The Double Twist: From Ethnography to Morphodynamics 9781442681125

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THE DOUBLE TWIST fROM eTHNOGRAPHY TO mORPHODYNAMICS

ANTHROPOLOGICAL HORIZONS Editor: Michael Lambek, University of Toronto This series, begun in 1991, focuses on theoretically informed ethnographic works addressing issues of mind and body, knowledge and power, equality and inequality, the individual and the collective. Interdisciplinary in its perspective, the series makes a unique contribution in several other academic disciplines: women's studies, history, philosophy, psychology, political science, and sociology. For a list of the books published in the series, see page 317.

The Double Twist From Ethnography to Morphodynamics

Edited by PIERRE MARANDA

U N I V E R S I T Y OF TORONTO PRESS Toronto Buffalo London

www.utppublishing.com University of Toronto Press Incorporated 2001 Toronto Buffalo London Printed in Canada ISBN 0-8020-3524-8 (cloth)

Printed on acid-free paper

National Library of Canada Cataloguing in Publication Data Main entry under title: The double twist: from ethnography to morphodynamics (Anthropological horizons) ISBN 0-8020-3524-8 1. Levi-Strauss, Claude. 2. Structural anthropology. 3. Mythology. I. Maranda, Pierre, 1930- . II. Series. GN362.D68 2001

306

C2001-930596-6

Every effort has been made to obtain permission to reproduce the illustrations that appear in this book. Any errors or omissions brought to our attention will be corrected in future printings. This book has been published with the help of a grant from the Humanities and Social Sciences Federation of Canada, using funds provided by the Social Sciences and Humanities Research Council of Canada. University of Toronto Press acknowledges the financial assistance to its publishing program of the Canada Council for the Arts and the Ontario Arts Council. University of Toronto Press acknowledges the financial support for its publishing activities of the Government of Canada through the Book Publishing Industry Development Program (BPIDP).

Contents

Contributors

vii

Acknowledgments

ix

Introduction 3 Pierre Maranda Part One: Ethnography and the Canonical Formula 1 Hourglass Configurations 15 Claude Levi-Strauss 2 Analogy and the Canonical Formula of Mythic Transformations Luc Racine 3 Is the Canonic Formula Useful in Cultural Description? Eric Schwimmer

33

56

4 Mapping Cultural Transformation through the Canonical Formula: The Pagan versus Christian Ontological Status of Women among the Lau People of Malaita, Solomon Islands 97 Pierre Maranda

vi

Contents

Part Two: Analyses, Tests, and Contextualizations of the Canonical Formula 5 Hesiod, the Three Functions, and the Canonical Formula of Myth 123 Lucien Scubla 6 Classical Myths and Transformation: Computer Observation of the Levi-Strauss Formula at Work 156 Sdndor Dardnyi 7 Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 177 Christopher A. Gregory Part Three: The Logico-Mathematical Status of the Canonical Formula 8 The Set of Canonical Transformations Implied in the Canonical Formula for the Analysis of Myth 199 Alain Cote 9 On Some Philosophical Dynamic and Connectionist Implications of the Canonical Formula of Myth Seen as Space Categorization 222 Andrew William Quinn 10 A Morphodynamical Schematization of the Canonical Formula for Myths 267 Jean Petitot Conclusion

313

Contributors

Alain Cote, Researcher, Cap-de-la-Madeleine, Quebec Sandor Daranyi, Science and Technology Attache, Embassy of the Hungarian Republic, Helsinki, Finland C.A. Gregory, School of Archaeology and Anthropology, Faculty of Arts, Australian National University, Canberra, Australia Claude Levi-Strauss, Academie franchise and Professeur honoraire, College de France, Paris Pierre Maranda, Department d'Anthropologie, Universite Laval, Quebec Jean Petitot, Ecole polytechnique and Ecole des Hautes Etudes en sciences sociales, Paris Andrew William Quinn, Department de Philosophic, College de Limoilou; Faculte de Philosophic, Universite Laval, Quebec; Chercheur associe, Centre d'Etudes en Lettres, Arts et Traditions (CELAT) Luc Racine, Departement d'Anthropologie, Universite de Montreal Eric Schwimmer, Departement d'Anthropologie, Universite Laval, Quebec Lucien Scubla, Centre de Recherche en Epistemologie Appliquee, Ecole Polytechnique, Paris

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Acknowledgme

I address thanks to Claude Levi-Strauss, Sandor Daranyi, Solomon Marcus, and Lucien Scubla, who made judicious comments on an earlier draft of my introduction, on which I draw more extensively in the conclusion. I also wish to thank the Faculte des Sciences Sociales and the Departement d'Anthropologie of I'Universite Laval for their financial support in the preparation of this manuscript, and the Aid to Scholarly Publications Programme of the Humanities and Social Sciences Federation of Canada for a grant that made possible the publication of this book. And thanks also to Ms Edith Grondin, who prepared the final version of the manuscript, and to Virgil Duff and Chris Bucci of the University of Toronto Press, for their helpful and friendly cooperation.

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THE DOUBLE TWIST

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Introduction Pierre Maranda

In 1955, Claude Levi-Strauss proposed a mathematical-looking 'Canonical Formula' (hereafter CF) for the structural analysis of myth (LeviStrauss 1955: chap. 11). It purported to investigate the dynamics of a basic semiotic structure: what we would now call the 'generative engine' of myth. This book focuses on, and offers tests and developments of, Levi-Strauss's pioneering endeavour. While some scholars consider the CF 'a meaningless abracadabra,' others have acclaimed it 'an intelligent formula' (see the first sections of chapters 5 and 10 for sources). For the mathematician and semiotician Solomon Marcus (1997), 'Levi-Strauss' explicit mathematics is poor, but the mathematical potential of his ideas is tremendous.' Readers interested in the history of the CF - applications and criticisms as well - should refer to Lucien Scubla's Lire Levi-Strauss (1998), a book that offers a full and diversified coverage of that topic. Our collection of essays complements in a substantial way the one published in French as a special section of an issue of L'Homme (Desveaux and Pouillon 1995), to which four of the ten authors of the chapters in this book contributed, namely, Alain Cote, Jean Petitot, Luc Racine, and Lucien Scubla. Here, they carry further and develop in much greater depth their dealings with the CF. One need not restate, after so many scholars, the importance of LeviStrauss's work. Its impact on modern thought remains foremost. Leaving aside his ground-breaking Elementary Structures of Kinship (1949/ 1964), suffice it to mention that the structure of his prodigious fourvolume Mythologiques (1964-71) initiated a 'hypertext' mode for the

4 Pierre Maranda

analysis of myth. His conceptual apparatus, and the method to implement it, have radically renewed the study of mental representations and anticipated implicitly many basic developments in the neurosciences. As for his CF, it has remained a guideline throughout his whole work. The CF looks somewhat cryptic to most scholars - mathematicians, social scientists, and humanists alike. It reads (Levi-Strauss 1955: chap. 11)

The chapters of this book explicate the CF and show it at work through detailed applications (Part One), through analyses and tests (Part Two), or through logico-mathematical analysis (Part Three). For a summary, let us say here that the formula deploys a mediation mechanism through an operator, the term 'b/ repeated in the second and third components of the formula. This mediation aims at overcoming oppositions given as definitely antithetic at the outset. The 'functions' f x and f y represent the irreconcilable oppositions. But those abstract vectors need actualization, hence the 'terms' a and b, which give them a problem-specific embodiment - for instance, knowledge vs ignorance, barrenness vs fertility, and so on. Such oppositions set a cognitive problem in binary terms. But the term b, polysemic, can subsume both 'functions.' With b the problem-solving equation switches from the binary to the ternary mode. And b's 'mediation' generates the 'double twist/ a double permutation, since (1) a undergoes an inversion (a 1) and at the same time loses its 'term' status to end up as the 'function' that specifies y which (2) also changes status and becomes actualized as a 'term'; thus, once inverted, a becomes the 'function' of y's supremacy. Accordingly and for example, the inversion of 'ignorance' works out as the determinant and consolidating condition of acquired 'knowledge.' Uses of the CF generally fall in two broad and different categories. Levi-Strauss has conceived and applied it originally to set out transformation relationships between variants of myths (in this volume his paper focuses on transformation relationships between architectural structures in connection with mythology). On the other hand, Maranda, Petitot, Mezzadri, and others have applied the CF to single cases in order to map out their 'generative formulas.' Thus, the CF can serve on both 'macroscopic' (comparative) and 'microscopic' (monographic) levels, and its fields of application vary accordingly. They also vary accord-

