the free fluctuations of the painter, which fall under the second step of Dondero’s analysis (Sect. 1.6.1.3) are vectors of disorientation, while the resulting gestural adjustments, which fall under the third step of the same analysis, are vectors of orientation. In short, the liberation of figures

183

Deleuze (1981), p. 90, cit. in [68, p. 86]. [68, p. 86]. 185 Ibid. p. 86. 186 [52, p. 13], cit. in [12, p. 21]. 187 Ibid. 184

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(Sects. 1.5.3, 1.6.1.1, 1.6.1.2) in Deleuze’s sense can be described as a double movement of disorientation/orientation within which orientation is identified as an internal articulation of disorientation. This is an efficacious synthesis, especially since it eludes the risk of thinking free bodily fluctuations and gestural adjustments as slots of a sequential chain. It is, however, some of Châtelet’s pages that give in our opinion one of the more exciting intellectual insights about such a double movement. We refer to the last paragraph of EM’s first chapter.

1.6.3 Between Complex Analysis and Algebraic Topology: First Remarks Cauchy and Poisson’s virtual cut-outs188 —this is the title of the aforementioned paragraph—includes some of the more important topics of Châtelet’s philosophy of mathematics: from the virtual 189 to the gesture,190 up to the carnal nature191 of diagrammatic experiments. Thus it is not by chance that it has been examined by many outstanding scholars (Alunni,192 Batt,193 De Freitas & Sinclair,194 Roy,195 etc.). The double movement of disorientation and orientation concerns this paragraph insofar as it highlights a whole series of “thought experiments” (expériences de pensée196 ) which on the one hand subvert some stable acquisitions of mathematical knowledge, but which on the other disenclose new and unexpected planes of formal reasoning. We are referring to some Châtelet’s remarks about the nature of geometrical points. These observations condense several and important theoretical results of complex analysis and algebraic topology. In essence, however, they can be described in an intuitively clear form.

1.6.3.1

What is a Point?

Scientific papers such as “L’enchantement du virtuel” and “La philosophie aux avantpostes de l’obscur”, or interviews like “Mettre la main à quelle pâte ?” and “La mathématique comme geste de pensée” highlight very well the intuitive core meaning of preceding topics. Let’s take, for instance, some passages of the last text. By 188

[52, pp. 32–36]. Ibid. p. 32. 190 Ibid. p. 33. 191 Ibid. p. 34. 192 Cf. [76, pp. 175–176]. 193 Cf. [56, pp. 39–41]. 194 Cf. De Freitas and Sinclair [8, pp. 67–68]. 195 Cf. Roy [77, pp. 69–93, 105–107]. 196 Châtelet (2010) p. 167. 189

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responding to questions posed by French philosopher Catherine Paoletti, Châtelet gives a clarifying example of tensions between disorientation and orientation in mathematics. He consider, in particular, how points have been conceptualized in different ways over time. The set of these conceptualizations shows, indeed, that a certain idea of point has been radically subverted in favor of new and heuristically fruitful visions of the same mathematical entity. According to this idea, which Châtelet calls “usual” (usual197 ) and which corresponds, roughly, to the perspective of the classical geometry, a point is “la plus petite portion d’espace concevable”.198 Historically, there have been several ways of presenting such an idea: from the point as “that which has no part”199 to the point as a “position, a semeion, on a line”,200 or still, from the point as a “submanifold of dimension zero”201 to the point as a “compound concept […]identified via its numerical coordinates”.202 All these presentations, which differ in their degrees of formality and in their ways of describing a “piece” of mathematical reality, are subverted in the sense of being led back to an integrally general vision of point of the past. It is what the author calls in the same interview “une espèce de concentration circonstancielle”.203 This expression, which does not shine for its clarity, is explained by Châtelet a few lines later when he observes that since a certain moment “l’on comprends que le point n’est plus x, y, z mais une sèquence des sphères dont le diameter tends vers zéro”204 (see Fig. 1.3). The general vision which is thus subverted is the one that makes points the outcomes of mere acts of designation or of “pointing gesture[s]”.205 Letters such as “x”, “y” and “z” are in fact the main symbolic trademark of such acts or gestures. On the other hand, seeing a point such as the “point focal d’un ensemble de petites sphères qui convergent vers lui”206 allows to transcend the limits of any designative ontology. Although a point is identifiable from an act of designation, the new way of presentation shows that the mathematical ontology of points is richer than the one that makes them the simple output of a pointing gesture: such a result is something that is intrinsically static, namely an element that entirely resolves itself within the circumscribed boundaries of a symbolically encoded position (or place). On the contrary, a sequence of spheres whose diameter tends to zero is something of intrinsically dynamical. It is, in other words, a process which makes a point

197

Ibid. p. 180. [15, p. 32]. 199 [78, p. 147]. 200 [79, p. 182 n. 3]. 201 Gunning (1990), p. 49. 202 [71, p. 178], see also Fleury (1986), p. 174. 203 Châtelet (2010), p. 180. 204 Ibid. 205 [71, p. 178]. 206 Châtelet (2010), p. 180. 198

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Fig. 1.3 “une sequence des sphères dont le diamètre tends vers zéro”. The picture is taken from: https://catlikecoding.com/unity/tutorials/basics/jobs/

a “locus of contraction” or “une puissance de compression”.207 Moving therefore from one conceptual framework to another, it is possible to find the main phases of the double movement of disorientation and orientation which characterizes a diagrammatic gesture in Châtelet’s sense. More precisely, we find the dialectical inter-play in which the subversion of a certain codified representation (the point as a result of an act of designation) contains the emergence of a radically new theoretical plane (the point as a locus contraction or “puissance de compression”). But there is more.

1.6.3.2

Between Contraction and Dilatation

According to the French mathematician, a point it is not only a “locus of contraction” but also, and at the same time, a “puissance de déchaînement de virtualités”.208 Put in more metaphorical terms, it is a “bombe[…] implosive[…] ou explosive[…]”.209 If a sequence of spheres whose diameter tends to zero is a good mathematical image of the “implosive side” of points, the explosive one could be described in terms of their mobility. In truth, the image of the spheres could be also a good presentation of the latter side, or at very least of a particular aspect of it. In fact, if “déchaînement de virtualités” is meant in the epistemic sense of “deployment of new ways of reasoning”, and if the metaphor of explosion primarily refers to such sense, then seeing a point as a sequence of decreasing spheres could not only reveal the “implosive” side of points, but also show the way in which such a side involves an “explosion” or, plainly speaking, an “orientation” in the sense of 207

Ibid. Ibid. 209 Ibid. 208

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Fig. 1.4 “the potentiality of points”. The picture is taken from Châtelet (2010)

the paragraphs Sects. 1.6.2 and 1.6.3. It is, however, very difficult to establish what is primary in Châtelet’s thinking. The metaphor of explosion admits, besides an epistemic reading, an ontological one. According to the latter, points are “explosive” in the sense of constituting the exact complement of a “locus of contraction”, that is, a “locus of dilatation”. And we say “dilatation” to indicate everything in the point that potentially transcends and expands its ordinary dimension. Here’s why we proposed to link the “explosion” of points to their mobility.

1.6.3.3

Becoming-Line

Some passages of L’enchantement du virtuel clarify this aspect with great accuracy. After having again criticized the epistemological presuppositions of any designative conception of points,210 Châtelet presents them in terms of “puissances d’explosion de droites”.211 In the same text, indeed, one can also read that points are the potentiality of an “intersection de droites [and] […] d’intersection de courbes”212 (see Fig. 1.4). It is, however, the first way of presentation that perfectly fits into a vision of points as “loci of dilatation”. By linking them to potentialities of explosion of straight lines, the author shows exactly in what sense they are able to transcend their ordinary dimensions: a straight line is a manifold of dimension 1. A point is therefore virtually beyond its ordinary dimension insofar as it subsists, as a “puissance d’explosion de droites”, in its “becoming-line”. Now, mobility is the generative matrix of such a becoming: “points in movement” are, in fact, straight lines, or rather, germs of straight lines. It is from the comparison with Hermann Grassmann’s theoretical works that

210

Cf. Châtelet (2010), p. 135. Châtelet (2010), p. 135. 212 Ibid. 211

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Châtelet derives and develops his main insights into such an important topic of mathematical thinking.

1.6.3.4

Germs of Straight Lines

Some pages of La chimie spatiale de Grassmann give, in our opinion, an excellent overview of such insights. The generative work of “mobile points”,213 on which Alunni has dwelt extensively,214 is described by Châtelet with particular reference to two main concepts of its theoretical imaginary: that of “éclosion du point mobile”215 and that of “dimension qui fait allusion à une autre”.216 As regards the second concept, it is “allusion” the key word. We previously examined “allusivity” (Sect. 1.6.1.2), which is a paronym of “allusion” and which refers to germinal potentialities of the “not yet” in the “already” (Sect. 1.6.1.2). Of course, “allusion” also does have this meaning. In any case, the geometrical framework in which it is inscribed, i.e. the theory of dimension, seems to further specify its contours. Indeed, when we spoke of allusivity, we were very sparing with examples. We used such a word only to mean a virtual and unspecified deployment of new mathematical entities (or forms) and new ways of reasoning (Sect. 1.6.1.2). Talking instead about dimensions alluding to other dimensions allows one to anchor such a deployment in a precise mathematical content: the “becoming-line” of mobile points (Sect. 1.6.3.3). This becoming is allusive, in the sense of paragraph Sect. 1.6.1.2, insofar it reveals how points are germs of straight lines, that is to say, germs of geometrical entities whose dimension is greater than zero. Thus, the concept, or rather, the intuitive image of the disenclosure (éclosion) of mobile points is fully contained within. Moving, namely shifting “d’un point à un autre sans lacune”217 and in a set direction, the mobile point disencloses itself in the sense of generating a straight line, or, at least, a finite linear segment “porté[…] par une même droite”.218 We thus find, in essence, the main argumentative steps justifying the theoretical nexus between the mobility of points and their virtually explosive nature. It must be noted, however, that Châtelet’s remarks go even deeper.

213

Cf. Châtelet (2010), p. 189. Cf. [15, pp. 31–38]. 215 Châtelet (2010), p. 190. 216 Ibid. 217 [80 p. 180]. 218 Ibid. 214

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1.6.3.5

33

Dasein and Mobile Points

After having observed that for Grassmann, a segment “AB”219 is a “chemin parcouru dans un sens opposé à BA”,220 the author details the generative work of mobile points with particular reference to a notion, that of orientation, which has already emerged several times. In previous paragraphs (Sects. 1.6.2, 1.6.3.1 and 1.6.3.2), we spoke of this notion to refer to the production of new concepts involved by the radical subversion of stable and seemingly immutable cores of knowledge. Such an epistemic sense of “orientation”—in which the epistemic is however intertwined with the ontological—is invariably present in Châtelet’s thinking. Even the remarks devoted to the mathematician from Stettin bear a trace of this particular usage of the word. For instance, among these, there is one in which you read that “[l]’orientation a à voir avec l’éclaircissement des virtualités de l’être”.221 In fact, seen in this light, the word “orientation” would seem to come close to the dual meaning, also ontological and epistemological, of “explosion” (Sects. 1.6.3.2 and 1.6.3.3). Nevertheless, besides such affinities, it is precisely observations of this nature that deepen the semantic potential of “orientation” in a sense of the term that we propose to call carnal and that just concerns the generative work of mobile points. Indeed, albeit expressions such as “éclaircissement” and “virtualité de l’être” respectively refer to the epistemic and the ontological moment of orientation, the previous passage is inscribed within in a larger theoretical framework concerning a possible gestural source—and in this sense “carnal”—of the geometry of mobile points. We refer in particular to those lines of La chimie spatiale in which Châtelet discusses the human being’s primordial experience of inhabitation/appropriation of space, or “Dasein”.222 This experience is presented in the text as inextricably connected to two fundamental ecological operations: moving in space223 and the ability of qualitatively distinguishing between a left-hand and a right-hand side.224 By linking the notion of orientation back to the nexus of these operations, and by presenting it in terms of a gesture of orientation (geste d’orientation225 ), the author seems to identify in the mobile corporeality of Dasein the matrix, or one of the possible matrices, of the conceptualizations underlying the generative work of mobile points. In other words, he seems to suggest that the becoming-line of mobile points enacts on a formal plane “manières de se mouvoir”226 constitutively belonging to the affordances of Dasein. We say “seems”, however, because Châtelet is far from justifying analytically a possible link between these two orders of facts. 219

Châtelet (2010), p. 188. Ibid. 221 Ibid. 222 Ibid. 223 Ibid. 224 Ibid. 225 Ibid. p. 187. 226 Ibid. p. 189. 220

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1.6.3.6

F. La Mantia

Still on Mobile Points: For a Carnal Philosophy of Mathematics

As it is well known, the history of philosophy of mathematics of the last 30 years has been marked by a so-called embodied turn which has its roots in certain fields of contemporary cognitive sciences and that has explicitly questioned whether or not “the system of mathematical ideas”227 can be rooted in bodily experiences or, in a larger sense, in the sphere of “everyday life”.228 Whatever one may think, such an embodied approach to mathematics would have all the requirements for highlighting the empirical conditions of the constitution of an eventual bond between the mobile corporeality of Dasein and the presumably unconscious conceptualizations from which the geometry of mobile points develops. In fact, we learn, again from Châtelet, that “la distance de A à B n’est que le résidu d’un mouvement de translation qui a dû être décidé dans un sens ou dans l’autre”.229 The generative work of mobile points would depend in short on a “decision”, or rather on a “gesture of orientation” that ultimately would belong to a particular mobile corporeality: that of “géométre qui parvient […] en quelque sorte à s’incarner dans ce point mobile”.230 Now this “incarnation”231 would fit perfectly into the kind of embeddings envisaged by an embodied philosophy of mathematics: it could be considered, that is, as a particular outcome of the primary interplay that embodied perspectives on cognition postulate between conceptual system and bodily experiences. Besides, if you keep reading the dense and beautiful pages of La chimie spatiale, it is possible to come across some remarks that, at least in principle, support the compatibility of some aspects of Châteletian philosophy of mathematics with embodied perspectives. For instance, the assertion that “[c]ette ‘incarnation’ suppose l’expérience de l’orientation dans un tel ou tel sens”232 brings the mobile corporeality of Dasein to the fore. According to the just mentioned assertion, that is, the geometer can make him or herself a mobile point, namely embody it, and in this sense, access its deepest formal meanings, to the extent that he or she can draw on a implicit background of bodily experiences (e.g. movements to the left, movements to the right) in which the unconscious germs of such a mathematical entity would be rooted. Therefore, it is precisely from such an assertion, or from other similar ones, that it is possible to guess margins of compatibility between an embodied philosophy of mathematics and that of Châtelet: if, in fact, presupposing “l’expérience de l’orientation” means to root the geometry of mobile points in bodily movements which one makes experience in everyday life, then such a presupposition could be integrated into the main assumptions of an embodied philosophy of mathematics.

227

[81] p. XIV. Ibid. 229 Châtelet (2010), p. 189. The italics are in the text. 230 Ibid. 231 Ibid. 232 Ibid. 228

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1.6.3.7

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A Carnal Philosophy is not Exactly an Embodied Philosophy

On the other hand, however, we find that the philosophical project of Gilles Châtelet cannot be integrally assimilated to an embodied philosophy of mathematics. What is missing in the former, and what is massively present in this latter philosophy as well as in the neuroscientific approaches from which it is inspired, is the explicit identification of cognitive mechanisms reputed to import “modes of reasoning from the sensory-motor experience”.233 These mechanisms, known as “conceptual metaphors”,234 have become more and more central in embodied philosophies of mathematics, whereas they are not, it appears, at all covered by Châtelet at all. This is why we have been particularly cautious in drawing any conclusion about what the author would seem hint at regarding the relationships between mobile corporeality of Dasein and the geometry of mobile points (Sect. 1.6.3.5). Of course, it is nevertheless undeniable that the body—and, in a stronger phenomenological sense, the flesh (la chair)—is one of the main theoretical epicenters of Châtelet’s philosophy of mathematics. Again in the interview with Paoletti (Sect. 1.6.3.1), you can read that “il s’agit de comprendre les mathématiques comme chair”235 and that such a point of view “permet […] de comprendre les rapports entre physique et mathématique”.236 Regardless of how one wishes to evaluate the last assertion, this certainly would argue in favor, if not of a direct assimilation, at least of a more than significant affinity between the two theoretical orientations. The theme of flesh, namely the reference to the body not as mere “physical object” (or “corps physique”237 ) but as bio-praxeological locus, as “Leib”,238 that is, as unfolding and enclosing within itself possibilities of action,239 is indeed massively present in the more recent developments of the embodied paradigm. And that, at least in principle, should once more confirm possible affinities between the perspectives we are discussing here. But the centrality of such a theme, though fundamental, it is not enough to bridge the big gap that separates, as we have already said, Châtelet from the embodied paradigm in the philosophy of mathematics.

233

[81, p. XII]. Ibid. 235 Ibid, p. 179. 236 Ibid. 237 Ibid. 238 [82, p. 284]. 239 Or rather of perception in action and of action in perception. 234

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1.6.3.8

F. La Mantia

Against the Neural Reductionism: The Carnality of Mathematical Eidos

Besides conceptual metaphors (Sect. 1.6.3.7), the main watershed between the two research programs is represented by the naturalistic option for which the cognitive sciences have advocated since their founding. This option has resulted in a massive use of the explanatory methods of the neuroscientific discourse. From the point view of an embodied philosophy of mathematics, a possible grounding of mathematical concepts in our daily lives “must somehow be characterized in the neural structure of our brains”.240 For Châtelet, on the contrary, the eidos of mathematical rationality, though carnal, is in large part external to the level of neural constitution. The French mathematician, in short, while recognizing that mathematical practice, like any human practice, is a form of manifestation of brain activity, refuses to reduce the understanding of mathematical rationality—its eidos, specifically—to the understanding of the cerebral dynamics underlying it. One could say then that Châtelet’s carnal philosophy comes close to a sort of weak embodied philosophy of mathematics—and we say “weak” in the same sense in which Lakoff and Núñez use this adjective. The two cognitive scientists distinguish, in fact, between weak and strong approaches to the embodiment of mathematics.241 “Strong” approaches in particular are said to be all those approaches aiming not only to give a “biologically based account of the [cognitive] mechanisms by which they [i.e. the mathematical concepts] are created, learned, represented, and used”,242 but also to explain their heuristic potential in terms of neural correlates of such mechanisms. It is from these explicative claims that Châtelet distances himself. The thesis underlying the strong embodied approaches is that mathematics is an intellectual “enterprise that is [not] equipped to answer scientific questions about mathematic cognition”.243 Therefore only through a refined analysis of the cerebral activity—a task historically pertaining to neurosciences and cognitive sciences—it is possible to have adequate theoretical tools to understand “cognitive mechanisms involved in conceptualizing and learning […] definitions, axioms, theorems, and proofs244 ”. For his part, the French mathematician wonders instead what exactly means conceptualizing and learning—in one only word, “comprendre245 ”—in mathematics. The question is raised, it seems to us, with the specific intent of disqualifying the main empirical assumptions of strong embodied approaches from an epistemological point of view. The author, in fact, continues by observing that to know “que si on fait telle multiplication ou telle addiction […] cela illumine telle partie ou telle autre du cerveau

240

[81, p. 347]. Cf. Ibid. 242 Ibid, p. 349. 243 Ibid. 244 Ibid. 245 Châtelet (2010), p. 181. 241

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[…], cela ne répond pas à la question !”.246 On the other hand, one might observe that such skepticism towards the explanatory potentials of neuroscientific inquiry is what prevents Châteletian analyses from being compatible with any naturalistic point of view. This, for example, is what a fine mathematician such as Giuseppe Longo seems “to reproach” to Châtelet for when, while declaring himself in admiration of Châteletian enterprise,247 he claims that “it is necessary to “naturalize” this concept of gesture much more than what has been done by Châtelet”.248 And this insofar as “a limit to his thought lies in a refusal of the animal life experience which precedes our intellectual life, and also in the lack of an understanding of the biological brain as part of the body”.249 These are undoubtedly important remarks, of which we also share the proposal that animates them, namely the idea that the “body allows gesture among humans, not only in a historical but also in an animal dimension”.250 We believe, however, that Châtelet’s skepticism, so to speak, is far from anti-naturalistic. In fact, the question from which the French mathematician takes his cues is accompanied by a declaration of materialist faith that is inscribed within, and not outside, the boundaries of a naturalized epistemology. Nevertheless, this inscription does not prelude reductionist solutions. In other words, if Châtelet can affirm to be “matérialiste”251 and not at all dispute that there is “« détection», trace matérielle de la pensée”,252 he can, on the other hand, observe that the incontestability of this empirical fact is not enough to explain the conceptual work enactivated by the mathematical practice (and/or gesture). And this is because—according to the French mathematician—the possibility of locating areas of the human brain deputed to the execution of operations or to the learning of theorems and definitions “ne permet pas de comprendre ce qui fait que telle chose fasse surgir telle autre”.253 Now, if such a theoretical posture were evaluated to be anti-naturalistic, to the extent that it would be considered to unduly enfranchise mathematical heuristics from neuro-biological analysis, then yes, one could reasonably assume that Châtelet’s attitude is radically anti-naturalistic. But in the French mathematician’s remarks we find nothing inappropriate. If anything, they are reflections, in our opinion well thought out ones, that invite to distinguish between different explanatory planes.

246

Ibid. Cf. [83, p. 105]. But see also [84, p. 159]. 248 [83] p. But see also [84, p. 159]. 249 Ibid. 250 Ibid. 251 Châtelet (2010), p. 181. 252 Ibid. 253 Ibid. 247

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1.6.3.9

F. La Mantia

Still Against the Neural Reductionism, or Which Platonism?

Producing new concepts, relating them, and even creating “des possibles qui ne sont pas là!”,254 are undoubtedly all brain-rooted practices, but such a neural rooting, if it tells us anything important, what it tells us about the brain, not about the eidos of mathematical rationality. We know that such an assertion could be read as “Platonist”, and perhaps it is. Nevertheless, we do not intend to commit ourselves in favor of a particular philosophy of mathematics, nor to enter the vast and complex galaxy of theoretical perspectives about the ontology of mathematical objects. Longo, for his part, has worked hard these years to develop a philosophically reliable critique of Platonism, or, at least, of what he himself has not hesitated to present as its naive forms.255 Châtelet, on the other hand, would have at least pronounced one in favor of a Platonist conception of mathematics.256 One could really set the conditions for a Platonic reading of what we have just argued. But this would cause us precisely to fall in that “quagmire” of visions and perspectives that we would like to avoid examining because they are far from the specific objectives of our Introduction. Indeed, if the explanatory autonomy of the mathematical eidos was interpreted in a Platonic sense, one would need to specify what kind of Platonism it involves. It could be, for instance, an ontological Platonism, that is to say, a vision of mathematical entities as mind-independent objects, or a transcendental Platonism in Petitot’s sense (henceforth, PTP), or still a dynamical Platonism in Lautman’s sense (henceforth, LDP). If we exclude the first variant, against which Longo seems to have directed his main polemics, the last two ones could in fact offer an optimal framework for investigating the main scope of such an autonomy. From PTP we learned in particular that “les idéalités mathématiques”,257 although dependent on particular cognitive gestures (or “synthèses noétiques corrélatives”258 ), retain a significant margin of formal autonomy, namely they aren’t “cognitivement réducibles”.259 Thanks to LDP, moreover, we understood that such idealities or “êtres mathématiques”260 aren’t “êtres statiques”,261 but rather generators of other mathematical beings,262 and that in such a generativity lies the “puissance infinie d’expansion”263 of all mathematical theories. As you can guess, then, the explanatory autonomy of mathematical eidos, even if inscribed within a Platonic framework, shouldn’t necessarily fall into a form 254

Ibid. p. 182. Cf. [83, p. 103]. But see also [84, pp. 137–138]. 256 In this regard see [15, pp. 17–19]. 257 [85, p. 14]. See also Petitot [86, p. 223]. 258 Ibid. 259 Ibid. 260 [87, p. 223], cit. in [76, p.165]. 261 Ibid, p. 226, cit. in [76, p. 164]. 262 Cf. Ibid, p. 187, cit. in [76, p. 172]. 263 Ibid, p. 131, cit. in [76, p. 163]. 255

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of ontological Platonism. But there is more, if we really want to get into the study of such topics. On the one hand, PTP seems to be consistent with a naturalized approach to mathematics; on the other hand, PDL is carnal in a sense of the word that comes close to Châtelet’s perspective. In fact, insofar as PTP excludes that mathematical idealities are “entités séparées et indépendantes”264 from the “actes, [les] états et [les] processus mentaux”265 which establish them as such, it roots a fundamental aspect of mathematical practice—the so-called “transcendental constitution of mathematical objectivity”266 —in the brain’s neuro-cognitive activity. As for LDP, instead, after having remarked that “les schémas logiques [les idées (…)] ne sont pas antérieures à leur réalisation dans une théorie”,267 it presents some mathematical properties, such as “la non-commutativité de certaines opérations abstraites de l’algèbre”,268 in terms of formal features which are susceptible to be symbolized269 “sur le plan de l’expérience”270 by key aspects of mobile corporeality of Dasein (e.g. “la distinction de la gauche et de la droite”271 ). Hence, research programs such as PTP or LDP could allow the explanatory autonomy of mathematical eidos to perfectly fit into an embodied framework without, however, consenting to any form of neural reductionism. But to fully justify this last point would require many more details. First, it would be necessary to explain in what exactly the cognitive non-reducibility postulated by PTP consists, and how such a non-reducibility works with the transcendental constitution of mathematical objectivity. Second, it would be necessary to clarify how the infinite potential of expansion of mathematical theories postulated by LDP unfolds. Therefore, it would deal with theoretical questions such as the “normativité du formel en tant que tel”272 or the “generativity of mathematical beings” that would imprint an abrupt deviation from the theoretical path we set out to terminate. This is why we prefer to limit ourselves to interpreting what we believe to have understood of Châtelet’s discourse as an ontologically neutral claim regarding the explanatory autonomy of the mathematical eidos with respect to the level of neural constitution. We wish, in short, for this eidos to be examined in terms of methods and concepts that exclusively belong to a philosophical inquiry on mathematics, and yet not to be unbalanced neither towards the Platonist pole nor only towards the anti-Platonist one. It could also be objected that this search for a neutral balance between opposing views of the mathematical entity is nothing a pious illusion, since there are always

264

[85, p. 12]. See also Petitot [88, p. 221]. Ibid. 266 Ibid, p. 221. See also Petitot [88, p. 185]. 267 [87, p. 226], cit. in [76, p. 166]. The italics are in the text. 268 Ibid, p. 269, cit. in [76, p. 176]. 269 Cf. Ibid. 270 Ibid, p. 267, cit. in [76, p. 175]. 271 Ibid. 272 [86, p. 14]. See also Petitot [88, p. 223]. The Italics are in the text. 265

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reasons that can push the focus of analysis in one direction rather than the other. But it is this neutrality, however illusory, that we will stick to in these pages.273

Return to Mobile Points: Becoming-Line as Becoming-Arrow Let’s return to the geometry of mobile points (Sects. 1.6.3.3–1.6.3.6). The only thing that these relations allow to assert with relative certainty is that for Châtelet, and even earlier for Grassman, mobile points are more properly germs of arrows or oriented segments. Indeed, if the geometry of mobile points is unconsciously modeled by an implicit knowledge of space which presumably inhabits the mobile corporeality of “Dasein”, then the “becoming-line” of such points is the form of manifestation of primordial gestures of orientation that unbalance (orient, specifically) their development in one direction or another. This is why Alunni can rightly say that “Le «point mobile» de Grassmann […] fait du segment une trajectoire”,274 and this is why Châtelet can assert that “il y a quelque chose de profondément interne au point, […]”,275 namely “une petite flèche qui est là et qui jaillit hors du point”.276

1.6.4 From Grassmann to Klee An excellent speculative archeology of such formal intuitions is offered by Swissborn German artist Paul Klee. We don’t know if he was aware of Grassmann’s works, including the seminal Ausdehnungslehre, but it is sufficient to leaf through the pages of his wonderful Note-Books to realize how pertinent are the observations he devotes to the “genesis of form” and to “motion […] [as] the root of all growth”.277 From them emerge at least three thematic nuclei that are closely connected to the geometry of mobile points: a. The “cosmic”278 nature of point; b. The point “(as agent)”279 ; c. “The point, dynamically seen”.280

273

For a sharper take on these issues, I refer to the essay Diagrammatic Eidos and Dynamical Platonism contained in this volume. 274 [15, p. 32]. 275 Châtelet (2010), p. 141. 276 Ibid. 277 [89, pp. 18–37]. 278 Ibid, p. 19. 279 Ibid, p. 24. 280 Ibid, p. 22.

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1.6.4.1

41

Movement as an Absence of Movement and as an Original Tension

The first nucleus revolves around the germinal potentialities of points: “The point as a primordial element is cosmic. Every seed is cosmic”.281 As we have seen, the cosmogonic force of points is expressed by Châtelet as a becoming-line or a becoming-arrow, or, more generally, as a “puissance de mouvement, […]”.282 Klee, for his part, seems to be totally sympathetic to this point of view. After having rejected the vision of points as zero-dimensional submanifolds,283 the author points out the paradoxical omnipresence of the movement in the formal constitution of such entities. We say “paradoxical” because for the German painter movement is already present in the points in the form of… absence of movement: “Der Punkt […] die Bewegung Null ausführt”.284 This way of presentation, however, captures only one aspect of the constitutive mobile nature of points. According to Klee, in fact, movement is primarily present in them as an original “tension”,285 namely as the willingness of each point to displace, to head towards other points. This is why he can assert that “[t]ension between one point and another yields a line”286 and that “[t]he primordial movement, the agent, is a point that sets itself in motion”.287

1.6.4.2

“Spannungswille”

If, for a moment, we go back to Châtelet, many of the considerations developed by Klee are faithfully mirrored in some of the French mathematician’s most important remarks. This is the case, for example, of what the artist says about the movement as a primordial tension of points (Sect. 1.6.4.1). This tension, that Klee also presents as reciprocal voluntariness of attraction inscribed in and between points, is well highlighted by Châtelet when he discusses some aspects of the Leibnizian philosophy of mathematics. We are thinking in particular of the pages of L’Enchantement du virtuel concerning points as “sources de choses”288 or “créateurs de possibilités”.289 281

Ibid, p. 19. Châtelet (2010), p. 135. 283 Cf. [89, p. 19]. 284 [90, p. 19]. N.B.: in this case we prefer to quote the German text, since we find the English translation of the same passage to not be entirely faithful to the original. In [89, p. 22] we read in fact that “The point […] carries out no motion”. In this sense it is decidedly preferable the Italian translation of the same passage that we report for sake of completeness: “Il punto […] compie il movimento zero” (cf. [91, p. 19]). 285 Ibid. 286 Ibid. 287 Ibid, p. 105. 288 Châtelet (2010), p. 134. 289 Ibid. 282

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Fig. 1.5 “intervalle”. The picture is taken from Châtelet (2010)

Although it is an argument of which we have already given a first outline by analyzing the “becoming-line” of points, the references to Leibniz are capital to understanding what exactly the nature of points as agents consists of. Of course, it has to be said that Châtelet could have been a little more specific from a textual perspective: although very interesting, his reflections presuppose a knowledge of passages and pages by the great German philosopher and mathematician that it would have been good to make explicit with shrewd clarification notes. But, if we disregard this important flaw of structure, setting aside the claims of a certain philological prudery, these are nonetheless enlightening remarks that clearly thematize what is then the second conceptual core of Klee’s speculations: the agentiveness of points, precisely. In fact, when Châtelet observes that “Pour Leibniz […] deux points n’existent pas béatement, mais représentent la possibilité […] de remplir tel trou avec tel […] (intervalle)”290 (see Fig. 1.5), it is the reciprocal voluntariness of the attraction of points that he is discussing, that is to say, their agentiveness. In other words, the passage in question alludes to something surprisingly similar to the “Spannungswille”291 that Klee attributes to points as agents, or, in Châtelet’s words, to points as opérateurs.292 It is in short an internal dynamical force of change by virtue of which each point is in principle a matrix of tracing gestures, or, more precisely, of hand movements which generate lines or oriented segments.

1.6.4.3

Arrows as “Growth of Energy”

Nevertheless, because they are more properly oriented segments (Sect. 1.6.3.8), it is in terms of arrows that it is possible to efficaciously figure Kleeian Spannungswille. If you continue reading the Notebooks, the use of arrows is systematically implemented by the author to put into form (to figure, specifically) the dynamic operativity of points with respect to the line (see Fig. 1.61 ), that of lines with respect to the plane (see Fig. 1.62 ), and that of plane with respect to the three-dimensional bodies (see Fig. 1.63–4 ). This brings us to the third and last thematic nucleus of Kleeian reflection, namely the “point, dynamically seen” (Sect. 1.6.4). In fact, to the extent that “the growth of energy [i.e. the Spannungswille of points] […] determines direction”,293 it is the arrow the latter’s dynamical form.

290

Ibid, 144. [89, p. 23]. 292 Châtelet (2010), p. 186. 293 [89, p. 22]. 291

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Fig. 1.6 “the dynamic operativity of points”. The picture is taken from [90]

1.6.4.4

Interconnections

What, then, is a point? This question, already raised elsewhere (Sect. 1.6.3.1), actually has as for object a wider domain than that of pure (and far from simple!) geometrical points. This is why, having almost reached the end of the analysis, we are proposing it again. The amount of topics addressed in the previous paragraphs is such that they cover all the main topics of a philosophy of the diagrammatic gesture. Reflecting on points, and in particular on their explosive potential (Sects. 1.6.3.2– 1.6.3.4), the themes of gesture, virtual and heterogenesis were addressed. The first one emerged while we were discussing the relationships between the geometry of mobile points and the mobile corporeality of Dasein (Sects. 1.6.3.5–1.6.3.8). Moreover, also thanks to Klee’s remarks, it has been possible further specify its scope by showing how the Spannungswille of points (Sects. 1.6.4.2 and 1.6.4.3) is a matrix of tracing gestures (Sect. 1.6.4.2). As for the second and third theme, namely the virtual and the heterogenesis, they have dominated our inquiries ever since we considered the potential of points in the form of their becoming-line (Sects. 1.6.3.3–1.6.3.5 and 1.6.3.9) or of their becomingarrow (Sect. 1.6.3.9). Thus, given the wealth of thematic references, wondering about the nature of the point implies an answer that is able, while not bypassing the initial question (what is a point?), to take into account the interconnections between the virtual, the gesture and the heterogenesis. Perhaps, but it is a hypothesis to be verified, an answer to these theoretical expectations has been provided by the English mathematician Brian Rotman.

1.6.4.5

The Diagrammatic Life of Points

In his Foreword to Mathematics and the Body, Rotman asserts that “[a] point is the simplest example of a diagram”.294 We believe this to be the answer that is right 294

[92, p. xvi].

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Fig. 1.7 “free lines”. The picture is taken from [90]

for us. In fact, assimilating points and diagrams, the author creates the conditions to conceive of points with respect to the aforementioned connections (Sect. 1.6.4.4). In a Châteletian perspective, to which Rotman seems to adhere, saying “diagram” means to say at the same time “virtual”, “gestural” and “heterogenetic”. In particular, it means to say “virtual” insofar as a diagram—far from being a simple illustration or inscription—is a reservoir of germinal forms (or figures, in the Deleuzian sense of term (Secs. 1.5.3, 1.6.1.1, 1.6.1.2). Finally, it means to say “gestural” and “heterogenetic” because, on the one hand, this reservoir provisionally captures (or “transfixes”) modalities of movements (or gestures) that put it into form, and, on other hand, because it preludes other gestures and thus other forms (or “plastic unities”). Points, therefore, would be diagrammatic insofar as they form: (a) reservoirs of germinal lines and arrows; (b) “snapshots” of specific gestures of pointing (whose symbolic trademark are letters like “x”, “y”, “z”) and of contraction (whose geometrical trademark are sequences of spheres like ones converging to point); (c) matrices of other gestures (e.g. the gestures at the root of lines and arrows) and of other forms (e.g. lines, arrows, but also the infinite variety of curves, or “free lines”,295 that point generate as soon as they set themselves into motion (Fig. 1.7). Lines, arrows and “free lines” would therefore be the main forms in which the diagrammatic life of points unfolds itself. These are forms that are surprising for the elements of novelty of which they are bearers, and that are captivating—especially the magnificent squiggles by Klee—for their extraordinary beauty. And yet, there is more: returning to Châtelet, and in particular to some pages of EM, the diagrammatic life of the points shows a completely unexpected aspect, perhaps the most heterogenetic. It is to this aspect that we will devote the last notes of commentary before heading to the conclusions of our chapter.

295

[89] p. 105.

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1.6.5 Points as Virtual Holes and Zones In a crucial passage of EM you can read that a point is also “that around which one can excavate and which can be given density”.296 Likewise, this passage is echoed at least in the line of “Mettre la main à quelle pâte ?”. There, in fact, it is asserted that a point can become “un trou virtuel, une cavité en pointillé, un germe de cavité”.297 Many other quotations would merit to be reported.298 The ones already mentioned, however, are enough to show a further aspect of the diagrammatic life of points. In short, according to what has been just said, points are not only germs of arrows (cf. Sect. 1.6.3.9) but also germs of holes. Indeed, to the extent that a point “can be given density”, the diagrammatic forms (or heterogenetic virtualities) examined by Châtelet are not reduced to the nevertheless rich duality of “becoming-arrow/ becoming-hole”. There is also, we find, a becoming-zone of points which is an integral part of their diagrammatic life and that therefore enriches the already dense variety of gestures analyzed so far. Indeed, besides the gestures of orientation (Sects. 1.6.3.1, 1.6.3.3 and 1.6.3.5), the geometry of mobile points involves gestures of excavation (points as virtual holes) and propagation (points as virtual zones) that expand the range of associated carnal experiences.

1.6.5.1

Gestures of Excavation

Let’s first of all examine the gestures of excavation. The first thing to say is that such gestures, as well as those of orientation and propagation, once again confirm the epistemological limits of what we have elsewhere called the “designative ontology” of points (Sect. 1.6.3.1). In the perspective unfolded by this ontology a point—as it has been already said—is reduced to a sort of “placeholder” (Sect. 1.6.3.1). Instead, from the point of view we are introducing here, a point, far from being what simply indicates a position299 —is rather something that is available to be extracted from its milieu (from complex plane C, in the simplest case) and thus to be figurated as “that around which one can excavate” (Sect. 1.6.5). So it is in this sense that points begin to live as “potential holes” and to emerge as outcomes of gestures of excavation. Châtelet for his part clarifies very well the meaning of these interrelated aspects by recalling the role played by French mathematician Antoine Louis de Cauchy with regard some important innovations in mathematical analysis. We owe to the latter, as well as to his contemporary physicist Siméon-Denis de Poisson, a formal interpretation of points which transcends the positional codifications of any designative ontology and which therefore “would no longer be happy to 296

[52, p. 35]. The italics are in the text. Châtelet (2010), p. 167. The italics are in the text. 298 Cf. Ibid, p. 168. 299 Cf. Ibid. 297

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indicate points without regard to their potential extraction”.300 Put in more concrete terms, it is thanks to the important insights of the aforementioned scholars that any point of C, namely “P0 ”,301 begins to be thought of as something which is in the condition of “[making] an opening in the plane and [becoming] the visible scar of the incision”.302

1.6.5.2

Mathematical Surgery

Thus, “mathematical surgery”303 was born. The expression is metaphorical and refers to a particular way of looking at points, namely the idea of envisaging them not in themselves but according to the “modalit(ies) of motion”304 that they are able to arouse. One of them, as we have seen, is their extraction (cf. Sect. 1.6.5.1): this is why we can speak of “mathematical surgery”. Extracting a point, taking it from the complex plane in which it was immersed is a way of operating that we may qualify as “surgical” insofar as it involves, albeit ideally, cuts, or rather, “incisions” which recall gestures made by a surgeon. Of course, one could argue that the surgical metaphor—and, more precisely, the assimilation it implies—is too bold to be valid: points are abstract entities as are the “formal niches” (lines, planes and spaces) within which they are placed. How then, albeit metaphorically, can we assimilate the modalities of movement they elicit to the modalities of the surgeon, that is, to the cuts and incisions he makes on the “living flesh” of the patient? In short, the proposed analogical juxtaposition would fail insofar as the domains compared—that of points and that of living flesh—are too distant to be in any way assimilated. On the other hand, however, it is precisely the carnal nature of the mathematical eidos (Sects. 1.6.3.5–1.6.3.8) that strengthens the proposed metaphor. In fact, as the examples referring to the mobile corporeality of Dasein show (Sects. 1.6.3.5 and 1.6.3.6), the margins of possibility for a gestural grounding of mathematical experience are such that even the most intangible abstract entity can present aspects that enact on the formal plan specific forms of the sensible experience. In the case of the becoming-arrow of points these forms concern certain “manières de se mouvoir” in one direction or another (Sect. 1.6.3.5); in the case of their becoming-hole, instead, they presumably concern the ancestral memory of primordial practices of extraction which manifest themselves as cuts or incisions. We say “presumably” because Châtelet is even less explicit on the subject than he had already been about the relationships between the geometry of mobile points and the mobile corporeality of Dasein (Sect. 1.6.3.5).

300

[52, p. 32]. Ibid, p. 33. 302 Ibid, p. 34. 303 Cf. Châtelet (2010), p. 167. But see also Ibid, p. 145. 304 [52, p. 34]. 301

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In the latter case, the French mathematician seemed to suggest, albeit timidly, the possibility of a bodily rooting of unconscious conceptualizations underlying the geometry of mobile points (Sect. 1.6.3.5). This, at least, is what could be inferred from the articulated chain of reasonings that we tried to reconstruct in the previous paragraphs (Sects. 1.6.3.5–1.6.3.8). This chain would seem to identify in some primordial gestures of orientation a possible matrix of the becoming-arrow of points Sect. 1.6.3.5). As for their becoming-hole, instead, the remarks of Châtelet are so “frayed” that they do not allow an analogous work of reconstruction. The references to the metaphor of the “mathematical surgery” are not, in our opinion, formulated with sufficient clarity to identify in the aforementioned cuts and incisions a possible matrix of the becoming-hole of points. Nonetheless, when we read that “la géométrie devient en quelque sorte charnelle”305 and that “le doigt de Cauchy pénètre dans le plan et dégage une cavité autour du point”,306 we realize that metaphor works even without the support of explicit argumentative chains.

“Procès De Concretization” Observations of this tenor are in fact the outcome of a “procès de concrétisation”307 which aims to overcome the hiatus between the sensible and the intelligible within which many of our most common judgments are still forged. In the case we are considering, therefore, the metaphor of “mathematical surgery” is right: it is not by chance that the process in question aspires to the institution of an ontological framework in which points are at the same time ideal and material, virtual and actual, impalpable and carnal. Thus, the ideality of incisions to which they are subjected as becoming-holes it is not opposed to the physicality of the concrete ones, but it is available to present itself and to be grasped in the sensible form of material cuts. In order to understand what this availability consists of, it is necessary to look at the wider spectrum of gestures that the diagrammatic practice of mathematics mobilizes in its very process of constitution. Drawing lines, prolonging them “in dotted lines”, compacting them into closed shapes or, why not, dispersing them in a trail of points filled with virtuality are all actions that the mathematician can imagine with the eyes of the mind. On the other hand, however, the mathematician can also put them on paper, visualizing them by means of ink spots, pencil traces and other material supports that can be used by the “raison graphique”308 of the hands. Now, what it is true for such actions will be even more true for the ideal incisions supporting the extraction of planar points: they too can be put on paper and present themselves in the form of concrete incisions, that is to say, in the form of tears in the sheet enacting the 305

Châtelet (2010), p. 167. The italics are in the text. Ibid. 307 Ibid. 308 Ibid, p. 75. The italics are in the text. 306

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becoming-hole of planar points and making of every mathematical point a material point. This is why De Freitas and Sinclair can assert that “[f]or Châtelet, the diagram constitutes the point as a material bump in the surface of the page”.309 Consequently, considered in its most general form, the availability of the formal idealities to be put on paper (or on any material support the diagrammatic practices elect as their applicative field) consists in a particular type of transitions: those experienced by the mathematician by proceeding from the eyes of the mind to the “raison graphique” of the hands. It is in these transitions that the carnality of the mathematical eidos resides.

1.6.5.3

Latent Materiality

Then, speaking of the “carnality of the mathematical eidos” doesn’t mean to deny the abstract ideality—the same ideality which allowed a very fine mathematician such as Alain Connes to affirm that “la suite des nombres entiers […] a une réalité plus stable que la réalité matérielle qui nous entoure”310 —but it does mean, if anything, to point out that such ideality has in itself a latent materiality. We say “latent materiality” to refer once again to the willingness of mathematical beings to take root in the graphic reason which emerges from the hand movements of the mathematician and which has in the excavation gestures one of its privileged forms of manifestation. Besides them, there are also other kind of gestures, the ones underlying the becoming-zone of points (Sect. 1.6.5). Before discussing this, however, we need to spend a few more words on one last important detail of the becoming-hole of the points.

1.6.5.4

Virtual Cut-Outs

The mathematical surgery of planar points involves not only their extraction (Sect. 1.6.5.1), but also the virtual cut-out of small spaces (or neighborhoods) “surrounding” them. Therefore, the gestures of excavation are composed by a double work of extraction and cutting-out which contributes to the becoming-hole of the points. Still discussing Cauchy, Châtelet indeed remarks that the French mathematician “a construit un circuit (around a point) […]”311 and that “il a imaginé qu’on enlevait le point, et qu’en tirant ça se reserrait”312 (Fig. 1.8). Moreover, in the same spirit, he remarked that “Cauchy […] dessine un cercle en pointillé autour du point”313 (Fig. 1.9).

309

[8, p. 68]. Connes in [93, p. 28]. 311 Châtelet (2010), p. 143. 312 Ibid. 313 Ibid, p. 167. 310

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Fig. 1.8 “un circuit (around a point)”. The picture is taken from Châtelet (2010)

Fig. 1.9 “un cercle en pointillé autour du point”. The figure is taken from Châtelet [52]

Furthermore, he remarked that “L’idée géniale de Poisson ce fut […] de tracer un cercle autour du point, puis le retirer”.314 All the citations reported here show the extraction of planar points as a gesture which works in parallel with another gesture: the one that cuts out the spaces (or neighborhoods) surrounding the extracted points (Sect. 1.6.5.4). Châtelet’s remarks present the becoming-hole of points with respect to the gestures which extract them and with respect to the ones which generate (or “stir”315 ) the neighborhoods “of such points”.316 This “pair work” is important for at least two reasons. First, because it involves at least a third gesture: the one underlying the becoming-zone of the planar points. Second, because the nexus between gestures of excavation (i.e. extraction+cut-out) and gestures of propagation is heterogenetic to the maximum degree.

314

Ibid, p. 145. Cf. [52, p. 36]. 316 Ibid. 315

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Fig. 1.10 “from point P0 arise arrows oriented toward the borders of the neighborhood within which P0 lies”. The picture is taken from Châtelet [52]

1.6.6 Gestures of Propagation: First Remarks The gesture of propagation is what we have just called the “third gesture” (Sect. 1.6.5.4). It is involved by the double work of extraction and cut-out achieved on the planar points (cf. Sect. 1.6.5.4). More specifically, it is the cut-out gestures that trigger the underlying gestures of propagation. A diagrammatic indicator of such a gestural activation is provided by a graphic detail of the Fig. 1.8 (Sect. 1.6.5.4): from point P0 arise arrows oriented toward the borders of the dotted cavity (or neighborhood) within which P0 lies (see Fig. 1.10). The gesture of cut-out, in short, one on which depends the genesis of neighborhoods of P0 (Sect. 1.6.5.3), triggers P0 by setting it in motion. Indeed, it is as if the emergence of one of such neighborhoods “attracted” P0 in direction of the internal edges of the cavity surrounding it. The arrows would then provide a diagrammatic snapshot of this movement of attraction. Once again Châtelet’s remarks nudge us toward this interpretation.

1.6.6.1

“a (Dialectical) Continuity of Gestures”

When he asks how it is possible to “bring to life ‘just any’ point chosen in the interior of317 ” a certain domain (e.g. the D domain of the Fig. 1.9), French mathematician remarks that it is necessary to “make this ‘just any’ singular by clearing a little space around P0 ”.318 As should be clear, then, Châtelet answers the question by referring precisely to one of those cut-out gestures which generate neighborhoods (or “little spaces” around) of P0 . (Sect. 1.6.6). Of course, by mentioning this passage we are still far from having explained in what sense the genesis of a neighborhood of P0 sets this point in motion. What the author says next, however, confirms our working hypothesis. After having asserted, in fact, that is by means of a gesture of cut-out that a ‘just any’ of the points of the plan become a singular (or “remarquable”319 ) 317

Ibid, p. 34. Ibid. The italics are in the text. 319 Châtelet (2010), p. 167. 318

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point, Châtelet suggests that from the cut-out gesture, a further gesture descends: one of “propelling oneself”320 in the newly generated neighborhood of P0 , that is to say, one of “giv[ing] [the] point thickness and hav[ing] it irradiate out by extracting a small disc capable of revealing a geometric matter”.321 We thus find a nexus, actually more imagined than argued, that shows the way in which from the emergence of a neighborhood of P0 one can achieve the setting into motion of the same point. In fact, if the gesture of cut-out is followed by a gesture of propulsion, and if this second gesture involves, among other things, an irradiation of the thickened point, then from the emergence of a neighborhood of P0 follows its putting in motion. On the other hand, however, on the ground of what has been quoted, one could also hypothesize the inverse nexus. Insofar as it is a matter of “giv(ing) a point thickness” and of making it radiate along planar directions that have as their only halting condition the extraction of a disk, one could assume that it is the cut-out gesture that descends from the gesture of propulsion. It could also assumed that it is the propagation movement of the point that generates the borders of the neighborhood that sets it in motion. How are things then? We do not know. It cannot be ruled out however that the two types of gestures imply one other. Châtelet’s, after all, was a highly dialectical scholar, that is, a thinker sensitive to the interdependencies between different aspects of mathematical reality. He would therefore have been more than willing to hypothesize that propulsion gestures follow from cut-out gestures as cu-tout gestures follow from propulsion gestures. Hence, there is a possible bi-directional solidarity between virtual cut-outs and propagations, or, rather, “a (dialectical) continuity of gestures”322 which allows to connect the becoming-hole of the points with their becoming-zone, and vice versa. But how can a point become a zone? How can it be thickened until it changes into a disk or a cavity?

1.6.7 Loops and “Turning Around” We asserted that the nexus between gestures of excavation and gestures of propagation (or propulsion) is heterogenetic in the highest degree (Sect. 1.6.5.4). In saying this, we were referring to the emergence of radical alterities which perturb the continuity (Sect. 1.6.5.4) of those gestures and which imply therefore that “on saute dans un espace tout à fait différent”.323 This space is one of “des deformations continues de lacets”.324 It is from such deformations that the becoming-zone of points begins. Let us recall in this regard that a loop is assimilable to a sort of segment (or edge,

320

[52, p. 34]. Ibid, p. 35. The italics are in the text. 322 Ibid, p. 34. 323 Châtelet (2010), p. 168. 324 Ibid. 321

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Fig. 1.11 Loops. The picture is taken from [75]

according to the lexicon of graph theory) which “connects a vertex to itself”325 (see Fig. 1.11). If we now imagine a point—let’s call it again P0 —and if we install on the plane loops having P0 as their own center, the paths that can be travelled on each loop will provide a first intuitive figuration of the aforementioned becoming-zone. With respect to such paths, in fact, P0 acquires, so to speak, a thickness and becomes, in this sense, a zone. More precisely, it becomes exactly what Cauchy wanted it to be: something that—far from being a mere place-holder (Sects. 1.6.3.1 and 1.6.5.1)—is virtually inscribed in the plane as that around which one turns and turns again and again. This “turning around” (tourner autour 326 ) shows the thickness of P0 since it never crosses through P0 . The latter, if anything, acts as a virtual obstacle to the enclosure of the loops, namely to their point-like reduction. We say, however, “virtual obstacle” because it is not the “being a point” of P0 that hinders this reduction, but its becoming-hole (Sects. 1.6.5, 1.6.5.1–1.6.5.5). Indeed, when Cauchy takes P0 out of the plane C (Sect. 1.6.5), what remains in C is a hole: something that by constitution cannot be crossed without the solution of continuity and which therefore “nous oblige à contourner”.327 Seen in this light, i.e. envisaged such as a “becoming-hole”, P0 is then “une certaine manière d’empêcher les lacets de se refermer”.328 The heterogenesis of the diagrammatic gestures examined so far shows then its most surprising aspect.

1.6.7.1

Gestures that Liberate New Plastic Unities

The nexus between gestures of excavation and gestures of propagation involves a “theoretical jump” in the algebraic topology of loops. This jump is in the highest 325

Bender and Williamson [94, p. 4]. Roy [77, p. 79]. 327 Ibid, p. 107. 328 Châtelet (2010), p. 143. The italics are in the text. 326

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Fig. 1.12 “points such as centers of irradiation”. The picture is taken from [90]

degree heterogenetic because “Cauchy and Poisson, of course, did not know about algebraic topology”.329 And yet the most striking novelty of this heterogenesis does not lies so much in the radical fractures that it entails, but rather in the continuities that it paradoxically restores. If the becoming-zone of points inaugurates “gesture[s] that liberate[…] new plastic unities”,330 among which “the turning around something”, there are previous unities that resurface exactly where the produced fractures are installed. We are thinking of the gestures of extraction and the becoming-hole of points that they entail. In fact, insofar as the thickening of the points leads to their becoming-hole (Sect. 1.6.7), “the turning around something” presupposes the extraction of points. This re-establishes interrupted continuities, bringing us back to a gestural plane that has already been consolidated.

1.6.7.2

“Germes D’entourages”: Châtelet and Klee

But there is more: if the becoming-zone of points actually leads to their becominghole, and if therefore the two becoming are one, it is the point that becomes a hole, that is, a zone, which triggers the gesture of “turning around”. It is the point as a virtual hole that generates the loops. So, we find a last diagrammatic form of the point, the one that makes it a “germe d’entourages possibles”.331 In order to grasp this last aspect of the diagrammatic life of points, it may be useful to return once again to Klee. His Notebooks present planar points as germs of loops. The splendid drawings reported in both the first and the second tomes of Das Bildnerische Denken show points such as centers of irradiation of circles whose radii become progressively larger by growing “from the inside out in pure progression”332 (Fig. 1.12). The Swiss-born German artist places other images alongside these figurations, perhaps even more significant and in line with Châtelet’s perspective. The points are 329

[52, p. 36]. Ibid. 331 Châtelet (2010), p. 168. 332 [89, p. 30]. 330

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Fig. 1.13 “the point as a site of [u]nequal tensions”. The picture is taken from [90]

Fig. 1.14 “loci of growth irradiating from the center”. The picture is taken from [90]

also presented by Klee as pivots around which a line revolves (Fig. 1.13), or as sites “of equal tension in all directions”333 (Fig. 1.14). The manners of presentation just mentioned, moreover, might allow for further variants. Not surprisingly, the points are figured elsewhere as sites of “[u]nequal tensions”334 (Fig. 1.15), or, furthermore, as loci of growth irradiating from the center (Fig. 1.16). Differences aside, which are far from negligible and on which there would be much to say, the cases mentioned above, however, are nothing more than re-propositions of the Châteletian idea of the point as a “germe de tout en espace d’entourages possibles”335 (Fig. 1.17).

333

Ibid, p. 193. Ibid. 335 Châtelet (2010), p. 168. 334

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Fig. 1.15 The picture is taken from [90] Fig. 1.16 “the point as a “germe de tout en espace d’entourages possibles”. The picture is taken from [90]

1.6.8 Loci of Fulguration The scientific survey that has just been pursued allows us to capture the diagrammatic life of points in terms of “fulguration”,336 which is a Leibnizian word337 that Châtelet employs several times in the course of his reflections. Such a frequent use is largely explained by the fact that this word is well-suited to denote the unfolding of the “virtualités géométriques”338 we examined. As for 336

Ibid, p. 141. The italics are in the text. See at least [95, pp. 80–81]. 338 Châtelet (2010), p. 82. 337

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Fig. 1.17 The point as a “germe de tout en espace d’entourages possibles”. The picture is taken from [52]

“fulguration”, it is a key term of Châteletian philosophy of mathematics because it is suitable to refers to the multitude of latent heterogenesis (or becoming-other) which constitute the diagrammatic repertoire of the points. Indeed, its primordial usage context is theological: Leibniz uses “fulguration” mainly to designate the continuous and instantaneous poietic activity of the Divine substance.339 Nevertheless, it is possible that this term transits easily from the original theological context to the mathematical one we are dealing with. It is Châtelet himself who allows this transition. When he reflects oon the becoming-arrow of the points (Sect. 1.6.3.10), “fulguration” is the term that he uses to designate the germinal potentialities of the mathematical entity. We quote, for the sake of clarification, the passage testifying to this particular usage of the term: “il y a bien quelque chose de profondément interne au point, mais précisément en tant que c’est interne, ce n’est pas un point pris comme x = 1, mais une petite flèche qui est là et qui jaillit hors du point: c’est ce que j’appelle une fulguration”.340 We have already commented large excerpts from it (Sect. 1.6.3.10), but, reported in its entirety, we find it confirms the semantic shift at stake. By locating in the points “quelque chose de profondément interne” the French mathematician transfers to the mathematical eidos the internal productivity of Leibnizian substance. This transposition is certainly bold: Châtelet’s points are not Leibniz’s God. On the other hand, however, like Leibniz’s God they are presented as animated by an continuous generative process that makes them germinal matrices. In this sense, therefore, they are Leibnizian loci of fulguration.

339 340

Cf. [95, pp. 80–81]. Châtelet (2010), p. 141. The italics are in the text.

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1.6.8.1

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“Quotient De Polynôme”

Even more interesting, however, is the type of formal considerations that the author develops from this theological-germinal vision of points. After having described in terms of fulguration “la petite flèche […] qui jaillit hors du point” (Sect. 1.6.8.1), Châtelet focuses on one of the possible algebraic forms of the virtual in mathematics. This form concerns again the germinal nature of points and thus their diagrammatic life. We are thinking of what the author says about the point envisaged as a “quotient de polynôme”.341 X = 1/x − 1 The aforementioned expression actually occurs a few pages before the considerations devoted to the “flèche […] qui jaillit hors du point”. It is, however, in the subsequent pages that the geometric-algebraic idea of the point as a quotient of a polynomial is made explicit in its technical details. The background context is anyway the same: the critique of the designative ontology of points (Sects. 1.6.3.1, 1.6.3.3 and 1.6.5.1). Continuing this critique, and observing for instance that “la désignation assassine toute virtualité”,342 Châtelet ironically asserts that he can take “x = 1, […] rester des siècles planté devant”343 and no longer have “rien à faire!”.344 But, the scholar again observes, there is a lot to be done: to correct the previous equation by putting x = 1/x − 1345 [A]. Observations of the same tenor can be found moreover in EM. We refer in particular to the pages that deal with Cauchy’s and Poisson’s profound innovations. Just as he remarks that “Cauchy and Poisson […] provided an answer to the problem: how to make a sensible point out of a point”,346 he invites the reader to envisage “the fraction f (x) = 1/x − x0 ”347 [B]. [A] and [B], with the exception of some differences, are equivalent. They provide, in our opinion, a good algebraic representation not so much and not only of “a singular point in a complex plane”,348 but, above all, of the point as a Leibnizian locus of fulguration (Sect. 1.6.8).

341

Ibid, p. 135. Ibid. The italics are in the text. 343 Ibid, p. 146. 344 Ibid. 345 Cf. Ibid, p. 142. 346 [52, p. 32]. 347 Ibid, p. 33. 348 [8, p. 68]. 342

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“Faire Fleurir Les Points” Let’s us focus on the expression on the right side of [A]. The fraction 1/x − 1 (henceforth, [Ar ] is a germinal matrix (Sect. 1.6.8) insofar as the x in the denominator can assume infinite values. For instance, this is the case for f (x) = 1/x − 1, ∀ x ∈ Z+ . If we dispense with x = 1, [Ar ] will be comparable to a root in the botanical sense of the term. That is, excluding the substitution under which “1/x − 1 […] n’est pas défini”,349 our algebraic fraction will generate an infinity of points for as many elements of Z+ as there are. Thinking of a point as a quotient of a polynomial is equivalent then to recognizing that “un point n’est pas absolument rien en soi”.350 This is why “Construire des mathématiques, c’est faire en quelques sorte «fleurir les points»”.351 The great theoretical legacy of Châtelet is contained in these last assertions. By saying that points are nothing in themselves the author brings his philosophy of the virtual to its final consequences. Further stating that doing mathematics means “making the points flourish”, he endows this philosophy with vegetal metaphors which contribute to an even more significant vision of the virtual in mathematics. The algebraic concept of the quotient of a polynomial and the image of blossoming provide a small-scale representation of assumptions that are much broader in scope..

Flower Roots and Algebraic Fractions: The Virtual The virtual—a concept which is often abused in current debates—has, for Châtelet, very accurate features: it is neither the possible nor the real.352 Flower roots and algebraic fractions are some of the instruments by which the French mathematician seeks to justify the differences at stake. According to what we can read in L’enchantement du virtuel, the possible and the real are affected by a sort of permanent reversibility which is not found at all in the virtual. The latter is animated by a kind of “puissance irréversible”353 that makes it “l’une des choses les plus décisives et les plus implacables qui soient”.354 Quotients of a polynomial effectively capture this irreversibility: insofar as they are germinal matrices (Sect. 1.6.8.1.2), the internal productivity of points imposes itself as an irrevocable aspect of the mathematical eidos. As soon as they are thought in terms of fractions like [Ar ] points unfold potential infinities of points from which it is no longer possible to retreat. As to the vegetal metaphors, they do nothing but intuitively present the ineluctability of such an unfolding. A root may even not

349

Châtelet (2010), p. 142. Ibid. 351 Ibid. 352 Cf. Ibid, pp. 138–139. 353 Ibid, p. 139. The italics are in the text. 354 Ibid. The italics are in the text. 350

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to sprout, and in this “to may not” lies all the “ambiguïté [du] possible”355 : its reversibility. On the other hand, however, if we abstract from this ambiguity, every root subsists as such exclusively with respect to being able to sprout. It is only this affordance that is part of its virtualities, namely of the sprouts that make it a root. Seen in this light, the image of points “in bloom” is a well-suited description of their internal generativity, or rather, of their fulgurations (Sect. 1.6.8). It would remain the real. The latter, according to Châtelet, “donne l’impression du changement permanent”.356 Even in the real, therefore, the root is exposed to the revocability of the “to may not”. If everything changes in the real, the generative resources of the root can deteriorate, become sterile and therefore produce nothing. Not even the real is then part of the root’s virtual. On the other hand, however, it shows its most unexpected aspect: that of its fragility. The virtual of the root—like the virtual of everything else—is fragile, that is, as it is exposed to the weight of “to may not”. Nonetheless, it is this fragility that leads to irreversibility: precisely because it is subject to the real weight of “to may not”, the virtual, if it subsists as such, cannot but explode or flourish. In this sense, talking about points as quotients of a polynomial or as “sprouting roots” does not make much difference: they are both ways of grasping the virtual that “makes the points live”.

A Last Little Note Before Moving to on the Conclusions The sense—if there is a sense—of all that we have tried to say lies then in the just mentioned “to make live”: this formula has here been mainly referred to the points, but it can be extended to the totality of the mathematical eidos. To make the eidos live means in the last instance to grasp it in its becoming other, in the heterogenesis that crosses it and that transpires in the history of human beings, in the vicissitudes of their gesticulating flesh and in the figures that they release in the form of dotted diagrams. All of this, however, has to be demonstrated and it will be up to future research to take on the burden of proof.

1.7 Conclusions: A Philosophy of Diagrammatic Gestures This introductory essay had an ambition: that of providing an update on the main topics of study in diagrammatic gesture theories today. The scientific path we have walked so far allowed us to highlight the subjects as follow: a. Emergence of diagrammatic gestures in mathematical education (Sect. 1.4); b. Role of Châteletian philosophy in the analysis of such gestures (Sect. 1.5); 355 356

Ibid. Ibid.

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c. Identification of the aforementioned gestures in terms of “becoming-other” and of “pressures of the virtuality” (Sect. 1.5.1); d. Role of the notion of “diagrammatic alteration” with respect to the differences between gestural praxis and algorithmic performance (Sect. 1.5.2); e. Deleuzian heterogenesis as main feature of diagrammatic gestures (Sects. 1.5.3– 1.5.5 and 1.5.5.1); f. Differences between transformation and deformation Sects. 1.5.6, 1.5.7 and 1.6); g. Pictorial experience as a privileged locus of dissimilation between transformation and deformation (Sects. 1.6.1 and 1.6.1.1); h. Role of so-called “free fluctuations of the body” in mathematics and in the visual arts Sects. 1.6.1.2c and 1.6.1.3); i. Analysis of such fluctuations in terms of orientation and disorientation (Sect. 1.6.2); j. Transition from the intertwined domains of visual semiotics and the philosophy of mathematics towards a more detailed analysis of mathematical topics (Sects. 1.6.2 and 1.6.3); k. Multiform nature of the point (Sects. 1.6.3.1–1.6.3.4); l. Corporeality in the geometry of mobile points (Sect. 1.6.3.5); m. Relationships between the geometry of mobile points and the carnal nature of the mathematical eidos (Sects. 1.6.3.6–1.6.3.8); n. Mathematical Platonism and critiques of neural reductionism (Sect. 1.6.3.9); o. Comparison between the Châteletian philosophy of mathematics and some Paul Klee’s remarks about the germinal nature of points (Sects. 1.6.3.10, 1.6.4, 1.6.4.1–1.6.4.4); p. The “diagrammatic Life” of points (Sects. 1.6.4.5 and 1.6.5); q. Deepening of the gestural dimension of the aforementioned life (Sects. 1.6.5.1, 1.6.5.2, 1.6.5.2.1, 1.6.5.3, 1.6.5.4, 1.6.6, 1.6.6.1, 1.6.7 and 1.6.7.1); r. Further comparisons between Châtelet and Klee in light of diagrammatic life of points (Sect. 1.6.7.2); s. Final remarks about the Leibnizian background of Châteletian philosophy of mathematics (cf. Sects. 1.6.8 and 1.6.8.1) and about the status of the virtual in the diagrammatic life of points (Sects. 1.6.8.1, 1.6.8.1.2 and 1.6.8.1.3). The development of all topics has been prepared by the preliminary analysis of the notions of diagram and gesture. As for the first, outcomes obtained by the etymological study of the term “diagram” have highlighted a rich variety of technical meanings: “figure”, “letter”, “interplay between figure and letter”, “Interplay between figure and formula”, and so on… As for the second notion, the etymology of “gesture” has allowed us to focus on a semantic potential which refers to the fact of carrying out an action, or, in other terms, to the fact of accomplishing something. Etymology, however, has only been the point of departure of our research. After having acquired the basilar meanings of each term, the surveys conducted here have been developed toward larger and more articulated domains of reference. With respect to the diagram, the comparison between Deleuzian aesthetics and Châteletian philosophy has enabled to see the heterogenesis as a peculiar feature of any diagrammatic

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construction. In this sense, the diagram, far from being equated with a figure (or with the interplay between the latter and something else), has been identified as a germ of unexpected alterations. With respect to the gesture, a series of “cross-readings” in areas such as semiotics and anthropology has permitted to envisage gestural praxis not only in terms of the contexts of regularity that make every gesture an act of a certain kind, but also in terms of agency potentials that make every gesture a matrix of other gestures. In this sense, the gesture—far from being reduced to an act—has been presented as something that is characterized as possessing a “potential of irritation”, that is to say, as possessing an internal potential for change which is in some way akin to the diagrammatic heterogenesis. Therefore, all in all, this introductory essay has tried to think the nexus between the germinal resources of the diagram and the praxeological resources of the gesture. We believe we have succeed in doing so by systematically talking about “diagrammatic gestures” (1), by presenting diagram and gesture as “dynamical units of change” (2), and, finally, by linking these units back to an “ontology of the provisional” (3), that is, to that universe of virtualities which is the main trademark of what we called here “the philosophy of diagrammatic gestures”.

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Part I

Diagrams and Gestures: Mathematics

Chapter 2

The Gestural Construction of Musical Time Guerino Mazzola

2.1 Introduction This chapter presents a mathematically conceived method of gestural time construction in music. In particular we adhere to the prominent thesis that in music, time is not only inherited from the physical reality, but it is essentially the result of the composers’ and musicians’ constructive activity. This approach is in particular forwarded by Jonathan Kramer’s concept of a “vertical time”. We more precisely model Kramer’s approach in the framework of musical gesture theory. The main thesis which we forward is that the construction of vertical time can be derived from an interaction of musical gestures, and this will be done based upon the existence of projective limits of gestural diagrams. Our analysis of musical time gets off the ground with a summary of philosophical ideas related to gestural temporal constructions. We then summarize cultural roots of such approaches, followed by a view upon physical and neuroscientific aspects. We then focus on the musical and music-theoretical perspectives of musical time construction, converging to the theory of gestural interaction in performance. The final sections discuss our main thesis of projective limits of gestural diagrams for time construction, together with the challenge of their realization in imaginary space-time.

G. Mazzola (B) School of Music, University of Minnesota, Minneapolis, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_2

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2.2 Time in Philosophy and Cultures The concept of time has always been a critical topic of philosophy in all cultures. We give a summary of these approaches to show that our main topic, namely the gestural genealogy of time, is a universal philosophical perspective despite its often subcutaneous presence.

2.2.1 Philosophy In classical Greek philosophy, time is a double concept: Chronos versus Kairos. Chronos signifies the flowing time of the universe, while Kairos is the balance of the presence connecting past to future. For Plato, time was a projection of his Platonic ideas, which live outside of temporal categories, in the existential cave of human existence (Fig. 2.1). For Plato Chronos was (and also for Isaac Netwon) the a priori empty container that exists independently of its content, the world’s objects. But chronos time was not the fundamental existence, the world of humans was merely a projection into the existential cave existence deduced from atemporal ideas. In Plato’s philosophy Chronos time is only an attribute of our existence, not a ontological entity. The youngest son Kairos of Zeus is the second of the Greek gods of the time. He represents the right moment in Greek mythology. Kairos is described as a man with little wings on his feet, always scurrying around on tiptoes, never standing still (Fig. 2.2). Very striking is Kairos’ mop of hair and his bald skull, as well as the razor sheep in his hand. Kairos said “seizing the opportunity by the hair”, which means that one should be prepared for the right moment, then one can grab the hair. Kairos gives Fig. 2.1 Chronos

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Fig. 2.2 Kairos

time a completely new dimension. With courage, one goes into joyful action. The key to happiness lies in Kairos and thus the right moment, the responsible courageous action. Kairos goes way beyond Plato’s chronological concept of time, who considered Kairos as a merely rhetorical technique, not as an ontological dimension, he condemned its usage. In other words, Kairos is the first conceptualization of presence, the only aspect of time which we experience. The opposition of Chronos versus Kairos is an early understanding of the two ontologies of time: Chronos is, similar to Newtonian time, that line where we are embedded in a dramatically passive way. This is characteristic of the platonic cave, where we exist as shadows of atemporal platonic ideas. For Plato, the Kairos time was not an ontological category, he did not understand that the presence in Kairos time is ontologically substantial, not just a rhetorical construction. In our thesis of musical time being a gestural construction, Kairos is a dancing reificton of time, one’s existence in time unfolds from a balanced dancing presence, that delicate inner substance as embodied from one’s active gestural existence. Existence in the Latin etymology means ex sistere, “to step out” from the simple being that would also be shared by a dead object. Aristoteles was the first Greek philosopher who stated an explicit and concise definition of Time. Opposed to Plato, Aristoteles considers time asbeing the amount of motion on a continuum of pre and post other than the motion itself. Time for Aristoteles was a global order in which all things are related. Aristoteles, Gottfried Leibniz, and other philosophers argued that time does not subsist independently of the temporal events. This theory is termed “relationism with respect to time,” since from this point of view, all time talk reduces to talk about temporal relations among things and events. In this understanding of time, only the events among objects realize time, it doesn’t exist in empty space without events and objects. The news here is that time is the result of a construction among general events. This shift from an abstract

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Fig. 2.3 Kant

Platonic chronos to a secondary category that is derived from events is remarkable and will be in the focus of many philosophical and cognitive approaches. Our main thesis is about the construction of time in music, a radical extension of Aristotelian ideas. However, Aristoteles agreed on the ontological status of Kairos with Plato as a merely rhetorical device. The interesting impact of Aristoteles’ time concept is the introduction of events as substantial components of temporal experience. Despite of Aristoteles’ agreement with Plato’s (dis)qualification of Kairos as a merely rhetorical category, we perceive the introduction of an explicitly gestural understanding when basing time on event-related phenomenology. For Immanuel Kant, and opposed to Newton, time was not an external parameter, but an a priori form of our perception/representation, similar to space. It is relevant here to understand the conditio humana of this approach (Fig. 2.3). Kant, in his Kritik der Reinen Vernunft [1], writes: Die Zeit ist nichts anderes, als die Form des inneren Sinnes, d. i. des Anschauens unserer selbst und unseres inneren Zustandes.

This approach is in neat contrast to Newton’s and Plato’s idea of time being a “container” of things out there. For Kant, time is a human a priori way of viewing the perceived empirical reality. He takes the same approach for our concept of space. It is a building block of our “inner state”, which is not empirical, but a conditio sine qua non of human world perception. Space is specified as an a priori view of external reality, while time is present in external and internal human perception. He concludes: Alle Erscheinungen überhaupt, d. i. alle Gegenstände der Sinne, sind in der Zeit, und stehen notwendigerweise in Verhältnissen der Zeit.

This statement raises questions. First, the Platonic idealism of the divine Chronos idea is transferred into the realm of human a priori. Time is not something out there, but a genuinely human entity. The external world has no time, but is given a temporal

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reality as a human entity. And the Aristotelian understanding of time as a relationship among events is altered to the claim that this relationship is not among events, but among their projection in our cognitive instance. The major question here is about the genealogy of time in the human a priori. How can we make time? For Kant, this question is missing, the qualification of an a priori entity eliminates his genealogical approach. Time and space are simply there and don’t have to be constructed. Kant’s categorical step is the transformation of time’s ontology from the idealistic Platonic sky into the human ontology. For him, time remains a given entity, it is not a human construction. But it is inside human existence, it no longer belongs to a super-human ontology. The shift to the internal, however a priori, ontology of humans, is the opening of a (however not enunciated by Kant) program that should investigate the mechanism of human time reification. The logical consequence of Kant’s shift is in fact this program if we acknowledge that our a priori of time could be a human process, and not a God-given implantation. Unlike Plato and Newton, Paul Valéry does not argue that time is an empty container possibly filled with events. Similar to Kant, but delving deeper in human conscience as a basic element of the notion of time, Valéry considers “time as being production” (see also, [2]. For Kant time is construction—however, he does not make explicit how the genealogy of this construction unfolds to the form of our experience (Fig. 2.4). By affirming that time is production, Valéry highlights the importance of human conscience in the production of time and how different states of mind, such as attention or surprise, might affect how time is modeled. “Time and consciousness are intimately related” [3] points out to the underlying function of the individual’s conscience in producing one’s own time judgments as a “residual mental phenomenon” [3] coming from the different states of mind and perception that inflect what we call time. But if for Valéry time is unmetrical and dependent on the perspective of the subject that produces it—here a subject taken into all its parts, mind and body with all its components—how is time then perceived? Through Valéry’s Cahiers [3], he states that the construction of time finds its underpinnings in the perceptions of difference: “time is the generic name in all the perceptions of difference” (or Fig. 2.4 Valéry

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“time is a collection of different portions, each one containing different things that coexist”). Time is therefore the mental phenomenon that is produced as a residue of perceptions of difference. Awareness of difference is based on a duality of things that are recognizable as a single entity but presenting, nevertheless, a change in its existence [3]: Le temps — en somme — est ce qui peut-e ⌒ tre construit avec n’importe quelle diversité d’éléments sous certaines conditions comme par exemple d’être à une dimension — C’est la substitution de n’importe quoi à n’importe quoi avec distinction ou asymétrie des deux termes ou quelque chose qui les distingue quels qu’ils soient (fussent-ils identiques).

This one-dimensionality of the changes in time relates to the Aristotelian idea of time as being the amount of motion spent in a single axis without, however, considering the motion itself. The main point is here an unbalance of objects that are experienced as being the same but change their positions in this one-dimensional continuum, inflecting the perspective of the observer: “Time is a valuation, a function of two or n states that declares them as successive.” This thought is also close to John McTaggert’s “A series”—see our discussion of McTaggert below. Valéry goes beyond the formal understandings of time. He argues that time is body-dependent, embodied, poietic (and also aesthesic). The body is not only seen as a traditional cognitive entity, as a brain (acting as a master) with its mental representations, governing a body (playing the role of a vassal). The body is taken in its full ramifications, with the muscles and the limbs—the gestures, in short—acting as building blocks not only of human experiences in general, but also of the form in which we perceive and produce time. Time cannot be generalized but as stated by [2], “instead of assuming a universal and absolute time to find it applicable in individually localized set-ups later, Valéry’s understanding of time always starts from the closure of an observably local functional cycle that is capable of undergoing transformations and modulations.” Valéry’s thoughts also account for the multiplicity of possibilities for the construction of time in music, by considering the plurality of ‘times’ that can arise from different contexts or local systems—such as different music compositions or different interpretations of the same piece—through the actions of the musicians’ minds and gestures. The fixity of time is rejected in favor of a time that is ready to be newly invented, produced and recreated, “There is no such thing as time in general. There are only the times belonging to systems” [3]. Valéry’s approach to time is a major step from Kant’s abstract time a priori to a proper human construction. The non-existence of “time in general” implies that the individual human becomes the generator of time. Valéry introduces two new thoughts: time is an individual human construction. But a third new perspective is that Valéry also delegates the responsibility for this temporal construction to a mechanism of the totality of human embodiment. To our knowledge, this is in the history of philosophical ideas the first moment, where this radical perspective of time’s ontology is set forth. What is however missing in this approach is an explicit description of how time is derived from the embodied construction.

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Fig. 2.5 McTaggart

In John M. E. McTaggart’s The Unreality of Time [4] he distinguished two ordering of temporal events. First, time can be ordered by the relation of being earlier than. He calls the ordering generated by this relation the B-series. The second ordering is generated by specifying a moment within the B-series as the moment of presence. McTaggart calls this ordering the A-series. Following McTaggart, for time to be a reality both series must coexist. His argumentation against the existence of time is that reality of time means that change takes place. And that this must be reified within the A-series as a succession of changing moments of presence. But he argues that the A-series does not exist since nothing is really ever present, past or future. He argues that being present, being past, and being future are incompatible specifications. This argumentation is a corollary of McTaggart’s concept of the ontology of presence. Presence does not subsist, it disappears ontologically when it is over. For our present study this position matters as far as it evokes the fundamental question of the time’s ontological status, and more precisely the status of presence (Fig. 2.5). Edmund Husserl in his Vorlesungen zur Phänomenologie des inneren Zeitbewusstseins analyzes our phenomenological perception of time as an activity that transcends the mere sensual input. The perception of a melody is generating a temporally ordered succession and relation among the melody’s tones. This performance is added to the sensual input as a temporal dimension. And Henri Bergson proposes two forms of time: pure and mathematical time. The former shares a real duration and differs from the latter that is perceived via distinct units. Pure time or lived consciousness is continuous and indivisible. Mathematical time can be divided into discrete units, seconds etc.; these units do not comprise the flow of pure time. Bergson also argued that time as measured by the hands of a clock is a spatial abstraction. But lived consciousness is a bundle of space-time continuity. It is indivisible. The flow of real time can only be known by intuition. According to Bergson, pure time cannot be analyzed mathematically. Any measurement of time presupposes a breaking or disruption of time. Real duration can only be experienced by intuition. Maurice Merleau-Ponty stresses the body as the center of all human experiences. Time, hence, depends not only on our imposition of certain qualities that define its

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Fig. 2.6 Merleau-Ponty

perceived passage, but also on how this time is embodied, how it is lived through our bodily interactions and gestures (Fig. 2.6). In Merleau-Ponty’s perspective, Husserl’s understanding accounts for the very failure of the phenomenon’s comprehension. Since they place an abstract consciousness—somewhat similar to the Cartesian cogito—at the center of the phenomenon, they take for granted the role of the subjects and their concrete position in relation to the events perceived by them. This does not account for a negation of consciousness, but for a necessity of understanding the consciousness as interacting with its medium and having a much greater role in temporality than just gathering cues from the environment or measuring the time of the world against a conceptual time, which remains inherently imprisoned in a consciousness that is detached from the world with which this consciousness is interacting. Merleau-Ponty advocates for an embodied consciousness, in incessant communication with the subject’s body. There cannot be, in Merleau-Ponty’s vision, a consciousness without a body, for the subject exists in the world precisely because of the body that allows itself and the consciousness to interact with their environment. The boundaries between mind and body become blurred in favor of an idea of communication. Data are not anymore sealed in the consciousness, because incommunicable mental representations lose their meaning and their connection to the world that made them possible. Merleau-Ponty argues that time has to be understood from the perspective of an embodied consciousness inhabiting a world, not only as pure dimension of the mind: “let us no longer say that time is a ‘datum of consciousness’, let us be more precise and say that consciousness deploys or constitutes time”. Merleau-Ponty’s phenomenology of embodiment also opposes his conception of time to that of Kant, whose understanding of time placed the phenomenon as one mere category of internal a priori perception. Kant, in Mearleau-Ponty’s view, thinks of time as being solely this ‘datum of consciousness’, a consciousness that constructs temporality disregarding the external world or transforming it through mental operations that cause the world, once having been transformed into mental

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representations, to lose its meaning. Merleau-Ponty (in agreement with Heidegger) on the other hand, argues that humans do not create temporality, but they temporalize the world through the interactions of their situated embodied consciousness. If time were merely a category of perception, as Kantian time is, then the experience of time would be homogenized, a perspective of time that even implies, for Merleau-Ponty, a negation of time altogether. Merleau-Ponty’s thought is similar to Valéry’s conceptions, especially regarding the subjectivity of time. For Valéry time is a production, derived from the subject’s individuality, and always undertaken afresh. Merleau-Ponty, however, goes deeper in the quest for an embodied construction of meaning, placing the body at the center of phenomena. Perception is no longer imprisoned in the cogito, but residing in the body, for there can be no interaction with a world without a body which inhabits it. Here, the cartesian res extensa of the body is replaced by a substantial role of the body in the constitution of our very existence. It is no longer “cogito ergo sum” but, as we may say, “corporeo ergo sum”, and then the experience follows from the embodiment. Within a musical note a ‘micromelody’ can be picked out and the interval heard is merely the final patterning of a certain tension felt throughout the body”. Merleau-Ponty is trying to illustrate that we sense things, with the whole constitution of our bodies, before we can make sense of things: “Time is, therefore, not a real process, not an actual succession that I am content to record. It arises from my relation to things”. But how does this time emerge from my relation to things? Merleau-Ponty’s approach to describe time is a quintessential demonstration of the phenomenological approach of doing philosophy: he does not seek for an explanation, but rather looks for a description of what time is. For Merleau-Ponty, the process of objectivizing time (effect its measurement by means of clocks) is not enough to account for its deep relation with our consciousness, as real time means more than just awareness of its passage. Time becomes the very medium where consciousness and being are deployed. The relationship between time and the subject is deeper than just recording its flow: time and the subject are inextricably united, one needs the other in order to exist, their relation is conceived from within, the subject needs to be temporal, as “virtue of an inner necessity. “Time is a dimension of our being”, a being that is integrated with the world our body occupies, where my gestures define the meaning of our interactions, indissolubly with our consciousness. For Merleau-Ponty, we are time and time is us, time is in us, and Heidegger’s statement, also quoted by Merleau-Ponty at the beginning of his chapter on temporality, is very appropriate to summarize much of his ideas about time: “The sense/meaning of being is temporality”. Valéry’s individual character of time, its construction and its embodied reification are inherited by Merleau-Ponty. But he goes deeper into the existential role of time: Time is characteristic for our existence, and for its identification. It is a thoroughly individual category. Uniformization of the time concept negates time from the beginning. Merleau-Ponty performs his own a priorism in that his time philosophy connects time to individual existence as an identifier: time is embodied existence. In opposition to Valéry, who maintained that someone who tried to reduce thinking

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to neuroscience is an “imbécile” (an idiot), Merleau-Ponty attempted such a reduction. However, Valéry and Merleau-Ponty don’t describe how their embodied time concept creates time, they give us a qualitative theory, whereas the concrete gestural mechanism that leads to time remains arcane.

2.2.2 Cultures In Chinese philosophy, as deployed in Confucian and Taoist traditions, a clear absence of the topic of time is manifest. The philosopher François Jullien [5] discusses this difference without classifying it as a deficiency, but as a categorically different understanding of time, not as an abstraction from events, but being engrained in these processes (Fig. 2.7). Jullien analyzes the difference between European and Chinese approaches to temporality and concludes that the principle of Yin and Yang of an incessant processuality between these two poles implies that “time” is not a concept and is replaced by “process”. In Chinese, the verb “to be” is absent, it only appears in the function of a copula. In Chinese language, verbs cannot be conjugated, they have no temporal specification in their grammatical structure. Jullien refers to Saint Augustin as the first among European philosophers who has understood that our temporal conceptualization is fundamentally based upon our language. The tripartition past, present, and future is a language principle and our thinking of time is derived from this linguistic base. Jullien recognizes that Saint Augustin’s analysis of time was precisely driven by his Latin language. He asks a critical question about the European approach to time: Could it be that the European philosophers have never really investigated the very concept of time and instead only operated their temporal analyses to describe other items that use time as a regulatory parameter? This observation is important for Fig. 2.7 Jullien

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Fig. 2.8 Granet

our own approach to time, since we shall not just apply time to construe the musical concept space, but we shall most importantly investigate its very definition. Chinese do not oppose the a-temporal to the temporal. Jullien refers to Marcel Granet’s fundamental work on Chinese thinking [6] as being the first (non-Chinese) instance of having recognized this radical difference to European approaches to time (see especially Second Book, Chap. 1: “Time and Space”). His conclusion coincides with Granet’s insight, namely that China was led to think not of “time”, but of process (Fig. 2.8). And this process was always a process of events involving objects of a given (although often complex) reality. Granet states that: “Aucun philosophe chinois n’a voulu voir dans le Temps un paramètre”. This difference is significant since the abstract time concept is replaced by a temporality that is induced by or derived from processes, i.e., in particular, a temporality that is a secondary effect being constructed from processual interaction. This is relevant to our own approach to musical time. In fact, there are deep relations between language and time concepts in cultures. Benjamin Lee Whorf’s famous treatise Language, Thought and Reality [7], discusses different languages, especially the North-American Hopi language, and sets forth the idea that language is a strong creator of realities, not only their faithful reflection. He writes: “Thus one of the important steps for Western knowledge is a re-examination of the linguistic backgrounds of its thinking, and for that matter of all thinking.” Whorf argues that the linguistic construction is also strongly one of understanding and shaping the reality, from mathematics to physics to the humanities. Especially when it comes to time, different languages have divergent temporal structures (Fig. 2.9). Language is strongly related to temporal concepts and the associated understanding of reality. But we do not have to follow Whorf in his thesis that language is causing our way of thinking. Nevertheless, one should become aware about the strong relationship between language and reality control. This relationship is also stressing the aspect of embodiment in temporal conceptualization, the index that events and in particular gestures are core in the creation of temporal culture.

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Fig. 2.9 Whorf

2.3 Time in Physics and Neurosciences 2.3.1 Physics Newton’s original document about time and space can be found in the manuscript fragment “De gravitatione et aequipondio fluidorum et solidorum …”, written around a decade before Newton’s famous Principia [8]. He writes: “Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time…” (Fig. 2.10). In particular humans cannot perceive absolute time, it is a mathematically abstract reality. And it is unique, absolute time is a singular divine background structure that does therefore not involve physical events, which are embedded in this absolute, immutable, and non-empirical frame. Newton’s space-time concepts were criticized by the point of view that all inertial frames of reference are equivalent and related by Galilei’s relativity theory (which nevertheless keeps time invariant). Already Gottfried Leibniz, whose approach was criticized by Newton, maintained that without bodies and events, space and time are meaningless. In the nineteenth Fig. 2.10 Newton

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century, Newton’s position was questioned by physicist Ernst Mach, who stated that space-time is a reference to the “whole universe”, and not to a transcendental ontology. For our concerns, Newton’s position is the extreme of a European approach to time as an autonomous ontological dimension, an immutable frame, where all events are embedded a priori. The difference to Kant’s time concept is that his a priori of time is an internal human perspective, while for Newton, this is an external objective reality that humans cannot influence. Einstein’s special relativity performs two radical changes from Newton’s approach to time: First, he views time and space as connected within a four-dimensional “spacetime”. Second, the transformation between different inertial frames of reference is described by the Lorentz transformation that connects time transformation with the relative velocity of two such frames (Fig. 2.11). This relativity revolution however does not change radically the ontological status of time, it simply connects time to space and allows for a 4-dimensional transformation of space-time. But the ontology of this space-time remains an external frame, much in the spirit of Newton. It relates to the inertial frames of reference, but it is nevertheless detached from the objects and events in such frames. This ontological situation is preserved for time dilation in general relativity, which is caused by gravitational fields; however, now the presence of gravitating masses influences the time’s values. But it does so in a Newtonian transcendence, no human action is involved, time is simply there as a given world parameter. In Quantum Mechanics we observe a double reality between the Hilbert space H, plus observables (linear operators on H) on one side, and their projection, via Fig. 2.11 Einstein with his friend Gödel

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eigenvalues of observables in experimental measurements, on the other. In Quantum Mechanics, time is not an observable, there is no experimental access to time. It is, similar to Newton’s “divine” time, a given ‘absolute’ variable, for example used in the description of the wave function Ψ: R4 → C of a single particle (three real space coordinates, one time coordinate). The Hamilton operator H, an expression for the observable’s energy of an observable Ψ, describes the temporal behavior of the observable by the Schrödinger equation i.d/dt Ψ(t) = H Ψ(t). Also, in contrast to Relativity Theory, Quantum Mechanics does not integrate time and space. Physicist Leo Smolin in his book Time Reborn [9] defends time as a non-fiction (contradicting Einstein), and he discusses the problem of consciousness as a fundamental qualia, where time becomes a core characteristic. He stresses the difference between the typical physical description of neuronal processes and the ontology (what really is, exists) of consciousness. The physical reality and the psychological, the description and the essence of being, are Smolin’s “big questions”. In his final chapter, he stresses the central role of time in human consciousness (Fig. 2.12): One further aspect of consciousness is the fact that it takes place in time. Indeed, when I assert that it is always some time in the world, I am extrapolating from the fact that my experiences of the world always take place in time. But what do I mean by my experiences? I can speak about them scientifically as instances of recordings of information. To speak so, I need not mention consciousness or qualia. But this may be an evasion, because these experiences have aspects that are consciousness of qualia. So my conviction that what is real is real in the present moment is related to my conviction that qualia are real.

We argue that the construction of time is a core substance characterizing this difference in that time is not an illusion, but a fundamental construction of our consciousness. To be clear: a construction is not an illusion, but a fundamental motor of consciousness. Leo Smolin’s approach to the qualia ontology of time (and, by the way, also Roger Penrose’s thoughts about consciousness in physics) in human consciousness opens up a window to the question of how time could be a constructive entity. Our approach in this text deals with the description and investigation of this construction mechanism. This is our main statement and answer to Smolin’s perspective. The impact of this situation in physics is that the quasi-objective reality of physics is missing a fundamental understanding of the nature of time. Penrose and Smolin are by no means exotic physicists, they are only the first to realize that physics is Fig. 2.12 Smolin

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more than a bunch of objects and experiments floating around in a divine spacetime theater. And it is not clear, whether the conflict between Relativity Theory and Quantum Mechanics could be related to the present deficient time concept in physics altogether.

2.3.2 Neurosciences Neuroscience and Experimental psychology have, similar to physics, challenged some of our intuitive notions of a constant and linear passage of time, and have begun to elucidate some of the mechanisms by which the brain may represent time. At least over a short time frame, our temporal perception of events is far from veridical, and multiple timelines are capable of dynamic recalibration. This is incompatible with the notion of a unitary centralized and dedicated clock, from which all timing is performed. Alternative accounts for multiple centralized clocks and distributed timing mechanisms were proposed. These models are not mutually exclusive, and timing may also be achieved through a combination of centralized and distributed processing. Regardless of the exact timing structures and mechanisms, the fundamental question remains whether neural processing is ever exclusively dedicated to the problem of timing. Neural activity that may appear as such can always be reframed to be coding some other process that occurs in time, rather than time itself. In view of our main hypothesis about the gestural construction of musical time, we summarize investigations of the effects of dance therapy on Parkison’s patients. Dance therapy was applied by a professional dance therapist to these patients, especially rhythmical dance akin to Tango. The effect of this therapy was measured by means of standardized criteria of the Movement Disorder Society Unified Parkinson’s Disease Rating Scale. The involvement of music in dance therapy offers an interesting perspective for future investigations on the pathology of Parkinson’s effects on time perception, since time is a characteristic dimension in music, and even more so in its rhythmical structures. Dance therapy activity requires planning movements, following music and signals, and remembering choreography. These music/ dance (dancing and listening to music with other people) exercises improve patients’ conception/sense of time via physical body movements. Since dance therapy helps patients with Parkinson’s disease in their movements, we conclude with the following hypothesis: The Parkinson patients’ bodily, emotional, cognitive, and social integration and wellbeing generated by dance therapy influences their neurological dopamine processing.

There is strong empirical evidence of Alzheimer patients showing a distortion of time perception. This is more precisely manifest when these patients are given verbal instructions and then tested about their time judgments. For example, such tests asked participants with Alzheimer to read either 5, 10, 20, or 40 digits appearing one at a time. At the end of each sequence, participants had to judge the elapsed time.

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Results showed inaccurate time estimations in Alzheimer participants, especially for long intervals. The most distinguishing characteristic of human memory is mental time travel, or the state of autonoetic consciousness permitting the episodic reliving of past experiences. By inducing the feeling of subjective time, mental time travel is likely to depend on time perception, which may explain why ‘remember’ responses, reflecting autonoetic reliving, were significantly correlated with time perception. It is worth noting that no significant correlations were found between time perception and ‘Know’ responses, reflecting a dissociation between time perception and noetic consciousness. Time distortions may impair mental time travel, and consequently may contribute to an episodic memory impairment in Alzheimer. In Alzheimer, this reliability is degraded due to memory impairment potentially resulting in time distortions and longer decision times, collectively aggravating patients’ feeling of being ‘stuck in time’. The investigations of Manuela Kerer et al. [10] focused on mild cognitive impairment (MCI) and early-stage Alzheimer (AD) patients, who were compared to healthy individuals’ response to musical tests “in terms of verbal memory of music by the identification of familiar music excerpts and the discrimination of distortion and original timbre of musical excerpts (Fig. 2.13).” The results showed that “MCI and Alzheimer’s patients showed significantly poorer performances in tasks requiring verbal memory of musical excerpts than the healthy participants. For discrimination of musical excerpts, MCI and AD patients surprisingly performed significantly better than the healthy comparison subjects.” In conclusion, this research supports “the notion of a specialized memory system for music.” Fig. 2.13 Kerer

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2.4 Time in Music and Its Theory 2.4.1 Musical Time Constructions It is remarkable that the introduction of the new clock technology at the end of the thirteenth century goes perfectly parallel with the invention of mensural notation in music. Its first accomplished presentation was written by Franco of Cologne (ars cantus mensurabilis, around 1250). He distiniguishes four durations: duplex longa, longa, brevis, semibrevis. This notational progress was motivated by the necessity of having a precise collaboration among different, quite autonomous voices. Therefore in music we also have an augmented desire of well-defined collaboration. The evolution of European music sub specie aeternitatis, under the view of an eternal time line, which was also working following a divine clockwork, culminates in Johann Sebastian Bach’s perfect machinery of the Wohltemperiertes Klavier. Listening and watching Glenn Gould’s performance of Bach reminds us of God’s marionette hitting the piano’s keys as driven by a higher string mechanism. Although this European time technology is driven by gestural mechanical devices, such as Maelzel’s metronome, the concept of time that it generates is a strong abstraction. Time is framed into a succession of time points (tic tac) that define start and end, without being concerned with the existential expressivity while the transition from start to end is happening. It is the opposite of the negative time, Ma in Noh theater. This spirit in European music has created an important tension between mechanical/ metronomic time, and the time which stems from free gestural expressivity. This is manifest, for example, when we compare Glenn Gould’s performance of Beethoven’s Sonata op. 57 Appassionata to Vladimir Horowitz’s rendition. Gould plays it as if it were a mechanical construction à la Bach, he misses Beethoven’s underlying gestural approach, which is represented in Horowitz’s stringent performance. It is also manifest in the same way when we compare Gould’s interpretation to Maurizio Pollini’s of Beethoven’s Sonata op. 106 Hammerklavier. This tension and conflict remains omnipresent in European composition and performance to the present day. It has been traced in a quasi-ironic way in György Ligeti’s composition Poème Symphonique for 100 metronomes, but also in the antimetronomic compositions of Alexander Scriabin, for example his last poem op. 72 Vers la Flamme, which starts from large, slow durations and ends up with microscopic and fastest trill durations. Ultimately, this conflict stems from the question of who defines time in music. Bach’s mechanical/metronomic style expresses the divine clockwork time, whereas Beethoven’s gestural expressivity insists on the human autonomy in the construction and perception of musical time. The European introduction of musical time as a mechanical entity, enabled by the introduction of mechanical clockworks, was a strong contribution to the Newtonian divine time as an abstract reality where human activity would be embedded as an organization within an external ontology. This strong direction in musical time construction was however put into question by many important composers, such as Beethoven, Scriabin, or Ligeti. This conflict between an objective external and

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an individual internal approach to musical time is still virulent. But in the twentieth century, especially with the introduction of improvisation in jazz and nondeterministic musical creativity, the ‘counterpoint’ of mechanical against individual, between divine and human time conceptualization has created in the Western musical culture a tension that developed an appropriation of musical time by humans without any essential reference to ‘God’s time machine’. The Indian time culture in music is loaded with a deep semiotic weight that connects time to categories of divinity. The tala system of rhythm stresses duration against accents of loudness. Time is not structured by other parameters (such as loudness), it is an immanent existential category. The complexity of metrical subdivisions enforces a strong focus on duration qua duration, time is an auto-referential dimension. It is however very different from the neighboring Chinese approach since its quantitative complexity steers away from the inner quality of Chinese temporality. But at the same time it is not the European extremism that stresses “on” and “off” without being ‘interested’ in the inner vibration. The realization of the Indian temporal musicianship is in fact not an abstract description of time intervals on the time line, it is a focus on the musician’s complex performing gesture. Musical time concepts in African cultures are often developed and investigated under the umbrella of rhythm. Kofi Agawu confirms that “rhythm is heart and soul of African music and its association with dance” [11]. He however criticizes that scholars are reducing this understanding to a “bodily knowledge”. He proves in his discourse that African rhythm expresses a “rational complexity”. He stresses the difference to European rhythms in the emphasis and idiomatic preference. He then exhibits three prominent features: timelines, polyrhythms, and lead drummer’s narrative (Fig. 2.14). For our concern here, the cultural aspect of musical time is the focus. The first feature, timelines, is characterized by a multitude of musical times: reversed, discontinuous, sliding, circular, spiral. Often time-reckoning is related to social activities. As opposed to African languages, where time concepts rather sparse, musical time shows a sophisticated differentiation. Agawu concludes that musical time in Africa is not a microcosm of ordinary time, it is its own time realm. This entails that the African culture of musical time in its specific and intense social reality generates a different time reality, a specifically musical one, and one that is differentiated in three modes of rhythmical expressivity: in speech mode, where the rhythm is not metrical, but driven by the semantic vector, in signal mode, Fig. 2.14 Agawu

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where speech is stylized in time, and in dance mode, where the rhythm of the body is dominant. The social aspect of these modes is that many skills are combined in one philosophy, combining laymen’s and professional’s capabilities in an intelligent design of musicians’ collaboration. This is explicit in the polyrhythmic structure of African musics, where many, eighteen or more, rhythmical lines are combined to a polyrhythmic maze. Following Agawu’s analysis, this complex collaborative structure of time lines, polyrhythms, and lead drummer’s dimensions is a social construct of polyphonal, polytheistic, polycentric, or polygamous social organization. And moreover, this social construct is also strongly gestural, as realized by dancing, drumming, and singing movements.

2.4.2 Summary of Mathematical/Musical Gesture Theory In Mazzola’s musical gesture theory [12], a gesture is a representation of a directed graph (the gesture’s skeleton) as a system of continuous curves in a topological space (the gesture’s body). This enables the construction of topological spaces whose points are gestures, and therefore of hypergestures, i.e., gestures whose curves are curves of gestures. A gesture g is a morphism of directed graphs g: Γ→ XI , whose domain is called the gesture’s skeleton, whereas its codomain is the gesture’s body, namely the directed graph of continuous maps I→ X from the real unit interval I to a topological space X (more generally, X would be a topological category). The set Γ@X of gestures with these skeleton and body is canonically a topological space (category), so it is possible to consider hypergestures, i.e., the space Δ@Γ@X of gestures with skeleton Δ and body Γ@X. The Escher Theorem states that the order of hypergestural spaces does not matter, i.e., if (Δi )I=1,…,n is a sequence of directed graphs, then Δ1 @ Δ2 @ … Δn @X is homeomorphic to Δp1 @ Δp2 @ … Δpn @X for every permutation p of the index set {1, 2, …, n}. For two gestures g: Δ → X, h: Γ → Y, a morphism f:g → h is a pair of morphisms f = (t,k ) t: Δ → Γ, k: X → Y with kI . g = h.t, where kI is the canonical digraph morphism XI → YI . This defines the category Gest of gestures. If Δ is a digraph, one can associate to Δ in a natural way (natural in terms of category theory) a canonical topological space |Δ| such that any gesture g: Δ → X corresponds to a continuous map |g|: |Δ| → X with the following property: Consider the canonical gesture d(Δ): Δ → |Δ|, then g = |g|.d(Δ). The map |g| is called the natural map of g. Gestures are important concepts in many music-theoretical treatises, such as the research by Roger Sessions, Theodor Wiesengrund Adorno, and Renate Wieland.

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2.5 Kramer’s Time Categories In what follows, we want to propose some interpretations of Jonathan D. Kramer’s conceptual time variety [13] in the framework of mathematical gesture theory. The main purpose of this action is not to put his approaches into question, but to show how strongly his conceptual sketches are congruent with mathematical conceptualization. In Kramer’s impressive account on time in music, we are given a spectrum of different time concepts. Kramer however stresses that he does not present strict definitions, he states: “Time must ultimately be taken as undefinable.” and then concludes: “They (the categories of time) are more suggestions for listening than rigorous theoretical formulations.” Nevertheless, Kramer proposes quite precise conceptualizations, also traced in a clear and even mathematically oriented glossary at the end of his book.

2.5.1 Kramer’s Time Variety It is important to understand how Kramer positions musical time as an ontological locus. He writes: “In the following chapters I postulate many types of musical time. To do so becomes possible once I accept the notion that music creates time. Time itself can (be made) move, or refuse to move, in more than one ‘direction’: not an objective time out there, beyond ourselves, but the very personal time created within us as we listen deeply to music.” This constructivist position is essential in our approach, too. Kramer distinguishes “ordinary” physical time from its musical counterpart: “The difference between ordinary lived time and musical time is, according to Clifton, the difference ‘between the time a piece takes and the time which a piece presents or evokes’. All the species of time discussed in this book are experienced and understood simultaneously with ordinary time.” Kramer proposes four categories of time: Multiply-directed, gestural, moment, and vertical. Here are the definitions in his glossary: • gestural time: species of multiply-directed musical time in which the conventional meaning of gestures (beginnings, endings, structural upbeats, etc.), rather than the literal successions of events, determine the logic of continuity • moment time: temporal continuum of a moment-form composition • multiply-directed (linear) time: temporal continuum in which progression is seemingly in several directions at once • vertical time: temporal continuum of the unchanging, in which there are no separate events and in which everything seems part of an eternal present. In what follows, we discuss Kramer’s time concepts from his glossary and also from other locations where he defines time qualities. Gestural time is a special multiply-directed time, where gestural determinants override the literal succession of events. If we insert our definition of a gesture, it has its parametrization, which is essentially a linearization on each arrow of the defining

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digraph. We would define gestural time as this type of parametrization. The gestural parameter is not a physical time, the body of a gesture only would have physical time as a parameter in a space-time. If a hand moves in physical space-time, its gestural parametrization is different from the tie coordinates of the gesture’s curves. Moreover, if we deal with hypergestures, the Escher theorem yields a variety of gestural parameters. In fact, if a gesture g is in the space Δ2 @ Δ1 @X, the parameters of Δ2 can also be replaced by the parameters of g, when interpreted as living in the homeomorphic space Δ1 @ Δ2 @X. This can be read as an instance of a multiply-directed time: gestural parameters are multiply-directed. Therefore we interpret Kramer’s gestural time as the parameter set of a gesture that is given by its defining digraph. Moment Time. We shall not discuss this type of musical time for two reasons: to begin with, it plays a minor role in Kramer’s theory, and second, it is absorbed in the central category of vertical time. Linear/Absolute Time. Linear/Absolute time is the traditional external time frame imposed by the metronome and referring to physical reality. This is the opposite of Kramer’s focus, it is not a time that can musically be constructed, it is absolute, disjoint from musical creativity. Multiply-directed (Linear) Time. Multiply-directed time will not be defined in general here, but we have already given its interpretation for the case of gestural time. It means that the gestural parameters are multiple (even on one determined digraph, we have several arrows with their associated parameters), and this in a dramatic variety when considering hypergestures and their variants that are defined by the Escher theorem. Vertical Time. This is the musical time par excellence, very difficult to conceive, but it is the type of time we want to interpret in mathematical gesture theory. Let us give some references to Kramer’s conceptual sketches. Two citations seem central: “But some new works show that phrase structure is not a necessary component of music. The result is a single present stretched out into an enormous duration, a potentially infinite ‘now’ that nonetheless feels like an instant. In music without phrases, without temporal articulation, with total consistency, whatever structure is in the music exists between simultaneous layers of sound, not between successive gestures. Thus, I call the time sense invoked by such music.” The second citation describes the way one listens to vertical music: “Listening to a vertical musical composition can be like looking at a piece of sculpture. When we view the sculpture, we determine for ourselves the pacing of our experience: We are free to walk around the piece, view it from many angles, back and view the whole, contemplate the relationship between the piece and the space in which we see it, close our eyes and remember, leave the room when we wish, and return for further viewings.” This means that vertical time creates an imaginary space where time takes the shape of a spatial category, with multiple dimensions that are perceived as if we were observing a sculpture. This perceptual work is the ‘performance’ of an interpretative trajectory of one’s senses/ eyes, which move on the sculptural surface similar to what Paul Klee described as the trajectory of one’s eyes when looking at a visual object.

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This category is a very difficult one, and we could not identify the mechanism that would generate vertical time in Kramer’s book. It seems however plausible that vertical time is generated by a collaborative effort of gestural components. The dynamic perception of a sculpture is in fact the integral of a variety of gestural trajectories. In other words, we could argue that vertical time is the result of a deep interaction of gestural time perspectives.

2.6 The Gestural Construction of Musical Time 2.6.1 Distributed Identity in Musical Performance Musical performance, be it traditional performance of a given compositional score or performance in improvised music, such as free jazz, generates an intense interaction of the musicians’ gestures [14]. It has been proved in our research about the quality of free jazz interaction, that the quality of a performance is a function of these gestures’ interaction. More precisely, such an interaction, if successful, generates an elevated existential state, where, as they say, “the music plays you”, and not “you the music”. Viewed from the theory of musical gestures, this state can be related to diagrams D of gestural morphisms: Musicians “throw” gestures to each other and thereby generate an artistic identity that is shared by the orchestra’s musicians. This diagrammatic musical structure was recognized as the generator of what is now called a “distributed identity”.

2.6.2 Limits of Gestural Diagrams In mathematical category theory, one considers co- and contravariant functors of very different types, stemming from mathematically interesting constructions. Such functors are not conceived as being related to particular spaces, i.e., objects, in the given categories. This means that such functors are only defined by a specific behavior and not by the points of a space that would “materialize” such a behavior. For example, if A, B are two objects in a category C, one may consider the contravariant functor C(A, B): X ~> X@A ⌴ X@B mapping object X to the coproduct (disjoint union) of the morphism sets X@A and X@B (we denote by X@Y the Hom(X, Y) set in a category). This is the set of X-valued points in A or in B. But in general this is not the set of X-valued points in a determined space in C. The same construction would be possible when defining L(A, B): X ~> X@A x X@B, the mapping of X to the cartesian product of the two sets X@A and X@B, and here again, this set is not, in general, associated with the set of X-valued points in a space of our category. A special case of such functors is given by the definition X → X@Z, the set of X-valued points in a space Z. Such a functor, denoted by @Z, is said to be

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represented by space Z. A functor which is isomorphic to a represented functor is called representable. In our above examples, if they were representable, we would have representing spaces C(A, B), L(A, B) such that X@C(A, B) ≃ X@A⌴X@B and X@L(A, B) ≃ X@A x X@B. In the category Sets of sets, such representing sets do exist, namely C(A, B) = A ⌴ B, the disjoint union of sets, and L(A, B) = A x B, the cartesian product of sets. In our context, we are interested in the following type of functors. Let us first discuss the situation for sets. A diagram D of sets is a map D: Δ → Sets of a directed graph Δ into the category Sets that sends every vertex v of Δ to a set D(v), and every arrow a:v → w to a set function D(a): D (v) → ∏ D(w). The (projective) limit of this diagram is the subset of the cartesian product v D (v) of the diagram’s vertex spaces such that for any two elements x ∈ D (v), y ∈ D (w), we have D (a)(x) = y for any function that is associated with an arrow a: v → w. This limit is denoted by Lim D. We have all projections pv : Lim D → D(v), and the corresponding commutative triangles for all arrows a: v → w. The universal property of this limit is that in the category Sets, we have a functor X ~> Lim D(X), where D(X) is the diagram of function sets X@ D(v), for the vertices v of D, and the canonical transition maps D (X)(a): X@ D(v) → X@ D(w) given by the original maps D(a). We then have an isomorphism of functors Lim D (X) ∼ = X@Lim D, i.e., X ~> Lim D(X) is represented by the limit set Lim D. A simple example for sets is the diagram D defined by a single set map f: A → B, representing the arrow digraph 0 → 1. The limit of this diagram is the classical graph of f. It consists of all pairs (u,v) ∈ A x B such that f(u) = v. The general meaning of limits is to represent collections of spaces that are connected by characteristic transition maps. The situation for limits in a more general category C is the following: We are again given a diagram D: Δ → C, now into the category C instead of Sets. The transition maps are now morphisms in C. We again have the functor X ~> Lim D(X), where D(X) is the diagram of morphism sets X@ D (v), v the vertices of D. But now, it is not true that this functor is representable by an object of C in general. The following paragraph is dedicated to prove that in the category Gest of gestures, limits do exist. Given a diagram D: Δ → Gest of gestures, we have the associated limit in the category DIG of digraphs. We should recall here that DIG is a topos and therefore admits limits (even for infinite diagrams). This means that we have a canonical limit morphism given by Lim D Δv → Lim D XI v . The codomain digraph LimD XI v is the limit of the digraphs XI v , which are essentially the spaces I@Xv , so we are looking at the limit of function sets for I-valued functions with values in the members of a diagram Σ → Top of topological spaces Xv . But it is known that limits exist in the category Top. This means that we have LimD XI v ∼ = (LimD (Σ → Top))I . Therefore the limit of D is given by the digraph morphism LimD Δ → (Lim (Σ → T op))I and this gesture is the limit of the given diagram of gestures in the category Gest. Applying the natural map associated with LimD Δ → (Lim (Σ → Top))I , we get a map

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|LimD Δ| → (Lim (Σ → T op)) of topological spaces, whose domain is the limit topological space generated by the diagram D of topological spaces | Δi |.

2.7 Construction of Musical Time from Diagram Limits It is this type of points which we now define as being the temporal arguments derived from the given gestural collaboration (diagram). This definition has two conceptual resources: the gestural distributed identity described by the diagram D and the limit of this diagram. The former is an empirical fact, while the latter is an a priori construction from pure mathematics and could be viewed as a Kantian component of our construction. The next and final step in our conceptualization of Kramer’s vertical time and similar non-physical (musical) temporalities will be to consider space-time environments that may comprise the above diagrammatically conceived temporal arguments. But the decisive point here is that time is not given outside of a gestural interaction, it appears as a construction that follows from a gestural collaboration, which also means that it vanishes after such a collaborative artistic maneuver.

2.7.1 Imaginary Time In Stephen Hawking’s physical model of the Big Bang, time is a complex number t + i.s, where t is the traditional time value and i.s is the imaginary component. It is not clear at all whether and how this imaginary component could be a part of human consciousness. We conjecture that our presence of consciousness, where thoughts are built and processed, could happen in that imaginary direction, so that the classical real space-time is complemented by an imaginary time-space defined by imaginary time. If physicists are entitled to introduce new time dimensions, there is no reason to prevent artists from doing the same and claiming that creative human consciousness is hosted in such a time-space that is “orthogonal” to the physical one. This means that at any classical physical time t, we would have an entire time-space defined by an imaginary time i.s plus some space coordinates attached to that time. Based upon such arguments for an extra time dimension, we argue that the concept of presence in time-critical arts requires such an imaginary time and space. It is in this realm where the transitional processing of past music to future music, the planning of gestural strategies, the body-instrument-sound interface are all displayed and organized. It is a quite dramatic change of understanding of what happens in the artistic presence, since it eliminates the mystification of spontaneity and unpredictability in creative performance. These attributes have had a great influence on the non-understanding of creative performance, from classical music to free jazz. Improvisation, creative performance—all of that has been boiled down to these negative

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concepts: Spontaneity and unpredictability are just negations of any sort of positively defined artistic shaping; they are the “emergent properties” of creativity mysticism, creativity by negation. Telling a musician to be spontaneous is of no help whatsoever: It is just a recommendation to rely on nothing that could be conceived of in the artistic-shaping activity. Let us recall the gestural construction of musical time as being realized with the limit of skeletal spaces of diagrams of gestures. The question here is about the realization of such limits in our space-time reality. To begin with, the imaginary time space-time iR ⊕ R3 is a four-dimensional vector space where imaginary temporal instances can survive. From the above discussion, we learn that representing the gesturally constructed time needs a representation of a limit digraph skeletal space. Such as space is a colimit of line spaces (the real number space I = [0, 1]). But to represent such a space, it evidently sufficient to have a three-dimensional real space. Such a space is available in iR ⊕ R3 . We therefore may realize the temporal parameters in the space |LimD Δv | within the imaginary time-space iR ⊕ R3 . What we are stating is that the temporal space can be realized within iR ⊕ R3 . Moreover, an evaluation of points p: |Δ| → |LimD Δv | can be achieved with values in iR ⊕ R3 . The specific nature of such points will reflect the management of temporal data derived from the gestural collaboration in a distributed identity. And let us also stress that the gestural time construction is by no means the simulation of a linear time concept. The temporal parameters in the limit space can easily be a digraphic combination of line segments. The vertical time in the sense of Benjamin Kramer isn’t linear either, it is a mental construction that expresses the complexity of our time deployment, including multiple threads between shared startand endpoints, or loops. It is also important to understand the qualitative difference between vertical time and the gestural parameter, which is essentially the unit interval domain I = [0, 1] ⊂ R. One could argue that this domain already defines the linear character of time. It is true that the digraph parameter space I has a linear nature. But this is an abstract ordering of gestural curves. It may be applied to purely spatial gestures without any temporal connotation. This could be what Kant used in his description of time as an “inner a priori sense”. But what generates our concept of time stems from the limit of a gestural diagram, not from the gesture as a structure per se. The special case of a singleton diagram of one gesture cannot be taken as a typical example. A single gesture does not constitute a valid temporal entity, it resembles the empty time rather than what our experience of time looks like. One single gesture is the conceptual germ, not the full-fledged time ontology.

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2.7.2 Modeling Vertical Time The gestural time is however not a concept that unites the musical instances to generate an identity, that big picture, which Kramer draws when invoking the integrated perspectives of a sculpture when uniting the movement around the sculpture’s presentation. Keeping Kramer’s metaphor in mind, vertical time is an integral of gestural approaches, of components of comprehension. It is more than the sum of those gestural perspectives. The integrated information results from the gestural interaction, from the comparison and relationship among all of the sculpture’s perspectives. This concept therefore asks for a precise shaping of such a network of gestures and therefore a construction of vertical time. It is more than the gestural architecture; vertical time must be a consequence of the gestural network, not the network alone. In view of our discussion of diagrams of gestures as related to the construction of a distributed identity in free jazz [14] and other music performance, we propose to define vertical time as being the topological domain space of the limit of a gestural diagram, and, more generally, as the morphism of a digraph’s topological space into that limit domain. A first small example of such a vertical time construction from the limit of an elementary ‘linear’ gestural diagram is shown in Fig. 2.15. This limit is the diagonal line in this figure’s square to the right. It shows the pairs of corresponding values from lines c and c’. This time construction is an elementary ‘communication’ between the two involved simple lines. Ontologically it is more than the single lines, it describes a correspondence of gestural parameters that generates a temporal collaborative quality. A second small example of such a vertical time construction results from the limit of an elementary ‘circular’ gestural diagram, see Fig. 2.16. This example shows the limit as being the red line in the torus of this figure. It is essentially the same ontological construction as in the previous example, except that the loops now connect in a torus, cartesian product of two circles, instead of a square as before.

Fig. 2.15 A simple limit

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Fig. 2.16 A circular limit

An example of such a vertical time construction from the limit of a simple, but slightly more involved gestural diagram is shown in Figs. 2.17 and 2.18. We start with a rather simple diagram D of three gestures, which are connected by three morphisms f, g, h as shown in Fig. 2.17. The diagram is composed of three gestures and three connecting morphisms f, g, h, as defined by the images of their arrows. The limit of this diagram, when transformed in the equivalent display of a continuous map from the topological space associated with the limit of the three digraphs is shown in Fig. 2.18. This representation requires some comments. First, the limit of the two loops a, a’ is shown as a line on the torus, which is the cartesian product of two circles. Then, the image of line b on a’ identifies with the line on the tube that is attached to the inner circle of the torus, corresponding to the limit of b and a’. To this line (b ~ a’) we attach the line c ~ c’ corresponding to the limit in the cartesian product of c and c’. We also show the identification of these points with points from the third digraph (c’ ~ c”, a” ~ a”’). This configuration of three lines is the limit of our diagram’s topological domain. It is the time-line that is generated following the limit construction. As for every digraph, this line can be represented without overlapping parts in R3 .

Fig. 2.17 A diagram of gestures

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Fig. 2.18 The vertical time space of the above diagram

It is difficult to compare our construction of vertical time to Kramer’s concept because he does not present a precise conceptual definition (and he admits it). Our main concern is to apply our approach to concrete musical situations and to show that it effectively helps understand human time construction in a deeper way. It is however evident that our construction generates a non-linear time reality, and it also takes care of the relational nature of human time construction: a projective limit is the perfect mathematical framework for complex relational configurations.

2.8 Conclusions Our method models Kramer’s vertical time concept and captures the gestural interaction of a performative distribute identity. This time setup is nonlinear and can be realized on the imaginary time reality.

References 1. Kant, I.: Kritik der reinen Vernunft. Meiner, Hamburg (1956) 2. Ustun, B.: Time is production: process-art, and aesthetic time in Paul Valéry’s Cahiers. Humanities 7(4) (2018). https://doi.org/10.3390/h7010004 3. Valéry, P.: Cahiers I-IV (1894–1914). In: Celeyrette-Pietri, N., Robinson-Valéry, J. (Eds.). Gallimard, Paris (1987) 4. McTaggart, M.E.: The unreality of time. Mind 17, 457–473 (1908) 5. Jullien, F.: La Pensée Chinoise: En Vis-à-vis de la philosophie. Folio, Paris (2019)

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6. Granet, M.: La Pensée Chinoise. Albin-Michel, Paris (1936) 7. Whorf, B.L.: Language, Thought and Reality. The Theosophist, Madras (1941) 8. Newton, I.: 1726. University of California Press, Philosophiae Naturalis Principia Mathematica (2016) 9. Smolin, L.: Time Reborn. Mariner Books, Boston (2020) 10. Kerer, M. et al.: Explicit (semantic) memory for music in patients with mild cognitive impairment and early-stage Alzheimer’s disease. Exp. Aging Res. Int. J. Devoted Sci. Study Aging Process (2013) 11. Agawu, K.: Representing African Music. Routledge, New York (2003) 12. Mazzola, G., et al.: The Topos of Music III: Gestures. Springer, Heidelberg (2017) 13. Kramer, J.D.: The Time of Music. Schirmer, New York (1988) 14. Mazzola, G., Cherlin, P.B.: Flow, Gesture, and Spaces in Free Jazz—Towards a Theory of Collaboration. Springer, Heidelberg (2009)

Chapter 3

Gilberte’s Gesture and the Commutative Combray Jean-Claude Dumoncel

habent sua fata libelli (2205) parfois l’attention éclaire différemment des choses connues pourtant depuis longtemps et où nous remarquons ce que nous n’y avions jamais vu (1721) comme dans un problème plus simple qui initie à des difficultés plus complexes mais de même ordre (2337)

3.1 Gilberte’s Gesture in the Commutative Combray In Remembrance of the Past our pair1 obtains two main illustrations.2 Proust [1] alludes to the «chansons de geste» (937, 1965) and to some cycles (2057, 2153). So that the pair permits us to see in the novel two great cycles: Gilberte’s cycle and «l’histoire d’Albertine» (1531), which are therefore two «chansons de geste» in a new meaning. The first extends on the whole duration of the novel. It begins, as a kind of «singnpost» at Tansonville, with a sign emitted by Gilberte, her “indecent gesture”: (119) elle laissa ses regards filer de toute leur longueur dans ma direction, sans expression particulière, sans avoir l’air de me voir, mais avec une fixité et un sourire dissimulé, que je ne pouvais interpréter d’après les notions que l’on m’avait données sur la bonne éducation, que comme une preuve d’outrageant mépris; et sa main esquissait en même temps un geste indécent, auquel quand il était adressé en public à une personne qu’on ne connaissait pas, le petit dictionnaire de civilité que je portais en moi ne donnait qu’un seul sens, celui d’une intention insolente. 1 Thanks to Charles Alunni, Francesco La Mantia and Fernando Zalamea for their invitation to the program Diagrams and Gestures towards which this paper is intended. 2 All references to A la Recherche du Temps perdu in this paper are to the nrf Quarto edition (1999) by page number. For a complete bibliography, see our final References.

J.-C. Dumoncel (B) CNRS, Centre d’études Théologiques de Caen, Caen, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_3

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But, at the level defined by the titles of the successive tomes paving the novel: (113) il y avait autour de Combray deux ‘côtés’ pour les promenades, et si opposés qu’on ne sortait pas en effet de chez nous par la même porte, quand on voulait aller d’un côté ou de l’autre: le côté de Méséglise-la-Vineuse, qu’on appelait aussi le côté de chez Swann, parce qu’on passait devant la propriété de M. Swann pour aller par là, et le côté de Guermantes.

In “Combray» these two sides are confined “dans des vases clos et sans communication entre eux» (114). But in Le Temps retrouvé, which is also a Gilberte retrouvée, she overcomes the incommunicableness: (2125) Si vous voulez, nous pourrons tout de même sortir un après-midi et nous pourrons alors aller à Guermantes, en prenant par Méséglise, c’est la plus jolie façon.

This means that Gilberte obtains a commutative diagram of Combray (Dumoncel 2018a):

Méséglise ⎯⎯⎯⎯→ Guermantes ↑ ↑ (Tansonville) | | | (porte M ) ⎯ Combray → (porte G) Sed contra, according to Deleuze,3 «Impossible de faire comme dit Gilberte» (150). In Remembrance of the Past, Gilberte is a kind of divinity promoted master of ceremonies: she opens the ball and closes it. But her function is fully fulfilled only when, in an ultimate presentation at the scale of the whole book, answering to Marcel who asks for a «succédané» of Albertine (2388), she presents her own daughter, Mlle de Saint-Loup. Now, Mlle de Saint-Loup requires her own diagram. She is a «star»: (2387) Comme la plupart des êtres, d’ailleurs, n’était-elle pas comme sont dans les forêts les «étoiles » des carrefours où viennent converger les routes venues, pour notre vie aussi, des points les plus différents? Elles étaient nombreuses pour moi, celles qui aboutissaient à Mlle de Saint Loup et qui rayonnaient autour d’elle.

3

Unless otherwise stated, all references to [2] are to Proust et les signes [2], by page number.

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We have the following diagram, where the more shining stars are only entered: Côté de Méséglise (2067) « dans les choux » @ Balbec → Elstir ⏐ ↑ mother = Gilberte kisses (1886) Albertine → Mlle Vinteuil → sonate & septuor ↓ Mlle de Saint-Loup ↑ father = Saint-Loup ⏐ Côté de Guermantes → Oriane

The junction of the Tansonville and the Guermantes «côtés», impossible for Mme de Saint-Loup, becomes then possible on Mlle de Saint-Loup, even if it is in taking «transversales» in the diagram [2, p.150, 1, p. 2387]. The second hypostasis of the pair , in Proust, unifies A l’ombre des Jeunes Filles en Fleurs and Le côté de Guermantes. The link is «un geste» (1020) but «this gesture» (1020) undergoes a mutation of its modal status by reason of a corresponding change in Albertine (1026), between Albertine at Balbec and Albertine in Paris. At Balbec: (727-729) J’avais appris qu’il n’était pas possible de la toucher, de l’embrasser, qu’on pouvait seulement causer avec elle.

But in Paris: (1018-1033) savoir qu’embrasser les joues d’Albertine était une chose possible, c’était pour moi un plaisir peut-être plus grand encore que de les embrasser.

In standard symbolism, conceived as a Begriffsschrift (with ¬p for «not p», ◇p for «it is possible that p» and @ for a generalized «at»):

@Balbec ¬◊ I kiss Albertine. @Paris ◊ I kiss Albertine. And this mutation of the modal status has also its diagrammatic aspect: (1020) un soir déjà lointain où nous formions un couple symétrique mais inverse de celui de l’après-midi actuelle, puisque alors c’était elle qui était couchée et moi, à côté de son lit.

We must distinguish here (as in an obsolete vocabulary about vectors) between two states of the diagram. We have, first, the free diagram

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It only indicates the laying and standing positions—and ↑. Then, second, we have the bounded diagrams obtained when the free diagram is differently and successively decorated by the dramatis personae:

But what is the etiology of the modal mutation? Proust provides it by an interplay of Boolean operations, each of which asks for its representative diagram. We must begin by what happens in the Balbec «scene»: 1. «Albertine in his bed»: (728) Dégageant son cou, sa chemise blanche changeait les proportions de son visage qui congestionné par le lit, ou le rhume, ou le dîner, semblait plus rose; je pensai aux couleurs que j’avais eues quelques heures auparavant à côté de moi sur la digue, et desquelles j’allais enfin savoir le goût.

2. Marcel allait «savoir l’odeur, le goût, qu’avait ce fruit rose inconnu»:

In the review of Vendredi reprinted in the Logique du sens, Deleuze evoques: (357) ce monde possible nommé Albertine.

This means that, according to Deleuze, Albertine is a possible world. This is a paradox. But, after all, a paradox may be true. And we shall now demonstrate that this Deleuzian paradox is true. In his book based on his John Locke Lectures of 1964, Hintikka writes that in such and such «states of affairs» or «courses of events», «the relevant ‘possible

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worlds’ are described by certain conjunctions which we shall call, following a timehonored precedent established by Boole, the constituents of propositional logic4 ». In support of this fatherhood, Hintikka cites An Investigation of the Laws of Thought, the book by Boole published in 1854 at the epicentre of the mathematical renewal that formal logic was to undergone, for the Sect. 11 in its Chap. V. In this page, the alleged conjonctions, under the name of «constituants», are defined as the logical products of the form xy, xyz (etc.). And [3] stipulates: «Of the constituent xy, x and y are termed the factors». Accordingly to Hintikka’s genealogy, therefore, each of these conjunctions describes a «possible world». So that the disjunction of these conjonctions will represent the whole set of «the relevant ‘possible worlds’» which, in Boole’s Begriffsschrift, will be writen by additions of multiplications, such as xy + xyz (etc.). In Proust, Albertine is a fruit. More precisely, Albertine is exactly described by the Boolean product xyz, where the factors x, y and z take respectively the following values: x = the color of the fruit Albertine = rose y = the odour of the fruit Albertine z = the savour of the fruit Albertine.

So, when Deleuze will speak of «ce monde possible nommé Albertine», he will simply provide the legend of diagram [2]: Albertine is here the possible world defined by the Boolean product of her odour AND flavour AND colour. 3. But Albertine Causes «un son précipité, prolongé et criard»:

So, the sound produced by Albertine, hasty AND lengthy AND crying, had as result that Marcel will not know the odour of Albertine, NEITHER her flavour, NOR her colour. This illustrates the fact that, in Boolean Algebra, x AND y AND z encounters its contrary in NON x NEITHER y NOR z. So that, if NON x NEITHER y NOR z is true, then x AND y AND z is false. Following the Boolean Begriffsschrift reminded by Hintikka, the possibility of the kiss to Albertine is therefore of the form f (1, 1, 1) xyz

4

Jaakko Hintikka, Logic, Language Games and Information, Clarendon Press, Oxford, 1973, p. 152.

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So, it will be writen f (1, 1, 1) color AND odour AND savour of Albertine

But, by ringing the bell, Albertine has converted this possibility into an impossibility: f (0, 0, 0) xyz

This means: NEITHER x, NOR y, NOR z. «Non, non Marcel, tu n’auras pas ma rose». And we must exhibit now what happens in the Paris «scene» (1029): 4. Albertine «entra souriante, silencieuse, replète, contenant dans la plénitude de son corps, préparés pour que je continuasse à les vivre, venus vers moi, les jours passés dans ce Balbec où je n’étais jamais retourné» (1018)

So that (1020) avaient été détruites ces résistances contre lesquelles je m’étais brisé à Balbec.

Inside the sound produced at Balbec, in Paris, the hasty resistance was destroyed by the smiling Albertine AND the lengthy resistance was destroyed by the silent Albertine AND the crying resistance was destroyed by the stoutish Albertine. And this effect of diagram [4] on diagram [3] is reinforced by a parallel fact: 5. «Sur chaque trait rieur, interrogatif et gêné du visage d’Albertine, je pouvais épeler ces questions: ‘Et Mme de Villeparisis? Et le maître de danse? Et le pâtissier?’» Quid of Mme de Villeparisis? [5] On Albertine’s face Marcel may read

AND « Quid of the dancing-master? AND « Quid

At last, the explanation [4] is explained in its turn: 6. (1029-30) Sans savoir si j’avais à faire honneur et savoir gré de son changement d’attitude à quelque bienfaiteur involontaire [= 1403 le séducteur inconnu] qui,

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un de ces mois derniers, à Paris ou à Balbec, avait travaillé pour moi, je pensai que la façon dont nous étions placés était la principale cause de ce changement. C’en fut pourtant une autre que me fournit Albertine; exactement celle-ci: «Ah ! c’est qu’à ce moment-là, à Balbec, je ne vous connaissais pas, je pouvais croire que vous aviez de mauvaises intentions».

[6] I can kiss Albertine because

OR the difference Albertine

Marcel

\ Marcel

Albertine

OR @Paris Albertine believes to know Marcel

Leibniz defines necessity as truth in all possible worlds. This is the Leibniz bridge between a modality on the left side and a logical quantity on the right side. Peirce, in On the Algebra of Logic: A Contribution to the Philosophy of Notation (1885) writes: (3.393) Here, in order to render the notation as iconical as possible we may use Σ for some, suggesting a sum, and Π for all, suggesting a product. Thus Σ i x i means that x is true of some of the individuals denoted by i or

Σ

xi = xi + x j + xk + et c.

i

In the same way, Πi x i means that x is true of all these individuals, or

Π i

xi = xi x j xk , etc.

In other words, quantification theory is a stenography of Boolean calculus. And in [4, pp. 214–5] appears the symbol ∃. But, in a novel, neither ∃ nor Σ is available. Yet Proust has at his disposal a conjonction specifically appropriated to his writing, encapsulated in a rule «dite des trois adjectifs» which «revêtait dans les billets de Mme de Cambremer, l’aspect non d’une progression mais d’un diminuendo»: (2945-946) Mme de Cambremer me dit, dans cette première lettre, qu’elle avait vu SaintLoup et avait encore plus apprécié que jamais ses qualités «uniques – rares – réelles », et qu’il devait revenir avec un de ses amis (précisément celui qui aimait la belle-fille), et que si je voulais venir, avec ou sans eux, dîner à Féterne, elle en serait «ravie – heureuse – contente».

Inside this promotion of the AND, as we see, Proust takes good care to slip a typical OR. So that we obtain a new diagram:

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With Saint-Loup [7]

OR

Mme de Cambremer is delighted AND happy AND pleased

without Saint-Loup

Meredith and Prior wrote [5]: (Lp)x = ∀y (Uxy → py) (Mp)x = ∃y (Uxy & py)

This means that, on the Leibniz bridge, upon the right side, a footbridge Uxy was added, which will be called the Prior footbridge. And since we are in an abstract, uninterpreted, «calcul», Uxy may be viewed as «y is relevant to x». So that diagram [7] reads: in all possible worlds relevant to the situation, Mme de Cambremer is delighted and happy and pleased. So that, according to Meredith and Prior (with ☐ = L for «necessarily»): ☐ Mme de Cambremer is delighted and happy and pleased.

From a Boolean point of view, the «possible worlds» are simply the constituants of a disjunction, each of which is a conjunction of the propositions happening to be true in this constituant. So, in Proust, we find the anticipation of a modality defined by mean of a relevant relation added on the Leibniz bridge as a footbridge from possible world to possible world. Since AND and OR are Boolean operators (Booleans by abreviation), we are here, in Proust, on his Boolean register. On this register the fact that, at Balbec, a possibility was eliminated by a sound at once hasty AND lengthy AND crying, and then that this possibility was established in Paris by an Albertine at once smiling AND silent AND stoutish acquires its full meaning. The Boolean destiny where the kiss to Albertine, impossible at Balbec, becomes possible in Paris, means that, in Proust, the Leibnizian definition of modalities by a quantification over possible worlds has given place to an explanation of modalites by the Boolean operators acting on possible worlds or situations. This is the Boolean Begriffsschrift of modalities by Proust. But, as Gilberte’s cycle, «l’histoire d’Albertine» finds its end only at the end of the novel; what, for poor Albertine, means post mortem. Andrée tells the tale: (2067) elle a été forcée de vous quitter par sa tante qui avait des vues pour elle sur cette canaille, vous savez, ce jeune homme que vous appeliez «je suis dans les choux », ce jeune homme qui aimait Albertine et l’avait demandée…

Marcel speculates: Je suis dans les choux (= peut-être) (2067) le déniaiseur grâce auquel j’avais été embrassé la première fois par elle.

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Now, with «je suis dans les choux», the Albertine diagram is enriched of a new arrow, traveling in a new branch of modal logic: hybrid logic, the one where the emblematic symbol is @. In «Arthur Prior and Hybrid Logic» Patrick Blackburn tells us that Arthur Prior was «the first person to see» how to hybridize logic: «sort the propositional symbols, and use formulas as terms» [6]. Let’s do this right away. Take a language of basic modal logic (with propositional symbols p, q, r, and so on) and add a second sort of propositional symbol. The new symbols are called nominals, and are typically written i, j, k, and l. Both types of propositional symbols can be freely combined to form more complex formulas in the usual way. And now for the key change: insist that each nominal be true at exactly one point in any model. A nominal «names » a point by being true there and nowhere else.

So, we saw that Proust has taken the proposition «je suis dans les choux» and used it as a name. In the Proustian universe, «je suis dans les choux» is true at the point Octave and at no other. So that «je suis dans les choux» is a Proust nominal. And that «the first person to see» how to hybridize logic, rather than Arthur Prior, was Marcel Proust. Between the Gilberte cycle and the Albertine cycle, the comparison reveals a convergence and a difference. The convergence is that, in the two cycles, we have the transition from impossibility to possibility. (The third main gesture of the novel is (1033) the mother’s exorcist gesture by which she delivers Marcel from his nervous love for Oriane). The difference is a kind of diametral opposition. In order to kiss Albertine, Marcel must, in default of being presented by Saint-Loup, receive from Elstir un bon pour des plaisirs futurs, and then, from Albertine, un bon pour un baiser; but, in order to inflame Marcel, an indecent gesture from Gilberte suffices. Men, in order to reach women, must take the mundane way (including the pimps and procuresses); the young girls, for the inverse trajectory, take, as Gilberte, the Tansonville shortning. And now that we have followed Gilberte and Albertine in their characteristic cycles, we may arrive at the sentence where their fates are linked. Albertine says of Gilberte: (1886) je crois qu’elle me ramena une fois et m’embrassa.

Moreover, for our inquiry, Gilberte and Albertine share a more important common rôle. Amid the multifarious «worlds» meeting in the Proust universe, [2]has discerned the hierarchy or scale raised by the superposition of four worlds (or «rings») which constitute the Proustian Spectre where the whole of his universe is diffracted (and, in Proust et les signes, are enumerated, pp. 12–21): (12) Le premier monde de la Recherche est celui de la mondanité (13) Le second cercle est celui de l’amour (18) Le troisième monde est celui des impressions ou des qualités sensibles (21) At the top there is «le monde de l’Art ».

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Now, amidships these four worlds, Gilberte and Albertine are the amphibious. Moreover, they are also the two Ariadnes who will in turn guide Marcel in the four corresponding labyrinths. The alluded labyrinths crosses parallels and transversals: (1802) une jeune fille entretenant avec Mlle Vinteuil des relations parallèles à celles de Charlie et du baron, avait mis au jour toute une série d’œuvres géniales… D’ailleurs, à ces œuvres, tout autant que les relations de Mlle Vinteuil avec son amie, avaient été utiles celles du baron avec Charlie, sorte de chemin de traverse, de raccourci, grâce auquel le monde allait rejoindre ces œuvres…

3.2 The Elements of a Modal Logic 3.2.1 Basic Modal Logic and Its Proustian Challenge The «Basic Modal Logic» by Benthem [7] is centered on a «decorated graph»:

Diagram [8] In a decorated graph, the graph is a constellation of points (here figured by numbers) and of arrows leading from point to point; the decoration is provided by some propositions p of a language L, which are true at some points on the graph and false elsewhere. Here, the decoration is minimalist, since it is limited to the unique proposition p, true at 1 and 2, false at 3 and 4. On this graph, van Benthem leaves as an exercise the demonstration5 that ◇☐◇p is true at 1 and 4 but false at 2 and 3. The required proof follows from what van Benthem names «the well-known truth definition»: M, w ⊢ ☐p ⇔ ∀v∈W, wRv → M, v ⊢ p. M, w ⊢ ◇p ⇔ ∃v∈W, wRv ∧ M, v ⊢ p.

Here we recognize that, as in the Meredith and Prior draft, the Leibniz bridge between modality and quantity is increased by the footbridge of the relation R between possible worlds. And this truth definition is obtained from a Kripke model [9]: M =

5

The elementary demonstration is available in [8].

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In such a model, says van Benthem, W has the name W, reminiscent of «Worlds», only «for its nostalgic mood» since, on the graph, the points w may represent anything; likewise, R may be any (binary) relation. The pair is the frame of the model. And the so-called «valuation» V is any mapping from the propositions p of L to the subsets of W where they are (supposed to be) true. In Husserlian terms, derived from Aristotle, the frame of the model describes its ontology, and the valuation stipulates its apophantics. Such is the «Basic Modal Logic» offered by van Benthem. Because in it any W, R and V do the trick, this basic modal logic may and must be said «abstract». Pale epithet however. In the Proustian lexicon, it is mostly idealistic: in the Russell’s antagonism between Ptolemaïc and Copernican, it is ptolemaïc: it works as if the diagram was making the modalities. Undoubtedly, abstraction has its virtues. When Marcel says to Robert that his military theory entails that the battles are conceived as «des exemples interchangeables—Ah ! interchangeables, très exact ! excellent ! tu es intelligent» says Saint-Loup (835). And what may be the meaning of a modal logic rendered «realistic» or «Copernican»? In order to understand the point, we must start from a fresh start in a parallelism between Proust and Leibniz. Proust observes: (631) l’aristocratie est une chose relative. Et il y a des petits trous pas chers où le fils d’un marchand de meubles est prince des élégances et règne sur une cour comme un jeune prince de Galles.

And Leibniz, in his Nouveaux Essais: (IV xvii §18) si nous étions vains, nous pourrions juger comme César qui aimait mieux être le premier dans une bourgade que le second à Rome.

This baroque «vanity» by Leibniz has a hidden connection with the Leibniz’s concept of «possible worlds». At least since Carnap in Meaning and Necessity (Sect. 3.2), the definition of necessity as truth in all possible worlds is regularly credited to Leibniz. But the textual source of this genealogy was only discovered by Kalinowski [10, etc. pp. 136–237]. It lies in one of the Opuscules et fragments inédits edited by Couturat in 1903 (pp. 16–24). Here we read, with the key sentence in bold and its translation by Kalinowski: Hinc jam discimus alias esse propositiones quae pertinent ad Essentias, alias vero quae ad Existentias rerum; Essentiales nimirum sunt quae ex resolutione Terminorum possunt demonstrari; quae scilicet sunt necessariae, sive virtualiter identicae; quarum adeo oppositum est impossibile, sive virtualiter contradictorium. Et hae sunt aeternae veritatis, nec tantum obtinebunt, dum stabit Mundus, sed etiam obtinuissent, si DEUS alia ratione Mundum creasset. «Et celles-ci sont de vérité éternelle, car non seulement elles tiendront aussi longtemps que le monde durera, mais encore elles tiendraient si Dieu avait créé le monde selon quelque autre plan.»

As Kenneth Konyndyk writes in his Introductory Modal Logic [11, 2.1.b]: Leibniz introduces possible worlds as different ways God could have created the universe, or, to put it a little differently, possible worlds are alternative universes God could have

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created. Leibniz observes that God could have created different individuals from the ones that presently populate this world, he could have change the natural laws, and he could have decided not to create at all. So a possible world is a total way things could be or could have been, and a necessary truth is a proposition that is true in every possible world.

So that Caesar, even if the first in his village (a village becoming Roussainville or Martinville in Proust), is always the second in the Leibnizian best possible world. An so that, in the key sentence by Leibniz, the use of a word in capital letters acquires a formal power: in Leibniz, this word is DEUS, but this may be seen as a particular answer to a more general question: who deserves the capital letters in the definition of necessity and possibility? According to Leibniz, God is required as «the being who could produce the possible» (Phil. III, 572). According to Proust, the mistake in this proposition is the definite article «the». Proust will empower the different regions who were maintained by Leibniz in their provincial status, so that the production of the possible will be democratically devided between different jurisdictions. Yet, in a scale invariance, a principle remains the same from Leibniz to Proust: since there are laws in force, the shakespearian «question» is: who makes the law? In each case, this is depending of such and such «position dominante» (1316). Oriane «fait la pluie et le beau temps» (1125)—in the faubourg Saint-Germain. Oriane «peut tout sur le général Saint-Joseph» (1064)—but not on Moltke. So that the power is stretched between the two extremities of the Deleuzian spectrum where the four Proustian worlds are superimposed. In the realm of snobbery, Charlus is the arbiter of elegance. But in the register of art it is the triumvirate of Bergotte, Elstir and Vinteuil, “ces grands rénovateurs du goût” (1781). Between the two extreme realms, Gilberte and Albertine are the tutelary deities. This thesis, where the modalities depend on a legislative power of varying scope, will be called the “potential conception of modality”. In the series of the Deleuzian “conceptual personages” (Empedocles jumping into Etna, Diogenes coming out of its barrel, Descartes meditating in his “poêle”, etc.) we need to define a new one: the Proustian sovereign of the possible. Since in Le Dominateur et les Possibles, the first book on the Master Argument of Diodorus, Pierre-Maxime Schuhl thanked Deleuze for helping him correct the proofs, this personnage will be also said the Modalities Master, not without his grotesque parody, «le metteur en scène et régisseur Aimé» (1393). The concept of Modalities Master entails a new concept of «possible world». The point is that a world relevant for Charlus may be without relevance for Albertine dreaming of donkey rides, and that the worlds relevant for Odette may be without relevance for Swann and, a fortiori, for Vinteuil. Though Sorbonae nullum jus in Parnasso is false, Oriane doesn’t rules in Elstir’s studio. So that each Sovereign of the possible has his relevance territory. And, as we saw, the Uxy of Meredith and Prior finds its most natural interpretation in «x has a relevant alternative in y». After all, as Leibniz, Nicholas Rescher6 distinguishes metaphysical and physical modalities; Proust, master at Combray and Balbec, may distinguish Bergottian and Albertinian modalities. 6

Personal communication.

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According to Deleuze the Sextus Tarquin tale in Leibniz is «une source de toute la littérature moderne» (Cinéma 2, 171). If this is true, then this tale is among others a source of the Recherche. We have no positive proof that Proust knew it. Yet one clue for this genealogy is, amid Oriane’s witty words, her «Taquin le Superbe» (1119) about Charlus. But much more decisive is the isomorphism between this fable and one of the most intriguing episodes in the novel (684–685), when Albertine’s beauty mole successively occupies several positions on her face. Indeed, by this transport, Proust distinguishes several possible Albertines, as Leibniz labelled different Sextus as Sextus at Corinth, Sextus in Thrace and Sextus at Rome = the real Sextus. So that we have likewise (685) Albertine with beauty mole on cheek below the eye (686) Albertine with beauty mole on the chin (688) Albertine with beauty mole on the upper lip below the nose = real Albertine.

This isomorphism, in the Leibnizian corpus as a whole, cuts out a narrative model of modal metaphysics, transposable into a novel, and so provides a halfway house between the monarchical universe of Leibniz and the anarchic archipelago depicted by Proust.

3.2.2 Modal Logic in the Mood of Miss Anscombe and P.T. Geach In a letter from Geach to Prior dated April 15, 1960, with the Lukasiewicz symbols L for «necessarily» and M for «possibly» in the S4 and S5 systems of C.I. Lewis, we read: Here is a thing that might amuse you, since you combine an interest in modal logic and in SF. I am sure you are very familiar with the SF stories in which there are parallel worlds that you can reach by a machine. It occurs to me that the systems between S4 and S5 (inclusive) can all be characterized in terms of different suppositions as to the possibilities of world-jumping. L = ‘in the world we are in, and in every world we can jump from it, it is the case that…’ M = ‘in the world we are in, and in some world we can jump from it, it is the case that…’ Then the S4-reduction just means that my being able to jump to a world that either is the world W, or is a world I can jump from W, is to count as being to jump to W. The S5-reduction means that I can jump from any world to any world. Other hypotheses as to my jumping abilities, e.g. that I can jump from W’ to W if I can jump from W to W’, would give intermediate logics.

[12, p. 120] adds: «Geach used the term ‘Trans World Airlines’ for this voyaging between worlds». Briefly TWA. [13, p. 42] pointed: «It was later suggested by Geach that we might take a, b, c, etc., to name worlds, and Uab to mean that world b

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is ‘accessible’ from world a…». Geach, Miss Anscombe’s husband, has imagined a maximally packed storytelling: «Smith committed seven burglaries, then Smith committed a murder, then Smith was hanged» [14, p. 71]. At the end of Remembrance of the Past, we read: (2126) je me dis que la vraie Gilberte, la vraie Albertine, c’étaient peut-être celles qui s’étaient au premier instant livrées dans leur regard, l’une devant la haie d’épines roses, l’autre sur la plage. Et c’était moi qui n’ayant pas su le comprendre… avais tout gâté par ma maladresse. Je les avais «ratées »…

Geach teaches us how to sum up A la Recherche du Temps perdu in a single and laconique sentence: Marcel manqua Gilberte, puis Marcel manqua Albertine, puis Marcel retrouva le temps perdu en écrivant un roman.

G.E.M. Anscombe and P.T. Geach, together, had three sons and four daughters. Prior specifies [13], 70–71: «Miss Anscombe has a… logic…of ‘and then’ or more precisely ‘It was a case that p and then it was a case that q’». This Begriffsschrift by Miss Anscombe provides the formal frame for the common denominator of Gilberte’s and Albertine’s cycles, i.e. the modality mutation: It was a case that ¬◇Marcel kiss Albertine. and then it was a case that ◇Marcel kiss Albertine.

3.2.3 The Components of a Modal Logic In Proust we may now define a kind of Kripke model M = . It will take the form of a Proustian triad P = where W is a set of Proustian worlds w, A is the access (or accessibility) relation into a world, and D (as «discourse») the characteristic saying spoken in such and such situation. Each characteristic episode in the Recherche will have its Proustian triad P. As a paradigm, in the mythical episode of the maternal kiss denied and then granted, W is the couple formed by Marcel’s bedroom and the garden where the dinner with Swann ends (35), «lieu de plaisir où l’on n’est pas» (33), A is the access to the maternal kiss, and D gathers a entrusted to Françoise with the two successive verdicts: «Il n’y a pas de letter réponse» (from the mother) and then «couche pour cette nuit auprès de lui» (from the father). In one of the idyllic episodes on a cliff overlooking the sea (712–713) W gathers Albertine who says «Qu’est-ce qui a un crayon?», Andrée who provides it, Rosemonde who offers paper, and this paper itself; A is the access to what Albertine writes, forbidden to her compagnes because reserved to Marcel; and D = «Je vous aime bien». Etc. 1° W: The Proustian space of Possible Worlds. It is reminiscent of Leibniz’s «pays des possibles»: (834) Tu te rappelles ce livre de philosophie que nous lisions ensemble à Balbec, la richesse du monde des possibles par rapport au monde réel.

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As the pyramid at the end of the Théodicée, in Proust W is infinite: (1671) Le champ infini des possibles s’étend.

Paradoxically, «le pays des possibles» is more accessible to the Narrator than the real world: (1620) par nature le monde des possibles m’a toujours été plus ouvert que celui de la contingence réelle.

However, the existence revealed itself to Marcel as it will do to the Roquentin of La Nausée: (1959) l’existence (et non la simple possibilité) venait de m’être révélée.

In the fork of Aristotle’s future contingents, the Hercules of the Stoics between Vice and Virtue hesitates: (972) vous êtes comme Hercule… au carrefour de deux routes.

This aside, the Proustian «worlds», as well as the points on a van Benthem’s graph, may be of an unlimited variety. There is firstly (1796) ces mondes que nous appelons les individus

With three crucial cases: (1675) qu’il s’agisse d’une femme, d’un pays, ou encore d’une femme enfermant un pays.

But in A la Recherche du Temps perdu, times are also worlds (incarnated as countries): (1663) chaque jour était pour moi un pays différent.

There are also (2357) plusieurs Mme Swann, séparées par l’éther incolore des années, et de l’une à l’autre desquelles je ne pouvais pas plus sauter que si j’avais eu à quitter une planète pour aller dans une autre planète que l’éther en sépare.

And with the times, places are also promoted as possible worlds: (1959) au lieu d’être dans deux ou trois endroits possibles elle était en Touraine.

2° A: From Inaccessibiliy to Accessibility and access. Love is its best dramatisation: (1893) dans les yeux d’Albertine… je sentais comme un éclair de chaleur… dans des régions plus inaccessibles pour moi que le ciel.

It is a law: (1892) On n’aime que ce en quoi on poursuit quelque chose d’inaccessible.

With only mundane harmonics:

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(1095) un salon si peu accessible; (1543) les lieux officiels où les profanes n’entrent pas.

Proustian space has its characteristic obstacle: (1057) une barrière; (2143) une Normandie qui serait absolument insoupçonnée des Parisiens en vacances et que protège la barrière de chacun de ses clos.

In such a universe you have to get your (1056) dignus est intrare.

You have to pronounce (1602) «le Sésame », (2009) «le mot de passe »

In Proust, the sovereignty over the possible has its Queen: the jeune fille (1651) capable d’accéder à tant de possibilités diverses dans le courant vertigineux de la vie.

And the universal problem is the communication of worlds (2013), where Hermes has delegated his power to (847-8) les Vierges Vigilantes, les Danaïdes de l’invisible qui sans cesse vident, remplissent, se transmettent les urnes des sons, les Demoiselles du téléphone. (1676) je me saisis du récepteur du téléphone, j’invoquai les Divinités implacables, mais ne fis qu’exciter leur fureur qui se traduisit par ces mots: «Pas libre. »

When Proust’s logic will become symbolic, among its symbols we will find the icons In the country we like to be able to say (1485) Les communications sont on ne peut plus faciles.

Thanks to the (1348) petit train d’intérêt local.

In the city we want to have (1621) large communication avec la rue;

And at home (1439) franchir sans agoraphobie l’espace creusé d’abîmes qui va de l’antichambre au petit salon.

At Venise we have (2077) le Grand Canal, où la mer se prête si bien à la fonction de voie de communication.

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Between the worlds there are bridges: (1453) Pons cui aperit. (2864) jeté au milieu des champs semés de boutons d’or où s’entassaient des ruines féodales, le petit pont de bois nous unissait, Legrandin et moi, comme les deux bords de la Vivonne. (1398) quand dans la salle du casino deux jeunes filles se désiraient, il se produisait… une sorte de traînée phosphorescente allant de l’une à l’autre.

From flower to flower flies (1213) un gros bourdon.

A duke corresponds with a viscountess (1114) par pigeons voyageurs.

For Geach’s TWA flights, Proust has his aeroplane (441; 1532 sq.; 1908, 1681sq). Proustian communication has as well its intermediaries and its Do It Yourself : (2005) Andrée n’était pour moi… qu’un chemin de raccord… qui me reliait indirectement à Albertine. (2009) même une syllabe commune à deux noms différents suffisait à ma mémoire – comme à un électricien qui se contente du moindre corps bon conducteur – pour rétablir le contact entre Albertine et mon cœur;

3° D: À la Recherche du Temps perdu in eight emblematic speeches (1358) «Non, ce soir je ne serai pas libre», (1772) «Gilberte ne viendra pas», (713) «Je vous aime bien», (729) «Finissez ou je sonne», (1676) «Pas libre», (1772) «Mlle Vinteuil est invitée», (1919) «Mademoiselle Albertine est partie», (2386) «Si vous permettez, je vais aller chercher ma fille pour vous la présenter.»

3.3 Deleuzian Modalities In 1964, with Proust et les signes, Deleuze introduces the concept of possible world in the Proustian commentary: • Dans la mesure où l’être aimé contient des mondes possibles (Mlle de Stermaria et la Bretagne, Albertine et Balbec), il s’agit d’expliquer, de déplier tous ces mondes (167). • L’amant jaloux développe les mondes possibles enfermés dans l’aimé (110). • Aimer, c’est chercher à expliquer, à développer ces mondes inconnus qui restent enveloppés dans l’aimé (14). • D’Albertine aperçue, Proust dit qu’elle enveloppe ou exprime la plage et le déferlement des flots.

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In the 1967 review of Vendredi, the two propositions are both united and paradigmatically exemplified par Albertine (Logique du sens, 357): • L’amour, la jalousie, seront la tentative de développer, de déplier ce monde possible nommé Albertine. The concept of modality appears much later: même quand Wittgenstein envisage des propositions de frayeur ou de douleur, il n’y voit pas des modalités exprimables dans une position d’autrui.7

Here we read: Autrui, c’est un monde possible, tel qu’il existe dans un visage qui l’exprime, et s’effectue dans un langage qui lui donne une réalité. En ce sens, c’est un concept à trois composantes inséparables: monde possible, visage existant, langage réel ou parole.

If we juxtapose the whole of what Deleuze says in modal logic with the basic modal logic in van Benthem, we may at least establish a bijection between the Deleuzian triad and a Kripke model M = :

So that we may define a Deleuzian structure or Deleuzian triad (briefly a Deleuzienne). Dz = Dz = . In a Deleuzienne, W is a set of possible worlds w in the generalized meaning where China and Albertine are also possible worlds (and therefore «worlds» simpliciter), E is a relation of «expression» borrowed by Deleuze from Leibniz, and LP is a set of propositions from a language L or an ana of talks P by inhabitants of the possible worlds.

3.4 The Boxes and Vases of the Recherche Du Temps Perdu In the augmented edition of Proust et les signes published in 1970, the main augmentation is the chapter «Les boîtes et les vases» where the terms of the title are the names of two «figures fondamentales»: 7

Qu’est-ce que la philosophie?, 22–23.

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(P&S 142) La première est dominée par l’image des boîtes entr’ouvertes, la seconde par celle des vases clos. La première (contenant-contenu) vaut par la position d’un contenu sans commune mesure; la seconde (parties-tout) vaut par l’opposition d’un voisinage sans communication.

Therefore, a «box» is a container with an incommensurable content, an a «vessel» or «vase» is an uncommunicating vase. We shall also speak of overflowing boxes and uncommunicating vessels. The decisive importance of this Deleuzian distinguo follows from the fact that it selects immediately a constellation of paradigms purveyed with the higher dramatic and architectonic significance: For the boxes, with their incommensurable content, the arch-formula is: (1675) une femme enfermant un pays.

For example, already from the first edition of Proust et les signes: (P&S 14) Albertine enveloppe, incorpore, amalgame «la plage et le déferlement du flot ».

And for the uncommunicating vessels, the two sides of Combray and the «plante qui s’appelle le vanillier» (1142) where (1228) L’organe mâle est séparé par une cloison de l’organe femelle. (1142) Celle-là produit bien des fleurs à la fois masculines et féminines, mais une sorte de paroi dure, placée entre elles, empêche toute communication.

Among the litteral «boîtes» and «vases» in the Proustian text we find: (2077) comme une surprise dans une boîte que nous viendrions d’ouvrir (1327) notre corps, semblable pour nous à un vase où notre spiritualité serait enclose / dans un domaine inconnu.

We must add a morphologic remark which opens the following breakthrough: in the Deleuzian distinguo between boxes and uncommunicating vases, the boxes are insulary entities while, as the communicating vases, the uncommunicating vases go in two. This corresponds to a decisive step.

3.5 Proustian Tierceity: Peirce in Proust Here we reach what is probably the knot of the whole modal logic builded by Proust. This knot is knotted by a correspondance established between, on the one hand, the Proustian boxes and vessels, and, on the other hand, the transition from the dyadic relationship, where an X actor finds access to a box B, to the triadic relationship where an X actor makes a V vase communicate with a W vase or, in other words, where X gives V access to W. In the Recherche, the don (giving, gift) is the paradigm of the triadic relation. It has two main hypostases, corresponding to the two most desirable gifts: Gilberte and Albertine. Firstly Swann gives Gilberte to Marcel. This is the traditional giving: the father gives his daughter to the pretender:

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(422) maître de sa fille, il me la donnait.

As Proust reminds us, an archetypal Father is the conjugal «master» of his daughter. But, when Elstir introduces Marcel to Albertine, he gives him only (684) un ‘bon’ valable pour des plaisirs futurs.

Elstir gives Marcel a ticket for Albertine. And the don of this ‘bon’ must be renewed by Albertine herself: (1027) MARCEL: Il me faut un «bon pour un baiser » ALBERTINE: Vous m’amusez avec vos bons, je vous en referai de temps en temps. (1313) Je peux prendre un bon, Albertine? – Tant que vous voudrez.

So, the kiss tickets are the keys of the Recherche romance. In this book, the performance of gifts is innumerable: Gilberte gives Marcel an agate ball: (324) je vous la donne, gardez-la comme souvenir.

Just after refusing a kiss (735) Albertine «me donna un petit crayon d’or ».

At the Opera House the ladies gave sweets to the gentlemen: (777) Elles se penchaient vers eux, leur offraient des bonbons.

At the seaside we find (741) ces jeunes filles qui, étendues sur la falaise me tendaient simplement des sandwiches.

Rachel is a triple donor: (876) elle m’offrit du champagne, me tendit une de ses cigarettes d’Orient et détacha pour moi une rose de son corsage.

A good matchmaker knows what to give: (1540) Mme Verdurin leur donnait alors deux chambres communicantes.

There are other kinds of gift: (1047) Mme de Stermaria se donnerait dès le premier soir.

And (Jn. 15.13): (2167) donner sa vie.

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According to C.S. Peirce, giving is the paradigm of the triadic relation, as the territory of Tierceity . Deleuze is well aware of this pivotal point: «la tiercéité, ce qui est trois par soi-même, ne se laisse pas ramener à des dualités: par exemple, si A ‘donne’ B à C, ce n’est pas comme si A jetait B (premier couple) et C ramassait B (second couple)…» [Cinéma 2, pp. 266]. And Tierceity, by itself, will qualify every triadic relationship, in general, for its Proustian function of creating communication between ci-devant uncommunicant vases. It is under this law that, since «Combray», Charlus watches Odette for Swann; that, more generally, (1520) «d’autres s’occupaient de surveiller pour moi»; that (2135) Robert entrusts Gilberte to Marcel («c’était en me confiant Gilberte qu’il repartait pour Paris»). Under the aegis of Tierceity, the fact that, as well as «x gives y to z», «x takes y to z» is a triadic relation will widen the triadic territory to the opposition of the captative and the oblative. This is what gives Morel’s dream its exact place: (1515) mon rêve, ce serait de trouver une jeune fille bien pure, de m’en faire aimer et de lui prendre sa virginité.

Thank God, as Sartre says, another relationship is framed at Albertine and Marcel in the household: (1685) faire la paix. To make peace, per se, is only a dyadic relation. But Marcel adds a crucial point: «Aucun de nous deux n’en avait pris encore l’initiative» (1685). This avowal leads us to the triadic relation containing the decision on the situation: X takes the initiative before Y to make peace with Y. With its generalization from (Mt. 5.9): X is a peacemaker between Y and Z. The Proustian function of triadic relations has a decisive corollary on the Deleuzian theory of possible worlds. When Deleuze says that «Aimer, c’est chercher à expliquer, à développer ces mondes inconnus qui restent enveloppés dans l’aimé», the objective relation by which Albertine envelops Balbec is contrasted with the subjective relation by which Marcel develops Albertine. But Marcel finds Montjouvain in Albertine is triadic as well as Albertine hides Montjouvain from Marcel. Triadicity, therefore, is the universal chiffre of the Proustian personnage and the triadic relationship his logical monogram. Up to Albertine’s apotheosis: (1967) c’est d’[Albertine], comme d’un vase, que je pouvais… recevoir [toutes choses]

This is also a Triadicity apotheosis (∀ at the heart of the gift relationship): Marcel can receive all things from Albertine. But Albertine’s gift has two faces: (1595) Elle m’offrait justement – et elle seule pouvait m’offrir – l’unique remède contre le poison qui me brûlait, homogène à lui d’ailleurs; l’un doux, l’autre cruel, tous deux étaient également dérivés d’Albertine.

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Incipit Pharmakos. Here it is in its theoretical rank, recently reassesed by Camille Tarot [15], 2021). With the Pharmakos amid the Triadicity, and therefore among the Sovereigns of the Possible, the Proustian metaphysics of modalities has acquired its axiological tip. Among the Sovereigns of the Possible, Albertine is the most powerful. In her is celebrated the marriage of heaven and hell. The axiological charge of the Pharmakos is transmitted to the scapegoat (bouc émissaire). (2081) «le premier bouc émissaire serait Paléologue », predicts Norpois.

But the Proustian universe has his own scapegoats: a mundane one, Saniette; and a political one, Dreyfus. So that the scapegoat operates a Platonic division of the Proustian worlds, between societies with scapegoat (such as the «petit clan» Verdurin) and societies without scapegoat (such as the «petite bande» of young girls). Yet, in Proust, the two-sided object, which concentrates the maximum difference, has unlimited repetition power: (1880) car tout doit revenir, comme il est écrit aux voûtes de Saint-Marc, et comme le proclament, buvant aux urnes de marbre et de jaspe des chapiteaux byzantins, les oiseaux qui signifient à la fois la mort et la résurrection.

However, amid the Queens of the Possible, Tierceity not only fathered Albertine, but also Gilberte. And (87) Le plus souvent maintenant quand je pensais à elle, je la voyais devant un porche d’une cathédrale, m’expliquant la signification des statues, et, avec un sourire qui disait du bien de moi, me présentant comme son ami à Bergotte.

Adicity (now «arity»), according to Peirce, has a scale. And on this ladder Gilberte climbs two rungs. In Proust, the metonymic cathedral is the Amiens of Ruskin, containing its labyrinth. So that Gilberte rises from dyadic to triadic as a new Ariane: Gilberte explains Amiens to Marcel. Then Gilberte climbs from triadic to tetradic: Gilberte introduced Marcel to Bergotte as Gilberte’s friend.

3.6 The Construction of Proust’s Landscape Since Un amour de Swann, we know that the Proustian universe is «la carte du Tendre» where «le plus émouvant des romans d’amour» est «l’indicateur des chemins de fer»: (239) Il passait ses journées penché sur une carte de la forêt de Compiègne comme si ç’avait été la carte du Tendre. (237) il se plongeait dans le plus enivrant des romans d’amour, l’indicateur des chemins de fer.

But it is also a Pascalian mad asylum: (1365) ce lieu jadis si terrible ou maintenant j’entrais dix fois par jour, ressortant librement, en maître, comme ces fous peu atteints et depuis si longtemps pensionnaires d’un asile que le médecin leur en a confié la clef.

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This key opens all the boxes of a Pythagoras table: Deleuzian scale of worlds

Soveraign of possibilities ◇ Tutelar character

Mondanité

Arbitre des élégances

Oriane de Guermantes, Charlus

Amour

Ariane-pharmakos

Gilberte, Albertine

Impressions and réminiscences: Écrivain «petit pan de mur jaune» Tasse de thé, madeleine Clochers de Martinville

Philosophe norvégien Bergotte Marcel /= Narrateur

Art

Elstir, Vinteuil

Rénovateur du goût

By the way (1719) Les clefs deviennent inutiles quand celui qu’on veut empêcher d’entrer peut se servir d’un passe-partout ou d’une pince-monseigneur.

Vainly Charlus proclaims (1240) «On n’entre dans ces salons que par moi». (1544) M. de Charlus vivait dupé comme le poisson qui croit que l’eau où il nage s’étend au-delà du verre de son aquarium qui lui en représente le reflet, tandis qu’il ne voit pas à côté de lui, dans l’ombre, le promeneur amusé qui suit ses ébats ou le pisciculteur tout puissant qui, au moment imprévu et fatal… le tirera sans pitié du milieu où il aimait vivre pour le rejeter dans un autre

Éternel (1219) Mané Thécel Pharès. (1280) Saint-Loup «alla jusqu’à me faire l’éloge des maisons de passe. ‘Il n’y a que là qu’on trouve chaussure à son pied, ce que nous appelons au régiment son gabarit’ ».

We leave to the reader the cutting of the cardboard wheels of Lulle (underestimated by Descartes, exactly estimated by Leibniz) representing this eulogy. The same wheels will be able to spin in the hands of «la Patronne»: (1541) Mme Verdurin cherchait «à savoir si on pouvait inviter ensemble le prince et M. de Charlus, si cela corderait »

On the level of love we now know the exploits of Gilberte and Albertine. At the level of impressions and reminiscences, Deleuze decides [2 p. 9]: les clochers de Martinville… l’emporteront toujours sur la madeleine.

3.7 Rally Paper on the Tender Map At the level of worldliness, Mrs. Verdurin is the champion, with Mme de Maintenon as the precedent: = princesse de Guermantes «par un 3e mariage» = duchesse de Duras; = (2330) Sidonie Verdurin («née des Baux »).

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Marcel has only some small successes on his small territory: (682) quelques jours après le départ de Saint-Loup, j’eus réussi à ce qu’Elstir donnât une petite matinée où je rencontrerais Albertine.

But the rally paper party is also a masked ball. And in a masked ball, what Quine calls referential opacity becomes a law. In Proust this referential opacity has its paradigm: (868) Madeleine prend Jésus pour le jardinier.

It also has its vegetal archetype: (1224) le volubilis jette ses vrilles là où se trouve une pioche ou un râteau.

This means that the volubilis takes a pickaxe or a rake for a tutor. Sometimes the misunderstanding is simulacrum: (1045) je ferais comme d’autres qui, ne pouvant pénétrer dans un couvent, du moins, avant de posséder une femme, l’habillent en religieuse.

But the misunderstanding has its Proustian apex in the lesson given to Marcel by Charlus: (1171) vous offrez à votre derrière une chauffeuse Directoire pour une bergère Louis XIV. Un de ces jours vous prendrez les genoux de Mme de Villeparisis pour le lavabo et on ne sait pas ce que vous y ferez.

The diagram is obvious:

bergère Louis XIV ⎯⎯⎯⎯⎯→ genoux de Mme de Villeparisis ↑ ↑ Derrière de Marcel ⎯ un de ces jours → Marcel ↓ ↓ chauffeuse Directoire ⎯⎯⎯⎯⎯⎯⎯→ le lavabo To pierce the masks, however, Proust saw that the champion is the Idiot of Dostoïevski: (1887) «Tu n’es pas telle» dit Muichkine à Nastasia.

Proust’s Rediscovered Time is firstly the remembrance of reminiscences. Proust himself marks in this moment the Terminus a quo and the Terminus ad quem: Terminus a quo (goût de la madeleine trempée, bruit métallique, sensation du pavé) Terminus ad quem (chambre de ma tante Octave, wagon du chemin de fer, baptistère de Saint-Marc)

Between the two, he imagines, out of the Thousand and One Nights,

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(2264) un docile génie prêt à le transporter au loin.

On Geach’s TWA, this genii drives Marcel’s flying carpet. But there is a supernumerary reminiscence. Its terminus a quo, a «serviette», «déployait, réparti dans ses pans et dans ses cassures, le plumage d’un océan vert et bleu comme la queue d’un paon» (2264). In other words, it is to Balbec what the origami was to Combray (46–47). Its terminus ad quem is the Grand Hôtel at Balbec, where (2268) Je n’avais, pour rejoindre Albertine et ses amies qui se promenaient sur la digue, qu’à enjamber le cadre de bois à peine plus haut que ma cheville.

Here a category-theoretic distinguo is required. In the previous reminiscences, all the terms, «chambre de ma tante Octave, wagon du chemin de fer, baptistère de Saint-Marc» as well as «goût de la madeleine trempée, bruit métallique, sensation du pavé» are just as many Proustian «worlds» and so, in a category, just as many points or «objects». While rejoindre Albertine et ses amies qui se promenaient sur la digue is an arrow from the world Marcel to the world Albertine or «petite bande». After the remembrance of reminiscences, impressions and reminiscences are subjected to an operation of clarification, with the aim of «les rendre claires jusque dans leurs profondeurs» (2270). This ultimate operation is explained long in advance in two great and contrasted metaphors. The first metaphor of writing shows the Duchess of Guermantes who, faced with small worldly miseries which the Courvoisier would think die of shame, plays with the difficulty, making it on the contrary the material of one of his famous witty words: (1105) on était obligé de l’envier d’avoir manqué de chaises, d’avoir fait ou laissé faire à son domestique une gaffe, d’avoir eu chez soi quelqu’un que personne ne connaissait, comme on est obligé de se féliciter que les grands écrivains aient été tenus à distance par les hommes et trahis par les femmes quand leurs humiliations et leurs souffrances ont été, sinon l’aiguillon de leur génie, du moins la matière de leurs œuvres.

This metaphor has its immediate diagram:

On another register «j’avais déjà tiré de la ruse apparente des fleurs une conséquence sur toute une partie inconsciente de l’œuvre littéraire» (1210). Hence the second metaphor, which bathes in the inebriation of pollination, with (1210) l’arrivée, presque impossible à espérer (à travers tant d’obstacles, de distance, de risques contraires, de dangers) de l’insecte envoyé de si loin en ambassadeur à la vierge qui depuis longtemps prolongeait son .attente. Je savais que cette attente n’était pas plus passive que chez la fleur mâle, dont les étamines s’étaient spontanément tournées pour

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que l’insecte pût plus facilement la recevoir; de même la fleur femme qui était ici, si l’insecte venait, arquerait coquettement ses «styles » et pour mieux être pénétrée par lui ferait imperceptiblement, comme une jouvencelle hypocrite mais ardente, la moitié du chemin.

But the fertilization is that of Marcel by Albertine, and (2301) elle m’avait fécondé par le chagrin.

A rally paper is a game, where the players are the snobs, the lovers, the steeples and the artists. But what sort of a game? A competition or a cooperation game? Proust invents a third possibility: the paradoxical cooperation. Albertine fertilizes Marcel. But par le chagrin. Albertine’s modus operandi leads us to a paradoxical competition: (1592) Vous vous rappelez que je vous ai parlé d’une amie plus âgée que moi qui m’a servi de mère, de sœur, avec qui j’ai passé à Trieste mes meilleures années et que je dois d’ailleurs dans quelques semaines retrouver à Cherbourg, d’où nous voyagerons ensemble (c’est un peu baroque, mais vous savez comme j’aime la mer), hé bien ! cette amie (oh pas du tout le genre de femme que vous pourriez croire !), regardez comme c’est extraordinaire, est justement la meilleure amie de la fille de ce Vinteuil, et je connais presque autant la fille de Vinteuil. Je ne les appelle jamais que mes deux grandes sœurs. Je ne suis pas fâchée de vous montrer que votre petite Albertine pourra vous être utile pour ces choses de musique, où vous dites, du reste avec raison, que je n’entends rien.

For we discover here that Albertine is engaged in the role of Ariane—that one that Marcel had dreamed of for Gilberte.

3.8 Double Conclusion Our study calls for a double conclusion: on logic and on its role in the novel. On logic, the continuous bass of AND with OR at Proust provides the most obvious benchmark. On this bass, when the kiss to Albertine goes from the impossible to the possible, this means that, in Balbec, Albertine is firstly represented as a possible world by a conjunction AND, and secondly made impossible by a NAND; then, in Paris, the three conjuncts of this NAND are negated, so that the initial possibility is restored. While the modal logic of Leibnizian ancestry built after Proust will define modality by logical quantity, Proust defines it directly by the Booleans. In Proust we therefore firstly find at least an anticipation of what will become of modal logic in the second half of the twentieth century. But this Boolean core is obtained at the price that Albertine is identified to the metaphorical conjunction given by the equation Albertine = her flavour AND her savour AND her colour. In a novel, such an allegory is available, especially if it allows a pioneering calculus in logic. But it can only be tolerated in very measured doses. That’s why, on this hard core, Proust articulates another dimension of his logic. It has also a Leibnizian model, when Leibniz recalls that Caesar preferred to be the first in a village rather

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than the second in Rome. But in Leibniz Caesar will always be second in the best possible worlds. Instead, Proust proceeds to a democratic distribution of possibilities, in affinity with the locality of modalities in the Egocentric Logic of A. N. Prior. Proust’s hidden logic, however, retains an enigmatic character. It calls for a foundation. What does a modal logic do in a novel? The answer takes us from Proust et les signes to Foucault’s problematic prognosis: «Un jour, peut-être, le siècle sera deleuzien». This prognosis, like an amphisbene, must be taken by its two ends: the problem end and the end solutions. According to Nietzsche, the philosopher is the doctor of civilization. Now medicine, like the amphisbene, has two ends: etiology and therapy. So that, on the Foucault’s amphisbene, medicine is the adequate Janus. On the problem or etiology side, the incipit of the twenty-first century teaches us, through its cumulation of catastrophes, that the century is, already, Deleuzian. This is because, for the Nietzschean function of the philosopher, Deleuze is now the only pretender, from literary criticism to geopolitics. On the solutions or therapy side, we are in a novel where Morel follows «des cours d’algèbre» while declaring «Ça dissipe ma neurasthénie» (1565). So that, in our therapy, the inventory of our pharmacopoeia asks for remembrance of mathematics. The history of mathematics is a curve passing through three main inflection points, each located in an inflection epoch. The first two epochs are easy to name. These are the days of Thales, Pythagoras and the Thaetetus, then the time of Descartes, Pascal and Leibniz. In other words, these are the epochs of consubstantiality between philosophy and mathematics. But the third epoch has a more complicated nature. In order to describe it we must imagine Columns of Hercules between which three naves follow one another. Hercules’ columns are Poincaré and Cantor, and the three naves, narrates Fernando Zalamea, «Riemann, Galois et Grothendieck, pour ne nommer que des figures indispensables» [16, p. 74], cf. p. 99 for the devils in the detail. What happened between the first two epochs and the third? Auguste Comte, last founder of religion, passed by the way. So that, in the Encyclopedia of Diderot & D’Alembert, was cut up the positivist Encyclopedia (this masterpiece where the function y = 1/ x enters into the philosophy of History with y as Extension and x in Intension). It’s the culture fracture, producing the “two cultures” according to Snow. Proust, in his casting, master of counter-employment roles, puts in Charlus’ mouth the civilizational stakes of the novel: (964) Dans le fond de notre tonneau, comme Diogène, nous demandons un homme.

What we are talking about is therefore the formation of the twenty-first century man. In case of fracture the doctor becomes the surgeon who reduces the fracture. And this is where Proust exercises his logic. Remembrance of the Past is not a quelconque-level novel. And we have to compare what is comparable. According to John Rajchman in The Freedom of Philosophy, «dans le domaine de la pensée Finnegans Wake» has «un rôle plus déterminant que la logique de Russell et de Frege». But, in Finnegans Wake we find the same ∃ as in the Principia Mathematica, and in his edition of Ulysses, Jacques Aubert registers that Joyce read the Introduction to Mathematical Philosophy of Russell where he found a Peirce neologism then recycled in his novel. However, these Joyce’s mathèmes are microscopic amphibians,

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simple floating beacons in the culture fracture. While in Remembrance of the Past the modal logic is a full scale and full-fledged system which pervades the total extension of the novel and of its universe. So that Marcel Proust is a doctor who has reduced the culture fracture. And since, in the Proustian universe, Dreyfus is the main scapegoat, we shall demonstrate, as an «ecthesis» of this general thesis on civilizational medicine, that the Dreyfus affair, in Proust, becomes a case of his modal logic: (862) Mme Sazerat, seule de son espèce à Combray, était dreyfusarde. Mon père, ami de M. Méline, était convaincu de la culpabilité de Dreyfus. Dès qu’elle le sut antidreyfusard, elle mit entre elle et lui des continents et des siècles. (...) elle n’eût pas songé à une poignée de main et à des paroles, lesquelles n’eussent pu franchir les mondes qui les séparaient.

Mrs Sazerat, such as a Hestia of Tansonville, sovereign of her gestures (including the shakehands), in our diagram of the Proustian universe, put a kyrielle of possible worlds between herself and papa. Therefore the Dreyfus affair is a case of the Proustian logic. Q.E.D.

References 1. Proust, M.: A la Recherche du Temps Perdu. Quarto, nrf, Paris (1999) 2. Deleuze, G.: Proust et les signes. Presses Universitaires de France; 2édition, augmentée, 1970, Paris (1964) 3. Boole, G.: An Investigation of the Laws of Thought. Walton & Maberly, London (1854) 4. Peano, G.: Studii di logica matematica. Atti Reale Acad. Sci. Torino 32, 565–583 (1897) 5. Meredith, C.A.: Interpretations of different modal logics in the ‘property calculus’. Mimeograph, Department of Philosophy, University of Canterbury, Christchurch, New Zealand, August 1956. Recorded and Expanded by A. N. Prior (1956) 6. Blackburn, P.: Arthur prior and hybrid logic. Synthese (2006) 7. Benthem, J.V.: Modal Logic. In: Jacquette, D. (Ed.) A Companion to Philosophical Logic. Blackwell, Oxford (2002). 8. Dumoncel, J.-C.: La Mathesis de Marcel Proust. Classiques Garnier, Paris (2015) 9. Kripke, S.: Semantical Considerations on Modal Logic. Acta Philosophica Fennica, 16, 1963, 83–94, reprinted in Leonard Linsky (ed.), Reference and Modality, Oxford, 1971 (1963) 10. Kalinowski, G.: Sémiotique et Philosophie. Hadès-Benjamins, Paris-Amsterdam (1985) 11. Konyndyk, K.: Introductory Modal Logic. University of Notre Dame Press (1986) 12. Copeland, J.: The genesis of possible worlds semantics. J. Philos. Log. 31, 99–137 (2002) 13. Prior, A.N.: Past, present, future. Clarendon Press, Present and Future, Oxford (1967) 14. Geach, P.: Mental Acts, their Content and their Objects. Routledge and Kegan Paul, London (1957) 15. Tarot, C.: Le symbolique et le Sacré: Théories de la Religion. La Découverte, Paris (2008) 16. Zalamea, F.: Philosophie synthétique de la mathématique contemporaine, 2009, traduit de l’espagnol et de l’anglais par Charles Alunni. Hermann, Paris (2018)

Chapter 4

Existential Graphs as an Outstanding Case of the Use of Diagrams in Mathematics Arnold Oostra

Diagrams play a far underestimated role in present-day mathematics, save for a few exceedingly rare exceptions. Although it would seem less expected, one of the realms where diagrams may thrive in the future is that of mathematical logic. A suitable context for certain diagrams, which are so plain that they may be seen simply as gestures, provides a diagrammatic presentation of mathematical logic, classical and beyond. These diagrams, called existential graphs by their creator, Charles S. Peirce, were labeled by him as the logic of the future, and probably rightly so. In this paper we work out different systems of existential graphs starting from some extremely simple diagrams, strokes, or gestures, thus emphasizing the power of the basic ideas involved. Section 4.1 is a short reflection on the use and misuse of diagrams in mathematics. In Sect. 4.2 we give a brief fundamental and historical introduction to existential graphs. In Sects. 4.3, 4.4, and 4.5 we present the classical graphs emerging from considerably basic gestures. Section 4.6 displays a system of existential graphs for a well-known non-classical logic, while Sect. 4.7 shows a way back from geometry to logic.

4.1 The Role of Diagrams in Mathematics Undoubtedly, one of the paradigms in the use of diagrams in mathematics is Euclid’s Elements. Almost all propositions are accompanied by a diagram, which in most cases serves as a guide to the argument. In some cases, however, the proof is strongly attached to the diagram because the conclusion is derived directly from it. Over time, a progressive de-visualization of geometry and mathematics became evident. In the seventeenth century Descartes translated geometric plane curves into A. Oostra (B) Departamento de Matemáticas y Estadística, Universidad del Tolima, Ibagué, Colombia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_4

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algebraic equations, and, at the same time, projective geometry was born, in which non-graphical objects such as the point at infinity occupy an important place. In the nineteenth century, the emergence of non-Euclidean geometries probably caused some distrust towards diagrams and certainly prompted the study of formal deductive systems. In 1899 Hilbert published Foundations of Geometry and, at that point, the objects of study of geometry became completely abstract things whose relationships were defined by the axioms. Hence, the very few diagrams in Foundations can be eliminated without the argumentation losing any of its rigor. This formalist tradition was taken up by Bourbaki, a group of predominantly French mathematicians who wrote a large series of textbooks known as Elements of Mathematics. Their writings are known for the scarcity of illustrations, in fact, V. I. Arnol’d wrote: What did Barrow’s lectures contain? Bourbaki writes with some scorn that in his book in a hundred pages of the text there are about 180 drawings. (Concerning Bourbaki’s books it can be said that in a thousand pages there is not one drawing, and it is not at all clear which is worse). ([1], 40)

For better or for worse, this seems to be the main attitude of present-day mathematicians towards diagrams: They are a useful help to clarify some idea, but they have no mathematical value, since the rigor lies in the formal, verbal, and algebraic proof. There are even flawed proofs in basic Geometry—like all triangles are isosceles— based upon an erroneous figure, and the conclusion that always follows is: “See? We cannot trust diagrams”. Thus, diagrams are banned and shunned in mathematical practice—although surely every mathematician draws some secret sketches here or there. But obviously, the fact that a certain figure leads to wrong conclusions does not imply that all diagrams lead to errors and that, therefore, all visual representation should be banished. We may only conclude that some diagrams are incorrectly constructed or interpreted. If we would accurately describe a certain type of figures and specify strict rules of formation, transformation, and interpretation for them, no wrong conclusions could possibly follow. In fact, some such graphical contexts have already been defined in mathematics and are in use, although extremely limited. In homological algebra the use of diagrams is quite common, and even there is a famous result known as the snake lemma because of the figure that always accompanies it. The history of those diagrams is fascinating ([10], 29): The fundamental idea of representing a function by an arrow first appeared in topology about 1940, probably in papers or lectures by W. Hurewicz on relative homotopy groups. His initiative immediately attracted the attention of R. H. Fox and N. E. Steenrod, whose 1941 paper used arrows and (implicitly) functors; see also Hurewicz-Steenrod [1941]. The arrow f : X → Y rapidly displaced the occasional notation f (X) ⊂ Y for a function. It expressed well a central interest of topology. Thus a notation (the arrow) led to a concept (category). Commutative diagrams were probably also first used by Hurewicz.

In fact, then, according to Mac Lane, this notation from homological algebra led to the birth of category theory, a field whose content is all we can express with arrows.

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According to its founders, “category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams and arrows” ([10], 1). The rules that govern these diagrams are the very precise axioms of a category, and this is a context in which correct drawings lead to valid conclusions. In fact, in category theory we can carry out many rigorous proofs using only diagrams. A more recent context in which diagrams play a central role are dessins d’enfant, which are remarkably simple connected graphs with alternating black or white vertices. Drawn on a Riemann surface, “there is an amazing relationship between these dessins and deep arithmetical questions” ([25], 788). Following Grothendieck, the modern inventor of these graphs, “to such a dessin we find associated subtle arithmetic invariants, which are completely turned topsy-turvy as soon as we add one more stroke” (Op. cit., 789). Once again, an amazingly simple system of diagrams with adequate rules provides powerful rigorous mathematical results. Here we could ask ourselves about the possibility of a general theory of mathematical diagrammatic systems. Undoubtedly, one of the fundamental characteristics of such must be their extreme simplicity. But also, its potential to obtain a whole body of results from little basic information, thus reflecting the axiomatic thinking so common in mathematics. In this paper, however, we will limit ourselves to expanding on just one system of diagrams, which in the future could probably become an archetype of the sought systems.

4.2 Peirce’s Existential Graphs Charles Sanders Peirce was one of the pioneers of the mathematical study of predicate logic, called in his time the logic of relatives. In papers written between 1879 and 1885, he developed the theory of quantification with almost the same content as the first-order logic in standard use in mathematics today, only with slightly different signs. However, as early as 1882, he wrote in a famous letter: “The notation of the logic of relatives can be somewhat simplified by spreading the formulae over two dimensions” ([20], 4.394). Contrary to what is customary in various contexts of mathematics, such as Hasse diagrams for partially ordered sets, Peirce’s basic idea is to represent the subjects with lines and the relatives with nodes. Even in these early experiments, there were nodes with one, two, or three lines, representing monadic, dyadic, and triadic relatives. On the other hand, the lines could branch out to have two, three, or four ends connected with nodes. About fifteen years later, and after much experimentation, Peirce came to the oval as a sign of denial [2]. This simple diagram also marks off two different regions and, hence, points out a direction, as we will see later in detail. Thus he arrived at the system of existential graphs as a diagrammatic representation of logic. In 1896, the year existential graphs were born, Peirce wrote:

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We now have an apparatus capable of analytically expressing every proposition which can be analytically expressed by the “general algebra of logic”, so-called, and it thus involves a virtual analysis of the proposition which I will not lengthen this paper by explicitly developing; for I hope that others will think the matter out for themselves. I will however call attention to the fact that this system is far more perfect than logical algebra in being more analytical, and analysis is the chief thing in logic. It also supplies a far more fundamental list of rules for logical algebra than has ever been given before. This system is also more perfect than any algebra of logic yet devised in that it is not encumbered with a mass of formal theorems which have no meaning except the equivalence of two ways of writing the same thing. ([21], 289).

Starting from amazingly simple gestures—the line, the oval—and giving them an appropriate contextual meaning, Peirce built a robust system of representation for mathematical logic. In the next sections, we will tell again the tale of Peirce’s existential graphs, emphasizing their development from plain diagrammatic strokes.

4.3 Classical Alpha Graphs Simple stroke: an oval. In this section we introduce the most basic system of existential graphs, starting from the elementary gesture of enclosing. Classical existential graphs are twodimensional diagrams drawn upon a plane surface. We attach a logical meaning to every graph, and the first convention we make is that marking a graph on the surface signifies asserting its meaning, hence the name sheet of assertion given to this underlying surface. For instance, to write a sentence on the sheet means to assert it. As is customary in mathematics, we abbreviate the sentences with letters. To write two or more propositional letters on the sheet means to assert all sentences thus represented. In general, if we juxtapose two graphs on the sheet, we assert the meaning of both, hence obtaining naturally a graphical representation of conjunction. Since the diagrams are two-dimensional, and since the relative position of the graphs is immaterial, this conjunction is automatically commutative and associative. At this point, the need for a way of representing negation becomes apparent. Simply crossing out the false propositions would not suffice, since we could not deny the conjunction of two incompatible propositions. Following Peirce’s much sought after and masterful solution, the second convention we make is that enclosing a graph on the sheet signifies denying its meaning. Because of the opposition it represents, the simple closed curve we use to surround a graph is called a cut. The shape of the cut is irrelevant, but its preferred and simplest appearance is the oval. In mathematics, a simple closed curve is also known as a Jordan curve, because of the celebrated curve theorem stated, and first proved, by the French mathematician Camille Jordan in 1887. Following this result, a cut divides the sheet of assertion into two regions, the bounded interior of the curve and the unbounded exterior. Hence, by the two basic conventions, we may write any given sentence on the outside or on the

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inside of the cut, i.e., we may assert or deny it. In this way, the proposed interpretation of these graphs leads to a certain immanent principle of excluded middle. If enclosing means denying the content of a cut, then a cut around another is the double negation of the graph contained in the innermost. In a classical logic setting, a double negation amounts to the same as an assertion, and thus we find two different graphs with the same logical meaning. Even better, we might think of graphical inference rules that would permit us to transform one graph into another, with logical coherence. A first rule that we could state is: A double cut, without any graph between them, may be drawn around or removed from any graph on any area. Thus, the basic interpretation of the existential graphs naturally gives rise to graphical rules of inference, which constitutes a true diagrammatical logic. But other combinations with cuts may arise. When we juxtapose two enclosed sentences, we assert the negation of both. By contrast, tracing a cut around two juxtaposed propositions means denying their conjunction, or to assert their discrepancy, without judging which of them is false. Furthermore, if we enclose the whole graph composed of a sentence juxtaposed to an enclosed one, we are asserting the discrepancy of the former with the negation of the latter, or, what amounts to the same, we assert the material implication of the second from the first. Note that here again we have two cuts, one inside the other, but this is no double negation since there is a letter in the region between them. In this way, we arrive at the existential graphs representing the basic connectives (Fig. 4.1). Letters A, B in this list stand for sentences but we may also substitute them for other graphs. We obtain the graph of the disjunction by denying the simultaneous negation of two sentences or graphs. Again, this disjunction is automatically commutative. Although it is not entirely immediate, we can deduce its associativity with the rule of double negation. By contrast, implication is clearly not commutative. No matter how much we bend or twist the diagram, always the antecedent will be between the two cuts, and the consequent inside both. The implication graph has an outer region and an inner region, which in turn clearly specify an inward direction “from the outside to the inside”. What can we say when we combine various cuts? Since a cut always surrounds something, multiple cuts never intersect. If we start with a graph that has only one Fig. 4.1 Alpha graphs for the basic connectives

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Fig. 4.2 The inward direction in alpha graphs

cut, we have two areas, one bounded and one unbounded. If we then draw a second cut, this could be on the outside of the first to signify two juxtaposed negations, or it could be on the inside of the first to signify an implication or a double negation. In both cases, we obtain three areas, one of them unbounded. In general, a graph with any finite number n of cuts marks out n + 1 regions, only one of them unbounded. This unbounded area is the sheet of assertion on which the graph is drawn. But it is also the outermost region, which signals the beginning of the inward direction. Every cut not enclosed on the sheet marks a first step inwards, then any cut inside this area a second, and so on, always further inward (see Fig. 4.2). In this way, the cut is not only a symbol of negation but also a “collectional sign” ([2], 541). It allows us to gather different sentences or graphs that we would like to deny. Moreover, it fulfills the function of indicating an order of application of the logical operations. The inward direction of the graphs brings with it a surprising logical consequence. If a sentence materially implies another, then it certainly implies the conjunction of both. In the context of existential graphs, this means that we may copy a graph that is in the antecedent of an implication and transfer it inwards to the region of the consequent. Thus, the two graphs on the left of Fig. 4.3 have the same logical meaning. On the other hand, if a sentence is asserted on the sheet but its conjunction with another is denied, then the second one is false. Hence, the two graphs on the right of Fig. 4.3 have the same logical content. In this way we arrive at a new pair of graphical entailment rules. Inwards iteration: Any graph may be iterated in its own region, or in any area contained in this. The only exception is that, to avoid a certain “logical inbreeding”, the contained area is not part of the graph to be repeated. On the other hand, “deiteration” from the outside: Any graph may be erased if a copy of it persists in the same area or in any area around this. Thus, the inward direction of the existential graphs naturally gives rise to graphical rules of inference. Fig. 4.3 Instances of iteration and deiteration

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When we move inwards across various cuts of an existential graph, starting from the unbounded exterior, the first cut means a negation and marks the passage from assertion to denying. Going across a second cut means denying with respect to the negation and, in a classical logic scenario, it is again a passage from negation to assertion. And so on, hence it makes sense to call an area surrounded by an even number of cuts even, and odd one surrounded by an odd number of cuts. Loosely speaking, even areas are regions of assertion and odd areas of denial. This conception suggests a final pair of graphical entailment rules. Asserted information may be safely forgotten, this is the traditional conjunction elimination. In graphical form we obtain the rule of erasure from even: In an even area, any graph may be erased. On the other hand, in a denying area we may add freely any information, hence the rule of insertion in odd: In an odd area, any graph may be scribed. The letters and cuts on the sheet of assertion, with their given logical interpretation, are known as Alpha graphs. The system is completed with the above-mentioned graphical rules of transformation, which we can list as follows: 1. 2. 3. 4. 5.

Erasure in even Insertion in odd Iteration inwards Deiteration from outwards Double cut

The upper part of Fig. 4.4 shows a completely graphical proof of modus ponens, where from A ∧ (A → B) we derive B. The lower part is a proof of the law of excluded middle, where from the empty sheet of assertion, that is without premises, we deduce A ∨ ¬A. With this interpretation and these rules of transformation, the system of Alpha graphs is logically equivalent to classical propositional calculus. For formal proofs see, for example: [5, 22, 23, 26].