Introduction 5 ing to a vast range of disciplines in the humanities: philosophy, literature, the analysis of rituals, and so forth (see the bibliography in Scubla 1998). And the CF also impacts mathematics (cf. Marcus 1997), especially in connection with morphogenesis (Petitot, this volume). The ten essays below represent three different approaches and thus form three parts. The ethnographically based 'Ethnography and the Canonical Formula/ papers in Part One, apply the CF to original, live field data. The authors use the CF as a 'metaphor' of sorts, much in the same way as Levi-Strauss has sometimes referred to it. Part Two, 'Analyses, Tests, and Contextualizations of the Canonical Formula/ is a transition toward the more formal and mathematical approaches of Part Three. Here, the authors apply the CF to archive materials. Much more theoretically and epistemologically oriented, the papers in Part Three, The Logico-Mathematical Status of the Canonical Formula/ unfold on a high conceptual level and contribute to the elucidation of the internal mechanism of the formula. Part One begins with a chapter by Levi-Strauss himself that takes up comparative architectural data, the dynamic structure of which he maps onto his formula. Chapters 2,3, and 4 bear mainly and monographically on Melanesian cultures, with the addition of a case from the Andes in chapter 2. In Part Two chapters 5, 6, and 7 focus on classical myths and, because of their more formal orientations, lead to Part Three. The latter consists of three different types of analyses of the structure of the CF itself: they narrow down on its very nature. Several contributors pay considerable attention to the use of the CF by Levi-Strauss in his book The Jealous Potter (1985). Their diverse tacks provide multidimensional appraisals of the way the creator of the CF has handled it. Chapter 1, 'Hourglass Configurations.' After having used the CF in diverse writings and having stated that it had constantly guided his approach to the analysis of myths, Claude Levi-Strauss applies it this time to indigenous architecture. Starting with the 'X' structure of Ise Shrine in Japan, he sets it in regard to the 'hourglass'-shaped axial Mountain (Mount Meru / Sumeru) in India. There follows a series of other similar architectural structures that Levi-Strauss documents for Siberia, America, Indonesia, and Fiji, with references to K'un Lun (the Chinese cosmic mountain). Levi-Strauss takes into account the constraints imposed by the building materials and shows how transformations relate to them. And he raises the question, 'How does one explain these often striking resemblances?' He answers it by a query: 'Should

6 Pierre Maranda we search no further than the irrepressible tendency of the human mind to pour the profusion of its available vacant significations into all the forms it perceives or conceives, by reason of the permanent gap between its impatience to understand everything and its limited knowledge?' This cognitive dynamics would find support in the fact that the CF would generate not only myths but buildings as well. LeviStrauss concludes this first chapter with the following words: 'In this way one can simultaneously apprehend the formula as a conceptual relation and see it, here and there in the world, stamped in a geometry that human endeavour has envisioned.' This reference to geometry and the geometrization of the CF itself - comes up in other chapters (2, 5, 6, 8, 9, and 10). Chapter 2, 'Analogy and the Canonical Formula of Mythic Transformations. ' In this chapter Luc Racine shows first that reducing the CF to a simple analogy distorts it and lessens its modelling range considerably - a point that Scubla (chapter 5) also makes from a different angle. Racine clarifies most usefully a theoretical point, thus offering a pedagogical explanation of the CF that guides readers not familiar with it through the unfolding of its originality and of its explanatory power. Then Racine discusses a broadening of the CF as applied to nonmythic data, that is, ritual, by Mark Mosko. He comments on the latter's handling of the CF and shows that his reduction disserves the Melanesian (North Mekeo, Papua New Guinea) data Mosko models with his impoverished application. Racine gives further weight to his argumentation by using the CF to explore Andean symbolic classifications (Chipaya of Bolivia), an analytic operation that yields a most innovative interpretation of the village church. In that respect, LeviStrauss's application (chapter 1) to a corpus of architectural data exemplifies a 'macroscopic' use of the CF, whereas Racine's presents a 'microscopic' application. Chapter 3, 'Is the Canonic Formula Useful in Cultural Description?' Eric Schwimmer tackles a basic controversial issue, namely, the ability of structuralism to cope with history. Focusing on the CF, and contrary to many critics, Schwimmer does not think the formula to be 'nonhistorical.' In this author's judicious words, 'logical "operators" are never non-historical because they cannot exist in a historical void; they require a historical subject to "operate" them.' And Schwimmer continues by describing his own use of the CF to analyse his field data, Orokaiva materials from Papua New Guinea. After defining his basic concepts - mythical thought, myth as a form of oral literature, and

Introduction 7

mythic mode - Schwimmer contextualizes them in the semiotic framework of 'mutations in the semiosphere' and of the 'theory of invention/ This enables him to differentiate between two aspects of the CF, namely, its structural aspect and its capacity to model transformative events in the history of Orokaiva oral literature. Then he applies very carefully the formula to model in depth such transformative events, casting a new light on empirical data. Schwimmer concludes that the CF, better than Northrop Frye's 'mythic mode/ reaches beyond the mere analysis of myth: it can and does deal with Orokaiva identitybuilding processes and artistic invention. Chapter 4, 'Mapping Cultural Transformation through the Canonical Formula: The Pagan versus Christian Ontological Status of Women among the Lau People of Malaita, Solomon Islands/ Like the preceding one, this chapter remains in the Melanesian area. Here, the present author explores a change of gender relationship over time. Traditionally, pagan men derived their ontological status from that, more fundamental, of women. They mapped themselves onto the other gender: metaphoric women, they remained ontologically marginal, shadows of a sort to which their patrilineal social organization could not really confer 'telluric' substance except through elaborate funeral rituals mimicking their women's biological, 'real/ birth-giving power. With the advent of Christianity, men have found a way to overcome the traditional genicocentric world view, invert their relative position, and claim ontological pre-eminence. Thus, they neutralize the chtonian powerful and dangerous - semiotics of women. Two successive applications of the CF may serve to model the diachronic transformation. The first one maps the position of men as traditionally excluded from a basic relationship to the 'low/ that is, to Earth: pagan, they must remain 'virtual women/ The second application maps the androcentric transformation made possible through Christian ideology, which deprives women of their ontological power in order to give supremacy to the 'high/ that is, the celestial connection. Thus, the CF displays the deep, historical transformation of the ontological status of genders in an island of the Solomons archipelago. With Part Two, chapters 5, 6, and 7, the reader moves from contemporary ethnographic data and Melanesia to classical myths, which their authors take them in order to investigate and test different dimensions of the CF. In so doing, they offer a transition toward the third, more theoretically oriented, part of this book.

8 Pierre Maranda Chapter 5, 'Hesiod, the Three Functions, and the Canonical Formula of Myth.' Lucien Scubla - the author of an extensive historical review of the CF (Scubla 1998, part II) - starts by reviewing diverse evaluations of the CF and comes up with a balanced conclusion. On the one hand, difficulties occur in the definition of the formula's components, and in often idiosyncratic and arbitrary interpretations that fail to convince. On the other hand, positive assessments stem from the morphodynamic process to which the CF relates (cf. Quinn, chapter 9 and Petitot, chapter 10), from the major issues it raises for the future of structuralism and anthropology, from its eventual clarification of the genesis of myths, and from the further developments it makes possible with respect to the construction of social identities (on that aspect see also Schwimmer, chapter 3). After clearing up the greater part of the obscurities inherent in the literal structure of the CF, Scubla applies it to the structure of the Hesiodic myth of races, which he takes up after the Classical Greece scholar JeanPierre Vernant. He does so by emphasizing that the CF allows one to deal with the Hesiodic myth as a whole, and that it makes it possible to formulate a refutable prediction of the existence of 'anti-kings/ a point indeed corroborated by Vernant. Scubla concludes by referring to Petitot's earlier work on the CF. The latter's implicit rejoinder in this book shows that his present contribution gives the CF a deeper and fuller meaning than any other previous treatment. Chapter 6, 'Classical Myths and Transformation: Computer Observation of the Levi-Strauss Formula at Work.' Sandor Daranyi's chapter bears on texts related to Attis, a Phrygian deity from Asia Minor whose cult was imported into Greece and Rome. Daranyi's analysis demonstrates the correspondence of the geometric interpretation of the CF with the criteria for the spatial distribution of text comparisons. He makes his point by resorting to the classification of myth variants about Attis. As for statistical tools, Daranyi used three versions of hierarchical cluster analysis, block clustering, principal-component analysis, maximumlikelihood factor analysis, and Kaiser's factor analysis. Such a geometrization of the CF in connection with factor-analysis scores leads to visualizations of structures that allow for theoretical developments. Among them, let us mention the furtherance of a coherent theory of text variation congruent with the multivariate classification of texts, with the mapping of their contents, and with the extraction of paradigms from discourse universes. With respect to the use of mere statistical methods, Daranyi's cautious concluding remarks leave open the question of their bearing in the analysis of texts.

Introduction 9

Chapter 7, 'Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic/ Christopher A. Gregory sees the CF as a 'partial rehabilitation of the Ramist tradition of thought that flourished in the sixteenth and early seventeenth century/ He argues that Levi-Strauss's binary logic is Ramistic, not Boolean, and therefore pre-Cartesian. (In chapter 9, Quinn makes a similar point but from an entirely different tack.) For Gregory, the CF is 'an important and original contribution to the theory of logical similitude/ Along those lines, he contrasts commonplace logic and axiomatic logic. The former, based on commonplace opposition, works along polysemic axes. To illustrate his point, Gregory compares those two kinds of binary logic by analysing the story of the mongoose and the snake, taken from the Indian Classic The Panchatantra, as well as examples from Australian aboriginal cultures. This view, according to which the CF is 'a complicated five-tiered variation' of commonplace logic, leads to Gregory's diagram showing 'the five levels of logical analysis that the formula operates on/ And he concludes: '[I]f Levi-Strauss's work has solved the problem of "their" logic he has posed the problem of "ours." European axiomatic logic is logical but its history is not/ The chapters by Scubla, Daranyi, and Gregory bridge parts One and Three. Now, with chapters 8 to 10, we move to more theoretical - epistemological and mathematical - considerations of the CF. Cote explores the forty-eight possible expressions of the CF to narrow them down to Levi-Strauss's own applications. Quinn investigates the formula in terms of connectionist philosophy and cognitive grammars, and his contribution leads to Petitot's. The latter recasts the CF, a 'structural equation,' in the framework of Rene Thorn's morphogenetic theory as Petitot himself has developed it. Chapter 8, 'The Set of Canonical Transformations Implied in the Canonical Formula for the Analysis of Myth.' Alain Cote's analytic purpose aims at elucidating the nature of the CF. In order to do so, he defines the set of canonical expressions, that is, the class of statements to which the CF belongs. He finds that it includes 144 expressions, 48 of which can actually be kept as meeting the dynamics of the CF requirements. He then notes that 'it is much more than the three canonical expressions that explicitly appear in the works of Levi-Strauss/ which he has reviewed in his paper. He concludes that the set of 48 possible expressions may, after all, 'only point to different ways ... to describe the same set of relationships,' a set that corresponds to 'the idea of crossing a