Fig. 4.4 Two graphical proofs with alpha graphs

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4.4 Beta Graphs Added simple stroke: a line. When we include the elementary gesture of connecting, we come to the heart of existential graphs, a realm where there is more than just Alpha. To represent predicates, we first need a way to graphically symbolize a subject. This is attained with a new component of existential graphs that is a line, which eventually may branch out. Drawing a line on the sheet of assertion signifies stating the existence of a subject, and from this convention comes the generic name of existential graphs that is given to this system of diagrams. Furthermore, connecting different lines means asserting that these subjects are the same, thus identifying them, hence the name line of identity given to this item. At any end of a line of identity we may attach a predicate, again abbreviated by a letter. Writing a predicate attached to a line means asserting the symbolized predicate with respect to the existing subject. Moreover, we may include relations or predicates about different subjects, simply joining various lines to the same letter. Thus, in this setting, every letter has a predetermined number of lines attached, and if there are more than one, then an order is settled a priori between them. Contrary to the Alpha case, here multiple possibilities arise in the interplay between lines and cuts. When we surround a line completely by a cut, we deny the existence of the subject altogether. But we might also deny only a predicate, enclosing its letter but leaving part of the line outside the cut. With this we assert that the subject does exist but that the predicate in question is false about it. Here we assume that a line can cross a cut, and again the inward direction becomes quite significant. In the simple case mentioned, there exists (on the outside) a subject that does not fulfill the predicate (on the inside). The reading of the sentence is carried out in the inward direction. An unbranched line of identity with a monadic predicate at each end gives rise to various combinations with cuts. Surprisingly, four of these existential graphs exactly match the four basic categorical propositions of Aristotelian syllogistic logic (see Fig. 4.5). Fig. 4.5 The square of opposition with existential graphs

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We might read the existential graph on the top left of Fig. 4.5 with the inward direction as: It is false that there exists an S that is not P. This way we arrive at the standard reading: All S are P. The system of Beta graphs comprises identity lines, letters with the suitable number of lines attached, and cuts, all on the sheet of assertion, with their given logical interpretation. The graphical rules of transformation that complete the system are the same Alpha rules, only extended with the following adaptations to the line of identity: 1. Erasure in even. In an even area, any line of identity may be cut. 2. Insertion in odd. In an odd area, two lines of identity may be joined. 3. Iteration inwards. A branch with a loose end may be added to any line; any loose end of a line may be extended inwards through cuts; when there are lines of identity involved in the graph to be iterated, they must correspond exactly to those of the original graph. 4. Deiteration from outwards. A branch with a loose end may be removed from any line; any loose end of a line may be retracted from the outside through cuts; when there are lines of identity involved in the graph to be deiterated, they must correspond exactly to those of the outside copy of the graph. 5. Double cut. The application of this rule is not prevented by the presence of lines that cross both cuts, that is, that pass from outside the outer cut to inside the inner. Figure 4.6 shows a completely graphical proof of syllogism Darii: All M are P; some S are M; hence, some S are P. The system of Beta graphs is logically equivalent to first-order logic with equality over a purely relational language. For formal proofs see [22, 26].

Fig. 4.6 Proof of syllogism Darii with beta graphs

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4.5 Modal Gamma Graphs Added simple stroke: a dashed oval. A subtle change in gesture opens a wide spectrum of new possibilities with existential graphs. If a closed curve indicates the negation of its interior, what does a closed curve drawn with a dashed line mean? A feasible interpretation would be a possible negation of its interior, here “interior” refers to the interior of the suggested continuous Jordan curve with the same “shape”. Although other modal interpretations are admissible ([9], sec.5), we work out some consequences of the basic alethic modal reading. We call a cut drawn with a dashed line a broken cut and enclosing a graph on the sheet with a broken cut means asserting that it is contingent, or non-necessary. The combination with continuous cuts brings forth graphs to express the basic modalities. In a classical setting, a double cut consisting of a continuous cut around a broken cut expresses the necessity of its interior and, dually, a double cut with a broken cut around a continuous one expresses the possibility of its interior. In Fig. 4.7 we show four basic alethic modalities with their representation in existential graphs and their customary logical notation. Being just a special kind of cut, the broken cut also divides the sheet into two regions. Hence, to determine whether a given area is odd or even, we count continuous and broken cuts alike. Moreover, the rules of erasure and insertion apply in the same way as for Alpha graphs, for instance, we may draw any graph in an area enclosed only by a broken cut on the sheet. Furthermore, a natural adaptation of these rules of transformation to the broken cut is: A continuous cut within an even area may be “half erased” into a broken cut, and, on the other hand: A broken cut in an odd area may be “completed” into a continuous cut. With these permissions we may transform a double necessary cut into a double continuous cut, and these again into a double possible cut. Since a double continuous cut may be drawn or removed freely, the outcome is, first, that from A is necessary we may conclude that A is true; and

Fig. 4.7 Basic modalities with existential graphs

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Fig. 4.8 Proof of a modal axiom with gamma graphs

second, that from A is true we may derive that A is possible. These inferences are common in many modal logics. Another usual feature of modal logics is the rule of necessitation, which states that all tautologies are necessary. Since all tautologies are graphically derived from the empty sheet of assertion, in the context of existential graphs this rule may be stated as a case of double cut: An empty double necessary cut may be drawn or removed on any area. What about iteration? In the Alpha graphs we derived the first example of iteration from this consideration: When a sentence materially implies another, it also implies the conjunction of both. But in modal logic, if one sentence implies that another is necessary, this does not imply that both are necessary. Because this would mean in particular that the antecedent implies it is necessary, which is not true for all propositions—although it is for a necessary one. Surprisingly, then, iteration and deiteration through broken cuts are not allowed for all graphs. Moreover, if a given asserted graph is iterable through broken cuts then it is also necessary. At this point, a series of options opens. For different selections of graphs that are iterable and deiterable through broken cuts, we obtain diverse systems of existential graphs that match various modal logics. In this way, the broken cut, which initially posed a problem for iteration, turns out to open a variety of alternatives. It appears that our first choice for Gamma rules of iteration and deiteration is this: Only necessary graphs may be iterated and deiterated through broken cuts. Figure 4.8 shows a graphical proof of tautology ☐A→☐☐A, which is characteristic of Lewis’ modal logic S4 ([8], 53). We may carry out the steps marked with an asterisk by the Gamma variations of the rules, specifically adapted to the broken cut. The system of Gamma graphs with these rules of iteration and deiteration is logically equivalent to modal logic S4, for a formal proof see [26]. There are two other systems of Gamma graphs, given by certain rules of iteration and deiteration, which match known modal logics ([8], 58, 134): • S4.2. Only necessary or possibly necessary graphs may be iterated and deiterated through broken cuts. • S5. Only graphs whose minimal components are all inside a broken cut belonging to the same graph, may be iterated and deiterated through broken cuts. This variety of alternatives highlights the central role that the iteration and deiteration rules play in all systems of existential graphs. In addition, the filter of the broken cut regulates the rules of transformation and, therefore, the logic of the whole system. Once again, we ratify the power of the diagrams involved.

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If a broken cut on the sheet of assertion is interpreted as a possible negation, giving rise to graphical systems of modal logics, perhaps we could interpret a broken identity line on the sheet as the possible existence of a subject. This is not the same as enclosing the line of identity in a double possible cut, due to the greater geometric freedom of the line. This apparently unexplored way of inquiry might lead to new systems of existential graphs and even new logics.

4.6 Intuitionistic Existential Graphs Simple stroke: a limaçon. Surprisingly, the gesture of pasting an oval into another with one continuous stroke breaks the logic. In the original system of Alpha graphs, there is a specific sign only for the negation, which is the cut. We attain the conjunction without any sign, just by the juxtaposition of the conjuncts. The other usual connectives of propositional logic: implication and disjunction, are defined in terms of conjunction and negation. This is a characteristic feature of classical logic. In a purely diagrammatic context, it makes sense to think of a special sign for other propositional connectives, at least for implication and disjunction. Perhaps this could guide us to a deeper understanding of the interaction between the propositional connectives, or even to other logics. There are logical systems, such as intuitionistic logic, where these connectives cannot be expressed in any way in terms of negation and conjunction [4, 6, 7, 11]. Therefore, for a graphical representation of such logics, we imperatively need other signs. As a fundamental diagram for implication, we propose a slight variant of the Alpha graph, in which the inner cut touches the outer cut at a single point. Thus, there is a strong geometric connection between the two cuts, the exterior one that contains the antecedent and the other cut, and the interior one that contains the consequent. In fact, topologically, this is a connected set. We may see this compound sign as an individual curve or limaçon, and we can draw it with a single continuous stroke (Fig. 4.9). This new sign for implication is called a scroll, and its inner cut loop. It is particularly important that the loop is inside the cut. Two cuts connected by a point but each one outside the other would not suit as a sign for implication, because of the symmetric positions of the two regions: There would be no way to determine which is the antecedent and which is the consequent. In fact, this sign consisting of two connected cuts is used elsewhere ([3], 10) for disjunction, which is a commutative connective. Fig. 4.9 A new sign for implication

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Here we might wonder if there is any logical relationship between this new diagram for implication and the classical Alpha graph. If the graph in Fig. 4.9 symbolizes A → B, then its conjunction with A and ¬B would lead to a contradiction, and we should deny this conjunction. Therefore, the newly introduced sign symbolizes a strong implication, from which we might derive the classical one. In this setting, we can naturally derive the sign for disjunction from the diagram we choose for implication. If a scroll made from a cut and a loop means the implication with antecedent in the outer area and consequent in the loop, then a multiple scroll with two loops might signify the implication whose antecedent is in the outer area and whose consequent is one of the loops, or the disjunction of the two. Notice that in the classical Alpha graphs, such an implication of a disjunction is drawn, minus a double cut, as a cut that contains the antecedent besides two cuts with the disjuncts. Again, now it is as if we glued the two inner cuts as loops in the outer cut. And so on, for three or four or more loops. If a scroll with two loops represents the implication with antecedent the content of the outer area and consequent the disjunction of the loops, then it follows immediately that such a scroll with an empty outer area is the disjunction of the contents of the loops. In this way we arrive at the graphs for the disjunction of two, three or more sentences as a cut with the appropriate number of loops inside. Figure 4.10 shows the new graphs representing the basic connectives. We can even recover the cut as the sign for negation in this setting. If a scroll with multiple loops means implication with the antecedent in the outer area and consequent in some of the loops, then a cut, seen as a scroll without loops, means implication with no choice of consequent, or an absurd consequent, which is exactly the negation of the given antecedent. Moreover, we might choose the empty cut as a sign of the absurd. Actually, in classical Alpha graphs, this is an explosive graph

Fig. 4.10 New graphs for the basic connectives

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from which we can deduce everything, and this amounts to absurdity in many logics. Then, the cut that surrounds a graph can be seen as an implication whose consequent is the empty cut, as we show in Fig. 4.10. When it comes to deciding whether an area is odd or even, we will count cuts and loops equally, whence an area is even if it is surrounded by an even number of cuts or loops, and odd otherwise. For instance, in a scroll not enclosed on the sheet, the outer area is odd and the areas inside the loops are even. This has a precise and coherent mathematical explanation, but here we take it as an arbitrary convention. As we did with the Beta and Gamma graphs, we can extend and adapt the basic rules of transformation to these new graphs. In this case, we state the rules in length and add the special clauses referring to loops. Perhaps the most significant change is the double negation, which we replace entirely for a rule of scrolling, in the sense of ornamenting with a scroll. 1. Erasure in even. In an even area, any graph may be erased. Any loop within an even area may be eliminated with its contents. 2. Insertion in odd. In an odd area, any graph may be scribed. In an odd area limited externally by a cut, a loop containing any graph may be added to this cut. 3. Iteration inwards. Any graph may be iterated in its own area, or in any area contained in this, that is not part of the graph to be repeated. Any loop may be iterated, with its contents, on its own cut. 4. Deiteration from outwards. Any graph may be erased if a copy of it persists in the same area or in any area around this. A loop with its contents may be erased if another loop with the same contents persists on its cut. 5. Scrolling. A scroll with empty outer area may be drawn around or removed from any graph on any area. Now we can prove graphically, but with all formal strictness, that the new implication is stronger than the classical Alpha implication (Fig. 4.11). In an informal but very geometric way, this means that we may loosen a loop from its cut. In the same way, but with some more steps, we can prove that the new disjunction is stronger than the classical Alpha disjunction. Furthermore, we may loosen one loop from a disjunction, going from A ∨ B to ¬A → B, and then the other, going from ¬A → B to ¬(¬A ∧ ¬B). However, neither of these loosenings is, in general, reversible. Because, if we could prove the reverse joining, then the graphs

Fig. 4.11 Proof of strong implication with new graphs

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would be completely equivalent to the classical Alpha graphs. And the interest of these new graphs is precisely their novelty and difference from the original ones. If we remove A throughout the proof shown in Fig. 4.11, then from B surrounded by a scroll with an empty outer area, which by the rule of scrolling is equivalent to B, we get the double negation of B. However, the converse is, in general, not feasible. Thus, this new system of graphs leads us to a logic without the principle of double negation. With the mentioned rules of transformation, the system of these new graphs is logically equivalent to intuitionistic propositional calculus—for a formal proof see [17, 18]. Hence, we can correctly call this graphs intuitionistic Alpha graphs.

4.7 Existential Graphs on Non-planar Surfaces Simple stroke: an oval. In this section, we muse on what might happen if we change the context of the most elementary gesture. Up to this point, we have developed all systems of existential graphs upon the sheet of assertion, which is tacitly assumed to be a plane in the mathematical sense, that is, a flat or level surface thus extending unlimitedly in all directions. But there are other surfaces that locally are like a plane but not globally, such as the sphere, the cylinder, or the torus. Plane geometry, which was probably first invented for small flat portions of the earth’s surface, when ideally extended on the plane became Euclidean geometry. But the geometry of the entire earth’s surface, ideally developed on the sphere, is non-Euclidean. In general, in mathematics, a surface is a generalization of the plane in the sense that it is locally like the plane, but whose curvature is not necessarily zero, that is, a surface is not necessarily flat. As an exploratory extension of previous systems, we might think of drawing existential graphs on surfaces that are not necessarily flat. Just as geometry changes when the plane is replaced by a different surface, if existential graphs are drawn on a non-planar surface, then perhaps a suitable interpretation is possible and the outcome is some logical or pre-logical system, which most surely corresponds to some non-classical logic. This idea, if successful, can lead the way to a surprising new interplay between logic and geometry. In one direction, given a logic, we may ask whether there is an adequate system of existential graphs for it. As we have seen, in many cases the answer is yes. Now, in the other direction, given a surface, we may ask if there is a logical system that corresponds to the existential graphs drawn on it. In the following paragraphs, we will study some illustrative cases and reflect on the difficulties found in this still incipient program. To begin with, we consider the sphere. Writing a propositional letter on the sphere might be interpreted as asserting the proposition and writing two or more could mean the assertion of all. In this way, we maintain a graphical representation of conjunction on the sphere. As on the plane, this conjunction is naturally commutative

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and associative. However, the drawing of an oval on the sphere does not enclose a region since there is no longer an inside and an outside. It is well known that the Jordan curve theorem on the sphere is valid insofar a simple closed curve divides the sphere into two regions. However, both components are bounded and, furthermore, they are completely indistinguishable. Therefore, we cannot interpret the cut as a negation of one region with respect to the other. Although it separates certain letters from others, in this context the oval is not a “collectional sign” [2] since it does not define a direction for the application of logical operations. However, we might interpret a cut drawn on the sphere as a contrast, an antagonism, or an antithesis of the propositions written on the opposite sides. In that situation, the sentences are considered contrary, although in no way could we mark one of them as “true” or “false”. What about the combination of various cuts? Assuming a certain duality principle, we might agree that doubly opposite propositions are compatible or congruent. When we draw on the sphere two cuts that do not intersect, we obtain three areas, all bounded although one is different from the other two because it takes the shape of a cylinder (Fig. 4.12). In general, a graph with a finite number n of cuts drawn on the sphere marks out n + 1 regions, all of them bounded but with some different shapes. Among the letters written on the sphere now emerges a surprising relation: two of them are opposite if between them there is an odd number of cuts, and compatible if, on the contrary, the number of cuts between them is even. To complete a certain logical or pre-logical system on the sphere, rules of transformation are required. The double cut rule seems naturally valid in this context because drawing or erasing two cuts without any letter or cut between them does not alter the defined relations of opposition and compatibility. But the other rules of transformation on the sheet depend strongly on the inward direction determined by the inside and outside of the cuts, a direction that does not exist on the sphere, and on the parity of the areas defined by the outermost area, which also does not exist on the sphere. Future research should define rules of transformation for writing or erasing graphs on the sphere, thereby hoping to obtain a system of Alpha graphs for the pre-logic of opposition on the sphere. Fig. 4.12 An alpha graph on the sphere

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Fig. 4.13 An alpha graph on the cylinder

In this way, the simple gesture of drawing an oval depends largely on the context where it is performed. If it is an unlimited flat plane, this trace determines an inward direction and brings with it a whole diagrammatic system for the complete classical propositional logic. In contrast, if the drawing surface is a sphere, this stroke determines only a separation, and we hope that it will eventually lead to a graphical system for a pre-logic of opposition. Next, we ponder Alpha graphs on the cylinder, that is a tube that extends indefinitely in both directions. Unlike the plane and the sphere, on the cylinder there are two different types of simple, closed curves or Jordan curves (Fig. 4.13). For closed curves that do not surround the cylinder, we obtain the same result as on the plane: The curve divides the cylinder into two regions, the bounded interior of the curve and the unbounded exterior. Therefore, such a curve encloses a bounded area of the cylinder. But on the other hand, a closed curve might go around the cylinder. These other curves divide the cylinder into two regions, but both are unbounded and, as with the sphere, the two parts are indistinguishable. Hence, a curve of this second type has no interior or exterior. If we restrict the graphs on the cylinder allowing only cuts that do not surround the tube, any of these cuts clearly determines an inside and an outside, hence we recover the inward direction and the parity of the areas. Taking up the same rules of transformation established on the plane, this new system of Alpha graphs expresses the same logic as the original ones, which is classical propositional logic. Therefore, the novelty of the graphs on the cylinder lies entirely in the cuts that go around the cylinder. Perhaps we might interpret them as opposition or contradiction between the truths represented by the two unbounded ends of the cylinder. Again, in a classical setting, we might think that an even number of surrounding cuts means that those truths are congruent or compatible, while an odd number would mean that they are opposite or contrary. Future research could specify the natural rules of transformation for this type of cuts and thus for arbitrary Alpha graphs on the cylinder. As a final example in this article, we consider the Alpha graphs on the torus. This surface is a cylinder that turns back on itself, or a surface of revolution that is generated by rotating a circle around an axis that is coplanar with it. On the torus, there are infinitely many different types of simple closed curves (Fig. 4.14). A closed curve that does not surround the “tube” nor the “hole” of the torus divides it into two regions, which are both bounded. However, one of these areas is isomorphic to the interior of a Jordan curve on the plane, while the other contains the hole of the torus. Therefore, we can clearly distinguish the inside and the outside of such a curve. But,

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Fig. 4.14 Different cuts on the torus

on the other hand, there are Jordan curves on the torus that surround the tube and, surprisingly, they do not divide this surface into two areas, since its complement is isomorphic to the cylinder. There are also simple closed curves that surround the hole of the torus, even curves that simultaneously turn a finite number of times around the tube and a finite number of times, the same or another, around the hole. However, the exterior of any curve like this is again isomorphic to a cylinder and these curves do not divide the torus. If we restrict the graphs on the torus allowing only cuts that do not surround the tube or the hole, any of these cuts clearly determines an interior and an exterior. Once again, we recover the inward direction and the parity of the areas and thus, with the same rules of transformation, we obtain on the torus the Alpha graphs for classical propositional logic. Moreover, we can dare to propose the conjecture that, on any surface, if we restrict ourselves to those Jordan curves that divide the surface into two regions, only one of which is isomorphic to the interior of a Jordan curve on the plane, then we have the inward direction. Hence these graphs, with the same known rules of transformation, are Alpha graphs for classical propositional logic. Therefore, this logic remains valid on many different surfaces. Thus, the novel feature of the Alpha graphs on the torus is the possibility of drawing a cut that does not divide the surface, so we can hardly interpret it as a negation or even a sign of opposition. In fact, its only effect seems to be that it reduces the remaining surface to a cylinder, with the logical questions already raised about this surface. Anyway, still indeterminate rules of transformation that match a suitable interpretation might lead in the future to a full system of Alpha graphs on the torus. For further indications on graphs on non-planar surfaces, see [16].

4.8 Concluding Remarks Existential graphs constitute a formal system of diagrams, with precise rules of formation and transformation, which allow us to execute rigorous mathematical proofs. Duly interpreted, they give graphic versions of various mathematical logical systems, completely equivalent to the known algebraic presentations, both syntactic

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and semantic. Moreover, existential graphs open the path to a completely new way of doing logic and mathematics, in which diagrams are absolutely essential. In this paper, we have shown how we can obtain every known system of existential graphs, including intuitionistic existential graphs, from an amazingly simple drawing or stroke, enriched only with a suitable logical interpretation that in most cases is basic and natural. This emphasizes the enormous power of a simple gesture performed in an appropriate environment. However, we could also see that the immediate context of the gesture—in this case, the sheet of assertion—turns out to be especially important. Because a seemingly small change in the surface where we trace the drawing might completely shatter its interpretation or, on the contrary, open the way to whole worlds of new possibilities. May the inquiry into these possibilities continue and may a wide place for true diagrammatic thinking open in our universe of ideas.

References 1. Arnol’d, V.I.: Huygens and Barrow, Newton and Hooke. Birkhäuser, Basel (1990) 2. Bellucci, F., Pietarinen, A.-V.: From Mitchell to Carus: fourteen years of logical graphs in the making. Trans. Charles S. Peirce Soc. 52(4), 539–575 (2016) 3. Bellucci, F., Chiffi, D., Pietarinen, A.-V.: Assertive graphs. J. Appl. Non-Classical Logics (2018). https://doi.org/10.1080/11663081.2017.1418101 4. Van Dalen, D.: Intuitionistic logic. In: Goble, L. (ed.), The Blackwell Guide to Philosophical Logic. Blackwell (2001) 5. Fuentes, C.: Cálculo de secuentes y gráficos existenciales Alfa: Dos estructuras equivalentes para la lógica proposicional. Undergraduate Thesis, Universidad del Tolima, Ibagué (Colombia) (2014) 6. Goldblatt, R.: Topoi. The Categorial Analysis of Logic. Elsevier, Amsterdam (1984) 7. Heyting, A.: Intuitionism. An Introduction. North-Holland, Amsterdam (1971) 8. Hughes, G.E., Creswell, M.J.: A New Introduction to Modal Logic. Routledge, London and New York (1996) 9. Ma, M., Pietarinen, A.-V.: Gamma graph calculi for modal logics. Synthese 195, 3621–3650 (2018) 10. Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1971) 11. Oostra, A.: Álgebras de Heyting. Universidad Pedagógica Nacional, Memorias del XIV Coloquio Distrital de Matemáticas y Estadística, Bogotá (1997) 12. Oostra, A.: Los gráficos Alfa de Peirce aplicados a la lógica intuicionista. Cuad. Sist. Peirceana 2, 25–60 (2010) 13. Oostra, A.: Gráficos existenciales Beta intuicionistas. Cuad. Sist. Peirceana 3, 53–78 (2011) 14. Oostra, A.: Los gráficos existenciales Gama aplicados a algunas lógicas modales intuicionistas. Cuad. Sist. Peirceana 4, 27–50 (2012) 15. Oostra, A.: Representación compleja de los gráficos Alfa para la lógica implicativa con conjunción. Boletín Matemáticas 26(1), 31–50 (2019) 16. Oostra, A.: Existential graphs on nonplanar surfaces. Rev. Colomb. Matemáticas 53(2), 205– 219 (2019) 17. Oostra, A.: Equivalence proof for intuitionistic existential alpha graphs. In: A. Basu et al. (eds.), Diagrammatic Representation and Inference – Diagrams 2021. Lecture Notes in Computer Science, vol. 12909, pp. 188–195. Springer, Cham (2021) 18. Ortiz, J., Segura, J.: Gráficos Alfa intuicionistas. Undergraduate Thesis, Universidad del Tolima, Ibagué (Colombia) (2018)

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19. Peirce, C.S.: Collected Papers of Charles Sanders Peirce. Hartshorne, C., Weiss, P., Burks, A.W. (eds.), Harvard University Press, Cambridge (Massachusetts) (1931–1958) 20. Peirce, C.S.: Writings of Charles S. Peirce. The Peirce Edition Project, Indiana University Press, Bloomington and Indianapolis (1982–2010) 21. Peirce, C.S.: Logic of the Future. In: Pietarinen, A.-V. (ed.), History and Applications, vol. 1. De Gruyter, Berlin (2020) 22. Roberts, D.D.: The Existential Graphs of Charles S. Peirce. Mouton, The Hague (1973) 23. Taboada, J., Rodríguez, D.: Una demostración de la equivalencia entre los gráficos Alfa y la lógica proposicional. Undergraduate Thesis, Universidad del Tolima, Ibagué (Colombia) (2010) 24. Zalamea, F.: Peirce’s logic of continuity. A Conceptual and Mathematical Approach. Docent Press, Boston (2012) 25. Zapponi, L.: What is… a Dessin d’Enfant? Not. AMS 50(7), 788–789 (2003) 26. Zeman, J.J.: The Graphical Logic of C. S. Peirce. Ph.D. dissertation. University of Chicago, Chicago (1964)

Chapter 5

54 Gestures on Higher Mathematics, and Their Use for a Diagrammatic Approach to the Question “What Is Mathematics” Fernando Zalamea

We synthesize in this text the findings obtained in our Philosophy of Mathematics Seminar, held at Universidad Nacional de Colombia, Bogotá, AugustDecember 2018. Thanks to multivalent (logical) and multidimensional (geometrical) approaches, twelve meetings during the semester addressed the question What is Mathematics? The experiment was realized through my very body (FZ), only using arm, hand and finger gestures, just accompanied with my voice (as some sort of basso continuo). No possibility to draw on the blackboard was allowed (a very stringent limitation for a mathematician), nor any written previous account of the acting performances was permitted. The result produced a few surprises, which we survey in this account. In fact, it became possible to explain, with very simple gestures, many apparently complex features of mathematics, both modern (1830–1950) and contemporary (1950-today), profiting from the conceptual imagination of Galois, Riemann, Poincaré, Cantor, Hilbert, Gödel, Grothendieck, and many other mathematicians of the highest order, transposed to body language. Section 1 explains 9 basic hand and finger gestures, which try to capture some fundamental dialectics of mathematical thought. Section 2 expands those gestures with more complex arm movements, six for each basic gesture, summing up to the 54 gestures on higher mathematics mentioned in the title of this chapter. Section 3 analyzes the gestures thus obtained, and proposes a typology of them. Finally, Sect. 4 puts at work all these gestures in order to try to explain, in a diagrammatic, categorytheoretic way, through representable functors, the extremely complex question What is Mathematics?

F. Zalamea (B) Departamento de Matemáticas, Universidad Nacional de Colombia, Sede Bogotá, Colombia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_5

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5.1 Nine Basic Hand and Finger Gestures for Some Fundamental Dialectics in Mathematics Many partial definitions can be provided for mathematics, according to the diverse perspectives, dualities, dialectics, that we choose to emphasize. Following Peirce’s categories –(1) Firstness: immediacy, first impression, freshness, sensation, unary predicate, monad, chance, possibility; (2) Secondness: action-reaction, effect, resistance, binary relation, dyad, fact, actuality; (3) Thirdness: mediation, order, law, continuity, knowledge, ternary relation, triad, generality, necessity, see [18], 54– 67)–, some standard, and non-standard, “definitions” for mathematics may be the following: • Def. 1.1. Mathematics is the precise study of varieties of space and number • Def. 1.2. Mathematics is the precise study of mediations between the continuous and the discrete • Def. 1.3. Mathematics is the precise study of infinity and its (de)generations • Def. 2.1. Mathematics is the precise study of borders of negation • Def. 2.2. Mathematics is the precise study of potentialities of structural beauty • Def. 2.3. Mathematics is the precise study of transits and obstructions possibility-actuality-necessity • Def. 3.1. Mathematics is the precise study of founding dialectics One/Many • Def. 3.2. Mathematics is the precise study of foldings and unfoldings ideal/real and abstract/concrete • Def. 3.3. Mathematics is the precise study of structures and their deformations. In each of these partial approaches, a basic polarity (Secondness) is observed (1.1: space/number; 1.2: continuity/discreteness; 1.3: infinite/finite; 2.1: negativity/ positivity; 2.2: beauty/truth; 2.3: possibility/necessity; 3.1: One/Many; 3.2: ideality/ reality; 3.3: structure/deformation), and the many mediations (Thirdness) between them –or mixtures, following Lautman (1935–42) [13]– become the main object of study of the mathematics involved. [12] develops a thorough study of the triadic classification in play, and [16] puts it at work in the realm of mathematics. These projections on the question “What is Mathematics?” involve in turn many powerful configurations of the hand, which capture in part the previous definitions (for all the following descriptions, see Fig. 5.1): • Gest. 1.1: dialectics between covering (space) and interlacing (number) finger pointers • Gest. 1.2: dialectics between vertical fingers (discreteness) and a horizontal hand (continuity) • Gest. 1.3: dialectics between an open thumb (generation of infinities) and a closed hand (finite levels) • Gest. 2.1: dialectics between a concave hand (negative penumbrae) and a plane hand (positive light)

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Fig. 5.1 9 basic hand and finger gestures which capture fundamental dialectics in Mathematics

• Gest. 2.2: dialectics between a heart with two hands (beauty) and a tunnel of fingers (truth) • Gest. 2.3: dialectics between a vertical hand (actuality, necessity) and a diagonal hand (possibility) • Gest. 3.1: dialectics between scattered fingers (Many) and an upper projective finger (One) • Gest. 3.2: dialectics between a closed fist (real obstructions) and a soft hand (ideal transits) • Gest. 3.3: dialectics between ordered fingers (structure) and chaotic fingers (deformations). In our actual Seminar (2018), these hand and finger gestures were first produced in class, and then some photos (as shown in Fig. 5.1) were taken to freeze the moment. The process of thinking was first enacted by the body (dynamic film), in an unclear,

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vague and free disposition, and only afterwards it was drawn by the mind (static photo). In this way, some sort of natural impulses reflected very general mathematical ideas, and those very processes, movements, or impulses, were the ones which afterwards helped us to obtain some sort of typological (Sect. 3) and diagrammatical (Sect. 4) understanding of the main question “What is Mathematics?”.

5.2 Fifty-Four Gestures on Higher Mathematics After capturing synthetically, through the nine hand gestures Gest. 1.1–3.3, some central dialectics of mathematical thought, the main bulk of our Philosophy of Mathematics Seminar (2018) consisted in creating arm performances to develop each of those compact hand gestures. In a sense taken from Riemann surfaces and algebraic topology, I tried to enact covering surfaces for each simple gesture 1.1–3.3, in order to expand the gestures with many ideas of modern and contemporary mathematics. The “performances”, or “enactments”, consisted each of fifteen minutes (15’) in which my arms, hand, and fingers, were twisted in many ways to try to materialize through the body many general mathematical concepts. A back-and-forth between the concrete and the abstract was thus “choreographed” and really improvised, with very little previous preparation (unfortunately, I am a non-existent dancer, and my movements only used the upper part of my body). On the other hand, after each session, which consisted of six “covering surfaces” for each basic gesture 1.1.-3.3, I had the opportunity of sketching the movements shown to the class. The results of these nine sketches are included below in Figs. 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 5.10. An important mandate of the Seminar was to use concepts and examples coming from “higher mathematics”, that is, non-elementary mathematical achievements, which could be simply explained through the double narrative of voice and gestures. This mandate sprang from three main sources: (i) a recognition that the philosophy of higher mathematics (mathematical logic after Gödel, algebraic number theory after Galois and Dedekind, ordinals and alephs after Cantor, algebraic topology after Poincaré, abstract algebra after Emmy Noether, complex variables and analytical number theory after Riemann, set-theoretic topology after Hausdorff, functional analysis after Hilbert [11], categories after Grothendieck, etc.) produces new perspectives and new problems not present in elementary approaches [17], (ii) a recognition that the French School in the philosophy of mathematics [3, 14], Cavaillès 1937–40 [4], Lautman 1935–42, [5, 9] situated mathematics as a paradigm of historical science, well rooted in precise time frames in which creators acted, with the consequence that many exponents of the School studied carefully mathematical gestures, (iii) a recognition of the seemingly contradictory “equation” deep = simple occurring in higher mathematics, following Grothendieck’s ideas that deep mathematics must always be at the same time simple (for introductions to Grothendieck, see [19, 20]). In what follows, we will then expand through space the basic gestures Gest. 1.1–3.3, taking into account some fundamental concepts at the heart of modern mathematics (the “ramifiers”: Galois, Riemann, Poincaré; the “founders”: Cantor, Hilbert, Gödel) and

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Fig. 5.2 Covering gestures for Def. 1.1 and Gest. 1.1

contemporary mathematics (Grothendieck, Lawvere, Connes, Gromov), and taking as well into account other telling conceptual examples (Dedekind, Hermite, Zermelo, Brouwer, Leray, Cartan, Cohen, Kripke, Drinfeld, Shelah). The following Figs. 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 5.10 are presented in Spanish, as direct results of trying to record the gestures previously enacted in class. Their immediacy may be their main interest, showing products of the body which only afterwards became processed by the mind. In that sense, maintaining the original sketches in Spanish (“dedo” = finger, “mano” = hand), and exhibiting explicitly the

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Fig. 5.3 Covering gestures for Def. 1.2 and Gest. 1.2

hand writing of diagrams and colors on the paper, show some of the freshness and simplicity expected from gestures. Figure 5.1 extends Def. 1.1 (space/number) and Gest. 1.1, thanks to examples: • 1.1.1. Galois theory (arm back-and-forth: Galois connection; inversion of the hands: fields versus groups) • 1.1.2. Riemann surfaces (distributed fingers: sectors of the plane; overlapped hands: leaves of the surface) • 1.1.3. Poincaré’s homology and homotopy (tangential hands: homology; curved fingers: homotopy)

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Fig. 5.4 Covering gestures for Def. 1.3 and Gest. 1.3

• 1.1.4. Hilbert spaces ( fingers grid: orthonormal basis; extended arms: basis generation) • 1.1.5. Leray-Cartan’s sheaf theory (pointing finger: projection; unfolded hands: sections) • 1.1.6. Grothendieck toposes (pointing fingers: sheaves; connecting hands: exactness in the topos).

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Fig. 5.5 Covering gestures for Def. 2.1 and Gest. 2.1

Figure 5.3 extends Def. 1.2 (continuity/discreteness) and Gest. 1.2, thanks to examples: • 1.2.1. Riemann-Roch theorem (hand cut by fingers: genus; hand deployed by fingers: meromorphic harmony)

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Fig. 5.6 Covering gestures for Def. 2.2 and Gest. 2.2

• 1.2.2. Riemann’s Zeta function (inverted fingers: terms 1/ns ; zigzagging arm: summation) • 1.2.3. Cantor’s continuum (discrete fingers: rationals; overlapping hands: real completion) • 1.2.4. Brouwer’s continuum (soft moving arm: primordial continuum; cutting fingers: discrete two-oneness)

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Fig. 5.7 Covering gestures for Def. 2.3 and Gest. 2.3

• 1.2.5. Leray-Cartan’s sheaf theory (soft moving hand: upper continuous space; vertical fingers: discrete fibers) • 1.2.6. Connes’ arithmetical topos (curved fingers: tropical site; covering hands: topos over the site).

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Fig. 5.8 Covering gestures for Def. 3.1 and Gest. 3.1

Figure 5.4 extends Def. 1.3 (infinite/finite) and Gest. 1.3, thanks to examples: • 1.3.1. Dedekind’s infinity (arm back-and-forth: bijection; small fingers circle: the part against the whole) • 1.3.2. Cantor’s ordinals and alephs ( finger iterations: ordinal generation; hand levels: alephs)

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Fig. 5.9 Covering gestures for Def. 3.2 and Gest. 3.2

• 1.3.3. Zermelo’s axiom of choice (pointing fingers: disjoint elements; arm circulating hands: choice function) • 1.3.4. Hilbert’s On the Infinite (1925) ( fingers grid: recursive constructions; soft upper hand: the continuum)

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Fig. 5.10 Covering gestures for Def. 3.3 and Gest. 3.3

• 1.3.5. Gödel’s infinite logics Int–Clas (open fingers: Brouwer algebras; upper hand: Boolean algebra) • 1.3.6. Shelah’s Main Gap theorem ( fingers grid: order, stability; chaotic hand: non-structure).

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Figure 5.5 extends Def. 2.1 (negative/positive) and Gest. 2.1, thanks to examples: • 2.1.1. Galois theory (thumbs up: adjunctions ascent (+); thumbs down: subgroups descent (–)) • 2.1.2. Poincaré’s conjecture (triadic fingers: homology obstruction (–); circular fingers: homotopy transit (+)) • 2.1.3. Cantor’s diagonal argument (hand waving: non-bijection (–); hands ramification: power multiplicity (+)) • 2.1.4. Gödel’s incompleteness ( fingers grid: recursive constructions (+); growing hands: non-recursion (–)) • 2.1.5. Shelah’s Main Gap ( fingers grid: structure (+); chaotic dancing hands: non-stability (–)) • 2.1.6. Connes’ non-commutation ( fingers alternation: commutation (+); fingers twist: non-commutation (–)). Figure 5.6 extends Def. 2.2 (beauty/truth) and Gest. 2.2, thanks to examples: • 2.2.1. Riemann’s conformal harmonics (open hand: arbitrary complex domain; circular hand: open disc) • 2.2.2. Poincaré’s automorphic world (parallel fingers: Euclidean truth; wide fingers: non-Euclidean beauty) • 2.2.3. Grothendieck’s dessins d’enfants (sharp fingers: true combinatorics; dancing hands: Riemann beauty) • 2.2.4. Hilbert’s metamathematics (plane hand: true arithmetic; curved hands: mathematical beauty) • 2.2.5. Gödel’s incompleteness ( fingers grid: proof control; moving hands: beauty beyond proof) • 2.2.6. Cohen’s forcing (discrete fingers: ground countable true model; curved hands: real beauty). Figure 5.7 extends Def. 2.3 (possibility/necessity) and Gest. 2.3, thanks to examples: • 2.3.1. Galois’ non-solvability (diagonal descending hands: obstruction; horizontal hands: stratification) • 2.3.2. Hermite’s transcendence of e (upper hand: transcendent world; lower hands: algebraic numbers) • 2.3.3. Poincaré’s homology (icosahedron hands: impossibility; curved hands: possibility, then necessity) • 2.3.4. Gödel’s incompleteness (upper hand: possibilities beyond; constructive hands: formal necessity) • 2.3.5. Gödel’s completeness (moving arm of all hands unfolded: sufficiency of first-order semantics) • 2.3.6. Kripke’s modal models (one finger: possibility in one world; all hands: necessity in all worlds).

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Figure 5.8 extends Def. 3.1 (One/Many) and Gest. 3.1, thanks to examples: • 3.1.1. Cantor’s definition of a set (inverted hand: set—One; fingers below: elements—Many) • 3.1.2. Hilbert’s basis strategy (lower hand: basis—One; unfolded upper hands: developments—Many) • 3.1.3. Gödel’s triad LOG-FOR-INT (three fingers: foundations—Many; tetrahedron hand: Gödel crossing—One) • 3.1.4. Grothendieck’s thought (wrist: abstract categories—One; deployed hand: concrete categories—Many) • 3.1.5. Lawvere’s ubiquitous adjunctions (two inverted fingers: adjunction—One; open hand: ubiquity—Many) • 3.1.6. Gromov’s h-principle (hands: metrics everywhere—Many; arm: rooting h-principle—One). Figure 5.9 extends Def. 3.2 (ideal/real, abstract/concrete) and Gest. 3.2, thanks to examples: • 3.2.1. Galois’ ambiguity (three fingers: indiscernible real roots; hand movements: Galois ideal automorphisms) • 3.2.2. Riemann surfaces (three fingers: multiple complex roots; hand covers: ideal leaves of the surface) • 3.2.3. Hilbert’s Program (plane hand: ideal basis; deployed hands: real mathematical regions) • 3.2.4. Grothendieck’s “rising sea” method ( fist: hard real “nut”; waving hands: soft ideal liquid medium) • 3.2.5. Shelah’s Main Gap ( fingers crossed: concrete order; overlapping hands: abstract chaos) • 3.2.6. Connes’ arithmetical topos (arm embedding of “real” gesture 1.2.2 into “ideal” gesture 1.2.6). Figure 5.10 extends Def. 3.3 (structure/deformation) and Gest. 3.3, thanks to examples: • 3.3.1. Galois’ ambiguity (three fingers moving: automorphisms; curved hand: Galois group structure) • 3.3.2. Riemann’s complex variable techniques (open hand: ampli-twist; closed hand: minimum angles) • 3.3.3. Poincaré’s homotopy ( fingers: homotopical deformations; hand: fundamental group structure) • 3.3.4. Gödel’s incompleteness (bottom hand: PA axiomatics; finger breaking: deformations beyond PA) • 3.3.5. Grothendieck’s n-categories (ascending arm: level multiplier; cutting hands: categorical strata) • 3.3.6. Drinfeld’s quantum groups (concave hand: Hopf algebras; agitating arm: parameter deformations).

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5.3 A Typology of Mathematical Gestures The many examples presented in Sect. 2 show that some simple gestures can capture conceptual features of advanced mathematics. Intellectual imaginative deepness is thus enacted in bodily material simplicity. Assuming the hypothesis that mathematics is a historical science by excellence (Cavaillès 1937–40; Lautman 1935–42), and that the ideal concepts of mathematics have been created in very real material contexts, in our Seminar (2018) we let our body fluctuate freely, to allow gestures which tried to catch faithfully the transits between ideality and reality, constitutive of the discipline. In Fig. 5.1, we observe a preponderance of complete hand couplings (7/9), over partial hand-finger couplings (2/9). These last two, nonetheless, represent some crucial mathematical dialectics: gestures 1.2 (continuous/discrete) and 3.1 (One/ Many), enact through the index finger, depending on the context, a local discrete quantity or a global qualitative unity. The other seven “complete” hand gestures underline a need to represent the complex dynamics at play. In those basic complete gestures, (i) a first counterpoint is obtained between gestures 1.1 (space/number) and 3.3 (structure/deformation), the first interweaving fingers, the second separating soft and crisped fingers. In a similar vein, (ii) gestures 1.3 (infinite/finite) and 2.1 (negative/positive) oppose planar hand palms (number growth) and concave hand palms (negative curves). Finally, (iii) gestures 2.2 (beauty/truth), 2.3 (possibility/ necessity) and 3.2 (abstract/concrete) constitute a natural triad along a general softness problematic produced by the hands, either in the form of a heart, in a diagonal ascension, or in a superior cover. In this way, mathematics expresses some of its major forces: (i) richness of structuration and deformation of space and number, (ii) richness of the understanding of negation, (iii) richness of abstract generalizations where many particular tensions can be solved and softened. The nine basic gestures 1.1–3.3 constitute thus some sort of ramified blow-up (both a multiplication in the realm of differential geometry, and a reference to Antonioni’s Blow-Up), in which the body is expanded through gestures in order to be able to enact conceptual thinking. In some way, real hand gestures caress ideal conceptualizations. In this way, an ever growing back-and-forth between matter and spirit acquires new weight in an extended reason dialectics. The 54 extended gestures (Figs. 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 5.10) constructed over the basic 9 gestures show some fundamental patterns: • inversion gestures, following an arm pendulum (1.1.1, 1.2.1, 2.1.1, 2.1.3, 2.2.2, 3.1.5, 3.2.1, 3.3.3) • transference gestures, following arm extensions into hands (1.1.3, 2.1.2, 2.2.1, 2.2.3, 2.3.3, 2.3.6, 3.1.4, 3.2.2, 3.3.2, 3.3.5) • elevation gestures, following hand coverings (1.1.2, 1.1.4, 1.1.6, 1.2.6, 1.3.2, 1.3.4, 1.3.5, 2.1.4, 2.2.4, 2.2.5, 2.2.6, 2.3.2, 2.3.4, 3.1.2, 3.2.3, 3.2.6, 3.3.4) • back-and-forth gestures, following hand ascents/descents (1.1.5, 1.2.5, 1.3.1, 2.3.1, 3.1.1, 3.1.6, 3.3.6)

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• equilibrium gestures, following mirroring fingers and waving hands (1.2.2, 1.2.3, 1.2.4, 1.3.3, 1.3.6, 2.1.5, 2.1.6, 2.3.5, 3.1.3, 3.2.4, 3.2.5, 3.3.1). The structure of the gestures confirms the dynamical characteristics of mathematical doing [7, 8], or effective mathematics ([1], 287). In fact, three precise operations—inverting, transferring, elevating—produce at the end a multi-level back-andforth equilibrium. A global systemic gesture ([2], 208) emerges, which encompasses many local gestures. The examples most used in our seminar included Grothendieck (8), Gödel (7), Riemann (6), Galois (5), Poincaré (5), Cantor (5), Hilbert (5). In each case, the gestures tried to synthesize deep ideas in a simple way, according to Grothendieck’s Récoltes et semailles recommendation—to look at things with “childish eyes”—[10], for a detailed study, see ([19], 387–430). Beyond the eventual interest of the above typology of signs, the body experience advanced in our Philosophy of Mathematics Seminar (2018) produced some useful general observations: (A) accessibility: some core ideas of higher mathematics were shown to be accessible to a wide public (thanks to Natalia Palomá, a high-school teacher who attended the Seminar, we had the wonderful opportunity to have young students (ages 12–15) present at the classes); nevertheless, in many times it was repeated that gestures just introduced the ideas, and that mathematics required very strong discipline, many hours, and extremely hard work, to really understand the techniques involved, beyond the apparent simplicity of gestures; (B) freshness: even for mathematical students at the end of their careers, for Master students, and for colleagues present at the Seminar, the body play turned out to be fairly surprising for everybody (beginning with the very performer); something strange, vague, not very well defined (as must happen with deep synthetic gestures) helped to view in new ways many technical achievements apparently well known; (C) liberation: escaping from our apparently mandatory blackboard for mathematicians, the enactment of abstract conceptions through only gestures allowed an unexpected freedom in “thinking about mathematical thinking”; the functional iteration thought(thought) materialized through the body produced a wonderful liberty, not usually dealt with in the mathematician’s extremely careful writing processes. All in all, the mathematician’s desire to escape appearances, to look beyond our “blind eyes” [15], to capture transcendent penumbrae beyond material phenomena, was in part enacted in the Seminar, where, in an unknown sense, the material gestures came as projections of some hidden spiritual harmonics. We lived the fact that the fluctuations between Body and Mind could not be solved through a fixed dimension of the pendulum, but required its full swing.

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5.4 What is Mathematics?: A Diagrammatic, Category-Theoretic, Approach “What is Mathematics?” has always been an exceedingly difficult question. A very nice approach is presented in the classic [6], trying to explain, through a review of examples, what are some of the main characteristics of mathematical practice [7, 8]. Nevertheless, a more global or systemic approach [2] lacks in the literature. A way to approach this question is to use Category Theory (CT). Two main orientations of (CT) may be used here: (1) the possibility to define exactly the back-and-forth between universal realms (in abstract categories, thanks to ∃! definitions) and particular realizations (in concrete categories, that is, in classes of mathematical structures), (2) the use of Yoneda’s Lemma (which may well be understood as the “heart” of (CT), see Fig. 5.11), where one observes the emergence of “ideal” phantasmata (presheaves) beyond “reality” (representable functors). The representable functor hA captures the complete “aura” of object A, looking at all the morphisms which depart from A: an internal, analytic, knowledge of A (through elements, Set Theory ∈ -decomposition) is subsituted by an external, synthetic, knowledge (through arrows, Category Theory → -composition) (for a nice gestural reading of Yoneda, see [2]. Thus, to understand an object (or a concept), we have to understand its contextual effects, something very close to Peirce’s Pragmaticist Maxim [18]. Figure 5.12 offers such a contextual understanding of the question “What is mathematics?”, profiting from definitions Def. 1.1–3.3 and from the 9 basic gestures Gest. 1.1–3.3. Hidden on the back of the diagram, in some sort of Riemann surface deployed in the verso of the page, one can imagine the 54 extended gestures 1.1.1–3.3.3. As a result, the multiplicity of mathematics is preserved, but it becomes integrated in a unitary handling, characteristic of Category Theory. The diagrammatic perspective offered in Fig. 5.12 allows for the many strata present in the alternative “definitions” Def . 1.1–3.3, but correlates them in the unique “meta-representable” functor h{What is Mathematics?} . From the point of view of gestures, the photos in Fig. 5.1 act as “real” representations, while the diagrammatic sketches in Figs. 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 5.10 act as phantasmata, in that vague, unclear, and free disposition in which the body enacts the “ideal” forces of mathematical creativity.

Fig. 5.11 Yoneda’s Lemma: the heart of (CT). Lucis et umbrae: concrete representable functors (hA ) versus abstract presheaves (functors C → Sets)

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Fig. 5.12 What is mathematics? A category-theoretic reading of alternative definitions of “mathematics” and their gestural perspectives

References 1. Alunni, C.: Albert Lautman et le souci brisé du mouvement. Revue de synthèse, 5e série, 2005(2), 283–301 (2005) 2. Alunni, C.: Le Lemme de Yoneda : enjeu pour une conjecture philosophique? (variations sous forme-pro-lemmatique, mais en prose). In: Andreatta, M., Nicolas, F., Alunni, C. (eds.), A la lumière des mathématiques et à l’ombre de la philosophie. Dix ans de Séminaire MamuPhi, pp. 195–211. Ircam, Paris (2012) 3. Brunschvicg, L.: Les étapes de la philosophie mathématique. Albert Blanchard, Paris (1912) 4. Cavaillès, J.: Méthode axiomatique et formalisme, et divers écrits. Hermann, Paris (réed. 1994) (1937–40) 5. Châtelet, G.: Les enjeux du mobile: mathématiques, physique, philosophie. Seuil, Paris (1993) 6. Courant, R., Robbins, H.: What is Mathematics? An Elementary Approach to Ideas and Methods. Oxford University Press, Oxford (1941) 7. De Lorenzo, J.: Introducción al estilo matemático. Tecnos, Madrid (1971) 8. De Lorenzo, J.: La matemática y el problema de su historia. Tecnos, Madrid (1977) 9. Desanti, J.T.: Les Idéalités mathématiques. Seuil, Paris (1968) 10. Grothendieck, A.: Récoltes et semailles. Unpublished Manuscript (1986) 11. Hilbert, D.: On the infinite. In: van Heijenoort, J. (ed.), From Frege to Gödel. A Source Book in Mathematical Logic 1879–1931, pp. 367–392. Harvard: Harvard University Press (1925, 1967) 12. Kent, B.: Charles S. Peirce. Logic and the Classification of Sciences. McGill - Queen’s University Press, Montréal (1987) 13. Lautman, A.: Essai sur l’unité des mathématiques, et divers écrits. Hermann, Paris (réeds. UGE 1977; Vrin 2006) (1935–42)

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14. Poincaré, H.: L’invention mathématique. Institut Général Psychologique, Paris (1908) 15. Tarkovsky, A.: Sculpting in Time. University of Texas Press, Lubbock (1984) 16. Zalamea, F.: Signos triádicos. Lógicas, literaturas, artes. Nueve cruces latinoamericanos. UNAM—Mathesis, México (2006) 17. Zalamea, F.: Filosofía sintética de las matemáticas contemporáneas. Universidad Nacional de Colombia, Bogotá. (Extended translations: Synthetic Philosophy of Contemporary Mathematics, Sequence, 2012; Philosophie synthétique des mathématiques contemporaines, Hermann, 2018) (2009) 18. Zalamea, F.: Peirce’s Logic of Continuity. Docent Press Boston. (Extended translation of El continuo peirceano, Universidad Nacional de Colombia, 2001, and Los gráficos existenciales peirceanos, Universidad Nacional de Colombia, 2010) (2012) 19. Zalamea, F.: Grothendieck. Una guía a la obra matemática y filosófica. Universidad Nacional de Colombia, Bogotá (2019) 20. Zalamea, F.: Modelos en haces para el pensamiento matemático. Universidad Nacional de Colombia, Bogotá (2021)

Part II

Diagrams and Gestures: Philosophy

Chapter 6

The Diagram: Demon of Proof Charles Alunni

I dedicate these pages to Aldous Harding, Billy Nomates, Amy Taylor and St. Vincent for their embodiment of the gesture as I see it in philosophy.

If the question of the image in science implies, as underlying its possible status, the question of writing, this applies a fortiori (if not a priori) to the diagram as well. This is why I will first deal with the Galilean introduction of the writing of science, then with its contemporary expansion by a procedure of compaction; then, finally, with its probatory stabilization in a diagrammatical writing that will be seen as an « anticipatory memory» of the gestures of science and its issues, a relay-station for the creative imagination that ceaselessly gives new impetus to the operative field of scientific practice. I’d like to make it clear from the start that for reasons of context I will stick to an imaginative approach and not a technical one, even if the examples I’m presenting have a notorious theoretical complexity. By technical approach, I mean the computational and formal deployment of the diagram within the symbolic context of its inscription. Here I show some examples:

C. Alunni (B) École Normale Supérieure, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_6

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Diagrammatic Morphisms and Applications AMS 2000 Convolution & Quasi-Hopf Algebra

Saunders Mac Lane Categories for the Working Mathematician Proof of the Diagram Lemmas

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Robert H. Wasserman, Tensors & Manifolds Multilinear Mappings and Dual Spaces Orthogonal complement theorem

Louis H. Kauffman Knots and Physics I also see the image and the diagram as being immersed in the « element» of proof. I use “element” here in the sense of the Hegelian “Medium”: But this transparency of the finite that lets only the absolute transpire through it ends up in complete disappearance, for there is nothing in the finite which would retain for it a difference over against the absolute; as a medium, it is absorbed by that through which it shines.1

1

[30], 468.

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6.1 The Galilean Proof of the « Codex Naturae» Galileo says that. [Natural] philosophy is written in that great book which ever lies before our eyes – I mean the universe – but [we] cannot understand it if we do not first learn the language and grasp the symbols in which it is written. [That] the book is written in a mathematical language whose symbols are triangles, circles, and other geometrical figures (irrespective of any alphabetical repository).2

It is this metaphor of the “libro della natura” that will condense a decisive mutation between Cinquecento and Seicento. The book, which appears in this period of thematization of the act of seeing, is itself the material condition of possibility (then of circulation) of these transformations:

The space is all at once limited and infinite, material and fictional.3 For Galileo, as for all of physico-mathematical posterity, mathematics are not to be spoken or read (as one reads a text): in this sense mathematics is not a language. Jean Cavaillès writes: “By essence, language has nothing to do with mathematics”. It is more appropriate to speak of an algorithm conceived as an “algebra”, that is, as a group of operating rules proper to a calculation, a sequence of actions necessary to the realization of a calculation. In this sense, thought about “action” will necessarily be thought about action,4 since, as Cavaillès specifies in Méthode axiomatique et formalisme, “what is perceptible, immediate consciousness, is not abandoned: to act on it is not to take leave of it (any abstract object obtained, for example, by thematization, is a gesture on a gesture)”.5

2

[26] & 1641. SEE [3], passim. 4 “Die Mathematik ist mehr ein Tun denn eine Lehre”, [18]. 5 [19], 178. 3

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6.2 The Phenomenon of Compaction What about the twentieth century? “To understand is to seize the gesture and to be able to continue” affirms Cavaillès in his thesis Axiomatic method and Formalism. The twentieth century has probably consisted in a resumption, a continuation and an aggravation of the Galilean gesture as it is understood in all its complexity, that is to say both in its radicality as well as in its solidarity with metaphysics. Equally essential for us is the question of ideo-notational (and ideo-national) elaborations and, specifically, the analysis of the phenomenon of compaction (with effects of condenser and of formal induction)6 : thus, when we immerse equations into mathematical formalisms that are more and more powerful, we witness a spectacular compression of them; this is certainly not indifferent for the realm of thought. It’s to note that in the PhD thesis of Saunders MacLane, Abgekürzte Beweise in Logikkalkul [1934], his intention was to compact formal proofs by abbreviation (or compaction) of sequences of paths. The future theoretician of Category theory offered to organize the proof and the discovery of proofs: We can construct methods of abbreviation more general and more deep founded upon the concept of plan of proof […] which determines efficiently the individual paths of the proof.7

His thesis supervisor was nothing less than Hermann Weyl. Maxwell’s equations for electromagnetism are another example, which account for, among other things, visible light:

See The Works of Gaston Bachelard: in particular [15], 20142 , and [16] on the concept of “formal induction”. See also [1], passim. 7 MacLane, Abgekürzte Beweise in Logikkalkul [1934], in 1979, 1–62. 6

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I don’t speak of the version by Sir Roger Penrose:

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Penrose Maxwell Equations But what is at stake here is the hand and what it holds – what fills the hand (manipulation + handling) – for thought. Ludwig Wittgenstein, in The Blue and Brown Book, indicates that. We may say that thinking is essentially the activity of operating with signs. This activity is performed by the hand, when we think by writing […] thinking is an activity of our writing hand.8

Even more radically, Wittgenstein asserts in Vermischte Bemerkungen: Ich denke tatsächlich oft mit der Feder, denn mein Kopf weiß oft nichts von dem, was meine Hand schreibt (“I think with my pen”).9

Echoing this device, and with the aim of extending its resources and general philosophical scope, I will appeal here to the one that Alain Badiou will have baptized with a wonderful formula: “The Prince of sleeping gestures”.10 In the Enjeux du mobile,11 we find a passage which finally offers an overall perspective, a singular “foundation of thought” which testifies to its affinities with our present work: “At the level of fundamental discoveries, the work of a physicist is comparable to that of the painter who has to smash figurative data”.12 In Les Enjeux, two chapters are particularly important for our purposes: 1) Chapter 4, entitled Grassmann’s capture of the extension. Geometry and dialectic, and chapter 5 devoted to electrogeometric space. (1) In Chap. 4 dedicated to Grassmann’s work Die lineale Ausdehnungslehre of 8

Wittgenstein, 1980, 6 and 16. In 1984, 473 (and Oxford Blackwell, 1977, 29). Now in Wittgenstein, 1980: “I really do think with my pen, because my head often knows nothing about what my hand is writing”, 17c [1931]. 10 Alain Badiou’s relationship with Gilles Châtelet and his work is entirely devoted to a detailed philosophical analysis of Figuring Space. On the essential link of Châtelet’s work with Naturphilosophie, with the work of the first Schelling and with Romanticism, see the very interesting article by [40]. The author also confronts the fundamental question of Gilles Châtelet’s so-called integral “Deleuzism”. 11 [21]. English version, [22], note 32, 71. 12 Châtelet, idem. See on this point the Presentation by Francesco La Mantia. 9

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1844, the focus of attention is on “Grassmann’s products”, and more precisely on the “progressive” product and the “regressive” product which generate, through the association of multiplication and orientation, a new dimension or “next dimension” of space. This is the very heart of a very particular “diagrammatical dialectic” (or dialectic of articulation)13 which, as we will see, proves to be perfectly related to our modeling of a “capsule of compactification” (see diagram above). And this analysis of Grassmann’s work is crucial because it quite simply brings into play all the concepts practiced by Châtelet. At point C. Exterior product (aüßere Multiplikation) and intensive/extensive dialectic, Châtelet gives a diagrammatic interpretation of this product with permutation of roles and fibration of a spread out rectangle (Fig. 16 and 17) which we reproduce here:

13

Note the absolutely decisive Schellingian inflection in Châtelet: “Dialectic is here to be understood as ‘the intelligence that separates and, by very reason of this separation, introduces an order into things and imposes a form on them’ “, op. cit., 106: it is this fragment of Naturphilosophie that will allow him to decode the mathematical, physical and philosophical issues of the “mobile”.

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Châtelet sees this distinction in the order of enunciation,14 I1 then I2 or I2 then I1 which reproduces exactly the two possible decisions to streak the rectangle vertically or horizontally. He notes that “the two modes of ‘fibration’ of the spread-out rectangle are therefore exactly grasped by the notations I1 ∩ I2 or I2 ∩ I1 . […] The order of the symbols leads the hand and prescribes an effective construction of the “figure” […] Here, the hand is implicated in the product”.15 It can be noted that this is the particular form of a general law of signs that Grassmann intended to deduce geometrically and which will become the fundamental rule of non-commutativity. The second decisive point is developed in paragraph E. entitled The regressive or revolving product,16 which previously required going through the mediation of the important notion of “oriented envelope”.17 With the regressive product, the geometer will become master of all possible dimensions and will be able to “explain space”. Grassmann’s penetration allowed him to sense that this naive prioritizing of the expansion, based on the degrees of independence, had to be balanced out by a symmetrical process: a multiplication that would show the spatial contraction […] Grassmann therefore defines a regressive product that envelops the intersection in the way that the progressive product enveloped the union. We have just seen that the oriented envelope E worked like a horizon starting from a point-origin. Grassmann brilliantly sensed the necessity of balancing this situation: 0 must now be seen as a horizon starting from E, and that through a symmetry that no longer switches over the figures of space but involves the very dynamic of thought.18

Châtelet notes how Grassmann had “a very sure dialectical sense” and how any movement of the mind must be controlled and rebalanced by a device of symmetrization, which is part of the tradition of Naturphilosophie and the “balances of Being” (see the young Schelling and a certain Hegel). Then he adds: After the dialectic of the mobile point and that of the deformations of opposed volumes which made the rise through the dimensions possible, a new stage asserts the regressive product that allows the descent from a system of step n as far as the point. The same concern for symmetry is behind the inclination to articulate the degrees of dependence in the way that the degrees of independence were articulated.19

14

Francesco La Mantia and I reserve for a forthcoming publication the deployment of this question of “the order of enunciation” put forward by Châtelet. 15 [22], 130. 16 Châtelet, Ibid., 138. 17 Châtelet, ibid. 137–138 (Fig. 22). For Châtelet, “With an ordinary point, the hand that draws completely ignores the look that enve1ops”, 66 and “the gesture envelops before grasping and sketches its unfolding long before denoting or exemplifying”. In a word, “The diagram […] gives priority to the envelopment”, 46. 18 Châtelet, ibid., E. The regressive or involving product [eingewandtes Produkt], 138. 19 Châtelet, ibid., 139. We will note here, at the very heart of the Grassmannian device, the spectral presence of the Lautmanian dialectic. See especially his notion of “ascent to the Absolute” and the symmetrical operation of “descent”. See our Appendix on Dynamic Platonism below.

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If the external product erased the dependence of the factors, the regressive product, conversely, appreciates only it. Thus, Châtelet shows how the involving product rebalances in intuition the progressiveness of the exterior product (Fig. 23).

This diagram will finally be completed by the notion of oriented complement *. This complement “possesses a remarkable quality: the operation * switches the liaisons ∩ and • and induces a correspondence that exactly balances the diagrams in Fig. 23 […] which transforms the ascent of the degrees of independence into a descent of the degrees of dependence, and vice versa (see Fig. 27).

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By arranging the liaisons ∩ and • like two symmetrical wings of a single motion of thought, the operation * emerges as the keystone of a magnificent edifice”.20 It is easy to see the deep structural affinities of these different processes, both dynamic and symmetrical, with the dialectic, itself diagrammatic, of the “cell of compactification”. To the pulsations of a logic of complicatio (envelopment/enfoldment) and of explicatio (development), of contractio (reduction) and contractus (what is reduced), of dilation or centripetal expansion and of expansive or centrifugal reduction, correspond the beats of the grassmannian dialectic between explanatory external product and implicating regressive product. Except that this isomorphism is somewhat complicated by a sort of Möbiusian “torsion” affecting the compacted cell (isomorphism with torsion). (2) Moreover, we also find in Châtelet, and in a way that is both explicit and united, a second element which is the “phenomenon of compactification”.21 It is in his study of Hamilton’s Lectures on Quaternions that this phenomenon is described in connection with a double experience of space: the scalar experience of dilations and the vector experience associated with “perpendicular torsions”.

20

Châtelet, ibid., 143.144. Châtelet already thematizes the phenomenon of compactification with regard to the perspective effects of the painting where he speaks of “this compactification [which] domesticates the inaccessible”, op. cit., 49.

21

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Thus, Châtelet quotes a key text by J. C. Maxwell on this subject: “Quaternions […] constitute a mathematical method and a method of thought. They constantly prompt us to form a mental picture of the geometric structures represented by the symbols, so that in studying geometry by this method our minds are challenged by geometric ideas”.22 And even: For strictly physical reasoning (which is not to be confused with calculations), it is desirable to avoid explicitly introducing the Cartesian coordinates […] We must fix our attention on a point in space and not on the three coordinates. In the same way, one must consider the magnitude and the direction of a force instead of its three components. It is a way of contemplating geometrical and physical quantities that is more original and natural than the other (Cartesian), but it was not until Hamilton and the invention of quaternions that these ideas experienced their full development […] I am convinced that the introduction of these ideas will be very useful in electrodynamics […] where a small number of expressions written by Hamilton’s method replace many ordinary equations.23

It is by a kind of mathematical “breaking of symmetry” as access to an enveloping formal dimension (namely a more powerful formalism), breaking with the dimensions posed by Descartes, that this compactification is carried out in “small number 22 23

[33], 138. [34], 9.

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of expressions”. From then on, geometry and physics are balanced in this precious “point” understood as a field. It is moreover by the famous operator Nabla ∇ that Hamilton will capture in a single symbol the articulation of the dualities of Chasles and Ampère, thus making manifest the conversion between the axial-magnetic point of view and the polar-electric point of view.24 That’s how Châtelet will be able to speak of “diagrammatic asceticism”.25

6.3 On the Diagram as Proof by Image A diagram can immobilize a gesture, put it to rest, long before it buries itself in a sign, and that is why contemporary surveyors or cosmologists love diagrams and their peremptory power of evocation. They seize gestures in mid-air; for those who know how to be attentive, they are the smiles of being.26

At this point I’d like to clarify what differentiates my approach from that of Gilles Châtelet. I would formulate the difference by saying that, for me, it is a question of an even more radical retracing of the diagrammatic act of the drawing of a lineexpressed in its pure geometrical representation—to the letter bound in the discursive economy of the formula. Any formula already constitutes a complex diagram that only has meaning at the dotted center of a theoretical context and a conceptual Gestalt.27 I would therefore position the letter or the gram well upstream of any diagrammatic act. This is what is made decipherable by a genealogical return to the unprecedented and founding metaphor of the Libro della natura and its diverse prolongations. From this perspective the “standard” diagram now appears as a manifestation of the articulated and dialectical structure of the gram. This “vital” and “initial” pulsation of the gram, the act that induces its potential mobility (what Châtelet designates as “the stakes of the mobile”) is subsumed in the original sense of the prefix dia-: “dividing”, then crossing, “passing through”: • δια—γραφω = to design; to describe || to record; to assign || to erase. • δια—γραμμα (τo) = 1° drawn figure || 2° register || 3° decree. • status of δια-: prep. meaning through. Here, what pierces (in) the gram and crosses the writing (and the “formula”). 24

[22], 256 and appendix III, 278–280. Châtelet, ibid., 180. 26 [21], 33 (my translation). The English translation by R. Shore and M. Zagha gives: “A diagram can transfix a gesture, bring it to rest, long before it curls up into a sign, which is why modern geometers and cosmologers like diagrams with their peremptory power of evocation. They capture gestures mid-flight; for those capable of attention, they are the moments where being is glimpsed smiling”, Châtelet., 1999, 10. 27 On this point, see [28],[29]. 25

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This “division” operates as a “difference potential” engendering the tension and the “autonomy of its center of indifference”: this is what gives its reserve to the act, in movement and without ever exhausting it.28 This evokes a new question: how can we imagine, “diagrammatically” speaking, a diagramma that is increasingly purified of its symbolic forms, these forms having been immersed in increasingly powerful formalisms? It is the reinterpretation of a geometrodynamical diagram by the physicist John Archibald Wheeler, inventor of the term big-bang, that led me to this analogically similar “image”:

Formal compactification capsule Path exterior differential. Universal operator 2-form of space-time Dual operator assuming a metric Cohomology operators

dF = 0 d*F = 4 *J

Inspired by John Archibald Wheeler Here the diagram is applied to Maxwell’s equations compacted by the so-called pforms of differential geometry. What is interesting here, as we briefly saw, is the indication of a logic of complicatio (envelopment/enfoldment), of explicatio (development), contractio (reduction) and contractus (what is reduced)—contrahere meaning all at once to draw together, tighten, derive, and to be linked. Here a dilatation or centripetal expansion responds to an expansive or centrifugal reduction. Technically, the model of this dialectical pulsation of the compaction diagram is that of the expansion of the physico-mathematical formula in an indexed

28 On the “positive” notion of “epistemological act” (transcendent induction, dialectical generalization and epistemological synthesis), see [17], 25. Cf. the developments of Marrone and Patras, in Alunni (ed.), 2022.

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system of coordinates (“index expansion”) versus the contraction in an “abstract notation” (as “index-free notation”).29

6.4 Image I’ve used the notion of the “image”, but with a meaning I should clarify. It’s the “image” in the mathematical and Set theoretical sense of the term – as a refined and “abstracted” extension of the Venn diagrams that I have in mind, not the image as “illustration”:

The image is in the diagram and the diagram envelops the image as one of its functional elements. The diagram inscribes the image as its nucleus kernel—here we can take the word “kernel” in its strict mathematical sense30 : 29

Misner, Thorne, Wheeler, 1973. See § 8.3, Three aspects of geometry: pictorial, abstract, component, 198–199, Box 8.3 and 223–224. For the diagram inspired by Wheeler, see 1183. Wheeler speaks also of Gymnastics for technics of indexation, 85. 30 Desanti declared in 1999, 138: “The relationship between ‘fruit kernel’ and ‘homomorphism kernel’ is not a simple and arbitrary homonymy. […] ‘the kernel encloses and maintains a virtual tree conforming to its species’. It is this function of ‘signal’ of invariance, within a field in the making, which seems to me to authorize the extensions of meaning to the fields most foreign to our usual natural experiences at the start. Thus, the use of the word has been accepted to designate, within the atom, the region where its charge is concentrated, within the cell the region where its genetic

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Homomorphic sequences and structure of homotopy groups Exact sequence theorem Wang RONG, Chen YUE, Differential Geometry and Topology in Mathemarical Physics, World Scientific, Singapore, 1998. As for the image, you could say it develops the diagram. So, here we have the juncture image/diagram and the double invaginating structure of envelopment / development. material is concentrated and preserved: always in accordance with a requirement of invariance, each time specific, and identifiable according to appropriate procedures. In the language of algebraists, a kernel takes on the meaning of: invariant”.

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6.5 Figure There is more in a diagram than in a figure. While the figure is bound to its doleful illustrative function, the diagram attempts to articulate an “interpenetration of image and calculation”. One could say that the diagram propels calculation with its “allusive stratagems” and that, contemporarily, the calculation ensues from and persues the diagrammatic image that is orientation towards proof. If the calculation braces and is a tracing of the image, conversely, (and with the same cohesive gesture), the (diagrammatic) notation contaminates the calculation. The diagram differs from the figure by the machinery that makes it work. It is a machinic power in its own right.31 It is not a simple coding or information gathered in one singular place that simply illustrates or summarizes a model; it is a deployment of virtual gestures. The diagram is a covariant structure, that is to say, it is independent of benchmarks and valid in all possible worlds. In this sense, the diagram conveys its own syntax rules. It is an “‘indicator of required relationships” as Jean-Toussaint Desanti so elegantly puts it. Here we can quote Charles Sanders Peirce: The diagram represents a defined Form and Relation. This relation is ordinarily a relation that exists, as in a map, or else a relation that is intended to exist, as in a plan.32

The diagram combines an interpretation of the signs depicted in a schema, and how these signs work together by structurally connecting the specific intentional acts that produced them. This diagrammatic “indication” of sequences of actions (as sequences of potential acts) is always characterized by a very high degree of plasticity so that it is never fully revealing: it always proceeds by indices and indexations whose operative (and manipulative) dialectic is extremely subtle. Far from solidifying a reality, the “indication” envelops it in an indetermination that is not a lack of determination, but a promise of virtualities to come.33 We see what differentiates the diagram from the figure: if the figure is able to “illustrate” in a static way, the diagram establishes itself as a purely dynamical “figure-calculation”, simply because it mobilizes both image and calculation. Because they are self-reactivating, diagrams are veritable “multipliers of virtualities”: There is an operative power that is completely specific to diagrams. They do not limit themselves to visualizing algorithms or to encoding and compacting “information” in order to restore it in the form of models or “paradigms”. The diagram is precisely this swarm of virtual gestures: indicating, knotting, extending, striating the continuum. A simple bracket, the tip of an arrow and the diagram leaps over the figure and forces us to create new individuals. The diagram haughtily ignores all the old oppositions like “abstract-concrete”, “local-global”, 31

We will compare the idea of diagram as a “machinic power” to Wheeler’s idea of “machine with slots” concerning “geometric objects” and operators. See Misner, Thorne, Wheeler, 1973, Box 10.3. Covariant derivative viewed as a machine. Connection coefficients as its components. A. The Machine View, 254–255. 32 [36]. About the notion of a plan, see McLane and his “plan of proof” (above note 7). 33 SEE [24].

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“real-possible”. It guards in reserve the fullness and all the secrets from the depths and horizons that its magic, nonetheless, keeps constantly on the alert.34

6.6 Conclusion—The Question of the Diagrammatic “Proof” One could argue that the diagram is the organon of new evidence. In a diagrammatic demonstration, the changes focus on sub-parts of diagrams to which we apply results obtained elsewhere (which is a standard procedure in mathematics). The motor of the demonstration remains the same since it consists in using equivalent expressions defined by the equivalence of two diagrams and elementary algebraic manipulation: addition, subtraction, multiplication and division. But rather than speaking of “diagrammatic evidence” stricto sensu, it might be more appropriate to speak of a diagrammatic orientation of the demonstration: the diagram as demon of monstration, of evidencing or showing, the demon of proof! Thus, on a scale of grams (diagramma), measuring the interval between two or more grams according to a new analogical dimension and new connections between the grams, comes down to orienting oneself diagrammatically in thought and thereby providing insight into the solution as well as establishing the machinic expression of prospective virtualities which will deliver up the solution.

References 1. Abramo, M.: Gaston Bachelard e le fisiche del Novecento. Alfredo Guida Editore, Napoli (2002) 2. Abramo, M.: Il razionalismo “induttivo” di Gaston Bachelard, Ph.D., Università degli studi di Messina (2019) 3. Alunni, C.: “Codex Naturae et Libro della Natura chez Campanella et Galilée”. In: Annali della Scuola normale superiore di Pisa. Classe di Lettere e filosofia, Serie III, vol. XII, 1, Pisa (1982) 4. Alunni, C.: “Diagrammes & catégories comme prolégomènes à la question : Qu’est-ce que s’orienter diagrammatiquement dans la pensée ?”. In: Batt, N. (ed.) Penser le diagramme de Gilles Deleuze à Gilles Châtelet, in Théorie, littérature, enseignement, no 22 (automne 2004), pp. 83–93. Presses Universitaires de Vincennes, Saint-Denis (2004) 5. Alunni, C.: “Le lemme de Yoneda: enjeux pour une conjecture philosophique”. In: À la lumière des mathématiques et à l’ombre de la philosophie. Dix ans de seminaires mamuphi. Moreno Andreatta, François Nicolas, Charles Alunni eds., Paris: Delatour, Ircam/Centre Pompidou, “Collection Musique/Sciences”, pp. 195–211 (2012) 6. Alunni, C.: “De l’écriture de la mutation à la mutation de l’écriture : de Galileo Galilei et Leonardo da Vinci au ‘technogramme’”. In: Nicolas Fr. (ed.) Les Mutations de l’écriture. Paris: Publications de la Sorbonne, “LogiqueLangageSciencePhilosophie”, pp. 123–137 (2013) 34

[20], 111–112. Available online https://www.unebevue.org/images/ubweb/ubweb29/diagramme/intuition.pdf.

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7. Alunni, C.: “Maximilien Winter et Federigo Enriques: des harmonie exhumées”. In: Charles Alunni & Yves André (eds.) Federigo Enriques o le armonie nascoste della cultura europea. Tra scienza e filosofia. Pisa: Edizioni della Normale, pp. 101–147 (2015) 8. Alunni, C.: “Guido Castelnuovo & la teoria della relatività. Prolegomeni alla sua epistemologia”. In: La Matematica nella Società e nella Cultura, pp. 353–370. Rivista della Unione Matematica Italiana (I), Agosto, Bologna (2015) 9. Alunni, C.: “Philosophie et mathématiques. Présentation” and “Gaston Bachelard face aux mathématiques”. In: Revue de Synthèse, “Philosophie et Mathématique”, Charles Alunni ed., Paris: Lavoisier, Tome 136, 6e série, N° 1–2, 2015, pp. 1–8 and pp. 9–32 (2015) 10. Alunni, C.: “Gian-Carlo Rota & Gilles Châtelet: une combinatoire”. In: Revue de Synthèse, “Philosophie contemporaine de mathématiciens: Évariste Galois, Gian-Carlo Rota, Gilles Châtelet”, Charles Alunni, Yves Andr´e & Catherine Paoletti, eds. The Hague: Brill, pp. 19–50 (2017) 11. Alunni, C.: “Actualit´e de Jean- Toussaint Desanti. Mathesis en suspens”. In : Revue de Synthèse, “Philosophie contemporaine de mathématiciens : Évariste Galois, Gian-Carlo Rota, Gilles Châtelet”, Charles Alunni, Yves Andr´e & Catherine Paoletti, eds. The Hague: Brill, pp. 337–346 (2017) 12. Alunni, C.: “Évariste Galois. La copie d’Évariste Galois au concours d’entrée à l’école préparatoire”. In : Revue de Synthèse, “Philosophie contemporaine de mathématiciens : Évariste Galois, Gian-Carlo Rota, Gilles Châtelet”, Charles Alunni, Yves Andr´e & Catherine Paoletti, eds. The Hague: Brill, pp. 367–444 (2017) 13. Alunni, C.: Spectres de Bachelard. Gaston Bachelard et l’école surrationaliste. Hermann, “Pensée des sciences”, Paris (2018) 14. Alunni, C.: “Gaston Bachelard, encore et encore”. In : Métaphysique de la mathématique, Charles Alunni, C. et Ienna, G. eds., Paris: Hermann, “Pensée des sciences”, pp. 7–22 (2021) 15. Bachelard, G.: La Valeur inductive de la relativité. Paris, Vrin, [20142 ] (1929) 16. Bachelard, G.: L’Expérience de l’espace dans la physique contemporaine. Alcan, Paris (1937) 17. Bachelard, G.: L’Activité rationaliste de la physique contemporaine. Puf, Paris (1951) 18. Brouwer, L.E.J.: “Mathematik, Wissenschaft und Sprache”. In: Monatshefte f. Mathematik u. Physik, Bd. 36 (1929) 19. Cavaillès, J.: Méthode axiomatique et formalisme [1937]. In: Huisman, B. (ed.) Œuvres Complètes de Philosophie des Sciences. Herman, Paris (1994) 20. Châtelet, G.: “Intuition géométrique – Intuition physique”. In: Clemens, R.R., Longuine, P., de Turckheim, E. (eds.), Selected Papers on the Teaching of Mathematics as a Service Subject (CISM, Courses and Lectures, IV, n° 305). Springer, Vienne-New York (1988) 21. Châtelet, G.: Les Enjeux du mobile. Mathématique, physique, philosophie. “Des Travaux”, Seuil, Paris (1993) 22. Châtelet, G.: In: Shore, Tr.R., Zagha, M. (eds.) Figuring Space: Philosophy, Mathematics, and Physics. Springer, London (1999) 23. Châtelet, G.: L’Enchantement du virtuel. Mathématique, physique, philosophie. Paris: Éditions rue d’Ulm, Charles Alunni & Catherine Paoletti ed. (2010) 24. Chiereghin, F.: Relire la Science de la logique de Hegel. Récursivité, rétroactions, hologrammes. Paris, Hermann, « Coll. “Pensée des sciences” (2020) 25. Desanti, J.T.: Philosophie: un rêve de flambeur. Variations philosophiques, vol. 2, Grasset, Paris (1999) 26. Galilei, G.: The Assayeur (1623) 27. Galilei, G.: Letter to Liceti (1641) 28. Hanson, N.R.: Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science. Cambridge University Press, Cambridge (1958) 29. Hanson, N.R.: The Concept of the Positron. Cambridge University Press, Cambridge (1963) 30. Hegel, G.-F.: The Science of Logic. Cambridge University Press, Cambridge (2010) 31. MacLane, S.: “Abgekürzte Beweise in Logikkalkul” [1934], in Saunders Mac Lane Selected Papers. Springer, New York (1979)

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32. Marrone, S., Patras, F.: “L’épistémologie mathématique de Gaston Bachelard”. In: Alunni, C. (ed.), Gaston Bachelard et la philosophie des sciences aujourd’hui in Bachelard Studies/ Études Bachelardiennes/Studi Bachelardiani. Mimesis, Sesto San Giovanni (2022) 33. Maxwell, J.C.: “Quaternions”. Nature (1873) 34. Maxwell, J.C.: In: Thompson, J.J. (ed.) A Treatise on Electricity and Magnetism [1873], 3e ed. Oxford University Press, Oxford [1892]; reprint New York (1954) 35. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman and Company, New York (1973) 36. Peirce, C.S.: “Prolegomena for an apology to pragmatism” [1906]. In: Eisele, C. (ed.), The New Elements of Mathematics. La Haye, Mouton Publishers (1976) 37. Wittgenstein, L.: The Blue and Brown Book. Harper & Row, New York (1980) 38. Wittgenstein, L.: Culture and Value. Chicago University Press, Chicago (1980) 39. Wittgenstein, L.: Werkausgabe in 8. Frankfurt, Suhrkamp Verlag, Bänden (1984) 40. Woodard, B.: “Parameter Breach: Châtelet and the Ground of the Diagram”. In: Differential Heterogenesis, La Deuleuziana, Online Journal of Philosophy (2020) (ISSN 2421-3098), N. 11/2020

Chapter 7

Gesture, a New Tool for a Different Vision of Synthetic Reasoning Giovanni Maddalena

Gestures have usually been thought of as mere rhetorical instruments. We human beings accompany our discourses with bodily movement, especially of our hands, which helps us clarify meaning or heighten emotions. This is the common meaning of “gesture,” which involves a more primitive and sometimes more spontaneous form of spoken language. The good orator is able to master gestures in support of his spoken language. Gestures are a reminder of ancient forms of conversation that were shoved aside by the rise of spoken language. This is the ‘parallelist’ view of gestures featured in ancient rhetorical studies and in the modern way to look at them, especially in studies of regional gestures [3, 4] and in psychological studies [8, 32]. In an inchoate way, Vico [31], and, in a more sophisticated way, Mead [22], presented an alternative view of gestures: they are the fundamental mechanism of human thought and not an expression of thought parallel to spoken language. As an heir to the pragmatist tradition, Mead refuses to think of body and mind as two distinct entities. He thinks of them as belonging to the same social relationship or, as he calls it, “conversation of gestures”. This conversation of gestures creates thought, gesture is the initial way in which we think, not a primitive expression of our thinking. We cannot think but through gestures. Spoken language emerges from this conversation of gestures as a peculiar kind of gesture, one that calls upon the intervention of a third element: when the answer to a stimulus obliges the utterer to share the same responsive habits of the receiver, gesture has become word. Because of this continuity between gesture and word, we call this view ‘continuism’. In recent times, authors like Kendon [17] and McNeill [21] have developed the study of gestures in a scientific way, supplementing the two views we have mentioned with neuroscientific research. We can say that the contemporary heirs of parallelism and continuism have arisen from research underlining the connection between gestures and both the Broca’s area and mirror neurons. The connection G. Maddalena (B) University of Molise, Campobasso, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_7

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with the Broca’s area stresses the relevance of gestures to spoken language, and the connection with mirror neurons stresses the cognitive role of gestures [5]. Kendon and McNeill also helped to establish a classification of gestures called “Kendon’s continuum”. Passing from the less to the more symbolic, we have gesticulation, language-like gestures (like the rotating finger that says “after” or “rewind” or “say it again”), pantomimes (like mimicking a stomachache), emblems or Italian gestures (conventionalized pantomimes like the well-known finger purse or chin flick), and signed languages (i.e., languages for deaf people). By looking at human experience, it is important to complete Kendon’s continuum with at least two other kinds of gestures. The first is speech acts, those performative and effective words and formulas first described by Austin [1]. In the continuist view, in which language is a kind of gesture, speech acts should clearly be included in any classification of gestures. Second, the synthetic gestures identified by the French philosophers of mathematics Jean Cavaillès, Jean-Toussaint Desanti, and Gilles Châtelet should also be included on the continuum. When we are doing mathematics using diagrams and drawings, including those special drawings we call “numbers”, we are engaged in understanding and creating by our doing. If we look closely at their structure, synthetic gestures explain the true nature of any gesture and are able to shed light on an aspect of our entire epistemic enterprise that is usually left in obscurity. It is the intent of this paper to take a closer look at synthetic gestures, in particular those used in logic and mathematics, and their impact on the general picture of how our reasoning functions. In order to do so, we have to start from a critique of the concept of synthesis (1) and a semiotic study of Peirce’s existential graphs (2). Then we will shift our focus to the structure of gestures and their classification (3–4). Finally, we will propose a different general image of our reasoning (5).

7.1 A Critique of the Concept of Synthesis and a Different Proposal Our usual image of synthesis still relies on Kant’s definitions of synthetic and analytic and on his subsequent distinctions between theoretical, practical, and aesthetical judgment. Following Robert Hanna’s reading of Kant, I agree that Kant’s view of syntheticity is based on analyticity [12, 191–239]. When we examine how Kant built up his certainty about synthetic a priori judgments, we see that he grounded them on an analytic subdivision of a unique space and a unique time in transcendental aesthetics. He moved from a conceptual composition/decomposition for analytic judgments to an aesthetic composition/decomposition for synthetic a priori judgments. These latter judgments work—that is, they are able to keep universality and necessity—because they grasp something new and non conceptual by referring to the unique space and time singled out by transcendental intuitions. This is a specific form of universality and necessity that refers to a limited transcendental experience of the universe. Using a metaphor, we can say that synthetic a priori judgments and

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reasoning work like moves on a chessboard that are determined by aesthetic intuition. I call this general schema the “part/whole schema,” and I believe that it is still the most accepted view of how we reason. As is well known, Kant used this schema to paint a sophisticated fresco that includes transcendental deduction, transcendental schematism, and the free play of our faculties.1 Since, according to Kant himself, we can sum up the problem of the First Critique with the question, “How are synthetic a priori judgments possible?”, his pivotal construction of those judgments is the keystone of his edifice, which has not really been challenged by later developments. When someone attacked Kant’s view of reasoning, he/she usually rejected the epistemic power of reasoning completely, as did Nietzsche, but in doing so, he rejected all of the benefits that analytic picture bestowed on contemporary philosophy of knowledge, benefits that contributed to actual improvements in science. Others transformed the Kantian model without changing this fundamental kernel. As Cavaillès [6] explained, Kant’s successors reformulated his “philosophy of conscience”— the name he gives to Kant’s transcendental philosophy—by relying on demonstration (Bolzano), intuitions (Brunschvicg), axioms (Hilbert), and phenomenological constructions (Husserl). None of them changed the rationalist understanding of the way in which we grasp non conceptual reality in our syntheses, nor did they challenge the part-whole schema that is at the bottom of this understanding. In this view, any knowledge, including mathematical knowledge, must rely on this logical schema, which presides over any other form of reasoning. The ‘logicist’ hypothesis about the foundation of mathematics was the heir of this tradition of thought. For this reason, our task is to rethink syntheticity without losing the results of analyticity. A hint towards a different conception of syntheticity comes from Charles S. Peirce, one of the founding fathers of American pragmatism. Peirce focused on the problem of syntheticity insofar as it is a problem of how we broaden our knowledge beyond concepts. He concentrated his efforts on induction, abduction and semiotics, together capable of accounting for the ampliative property of our reasoning. He referred to the analytic/synthetic definition given by Kant only occasionally after the work of his early career, but this work made a strong push towards a reformulation of this old view of reasoning.2 In fact, in one passage he says that the analytic identity A = A, usually considered as the keystone of precise reasoning, is only a degenerate form of the real identity A = B. A = A is the static correspondence drawn from a set theory definition of multitudes and is a simplification of a “more primitive” form of relation (NEM IV: 325–328). Identities are always passing through changes. From here our new understanding of syntheticity can begin. We are looking for a synthetic identity A = B, of which the analytic A = A is only a derivative form. When we look for the A = B identity we have to include the concept of change in our reasoning, which means that the new definition of synthesis must involve a recognition of an identity that occurs through changes. If we define analysis from the same point of view, considering it to be a derivative form, we will conclude that analysis is the losing of the identity A = B through changes. Our experience confirms 1 2

For a different picture of Kantism within the philosophy of gesture, see [2]. For a good summary of Peirce’s take on syntheticity, see [7]. See also [14] and [28].

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this new perspective: when we synthetize, we acquire new knowledge about the same object, which we gain only by moving through changes. When we analyze an object, we have to stop this process at a moment in time and break it into pieces. We always apply this theoretical step in academic studies, whether we are discussing literature or physics—we take our subject matter apart in order to know it better and then to subsequently recompose and use it. However, our understanding is also often “synthetic” in the way we have defined it. We learn our mother tongue and then foreign languages, but also mathematics and physics, as well as how to ski and swim, by doing something through time; namely, by passing through changes. To be fair, following André De Tienne’s suggestion,3 a third option is possible: we can be blind to any identity through changes. This is the realm of vague reasoning, which happens when we are in awe, when we are passing from an analytic to a synthetic perspective, or when an analytic problem reaches a paradoxical outcome. This third option, vagueness, is not the topic of this paper but it is nevertheless worth further study. The new paradigm of reasoning, thus, is tripartite: synthetic, analytic, and vague. It is important to define “change”; otherwise, the new paradigm would fall apart. That is why Peirce’s studies are so important. The great logician dedicated much of the second part of his life to understanding change and evolution as continuity. After many years and coming to many different understandings of it, he arrived at a conception in which continuity escapes any metrical measurement or any possibility of reaching it by Cantor’s progression of sets [18, 37–192].4 Peirce called Cantor’s solution “pseudo-continuity” and described his own continuum, in which Fernando Zalamea has identified four properties: reflexivity, generality, modality, and transitivity. Each of these four properties underlies one aspect of the relationship between the parts and the whole of a continuity and rejects the usual treatment of these objects according to composition/decomposition.5 Real continuity is a transition among modalities in which possibilities can become actualities according to a general regularity. In a Peircean continuum, there must always be room for all three modalities—possibilities, actualities, generalities—in the same point, which prevents it from being metrically measured. Francisco Vargas (2020) confirmed Zalamea’s description of a Peircean continuum from a mathematical point of view, offering a splendid demonstration based on topology, one of the strategies that Peirce himself tried to explore.6 In brief, change is the continuity between A and B, or, put differently, A and B belong to the same continuity. Synthesis happens when we recognize that A has become B. If we take in this sense the celebrated example of 7 + 5 = 12 [15: B15-6], we understand that Kant was indeed right in saying that the concept of twelve is not comprehended in the concepts of 7 or 5 or the plus sign. It is something new, and at 3

Private conversation. See also [13, 23]. 5 [34] subsumes transitivity under modality, but I prefer to make it a separate characteristic of Peirce’s continuum because of transitivity’s fundamental role in explaining “change.”. 6 See also [13, 23]. 4

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the same time it is an identity. However, Kant did not consider the possibility that this novelty is something that happens through the actions that move the stones or the fingers that we use to calculate or through the drawing of the numbers on a board or in our minds. He passed immediately to the transcendental aesthetic foundation, in which he could find his pacifying part-whole scheme. If he had remained at the example, he would have discovered a more fallibilist notion of synthesis that is closer to our real experience. When one accepts this fallibilist view, he/she realizes that 7 + 5 = 12 is not a special case, but rather a typical case of a posteriori synthesis, in which we recognize a change that occurs to our subject by the doing of a certain action. At the same time, he/she realizes that this acknowledgment of syntheticity does not exclude belief, certainty, and, in the long run, truth. In fact, the habit of addition has been repeatedly verified throughout history. As we will see, the same recognition happens when we understand a physical truth by means of an experiment, a moral event like the formation of a family through a wedding or an artistic event like the meaning (or lack of meaning) of a performance. Synthetic reasoning works by means of an action; it is by doing that we understand something new. Of course, this also means that when we say something simple like “the snow is white” we are passing from an unclear experience of something to this general statement by performing some action. Therefore, the fundamental task we must undertake is to understand the nature of the action implied in synthesis and how it works.

7.2 Existential Graphs Peirce studied a version of synthetic reasoning understood as recognition of an identity that occurs through change, even though he did not approach his discovery/ invention from this standpoint. In the last decade of the Eighteen century he invented an iconic logic for what we now call propositional, first order, and modal logic. He called the entire logical system “existential graphs” and the three parts respectively alpha, beta and gamma graphs. A few examples will illustrate this idea (Tables 1, 2 and 3). Peirce did not formulate or use our definition of synthesis, but he provided all the essential phenomenological and semiotic observations to build up tools for that kind of synthesis. This happens in the gamma graphs, in which he uses all of the semiotic characteristics of all the graphs. Let us try to list them briefly.7 1. The sheet of assertion of the alpha and beta graphs involving propositional and first order logic is a continuum in the Peircean sense. 2. While describing beta graphs, that is, first order logic, Peirce informs us that the line of identity, which is a quantifier (universal or existential according to the way in which it is enclosed in the cuts), is a continuum that moves on the continuum of the sheet of assertion. 7

For studies on Existential Graphs, see [26, 27, 29, 34].

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Table 7.1 Examples of propositional logic (alpha graphs)

Table 7.2 Example of first order logic (beta graphs)

Table 7.3 Physico-chemical properties of SMM and GU

3. The line of identity can work as a quantifier because it is a “perfect sign”; namely, an “equal blending” of icons, indexes, and symbols. 4. In gamma graphs—modal logic—the sheet of assertion must be imagined as a multidimensional (plastic) solid. Zalamea has suggested using Riemann’s surfaces for this task. 5. The line of identity in gamma graphs represents becoming, understood as a transition among possibilities, actualities, and necessities. Therefore, the line of identity becomes a line of ter-identity, a line of identity with a loose end, apt to represent some unrealized possibility. From the graphs, we understand that there is a logical tool that can represent reality as change; namely, as a continuous transition among modalities and including any reasoning that happens within this transition. In our idiom, we can call this tool a

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“mathematical gesture” or a synthetic gesture, connecting this logic with the French tradition aforementioned. The graphs are really a tool that unites theory and practice in a moving picture so that we can understand that reasoning itself is in its turn a transition. Moreover, we understand that this representation is possible because of the semiotic structure of our actual scribing. Finally, and most importantly, reasoning happens while we are scribing our graphs: there is no gap between reasoning and the representation of reasoning. In other words, in the graphs we understand something new by doing the action of scribing: we synthetize by drawing a line on a multidimensional plastic continuum. The result is striking, but Peirce focused on the deductive consequences of the graphs. He was in awe at this magnificent tool because it could easily accomplish all of the operations he had thought would be solved using symbolic logic. His first attempts at symbolic logic were made at the end of 1870s, while the existential graphs came in at the end of the 1890s, and their most mature formulation was explicated at the Lowell Lectures held during the fall of 1903. Peirce did not see that his chef d’oeuvre was opening up a completely different view of synthesis. How could this happen? If we single out Peirce’s mathematical, semiotic, and phenomenological descriptions of the graphs, we can generalize a kind of action that is apt to synthetize.

7.3 The Structure of Gestures The equivalent of the drawing of the line of identity on a plastic continuum is, in our everyday reasoning, a “gesture”. As I said, in using this term, my theory will confirm an important idea of twentieth-century French philosophy of mathematics. In fact, in 1939 Cavaillès used the word “gesture” to provide a keen picture of the rising development of mathematical reasoning following Gödel’s insights about the structural incompleteness of founding mathematics upon logic. Cavaillès could not exploit Peirce’s semiotic studies on existential graphs, which today enable us to depict a more precise sketch of how synthetic reasoning works. However, he contrasted the idea of mathematics as event, a position maintained by his friend Albert Lautman, with the continuous tradition of constructing mathematical realities through our meaningful gestures. He defined mathematics as “attraper le geste et pouvoir continuer” [6, 186]—“to grasp the gesture and be able to continue.” He did not further deepen the concept of gesture because he was shot by the Nazis a few years later, but our research into Peirce’s phenomenology and semiotics allows us to accomplish the task he left unfinished and also to overcome the distinction between mathematical and non-mathematical gestures, a distinction that Cavaillès still maintained because of a Kantian prejudice that mathematical certainty had a special place. However, the synthetic power of mathematical gesture is at work also in any other meaningful activity. What is a gesture, in general? A gesture is any performed act with a beginning and an end that carries on a meaning (from gero = I bear, I carry on). But, we are not talking about just any gesture—some gestures are only reactions. All gestures carry

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meaning but they do not necessarily serve to recognize an identity fully; we know that we must find those actions that recognize an identity A = B. So, every gesture is meaningful, but not every gesture shows a process of complete synthetic reasoning; that is, not every gesture causes us to know something new and nonconceptual. In order to understand “syntheticity,” we have to look for a “complete” gesture, a gesture that respects all the characters we find in Peirce’s logic of existential graphs. After we find a complete gesture, we might consider all other gestures, including incomplete ones that display a different or weaker form of syntheticity. Given the Peirce-Zalamea-Vargas mathematics of continuity for explaining the nature of “change” within which our gestures work, the structure of a complete synthetic gesture is phenomenological and semiotic. Let us start with phenomenology. A gesture must show all of the phenomenological characters identified by Peirce and called firstness, secondness, and thirdness: complete gestures involve a vague idea or a pure feeling (firstness), a physical act implying a reaction, that is, involving two objects or subjects (secondness), and a general habit of action that we can replicate in various circumstances (thirdness). Turning to semiotics, Peirce called the line of identity “the perfect sign” because of its equal blending of icons, indices, and symbols. Following this insight, we should call a gesture that has all the semiotic elements blended together proportionally or densely, a “complete gesture”. “Dense” is the way in which I translate Peirce’s “equality” of blending not metrically. For a gesture to be complete, it must have a general meaning (symbol) obtained through a reference (index) that embodies different possibilities of forms and feelings (icon). The three semiotic characteristics describe what a complete gesture is and should be: creative because of its possible forms and feelings, singular in its individuality, and recognizable for its unity and conformity to an established pattern that the gesture itself tends to bring about. We should complete this semiotic level with a tripartition from the point of view of determinativeness (according to Peirce, this is the stechiological level, the doctrine of elements) (EP2: 350–3). This level is based on the divisions among vagueness (indeterminativeness by the utterer), determinativeness, and generality (indeterminativeness by the interpreter). A complete gesture, like the line of identity, must display a path that teleologically determines meaning, moving from vagueness to generality, passing through a determinative act (Table 4). As I mentioned, clear examples of complete gestures are present not only in mathematics and in logic. We find examples in every field of human endeavor: liturgies in every religion, public and private rites, public and private actions that establish an identity, and, of course, artistic performances and experiments based on hypotheses. In these types of complete gestures, we observe the general structure of complete gestures and the pattern of syntheticity as I have described it: “recognition of identity through changes”. If you think of forms of “baptism” in any religion, or ceremonies like coronations, oaths, or funeral services, then you have a good image of what a “complete gesture” is in religion and public rites. As anthropology explains, ritual gestures fill up everyday life in relation to education, love, death, and other things. A gesture is not just one action among many others; rather, it is an expression of meaning embodied

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Table 7.4 The dynamics of gesture (stechiology)

in one person at a singular moment, and it tends to become a habit for the person and eventually for the generalized person—the people—thus becoming a tradition. A good example of private “complete gestures” is more difficult to offer because privacy prevents acts of sharing. However, I suggest one public–private “complete gesture,” which is to be understood educationally. Some Italian mountaineers have perpetuated an old habit: if a young person causes a stone to roll during a hike due to inattention, he/she has to carry the stone for some meters so that he/she can remember how dangerous his/her inattention could have been to other people. This is a private– public “complete gesture” created to link the person, the stone, and the educational purpose in an original way, that is, in an individual (in an experience, Dewey used to say)8 action that is evident and generalizable. At the scientific level, an experiment is clearly a “complete gesture” the first time it is performed. If you think of Rutherford’s gold foil experiment, you can see that the invention of this experiment links physical atoms and the golden device in an original way to a determinative moment in order to determine the structure of atoms. Writing a play is a “complete gesture” that links imagination to specific forms of experience, to the actual mechanism of the plot, and to the end or the general purpose of the story (obviously an absence of purposefulness is a purpose). When mathematical thinking is creative, linking global and local, universal and particular—as Zalamea describes it in his RTHK model, which generalizes approaches that originated with Riemann, Grothendieck, Lawvere, and Kripke [36]—it is a gesture that shows the movement from vagueness to generality passing through the determinativeness of the actual scribing, whether mental or physical. In all the above examples there is a phenomenological path that establishes a threefold relation in which two objects are related according to a general law (to which they tend in a teleological way) in order to embody some possible ideas in a determinative way. In turn, this embodiment enhances the general law itself and the path of determinativeness that proposes or fosters a habit of action. In addition, this relation singles out the objects (by means of indices) according to an interpretative 8

See [10].