10 Pierre Maranda boundary' (a point Levi-Strauss has indeed emphasized when comparing the myths, arts, and rituals of neighbouring societies that resort to structural inversions to better state their identities). As for the 'idea of crossing a boundary/ Cote's approach presents the indisputable merit of posing in rigorous terms the examination of the purpose and 'mechanics' of the 'double twist,' which Petitot develops and operationalizes further in chapter 10. The other consequences Cote derives from his formal analysis, especially with respect to the study of myth and to his fostering of structuralism, lead to questions that Quinn and Petitot address in different ways and echo in the subsequent chapters. Chapter 9, 'On Some Philosophical Dynamic and Connectionist Implications of the Canonical Formula of Myth Seen as Space Categorization.' Andrew Quinn's connectionist philosophy focuses on the space paradigm of cognitive grammars and reads the operation performed by the CF as 'a formalization of the natural order and a universal categorization of "space," or the universe.' He sets his analysis of the CF on a Kantian background and in the context of the Classical Cognitivist (CC) / Connectionist (CN) paradigms and situates the development of the CF in terms of the latter. This 'architecture/ both 'modal' and 'cognitive' leads him to investigate the isomorphism between the mind's topological structures and representations of the world through the fundamental problem implied in categorizations. This isomorphism would not only be generated by the mind's dynamic structures: it attains a logical expression at middle-range levels of representation, where connections between the world and perception occur. Myth thus works out as a high level of categorization. Quinn opposes the formalization by the CF to logistic conceptions. In his view, as in Petitot's (chapter 10), the CF offers a genuine mathematical formalism, a formalism that operates along the lines of three important theoretical directions: (1) the components of the geometric structures of space; (2) connectionist models in affinity with Rene Thorn's morphogenetic theory (Petitot's chapter carries further the morphodynamic approach to which Quinn refers), and (3) some postulates of cognitive grammars. The implementation of this formalism in terms of dynamic processes implies a deep and infinite variety of virtuality. One cannot specify entirely such processes, but one can explicate them using a number of notions about semio-narrative structures. This approach could offer a way out of the aporia arising in connection with the structural isomorphism of mind and the world.

Introduction 11 Chapter 10, 'A Morphodynamical Schematization of the Canonical Formula for Myths.' Whereas Cote's' approach to the CF draws on classical mathematics and logic, and while Quinn's hinges partly on morphodynamics, Jean Petitot's analysis rests on his own developments of that mathematical theory. He begins by making an important epistemological point: his discussion of the debate around the CF clarifies a basic issue, that is, the conditions under which real modelling occurs versus data distortions to fit the model - compare Scubla's evaluations of inept applications of the CF (1998, and this volume, chapter 5). Another significant contribution of Petitot's resides in the operationalization of Levi-Strauss's theory of the interplay between syntagmatic chains and paradigmatic sets. Petitot takes up the topic through his triangulation of the contributions of Vladimir Propp, AJ. Greimas, and Levi-Strauss. If one takes the CF as a 'structural equation,' a major aspect of Petitot's text resides in his elegant development of the mediation process - an operation at the core of the CF, expressed by its constituent term 'b'. Unfolding an heretofore unrecognized property of 'b,' he shows how the 'double twist' (the fundamental, most crucial, and often misunderstood inner dynamic of the CF) works. Petitot thus explicates the transformation of the y function into the final term of the formula through the operator 'b,' which hypostazises the basic mechanism of morphogenesis. By internalizing the deed to accomplish and by assuming it as its own intentionality, b causes the operation itself to become a value. One then moves on the cognitive level with the valuation of knowledge through its generating formula. The essays in this book explore a frontier. They attempt to generate understanding of cultural data by grasping the basic operations that produce them. The several convergent interpretations or readings of the CF that recur through these chapters each throw a light on it from a different angle. At first sight, these readings may seem redundant but they actually contribute, each in its own way, to a better understanding of what Petitot calls 'an intelligent formula' and of its diverse possible applications. Levi-Strauss has opened up a way: his non-linear approach - the 'double twist' he sees at work in the construction and the constitution of social representations - digs deeply into still unmapped grounds. We, his co-authors, hope that our modest contributions may foster research on one of the most fundamental issues the human mind has to cope with, namely, its own operations.

12 Pierre Maranda References Desveaux, E., and J. Pouillon. 1995. 'Rencontre autour de la formule canonique.' L'Homme 135: 7-63. Levi-Strauss, C. 1949/1964. Les structures elementaires de la parente. Paris: Presses Universitaires de France. New revised edition, Paris, The Hague: Mouton 1964. - 1955. Structural Study of Myth. In T.A. Sebeok, ed., Myth: A Symposium. Bloomington: Indiana University Press. Reprinted in French as chapter 11 of Anthropologie structurale, Paris: Plon, 1958. - 1964-71. Mythologiques. 4 volumes. Paris: Plon. - 1985. La potiere jalouse. Paris: Plon. Marcus, S. 1997. The Logical and Semiotic Status of the Canonic Formula of Myth.' Semiotica 116 (2/4): 115-88. Scubla, L. 1998. Lire Levi-Strauss. Paris: Odile Jacob.

PART ONE:

Ethnography and the Canonical Formula

The four first chapters, empirically oriented, apply the CF to field data, mostly Melanesian. Levi-Strauss's chapter conjugates cross-culturally vernacular architectures - among others, Fijian - to mythologies. Racine discusses an application of the CF to Papuan ritual and also to Chipayan (Bolivia) religious architecture, while Schwimmer tests the usefulness of the CF to describe historical change against his Papuan field data. Maranda uses the CF to contrast another basic culture change, the passage from pagan to Christian ontologies of the Lau people of Malaita, Solomon Islands, Melanesia.

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1 Hourglass Configurations Claude Levi-Strauss (Translated from the French original by Robbyn Seller, Anthropology, McGill University, Montreal)

It was in 1977, in Japan, facing the Ise Shrine, that the reflections I am sharing here took shape in a somewhat disorderly fashion. I was struck, as would be most people, by the roof frames, of which the principal rafters cross in an X and jut out past the ridge (see figure 1). The Izumo Shrines, also of archaic style, have a similar appearance, but due to crossbeams which are not part of the structure but are affixed to the roof as a decoration. This is reminiscent, of course, of islands in the South Seas where the roofs of certain houses resemble those of Ise: a further indication of the links that existed between Japan and that area of the world, already manifest when one compares their myths.1 However, to apprehend what this kind of structure could signify to the Japanese themselves, we must let their ancient texts speak. According to the Kojiki, a ritual formula accompanied the construction of a palace or a shrine: 'Root the posts of your palace firmly in the bed-rock below and raise high the crossbeams unto the upper world.'2 In this manner, the shape of the roof frame, which one might say recalls that of an hourglass, reproduces the form of the universe. The part below the roof ridge corresponds to the earthly world, the part above it to the heavenly world, which rises up to the 'plain of the highest heaven' inhabited by the gods. This representation of the cosmos comes to us, by way of China, from India. It may have originated in Mesopotamia, but this will not be my concern; rather, I will consider its extension in the opposite direction. Paul Mus has often evoked, in its Indian form, the axial Mountain that carries the lower stories of the divine worlds, while more immate-

16 Claude Levi-Strauss

Figure 1 Ise Sanctuary (after Aston 1896: vol. 1,133).

rial worlds float above its peak.3 'The first feature to be considered' of this axial Mountain, centre of the world, Meru or Sumeru in Sanskrit, writes Rolph Stein, 'is the mountain's shape [...] it is a pointed cone emerging from the sea and carrying on its summit another, inverted cone that represents the abode of the [...] Gods [...] The whole thing resembles an hourglass. Wide above and below, it is narrow in the middle'4 (see figure 2). A feature of religious architecture thus refers to a cosmology. These hourglass forms, in their application to architecture or to movable objects imbued with symbolic meaning, are also found in the New World that Orientalists have left out of their investigations. Before coming to America it is appropriate to make a stop in eastern Siberia, en route to the Bering Strait. In his investigation of the relationship between architecture and religious thought, Stein has mentioned the Koryak, who reside in wooden houses, roughly hexagonal in shape, with roofs in the form of a funnel or an inverted umbrella (see figure 3). Like other commentators, he has reiterated Jochelson's utilitarian explanation, in which this unusual structure functions to protect the entrance hole in the roof from snow, or to break the force of the

Hourglass Configurations 17

Figure 2 Mount Sumeru as a seat (after Stein 1987: 232).

wind during a blizzard so as to prevent snow from covering the house.5 Stein remarks in a note, however, that this inverted umbrellashaped roof evokes the form of K'un-Lun (the Chinese name for the cosmic mountain) and that of Mount Sumeru.6 Whatever the practical utility of such an appendage, one must not exclude the possibility that it may also be imbued with symbolic meaning. This appears even more credible because other parts of the house or furnishings, such as a notched central post used as a ladder, the hearth, a fire-drill, and so on all have symbolic value, and because, for the entire region of the Far East, the house and each of its parts have symbolic significance, as Stein has admirably shown. Crossing over to America, one would be tempted to connect the Koryak house with, on the one hand, the Far Eastern hourglass forms of

18 Claude Levi-Strauss

Figure 3 Koryak house (after Fitzhugh and Crowell 1988:201). Arctic Studies Center, Smithsonian Institution. Reproduced by permission.

which Mount Sumeru is the prototype and, on the other, with temples of the Kogi, who live in the Santa Marta sierra in Colombia. According to G. Reichel-Dolmatoff, who devoted years of study to the Kogi, their culture is unique among Amerindian cultures. Their religious buildings bear witness to this singularity. Seven or eight metres in height, of conical form crudely rounded out at the base, the thatched frame extends upward at the roof ridge into a boat-shaped construction from which a trellis made of poles tied together spreads outward, resembling an inverted umbrella (see figure 4)7 However, we will not belabour the comparison with hourglass forms from the Far East. While the Kogi distinguish nine cosmic layers, they conceive of them as contained within two hives joined together at the base. Each hive comprises four stories, and ours is situated at their junction. Erected on the ground that is the abode of humans, the temple represents therefore the hive that contains the upper stories. The other, inverted, is located symbolically below it. The Kogi cosmos is shaped like an egg rather than an hourglass. I shall return to this point. Reichel-Dolmatoff makes a sexual interpretation of the boat-like structure that crowns the temple. It would symbolize the vagina of the goddess-mother, of which the temple would then be the womb. It is worth noting, however, that the hourglass form, image of the cosmos, reappears for the Kogi in the symbolism of the narrow-waisted gourd (in a figure 8, Reichel-Dolmatoff says), a Kogi accessory used to carry the lime added to coca leaves before chewing them. According to our author, this gourd represents both the cosmos and the womb, while the spatula used to take the lime symbolizes the phallus.

Hourglass Configurations 19

Figure 4 Kogi temple (after Reichel-Dolmatoff 1990: plate VII). Reproduced by permission of Brill NV.

20 Claude Levi-Strauss

In the context of China, Stein has also brought the cosmic symbolism of the narrow-waisted gourd to our attention, emphasizing an analogy of form with Mount K'un-Lun, 'that is, a mountain or a composite structure built of two superposed cones that recall the two spheres of the gourd/ The resemblance with the Kogi increases due to the fact that, according to Stein, 'a closed world in the shape of a gourd is a site dominated by the presence of a female principle/ In China and Vietnam Holy Mothers preside over sites in the shape of a calabash.8 The hourglass form reappears in another kind of domestic object among the Desana, natives of the Amazon region of Colombia, to whom Reichel-Dolmatoff has also devoted considerable attention. They see, in an ingeniously twisted construction of sticks or slats used to support clay receptacles, a cosmic model of the upper and lower worlds, with ours represented by the narrow part. Viewed from above, the object has the appearance of a hollow vortex (see figure 5). This motif evokes the idea of transformation, which indigenous thought associates with whirlpools, birth, rebirth, and, more generally, with female fertility.9 The same author also describes crude stands of clay, placed in the hearth, that have roughly the form of an hourglass ('reloj de arena/ wrongly translated into French as 'clepsydre'). These stands unite the two cosmic levels. Moreover, they symbolize the sexual organs (see figure 6).10 Reichel-Dolmatoff's Desana informant was an extremely peculiar character, distanced from his birth environment and exposed throughout his existence to very diverse influences. It is therefore desirable that these interpretations of the hourglass form, akin to Far Eastern concepts, be corroborated by other sources. They are, in fact, for the Tanimuka, neighbours of the Desana. M. von Hildebrand (then close to Reichel-Dolmatoff, but whose observations are of independent value) noted and sketched a wicker object called a hanea, in the shape of inverted cones that would symbolize a whirlpool because of the twist imprinted in it (see figure 7). The Tanimuka, we are told, confer much importance to rapids and whirlpools, where, according to them, humans and animals originated; it is, therefore, the site of the transition from one world to the other, and in this sense resembles a vagina. The shaman performs before the hanea to expel sickness and to facilitate childbirth. For the latter, he dilates in his mind the narrow area of the hanea, of which the upper part corresponds to the uterus. The spiral aspect of the hanea is said to evoke changes of state.11 In America, the form and symbolism of the hanea are therefore not simply the same as those attributed to Mount Sumeru by China and

Hourglass Configurations

21

Figure 5 Stand made of basketry, Desana Indians (after Reichel-Dolmatoff 1987: plate XXXV). Reproduced by permission of Brill NV.

22 Claude Levi-Strauss

Figure 6 Stands made of basketry and clay of the Uaupes region (after Briizzi Alves da Silva 1962:182).

India. Here and there, the idea of a vortex is also present. Indeed in Far Eastern conceptions the Sumeru can be viewed from two apparently incompatible perspectives. Ropes in the form of snakes, coiled around its middle, serve to rotate the mountain on its axis. On the one hand, 'it is unchangeable and firm, immovable ... yet on the other hand, the very motifs of a wheel and an upright axis and the mythical theme of churning suggest a turning movement/ Some Japanese sculpture depicts Mount Sumeru as a dragon coiled in a spiral around a vertical sword, and 'thus are combined immovability and spiral movement.'12 One of the mnemotechnic signs used by the Cuna of Panama has the shape of an hourglass. Carlo Severi notes to this effect that 'not only the Cuna, but also numerous Amerindian populations (the Aruac of Santa Marta, the Tanimuka and the Tatuyo, to cite but a few tribes from the Columbian region) attribute the meaning of "birth and transformation" to this sign, undoubtedly an indication of a "passage" from one state to another within the universe.'13 (He thus makes his own the hypothesis I suggested to him, based on Tanimuka and Desana material, in the hope that he could verify it with his Cuna informants.) For want of a native gloss, and as conceivable as the interpretation appears in the context, it is nevertheless advisable to be prudent. In America and elsewhere in the world, the hourglass figure can take on other

Hourglass Configurations 23

Figure 7 Hanea (after von Hildebrand 1979:132a, fig. 13).

meanings: a human being schematized as a torso, a figure of a bird, its wings and tail spread out, and so forth. It is worth noting that the Desana extend the same sexual and cosmic symbolism to the frame of their large collective dwellings that they bestow upon their wicker stands for vases and hearth pottery. Stein notes that in China 'images of K'un-Lun are taken primarily from imperial palaces, but [...] they also go back to vessels or various kinds of hearths.'14 In the Desana house, a pole called a gumu, in spite of its

24 Claude Levi-Strauss

horizontal position, is perceived as a ladder that cuts through all the layers of the cosmos. At the same time, it represents a phallic tube that unites the back of the house, chthonic and female, with the front, solar and male.15 Many examples from India and China portray this conversion of the vertical axis to a horizontal axis and the reverse. Likewise, we can understand the connections between 'Visnu's three strides' and those of Buddha; or, yet again, the nine stories of the K'un-Lun, which are sometimes interpreted as the meanderings of a path.16 In Indonesia, the Atoni of Timor bestow symbolic meaning upon the tiny vertical space between two horizontal ridge-poles, located one beneath the other. The name of the upper pole calls forth the notion of solar heat; the name of the other, that of heat from the fire. On the vertical axis, the upper pole therefore protects the inhabitants from the sun just as, on the horizontal axis, the door on the eastern side prevents the sun from entering the house. This double encoding, horizontal and vertical, applies to the house as a whole and to each of its parts. All that is situated completely or relatively towards the outside or the top of the house is imbued with the male principle. All that is completely or relatively located towards the interior or the bottom is imbued with the female principle.17 Perhaps one could explain in this way the unusual form of Fijian temple roofs as they were described in the nineteenth century. The ends of the ridge-pole project for a yard or more beyond the thatch, having the extremities blackened, and increasing with a funnel-shape, and decorated with large white shells' (see figure 8).18 One would willingly recognize in this, converted to a horizontal position, the hourglass form of Mount Sumeru. A century later Quain observed a purely decorative structure that all the houses, according to him, must include to be considered finished, that is, long poles made of black, porous sago palm that extend past the roof covering at each end of the ridge. Quain adds: 'Europeans have frequently speculated as to [this structure's] esoteric significance; no one has ever discovered its "meaning."'19 Here, then, are two apparently dissimilar types of roof extension, between which one might query if there is not a relation of some sort, and whether it would not be appropriate to place them among the others I have considered. Hourglass forms, already noted and abundantly documented in the Far East, also exist in America, and apart from a few shifts in the cosmic levels they are supposed to represent, their symbolism is the same

Hourglass Configurations

Figure 8 Fijian temples (after Williams 1970:158,188).