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path (symbols) that helps determine and transform a “form” connected to the two objects. Also in this case, determinativeness proposes or reinforces the interpretation or meaning.

7.4 Incomplete and Complete Gestures In complete gestures, phenomena and signs blend densely into one another. What does dense blending mean? At present, it is only possible to propose a negative description of dense blending by describing the meaning of loose blending from both the phenomenological and semiotic points of view. The list of disproportions coincides with the list of incomplete gestures, actions that carry out a meaning but do not reach the full syntheticity of the identity A = B; that is, the evident and generalizable expression of a universal in a singular action. These incomplete gestures can happen in any field (Table 5). Starting from phenomenology, we can say that when secondness and thirdness are weak (it is impossible that they are missing completely; otherwise, we are no longer speaking about gestures), gesture remains an ideation (I). When only thirdness is weak, gesture is an exhibition (I-II). The object of experience is shown in a singular point, but no further habit of action develops from it. When secondness is weak, a gesture becomes a projection (I-III). When firstness and thirdness are weak, a gesture Table 7.5 Phenomenological structure of complete and incomplete gestures

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becomes a clash or a reaction (II), a single response that does not entail new ideas or feelings and whose further realization is not possible. When firstness is weak, a gesture based on secondness and thirdness becomes a habit whose exercise does not involve novelty, a “habit” in Wittgenstein’s sense of the word (like driving a car after one has learnt to do so). I will use the Kantian word of schematization (II-III) to describe this. When both firstness and secondness are weak, a gesture becomes an abstraction (III) (Table 6). The list of incomplete gestures also has a semiotic side. Here, we also find Kendon’s continuum of gestures. When indices and symbols in a gesture are weak, the gesture becomes a stream of images (imagination, IC). When symbols are weak, gestures are information (IC-IN); they work like living captions. Gesticulations and language-like belong to this category. When indices are weak, the result is formal models (modeling, IC-SY). When icons and symbols are weak, a gesture is only indication (IN) or communication of reference. Most gestures that are per se pantomimes or emblems share this semiotic function. When icons are weak, we have plain repetition (IN-SY). Some emblems, including bad gestures, have this semiotic function. When both icons and indices are weak, we have a conceptualization (SY). Because signed languages are languages with a highly symbolic conventionalism, they also fall under this heading. None of these incomplete gestures has a “negative” role. They are still actions with a beginning and an end and they bear a meaning. In-completion means only Table 7.6 Semiotic structure of complete and incomplete gestures

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that the way in which they are gestures does not reach full synthetization, or embodiment of universals, or recognition of identity through change. In a way, they cannot fully embody universals into singulars that propose or enhance generals. However, sometimes we need these incompletions, and sometimes we want to communicate an incomplete meaning.

7.5 A More Interesting Image of Our Reasoning The list of complete and incomplete gestures will change significantly as additional studies examining a wider range of signs are conducted. At this stage, I proposed only a suggestion so that we can better understand the role of gestures in many of our epistemic activities. Conceived in the way we explained, gestures are not a parallel or complementary or initial way of getting at meaning; rather, they are a fundamental tool for the whole of our reasoning. Since they are actions, they show that we learn by doing. They do not insert practice into the theoretical; rather, they are, as an essential part of their activity, a theoretical path. Or, rather, they are cognitive paths because a philosophy of gesture like the one I have described is capable of overcoming the division between theory and practice. Moreover, gestures are tools showing the true nature of syntheticity, rescuing it from a crypto-analytic reading. We could say that they are the instruments of syntheticity understood as recognition of identity through change, just as logical analysis is the instrument of the analytic part of our reasoning. As we mentioned, synthetic reasoning must also imply some kind of vague reasoning, even though we haven’t yet found adequate instruments that could guide us into it. We can summarize this new concept of syntheticity by inserting it into a general image of the functioning of our reasoning operating as a pendulum (Table 7). Table 7.7 The swinging pendulum of our reasoning

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Dynamically described, on the one end of the pendulum, we embody vague experiences in gestures that develop meaning to the point of reaching a general, universal form. On the other end of the pendulum, we disembody vague experiences into unending analyses. Vagueness plays the double role of being the passage between synthesis and analysis (and vice versa) and the fertile emergence of experience. It would be possible to begin studying this part of our reasoning by conducting topological research on borders. That is why Roberto Perry called this discipline that studies vague reasoning “horotics” (horosis = border), and André De Tienne has pointed out the necessity of a horosemiotics (De Tienne 2019). A common sense approach to vagueness could be ventured through awe (Schelling, Florensky), antinomies (Peirce, Cantor), empathy (Simenon, Fellini), and bullshitting (in the negative sense described by Frankfurt [11]. Returning to synthesis and to its special role in mathematics (and in other forms of reasoning), this image allows us to understand how much the doing of mathematics is relevant to its creative side. As Euler used to say, “There is more mathematics in my pencil than in my head”. Synthetic gestures are the means by which mathematics grasps reality. The possibility of thinking of mathematics as a kind of synthetic gesture allows for a more creative view of the discipline as a whole. Mathematics, just like any other creative, gestural enterprise, can grasp and develop reality through its own gestures. Usually it can, in its process of development, yield a better, keener image of the functioning of reality. In a broad vision, we can say that through this new conception of synthesis and gestures, mathematics again finds its place within philosophy. In fact, if the modern mathematics beginning with Descartes reduced the contribution of mathematics to measurement, exactness, and evidence, the ancient way to look at it was as a form of philosophy, a discipline allowing the creation of models for understanding the world. So it was with Plato or Pythagoras and with the notion of the trivium and quadrivium in the Middle Ages, and so it is with Peirce or Grothendieck.9 Some contemporary mathematics provides images of the metaphysical functioning of reality, and some provides images of the functioning of reasoning. Obviously, these contemporary conceptions do not discount any of the important results of metrical and mathematical analyses, but they show that these analyses are only one part of the pendulum of our reasoning. These analyses have some important functions and roles, but we should not forget the other more creative aspect of mathematics from which they were born. Finally, we can rebuild a more complete continuum of gestures by understanding that symbolic languages, whether natural or mathematical, are a kind of synthetic gesture and must be thought of, and taught, as such. If this is kept in mind, even analytic research and development can be understood better. Cavaillès was right: mathematical creation and tradition become clearer if we understand them as “attraper le geste et pouvoir continuer”. More generally, the continuity of gestures

9

For recent studies on Grothendieck, see [35].

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from gesticulation to synthetic creative gesture can revive a conception of knowledge that avoids the dualism between body and mind, and—even more important— between hard and soft sciences. All of the sciences move along the pendulum of our reasoning, using their own gestures to comprehend reality synthetically and their own logical paths to break up reality analytically.

References 1. Austin, J.L.: How to do things with words. Harvard University Press, Cambridge Mass (1962) 2. Baggio, G.: Lo schematismo trascendentale e il problema della sintesi tra “senso”, “segno” e “gesto”. Un’interpretazione pragmatista. Spazio filosofico 21(1), 83–99 (2018) 3. Bonifacio, G.: L’arte de’ cenni, Francesco Grossi, Venezia (1616) 4. Bulwer, J.: Chirologia: or the Natural Language of the Hand, and Chironomia: or the Art of Manual Rhetoric. Tho. Harper, London (1644) 5. Campisi, E.: Che cos’è la gestualità. Carocci, Roma (2018) 6. Cavaillès, J.: Œuvres complètes. Hermann, Paris (1994) 7. Chevalier, J. M.: Peirce’s relativization of the analytic vs. synthetic dichotomy. Blityri IX(2), 75–104 (2020) 8. De Jorio, A.: La mimica degli antichi investigata nel gestire napoletano. Stamperia e Cartiera del Fibreno, Napoli (1832) 9. De Tienne, A.: Prolegomenon to horosemiotics. Semiotic ramifications of a peircean borderline distinction. Semiotics: New Front Semiot 1–14 (2019) 10. Dewey, J.: Art as experience. in the collected works of John Dewey, 1882–1953. Later Works, vol. 10. Southern Illinois University Press, Carbondale and Edwardsville (1934) 11. Frankfurt, H.G.: On Bullshit. Princeton University Press, Princeton (2005) 12. Hanna, R.: Kant and the Foundations of Analytic Philosophy. Clarendon Press, Oxford (2001) 13. Hâvenel, J.: Peirce’s Clarifications on Continuity. Trans Charles S. Peirce Soc 44(1), 86–133 (2008) 14. Hintikka, J.: C.S. Peirce’s ‘first real discovery’ and its contemporary relevance. Monist 63, 304–315 (1980) 15. Kant, I.: (1781, 1787) Critique of Pure Reason, Unified Hackett Publishing Company, Indianapolis, Cambridge (1996) 16. Kant, I.: Critique of Practical Reason. Longsman and Green, London (1898) 17. Kendon, A.: Gesture. Visible action as utterance. Cambridge University Press, Cambridge (2004) 18. Maddalena, G.: Metafisica per assurdo. Soveria Mannelli, Rubbettino (2009) 19. Maddalena, G.: The philosophy of gesture. McGill-Queen’s University Press, Montreal (2015) 20. Maddalena, G., Zalamea, F.: (2012) A New Analytic/Synthetic/Horotic Paradigm. From Mathematical Gesture to Synthetic/Horotic Reasoning. Eur J Pragmatism Am Philos VI:208–224 21. McNeill, D.: Gesture and Thought. University of Chicago Press, Chicago (2005) 22. Mead, G.H.: Mind, Self, and Society. University of Chicago Press, Chicago and London (1934) 23. Moore, M.: The Genesis of the Peircean Continuum. Transactions of the Charles S. Peirce Society 43(3), 425–469 (2007) 24. Peirce, C.S.: The New Elements of Mathematics (NEM), C. Eisele (Ed.). The Hague, Mouton, (1976) 25. Peirce, C.S.: The Essential Peirce, (EP2), The Peirce Edition Project (Ed.). Indiana University Press, Bloomington- Indianapolis (1998) 26. Pietarinen, A.V.: Existential graphs: what a diagrammatic logic of cognition might look like? Hist. Philos. Logic 32(3), 265–281 (2011) 27. Roberts, D.D.: The existential graphs of Charles S. Peirce. Mouton, The Hague-Paris (1973)

7 Gesture, a New Tool for a Different Vision of Synthetic Reasoning 28. 29. 30. 31. 32. 33. 34. 35. 36.

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Shin, S.: Kant’s syntheticity revisited by Peirce. Synthese 113, 1–41 (1997) Shin, S.: The iconic logic of Peirce’ s graphs. The MIT Press, Cambridge Mass. (2002) Schmitt, J.C.: La raison de gestes dans l’Occident médiéval. Gallimard, Paris (1990) Vico, G.B.: La scienza nuova. Gaetano e Staffano Elia, Napoli (1744) Wundt, W.: Elemente der Völkerpsychologie. Grundlinien einer psychologischen Entwicklungsgeschichte der Menschheit. A. Kröner, Leipzig, (1912) Zalamea, F.: Filosofia sintetica de las matematicas contemporaneas. Universidad Nacional de Colombia, Bogotà (2008) Zalamea, F.: Peirce’s Logic of Continuum. Docent Press, Boston (2012) Zalamea, F.: Grothendieck. Universidad Nacional de Colombia, Bogotà (2019) Zalamea, F.: Modelos en haces para el pensamiento matematico. Universidad Nacional de Colombia, Bogotà (2020)

Chapter 8

Diagrammatic Gestures of Friendship in Plato’s Meno Rocco Gangle

Socrates: …And if my questioner were one of those eristical and contentious wise fellows, I would say to him, “I have given my answer. If what I say isn’t correct, your function is to demand an account and then refute it.” But if people want to engage in discussion with one another as friends, the way you and I are doing now, they must answer more gently, as it were, and in a more dialectical manner. And it is perhaps more dialectical not to answer with the truth alone, but also to do so in terms which the answerer agrees he knows. Meno, 75c-d

8.1 Introduction Plato’s Meno provides perhaps the best known dramatization of diagrammatic reasoning in the canon of Western philosophy.1 In the context of inquiry into the question of how virtue is acquired and more particularly whether or not it can be taught, Socrates introduces the thesis of the immortality of the soul and aims to verify this thesis with a demonstration. He guides a geometrically untrained household slave in the construction of a series of diagrams that lead the slave progressively from a cognitive state of initially confident ignorance to that of critically self-aware and thus genuine puzzlement and then finally to a condition of well-grounded practical knowledge: How, given a square, to construct a new square twice the original

1

All citations from Meno follow the standard Stephanus numbers using the translation by C.D.C. Reeve available in [9]. It is impossible here to do justice to the immense literature on the Meno, much less Plato’s thought as a whole. Especially helpful in developing the present interpretation has been [5], still the best single monograph on Meno. R. Gangle (B) Endicott College, Beverly, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_8

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square’s size. Under the guidance of Socrates, the concrete diagrammatic constructions manipulated by the slave serve in this way as self-explicating means for awakening a general diagrammatic capacity, already implicit yet unactualized in the slave’s ordinary understanding, to construct new figures of a definite type on the basis of determinately given materials. The episode of diagrammatic learning as recollection (anamnesis) in Meno is typically interpreted in terms of its contributions to Platonic epistemology and psychology. Certainly, these issues are crucial. But the dialogical and diagrammatic methods employed in the episode also illustrate a dynamic and complex intersubjective situation in which different social and dialogical roles are expressed and reflected at multiple levels of the text. This field of social-discursive relations, quite apart from any direct bearing on philosophical problems of knowledge or metaphysics, dramatizes the very process of philosophy in the Socratic mode as a collective practice of inquiry. At the heart of these relations and the philosophical practice that develops out of them are the diagrams that come to be partially mastered by a slave under the guidance of Socrates. The diagrams themselves, their experimental usage, and the social context in which that usage becomes philosophically meaningful all consist of concrete systems of relations. Understanding the interplay of these systems of socialdiscursive and diagrammatic-experimental relations in the dialogue sheds light on how the Socratic practice of philosophy at once depends upon and reproduces a certain kind of community, one that organizes itself around a form of inquiry pursued according to a certain image of friendship (philia). In the epigraph above from the opening section of the Meno, Socrates distinguishes between two modes of discursivity: One, appropriate for talk with “eristical and contentious wise fellows” (one is reminded here of the modern colloquial term “wiseguy”), follows the strict logical protocols of question-and-refutation (elenchus) and aims only at truth in the sense of argumentative victory; the other, proper to philosophical discourse among friends, is said to proceed more dialectically and with more gentleness (praos).2 This latter type of discourse aims not simply at truth in the sense of the enunciation of merely correct claims, but more essentially works at framing such claims in ways that are in agreement with the particular situation, aptitudes and background knowledge of one’s interlocutors. This way of proceeding in and through friendly discourse generates newly enriched social and epistemic relations out of those already present in whatever context. If this discursive mode based in gentle agreement among friends best characterizes genuine philosophy for Socrates, and if such philosophical discourse is not simply to remain in circulation among those who have already mastered it, it is worthwhile to seek signs within the Socratic dialogues of just how to initiate such friendly inquiry, how to establish the agreement that makes such gentle discourse possible. It seems the diagrammatic episode with Meno’s slave signifies in part where and how it makes the best sense to look for such marks of philosophical inception.

2

THE theme of gentleness in philosophy has been analyzed insightfully in [2], including a discussion (p. 28) of the subtle distinction between the terms proates and proas in ancient Greek.

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The following chapter analyzes the diagrammatic interlude at the heart of the Meno with respect to its synthetic function across multiple levels of the text. It is argued that the geometrical diagrams used to guide Meno’s slave to the correct solution of the given geometry problem may also be viewed as components of higher-order diagrams consisting not only of geometrical but also social relations with increasingly selfreflective structure. The dialogue as a whole may then be understood as an exposition of and reflection on the ultimate philosophical coherence of these progressively complex levels of diagrammatic relation, in particular as they instantiate the unifying pattern of inquiry pursued in friendship that is their common gesture. The chapter consists first of a brief recapitulation of the dialogue’s dramatic structure, focusing on the central role of the diagrammatic interlude with Meno’s slave. This interlude itself is then examined in detail in order to bring to light the process of reasoning exhibited by the sequence of the three diagrams constructed by the slave with Socrates’ guidance. Next, this process of reasoning is interpreted semiotically as it functions at three distinct levels, two within the text and one inclusive of the text. Finally, this semiotic interpretive framework is used to show how the interplay among these three levels may be better understood in terms of a structural connection between concrete diagrams like those used by Socrates and Meno’s slave in the dialogue and collective inquiry in friendship (philia) as the medium through which Socratic philosophy constitutes itself.

8.1.1 The Dramatic Structure of the Meno The dialogue falls naturally into three main sections. The first Sect. (70a–80e) consists of the abrupt opening in which Meno poses to Socrates the question determining the dialogue as a whole, namely the question of how virtue (arete) is acquired: “Can you tell me, Socrates, is virtue something acquired by teaching? Or is it something acquired not by teaching, but by practice? Or is it something acquired neither by practice nor by learning, but something human beings possess by nature or in some other way?” (70a). In a pattern familiar to readers of Plato’s other Socratic dialogues, Socrates in response proceeds to call into question the assumptions built into Meno’s question itself, in particular the assumption that the thing to be discussed concerning its mode of being acquired—virtue—is sufficiently known in itself that its mode of acquisition may be inquired about at all. Meno, a student of the Sophist Gorgias, seems more interested in rhetorical gamesmanship than actual inquiry, but he attempts to placate Socrates with a series of accounts of “virtue”: “for each of the affairs and stages of life, and in relation to each particular function, there is a virtue for each of us” (72a); “to be able to rule people” (73c); “to desire beautiful things and have the power to get them” (77b). In the course of demonstrating what is insufficient in each of these different accounts, Socrates insists on properly defining virtue in the sense of discovering what is one in essence among the many concrete cases or instances of virtue by selecting the relevant features common to all such cases, a discussion that significantly uses

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“shape” (schema) as the analogous example proposed by Socrates to clarify the problem. No clear definition of virtue is forthcoming, however, and a typical Socratic aporia results. It appears the conversation up to this point has gone nowhere: “So it seems to me that you need to start again with the same question, my friend” (79c). Meno accuses Socrates of “using sorcery” on him so that he is “full of puzzles” and made numb as though by a “stingray” (80a), to which Socrates responds that he himself makes no claim to knowing what virtue is, entreating Meno, “I want to consider the matter with you and join you in inquiring about what it is” (80d). To this, Meno retorts that such inquiry is, in general, impossible: “For what sort of thing, from among the ones you do not know, will you take as the object of your inquiry? And even if you do happen to bump right into it, how are you going to know that it is the thing you did not know?” (80d). The second Sect. (81a–86b) begins with Socrates answering Meno’s sophistical objection to the very possibility of philosophical inquiry by introducing the theme of the immortality of the soul, recounting the claim made by “those priests and priestesses who have made it their concern to be able to give an account of their practices” (81a) that the soul dies only to be born again and never ceases to be. The bulk of this section of the dialogue then consists of what we will call the diagrammatic interlude in which Socrates solicits the help of Meno’s slave to illustrate the truth of the claim that the soul is immortal, or more precisely to illustrate one important consequence of this claim, namely the consequence that knowledge is already implicit in the soul because of its wide experiences in previous lives and that therefore knowledge need not be externally “taught” but only internally “recollected,” albeit perhaps with external facilitation. The full account of this diagrammatic interlude will be the focus of the detailed analysis in the following section. The third and final part of the dialogue (86c–100c) returns to the question of the teaching and learning of virtue. Having demonstrated by the slave’s finally successful geometrical inquiry that the process of gaining knowledge is actually that of internal recollection, not external teaching, Socrates is drawn back to the previous discussion by Meno seemingly as if the diagrammatic interlude had never happened. Socrates is happy nonetheless to proceed in the inquiry so long as Meno will “agree to investigate from a hypothesis whether [virtue] is acquired by teaching or whatever” (86e), that is, by inquiring into what would follow consequentially from the branching possibilities of virtue being a kind of knowledge or not. Knowledge is shown to be what makes a thing either beneficial or harmful in any particular situation, and the conclusion is thought to follow that virtue must therefore be a kind of knowledge and thus teachable. But Socrates suggests that if virtue is taught then there must be teachers of virtue, and the topic of discussion thus shifts to the status of the nouveaux savants of Athens, the Sophists. Here, Socrates’ later legal accuser Anytus enters the fray, dismissing the Sophists and arguing for what seems to be a kind of xenophobic communitarianism (92e). After Anytus’ angry departure, the dialogue concludes enigmatically with the thesis that since merely true belief and not knowledge is the actual basis of the correct political guidance of the city, and since virtue primarily concerns such guidance, virtue itself must be thoroughly contingent and only “comes to be present by divine dispensation” (99e). But this conclusion itself remains somewhat dubious,

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since it remains dependent upon whether, in Socrates’ words, “we have inquired and spoken correctly throughout this entire discussion” (99e). The present reading takes the diagrammatic interlude as its interpretive fixed point, but it must be admitted that the overall argumentative and dramatic roles of the interlude in the dialogue are fuzzy at best. From a purely rhetorical and logical point of view, the diagrammatic interlude is intended to verify the hypothesis of the immortality of the soul. This hypothesis itself is introduced, it seems, only to answer Meno’s sophistical argument that inquiry is impossible. And surely, from the very beginning the hypothesis has a certain mystical dubiousness about it. It is proposed by Socrates initially on the basis of the authority of “priests and priestesses” and is supported first off by a fragment of a poem about Persephone, the unquiet queen of Hades (81b-c). Although indeed the content (the inner form) of the subsequent geometrical demonstration with Meno’s slave conveys a certain spirit of mathematical rigor, the ostensive argumentative force of the episode depends only on its sheer existence (its outer form): It is proposed to be no more than a concrete instance of knowledge secured without teaching. The fact of knowledge obtained in this way is supposed to support the claim of the soul’s immortality since the new content of knowledge, not donated or imported from without, can only have already been present within the soul in some latent state, and this latency (the logic goes) can only have been the result of experience in some previous life. At best, there are some glaring gaps in this argument that need to be filled. In the last analysis, the geometrical demonstration does not seem especially persuasive from either a logical or a rhetorical standpoint as evidence for metaphysical claims about the soul and reincarnation. Indeed, it appears that by the end of the diagrammatic interlude its argumentative force has been, in a sense, inverted. Whereas the notion of the soul’s immortality was first introduced to make plausible the conception of knowledge as recollection (if the soul is immortal, then knowledge can be recollection, and inquiry in general is therefore possible), once Socrates turns back to Meno to continue the conversation about virtue he summarizes the lesson learned to be, rather, that “if the truth about the things that are is always in our soul, the soul is immortal” (86b). From this latter implication follows the practical injunction “So you should confidently try to inquire about and to recollect what you do not happen to know at present—that is, what you do not remember” (86b). Now, it seems, since the soul is immortal, inquiry is not only possible but obligatory! If its argumentative status is dubious at best, the dramatic function of the interlude is perhaps more clear. The question of how virtue is acquired which opens the dialogue remains quite narrow in focus and seems primarily proposed by Meno as an exercise in sophistical discourse. Although in principle it touches upon metaphysical matters, it is addressed in rather pedestrian terms. When Socrates introduces the theme of the soul’s immortality, however, the scope and philosophical resonance of the dialogue suddenly deepens. All at once, it seems, the discourse moves within a greatly expanded space of endlessly recirculating souls and a timescale of indefinitely many lifetimes. It is as though the dramatic staging of the dialogue has expanded from the local human scale of interacting speakers to a cosmic perspective enveloping uncountable eons. And yet to justify the content of the astonishing claim

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presented at this cosmic scale the dialogue itself moves in the opposite direction, for the first time becoming entirely concrete. To demonstrate the immortality of the soul Socrates turns to a human being of negligible social status, a slave, as though the very lack of social standing, the slave’s minimal humanity, would illuminate the point most clearly. And in relation to the slave, Socrates does not directly engage the theme of the immortality of the soul, but instead mediates the argument through a specific elementary problem in geometry which is itself concretized in the drawing of a diagram in the dirt. It is crucial in this regard to remain attentive throughout the analysis of the dialogue’s dramatic form to the immediate relational framework of social and political givens that determines (but in what ways and to what degrees in each case?) the roles of the participants. On the one hand, for instance, the slaveboy’s subordinate status is essential to the argumentative purpose made of him by Socrates. It is the very fact that an individual of such status is capable of generating or recognizing geometrical truth from within himself that is marshalled as evidence for the theory of the immortal soul and the soul’s knowledge sketched out by Socrates. Yet this very usage of the slaveboy’s reasoning as relevant only as a function of his unequal status works partially to undermine that status, since the capacity for geometrical truth establishes at the level of such truth a certain at least psychological (and under the assumption of the immortality thesis, psycho-ontological) equality between the slaveboy and the primary interlocutors of the dialogue, Socrates and Meno. On the other hand, the relation of epistemological independence established by the dialogical and diagrammatic interactions between Socrates and the slaveboy is itself thematized within the explicitly “friendly” dialogue between Socrates and Meno, who remain social equals at least to the extent that they pursue truth dialectically together. The demonstrative use of the diagrammatic interlude and its attendant social relations is thus itself expressed as a social relation, one that represents certain features of the interactions between Socrates and the slaveboy to underwrite a philosophical argument among friends and equals. We will return to the themes of friendship and philosophy’s partial overcoming of the gap between concrete and ideal community after examining the diagrammatic interlude in detail.

8.1.2 The Diagrammatic Interlude The original written text of the Meno, so far as we know, did not include drawings of diagrams, although commentators have for centuries contributed their interpretations of the diagrams at stake.3 Without authoritative direct evidence, that is, concrete instantiation, the geometrical structure of the diagrams drawn by Meno’s slave must be inferred from the verbal directions given by Socrates and from the questions and answers present in the discussion. There is something of an interpretive irony here 3

SEE, for instance, the diagrams inscribed in margins of the tenth century CE manuscript reproduced in [4: 81].

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in that the reader’s primary evidence for reconstructing the diagrams at issue are indexical linguistic expressions inside a fiction that strictly speaking do not refer to anything at all. They are empty expressions: “this square here”; “this length of this line”. What is at stake is thus the gesture of indication that presupposes a context of common reference. The reader is led to construct (or reconstruct) the diagrams at work in the dialogue solely on the basis of the indexical expressions used not to describe them but only to indicate their relevant parts and relations within the dialogue’s social-discursive context.4 For present purposes, we will take for granted the verbal details from which the actual diagrammatic constructions must be inferred. The diagrams presented below are thus only drawn so as to suggest reasonable interpretations of the stages through which Socrates guides Meno’s slave from ignorance to knowledge. It is important to note that the dialogue itself suggests something more like a continuously deformed diagram, one that is partly erased and rebuilt, rather than a discrete sequence of three (or four, or five) separate diagrams. Nevertheless, it is convenient to treat the diagrammatic process represented in the dialogue as though it consisted of a sequence of static, finished diagrams. It should be kept in mind throughout that the diagrammatic interlude is meant to serve an illustrative (and thereby rhetorical) purpose: The aim of the interchange between Socrates and Meno’s slave is primarily a second-order relation between Socrates and Meno. Socrates intends to demonstrate the truth of the thesis that knowledge is recollection, and his demonstration is to consist of a concrete instance of recollective knowing (that of Meno’s slave). For simplicity’s sake, we designate the practical problem posed to the slave of how to construct a square twice the area of a given square with the shorthand notation: s = S/2. Given a square (represented by lower-case “s”), the problem is to construct another square (upper-case “S”) such that the area of S is twice that of s.5 Three different diagrams are discernable in the discussion, each corresponding to a different hypothesis for solving the s = S/2 problem. Each of the three diagrams is constructed on the basis of what we may take to be a fixed original square, which we may assume remains indentifiable throughout the various reconstructions. This square plays the generic diagrammatic role of “any square whatsoever” since it is treated throughout solely in terms of its properties that it shares immediately with any other square. Thus it is legitimate to count its sides (since all squares share this property), but it would be illegitimate to ask if one of its sides is shorter or longer than Socrates’s arm (since this property depends on the contingent circumstances of its construction). The relevant properties are thus the intrinsic determinations of the square that do not directly depend upon any relations to other aspects or elements 4

AMONG the many studies of links between Platonic philosophy and Greek mathematical practice, noteworthy here is the work of [6] who has amply demonstrated the essentially indexical (and not symbolic) nature of Greek letters as employed in the Euclidean geometrical tradition, and [12] who has in a different way shown how the combinatorial concept of stoicheia (meaning both alphabetic letter and general element) unifies different models of education at work in Plato’s dialogues. 5 It is hoped that the reader will forgive the convenient notational confusion here of the name of each square with its area.

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Fig. 8.1 Doubling the length of the sides (Areas 1:4)

of the world in which the square appears, only those internal to the square itself or constructible solely on its basis. The first diagram corresponds to the hypothesis construct S by doubling the length of the side of s and at Socrates’ prompting is effectively constructed in two steps (82b– 83b), resulting in the diagram shown in Fig. 8.1. The first step subdivides the initially given square into four equal parts, each being a square having one fourth the original square’s area. This fact is available to direct visual inspection and is verifiable by counting. The second step performs essentially the same operation of quartering but in a dual manner, scaling up rather than down. Instead of subdividing the square into four parts, it quadruples the initial square (generating a new square S with side lengths twice that of s). This second operation reproduces the same diagrammatic shape of a fourfold “square of squares” but now at an external rather than internal level of organization relative to the initial square.6 At this new level of organization, the original square is now a part rather than the whole, and it subdivides the new larger square. The slave had proceeded under the hypothesis that the construction of this larger square would solve the s = S/2 problem, his reasoning being that a square with sides twice as long should have twice the area. But the diagrammatic construction shows not only that but also how such reasoning fails. The even mildly sophisticated mathematical eye will immediately recognize the wrongness of the slave’s first hypothesis, but it takes a greater sophistication—one involving a kind of empathetic conjectural ignorance and creative imagination— both to understand the naturality of its wrongness and to find an appropriate method for making this wrongness demonstrable. To do so, one must be more concerned 6

The iteration of one and the same scaling/subdivision operation both internally and externally calls to mind the geometrical structure of the divided line from Republic. In the divided line, the operation is applied twice internally (“Represent them, then, by a line divided into two unequal sections. Then divide each section … in the same proportion as the line” (509d)) but the resulting diagrammatic structures displaying geometrical ratios (a:b:: b:c) are analogous.

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with seeking “terms which the answerer agrees he knows” than with finding “the truth alone” (75d). The mistake of Meno’s slave is essentially that of making a bad inference from one type of part/whole relation to another. The implicit principle governing the hypothesis is that differences of side lengths (one part of the square) will correspond directly to differences of areas (another part of the square) as a square (the whole) varies in size. That this differential correspondence fails is made evident when the double fourfold diagram subtly suggested by Socrates is actually drawn. Constructing the larger square necessarily entails constructing all of its component parts, including its internal quarter squares, each of which, once constructed, is visibly equivalent to the original square (with its own component internal quarter squares). The resulting area is thus shown to be four times that of the original when the side length is doubled. The wrongness of the hypothesis is made visible to the slave by his own diagrammatic elaboration of its basic principle. The second diagram, a revision of and experimentation with the first, aims to correct the first’s deficiency. It corresponds to the new hypothesis construct S by extending the length of the side of s by the ratio 2:3. It is clear that the operation of doubling the side length of the original square generated a square with too large an area. Equally clear is that changing nothing of the original square would be insufficient to double its area, a square’s area being merely equal to itself. Between two and one, then, it seems reasonable to try the compromise of one-and-one-half, to split the difference so to speak. In other words, since the length of the sides of the original square has now been reconstructed by the quartering operation in the first diagram to be (visibly) two units, the slave proposes to construct a square of three of these units on each side, multiplying the side-length of the original square by a factor of 3/2, as shown in Fig. 8.2. Because of the relations made visible by the operations used to construct the first diagram, the simple work of counting shows that this hypothesis fails as well. The larger square in this case may be seen to be composed out of nine, not eight, of the smaller unit squares. Fig. 8.2 Sides in a ratio of 2:3 (Areas 4:9)

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At this point, Socrates steps momentarily out of the frame of the interlude to reflect on its meaning. He redirects his attention and his words to Meno and points out that although the s = S/2 problem has not been solved, epistemic progress has nonetheless been made (84a). Meno’s slave has become “puzzled,” just as Meno himself had been puzzled in the first section of the dialogue when questioned by Socrates about the essence of virtue. By way of his exchange with Meno’s slave, Socrates has now demonstrated to Meno the utility of the numbing power of the “stingray”. To be made numb with a sting is to be made aware of one’s numbness, conscious of one’s lack of feeling. This form of knowledge as self-awareness represents the kind of meta-epistemic state that may be present both in the presence and the absence of substantive knowledge of whatever external domain. It is a self-reflexive knowing of one’s own actual epistemic state with respect to that domain. Because of the concrete mediation of the diagrams he has drawn and observed, Meno’s slave now knows that he does not actually know, which indeed marks a kind of epistemic advancement—but he also has no idea how to proceed further. The way of inquiry appears blocked. Breaking through the aporia at this point requires more than merely retrying the same kind of strategy that failed in each of the previous constructions. What is required is a new type of diagrammatic construction corresponding to a different strategic hypothesis: Construct S on the diagonal of s. Instead of inquiring on the basis of the question What ratio of the side of the given square should the larger square have? the already drawn diagram itself suggests new possible interrogations, in particular Where in the diagram can I draw a new kind of determinate line? Having selected any corner of any square, three corners remain distinct from it, only two of which are joined to the fixed corner by one of the square’s sides. The third, opposite, corner may be joined to the fixed one by a new kind of line with a different kind of length (one not generated as a whole number ratio of the side of the square). Having constructed this line, which as Socrates will explain is called “diagonal” (85b) by the geometrical specialists, the solution to the s = S/2 problem quickly becomes evident. The previous diagrams in Figs. 8.1 and 8.2 with their wrong answers now serve a positive, constructive purpose, as the quadrupling of the original square now forms the basis for reproducing three additional diagonal lines, each of which divides its determining square in half, as in Fig. 8.3. A new sort of counting by Meno’s slave then verifies the correctness of the resulting construction: Two halves of two squares is equal to one of the original square, and this new figure of a triangle equal in area to the original square may be found reflected symmetrically in the larger figure and thus counted twice. The new square constructed by the four diagonals is thus twice the area of the original square, as desired. Implicitly, the slave has learned that whole number ratios of the areas of figures need not correspond to whole number ratios of their sides. Side-length-ratios and figure-area-ratios form distinct (although not independent) systems of relation, and the right diagrammatic gesture of construction—which constitutes in a sense a basis for the community of the figures themselves, the generation of one figural type from another—provides the means for translating between them. From out of his own reflective understanding of what he draws (although only as instigated by following the gentle guidance of Socrates’ suggestions as to what he

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Fig. 8.3 Construction along the diagonal (Areas 2:4)

might consider drawing), the slave has come to possess the practical knowledge of how to construct a square twice the area of a given square. The logic of the diagrammatic discovery of Meno’s slave amounts to successive stages of (A) mastering certain constructive operations on a given square and (B) understanding by way of visual inspection the ratios of the areas of the new shapes generated by those operations to the original square. What has come to light by following the slave’s rather clumsy, though ultimately effective, reasoning in such detail? The correct answer to the s = S/2 problem (if one is concerned with “the truth alone”) is obvious once one has seen the solution. How one may facilitate the discovery and understanding of this solution (if one cares to proceed “in terms which the answerer agrees he knows”) is, however, less obvious and depends not only on examining features of the relevant diagrams but also attending to the socially conditioned roles and cognitive aptitudes and habits of those who make use of them. One must reason not only correctly in the latter case, but also in the spirit of friendship. On the one hand, treating the details of the slave’s diagrammatic hypotheses, operations and observations in the episode demonstrates how multiple steps of implicit reasoning can be compactly represented in a unified way with diagrammatic tools and also how varied the abstract notions involved can actually be once they are made explicit. On the other hand, from the welter of these details emerges more discernibly the high-level principles that govern the process of diagrammatic experimentation and learning as a whole. Overall, the analysis shows that there are two primary axes or dimensions of the diagrammatic reasoning process. The first is practical: Learning how to make certain geometrical constructions. The second is theoretical: Discovering (that is, seeing) properties of the figures resulting from the constructions. Both of these dimensions are organized by an intrinsic tendency toward generalization, a development from the concrete particular case to the abstract general type. Meno’s slave constructs this diagonal here and thereby learns how to construct a diagonal in general under relevantly similar conditions. He sees that this diagonal here divides

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the area of this square here in half and thereby understands that any such diagonal will similarly divide any such square. In this way, the reasoning process itself of Meno’s slave serves as a concrete case that demonstrates (both for Meno who observes it directly as well as for Plato’s readers who reflect distantly upon it) how the interplay of practical and theoretical considerations in diagrammatic reasoning in general enables inferences to general truths on the basis of concrete instances. The next and final section aims to clarify in terms of C.S. Peirce’s theory of signs the structure of relations that underwrites this sort of reasoning and to articulate how such a structure can be seen to recur at multiple levels of the text and used self-reflexively to characterize Socratic philosophical practice in general.

8.2 Diagrammatic Semiotics of Philosophical Friendship The semiotic theory of Peirce considers signs in general as triadic relations determining the structural roles of three terms: Representamen, object and interpretant.7 Formally, but perhaps somewhat confusingly, Peirce explains: A Representamen is the First Correlate of a triadic relation, the Second Correlate being termed its Object, and the possible Third Correlate being termed its Interpretant, by which triadic relation the possible Interpretant is determined to be the First Correlate of the same triadic relation to the same Object, and for some possible Interpretant (1998: 290).

Elsewhere, making use of the term “sign” doubly to designate both this triadic relation as a whole as well as to pick out its structurally mediating part called the representamen above, Peirce states, “I will say that a sign is anything, of whatsoever mode of being, which mediates between an object and an interpretant; since it is both determined by the object relatively to the interpretant, and determines the interpretant in reference to the object, in such wise as to cause the interpretant to be determined by the object through the mediation of this ‘sign’” [8: 410]. Or again, in an admirably simple formulation from the third of his 1898 Cambridge lectures, “a sign is a thing which represents a second thing to a third thing, the interpreting thought” [7: 146]. In Peirce’s semiotics, every sign involves these three components, and the signifying process itself can only be fully understood on the basis of the irreducibly triadic relation that binds them into a single coherent structure. The different types of signs classified by Peirce in various settings derive from the ways each term in the triad is characterized in itself and how each is related to the others. In particular, a diagrammatic sign consists of a complex interplay between indexically determined elements of the representamen and iconic relations holding between the representamen and the object. Precisely which elements and relations of these types determine the diagram must be specified (whether explicitly or implicitly) by the community of inquiry who make use of the diagram, even if such a community consists of no more 7

GENERAL studies of Peirce’s semiotics may be found, among others, in [10] and [1], the latter being especially notable for locating Peirce’s theory in historical context. Particular analyses of Peirce’s semiotics as applied to diagrams is available in [13].

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than a single mind. This act of specification itself is a kind of rule-based agreement or convention, and thus in Peirce’s terms a type of symbol. In this way, a diagram internally synthesizes iconic, indexical and symbolic aspects in a single semiotic relation. Thus, diagrammatic signs are simply general types of relational structures with substructures conforming to certain determinate roles, and the constituent parts of a diagrammatic sign are determined only up to the indexical, iconic and symbolic roles that those parts in fact play. What determines the diagram as such is a triadic relation between (1) a concrete system of relations specified and exhibited directly in some medium; (2) a domain of intended interpretation characterized by systems of relation structurally akin to the relations exhibited in (1); and (3) a context of interpretation in which the mapping of relational systems (1) and (2) takes on significance for some practical or theoretical concern. These three components of the diagram are what correspond to the Peircean triad of representamen, object and interpretant in the typical case of the diagrammatic sign.8 In order to apply the Peircean schema to the diagrammatic interlude of Plato’s Meno and its multi-scale role in the dialogue as a whole, it will be convenient to augment the triadic sign-structure with a fourth term, which will be called here the facilitator. Like the terms representamen, object and interpretant in Peirce’s general semiotics, this new term will be characterized only up to its structural role in the new four-term relation that results from its adjunction to the triadic sign. A facilitator, which is in general not essential to any sign but may be present and effective for a given sign under contingent circumstances, is defined as that which brings the interpretant of a sign into definite relation with the representamen, bringing certain features of the representamen to light so as to encourage the strength and clarity of the interpretant’s relation to the relation of the representamen to its object.9 The facilitator thus stands in a relation to the relation of interpretant to representamen that is analogous to the relation of the interpretant to the relation of the representamen to its object within the triadic sign, although the former remains relatively external to and merely instrumental for the semiotic process whereas the latter is internal to and indeed constitutive of it. The role of the facilitator is thus close in spirit to the well-known maieutic philosophical role that Socrates ascribes to himself.10 The diagram in Fig. 8.4 represents the Peircean triadic sign as a triangle of three arrows. This triangle should be understood on analogy with mathematically commutative diagrams in the sense of category theory, namely, the relation of interpretant to object can be thought of as the composition of the relations of interpretant to

8

A more formal treatment of diagrammatic semiotics using the tools of category theory may be found in [3]. 9 This may not after all be as much of an innovation to Peirce’s conception of semiotics as it might otherwise appear. Peirce himself frequently emphasizes that the interpretant should be understood as being itself a sign for a subsequent interpretant. The facilitator simply puts this self-scaling relation of semiotic referral prior rather than subsequent to the accomplished semiotic relation, and by the same token makes its role that of an external catalyst rather than an internal iteration or elaboration. 10 The word maieutic derives from the Greek maieutikos (pertaining to midwifery). See Plato, Theaetetus (149a-c).

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Fig. 8.4 Augmenting Peirce’s semiotic triad

representamen and representamen to object.11 What is important is to conceive of the synthetic unity of this compositional triad. As Peirce repeatedly points out in various contexts, the triadic sign relation is not a mere sum of two or three relations, but the fabric of their internal coherence as a single unified structure. To this triangle or triad of arrows has been added the term of facilitator with appropriate relationarrows linking it to the interpretant and the representamen respectively. By way of the relations signified by these arrows, which are not independent but synthetic, the facilitator brings the interpretant into relation with definite features and aspects of the representamen that make the latter’s relation to its object more salient for the interpretant. By mediating between interpretant and representamen, the facilitator thus facilitates the triadic sign-relation as a whole. The triadic structure of the diagrammatic sign as augmented by a facilitator may be discerned at three succesive levels of Plato’s Meno. Figure 8.5 pictures the hierarchy of diagrammatic sign-structures at these three nested levels with each semiotic triad together with its specific facilitator marked off by a rounded rectangular border. What should be noted is how an entire sign-structure together with its facilitator serves at each of the successive second and third levels as a complex representamen for a higher-order diagrammatic sign. Thus, the entire diagrammatic interlude with the slave (represented by the system of terms and arrows in the innermost box) functions as the representamen for Meno (the interpretant) at the next level up. Similarly, that complex sign-relation as facilitated by Socrates serves as the representamen for the reader of Plato’s dialogue, with Plato inserted here as facilitating author. In each of the three sign-relations the content of the object-term remains indeterminate and is represented only by a black dot. This is meant to suggest that there is no simple and straightforward answer to the questions “What does the slave learn from the sequence of geometrical diagrams?”, “What does Meno learn from the diagrammatic interlude?”, “What does the reader learn from the dialogue as a whole?” As is appropriate to the core intentions of Socratic discourse, in each case the object of the diagrammatic sign would seem to be best characterized as a sort of open cone of ever-widening significance and unfinished philosophical self-understanding. What is being shown at each level is something that extends in a continuous manner into, across and beyond the higher levels that include it. In the last analysis, the object of 11

A familiar type of such relational composition in mathematics is the composition of functions: for instance, the functions f: A → B and g: B → C compose to form gf: A → C.

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Fig. 8.5 Three nested diagrammatic signs

all the levels of diagrammatic semiotics in the dialogue could be understood to be one and same: The practice of Socratic philosophy itself. The diagrams constructed and reconstructed by Meno’s slave at the center of the dialogue serve to hold these three levels of diagrammatic relation together and to align them thematically.12 Ultimately, the point of the diagrammatic demonstration in the Meno is threefold: (1) by way of diagrammatic reasoning, it leads Meno’s slave to discover a method of geometrical construction; (2) by way of reflection upon the method of diagrammatic reasoning, it teaches Meno a surprising feature of knowledge as grounded in the metaphysics of the soul; and (3) by way of recognition of what they share with the community of inquiry consisting of Socrates, Meno and Meno’s slave, it teaches the readers of the Meno how to understand their own participation in the locus communis of philosophy as pursued “in a more dialectical manner” under the sign of friendship. Meno’s unnamed slave learns as he draws, works with and observes the sequence of diagrams, and he thereby teaches both Meno (within the dialogue) and Plato’s readers (external to it) how to observe and also to reconstruct what learning, teaching and in the last instance philosophy really are. In conclusion, we may summarize three primary characteristics of the gesture of philosophical friendship in Meno that arise out of the structure and usage of the diagrams at its heart and by way of the diagrammatic semiotics that organize it at all levels. Each of these characteristics of the gesture of friendship takes a general object = X in some determinate respect and at the same time takes itself or gives itself as that object in that respect, thus as philosophical friendship in that modality of object. Thus, the general syntax governing all three characteristics is that of philosophical friendship operating actively in some mode both generally (vis-a-vis an object in general) and self-reflexively (vis-a-vis the idea or essence of friendship itself). 12

It is noteworthy that the diagrammatic interlude appears more or less in the middle of the text of the Meno. Similarly, the diagrammatic interlude of the Republic consisting of the allegory of the cave and the divided line appears more or less at the center of that dialogue. In general, the Meno mirrors the Republic in miniature both thematically and structurally.

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First, friendship facilitates (its) discovery. The role of facilitator is partly internalized to the overall semiotic structure of the diagram. The representamen becomes a diagram once its relevant structural features have been selected and its irrelevant contingencies disregarded. This conventional or, in Peirce’s terms, symbolic aspect of the diagram represents a twofold agreement: It is first of all an agreement among the community of diagram-using minds (which may, as Peirce often insists, be the community of a single individual with instances of mind distributed across different intervals of time) as to what aspects of the available system of concrete relations are to be treated diagrammatically; and it is also an agreement (one based on and inseparable from the previous agreement) between that chosen subsystem of relations and some target domain, the semiotic object of the diagram. It is on the basis of this double agreement that new information about the object may be discovered by investigating and experimenting with the representamen. The Socratic practice of philosophy gathers a community of friendly inquiry that takes whatever social-discursive materials it has available at hand as an opportunity to discover and understand general truths on the basis of the relations instantiated diagrammatically in those materials. In particular, the Socratic community can take itself as such a concrete system of relations that instantiates the form of friendly inquiry in a way that may become the source of discovery and understanding of general truths about this form of philosophical inquiry itself. In this way, such a philosophical community takes itself as a diagram for learning constructively what philosophical friendship is and can become. Second, friendship demonstrates (its) relations. The explication of any diagram makes essential use of indexical and ostensive gestures. One points to concrete elements of the diagrammatic structure and thereby shows, that is, makes directly evident to oneself and others, the relevant relations obtaining among those elements. The Socratic practice of philosophy necessarily incorporates such demonstrative gestures, since it is this effort to make truth visible with and for one’s interlocutors that grounds the gentle and friendly mode of discourse that Socrates distinguishes from the merely epistemic concern with “truth alone.” When lifted to the level of the philosophically inquiring community, the demonstrative character of diagrammatic reasoning becomes a self-reflexive gesture that continually invites the critical and corrective input of indeterminately many other potential participants in what thereby becomes a generic and diagrammatic philosophical community. This Socratic philosophical community (at once actual and virtual) shows itself in and to the ambient social-discursive relations of the present polis by way of concrete performances of the art of friendly inquiry that aim to reconstruct those relations as philosophical. The very form of philosophical inquiry in such a diagrammatic mode implies the potential extension of its gestures of gentleness, friendship and diagrammatic discovery to any new, even thoroughly untrained, participants who would be willing to inquire in the same way. Finally, friendship dissolves (its) closure. No diagram is ever complete. Every diagram expresses a limited whole consisting of determinately selected relations relative to some intended domain of interpretation, and both the system of selected concrete relations determining the diagrammatic representamen and the system of

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relations epistemically targeted in the corresponding object domain may always in principle and in fact be embedded in larger, more encompassing systems of relations. There is always more to be shown and discovered. It is impossible to exhaustively elaborate any diagram in concrete terms because, at a minimum, its constitutive relations always imply their meta-relations and thus produce a potentially infinite hierarchy that exceeds any final closure within definite limits. This is true even for a fixed finite collection of initially given relations, no matter how simple, since even the basic relation of self-identity is indefinitely iterable (the identity relation I(X) of X to X is itself identical to itself via the higher-order identity-relation I(I(X)), and so on).13 The diagrammatically oriented philosophical community commits itself to the unending inquiry into the systems of relations in which it finds itself, the infinite horizon of “what you do not happen to know at present” (Meno 86b). And when taking itself under its own purview, the philosophical community by its very nature necessarily remains open to its unlimited further extension, its infinite diagrammatic potential for ongoing elaboration of and participation in the form of friendly inquiry it itself is. It should be emphasized that these expressions of friendship carry no moralistic intent. They are meant only to represent the formal features of the multi-level relations that evolve naturally out of the concrete usage of geometrical diagrams at the center of the Meno and that provide a possible image of philosophical inquiry in general. Whether the practice of philosophy this image suggests is a good one or not remains an open question. But if it is broadly correct as an account both of how such inquiry proceeds and how it represents itself to itself, the coordination of the structure of diagrammatic reasoning and the Platonic/Socratic image of philosophy may perhaps hold some explanatory power with respect to the actual historical development and reception of Plato’s ideas.14 Under the slogan every diagram is a diagram of diagrammaticity we might capture the self-participating, multi-scale formal nature of diagrammatic reasoning and its role in establishing as well as exemplifying the relation of friendship (philia) in the Socratic image of philosophy. The practice of such a philosophy can only emerge from within a discursive arena that suspends any complete polarization by the binary values of self and other, and remains attuned instead to the essential role and enduring presence of a level of “pre-individual reality” grounding the continuous variabilities of what has been called “transductive order” [11: 357, 359]. What is unique about diagrammatic reasoning is not just that it depends upon and exploits indexical elements of the environment in order to fix parameters of some domain of inquiry. In 13

The recent development of homotopy type theory as a powerful and elegant framework for mathematical foundations exploits this fact in a pluralist and differential mode, which is captured nicely by the formal concept of ∞-groupoids [14] (Univalent Foundations Program 2013). 14 For instance, it might help to account for the successful historical integration (here and there, now and then) of Platonic philosophy with broader socio-cultural traditions like Christianity and Islam that combine the principle of an open universalistic form of community with the centrality of a concrete, diagrammatically interpreted event or text. Perhaps the coherence of the Peircean ideal of a community of scientists interacting experimentally with nature and the Platonic/Socratic image of philosophical practice could be conceived on this basis as well.

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addition, the process of diagrammatic reasoning makes use of background relations that structure the environment within which those indexical markers have been set. Implicit relations are inevitably concomitant to those that determine the diagram explicitly. There is always space for further diagrammatic elaboration. In contrast, then, to the conventional understanding of “world” as a transcendental unity and therefore an inevitably global structure, the concept of the diagram as a small (that is, locally specified) structure embedded in indefinitely many possible larger structures suggests the possibility of conceiving an open horizon for philosophy as grounded in merely local and thereby plural and variable worlds. Philosophy in the Socratic mode would entail friendly participation in the collective exploration of that unlimited variable plurality on the basis of the potential diagrammatic relations to be constructed and thus discovered here in this world.

References 1. Deely, J.: New Beginnings: Early Modern Philosophy and Postmodern Thought, University of Toronto Press (1994) 2. Dufourmantelle, A., Payne, K., Sallé, V. (trans.): Power of Gentleness: Meditations on the Risk of Living, Fordham University Press (2018) 3. Gangle, R.: Diagrammatic Immanence: Category Theory and Philosophy, Edinburgh University Press (2016) 4. Kennedy, D.F.: “Metaphysics and the mathematical diagram: Geometry between history and philosophy”. In: Michelakis, P., (ed.) Classics and Media Theory. Oxford University Press (2020) 5. Klein, J.: A Commentary on Plato’s Meno. University of North Carolina Press (1965) 6. Netz, R.: The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History, Cambridge University Press (1999) 7. Peirce, C.S., Ketner, K.L. (ed.): Reasoning and the Logic of Things: The Cambridge Conferences Lectures of 1898, Harvard University Press (1992) 8. Peirce, C.S. (ed.): Peirce Edition Project, The Essential Peirce: Selected Philosophical Writings, vol. 2 (1893–1914). Indiana University Press (1998) 9. Plato, Reeve, C.D.C. (trans.): A Plato Reader: Eight Essential Dialogues (2012) 10. Short, T.L.: Peirce’s Theory of Signs. Cambridge University Press (2007) 11. Simondon, G., Adkins, T. (trans.): Individuation in Light of Notions of Form and Individuation, University of Minnesota Press (2020) 12. Smith, C.: Dialectical methods and the Stoicheia paradigm in Plato’s trilogy and Philebus. Plato Journal 19, 9–24 (2019) 13. Stjernfelt, F.: Diagrammatology: An Investigation on the Borderlines of Phenomenology, Ontology, and Semiotics, Springer Press (2007) 14. Univalent Foundations Program (multiple authors): Homotopy Type Theory: Univalent Foundations of Mathematics, (2013) Creative Commons, available at homotopytypetheory.org/book/

Chapter 9

The Act of Writing Vilém Flusser and Charles Alunni

9.1 Introductory Note Vilém Flusser was a Brazilian Czech-born philosopher, writer and journalist. He lived for a long period in São Paulo (where he became a Brazilian citizen) and later in France, and his works are written in many different languages. His early work was marked by discussion of the thought of Martin Heidegger, and by the influence of existentialism and phenomenology. Phenomenology would play a major role in the transition to the later phase of his work, in which he turned his attention to the philosophy of communication and of artistic production. He contributed to the dichotomy in history: the period of image worship, and period of text worship, with deviations consequently into idolatry and “textolatry”. His writings reflect his wandering life: although the majority of his work was written in German and Portuguese, he also wrote in English and French, with scarce translation to other languages. More precisely, as a thinker in exile, a migrant, and a polyglot, Flusser wrote and published in four languages, repeatedly translating his own texts, often altering them in the process, and preferring the essay genre. Because Flusser’s writings in different languages are dispersed in the form of books, articles or sections of books, his work as a media philosopher and cultural theorist is only now becoming more widely known. The first book by Flusser to be published in English was Towards a Philosophy of Photography in 1984 by the then new journal European Photography, which was his own translation of the work. The Shape of

Vilém Flusser is a deceased author with a life span (1920–1991). V. Flusser (Deceased) · C. Alunni (B) École Normale Supérieure, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_9

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Things, was published in London in 1999 and was followed by a new translation of Towards a Philosophy of Photography.1

Photograph by Ralph Hinterkeuser

Flusser is among others the author of two fundamental books related to our topic. First of all, Gestures: Attempt toward a Phenomenology2 which incorporates a somewhat different version of the text that we publish here and which is an “original”, written directly in French. Gestures was first published in 1991 in German as Gesten: Versuch einer Phänomenologie.3 Apart from two introductory chapters and a final theoretical consideration of a possible, all-encompassing interdisciplinary theory of gestures, it contains sixteen phenomenological descriptions of different fundamental human gestures (The Gesture of Writing, The Gesture of Speaking, The Gesture of Making, The Gesture of Loving, The Gesture of Photographing, The Gesture of Filming, The Gesture of Turning a Mask Around, The Gesture of Planting, The Gesture of Shaving, The Gesture of Listening to Music, The Gesture of Smoking a Pipe, The Gesture of Telephoning, The Gesture of Video and The Gesture of Searching). For Flusser, Gestures are not just reflexes; they articulate a moment of freedom and are in this sense not fully explicable in causal terms. There are seven different versions of “The Gesture of Writing” in four different languages, 5 making it a good example of the difficulty that accompanies any search for an “original” text, of deciding which of many existing versions was “first,” to say nothing of which should be considered “authentic,” “definitive,” or even “complete.” It is as if we are dealing with seven different originals.

1

Flusser’s archives have been held by the Academy of Media Arts in Cologne and are currently housed at the Berlin University of the Arts. 2 Flusser, V., Gestures, 2014. 3 Flusser, V., 1997.

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Flusser on one of his many travels to West Germany, late 1980s

Here, a central concept for Flusser is that of “linear writing”: When linear writing was invented, it constituted a break with the scenic structure of the universe. Up until this point, images primarily mediated between human and world. For this reason, the world was experienced as a context for scenes in which imagination constituted the ability to orientate oneself in the world: the ability to decode the meaning of images. Linear writing dismantled images into lines (it ‘developed’ surfaces into lines) and the world assumed a processual structure. Conception was added to imagination as the ability to orientate oneself in the world. Conception is the ability to decipher the clear and distinct symbols of texts (letters and numbers). The world was no longer merely imaginable but also understandable. This is ‘historical consciousness’. ‘History’ was invented alongside linear writing. Historical consciousness is the awareness of readers and writers.4

It is worth noting the historico-anthropological consequences that Flusser draws for the near future. He stipulated in Does Writing Have a Future?, “We will have to learn to rethink our entire history. Backwards and forwards”.5 The future reader is not limited to knowledge that has been accumulated in linear time, but he can broaden the appropriation of understanding by learning more flexibly, by creating nets and making connections, by analyzing conjunctures that defy a strictly linear examination. The present situation, analyzes Flusser in “Line and Surface,” published in the volume Writings, “does not look like the result of a linear development from image to concept, but rather like the result of a sort of spiral movement from image through concept to image.”6 A second book titled The Shape of Things: A Philosophy of Design,7 was first published in German in 1993. This collection of essays speaks to Flusser’s interests in art, design, and creativity, the latter serving as a potential leitmotif for his thinking 4

Flusser, V., 2011, p. 199. . Flusser, V., 2011, p. 146. 6 Flusser, V., 2002, p. 31. 7 Flusser, V., The Shape of Things: A Philosophy of Design, Reaktion Books, London, 1999. These essays were taken from Vom Stand der Dinge, Braunschweig, Bollmann Verlag GmbH, 1993, and from Dinge und Undinge, München, Carl Hanser Verlag, 1993. 5

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and writing in general. Flusser was always more interested in poiesis, in making art and realities, than in mimesis. Comparable to Gestures, the philosopher here explores possibilities for different approaches to design, including everyday objects such as the typewriter and tools, proposing to read “Design as Theology.” Finally, another important point for Flusser, a theory of translation. In an interview with Hans Joachim Lenger in Hamburg in 1990, published in Zwiegespräche, Flusser sums up his lifelong interest in the subject: “Perhaps everything I do is an attempt to elaborate a theory of translation. But I am not going to live long enough to do that”8 (ZG, 149). I can reformulate the thought in another language, to grasp my problem from a different side. I can thus make the meaning of the thought “richer” because the problem to be thought about will be illuminated by projectors of different structures (I am convinced that translation is a powerful epistemological instrument). This is why I start translating my German text into, for example, the Portuguese language.9

9.2 Text. English translation by Charles Alunni It is an action by which a material is put on a surface (for example: chalk on a blackboard, or ink on a sheet) to form designs (for example letters). The tools of this action (for example: the pencil or the typewriter) are instruments for adding one material to another. One could therefore suppose that the gesture of writing is a constructive act, if by “con-struction” we mean: adding various materials to form a new structure. In fact, the opposite is the fact. By its “essence” (eidos), the gesture of writing is an act of excavation, of engraving, and the Greek verb “graphein” is always the witness to this. His current technique hides this essence. Some thousands of years ago people began to scratch the surfaces of Mesopotamian bricks, and this is the origin of writing for our tradition. It is a de-structive gesture, an act that removes. To write is to make holes. It is in-writing, and it is not over-writing, although technology now denies it. A written text is not a formation on a surface, but an in-formation in a surface. It is a penetrating gesture, negative, by its origin, and by its intention, although by its technique the gesture is its own opposite. Of course, we are not aware of this fact during the act. We do not think of the act of describing, but of what we write, because writing for us is a habit, and habits are acts that we do without thinking about it. Indeed, writing is more than a habit, it is almost a skill. It seems that there are writing centers in our brains. We are therefore born with the ability to write, as birds are born with the ability to make nests. But this comparison is certainly wrong. Writing cannot be in our genetic program, because it is a cultural gesture that must be learned in order to be able to do it. Writing must be in our cultural program. Nevertheless, we must learn to write in order to become what we are by our nature: men born in the twentieth century. As we have to learn to walk 8

. Flusser, V., 1996, p. 149. Flusser, V., “The Act of Writing” (“Le geste d’écrire”), here p. 52. For a generic philosophical vision of this question, cf. Alunni, C., 2013. Available on https://journals.openedition.org/rsl/293

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and talk. Or maybe this is an exaggeration? Perhaps there is a hierarchy that orders the passage between nature and culture in man, according to which walking is more natural than talking, speaking more natural than writing, and breathing more natural than walking? Or maybe still, wanting to distinguish between nature and culture is an ontological error when it comes to man? In any case, we do not think about the act of writing during the action, because for us it is a “natural” gesture. We just write. But if we think about it, the gesture becomes complex. We need a lot of data to be able to write. From an empty surface (for example a sheet of paper). A tool to put material on this surface (a typewriter, for example). Shapes to put on the surface (e.g. letters of the alphabet). These forms can be stored in our memories (the case of the pencil), or in the memory of the tool (the case of the machine). We need rules that give meaning to shapes (in the case of letters, this meaning is a series of sounds in a spoken language). We need rules that order the shapes on the surface (“orthography”). We need a spoken language which is signified by the forms. We need rules that order this language (the “grammar”). We need to have something to articulate in that language (for example, an “idea”). We need a motive to articulate this thing. And this list of essential data is not complete. And what is even worse: the indispensable data are of varied ontological provenance. The typewriter is not of the same reality as spoken language, and the latter is not of the same reality as grammar. Writing is a complex act, especially because of the complexity of the ontological layers it mobilizes. The structure of the gesture is linear. But this is a specific linearity. We start, in thesis, with the upper left corner of the surface, we make a line until we arrive at the upper right edge, we jump to the left to start the gesture again a little lower, and we repeat this movement to the lower right corner. It is an apparently accidental structure: it has been imposed on the gesture by the accidents of our history. It could be different, and indeed it is in other civilizations. Nevertheless, this structure, which is the result of accidents as despicable as the quality of the mud in Mesopotamia, orders a whole dimension of our being-in-the-world: it orders our linear, logical, historical, scientific thoughts. Because we are programmed for these types of thoughts by our writing, and, conversely, these thoughts are programmed to be written according to the structure that I have just described. The slightest change in this structure would undoubtedly change these types of thoughts. But, of course, the reverse is also true: any structural change in our thoughts implies a change in the structure of writing. Maybe it is happening now. The typewriter is programmed for such a structure. It cannot easily make diagonal or spiral lines on the surface for example. The pen can do that. Apparently we are more “free” with the pen than with the machine. The structure that programs our linear thoughts further determines whether we strike. It is more “material” than in the case of the pencil. But is this “freedom” despising our programs? To be free, shouldn’t we assume the program, to change it later? If I strike diagonal lines (by manipulating the paper, for example), have I not written in a more “revolutionary” way? The machine is a more “creative” instrument than the pen, precisely because it materializes the program of my gesture of writing, and thus makes it more conscious

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and better visible. Concrete poetry (which is a gesture of writing in search of new structures) is proof of this. The pen and the machine are tools that lay material on the surface. But the pen (and pencil and brush) still retains the pointed shape of the original Mesopotamian staff. We see, when we look at a pen, that writing is, by its intention, the act of engraving. In the machine we no longer see it. It does not look like a stick, but like a piano: it is made up of hammers. When we hit a text, we hammer a surface. To “hit” is to hit against. To type is to recapture the Mesopotamian experience of writing. Writing again becomes a penetrating gesture that denies. Only, you deny the surface with your ten fingers, and not with a single point. And that is the gesture of writing: denying the surface in order to penetrate it. You can feel it concretely, when you type. But this gesture which denies the surface in order to penetrate it, is this not the phenomenalization of thought? To think, is it not to want to deny surfaces in order to penetrate them? When you type, you make the gesture of thought. We think concretely when we type. At least, that’s my experience. But we can see this objectively too, when we look at what happens during the act of writing. It is not a continuous movement. It stops periodically. And these stopping phases are as characteristic of the gesture of writing as are the active phases. One cannot observe, of course, what happens during the phases of concentrated suspension, except by introspection. But we can observe that these phases are essential for the gesture, because when they are absent, as in the case of an illiterate child or a chimpanzee who hits a machine, the gesture is not that of writing. We can therefore observe that the phases of concentrated stopping distinguish the gesture of writing from the gestures of children and chimpanzees with a typewriter (and with a pen), because it is the stopping phases that seem to be responsible for the choices letters to type. We observe that when the stopping phases are absent, the letters are typed in a random way, and we can therefore conclude (without being able to observe it from outside), that the stopping phases are the moments of choice. We can distinguish two situations with respect to these stopping phases. In one, the writer stops to look at a text that is placed next to the machine, with a look that is called “the gesture of reading”. It is obvious that he chooses the letters to type according to this pre-existing text. In the other situation, the writer stops to look at the “void”; One could say that he chooses his letters according to an invisible text. Why not say that such a reading of a nonexistent text is “the gesture of thinking”? Why not say that “typing” is, in this case, the gesture that phenomenalizes (makes visible and “realizes”) the gesture of thinking? I propose to eliminate the first situation from the consideration of the gesture of writing. In this case, it is a transcription. The text to be written, in this case, is a copy. The choice of letters to be typed is, in this case, imposed on the typer: he is not “free”. The person who makes this gesture (the “typist”) functions according to a program of others. He is “alienated” by his own act. He does not write, he is someone else’s typewriter. I therefore propose that we consider the gesture of writing as a gesture during which a surface is informed by letters chosen during phases of stopping that we can call “thinking”. Having thus eliminated from the gesture of writing illiterate children, chimpanzees and typists (children and chimpanzees because they make

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random movements, and typists because they make determined gestures), we are left only the writer. According to this proposition, the writer is the only one who writes, because his gesture is neither random nor determined: he is “free” (with all the reservations that are imposed by the dialectic of freedom). The following problem arises: the gesture consists of two phases. During one, the visible phase, letters are tapped against a surface. During the other, invisible phase, the letters are chosen. The first phase can easily be described. But how to describe the second? Only if one describes one’s own praxis. That’s what I propose to do. In order to be able to write, I need to have all the essential data that I have listed, but this list has two horizons, an external and an internal one. These are the only horizons that matter for my decision to write. I need, on the one hand, a typewriter and a piece of paper, and on the other hand, I need something to express. The other essential data (alphabet, spelling, spoken language, etc.) are always available to me. But if I don’t have something to express, my writing tools are useless. And if I don’t have a machine and paper, the thing to be expressed is repressed. This is a trivial observation, of course. But this banality hides the tragedy of the gesture of writing. Because to live, I have to write. “Scribere necesse est, vivere non est”. To be able to do this, I need this coincidence, strictly speaking unlikely, of a machine and a thing to express. Or am I exaggerating this drama? We can ask three questions. (a) Can’t one write without having something to express? (b) Can’t we express the thing by gestures that are not that of writing? And more radically (c) Can’t we live without writing? As for (a), of course we can do it. We do it everywhere. This gesture of writing without expressing anything is responsible for the inflation of texts in which we are immersed. And this inflation renders all true written expression futile. Because how to recognize it in the crowd of texts? This is why “alphabetic communication” is in crisis. The gesture of writing is no longer very useful, precisely because it is easy to write without expressing anything. As for (b), no, it cannot be expressed by other gestures. One cannot speak, or make a film, or paint the thing which urges to be written. Because that thing is a thought programmed to be written down. If you articulate it in a different way, it gets lost. This is why abandoning writing, or even just diminishing the importance of writing in our culture, would lead to the loss of this type of thinking. Because there is a circle there that is not necessarily vicious: writing programs this type of thought, and this thought is realized by and in writing. The circle is not necessarily vicious, for both writing and thought can grow during circulation. This, in my opinion, is the history of Western thought. But maybe I’m wrong. Because we see everywhere the tendency to replace writing with other “means of communication”. I am wrong, perhaps, because of my attitude towards question (c). I cannot live without writing. This necessity is independent of any reasonable consideration. I know that writing is becoming an increasingly futile endeavor. I know that there are incomparably more effective “means of communication”. But sometimes things arise within you that want to be written down. And it is precisely when this happens that I am truly alive. I know that these things have been programmed into me by the accidents of my history. Nevertheless, I am in the world because of

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this program and according to this program. If I repress things that want to be written down, or if I try to express them with other gestures, I am not really in the world. For to be in the world is to realize one’s program by changing it. That is to say, by changing oneself and changing the world. And I can’t change myself and the world without writing. And I suspect there are others who are like me. For them, as for me, the gesture of writing is the decisive gesture, because it gives meaning to life. Therefore, the dramaticity of the gesture of writing is not an exaggeration. To live, I need a machine and something to express. If I don’t have a machine (for lack of time or for other economic and social reasons), my life has no meaning. At the limit, this can lead me to suicide. And if I have nothing to express, I no longer live: I am a living dead. Both things happen to me, and I suffer from the two opposite and complementary sufferings. But sometimes the coincidence of the machine and the thing to be expressed also happens to me. So I am, I realize myself: I write. To write is to express something. “Express” is a relative term. It is to press from somewhere towards something. To write is to express something towards a surface. But not directly. There are, between the thing that is pressing and the surface to be printed, other obstacles. You have to penetrate them, in order to reach the surface. Each obstacle that stands between the thing that is pressing and the surface has its own structure and its own “materiality”: it is “objective” in its own way. It therefore offers its own resistance to pressure towards the surface. It is sometimes said that writing is “clarifying one’s thoughts”. This is an inexact way of speaking. To write is to realize one’s thoughts under the objective resistance of the obstacles that stand between the thing that puts pressure and the surface to be printed. Writing is a progressive objectification of thoughts. Thought, as it is imprinted against the surface, is the result of a whole series of dialectical processes on a whole series of ontological levels. That’s why I hesitate to say that the thing that presses is a thought. It is not until it hits the surface. The thing that exerts pressure is a tendency towards thought which is realized by the gesture of writing. Having something to express (having “ideas”) is not worth much. It is only when one writes that one verifies, under the resistance of various obstacles, whether the thing to be expressed is a thought (It often happens, alas, that one verifies the contrary). Everyone has “ideas” behind their heads. It is only by the gesture of writing that we can see if he is thinking. Because the gesture of writing is the act by which one thinks (If by “thought” one means: ordering elements according to the structure of writing). The obstacles that stand between “virtual thought” and the surface against which it will become “effective” are of various orders: the rules of logic, the grammar of a spoken language, the denotations and connotations of the words of this language, their sound quality, their rhythm, the rules that order the letters of the alphabet, the structure of the typewriter. And there are others. It would be necessary to describe each obstacle, and the resistance it offers to the pressure of virtual thought towards the surface of the paper in the machine, to grasp the gesture of writing. I suspect that such a description would be a “phenomenology of linear thought”. I cannot attempt it here, as it would break the limits imposed on a collection of essays on gesture, as is the case here. I will therefore focus attention on a single obstacle: the passage of thought from a spoken language to the sheet of paper. Here are my

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justifications: (a) it is the last step in the gesture of writing, and it involves, in a way, all the others. (b) it shows the difference between the linearity of a spoken language and writing, and gives proof that linear, scientific, historical thought does not have the structure of spoken language but that of writing. (c) my own praxis of this final step is more complex than usual and can therefore serve as an extreme example for a more “normal” praxis. I am programmed by various spoken languages. This does not imply that I can choose among the languages to express myself. I am not free in the mercantilist sense of the term in relation to languages. Any language stored in my memory has its own structure and its own “materiality”, and therefore its own function in relation to the thought to be expressed. So I am programmed by each language with a different intensity and depth. I am therefore led to express certain thoughts in one of the languages rather than in another. But the various functions of languages mesh. This imposes a specific strategy on me for writing. One important fact must be emphasized before describing this strategy. I don’t speak the thought out loud when I write. The “spoken” language remains silent during the act of writing. The written text will therefore not represent a really spoken language, but a “sotto voce” language, that is to say a language not entirely realized. This fact is important, because I am convinced that the thought would be different if I thought it “aloud”. It would be subject to phonetic, “musical” rules, which are present, of course, in the silent formulation as well, but in a less effective way. This is why I am wary of transcriptions of texts based on tape recorders. Thinking aloud does not have the same structure as written thought. Perhaps there is also a difference in velocity. We think faster when we think low. It is because when we think low, we do not think yet. We prepare the written thought. The linguistic formulation of a thought during the gesture of writing is only the penultimate step in the process of articulating thought. It is the last step during the gesture of speaking. Again: during the gesture of speaking, the whole body is possessed by the spoken language: the face and the hands participate in it. The written text, on the other hand, has become autonomous from the gesticulating body. In short, I do not speak when I write, and written thought does not represent spoken thought. It is much more clearly linear. I therefore silently formulate a virtual thought in one of the languages at my disposal, and it is the thought itself which chooses the language which is appropriate to it (At this point it is difficult to make the distinction between me, the thought and the language, because these are three abstract entities. Concrete reality is the meeting point between these three abstractions). Most of the time this language is German, but often also Portuguese or English. I will say (without wanting to insist on it too much) that my “philosophical” thoughts tend towards German, the “political” thoughts towards Portuguese, and my “scientific” thoughts towards English (My “sentimental” thoughts tend towards Czech, which is my mother tongue, but I have lost the command of this language). The process of this rapid formulation, so rapid that I am in danger of not being able to control it. The thought formulates itself in the tongue and causes a whole chain of other thoughts to form the branches of a tree that threatens to grow without limit. This growth is caused by the “content” implicit in the thought, by the associations of the words of the language, by the

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rules of the language, by the rhythmic and sound quality of the language, by its “unconscious”, and surely also by other factors unknown to me. It would be easy for me to let myself be carried away by this adulterated mixture of thoughts, language and unconscious (by this “Joycian river”), and I know its beauty and its seduction. But this is one of the temptations (perhaps the strongest) that I must resist during my act of writing. Because it seems to me that for me, writing is precisely not allowing this uncontrolled growth. To write, for me, is to try to diachronize the synchronicity of the tree of thought, to resolve its agreements. Develop the thought, instead of letting it flow. Because for me, to write is to make lines. That’s why I feel like “automatic writing” is a contradiction in terms. But this is perhaps a prejudice, and I wouldn’t want to elevate it to a rule for any writing practice. Thus, I make a series of negative choices when I type the thought I have formulated in the language appropriate to it. I eliminate all the “associations” that the thought provokes in the language and in me, to give the thought a single direction: the one towards which it presses by its “central intention”. Of course, this direction is not given by me, but by the “meaning” of the thought (we will talk about it later), by the grammar of the language, and by the linearity of the typewriter. Nevertheless, this direction is taken by thought thanks to my elimination of associations. I allow, by writing it, that thought can develop. I give him the floor, and, what is even more important, the letters. And this is how it becomes an effective thought. The result will be a text, that is to say, an effective thought because developed along the lines of writing. Let’s assume that this text, which I now have in front of me, is in German. The question arises: what does it mean? This question arises, for me, only after typing the text. Before, I naively accept that the thought that I am going to develop signifies a “problem”, that is to say, an obstacle that stands in front of me. I accept it, because I feel that it was the problem (and my need to solve it) that provoked my desire to formulate thought through the gesture of writing. But now, having the German text in front of me, I realize the complexity of the relationship between the thought and the problem signified by it. For the German text does not signify the problem “as it stands before me,” but it signifies the problem “as it appears in the universe of German discourse.” Having formulated the thought in German, I also manipulated the problem to make it thinkable in German. It is not a question, here, of the Kantian discovery: the thought thing assumes the categories of thought. It is a much more radical “discovery”: the thing thought of assumes the categories of the language in which the thought was formulated. I am ready to believe that I cannot go beyond the limitations of “pure reason” to grasp the “thing in itself”. But surely I am not limited by the categories of a single language. Because I can reformulate the thought in another language, to grasp my problem from a different side. I can thus make the meaning of the thought “richer” because the problem to be thought about will be illuminated by projectors of different structures (I am convinced that translation is a powerful epistemological instrument). This is why I start translating my German text into, for example, the Portuguese language. When I start to rephrase the thought in silent Portuguese, I lose control again. But this time it is not only because of the associations caused by this language, but also

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because the thought refuses to be reformulated as I want it to be. It escapes me, and I have to rethink it. That is to say, I have to remember the process by which I formulated the thought in German. By doing this, I remember all the German associations that I eliminated. And as I remember now in Portuguese, these associations mesh with the Portuguese associations. At this point I am the victim of a galloping thought that threatens to escape in all directions. And I also have the sensation of the unpredictable adventure of the act of thinking. This is the moment of joy in the gesture of writing: I circle the problem. I discover unexpected sides to it. The problem becomes multidimensional. My thought is rich in meaning. But unfortunately, I do not control it, and therefore it is not “good”. Because it is a thought that flows in two languages at the same time, and one cannot write in two languages at the same time. Thought is not “good” because it is not effective thought. It will become so only after the act of writing. My joy is premature, because only now does the real work begin. You must type the Portuguese text. And to do that, you have to make negative choices again. The Portuguese text that I type is a translation of the German text, that is, the German text is its reference system. But I don’t translate like a “normal translator”. The German text is not the “meta-text” but the “pre-text” of my Portuguese text. I am not trying to be faithful to the German text, but to go beyond it. Of course, the Portuguese text will be as linear as the German text, because it too will be a written text. In this sense it will not exceed its pre-text. Nevertheless, there will be, so to speak “between the lines” of the Portuguese text, vestiges of the German lines. It will be a kind of palimpsest. The thought will not take quite the same direction as that taken in the German text. But nevertheless, the German leadership will somehow be there. For me, that is the gesture of writing: making palimpsests. This process of translation must be continued if the purpose of the gesture of writing is to imprint a “highly significant” thought on a surface. The Portuguese text must be translated into English, French, etc. But what is even more fascinating is the attempt to translate the text back into the first language, after passing it through different languages. One will find that the second German text (for example) will be radically different from the first, although the thought expressed by the text is the “same.” Which is empirical proof of how much the gesture of writing changes the thought it expresses, as well as the surface on which it imprints it. But here we are faced with a movement which is, theoretically, an infinite regress. Because if the gesture of writing is the movement of translating and retranslating texts (as it is in my praxis in an explicit way, and very probably in the praxis of any writer in an implicit way), then it can go on forever without ever coming to an end. Unfortunately, such an infinite progression of the gesture of writing towards perfection is only theoretical. Because the progression of the gesture exhausts the thought, and, after a critical point, the gesture becomes repetitive. Then, a melancholy discovery imposes itself: if the thought to be written is exhausted by the gesture after a “reasonable” time, it is not very significant, and if it is, it cannot be written in a “reasonable” time. Which imposes a choice on the writer, which is perhaps the most fundamental choice, but also the most difficult to make: to write in order to be read (i.e., in a “reasonable” time), and therefore write relatively insignificant thoughts; or else write his “richest”

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thoughts and run the risk of never producing a thought-exhausting text. As for me, I try to escape the choice in the following way. I am constantly attacked by a considerable number of “virtual” thoughts that are pushing to be written down (which has no intrinsic value, because an unwritten thought is absolutely nothing and worth absolutely nothing). I am unable to judge these “vague ideas” as to the richness of their meanings until I write them down, because they all seem to me to be “great” and capable of solving all problems. I therefore try, desperately, to write them all by the method I have just spoken of (That is why writing is living for me). The vast majority of these “brilliant ideas” are quickly exhausted by writing and show that they are banal ideas. The texts thus written are then thrown away, or they become letters sent to my friends, who are thus the victims of this genius. A small minority of my ideas resist the method of translation and re-translation for a few days, or even weeks. These are the texts that I publish. And there are two or three ideas (for example precisely the idea in relation to the problem of translation) that I cannot manage to exhaust, despite my efforts which have already lasted for years. That’s why I don’t know what a “good writer” is. Is he someone who exhausts his thoughts well? Or can we not say that having thoroughly exhausted one’s thoughts (having expressed them well logically, aesthetically, etc.) is proof that these are relatively poor thoughts? The great writers of the past (who are our models) cannot solve the question. Because, of course, they wrote rich thoughts in an admirable way. But one wonders if it is not better to struggle with thoughts that cannot be written admirably, so rich are they? Isn’t that the reason why Wittgenstein, for example, stopped writing? That is to say, to make texts publishable? Because “to write” is the gesture that fights against a thought with a typewriter, and if the machine is the winner, it is because the thought was weaker. And can we not, also, publish a provisional text (as is the present case) by which thought has not yet been overcome, and in which thought is therefore not yet effective? A “trial” in the true sense of the term? Is it being a “bad writer”? Indeed, this is the way I, in most cases, escape choice: I write essays. Let us return, at this point of the discourse, to its starting point. It is a question here of the effort to grasp what is called the “essence” of the gesture of writing. We have found that it consists of two phases: one is a visible movement, the other a stoppage of movement during which invisible things are happening. During the visible phase a surface is imprinted with letters by a method invented thousands of years ago in Mesopotamia. During the invisible phase, the letters to be printed are chosen according to a language or several languages that are manipulated to express, “sotto voce”, a virtual thought. There is a “feedback” between the two phases. The surface is imprinted according to the thought that has become linguistic, and the thought is structured according to the structure of the writing. Through this “feedback” the thought is realized on the surface in text form, and the surface is informed by the thought in text form. It is a form and information that cannot be realized except by the gesture of writing. This form and this information is “historical and scientific linear thought”. Any written text has this form and contains this information, even if it denies it. And no other human product has this form and contains this information. That is to say, the gesture of writing is the gesture by which this form and this

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information are produced. It is thanks to the gesture of writing that the problems in which we are immersed (the world) have this form and can be solved in this form. So the gesture of writing aims to make the world “readable”; it is the gesture of a specific thought, which is the characteristic thought of our culture. This is not the goal we usually associate with the act of writing. We don’t usually say that we write to make the world readable, but to write a text to be read by others. Is there a complementarity or a contradiction between these two goals? Of course, we can say that there is a sequence between them: we write to make a text to be read by others, so that they can read the world. Or is it the opposite sequence? Do we write a text that makes the world readable, and this text is intended to be read by others? The fact that the sequence of goals is reversible shows that these are two goals that arise at two different existential levels. It is not existentially true that the surface on which one writes is a “means” by which the writer seeks to establish communication with others. Writing is not the gesture by which one wants to pierce a surface in order to touch another. On the contrary, it is the gesture by which one seeks to engrave shapes on a surface without piercing it. The writer does not write towards others, but against a surface, and the structure of his gesture is proof of this. It is therefore obvious that the “committed writer”, that is to say the one whose goal of writing is to change others and not a surface, is not a true writer. His attention is not absorbed by the resistance offered to his gesture by the different surfaces, but part of his attention is directed towards the effect of his gesture. “The committed writer” is therefore an actor (“Dodos”) who performs the gesture of writing. He is always on the stage of public events, while the “real” writer is in the solitude of his opposition to a surface. These are two different existences: the actor is the man who acts, and the real writer is the man who writes. But the fact that the two purposes of writing (to make a surface to imprint a thought on it, and to make a surface to be read by others) are neither complementary nor contradictory, but characterize two different existences, does not justify the clear distinction between these two existences, that of the actor and that of the writer. It is true that if I am an actor I approach the gesture of writing with a different attitude from that of a “real writer”, and this must be recognizable in the text that I will write. So to speak, the text written by an actor “directs” towards the reader (is a “letter”), and the text written by a “real writer” “focuses” on its theme (is a “treaty”). This is true, and perhaps all literature should be divided, according to this existential attitude, into two categories: that of “letters” and that of “treaties”. Nevertheless, the two purposes of writing are implicit in the act of writing. It is a gesture that prints on a surface, by the fact that it scratches on a surface. It is therefore a gesture of “work”. But it is also a gesture that follows a program whose purpose is to make conventional signs. It is therefore a gesture of “communication”. That is to say, the one who writes to communicate (the actor) is obliged to work on a surface and thus becomes, in spite of himself, a writer. And he who writes to impress ideas on a surface (the writer), is obliged to follow a communicative convention, and thus becomes, in spite of himself, an actor. The writer who withdraws to be alone with his thoughts is a romantic myth: the convention that established the letters of the alphabet as means of communication is still with him. It is therefore important to note the difference between writing to

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inform a surface, and writing to inform others, because it is recognizable in the texts, and it is really about two different existential attitudes in relation to the gesture of writing. But it is also important to note how the gesture of writing “privatizes” the political act by concentrating it against a surface, and how it “politicizes” the act of thinking by channeling it into a communicative convention. Of course, one can argue that any gesture of work has a communicative dimension. We make shoes to sell them to others (“publish” them), and we make highways for the circulation of cars. And we can also argue that any communicative gesture has a work dimension. When we speak, we work a language and produce sound waves, and when we point, we change the position of our body. But such an argument diminishes the dramatic tension between concentration on thought and concentration on communication that characterizes the gesture of writing, and, in my view, only the gesture of writing. And by diminishing it, such an argument falsifies this tension. I am convinced that in no gesture (neither in that of the shoemaker, nor in that of the orator, nor in that of the painter or the mathematician) is the clash between withdrawal into oneself and public commitment as violent as in the gesture of writing. Because in any other gesture one of the two aspects prevails over the other. But the gesture of writing is totally and radically the gesture of the thinker (as totally as Rodin’s gesture of the thinker). And it is also totally and radically the gesture of communication (as totally as is the gesture of the speaker). When we write, we are totally immersed in our thoughts, even when the purpose is to change others. And when we write, we are totally dedicated to publishing, even when we don’t want to publish the text we are writing. The two existential attitudes in relation to the gesture of writing are annihilated by the gesture itself: when one writes to realize one’s thoughts, one “falls” into political engagement, and when one writes to engage politically, one “falls” in solitary thought. Every time we start writing, we suffer from this dramatic tension, even if we try to deny it later. We are always a “real writer”, even when the purpose is to write “letters”, and what for me is the drama, we are always a “committed actor”, even when we want to write “treaties”. The gesture of writing condemns us to think and to publish, and to say that one thing implies the other, or that one aims at the other, or that it is as it should be, or that it is banally like that, is to want to deny the dramatic nature of the condemnation to live both in private and in public. So, in the case of my own praxis, I approach the gesture of writing with the intention of imprinting thoughts on a surface, and thus of realizing them, but I am condemned, by the gesture of writing, to do my best to publish the text. That is to say that the last language in the game of translations and retranslations that constitutes my writing strategy is imposed by the social situation in which I find myself. This is the language of the publication. This poses a problem that I have only been aware of for a few years. Because this last language was, for me, for years, Portuguese, and, exceptionally, English. Two languages of which I have the mastery. But lately, the language in which I am obliged to publish has become, for social reasons (as befits a social imposition), French, a language which I do not dominate. I am therefore obliged to “write my final text” by handling material that I do not control. Well I realized that my situation is not exceptional. This is the situation in which all those

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who write find themselves. The last phase of the gesture of writing is always a redaction of the text to make it readable to others. So, a manipulation of a material that we cannot control. My special case makes this general fact a bit more obvious, but the situation is still typical. All writers write their final text in their own “French”. That is to say, in the language that does not belong to him, but belongs to others. This fact often remains hidden even from the eyes of the writer (as it has remained hidden from my eyes for years), and it justifies my speaking about my specific situation. The last phase of the act of writing is that of editing. In this phase the elements that constitute the gesture of writing change position. Thought is no longer a tendency pressing towards a surface. It has now become a printed surface (the text is “correct”), so it has become my “object”. When I write, I no longer think about the thought, but I think about those who are going to read it. The surface on which I tap is no longer a resistance to my thought, thanks to which the thought will become effective. It is now a “manuscript”, i.e. a letter to the editor. The language signified by the letters I type is no longer a logical, aesthetic and epistemological structure in which I formulate thought in order to realize it. It has become a means of communication. The letters I type are no longer the forms for which thought was programmed, and the lines I type are no longer the rules that order thought. The letters and the lines they form have become a code to be deciphered by others. In short, when I write, I am outside of myself, at an ironic and critical distance from my real act of writing. It is not, in writing, an “internal” criticism of the gesture of writing. For such an internal critique is precisely the moment of choice by which the movement of the gesture is periodically interrupted. It is, in writing, an external and ironic criticism of the gesture of writing. The editor is the “enemy” of the writer. Therefore, the writer becomes his own enemy in the last phase of his gesture of writing, and this by the very structure of this gesture. I have described, in this essay, my gesture of writing to me. But it is not an autobiographical confession, which has no interest for those who will read it, nor, for that matter, for me. Because my gesture of writing is typical. It is a gesture for which we are all programmed by our culture. It is one of the ways in which we are all in the world. We are all programmed to be writers. That is to say, to think according to specific rules, and to materialize one’s thoughts by scratching lines on surfaces. Of course, there are other ways of being in the world in our cultural program. We cannot achieve them all in our lifetime. Most of the virtualities in our program remain virtual. We are all virtual writers, but not all of us actually are. Nevertheless, we all know how to write. That is to say, there are those who write without being writers. They are numerous. Their gesture of writing is not as I have described it here. They do not realize the potentialities inherent in writing through this gesture. They only use it as a means of communication. But as a means of communication, the gesture of writing is no longer very effective. There are, now, means much more powerful, much more flexible, and much easier to handle. This is why the gesture of writing is becoming archaic. In the age of tape recorders and video recorders, computers and data systems, telephones and television, writing is rapidly becoming an outdated gesture. We finally passed the Mesopotamian bricks. The gesture of writing will be eliminated from our program, the “art of writing” will be forgotten.

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But there are those who write as I have described it here. That is, those who try to realize the program inherent in writing, without ever being able to exhaust it. Writers. After the elimination of the act of writing from our collective memory, there will be no more writers, and no one will feel their absence. At present, and provisionally, they still exist. And, as with any extinct species, they cannot existentially agree with “evolution” (although they can intellectually). No, because they know that with the art of writing a whole form of thought and being in the world will be lost. They may agree that such a form is not worth preserving. But quite simply because writing, for them, is living.

References 1. Alunni, C.: “On the translatability of knowledge”. Revue Science/Lettres 1 (2013). Available on https://journals.openedition.org/rsl/293 2. Finger, A., Guldin, R. and Bernardo, G., eds.: Vilém Flusser. An Introduction. Minneapolis London: University of Minnesota Press, Electronic Mediations, volume 34 (2011c) 3. Flusser, V.: Vom Stand der Dinge. Verlag GmbH, Braunschweig, Bollmann (1993) 4. Flusser, V.: Dinge und Undinge. Carl Hanser Verlag, München (1993) 5. Flusser, V.: The Shape of Things: A Philosophy of Design, Reaktion Books, London (1999). These essays were taken from Vom Stand der Dinge, Braunschweig, Bollmann Verlag GmbH, 1993, and from Dinge und Undinge, München, Carl Hanser Verlag, 1993 6. Flusser, V.: Gesten: Versuch einer Phänomenologie [1991], Frankfurt Fischer (1997) 7. Flusser, V.: Gestures. University of Minnesota Press, Translated by Nancy Ann Roth, Minneapolis London (2014) 8. Flusser, V.: Writings, ed. Andreas Ströhl, trans. Erik Eisel. Minneapolis: University of Minnesota Press (2002) 9. Flusser, V.: Does Writing Have a Future? University of Minnesota Press, Minneapolis (2011) 10. Flusser, V.: “Towards a Theory of Techno-Imagination”, vol. 2. Philosophy of Photography (2011b) 11. Meulen, S., 2010, “Between Benjamin and McLuhan. Vilem Flusser’s Media Theory”, in New German Critique vol. 37 iss. 2. 12. Roth, N.A.: “A note on ‘The Gesture of Writing’ by Vilém Flusser and the gesture of writing”. New writing. Int. J. Pract. Theo. Creat. Writ. 9(1), 24–41 (2012)

Vilém Flusser (1920–1991) was a Brazilian Czech-born philosopher, writer and journalist. He lived for a long period in São Paulo (where he became a Brazilian citizen) and later in France, and his works are written in many different languages. His early work was marked by discussion of the thought of Martin Heidegger, and by the influence of existentialism and phenomenology. Phenomenology would play a major role in the transition to the later phase of his work, in which he turned his attention to the philosophy of communication and of artistic production. He contributed to the dichotomy in history: the period of image worship, and period of text worship, with deviations consequently into idolatry and “textolatry”. His writings reflect his wandering life: although the majority of his work was written in German and Portuguese, he also wrote in English and French, with scarce translation to other languages. Because Flusser’s writings in different languages are dispersed in the form of books, articles or sections of books, his work as a media philosopher and cultural theorist is only now becoming more widely known. The first book by Flusser to be published in English was Towards a Philosophy of Photography in 1984 by the then

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new journal European Photography, which was his own translation of the work. The Shape of Things, was published in London in 1999 and was followed by a new translation of Towards a Philosophy of Photography. Flusser’s archives have been held by the Academy of Media Arts in Cologne and are currently housed at the Berlin University of the Arts. See http://www.flusserstudies.net/node/208.

Chapter 10

The Diagram on Stage: Movement, Gesture and Writing Catherine Paoletti

Form fascinates when one no longer has the force to understand force from within itself. That is to create. Jacques Derrida, Wrinting and Difference. Substance. It is not formless but in the process of taking shape. The proof, it is directed towards language. Claude Maillard, The Scribe. Write the dance. Following traces. Topology of the invisible. Angelin Preljocaj, Topology of the Invisible.

Gilles Châtelet highlighted that the diagram as instrument, object and place of thought (or site with Alexandre Grothendieck) gives access to the invention of new spatial configurations, new spatialities, therefore to the potential for movements in space, and I will add, on the motive of writing. Mobile which is obviously to be understood as both an adjective and a noun. For it is also a provocative summons (herausfordemdes Stellen), “a provocation to mobilities and to churn out compact signifying units”: it is “a language of plastic tradition which stammers thanks to the fact that the being present comes to appear in order to make appear the absent present”.1 Everyone knows the importance of the notion of site in contemporary art, which has also led to performance. The potential of in situ work allows “to introduce within it, an understanding of the movements that may occur in the future. […] It is finally the sine qua non condition to show that […] the work in situ, and it alone, opens the field to a possible transformation, precisely of the place”.2 Daniel Buren is certainly lacking in not having thought of thinking of these movements in diagrammatic terms.

1 2

See Gilles Châtelet Archives (Ecole Normale Supérieure), typescript 5 Θ 11bis. Buren [5], p. 80.

C. Paoletti (B) École Normale Supérieure, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_10

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Fig. 10.1 Mycenaean Hydria, Archaeological Museum of Naxos (creative commons Zde)

That is to say in terms of situations because “all existing or impossible situations, wanted or not, can be manipulated and deformed (metamorphosed) until giving rise to a re-writing of their structure as a diagram” as René Guitart says in his text: “Figures, letters and proofs: pulsation and figurations instead of writing”.3 We will start from this old definition of the diagram, taken from an old Larousse and noted by Thibault Damour in his article on “The unreasonable efficiency of diagrams”4 : “in the figurative sense. Determination of the various causes which intersect in different ways”, and whose generality is ultimately not totally foreign to what precedes and what is to follow. In its etymology, diagram comes from the Greek dia-graphein which inscribes the line and what can be written in its intertwining; as well as choreography: khoreía (χoρε´ια) inscribed with the chorus and the graph, ´ of the writing and the “chorea” which was the first round, the the graphée (γραϕη) circle of the round as the first dance identified, and whose trace was found in Naxos on a Mycenaean vase, dated to 1200-1100 BC. A two-handled earthen pot, called a “hydria” (Øδρ´ια), which was used to carry water (see Fig. 10.1). A diagram can transfix a gesture, bring it to rest, long before it curls up into a sign, which is why modern geometers and cosmologers like diagrams with their peremptory power of evocation. They capture gestures mid-flight; for those capable of attention, they are the moments where being is glimpsed smiling.5 A definition which, in many respects, is not unrelated to dancing, if we follow the intuition of Marin Mersenne, in his Universal Harmony of 1636 which already explained that ballets could represent figures scientific par excellence, those described by the celestial bodies:b One can make Ballets which will represent and teach Astronomy, particularly if it is permissible to express by singing part of the science which one can represent and teach. For example, we will represent the distance of Saturn from the Sun by a dance of 10 steps, especially since

3

Guitart [27], pp. 59–80. Alunni et al. [3], p. 233. 5 Châtelet [6], p. 10. 4

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Fig. 10.2 a, b Bibliothèque nationale de France, département Réserve des livres rares, RES M-V303 (1). http://catalogue.bnf.fr/ark:/l2148/cb30432725f

it is ten times farther from it than the earth […]. That we can represent the distances of other stars and their daily and annual movements, and everything that appears in the sky.6

Thus, sketched out what will be deployed, particularly as a diagrammatic writing of dance and which is developed from the outset through the search for a system of notations. The term choreography itself appeared for the first time in 1700 to precisely designate a system of dance notations that the dancing master Raoul-Auger Feuillet had developed in his treatise Chorégraphie, ou l’art de décrire la danse par caractères, figures et signes démonstratifs (see Fig. 10.2). And we must not lose sight of the issue of ratings is “this ability to train and mobilize new intuitions. Notation is closer to discipline than to rule”.7 From that time, this system of notations developed by Feuillet will be very successful and will allow the distribution throughout Europe of the French repertoire. For an in-depth study of the Feuillet system, I refer to the article by Évelyne Barbin: “Choréographie et Cinétographie: une mutation de l’écriture de la danse”,8 which compares it with the kinetography of the Hungarian choreographer and theoretician of dance, Rudolph Laban, who will establish a system of writing movement in space, including dance, described in 1928 in his work entitled Kinetografie.

6

Mersenne [20], p. 120. Châtelet [7], p. 169. 8 Barbin [4], pp. 59–80. 7

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Fig. 10.3 Laban’s lecture on his notation system, 1928 © Suzanne Perrottet ARR

His gesture, by his own admission, is to seize, and make visible by a system of notations what we do not see: “we do not see the movement, we only see forms changing their location”9 (Fig. 10.3). What characterizes this system is that it is constituted as a writing of the body captured as the flow of movement, and no longer as has been the case since the fourteenth century, including Feuillet, where “the accent [was] placed on the representation theatrical: the precise measurement of time, the exact recall of details, gestures and steps, the harmonious coordination of the movements of the arms and the body; the rules of bearing, the imaginative variability of certain details; and, finally, the importance of the ground paths that the dancer follows”10 (Figs. 10.4, 10.5 and 10.6). What Feuillet designates as “path”, that is to say the line on which we dance, and on which we can write the steps and the positions. For Laban, movement is no longer considered as a trajectory between two points, but as transport in a space–time where gravity and forces (Laban speaks of “efforts”) are at stake, and where heaviness is metamorphosed into lightness because “dance transmutes the heavy into the light”.11 Laban adds that “The flow of movement is 9

Laban [15], p. 49. Laban, ibid., p. 49. 11 Cf. Gilles Deleuze about Nietzsche [22, p. 222] (cf. Thus spoke Zarathustra, III, “The Seven Seals”, 6: “Everything that weighs must become lighter, every body become a dancer, every bird spirit”). 10

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Fig. 10.4 The “presence” of the body on stage in the Feuillet system

toe

first +

second ++

third +

fourth +

fifth +

hell ankle

Fig. 10.5 The foot and its five good positions

Fig. 10.6 Representation of the dance path (to return to a path on which one has already walked)

controlled as it goes inward, beginning at the extremities and progressing towards the center of the body.”12 Its notation system is based on five main principles and a three-dimensional typology (plane-space–time).13 The first three principles qualify the movement while the following two express the movement in its relation to the positions of the body: ● the directionality (rotations and paths) refers to the representation of the directions of movements, by signs and symbols (Fig. 10.7). 12

Laban, La maîtrise du mouvement, op. cit., p. 46. Cf. Knust [13], pp. 73–96. Albrecht Knust, dancer and choreographer, student of Laban, will work on the development of the notation of the movement, of which he will be one of the main propagandists. He wrote the first kinetography manual in 1937: Abriss der Kinetographie Laban which he considered his «Handbuch der Kinetographie Laban» (Encyclopédie de la cinétographie Laban), which was not published until 1956, and that he will not stop developing to lead to his monumental dictionary that will only see the light of day after his death (1979). 13

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circular path

front

front

left

right

left

right

on the left turn left

turn right

back

back

left

right

on the right

back

Fig. 10.7 Representation of movement directions

high

average

low

Fig. 10.8 Representation of pitch (height) and duration

● the spatialization (top–bottom-intermediate) by symbols, represents the representation of the levels of movement (height). ● the temporality (duration of the movement) is concretized in its duration relative to the length of the sign (Fig. 10.8). ● the amplitude and the dynamics (of the movement of the different parts of the body) makes it possible to visualize which dancer is performing and which parts of the body are at stake (Fig. 10.9). ● finally, the intensity and the relationship to other parts of the body (Fig. 10.10).

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right

moves legs upper body

an example

arms

Fig. 10.9 Representation of the body: the staff

It is indeed for Laban to translate the movement from a system of articulated notations which integrates time and rhythm in the sense of a writing which would make visible what is not seen but which unfolds and that the Laban notation formalizes parallel to the musical staff.14 In this sense, there is an obvious qualitative jump between the notation of Feuillet, more iconic, in the sense of Pierce, and the diagrammatic elevation, more allusive of Laban, which translates moreover in his concept of kinesphere as integrated space, “which is the space that the body carries with itself, better, where it is constructed, is constituted, [and which] is the meeting of all the events possible motors, which distinguish weight transfer modes and tensional qualities or orientations”.15 Of course, Laban will classify them through his “scales”, but he will never designate either their closure or any codification. The space carried by the body is circumscribed inside the Platonic and Archimedean solids: hexahedron, octahedron and icosidodecahedron, this polyhedron with twenty triangular and twelve pentagonal faces (see Fig. 10.11). The kinesphere dots to the sagittal orientation of the gesture that geometrizes the dancer’s body. Mallarmé will be sensitive to this rise of abstraction specific to choreographic modernity, when he asserts in “Crayonné au théâtre” that “the dancer is not a woman who dances, for these juxtaposed motifs that she is not a woman but a 14

Ann Hutchinson Guest emphasizes in this regard that these “patterns are easily recognizable by those who have learned to read them, just as those created by musical notes are by musicians” Fig. 10.10, 1990, pp. 203–14. See also his work on Laban notation, 1970. 15 Louppe [16]3 , p. 213.

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Fig. 10.10 Music and movement in Laban’s writing. Serenade, Balanchine Ballet, 1950 ARR

metaphor […] sword, cup, flower, etc., which she does not dance, suggesting by the prodigy of shortcuts or dashes, with a bodily writing”16 : she condenses “elementary aspects of our form “. It is certainly for identical reasons that Paul Klee, Wassily Kandinsky and Oscar Kokoschka will share so many artistic affinities with the dancer

16

Mallarmé [19]2 , p. 304. «À proprement parler, pourrait-on ne reconnaître au Ballet le nom de Danse; lequel est si on veut, hiéroglyphe» (“Strictly speaking, could one not recognize the name of Dance in the Ballet; which is, if you will, hieroglyph”).

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Fig. 10.11 Metal icosahedron from Miki Tallone, Monte Vérità, Laban Training Area (creative commons Pakeha)

Gret Palucca.17 Kandinsky will diagram this dancing body, to reveal its structure and lines in his article “Tanzkurven”, as well as in his book: Dot, Line, Plan, both published in 1926 (see Fig. 10.12).18 Starting with the dot of the point of the dancer whose rapid run on the points scores dots on the ground. Kandinsky points out that the dancer uses the dot also in his high jumps when the head is clearly aimed at the point, or when he bounces on the floor. But he insists on what characterizes the jumps of modern dance in relation to those of classical ballet which are a vertical one, while the modern jump forms, as is the case with Palucca, “a surface endowed with five angles and five dots: the head, the two hands and the two tiptoes, while the ten fingers form ten smaller points”.19 Likewise, the positions of brief and rigid immobility can be conceived as dots, without forgetting an active and passive dotted line that he envisages as related to the musical form of the dot. The dot is therefore indeed the place of the inscription of a tension, potential of the passage to the line, because “the line is a dot that went for a walk”, as Klee would say. It is not insignificant to remember that this sign has existed since the dawn of humanity, and that it is found in multiple configurations in prehistoric caves, opening up to semantic compositions whose distribution we clearly perceive is not random 17

We must also keep in mind all the contemporary technical developments relating to the analysis of movement: from Étienne-Jules Marey’s chronophotography to Eadweard Muybridge’s zoopraxiscope to Thomas Edison’s kinetoscope which will lead to the invention of the cinematograph. 18 The theme of the dot had previously been addressed within the Imperial Academy of Sciences of Saint Petersburg, the current Russian Academy of Sciences, in particular by the mathematician, philosopher, engineer and theologian Pavel Aleksandrovich Florenski, in his essay: “Symbolarium. Il punto”, (1922–1923), in [●●●], pp. 217–262, and whose inverted perspective particularly interested the painter David Hockney. See also, Florenski [25], and Zalamea [35]. 19 See Kandinski [11], p. 77. The dancer’s jump is represented by an arc (that of the legs) supported by two other arcs (those of her arms); her body is a sphere which rests on the arc of her legs, and the positions of rigidity are represented by 5 dots: feet, head and hands; the two dots of the hands being, themselves, one and the other surmounted by an arc of 5 finer dots which represents the spacing of the 5 fingers.

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Fig. 10.12 Wassily Kandinsky, “Tanzkurven: Zu den Tänzen der Palucca”, Das Kunstblatt, Postdam, vol. 10, n° 3, 1926, pp. 117–120

and testifies to a spatial distribution,20 but, even more, which confirms the signwall(surface)-space relationship.21 It should be emphasized that all of these signs use, in order to construct and transmit themselves, stable symbolic patterns, over long periods of 20,000 years. The prehistorian Jean Clottes highlighted their ideographic value with regard to the animal figures they accompany. On some hundred and fortysix sites located in France, twenty-six different signs have been listed, which could well be the prelude to our alphabet.22 But let’s come back to what constitutes the potentiality proper to the dot: from the two dots, tension of an orthogonal and cardinal suspension—because the dot is axis and pivot, promise of horizon–, up to the infinite gap of the three dots. Condensation point, because we must not lose sight of the fact that the dot “is the central axis which represents the pivot point from which the conquest of infinite dimensions becomes possible. By deliberately forcing the metaphor, the dot is the power of compression and simultaneously the power of unleashing virtualities”.23 Landmarks, cardinal points, Braille dots to be grasped gropingly, in the embrace of the present night or on tiptoe. Three dots, the dot opens to the line: because it takes sharing and separation, line and border to seize “in sharing” the sharing which opens to symmetries and translates the gesture into movement. 20

Petzinger von G. [31]. Robert [33]. 22 Petzinger von G., op. cit. 23 Châtelet, «La Mathématique comme geste», in Alunni, Paoletti [2] , p. 180. More generally, see Gilles Châtelet, Figuring Space, pp. 10, 32, 35, 54, 57, 95, 136, 161, 162, 166, 167, 175, 182. 21

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To interfere in the movement of the mind in the process of knowing, [we could just as well say: to create or to write], to understand that the part is not in the Whole (the Whole is not a container !), knowing how to draw dotted lines […], this is indeed the fascinating dimension of symmetry, as a way of resuscitating the part in a phosphorescent fragment, as if all the vivacity of the Infinite, of the totality, were mobilized to invite us to penetrate further, to “dig in”, like the interlacing of the Moorish window leads into the labyrinth of its spirals…24 The artistic avant-gardes, from Malevitch to Francis Bacon, via Marcel Duchamp, have also diagrammatized forms and movements to think up new gestures and go up in abstraction. “The movement of the form in a given time leads inevitably to geometry and mathematics” Duchamp confided to Pierre Cabanne.25 Gilles Deleuze will have been the first to highlight in Francis Bacon’s painting the function of the diagram as potential for the emergence of form (by effect of symmetry, purification, staging, destruction of degrees, dialogue, calculation/image), as evidenced by most of his paintings. Francis Bacon apprehends it as the means of implanting “possibilities of facts of all kinds”: it is the prerequisite for the gesture of the painter of which he partially retains the trace which becomes structure in the painting. Moreover, the diagram “induces or distributes in the painting the informal forces with which the deformed parts are necessarily related, or to which they serve precisely as ‘places’”.26 As such, the diagram is a “scene” whose axes (lines, points, arrows, circles, volumes, letrasets, etc.) structure the mobile to capture the liveliness of the movement. But the diagram, like symmetry, is not an image: it drains other images, sets up devices that make the jump to be made tangible, it allows thought experiments to be put into symmetry, because symmetry opens up to the effect of balance, of polarization of forces, and where its very ™πoχη´ suspends the attention; because duality articulates spectra of degrees which makes possible, as Gilles Châtelet says, a superior predation. The diagram then makes it possible to capture this reverberation effect. One could go so far as to say that it is indeed ideography which makes possible, by a jump and a condensation, the materialization of the idea, of the idea in writing. We must not lose sight of the fact that the art of staging is also a square, blank, to be filled in or not. Space to cross or not. Space of Quad 27 by Samuel Beckett, Kasimir Malevitch’s Quadrangle or Claude Maillard’s Vertigo Machines28 : a white square, topology of the invisible and writing scene. Like the Black Square on a White Background by Malevich—because it is also a question of light—whose detailed analysis of its surface under different types of light (grazing, under ultraviolet rays,

24

Châtelet, «L’univers de Roger Penrose: un royaume dont le prince était un enfant», in op. cit., p. 246. 25 Duchamp [13], p. 28. 26 Deleuze [23], p. 148. 27 Also called “ballet for four people”, it was broadcast for the first time by the Süddeutscher Rundfunk, under the title Quadrat I + II, on October 8, 1981, performed by four dancers from the Stuttgart Preparatory Ballet School. 28 Maillard [18].

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Fig. 10.13 Milda Vikturina & Alla Lukanova, 1990 ARR

in radiography of X-rays and scanning electron microscopy) revealed the palimpsest that supported its whitewashing (see Fig. 10.13).29 The radicality of its suprematism which will culminate with the “objectless” world of white on white, then of the white canvas, is well and truly preceded by the engramming of its trace and its weft. Apart from the fact that the sequence which preceded it was that of the black square (1915), the spectrography of the canvas finally revealed its diagrammatic palimpsest: a white frame on a black background where the white edge is not a background but also the frame of two other invisible paintings: an alogical composition and a suprematist painting, themselves covered with a black square30 bordered by a white desert. It should be remembered that Malevich used the term diagram or graph—a term found in Francis Bacon—to evoke the dynamics of his Suprematist operation. “My attention was drawn to the results of the study of nature and physics in which the colorful rainbow stands out with particularly vivid outlines against a background of dark clouds. This brought into me a whole series of thoughts, and the result of my meditations was put into the form of a graph of 29

Cf., Vikturina and Lukanova [34]. This black that Malevich had introduced “as a fifth dimension in art”. Ekaterina Voronina, Irina Rustamov and Irina Waqar, curators at the Tretiakov Gallery, will also manage to decipher another inscription under the blackness of the painting, an allusion to the title of the painting by the poet Paul Bilhaud (1854–1933): Le combat des nègres dans une cave la nuit, a blackboard in a gilded frame, which will be hung on the occasion of the first exhibition of Incoherent Arts in 1882. This monochrome will have inspired Alphonse Allais (1854–1905) the following year to hang a simple white bristol titled: First communion of young chlorotic girls in snowy weather, which will be part of his series of seven “monochroidal” paintings. A wink from Malevitch to the author of the blackboard and perhaps to its variations by Alphonse Allais, which will be published fifteen years later, with Le combat des nègres dans une cave la nuit by Bilhaud, in his Album primo-avrilesque [1]. “The painter in whom I idealized myself was the genius who sufficed for a canvas with one color: the artist, dare I say, monochroidal”, wrote Allais in his preface. See also Rosenberg 2014. 30

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the movement of color through the centers of human culture”.31 Until his ultimate jump to the “zero of form” and to architecture which can only confirm his deliberate gesture towards what could be described as diagrammatic elevation. Whether the score is played alone or with others, on a single or multiple spaces, it is always a question of reducing or increasing the abyss or the chaos by gestures that model the body of time to bring out possible worlds, written on a stage where new mental constellations appear, revealed by new diagrammatics. If as the choreographer, dancer, Angelin Preljocaj writes: “writing is a perspective of movement”, from the dance scene to that of writing, there is only one step: “the step of time and the time of the step”, will have answered Claude Maillard.

References 1. Allais, A.: Album primo-avrilesque. Ollendorff, Paris (1897) 2. Alunni, C., Paoletti, C. (éds.): L’Enchantement du virtuel, Paris: Éditions “Rue d’Ulm”, coll. «Pensée des sciences» (2011) 3. Alunni, C., André, Y., Paoletti, C. (éds.): “Philosophies contemporaines de mathématiciens: Évariste Galois, Gian-Carlo Rota et Gilles Châtelet”. Revue de Synthèse, t. 138, 7e série, 1–4 (2017). https://doi.org/10.1007/s11873-000-0000-11 4. Barbin, E.: “Chorégraphie et Cinétographie: une mutation de l’écriture de la danse”. In : Nicolas, F. (éd.) Les mutations de l’écriture, Publications de la Sorbonne, Paris, coll. “LogiqueLangageSciencePhilosophie” (2013) 5. Buren, D.: À force de descendre dans la rue, l’art peut-il y monter? Paris: Sens&Tonka, coll. “11/24” (1998) 6. Châtelet, G.: Figuring Space. Philosophy, Mathematics and Physics, Kluwer Academic Publishers (Trans. by Shore, R., Zagha, M.); Springer Science+Business Media Dordrecht, vol. 8. Introduction Kenneth J. Knoespel, “Diagrammatic wrighting and the configuration of space” (1999, 20102 ) 7. Châtelet, G.: “Mettre la main à quelle pâte ?” Alunni, Paoletti 2011, 163–176 (2011) 8. Châtelet, G.: “La Mathématique comme geste de pensée”. Alunni, Paoletti 2011, 177–182 (2011) 9. Châtelet, G.: “L’univers de Roger Penrose: un royaume dont le prince était un enfant”. Alunni, Paoletti 2011, 215–232 (2011) 10. Damour, T.: “La déraisonnable efficacité des diagrammes”. Alunni, André, Paoletti 2017, 231– 260 (2017) 11. Kandinsky, W.: Point ligne plan. Pour une grammaire des formes. Contribution à l’analyse d’éléments picturaux. Denoël & Gonthier, Paris (1970) 12. Knust, A.: Abriss der Kinetographie Laban. Albrecht Knust Publisher, Das Tanzarchiv, Hamburg (1956) 13. Knust, A.: “An Introduction to Kinetography Laban (Labanotation)”. J. Int. Folk Music 11, 73–96 (1959) 14. Knust, A.: Dictionary of Kinetography Laban (Labanotation). Estover, Macdonald and Evans, London (1979) 15. Laban, R.: La maîtrise du mouvement. Actes Sud, Arles (1994) 16. Louppe, L.: Poétique de la danse contemporaine. Contredanse, Bruxelles (20003 ) 17. Malevitch, K.S.: Essays on Art, 1915–1933. Troels Andersen (ed.), Rapp & Whiting, London [1969] (1993) 31

Malevich [17], 3, I/42, p. 97.

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18. Maillard, C.: Machines vertige, en cours de scènes et d’actes. Le Temps du Non, s.I./France (1993) 19. Mallarmé, S.: Œuvres Complètes. Mondor, H., Jean-Aubry, G. (éds.). Paris: Collection Bibliothèque de la Pléiade 65, [1945] (1979) 20. Mersenne, M.: Harmonie universelle. Sebastien Cramoisy, Paris (1636) 21. Nicolas, F. (éd.): Les mutations de l’écriture. Paris: Publications de la Sorbonne, coll. “Logique Langage Science Philosophie” 3 (2013) 22. Deleuze, G.: Nietzsche et la philosophie. Puf, Paris (1977) 23. Deleuze, G.: Francis Bacon: logique de la sensation. Seuil, Paris [1984] (2002) 24. Duchamp, M.: Entretiens avec Pierre Cabanne. Allia, Paris (2014) 25. Florenski, P.A.: Il simbolo e la forma. Scritti di filosofia della scienza. Bollati Boringhieri, Torino (2007) 26. Florenski, P.A.: “Symbolarium. Il punto”. In: Stratificazioni. Scritti sull’arte e la tecnica, trad. di Valentina Parisi, pp. 217–262. Diabasis, Reggio Emilia [1922–1923] (2008) 27. Guitart, R.: “Figures, lettres et preuves: pulsation et figurations au lieu de l’écriture”. Nicolas 2013, 141–156 (2013) 28. Hutchinson Guest, A.: Labanotation, or, Kinetography Laban: The System of Analyzing and Recording Movement. Theatre Arts Books, New York (1970) 29. Hutchinson Guest, A.: “Dance notation”, Theater, Theatricality and Architecture, pp. 203–214. Perspecta 26, Cambridge (MA) (1990) 30. Nietzsche, F.: Ainsi parlait Zarathoustra. 2 vols. Trad. fr. et préface de Geneviève Bianquis. Aubier Flammarion, Paris. Trad. de: Also sprach Zarathustra, Chemnitz, pp. 1883–5 (1969) 31. Petzinger, G. von (s.d.) “Geometric Signs & Symbols”. In: Rock Art. A New Understanding. http://www.bradshawfoundation.com/geometric_signs/geometric_signs_france.php 32. Robert, É.: “Signes, parois, espaces. Modalités d’expression dans le Paléolithique supérieur ouest-européen”. In: Clottes, J. (éd.) L’art pléistocène dans le monde = Actes du Congrès IFRAO (Tarascon-sur-Ariège, septembre 2010), Tarascon-sur-Ariège: Sociét´e préhistorique Ariège-Pyrénées, pp. 1941–1958 (2012) 33. Rosenberg, R.: “De la blague monochrome à la caricature de l’art abstrait”. In: Le Men, S. (éd.) L’art de la caricature, pp. 27–40. Presses universitaires de Paris-Ouest, Paris. (2011) 34. Vikturina, M., Lukanova, A.: “A Study of Technique. Ten Paintings by Malevich in the Tretiakov Gallery”. In: D’Andrea, J. (ed.) Kazimir Malevich, Armand Hammer Museum, Los Angeles: Washington Press, (Washington, DC, 16 September-4 November 1990; Los Angeles, 28 November 1990–13 January 1991; New York, 7February-24 March 1991). Los Angeles, pp. 187–97 (1990) 35. Zalamea F: Modelos en haces para el pensiamento matemático. Bogota: “Colleción Obra Selecta”, Universitad Nacional de Colombia (2021)

Part III

Diagrams and Gestures: Linguistics and Semiotics

Chapter 11

Meta-Morphosis: Kinesis and Semiosis in Language Concerning a Theory of Enunciation Dominique Ducard

My observations in this article will be restricted to the theory of enunciation developed by Antoine Culioli (1924–2018),1 one of the most eminent figures in French linguistics, whose thinking has not acquired the international renown due to it—this despite the fact that English-speaking linguists, as well as those of other languages, have entered into dialogue with Harris, Chomsky, Langacker and Talmy, to name but a few. The Théorie des Opérations Prédicatives et Énonciatives (TOPE), to give it its precise name, remains unrecognized partly because its author has always refused, for reasons of scientific scruple, to issue a synthetic and didactic version which might serve as a benchmark; and partly because acceptance of the notion of enunciation is problematic, beginning with its translation into English. The questions posed in the argument of the editors of this volume, on the relationship between diagram and gesture, referring to “the hypothesis of a diagrammatology of enunciative forms in linguistics [1–37]”, is at the heart of language theorization in TOPE. My contribution will therefore attempt to respond to the general objective of this collective publication, which is to demonstrate, insofar as I am concerned with a defined and circumscribed area of language sciences, “the heuristic role played by the concept of diagrammatic gesture in the domains of mathematical sciences and of the humanities”. My hope is that it will also enable Culioli’s compelling and original thinking to become more widely known; that it will encourage researchers to delve into it more deeply; and that it will promote interdisciplinary collaboration, which enunciation theory itself requires in accordance with its founding scientific principles. 1

See Culioli’s publications listed in the bibliography. Also of note are works in English, Cognition and representation in linguistic theory, edited by Michel Liddle (John Benjamins Publishing Company, Amsterdam/Philadelphia 1995); and in Italian, L’arco e la freccia: scritti scelti, translated and edited by Francesco La Mantia with a preface by Tullio di Mauro (Bologna, Il Mulino, 2014). D. Ducard (B) Emeritus Professor for Sciences of Language, Université Paris-Est, Créteil, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_11

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11.1 A Theory of Enunciation Based on a Linguistics of Operations The term enunciation, in theory, and particularly in the Francophone area of linguistic and semiotic studies, has experienced a remarkable fate, both in its interpretation and by extension to different domains. Culioli was always concerned to lay down its limits by means of a definition that belonged to his theorization of language activity. In this definition, the purpose of enunciation theory was to study the cognitive-affective activity of language through the diversity of languages, texts and situations. The word through indicates that the forms of language (or languages) in texts have to be examined when seeking to account for the operations and representations that underlie signifying activity. Language is fundamentally seen as an activity (Tätigkeit for Humboldt or Energeia), that is to say as process and action. Access to this activity is through texts (oral and written) resulting in a series of operations through which an enunciator constructs a combination of markers (signifying forms). This is combined with a system of location enabling another enunciator—or co-enunciator—to adopt a corresponding locator system and to reconstruct, which is to say recognize and understand, a complex representation (significations, referential values) from the arrangement of markers; this representation may or may not coincide with that of the original enunciator. This linguistics of enunciation therefore places the notion of operation, together with that of representation, at the heart of the theory, with operation being the corollary of activity: “the linguistics of enunciation can only be a linguistics of enunciative activity, that is to say a linguistics of operations (whatever these operations may be).” (idem: 45) The notion of operation was never really defined by Culioli—perhaps because of an overlap between the different meanings ordinarily given to it—but it is possible to relate it to dictionary definitions (based on French) which distinguish (1) a general use of the term to signify an action or a series of actions aiming to achieve a goal, with the teleonomic value assumed by any intentional act; (2) a particular usage when it indicates the activity of a function (physiological, psychic: interiorized action); or (3) a specific use, in logic, to determine the value of a relationship linking several binary variables. To these I would add (4) operations which are seen by Jean Ladrière, a philosopher of science and an epistemologist, as “transformations governed by formal schemata”.2 It is possible to connect these meanings with the different methods of approaching enunciation, according to whether (1) attention is focused on the enunciative activity of speakers as an activity of production and interpretative recognition of interpretable forms (texts), in which a verbal act is to say to others on the subject of- in view of-; (2) language activity is seen as one of symbolic representation, or more specifically a cognitive-affective activity, which then poses the question of the mental operations underlying the texts that are produced; (3) the predicative and enunciative operations, reduced to the establishment of relationships, are analysed by location and 2

See Jean Ladrière, “Système, épistémologie”, Encyclopoedia Universalis. http://www.universalisedu.com/encyclopedie/systeme-epistemologie/ (accessed 7 May 2015).

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regulated organization, and to the assignment of values (semantic, referential and pragmatic). Meaning (4) returns us to what happens at the level of the metalinguistic representation system (conceptualization, formalization, schematization and notation). These distinctions partly cover the epistemological model of the three levels or degrees upon which the procedure followed by enunciation linguists is based, at least from the perspective of a linguistics that aims to link signifying forms to their cognitive and affective determiners. Through a process of controlled observation and methodical analysis, in which manipulation and reasoning are linked, and intuition and imagination play a major role, the linguist attempts to register, by means of metalinguistic representations, those representations and operations that are not directly perceptible but are represented by linguist markers whose trace they preserve. The metalinguistic representative of first-degree representations is the notion, which indicates “a complex system of representation structuring physicalcultural properties of a cognitive order” [19: 100]. Notions have a dual epistemological status: on the one hand they are “mental objects”, or “cognitive constructs, representing chains of construction, states of things, inter-subjective regulation and programmes of adaptation and action in relation to the world and to others” [21: 203]; and on the other, they are representations captured in linguistic units and categories, which vary according to the formal systems that are languages: “These representations evade words, but they are caught in a trap, and the lexemes organize notional domains, veritable systems of complex representations that are both stable and labile” [22: 39]. Notions are then treated as lexical and grammatical notions. Three orders of representation—of language, linguistic and metalinguistic—thus become interconnected. Without further developing the elements of the theory, it is important for my purposes to highlight the relationships established between accessible material forms and inaccessible abstract forms, which pass via metalinguistic constructions: “The objective of the approach is to construct abstract forms which will enable us to establish as fruitful a relationship as possible, with a qualifier to be defined, between accessible forms and inaccessible ones” [21: 203–204]. These metalinguistic constructions are thought of, by presupposition, as analogous to the abstract forms characteristic of cognition—including the affective element–, as a system and activity.

11.2 Schematic Form, Graph, Diagram3 The method adopted by Culioli and the principles that form the basis of his research pose a certain number of questions. For example, when using a scientific approach, how is it possible to maintain a position that belongs partly to a phenomenology and a hermeneutics of language, while giving oneself the formal means of explaining 3

In what is to follow, I partly return to two previous articles which I devoted to formalization in Culioli’s linguistics [28, 29].

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the interpretation? What metalanguage should be adopted in order to describe, using signs that can be manipulated, the relationship between linguistic forms and relationships of a notional order? What model should be adopted in order to simulate language operations? The metalanguage of representation was initially thought of as “a metalanguage of calculation” [19: 23], with which it is possible to carry out controlled operations, avoiding an approach that is too intuitive and subjective, and to generalize by moving from the local to classes of phenomena and then to the global.4 Abstract forms then needed to be developed, by inventing a formal language adapted to natural languages. Recourse to logic—“science of the formal, which is to say of the form and catenation of the rules of writing” [23: 53]—in fact proved simplistic and inadequate, unless one restricted oneself to the metatext of logicians and to phenomena that were limited “to a finite, normed and calibrated world”, whereas the linguist has to “produce theories on that which is deformable and exceeds categories” [19: 12]. Culioli states that he passed through several phases, of which the first was devoted to collecting, sorting and organizing data, in several languages, the initial result being “a kind of corpus of doctrine”. This period of systematization of problems and of axiomatization, which led to theoretical propositions and a conceptualization of metalinguistic objects (primitive relationship, lexis, notional domain, markers and categories, space of reference, and so on)5 was followed, or rather accompanied, by what he called the “transition to a ‘formulaic’ or ‘formulary’ stage” [23: 71], according to the ideal of a metalinguistic model in which “everything that is relevant must be capable of being represented in writing” [23: 55]. He therefore went on to create—inspired especially by set mathematics and combinatory logic (Curry)— symbols and an abridged notation, in collaboration with mathematicians, computer scientists and logicians, particularly Jean-Pierre Desclès. It is not possible here to enlarge on this system, which has notably been applied to the primitive relationship of agency (capacity to act), as well as to enunciative location, with more sporadic uses limited to case studies. However, it is interesting to mention the two stumbling blocks encountered by linguists in their formalizing procedure, which pose the question of scientific communication and transmission. These are: (1) a tendency for linguists to reverse the process by prefacing the actual study of a problem with an exposition that is “often pseudo-theoretical” and “formulaic”; and (2) the avoidance of shared discussion, conditioned by theoretical and practical knowledge of the formal system. Added to these two acknowledged difficulties is a reassessment of the role of reasoning in everyday language, as against calculation and formulae. The presentation of analyses, in formulaic style, gives rise to restrictions which limit the force of the argument. Linguists, who should not dream they are mathematicians, can always turn towards mathematicians to formalize an 4

“We can therefore say that, with texts, we are dealing with (mental) representations that are fixed (materialized and stabilized) through the intermediary of signs, for which we shall provide a metarepresentation, so that we may manipulate them—manipulate them with a view to searching for consistencies and rules of good formation, in order to arrive at a calculation” [19: 12]. 5 It may be useful to refer to the Lexique élémentaire de la Théorie des Opérations Prédicatives et Énonciatives d’Antoine Culioli compiled by de la Mantia [36].

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axiomatized linguistics, on condition that they do not succumb to the illusion that calculation is all-powerful, or lapse into the fabrication artefacts. In fact there are essential linguistic questions which, in their relationship to language, largely escape any form of logic: these are temporality and aspectuality; referentiation and referential value; modalization and intersubjective relationship, together with the adaptation and regulation of language activity in its relationship to the cultural and social world; and the whole notional domain, with its links to the cognitive and affective functions. More generally, as Culioli states, “It is this shifting complexity, linked to the epilinguistic activity of subjects (where epilinguistic corresponds—all things being equal!—to epigenesis),6 which, together, lead to deformable forms and stabilized instability (etc.), consequently preventing any appeal to a logical-algebraic device. We are not operating here on what is categorized and fixed, but on dynamic processes, where geometry is of a topological order” [14: 18]. Starting from his 1968 inaugural article on formalization in linguistics,7 Culioli [19: 27] began to introduce the cam diagram8 —a term suggested by the psychologist François Bresson–, based on Freud’s analysis of a case of obsessional neurosis known by the name of “Rat Man” (see Fig. 11.1).9

Fig. 11.1 The cam

6

Epilinguistics, which I am renaming epilanguage, refers to an unconscious, permanent activity, corresponding to a process in which forms are generated by variation-selection, in a similar way (but for language) to the epigenetic theories described by Courrèges, Changeux and Danchin (see Jean-Pierre Changeux and Antoine Danchin, “Apprendre par stabilisation sélective de synapses en cours de développement”, L’unité de l’homme 2. Le cerveau humain, Seuil, Paris, 1974). The neuronal model, known as the Darwinian model, of epigenesis through selective stabilization, has been extended, in conjectural fashion, to mental activity in general; it is based on a dynamic and interactive pattern of variation-selection, with a diversity generator and a selection system determined by the environment. 7 The article first appeared in Cahiers pour l’analyse, 9, 1968, Paris, Seuil. 8 See La Mantia [36, 35]. 9 Sigmund Freud, “Remarques sur un cas de névrose obsessionnelle (L’homme aux rats)” [Notes upon a case of obsessional neurosis], Cinq psychanalyses (Paris: P.U.F., 1954) pp. 199–261.

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The diagram gives rise to some comments on the cam’s structure, featuring a cycle which, between one point and the other, is relaunched by virtue of the last point being projected onto the initial point (a), and it is illustrated by being applied to the interaction of pronouns. “The advantage of this representation is that it forces decisions to be taken, and therefore a problem to be posed: so what will be the departure point? Why analyse the system of pronouns in this way? It is important to understand also that the diagram is not a plaything, an illustration to hold the attention, but a tool with its own strict rules for use” [23: 28–29]. Subsequently, Culioli would tend to speak of a graph, referring to graphic representations that delimit zones, connect points by arcs or lines, and orient, by means of arrows, the relationships thus established, which all exemplify movement. He thus follows his declared procedure, as against a static linguistics of classification and formalization—not by means of formal logic but by constructing spatialized abstract forms to support and encourage coherent reasoning, enabling the transition from one form to another to be understood, along with the interaction of significations and co-enunciation. Another type of metalinguistic representation, inspired by topology, was therefore developed to schematize aspectual-temporal phenomena, with notions of interval and of open and closed, or for use in the study of linguistic markers, of various kinds and in different languages. The linguistics of enunciative operations then developed in terms of access paths, obstacles, detours, loops and orientations, with transitions, zone changes, thresholds and limits, stops, gaps, hiatuses to be filled and objectives, with successes and failures, in an abstract and dynamic space equipped with tensors and vectors. I will provide a simple illustration using the metalinguistic representation of the marker seulement—its schematic form–, constructed according to an abstraction procedure based on the gloss, with a reasoned observation of its formal properties, and by graphic representation of the operations at work in the construction and interpretation of utterances [14: p. 25] (see Fig. 11.2).

Fig. 11.2 Schematic form of seul/seulement [only]

The diagram, which is a dynamic one, delimits the zones of a notional domain by vertical lines, with an Interior I (non nul)[not nothing], an organizing Centre C and an Exterior E (nul) [nothing], indicating an orientation or counter-orientation by means of arrows. The marker of the notion seul or seulement, which is the starting point, includes a dual property: there is an x (non nulle [not nothing] value) and pas plus [no(t) more] than x. x is therefore situated in I (Interior of the notional domain),

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and not in E (Exterior: Ø vide [empty]) and on the Frontière [Border] edge, it is oriented towards C (Organizing Centre of the domain), with a counter-orientation towards the complementary zone E. It is not possible here to provide all the theoretical elements which would enable the metalinguistic terms to be explained and the reasoning leading to the schematic form to be understood. And I shall not repeat the detailed analysis made of the marker seulement, whose different values emerge from a set of utterances grouped into three families, depending mainly on whether they are intended to measure distances between landmarks and the marking of zones, to construct the existence or non-existence of a state of things, or for teleonomic categorization (coordinating seulement), with orientations towards I or E. An initial analysis focuses on utterances of the type10 J’ai seulement dix euros en poche [I only have ten euros in my pocket]; Il a seulement 16 ans, il ne peut pas entrer au club [He is only 16, he can’t enter the club]; Il est seulement trois heures, nous avons encore largement le temps de finir/je commence à m’impatienter [It is only three o’clock, we still have plenty of time to finish/I’m beginning to get impatient]; Je viens seulement d’arriver, je ne sais pas ce qu’il s’est passé/vous m’excuserez de mon retard [I’ve only just arrived, I don’t know what has happened/please excuse my lateness]; the second involves utterances such as Entrez seulement! [Just come in!] or Essaie seulement (et tu le regretteras)! [Just try (and you’ll be sorry)!]; Si seulement j’étais reçu au concours! [If only I could pass the exam !]; Si seulement tu savais [If only you knew]; Elle nous a quittés sans seulement nous prévenir de son départ [She left us without even telling us she was going]; the third is based on utterances such as C’est un très bon joueur, seulement il est imprévisible [He’s a a very good player, only he’s unpredictable]; J’aime beaucoup les voyages touristiques, seulement je n’ai que peu de temps pour les loisirs [I love tourist trips, only I don’t have much time for leisure pursuits]; Il a très bien réussi dans sa vie professionnelle, seulement il a beaucoup travaillé pour parvenir à ses fins [He’s been very successful in his professional life, only he’s had to work hard to get there]. Let us consider the third utterance (Il est seulement trois heures, nous avons encore largement le temps de finir/je commence à m’impatienter) which is easily interpreted: a time is given with a dual quantitative value (3 o’clock and no later than 3 o’clock), which is the locator t of an indeterminate t’(t + n) state that has a different qualitative value in accordance with the context that follows seulement. As for the utterance of a desire in the phrase Si seulement j’étais reçu au concours! it posits the potential existence of a failure (E), which is envisaged by conjuring up a fictional success (I), centred on an imaginary point oriented towards a plus (case 2 in the schematic form). The tension between two possible states and the focusing on a point of attraction may correspond to an anxious expectation, somewhere between hope and fear. The expression of desire is based on the construction of a fictional state (non nul) as being real, with the value of the seulement preserving the orientation towards the complementary state (nul), held in reserve (possible disappointment).

10

My examples are in the spirit of those analysed by Culioli but are not reproduced literally.

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Let us ‘only/just’ say that the schematic form results from a reduction and is presented as an invariant form displayed in a regulated interplay of variations, according to the occurrences of the notion in textual sequences; in these sequences, all terms interact in harmony with their contexts and situations, that is to say the referential space and the field of inter-subject forces in which each enunciative act is immersed. “The (abstract) form is invariant: what produces the variation is the structured domain (the ‘milieu’) in which we immerse the form and the interactions that result from it” [14: 26]. We can apply to the schematic form what Jean-Toussaint Desanti says of the kernel, in its mathematical sense of “kernel of homomorphism”; this is also valid for the kernel of a fruit and its meaning can be extended to other domains, in its “function as a ‘signal’ of invariance, within an evolving domain”.11 The graphs of schematic forms are not only ways of representing and understanding predicative and enunciative operations; they appear as outlines of the gestuality that is internal to language activity and become, with the hypothesis of mental gesture, what I shall suggest are diagrams of the movement of thought in language.12 It is a viewpoint confirmed by Culioli’s statement that: “[the] schematic form gives us the means to construct the identification of mental gestures”.13

11.3 Diagram and Mental Gesture 11.3.1 Thinking with/by the Diagram Some references to philosophers made by Culioli will enable us to understand the link between the graph and the diagram as a figure of the mental gesture, in a theory of knowledge—starting with Bachelard [3, 4], who considered abstraction to be characteristic of the scientific mind, and whose aim was to “Make representation geometric” in order to arrive at an “intermediary zone” between laws and facts. In his teleological conception of how the scientific mind is formed, he distinguishes three stages: the concrete stage, the concrete-abstract stage and the abstract stage, to which he adds, in accordance with the psychology of scientific patience, three stages of the soul. The last of these, “the soul desperate to abstract and reach the 11

“The word is accepted, for example, to describe the area within the atom where its charge is concentrated, and within a cell, the area where its genetic material is concentrated and preserved: always in accordance with a demand for invariance, each time specific and locatable according to appropriate procedures. In the language of algebraists, a kernel acquires the meaning of invariant” (Desanti 1999: 138–140). 12 My hypothesis may be compared with that of Per Aage Brandt on the dual status, semiotic and cognitive, of the diagram: “The meaning of the diagram consists, so to speak, in a mental graph that is identical—at least partially—to the graph that we occasionally draw or sketch physically. It might therefore be said that the diagram is an indicator of the cognitive mechanism, which schematizes graphically while a thought is developing” [7]. Brandt’s perspective is semiotic, while Culioli’s work belongs to formal and empirical linguistics. 13 “Autour d’un objet” (unpublished), TOPE seminar, 17 March 2003.

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quintessential”, marks the fulfilment of abstraction, which is “a duty, the duty of scientists, at last refining and possessing the world’s thought” [4: 20–21]. When the destiny of the scientific mind is fully realized: “In the purity that a psychoanalysis of objective knowledge gives it, science is the aesthetic of the intellect” [4: 21]. In Culioli’s theorization, schematization is a formal representation, in a metalinguistic system, of mental representations and operations which, according to Bachelard, participate in geometric thinking—but are present and active in language activity, without being directly observable. In the introduction to the series of texts by Gilles Châtelet, collected and edited by Charles Alunni and Catherine Paoletti [1]—to which I shall return later—Châtelet demonstrates, by referring to the negativity of the concept in Hegel, that the movement of thought by reflective abstraction is the very thought process that deepens itself and in itself, by “rotation”14 and “(un)screwing”. The highest form of abstraction is achieved “through the categorical transfer and amplification of intuition” (Ibid.: 10). Dagognet [25], in his studies of iconographies in the history of arts and sciences, has highlighted the heuristic function of “diagrammatic schematization”, for which he emphasizes the two operations of inscription and abbreviation. Scientific procedure involves a search for a system of transcription that can introduce a new intelligibility. Through formalization, “The symbol finally ceases to be a means of fixing, recalling or intensifying: it is constantly becoming more of an “ideal corpus” that can be directly manipulated” [25: 124]. The schematic form is therefore much more than a synthetic and figurative notation, designed for better understanding or to focus ideas: it stimulates thought and suggests new avenues for observation and reasoning. It does not permanently close a demonstration any more than it appears as a simple illustration; it intervenes in the process of an analysis to envisage its result, while at the same time opening up other perspectives. In the work of Desanti [26] we find a definition of the diagram that supports demonstration. In Réflexions sur le temps. Variations Philosophiques I, he refers to Husserl’s diagram of time, in § 10 of the Leçons pour une phénoménologie de la conscience intime tu temps (1905), which is reproduced here (see Fig. 11.3).

Fig. 11.3 J.-T. Desanti: “dynamic” reading of the “diagram of time” according to Husserl

14

Gilles Châtelet quotes this “axiom” of Hegel [Wissenschaft der Logik]: “Negativity is the rotation point (Wendungspunkt) of the movement of the concept” (op. cit., p. 30).

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The figure is shown as a “graphic representation of the manner in which awareness of the duration of the passing time object is formed” [26: 135]. The philosopher then asks “how should one read this diagram?” which is not a representation of Time but a representation of “the connection of intentional acts specific to the modalities of forming an awareness of a time object” [26: 136]. The graphic figure can be understood in itself, as a “simple and well defined geometric ‘being’”, a “static figuration”, or as representing something else. In this case it is appropriate to phenomenologically interpret and determine the modes of correspondence between the points of the two lines. It is on this condition that it may be considered a diagram, which is to say an “‘indicator’ of demands for relations”. In other words, the diagram relates to an object that is represented by virtue of the correlations which the interpreter establishes between the figure’s components. A diagram is a form of relations. Peirce, in his construction of a “System of diagrammatization by means of which any course of thought can be represented” [42: 492], is thinking along the same lines: “(…) it [reasoning] must be chiefly concerned with forms, which are the chief objects of rational insight. Accordingly, Icons are specially requisite for reasoning. A Diagram is mainly an Icon, and an Icon of intelligible relations” (Ibid.: 497). And when the philosopher Langer [38] dismisses the figurative character of the diagram in favour of the indication of relations, defining it as “only ‘the image’ of a form” she invites us to distinguish in the diagrammatical figure, graphic notation on the one hand and the form of a thought on the other. Moreover, Culioli says of his modelling that it is “halfway between ‘a drawing’ and something that has formal properties”.15 Let us therefore place the diagram on the side of thought form, a dynamic system of relations of which the schematic form is a representative; and the graph on the side of graphic notation, in what English-language historians of science call “inscription devices” [44]. This distinction is reminiscent of the difference established in Greek Antiquity (Plato, Aristotle) between the geometrical object that the theorem aimed to construct and its material representation, with two words to indicate the figure: one for the figure as “geometric object” (eidos, form as opposed to matter or substance hulè; “real” form as opposed to form-appearance, dream, phantom: morphè) and another for the figure-drawing (skhêma: form as exterior figure, geometric figure, gesture).16 The notion of diagram relates to an experience of thought, like that referred to by Culioli when he says, alluding to Taoist philosophy, that he has to become language in order to represent it, and like Chinese painters when they say they have to become the reed to be able to paint it: “Just as the Chinese painter becomes a reed in order to paint a reed, to have the correct gesture, I had to try to make myself language, text, in such a way as to be able to capture something which, otherwise, would escape

15

Oral seminar, 22/03/2006. I would point out, in relation to this, and echoing what has been said of the role of schematic form, that the history of science reports cases of mathematical prefiguration, visual figures giving access, after the event, to mathematical realities that were unknown at the time when these graphic productions appeared in texts (see Peiffer, already cited, also [39]).

16

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me” [15: 116].17 I would link this experience to the one described and analysed by the philosopher of science Gilles Châtelet—another major reference for Culioli—in his studies of mathematicians and physicians (Grassman, Faraday, Maxwell): “The thought experiment taken to its conclusion is a diagrammatic experiment in which it becomes clear that a diagram is for itself its own experiment” [9: 36] [English edition, Figuring Space: Philosophy, Mathematics and Physics, Introduction, 2000, p. 12]. For this, one should be prepared to learn, “to put oneself into such a state where the connection between things reverberates in the connection of the mind”.18 To understand the movement of abstraction represented by the diagram as experience, one needs to understand the gesture it attempts to grasp. The gesture has characteristics that make of it a dynamic operator of knowledge, rooted in the body and its images, a purveyor of meaning by analogy, an indicator of potentialities, and a source of allusions. According to Châtelet, it is not substantial: it is inaugural (it opens up to problems); it is a modality of “to move”; it is elastic, it envelops and outlines its development; it leads to other gestures. It thus enables “all this talking with the hands (which it would perhaps be better to call this talking in the hands”) to be revealed; it is a “symbolic practice which is prior to formalism, this practice of condensation and amplification of the intuition” [9: 34] [English edition, Introduction p. 11]. This way of seeing and conceiving, through an intuitive grasp of the diagrammatic nature of phenomena, tends to erase the opposition between subject and object, and places the observer in an inside-outside, neither pure interiority nor complete exteriority. The theorization of language activity, with the hypothesis of the mental gesture, seeks to join up these different levels by responding, through the graph of the schematic form, to the demand for relations that is the diagram, according to Desanti; and to the experience of thought that is the gesture, according to Châtelet.

11.3.2 Diagrammatization of Thought and Mental Gesture The notion of mental gesture is based on the hypothesis that language activity is the product of symbolic activity through gestures, according to a process that transmutes an interiorized sensorimotricity into mental representations that are traceable in linguistic forms. In the 1983–1984 seminar Culioli spoke of grammatical notions as “representations of the order of bodily activity” [16: 23], which appeared curious at first. He also pointed to the filtering of culture over praxes, in the system of representations. He opened the 2005 seminar19 with the “speculative” hypothesis of language as product, the transformation of a symbolic activity through gestures, with a reference to the 17

Culioli was then alluding to Taoist philosophy and to the idea that artists seek to grasp the intimate form of their subject, its internal structuring principle (the li in Chinese). 18 Idem, p. 30. [English edition, Introduction, p. 8.]. 19 Oral seminar at the École Normale Supérieure, Paris.

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Platonic khôra (Timaeus), that intermediary place of transition from the corporeal to the symbolic, a receptacle—a void in waiting–, indicating what is situated between the sensible and the intelligible. He then referred to the work of the Indianist Charles Malamoud,20 Cuire le monde. Rite et pensée dans l’Inde ancienne (1989) [Cooking the World: Ritual and Thought in Ancient India], in particular to the chapter entitled “Les Chemins du couteau” [Paths of the Knife: Carving up the Victim in Vedic Sacrifice], and alluding to a study by Jean-François Billeter on Zhuangzi (2004).21 Charles Malamoud’s study deals with Vedic treatises concerning the sacrificial act, in particular the techniques used for dismembering the sacrificed animal. His conclusion on the reference made to this is enlightening: “The treatises on sacrifice are like a laboratory of discursive thought. With rites as their working material, the authors of the ´ Br¯ahmanas and the more technical manuals of the Srauta-S¯ utras provide an outline and, in some cases, even a precise formulation of some of the essential categories in Grammar” [41: 179].22 The reference to Zhuangzi is directed more specifically at a chapter in the works of the Chinese philosopher which, in Jean Lévi’s French translation [40: 31–33] is entitled “Principes pour nourrir sa vie ou l’hygiène du boucher” [Principles for Nourishing One’s Life or the Butcher’s Hygiene]. A story tells of Prince Wen-Houei’s surprise when faced with the spectacle of the cook, Ting, carving up an ox with surprising skill; upon which Ting replies to the prince that he is not so much interested in “technical skill as [by] the intimate being of things” (the li, see footnote 17). His experience has meant that he has constructed an internal representation of the “essential elements” and that he has acquired “an intuitive rather than a visual understanding” (Ibid.: 31). The establishment of a link between the sensory and the symbolic is in fact based on a reading of the Stoic philosophers, who were frequently mentioned in the oral seminar. During a round table event and a discussion with linguists who were adherents of Guillaume,23 Culioli gave a quotation from the philosopher and logician Claude Imbert, taken from an article on Stoic logic. This quotation evoked 20

The work of Charles Malamoud focuses mainly on Vedic ritual, based on his reading of the relevant philological and anthropological treatises. He attended courses given by Émile Benveniste from 1956 to 1962 and was a regular participant in Culioli’s seminars. 21 These two references are taken up and commented upon in a 2012 interview [24]. 22 The author continues as follows: “In a more general sense, the texts are an update of the elementary and fundamental questions suggested to, or rather imposed upon, the authors by their self-attributed task of providing an explanation for ritual acts: What does it mean to begin? What does it mean to proceed to the next step? How is one to understand that the same act can be both single and multiple? What is meant by ‘too much’ and ‘not enough’? What is the relationship between parts and wholes? What does it mean to measure? How are repetition and difference to be understood? How does one tell creative reiteration apart from harmful redundancy? […] It is tempting to think that Vedic theoreticians considered rites to be the systems of acts where such concepts operated with the greatest clarity and purity and that it was for this reason that they devoted so much ingenuity and pertinacity to their analysis and glorification” [41, 180]. 23 Gustave Guillaume, a French linguist who had a highly unusual career, at the margins of university institutions. He was the founder of “psychomecanics”, whose purpose was to analyse the operations at the root of linguistic signs—from language in its virtual state to the effectiveness of speech—in terms of cinetisms. A trend developed around his body of work, which was immense, although only a small part has been published. There are very few followers of Guillaume’s theory today but,

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the image of the octopus which, for the Stoics, served to illustrate “the unity of the soul in the diversity of its actions” [32: 234]. I will just give the last lines of it here: “Finally the physical model [of the octopus] shows that sensory activity does not end with the representation of the perceived object for oneself, but it persists, through the soul’s natural inclination, right up to its utterance. Enunciation, in its turn, is a representative activity which reduces that of the senses, for oneself and for others” (Ibid., 234, cited in Culioli [22: 43]). Over and above the variety of readings which contribute to Culioli’s thinking and the different realms we are invited to enter in order to understand language in all its complexity, these references converge on the principle of an imaginative activity through abstraction, which operates on the basis of sensations, perceptions, affections and actions in the world—a mental process of schematization, thinking by diagrams, such as Peirce describes and formalizes by means of graphs. Culioli is in agreement with Peirce’s idea that the symbol incorporates,24 and he quotes the following passage, commenting on it by stating that “The diagram represents a state of the body, and this is dynamic” [24: 143]: When I was a boy, my logical bent caused me to take pleasure in tracing upon a map of an imaginary labyrinth one path after another in hopes of finding any to a central compartment. The operation we have just gone through is essentially of the same sort, and if we are to recognize the one as essentially performed by experimentation upon a diagram, so must we recognize that the other is performed. The demonstration just traced out brings home to us very strongly, also, the convenience of so constructing our diagram as to afford a clear view of the mode of connection of its parts, and of its composition at each stage of our operations upon it [43: 420].

Culioli does not directly attempt to use this process but his observations and reasoning lead him to create a link between “the symbolic activity of a cognitiveaffective order of language” and the dynamic schema of the body proper, which is our primary system of reference. He thus declares: “What I know – well, what I believe I know about the matter (one should always have a certain degree of scepticism towards one’s own statements!) is that there are indeed operations that possess an abstraction enabling them to be applied to conduct in space”. Or again: “I am going to be dealing here with modes of representation that are actually of the order of cognitive functions, such as categorization, perception, or situation in a reference frame, which means what we do as well when we move within spaces, when we represent ourselves, let us say as in a room, and on a path in this room which will enable us to reach a particular place” [17: 178–179]. In an explanatory note during an interview in which he explains his thinking and methods (ibid.: 176), he gives the example of the way in which concession is constructed, based on a principle of causality for which we have a representation through our experience of the world. The relation of agency, which is a mode of causation, with an agent who acts with a recently, there has been a resurgence of interest in it, from a cognitive and philosophical point of view. 24 “A symbol incorporates a habit, and is indispensable to the application to any intellectual habit, at least. Moreover, Symbols offer the means of thinking about thoughts in ways in which we could not otherwise think of them” [43: 414].

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view to achieving an effect, depends on this principle, and the concessive appears as a relation of “anti-causality”, a relation in which a driving force is frustrated.25 Although he regularly returned to the question of corporeality and spatiality in language during studies of markers and utterances, particularly in his oral seminar, he always positioned himself within an interdisciplinary perspective, inviting people to turn to cognitive psychology and the neurosciences, as well as the philosophy of language and phenomenology, psychoanalysis and cultural anthropology—or the study of techniques and craftsmanship, all of them areas to be investigated “for any discussion on the ontogenesis of representations” (Ibid.: 40); this was without neglecting the attention that should be given to our own most everyday verbal and gestural conduct. And although, at an early stage in the theory’s development, there was the issue of corporal schema and bodily image, it is only in the last ten or so years that mental gesture has become central. It is an expression that links the corporeal—what is perceived by kinaesthesia and proprioception—to the mental (mind in English, or mens in Spinoza) and relates to a process that Marcel Jousse [33, 34], in his anthropology of the gesture, called intussusception (from the Latin suspicere, to amass, gather; and intus, from a movement which leads to the inner self), in order to describe the mental integration of gestuality. After many years, I arrived at mental gestures. I recall what it involved: it was about introducing the relationship between, on the one hand, our sensorimotor activity and, on the other, our gestures with a view to action; through this, we deploy our Représentations (representations, of a mental order, therefore), an action that is embedded within existing situations (hence Référenciation [referring], in the sense of “situating something”, rather than seeking a referent) and it is always accompanied by Régulation [regulation] – in relation to what one is in the process of doing, the comments one is making about it, controversies with others, etc. [18: 8]

As a conclusion to his speech at the end of a symposium devoted to his work, Culioli summed up his scientific activity which “posits an essential link between linguistics (language, languages, etc.), anthropology, philosophy, the domain of the psyche, biology and formal disciplines” [27: 368], declaring that “Gestures, actions, values, stories, representations, activity of a physical and mental order—this is the area mapped out by my programme of work.” (ibid.: 372). This programme is akin to other conceptions and models of language which presuppose a close link between bodily activity and symbolic activity,26 and can be compared with the numerous studies based on the premise of embodied cognition [45], according to which cognition depends on experiences linked to a body endowed with a variety of sensorimotor capacities, in an environmental, biological, psychological and cultural context, based more specifically on the idea of embodied action or enaction.

25

“And in the case of the concessive, it is precisely a kind of counter to this: you do everything that you can, you make all the efforts you want, the fact remains that it does not prevent it coming to nothing; whereas, normally, your efforts should lead you to” [16: 177]. 26 Some of these are presented in Ducard 2020.

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11.4 Questions by Way of Conclusion The hypothesis of mental gesture remains contingent on a fundamental questioning of this process of transformation or transmutation of sensorimotor activity, through the imagination, into an activity of symbolic representation, in the signifying activity of language. The theory of enunciation, in its version of a linguistics of predicative and enunciative operations, is not a genetic theory seeking to explain the origin of linguistic signs by mental patterns for which they might be forms of expression. It does not offer a theory about the origin of language, despite the fact that many studies [36–12] reinforce the idea of a gestural foundation for verbal language. The model I have presented, in which metalinguistic representations are representative of representations and of language operations of a cognitive-affective order, constructed from a study of the empirical forms produced, should be seen as a model of form generation. Linguistic markers used in the act of production and interpretative recognition of texts are described by Culioli as sensors or triggers of meaning; this meaning has to be sought in notions that are labile and deformable according to the contingencies of utterance, and also linked to experiences of the world (events, objects, states of things, subjective states). And these notions are “caught in the trap of words”, in a language that is a virtual system of forms. I have referred to the theories of embodied and situated cognition, which can be linked to the hypothesis of mental gesture in language activity. From this perspective, I have been particularly interested in the work of Lawrence Barsalou27 and his laboratory of psychology and neuroscience (Barsalou Lab, University of Glasgow), whose model of human conceptual processing fits with the idea of a diagrammatic nature of abstract forms in language. The core thesis of this theory of knowledge is that the conceptual system is grounded in multimodal simulation, situated conceptualization,28 and embodiment. The concepts or perceptual categorizations constitute the Perceptual Symbol System, and perceptual symbols, largely nonconscious, are schematic representations that are multimodal, across multiple sensory modalities, sensorimotor, proprioceptive, and introspective. A concept is a simulator and a simulation is a physical experience or imagined experience of a simulator. A theoretical model by the name of LASS: LAnguage and Situated Simulation theory, integrates the linguistic approach and the modal approach, by coupling the system of modal simulations anchored and situated in the body with the system based on linguisticcontext vectors: “Thus, activated linguistic forms serve as pointers to simulations that are potentially useful for representing the cue word’s meaning” [5: 250].29 This 27

Lawrence Barsalou is Professor of Psychology at the University of Glasgow, carrying out research at the Institute of Neuroscience and Psychology. 28 “From this perspective, a category is typically simulated in diverse situations. Depending on the situation currently relevant, a different situated simulation is produced. […] One proposal is that the brain is a situation-processing architecture whose primary function is to capture and later simulate situated conceptualizations” [6: 15–16]. 29 Barsalou’s model is along similar lines to one of the hypotheses put forward by Per Aage Brandt and Ulf Cronquist “That diagrammatic signs are the active components of the abstract mental

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returns us to the idea of linguistic markers as triggers and sensors of representations (see above). This model does not answer all the questions posed in relation to the abstraction process, notably those regarding the construction of types of situations, through memorization of similar recurring situations and through the formation of frameworks or patterns. Neither is it satisfactory where interaction between the two systems—the simulation system and the linguistic system—is concerned: “In general, we assume that linguistic forms provide a powerful means of indexing simulations (via simulators), and for manipulating simulations in language and thought”. (Ibid.: 251) And, in the authors’ view, the general idea according to which structures and symbolic operations emerge from continuous interactions between language and simulation systems still needs to be documented by experimental research. In LASS theory “linguistic system” is used to distinguish between linguistic forms and situated simulations, and does not correspond to the linguistic representation of meaning that is largely represented in situated simulation. Linguistic forms thus relate partly to simulations and partly to other linguistic forms, by association. The relationship between the simulation process and linguistic forms is therefore not clearly established. My thinking, in the wake of the linguistics of predicative and enunciative operations, leads me to consider language as a dynamic process in which sensations, perceptions, actions, mental states and affects are interpreted and then configured into the formal system of a language. Based on the principle that signifying forms are markers of notions and that texts bear traces of the representations and mental operations that produce them implies that language itself is a system of simulation. Autonomous, complex and self-regulated, it is at the origin of a superior order of consciousness according to the biologist Edelman [30, 31]30 : a consciousness of the consciousness of our experience of the world and consciousness of one’s own activity, at different levels of consciousness. Peirce enables us to make a link between this speculation and the question with which we are concerned. Referring to the semiotic triad (Icon, Index, Symbol) he says that words, generally, cannot be mere Symbols, through a relationship of conventional denotation with Objects, and Index, through connection with the speech act at a specific time and place, but they are also Icons: “The arrangement of the words in the sentence, for instance, must serve as Icons, in order that the sentence may be understood. The chief need for the Icons is in order to show the Forms of the synthesis of the elements of thought. For in precision of speech, Icons can represent nothing but Forms and Feelings” [42: 512–513]. Thus the diagram, as it is represented in figuration in working memory that accompany perception and deliver the schematizing ideas which invest and connect our categories and construe proper descriptive, narrative patterns of inferential thought and programs of sensorimotor and social action” [8, abstract]. 30 In his theoretical model of brain functioning, the neurobiologist Edelman [30, 31] distinguishes primary consciousness, involving a “scene” in response to objects and events and connected by memory to a former experience associated with values and awareness of a superior order. This implies semantic aptitudes, with which certain primates are endowed, enabling a symbolic activity of representation to take place, made possible by language, and thus creating the potential for a consciousness of consciousness.

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the graph of the schematic form, contains an image of the mental gesture, which is itself an image of a complex and dynamic process. It is an image in the mathematical sense31 and by analogy, that is to say an image through the language function whose antecedents, still in the mathematical sense,32 are representations and operations linked to our sensorimotor activity and captured in linguistic forms. Consciousness of consciousness and image of image, the reduplication and reflexivity that belong to language activity, therefore justify the term—meta-morphosis—chosen for the title of this paper, in reference to the process of “symbolic transmutation” invoked by Culioli.

References 1. Alunni, C.: Des Enjeux du mobile à L’Enchantement du virtuel–et retour. In: Alunni, C., Paoletti, C. (éds.) Gilles Châtelet, L’enchantement du virtuel. Mathématiques, physique, philosophie. Éditions Rue d’Ulm (2010) 2. Armstrong, D.F., Stokoe, W.C., Wilcox, S.E.: Gesture and the Nature of Language. Cambridge University Press (1995) 3. Bachelard, G.: La formation de l’esprit scientifique, Paris, Vrin (1970) 4. Bachelard, G.: The Formation of the Scientific Mind. Introduced, translated and annotated by Jones, M.M. Clinamen Press, Manchester (2002) 5. Barsalou, L.W., Santos, A., Simmons, W.K., Wilson, C.D.: Language and simulation in conceptual processing. In: De Vega, M., Glenberg, A.M., Graesser, A.C. (eds.) Symbols, Embodiment, and Meaning, pp. 245–283. Oxford University Press, Oxford (2008) 6. Barsalou, L.W.: Situated conceptualization: theory and applications. In: Coello, Y., Fischer, M.H. (eds.) Foundations of embodied cognition: perceptual and emotional embodiment. pp. 11– 37, Routledge/Taylor & Francis Group (2016) 7. Brandt, P.A.: La pensée graphique.–Pour une sémiotique des diagrammes. Visible No. 9, Visualisation et mathématisation, Maria Giulia Dondero et Sémir Badir dir., Limoges, PULIM (2012) 8. Brandt, P.A., Cronquist, U.: Diagrams and mental figuration. A semio-cognitive analysis. Semiotica, N° 229 (2019). En ligne: https://www-degruyter-com.ezproxy.u-pec.fr/view/journals/ semi/2019/229/article-p253.xml?tab_body=abstract 9. Châtelet, G.: Les enjeux du mobile. Mathématiques, physique, philosophie, Paris, coll. Des travaux, Seuil (1993) 10. Châtelet, G.: Figuring Space: Philosophy, Mathematics and Physics. Translated by Shore, R., Zagha, M. Kluwer Academic Publishers, Dordrecht, Boston, London (2000) 11. Corballis, M.C.: From Hand to Mouth, the Origins of Language. Princeton University Press, Princeton and Oxford (2002) 12. Corballis, M.C.: Evolution of language as gestural system. Marges linguistiques–N°11, mai 2006, M.C.M.S. éditeur, pp. 218–229 (2006). Disponible en ligne: http://www.marges-lingui stiques.com 31 Let us remind ourselves of this simple definition of the image in mathematics: “In mathematics, we say that y is the image of x through the function f if y = f (x). By extension, the set of y elements for which there is an antecedent in E are called the image of a part E through an f function. For each y of the image set we can find an element x of the definition set, such as y = f (x)”. (https:// www.techno-science.net/definition/10532.html). 32 “In mathematics, given two non-empty sets E, F and a map, we call any element x of E such as f (x) = y the antecedent (by f ) of an element y of F”. (https://www.techno-science.net/definition/ 10523.html).

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13. Culioli, A.: Pour une linguistique de l’énonciation. Tours et détours, Tome IV, Limoges, Lambert-Lucas (2018) 14. Culioli, A.: Pour une théorie des opérations énonciatives, T. 3, HDL, Ophrys (1999b) 15. Culioli, A.: Nouvelles variations sur la linguistique. In: Vivre le sens, Paris, Seuil, pp. 113–145 (2008) 16. Culioli, A.: Notes du séminaire de DEA 1983–1984, Université de Poitiers (1985) 17. Culioli, A.: Variations Linguistiques, Klincksieck, Paris (2002) 18. Culioli, A.: Gestes mentaux et réseaux symboliques : à la recherche des traces enfouies dans l’entrelacs du langage. In: Faits de langue, n°3, Paris, Ophrys, pp. 7–31 (2011) 19. Culioli, A.: Pour une théorie des opérations énonciatives, T. 1, HDL, Ophrys (1990) 20. Culioli, A., Normand, C.: Onze rencontres sur le langage et les langues, HDL, Ophrys (2005) 21. Culioli, A.: Table ronde. Intervention d’Antoine Culioli. In: Lüdi, G., Zuber C.-A. (éds.) Contributions à l’École d’été de la Société Suisse de Linguistique, Sion, 6–10 Sep. 1993, ARBA 3 (1995). Disponible en ligne: https://emono.unibas.ch/catalog/book/33 22. Culioli, A.: Rapport sur un rapport. In: Joly, A. (éd.) La psychomécanique et les théories de l’énonciation. Presses Universitaires de Lille, pp. 37–47 (1980) 23. Culioli, A.: Pour une théorie des opérations énonciatives, T. 2, HDL, Ophrys (1999a) 24. Culioli A., Ducard D.: Un témoin étonné du langage. In Espace théorique du langage. Des parallèles floues, Cl. Normand, E. Sofia dir., Bruxelles, academia, pp. 129–172 (2012) 25. Dagognet, F.: Écriture et iconographie, Paris, Librairie philosophique J. Vrin (1973) 26. Desanti, J.T.: Réflexions sur le temps. Variations philosophiques 1, Paris, Grasset (1992) 27. Ducard, D., Normand, C. (eds.) Antoine Culioli, un homme dans le langage, Paris, HDL, Ophrys (2006) 28. Ducard, D.: Le graphe du geste mental dans la théorie énonciative d’A. Culioli. In: Les linguistes et leurs graphiques, Cahiers Parisiens/Parisian Notebooks, vol. 5, pp. 555–576. The University of Chicago Center in Paris (2009) 29. Ducard, D.: La formalisation dans la théorie des opérations énonciatives: formes, formules, schémas. In: Dossiers d’HEL, SHESL, Écriture(s) et représentations du langage et des langues, vol. 9, pp. 113–122 (2016). Disponible en ligne: http://htl.linguist.univ-paris-diderot.fr/hel/dos siers/numero9ethal-01304863 30. Edelman, G.M., Tononi G.: A Universe of Consciousness. How Matter Becomes Imagination. Basic Books (2000) 31. Edelman G.M.: Wider than the Sky. The Phenomenal Gift of Consciousness. Yale University Press (2004) 32. Imbert, Cl.: Les Stoïciens et leur logique, Paris, Vrin (1978) 33. Jousse, M.: L’Anthropologie du geste. Gallimard, Paris (1974) 34. Jousse, M.: L’Anthropologie du geste, II La manducation de la parole, Paris, Gallimard (1975) 35. La Mantia, F.: Et la structure en came? Notes pour une diagrammatologie énonciative. In: La Mantia, F. (éd.) Pour se faire langage. Lexique de base de la théorie des opérations prédicatives et énonciatives d’Antoine Culioli. Louvain-La Neuve, Éditions Academia (2020) 36. La Mantia, F.: Pour se faire langage. Lexique de base de la Théorie des Opérations Prédicatives et Énonciatives d’Antoine Culioli, Louvain-La Neuve, Éditions Academia (2020) 37. La Mantia, F.: From Topology to Quasi-Topology. The complexity of the notional domain. In: La Mantia F., Licata, I., Perconti, P. (eds.) Language in complexity. The emerging meaning. Berlin, Springer (2017) 38. Langer, S.: Philosophy in a New Key, A Study of the Symbolism of Reason, Rite and Art. Harvard University Press, Cambridge (1942, 1979) 39. Lenoir, T. (ed.): Inscribing Science. Scientific Texts and the Materiality of Communication. Stanford University Press. (1998) 40. Les Œuvres de Maître Tchouang, traduction de Jean Levi, L’Encyclopédie des nuisances (2006) 41. Malamoud, C.: Cooking the World: Ritual and Thought in Ancient India. Translated by David White, Bombay, Calcutta, pp. 169–180. Madras, Delhi, Oxford University Press (1996). “Paths of the Knife: Carving up the Victim in Vedic Sacrifice”

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42. Peirce, C.S.: Prolegomena for an apology to pragmatism. The Monist, vol. 16, no. 4, pp. 492– 546. Oxford University Press (Oct. 1906). Published by Oxford University Press: https://www. jstor.org/stable/27899680 43. Peirce, C.S.: Collected papers of Charles sanders Peirce. In: Hartshorne, C., Weiss, P. (eds.) The simplest mathematics, vol. IV. The Belknaf Press of Harvard University Press, Cambridge, Massachusetts (1960–67) 44. Pfeiffer, J.: Rôles des figures dans la production et la transmission des mathématiques. In: Images des mathématiques 2006, Paris, CNRS éd. (2006) 45. Varela, F.J., Thompson, E., Rosch, E.: The Embodied Mind: Cognitive Science and Human Experience. MIT Press, Cambridge, MA, USA (1991)

Chapter 12

Fluid Formalism Lionel Dufaye

In his article on Mental Gesture, Antoine Culioli, quoting René Thom, points out that “any scientific endeavor is based on an aporetic situation which the scientist tries to find a way out of” (2011: 64).1 The aporetic problem at stake in the present case could be formulated as follows: How can metalinguistic representations that have to be fixed on paper, at some point, be it by a diagram, a formula or a definition, account for the ever “overflowing”2 symbolic activity at work through linguistic markers? Indeed, since the metalinguistic level aims mostly at representing markers and operations as part of a given language-state, to what extent can it allow for the interpretation of diachronic evolution and semantic plasticity and impermanence? To quote Culioli: Symbolic meansthat there is always something that turns into something else; but that transformation, that metamorphosis has some form of coherence. The problem is how to account for it. (2011: 68)

Symbolic activity, which we understand is not to be confused with linguistic activity per se, is at the core of the epilinguistic process that “formalizes” thought into language: Formalization is a tool of the mind, like algebra [ … ]; it “corporifies” ideas to borrow Lordat’s words once again. (Laplane [10: 115]3 )

1

Our translation for this quote and the following ones by A. Culioli. In French, Antoine Culioli uses the word débordant. 3 Our translation and the following ones by D. Laplane. 2

L. Dufaye (B) Université Gustave Eiffel-LISAA EA4120, Champs-sur-Marne, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_12

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But there appears another paradox, in the sense that language, being made up of discreet markers, de facto fragments the diffuse process of thinking into discontinuous units. A consequence is that words should not be equated with the notions they point to: The words of a text do not carry meaning in themselves, they are triggers of mental representations. (Weinberg (in Dortier [5: 335]))

Or, as Culioli puts it: Words are a form of condensed version of notional representations. They capture notions: with a word one can access a notion. The word refers to the notion as a whole but that relation is not symmetrical: a notion is only partially captured by a word. So again, this is not a one-to-one relationship. (Culioli [2: 16])

Following that line of reasoning, according to which “markers represent representations” (Culioli ibid),4 one can surmise that the metalinguistic representation of markers should also not be equated with the notions that they refer to. Actually, even beyond the issue of lexical and morphematic discreetness, it is not unreasonable to assume that pre-assembled phrases and structural specificities contribute to shape meaning into imposed patterns. For the sake of demonstration, let’s imagine that someone falls asleep while reading a book or while watching a movie. They can describe what happened by saying I read myself to sleep, whereas *I watched myself to sleep is impossible. As we know, resultative structures accept only one NP in object position; however, contrary to an ambitransitive verb like read, with watch the object NP is mandatory, but of course *I watched a movie myself to sleep is ill-formed, since there would be not one but two NPs in object position. For the same reason, even though one can say that they read themselves to sleep, resultative structures do not make it possible for them to specify what they were actually reading, unless restructuring the syntax completely: *I read War And Peace myself to sleep. As Laplane observes: Language is likely to freeze the fluidity of thought in its gel, precisely because it fixes it through its constraints, which are also habits. Language should no longer be seen as a means to express thought, but as a frame to define and organize it. [ … ] Good writers are those who can develop their own use of language. Great thinkers are those who can detach themselves from its formalism. (2000: 118)

On the face of it, the problem of metalinguistic modelling would seem to rest on a twofold paradox: it aims at representing non-discreet mental activity (Level 1 in Culioli’s theory) through the use of discreet concepts (level 3) based on the observation of discrete markers (level 2). In agreement with Culioli’s approach, we will posit that, for it to account for semantic plasticity, metalinguistics has to rest on discreet dimensions from which dynamic relations can be drawn: Fixist or physical approaches would map on langage descriptions such as those we find in physics. To say that we are dealing with a network means that we have a network system, whatever its meaning may be and it can be represented as something like this: 4

«Les marqueurs sont les représentants de représentations.»

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Each of these elements stand for something; then we establish relations and ponderations among them. The simplest example — which is not even a network! — consists in defining an occurrence with Qnt—Qlt5 dimensions, with either equiponderate value, or Qnt/Qlt (Qnt preponderant over Qlt), or inversely Qlt preponderant over Qnt. (64–65)

Although componential in its essence, this conception has little to do with a Katz and Fodor type of semantics (1963); the components that it rests on are not predefined semes (e.g. animate, female, adult, …) but conceptual primitives drawn out from trans-linguistic and trans-categorial analyses. Although connexionist in appearance, it also has little to do with [1] semantic organization, as Culioli insists, it “is not even a network!”. Contrary to a static meshwork of interwoven notions, Culioli’s model is conceived as a modular representation of primitives allowing for dynamic operations between them. Mental gestures are to be construed as active processes whereby meaning is not so much “retrieved” as it is “built”.6 In our view, for it to be in keeping with this approach, the concept of schematic form should be thought of as a modular configuration implying a set of abstract, fundamental features, rather than a synthetic “invariant” formula (precisely because “variance” needs to be accounted for). It is trite to mention that etymology and language change teach us that one same word base in two different languages will take different paths,7 but the interesting conjecture that can be drawn from that is that although one cannot prospectively assess how a word will evolve, one may account retrospectively for why it evolved the way it did. In other words, language change may be unpredictable and random a priori, but it is nonetheless analysable and consistent a posteriori. It is through the “traces” [3: 68] left by markers that the linguist sets out to investigate the semantic consistency in actual use. For instance, drawing attention to the Catalan word gairebé, meaning almost, Culioli explains: Gairebé is made of gaire, which is the same word (except for the meaning, of course) as guère; and bé which means bien. [ … ] Gaire comes from a Germanic word (Frankish *waigaro) which meant “resist, refuse, not give in”; so a possible gloss, widely accepted actually, would be: to have the strength. In Old French the word became gaire “strong, much”, then in the negative form ne…guère: “little, fairly little”. All of this is the result of symbolic activity. Words are not set in stone once and for all; they can be deformed, their initial value stretching into a different shape. (2011: 65–66)

5

Qnt stands for Quantity, i.e. the extensional or existential value of an occurrence, whereas Qlt stands for Quality, its notional or appreciative values. 6 Although the apprehension of semantic operations as mental gestures paralleling corporal activity was developed by Culioli in the course of the 2010s, its seed seemed to have been there as least since the seminal notes published in 1985: [ …] notions are representations pertaining to corporal activity. Culioli [2: 23] (les notions qui sont des représentations de l’ordre de l’activité du corps). 7 Sometimes radically so, for instance, although complete opposite, black and blank, from French blanc meaning white derive from the same Indo-European root *bhel-.

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One among many, this example highlights the fact that there is no anticipating that the co-occurrence of two words meaning much and well can come to mean “almost”, which shows that, on the epilinguistic level, symbolic activity comes up with a language-specific means of formalizing infralinguistic activity, exploitating semantic ductility to stretch meaning into new uses. Thus, metalinguistics requires a type of representation that makes it possible to specify from which semantic dimension any such derivation occurs. To do so, we posit that meaning has to be configurable in such a way that semantic variables can be singled out so as to make it possible for us to identify which specific dimension was foregrounded to extend the use of the marker. A further reason for elaborating schematic forms that can be broken down into discreet features pertains to the question of local synonymy. If one given context allows for two markers to alternate relatively freely (e.g. Elephants MAY be dangerous and Elephants CAN be dangerous) whereas another context prevents the same two markers from alternating or else brings forth radical differences (e.g. MAY the force be with you and *CAN the force be with you), one can hypothesize that one facet of their schematic form is shared while others are not. Deschamps and Dufaye [4] apply this principle to propose an articulation of modal auxiliaries in English based on modular operations: The assumed representation must first account for all the possible values generally attributed to each modal in all kinds of different contexts. It must also allow us to deal with the similarities, quasi-equivalences, compatibilities and incompatibilities of the modals and nonexisting patterns, that is it must explain why commutations between modals are permitted (for instance SHALL and WILL, CAN and MAY, WILL and CAN, SHALL and MUST, WILL and MUST) as predicted by the theoretical model itself, or prohibited (SHALL and CAN, SHALL and MAY, CAN and MUST, MAY and MUST…). (Deschamps and Dufaye [4: 130])

Based on this assumption, the authors propose to define each modal by a set of two operations—a Quantitative (i.e. existential) one and a Qualitative (i.e. notional or subjective) one—either one of which is shared by at least another modal.8

When two connected modals appear in a context where the shared operation is compatible, an alternation is conceivable with related modal values—related though never quite synonymous in the sense that the second operation preserves a kind of semantic ponderation.9 As such, this schematic type of representation can be seen as 8

I and E stand for Interior (validation of the process) and Exterior (validation of the process). Each modal takes into account one or both scenarios. When a scenario is closed off (MUST and SHALL) it is conceivable but considered undesirable or incompatible. 9 See Deschamps and Dufaye [4] for detailed illustrations.

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a diagram. However, Culioli insists that it is only under the form of a dynamic graph that this sort of representation can possibly modelize markers as the trace of mental gestures: We are dealing with something that is not only deformable, but also malleable, etc. The first problem we face is that of the graph. It is oriented, therefore it is not a diagram. Our graph became more and more complex as it evolved. Its purpose was to resolve the problem of fixism that we exposed in our 1968 article.10 I wanted to avoid cases where we consider a word and say: “That word means what it means”. (70)

It might not be an unfitting comparison to say that, in the same way Chomsky reintroduced mentalism by positing syntactic movement between a deep and a surface structure, Culioli’s theory on mental gesture posits cognitive movement among primitive components. With regard to Deschamps et al.’s representation, it is through a graph based on such diagrams that the logic behind modal operations and values can be “visualized”. If we come back to the previous example where MAY and CAN alternate relatively freely: Elephants MAY /CAN be dangerous, the reason they appear to be interchangeable is because both operations are centered on the same quantitative scenario, namely the expression of both positive and negative validations in a generic context (or so called “sporadic modality”). Even in that context the two markers are never entirely synonymous, of course, in the sense that they correspond to different “gestures”:

While MAY rests on a notional representation of positive and negative scenarios as both conceivable to envisage occurrential situations, CAN rests on situational experience or knowledge to predicate a potential notional property. Accordingly, An elephant can be dangerous (QLT oriented) is fine since an elephant is read as a notional occurrence, whereas ?An elephant may be dangerous (QNT oriented) would call for a form of situational anchoring: An elephant may be dangerous if you creep up on it. To bring theory and practice together, we shall take this discussion on mental gestures further with an analysis of the prepositional/adverbial particle OVER. Cases of semantic proximity such as Fido slept with his paw over my leg/Fido slept with his paw on my leg or The night is over/ The night is through will serve to illustrate how meaning is built by gestures linking one dimension to another within a network of componential semantic features. 10

«La formalisation en linguistique», Cahiers pour l’analyse, n° 9, Paris, Éditions du Seuil, 1968, pp. 106–117. Repris dans Culioli, Antoine, Pour une linguistique de l’énonciation [ci-après PLE], Formalisation et opérations de repérage, Paris, Éditions Ophrys, t. 2, Coll. “L’homme dans la langue animée par Janine Bouscaren», 1999, pp. 17–29.

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For the purpose of brevity, we shall not go into the heuristic intricacies or past experiences of other markers that led us to hypothesize that the various uses of OVER might be apprehended through a minimal set of three semantic components. Namely a topological component, a Qualitative component (henceforward QLT), Quantitative component (henceforward QNT), whose operations can be summed up as follows: α. Topological: localization relative to the locator’s surface (or “closure”). β. QNT: extensional path γ. QLT :asymmetrical valuation

The three components will be addressed alternatively, for each implies specific synonymic11 areas, and for the purpose of argumentation we shall start with the QLT dimension (3d component in our list). Component γ: asymmetrical valuation As we know, in spatial contexts, OVER mostly localizes the locatum in an upper position relative to the locator12 : 1. The sign over the door said “Exit”. Yet the up-down axis is not semantically primitive but contextually derived from a more fundamental QLT alterity: X over Y implies an asymmetrical valuation in which the locatum X is associated to a superior value. This asymmetry may apply to any form of relation implying a hierarchy of values: salience, intensity, preference, ponderation, gradience, or event spatial configuration.13 For instance: To go over the speed limit, to produce over a certain amount, to choose one thing over another… In spatial contexts, the asymmetry, which consistently gives prevalent value to the locatum, corresponds to a high (locatum)/low (locator) positioning. Component α: localization relative the locator’s topological surface (or “closure”) Just as the primitive QLT component γ can apply to either spatial and non-spatial values, topological configurations, although somehow at the core of spatial representations, can also further apply to non-spatial uses. Let’s take a spatial context to begin with: A swarm of flies gathered over the pond for the sake of illustration— here, the locatum (i.e. a swarm of flies) is in the neighborhood of the closure (i.e. the 11

It goes without saying that ‘synonymic’ is not to be understood as semantic ‘sameness’ (which is why we preferred it to the more colloquial adjective ‘synonymous’); in our approach, ‘synonymy’ is used for want of a better term to describe cases where possible substitutions have a similar enough meaning. 12 “Covering” values such as She placed the plywood over the hole in the ceiling imply that we take component 3 into account. 13 Metaphorical uses like I’m down, I feel low, I hit the bottom This whole operation is going south, I’m under the weather… … as contrasted with I have the upper hand, I’m at the height of my fame, I’m at the top of my game… reveal explicitly how this axis is valued.

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surface) of the locator (i.e. the pond). Put differently, we can say that the locatum is “in the Exterior of” the topological space derived from the locator, and yet it is still in its reference field. To better understand this compound localization, one may contrast OVER with THROUGH et ACROSS, for which Lindstrombreg provides the following comparison: If I hear that you walked through a stream, I assume you got your feet wet. If I hear that you walked across a stream, I do not know whether you did or did not [. . .]. If I hear that you walked (or rather jumped) over a stream, I can safely assume that your feet remain dry. (Lindstromberg [11: 127])

To rephrase this in more theoretical terms: – Unlike THROUGH, with OVER the locatum is in the Exterior of the domain defined by the locator: the existence of the compound THROUGHOUT is evidence that THROUGH refers to the interior of the domain, whereas *OVEROUT does not exist; – Unlike ACROSS, with OVER the locatum is localized relative to reference field of the locator: with ACROSS neither the Interior nor the Exterior is relevant, simply the dissociation of the locatum and the locator: She sat across from me (=There was a gap between her and me). In terms of localizing operations, THROUGH can be described as an identification, OVER as a differentiation, ACROSS as a disconnection. In the form of a topological space we get:

Let’s compare OVER to another marker which shares component γ, namely ABOVE. Their shared component (asymmetrical valuation) may make their meaning in spatial contexts hardly discernible: The sign over/above the door said “Exit”. ABOVE has often been analyzed as the expression of a gap or an absence of contact (e.g. [7, 11, 12]), contrary to OVER which is not incompatible with a representation where there is contact between the locatum and its locator. The difference here again is that OVER expresses differentiation (i.e. outside but relative to) while

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ABOVE expresses disconnection. Whereas The sign above the door is a mere localization of the sign in a higher position relative to the door (shared component α), The sign over the door refers to a more holistic relationship between the sign and the door as forming a cohesive, or expected, representation. Accordingly, in the phrase over and above the relative alterity expressed by OVER is subsequently amplified by ABOVE; the same gradation effect applies to over and out, or out and away, sequences which cannot be reversed: *above and over, *out and over, *away and out. Lindstromberg provides another comment which reinforces this view: (2) Dale’s flat is somewhere above Jill’s, in the same building. That is, if Jill’s flat is on the 3rd floor, then Dale’s must be on some higher floor. It may be directly up from Jill’s or on the opposite side of the building. If, on the other hand, one says, (2.1) Dale’s flat is somewhere over Jill’s... then that would suggest that Dale’s flat was directly up from Jill’s, on some higher floor. (Lindstromberg [11: 112–113])

With ABOVE the locatum (Dale’s flat) is localized relative to a scale independent from the locator (Jill’s flat); conversely, with OVER the locatum is localized within the extension of the locator, and thus remains within its reference space. These two modes of localization (i.e. OVER = differenciation vs. ABOVE = disconnection) provide a key to interpret the non-spatial forms of alternations between the two markers. Eric [6] quotes Michael Swan14 : ABOVE is used in measurements of temperature and height, and in other cases where we think of a vertical scale. The temperature is well above zero. We usually use OVER, not ABOVE, to talk about ages and speeds, and to mean ‘more than’. You have to be over 18 to see this film.

In keeping with Gilbert’s analysis, who also describes ABOVE in terms of disconnection and OVER in terms of differenciation, one can surmise that temperatures are calibrated relative to external scales (Celsius, Fahrenheit), whereas age corresponds to an intrinsic property, possibly associated with qualitative predicates such as being allowed to, being of age, having the duty to, etc.: 2. Children over the age of 12 must have full-price tickets. Indeed, differentiation allows for the expression of alterity within cohesive representations; for that reason, OVER (unlike ABOVE) is prolifically used to derive notions: overhead, overspeed, overreact, overdose, overtime, overwork, overrated; or set phrases expressing prototypical situations: To go arse over tit/head over heels/ hand over fist; or else compound adverbials: over the counter, over the top … The 14

Swan, M., (1995), Practical English Usage, Second Edition, Oxford University Press: 4.

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following example illustrates the difference between differentiation/relative alterity (OVER) and disconnection/absolute alterity (OUT): 3. “MILO, Air One, Air One, follow the Senator, roger? I’ll take the sport boat, you take Baynard, over.” AIR ONE, That’s a roger. Over and out.

OVER expresses discontinuity (end of the message) within a homogeneous domain (a communication). In contrast, OUT expresses the exit from that domain and consequently radical disconnection (end of the communication itself). Thus, OVER is suitable to express notional, cohesive linking between the locatum and its locator, and not just a positional relation between notionally distinct elements. A structured network of semantic values is emerging from those alternations. Both OUT, ABOVE and ACROSS have been analyzed as markers of disconnection relative to the Interior, but only ABOVE shares the QLT component γ (asymmetrical valuation) with OVER, and only ACROSS shares component β (extensional path), as we shall see it in the next section. If we note component types α, β, γ …, and component subtypes 1, 2, 3…, we can put forward a heuristic diagram where each marker is defined by a conjunction of discrete component subtypes:

In the last section of this article, we shall try to make the case that mental gesture operates by either pondering certain subtypes and/or by establishing new links. But first, a few words should be said about the third component type β. Component γ: Extensional Path In an example like The dog ran over the lawn„ two semantic components are at stake: a localization relative to the locator’s closure (i.e. the locatum (car) is on «the outside»); the locatum moves along the locator’s closure (its surface). This latter scanning value is manifested in a variety of uses of OVER: 4. He would wander over the moors, losing all track of time. 5. Mind you don’t spill coffee over my best tablecloth. 6. Come over here.

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The extensional component accounts for the frequent—and sometimes mandatory— cooccurrence of OVER with the totalization quantifier ALL, which somehow reinforces the path by scanning the entirety of the locator’s surface: He had blood all over his face. As it appears if we substitute ON for OVER, without the scanning operation conveyed by OVER, the totalization quantifier is incompatible: *He had blood all on his face. It is also noteworthy that ALL is likely to introduce an appreciative modality: 7. Dogs were barking all over the place. 8. If I can order something with garlic on the first date and she’s all over it, she’s a keeper. The hyperbolic value that that modality may take on overshadows the asymmetrical valuation described in spatial (and other) uses: 9. She jumped over the table: The qualitative asymmetry is preponderant (with an “on top” value). 10. She jumped all over the table: The extensional path is foregrounded with a hyperbolic value. Based on the same model, OVER and ON would appear to share two semantic components, but differ as far as the extensional dimension is concerned.

By foregrounding the extensional scanning, ALL both backgrounds locating and Qualitative components and explicitly blocks alternative extensional values. Beyond the specificity of micro-analyses, such examples serve to illustrate the more general principle that meaning is built through co-ocurrences and interactions that induce dynamic salience within isomeric relations. It is thanks to the contrastive study of markers within a paradigm of more or less synonymous contextual values that semantic structures emerge, thus making it possible for us to set a network under the guise of a diagram. Such diagrams, however, are at best a static cartography of the wefts and warps of the semantic fabric markers are made of. It is the representation of operations on such networks that account for semantic plasticity and ductability. Only graphs can indeed provide a representation of dynamic meta-operations (i.e. operations on components) such as the ponderation of certain components, the stretching of certain dimensions or else the local interaction with components induced by other

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markers. The distinction between static diagrams and dynamic graphs somehow echoes the concepts of khrema (what is available) and pragma (what we make of it).15 To reiterate our main point, we hypothesize, alongside Culioli’s theory on mental gestures, that only a two-tier mode of representation is able to somehow modalize the shifting nature of language through fluid formalism.

Bibliographical References 1. Collins, A.M., Quillian, M.R.: Retrieval time from semantic memory. J. Verbal Learn. Verbal Behav. 8, 241–248 (1969) 2. Culioli, A.: Notes du séminaire de D.E.A. 1983–1984, éditées par le Département de Recherches Linguistiques: Université Paris VII (1985) 3. Culioli, A.: Gestes mentaux et réseaux symboliques: à la recherche des traces enfouies dans l’entrelacs du langage. In: Pour une linguistique de l’énonciation IV, Tours et détours. Lambert Lucas, Limoges, pp. 61–89 ([2011] 2018) 4. Deschamps, A., Dufaye L.: For a topological representation of the modal system. In: Modality in English, Proceedings of the 2d International Conference, University of Pau, Mouton de Gruyter (2009) 5. Dortier, J.-F.: Le Cerveau et la pensée. Sciences Humaines éditions, Auxerre (2012) 6. Gilbert, E.: Across, by et through, Considérations sur les conditions de représentation métalinguistique des prépositions. Anglophonia, n° 14, Presses Universitaires du Mirail, pp. 37–61 (2003) 7. Gilbert, E.: Remarques autour de quelques prépositions. Actes du colloque de Cerisy 2005, Antoine Culioli, Un Homme Dans la Langue. Ophrys, Paris (2006) 8. Katz, J.J., Fodor, J.A.: The structure of a semantic theory. Language 39, 170–210 (1963) 9. Kreitzer, A.: Multiple levels of schematization: a study in the conceptualization of space. Cogn. Linguist. 8, 291–325 (1997) 10. Laplane, D.: La Pensée d’outre-mots, la pensée sans langage et la relation pensée-langage. Institut d’édition Sanofi-Synthélabo, Paris (2000) 11. Lindstromberg, S.: English Prepositions Explained. John Benjamins Publishing Company (1997) 12. Stunell, K.: The Construction of Sense in English Particle and Prepositional Verbs: Focus On OVER. Thèse de doctorat (Geneviève Gillet-Girard dir.). Université Sorbonne Nouvelle-Paris 3 (2010)

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Culioli [3: 41].

Chapter 13

But What About the Cam Structure? Notes for an Enunciative Diagrammatology Francesco La Mantia

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

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.

13.1 The Story of a Gap After reading Les mots de la linguistique, the sole essay, to my knowledge, in which the metalinguistic status of Culioli’s theory had been assessed, I became convinced that some measure of clarification was necessary. The book, rather excellent I must say, nevertheless left me with a feeling of dissatisfaction that increased as I continued to read Culioli’s articles and seminars. F. La Mantia (B) Dipartimento Di Scienze Umanistiche, Università Degli Studi Di Palermo, Palermo, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_13

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In particular, rereading Marie-Line Groussier’s and Claude Rivière’s work several times over, I wondered why the glossary lacked material regarding multiple specific points. But most of all, I wondered why there was nothing to be found regarding the notion of the cam structure. This topological object, which Culioli introduced in 1968, and which had exercised such a great fascination upon theorists of the calibre of Jacques Lacan, Louis Althusser, and René Thom, well, this object was simply left without mention by Groussier and Rivière. This is why I have decided to write these few notes. And this, not so much to fill a void but, rather, to propose a reflection at its margins, and, to a certain extent, to show its elusive character. Excusatio non petita? Maybe so. In any case, the metalinguistic question of the cam structure remains open. In order to better understand this stalemate, I think it necessary to first adequately visualise what Culioli meant by “cam structure”.

13.2 The Formal Enigma of the Cam Structure I reproduce below the figures found in La formalisation en linguistique, which is, if you will, the foundational text of the theory of predicative and enunciative operations.

As we can see, they are mostly curvilinear structures that, so to speak, intrigue both the eyes and mind. Beyond their aesthetic appeal, each figure constitutes, I believe, a little formal enigma. For my part, I will limit myself to the examination of the first figure, the others exceeding the objectives of these humble notes—they will be addressed, I hope, in future research work.