25

26 Claude Levi-Strauss

everywhere. Everywhere as well, right down to the details, these symbols have developed along the same lines. How does one explain these often striking resemblances? Do they hark back to an old common palaeolithic substratum of Asiatic populations, some of which migrated to America? Are they the fruit of more recent trans-Pacific contacts between the Old and the New Worlds? Or should we search no further than the irrepressible tendency of the human mind to pour the profusion of its available vacant significations into all the forms it perceives or conceives, by reason of the permanent gap between its impatience to understand everything and its limited knowledge? If the forms under consideration are simple, even rudimentary, they would have every chance of being widely shared, and would lend themselves admirably to symbolize correspondences or oppositions between equally simple conceivable contents. These contents, consequently, would hold every chance of being shared among a good number of cultures or civilizations. If so, the alleged depth psychology would be in fact the most superficial: a shallow psychology, one might say, that offers no real substance for reflection. Our present knowledge of America's past does not allow us to choose between these hypotheses. I will leave them in abeyance to go on to another problem in this particular case: that of the relation between symbolic forms and the materials available to represent them figuratively. One can construct hourglass forms with many materials without encountering any difficulty: wood, fibres, or clay. We have come across examples using these materials in the frames of buildings and in wicker and clay objects. By contrast, it would be extremely difficult if not impossible to construct hourglass forms in stone.20 Monument architecture achieves only a partial representation of Mount Sumeru; it reduces it to the temple-mountain, leaving out its inverted heavenly counterpart. What may be 'said' in wood or in wicker cannot be 'said' in stone. The synthetic, global representation of Mount Sumeru as an hourglass figure21 would necessitate a radical structural transformation if it were to be constructed in masonry. Indeed, that is what the builders of Barabudur, a Buddhist monument in Java that dates back to the twelfth century, succeeded in doing. Heine-Geldern had noted already in 1930 that although the shape of this building is obviously that of a mountain, it is not conceived as a symbol of Meru, but rather as a representation of the upper heavens.22 Paul Mus has carried the analysis further and shown that in fact both

Hourglass Configurations 27

Figure 9 Cosmological scheme of the Barabudur (after Mus 1990:112).

mountain and heavens are represented by Barabudur. To apprehend the monument in all its complexity, he explains, one must imagine that Meru proper, buried at the base of the Javanese stupa, is topped by the four stories of the heavenly mountain encased in a spherical cap (see figure 9). The schematic image of the whole, writes Mus, is therefore that of a dome, the image of heaven that ideally conceals in its interior a pyramidal mountain, Meru.23 What, then, is the relation between this image and that of the hourglass which, as we have seen, is also fit to represent the two ontologies, earthly and heavenly, of the cosmic mountain? To switch from one image to the other, it is sufficient but necessary to transform the two distinct elements in (B), identical in shape (they are two pyramids) but different in their position in space (upright or inverted), into (A), two identically positioned elements (both are upright) but which differ in shape (a pyramid and a dome). These two

A

B

28 Claude Levi-Strauss

propositions are not homogeneous, however, because they strike an equivalence between meanings - which alter with a change in position - and shapes - which change with the opposite result, that is, keeping the meaning constant. One proposition emphasizes form and the other function. Together, therefore, they constitute a complex proposition, reducible to the formula applied elsewhere by me to mythical transformations, that have the same type of structure:

'•earth *heaven ^earth ^pyramid'1 (pyramid)

(dome)

(dome)

(heaven)

In the above formula, it should be noted that f and f^aven *° symbolic space because, in this case, there is interference of the architectural and the cosmological planes. One must remember that the dimension to which the formula refers is the 'here-below' of the mason in the one case, and the 'heaven-above' of the theologian in the other. We will recall that, unlike the Hindus and the Chinese, the Desana do not represent their cosmos as pyramids joined at the summit, but as hives joined at the base (see above, p. 6):

Viewed in relation to Barabudur, the Desana vision of the cosmos is the symmetrical and inverse transformation of the hourglass configuration. To travel, so to speak, from Sumeru to Barabadur, the heavenly pyramid must be tipped over and transformed into a dome while the earthly pyramid stands unchanged. To go from the Desana to Barabudur, the chthonic dome must be tipped up and transformed into a pyr-

Hourglass Configurations 29

amid, the heavenly dome remaining unchanged. The three states, therefore, belong to the same set. On the other hand, one cannot overemphasize the fact that the transformation of the figure of domes opposed at the base (of the Kogi type) into that of triangles opposed at the vertex (of the Sumeru type), and vice versa, is not amenable to a representation by my canonical formula. This is a trivial example of a four-term analogical relation, to which certain commentators believed the canonical formula was reducible. The geometric representation is of interest, in part, because it allows us to recognize and dispel this confusion by a simple examination of those figures. What can we retain from the preceding considerations? Two points, it seems to me, that, although without apparent relation, are mutually supportive. In the first place, a mode of representing the cosmos for India and the Far East, well known and long studied, also exists in America with similarities down to the details. Second, the canonical formula that I have proposed to illustrate mythical transformations can be applied to other fields, notably to architecture. R. Bucaille and F. Chergui have already done so in regard to dovecotes in Limagne;24 however, in the present case, it is remarkable that the double transformation illustrated by the formula is initiated by technical constraints. The passage from one material to another thus plays the same role as do changes of a linguistic or cultural nature in other contexts: it always involves the crossing of a threshold. Finally, and most significantly, the fact that the architecture reflected upon here is a religious architecture particularly favours the application of this formula, as the forms considered are imbued with symbolic meaning. A conscious link exists between two fields that intersect. In this way one can simultaneously apprehend the formula as a conceptual relation and see it, here and there in the world, stamped in a geometry that human endeavour has envisioned. Notes 1 C. Levi-Strauss, 'La place de la culture japonaise dans le monde/ Revue d'esthetique 18 (1990): 12-14. 2 Kojiki trans, with intro and notes by D.L. Philippi (Tokyo: University of Tokyo Press, 1968), chap. 24 (14), 27 (3), 39 (18); see also Nihongi, trans.

30 Claude Levi-Strauss

3 4

5

6 7

8 9

10

11 12 13

W.G. Aston, 2 vols. (London: Transactions and Proceedings of the Japan Society, 1896), 1:132-3. P. Mus, La lumiere sur les six voies, Travaux et Memoires de 1'Institut d'Ethnologie, vol. 35) (Paris, 1939), 42,54,172-4,284f. R.A. Stein, Le monde en petit (Paris: Flammarion, 1987), 232 (English trans., P. Brooks, The World in Miniature [Stanford: Stanford University Press, 1990], 246). Ibid., 163f. (Eng. trans., 169); W.W. Fitzhugh and A. Crowell, Crossroads of Continents: Culture of Siberia and Alaska (Washington: Smithsonian Institution Press, 1988), 33,200,201. Stein, Le monde en petit, 318 n.93 (Eng. trans., 323 n.60). G. Reichel-Dolmatoff, Templos Kogi Introduccion al simbolismo y a la astronomia del espacio sagrado,' Revista Colombiana de Antropologia 19 (1975); The Great Mother and the Kogi Universe: A Concise Overview,' Journal of Latin American Lore 13 (1) (1987); The Sacred Mountain of Colombia's Kogi Indians, Iconography of Religion, IX/2, State University Groningen (Leiden: Brill, 1990). Stein, Le monde en petit 77, 70,83ff. (Eng. trans. 70,62,77). G. Reichel-Dolmatoff, Shamanism and Art of the Eastern Tukanoan Indians, Iconography of Religion, IX/I, State University Groningen (Leiden: Brill, 1987), plate XXXV. G. Reichel-Dolmatoff, Desana (Bogota: Universidad de los Andes, 1968), 81 (French trans., Desana [Paris: Gallimard, 1973], 137); Shamanism and Art, 16. Same objects illustrated in A. Briizzi Alves de Silva, A Civilizagao Indigena do Uaupes (Sao Paulo, 1962), 182. M. von Hildebrand, 'Cosmologie et mythologie Tanimuka,' doctoral thesis, Universite de Paris VII, 1979, figure 13 and pp. 131^4. Stein, Le monde en petit, 232,235-6,238-41 (Eng. trans., 253,258). C. Severi, La memoria rituale (Firenze: La Nuova Italia, 1993), 149 n.15,155; R. Seller's English trans.

14 Stein, Le monde en petit, 220. 15 G. Reichel-Dolmatoff, Amazonian Cosmos (Chicago: University of Chicago Press, 1971); Desana (French trans.), 135. 16 P. Mus, Barabudur, reprint of Ecole franchise d'Extreme-Orient edition, Hanoi, 1935 (Paris: Arma Artis, 1990), 535; Stein, Le monde en petit, 216,218, 229; G. Dumezil, Mythe et epopee I (Paris: Gallimard, 1968), 235-6; Le roman desjumeaux (Paris: Gallimard, 1995), 318-23. 17 C.E. Cunningham, 'Order in the Atoni House,' Bijdragen tot de Taal-, Land-en Volkenkunde 120 (1964).

Hourglass Configurations

31

18 T. Williams, Fiji and the Fijians (London: Hodder and Stoughton, 1970), 73, 158,188 (New York: D. Appleton and Co., 1859, p. 65). 19 B. Quain, Fijian Village (Chicago: University of Chicago Press, 1948), 169. 20 The same problem arises for the vortex figure. The Chinese used to build rotating wood pagodas and libraries. In masonry or other inexpedient material, how would one go about it? Based on information provided by a 19th-century traveller, Stein reproduced a reduced model of Mount Sumeru from the 12th century, made of cast iron and in two parts, one of which could pivot on the other, but weighing nonetheless 300 kg, too heavy to be moved by the arm of a man. Another solution is to have the faithful walk around the monument rather than watching it turn: the idea of movement is saved. Movement has also been suggested by displacing the angles of the building by 45 degrees in relation to the cardinal points, or by multiplying the number of intermediate figures to create in sculpture, according to the expression of Paul Mus, a sort of 'solid cinema' (Stein, Le monde en petit, 245-7; Mus, 'Une Cinema solide/ Arts asiatiques 10 [1964], fasc. 1,21-34; Annuaire du College de France, 1964: 336-7,1967: 288). 21 Lowie, who was interested in hourglass forms in another context, concluded, so he says, that in order to understand a decorative art, one must grasp each figure as a whole without trying to reduce it to a combination of simple geometric abstractions such as rectangles or triangles. His critique is covertly directed at Kroeber, who reduces the hourglass figure to two triangles opposed at the summit, while he himself treats triangles as derived forms that he calls half-hourglasses (R.H. Lowie, 'Crow Indian Art,' Anthropological Papers of the American Museum of Natural History 21 (4) [1992]: 279). It seems that the most structuralist may not always be the one we think! 22 P. Mus, Barabudur,55, 111. 23 Ibid., 107,117, and passim. 24 R. Bucaille and F. Chergui, in R. Bucaille et al., Pigeons de Limagne, (Clermont-Ferrand: Universite populaire, 1987), 61-83.

References Briizzi Alves de Silva, A. 1962. A Civilizafao Indigena do Uaupes. Sao Paulo. Bucaille, R. 1987. Pigeons de Limagne. Clermont-Ferrand: Universite populaire. Cunningham, C.E. 1964. 'Order in the Atoni House.' Bijdragen tot de Taal-, Landen Volkenkunde 120.

32 Claude Levi-Strauss Dumezil, G. 1968. Le roman desjumeaux. Paris: Gallimard. - 1995. Mythe et epopee I. Paris: Gallimard. Fitzhugh, W.W., and A. Crowell. 1988. Crossroads of Continents: Culture of Siberia and Alaska. Washington: Smithsonian Institution Press. Kojiki. 1968. Translated with an introduction and notes by Donald L. Philippi. Tokyo: University of Tokyo Press. Levi-Strauss, C. 1990. 'La place de la culture japonaise dans le monde.' Revue d'esthetique 18. Lowie, R.H. 1992. 'Crow Indian Art.' Anthropological Papers of the American Museum of Natural History 21, pt 4. Mus, P. 1939. La Lumiere sur les six voies. Travaux et Memoires de ITnstitut d'Ethnologie, vol. 35. Paris. - 1964. 'Une Cinema solide.' Arts asiatiques 10. - 1990. Barabudur. Reprint of Ecole franchise d'Extreme-Orient edition. Paris: Arma Artis. Nihongi. 1896. Translated by W.G. Aston. 2 vols. London: Transactions and Proceedings of the Japan Society. Quain, B. 1948. Fijian Village. Chicago: University of Chicago Press. Reichel-Dolmatoff, G. 1968. Desana. Bogota: Universidad de los Andes. - 1971. Amazonian Cosmos. Chicago: University of Chicago Press. - 1975. Templos Kogi Introduccion al simbolismo y a la astronomia del espacio sagrado.' Revista Colombiana de Antropologia 19. - 1987. 'The Great Mother and the Kogi Universe: A Concise Overview.' Journal of Latin American Lore 13 (1). - 1987. Shamanism and Art of the Eastern Tukanoan Indians. Iconography of Religion, IX/1, State University Groningen. Leiden: Brill. - 1990. The Sacred Mountain of Colombia's Kogi Indians. Iconography of Religion, IX/2, State University Groningen. Leiden: Brill. Severi, C. 1993. La memoria rituale. Firenze: La Nuova Italia (Robbyn Seller's English translation). Stein, R.A. 1987. Le monde en petit. Paris: Flammarion (English translation, Phyllis Brooks, The World in Miniature. Stanford: Stanford University Press, 1990). von Hildebrand, M. 1979. 'Cosmologie et mythologie Tanimuka.' Doctoral thesis, Universite de Paris VII. Williams, T. 1970. Fiji and the Fijians. London: Hodder and Stoughton.

2 Analogy and the Canonical Formula of Mythic Transformations Luc Racine (Translated from the French-language original by Richard Ashby)

Can the canonical formula of mythic transformations be reduced to an analogy of the type 'a is to b as c is to d'? This is the central question to be addressed in the present paper. The discussion is based on the two cases of the application of this formula in which Claude Levi-Strauss manifestly attempted to be as clear and pedagogical as possible in this regard. Thus, it will be possible to determine whether analogy and mirror symmetry, itself a substructure of the former, are related to and premised by the formula. There is no question here of reducing the formula to these structures, given that this would limit its scope with regard to the analysis of mythic transformations. Be it a matter of myths, rituals, or masks, the formula attempts to account for an extremely important phenomenon from a comparative perspective: whenever one crosses a linguistic, cultural, or other boundary, one notes the presence of a kind of particular transformation that only partially resembles transformations linking the various elements of a single domain. The transformation related to f x (a) : fy(b) puts elements belonging to the same domain in relation with one another, while the transformation related to f(x)b : fa-i(y) articulates elements belonging to different domains; f(x)b, like the first two members of the formula, belongs to a first domain and fa-i(y) belongs to a second. It is for this reason that the relation of the first to the second member cannot be considered as identical to that which relates the third to the fourth, with this latter relation introducing an opposition that distinguishes the last member from the three others. It is only by ignoring one of the three oppositions characteristic of the formula, thus removing much of

34 Luc Racine

[A (dl, hi)] -> [C (dl, h2)]

4

I

[B (62, hi)] -> [D (d2, h2)]

Figure 1 (Note: h is marked by —> and rf by -I)

its interest for comparative analysis, that it is possible to consider it as a simple analogy. Second, this paper will discuss a variation of the canonical formula used by Mosko (1985, 1991) in the analysis of ethnographic data that he labels 'non-myth' (social rituals and representations with regard to conception, eating, etc.). It will be shown that, contrary to LeviStrauss's formula, Mosko's version of it can be reduced to a combination of analogies. The Analogic Square Before comparing the canonical formula, discontinuous analogy, and mirror symmetry, let me first define the latter two terms. In the sense in which it is used here, an analogy supposes the classification of four distinct terms (noted as A, B, C, and D) as a function of two distinctions or intersecting binary oppositions (logical multiplication) (Cote 1989; Perelman and Olbrechts-Tyteca 1958; Racine 1986, 1989, 1990). The first of these distinctions, which I express as dl/d2, differentiates A from B as it does C from D, while the second, hl/h2, differentiates A from C as it does B from D. Thus, according to the second distinction, A resembles B as C does D, with the first two terms being characterized by hi and the two others by h2. On the other hand, according to the first distinction, A resembles C as B does D, with the first two terms being dl and the two others d2 (see figure 1). The passage from dl to d2 is denoted by d and the inverse passage by d~ ; similarly, h is the operator transforming hi into h2, and h that which transforms h2 into hi. So defined, the analogic square is an extended category in the sense discussed by Lorrain (1975), that is, a structure weaker than a Klein group, which assimilates the h and h~l operators on the one hand,

Analogy and the Canonical Formula of Mythic Transformations

35

and d and d~l on the other (Racine 1989:16-21). A classic illustration of the analogic square is given by the proposition that youth (A) is to old age (B) what the morning (C) is to the evening (D), with hl/h2 meaning life/day and dl/d2 that of beginning/end. Several ethnographic illustrations of the square can be found in Racine (1986,1989,1990) and in Cote (1993). Analogy does not account for all possible relations between the square's four terms. In particular, the intersecting correspondence (per diametrum) of pairs of terms cannot be characterized as analogic. The relation A is to D what C is to B (or B is to C) is not analogical but chiastic: A is transformed into D by the play of the d and h operators, while C is transformed into B by the play of d and h~l (in the case of a double chiasmus, B is transformed into C by the play of d~l and h}. Mirror symmetry is a substructure of the analogic square, relating only three distinct terms with one another, with the first differing from the second as a function of a first distinction and the second from the third as a function of a second distinction. The square has four isomorphic structures of this kind, with each term capable of serving as mediator. In the case of A, C, and B, for example, one can determine whether C differs from A as a function of hi /h2 and resembles it as a function of dl/d2 (with C and A both being dl), while A differs from B as a function of dl/d2 and resembles it as a function of hl/h2 (with A and B both being hi). The d~l operator transforms B into A and h~l transforms C into A: Bd-J -» A [ A / A' and B / B' ] -> [ A : A':: B : B' ].

As one can see, the framework of this analogical relation is the 'opposition square' of the Greek logicians, which underlies Greimas's 'semiotic square.' But however well suited to the mediating function of myth the analogical scheme may be, we must note that 1. The analogical relation is a balanced relation that lends itself to the description of justice according to Aristotle, but that can hardly be used as a model for the unbalanced relation (1984:13) that Levi-Strauss has always associated with the canonical formula. 2. The analogical relation represents a static equilibrium bearing no mark of the dynamic process that has engendered it. 3. The analogical relation is symmetrical: one can equally well write 1 : 2 : : 3 : 6 or 3 : 6 : : 1 : 2, or represent the structure of the Oedipus myth with A : A': : B : B' or B : B': : A : A.' Analogy therefore reflects only the synchronic or achronic aspects of myth.