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So, the first figure. Culioli tells us that he called it a “cam” following a suggestion by François Bresson, and that the French word “came” is “un terme général pour renvoyer à une catégorie”.1 What the author stated a few lines later, that “certains des points considérés ici sont bien connus des mathématiciens”,2 clarifies the very technical sense in which both the words “category” and, consequently, the word “cam” are employed. Indeed, when mathematicians discuss “categories”, they are usually referring to abstract universes which are composed, on the one hand, of certain collections of objects, and, on the other hand, of certain collections of morphisms acting upon these objects, the morphisms being in turn subject to a whole series of rules upon which I will not dwell here for the sake of simplicity. The word “cam”, therefore, as a “terme général [qui renvoie] à une catégorie” [a general term which refers to a category], would seem to specify within the economy of Culiolian discourse a particular type of category in the mathematical sense of the term. However, upon closer inspection, this clarification complicates the conceptual framework of reference. In other words, if “cam” refers to a category in the mathematical sense of the term, then Culioli should have attended to at least two things: (1) giving, in advance, a much more thorough definition of “category” than the one I have just given; (2) explicitating the formal rules characterising the particular type of category referred to by the general term “cam”.3 However, if we return to the text of 1968, it appears that the author does neither of these things. And, of course, what astounds me is not so much the fact that he had not formulated a definition of the category—as such definition may be found in any essay on the topic of category theory (cf. for instance [2, 3])—but that he failed to actually specify the formal laws characterising a cam structure. Indeed, regarding this aspect, Culioli had always insisted upon the fact that the cam structure “n’est pas un jouet, une illustration pour soutenir l’intuition, mais un outil avec ses règles formelles d’emploi”.4 And yet, one could point out—and quite justifiably so—that the author never formulated these rules anywhere whatsoever. Naturally, what I have just said could come across as extreme, and, consequently, incur the risk of approximation, or worse, of falsehood. These are possible risks. But there are, I believe, at least three reasons which are strong arguments in favour of my pessimism: (1) notwithstanding Culioli’s unpublished texts—of which I do not know 1

Culioli [1]. 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]. 2

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whether they attend to the formal aspects of the cam structure or not—all (and I indeed say “all”) of his scientific production addresses this very delicate epistemological question; (2) when asking Jean-Pierre Desclés, who is both a mathematician and former collaborator of Culioli’s, if he was aware of any works by the latter having precisely for object the formal rules of the cam, he replied that “Culioli n’a jamais reparlé de la structure en came”,5 at least not in the terms I would have wished him to; (3) for a very great number of researchers having studied Culioli’s thinking, the cam structure remains a veritable mystery. Regarding this last point, I would like to quote a passage from Onze rencontres sur le langage et les langues where Claudine Normand voices the feeling of incertitude which surrounds the epistemological status of the cam structure: NORMAND: (…) Mais il y a plusieurs choses sur lesquelles je voudrais revenir. Il y a des choses qui m’intéressent particulièrement, ce sont les moments de votre itinéraire (…). Et aussi ce qui peut être, c’est de voir pourquoi il y en a dont vous ne parlez plus; est-ce que c’est parce que vous considérez que c’est acquis et que ce n’est pas la peine, la came, par exemple, ou bien… CULIOLI: elle est toujours là, comme je l’ai dit mardi dernier. NORMAND: Oui, vous l’avez dit mais depuis cinq ans que je suis votre séminaire, c’est la première fois que j’entendais le mot; donc ça veut dire que c’est acquis, voilà, vous n’éprouvez plus le besoin d’en parler? CULIOLI: Eh bien oui, c’est acquis, pas de doute.6

It immediately appears as though Culioli’s last words may be capable, maybe not of satisfying, but at least of mitigating the legitimate concerns expressed by Normand. But if we reread these few excerpts from Onze rencontres, as well as a few others from the same book where the cam structure is discussed, we will find that the doubts that beset Normand remain. Nowhere in the text may we find an answer to the linguist’s perfectly legitimate questions, meaning that nowhere is there any mention of the formal rules characterising the cam structure. So, what can be done? The first thing, presumably, would be to return to La formalisation en linguistique.

13.3 Spiralling Involutions and Topologies The few lines Culioli’s essay devotes to the cam are highly synthetic. In short, one can read that “il ne s’agit pas ici d’une involution: le schéma n’est pas à deux dimensions et a¯ 1 amorce une spirale, puis se projette en a¯ et le cycle recommence”.7 To begin, two remarks. First, we must be clear regarding the term “involution”. 5

Desclés, J.-P. (2019-04-09). Letter to Francesco La Mantia [personal communication]. Culioli and Normand [4: 73–74]. 7 Culioli [1]. 6

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In mathematics, this word generally designates an endomorphic operator which “composed with itself gives the identity”,8 that is, a morphism which has for domain and co-domain the same object (let’s say a type f morphism: A → A) and which, applied to itself two times in succession, yields its same initial value. For example, if we set A = Z and if we then define f by stating f (a)a ∈ A = –a, we get f ( f (a)) = a. By adopting a symbolism widely used in category theory, we can also write f◦f = 1A . But beyond the visual tracings that allow us to grasp the formal meaning of an involution, what we need to remember in order to understand a constitutive aspect of the cam is the fact that except for fixed points, an involutional operator is an endomorphism of period 2,9 or a morphism which, after two applicative steps, returns to the same initial value. Additionally, it may be useful to recall a last formal detail. I refer in particular to the fact that, within the framework of category theory, we can visualise an involution through an elementary semiography (or “diagrammatisation”10 ) which in turn rests upon very simple geometric intuitions. Indeed, if we follow a graphical convention which is very widespread among categoricians, it is common to associate planar points with elements of the domain (and co-domain) of f and directed arrows (or “planar vectors”) with the applications of f . This makes it possible in particular to achieve the following diagrammatisation (see the figure below):

Well, as we have just seen, this diagrammatisation allows us to visualise both cycles of length 2, represented by pairs of bi-directional arrows, and a fixed point, represented by a single arrow whose starting point coincides with the end point. And this is precisely the characteristic of a diagrammatisation having for object an endomorphism of period 2 which has Z for domain and co-domain. However, a cam in Culioli’s sense is anything but the diagrammatisation of an endomorphism of period 2 (without fixed points). In other terms, it is a diagram which does not instantiate second order cycles—and this, despite, as we will see later, that the cam presents particular affinities with some 8

Lawvere and Schanuel [5: 263]. 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). 9

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types of cycles (cf. § 6.1). However, this being said, it is an established fact that this topological object is irreducible to a cyclical structure of period 2. It is in such a sense that the assertion according to which “il ne s’agit pas ici d’une involution” [it is not an involution] and, I would add, it is in a similar sense that we must also understand what completes this first assertion, that is, the remark that “le schéma n’est pas à deux dimensions” [the schema is not two-dimensional]. Indeed, if we pay attention to the topology of the cam, the spiral it appears to form is constructed between the third and second dimension, or, more specifically, through a “parcours qui se fait simultanément dans la troisième dimension (le long de la verticale) et dans les deux premières dimensions (parcours circulaire dans le plan)”11 (see the figure below).12

And this trajectory, of course, is what makes it possible to diagrammatise the shifts typical of the cam, that is, the incessant transitions “d’une courbe descendante à une courbe montante”.13 Therefore, in this regard, the aspect to remember about this spiralling topology is that there is a projection of a¯ −1 onto a¯ , this projection being, if you will, the geometric trace of the shifts (and dynamics) that affect the cam structure. But what is the metalinguistic significance of all of this? Such is the question which allows me to introduce my second remark.

11

Victorri, B. (2019-12-20). Letter to Francesco La Mantia [personal communication]. I would like thank Jean-Pierre Desclés for kindly donating this image. 13 Coulardeau [8: 9]. 12

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13.4 The Metalinguistic Signification of the Cam Three letters mark the main applicative steps of the cam structure: a¯ −1 , a, a¯ . But regarding these letters, Culioli never said a thing. However, some very recent works have attempted to analyse this elementary notation. Among them, there is mainly an article by Nicolas Ballier which took care of explicitating the relationships underlying this type of formal sign. In his What Linguists Do When They Write Something: The Art of Stenography,14 the author first looked at the relative positions of each letter. In particular, he examined the positions of the signs forming the virtual pair “(¯a−1 , a¯ )”. Even more specifically, he noted in this regard that “Point a¯ −1 is actually exactly above point a¯ ”15 and that “The a¯ −1 versus a¯ indications are meant to show that the quasi-circular path is not a single horizontal plane, but a 3D representation where the start and the end point are not on the same plane”.16 Now, leaving aside the few slight inaccuracies committed by the linguist in determining the exact dimension of the cam,17 it is interesting to see what Ballier said about the respective metalinguistic roles of a¯ −1 and a¯ . By focusing on a specific use of the figure, i.e. on the one which interests the diagrammatisation of the possible values of the pronoun “il” (which, in French, is the masculine singular third-person pronoun), the author states that the cam shapes “the variation of meaning of certain markers”.18 From this point of view, therefore, a¯ −1 and a¯ , the one being the reverse of the other, would each represent a value of “il”, susceptible to contradictory interpretations in relation to the value that the other sign will represent. This is also what the author emphasises when, in addressing the relations between a¯ −1 , a, and a¯ , he observes that “The abstract a¯ versus a versus a¯ −1 suggests that the range of possible meanings might even include contradictory interpretations”.19

14

Ballier [9: 43–72]. 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. 15

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As the reader can easily refer to Ballier’s analyses,20 I will focus less on this particular metalinguistic use of the cam than on one that is more general, but at the same time more problematic. I refer to the last diagrammatisation that Culioli discussed in La formalisation en linguistique. Indeed, after having examined the possible values of “il”, and after having extended his own analyses to the “pronoms incorporés du français”,21 the Corsican linguist raised the question of the relationship between lexis, negation, and assertion, once again using the cam (see figure below).

However, if the generality of the schema emerges with regard to the categories it puts into relation—lexis, negation, and assertion, each covering a very wide range of cases—its problematicity becomes apparent as soon as we start to reflect on the substitutions allowing the transition from

to

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 il 1 ) to non specific (noted as il 2 ). 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 (il 1 ) or beyond any specific or generic uses (il 2 ). In Il 2 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, il 1 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 il 1 comes the so-called impersonal il 2 (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 (Il 2 est arrivé trois personnes)”. For the impersonal mode in Romanian, cf. Vilkou-Pustovaja [38: 287]. 21 Culioli [1].

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In some ways, it could be said that the second figure is a particularisation of the first. This is probably true of we consider a¯ −1 , a, and a¯ : These are formal symbols, that is, signs which can refer to anything, whereas “lexis”, “aff.” (for “affirmative assertion”), and “neg.” are metalinguistic labels which, although general, cover a much narrower field than those covered by each of the letters mentioned. However, making such a connection also reveals a slight incongruity in Culioli’s theoretical edifice. To get an idea of this, it suffices to look at the diagrammatisation of the relations between the labels in question.

13.5 Lexis, Assertion, Negation At the start point, which is also the end point, we have the “lexis”. And it is indubitably the correct position. Indeed, as we know, this key term in Culioli’s theory of enunciation designates what we could all a generator of predicative relations or, if you will, “une forme génératrice d’autres formes dérivées”.22 Now, without going into the technical details of this metalinguistic construction, it should be noted that the characteristic of this form is that it is neither affirmative nor negative. It is therefore for this reason that it is posited at the “start/end” of the cam, this point being on a plane which is a different plane than the one with the points representing affirmative assertion and negative assertion, respectively. Which is more than enough for understanding how the transition from the first figure to the last reveals, if I may say so, a metalinguistic opacity in Culioli’s diagrammatisation. This opacity resides in the play of substitutions which accompanies precisely such a transition: if we return to the initial figure, we will see that it is the letter a¯ −1 which is in correspondence with the “lexis”. And it is this correspondence which, in my opinion, is problematic: a lexis being neither affirmative nor negative, I think the letter a should have been posited at the “start/end”. And this inasmuch as, conversely to a¯ −1 and a¯ , a is an unmarked sign, that is, a grapheme which, being devoid of any distinctive mark, is perfectly suited to the

22

Culioli [11]. In italics in the original.

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indeterminacies of the lexis.23 In short, if I may generalise, I find it would have been more efficient to distribute the letters otherwise. Thus, making my own attempt at proposing a slightly modified version of the cam, I would say that the transition from one figure to the other could be visualised as follows:

Before delving into the relations between lexis, assertion, and negation, I wish to point out that the changes brought to the cam’s original schema by no means alter its metalinguistic significance: Although different, the new topological arrangement of letters is still capable of diagrammatising “the variation of meaning” which Ballier discusses in his article. Furthermore, I also wish to say that if I have insisted on this truism, it is not because I wanted to recant the cam in its primitive version, but rather because I would like to emphasise the little workaround, the positional modification suggested here, which safeguards its formal coherence.

13.6 A Topology of Ambivalence and Ambiguity: Freud and the Cam Structure In order to progress in the analysis, I would like to quote a few passages from La formalisation en linguistique:

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 a¯ −1 ) est la fin (ou fin-début, parce qu’elle se projette sur a), en surplomb, intégrant tout le mouvement de a à a¯ ”. 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.

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Ce modèle, d’une grande importance dans les langues naturelles, permet de mieux concevoir certains problèmes touchant à l’ambiguïté, l’ambivalence (au sens psychanalytique du terme), et d’une façon générale fait sans doute apparaître une propriété fondamentale du langage.24 Ici encore, de tels modèles permettent de résoudre des problèmes qui se posent à propos des langues ou du langage. Parmi ces problèmes, signalons l’ambiguïté de la lexis [mon père, mourir] (a) la mort de mon père (simple événement: «je considère que mon père meurt ») (b) désir («je souhaite que mon père meure »; «que mon père meure! ») (c) rejet («je ne veux pas que mon père meure »; «Je ne veux pas envisager l’idée que mon père meure ») (d) retour à la lexis, etc. On a, naturellement, reconnu ici la discussion par Freud du cas de l’Homme aux Rats: ce qui est important, c’est que, comme l’indique le diagramme ci-dessus, on a un chemin qui est un gros le suivant: [père, mourir ou ne pas mourir] → «l’idée que…» → «le souhait que…» → «le souhait que ne…pas» ou «le rejet du souhait que» → finalement «le souhait que…», par l’intermédiaire de [père, mourir ou ne pas mourir].25

Among the remarks made by Culioli, the latter are most likely those which aim to show the full heuristic power of the cam: the psychoanalytical theme of ambivalence, though mentioned in passing by the author, is probably the best key to understanding how it works. Indeed, the reference to Freud’s essay would have deserved to be addressed in its own right. Furthermore, it would have been fitting to introduce a few words on the second chapter of Totem and Taboo.26 It is indeed in this book that the subject of ambivalence is articulated in detail, accompanied by a whole series of anthropological remarks from which Culioli’s discourse might have benefited. However, it is also true that the example upon which Culioli bases himself is sufficiently clear to be understood. It is thus for this reason that it constitutes a unique opportunity to access the intuitive topology of the cam.

13.6.1 Shifts: Connectedness, Closed Curves, Unhingings In particular, as said before, it should first be reminded in this respect that it is the indeterminacies of the lexis which justify and require the shifts typical of this diagrammatisation. Being neither assertive nor negative, the “forme génératrice d’autres formes” [form which generates other forms] is “posée” [posed] upon a dimensional plane which is necessarily “décroché” [unhinged] from the plane upon which assertion and negation are “posées”. But “décroché”—as shown in the figure—does not mean “déconnecté” [disconnected]: the projection of one plane of the cam onto another makes it form a single piece, albeit shifted. 24

Culioli [1]. Ibid., p. 29. 26 Cf. Freud [12]. 25

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If truth be said, the cam would remain connected even if were devoid of the segment connecting the points representing lexis and assertion, respectively. In the literature, also, we can indeed encounter examples of the cam which do not diagrammatise the projection of the plane “au-dessus” [upper plane] onto the plane “au-dessous” [lower plane]. This is the case, namely, of the type of cam which Laurent Gosselin discusses in an article devoted to the phenomena of “polysémie contextuelle généralisée” [generalised contextual polysemy].27 In this text, as may be seen below, the cam is represented without the typical projections which characterise it:

And yet, as I have just indicated, this type of cam, though different than the one we first visualise in La formalisation en linguistique, is also connected. And this connectedness owes to the fact that it also constitutes a single entity. However, unlike the cam introduced by Gosselin, the one reported by Culioli is a topological entity that looks like a closed curve. In other words, while being spiral-shaped, Culioli’s cam has the peculiar property of being somewhat similar to a cycle, or more specifically, to a path (or “a continuous sequence of points”28 ) of which the extremities are identical.29 If truth be told, the extremities of the cam are identical and not identical. On the one hand, they are not, inasmuch as “la structure de la came est liée (…) à la relation de repérage par rupture”,30 this relation being diagrammatised by the shifts typical of the cam. 27

Gosselin [13: 94]. But also cf. Gosselin [14: 108]. 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 a¯ −1 is projected onto a¯ . 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]. 28

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On the other hand, they are identical, or at least identifiable with one another, in that, while being in a relationship of rupture, one of the two ends—let’s say, that of the “au-dessus” plane—introduces “éventuellement une projection [sur l’autre], en réduisant la structure à spirale à n’être que une structure cyclique”.31

13.6.1.1

The “Pulsations” of the Cam

Now, I know that saying that “the extremities of the cam are and are not identical” puts itself at risk of the cruellest of dialectics. But regardless of this dialectic, presenting itself as a co-presence of opposite and contradictory tendencies is, if you will, a morphodynamic characteristic quite specific to the cam. And if I say this, it is because it is necessary to consider this structure as a form in movement—such movement being a double process by which each extremity moves both away and towards the other. Indeed, by considering the cam in these terms, it is possible to easily diagrammatise the relations of identity and rupture which exist between a¯ −1 and a, or, if you prefer, between a and a¯ (cf. § 5). And this, inasmuch as it is but the closening of a towards a¯ that makes the spiral into a cycle (a = a¯ ), with this cycle itself initiating another spiral when a moves away from a¯ (a # a¯ ). In short, to use a cardiological metaphor, we could say that it is the pulsations of the cam that causes the extremities to be both identical and non-identical. This metaphor being justified by the contractions and dilations typical of the cardiac muscle which the cam’s movement mimics somewhat, all that is left to do is to point out a last formal detail: the space comprised between a and a¯ is an ever-changing space in which the projection of one extremity onto the other is but a local phase within a much more complex overarching process. This being said, I would like to emphasise the metalinguistic value of this particular local phase.

13.6.2 Germinal Powers and Enunciative Stabilisations In what concerns its cyclical structures, the cam, and more specifically, the path guaranteed by its own topology, diagrammatises the generative potentials of the

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 a¯ −1 et a: [¯a −1 # a] et à la relation de repérage par différenciation entre a et a¯ : [ a¯ /= a] et entre a¯ −1 et a¯ : [ a¯ −1 /= a¯ ]. 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 [¯a −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.

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lexis. In other words, it is thanks to the projection of the lexis towards assertion— this projection being a path—that we may visualise the morphogenetic activity of the lexis. This graphical convention is easy to justify considering Culioli’s words, that is, considering the fact that it is from such projection that “le cycle recommence”.32 It is indeed this cycle, or, more aptly put, the nexus of the “courbes descendantes et montantes” which represents in turn the provisional enunciative stabilisations of the lexis. And this, inasmuch as each point of the cycle refers to an enunciative landmark of the generating form. From this point of view, the indeterminations of the lexis are more properly conceived of in terms of “cristaux de devenir” [crystals of becoming].33 Which means that the lexis, which is neither affirmative nor negative, is nevertheless capable of forming both affirmative enunciables and negative enunciables, or ones of any variety comprised between the two. It is for this reason that Culioli could say that “au départ, [il y a] deux valeurs équipondérées, puis en trajet ordonné qui prépondère la valeur positive, suivie de la valeur négative, et retour à la représentation de la relation prédicative (avec les deux valeurs équipondérées)”.34 Which is precisely the case of “[mon père, mourir]”. All the enunciative stabilisations of this lexis and, above all, the ways in which they are ordered, seem to indeed confirm the germinal powers (the “cristaux de devenir”) 32

Culioli [1]. 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 a¯ −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. 33

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which work it from the inside. We start with [père, mourir-ou-ne-pas-mourir], and passing through all the enunciables that this lexis can shape, we return to [père, mourir-ou-ne-pas-mourir]. Which is exactly what the cam diagrammatises in detail. Now, three remarks are in order before concluding. Firstly, it should be noted that “[mon père, mourir]” is, if you will, the contracted form of “[père, mourir-ou-ne-pas-mourir]”. This is because the starting point of the generative cycle is identified by Culioli with the form which explicitates “les deux valeurs equipondérées” [the two equiponderant values]. Secondly, it should be reminded that this identification is justified on the basis that the term “mourir” in [“mon père, mourir”] is the trace of a notion, a notion being a particular type of content (or a “faisceau de propriétés bio-physico-culturelles” [bundle of bio-physicocultural properties]) that is representable, among other things, as a “couple polaire de prédicables”,35 and, in this particular case, as the [“mourir/ne-pas-mourir”] “couple polaire”. This explains the ambivalences and ambiguities of the lexis: the former become apparent if one reads the “ou” of “mourir-ou-ne-pas-mourir” as an inclusive vel involving the “présence simultanée”36 of these two values, whereas the latter do if one reads the same form as an exclusive aut itself involving “une alternative entre deux termes mutuellement exclusifs”.37 Thirdly, and more generally, it must finally be said that the twists and turns of the cam provide a diagrammatisation of the enunciative process, one which is approximative no doubt, but which, in broad terms, is realistic. And I say “realistic” for a very specific reason. There is something in enunciation—about enunciation, sometimes in the loose sense of “prise de parole” [engagement in speech], sometimes in the specific sense of “processus de constitution des énoncés”—that is well represented by the cam, or more specifically, by the peak of the cam. And this something is the instabilities proper to enunciation. The tip of the cam— which will later become the peak of the bifurcation—captures these instabilities inasmuch as it diagrammatises a germinal locus, this locus being a receptacle of enunciables awaiting stabilisation, or, likewise, to manifest over the course of the exchanges of interlocution. However, if I say “instabilities”, it is not to simply say “potentialities” by means of a different word. The change of word is significant,

35

Desclés [19: 14]. Le Goffic [20: 86]. 37 Ibid., pp. 84–85. 36

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because “instabilities” is far richer than “potentialities”.38 By speaking of “instabilities” of enunciation, I refer both to the fragilities of the “prises de parole” and to the alterations of the “processus de constitution des énoncés” [processes of constitution of the utterances]. Enunciation is unstable, in the sense of “fragile”, inasmuch as each act of speaking is liable to all manner of things that can put it in a state of suspension,39 be it through hesitations, sudden interruptions, or, in short, any sort of “arrêts du dire” [cessations of speech]. It is also instable in the sense of being “alterable”, inasmuch as each process of constitution of an utterance goes through pairs of notional polarities—the pair [“mourir-ne pas mourir”] being an exemplary case—which “compete” over the stabilised form of the utterance. Now, the peak of the cam diagrammatises the fragilities and alterations of the enunciation—in a word, its instabilities—because they imply, each in their own manner, a return to the germinal state of the lexis of which the peak is the topological representative. More specifically, there is a return to the lexis via enunciative fragilities, because each suspension of the act of speech causes a shift from the level of the utterance (énoncé) to the level of the enunciable, the latter belonging to the “cristaux de devenir” of this generative form. There is a return to the lexis via enunciative alterations, because the notional “tensions” to which each enunciation is subject refer to the generative dynamics of the lexis. And, in any event, the two types of return are indeed diagrammatised by the nexus of the cam’s ascending and descending constitutive curves. It is therefore in this sense that the cam is a diagrammatisation of enunciative work. This being said, it would be interesting to see what would be the level of detail of such a representation. However, the degree of generality of these notes, and especially the objectives I have set for myself, impose a different approach.

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).

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In what follows, I will return to one only word—“diagram”—which plays a key role in Culioli’s metalinguistic reflection.

13.7 The Diagrammatology of the Cam Indeed, as we have seen here, the notion of “diagramme” [diagram] is one of the expressions Culioli uses in order to designate the cam. The other is the notion of “schéma” [schema]; however, it indeed appears that for Culioli, it is but a synonym of the first locution. In any case, whether we say “schéma” or “diagramme”, it is clear that the metalinguistic repertoire of the 1968 text introduces two key terms of Culioli’s theory of enunciation. One only has to read a few texts published after La formalisation en linguistique to see this. In particular, if one considers the four volumes of Pour une linguistique de l’énonciation, especially the fourth, one sees that “diagramme” and “schéma”—but also “schème” [scheme] and “graphe” [graph]—are among the words which appear the most frequently in the writings of the great master of enunciative linguistics. It is undoubtedly to Dominique Ducard whom we owe the closest analyses of these four labels. In a whole series of articles having the linguistic epistemology of Antoine Culioli for object, he examined in great detail the relationships between “diagramme”, “schéma”, “schème”, and “graphe”40 shown throughout the teachings of Culioli. The objectives of this paragraph being much more modest than those set by Ducard, I will limit myself here to a few remarks on the notion of diagram. Of course, if I neglect the relationships (and differences) I have just mentioned, it is not simply for reasons of simplicity, but also for very specific textual reasons: La formalisation en linguistique, the short essay upon which I have based most of my

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.

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reflections, dates back to a period which could be described as auroral in Culioli’s thinking. It is for this reason that he does not thematise a whole series of formal distinctions—among which those discussed by Ducard—that came to Culioli’s attention only later.

13.7.1 “La Revanche De La Main” As correctly stated by Noëlle Batt, “Diagramme vient du latin diagramma lui-même emprunté au grec diagramma, issu d’une combinaison de deux autres mots grecs dia-graphein (inscrire) et gramme (ligne)”.41 But there is more, because at “l’origine de ces mots, [on retrouve] l’association de deux racines indo-européennes: grbh-mn; grbh—“gratter”, qui engendrera “tracer”, “dessiner”, “écrire” (…) et mn, qui donnera naissance à: “image”, “lettre”, “texte”».42 Le diagramme, c’est donc l’«Inscription (…) qui peut se faire lettre ou image, lettre et image”.43 In other terms, terms which proceed from a register perfectly familiar to Culioli, we could also say that the diagram is “la mise en œuvre d’une raison graphique”,44 this implementation being in turn “la revanche de la main qui (…) peut jouer sur tous les parcours permis par les entrelacs”.45 These are the words of Gilles Châtelet, a mathematician and philosopher of science who has devoted a great amount of work to diagrams, especially in the context of the exact sciences, and whom Culioli quotes several times throughout his writings.46 Now, if I cite Châtelet myself, it is because it is the manual exercise of this graphic reasoning that I specifically call “diagrammatisation”, the very gesture by which the cam is traced. But saying “diagrammatisation”—a term which I have, in fact, already employed in these notes—yet again incurs the risk of approximation. All the more so, since by associating the word “geste” [gesture] with “diagrammatisation”, the latter takes on a variety of acceptations that are difficult to reduce to a unitary meaning.47 Nonetheless, it is “diagrammatisation”, or rather, the verb “diagrammatiser” [to diagrammatise] which requires the greatest level of attention and the greatest

41

Batt [25: 6]. Ivi. 43 Ivi. In italics in the original. 44 Châtelet [26]. 45 Ivi. 46 Cf. for example Culioli and Normand [4: 75], Culioli [27: 41–42]. 47 On this matter, I can only mention two essential texts which are Citton [28] and Guérin [2]. 42

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efforts of clarification. The word “gesture” being graspable within its generality48 — a generality that enabled Culioli to gather under a single word both the “musculaire” [muscular] and the “mental”,49 and above all, the coalescences of the one and the other—it was first of all necessary to ask in what sense and to what extent the cam diagrammatises certain aspects of linguistic activity. A first answer, to which I have so far limited myself, is that the cam diagrammatises the aforementioned aspects in the sense that it enables to visualise them, that is, to give them a visible form. It is this loose but nevertheless fundamental sense of “diagrammatisation” as a graphical visualisation that is revealed by the word “schéma” in La formalisation en linguistique. If, as Ducard noted, “Dans la démarche scientifique d’Antoine Culioli, la schématisation est une représentation formelle, dans un système métalinguistique, de représentations mentales”,50 then the products of this schematisation are precisely the algebraic writings and topological images—in one word, the schemas—which inhabit the pages written by Culioli and which illustrate, each in its own manner, what was already present “sans être observable directement” [without being directly observable].51 By retaining this first aspect, we could then say that there is a representational aspect to diagrammatisation, and that it is this sense which must in turn be retained in order to understand the metalinguistic functioning of the cam. And yet, this diagrammatical functioning is not solely representational. Actually, as it must be clear, when employing this adjective, I am referring to the illustrative features of the cam, and, more specifically, to the fact that it is able to “présenter à nouveau” [present anew], i.e. “re-présenter” [to re-present] operations of language that can only be accessed indirectly. Now, the cam, in all likelihood, has a representational role in the sense I have just indicated, but this role does not exhaust the complexity of its diagrammatical function. And this, inasmuch as this complexity forms part of a much wider field of operation than that of mere graphical visualisation.

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.

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13.7.2 The Lesson of Gilles Deleuze and Gilles Châtelet: Diagrammatisation and Differential Deformations This domain, or rather, this field of diagrammatisation, which was anticipated by authors such as Gilles Deleuze and Gilles Châtelet,52 and which is at the source of research conducted by semioticians such as David Piotrowski and Yves-Marie Visetti, appears and imposes itself in the reflections conducted by Culioli over the last years of his scientific career. Now, in my opinion, it is precisely the words of Piotrowski and Visetti which enable to grasp this new sense of diagrammatisation and to appreciate its impact on Culioli’s linguistics. As we know, according to the authors of Connaissance sémiotique et mathématisation, diagrammatisation—in the renewed sense of the word— would not so much aim at “la simple représentation graphique d’un processus”53 than at setting into motion the “la différenciation d’un ensemble opératoire au sein d’une figure en devenir”.54 The distinction operated by the authors is therefore crucial: If we try to follow their reasoning, we will see that the transition from the first to the second sense of “diagrammatisation” opens up an epistemological perspective which is entirely different from the one brought to light by the representational reading of the cam. But the comprehension of this difference requires a preliminary clarification. In other terms, in order to delve more deeply into the diagrammatic field of action of the cam, it is first necessary to explain what Piotrowski and Visetti mean by “différenciation d’un ensemble opératoire d’une figure en devenir”. It is by reading Connaissance sémiotique et mathématisation that we may resolve this little hermeneutical mystery: indeed, in the same article where the aforementioned differentiation is discussed, it is written, a few lines later, that diagrammatisation “se profile comme une articulation fonctionnelle, canalisant l’advenue de toute une série d’autres figures”.55 By paraphrasing Châtelet once more, whom the two authors recognise as one of the true pioneers of diagrammatical thinking,56 we could say that “diagrammatiser” [to diagrammatise], in the non-representational sense of the word, is, foremost, to assume various modalities of movement and configurations of the hand when it is engaged in “la mise en œuvre de la raison graphique”. By moving in its own way, and, each time, by moving differently, the hand which traces the diagram shapes figures which are “other” than the starting figure. And it is precisely this shaping which “inaugure une série générique de variations”,57 that is, differential deformations which—as rightly shown by Noëlle Batt—are “orienté(e)s

52

On this matter, one may refer to the invaluable remarks by Batt [25: 8–24]. 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]. 53

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vers le non-encore-pensé”.58 Now, it is precisely this latter aspect of diagrammatisation which we find in the topology of the cam. In fact, the author of L’expérience diagrammatique never mentions this object. Furthermore, to justify, if you will, the “futuristic” features of non-representational diagrammatisation, she rather refers to the reflections which Deleuze devotes to the painting of Francis Bacon. It is indeed in a chapter of La logique de la sensation that Batt identifies the conceptual seeds that orient diagrammatisation “vers le pas encore pensé”, the “non-su”.59 However, if we examine the cam and, foremost, some of the reflections developed by Culioli throughout the Onze rencontres, we notice the formidable nexus between diagrammatisation and the “non-su” shaped by his theory of enunciation.

13.7.3 The Figural in the Metalinguistic and Vice-Versa The strictest thematisation of this nexus is to be found in the pages of Onze rencontres where Culioli and Normand revisit the topology of the cam and examine what this topology would have prefigured without its own author’s knowledge. More specifically, these are passages where the linguist connects the peak of the cam with the peak of the so-called “schéma de bifurcation” [bifurcation schema] (see figure below).

Now, it is this relation in the making which shares a lot of features with the processes of diagrammatisation in the non-representational sense of the term. Indeed, in the oracular style that was his, the great master of enunciative linguistics sought to suggest, quite convincingly, an unconscious filiation from the cam towards bifurcation. And this filiation is an excellent exemplification of the “vers le pas encore pensé” orientation which Batt assesses in relation to Deleuze’s and Châtelet’s diagrammatisation. Furthermore, it is precisely to Les enjeux du mobile, that is, to the capital work by Gilles Châtelet, that Culioli refers by thematising the relations between the cam and bifurcation. However, beyond this reference which, truth be told, Culioli does not deepen to its full merit, it is the reasoning developed with Claudine Normand which confirms the soundness of my hypothesis. 58 59

Batt [25: 21]. Ivi.

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Over the course of a series of very rich and stimulating exchanges, the two specialists outline a figural genealogy, ranging from the cam to bifurcation, which comes close to diagrammatisation in the sense of Batt. And this, for at least two reasons. Firstly, because the bifurcation schema is presented by Culioli in the terms of a figure which “serait sortie” from the cam in a completely unforeseen manner, this unpredictability being a typical feature of the diagrammatical “pas encore pensé” or “insu”.60 Secondly, because this unforeseen figure would be, according to Culioli, very different from the cam, this radical difference being in turn a typical feature of the “diagrammatique” [diagrammatic] in the sense of Deleuze-Châtelet. However, for reasons of clarity, I have included below the passages where the two aspects I have just mentioned are discussed: CULIOLI: Ensuite dans la came, je partais – puisque c’est «mort de mon père, n’est ce pas? – de: Que mon père meure! Quelle horreur! Que mon père ne meure pas!, et puis je remontais: mourir-pas mourir. Or, mourir-pas mourir, c’était en un sens poser déjà le problème de la bifurcation. Mais… NORMAND: …vous ne le voyiez pas comme ça? CULIOLI: Je ne le voyais pas comme… ce site, cette situation, ce point - dites-le comme vous voulez – ce poste d’observation d’où, à un moment donné, ça peut «bifurquer» dans un sens ou dans un autre. Pourquoi? Parce que je n’avais pas introduit une complexité supplémentaire: vous avez d’un côté la came (…) c’est à dire que c’est ordonné, mais d’un autre côté il faut ajouter la bifurcation (…). Et ça c’est très différent! (…) Là, je renvoie à la lecture du livre de Gilles Châtelet, les Enjeux du mobile, paru bien après, où il y a toute une discussion extrêmement brillante sur ce genre de problème.61

As we can see, despite the simplifications I have introduced, this short diagonal fragment condenses and revives all the particular motifs of a practice of diagrammatisation or, if you prefer, of a diagrammatical praxis which may be found, albeit in different manners, in the reflections respectively of Deleuze and Châtelet. Indeed, one should not believe however—and I will refrain from suggesting it— that the affinities reconstructed here have the value of a true identity. Despite being numerous, the similarities between Culioli, on the one hand, and Deleuze-Châtelet, on the other, do not suffice to overcome the hiatus between the one and the others. In this respect, two differences are manifest. The first is methodological: whereas the reflection by the two philosophers proceeds from and keeps with diagrammatical practices as such, Culioli’s theoretical undertaking approaches the matter of diagrammatisation on the basis of a major and deep understanding of the linguistic. We could then say that for Culioli, interrogating the practices of diagrammatisation is secondary to the metalinguistic analyses which these practices enable to

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].

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conduct. Which brings us to the second difference, the latter being epistemological and concerning in particular the status of representation. Indeed, it appears from Batt’s article that Deleuze’s and Châtelet’s diagrammatics is neither confoundable with nor reducible to the representational in the illustrative and dated sense of the word. Now, I am not in a position to assess the accuracy of what the author says concerning this crucial aspect of Deleuze’s and Châtelet’s thinking. All that I can say is that such is not the case with Culioli’s diagrammatics. As we have seen, the diagrammatisation practices underlying Culioli’s theory of enunciation present a whole series of features which are representational in the strong sense of the word. If I am now returning to this point, it is to emphasise that these features coexist in a state of coalescence with the non-representational aspects of the diagrammatical. And it is once again the passages from Onze rencontres which confirm all of this. I quote them here once more, but without any excessively simplifying cuts: CULIOLI: Ensuite dans la came, je partais – puisque c’est «mort de mon père», n’est ce pas? – de: Que mon père meure! Quelle horreur! Que mon père ne meure pas!, et puis je remontais: mourir-pas mourir. Or, mourir-pas mourir, c’était en un sens poser déjà le problème de la bifurcartion. Mais… NORMAND: …vous ne le voyiez pas comme ça? CULIOLI: Je ne le voyais pas comme… ce site, cette situation, ce point - dites-le comme vous voulez – ce poste d’observation d’où, à un moment donné, ça peut «bifurquer» dans un sens ou dans un autre. Pourquoi? Parce que je n’avais pas introduit une complexité supplémentaire: vous avez d’un côté la came c’est à dire que vous partez maintenant de IE, ou E renforcé, vous allez à I, vous allez à E et vous reprenez – c’est à dire que c’est ordonné; mais d’un autre côté il faut ajouter la bifurcation qui elle, vous donne come équi-possibles, d’un côté I et de l’autre E. Et ça c’est très différent! Parce que, dans un cas, vous avez un IE comme un germe, qui va vous donner I, puis E, puis ni l’un ni l’autre…et dans l’autre cas vous avez un point qui, en fait, n’est pas un point inerte, mais qui contient déjà I et E. Là, je renvoie à la lecture du livre de Gilles Châtelet, les Enjeux du mobile, paru bien après, où il y a toute une discussion extrêmement brillante sur ce genre de problèmes.62

These remarks show the full complexity of Culioli’s reasoning, such complexity consisting in cross-contaminating considerations belonging to two orders: the first, of a figural type, concern the virtualities set into motion by the gesture which traces the cam; the others, of a metalinguistic type, concern the intelligibility of the language ensured by this gesture. I am hence speaking of complexity in the sense of contamination, or, if you will, in the sense of a prolific heterogeneity, because the copresence relative to considerations of the one and to the other type is the main indicator of the double character of Culioli’s diagrammatics, that is, as being both representational and non-representational. And, most importantly, this doubleness, in turn, doubles. In other terms, in what concerns the shaping of the cam, there is something representational in the non-representational, and vice-versa. If I may explain: if one follows the parts of the reasoning having for object the virtualities of the gesture by 62

Ivi. The italics are in the original.

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which the cam is traced, that is, the movements to come—and which are thereby unknown—of the gesture underlying bifurcation, one will see that considerations of this type also contain observations on the generative potentialities of the lexis. In short, one will see that the non-representational aspects of the cam are intimately linked with its representational aspects. It is therefore in this sense that I say that regarding the shaping of the cam, there is representation in the non-representational. Besides, if I also say that concerning this shaping, the non-representational is in the representational, it is to highlight the possibility of a complementary connection, this connection meaning that at the level of the cam, considerations of a metalinguistic nature in turn contain observations pertaining to figurativity. Indeed, inasmuch as the author distinguishes between a cam representation and a bifurcated representation of the germinal powers of the lexis—these powers being noted using ie—we clearly see how the figural is in turn contained within the metalinguistic. And how, therefore, at the level of the cam, or better, at the level of its shaping, the non-representational is in the representational. To summarise, we could then say that the cam’s processes of diagrammatisation constitutes, at the same time, both the actualisations of virtualities leading to the topology of figures which are “other” (that is, bifurcation), and the schematisations of the germinal powers (that is, the enunciables), leading in turn to the provisional stabilisations of the enunciative activity.

13.8 Conclusion: Mr. Culioli’s Silence So, I now find myself facing the task of concluding these humble notes. The meaning of the present undertaking may be summarised as follows: after having noted all the difficulties linked to a formal definition of the cam structure, a definition which Culioli, despite his declarations, never gave, I embarked upon a project to shed some light aiming to ensure at least an intuitive comprehension of this topological object. Once this work accomplished, I was able to pursue a whole series of speculations concerning the Corsican linguist’s diagrammatical praxis, which allowed me to identify Culioli’s thinking regarding the form-in-the-making, one of great interest and of great heuristic power. And yet, the problem remains: without an adequate explanation of the formal rules characterising the cam, Culioli’s curved morphologies remain but a “toy”. Therefore, for want of anything better, I can only wonder about the reasons for his silence, or, as I mentioned earlier in the introduction, about the reasons for this metalinguistic void. All that I can say, actually, is that this silence of Culioli’s, I was able to experience it myself. May the reader forgive me if I take the liberty of rummaging through the drawers of my own personal memories but, I have to say, the cam structure was the first topic of discussion between Culioli and me. Being literally obsessed with the mysterious beauty of this spiralling curve, the first time I met the great master of the École Normale, I could not help myself but to ask him to tell about the formal rules underlying this mystery. Well, he answered,

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smiling, that “La structure en came? Mais ça, c’est acquis”.63 Having neither the courage nor the authority commanded by Claudine Normand, I was stunned by this answer, which, besides, I knew far too well. However, the story of this encounter does not end there. It is true that Culioli’s response to my question was “ça, c’est acquis”, but what is also true is that the father of predicative and enunciative operations began to explain to me the genesis of the cam structure, and he did so with a patience that is only to be found among true intellectuals. Naturally, he explained it to me in his own way: having retrieved a small red notebook from his bag, he began, on its pages, tracing lines, and then more lines, and, in doing so, he showed me shapes emerging from other shapes that sprung to life under spell of the commentaries he formulated as his pencil traced the figures on the paper. There was an artisan-linguist,64 as he often liked to moniker himself, engaged as he was in a metalinguistic game of plastic figurations which delighted the eyes and the mind. I wanted to ask him to give me his drawings, but the words remained stuck in my throat. Perhaps surprised by my lack of initiative, Mr. Culioli then took back his notebook and, asking if I had any more questions, he returned it to his bag. I will stop here, not because that was the end of the story—it continues, of course— but because this is precisely the slice of it I wished to emphasise. It is indeed in the story of the red notebook—if I may put it as such—that a possible explanation to Culioli’s silence may be found. However, before clarifying this point, I would like to quote him once more: Il s’agit en somme de multiplier les moyens. (…) Quant aux graphes, si je leur accorde une certaine importance, c’est parce qu’ils forcent à être cohérents avec soi-même. C’est pourquoi je crois à une certaine formalisation, non pas au sens de la logique formelle, mais au sens de construire des formes tel que l’on puisse parler sur le passage d’une forme à une autre forme.65

These are, as we know, the closing words of Gestes mentaux et réseaux symboliques. I have quoted them, because I find them perfectly fitting to the scenario I have just described. Indeed, what Culioli had shown me is precisely formalisation in a non-logicist—perhaps we could say enunciativist—sense of the word. But above all, what he had shown me was the answer to my question: the formal rules of the cam, the rules which I sought and which he never explicitated, were in the transitions 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].

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from forms to other forms, as he had traced on the pages of his little red notebook. I have reflected a lot about the meaning of this theoretical gesture, and I am not entirely certain to have fully understood. But most important, assuming that I had indeed understood, I wonder if, by renouncing to formulate such far-reaching rules, and by reformulating them through an explicit symbolic notation, we are not also renouncing a more complete knowledge of the cam and of everything it shapes. Unfortunately, I will never get the chance to address this question to Culioli or know what he would have answered, but most importantly, if he would have even answered at all! I am thus left with nothing but his silence and the full beauty of his diagrammatical gesture: the gesture by which the cam is traced. As for me, I volunteer to return to the gesture by which the cam is traced at a future time. Indeed, reconsidering this gesture, and, foremost, examining its presence in the fields of the humanities and of the physico-mathematical sciences,66 would afford me the opportunity to approach once again the diagrammatisations of the cam, and maybe, to remember what secrets were held in Culioli’s little red notebook—the one I never got to read.

References 1. Culioli, A.: La formalisation en linguistique. In: Cahiers pour l’Analyse, 9, Seuil, Paris, 106– 117. (réproduit dans Antoine Culioli Pour une linguistique de l’énonciation. Formalisation et opérations de repérage), pp. 17–29. Ophrys, Tome II, Paris (1968) 2. Guérin, M. Philosophie du Geste. Actes du Sud., Paris (2011) 3. Hazewinkel, M.: (edited by) 2013 Encyclopaedia of Mathematics, vol. III. Springer, Berlin 4. Culioli, A., Normand, C.: Onze rencontres sur le langage et les langues. Ophrys, Paris (2005) 5. Lawvere, F.W., Schanuel, S.H.: Conceptual Mathematics, A First Introduction to Categories. Cambridge University Press, Cambridge (2009) 6. Serge, A., Jean-Olivier, L.: Equivalence between order and cell complex representations. In: Proceedings of the Computer Vision Winter Workshop, preprint, pp. 1–21 (2004) 7. Alunni C.: Spectres de Bachelard: Gaston Bachelard et l’école surrationaliste, Paris, Hermann. (2018) 8. Coulardeau, J.: Pour une linguistique dynamique de l’anglais. Mémoire en vue de l’habilitation à diriger des recherches, texte repérable sur le site (2015). https://www.academia.edu/1432681/ MÉMOIRE_DHABILITATION_-_MENDE_2003 9. Ballier, N.: What linguists do when they write something: the art of stenography. In: Arigne, V., Migette, C. (éditeurs) Theorizations and Representations in Linguistics, pp. 43–72. Cambridge Scholars Publishing, Cambridge (2018) 10. Vilkou-Poustovaia, I.: Trancher le nœud gordien (de l’impersonnel en roumain et du sujet... de l’énonciation). In: Cahiers de l’ILSL, no. 12, pp. 281–306 (2000) 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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11. Culioli A.: Role des représentations métalinguistiques en syntaxe. Paris, Editions du Département de recherche linguistique de l’Université de Paris 7, 1–30 (1982) 12. Freud, S.: Gesammelte Werke, vol. 7, S. Frankfurt: Fischer Verlag, GmbH (trad. it. Opere 1912–1914. Totem e tabù e altri scritti, 1975). Paolo Boringhieri, Torino (1940) 13. Gosselin, L.: Le traitement de la polysémie contextuelle dans le calcul sémantique. Intellectica 22, 93–117 (1996) 14. Gosselin, L.: Temporalité et modalité. Postface de Bernard Victorri. Presses Universitaires du Septentrion, Lille (2005) 15. Gray, N.: A student’s Guide to General Relativity. Cambridge University Press, Cambridge (2019) 16. Deleuze, G., Guattari, F.: Mille Plateaux. Capitalisme et Schizophrènie. Les éditions de Minuit, Paris (1980) 17. Culioli, A.: Paris, 2003 Je veux! Refléxions sur la force assertive. In: Botella, C. (éditeur) Penser les limites. Écrits en l’honneur d’André Green, Paris, Genève, Delachaux et Niestlé (réproduit dans Antoine Culioli, Pour une linguistique de l’énonciation. Tours et détours). Lambert-Lucas, Tome IV, Limoges (2018a) 18. Lochak P.: Mathématiques et Finitude, Paris, Kimé (2015) 19. Desclés, J.-P. Schémes, notions, prédicats et termes. Mélanges offerts à Jean-Blaise Grize, pp. 2–27. Droz, Genéve (1997) 20. Le Goffic, P.: Ambiguïté et ambivalence en linguistique. Documentation et recherche en linguistique allemande contemporaine 27(2), 83–105 (1983) 21. Culioli A., Franckel J.-J.: Structuration d’une notion et typologie lexicale. Bulag 17, 28–35 (1991) 22. Authier-Revuz J.: Ces mots qui ne vont pas de soi. Boucles réflexives et non-coincidences du dire, Limoges, Lambert-Lucas (2012) 23. Ducard, D.: Le graphe du geste mental dans la théorie d’Antoine Culioli. Cahiers Parisiens 5, 555–576 (2009) 24. Ducard, D.: La formalisation dans la théorie des opérations énonciatives: formes, formules, schémas. In : Dossiers d’HEL, SHESL, 2016, Écriture(s) et représentations du langage et des langues, vol. 9, pp. 113–122 (2016) 25. Batt, N.: L’expérience diagrammatique: un nouveau régime de pensée. In: Batt, N. (éditeur) Penser par le diagramme de Gilles Deleuze à Gilles Châtelet, pp. 5–28. Presses Universitaires de Vincennes éditions, Paris (2004) 26. Châtelet, G.: Singularité, Metaphore, diagramme. In: Michèle, P. (éditeur) Passion des formes. Dynamique qualitative, sémiophysique et intellegibilité. À René Thom, Fontanay Saint-Cloud: ENS éditions (réproduit dans Gilles Châtelet, L’Enchantement du virtuel. Mathématique, Physique, Philosophie, édition établie par Charles Alunni et Catherine Paoletti, pp. 69–83. éditions Rue d’Ulm, Paris (1994) 27. Culioli, A.: Entretien avec Antoine Culioli. In: Biglari, A. (éditeur) Regards croisés sur le langage. Entretiens avec N. Chomsky, A. Culioli, M. Halle, B. Pottier, A. Rey, J. Searle, H. Walter, pp. 37–48. Classiques Garnier, Paris (2018) 28. Citton, Y.: Gestes d’humanités. Anthropologie sauvage de nos expériences esthétiques. Armand Colin, Paris (2012) 29. Châtelet, G.: Les enjeux du mobile: Mathématique, physique, philosophie. Les éditions du Seuil (trad. it. Le poste in gioco del mobile, a cura di Andrea Cavazzini, Milano-Udine: Mimesis), Paris (1993) 30. Piotrowski, D., Visetti, Y.-M.: Connaissance sémiotique et mathématisation. Sémiogenèse et explicitation. In: Piparo, F.L., Mantia, F.L., Paolucci, C. (éditeurs), Semiotica e Matematiche, VS. Quaderni di Studi Semiotici, Bompiani, pp. 141–170 (2014) 31. Dondero, M.G., Fontanille, J.: Des Images à Problèmes. Le sens du visuel à l’épreuve de l’image scientifique. Pulim, Bruxelles (2012) 32. Culioli, A.: Ceci n’est pas une conclusion. In: Ducard, D., Normand, C. (éditeur) Antoine Culioli. Un homme dans le langage, pp. 367–372. Ophrys, Paris (2006)

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33. Scubla, L.: À propos de la formule canonique, du mythe, et du rite. In: L’Homme, Tome, vol. 35, no 35, pp. 51–60 (1995) 34. Fogsgaard L.: Esquema copulativos de SER y ESTAR. Ensayo de semiolinguistica, Bern, Peter Lang Verlag (2000) 35. Brandt P.A.: Word, language and thought–a new linguistic model. Acta. Linguistica. Hafniensia. 50, 1, 102–119 (2016) 36. Portes, M.: La dynamique qualitative en psychanalyse. Préface de René Thom. Presses Universitaires de France, Paris (1994) 37. Thom, R.: Les racines biologiques du symbolique. In: Maffesoli, M. (éditeur) La galaxie de l’imaginaire. Derive autour de l’oeuvre de Gilbert Durand, Berg International, pp. 50–69 (1980) 38. Cadiot, P., Visetti, Y.-M.: Pour une théorie des formes sémantiques. Presses Universitaires de France, Paris (2001) 39. Alunni, C.: Diagrammes & Catégories comme prolégomènes à la question: Qu’est-ce que s’orienter diagrammatiquement dans la pensée?. In : Batt, N. (éditeur) Penser par le diagramme de Gilles Deleuze à Gilles Châtelet, pp. 83–94. PUV éditions, Paris (2004) 40. Amstrong, M.A.: Basic Topology. Springer, Berlin New-York Amsterdam (1997) 41. Culioli, A.: Rôle des représentations métalinguistiques en syntaxe. In: Collection ERA 642, Laboratoire de linguistique formelle de l’Université de Paris 7, Paris (réproduit dans Antoine Culioli Pour une linguistique de l’énonciation. Formalisation et opérations de repérage), pp. 95–114. Ophrys, Tome II, Paris (1982) 42. Culioli, A.: Nouvelles Variations sur la linguistique. In: Ginzburg, C. (éditeur) Vivre les sens, Seuil, Paris, pp. 113–145 (réproduit dans Antoine Culioli, Pour une linguistique de l’énonciation. Tours et détours), pp. 39–59. Lambert-Lucas, Tome IV, Limoges (2018a) 43. Culioli A.: Gestes Mentaux et Réseau Symboliques. À la recherche des traces enfouies dans l’entrelacs du langage », in Faits de langue. Les cahiers, n. 3, p. 7–31 (réproduit dans Antoine Culioli, Pour une linguistique de l’énonciation. Tours et détours, 2018a), pp. 61–89. LambertLucas, Tome IV, Limoges (2011) 44. Goldblatt, R. Topoi. The categorial analysis of logic. Dover Publications, New-York (2006) 45. Groussier, M.-L., Riviére, C.: Les mots de la linguistique. Lexique de linguistique énonciative. Ophrys, Paris (2000)

Chapter 14

Diagrams, Gestures, and Meaning. A Cognitive-Semiotic View Per Aage Brandt

14.1 What a Sign is. Peirce, Saussure, Biplanarity, and Criticality In the works of one of the classics of the theory of signs, in fact the founding father of Anglo-Saxon semiotics, the philosopher C. S. Peirce, an influential semantic classification among others distinguishes icons, indices, and symbols.1 In the Russian semiotician Roman Jakobson’s reformulation, icons are signs by similarity; indices, signs by contiguity; and symbols, signs by convention. The Peircean tradition considers the ‘semeiotic’ relation to hold between three instances, the Sign itself, or Representamen; its referential Object, i.e. the entity in the physical or social world to which the sign itself is (interpreted as) referring; and the Interpretant, i.e. the meaning or rather effect of the sign, in so far as it is interpreted by a new sign, and so on.2 The new sign can be any significant behavior responding to the sign. It is not clear what precisely defines the Interpretant as the meaning of the Sign, if something does. In Peirce’s work, diagrams do not count as a sign class but are considered to be a sort

1 See “76 Definition of the Sign by C.S. Peirce”, In Marty [8]. The list is here: https://arisbe.sit ehost.iu.edu/rsources/76DEFS/76defs.HTM. 2 In this model, the intentional meaning of signs is not considered important or even relevant; it is the response behavior of some affected instance that matters. This view facilitates the generalization of the model to become a bio-semiotics covering all of the living world, in so far as organisms affect or influence each other. Or even a semiotics of the physical universe. But it weakens its use correspondingly as an analytical tool for understanding the human (social and cultural) world, where intentional meaning constitutes an autonomous instance that can even be relatively independent of its contexts of creation and communication, namely when it is semiotically expressed.

Per Aage Brandt is a deceased author with a life span (1944–2021). P. A. Brandt (Deceased) (B) Centre for Semiotics, Aarhus University, Aarhus, Denmark © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_14

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of icons or of symbols, however difficult it then turns out to be to account for their specificity. In another classic of the theory of signs, the Swiss linguist Ferdinand de Saussure, particularly influential in European semiotics but most often ignored or rejected in the above-mentioned semeio-philosophical sphere, the sign is identified as a relation between essentially two interdependent aspects of signification, namely the signifier and the signified, both seen as conceptual entities and originally exemplified by the word in spoken language.3 Saussure generalized this model to hypothetically cover all signs; he thus saw the linguistic sign system as just one of many existing semiologies, but did not attempt to further classify sign types in this larger field of sign systems. It might be useful to say that the Peircean conception is materialistic, since it applies to all interpreted material things or acts; interpretation here means production of a reaction which can again produce a reaction—so, tautologically speaking, the sign is an entity that produces a sign “interpreting” it. The physical and living world is full of ‘signs’ in this sense. It is even all there is. The Saussurean conception, by contrast, is conceptualistic, meaning-oriented, since its model is defined by having two conceptual planes or levels, one identifying phenomena in the human world that signify, namely by referring to a system of distinct and contrasting expressions, and one identifying phenomena of all kinds that are signified by the former and further arranged in a system of distinct and contrasting meanings, or contents. There is a semiology, a semiotic system, or semiosis, in the Saussurean sense, if there is a conceptual system of interrelated signifiers whose compositions can refer to, or signify, meanings in a conceptual system of interrelated signified meanings.4 The simple Peircean semiosis is therefore everywhere to be found, whereas the Saussurean semiosis is highly structured and much more infrequent. Examples are: human languages and writing systems, mathematical writing, musical notation, games like chess,5 but not causal processes in nature, socio-historical events, biological or astronomical events, unless these are inscribed in human cultural codes such as philosophical or astrological systems, etc. It is, I think, a reasonable suggestion to limit the semiotic domain to intentional phenomena of expressive behavior, thereby leaving the Peircean pan-semiotics. This move leads us towards Saussure’s biplanary6 model of signs linking conceptualized signifiers to conceptualized signifieds. However, his systemic constraints seem too 3 Tullio de Mauro’s [9] critical edition of Ferdinand de Saussure’s Cours de linguistique générale (1916) has a substantial bibliography showing the importance of this ground-breaking course in linguistics. 4 The Danish linguist Louis Hjelmslev, inspired by Saussure’s suggestion, proposed to distinguish forms of expression and forms of content, both in the systematic sense of form [7]. 5 The elements of expression would then be the chess moves, not the chess pieces, as most comparative comments suggest. 6 Biplanarity refers to the two planes, or strata, of certain phenomena composed by a sensibilis (sensible, perceptible) part and a clearly separate intelligibilis (intelligible, conceptual) part that the former “means”. Mirror images, for example, are seen in the mirror but refer to events outside the mirror. This, however, does not make them signs, because the reference is not intentional (the relation is optical, mechanical, causal). Another example: when we count on our fingers, we mean numbers

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strong for including important human meaning-making forms of expression such as tonal music, pictorial and graphic art, and diagrams. In language, the signifying or expressive plane offers sound and graph systems based on distinct discrete units: phonemes and graphemes, yielding combinations that express word meanings; the signified or content plane holds grammatical structures that combine words into discrete sentences, and those into utterances. In music, the signifier plane offers discrete and combinable units, the tones, whereas the signified plane combines those into melodic and harmonic phrases whose shapes are of course less finite or firmly structured than those of the grammatical constructions of language. In pictorial signification, or “visual semiotics”, graphical and chromatic events on one plane (the “canvas”) matches figurative events on the other plane (the “representation”); but neither set of events offer or presuppose firmly structured systems, apart from some exceptions (such as religious icons). As to the category of diagrams—such as those that teachers improvise on blackboards and whiteboards in front of students, or those we draw on paper during discussions in order to quickly explain an idea to someone it concerns—it consists of biplanary phenomena composed of graphic figurations expressing something as anti-systemic as thoughts. If we therefore renounce on the strict Saussurean systemic requirements and just define signs by intentional biplanarity, we may reconsider the elementary semantic classes proposed by Peirce: icons, indices, symbols. There will indeed be iconic signs, namely images; these have richly variable grapho-chromatic expressions, conceptualized in perception, and infinitely variable figurative contents, conceptual by the nature of perceptual reality, and mapped to their graphical expressions by similarity; and they take place within frames that deictically manifest an intention-to-show.7 There will equally be symbolic signs, with socially coded expressions and ‘arbitrary’, conventional meanings that often have to be taught and learned, as opposed to iconic meanings. Writing systems, flag or horn signaling systems, specified telephone jingles… Their places, e.g. as signposts, are predefined and intentionally significant, and their deixis is an insistent calling on predefined receivers. The Peircean indices, however, such as traces, footprints, and symptoms, are admittedly biplanary (since what they mean is not what they show) but they are only semiotic if intentionally produced for communication, by suitably arranged “contiguity” (Jakobson).8 Weathercocks, thermometers and other instruments for rather than fingers, and yet we do not (necessarily) do this in order to intentionally communicate. Finger counting is only semiotic when it is gestural (addressing someone). 7 Visual semiotics is an autonomous branch of applied semiotics, one may say, and essentially focused on iconic semiosis. See Wildgen [16]. The journal Visible offers in its 2012 (#9) issue [15], “Images et dispositifs de visualisation scientifiques” a series of excursions into the realm of diagrams, mainly understood as Peircean “abstract icons”. The contrast between Maria Giulia Dondero’s contribution “La totalité diagrammatique en mathématiques et en art” and my own, “La pensée graphique. Pour une sémiotique des diagrammes” is striking. The texts of the issue however testify to a new—and long overdue—interest in the diagrammatic phenomenon. 8 The emblematic cross signifying Christianity, and the emblematic sickle and hammer signifying the agrarian and industrial working class, and communism claiming to represent it, exemplify arranged continuity (metonymic) from artefacts to their story; as we see, the result is symbolic.

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measurement are arranged but in a different sense: they are probes that let nature metaphorically “speak” in terms of quantitative values; those values can in turn be significant to human interpreters, like omens to priests. Strictly speaking, however, probes are indices, and indices are not signs, but just arranged manifestations of non-intentional, natural events. Criticality is an interesting criterion for “cutting things at their joints” in sign classification. A critical variation of the parameters of a point in the expressive plane is a variation that corresponds (by mapping) to a sensitive point in the content plane, where the content value changes. An uncritical variation of the point in the expressive plan hits an unsensitive point in the content plane and produces no change. There are critical points in the variation of the opening of the jaw that change the value of a vowel from /i/ to /e/ and to /a/. There are uncritical variations within the range of each vowel (in a given language). In iconic signs, for example in figurative paintings, almost all points in the expressive plane have sensitive counterpoints in the content plane; so, here, criticality is maximal. By contrast, on a page of alphabetic writing, only the black lines and the white blanks between words are critical, the rest is “empty” of significance; so, in this case, in writing, criticality is minimal. Icons have maximal criticality, whereas the criticality of symbols is minimal. Since criticality is evidently a measure of density of information, therefore icons, carrying a maximum, also are maximally informative and maximally close to percepts that are emotionally meaningful to us. Symbols are, inversely, minimally informative and, in that respect, mostly functional as succinct instructions, deontically and punctually meaningful. While traffic signs are symbolic, advertisements (even posted along the road) need to be at least somewhat iconic.9 The criticality of diagrams is typically intermediate, neither maximal nor minimal. The components of the graphical part of a diagram, apart from the possibly superimposed icons or symbolic labels, comprise: arrows, dependency strings, bounded containers (boxes, blurbs), dividing lines, channels, profiles, and maybe some more such “graphemes”.10 Since they are not explicitly coded, these are often much less systematic than symbols (such as letters and numbers), and can be much more schematic, or simplified, than the elements of any iconic drawing. They do not order you to do things, like symbols, and they do not look like anything you can have perceived, like icons; they just look like other diagrams. So what do they do or mean? (Fig. 14.1).

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Iconic, emotional meaning is illimited in time and space: referentially open-ended. By contrast, the relevance of symbolic meaning is limited in time and space: “You can or must do this here and now, before deadline or next instruction, and then it will be over!” The meaning of the symbol closes around the receiver’s present act. 10 It is everyone’s guess how many different basic diagrammatic forms, “diagrammemes”, there are—I suggest there are less than ten, universally. Their origin may be related to tools and weapons. Arrows to bows, dependency strings to ropes, dividing lines to swords and knives… All hand-held artefacts and in this sense, close to gestures.

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Fig. 14.1 A greenhouse effect diagram. From the French magazine Au fil de l’Yonne , 178, June 2021, p. 19. The diagram has flows and arrows, barriers, dividing lines, substance colors, and a host of superimposed emblematic icons and symbols (nouns, phrases, and sentences in writing)

14.2 Diagrams and Art In “Diagrams and Mental Figuration” [2], Cronquist and I argued that the specificity of diagrammatic semantics is to represent ideas and hypothetical solutions to problems as they appear in the mind, in scientific, philosophical, didactic, technical or otherwise intellectual contexts. Diagrams represent the way our inner eye apperceives thoughts. This is why words are welcome in diagrams, but not sentences, word meanings are conceptual and can easily form relational networks as we think or explain the constructions of our thinking to ourselves or to others. Thoughts are “figurative” in the sense, for example, that differences are mentally expressed by dividing lines, influences seen as flows between entities, qualities or components as boxes and blurbs, causation marked as vector arrows, quantities as lengths, proportions as angles… We use a sort of spontaneous qualitative geometry when we try to “figure out” how things work. We want to see, not what things look like but how they work. Diagrams are thus eminently epistemic, but neither deontic nor affective. The three basic sign classes, icons, symbols, and diagrams, are thus carriers of distinct meaning classes, which correspond to three major classes of sense-making in human societies.11 Roughly speaking: constitutive authorities (symbolic commanding), political 11

This comment refers to the socio-semiotic hypotheses concerning the stratification of activities in human societies, as sketched out in the last chapters of Brandt [1, 2]. There is, according to this idea, a basic, organic level of life dominated by the iconic meaning-making practices; then a political,

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Fig. 14.2 A Rudolf Steiner drawing

possibilities (diagrammatic planning), organic desires (iconic fantasizing). Signs of the three classes dominate respectively in the three areas of social life. A new suggestion that I would like to add to the panorama is that there is a link from the epistemic semantics of diagrams to the “spiritual” dimension of thinking that apperceives the diagrammatic expression of mental graphics as an opening into a supra-worldly world. The Goethe-inspired anthroposophist and artist Rudolf Steiner explained his thought by speculative drawings like the following, from a lecture (Fig. 14.2). Artists inspired by the theosophy of Mme Blavatsky (1831–1891) such as the Russians Kasimir Malevich and Vasilij Kandinskij during the same years invented an abstractly figurative pictorial art proposing works as the following (Fig. 14.3). The Swedish artist Hilma af Klint, equally spiritualistic, would paint diagrams such as the following (Fig. 14.4). Piet Mondrian, Paul Klee, and many others, joined the company with corresponding conceptions despite sometimes widely variable expressive results. Diagrams are mental windows. Klee’s rhythmic interest, as a musician, makes him hear and see the white-black-grey waltz pattern as an organic temporalization of numbers, as they pass through the body when dancing, for example, thereby investing the articulated line with white windows of escape, openness (Fig. 14.5).

epistemic, diagrammatic level, dealing with decisional, communal issues; and last, but not least, a superordinate level of deontic-symbolic organization dominated by sacred, violent practices: monetary, military, and religious forces. The social levels may have given rise to the semiotic differences, or the signs and the social differentiations may have evolved in parallel. Language comprises all functions and meaning levels: performative functions (symbolic), informative functions/dialogue (diagrammatic), and imaginative/narrative functions (iconic).

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Fig. 14.3 Malevich and Kandinskij

K. Malevich, Supremus #58. Yellow and black (1916).

Vasilij Kandinskij, Some circles, 1926.

The ‘abstract’ artistic painting originating in the second decade of the twentieth century was thus not, I would like to argue, a continuation of the iconic developments of the avant-garde of those years (Cézanne, Picasso, Braque…) but involved instead a new semiotic practice, a diagrammatic, non-iconic painting. It may “represent” thought, however in a special way: not its ideational content, not the processes thought of, but rather its very presence, the intuition of space that some more or less geometrical and repetitive shapes may invite, and a mystical sensation of disappearing into a reality deeper than those of material things given to perception. The

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Hilma af Klint, The Swan, No. 2, 1915.

Hilma af Klint, The Swan, no. 17, 1915.

Fig. 14.4 Hilma af Klint

P. Klee, Rhythmisches, 1930.

Piet Mondrian, Composition in yellow, red and blue, 1922

Fig. 14.5 Paul Klee, Piet Mondrian

human spirit (Geist) and the universal being may merge, these artists felt, when the diagram removes the particularities of things and circumstances. It is indeed characteristic of geometrical entities, as opposed to lifeworld entities, that they lack (hence eliminate) the contrast between the generic (a horse, any horse, the Horse) and the singular (this horse, my horse); for a circle, any circle, is in fact the circle, it exists nowhere, anywhere and everywhere at the same time, since it has no temporality and

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no locality.12 A geometrical form is a mathematical machine, which makes reality generic, erasing times and places. Paintings of this kind were therefore understood by their artists as magical windows, openings, uniting the inner and the outer, the subjective and the objective, in one total, spiritual universality, undisturbed by space and time. They are eminently metaphysical. Colors are no longer properties of things but, like the very blue sky, l’azur in the French poet Stéphane Mallarmé, become generic realities in themselves, beyond bounded earthly objects.13

14.3 Gesture The instance of enunciation is present in all signs. It marks, states, and qualifies the intentional relation between the enunciator and the enunciatee of any (intentional) semiosis. All signs come with enunciational features: icons have external marks such as the frame around the image, and internal marks such as the figurative viewpoint and the emotionally significant thickness of strokes. Symbols have, internally, their calligraphy and externally, the variable monumentality of the sign “posts”, plinths, supports, etc., and special locations where they are allowed to appear. Icons articulate our desires; symbols articulate our duties and forms of authority. Their enunciation is inter-personal, whether affective or deontic. And diagrams? By contrast, diagrammatic enunciation is impersonal, or third-personal, like our ordinary expressions of ideas of which we are the interested receivers: “It occurs/appears to me that…”, “it is clear/probable/evident that…”—this “it” is the impersonal voice of epistemic truth, or veri-diction.14 If we impersonate it, as we have to do while explaining its content to others, we must take on a theatrical role that represents the source of human truths: nobody or anybody, or reason in general. This is then the theatrical character gesturing to others when we express our occurring thoughts with our hands by “writing in the air” or on blackboards in front of students. The diagrammatic gesture is choreographic, the dancer is the born enunciator of diagrammatic meaning, which is why a ballet, in particular an abstract, nonnarrative ballet, can be seen as a display of geometry in the air and on the floor. Dancing, whether as artistic performance or as less pretentious rhythmic practice, correspondingly involves and triggers certain “spiritualistic” feelings, enhanced by the music.15 12

Leonardo’s Vitruvian Man, the drawing of a male body inscribed in a circle, creates a “universalization” of man by the genericity of the geometrical figure. 13 Architecture is eminently diagrammatical. Since Antiquity, the dimensions and shapes of buildings, especially of sacred constructions, temples, monuments of all kinds, have been created with the spiritual or otherwise mental force of diagrams in mind. Urbanization is another case. The geometries of Oscar Niemeyer’s Brasilia are illustrations of both cases. 14 Veridiction is a term and a developed concept in A. J. Greimas’ semiotic project; See Greimas and Courtés [6]. 15 May musical phrasing be a sort of “diagramming in the air”? If so, it would explain why it can be so forceful without manifesting iconic or symbolic properties (though such properties are of course

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The Trojaborg in Silkeborg, Denmark.

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A jewel made after Trinneberg Castle, Ulmejær, Sweden.

Fig. 14.6 Dance labyrinths. One might call this practice an archaic form of land art, cf. Robert Smithson’s Spiral Jetty (2005)

In the Bronze and Iron Age, stone-lined dance labyrinths (Trojaborgs) were common in Scandinavia. They were symmetric diagrams of flow patterns projected on the ground, forming complicated pathways that dancers followed during celebrations and ceremonies, according to popular belief, in order to be blessed with fertility and happiness. Here is an example from Silkeborg, Denmark,16 and one from Sweden, where hundreds of these stone-marked shapes are found (Fig. 14.6). Gesture research17 has studied iconic gestural forms, often related to conceptual metaphor structures, such as the shape of movements shown during speech. It has studied symbolic gestural forms including ritual performatives, benedictives, gestures of command, threats, and greetings. Diagrammatic gestures however remain remarkably unresearched, although continued rhythmic movements following patterns—dancing around trees, poles, or military parade marching, as well as other forms of demonstratively stiff locomotion in parades and processions—clearly qualify as being of this semiotic type. Epistemic air-diagram gestures, on the other hand, are quite common: they are all simply deictic!18 When we draw in the air with a hand to show someone where the ancient road was, we send out a dotted line for his eye to follow. And when, as kids, we draw a also available). The symphonic conductor’s hand-waving certainly qualifies as diagrammatic. See Orwell and Parikian [11] and Morten Schuldt-Jensen, “What is Conducting? Signs, Principles, and Problems”, in Brandt and do Carmo [4]. 16 The site Labyrinthia, Silkeborg, curiously adds after a general presentation of the ancient dance practices: “A modern use of Trojaborg labyrinths is for problem solving: Walk slowly to the centre and then out again. On your way in, find a solution to your problem by means of feelings or by intuition. On your way out, face the solution—don’t change your mind—and find out what the solution means to you in practice. When you are out again, you know exactly how to solve your problem!” Again, the diagram is connected to the epistemic meaning type, it is for problem solving. 17 The unsurpassed classic is McNeill [10]. 18 On deixis and semiotics, see the two chapters in Brandt [2]. What the present paper adds to those, and to the chapter on diagrams, is essentially this view of the foundational relation between deixis and diagram.

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line in the middle of the room to distinguish your side and my side, this dotted line is a projected diagram. Both examples are deictic and informative. The simple gesture of finger pointing towards an object in order to call the attention of someone to it, the gesture corresponding to using a demonstrative pronoun in language (“this bird…”), is elementary deixis.19 The imaginary dotted line from my finger to the bird is strictly speaking accompanied by another dotted line from my eyes to your eyes and further a third and a fourth line from your eyes both to my finger and to the bird out there. Attention-calling is quite a complicated diagram already, however elementary and trivial this gesture is. If, in an academic class, I call twenty students’ attention to a diagram I am drawing on the blackboard, I therefore create a complex deictic spider net of dotted lines in the air, an air diagram targeting a blackboard diagram, and both diagrams are in the making simultaneously. Which makes it possible to understand the actual graphical, drawn diagrams as traces of deixis-in-motion—signifying processes of pointing while sketching out lines, arrows, nodes, and what else the structure in question comprises. Diagrammatic ‘enunciation’ is thus, in the act, unseparable from the ‘enunciated’ content, and the resulting graphic diagram expresses enunciated enunciation (French: de l’énonciation énoncée) in every detail of its composition; this again distinguishes diagrams from icons and symbols, where the deictic presentation constitutes a distinct and separate dimension.

14.4 Perspective Semiosis, in whatever meaning mode, must be linked to cognitive mental activity, and an architecture of meaning is necessarily linked to the architecture of the mind. The semantic triad of iconic-affective, symbolic-deontic, and diagrammatic-epistemic meaning is not only related to bodily gesture, which covers the triad fully, but also to the structural registers of the human mind that gesture may even express more directly than other signifying practices, since it is inherent in the body’s expressive neuro-motor competence. It is straight-forward to hypothesize a register of desire, a libidinal component with a close connection to memory (episodic, semantic, and procedural),20 naturally connected to affective iconicity, whereas a distinct register of deontic motivations of agency, order-following and -giving, domination and submission, would be connected to symbolicity, probably the most visible and ‘concrete’ aspect of intersubjectivity and social life. We get—in the psyche as in societies’ institutions—a foregrounded register of symbolically driven practical agency, an agito, and a backgrounded register of iconically driven affective cultural themes, a

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Talmy [12] and Diessel [5] are my favorite cognitive linguistic specialists on demonstratives and deictics. The literature on the subject is otherwise immense. 20 Endel Tulving’s classification of human memory types [14].

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libido. The register that separates these two would evidently be that of the diagrammatically driven thinking and expressive-receptive epistemic activity, a new version of Descartes’ cogito, and in particular the expressive-receptive linguistic activity,21 closely connected to so-called ‘co-speech’ gestures, and eminently to the deictic ‘showing’ gestures we have mentioned. The series agito – cogito – libido may offer a comfortable terminological summary of the basic instances of a semiotic and mental stratification of possible importance to cognitive and social research on meaning. I act, I think, I love (thereby I am)—and I try to embody a coherent and integrated combination of these aspects of the art of living. The responsibility of my cogito, my diagramming mind and, in every minute, my gestures, for the coherence, is overwhelming; failures can be fatal, when desire and action coincide without mediation; whereas successes are in fact hardly noticed. Most diagrams are invisible. They are nevertheless pivotal to our mental and social life.

References 1. Brandt, P.A. Cognitive Semiotics. Signs, Mind, and Meaning. Bloomsbury, London (2020) 2. Brandt, P.A.: The Music of Meaning. Essays in Cognitive Semiotics. Cambridge Scholars, Newcastle upon Tyne (2019) 3. Brandt, P.A. L’analyse phrastique. Pour une grammatique. AIMAV, Bruxelles (1973) 4. Brandt, P.A., do Carmo, J.R. (eds.): Sémiotique de la musique/Music and Meaning. Presses Universitaires de Liège, Signata 6, Liège (2015) 5. Diessel, H.: Demonstratives, joint attention, and the emergence of grammar. Cogn. Linguist. 17, 463–489 (2006) 6. Greimas, A.-J., Courtés, J.: Sémiotique. Dictionnaire raisonné de la théorie du langage. Hachette Université, Paris (1979) 7. Hjelmslev, L.: Omkring sprogteoriens grundlæggelse. Akademisk Forlag, Copenhagen (1943, 1976) 8. Marty, R.: L’algèbre des signes : essai de sémiotique scientifique d’après Charles S. Peirce, vol. 24. John Benjamins, coll. Foundations of Semiotics, Amsterdam (1990) 9. de Mauro, T.: F. de Saussure, Cours de linguistique générale, critical edition, introduction, bibliography. (French translation). Payot, Paris (1972) 10. McNeill, D.: Hand and Mind: What Gestures Reveal About Thought. University of Chicago Press, Chicago (1992) 11. Orwell, B., Parikian, L.: Waving, Not Drowning. Classical Music Magazine. Amazon.co.uk, Marston Gate (2013) 12. Talmy, L.: The Targeting System of Language. The MIT Press, Cambridge, Mass. (2017) 13. Tesnière, L. Éléments de syntaxe structurale, 2ème éd. Klinksieck, Paris (1966) 14. Tulving, E.: Episodic and semantic memory. In: Tulving, E., Donaldson, W. (eds.) Organization of Memory. Academic Press (1972) 15. Visible No 9. In: Sémir Badir and Maria Giulia Dondero (eds.) Visualisation et mathématisation. PULIM, Limoges 16. Wildgen, W.: Visuelle Semiotik. Die Entfaltung des Sichbaren. Von Höhlenbild bis zur modernen Stadt. Transcript Verlag, Bielefeld (2013) 21

In language, every sentence is a diagram whose nodes are its lexical components. This intuition has guided grammarians of different schools trying to account for the syntactic “trees” that phrase structures create. Brandt [3] offers a “stemmatic” model inspired by Tesnière [13].

Chapter 15

Continuous, Discrete Diagrams and Transitions. Applications in the Study of Language and Other Symbolic Forms Wolfgang Wildgen

Abstract Peirce has shown that diagrams and diagrammatic reasoning are important in science and human thinking. Diagrams in science are like a shorthand for complex systems. There exists a choice between continuous and discrete diagrams. This choice has dramatic consequences for scientific modeling. The diagrams Peirce had in mind are founded in his logic of relations, whereas our focus is on topological and dynamic diagrams. Three fields of application are considered: technical diagrams in the context of architecture and engineering, mathematical diagrams, and diagrams in the study of language, visual and musical performance. Structural stability under deformation and variation enhances these diagrams’ abstraction power. Our analysis prioritizes continuous diagrams and qualitative dynamics (e.g., catastrophe theory). Discrete equivalents are considered based on vector calculus; this enables the construction of a cellular automaton. More specific applications concern the semantics of verbs and the coherence patterns of narrative texts.

15.1 The Notion of a Diagram and Diagrammatic Reasoning The proper starting point for a treatise on diagrams is the work of Charles Sanders Peirce. Diagrams are, in his view, mental images, thoughts, or signs on paper, on a blackboard, or actually on a computer or electronic media. In his classification of signs, diagrams first point to the relation between the sign-body (the “representamen” in Peirce’s terms) and its object. Peirce distinguishes three types of such relations: icon, index, and symbol. Diagrams stand primarily in an iconic relation to their object, i.e., mediating some similarity or shared quality.

W. Wildgen (B) University of Bremen, Bremen, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_15

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Moreover, diagrams may, in some cases, be the (causal) result of a natural process, e.g., in the case of a photograph or an electronic scan that makes visible selected features of an object; in this case, indexical cues are added. Finally, diagrams may even use rules (i.e., conventions) in their establishment and reading, thus involving a symbolic relation. Nevertheless, the dominating category in diagrams is the iconic one, i.e., diagrams are (in Peirce’s view) a kind of icon. Every picture (however conventional its method) is essentially a representation of that kind. So is every diagram, even although there is no sensuous resemblance between it and its object, but only an analogy between the relations of the parts of each.

Diagrams stand in an iconic relation to their objects; this means that between the appearances, the quality, the features of the object, and the (diagrammatic) sign, a (partial) mapping exists, which helps to identify the object, to which the sign refers or to select the sign which can “stand for” the object. Two further aspects have to be considered. First, the sign body can be simple or complex. Thus a single noncomposed sign is simple, and a compound sign, in the case of language, a morpheme (composed of phonemes), a word, a phrase, a sentence, or a text, are complex (to different degrees). A diagram has, in most cases, different components, i.e., it is complex, and specific relations exist that contribute to the meaning of the complex. These relations may be implicit. In this case, they are either specified by features of the components or are chosen from a limited set of general-purpose relations (cf. for nominal compounds, [26]: 135–138). In visual communication, the relevant components may be lines, surfaces, colors, etc.; the relations may be geometrical or due to the characteristics of color space. In musical composition, the components may be the single tones of a melody and their relations in a tonal system or the musical themes and their relations in a sonata. Peirce makes a further distinction in his triad: qualisign, sinsign, and legisign. We will mainly consider the case of sinsigns, e.g., spontaneous, single signs of iconic nature; in the language, we call the sinsign a token; if it is repeated and becomes a routine association, we call it a type. In the visual field, a picture may stand for different views of the same object or a recurrent category of objects, and then it is a type. The diagram as a legisign follows some rule of construction or for its reading. The aspect of construction will be important in the following. Peirce gives an outstanding example of using diagrams scientifically, the story of Kepler and his discovery of the laws of planetary motion. His admirable method of thinking consisted in forming in his mind a diagrammatical or outline representation of the entangled state of things before him, omitting all that was accidental, observing suggestive relations between the parts of his diagram, performing diverse experiments upon it, or upon the natural objects, and noting the results. Peirce [14]: 255.

In the following, we shall focus on three types of diagrams: 1. Technical diagrams in the context of architecture (building) and engineering (machines), 2. Mathematical diagrams,

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3. Diagrams in communication sciences (language, visual and musical performance). Technical Diagrams Diagrams are of common and central use in the technical sciences, e.g., in architecture, engineering, geography, and astronomy. A long history of millennia has given rise to elaborated technics of diagrammatic planning and control in the realization of technical artifacts. The use of diagrams in house-building is self-evident. The proportions and the measures can be fixed by a diagram of the ground- or the floor plan; others may specify the distribution of windows and doors and the sequence of floors, including the roof (its shape, inclination, etc.). We can assume that Egyptian architects already used diagrams for such purposes. Their usage normally presupposes an unwritten and un-pictured routine of technical realization. In the case of churches in medieval Romanic art, the use of diagrams on paper was minimal (if not inexistent). Mostly practices transmitted orally and by enactment were sufficient. The master builder used view lines following specific angles, rules of proper proportion, and knotted ropes to establish right angles (e.g., the “druid rope” with 12 equidistant knots). Applying the law of Pythagoras, right angles could be constructed. The ground floor was marked on the flattened surface scheduled for the church.1 Gothic cathedrals were such refined technical masterpieces that diagrams on paper and models for standardized stones or arches had to be used. Figure 15.1 shows a page in the portfolio of Villard de Honecourt (who lived around 1200) and the detailed delineation of the façade for the Strasbourg-cathedral (first half of fourteenth century). In the case of this cathedral, we know that the upper part does not follow the plan, i.e., in the technical realization of the plan, new considerations came to the foreground, or the plan was rather a program designed to persuade the clergy or urban authorities and was not coercive for later master builders who completed the cathedral. Beyond artistic criteria, a building must also be statically correct, i.e., the building should crumble neither during the construction nor centuries later. This important aspect had to be evaluated based on experience with previous constructions, which crumbled or resisted the forces of gravity.2 In constructing a machine, e.g., a racing car, the visual design may be important for the public. Primordial is that the car is competitive; it must run fast and without dropout and breakdown. Suppose the case of architecture, physical laws (statics) and the dynamics of wind and frost must be considered. In the first case, the racing engine must fulfill motor motion, aerodynamics, and stability criteria for a change in 1

See Boscodon [3]. The association “amis de Boscodon” reconstructed the techniques of medieval constructions starting from the abbey of Boscodon in France (founded in 1132). The first diagram, called “Gabarit” was traced on the gravel of the place prepared for the construction of the abbey. The dominant geometrical components were a circle, rectangles (two squares), and different angles (right angle and angles contained in the golden section, based on the pentagon). 2 The Catalan architect Antoni Gaudi devised special inverted models of the roof in his cathedral “Sagrada Familia” in Barcelona, where small sandbags simulated the forces of gravity. Such a three-dimensional model can be called a diagram, if we follow the definition given by Peirce.

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Fig. 15.1 Detail of the book of sketches by Villard de Honnecourt (first half of thirteenth century) and a draft of the facade of the Cathedral of Strasbourg (fourteenth century)

speed or direction. Therefore, the diagram has to consider more than visual appearances. It must respond to criteria of stability and dynamics valid for the object under construction. Modern technologies use computer-aided design (CAD) to build diagrams for technical purposes. Dynamical aspects, e.g., the aerodynamics of a racing car, can be simulated and diagrammatically represented using coded colors for simulated air resistance values. The code colors are symbols, but the distribution of colors on the surface of the racing car is a diagram (Fig. 15.2). Diagrams and Biological Archetypes “Archetypes” refer to Plato and his dialogue Timaeus, where he pleads for a geometric foundation of natural laws and even laws of the human soul. Plato’s treatise’s favorite geometrical building blocks are triangles, regular surfaces, and regular solids (the five Platonic solids). A long tradition of Platonism revived in the Italian Renaissance by Marsilio Ficino (1433–1499) and Giordano Bruno (1548–1600) reached a culminating point in the work of Johann Wolfgang von Goethe (1749–1832)and his “Morphologie überhaupt” (General Morphology; cf. [24]). Goethe became famous as a poet, novelist, and the author of dramas but in his “Farbenlehre” (Lessons on color) and the morphology of plants and vertebrates, he conceived the idea of “Urbilder” (primary images) underlying the huge variability of plants and their stages of development and those underlying the spinal column, including the cranium. Decennia, before the publication of Darwin’s treatise, Goethe tried to project the shapes of

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Fig. 15.2 Computational fluid dynamics (CFD) technology to improve aerodynamic performance

plants and vertebrae on an underlying primary image, e.g., the shape of a leaf or a spine. In the case of plants, he even suggested that a specific plant he had found in Palermo comes near to such a primary image. The idea of an (imagined) archetype is an example of a diagram that mirrors the constant and basic features of a class of biological forms. The unfolding of the biological diagram (archetype) to a multiplicity of specific morphologies can be observed in the development of plants and bodies. After Darwin’s treatise of 1859, the morphological dynamics in evolutionary time and growth processes could be accessed scientifically. Corresponding mathematical tools have been proposed in dynamic systems theory, and we will use these tools in the following. In modern differential topology, which is a generalization and elaboration of Greek and premodern geometry, the elementary catastrophes (cuspoids and umbilics) and the three symbolic genres correspond to regular surfaces (polygons ~ cuspoids), double-faced surfaces (dihedra ~ umbilics), and regular (Platonic) solids (polyhedra ~ symbolics); cf. ([28]: 49–56) and Wildgen [34]. The consequences for theoretical biology after Thom’s book of 1972 [19] are not the topic of this paper. We shall rather pursue his proposals for language and semiosis. Mathematical Diagrams In his article “Logic as Semiotic”, Peirce mentions “icons of the algebraic kind” ([13]: 106). He gives us as an example two algebraic equations containing letters, subscript numbers, and mathematical symbols (+, ·, and =). The letters are symbols, the subscript numbers indices, but the equation and the arithmetic operations (+, ·) form a diagram, i.e., an icon “in that it makes quantities look alike, which are analogous to the problem. Every algebraic equation is an icon, insofar it exhibits,

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employing the algebraic signs (which are themselves, not icons), the relations of the quantities concerned.” (ibid.: 107). Geometrical figures are visual diagrams insofar as lines, surfaces, intersections, parallels, and angles refer to possible real-world entities. However, they are also mathematical diagrams because they exhibit formal properties and basic laws, as those proved by Euclid, Archimedes, and many others throughout history. Moreover, a visual graph can be formulated algebraically in algebraic geometry, cf. Descartes’ geometry. In this sense, the visual mode is not a necessary feature of diagrams, although it is helpful in the case of applications (and teaching). In catastrophe theory, the calculus we shall apply in the second part of this article, the diagrammatic nature of the algebraic equations is more intricate. We take as an example the first compact elementary catastrophe called the cusp. The potential (gradient) is V = x4 /4. As this basic dynamic system is unstable under deformation, the classification theorem of Thom has derived a universal unfolding, which is structurally stable (cf. for details [23, 25]). The universal unfolding is V = x4 /4 + ux2 /2 + vx. It is a four-dimensional structure with the parameters: P (potential), x (internal variable), and u, v (external variables). The equation can be expressed graphically if we consider the first and second partial differentiation (relative to x): V' = x3 + ux + v, and: V'' = 3x2 + u. The critical points of the system are found if both derivations are equal to 0. A standard procedure for solving these equations leads to Eq. 27v2 + 4u3 = 0. It has the shape of a semi-cubic curve depicted in Fig. 15.3. We get an overview of the shape of the four-dimensional catastrophe called the cusp if we add to selected points in the graph depicted in Fig. 15.3 small twodimensional pictures of the values in the plane (P, x) at these points (Fig. 15.4). In this representation, we see the line of bifurcation, which separates the fields with one or two attractors (i.e., minima between the cusp line). As in Peirce’s example, the parameters: P, x, u, and v are symbolic units. The operations + , ·, = , 2, 3, 4 , and the two levels of partial differentiation V' , V'' are diagrammatic units. Moreover, the underlying classification theorem implies a larger “machinery” of diffeomorphisms and other topological operations in the definition of structural stability. Without this Fig. 15.3 The graph of the Eq. 27v2 + 4u3 = 0 in two dimensions

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Fig. 15.4 Overview graph of the four-dimensional catastrophe; in the center: space (u,v), in the small pictures of the periphery: space (P, x)

background, the algebraic equations would be rather trivial, and the far-reaching applications of catastrophe theory (see for an overview [17]) would not be possible. In the next section, both geometrical (topological) features of diagrams and the dynamics referred to in diagrams will be a central concern. Diagrams in Linguistics and Musical Analysis Suppose we follow Peirce’s consideration of diagrams in mathematics. In that case, algebraic grammars (cf. the tradition of generative grammar) and logical grammars (cf. the tradition of Carnap and Montague) are diagrammatical representations. They also show the two faces of geometrical representations, the visual and the formal. The phrase structure “trees” of sentence analysis and their formal (algebraic or logical) representations are manifestations of this duality. In traditional grammar (from Aristotle, through Roman (stoic) grammar, to medieval and modern school grammar), geometrical aspects are not pertinent but useful for didactical purposes. However, these grammars reduce language facts to linear combinations of phonemes (letters in early grammar) or morphemes and words (to morphology and the lexicon in modern terms). After 1930 and mainly in the second half of the twentieth century, syntax became the main issue of linguistics. Bloomfield, Harris, and linguistic behaviorism first denied the study of meaning scientific relevance. These “fathers” of modern linguistics considered meaning in language to be inaccessible. Later, Richard Montague saw meaning as a problem of logic (of possible worlds), and in Cognitive Linguistics (Langacker, Talmy, Lakoff), it was adapted to psychological or cognitive criteria. Although Langacker introduced the term “spatial grammar,” visual and geometrical diagrams were only considered easy didactic tools (cf. for a systematic analysis [29, 30]). In catastrophe theoretical semantics (cf. [23]), the spatial and dynamic nature of the entities referred to in linguistic utterances is not disregarded. Diagrammatic representations of meaning must be spatial and dynamic, even if the geometry and dynamics differ from those practiced in physics and mechanics. It

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Pitch distance

Time progression Fig. 15.5 The counterpoint movement as a type of point-to-point dynamic of differences and correspondent larger and smaller tensions

is qualitative and not quantitative, topological and not metrical. It retains abstract features of space and time. This transition was the message of René Thom’s semiophysics and his dynamical model of sentence meaning and the meanings of narratives; cf. Thom [19, 21] and Wildgen [23, 28]. We shall develop this argument and the topological and dynamic diagrams techniques in the second and third sections. Peirce’s diagrammatic logic was an early excursion in the same direction Cf. Peirce ([15]: vol. 4, book IV: §§ 347–584). In visual communication, shapes must be recognized under different angles, at different distances, and under changing light; i.e., visual recognition must be structurally stable under deformations. The same is true for musical gestalt, e.g., a melody. It may appear in different tunes, played on different instruments by different artists. Again structural stability under deformation is the critical feature. In Wildgen ([33]: 175–179), a dynamic diagram of the musical gestalt called “fugue” (cf. Bach’s “The Art of the Fugue”) is proposed. We can only present a short sketch here, The basic movement of the first and the second theme (Dux and Comes) is pursuit/ escape. The counterpoint technique plays an important role; it literally means point (note) versus point (note). It is a profile of differences (see [11]: 120, Fig. 15.7). There is an asymmetry because one voice is considered the leader or reference (base) voice, and the other is dependent on it and acts as a contrast. Before Bach’s work, this diagram applied to the cantus firmus and the discant voice (see [10]: 244). There are many restrictions to the allowed/reasonable versions of the counterpoint (Fig. 15.5).3

15.2 Topological and Dynamic Diagrams of Meaning in Language In the following section, some results exposed in Wildgen [28] and later research are summarized and used to specify the structure and use of topological and dynamic diagrams. In the first stage, continuous diagrams of meanings and the adequate ontology necessary for proper interpretations of these diagrams are introduced. In 3

For dynamic diagrams in visual semiotics see Wildgen [31, 35].

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the second stage, discrete diagrams of the spatial and temporal characteristics of (verbal and sentential) meanings are defined, and the cellular automata framework is sketched. This analysis shows the transitions of catastrophe theoretical semantics to discrete and combinatorial semantics (e.g., in feature and logical semantics). From Dynamic Models to Diagrams The construction and use of diagrams in catastrophe theory can conserve basic topological and dynamic characteristics and forget metrical details, variations in objects, or events under consideration. The crucial result in this field is the theorem by Whitney. It says that locally (in the environment of a point), we can only find three types of points (all other types become identical to these if perturbed): (a) regular points (Morse points); they do not qualitatively change under perturbation; we may say that they have a static identity (of self-regulation), (b) fold-points (a frontier line between a stable and an unstable domain appears), (c) cusp-points (two stable attractors conflict and one may appear or disappear). Thom’s classification expands this list in the domain of real analysis, and Arnold [1] presents a list for the more general case of complex analysis. First, however, it is important to note the basic difference between static and process stability in the present context. a. Static stability and the unstable points in its neighborhood. The prototypical (local) systems are the potential functions: V = x2 (one can add a function that contains more quadratic terms and constants). The gradient: V' = 2x = 0 defines the singularity of the unfolding. The stable system V = x2 has a minimum (V'' = 2 > 0) as its singularity. The dual of this function is V = −x2 , which is the prototype of an unstable singularity, V' = −2x = 0; V = −2 < 0; it is a maximum. Figure 15.6 shows the two dynamical systems and, as analogs, two physical systems (pendulums with damping).

Fig. 15.6 Basic dynamical systems

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Fig. 15.7 Configurations of conflict

The diagrams in Fig. 15.6 show the graph of the equations V = x2 and V = -x2 and physical analogs, the normal and the inverted pendulum. b. Process stability. Most dynamical systems are not structurally stable; they degenerate under small perturbations. Nevertheless, they can have a stable evolution called “unfolding” under specific conditions. These special cases can be called highly ordered instabilities or catastrophes. The minimum number of unfolding parameters gives the measure of degeneracy, and it is called the co-dimension. Figure 15.7 shows the conflict lines between stables regimes for the compact catastrophes: cusp (germ: V = x4 ), butterfly (germ: V = x6 ), and star (germ: V = x8 ). An even simpler picture is given by a diagrammatic representation of the stable attractors in the unfolding (⊕ = minimum, ◯ = maximum, — = vector field). cusp (A3 ): ⊕ — ◯ — ⊕ butterfly (A5 ): ⊕ — ◯ — ⊕ — ◯ — ⊕ star (A7 ): ⊕ — ◯ — ⊕ — ◯ — ⊕ — ◯ — ⊕ In the family of umbilics, the notion of a saddle (●) must be introduced (if we add a quadratic function, e.g., y2 to the family members, maxima become saddles; cf. Gilmore [7]: 119f). cusp (A3 ) + y2 : ⊕ — ● — ⊕ The consideration of saddle connexions becomes necessary in the derivation of four-valent diagrams in the case of the elliptic umbilic (D−4 ) (Fig. 15.8). Fig. 15.8 Dynkin diagram of the elliptic umbilic

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Fig. 15.9 Diagram of the compactified elliptic umbilic

If the elliptic umbilic is made compact, attractors ⊕ close the saddle connections. In this case, we obtain the maximal substructure with four minima (Fig. 15.9). This diagram is the basic type of a two-dimensional configuration with four attractors (cf. [25]: 204–212). The configuration with 1, 2, and 3 linearly arranged attractors and the configuration of four attractors in a two-dimensional (x–y) plane will be fundamental concepts in the following sections. If we consider linear paths in an unfolding, i.e., in the phase spaces sketched in Fig. 15.7, we can classify types of processes. In this chapter, only the most basic types will be used. The specific diagrams of such paths are called archetypal morphologies by René Thom. They are diagrammatical abbreviations of explicit dynamical descriptions. For example, in Fig. 15.10, the diagrams called EMISSION, CAPTURE, and (bimodal) CHANGE are derived from the catastrophe set of the cusp. The diagrammatic simplification eliminates the lines of (unstable) maxima, and the circles symbolize the bifurcation points (type ‘fold’: V = x3 ).

Fig. 15.10 The derivation of archetypal diagrams from the “cusp”

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Based on these conventions of diagram construction, a small list is defined (originally twelve members, in Wildgen [26], this list has been enlarged but remains moderate).

15.3 The Interpretation of Dynamic Diagrams in Semantics The starting point of catastrophe theoretical semantics were articles [18] and book chapters (in Thom [19]; in English 1975) which used an intuitive interpretation where forces (vector fields and attractors) were interpreted as animate agents (animals and humans). In the prototypical situation, one agent acts on an entity with less agency (matter, solid objects, living beings dominated by the agent). For example, we will show a scenario in which three agents interact. In Thom’s list of archetypes, it is called the diagram of “giving”. Later in this chapter, the situation with four entities will be considered. In catastrophe theoretical semantics, it is called the diagram of “sending”. The intermediate, symmetric scene is the most unstable in the three-agent scenario. Both agents concentrate their control on one target, and their control must be coordinated to secure a smooth exchange. Thus, if A releases his control before B takes the object, or if A holds the object tight although B seizes it, the character of the process is dramatically changed and degenerates to “A loses, drops the object” or “A and B compete for the object C”. Thus the unstable state of exchange is the “junction” of the process, the point of maximum co-ordination of the controls. On the other hand, it can be a metastable state if the object gains some autonomy, for example, if it lies on a table between A and B such that it is within reach of both but is not strictly controlled by either of them. This configuration corresponds to the transfer diagram (see [25]: 185). In Fig. 15.11, we distinguish five major phases separated by the sub-diagrams called “EMISSION”, “CAPTURE”, and “TRANSFER” (transition) between HAVE1 and HAVE2. The phases can be further subdivided by the dominant perspective (M1 or M2). The line of TRANSFER separates HAVE 1 and HAVE 2. Concerning the major agents M1 and M2, the diagram of giving is in disequilibrium. Agent M1 finishes “poorer”, and agent M2 “richer”. A symmetric configuration is found in the diagram of mutual exchange, which corresponds to a closed loop in the underlying control space of the catastrophe called “butterfly” (A5 ). Figure 15.12 shows this structure. In the first phase, M1, we may call him the patient, gets object 1 and “wins”, thus creating an asymmetry of possession; in the second phase, the attractor M2, now the patient, gets object 2 and “wins”. From a more general perspective, this figure represents two movements of a simple game.