Hesiod, the Three Functions, and the Canonical Formula of Myth 131

The second step overcomes these weaknesses by substituting the formalism of ordering for that of analogy. Through numerous examples taken from Pueblo mythology, Levi-Strauss demonstrates that mythical thought sets up what he calls ordered series of mediating functions in which each function derives from the preceding one through opposition and correlation (cf. 1955: 442, 1958: 251). Starting from two diametrically opposed terms, it inserts between them intermediate terms that attenuate this opposition, then weakens the new opposition in turn by means of closer terms, repeating this operation as many times as possible. This process comes down to constructing increasingly refined orderings within the limits set by the two terms of the initial contradiction, thus producing a continuous series of relations such as A < A' -> A < B < B' < A' -> A < B < C < C < B' < A', and so on,

in which A and A' are the opposite terms of the original pair; or, more precisely, a series such as A < A' -> A < B < M! < B' < A' -> A < B < Ml < C < M2 < C < B' < A',

inasmuch as Levi-Strauss postulates that, at each stage of the mediating process, mythical thought replaces a given binary opposition, not merely with a weaker one, but with a triad composed of such an opposition and of a median term. Thus, between life and death, Pueblo mythology introduces a first triad consisting of farming, hunting, and war, in which farming and war are the two polar terms and hunting an intermediate term that participates in both poles at once; then it inserts between farming and hunting another triad consisting of herbivores, carrion-eaters, and predators, whose structure is similar to that of the first one, since carrion-eaters, like herbivores, do not kill animals but, like predators, do eat meat (1955: 440). This calls for two observations. First: instead of dealing with symmetrical structures, we are now dealing with ordered series of terms and operations that correspond to oriented processes and the asymmetrical feature of which is underlined by Levi-Strauss by means of an appropriate symbolism (-», spear): (spear >sceptre) = kings : warriors, which enables us to simplify Mezzadri's canonical equation by replacing the expression 'D(kings): H(kings):: H(warriors): D(warriors)' by 'D(kings): H(warriors)' and, therefore, ultimately to write: D(kings): H(warriors):: D + H(produc.): Kings'1^). We are then immediately able to see that the reduced form of the myth of the races coincides exactly with the reduced form of the canonical formula that has been established in the previous section, namely, 19

If we assume that this coincidence is not accidental, two points that Mezzadri mentions only as plausible will need to be verified: (1) the ability of hubris to become an active element, that is, to change from a specific disorder generated by the kings, the warriors, or the producers to a generalized disorder itself producing new phenomena; and (2) the existence and nature of the 'anti-kings' this hubris is supposed to engender. These two points are hardly alluded to in the myth, even though it implies that, unlike the beings of the previous ages, the 'old iron age men do not practise any activity of which hubris or dike might only be a modality/ but that, through a kind of reversal, 'the decadent iron age men are no longer anything but what hubris made them' (Mezzadri: 53). That clearly is not enough to turn hubris into a generative power, and it leaves completely in the dark the status of anti-kings, who are admittedly mentioned neither by Hesiod nor by Vernant in his commentary on Hesiod.

146 Lucien Scubla

However, another text by Vernant, the mere title of which is highly significant, demonstrates that Mezzadri's double conjecture is likely to be justified. We are referring to an article entitled 'Ambiguity and Reversal' and subtitled 'On the Enigmatic Structure of Oedipus the King' (Vernant 1970), in which the famous Hellenic scholar examines a pair of terms that has precisely the same structure as the pair king/ anti-king in Mezzadri, namely, the pair turannos/pharmacos, or king/ scapegoat, 'two characters' who, he explains, 'in their opposition appear symmetrical and in some respects interchangeable' (ibid.: 1272). This 'kinship between the king and the scapegoat,' he reminds us, 'was inscribed in the religious and social practice of the Greeks ... The king discharges everything that is negative in his own character onto an individual who is like his mirror image. Such is the pharmacos: the double of the king, but a reverse double, similar to those carnival kings crowned only while the festival lasts, when social hierarchies are reversed and the order turned upside down: sexual prohibitions are lifted, theft becomes lawful, slaves take over from masters, women exchange clothes with men; then the throne must be occupied by the vilest, ugliest, most ridiculous, most villainous man. But once the festival is over, the counter-king is expelled or put to death, taking with him and thus purging the community of all the disorder he embodies' (ibid.: 1271-2). So we can see that Vernant describes a rite during which the simultaneous presence of a generalized hubris and of an anti-king enables the real king to regain his position: in other words, the very type of closed loop the Hesiodic myth is supposed to describe.20 But according to Mezzadri's analysis - or, more precisely, the canonical formula - hubris itself must explicitly engender the anti-king and consequently the restoration of social order: in other words, disorder must, under certain conditions, be capable of regulating itself and becoming a source of order. At the present day, this bold hypothesis can be neither corroborated nor dismissed in view of the available data, but it can be investigated further thanks to the canonical interpretation of myth.21 In the present state of things, we can only note that the theme of violence - a classical image of hubris - is omnipresent in the Hesiodic myth, even if it is approached from the point of view of dike. Seen from this angle, the first level is in fact that of ritual violence: the sacred king puts to death the sacrificial victims and pacifies conflicts

Hesiod, the Three Functions, and the Canonical Formula of Myth 147

(Vernant 1971: 21-2). The second level is that of the regulated violence of warfare: military power remains at the service of law, subject to the king's authority (ibid.: 30). The third level is that of productive labour, which prevents economic competition from degenerating into pure rivalry (33, 66). But should the king neglect his ritual duties, warriors abuse their strength, and producers be free to argue and quarrel, then the society unfailingly sinks into a state of generalized violence, which Vernant depicts in the very terms he uses to describe the ritual of the pharmacos: 'an upside-down world where any hierarchy, any rule, any value is reversed,' and before long 'a negative universe where there remains only disorder and evil in the purest form' (34). In other words, a state of extreme decomposition that nonetheless is supposed to bring about the closure of the system, that is, to regenerate the sacrificial order prevailing at the outset of the myth. To clarify matters, we would have to bring to light the process allowing the myth to close upon itself: to circle back from the iron age antiking - engendered by hubris - to the original figure of the golden age king - the guardian of dike - of whom we know the anti-king is the reversed image. Now, to demonstrate that this type of reversal is not foreign to Hesiod, Mezzadri rightly recalls that there are' "reversible" objects at the heart of the Prometheus myth: the green ferula concealing fire, the stomach turned inside out in the sacrifice,' and so on (1988: 55). But we must undoubtedly go much further and ask whether the king himself might not be, by nature, a 'reversible object': in other words, whether the terms 'king' and 'anti-king' might not designate two undissociable aspects of the same royal office (sometimes embodied by two distinct figures in the same society), since the two terms obviously do not refer to two antithetical ways of exercising sovereignty that may be found in different places. This is at any rate what is suggested by the canonical formula, which, when applied to the myth, distinguishes, as we have seen, three terms and not two: the golden king or good king, the silver king or wicked king, and the anti-king. Actually, all the collected data on traditional monarchies bear witness to the ambiguity of the royal office, already emphasized by Frazer. Like the priest of Nemi, who starts out as a sacrificer and becomes a sacrificial victim, the holder of the royal office is often a king and an anti-king in turn (or at the same time): a god living among men and the prisoner of his people, the source of all good and a scapegoat, the vie-

148 Lucien Scubla

tim of his subjects after being their predator, or the predator of his subjects after being their prey. Thus, in the ancient kingdom of Dahomey, the king was the great predator of his people, who talked to him in this way: 'Dada! Smite! Pillage! You are entitled to! You are the king and the natives of Dahomey are your creatures' (Le Herisse 1911: 7). But his own face bore the scars of the panther that had 'devoured' him during the coronation rite (ibid.: 10) - as if he had started out as the prey before turning into the predator. If such is the case, we can only express reservations about two propositions that Mezzadri does no more than outline at the end of his article. In order to clarify somewhat the anti-king's status, he invokes the figure of the 'tyrant/ which Plato often opposes to that of the king, in particular in the Politicus (Statesman). In this respect, he is in agreement with the historians of philosophy who generally see in the 'myth' of the Politicus a free commentary on Hesiod's text. But if we follow this path, we may well confuse the anti-kings properly speaking, engendered by the overall hubris of the old iron age, with the wicked silverrace kings or the wicked bronze-race warriors, of which more accurate replicas can be found in Plato's tyrants. Far from being brought together, anti-kings and bad kings should in our view be much more clearly distinguished than in Mezzadri's analysis. Likewise, we should reject the pendular model of the Politicus that, following Vernant (1971: 73), Mezzadri (1988: 54) is tempted to borrow from Plato: namely, the idea that, in order to return to its starting point, the myth should retrace the path that leads from the golden age to the old iron age. And, along with this idea, we should also probably reject the topological model of the Moebius strip that Mezzadri curiously tries to associate with it (1988: 54). In fact, there exist cyclical models that suit far better the 'catastrophic' closure implied by the canonical interpretation of the myth of the races. In particular, the 'cam structures' used by Antoine Culioli (1968:115) to describe certain linguistic systems, and which seem especially appropriate for our myth since they possess three singularities that correspond perfectly to the three positions held respectively by the law-king, the wicked king, and the anti-king. Moreover, the 'cam structure' is homologous to what Thorn calls the 'predation loop,' that is, a closed path around the organizing centre of the 'cusp' (see Petitot 1992: 309), in which it is possible to see, among other things, a mathematical model of the predator/prey relationship: in other words, the description of a cyclical process endowed with a