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Fig. 15.11 The phases of the TRANSFER diagram

Fig. 15.12 The energetic cycle of transfer

15.4 Ontologies Underlying the Interpretation of Dynamic Diagrams Peirce distinguishes in his early treatise on categories (1867/68) two major domains: BEING (or the center of consciousness) and SUBSTANCE (the immediately real). Three further subdivisions appear as moments of the basic move from BEING to SUBSTANCE4 : • Quality (reference to ground); cf. Peirce ([15]: § 1.555) This oven (substance) is black (quality). • Relation (reference to a correlate) Peter is taller than John (correlate) • Representation (that which refers to ground, correlate, and interpretant) [These signs express] “always some relation of an intellectual nature, being either constituted by the action of a mental kind or implying some general law.” (ibid.: § 1.563). 4

Peirce looks back on a long tradition since the categories of Aristotle. Prominent philosophers preceding Peirce who tried to elaborate the scheme of Aristotle were Kant with his table of categories (in the “Kritik der reinen Vernunft”) and Hegel in his “Wissenschaft der Logik”.

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Peirce argues that these categories mediating between SUBSTANCE and BEING have an irreducible semiotic status. The three categories exemplify the concepts of Firsthood, Secondhood, and Thirdhood. This basic ontological distinction is parallel to the concept of quantitative valency applied in the following sections, i.e., diagrams can point to one entity (Firsthood), two entities (Secondhood), or three entities (Thirdhood). We shall add a more complex configuration with four entities (“Fourthhood”). The categories introduced by Peirce reappear as Quality (and Motion), Action (Interaction), and Internal Action/Interaction in our classification. In philosophy, many different ontologies have been proposed; finally, a general, absolute, independent subdivision of ontological domains was abandoned. Nevertheless, humans (and animals with a degree of conscience) operate with ontological distinctions in specific domains. Thus the lexicon of nouns and verbs in different languages shows the effect of ontological categorization. Our proposal for the semantics of human languages can be considered an operative categorization without metaphysical ambitions.5 Therefore, we shall only mention two further proposals: Rudolf Carnap distinguishes in his “Der logische Aufbau der Welt” [4] four types of objects (structures, events, states, qualities) ordered on a linear scale defined by epistemological presupposition: 1. 2. 3. 4.

Objects in the own mind, physical objects, objects belonging to other minds, abstract objects (cf. culture, society, religion).

In the framework of ecological psychology, initiated by Gibson [6], the underlying scale has the steps: (1) the psychophysical transition from the phenomena to the mind, (2) processes external to the individual mind, including those in other minds, and (3) internal (perceptual, mental) processes. A major problem in ecological semantics concerns the place of qualities (qualia). We presume that this is a domain that transcends this scale insofar as it emerges from stratum 3 (internal action) due to external processes (on strata 1 and 2). The level of action and interaction is the fully deployed domain and thus manifests as the center of this stratification. This level was chosen in the example given above, the diagram of “giving”. Based on such (and similar) proposals, we suggest a list of ontological levels, which primarily is a heuristic tool leading to different types of interpretations of diagrams. The Basic Ontological Stratification Four major stratified domains are distinguished: 1. locomotion in space, 2. change in a quality space, 3. external action/interaction, and 4. internal action, which are further subdivided:

5

Cf. as an example the onto-semantic analysis of the lexicon of German verbs in Ballmer and Brennenstuhl [2].

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1. Locomotion in space–time 1.1 Interlocal locomotion (outside the neighborhood relative to some landmark). 1.2 Locomotion in the system’s neighborhood and its periphery, e.g., the movement of the limbs relative to a body, is called local. 2.1 Change on one categorical, mainly bipolar scale (in one dimension of the space of qualities). 2.2 Change in the phase-space of a dynamical system (from one phase to the other). 2.3 Change on a quantitative scale (at the ordinal, interval, or metrical level of measurement). 3. Action and interaction (the process in an action or interaction scenario) 3.1 External (physical, chemical, biological) action of an agent on an object or another (secondary) agent. 3.2 Change of possession. 3.3 Communicative action. Action and interaction stand ontologically between physical locomotion (1), which governs parts of them, and internal (intentional) processes (4), which direct the action. The effect is often a change of quality (2). These processes are typically mixed, i.e., the different roles in an action/interaction scenario operate on different strata. 4. Internal action/interaction (with internalized objects and targets) 4.1 Perceptual action (in the sensory system). 4.2 Mental action. This process is at least partially self-referential (in the brain). In domain 4, the processes are strictly internal within a body or a cognitive system; we cannot observe them directly in other people. However, these processes have perceivable traces (in the individual’s behavior), and we can linguistically label such a process and tell the event to our audience. The processes of domain 4 also have another peculiar property. They are the basis of the modality scale (cf. [28], Chap. 5.3). For every domain, we may distinguish maximum diagrams and partial diagrams. However, only the maximum diagrams will be enumerated to illustrate the dependence between domains and diagrams. A Short Description of the Principal Domains Although we use traditional labels from case theory (cf. [25], Chap. 1, [32]), the content of these labels is independent of recent traditions. We systematically depart from classical case theory because our primary criterion is dynamic configuration.

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The possible dynamic configurations are nested and hierarchically structured. We distinguish: 1. primary agents (they are the foundation of the process and do not disappear in the process); 2. secondary agents (they appear and disappear in the process). The dynamic “cases “defined by the configurational criterion are called: (a) A (Agent)—P (Patient) (primary roles) (b) I (Intermediary)—B (Binding force) (secondary roles). The label I summarizes a plurality of forces that are linearly intermediate between A and P. Depending on the domain of interpretation, it can be a path (interlocal locomotion), a metastable phase on a quality scale (quality space), an instrument (action space) or an object (change of possession). Role B (binding force) has a rather variable realization. Configurationally it is an intermediary force parallel to the primary sequence A–I–P. Therefore it calls for a second dimension in state space (cf. [23]: 85–92). It can be parallel to A (a helper of the agent), P (a beneficiary of the event), and I (a secondary instrument, a medium of exchange). The Domain: Locomotion in Space–Time Locomotion may be simple (linear) or include the transition through a frontier or several linearly arranged frontiers (on a path). The maximum configuration is one with three roles: A (agent), P (patient), I (intermediary force). A possible elaboration contains one or more domains on the path through which the intermediary force goes when it comes from the source and before it reaches the goal. Partial configurations have one or two roles (attractors) (Fig. 15.13). The Domain: Change in a Quality Space The configurations are similar to those described above, the difference being that partial diagrams are more frequent and elaborations with a third (intermediate) quality are rare. We can introduce two pairs; A versus non-A (privation of A) and A versus

Fig. 15.13 The maximal diagram of locomotion and partial diagrams

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CA (bimodal proportional opposition of A to its complement CA). In the first case, we consider only a partial scenario while the complementary state is undetermined. In a proportional opposition, both qualitative states are present; the change from one quality to the other is moved into the foreground. 1. Privation Stop to be A (become non-A), begin to be A (stop to be non-A); 2. Proportional Stop to be A (leave the domain A and become CA, i.e., change from A to CA), begin to be A (leave the domain CA and become A) (Fig. 15.14). We can easily see that the first diagram is a part of the second one. The Domain: Action and Interaction The maximum configuration is the diagram of transfer (or of instrumental action, which is the symmetric variant of it). Figure 15.15 shows the two diagrams. The two variants have the same thematic grid (A–I–P).

Fig. 15.14 Basic processes in a quality space

Fig. 15.15 The maximal diagrams of action and interaction

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Transitory agent / object

Agent

Patient Mediating force

Fig. 15.16 A section of the elliptic umbilic and the diagram of mediated transfer (“sending”)

The scenario where four forces interact (cf. the diagram “star” in Fig. 15.7) corresponds in sentential semantics to a configuration of four dynamic “cases”. The fourth agent is a binding force that enables a specific interaction between the other forces. It was labeled B at the beginning of this section. Figure 15.16 represents the elaborated diagram, which refers to a two-dimensional behavior space. The two-dimensional space of internal variables (x,y) has three attractors, A, B, and C, that correspond to the basic triad of dynamic “cases” A–I–P; the central attractor D stands for the case B that mediates the transition between A and P via the transitory agent I. The graph to the right is only an incomplete representation of this dynamically very complex situation. The fourth participant can be interpreted as a helper (i.e., a secondary agent in the tradition of narratology) or a beneficiary (a secondary patient). The elaborated configuration can be represented in a three-dimensional diagram regarding our topologic-dynamic description.6 The four-valent scenario can be fully realized in the scenario of instrumental sending: Example: (i) Albert (A: source) sends Imela (I: secondary agent) with British Airways (B: helper) to Paris (P: goal). The intermediary force can also be an object exchanged or a primary instrument. Examples: (ii) Andrea (A) sends a letter (I) to her friend (P) by airmail (B). (iii) Annabel (A) gives an interview (I) to the press (P) by telephone (B). (iv) Anne (A) propels the arrow (I) towards the tree (P) with a bow (B). The Domain: Communicative and Perceptual Action The configuration is similar to those already discussed. We can distinguish between emissive actions, where perceivable events are produced, and receptive actions, where such events are received. If both partial diagrams combine, we have a transfer of perceivable units, signal transmission; if this transmission is mutual and reciprocal, 6

Cf. Wildgen ([23]: 86–92) and Wildgen ([25]: 204–222).

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Message 1

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Message 2

B = Code

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Fig. 15.17 The basic diagram of communicative action

we have sign communication. A binding force is added if a symbolic instrument, a system of conventional signs, is put to work. Language as a system is such a symbolic instrument. The roles defined by their place in the configuration have somewhat different content. Figure 15.17 shows the basic configuration. If we take a closer look at the dynamics of the event, we notice important differences between communicative action (domain 3.3) and the basic domains 3.2 (change of possession) and 3.1 (physical action): – The sender does not lose the message if he emits it. Rather, he sends a duplicate; similarly, the receiver creates an analogous message using the information he receives and his knowledge. – The intermediary role B (the binding force, the code) is a necessary constituent for the transfer, which could not occur without it. Furthermore, this force is very rich and complicated. Whereas the Agent and the Patient are individuals, the language system has a social, supra-individual, and, therefore, abstract nature. In perception, the object received can be either a sign (cf. the partial interpretation of receptive action in 3.3) or a percept (some natural input to the sensory organs). The sensory inputs continually entering our sensory organs are the background of sign reception.7 At an intermediate level, our attention is focused on a specific percept; we see, hear, and smell something specific. The topological scenario is that of CAPTURE. These basic derivations from external processes become even more prominent if we analyze what is going on in mental action. The Domain: Mental Action The new phenomena at this stratum are: – The semantic closure of the mind on itself. This feature was emphasized by Maturana and Varela [9] and other theoreticians of the brain. – The self-referential nature of mental processes. 7

Cf. Petitot ([16]: Chap. 6: Attractor syntax and perceptual constituency).

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Fig. 15.18 The basic diagram of mental action

– The overwhelming importance of cognitive contexts, i.e., memory, knowledge disposition, attitudes, personality traits, and others. These basic characteristics are diagrammatically represented in a specific form of the maximum schema given in Fig. 15.18. The different phases of the process are labeled: e: egressive (emission): the mind produces an idea, an emotion, an attitude, i: ingressive (reception): the mind receives, retains, and stabilizes an idea, an emotion, an attitude, s: self-referentiality: the mind produces and receives (from itself) an idea, an emotion, an attitude. The diagrams labeled e and i are only partial pictures of s, which is complete. The actual processes can become part of the permanent structure of the mind, and parts of the permanent structure can, in turn, be actualized. This mode is called the resultative (r). The pure form of the resultative phase is the stock of persistent ideas and emotions in mind.

15.5 Discrete Dynamic Diagrams Using Vector Calculus The minimal dimensionality of space–time is given by one dimension of time and another of space. A one-dimensional vector field can represent motion in one dimension, whereas the time dimension remains implicit. Throughout this section, vectors on one space dimension are used for the construction of discrete diagrams of processes in space–time. If a vehicle moves in a plane, one can note the direction and the length of the movement by a vector -> w (w = way). For example, if the distance is measured in km and direction with a compass, a displacement of 6 km in the direction North-East can be described as in Fig. 15.19. On the other hand, suppose neither the direction nor the specific amount of movement is of interest. In that case, the two-dimensional frame (the plane) can be reduced to one-dimensional, and the unit vector of length 1 replaces the vector of length /w/. A basic feature of the vector notion is that the factor time is implicit because the vector describes the difference between the place of the moving element in t0 and t1 . If the amount of motion is restricted to unit-amount 1, a constant movement

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Fig. 15.19 The vectorial representation of motion (a) and its reduction to the unit-vector (b)

inside every cell of our discrete system is given. In this simplified version, one can only distinguish between rest (the vector length is zero) and uniform motion (the vector length is 1). If two vectors, w1 and w2, in a plane are considered, the sum and the difference between w1 and w2 can be computed geometrically by applying the parallelogram law. A model of the whole system may be based on (continuous) differential equations. However, one can simplify the model by radically replacing it with a model with a grid of discrete steps and with self-similarity, i.e., every piece of the system is identical to all the others, and the same rules apply to every piece of the system. This allows very quick and highly frequent applications of the same rules to all system elements. This type of mathematical model is called a cellular automaton (CA). Toffoli characterizes the CA as follows: In the cellular-automaton model of a dynamical system, the “universe” is a uniform checkerboard, with each square or cell containing a few bits of data; time advances in discrete steps and the “laws of the universe” are just a small look-up table, through which at each time step each cell determines its new state from that of its neighbours; this leads to laws which are local and uniform. Such a simple underlying mechanism is sufficient to support a whole hierarchy of structures, phenomena and properties. ([22]: 119)

Compared to a continuous dynamic model the following transformations are necessary: a. continuous space and time are replaced by a discrete grid, b. the system/state at each point remains a continuous variable of the same kind (e.g. real, complex, vector) as in the original equation, and c. derivatives are replaced by differences between state-variables that are contiguous in space and time. ([22]: 121) If, for example, we take two vectors as described above, the zero-vector (state) and the unit-vector (say /ui / = + 1), we have two basic values since every cell of the system may have the value of either 1 or 0. Graphically we can represent the zero vector as a blank cell and the (positive) unit vector as a shaded cell. To illustrate what a cellular automaton can describe, we build a model of expansion/reduction

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of a narrative on this basis. The starting point is a matrix and a local environment defined by the neighbors which touch it at one point (if not along a line that separates the cells). One can imagine a game where several narrative units (clauses containing an event or action) are given. Every participant must complete sequences of events and eliminate narrative units without proper followers. The rules of this game can be stated in terms of a CA restricted to specific environments. In our example, the set (8,4) of units on the diagonal from upper left to lower right will be submitted to special restrictions. Rules of the “narrative” game: 1. zero-vectors (the cell is 0) are not affected by the game, 2. if a cell is 1 and its relevant neighbors (in the set (8,4)) are 0, make it zero 3. if a cell is 1 and one of its neighbors (8,4) is 1, change the other neighbor to 1. Figure 15.21 shows different phases of play with a random starting condition. One can observe how, in (2), all narrative units without narrative continuity are eliminated; only the basic sequence is completed until the borderline of the matrix is reached. This example suggests that the temporal evolution of a narrative plot (e.g., if a story is retold repeatedly) can be modeled with the help of a cellular automaton. Example of a diagrammatic description of a narrative episode The unit vector allows for the addition and multiplication of vectors (these operations are diagrammatical, as the comments of Peirce in Sect. 1 have shown). In Fig. 15.22, these operations are illustrated. These operations allow for constructing a list of basic uni-valent, bi-valent, and tri-valent diagrams. They constitute the vocabulary of a cellular automaton describing processual sequences, their major regularities, and restriction (quasi-a grammar of processes in space–time). Cf. for details Wildgen ([27] and [28]: Part two: The Meaning of Oral Narratives). Based on the notion of unit-vector shown in Fig. 15.20, a set of basic diagrammatic units (the first stratum of the vocabulary of narrative syntax) can be defined. All units occupy a unit cell (length 1 × 1), i.e., a unit-square with a vector length on time (t) and space (r) = 1 are defined. For t, only positive integers are possible (time is not moving backward). In the first elaboration, half-steps are allowed with values of t (0, 1/2, 1). For r, we distinguish positive values (motion of the protagonist) with the values (0, 1/2, 1) and negative values (motion of the antagonist) with the choices: (0, −1/2, −1). This simplification leads to the vocabulary of basic diagrams represented in Fig. 15.23. Description: Diagram 1: Positive motion/action r (protagonist) Diagram 2: Transition from a positive motion/action to a stable state Diagram 3: Transition to the opposite direction Diagram 4: Transition between two partial, positive motions/actions

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Fig. 15.20 Global and local neighborhoods and two states of the automaton

Fig. 15.21 A simple narrative game

Fig. 15.22 Illustration of the operation of an “addition” of unit vectors and a “multiplication” with a constant

Diagram 5: No motion on r, a stable state Diagram 6: Transition from a state to a positive motion/action (protagonist) Diagram 7: Transition from a state to a negative motion/action (antagonist) Diagram 8: Transition between two independent stable states Diagram 9: Negative motion/action on r (antagonist) Diagram 10: Transition from a negative motion/action to a stable state Diagram 11: Transition from a negative motion/action to a positive one

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Fig. 15.23 Basic set of univalent vector cells in a diagrammatic grammar of narratives

Diagram 12: Transition between two partial, negative motions/actions. The set of 12 basic diagrams is exhaustive for this space–time matrix’s chosen level of differentiation. In Wildgen ([28]: 165–171), two further sets, the set of bivalent and the set of trivalent diagrams, are defined. Based on these sets of diagrams, a cellular automaton for narratives can be defined, in which central notions of a narrative syntax such as adjacency, coherence, and narrative skeleton can be defined. For example, in Wildgen ([28]: 196–200), a complex diagram describes the processual content of a narrative. The story is called “Calvin’s rock-war” and was analyzed with traditional tools in Labov’s article “The Transformation of Experience in Narrative Syntax” [8].

15.6 Conclusion: Diagrams, Indexes, and Symbols in Scientific Theories Diagrams and the iconic mapping they realize are an encompassing phenomenon. We have focused on the scientific and rational aspects of diagram construction. Major tenants of diagrams such as bodily rooted and manifested diagrams, gestures, and more primitive means of expression found in non-human beings have been neglected. They are the focus of other contributions to this volume, e.g., Chaps. 10 and 14. Diagrams have their specific methods of construction as soon as they become complex and expose a rich network of internal relations. At this level, their organization is linked to self-organization, economy, and a kind of “formal structure”

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following self-organized optimality. These features are, however, in most cases, unintentional and not rationally controlled. Nevertheless, they make up the germ, the foundation of any explanation or broader understanding of the underlying phenomena. They contribute to the functionality of diagrams and their communicative success or failure. This non-arbitrariness was clear to Charles Sanders Peirce around 1900. It separates his semiotic thinking from conventionalism and formalism, which began to dominate after 1910, cf. the “Principia Mathematica” of Russel and Whitehead 1910. Carnap and Bar-Hillel prepared the position of “language as algebra” in Chomsky [5] and “English as a formal language” in Montague [12]. However, this is only one bank of the river. The other bank is the rhetorical and quasi-literary endeavor creating networks of quasi-theories mixing eclectic elements of different provenience motivated by “deconstruction” in post-structuralism (cf. Derrida’s publications). In his “Lessons on the History of Science,” Peirce tells us: “The first quality required for this process [diagrammatic representation, W.W.], the first element of high reasoning power, is evidently imagination; and Kepler’s fecund imagination strikes every reader. But “imagination” is an ocean-wide term, almost meaningless, so many and so diverse are its species. What kind of imagination is required to form a mental diagram of a complicated state of affairs? Not that poet-imagination that “bodies forth the forms of things unknowne”,8 but a devil’s imagination quick to take Dame Nature’s hints. The poet-imagination riots in ornaments and accessories, a Kepler’s makes the clothing and the flesh drop off, and the apparition of the naked skeleton of truth to stand revealed before him.” ([14]: 255). The different types of signs and their combination or interaction constitute an important condition for realizing scientific theories. They respond to different demands on science: • Diagrams. They respond to the traditional condition of correspondence or mimesis. The theory must tell us something about the object, the world under analysis. This function is essentially descriptive or informative. If we have no prior knowledge about the object, the scientist can tell us something about it, and we may imagine the object and recognize it if in contact with it. For instance, the biologist describes a new plant or a new animal. If we meet an animal in nature or a zoo or find such a plant, we can recognize (and name) it. Diagrams should reproduce major constellations and dependencies for complex ensembles of things or events. These are the central ingredients in the formation of scientific theories. • Indexes. In many instances, we may have a description, an image, etc., but we are unsure if the object in question exists. It may be a fantasy; the information given may be faulty or even a lie, a fraud. We ask for proof. In this case, indexical cues, hints, and even demonstrations “ad oculos” are asked for. In the case of networks of signs or complex sign constructions, the choice of relevant relations may be false or unrevealing, obscure or irrelevant. The central structures reveal causal 8

A quote from Shakespeare: “And as imagination bodies forth//The forms of things unknown, the poet’s pen//Turns them to shapes and gives to airy nothing//A local habitation and a name”, William Shakespeare, A Midsummer Night’s Dream.

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links and a chain of cause and effect. Any causal attribution must be checked for its validity. The truth of a story depends essentially on the validity of causal hypotheses. • Symbols. In every act of communication, the partners must presuppose a system of conventions that control the exchange of information. This is already the case for simple icons or indexes, but it becomes overwhelming in the case of complex signs or if sequences or fields of signs are to be handled. Therefore the symbolic nature of sign communication is the dominating mode. The major risk of symbols is given by the fact that they presuppose conventions. In most cases, we did neither initiate nor control these conventions (we even, in most cases, are unaware of them). Therefore, we can never be sure that those persons or institutions responsible for the conventions were honest. In other exchanges, e.g., in the case of goods sold or bought, we must be careful to check not only the quality of the goods and the validity of the money but also respond to the question: is the object bought worth the money we have paid? A highly developed field of laws, their application, and the institutions that control them are necessary to make us trust the system of economic exchanges. In sign communication, we must more or less believe and trust the partners of symbolic exchange. The only things we can rely on (still with risk) are iconic and indexical cues. The fundamental arbitrariness of symbols has the consequence that although iconic and indexical signs never appear in isolation or in their pure form, they are extremely important. If the iconic mimesis and the indexical (causal) foundation of sign communication are not guaranteed, the richness of symbolic communication loses its value, becomes irrelevant, or an annoyance. Silence and the avoidance of any exchange would be the better choice. The evolution of human language and civilizations shows that we did not follow this route.

References 1. Arnold, V. I.: Catastrophe Theory. second edition. Springer, Berlin (1986) 2. Ballmer, T.T., Brennenstuhl, W.: Deutsche Verben. Eine sprachanalytische Untersuchung des Deutschen Wortschatzes, Narr, Tübingen (1986) 3. Boscodon.: L’art des bâtisseurs romans. La Géométrie et les Maîtres de l‘œuvre, Cahiers de Boscodon Nr. 4, Association des amis de l’abbaye de Boscodon, Crots (1988) 4. Carnap, R.: Der logische Aufbau der Welt, Berlin-Schlachtensee (1928) 5. Chomsky, N.: Syntactic Structures. Mouton, Den Haag (1957) 6. Gibson, J.J.: The Senses Considered as Perceptual Systems, Hughton Mifflin, Boston (1966) 7. Gilmore, R.: Catastrophe Theory for Scientists and Engineers. Wiley, New York (1980) 8. Labov,W.: The transformation of experience in narrative syntax. In: Labov,W., (ed.) Language in the Inner City. Studies in the Black English Vernacular. University of Pennsylvania Press, Philadelphia, pp. 354–396 (1972) 9. Maturana, U., Varela, F.: The Tree of Knowledge. The Biological Roots of Human Understanding, New Science Library, New York (1987) 10. Mazzola, G.: Geometrie der Töne. Birkhäuser Verlag, Basel, Elemente der mathematischen Musiktheorie (1990)

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11. Mazzola, G.: Geometric Logic of Concepts, Theory, and Performance, Birkhäuser, Basel (2002) 12. Montague, R.: English as a Formal Language. Reprint In: Thomason, R.H. (ed.) Formal Philosophy: Selected Papers of Richard Montague, pp. 188–221. Yale University Press, New Haven (1970/1974) 13. Peirce, C.S.: Philosophical Writings of Peirce. In: Bucher, J. (ed.) Dover Publications, New York (1955) 14. Peirce, C.S.: Selected Writings (Values in a Universe of Chance). In: Wiener, P.P. (ed.) Dover Publications, New York (1958) 15. Peirce, C.S.: Collected Papers of Charles S. Peirce, Vol. 1–8, Harvard University Press, Cambridge (Mass.) (1931–1960) 16. Petitot, J.: Cognitive Morphodynamics. Constituency in Perception and Syntax (in collaboration with René Doursat), Lang, Bern (2011) 17. Poston, T., Stewart, I.: Catastrophe Theory and its Applications. Pitman, London (1978) 18. Thom, R.: Topologie et linguistique. In: Haefliger, A., Narasinkan, R. (eds.) Essays in Topology and Related Topics, pp. 226–248. Springer, Berlin (translated as chapter 11. in Thom [20]: 192–213) (1970) 19. Thom, R.: Stabilité structurelle et morphogenèse. Interéditions, Paris (English translation: Structural Stability and Morphogenesis. Benjamin, Reading, 1975) (1972) 20. Thom, R.: Mathematical Models of Morphogenesis, Horwood (Wiley), New York (1983) 21. Thom R. : Esquisse d’une sémiophysique: physique aristotélienne et théorie des catastrophes. Interéditions, Paris [English translation: Semiophysics. A Sketch. Addison-Wesley, Boston 1990] (1988) 22. Toffoli, T.: Cellular Automata as an Alternative to (rather than an approximation of) differential equations in modelling physics. Physica 10D, 117–127 (1984) 23. Wildgen, W.: Catastrophe Theoretic Semantics. An Elaboration and Application of René Thom’s Theory, Benjamins, Amsterdam (1982) 24. Wildgen, W.: Goethe als Wegbereiter einer universalen Morphologie (unter besonderer Berücksichtigung der Sprachform). In: Annual Report 1982, University of Bayreuth: 235–277 (1982/84). Reprinted 1984 in: L.A.U.T. Preprint, Series A, Papier Nr. 125, Trier, 1984. internet resource in research gate (visited 18.05.2023). https://www.researchgate.net/publication/379 32863_Goethe_als_Wegbereiter_einer_universalen_Morphologie_unter_besonderer_Beru ecksichtigung_der_Sprachform_Vortrag_gehalten_bei_der_Tagung_Goethes_Beitrag_zur_ Naturwissenschaft_heute. 25. Wildgen, W.: Archetypensemantik. Grundlagen einer dynamischen Semantik auf der Basis der Katastrophentheorie, Narr, Tübingen (1985) 26. Wildgen, W.: Dynamic aspects of nominal composition. In: Thomas T. Ballmer and Wolfgang Wildgen (eds.) Process Linguistics. Exploring the Processual Aspects of Language and Language Use, and the Methods of their Description, pp. 128–162. Niemeyer, Tübingen (1987) 27. Wildgen, W.: The distribution of imaginistic information in oral narratives. In: Altmann, G., Hreblicek, L. (eds.) Quantitative Text Analysis, pp. 175–199. Wissenschaftlicher Verlag, Trier (1993) 28. Wildgen, W.: Process, Image, and Meaning. A Realistic Model of the Meaning of Sentences and Narrative Texts. Benjamins, Amsterdam (1994) 29. Wildgen, W.: Kognitive Grammatik. Klassische Paradigmen und neue Perspektiven. de Gruyter, Berlin (2008) 30. Wildgen, W.: Die Sprachwissenschaft des 20. Jh.s: Versuch einer Bilanz, de Gruyter, Berlin (2010) 31. Wildgen, W.: Visuelle Semiotik. Die Entfaltung des Sichtbaren. Vom Höhlenbild bis zur modernen Stadt, transcript-Verlag, Bielefeld (2013) 32. Wildgen, W.: En cas de catastrophe. Les systèmes casuels et la dynamique qualitative, Contribution to the «Colloque Petitot», Paris-Nanterre, 29 mai 2015. Estudos Semióticos 13(1), 2017 (internet journal) (2015) 33. Wildgen, W.: Musiksemiotik: Musikalische Zeichen, Kognition und Sprache, Königshausen & Neumann, Würzburg (2018)

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34. Wildgen, W.: Structures, archetypes, and symbolic forms. Applied Mathematics in Linguistics and Semiotics. In: Alberto Peruzzi et Silvano Zipoli Caiani (eds.) Structures Mères. Semantics, Mathematics, and Cognitive Sciences, pp. 165–185. Springer, Cham (2020) 35. Wildgen, W.: Morphogenesis of Symbolic Forms: Meaning in Music, Art, Religion, and Language, Springer Nature (Series: Lecture Notes in Morphogenesis), Cham (CH) (2023)

Chapter 16

Conclusions: As a Kleiner Narr in Trance Towards a Diagrammatic Model of Enunciation Francesco La Mantia

16.1 Summary Notes and Auspices The path conceived together with Charles Alunni and Fernando Zalamea aimed to provide an overview of the current debate surrounding the notion of diagrammatic gesture. It will be up to the reader to judge whether or not we have succeeded. Having articulated the text into three distinct sections (i.e. Mathematics, Philosophy, and Linguistics), we upheld the effort to focus on at least some of the fundamental orientantions regarding this debate. Whereas in “Almost an Introduction. From the Basilar Notions to the Legacy of Gilles Châtelet” the issues adressed mostly pertained to the philosophy and mathematics of diagrammatic gestures, in these closing comments we will instead seek to deal with the notion of diagrammatic gesture from a more semiotically and linguistically oriented perspective. In these pages, the ontology of mathematical entities will remain a firm benchmark of analysis. We will discuss it, however, always in view of the semiotic-linguistic issues that are the focus of this chapter. Our goal is to develop a philosophy of the diagrammatic gesture that is an indispensable tool for the scholar of languages theory. We thus aim to show that the notion of diagrammatic gesture has a great heuristic potential to illuminate some constitutive aspects of human linguistic activity. We will seek to prove it by analyzing a model of enunciation that we believe to have Although the empirical author of the essay is Francesco La Mantia, the contents exposed are the result of shared and fruitful conversations between all three editors. This is why the writer wishes to express his gratitude to Charles Alunni, who has discussed and examined in detail all the paragraphs of the introduction, and to Fernando Zalamea, who has carefully re-read the text. Special thanks go to Maria Giulia Dondero who has been an attentive and valuable reader. Of course, La Mantia is solely responsible for any error. F. La Mantia (B) Dipartimento di Scienze Umanistiche, Università Degli Studi di Palermo, Palermo, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1_16

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located whitin a key chapter of “A Thousand Plateaus”. In doing so, we will deal with diagrammatic gestures in a framework—that of enunciation, to be precise—that is strictly linguistic and semiotic.

16.2 A Diagrammatic Model of Enunciation: November 20, 1923—Postulates of Linguistics Let’s consider the statement ‘I swear!’”.1 According to Deleuze and Guattari, “It is a different statement depending on whether it is said by a child to his or her father, by a man in love to his loved one, or by a witness before the court”.2 Although correct, the remarks just reported are not particularly original. Their apparent simplicity, however, should not deceive. In fact, they serve as a prelude to a whole series of remarks which sketch the model of enunciation to which we referred in the previous section. A few lines later, in particular, we read that “[n]ot only are there as many statements are there are effectuations, but all the statements are present in the effectuation of one among them”.3 Now, it is precisely in this last remark that lies the above-mentioned model. We will propose here to call it “Diagrammatic”, and we will seek to examine it starting again from some observations of Paul Klee.

16.3 Kleiner Narr in Trance… and Others After having highlighted the internal germinal forces which set points into motion (for an analysis of which see sections §§ 6.4, 6.4.1, 6.4.2, 6.4.3 of “Almost an Introduction”4 ), the Swiss-born German painter develops his own research towards a key argument of Das Bildnerische Denken. We refer to what he presents in terms of “das Übereinanderlegen von Momentgefaßten Bewegungen”.5 This “Übereinanderlegen”—which, for our part, we would prefer to call “fulguration”—shares with the Diagrammatic model of enunciation (§ 2.) the same basic insight. Namely, the simultaneous co-presence of virtual elements. We say “elements” to mean both “verbal forms” and “visual forms”. In the first case, we refer to the several “I swear!” that each effectuation of “I swear!” would carry in itself, at least according to Deleuze and Guattari (§ 2.); in the second one, we refer to the lines, or, more generally, to the

1

Deleuze and Guattari [27], p. 94. Ibid. 3 Ibid. The Italics are ours. 4 Henceforth, AAI. 5 Klee [42], p. 130. 2

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Fig. 16.1 Kleiner Narr in Trance 1927. The picture is taken from Klee [42]

“Elementarformen”6 which swarm in Kleiner Narr in Trance, a work by Klee dating back to 1927 (see Fig. 16.1). But there is more. Among these “Elementarformen”, we will also include a third example: the triangle “bougeant ‘un peu’”7 by Châtelet (see Fig. 16.2). It too fits perfectly into the basic insight shared by Deleuze and Guattari. In any case, initially, it will be another starting point that we will adopt to discuss the main features of the diagrammatic model of enunciation.

6 7

Ibid. Châtelet [19], p. 140.

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Fig. 16.2 “Châtelet’s triangle”. The picture is taken from Châtelet [19]

16.4 What is Nothing in Itself: Points and… Utterances Perhaps the most important lesson we learned from the Châteletian philosophy of mathematics is that points are nothing in themselves (§§ 6.8.1.2. & 6.8.1.3. of “AAI”).8 Behind this assertion, there is a subtle philosophy of the virtual.9 Our hypothesis is that this philosophy operates, in other regards and in complete autonomy, whitin a Deleuzian-Guattarian model of enunciation.

16.4.1 Germs of Something Else It is the main idea underlying this model that confirms us in our opinion. If, as Deleuze and Guattari argue, in every utterance there are other utterances (§ 2.), then the utterance is nothing in itself. It is rather such as the point of Châtelet. It is something, viz., which subsists only as a germ of something else. The utterance, in short, is null, not because it is vacuous or crumbling, but because all its “being”—as well as the “being” of the point—resides in “another being”. This other is particularized both in the other points that every point generates as a quotient of a polynomial (§ 6.8.1. of AAI) and in the other utterances present in every utterance. In what sense, however, would other utterances be present in every utterance?

8

But see also De Freitas and Sinclair [24], p. 203: “The point is nothing in and of itself, it becomes something through its creative potentiality”. 9 § 6.8.1.3. of AAI.

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16.4.2 Enunciative Virtualities Let’s start with an objection. You could argue that it is not true that an utterance in itself is nothing (§ 4.1.). For instance, each one of the several “I swear!” (§ 2.) plays a precise performative role—and this is far from being nothing. We make this objection our own, but only because it allows to better clarify our hypothesis. Contrary to what one might think, there is no incompatibility between the “nothingness” of the single “I swear!” and the performative role that it plays in a specific enunciative context. There is no doubt that, whenever it is said, “I swear!” exhibits its own materiality, and that by virtue of it, a particular linguistic act is performed. But this act, or rather, the materiality that underlies it, it’s just a local and transient aspect of “I swear!”. In other words, it is the aspect that the statement takes on in the instant in which it is uttered. Now, the presence of other utterances in each utterance can be explained precisely from this empirical detail. What we call here the “nothingness” of the utterance refers to what others would call, perhaps more appropriately, its “depth” or “thickness”.10 Every utterance would be null (or deep or thick) in the sense of being irreducible to its local conditions of production. And this is not because behind it there is an ideal utterance (or type) detached from its tokens, but rather because every token—every “I swear!”—is crisscrossed by enunciative virtualities which exceed those conditions. It is in this sense, and only in this sense, that every utterance subsists as a germ of something else (§ 4.1.). And it is still in this sense that should be understood the provocative assumption that “all the being” of an utterance resides “in another enunciative being” (§ 4.1.). According to this perspective, then, every utterance is what it is by way of the virtualities swarming within it. This is why the presence within any utterance in a given utterance would firstly be explained by such virtualities. Deleuze and Guattari certainly move in this direction when they remark that it is possible “[to] place the statement [i.e. “I swear!] in continuous variation […] [and that] the line of variation is virtual”.11 The expression “line of variation” refers to the several “I swear!” swarming in every “I swear!”, that is to say, to “the continuum of ‘I swear!’ with the corresponding transformations”.12 However, it’s only through a preliminary analysis of 10

Cf. Robert [70], pp. 255–274. See also Bondì [12], pp. 42–43. The notion of enunciative “thickness”, which we merely mention here, would deserve a separate space of in-depth study that we reserve for future work on the subject. The reflections on the diagram conducted by Maria Giulia Dondero offer in this sense a valuable starting point in the direction of a capital concept of Jacques Fontanille’s semiotics: that of “enunciative praxis” (praxis énonciative). In this regard cf. the seminal Dondero [30], pp. 75–88. To give a more precise reference, we simply note that the notions of “protension” and “retension” through which Dondero schematizes the work of enunciative praxis present very strong affinities with the notions of reprise and relaunching that we discuss in Appendix 1 to this chapter. We would like to thank Maria Giulia Dondero for pointing out this significant theoretical insight. 11 Deleuze and Guattari [27], p. 94. The italics are ours. 12 Ibid.

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the Deleuzian-Guattarian notion of the virtual that it is possible to justify the mode of presence we are discussing.

16.4.3 Klee’s Jester and Châtelet’s Triangle Klee’s jester and Châtelet’s triangle (§ 3.) provide, in this sense, some precious clarifications. They both diagrammatize fulgurations, namely germinal potentialities (§ 6.8. of AAI) grasping a key feature of Deleuzian-Guattarian virtuality. We are referring to a peculiar articulation of the notion, that is, to the Deleuzian version of the virtual which is embodied in the intuitive figure of the “egg”13 and which corresponds to “a task to be solved”.14 The potentialities diagrammatized by the abovementioned objects give access to this figure by means of an elegant aesthetical solution: the “setting their ‘Elementarformen’ (§ 3.) into vibration”. Deleuze’s egg can be approximated in equivalent terms since it is organized through the same agent which is responsible for putting those “Formen” into vibration. In the final pages of Ideas and the Synthesis of the Difference,15 this organizing agent is indeed identified with “[…] movements”16 or “actualizing […] agencies”,17 that is to say, with elements (§ 3.) having the same nature as those movements that, by overlapping, put the “Elementarformen” of the jester and the triangle into vibration. It is a crucial aspect of the diagrammatic life18 which efficaciously highlights the relationships between enunciative tokens and their virtualities (§ 4.2.).

16.4.3.1

“Elementarformen”: Putting into Vibration

This aspect can be grasped firstly on the visual level. Thanks to a broad and wellestablished sense of “enunciation”, even the lines of which a drawing is composed— or a drawing itself—can be envisaged as utterances. So long as “enunciation” is not only made to refer to the “act of speaking”19 but also to those of “painting, photographing”,20 drawing, and so on…, the presence of utterances in a given utterance can be thought in terms of a “putting specific ‘Elementarformen’ (§ 4.3.1.) into vibration”. It is Châtelet’s triangle that provides the most immediate example in this regard (§§ 3. & 4.3.). If drawing it on a sheet of paper is the carnal expedient that amounts 13

Deleuze [26], p. 214. Ibid. p. 212. 15 Cf. Ibid. pp. 168–221. 16 Ibid. p. 214. 17 Ibid. 18 As for the concept of “Diagrammatic life”, see §§ 6.4.5 & 6.5 of AAI. 19 Dondero [31], p. 22. 20 Ibid. 14

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Fig. 16.3 “A detail of the Kleiner Narr in Trance”. The picture is taken from Klee [42]

to its visual enunciation, and if the picture imprinted on that sheet is the product of that enunciation, then the dotted lines which seem at once to emerge from it and to intersect it are “ready-to-be actualized” virtualities: germs of other triangles that “are […] contained in it”21 and which surface in the form of movements, of vibratory patterns which continuously recreate it.

16.4.3.2

“I Swear!”: Putting into Variation

On the verbal level proper, on the other hand, one can explain the same point by referring to what Deleuze and Guattari call “[placing] the statement in continuous variation” (§ 4.2.). Indeed, if the locution “line of variation” refers to several “I swear!” swarming in every “I swear!” (§ 4.2.), and if this line is virtual (§ 4.2.), then the presence of utterances within a given utterance can be thought in terms of this “putting-into-variation”,22 or, more properly, in terms of enunciative virtualities crisscrossing “I swear!” as other ready-to-be-actualized “I swear!”.

16.4.3.3

Châtelet’s Triangle and “I Swear!”: A Nexus

There is thus a nexus between Châtelet’s triangle and “I swear!”, as they both contain “crystals of becoming”.23 In other words, notwithstanding their irreducible differences, the “Elementarformen” of one and of the other are all likely “to bud”. The same can also be said about the “Elementarformen” of Klee’s jester. This is the case, for instance, with the lines that propagate along the lower left end of the jester (see Fig. 16.3).

21

De Freitas and Sinclair [24], p. 205. Alliez [2], p. 142. 23 Deleuze and Guattari [27], p. 106. 22

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Fig. 16.4 “A modeste conjectural scheme by F. La Mantia”

They show their ‘readiness’ to bud to the extent that they produce zig-zag configurations likely to expand at every point of junction, maybe in the form of dotted lines (see Fig. 16.4). It is in this ‘expansion’ that the virtual resides as “a task to be solved” (§ 4.3.). Deleuze’s egg (§ 4.3.), in short, can unfold as much in the form of dotted lines ` Whereas the former refer to the as in the form of ‘sayables’ (dicibles or λεκτα). further drawability of figures—namely, to the hand’s movements that put them into vibration—the latter refer to the further utterability of verbal forms—namely, to the movements of the mouth that put them into variation. Putting a visual forms into vibration and putting a verbal ones into variation therefore share the same gestural nature. Both such processes originate from specific modalities of movement,24 that is, from manual or phonatory gestures.

16.5 First Steps Toward a Diagrammatic Model of Enunciation Enunciation, be it verbal or visual, is thus diagrammatic in the very general sense that its products are involved in a process of becoming. As the examples just discussed show (§§ 4.3.1. & 4.3.2.), utterances are “beings in becoming” to the extent that the drawability of visual forms and the sayability of verbal forms are iterable operations. As the triangle is drawn and redrawn in the form of other triangles, and as “I swear!” is said and resaid in the form of other “I swear!”, forms change themselves by becoming other forms. The virtualities of “I swear!” and of Châtelet’s triangle show then in what sense an utterance is irreducible to its local conditions of production (§ 4.2.). Insofar as

24

Cf. Châtelet [18], p. 37.

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Fig. 16.5 “Châtelet’s triangle and its sketches”. The picture is taken from De Freitas and Sinclair [24]

such virtualities are germs25 of other utterances (§ 4.2.), each utterance containing them will be likely to ‘project’ onto other conditions of production (§ 4.2.), namely, it will be available to circulate in enunciative contexts other than its instantaneous (or immediate) context. In more concrete terms, each utterance will be likely to become other utterances. On the verbal side, this kind of metamorphosis is mirrored in the prosodic variations affecting the rhythmic-intonational qualities of the several “I swear!”. To put it simply: “I swear!” modifies itself into other “I swear!” because the voice of a child who says “I swear!” is different from that of a witness in court or a man in love saying “I swear!”. In the visual domain, instead, one can observe analogous metamorphic processes in the vibratory patterns which continuously recreate the triangle (§ 4.3.1.). Drawing and redrawing it, the dotted lines (§ 4.3.3.) configure themselves as many overlapping triangle “sketches” that vary in angular widths, segment lengths, and so on… (see Fig. 16.5). And the same goes for Klee’s jester: the lines crossing and recrossing at every junction point of zig-zag configurations (§ 4.3.3.) also vary in angular widths, curvature, and so on… Moreover, other changes occur on the whole figure: from the fingers of the right hand (see Fig. 16.6) to the trunk (see Fig. 16.7), from the arms (see Fig. Fig. 16.8) to the neck (see Fig. 16.9), until to the ends of the head and feet (see Figs. 16.10 and 16.11), an entire tangle of lines is put into variation by vibratory patterns that “verändern und vermehren gleichmäßig und ungleichmäßig”.26

25

NB: In the next paragraphs we will broaden the semantic potential of “virtuality”: this terms will be used not only in reference to the germs of future utterances but also to the residues of past utterances. On the reasons for this expansion see the first Appendix to this chapter. 26 Klee [42], p. 130.

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Fig. 16.6 “Kleiner Narr’s hand in Trance”. The picture is taken from Otto [60]

Fig. 16.7 “Kleiner Narr’s trunk in Trance”. The picture is taken from Klee [42]

Therefore, both Châtelet’s triangle and Klee’s jester are in a process of becoming just like “I swear!”. Hence, they contain “crystals of becoming” (§ 4.3.3.)—where “becoming” is a contracted form for “becoming-other”. All of this is therefore diagrammatic since the becoming-other is one of the main features of the “diagram’s mode of existence”27 (§ 5.1. of AAI). In fact, insofar as the enunciative becoming entails an internal change of visual and verbal forms, enunciation is a diagrammatic practice par excellence. There are, however, at least three issues that need to be clarified in relation to the latter point: a. the actual heterogenetic scope of the enunciative becoming; b. the immanent virtualities animating its diagrammatic eidos; c. the distinctive peculiarities of the selected visual examples.

27

Châtelet [18], p. 10.

16 Conclusions: As a Kleiner Narr in Trance Towards a Diagrammatic … Fig. 16.8 L: “Kleiner Narr’s arms in Trance”. The picture is taken from Klee [42]. R: “Kleiner Narr’s arms in Trance”. The picture is taken from Klee [42]

Fig. 16.9 “Kleiner Narr’s neck in Trance”. The picture is taken from Klee [42]

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Fig. 16.10 “Kleiner Narr’s head in Trance”. The picture is taken from Klee [42]

Fig. 16.11 “Kleiner Narr’s feet in Trance”. The picture is taken from Klee [42]

16.5.1 Visual Forms or Immaterial Types? Let’s start with the latter issue. One could argue that Châtelet’s triangle and Klee’s jester are not all equivalent: the former is a geometric entity; the second, a drawing, or rather a “visual line”28 which “crosses and recrosses its own track rhythmically”.29 In previous paragraphs (§§ 4.3.1. & 4.3.3.), we deliberately underestimated (and occulted) this difference: we presented both Klee’s jester and Châtelet’s triangle in the terms of visual forms—of drawings, to be precise. However, according to a fitting metaphysical prejudice, only the former is as such. Except for the author of EM, for whom mathematical objects are at the same time impalpable and material (§ 6.5.2.1. 28 29

Albertazzi et al. [3], p. 2. Wenhman [76], p. 23.

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of AAI), the vast majority of mathematicians would refuse to assimilate a triangle— or any geometric object—to a visual form. In fact, while the lines of such a form always have some thickness,30 the perimeter of a geometrical object is completely devoid of it. In short, the closed broken line enclosing the interior of the triangle would be an abstract object (if you’re a Platonist) or a mental creation (if you’re a Constructivist).31 It would be an immaterial type that, be it metaphysical or cognitive, does not coincide with its own visual enunciation. As such, then, Châtelet’s triangle would stand exactly at the antipodes of Klee’s jester. The latter being a drawing, it indeed coincides with its own enunciation: it resolves itself into the iterable gestures that shape it. Yet, Klee himself would seem to have doubts in this regard. In one of the most theoretically interesting pages of his Das Bildnerische Denken, he observes that “Der Bereich der […] bildnerischen Mittel ist zu begrenzen nach der ideellen Seite”.32 This is as if to say that something totally external to the enunciative praxis would need to intervene in the genesis of visual forms, and, in this particular case, that a purely ideational element (der ideellen Seite) would be grafted into the jester’s figuration (“Gestaltung”)33 without, however, allowing itself to be contaminated by

30

Cf. Albertazziet al. [3], p. 2. We beg the reader to tolerate the rough tenor of the abovementioned partition: Platonists and Constructivists in mathematics are far from constituting two philosophically homogeneous and unitary camps. Notwithstanding some relatively stable macro-differences, they present within them such a wide variety of perspectives as to interdict the slightest attempt to categorize them in a coherent way. The ontology of mathematical entities is the main “battleground” in which such perspectives are formed and diversified: while from a Platonic point of view, mind-independence is the metaphysical trademark of the mathematical realm, from a Constructivist one, it is instead its mind-dependence. That said, however, there are many positions that, both on the Platonic side and on the Constructivist one, move away from their respective initial assumptions. This is the case, for instance, of Lautman’s dynamical Platonism (with regard to which see § 6.3.9 of AAI and the first Appendix to this chapter) and Richman’s epistemological Constructivism (with regard to which see [13, 66, 67]: the former detaches the mathematical realm from a static and transcendent ontology by situating it in the a priori “expérience […] des problèmes” (cf. [48], p. 229, cit. in Alunni [6], p. 168) feeding the Dialectic of Ideas of mathematical thinking. The second commits itself on the exclusive legitimacy of mathematical constructive methods (computability in a finite number of steps, algorithmic proofs exhibiting instances, and so on… (cf. [13], p. 17]), but “remains agnostic on the ontology of mathematical objects” (cf. [78], p. 6). You could consider other examples, but the just mentioned ones are more than sufficient to confirm the heterogeneity of the theoretical attitudes at stake: dynamical Platonism would seem to be open up to a historical dimension of mathematical objectivity which is totally absent in classical (or ontological) Platonism (cf. at least [64], pp. 99–100, 144–151, [4], pp.195–198),epistemological Constructivism, even if it “is certainly not inconsistent with a Brouwerian ontology” (cf. [13], p. 24) could coexist, given its agnosticism, with classical Platonic ontologies, too. In sum, in light of these variants, and many others on which we are silent for brevity, the theoretical developments of Platonism and Constructivism appear much less obvious than one might expect on the basis of distinctions that, while well-founded, grasp only the macroscopic aspect of the examined problems. 32 Klee [42], p. 17. 33 Ibid. 31

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the pictorial materials (colors, chiaroscuro, lines, etc.) that it mobilizes, which would then make the jester never quite coincide with its own enunciation. But this conclusion, although persuasive, does not hold up to closer inspection. After having pointed out the role of the ideational elements in the figurative processes, the German painter adds that “Sie sind nicht frei von Materie”.34 It therefore follows that the jester, while being a mental creation (or an ideality, in Klee’s lexicon), is not constituted separately from the pictorial materials that put it on canvas. Conversely, for the reasons stated, Châtelet’s triangle would be assimilated precisely to one of those ideational elements that are located outside the materialities and the diagrammatic gestures that figurate them. It would be, in short, a pure ideality. This would then confirm not only the distinctive peculiarities of each example, but also the seemingly unbridgeable gaps separating them. The visual enunciation of the one, that is, would be radically different from the visual enunciation of the other. Only the first, the one underlying the figuration of the jester, would seem to form its own contents; the other, instead, would seem to merely express them, namely, to bring them to the surface without, however, intervening in their constitution. Nonetheless, what we called the “carnality of the mathematical eidos”35 would seem to greatly rescale the differences involved.

16.5.1.1

Ideal Materiality/Material Ideality

This locution refers to the sensible facet of mathematical beings (§§ 6.5.1. & 6.5.2. of AAI), i.e. to their latent materiality (§ 6.5.3. of AAI). It is keeping this aspect in mind of that we have been able to examine Châtelet’s triangle in terms of vibratory patterns (§§ 4.3.1. & 5.). And it is by deepening the main features of it that one may conceive a diagrammatically unitary idea of enunciation. From a carnal perspective, the visual enunciation of Châtelet’s triangle and the visual enunciation of Klee’s jester are substantially equivalent. The triangle, exactly like the jester, would be at one with the iterable gestures making it an material ideality or an ideal materiality, that is to say, a gestural form which is at once a germinal unity of concept and ink, of concept and line, and so on… As far as of Klee’s jester is concerned, it’s pretty obvious that such is the case. By virtue of its own visual enunciation, it subsists as unities of this kind: not only do its “Elementarformen” (§ 3.) “bud” in a recurring way (e.g. zig-zag configurations at the lower left end of the jester [§ 4.3.3.]), but firstly they surface as “Linie, Helldunkell, Farbe”36 which fix “des Begriffes”.37 As for Châtelet’s triangle side, things are instead a bit more complicated. Even though all conditions for confirming its gestural and germinal eidos have already been explicated in §§§ 4.3.2., 4.3.3 and 5., Platonist and Constructivist vulgates (§ 34

Ibid. For which see §§ 6.3.5, 6.3.6, 6.3.7, 6.3.8, 6.2.5.1 of AAI. 36 Klee [42], p. 17. 37 Ibid. 35

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5.1.) still risk trading this eidos for a pure ideality (§ 5.1.). In sum, despite Châtelet’s theoretical efforts to “concretize” mathematics (§ 6.5.2.1. of AAI), one might always be tempted to reduce the material ideality (or ideal materiality) of mathematical objects to a type/token dichotomy (§ 5.1.). It would seem to be the case, then, that the carnality of eidos alone is not enough to recompose the gap between the two kinds of enunciation.

16.5.1.2

Platonist Vulgates and Constructivist Vulgates

It should be said that at least one aspect of this carnality is consistent with the abovementioned gap. We are thinking of what we called in AAI the “availability of formal idealities to be put on paper (or on any […] support diagrammatic practices elect as their applicative domain)”.38 This putting onto paper (onto a blackboard, onto a screen, etc.) would have all the requisites for an expressive process in the sense of the section § 5.1.1: the hand drawing the triangle would merely make visible something that exists independently of the diagrammatic gestures that manifest it. The readiness of the eidos to be rooted into the raison graphique of the mathematicians (§§ 6.5.2. & 6.5.1. of AII) would thus have nothing do with the enunciative processes involved in the genesis of Klee’s jester. Not surprisingly, a Platonist mathematician wouldn’t find it hard to translate this readiness in terms of a type/ token dichotomy (§§ 5.1. & 5.1.1.): idealities could be linked back to “abstract objects”39 and their graphical pictures to spatio-temporally situated reproductions of such objects. In a classical Platonic framework, the visual enunciation of Châtelet’s triangle could be likened to the material token of something that is “outside space and time”40 (this is what is meant by “abstract”), and that, as such, would be an immaterial type par excellence (§ 5.1.). But things would not be any different either by situating this enunciation in a Constructivist framework. If you think of mathematical objects as mental creations (§ 5.1.), and thus of Châtelet’s triangle as a particular case of such creations, the visual enunciation of a triangle would still remain the material token of a type; a cognitive type, of course, but still a type (§ 5.1.). The gap between the two kinds of enunciation would then resurface to the extent that the visual enunciation of Klee’s jester is not the token of any type. The jester is at one with the iterable gestures shaping it (§ 5.1.). Its enunciation is therefore a token without a type, or, rather, a token crisscrossed by enunciative virtualities (§ 4.2.). The visual enunciation of Châtelet’s triangle, too, from a carnal perspective (§ 5.1.1), would seem to possess all the requirements to be such a token (§ 5.1.1.). But the very fact that the triangle can be put onto paper excludes this possibility. For the stated reasons, this aspect of the carnality of eidos falls within the bounds of two ontologies, the Platonic and the Constructivist, that are far from satisfying minimal standards of the diagrammatic model of enunciation (§§§ 2., 3. & 4.2.). 38

Cf.§ 6.5.2.1 of AAI. Brown [15], p. 10. 40 Ibid. 39

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In a diagrammatic perspective, it has been said, there are no such things as typeutterances, but only token-utterances crisscrossed by their enunciative virtualities (§ 4.2.). Conversely, these ontologies reintroduce on the enunciative level precisely what that model rejects: the existence of (metaphysical or cognitive) types detached from their tokens (§ 4.2.).

16.5.1.3

Potentials of Concreteness

Yet, certain ante litteram Constructivist philosophies of mathematics might represent an appropriate context for the diagrammatic model. We are thinking in particular of some significant trends in mathematical and natural philosophy in sixteenth and seventeenth centuries. For authors such as Barrow and Newton, or Clavius, “Geometrical figures […] were things to be drawn”41 to exist. Such a perspective is “Constructivist ante litteram” in a precise sense of the word “construction”: the one which makes it a synonym for “generation of curves by movement”.42 It is from this sense, called “mechanical43 ” or “fluentist”44 by some, that you can develop a diagrammatic approach to eidos. By setting the existence of curves and figures within a generative process, and by identifying this process with the diagrammatic gestures drawing them, the aforementioned authors come very close to the diagrammatic conception that the eidos of circles, triangles and squares resides in the gestures shaping them. This point of view would entail a meaning of “construction”45 that, “applied to mathematical philosophy”,46 is consistent with the diagrammatic model of enunciation. But this is a more apparent than substantive consistency. To fit this model, the mechanical or fluentist constructability of curves and figures should involve only tokens and virtualities (§ 4.2.). This is not the case, however: “generation of curves by movement” leads to the identification of universals.47 From drawn figures, one goes back to immaterial concepts which play the role of cognitive types, thus reintroducing the dichotomies rejected by the diagrammatic model (§ 4.3.). Not surprisingly, commenting on some of Clavius’ passages, the historian of science Peter Dear observed that “the universal circle is real because circles can always be drawn”.48 Such a “claim of constructability”49 , whether one agrees with it 41

Dear [23], pp. 8–9, cit. in Sepkoski [73], p. 85. Italics in the text. Sepkoski [73], p. 85. 43 Ibid. p. 84. 44 Moretto [58], p. 74. 45 [73], p. 84. 46 Ibid. 47 Cf. Dear [23], p. 219. 48 Ibid. Italics in the text. 49 Ibid. 42

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or not, deviates from the diagrammatic model of enunciation insofar as it presupposes that the constructed reality of the type (the “universal circle”) is something that is distinguished—presumably by “a process of mathematical abstraction”50 —from all its contingent enunciations (the “circles [that] can be always drawn”). Naturally, it is a good thing that one can envisage mathematical objects as qualitatively distinct from the accidentalities of their tokens. Whether one is a Platonist or Constructivist, there is “a clear cognitive gain”51 in doing so. Subtracting and removing these accidentalities, mathematical objects emerge with respect to those properties of conceptual stability and temporal invariancy that mainly characterize their eidos.52 But this eidos is not only made of abstractions. Rather, it is very much made of abstractions, but they are so sophisticated as to retain in them an irrepressible potential for concreteness. With “potential for concreteness” we refer to a whole series of aspects of the carnality of eidos that that are not reduced to either the classical Platonic or Constructivist universe.

16.5.1.4

Constituting Formal Idealities

It is by virtue of these aspects that Châtelet’s triangle can also fit into the diagrammatic model of enunciation (§§§ 2., 3. & 4.2.). Indeed, one of them allows seeing the triangle as a gestural form (§ 5.1.1.): as one of those unities that the abovementioned model proposes as an alternative to the type/token dichotomy. We are thinking, in particular, of the genetic role played by the paper and pencil— and, even early, by inscriptions on stone and clay tablets—in constituting formal idealities. We say “constituting” in the carnal sense that the ideality, whatever it is, is inseparable from the hand’s movements shaping it. If—as Longo observes—“we do not understand what is a line, do not manage to conceive of it, to propose it, even in its formal explicitation, without the perceived gesture, or even without drawing it on the blackboard”,53 then paper and pencils—or chalks and blackboards, if you prefer—are constitutive in the sense that has been stated. Hence, this aspect of the carnality of eidos makes the triangle a gestural form since it locates its form—its morphogenesis, more properly—in the diagrammatic gestures that draw or redraw the triangles. In this sense, the visual enunciation of Châtelet’s triangle forges its own contents exactly like the visual enunciation of Klee’s jester forges its own (§ 5.1). The triangle, too, in short, would be one shaped by iterable gestures. Far from being an immaterial type, whether metaphysical or cognitive, it would constitute itself as a form crisscrossed by enunciative virtualities.

50

Drees [32], p. 206. Marquis [54], p. 166. 52 Cf. Longo [51], p. 136. 53 Longo [50], p. 47. See Longo [51], p. 140. 51

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Assemblages of Gestural Forms

Therefore, considering in this respect the chosen visual examples (§§5. & 5.1.), the jester and the triangle will be undoubtedly equivalent in the terms of their diagrammatic life (§§ 4.3.1. & 4.3.2.). According to the genetic role that diagrammatic gestures and instruments play in constituting formal idealities (§ 5.1.4), there is a carnal sense by virtue of which the geometrical forms are exactly like the visual forms. Both of them are carnally conceived through assemblages of gestural forms constituting their idealities. But there is more: there is no contradiction between carnally conceiving the diagrammatic eidos and asserting its impalpability (§ 5.1.). This can be understandood by identifying the latter with the invisible side of the above-mentioned forms. Instead of thinking of mathematical abstractness and aesthetical ideality as an universe of types detached from their tokens (§§ 4.2., 5.1., 5.1.1., 5.1.2.), one can conceptualize the immateriality of eidos in terms of virtualities immanent to the diagrammatic (or “symboliques”)54 materialities respectively enacting mathematical and artistic forms. It is precisely from this result that one can reconnect with the penultimate issue of our lineup: the immanent virtualities animating the diagrammatic eidos of the enunciative becoming (§ 5.1).

16.5.2 Immanent Virtualities and “Texture Charnelle” Among these virtualities there is certainly the broken line forming the perimeter of Châtelet’s triangle. This line is immanent insofar as it is inseparable from the gestures drawing it (§ 5.1.3); it is virtual, however, since it “is never [fully] exhausted by”55 the materialities enacting it. It for this reason that the triangle vibrates (§§§§ 4.3., 4.3.1., 5. & 5.1.1.): its boundary is nothing more than the provisional halting point of an iterable gesture. It is the locus of transitory manifestations of hand’s movements expand the triangle in the form of other triangles. The abstractness of the line—of the boundary as a pure immateriality—is constituted within the iterability of the gesture shaping it: the more the line is drawn and redrawn, the more we see that we do not see that absence of thickness which is the formal trademark of its immateriality. We say that “we see that we do not see” in a very precise sense, i.e. to mean that the above-mentioned absence is grasped only through the “texture charnelle”56 of drawn and redrawn lines. The type, then, rather than being rejected (§ 5.1.2), is absorbed in the iterability of the carnal gesture. It is given, in other words, as an invisible fiber of the flesh. The more I draw lines, the more I experience their thickness. Namely

54

Alunni [5], pp. 129–131. The italics are in the text. Rotman [69], p. 35. 56 Merleau-Ponty [57], p. 195. 55

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“something” that, no matter how slowly and thin, causes “me to see that I do not to see” the line in its own ideality. The eidos is grasped not by abstraction but in the negative. In the first case it would be a “seconde positivité”,57 that is, something which exists separately from the carnal gesture that expresses it (§§ 5.1.1. & 5.1.2.). In the second one, instead, it would precisely be the invisible fiber of a constituting gesture (§§ 5.1.1. & 5.1.2.). It is to this fiber that we refer when we assert that “we see that we do not to see”. This invisibility is only possible because of the language: since I read (or heard during a lesson) that the Euclidean line is without thickness, I realize that the gesture that draws it, and that allows me to understand it (§ 5.1.3.), is the same gesture however that conceals it—that makes me “see that I do not see”. Whenever I perform it, in fact, only material lines are produced: visible (and thick) lines that a formal knowledge made up of language, Euclidean geometry, allows me to judge as such—as lines, precisely—and as concealments of something that I only grasp only through this knowledge. I say “concealments”, and not “approximations”, because the latter term would again suggest an image of the eidos as a second positivity. For instance, thinking of a series of progressively thinner lines such as an approximation of the ideal line is to plunge the diagrammatic eidos back into a dichotomic universe of types and tokens (§§ 5.1., 5.1.1., 5.1.2. & 5.1.3.). Insofar as the products of the iterable gesture are considered as imperfect and always perfectible reproductions of the eidos—as approximations of it—the line without thickness is not in the least touched by the modalities of movement that try to figurate it. These modalities express it without constituting it: they emerge as tokens detached from their type (§§ 4.2., 5.1.2. & 5.1.4.). Conversely, an immediate consequence of the immanence of eidos is that it is constituted by the diagrammatic gestures that shape it: that is, it is internal to their carnal texture. This carnal interiority is the result of a complex dialectic between gestures and concepts. The former, as constituent modalities of movement, contribute to the conception/understanding of the eidos; the latter, as emanations of a formal knowledge made of language, contribute to identifying in the products of the materialieties of those movements which simultaneously figure and conceal the eidos. The continuous to and from—the dialectic, in fact—between the two moments is possible because there is no solution of continuity between formal knowledge and carnal gestures. The former is constituted within the mobile experience of the living body (e.g. the hand that traces the line as an experience of the conception of the line itself [§ 5.1.3.]). The second is at the service of the formal knowledge it helps to forge (e.g. the geometric line that the movement of the hand simultaneously exemplifies and

57

Ibid. p. 194.

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F. La Mantia

conceals). Formal knowledge is therefore carnal because it buds within certain modalities of movement. Carnal gesture, on the other hand, is formal because it is subject to the eidos that it constitutes. The absence of thickness of the perimeter of Châtelet’s triangle is a particular case of such a dialectic.

16.5.2.1

Immanent Virtualities as a “Réserve Invisible”

Certainly, from a cognitive point of view, “the line without thickness [simply] does not exist”58 : “it is we who put it in the space”.59 But from the perspective of the just mentioned dialectic (§ 5.2), this inexistence is rather a particular way of “existing in”. The line without thickness is inexistent in the sense that it exists within the visible lines and the iterable gestures constituting the triangle. It is, so to speak, the “armature intérieure”,60 the “réserve invisible”61 from which gesture draws whenever the triangle is put into vibration. The type is thus absorbed into the iterability of gestures (§ 5.2.) because it moves from the inertial status of a second positivity (§ 5.2.) to the dynamical one of an invisible fiber of the flesh (§ 5.2.). According to the previous dialectical schemes, that is, the type is dissolved into the diagrammatic materialities and thus into the mutagenic potentialities of the gestures shaping them. Upon reflection, the same can be said for the types of traditional linguistic semantics. Even on the verbal level, the type can be absorbed into the iterability of gestures and their transformations. To achieve this, it is sufficient to dilute each type-utterance into the immanent virtualities (§§ 5.1.3.4. & 5.2.) that put its enunciative tokens into variation (§§ 4.2. & 4.3.3.). Each token (§§ 4.2. & 5.) has its own materiality and thus its own gesturality.62 In accordance with the theoretical move suggested here, the type reside in the “crystals of becoming” (§§ 4.3. & 5.) which animate each of these materialities/gesturalities. Rather than presenting itself as a form detached from its own tokens, it constituted their “pulpe même”63 : the heterogenetic substrate that makes each token the hidden carnal matrix of other tokens. If we stick to the phonic aspects of these tokens (§ 5.), the type so understood will reside in a certain readiness to sonic metamorphosis. An example of this readiness can be found in the prosodic variations that surface from one “I swear!” to the next (§ 5.). On the visual level, instead, a similar example, but one of quite different workmanship, can be found in the metrical and angular variations that emerge from one sketch of a triangle to the next (§ 5). The immanence of the virtual is thus a spectrum 58

Longo [50], p. 46. See Longo [51], p. 141. Ibid. 60 Merleau-Ponty [57], p.158. 61 Ibid. p. 161. 62 For more details about this point, see the section § 5.2.2. 63 Merleau-Ponty, p. 351. 59

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of metamorphosis: a cascade of modifications upon modifications that are the very substance of the diagrammatic materialities and of the related gestural forms. If the type is the readiness of tokens to change, this readiness is in the gestures that forge them. The gesture, be it manual or phonatory, can change: it can be remade in the form of other gestures that can in turn change, and so on.

16.5.2.2

Iterability as a Readiness to Metamorphosis

A word like “iterability” captures very well the sense of change that animates the life of the gestures. According to Derrida, “to iterate” does not simply mean “to repeat”. Noting that the root “iter, derechef, viendrait de itara, autre en sanskrit”,64 the author employs the deverbal to refer to the outcomes of a “logique qui lie la répétition à la altérité”.65 All the cases of diagrammatic enunciation (§§ 2., 3., 4., 5.) examined so far exemplify this connection very well. The triangle’s vibrations (§§ 4.3, 4.3.1, 4.3.3) and the variations of “I swear!” (§§ 4.2, 4.3, 4.3.1), are the result of iterative gestures par excellence. Repetition and otherness coexist in them because, from one remake to the next, their diagrammatic eidos is simultaneously preserved and modified. It is preserved, because the modalities of movement constituting it are recognizable from iteration to iteration; it is modified because, from one token to the next, these modes emerge in a ever-changing manner. This is the case with “I swear!”. From a carnal point of view, this utterance is at one with the phonatory gestures that shape it. Its phonic materiality is interwoven with movements of the mouth, tongue and vocal cords that are activated whenever it is put into form. These movements change and are preserved at the same time because, from proffering to proffering, they vary with each variation while preserving the movements of “I swear!”. However much they may differ in terms of timbre or tonality, these movements are part of an enunciative metamorphosis in which there is always something that repeats itself: something that is not immutable—as a type might be with respect to its tokens—but which nevertheless does not exceed a certain threshold of deformability. In short, something that, while altering itself in turn, does not transform itself to the point of changing the phonetic gestures of “I swear!” into the gestures of “I love you!”, or of “See you tomorrow!”, for instance. Although pulverized into a multitude of tokens without type, the diagrammatic eidos of “I swear!” is iterable insofar as it is available to repeat itself through phonatory gestures that are always different and yet always identifiable as gestures of “I swear!”. The metamorphoses of this utterance—like those of any other utterance— would thus be iterative insofar as the gestures that trigger them are a mixture of repetition and alteration.

64 65

Derrida [28], p. 375. Ibid.

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F. La Mantia

Fig. 16.12 Kleiner Narr in Trance 1928

The same can be said, moreover, for the gestures that shape Châtelet’s triangle: the modes of movement that set it into vibration are iterative because they are repeated in ever different ways and yet within (and not beyond) the limits of a certain threshold of deformability, one beyond which, for example, the triangle would become a circle or a square. Hence, two examples of diagrammatic iterability. Perhaps the most striking case of this iterability, however, is just the Klee’s jester. We say “it is just” because its enunciation (or “Gestaltung”)66 is that which apparently lends itself the least to being read in terms of iterability. Excluding the zig-zag configurations on the lower left end of the jester (see Fig. 16.3), the rest of the figure is more akin to a tangle of lines than the result of an iterable gesture. Yet, if we consider two other versions of the drawing (see Figs. 16.12 and 16.13), it will also be possible to retrace the practice of an iterable gesture from the tangle. An overview of the different tokens shows that the tangle is shaped by gestures that are repeated and modified at the same time. From figuration to figuration the lines that compose it will always appear differently and yet remain recognizable in their variations. The jester’s different enunciations will then serve as local stages of a single iterable gesticulation—the same gesticulation that, in its most general form, is configured as

66

Klee [42], p. 17.

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Fig. 16.13 Kleiner Narr in Trance 1929

a kind of “alteration through repetition”67 which lurks in the “potentia agendi”68 of gestures and in that readiness to metamorphosis (§ 5.2.1) which is perhaps the main trademarks of the virtual.

16.5.3 And the Heterogenesis? What about heterogenesis then? This question, which introduces the last step of our path (§ 5.1.), imposes itself because of a theoretical assumption that has informed the research carried out so far. In the Introduction we identified the work of the virtual with heterogenesis (§§ 5.1 & 5.2 of AAI). Moreover, on the grounds of this identification, we presented diagrammatic gestures as a particular case of heterogenetic praxis (§§§ 5.3, 5.4 & 5.5 of AAI). It should therefore have followed that enunciative gestures are also heterogenetic. Besides, as a special case of diagrammatic gestures, they would seem to inherit that characteristic aspect of “diagrammatic life” that is the “becoming-other”: heterogenesis, precisely (§ 5.1.). In this conclusive essay, however, the notion of heterogenesis has first been joined, and then supplanted, by the Derridean notion of iterability (§§§ 5.1.1, 5.2.1 & 5.2.2): enunciative gestures, from being heterogenetic (§§§ 4.3.1, 4.3.2 & 4.3.3), have become iterable or iterative (§ 5.2.2). The question is therefore

67 68

Cooren et al. (2008), p. 1370. Citton [20], p. 287.

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whether or not the two notions are assimilable within the limits of the diagrammatic model of enunciation. Hence, the initial interrogation.

16.5.3.1

Heterogenesis is not Iterability

Strictly speaking, iterability is not heterogenesis. The Derridean notion differs from the Deleuzian-Guattarian construct because of the role it accords to the invariance. The latter corresponds to limited thresholds of deformability (§ 5.2.2) that guarantee the recognition of a form (of a gesture, of an action) through its alterations. In “Signature, événement, contexte” we read that “A travers les variations empiriques du ton, de la voix, etc., éventuellement, d’un certain accent, par exemple, il faut pouvoir reconnaitre l’identité, disons, d’une forme signifiante”.69 The same text also notes that “qu’une certaine identité à soi de cet élément (marque, signe) doit en permettre la répétition et la reconnaissance”.70 On the other hand, instead, heterogenesis is as far removed from such conditions of identity preservation as it can be. It is in fact a process that, unlike iterability, has potentially unlimited thresholds of deformability. It is these thresholds that, according to Deleuze and his epigones, operate in Bacon’s gesture, making it a heterogenetic gesture par excellence (§§ 6., 6.1, 6.1.1 of AAI). What Derrida calls the “identité à soi” of iterable forms is not the immutability of a type (§ 5.2.2). On the contrary, it is something subject to alteration (§ 5.2.2). Nor can it be overlooked that already for Deleuze “[t]he interior of repetition is always affected by an order of difference”,71 i.e. by a “heterogeneity”72 by virtue of which one finds “the Other in the repetition of the Same”.73 But this Deleuzian aspect of repetition, as important as it is, does not attenuate the divergences between iterability and heterogenesis. Rather, it deepens them. In a manner perhaps even more sophisticated than “Signature, événement, context”, the passages quoted here from Différence et Répétition (henceforth, DR) appropriately capture the relations of interiority that link repetition to becoming-other. As it happens, however, the word “heterogenesis”—entirely absent in DR—identifies in the Deleuzian reflections consecrated to Bacon a production of radical otherness, that is, a totally unpredictable differential mechanism that in itself does not guarantee any form of preservation, even minimal, of identity. Hence, the significant gap that separates this differential mechanism from iterability.

69

Derrida [28], p. 378. Ibid. 71 Deleuze [26], p. 25. 72 Ibid. p. 23. 73 Ibid. p. 24. 70

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383

But…

Although heterogenesis does not guarantee any margin of invariance, the gap between iterability and radical otherness can be recomposed from a broader diagrammatic perspective. In an extended sense of the expression, the becoming-other of gestures can present varying degrees of intensity. In its highest degree, this becoming involves those abrupt and sudden “redistributions of relationships” (§ 6.1.1 of AAI) that cancel out every previous similarity of form by inaugurating entirely new planes of composition (§ 6.1.1 of AAI). In its lowest degree, on the other hand, it can assume the features of an iterative process (§ 5.2.2), and guarantee that variable “coalescence”74 of repetition and alteration that, albeit in different forms and measures, constitutes an intuition common to Deleuze and Derrida. “Becoming other” can thus be used to mean different generative processes. One can qualify them as “heterogenetic”, or, more appropriately, one can distinguish them into “heterogenetic” and “iterative”. In these pages, we have opted for the first solution (§§ 5. & 5.2.1), and have then adopted the second, which is by far more refined and more reliable (§§ 5.2.1 & 5.2.2). In reality, however, even this second criterion, if rigidly applied, remains unsatisfactory. Iterability is a potential source of heterogenesis. From the various “I swear!”, a radically different form may suddenly arise, i.e. a complex of gestures that subverts (even locally) the iterable continuity of the movements of “I swear!”. On the other hand, the aforementioned form, i.e. the gestural complex that enacts it, may in turn originate from a previous iterable continuity or be inscribed within other continuities of the same type. The case of slips of the tongue is in this sense exemplary.

16.5.3.3

Slip: Heterogenesis in Iterability and…

It might happen, or it may have already happened, that the person who is about to say “I swear!” unintentionally75 says something else: “I sweat!” for example. The involuntary transition from “I swear!” to “I sweat!” is heterogenetic because the phonatory gestures prefixed to the enaction of one form (“I swear!”) are modified to such an extent that they result in the phonatory gestures of a radically different form: “I sweat!” indeed. The iterable continuity of “I swear!” is thus subverted by a slip. Admittedly, the subversion is local, but the effect is already heterogenetic: although the phonatory gestures of “I swear!” are for the most part recognizable in the gestures of “I sweat!”, the variations operating on the last phonic segment entail a global reconfiguration of the utterance.

74 75

Whitehead [77], p. 28. This point deserving a long note, see Appendix II to this chapter.

384

F. La Mantia

By switching the “articulatory positions”76 of the “r” of “I swear!” into the positions of the “t” of “I sweat!”, the “phonic visage” of the form is transformed into the visage of a radically different one. It is in this sense that iterability is a potential source of heterogenesis (§ 5.3.2): circulating from one interlocutory field to another, “I swear!” is always available to mutate into something else, to become something else. It matters little whether, in this particular case, it is “I sweat!” that is the product of this becoming. Even if an iterable continuity of gestures unites the two forms, the meaning is radically changed. The defendant or the lover who says “I sweat!” believing they are saying “I swear!”, are the vehicle of a semantic heterogenesis: the form they pronounce, perhaps without even realizing it, is an “aleatory emergence”77 of the enunciation that radically alters the sense effects of “I swear!”. This is why we have spoken of a “phonic visage”. According to Federico Albano Leoni, a phonic visage is any “set of non-discrete perceptual saliencies held together by a meaning”.78 This is a definition that has the merit of bringing into focus the inescapable semantic dimension of every phonetic complexion: phonic materialities—or at least the ones that develop within the field of interlocution—are never pure sound, but rather “meaningful sound” or “sound meaning”.79 The heterogenesis of a verbal form then lies not only in the radical discontinuities between phonatory gestures, but to an equal extent in the abrupt changes of meaning that occur in conjunction with those discontinuities. The involuntary transition from “I swear!” to “I sweat!” is interesting because it shows how a local discontinuity on the sonic level has global repercussions on the semantic level. But first and foremost, it is as an example of heterogenesis within an iterable chain that it should be considered. “I swear!” becoming “I sweat!” mainly deserves attention for what it suggests about the heterogenetic potential of iterability.

16.5.3.4

…Vice Versa

On the other hand, from the transitory result of this becoming, it is as much possible to go back to the link of previous iterable chains as to the germ of future iterable chains. With regard to the first aspect, it suffices to observe that “I sweat!” is an attested form. Assuming that every slip is “la victoire d’un dit d’origine singulière sur le dit du discours”,80 the heterogenesis induced by this form draws upon a background of

76

Cf. Lieberman [49], p. 45. As regards the notion of aleatory emergence of enunciation, see at least Fenoglio [38], p. 169. 78 Albano Leoni [1], p. 181. 79 Cf. Merleau-Ponty [55], p. 93: “Le langage n’est pas son+, par ailleurs, signification. C’est son signifiant et la signification l’habite en tant que le son fait partie d’un système de pouvoirs”. Underlined in the text. 80 Fenoglio [38], p. 178. 77

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presumably prior enunciations—of what was a “déjà-dit”,81 as Jacqueline AuthierRevuz would say—in which “I sweat!” circulates as an iterable form of the English language. Put another way, if it is true that this form is in a position to suddenly replace “I swear!”, it is equally true that it fulfils its own heterogenetic role insofar as it comes from other enunciative universes. Compared to “I swear!”, in short, “I sweat!” is a “parole latente”82 that bursts into discourse because, first of all, it is a word that has already been said, and, as such, it is susceptible to further circulation specifically, to iteration. However, do not think that you can equate the slip with a special case of iterability. Everything is iterable, even forms that suddenly burst into discourse. As soon as they are iterated, however, they lose their heterogenetic power. When repeated or duplicated, the slip enters the sphere of discourse that is “maîtrisable, prévisible”.83 Inspired by this awareness, Irène Fenoglio has observed that “‘tricot de porc’ pour ‘tricot de corps’, répété deviendra un jeu de mots”,84 or, more generally, a “duplicable”85 form. The latent word thus shows the second characteristic aspect of heterogenetic forms: that by which they become germs of iterable chains. But this aspect has nothing to do with the reduction mentioned earlier. By noting that heterogenetic forms are links or germs of iterable chains, we meant to suggest something else: the enunciative impossibility of separating iterability and heterogenesis, even while distinguishing them. In the particular case we have been discussing, this inseparability is confirmed by the fact that the sudden substitution of “I swear!” for “I sweat!” shows a conflict between two iterable chains (or “forces d’énonciation”)86 : that of the discourse in progress during which “I sweat!” breaks out, and that of the presumably already realized discourse from which the same form comes. The latter, conflated with “I sweat!”, is therefore a trace of another discourse (discours autre)87 that enters into the discourse in progress, perturbating it.

16.5.3.5

From the “Potential Listener” to the Repentir

There is another aspect of the slip that confirms the inseparability of iterability and heterogenesis (§ 5.3.4). If perceived, it induces a reaction in the one who has committed it. Insofar as he or she is his or her own “potential listener”,88 the enunciator goes back on his or her own words: he self-corrects by restoring the continuity of iterable chain 81

Authier-Revuz [8], pp. 382–383. Fenoglio [37], p. 57. 83 Fenoglio [38], p. 178. 84 Fenoglio [37], p. 62. 85 Fenoglio [38], p. 178. 86 Cf. Fontanille [39] p. 32. 87 Regarding the notion of discours autre, we refer to the monumental Authier-Revuz [7, 8]. 88 Pêcheux [59], p. 92. 82

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F. La Mantia

Fig. 16.14 Portrait of Mr. and Mrs. Arnolfini 1434. The picture is taken from Borchet (2020)

that the sudden onset of heterogenesis had interrupted. This aspect of enunciative work, which we called “pathic” elsewhere,89 has an interesting counterpart in the visual sphere: the so-called repentir.

16.5.3.6

The Life of a Hand: On the Repentir

A key term in Fine Arts, the French repentir (repentance) designates the works of self-correction that the artist performs “en cours d’exécution et dont les traces demeurent plus ou moins visibles sur le tableau”.90 More generally, repentir is given whenever the modification made on the canvas results in gaps “entre deux positions, deux tracés différents”.91 This is the case, for example, of the Portrait of Mr. and Mrs. Arnolfini by the Flemish painter Jan van Eycke (Fig. 16.14). In the general economy of the painting, the right hand raised by Mr. Arnolfini is a form that reworks—that corrects, in fact—a previous form. Thanks to sophisticated X-ray analysis techniques, this form can be glimpsed as an another right hand “close to the man’s chest, relatively flat”92 (Fig. 16.15).

89

La Mantia [46], pp. 13–82. Damisch [22], p. 23. 91 Ibid. 92 Anonymous (2010), p. 239. 90

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Fig. 16.15 “A detail of the Portrait of Mr. and Mrs. Arnolfini”

A gap then arises between the two to the extent that the reworking of the one by the other induces a qualitative discontinuity on the dimensional plane. That is, through the repentir, one passes from the flat hand, all contained in the plane, to a hand of greater thickness and plasticity that inserts “an additional spatial plane between the observer and the figure”.93 This is a diagrammatically relevant enunciative phenomenon. Certainly, this alteration falls more within the realm of iterability than whitin that of heterogenesis: the pictorial subject, despite that it changes, is still a hand. Nevertheless, although it remains within the aforementioned limits, the case under consideration offers sufficient food for thought to develop six considerations that we will submit to the reader’s attention before presenting our conclusions.

Still on Repentir: Gesture as an Hesitation The first concerns an aspect of repentir that enriches the ontology of the gesture sketched in AAI. As the reader will recall, adhering to a theoretical proposal by Pierluigi Basso Fossali (§ 3.3.1.1 of AAI), we identified the gesture as a praxeological device capable of subverting and innovating the most established regularities of action and signification. According to this identification, the gesture would be characterized by an internal potential for change (§ 7. of AAI) of which the slip (§ 5.3.3) is undoubtedly one of the most radical and sudden manifestations. Repentir—this is the consideration—shows however that this feature does not exhaust the ontology of the gesture. On a pictorial level the aforementioned practice reveals at least another dimension of this ontology.

93

Ibid.

388

F. La Mantia

We are thinking of everything in the gesture that is constituted as hesitation,94 as a provisional halting of the action. This component is already present on the verbal plane. In order to realize this, it is sufficient to pay attention to the provisional halts of the speech induced by self-reflexive comments or reformulations: the enunciator going back on his or her own words (§ 5.3.5) is often subjected to a “syncope rythmique”95 —to a provisional halting of the speech, precisely. On the other hand, in the particular case with which we are concerned, this aspect is certainly present in the portrait of Mr. and Ms. Arnolfini (§ 5.3.6): the pictorial coexistence of the two hands, so well highlighted through the use of technology (§ 5.3.6), allows us to go back to a “choix entre deux solutions”.96 Like “une proposition en partie double”,97 the drawing enacts not only a self-correction (§ 5.3.6), but first and foremost an alternative between enunciative polarities that—like every alternative—drags with it a margin of uncertainty. The hesitation of the gesture is given in this particular case as a temporary suspension of the manual movements engaged in the genesis of the drawing. It is likely that the reworking of a hand by means of another is preceded by a moment of indecision and that it is precisely this indecision that leads to the aforementioned interruption. Besides, it is only by interrupting the movements underlying the hand flattened on Arnolfini’s chest that the reworking movements can begin to result in the genesis of another form: the hand detaching from Arnolfini’s chest.

Enunciative Processuality in Enunciative Products There is more, hence a second consideration: if one separates repentir from “toute idée de correction ou de retouche”,98 it confirms the fundamental core of any diagrammatic enunciation. Both in the preceding paragraphs and in AAI we identified this core in the eminently processual character of enunciative products. From a diagrammatic perspective, the utterance never ceases to be enunciated (s’énoncer), namely, to change into something else. One of the main catalysts of this immanent processuality is movement. While on the verbal level, it resides in the readiness of forms to allow themselves to iterate, sometimes to the point of heterogenesis, on the pictorial level, it is the repentir that inoculates its catalyzing germ. In this sense, the portrait of Mr. and Mrs. Arnolfini (§§ 5.3.6. & 5.3.6.1) is once again an example of rare clarity. Coexisting pictorially in the same drawing, the two hands are configured as “moments successifs d’une même action”99 : faintly hinted and scarcely visible to the naked eye, the hand pressed to Mr. Arnolfini’s chest is 94

Cf. Basso Fossali [11], p. 105: “En ce sens, le geste est là où l’acte s’interroge, hésite […] par rapport à un terrain de jeu déjà codé”. The italics are ours. 95 Fontanille [39], p. 33. 96 Damisch [22], p. 24. 97 Ibid. p. 23. 98 Ibid. p. 25. 99 Ibid.

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likely the beginning of this action; clearly visible and almost three-dimensional, the hand detached from Arnolfini’s chest is its fulfillment, and if not its completion—a phase far subsequent to the initial one. The space between the two enunciative figurations is thus retrained: from a bifurcation in which the pictorial gesture hesitates and reworks itself, it becomes a place of mobility, a transitional space where the intermediate phases between the initial and terminal phases of the action presumably teem. It is in this sense, then, that the repentir inoculates the germ of the movement: the reworked hand and the reworking hand—or, to use another terminology, the replaced hand and the replacing one—are the snapshots, respectively initial and terminal, of a single figurative process that is given as a “déplacement”100 or, rather, as the twisting of a hand that uncovers in the plane a germ of third dimension. This is why Damisch can say that “[l]e repentir sous sa forme classique […] a toujours eu rapport à la gestualité et à quelque chose, déjà, comme un déplacement”.101

Latency and Virtuality: A Return to the Heterogenesis This diagrammatic aspect of repentir makes it possible to find a heterogenetic dimension even where one would not expect it. Our third consideration begins with this very observation. The transitional space that separates and connects the snapshots of the figurative process (§ 5.6.3.2). is a generator of forms. As such, it is exposed to alterations of the enunciative gesture that operate on the pictorial plane in a manner not unlike the alterations that operate on the verbal plane and that modify, for example, the phonatory gestures of “I swear!” into the phonatory gestures of “I love you!” or “See you tomorrow!” (§ 5.2.2). It is a heterogenetic matrix. The Portrait of Mr. and Mrs. Arnolfini (§§ 5.3.6, 5.3.6.1 & 5.3.6.2) is not, however, the best example to substantiate this thesis. The alleged intermediate phases in which the movement of hand twisting is articulated have nothing heterogenetic about them: since they are iterations that do not cross a certain threshold of deformability (§ 5.2.2), the pictorial subject remains recognizable while changing with each shot (see Fig. 16.16). There are, however, a great many other instances of repentir that force the thresholds of deformability assumed by the iteration and that therefore result in heterogenesis.This is the case, for example, with Picasso’s paintings dating from the so-called “Blue Period” of his artistic activity. From the The Old Blind Guitarist to The Blue Room, traces of repentir can be found that result in a radical redistribution of figurative relationships. In the first painting (see Fig. 16.17), an old musician is depicted intent on playing a large brown guitar. In the second, instead, a young woman is portrayed bathing (see Fig. 16.18). In recent years, thanks once again to sophisticated X-ray analysis techniques, it has been possible to trace each painting back to earlier, more or less sketchy paintings 100 101

Ibid. Ibid. The italics are ours.

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Fig. 16.16 . Fig. 16.17 The Old Blind Guitarist 1903. The picture is taken from Lieberman (1961)

that have within them pictorial subjects that are radically different from the subjects visible to the naked eye. The image of a young mother with her child appears under the brushstrokes of The Old Blind Guitarist (see Fig. 16.19), while a mysterious male figure, with a bow tie and a ringed hand, emerges behind the woman in The Blue Room (see Fig. 16.20). This is a sign that in the initial composition of each subject, an enunciative course change must have arisen that altered the artist’s painterly gestures over the course of the process.

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Fig. 16.18 The Blue Room 1901. The picture is taken from Lieberman (1961) Fig. 16.19 “A detail of The Old Blind Guitarist”. The Picture is taken from Lieberman (1961)

However, these changes—these repentir—are not iterative like those operating in the Portrait of Mr. and Mrs. Arnolfini. Rather, they are heterogenetic changes, namely enunciative alterations that interrupt the enunciation of the initial subjects in order to initiate an entirely different one. But there is heterogenesis and heterogenesis. The one operating in the cases just examined is certainly not comparable to the heterogenesis that rages in Bacon’s paintings (§§ 6.1 & 6.1.1 of AAI). Although the space of transition from one pictorial

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Fig. 16.20 “A detail of The Blue Room”. The picture is taken from Lieberman (1961)

Fig. 16.21 “A detail of The Old Blind Guitarist”. The picture is taken from Lieberman (1961)

subject to another is traversed by radical diversity, there is in Picasso a margin of preservation of form that, however minimal, allows glimpses of local contiguities between completely different pictorial gestures. This is what can be appreciated in the The Old Blind Guitarist: the musician’s face matches perfectly with the young mother’s left shoulder (see Fig. 16.21). This seemingly insignificant detail suggests that the interruption of certain modes of movement nevertheless left certain figurative relationships intact. Although the gesture tracing the old musician is in a sharp discontinuity with the gesture from which the young mother takes form, and however much this discontinuity involves the inhibition of one gesture in favor of the other, there is something of the earlier figure that survives in the figure visible to the naked eye. This something leads us to say that the heterogenesis induced by repentir does not in this particular case involve a liberation of the Figure in Deleuze’s sense (§ 5.3 of AAI). In his valuable commentary on Bacon, the French thinker adopts this locution to refer to the genesis of forms that radically transgress all prior figurative constraints. For the reasons stated, if Painting is by constitution one such form, The Old Blind Guitarist is not at all or, at any rate, not entirely so. A similar argument applies to Picasso’s other

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Fig. 16.22 “A detail of The Blue Room”. The picture is taken from Lieberman (1961)

painting. Although it deserves a separate analysis, in The Blue Room, there is an overlap of pictorial subjects that exhibits local contiguities akin to the contiguities detectable in The Old Blind Guitarist. The young woman’s right side matches the back of the ringed hand of the mysterious male figure (see Fig. 16.22). Again then, the repentir-induced heterogenesis is not accompanied by a complete liberation of the figure in the Deleuzian sense of the term. Therefore, in the cases considered, it is as if heterogenesis is locally populated by unquenchable pockets of iterability. Yet, contemporary art history knows examples of purely heterogenetic repentir, as in the case with Jackson Pollock. Hence, our fourth consideration.

Heterogenetic Repentir: Jackson Pollock According to Hubert Damisch, although the repentir is an “[a]ffaire de lignes, et non de taches”,102 even “la pratique”103 of the American artist can be read as a field of experimentation with this pictorial technique. As the art historian observed, “[c]haque tracé jeté à la voile sur la toile ou le papier prenait figure de repentir par rapport à celui qui l’avait précédé”.104 Consider, for example, One n°31 (Fig. 16.23) or Levander Mist n°1 (Fig. 16.24). Looking at them, the eye gets lost in a tangle of “layers of overlapping paintings”,105 in a pictorial labyrinth where one cannot “follow the Ariadne’s thread whenever it may lead”.106 Yet, despite being stunned by this overlapping of layers, 102

Ibid. p. 27. Ibid. 104 Ibid. 105 Marin [53], p. 215. 106 Ibid. p. 217. 103

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Fig. 16.23 “One n° 31 1950”. The picture is taken from Emmerling (2016)

Fig. 16.24 “Levander Mist n°1 1950”. The picture is taken from Emmerling (2016)

and although “the Ariadne’s thread is always and forever found cut by another thread of another Ariadne”,107 the wandering eye encounters repentir taken perhaps to its most extreme consequences. We say “perhaps”, mitigating Damisch’s judgement, because these paintings, at least apparently, are devoid of the slightest figurative features. Without the possibility of recognizing in the tangle of layers a pictorial element that has the semblance of representation, and thus plays the role of a figure, no in-progress reworking can be grasped in the maze of splashes and squirts of color that are observed on the canvas. On the other hand, however, even this judgement can be tempered. First, because it 107

Ibid.

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Fig. 16.25 “Pollock’s dripping”. The picture is taken from Longo and Longo [52]

is not so certain that there is a total absence of figures in Pollock’s paintings. Second, because it is the movements of the artist (Fig. 16.25) that locate in the labyrinth an albeit faint presence of repentir. Leaving aside the first point, which I will nonetheless address in a footnote,108 it is the second remark that corroborates Damisch’s thesis. As it is well known, having begun in the first half of the 1940’s, the American painter’s gesture specializes in the use of dripping pours of color that imprint themselves on the canvas, reflecting the artist’s movements. Similar to a dance,109 the painterly gesture disseminates “fragments of traces, segments of threads”110 that overlap one another modifying what precedes them and being in turn modified by 108

Marin recognized a “figural rhythm” ([53]: 219) in the American artist’s paintings. According to this judgment, then, the horizon of the figure would not be so foreign to Pollock’s paintings. Between these and the latter, there would, if anything, be sufficient margins of compatibility, provided, however, that “figure” is understood in a sense radically different from its usual encodings. That is, in the case at hand, the word should refer neither to “the outward contour of a body” (ibid.) nor to “its representation” (ibid.) let alone to “a volume bounded by lines” (ibid.) Rather, it should be referred to the tracing (or “path”) that the artist’s gestural dance released on the canvas. “Rhythm” (ibid. p. 220) would, in this sense, indicate every transitory phase of this dance, that is, “the singular and momentary state of a […] flowing” (ibid.) of gestures caught in the moment. Given that the idea of representation plays a prominent role in the classical concept of the figure, there are at least two issues that Marin’s important survey carries along. The first concerns the status of representation in Pollock’s painting, while the second concerns the status of the same notion in the global history of abstract art. These are huge issues that need countless notes for further study. Within the limits of this this space, the reader is requested to adhere to the following minimal additions: (1) Pollock repeatedly stated that he was “a very representational some of the time, and a little all of the time” (Pollock in Rodman 1961, p. 82, cit. in [44], p. 144 n. 81), (2) with particular reference to the concept of the figure, he further observed that “when you’re painting out of your unconscious, figures are bound to emerge” (ibid.); (3) influential contemporary art critics such as Pepe Karmel (cf. [41]) and Kirk Varnedoe (cf. [75]) have shown, with acumen and rich detail, that abstract art has never quite ousted the order of representation outside itself. 109 Cf. Longo and Longo [51], p. 27. 110 Marin [53], p. 217.

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what will follow them. It is in this interplay of acted and undergone modifications that the practice of repentir resurfaces. It is a heterogenetic practice insofar as “each segment”111 unfolds an unpredictable range of “spots […] cloud[s] of drops, splashes of puddles, intersections with various exits”.112 The impact of the casting on the canvas is a generator of diagrammatic paths that implement completely unexpected changes. Pollock was never a fan of chance. In an interview with William Harris in April 1951, he declared that he “does not believe in chance, does not work by chance”.113 His aesthetic is in this sense completely opposite to Bacon’s. For the Irish painter, one must “make marks at random”114 ; for the American painter, chance simply does not exist. However, the two artists are less distant than one might think. Although very different, Bacon and Pollock share a common pictorial gesture: the dissolution of any invariance of form. They both experience and experiment with the impossibility of tracing back to what traces made on the fly (Pollock) or “involuntary signs” (Bacon) alter by the mere fact of appearing on canvas. There is therefore heterogenesis through a radical redistribution of relationships that is manifested in Pollock through “differences without repetition”115 and in Bacon as “deformations” (or as a liberation of the figure). In the case of Pollock, however, there is more. So here is our fifth consideration.

“Calculated Randomness” and Heterogenesis: Marin, Thom, Pollock Although a fervent critic of the idea of random movement, Pollock reserves a prominent role for accidentality. Were this not the case, the dripping drops would not be those generators of unpredictability that lurk on the canvas. Pollock’s accidentality, however, is not Bacon’s accidentality. Precisely because the impact of the drip on the canvas is determined by a gesture that, however immediate, is not accidental, the accidentality admitted by the American artist is anything but “the residue or smear of an out-of-control process”.116 Pollock’s tracings on the fly are therefore not comparable to Bacon’s involuntary marks. Whereas the latter are the product of an “out-of-control process”, albeit balanced by purposeful adjustments that are activated in reaction to the aleatory nature of the gesture, Pollock’s tracings give “the impression that the casting itself traces its own pattern in obedience to a certain pattern”.117

111

Ibid. Ibid. 113 Pollock [65], p. 83. 114 Bacon in Sylvester (1980), cited in Dupuis [36], p. 132. 115 Marin [53], p. 217. 116 Ibid. 117 Ibid. The Italics are ours. 112

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Fig. 16.26 “The Mandelbrot set”. The picture is taken from Devaney (1989)

If one is to give credence to this “impression”, as Marin is first willing to do, one can conclude that Pollock’s accidentality—unlike Bacon’s accidentality—is a “calculated randomness”.118 Observing the videos depicting the artist at work, one can then partially agree with René Thom when he observes that what “les esprits superficiels [would call] chaos”119 is in fact “lié [in Pollock] à la simplicité (d’une) loi génératrice”,120 of a calculus. Although in line with Marin’s observations, this judgement sacrifices the mutagenic potential of heterogenesis on the altar of outdated mathematics. Pollock’s unpredictabilities are assimilated by Thom to the unpredictabilities observed in the context of non-linear qualitative dynamics. This field of analysis is known for revealing that there are “mécanismes déterministes très simples121 ” at the origin of unpredictable behavior. The generative rule of the Mandelbrot set is among these mechanisms: it is a complex iterating quadratic map over which the jagged boundary of the fractal curve can be made to appear on any computer’s digital monitor (Fig. 16.26). From Thom’s perspective, the quadratic map is to Mandelbrot’s whole what Pollock’s gesture is to the “fouillis ramifié(s) […]”122 it produces on the canvas. The sacrifice of heterogenesis lies in this analogy. Although Marin’s remarks on the design within the casting might encourage its adoption, what the French mathematician suggests actually misses perhaps the most important detail of the Pollockian gesture: its capacity to modify itself in the course of the work, or, to put it better, its readiness to mutate the design that guides it from within into other designs. It is once again the videos depicting the artist in action that confirm this aspect: throwing himself onto his canvases, Pollock initiates a gestural dance according to a design that is given in the gesture and which is in turn transformed by the gesture. 118

Ibid. Thom [74], p. 6. 120 Ibid. 121 Ibid. 122 Ibid. 119

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This dance is heterogenetic to the utmost degree insofar, as Sarti reminds us, as heterogenesis gives itself rules by also giving itself the possibility of modifying them “en cours de route de manière spatiale et temporelle”.123 In this sense, the generative rule of the Mandelbrot’s set is anything but heterogenetic: from iteration to iteration, the formal structure of the map remains unchanged, just as the range of statistically self-similar forms that the curve opens up by generating itself remains unchanged. Because of the continual modifications to which it is subjected, Pollock’s gesture mutates along spaces of variation that—far from being already given or compressed into the form of an invariant rule—are constructed in real time through deformations that modify the gesture and the pattern that informs it. This is why in Pollock repentir rises to its most extreme heterogenetic forms.

Heterogenesis as a Horizon of Virtuality But do not believe that ‘pure’ heterogenesis is the exclusive prerogative of repetitions without differences à la Pollock (§ 5.3.6.3) or of deformations à la Bacon (§ 5.3.6.3). In reality, this becoming-other skirts the diagrammatic work of enunciation along everything that it rejects during the genesis of the utterance. The latter does not only involve shaping (mise en forme), but also the emergence of a provisional boundary between what is formed and what is excluded from the enunciated form. Heterogenesis proliferates on this border insofar as the rejection at the origin of the enunciative gesture outlines a horizon of virtuality—or of pure informality—in which everything can become everything and the opposite of everything. Linguists such as Michel Pêcheux and Antoine Culioli have intuited this horizon by giving it different names. Pêcheux called it “unsaid”124 ; Culioli, “epilinguistic chaos”.125 Behind each label lie specific insights and mostly related to the universe of verbal enunciation. Because of the richness of their theoretical implications, however, they are broad enough to be generalized to the work of enunciation tout-court (whether verbal or visual). Hence, our last consideration.

Invisible Heterogenesis Let us return to the repentir situated between one one and the other of Monsieur Arnolfini’s hands (§§ 5.3.6, 5.3.6.1, 5.3.6.2, 5.3.6.3). If we try to examine it through the notions mentioned, a horizon of virtuality will open up no that is less heterogenetic than the changes at work in Pollock’s or Bacon’s paintings. The only difference is that, 123

Sarti in Sarti and Pelgreffi [71], p. 5. Cf. Longo and Longo [51]: “Heterogenesis introduces the possibility of changing laws spatially and temporally”. 124 Pêcheux [63], p. 159. 125 Culioli in Culioli and Normand [21], p. 85.

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in this case, the heterogenesis at play is totally invisible. While Pollock’s “differences without repetition” (§ 5.3.6.2), or Bacon’s “deformations” (§ 5.3.6.2), leave a trace on the canvas, the supposed heterogenetic virtualities of Van Eyck’s painting are totally off-canvas and therefore invisible. To understand this first aspect, Pêcheux’s “unsaid” (§ 5.3.6.4.) may come our aid.

Radical Heterogenesis as Invisibility According to the French discourse analyst, the materiality of enunciation is not reducible to what the speaker presumes to consciously say. Alongside this sphere, the aforementioned materiality provides at least another: that which is constituted on what the conscious (or presumed) saying rejects outside of itself as “unsaid”. This unsaid, which perhaps will be said, or which perhaps has already been said elsewhere, sits on the outer edges of consciousness by silently skirting the transitions from the sayable to the said with which the genesis of the utterance is usually identified. Thanks to the extended use of “enunciation” discussed in § 4.3.1, the recognition of these mechanisms of rejection allows a glimpse of Van Eyck’s “off-canvas” virtualities (§ 5.3.6.5.). According to Pêcheux, “[l]’énoncé se constitue peu à peu”126 : it proceeds by way of local determinations that inform “the sayable” and that establish, in so doing, a boundary between “the said” and “tout ce qui aurait été possible au sujet de dire (mais qu’il n’a pas dit)”.127 If we think of the utterance as a visual configuration—such as a hand—and of its enunciation as a transition from the figurable to the figurative, the virtualities in question will coincide with the boundary that the enunciative figuration ideally draws by ousting the non-figurative outside of itself. The latter, exactly like Pêcheux’s unsaid, will be situated at the outer edges of consciousness skirting the diagrammatic work of enunciation as an invisible presence, as an unfigured figurable that laps at the edges of utterance without, however, reaching figuration. To say “off-canvas” in this sense means not so much to be outside the canvas as it is, if anything, to be outside the process of figuration releasing its figurables onto the canvas. One of these figurables is undoubtedly the flattened hand on Arnolfini’s chest (§ 5.3.6.): although hinted at and clearly visible only using X-rays (§ 5.3.6), this configuration was released onto the canvas by an enunciative gesture. This gesture in turn established a boundary between the figurative and the nonfigurative: between the visible hand and all that its genesis rejected as an invisible presence at the margins of figuration. It is at these margins, at the boundary delineated by them, that heterogenesis proliferates. Certainly, iterability also occupies this boundary: the hand that detaches itself from Arnolfini’s chest, before it was visible on the canvas, had to be at the very frontier 126 127

Pêcheux [63], p. 169. Ibid.

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that the gesture traced: that is, it too was one of the many invisible virtualities that lapped at the hand crushed on the chest. On the border, however, there is room for all forms of becoming-other, and so, too, for heterogenesis. This co-existence is explained not only by the coalescence that binds the several forms of becoming-other (§ 5.3.2.), but also by the extreme variety of the margins outside of consciousness. These are articulated along a line that proceeds from the preconscious to the unconscious128 and includes variable zones of proximity to conscious enunciative making. The iterable forms probably come from the zones closest to this enunciative doing: it is by virtue of this proximity that they restore the continuities interrupted by enunciative hesitations (§ 5.3.6.1) and the cracks, more or less marked, that they imprint upon the canvas. On the other hand, the properly heterogenetic virtualities are presumably lodged in the recesses of the unconscious: it is the greater distance from conscious making that makes them radically different from the figures released on the canvas. As is clear from looking at Van Eyck’s painting, in the “Portrait of Mr. and Mrs. Arnolfini”, these radical heterogeneities remain off-canvas: perhaps they haunt the enunciative gesture in the form of imperceptible micro-hesitations, but they never reach the threshold of figuration. The space that opens up between one hand and the other (§ 5.3.6.2) thus shows the most heteropoietic aspect of latency: that which is produced as invisibility, or, to put it better, as non-figurative figurability that remains as such, that inhabits the canvas by standing outside the canvas and that, if it emerges, emerges as a “foreign body” grafted onto the local determinations of the utterance. In the invisibility that remains as such, or that prepares to become a foreign body, the multifaceted and undulating games of the wildest heterogenesis are consummated. The concept of epilinguistic chaos provides the necessary theoretical support to grasp this aspect of heterogenetic invisibility.

Epilinguistic Chaos, Lalangue, and Seething Magma The term “epilinguistic chaos”129 designates in Culioli a “language reality”130 that is not confused with the productions of explicit verbal activity and that nevertheless guarantees these productions. Even if it is independent of the communicative constraints of interlocution,131 it provides verbal activity with the forms it needs. It is thus a “form generator”132 that lives outside of actual discourse and that is in this sense silent and wholly unconscious. 128

Ibid. pp. 167–168. Culioli in Culioli and Normand [21], p. 97. For more details see La Mantia [45] in Bisconti, Simone, La Mantia, p. 72. 130 Ducard [35], p. 161. 131 Cf. Culioli in Culioli and Normand [21], p. 91: « Avec l’épilinguistique on est par certains côtés […] “libéré” de la relation à autrui». 132 Cf. Ducard [34], p. 87. 129

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Now, the heterogenesis of the invisible can be read as a semiotic variant of epilinguistic chaos for two reasons. First, because the invisible virtualities that lap up visual enunciation (§ 5.3.6.6.) stand to figures as the silent forms of epilinguistic chaos stand to segments of actual discourse. Second, because these forms, just like heterogenetic virtualities, can change by radically altering themselves. On the verbal level, the radical alteration depends on the total absence of interlocutory constraints. On the visual level too, however, this absence plays a precise explanatory role. If by “interlocutory constraints” is meant the requirements guaranteeing the accessibility of a form, and if “form” refers as much to the products of verbal utterance as to the products of visual utterance, the removal of the requirements thus devised can result in a chaotic proliferation of entities, in a heterogenesis in which everything becomes everything and the opposite of everything (§ 5.3.6.5). Accessibility is a dialogic curb to chaos because it imposes respect for regularities that are functional to comprehension and communicability. If one is outside of these regularities—as epilinguistic chaos is—then any dialogical impediment on heterogenesis is dissolved. This is why it is invisible and silent. It is only by denying itself to an external eye and ear that heterogenesis is free to proliferate in the forms that suit it best. As formless forms, according to an oxymoron that captures the eccentricity of chaos with respect to any conscious figuration or discourse, the virtualities of heterogenesis are nevertheless the reservoir from which the diagrammatic gesture draws in its enunciations. There are those who, like Dominique Ducard,133 have likened these virtualities to what Lacan called lalangue. The juxtaposition is apt and captures a crucial aspect of the epilinguistic generator: the Lacanian lalangue is par excellence that which eludes the functional regularities of comprehension and communicability: the lalangue “is not a dictionary” and “serves for anything other than communication”. On the other hand, as Lacan himself states, “[l]e langage est, sans doute, fait de lalangue”.134 For the Parisian psychoanalyst too, therefore, that which evades the regularities of verbal activity by distorting its constraints and regularities is nevertheless its silent generator, “the network of multiple supports”135 with which the illusion of language is woven.136 The circle is thus closed off: whether we speak of epilinguistic chaos, of lalangue or of the rays of the world 137 —as Merleau-Ponty would have said, perceiving the same tensions from a different perspective—the virtualities of heterogenesis are assimilated to a seething magma of forms that pulsate through the utterances they nevertheless overpower and transcend. Enunciation, whether verbal or visual, is the diagrammatic device that draws upon this magma in the form of provisional concretions.

133

Cf. Ducard [33, 34]. Lacan [43] 135 Lacan [43] 136 Cf. Lacan [43], p. 11: “[…] le langage, ça n’existe pas”. 137 Merleau-Ponty [57], p. 327. 134

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Called “utterances” by certain, “segments of discourse” or “speech acts” by others, these concretions never actually cease to form or to be reshaped, changing into something else or vanishing into the magma from which they never cease to be drawn by the enunciative gesture.

16.6 Conclusions: For a Diagrammatic Philosophy of Enunciation Pour mon malheur, ou pour mon bonheur […] rien n’est plus étranger à ma nature que l’accomplissement. J’ai approfondi dans le moindre détail tout ce que je n’aurais jamais fait. Je suis allé jusqu’au bout du virtuel. E. Cioran

This chapter aimed to investigate certain significant developments of the idea of diagrammatic gesture. In particular, it was intended to investigate its main repercussions in semiotics and linguistics. The concept of enunciation was very useful in this sense. Taken in the loose and yet fundamental sense of the “constitution of the utterance”,138 it offered a solid test-bed for verifying the heuristic potential of an extension that contemporary semiotics and linguistics have only recently begun to consider. Admittedly, terms such as “gesture” and “diagram” have been used in both disciplines for several years. Moreover, expressions such as “enunciative gesture” and “diagrammatic practice” have experienced enormous success in the same domains, reaching a more than remarkable level of diffusion, especially in semiotics. Yet, the term “diagrammatic gesture” does not however seem to have gained the attention it deserves among semioticians and linguists. Except for a few sparse occurrences, it is absent from the main technical lexicons of the two disciplines. This absence is not only a terminological fact, or rather, it is, but, like all terminological issues, it is by no means ideologically neutral.139 If linguists and semioticians have so far rarely adopted the term “diagrammatic gesture”, there must have been some other reason. And this reason, of which we are at present unaware, must presumably have had to do with the difficulty of thinking gestures and diagrams together, of transfusing them into a symbiotic unit—the diagrammatic gesture, precisely—that could prove of some use for semio-linguistic reflection. While not directly addressing the question mark this supposition raises, the paragraphs preceding these concluding notes have attempted to take on such an explanatory task.

138 139

De Vogüé [29], p. 118. Searle [72], p. 218.

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They have attempted, that is, to show in what sense and to what extent the diagrammatic gesture could constitute a privileged key to access certain problems that are particularly felt in the context of contemporary theories of enunciation. To this end, an attempt was made to interrogate a model of enunciation that could profitably interact with the notion of diagrammatic gesture. The model in question, identified in a key chapter of A Thousand Plateaus, was investigated with respect to an argumentative program articulated as follows: a. Analysis of a diagrammatic model of enunciation; b. Exploration of this model in relation to Paul Klee’s theory of figuration and Gilles Châtelet’s epistemology of mathematics; c. Affinities and differences between Klee’s diagrammatic gesture and Châtelet’s diagrammatic gesture; d. Critique of the type/token dichotomy in the light of such gestures; e. Merleau-Pontian reading of the abovementioned critique; f. Examination of the relationships between iterability and heterogenesis; g. Deepening of these relationships in an pictorial context; h. Exposure of some case studies: Van Eyck, Picasso, Pollock i. Return to the diagrammatic model of enunciation j. Comparation between the invisible and the unsaid: Michel Pêcheux; k. A Culiolian reading of the heterogenesis: Epilinguistic chaos. These, in short, are the nuclei from which it was deemed possible to develop a discourse on the diagrammatic gesture that could profitably dialogue with the domains of semiotic and linguistic investigation. The theme of heterogenesis, already discussed in the introductory chapter, has accompanied the reflections articulated from one nucleus to the other of the path briefly outlined here. After all, it could only be so: becoming-other is intimately connected to the diagrammatic gesture. In the model of enunciation that we have tried to develop in these pages, heterogenesis has thus been identified with a processuality immanent to the constitution of utterances, that is, with a background of virtualities that “pulsate”140 in the provisional concretions of enunciation in the form of metamorphic potentialities. Investigating the logic of these potentialities, dissecting their manifold forms of manifestation, so as to arrive at an adequate understanding of the enunciative gesture as a particular case of diagrammatic gestures, are all objectives that have oriented the writing of these pages without, these, however, have been ever being fully achieved. Had this been the case, in fact, we would have, at the end of the analysis, a complete map of the diagrammatic relations linking the various universes of the enunciative gesture. Instead, all that could be obtained was reduced to a few references, perhaps stimulating in their heterogeneity, but decidedly insufficient to develop an exhaustive analysis of the issues at stake. It is for this reason then, that, taking leave of the reader, 140

Paolucci [61], p. 232. For the concept of pulsation see La Mantia [46] and the monumental Guitart [40].

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we entrust the germs of this analysis to a diagrammatic philosophy of enunciation that is entirely to be constructed.

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27. Deleuze G., Guattari F.: A Thousand Plateaus. Capitalism and Schizophrenia. University of Minnesota Press, Minnesota (1987, 1980) 28. Derrida, J.: Marges de la philosophie. Les éditions du Seuil, Paris (1972) 29. De Vogué, S.: Culioli après Benveniste. Langage 118, 105–124 (2006) 30. Dondero, M.G.: L’approche sémiotique de Charles Goodwin: langage visuel, énonciation, diagramme. In: Tracés. Revue de Sciences humaines, 16, pp. 75–78 (2016) 31. Dondero, M.G.: The Language of Images. The Forms and the Forces. Springer, Dordrecht (2020) 32. Drees, W.B.: The divine as ground of existence and of transcendental values: an exploration. In: Buckareff, A.A., Nagasawa, Y. (eds.) Alternative Concepts of God. Essays of the Metaphysics of the Divine, pp. 195–212. Oxford University Press, Oxford (2016) 33. Ducard, D.: De Culioli à Saussure, aller-retour. Historie Epistémologie Langage 40, 8–93 (2018) 34. Ducard, D. : De l’épi-métalinguistique à l’épi-langagier : coscience et inconscience de la langue. In: Dufaye, L., Gournay, L. (éds.) Épilinguistique, métalinguistique : Discussions théoriques et applications didactiques, Limoges : Lambert-Lucas (2021) 35. Ducard, D.: The semiotic chora and the inner life of language. In: Beardsworth, S.G. (eds.) The Philosophy of Julia Kristeva, pp. 155–174. The Library of Living Philosophers, Chicago (2020) 36. Dupuis, J.D.: Gilles Deleuze, Félix Guattari et Gilles Châtelet. De l’expérience diagrammatique. L’Harmattan, Paris (2012) 37. Fénoglio, I. : La notion d’événement d’énonciation: le lapsus comme une donnée d’articulation entre discours et parole. In : Langage et société, n. 80, pp. 39–71 (1997) 38. Fénoglio, I. : Les événements d’énonciation. Focalisateur d’interprétation psychanalytiques, matériau pertinent de l’analyse linguistique d’énonciation. In : Arrivé, M., Normand, C. (éds.) Linguistique et Psychanalyse, pp. 167–184. Hermann, Paris (2013) 39. Fontanille, J. : Corps et sens, Paris, PUF (2011) 40. Guitart, R.: La pulsation mathématique. L’Harmattan, Paris (2013) 41. Karmel, P.: Abstract Art : a Global History. Thames & Hudson, London (2020) 42. Klee, P.: Das Bildnerische Denken. Sechste Schwabe Verlag, Basel [1956] (2013) 43. Lacan, J.: Encore. Séminaire XXe . Les éditions du Seuil, Paris (1977) 44. Langer, S.K.: Mind: An Essay on Human Feeling, vol. I. The John Hopkins Press, Baltimora (1967) 45. La Mantia, F.: Hasard et figure du locuteur. In: Valentina Bisconti, Raffaele Simone, Francesco La Mantia Hasard et langue, forthcoming (2023) 46. La Mantia, F.: Seconda Persona. Enunciazione e Psicoanalisi. Quodlibet, Macerata (2020) 47. La Mantia, F.: Che senso ha? Polisemia e attività di linguaggio. Mimesis, Milano-Udine (2012) 48. Lautman, A.: Les mathématiques, Les Idées et le Réel Physique. Vrin, Paris(2006) 49. Lieberman, P.: The Origins of Language. MacMillan, New-York (1975) 50. Longo, G.: The cognitive foundation of mathematics: human gestures in proof and mathematical incompleteness of formalisms. In: Okada, G.L.M., Grialou, P. (eds.) Images and Reasoning, Tokyo: Keio University Press. (trad. it. La gestualità umana nelle prove e l’incompletezza del formalismo, in Longo Giuseppe (2021), pp. 127–163) (2005) 51. Longo, G.: Matematica e Senso. Per non divenire macchine. Mimesis, Milano-Udine (2021) 52. Longo, G., Longo, S.: Reinventare il corpo e lo spazio. In: Boi, L., et alii (a cura di) In difesa dell’umano, forthcoming, pp. 1–32 (2021) 53. Marin, L.: Lo spazio Pollock. In: Corrain, L. (a cura di) Semiotiche della Pittura, pp. 207–223. Meltemi, Roma (2004) 54. Marquis, J.-P.: Stairway to heaven: the abstract method and levels of abstraction in mathematics. In: Pitici, M. (eds.) The Best Writing on Mathematics, pp. 144–171. Princeton University Press, Princeton (2017) 55. Merleau-Ponty, M. : Le problème de la parole. Cours au Collège de France. Notes 1953–1954, Genève (2020) 56. Merleau-Ponty, M.: Le visible et l’invisible. Gallimard, Paris (1969)

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57. Merleau-Ponty, M.: Le Visible et l’Invible, Paris, Gallimard (1979) [1964] 58. Moretto, A.: Con Euclide e Contro Euclide: Kant e la geometria. Studi Kantiani 26, 71–91 (2013) 59. Nølke, H.: Linguistic Polyphony. The Scandinavian Approach. Brill, ScaPoLine, Leiden (2017) 60. Otto, P.: Van Eyck, Torino, Einaudi (2013) 61. Paolucci, C.: Persona, Bompiani, Milano (2020) 62. Pêcheux M. : Automatic Discourse Analysis. Rodopi, Amsterdam (1995) 63. Pêcheux, M.: Les inquiétudes du discours. Les éditions de Cendres, Paris (1990) 64. Petitot, J.: Per un Nuovo Illuminismo. La conoscenza scientifica come valore culturale e civile, a cura di Fabio Minazzi, Milano: Bompiani (2009) 65. Pollock, J.: Interviews, Letters, and Reviews, edited by Pepe Karmel. The Museum of Modern Art, New-York (1951) [1999] 66. Richman, F.: Intuitionism as generalization. Philos. Math. 5, 124–128 (1990) 67. Richman, F.: Interview with a constructive mathematician. Modern Logic 6, 247–271 (1996) 68. Resnik, M.: Frege and the Philosophy of Mathematics. Cornell University Press, Ithaca, New York (1980) 69. Rotman, B.: Becoming Beside Ourselves. Duke University Press, London (2008) 70. Robert, S.: L’épaisseur du langage et la linéarité de l’énoncé : vers un modèle énonciatif de production. In: Aboubakar Ouattara (éditeur) Parcours énonciatifs et parcours interprétatifs – Théories et applications, pp. 255–274. Ophrys, Paris (2003) 71. Sarti, A. & Pelgreffi, I.: L’hétérogenèse différentielle Multitudes 1, 154–163 (2020) 72. Searle, J.: La riscoperta della mente. Bollati-Boringhieri, Torino (1995) 73. Sepkoski, D.: Nominalism and Constructivism in Seventeenth-Century Mathematical Philosophy. Routledge, London (2007) 74. Thom, R.: Local et Global dans l’œuvre d’art. Le Débat 23, 73–89 (1983) 75. Varnedoe, K.: Pictures of Nothing: Abstract Art Since Pollock. Princeton University Press, Princeton (2016) 76. Martin, W.: Understanding Art. A Guide for Teachers. Sage Publication Inc., London (2003) 77. Whitehead, A.N.: Process and Reality. An Essay in Cosmology. The Free Press, New York [1929] (1978) 78. Wilson Phillip, L.: Quantum Immortality and Non-Classical Logic. forthcoming, pp. 1–12 (2020)

Appendix A

Diagrammatic Eidos and Dynamical Platonism Charles Alunni and Francesco La Mantia

One could caricature Lautman by saying that he is Hilbert plus Plato. But Lautman’s attempt to answer his fundamental question – which is a fundamental question – is much more complex. His Plato is not, except by name, the one we know, more or less deformed by the successive readings of his works. He is, if I may say so, a Plato “in Idea”, whose whole Platonism is to be undertaken ex nihilo, from the experience of the future of mathematics. For Lautman, as for modern commentators of Plato, “Ideas are not the immobile and irreducible essences of an intelligible world”, but they are themselves in becoming, linked to each other “according to the schemes of a superior dialectic that presides over their coming”. André Lichnerowicz, “Albert Lautman and mathematical philosophy”, Revue de Métaphysique et de Morale 83rd Year, No. 1 (January-March 1978), Paris, Puf, p. 32.1 He cannot think he who cannot feel […] All thought engages the body in some way. François Roustang, Il suffit d’un geste, Odile Jacob, 2003, p. 123.

A.1 Dichotomic and Diagrammatic Conceptions of the Eidos In the previous chapter, we asserted that the visual enunciation of Châtelet’s triangle satisfies the minimal standards of the diagrammatic model of enunciation: like the visual enunciation of Klee’s jester, it is a constitutive process, a morphogenetic activity which creates and recreates its products by means of specific vibratory patterns (§§ 4.3.1, 5., 5.1.1). This essay is the result of long and dense conversations between the two authors. It can therefore be considered the product of common reflections. In any case, Charles Alunni is the author of the paragraphs (Sect. A.2); Francesco La Mantia is the author of the paragraphs (Sects. A.1, A.3, A.4, A.5, and A.6). Special thanks go to Fernando Zalamea who, as always, has been an attentive and valuable reader. 1

Online sur https://www.jstor.org/stable/40901828.

© The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1

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Therefore, compared to Platonist and Constructivist vulgates (§ 5.1.2), the shift in perspective is radical: precisely because the triangle is constituted within such patterns, it subsists as a material ideality or ideal materiality (§ 5.1.2). Moreover, since this “coalescence2 ” of ideality and materiality is given in the iteration of specific hand’s movements (§ 5.1.4), the eidos becomes a virtuality of the flesh (cf. § 5.1.4): far from being a type detached from its tokens (§ 5.1.4), it lurks in the “potentia agendi3 ” of gestures and in the vibrations of forms (§§ 5. & 5.1.1). Hence, as such, it neither precedes the flesh nor is subsequently disentangled from it by abstraction. Diagrammatic eidos is neither an ante rem (Platonic stance) nor a post rem type (Conceptualist stance). It is rather an invisible germ inscribed in the flesh, in the fluidity of its movements and in the stability of its symbolic productions. It is this “something” that constitutes a genuine watershed between diagrammatic and dichotomic conceptions of the eidos. As long as the constructed reality of the universal circle is presented as distinguished from the circles drawn on paper (on screen, etc.), the eidos will have all the formal features of a type detached from its tokens. Little does it matter, in this case, if it is a cognitive type: even if mentally constructed, the circle derived by abstraction will be the “bearer” of properties that are on another plane than the circles materially produced. Of course, these properties are not exactly the properties of an ideal Platonic type (§ 5.1.2): the latter has a totally different ontological scope from the constructed reality of a cognitive type. Nonetheless, however, they both transcend the symbolic universe of diagrammatic materialities (§ 5.1.4): the first because it is an abstract object (§ 5.1.2); the second, because it is an object forged in a “process of mathematical abstraction4 ”. It is on these forms of transcendence that every dichotomic conception of eidos is based: whether is matter of Platonic or constructed idealities, the sphere of the “immaterial” will act as a “scissor” (as a dichotomy, precisely) which separates formal concepts from their material supports. A diagrammatic conception of the eidos emerges, on the other hand, when we move from the transcendence of the aforementioned idealities to the immanence of virtualities (§ 5.1.4) at once feeding and exceeding germinal unities of matter, concept and movement (§ 5.1.4). This appendix aims to show that there is at least one theoretical move by which a diagrammatic conception of the eidos can be settled. It is the one accomplished by authors such as Charles Alunni and Fernando Zalamea. The French philosopher of mathematics and the Colombian mathematician have helped to greatly reduce the gap between types and tokens by advocating the need for a new form of Platonism inspired by the work of the mathematician and philosopher of science Albert Lautman.

2

Whitehead [46, p. 26]. Citton [12, p. 287]. 4 Drees [17], p. 206. 3

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A.2 Lautman and “Dynamic Platonism” A kind of halt is necessary here and concerns the fundamental importance of the work of Albert Lautman in the question of the establishment of a “dynamic Platonism”. This is, in our opinion, the very first use of the notion (or rather of its conceptual equivalent), outside the specialized field of antiquists,5 which is applied to mathematics and physics. We are going to see, among other things, to what corresponds the transversal development of his Platonism to the “renewal” of which he participates resolutely in metaphysics.

A.2.1 Context Lautman mainly bases his argument on the work of Julius Stenzel, Zahl und Gestalt bei Platon und Aristoteles, Leipzig, 1924, d’Oskar Becker, “Die diairetische Erzeugung der platonischen Idealzahlen”, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, B 1, 1931, pp. 464–501 et de Léon Robin, Platon, Paris, 1935. It is at the conclusion of the Essay on the notions of structure and existence in mathematics that Lautman reveals his sources: Before concluding, we would like to show how this conception of an ideal reality, superior to mathematics and yet so willing to be incarnated in its movement, comes to be integrated into the most authoritative interpretations of Platonism […] All modern Plato commentators on the contrary insists on the fact that Ideas are not immobile and irreducible essences of an intelligible world, but that they are related to each other according to the schemas of a superior dialectic that presides over their arrival. The work of Robin, Stenzel and Becker has in this regard brought considerable clarity to the governing role of Ideas-numbers which concerns as much the becoming of numbers as that of Ideas.6

Who are these authors carrying a new vision of Platonic philosophy? Becker, Stenzel and Robin are just contemporaries of Lautman, and it took him an absolutely extraordinary philosophical flair to discover their “dynamic Platonism” which would come to be applied in the field of contemporary mathematics and physics.

A.2.1.1

Oskar Becker

Let’s start with Oskar Becker (1889–1964). He was a German philosopher, logician, mathematician and historian of mathematics. He was born in Leipzig, where he studied mathematics. His thesis under Otto Hölder and Karl Rohn (1914) is titled On the Decomposition of Polygons in non-intersecting triangles on the Basis of 5

The antiquists to whom Lautman refers, apart from the case of Becker which we consider below, have limited their field of philosophical investigation to ancient mathematics. Only Lautman will take into account the contemporary physico-mathematical world. 6 Lautman [28], p. 190.

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the Axioms of Connection and Order.7 He served in World War I and returned to study philosophy with Edmund Husserl, writing his habilitation memoir on Investigations of the Phenomenological Foundations of Geometry and their Physical Applications (1923). Becker is Husserl’s assistant, unofficially, then the official editor of the Annuaire de la recherche phénoménologique. In the thirties, Becker adhered to a “new Nordic paganism”, which he evoked in a letter addressed in February 1936 to Löwith, while the latter was in exile in Rome: “The ‘Spirit’ has become an anachronism, and Christianity is coming to an end. Young people no longer fight either against God or for the faith. It fights for a ‘new faith’, without continuity with Judeo-Oriental Christianity: the ‘simple and modest’ faith in Germany”.8 Löwith indicates, for example, that Becker’s interest in the study of races (Rassenlehre) was at first apolitical, and that it was linked to a penchant for naturalistic philosophy, for everything that touched on the unconscious and instinct, which he tried to define, contrary to Heidegger’s Dasein, as “original nature”.9 His main works are: 1. Über die Zerlegung eines Polygons in exclusive Dreiecke auf Grund der ebenen Axiome der Verknüpfung und Anordnung, Leipzig: 1914. 2. “Contributions Toward a Phenomenological Foundation of Geometry and Its Physical Applications”, extrait de Beiträge zur phänomenologischen Begründung der Geometrie und ihre physikalischen Anwendungen (Jahrbuch für Philosophie und Phänomenologische Forschung IV 1923, 493–560). Sélections trad. par Theodore Kisiel, dans Phénoménologie et les sciences naturelles, ed. Joseph Kockelmans et Theordore J. Kisiel, Evanston, Illinois: Northwestern University Press, 1970, p. 119–143. 3. Mathematische Existenz. Untersuchungen zur Logik und Ontologie mathematischer Phänomene (Jahrbuch für Philosophie und phänomenologische Forschung, Vol. VIII, 1927, 440–809. 4. “La philosophie d’Edmund Husserl”, trad. R. O. Elverton, dans La phénoménologie de Husserl, éd. R. O. Elverton, Chicago: Quadrangle Books 1970,–originally “Die Philosophie Edmund Husserls. Anlässlich seines 70. Geburtstags dargestellt” in Kantstudien. vol 35, 1930, 119–150. 5. “Eudoxus-Studien: I: Eine voreudoxische Proportionenlehre und ihre Spuren bei Aristoteles und Euklid”, Quellen und Studien zur Geschichte der Mathematik, Astronomie et Phyik B. II (1933), 311–330. [published in Jean Christianidis, éd. Les classiques dans l’histoire des mathématiques grecques, Dordrecht/Boston: 7

“Oskar Becker” (https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18606), on the Mathematics Genealogy Project website. 8 Becker thus developed in 1938, in an article published in the journal Rasse, the idea of a “Nordic Metaphysics” in a fairly standard Nazi style. According to Oskar Becker, “the rhythm of life, which constantly recurs in the Dionysian dithyrambs of Nietzsche, was identical to the will to power […] and one could say by applying it to today, concretely, in the sense understood by the youth: identical also to the marching rhythm of the brownshirts [of the SA]”, in Löwith [32], p. 46. 9 Löwith [32], p. 46–47.

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6. 7.

8. 9.

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Boston Studies in the Philosophie of Science, vol. 240, 2004, 191–209, with intro. of Ken Saito, 188–9.] “II: Warum haben Griechnen Existenz der Vierten Proportionale angenommen”, 369–387, “III: Spuren eines Stetigkeitsaxioms in Art of Dedekindschen zur Zeir des Eudoxos”, vol. 3 (1936) 236–244, “IV: Das Prinzip des ausgeschlossenen Dritten in der griechischen Mathematik”, 370–388, “V: Die eudoxische Lehre von den Ideen und den Farben”, 3 (1936) 389–410. “Zur Logik der Modalitäten”, dans Jahrbuch für Philosophie und phänomenologische Forschung, Bd. XI (1930), pp. 497–548. Grundlagen der Mathematik in geschichtlicher Entwicklung, Fribourg/München: Alber, 1954 (2nd ed 1964; original text published by Suhrkamp Taschenbuch Wissenschaft 114. Francfort: Suhrkamp, 1975). Dasein und Dawesen, Zeitschrift für Philosophische Forschung 21 (3):455–465 (1964). Lettres à Hermann Weyl, in Paolo Mancosu and T. A. Ryckman, “Mathématiques et phénoménologie: la correspondance entre O. Becker et H. Weyl”, Philosophia Mathematica, 3 e série, vol. 10 (2002) 174–194.

Becker’s work has been little studied in France, no doubt partly because of his heavy Nazi past.10 It is certainly the “phenomenological” part (Husserlian, then Heideggerian) that played an essential role in the “transversal” reworking that Lautman made of it. In Becker, the subjection of mathematics to the factual experience of Dasein drawn from Heidegger is central. Exemplary is his phenomenological treatment of the transfinite.11 The phenomenological foundation of the transfinite rests on the rejection of the actualist aspects of Cantor’s theses, of which he values on the contrary the suggestions of a phenomenological type, distinct from the directly set-theoretic formulations. These suggestions relate to the laws of the process of generation of ordinals, which Becker interprets in terms of successive levels of acts of a transcendental consciousness operating reflexively on its previous acts, in the manner of an iterated recognition. This converges with the Weylian reference of symbolism to 10

Some notable exceptions: Mancosu and Ryckman [33], p. 130–202. See also the remarkable thesis by Barot [6]. On Becker, see the chapters “D’un intuitionnisme bien tempéré à la tentation phénoménologique: H. Weyl et O. Becker”, and the important sub-chapter “La fondation phénoménologique du transfini par O. Becker: Mathematische Existenz [35]” (p. 387 sq.). It is this chapter that we are following here. Finally, Gérard [21], pp. 3–10, and Rashed and Auffret [38], pp. 3–14, (in particular the pages on the re-contextualization of Stenzel as a singular Platonic figure–which we will also follow). For these two authors, the question of the unwritten doctrines of Plato, at the very heart of the dynamic Platonism of Robin and Stenzel, merges with that of the latent role of mathematics in all the dialogues. Plato’s so-called “esoteric” oral teaching focuses on mathematics and metamathematics. 11 For the treatment of this very precise sequence by Lautman himself, see the “Report on the philosophical works undertaken by M. Lautman”, unpublished manuscript, dated March 1935 (Fonds ENS. Bouglé, National Archives, dimension 61 AJ, box 96), published for the first time by Fernando Zalamea in its Spanish edition, Albert Lautman, Ensayos sobre la dialéctica, estructura y unidad de las matemáticas modernas, “Biblioteca Francesa de Filosofía”, Bogota, Universidad Nacional de Colombia, 2011, p. 439–449: “Becker, who in his work Mathematische Existenz strives to phenomenologically ground the construction of the Cantorian transfinite by relying on the iterations of self-reflecting consciousness” (p. 441).

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the founding process of iteration, since “actual existence–in the ontological sense– requires phenomenological donation, ‘access’. After all, only phenomena ultimately ‘exist’”.12 Becker thus defends the idea of a regulated intuitive fulfillment of the intention of determining the transfinite by interpreting the figuration-symbolization of the stages of ordinal generation as being such a phenomenological given: […] when we think of a figure that we have reproduced, and this reproduction is itself reproduced once again, then we obtain figures of level ω, ω + 1, ω + 2, etc. In this perspective is currently given an intuitive representation, at least for the first transfinite numbers ω, ω + 1, ω + 2... It is clear that in a representation actually performed [actual] only a finite number (and even very small) of figures can thus be interlocked... But it happens, despite everything, that the first interlockings make the infinite process evident according to its pure ideal possibility. The "ideal" infinite complication of these interlocking figures is represented symbolically in only one of the figures described. This means, in a certain sense, that it is on the symbolic representation itself that the following representative procedure operates (the ω-th, the ω + 1-th). This “symbolic” [representation], however, is in no way abstract-conceptual; she is quite intuitive.13

In other words, Becker wants to extend the iterative process of successive reflections-thematizations operated by a consciousness that Husserl determined in the finite case in § 101 of the Ideen (essentially an eidetic variation operating on previous eidetic variations), to transfinite iterations, and it is for him the regulated character of the intuited symbolic writings which ensures this repeated recognition– in passing, we note that Weyl refers to Becker for the phenomenological analysis of the levels of constitutions, in the sense above, of space14 : If we look carefully at the process of iterating through reflection on itself, that is, in an entirely concrete way, then we see the following: I reflect the reflection that I have just done, then I reflect again on this second-level reflection, and so on… [At] ω-th level, the object (the theme considered) of my present act of reflection is the entire previous thoughts.15 We glimpse it here, it is the temporal structure of the series of acts which is fundamental, that is to say irreducible, it is this which is the true phenomenological basis. This temporal structure is always, when it is a support for reflexive acts, finite since it operates on finite series of symbols, even if it intervenes at a transfinite level of the iteration, each level then functioning as an internal horizon obtained by thematization of an earlier horizon. We must scrupulously distinguish the structure of the contemplation of the interlocking of intentionalities, from the structure of the interlocking itself. The first structure is of finite complexity while the second is of transfinite complexity.16

12

Becker (1927), p. 527. Becker (1927), p. 540–1. 14 Weyl [45], p. 129. 15 Becker (1927), p. 546. 16 Becker (1927), p. 547. 13

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A.2.1.2

413

Julius Stenzel

Let’s come to Julius Stenzel (1883–1935).17 Politically speaking, he is almost the opposite of Becker: a victim of Nazism. After graduating from the Königliche Wilhelms-Gymnasium in Breslau, Stenzel studied classical philology at the University of Breslau from 1902. After the state examination (1907), he finished his seminary year at the Königliches Gymnasium in Beuthen and worked on his thesis: De ratione, quae inter carminum epicorum prooemia et hymnicam Graecorum poesin intercedere videatur. With this thesis, he obtained his doctorate in 1908. On October 1, 1909, he was employed at the Royal Gymnasium in Breslau and four years later he was promoted to head teacher. During World War I, Stenzel served as a radio operator from 1916 to 1918. He was awarded the Iron Cross, Class II, and was promoted to lieutenant on June 2, 1918. After his return, he continued to work as a head teacher in Breslau, but at the same time pursued his habilitation, which he obtained in 1920 in Breslau for the subject of philosophy. From 1923, in addition to his school office, he held a position as a philosophy teacher at the university. On April 1, 1925, Stenzel left Breslau and went to the University of Kiel, where he had received a full professorship in philosophy. The same year, he was elected a corresponding member of the Göttingen Academy of Sciences. In 1931 he declined an appeal to the University of Basel. Stenzel was a member of a disciplinary committee that expelled from the university in 1930 certain Nazi students who had disrupted the service of liberal theologian Otto Baumgarten. After the National Socialists seized power, Stenzel was denounced by a student and granted temporary leave. Despite the political rehabilitation, he was transferred to the University of Halle on November 1, 1933, on the basis of Article 5 of the Professional Civil Service Restoration Act. He died there two years later after a short and serious illness. His Jewish wife, whom Stenzel had married in Breslau in 1910, emigrated to the United States with her son Joachim Stenzel in 1939 and lived in Berkeley. His mother committed suicide to escape the threat of deportation. Julius Stenzel was one of the most eminent scholars of Plato of his time. He was also a historian of mathematics. Together with Otto Toeplitz and Heinrich Scholz he led a seminar on ancient mathematics in Kiel and together with Otto Neugebauer and Toeplitz founded the sources and studies in the history of mathematics (1929). In 1924 he gave a plenary lecture at the International Congress of Mathematicians (ICM) in Toronto (“Vision and Thought in the Classical Theory of Greek Mathematics”). Most of his work is contained in the following books: 1. Sur deux notions du mysticisme platonicien: Zoon et Kinesis. Breslau: Imprimerie coopérative de livres de Breslau 1914. 2. Studien zur Entwicklung der platonischen Dialektik von Sokrates zu Aristoteles, Breslau: Trewendt & Granier 1917. 3. Zum Problem der Philosophiegeschichte. Göttingen 1921. Auch in: KantStudien, Band 26, S. 416–453. Phil. Habilschrift 1920. 17

See Günther [22], pp. 83–96.

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4. Zahl und Gestalt bei Platon und Aristoteles, Leipzig: Teubner1924. It is the central work in close connection with the transversal “Platonism” of Lautman. 5. Wissenschaft und Staatsgesinnung bei Platon, Kiel: Lipsius & Tischer, 1927. 6. Platon der Erzieher, Leipzig: F. Meiner 1928 (2. Auflage Hamburg 1961). Italian version, Platone educatore, Bari: Laterza, trans. it. Francesco Gabrieli, 1936, 19662 , 19743 . See in particular ch. IV, 1b on “Mathematics and the Doctrine of Ideas”. 7. Metaphysik des Altertums, Oldenbourg: München 1931 (online https://archive. org/details/meta physikdesalt00sten). 8. Die nationale Aufgabe des humanistischen Gymnasiums. In: Neue Jahrbücher für Wissenschaft und Jugendbildung (1933), p. 315–328. 9. Philosophie der Sprache, München: Oldenbourg 1934. 10. Staat und Geschichte, München: Oldenbourg 1934. 11. Platonismus einst und jetzt, Zürich: Rascher 1934. 12. Plato’s Method of Dialectic, transl. and ed. D. J. Allan, Oxford: Oxford at Clarendon Press, 1940. In Zahl und Gestalt bei Platon und Aristoteles, Stenzel tries to describe the internal organization of the system of Ideas with the help of a reconstruction of the dieretic processes of the Academy, which will then allow E. Taylor to describe the ‘Dyad ´ as an infinite sequence of approximations by excess indefinite’ (¢´oριστoς δυας) and by default of irrational quantities. This intuition is the basis of an interpretation of irrational magnitudes conceived as numbers, therefore of an anticipation still imperfect, on the part of Plato, of real numbers.18 Taking Stenzel’s defense in a way, O. Toeplitz19 devotes two articles to responding to this new interpretation, without however taking up Stenzel’s.20 Toeplitz considers it beyond doubt that we must attribute to Plato an extension of the notion of relation, logos, in a numerical direction (our rationals). [O]n henceforth we knew that the Platonic question was that of the arithmetization of ratios (logoi) and magnitudes, and that the question of principles was, thereby, that of the relationship of metamathematical structures to the mathematical objects revisited by this arithmetization. It was necessary to interpret Plato mathematically, because the pivot of his ontology was provided by logistics, or the theory of relations, both finite and infinite.21

Finally, to conclude on this point, I will note this final remark of the article by Marwan Rashed and Thomas Auffret which is very interesting, because to bring 18

See on this point the positive accounts of Stenzel by Taylor [41], pp. 419–440, Taylor [42], pp. 12–33. 19 In Frankfurt, in the 1920s, Max Dehn and Ernst Hellinger, students of Hilbert, founded a famous seminar on the history of mathematics, attended among others by Otto Toeplitz–himself a student of Hilbert. All three are creative mathematicians. Once appointed professor at Kiel during the same decade, Toeplitz set up a seminar for the reading of Greek mathematical and philosophical texts, which brought together philosophers, mathematicians and historians (quoted by Julius Stenzel, in the foreword to the second edition of Zahl und Gestalt, 1933). 20 Toeplitz [43], pp. 3–33 and Toeplitz [44], pp. 334–346. 21 Rashed and Auffret [38], p. 9.

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closer to the positions of Lautman and his successors as to the remarkable function of Ideas in the mathematical field22 : Is it not at the very least adventurous to bring Plato closer to much later mathematical theories? Isn’t it obvious that Plato, more than a thousand years before the invention of algebra, had neither the logical nor the technical means to glimpse something like the real number? It’s undeniable. And yet, on the quite restricted case of quadratic irrationals, Plato compared the ontological status of their infinite anthypheretic approximation to that of a finite logos, itself understood as a number.23

Apart from this contemporary debate on the work of Lautman, a look at the table of contents of Zahl und Gestalt bei Plato und Aristoteles would suffice to show its importance for those who claimed it. Finally Leon Robin (1866–1947).

A.2.1.3

Léon Robin

[A]ll Plato’s faces have their truth in each of the mirrors of tradition. Perhaps the most faithful of these mirrors was that of Plotinus [...]. Léon Robin, Platon, 1935, p. 239.

Historian of ancient philosophy, he was considered in his time as “the master of Platonic studies in France”. He studied at the Faculty of Letters of Bordeaux and at the Sorbonne before becoming an associate professor of philosophy. To access the title of doctor of letters, he defended his theses in 1908, the main one, The Platonic Theory of Ideas and Numbers according to Aristotle. In 1927 he was visiting professor at the University of Pennsylvania. Pupil and disciple of Octave Hamelin, he ensured with Léon Brunschvicg the dissemination of his ideas and the publication of his works; he notably edited Le Système d’Aristote, a course given by Hamelin at the École Normale Supérieure in 1904–1905. He published numerous books on Greek philosophy, and directed the translation of the complete works of Plato which was published in 1950 in the Bibliothèque de la Pléiade. Léon Robin notably translated Plato’s Dialogues into French. His main publications: 1. La Théorie platonicienne des idées et des nombres d’après Aristote: étude historique et critique, Paris: Félix Alcan, 1908, 702 p. 2. La Théorie platonicienne de l’amour, Paris: Félix Alcan, 1908, 229 p. ˙ sur la signification et la place de la physique dans la philosophie de 3. Etudes Platon, Paris: Félix Alcan, 1919, 96 p. 4. La Pensée grecque et les Origines de l’esprit scientifique, Paris: Renaissance du livre, « L’évolution de l’humanité. Synthèse collective», 1923, 480 p. 5. Platon, Paris: Félix Alcan, 1935, 364 p. 22 23

See the founding Rodier [40], pp. 479–490. Rashed and Auffret [38], p. 13.

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6. La Morale antique, Paris: Félix Alcan, 1938, 180 p. 7. La Pensée hellénique, des origines à Épicure: questions de méthode, de critique et d’histoire, Paris: Presses Universitaires de France, 1942, 554 p. 8. Pyrrhon et le scepticisme grec, Paris: Presses universitaires de France, 1944, 255 p. 9. Aristote, Paris: Presses universitaires de France, 1944, 324 p. 10. Les Rapports de l’être et de la connaissance d’après Platon, Paris: Presses universitaires de France, 1957. His influence on Lautman is significant. Its only appearance per se in the text, apart from its generic association with Becker and Stenzel, is in the conclusion of the Essay on the Notions of Structure and Existence in Mathematics, Sect. A.2, The Schemas of Genesis: We are simply going to show how, to some extent, and to take up the expressions that Robin made use of in regards to Plato, the process of connecting theory and experience symbolizes the connection of Ideas and mathematical theories.24

Lautman does not cite La Théorie platonicienne des idées et des nombres d’après Aristote: étude historique et critique of 1908, but it can in all likelihood be assumed that he knew of its existence and had read it. He leans on The Sophist and The Philebus, based on the work of Robin on what Aristotle says about the Pythagorean then Platonic Number Ideas, essentially in Metaphysics A, 8–9.

A.2.2 Lautman and Plato The Works of Albert Lautman were published in 1977.25 It should be noted that there again very few critics were interested in this absolutely innovative and central work for a philosophy of mathematics and physics. Here is a “near-exhaustive” list of these publications: 1. Mario Castellana, “La philosophie des mathématiques chez Albert Lautman”, Protagora, no 115, Florence: 1978. 2. Revue d’histoire des sciences, “Mathématique et philosophie: Jean Cavaillès et Albert Lautman”, Paris: PUF, tome XL-1, 1987 (with texts from Michel Blay, Hourya Bénis-Sinaceur, Gerhardt Heinzmann, Catherine Chevalley et Jean Petitot). Avec des Lettres inédites de Jean Cavaillès et Gaston Bachelard à Albert Lautman par Hourya Bénis-Sinaceur. 3. Emmanuel Barot, L’aventure mathématique de la dialectique depuis Hegel. Perspectives sur les visages contemporains du “problème de la dialectique” en épistémologie des mathématiques et de leur histoire, thèse de doctorat de philosophie soutenue le 6 novembre 2004 (voir tout particulièrement Sect. A.5 24 25

Lautman [28], p. 192. Lautman [26].

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4. 5.

6.

7. 8.

9.

10. 11.

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du chapitre IV intitulée “La métamathématique néo-platonicienne de Lautman: le mauvais élève”, p. 335–374). Charles Alunni, “Albert Lautman et le souci brisé du mouvement”, Revue de synthèse, 5e série, année 2005/2, p. 283–301. Charles Alunni, “Continental genealogies. Mathematical confrontations in Albert Lautman and Gaston Bachelard,” in Virtual Mathematics: the logic of difference, ed. Simon Duffy, Bolton: Clinamen Press 2006, pp. 65–99. Albert Lautman, Les Mathématiques, les Idées et le Réel physique, Vrin, Paris: 2006. Edition compiled by Jacques Lautman with introductions by Jacques Lautman and Fernando Zalamea. Emmanuel Barot, Lautman, coll. “Figures du savoir”, Les Belles Lettres: Paris, 2009. Philosophiques, “Albert Lautman, philosophe des mathématiques”, Volume 37, numéro 1, printemps 2010, Sociét´e de philosophie du Québec, available on https://www.erudit.org/fr/revues/philoso/2010-v37-n1-philoso3706/. Mario Castellana, “Les mathématiques et l’expérience selon Albert Lautman”, in Les mathématiques et l’expérience. Ce qu’en ont dit les philosophes et les mathématiciens, dir. Evelyne Barbin, Jean-Pierre Cléro, Paris: Hermann, 2014. Mario Castellna, La matematica come resistenza, Postface de F. Zalamea, Castelvecchi, Roma: 2016. José Pedro Arriaga Arroyo, “La teoría de Las Ideas. Una exposición esquemática de la fiIosofía de Albert Lautman”, Devenires. Año xxiii, Núm. 45 (enero-junio 2022), p. 41–63. This article is particularly interesting because of its synthetic analysis of Lautman’s philosophy through diagrams.

In this very limited bibliographical framework, a work will have made a break, because at the origin of a new interpretative course: it is a reissue of the texts published in 1974 (but supplemented here) under the title A. Lautman, Les mathématiques, les Idées et le réel physique, Paris, Vrin, 2006. The important introductory study by Fernando Zalamea offers a completely new path for the mathematical interpretation of texts based on the theory of categories. Also noteworthy is the Spanish edition, edited, introduced and translated by Zalamea.26 Also fundamental is the publication of the issue entirely devoted to “Lautman, philosopher of mathematics” of the Quebec journal Philosophiques, which presents a complete overview of the question. The issue is edited by Emmanuel Barot, whose importance I have already underlined for his work for Lautmanian studies. Finally, note the pioneering nature of the work presented by Mario Castellana in 1978. As Jean Cavaillès remarks in the first sentence of his report on Lautman’s theses, his work constitutes “a new attempt to define the inherent reality of mathematical theories: the most recent works are used and the result invokes Plato”.27 26

Lautman [29]. Not only does Fernando Zalamea offer, as in 2006, a categorical vision of the work (“Platonic dialectic and the dialectic of the mathematical theory of categories” [Sects. A.5 “Platonism” and A.6 “Category Theory”]), but he adds texts that have never been published in French, which makes the edition the most comprehensive so far. 27 Lautman and Cavaillès [27], pp. 9–11.

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The first reference to Plato, which has ten occurrences, is introduced by Lautman in “Mathematics and Reality”, Presentation to the International Congress on Scientific Philosophy held in Paris in 1935. This first text marks Lautman’s radical position vis-à-vis the Wiener Kreis, which makes it one of his points in common with Gaston Bachelard: The participation of the sensible in the intelligible in Plato permits the identification, behind the changing appearances, of the intelligible relations of ideas. If the first contacts with the sensible are only sensations and emotions, the constitution of mathematical physics gives us access to the real through the knowledge of the structure with which it is endowed. It is even impossible to talk about the real independently of the modes of thought by which it can be apprehended, and far from denigrating mathematics to being only a language indifferent to the reality that it would disparage, the philosopher is committed to it as though in an attitude of meditation in which the secrets of nature are bound to appear to him. There is therefore no reason to maintain the distinction made by the Vienna Circle between rational knowledge and intuitive experience, between Erkennen and Erleben. In wanting to suppress the connections between thought and reality, as in refusing to give to science the value of a spiritual experience, the risk is to have only a shadow of science, and to push the mind in search of the real back towards the violent attitudes in which reason has no part. This is a resignation that the philosophy of science must not accept.28

The second occurrence appears in a reference to Gustave Juvet’s text La Structure des nouvelles theories physiques [24], where “Juvet gives a magnificent comparison that allows one to understand what profound harmony can exist between a schematic structure and a material realization”: Placed at a great distance from a window, the eye can first distinguish two axes of symmetry. On approaching, it recognizes in each quarter of the structure two new symmetries; certain motifs are repeated five times around a center. From closer yet again, more subtle ornamentations are seen in these motifs. It is the same with physical reality and the mind that examines it. The symmetries of phenomena, their alternations observe certain invariants at a given scale. The description that we give actually preserves these invariants and mimics these symmetries and these alternations in a game that reflects the structure of a group. Can it not be said that, in its way, physical reality at this scale mimics the structure of the group, or as Plato said, participates in this group? The reference to Plato is particularly significant and beneficial. To study only the signs, we can in effect come to believe that science deals only with those signs and excludes any consideration of a reality that the symbolism would have as its function to describe. The rational idea that the mind penetrates the becoming of things by knowledge of the mathematical connections in which they participate appears to some to be as obscure as the mystical beliefs in the participation of the subject in the object for the primitives spoken of by Levy-Bruhl (1931) […] Placed in front of a purely tautological conception of mathematics, the philosopher should stop linking the discovery of truth in science to the spiritual progress of a consciousness in search of a real to know and dominate. Scientific philosophy, by its formalism, would thus have contributed to the rejection of philosophy as belonging to the exclusive cult of irrational attitudes. One may however wish for the philosophy of science a higher ambition.29

28 29

Lautman [28], p. 12. Ibid. pp. 25–26. On Gustave Juvet, cf. Alunni [1], pp. 215–251.

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The third occurrence appears at the end of the 1937 paper, “On the Reality Inherent to Mathematical Theories,” a speech to the important Ninth International Congress of Philosophy. Descartes Congress, Paris, 1–16 August 1937. We only wish to indicate here the Platonic conclusion that these researches seem to us to impose: the reality inherent to mathematical theories comes to them from their participation in an ideal reality that is dominating with respect to mathematics, but that is only knowable through it. These ideas are quite distinct from pure arrangements of signs, but they have no less need of them as mathematical matter that lends them a field in which the layout of their connections can be provided.30

A fourth occurrence in the following text, still from 1937, « The Axiomatic and the Method of Division»: It is therefore not Aristotelian logic, that of genera and species, that plays a part here, but the Platonic method of division, as taught in the Sophist and the Philebus, for which the unity of Being is a unit of composition and a starting point for the search for principles that are united in the Ideas. Another rapprochement with the Platonic dialectic is necessary […] The distinction that is thereby established within a same notion between the intrinsic properties of an entity or notion and its possibilities of action seems to be similar to the Platonic distinction between the Same and the Other that is found in the unity of Being. The Same would be that by which a notion is intrinsic, the Other that by which it can enter in relation with other notions and act on them. It is […] extremely important for the philosopher to prevent the analysis of Ideas and the search for notions that are the most simple and separable from each other, from appearing like the search for the most extended types. A whole conception of mathematical intelligence, issuing from Platonism and Cartesianism is, in effect, at stake in this distinction.31

It is now the turn of the Essay on the Notions of Structure and Existence in Mathematics (Chap. 6, “On the Exceptional Character of Existence”), where Lautman details and employs his close readings of Stenzel32 and Robin.33 [W]e would like to show how this conception of an ideal reality, superior to mathematics and yet so willing to be incarnated in its movement, comes to be integrated into the most authoritative interpretations of Platonism. Certain historical explications are indispensable on this subject, given the sense that the expression of Platonism in mathematics generally receives. In the open debate between formalist and intuitionist, since the discovery of the transfinite, mathematicians have become accustomed to summarily designate under the name Platonism any philosophy for which the existence of a mathematical entity is taken as assured, even though this entity could not be built in a finite number of steps. It goes without saying that this is a superficial knowledge of Platonism, and that we do not believe ourselves to be referring to it. All modern Plato commentators on the contrary insists on the fact that Ideas are not immobile and irreducible essences of an intelligible world, but that they are related to each other according to the schemas of a superior dialectic that presides over their arrival. 30

Ibid. p. 30. Ibid. p. 30, 41 et 42. 32 Ibid. p. 190, 191 on the Ideas–numbers. 33 Ibid. p. 190, on the constitution of bodies in the Timaeus. 31

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The work of Robin, Stenzel and Becker has in this regard brought considerable clarity to the governing role of Ideas-numbers which concerns as much the becoming of numbers as that of Ideas. The One and the Dyad generate Ideas-numbers by a successively repeated process of division of the Unit into two new units. The Ideas-numbers are thus presented as geometric schemas of the combinations of units, amenable to constituting arithmetic numbers as well as Ideas in the ordinary sense. […] Metamathematics which is incarnated in the generation of ideas and numbers does not give rise in turn to a meta-metamathematics. The regression stops as soon as the mind has identified the schemas according to which the dialectic is constituted. We can thus see how well our reference to Platonism is justified, as regards the relations that exist between mathematical theories and the Ideas that govern them. […] In any case, the neo-positivism of the Vienna Circle, like the idealism of the English metaphysical physicists, separate mathematics and reality quite distinctly, while the philosophy of physics essentially has as its task the problem of their union. We do not pretend to treat this problem here, which is quite different from those we have envisaged throughout the course of the preceding pages. We are simply going to show how, to some extent, and to take up the expressions that Robin made use of in regards to Plato, the process of connecting theory and experience symbolizes the connection of Ideas and mathematical theories. […] The philosophy of physics would therefore also amount to the Platonic conclusion to which we have led mathematical philosophy, as we conceive it. The nature of the real, its structure and the conditions of its genesis are only knowable by ascending again to the Ideas whose connections are incarnated by Science.34

The sixth major occurrence appears in the 1939 text, New Research on the Dialectical Structure of Mathematics,35 the first of a series called Essais philosophiques, created by Jean Cavaillès and Raymond Aron at Hermann publishers. As Lautman states in his Foreword: This essay consists of two distinct parts: in the first, developing the ideas of our principal thesis [27], relative to the participation of Mathematics in a Dialectic that governs it, we try to show in an abstract way how the understanding of the Ideas of this Dialectic is necessarily extended in the genesis of effective mathematical theories […] It may seem strange to those who are used to separating the ‘moral’ sciences from the ‘exact’ sciences, to see, reunited in the same work, reflections on Plato and Heidegger, and remarks on the law of quadratic reciprocity or the distribution of prime numbers. We would like to have shown that this rapprochement of metaphysics and mathematics is not contingent but necessary.36

In his Chap. 1, “The Genesis of the Entity from the Idea”, Lautman specifies: We do not understand by Ideas the models whose mathematical entities would merely be copies, but in the true Platonic sense of the term, the structural schemas according to which the effective theories are organized. This distinction between dialectic and mathematics leads us to a more precise analysis of the nature of the ‘governing’ (domination) relation that exists between dialectical Ideas on the one hand, and mathematical theories on the other. The most habitual sense of a governing relation between abstract Ideas and their concrete realization is the cosmological sense, and a cosmological interpretation of such a relation is based almost entirely on a theory of creation. The existence of a matter that is the receptacle of the Ideas is not implied by the knowledge of the Ideas. It is a sensible fact, known by some 34

Ibid. p. 190, 191, 192 and 193. Ibid. p. 195–219. 36 Ibid. p. 197. 35

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bastard reasoning, as Plato said, or by a kind of natural revelation, as Malebranche thought. The Ideas are then like the laws according to which this matter is organized to constitute a World, but it isn’t necessary that this World exists to realize in a concrete way the perfection of the Ideas.37

He concludes Chap. 2, “The Analytic Theory of Numbers”, with an essential point for the philosophy of mathematics: It is then in accordance with the general theory of geneses to also define the passage from the structure of the domain to the existence of representations as a genesis, since this passage from essence to existence takes place from the structure of an entity to the assertion of the existence of other entities than the one whose structure was originally in play. The result of this comparison is that no such difference separates the genesis of mathematical theories as a result of the Dialectic, from the geneses that are carried out, within mathematics, from structures to existence. One could say in Platonic terms that the participation of Ideas among themselves obey the same laws as the participation of Ideas in the supreme genus and ideal numbers; in both cases, mathematical philosophy essentially offers its services as the object to witness the eternally recommencing act of the genesis of a Universe.38

In 1939, in his correspondence with the mathematician Maurice Fréchet, Lautman further clarified his thinking: Your first question concerns the way in which particular mathematical theories, for example those I cite in my summary, seem to receive all of their meaning from the fact that they provide examples of solution to the problems that are not strictly mathematical but dialectical (as defined by Plato). I call notions the Whole, the part, the container, structure in the topological or algebraic sense, existence etc. I call Ideas the problem of the elaboration of relations between notions thus defined. So I conceive the Idea of a dialectical problem of the relations between the Whole and part as knowing if global properties can be inscribed in local properties. I even conceive of the Idea or the problem of knowing if the situational properties can be expressed as a function of structural properties, and it is to the extent that a mathematical theory provides a response to a dialectical problem definable but not solvable independently of the mathematics that the theory seems to me to participate, in Plato’s sense, in the Idea, in comparison to which it is in the same situation as the Response with respect to the Question, Existence with respect to essence. Even if, historically or psychologically, it is the existence of the response which suggests the Idea of the question (the existence of mathematical theories allow the identification of the dialectical problem to which they respond), it is in the nature of a question to be rationally and logically anterior to the response.39

A.2.2.1

A Last Appearance

Plato reappears again centrally in his last posthumously published text entitled, “Symmetry and Dissymmetry in Mathematics and Physics”.40 The original edition (1946) 37

Ibid. p. 199. Ibid. p. 219. 39 Ibid. p. 221–222. Here allusion is made to the philosophy of Martin Heidegger. On this point, see Alunni [2], pp. 65–99. 40 Ibid. pp. 227–262. 38

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is reprinted as an overture the biographical notice written by Suzanne Lautman, also a qualified philosophy teacher, for the Annuaire des alumni de l’École Normale Supérieure (Obituaries edition published in 1945). Lautman confronts problems of symmetry (Pasteur, Curie), modern mathematical physics (theory of Spinors), general relativity (Einstein, Cartan), non-commutativity, involution, as well as questions about the theory of differential equations and topology. In Chap. 1, “Physical Space”, Lautman writes, and this will be the last reference to Plato: In seeing the sensible thus defined by a mixture of symmetry and dissymmetry, of identity and difference, it is impossible not to recall Plato’s Timaeus. The existence of bodies is based there on the existence of this receptacle that Plato calls the place and whose function consists, as Rivaud has shown in the preface to his edition of the Timaeus (Plato 1932), in making possible the multiplicity of bodies and their alternation in a single place in the sensible world, just as the role of the Idea of the Other in the intelligible world is to ensure, by its mixture with the Same, both the connection and the separation of types. This reference to Plato enables the understanding that the materials of which the universe is formed are not so much the atoms and molecules of the physical theory as these great pairs of ideal opposites such as the Same and the Other, the Symmetrical and Dissymmetrical, related to one another according to the laws of a harmonious mixture. Plato also suggests more. The properties of place and matter, according to him, are not purely sensible, they are, as Rivaud goes on to say, the geometric and physical transposition of a dialectical theory. It is also possible that the distinction between left and right, as observed in the sensible world, is only the transposition on the plane of experience of a dissymmetrical symmetry which is equally constitutive of the abstract reality of mathematics. A common participation in the same dialectical structure would thus bring to the fore an analogy between the structure of the sensible world and that of mathematics, and would allow a better understanding of how these two realities accord with one another. The development of modern mathematical physics offers in this regard an extremely suggestive lesson.41

A2.2.2 Taking Stock: Towards a Dynamic Platonism The simple statement of the Lautmanian quotations allows any reader to take stock of a theory which not only is aware of the radical and irreversible revolution affecting Platonic studies (with Becker, Stenzel and Robin), which definitively internalizes it in the philosophical and explanatory field of modern mathematics, but which, in this way, broadens the field considerably. This is the “return effect” of the Lautmanian theory on Platonic philosophy. It should be an object of study for any Platonic philosopher or any specialist somewhat interested in the dialectical question in general.42 41

Ibid. p. 231. Apart from his relevant reflections on Lautmanian dynamic Platonism, Emmanuel Barot’s enterprise is also exemplary in its rapprochement with the Hegelian dialectic. On this question, see Barot [7], pp. 111–148. As part of a parallel work on Jean Cavaillès, we will also read with great interest Mélès [34], pp.153–182. See https://hal.archives-ouvertes.fr/hal-01224100/document. 42

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For detailed and highly enlightened comments, I refer here to a careful reading of the journal Philosophiques (already cited), with articles by Jean-Pierre Marquis, Fernando Zalamea, Hourya Benis-Sinaceur, Pierre Cassou-Noguès, Brendan Larvor, David Corfield, Emmanuel Barot, Yvon Gauthier, André Lebel and Mathieu Bélanger, each of the authors considering the question of Lautman’s Platonism from their own angle. As a representative sample of the issue, I will take a quote from Fernando Zalamea which mathematically expands and clarifies his earlier proposals put forward as early as 2006.43 In fact, at a difficult time, the young normalien stood up against a “resignation” from philosophy at the expense of language, wished “for the philosophy of science a higher ambition”, and imagined a program for understanding the modern mathematics – non-analytical, discrete or static – open to a continuous and dynamic synthesis of knowledge (which explains his deep penchant for a non-reified Platonism, open to movement, following Natorp and Robin). It is amazing that the logic of the sheaves shows – in agreement with the philosopher – that this continuous knowledge of the truth is possible. Thus, by reading Lautman, one manages to “listen to the voice of things” (Grothendieck) and to perceive some conceptual prefigurations which, later, will be translated technically into effective mathematics. According to a General Dialectic which takes into account an authentic pendulum oscillation, an analysis/ synthesis of the concept of sheaf can be carried out along some lautmanian lines.44

To conclude, I would add to what has been said that Lautman was rigorously engaged in a process of desubstantialisation of mathematical philosophy, in particular by a displacement and a complicatio of the metaphysically founding relation form/matter.

A2.2.3 Some Examples By way of example, and among many others, still from the year 1937, Jacques Herbrand’s theorem on “fields” offers itself for Lautman as an almost pure case of solidarity between a set of formal operations defined by a system of axioms and the existence of a domain where these operations are realizable. He notes in this regard: It seems that a certain restriction still adheres to this logical schema; the genesis only takes place in effect in one sense, the operations of the domain. Now, if, between the domain and the operations definable on it, a rigorous appropriation can be established, then one can seek just as well to determine the operations from the domain as the domain from the operations […] Our intention being to show that completion internal to an entity is asserted in its creative power, this conception should perhaps logically imply two reciprocal aspects: the essence of a self-realizing form within a matter that it would create; the essence of a matter giving rise to the forms that its structure designs […] In fact, the schema of geneses that we are going 43

For a critical analysis of Zalamea’s undertaking regarding this mathematization–translating the dialectic of Lautmanian Ideas and Patterns of Structures into the idiom of category theory–see David Corfield’s article “Commentaire sur Emmanuel Barot”: Lautman, from the journal Philosophiques already cited (p. 210). 44 Fernando Zalamea, “Mixtes et passages du local au global chez Lautman: préfigurations de la théorie des faisceaux” in Philosophiques, op. cit., p. 22.

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to describe within more complicated theories abandons the too simplistic idea of concrete domains and abstract operations that would possess in themselves for example a nature of matter or a nature of form. This notion would tend, in effect, to stabilize the mathematical entities in certain immutable roles and ignore the fact that the abstract entities that arise from the structure of a more concrete domain can, in their turn, serve as a basic domain for the genesis of other entities.45

We will speak here of the result of a certain philosophical axiomatic and of its self-application by “reflection”: what Lautman affirms of mathematical logic, of the theorems of existence in the theory of algebraic functions, or of the theory of representation of groups as different domains of transcendental investigation, is “returned” by symmetry to the philosophical device itself, which, initially, is located in a position of operation–and vice versa. This is an extremely powerful operator of dialectical interaction which, starting from the domain of physics-mathematics, will induce effects in the philosophical field and activity. The most impressive place where the Lautmanian approach appears almost “prophetic”, the domain of his greatest philosophical and intuitive inspiration, the dialectical coupling on which all the promises of his mathematical philosophy are focused, seem without doubt to touch on the problem of “symmetry and dissymmetry in mathematics and physics”.46 I would add that this text could be emblematic of the difference between the Cavaillès-Lautman binomial. If from his thesis Lautman had already shown his interest in mathematical physics–in particular through his careful study of the texts of Élie Cartan (on the “generalization of the notion of space”, “absolute parallelism and unitary theory”) or those of Hermann Weyl (on “Riemannian spaces”), but also by his reading of Arthur Eddington’s work, Espace, temps, gravitation, or by that of La Structure des nouvelles théories physiques by Gustave Juvet, it was at the end of his life, before falling in battle, that he directed all his epistemological activity on these questions of physics (and this, thanks to his exceptional mathematical background). His very personal and extremely original contribution mainly concerns questions of enveloping the notions of symmetry and dissymmetry. It operates its thematic starting by an analysis of the pioneering work of Louis Pasteur on cellular asymmetry by “enantiomorphy”, “at the origin of all the structural theories of contemporary stereochemistry”. Then he goes on to the founding work in physics of Pierre Curie: The mixture of symmetry and dissymmetry becomes for him a necessary condition of physical phenomena in general […] To any physical phenomenon is tied the idea of a saturation of the symmetry, of a maximal symmetry compatible with the existence of this phenomenon and which characterizes it. A phenomenon can only exist in an environment possessing its characteristic symmetry or a lesser symmetry. Therefore, if the absence of an element of symmetry is called an element of dissymmetry, it is conceivable how Pierre Curie could write: “Certain elements of symmetry can coexist in certain phenomena, but they are not 45

Lautman [28], pp. 147–148. This is the title of one of Lautman’s very last texts [30], first printed as a separate booklet in the “Actualités scientifiques et industrielles” in 1946, before joining other contributions in the project initiated in 1942 in Le Lionnais [31].

46

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necessary. What is necessary is that certain elements of symmetry do not exist. It is the dissymmetry that creates the phenomenon”.47

Lautman links this idea of “limited symmetry” to Plato’s Timaeus, and in particular to his theory of the receptacle conceived as a “place” that is the Chora. However, it is not a question here (and this is the prodigious speculative force of Lautman) of a ceremonial reference: This reference to Plato enables the understanding that the materials of which the universe is formed are not so much the atoms and molecules of the physical theory as these great pairs of ideal opposites such as the Same and the Other, the Symmetrical and Dissymmetrical, related to one another according to the laws of a harmonious mixture. Plato also suggests more. The properties of place and matter, according to him, are not purely sensible they are […] the geometric and physical transposition of a dialectical theory. It is also possible that the distinction between left and right, as observed in the sensible world, is only the transposition on the plane of experience of a dissymmetrical symmetry which is equally constitutive of the abstract reality of mathematics.48

A.2.2.4

Lautman and the Virtual Power of Schemes

Here, at the height of his force of conviction, it is as if Lautman made the gesture of his thought flash by pivoting at the very heart of the domain or field, both thematic and operational, of mathematical physics. It is here that it is illustrated as in the outline of a pure diagram of thought, where the virtual power of its schemes would come to be visualized in the eye of the mind of both mathematician and philosopher. Thought accords itself something like its own “scopic” perception of itself. It is at the very moment when he concretely invests the physico-mathematical domain that the omnipotence of his dialectical device is marked. Philosophy comes here to reveal its habitation of/in Science, thus revealing their double power of reciprocal suggestion: for here, Science thinks (and thinks itself ), as if inhabited by its philosophical specters. It is in such a moment of suspense, in this “between-two Worlds” that the factitious objections raised against a supposed “arbitrariness” of his Dialectic come to cancel themselves. It is as if the dialectical operation suddenly revealed itself, suddenly made itself readable to the eye of theory by its own “rise towards the absolute”, thus providing it with a kind of “universal covering surface”. This feeling only becomes sharper as the Lautmanian argument unfolds. By a last exemplifying course, and in the form of insistence, Lautman releases the mathematical center of the whole device: the operation of involution posed as “universal” operator and core of any dual structure (or principle of “duality”)–a notion which is approached in an identical way by Weyl in the case of the transformation called automorphism for a “zero-dimensional” space (i.e. reduced to the structure of the point)49 : 47

Ibid. p. 230. Ibid. p. 231. 49 On the treatment of the “point”, see above. 48

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It is necessary to emphasize the fashion in which the distinction between left and right in the sensible world can symbolize the non-commutativity of certain operations of abstract algebra. The fundamental property of symmetry with respect to a plane, applied once, gives a figure distinct in its orientation from the original figure, and repeated a second time, gives the original figure again. It is for this reason that symmetry is said to be an involutive operation. Let us now consider an algebraic operation concerned with two quantities X and Y, and which can be written (XY), the parentheses denote an ordinary product, or any other operation defined on the two variables. It is a non-commutative operation if (XY) = (YX) and the most fruitful non-commutativity in mathematics is that in which (XY) = –(YX). The operation (XY) is dissymmetrical, in X and Y, but it is easily verified that it defines an involutive operation, as does ordinary symmetry. The expressions (XY) and (YX) are said to be antisymmetric, and this word reflects well the mixture of symmetry and dissymmetry which is thus installed deeply in the heart of modern algebra. The whole theory of continuous Lie groups is based on the non-commutativity of the product of two infinitesimal operations of the group. This theory, which is closely associated with the theory of Pfaffian forms, expressions with antisymmetric multiplication, allowed Cartan to discover a profound analogy between the generalized Riemann spaces which play a part in the physic– geometrical theories of relativity and the space of Lie groups.50

These are very fine considerations which imply a deep knowledge of the structuring operations of algebras and which at the same time refer to an anchoring in the sensible world that has been well received by the most significant representatives of the Lautmanian legacy: Charles Alunni and Fernando Zalamea.

A.3 Dynamic Platonism The Platonism of Alunni and Zalamea is more properly a “Dynamic Platonism51 ”. This locution, which is certainly confusing when evaluated by traditional criteria, actually summarize the main cornerstones of a diagrammatic conception of the eidos (§ 5.1.5 of AAI). First, because Dynamic Platonism (henceforth, DP) aims to mitigate the main typifying tendencies of the Ontological Platonism (§ 6.3.9 of AAI): for DP, the mathematical realm “is not […] fixed to a […] supposedly transcendent world of Ideas52 ” (or abstract types), but rather, it consists of a “transitory ontology53 ” in Badiou’s sense54 : it is a web of “[…] processes55 ” (or theorization efforts) that work the eidos from within.

50

Ibid. p. 233. Zalamea (2009) [47], p. 279. 52 Ibid. 53 Ibid. p. 277. 54 Cf. at least Badiou (1998) [5]. 55 Ibid. p. 272. 51

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Second, because this web–especially in Alunni’s interpretation of it56 –is in close continuity with the gestural activity (or “raison graphique57 ”) of the working mathematician: the aforementioned web, that is, includes the diagrammatic practices involved in the constitution of more or less extensive fragments of eidos (§ 5.1.4 of AAI). In fact, as Alunni himself rightly observed, there is a “dimension constructive du platonisme dynamique58 ”. This dimension is in some ways reminiscent of the mechanical constructability of 16th and 17th-century mathematical philosophies (§ 5.1.5 of AAI). That is, even DP situates the constructability of curves and figures in the hand’s movements (or gestures) shaping them (§ 5.1.5). On the other hand, however, between the mechanical constructability proper and the constructability of DP there is a fundamental difference. While the former gradually detaches itself from the diagrammatic gestures by resulting in the genesis of cognitive types (§ 5.1.5 of AAI), the latter, instead, does not go beyond those gestures; rather, it settles there in the form of their virtualities. Consider, for instance, Châtelet’s triangle. For a philosopher à la Barrow or à la Clavius (§ 5.1.5 of AAI), this triangle would be but one of many triangles that have been, are being or will be drawn throughout the course of human history. As such, therefore, it would add to the already dense array of triangles over which the universal triangle “towers” by abstraction (§ 5.1.5 of AAI). For a dynamic Platonist à la Alunni (and, presumably, à la Zalamea), things are somewhat different. For them, Châtelet’s triangle is not just one of many triangles that have been, are or will be drawn in human history. It is also and especially the transitory point of convergence between triangles that have been already drawn and triangles that will be drawn: a figure that exists–or that emerges–only insofar as it is a reprise of previous figures and a relaunching of future figures. The image discussed in section § 5 of AII, and presented here again (Fig. A.1), makes this very clear. The triangle depicted therein is ontologically inseparable from its sketches: it is not resolved in the gesture that traces it, but it is prolonged in the gestures evoked by the dotted triangles, which, by the mere fact of being outlined, disclose that invisible reservoir of virtualities in which the carnal eidos of triangle lies (§ 5.1.4 of AII).

A.4 On the Double Status of the Virtual: Reprise and Relaunch It is in this sense that it emerges such as a reprise of previous figures and a relaunch of future figures. The virtualities unfolded by the aforementioned gestures have in fact a double status. They can appear: (a) as residues, or prefigures, of already drawn forms; (b) as germs, or embryos, of drawable forms. In the former case, it is as if the 56

Personal communication of the author. Châtelet [11], p. 75. 58 Alunni [3], p. 195. 57

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Fig. A.1 “Châtelet’s triangle and its sketches”. The picture is taken from [15]

Fig. A.2 “The dashed triangle is reprised by the triangle I draw hic et nunc”. The picture is a scheme by Francesco La Mantia

dashed triangle was available to be put into circulation (or reprised) by the triangle I draw hic et nunc (see Fig. A.2). In the latter, it is as if the dashed triangle emerges from the vibrations of the triangle I draw hic et nunc (Fig. A.3). Therefore, insofar as Châtelet’s triangle cannot be detached from its sketches, it subsists as a reprise/relaunch of virtualities (see Fig. A.4).

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Fig. A.3 “The dashed triangle emerges from the vibrations of the triangle I draw hic et nunc”. The picture is a scheme by Francesco La Mantia

Fig. A.4 “Châtelet’s triangle such as reprise/relaunch of virtualities”. The picture is a scheme by Francesco La Mantia

This is why it is the “provisional [and visible] cutting-out59 ” of an “indeterminate [and invisible] multiplicity60 ” of drawn or to be drawn triangles. As such, hence, Châtelet’s triangle is the exact opposite of triangles falling under the domain of the mechanical constructability. In this framework, every form is reduced to the gesture that draws it. Geometric objects then appear without thickness, namely without that invisible reservoir of germs and prefigures which extend their diagrammatic life either here or beyond their local conditions of production.

59 60

Ibid. Ibid.

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A.5 Differences Between Mechanical Constructivism and Dynamical Platonic Constructivism Therefore, precisely because it is reduced to such conditions, and precisely because these latter freeze the form in the instant of its own production, the triangle of mechanical constructivists is nothing but an inert object, one of the many rigid figures from which the work of typifying abstraction begins. Conversely, because it is made of reprises and relaunches, Châtelet’s triangle will instead be here and beyond those conditions. That is, it will emancipate itself from the inertia of the mechanical constructability by two kinds virtualities. We say “two kinds” because the virtualities of such a triangle is made up not only of germs but also of prefigures. For as should be clear, in fact, if the virtualities of the triangle (and gestures shaping it) lie not only in the triangles (and gestures) yet to come, but also in the triangles (and gestures) that have already been, then the virtual is not only “a task to be solved” (i.e. a germ), but also “a flap of pure past61 ” (i.e. a prefigure or a residue).

A.6 Deleuzian Philosophy of the Virtual and Dynamical Platonism This duality is a peculiar feature of the Deleuzian philosophy of the virtual.62 Yet, when applied to the Châtelet’s triangle, it admirably clarifies the gap between the mechanical constructability and Platonic-dynamic constructability of the eidos. In particular, it shows how this gap resides mainly in one trait, that of “mobility63 ”, of which the first form of constructability is totally lacking. Precisely because the mechanical constructionist “sees” in Châtelet’s triangle only one of the many inert triangles from which the work of typifying abstraction begins (§§ 3. & 4.), the eidos of the triangle then appears to him as entirely rigid and static. On the other hand, if he were to see in that same triangle the visible cut-out of an invisible multiplicity of virtualities, that same eidos would then appear to him as dynamic, as something that is in the process of becoming and that owes its processuality to the games of reprises and relaunches that make each triangle the recipe of prior (or residual) triangles and a reservoir of future (or embryonic) triangles. Summing up, DP is diagrammatic insofar as it dilutes the supposed fixity of abstract and cognitive types in the becoming of objects that incorporate “movement and transit64 ”, and that, in this sense, are at one with the diagrammatic gestures 61

Alunni (1968) [16], p. 137. I dealt with two Deleuzian meanings of “virtual” in La Mantia [25], pp. 190–198. 63 De Freitas and Sinclair [15], p. 205. 64 Zalamea (2009, 2019), p. 279. 62

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shaping them (§§ 5.1.1, 5.1.3 & 5.1.4 of AAI). It would remain to be understood then in what sense DP is still a form of Platonism.

A.7 Conclusions: Towards a Dynamical Platonism From a certain point of view, one might suspect that DP is not a form of mathematical Platonism at all. Indeed, one might wonder what would remain of this prestigious paradigm if, proceeding in the manner of DP, one were to eliminate those attributes of a-temporality and mind-independence that classically characterize mathematical entities. Admittedly, it was noted in the preceding paragraphs that these attributes are encoded within a historically determined form of Platonism, the so-called Ontological Platonism (henceforth, OP). It was further argued that OP does not exhaust the varieties of mathematical Platonism currently circulating in the literature. On the basis of these observations, DP could then be assimilated to a non-ontological form of Platonism. While correct, this assimilation is, however, too general. The historian of the philosophy of mathematics knows that there is a plurality of non-ontological Platonisms. If one were to accept to present DP as a particular type of non-ontological Platonism–which indeed it is–the identification problem DP poses to the historian of the philosophy of mathematics would arise again. Assuming this, the question would be to which type of non-ontological Platonism DP belongs. According to a well-known partition by Michael Resnik, besides OP, there would exist a Methodological Platonism (MP), which advocates the use of non-constructive mathematical methods,65 and an Epistemological Platonism (EP), for which “knowledge of mathematical objects is at least in part based upon a direct acquaintance with them, which is analogous to our perception of physical objects66 ”. Now, DP is not analogous to either MP or EP.

A.7.1 Dissimilarities Between DP and MP, and Between DP and EP The first dissimilarity is explained because of MP’s implicit metaphysical background. The use of nonconstructive methods, such as “the excluded third [or] impredicative definitions67 ”, implies work on infinite and mind-independent domains of

65

Cf. Resnik [39], p. 162. Ibid. 67 Ibid. 66

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abstract entities.68 DP rejects these domains as the products of caricatured and improper versions of mathematical Platonism (§ 2.). The second dissimilarity depends on an underlying gnoseological difference. Whereas for EP, it is possible to access mathematical ideals directly, for DP this access is mediated and “cannot be fixed in advance69 ”. Exploiting the partial analogy with the perception of physical objects, EP postulates an apprehension of mathematical idealities that shares the same requirements of immediacy with the observation of the physical world. Aware of the “relative correlations between concrete phenomena and their theorizations70 ”, DP rejects this analogy for two reasons. First, because the idea of direct apprehension is, both mathematically and empirically, a paleo-positivist myth disqualified by the most solid acquisitions of contemporary epistemological thought (including the “theory-laden” character of all observation). Second, because EP’s partial analogy presupposes an ontology of static and atemporal entities, which DP rejects on the basis of a transitory and dynamic ontology (§ 2). Whether one accepts it or not, the direct apprehension of idealities can work only on “absolute backgrounds71 ” of entities. It is only by presupposing universes of objects fixed once and for all, and being in this sense “absolute”, that EP can propose a direct apprehension of mathematical idealities: universes so conceived are hotbeds of “beings72 ”, that is, matrices of stable formal entities that can be distinctly grasped. If these universes, however, are set into motion, and therefore considered with respect to the historical work of the theories that generated them, as well as with respect to future theorizing that will modify them, then those objects will appear as “quasi-objects73 ” in Badiou’s sense74 : they are processes that tend toward being or stability without ever fully achieving them. It is in this sense that for DP, access to mathematical idealities can neither be direct nor established in advance, but only mediated by the historically situated and mutagenic work of theories. It therefore follows that even EP does not offer an adequate framework within which to place DP.

68

Cf. ibid. Zalamea [47], p. 270. 70 Ibid. p. 276. 71 Ibid. p. 274. 72 Ibid. p. 275. 73 Ibid. p. 278. 74 Cf. Badiou [5], p. 47. 69

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A.7.2 Dynamical Platonism as a Production of Partial Invariants: The Legacy of Alunni and Zalamea To understand in which sense DP is a form of Platonism, we need to turn elsewhere, to philosophical-mathematical reflections that draw upon the texts of the great phenomenological tradition: Husserl and Merleau-Ponty, primarily. To Husserl in particular we owe the formulation of a viewpoint that is highly compatible with DP. We are thinking of the judgement that placing mathematical idealities “in an ouranios topos or in a divine spirit [would amount to] an […] absurd metaphysical hypostatization […]75 ”. As can be seen, in what is reported, the German philosopher also keeps away from the fetters of OP. Nevertheless, he remains a Platonist–“moderate76 ”, according to some “intelligent77 ”, according to others–when he attributes to the aforementioned idealities a degree of invariance that distinguishes them from the “real objects78 ” of the concrete world. Despite being “without ouranios topos79 ”, mathematical ideals have for Husserl the singular property of being preserved in time: of being “omnitemporal” (allzeitlich). It is by adhering to a variant of this “omnitemporality” (Allzeitlichkeit) that DP remains within the bounds of Platonism. For Zalamea and Alunni, and even earlier for Lautman, mathematical idealities possess a coefficient of temporal invariance that is always local, i.e. relative to historically determined formal contexts. With greater care than Husserl, DP thus notes that the omnitemporality of mathematical entities is contextually and historically limited. An example that illustrates this limitation is the theorem on the sum of the interior angles of a Euclidean triangle. Ever since the theorem was enunciated, presumably several millennia ago, it has been proved that this sum is equal to 180°. This result has remained unchanged over time, being passed down to the present day from one generation of mathematicians to the next. As the result of a historical activity perpetuated over time, the theorem would therefore produce an omnitemporality in Husserl’s sense. However, this would be a limited omnitemporality since the temporal invariance of the result only applies within the confines of Euclidean geometry. Outside of this geometry, this invariance dissolves, giving way to other invariances that are produced and grow within the limits of other historically situated geometries. This is why Zalamea can assert that “mathematics–understood as the study of the exact transits of knowledge (“dynamics”)–constructs partial invariants (“static ideality”) in order to […] (incessantly mediate) between dynamics and statics80 ”.

75

Husserl [23], p. 369. Benoist [9], p. 149. 77 Benoist [8], p. 36. 78 Husserl [23], p. 369. 79 Benoist [9], p. 147. 80 Zalamea [47], pp. 344–345. 76

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Thus, if the production of invariants (or “static idealities81 ”) makes DP a Platonist, the partiality of these invariants, their sensitivity to the history of formal contexts and the plasticity of the gestures that those contexts mobilize (their “dynamics”), is entirely consistent with a transitory ontology à la Badiou (§ 2.) and with DP’s proposal to substitute this ontology for the static and timeless one of OP. Hence, in conclusion, the portrait of a form of Platonism yet to be appreciated, studied and understood.

81

Ibid. p. 344.

Appendix B

The Illusions of the Speaking Subject. A Note on the Lapse Francesco La Mantia

B.1 The Lapse: A Semiotic Perspective Jacques Fontanille rejected any reading of the lapse as a “segment de discours qui ne serait pas intentionnel” ([18]: 31). Irène Fenoglio, who, according to some (cf. [36]: 52), would have adhered to the view condemned by Fontanille, actually summarized the meaning of this critique. According to the author, both the man in the street and the language theorist would tend to see “trajets d’énoncés” as “projets d’énoncés” ([20]: 178). Both would cultivate the illusion of planned enunciation down to its most sophisticated and minute articulations. It might seem that the writer succumbed to the seductions of this illusion. In the main text, the slip has been equated with the product of “non-intentional saying”. One might suppose that this assimilation rests upon the same background of theories and insights that would corroborate the exchange highlighted by Fenoglio. But the assumption, however justified, is erroneous. One can recognize a linguistic role for intention without turning enunciation into a project. Fontanille is not of this opinion: for the Parisian semiotician, using intention as a criterion for demarcating between slips and discourse would entail the risk of identifying enunciation with a program: with a project, to be precise.

B.2 From Jacques Fontanille to Jacques Coursil While agreeing with the scholar’s reasons, and therefore distinguishing enunciative practice from the execution of a project, it is believed that the rejection of the intentional criterion entails dangers far more serious than those it intends to heal. For Fontanille, but especially for Jacques Coursil [13, 14], this identification would endow the locutor with predictive powers he or she does not have. If “tout était peu

© The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023 F. La Mantia et al. (eds.), Diagrams and Gestures, Lecture Notes in Morphogenesis, https://doi.org/10.1007/978-3-031-29111-1

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ou prou réglé d’avance” ([18]: 31), the speaker should be able to minutely predict the development of his or her utterances–in short, be able to anticipate their global form before uttering them. But this is not the case: speech, even the most fluent speech, is always exposed to the possibility of hesitations, abrupt interruptions or lapses that belie this ability and contradict the idea of enunciation as a planned activity. On the other hand, the ouster of intention from enunciation risks losing sight of a capacity that the speaker actually possesses: the ability to pick up in the exercise of speech that which deviates from his own intentions. The self-correction of slips of the tongue is activated from this form of enunciative sensitivity. It is precisely because the locutor senses a gap between saying and intending (or vouloir dire) that he or she has the ability to recover (se reprendre) and self-correct. Fontanille, however, would object to even this. Instead of recognizing in self-correction a feedback of intention, the author sees in the locutor’s reflexive gesture the indicator of a “commutation […] sans avenir” ([18]: 32). Going back on one’s own words, or a simple self-arrest, would signal expressive and content modification–a commutation–that does not have the force to propagate itself in the enunciation and hence is without future. To the parameter of intention Fontanille would therefore substitute the parameter of relevance (pertinence). Self-corrections, rather than attesting to the gap between the intentional and the nonintentional, would show a gap of another kind: a hiatus between complete and incomplete relevancies, that is, the distance between commutations capable of securing stability in discourse and commutations incapable of doing so. From the point of view assumed here, however, Fontanille’s move is a level shift that only apparently modifies the terms at play: one would be forgiven for wondering what it is in discourse that ensures strength and stability to a commutation. The answer provided by the semiotician is singular because it reintroduces what was initially rejected. Drawing on a distinction from his own metalinguistic repertoire, Fontanille observes that a commutation is incomplete to the extent that “le corps-chair, le Moi qui s’exprime […] manque le soutien du Soi […]” ([18]: 32). A slip would be without avenir because the instance that initiates enunciation on the sense-motor plane (the Me-Flesh) would not gain the assent of the instance that validates what is enunciated (the Self ).

B.3 Still on Jacques Coursil’s Perspective This duality, also known by the binomial of “speaker/locutor” (Ducrot) or of “locutor/ enunciator” (Culioli), recovers the parameter of intention insofar as the taking charge of what is enunciated, in order to function, demands a certain degree of conformity between what is said and what one would like to say. Barring lies, one is accountable for what is said only if that saying conforms to what one wants/intends to say. For that matter, it is Fontanille who grants the aforementioned conformity an albeit minimal degree of trustworthiness. Discussing the means available to the locutor to signal an aborted commutation in the bud, the scholar mentions the ritual locution “ce que je voulais dire”. Now, “ce que je voulais dire” is the form by which the Self

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distances itself from the Me-flesh, not only to mark an aborted commutation, but also to emphasize a dissimilarity between saying and intending (vouloir dire). Hence the rehabilitation of intention as the criterion of demarcation and as the suturing force of discourse. That this rehabilitation does not transform enunciation into a project, however, is Coursil’s explanation: although saying always accompanies an intention to say (vouloir dire), this intention is never able to predetermine the syntactic and lexical form of what will actually be said. Not being in this sense premeditated (cf. [15]: 45), enunciation will be irreducible to the execution of a project. Of course, intention to say could also be an illusion, indeed, it could be only an illusion. But this possibility, credited by Coursil and the French school of discourse analysis, does not invalidate the role of intention. Paradoxically, it reinforces it.

B.4 French Discourse Analysis: From Michel Pêcheux to Jacqueline Authier-Revuz From Michel Pêcheux to Jacqueline Authier-Revuz, there has always been a denial of any authenticity to this wanting. First, because the choices the locutor makes lexically, syntactically, and pragmatically are unconsciously underdetermined by complex socio-linguistic constraints called discursive formations [19, 37]. One only has to compare this text with a philosophical writing of the seventeenth century, or of even more distant epochs, to discern profound enunciative differences only superficially ascribable to the individual’s will to say. It is the discursive formations, not the speaker, that sketch the historically variable boundaries of what is said and how it can be said. Second, because the wanting to say, unbeknownst to the speaker, is traversed by enunciative materialities already produced “ailleurs, avant et indépendemment” ([4]: 395; [37]: 227). These materialities operate in the speaker as an interdiscourse autonomous from consciousness but capable of subjecting it to the sedimentations of history, memory, and the “déja-dit”. The most radical evidence of this subjection is its lack of evidence: the work of interdiscourse is so external to consciousness that consciousness does not even for a moment come to contemplate its possibility. The speaker subjects himself or herself to the interdiscourse by not recognizing this subjection, that is, by experiencing it in the form of its mirror opposite: “under the form of autonomy” ([37]: 227). Hence, the illusoriness of wanting to say which is echoed by the Lacanian adage “The speaker does not know what he is saying”. On the other hand, it has been observed that the idea “d’un sujet énonciateur porteur de choix, intentions, décisions” ([37]: 169) plays a role that is as fictitious as it is indispensable to the balances of interlocution.

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B.5 Conclusions: A Necessary Illusion The illusion of wanting to say would in this sense be a “necessary illusion” ([37]: 169). It is the recognition of this necessity that is missing from Fontanille’s observations. The Parisian semiotician can oust intention from utterances because he does not take into account the speakers’ firm belief in the autonomy of their actions. The author could justify this choice by observing that it would be a serious mistake to use subjective belief as a category of analysis. However, while accepting this criticism, it could be observed that an error no less serious would be the exclusion of the same belief from the analysis. This is why, while taking the utmost account of Fontanille’s cautions, it was felt here that we should read the slip not so much through the eyes of the linguist as through the illusions of the speaker. This essay is the result of long and dense conversations between the two authors. It can therefore be considered the product of common reflections. In any case, Charles Alunni is the author of the paragraphs 2, 2.1, 2.1.1, 2.1.2, 2.1.3, 2.2, 2.2.1; 2.3, 2.3.1, 2.3.2; Francesco La Mantia is the author of the paragraphs 1, 3, 4, 5, 6. Special thanks go to Fernando Zalamea who, as always, has been an attentive and valuable reader.

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