Hesiod, the Three Functions, and the Canonical Formula of Myth 149

'catastrophic' reversal in the course of which the predator becomes the prey. With the addition of a minus sign, this process corresponds rather well to the coronation ritual we just mentioned, in which a creature who starts out as the prey of his people becomes its predator. Moreover, it demonstrates that our interpretation of the canonical formula is compatible with the interpretation put forward by Petitot. Indeed, the geometry of the 'cusp' that underlies our model is strictly included within that of the 'double cusp' required, according to our colleague, by Levi-Strauss's formula (Petitot 1988, 1992: 394-405). This agreement is all the more significant in that we both try, although via different paths, to provide structuralism with the morphogenetic complement it needs to make a fresh start. Nevertheless, the fact remains that the double cusp is more complex than the cusp. Therefore, if Petitot is right, reducing the canonical formula to a cyclical circuit similar to the narrative circuit of the semiotic square amounts to weakening it (cf. Petitot 1985: 201-77; 1992: 330-85). Yet the opposite possibility must also be considered. One might ask whether the mathematician has not devised an unnecessarily sophisticated apparatus for Levi-Strauss's formula. If such is not the case, specifically anthropological reasons should be added to Petitot's formal arguments for the adoption of his model. This is the task we have undertaken in our Lire Levi-Strauss, a mere outline of which can be found in the present article. Here we may add that it was Petitot's work that made us understand the theoretical interest of the canonical formula in the first place. The idea of explaining Levi-Strauss's formula and thus the entire range of mythical structures by coupling two binary oppositions was all the

150 Lucien Scubla

more attractive to us in that (1) we postulate, together with Hocart, the isomorphism of myth and ritual; and (2) we seek to construct a general theory of ritual structure on the basis of two fundamental binary oppositions from which we infer, among other things, Dumezil's three functions. In any event, our excursion into the intricacies of the canonical formula has ultimately taken us full circle, since we encountered at the end, thanks to Vernant, the same Oedipus legend with which LeviStrauss first began to construct this theory of myth. And it is no accident that we also encountered the coronation rites with which this legend must be deemed to be inextricably linked, as Marie Delcourt demonstrated long ago in an admirable work (Delcourt 1944) that Levi-Strauss himself did not fail to mention in his pioneer contribution to the understanding of myth (1958: 238). Notes 1 L. Scubla, Lire Levi-Strauss: le deploiement d'une intuition (Paris: Odile Jacob, 1998), which is a condensed version of a thesis entitled 'Histoire de la formule canonique du mythe et de ses modelisations, Paris/ Ecole des hautes etudes en sciences sociales, 1996. 2 This article is the revised text of a lecture delivered at a Franco-Canadian anthropology symposium held at the CREA (Centre de Recherche en Epistemologie Appliquee) in Paris, 4 December 1992. An earlier and briefer version of this paper was published in 1995 (L'Homme 135). 3 Reading a paper by Marc Barbut on the notion of structure in mathematics, while writing his Mythologiques, Levi-Strauss thought he recognized in Klein's group the quadripartite structure he needed to formalize his theoretical intuition. Therefore, in volumes 3 and 4 of his major work, he abandoned the canonical formula in its standard form and used instead the sequence of terms x, - x, 1 /x, and -1 /x, which make up the orbit of Klein's group, but always in order to designate the same morphogenetic process, while being aware of the inadequacy of this modelization (as his words and drawing printed in the Marandas's book mentioned above show). Thus, it is hardly surprising that a few years later he should rehabilitate the original formula in his teaching at the College de France. 4 Levi-Strauss defines his canonical formula as 'the genetic model of myth (i.e. what engenders it while simultaneously giving it its structure)' (1958: 265).

Hesiod, the Three Functions, and the Canonical Formula of Myth 151 5 All these rituals can be derived from a great prototypic ritual centred on the figure of the king (Hocart 1927,1954); this detail is relevant to what follows. 6 This principle postulates that any symmetrical relationship is subordinate to the existence of an antisymmetrical relation, in other words that

(Vx) (Vy) [(Rxy and Ryx) => (3z) ((z * x) and (z * y) and R'xz and R'yz and not-R'zx and not-R'zy)]. Consequently, as both Hobbes's and Rousseau's theories of sovereignty would suggest, we cannot describe a society with the principle of reciprocity alone. If we let y = x, we see that (Vx) [(Rxx => (3z) (z * x) and R'xz and not-R'zx].

7

8

9

10

This formula expresses the impossibility of radical autonomy, of pure selfreference, or even of simple presence to self, an impossibility recognized by both Leibniz and Kant in their critique of Cartesian idealism. This suggests that the principle is a general one, with applicability well beyond the domain of social structure. The pangolin, a small ant-eater that plays a major part in many African rituals, is a land animal that has scales like a fish, climbs trees, lays eggs like a bird, nurses its progeny like a mammal, and gives birth to one baby at a time like a human mother. It combines characteristics that are not usually found together and that could have been considered incompatible. The 'three main processes' (1955: 440) explicitly differentiated by LeviStrauss after analysing the Oedipus myth correspond to the last three steps in the construction of the formula. 'The inability to connect two kinds of relationships is overcome (or rather replaced) by the positive statement that contradictory relationships are identical inasmuch as they are both self-contradictory in a similar way' (1955:434). We may note that the two examples of cyclical structures Levi-Strauss took from the Pueblo mythology are both quadripartite, like the canonical formula they seem to portend: [(A:x)::(B:Q::p:B)::(y:A)]and[(A:x)::(B:A)::(A:C)::(y:A)].

On the other hand, they do not differentiate between term and function, and they exhibit strong symmetries not found in the canonical formula and incompatible with it (see below). 11 This is true in English and French, but not in classical Greek (cf. Aristotle, Poetics, chap. 21).

152 Lucien Scubla 12 In this sense, one could say that the canonical formula introduces the mobilism of Heraclitus - evoked by the idea that myth is the resolution of a conflict - into the Parmenidean structuralism with which some commentators have reproached Levi-Strauss. Situated in its fashion 'between time and eternity/ the formula is in tune with the most promising trends in contemporary physics (cf. Ilya Prigogine and Isabelle Stengers, Entre le temps et I'eternite [Paris: Fayard, 1988]). 13 In an unpublished essay, Georges Soumelis suggested regarding the 'double twist' of the canonical formula in the light of the general case of a twist of degree n, where the 'pecking order' is construed as a simple twist of degree 1. This hypothesis has not been developed by its author, but could profitably be elaborated upon by others in the future. 14 This is precisely what Levi-Strauss suggests when he writes that he conceived of his formula, not as 'an algorithm making it possible to perform calculations,' but as 'a graphic representation which ... could facilitate the intuitive apprehension of a chain of relations' (1984:13); except that we would be more inclined to regard it as an elliptical representation that needs deciphering and that might contain virtually within itself more than what its author was conscious of having put into it. 15 It may be added that the failure of most of the attempts to apply the formula to empirical data might well be due to the impossibility of expressing the formula literally in terms of functional propositions, for it is as if these unsuccessful attempts had aimed at taking up such a challenge at all costs. 16 One may think of numerous myths in which a natural or cultural benefit springs from the corpse or tomb of an evil creature (or deemed as such): the nighthawk-woman in La potiere jalouse, who makes clay appear when she is smashed on the ground, or the witch in a Tereno myth, on whose grave tobacco grows (Levi-Strauss 1964:108), and so on. 17 In the reference text, Levi-Strauss does not explicitly allude to undifferentiation. However, when he examines the logic of the Pueblo myths, he notes the existence of an invariant scheme, which he describes in the following terms: '[T]he starting point and the point of arrival are the simplest ones and ambiguity is met with halfway' (1955: 438). 18 The process of mediation, understood as a process of undifferentiation, both reaches a peak and comes to an end in the fourth component of the formula. It can be reconstructed in three steps: f (a) -> fa(y) —»f a _! (y). Like its counterpart, the third component fx(b), the first stage, uniting two elements that were separated in the first side of the formula, represents a simple mediation. The mediation process is redoubled in the second stage, in which terms and functions exchange roles in a kind of 'mathematical carnival' or 'ritual masquerade.' The third stage, in which a is replaced by a"1,

Hesiod, the Three Functions, and the Canonical Formula of Myth 153 represents the cancelling and the reversal of the mediation process. Nevertheless, we must admit that this reconstruction, and all the more so its interpretation, are highly conjectural. In fact, the last two stages can be permuted. In that case, it is the substitution of a"1 for a that would bring about the interchange between term and function, which would then represent the reversal of the undifferentiation process. 19 This reduction should not conceal the fact that the Hesiodic myth displays an enriched version of the basic structure A : B :: M : X. Indeed, in its developed form the quadripartite structure of this myth is (D/H): (H/D):: M : X or (D/H) -> (H/D) -> M -> X, in which (D/H) represents an opposition, (H/D) the reversal of this opposition, M the erasure of this opposition, and X the reversal of this erasure, i.e., the genesis of the opposition. The significant point is the distinction made between the operation (D/H) —»(H/D) and the operation (D/H) —> M, i.e., between the reversal of the oppositions, which is generally a ritual operation, and the erasure of the oppositions, which corresponds to the loss of culture and the collapse of the social bond. 20 We recall that in the interpretation put forward by the Marandas the apotheosis of good (y changed into a term) requires the annihilation of an evildoer (a transformed into a"1). Therefore the term a"1 seems to play the same role in their analysis as in Mezzadri's, in which it more precisely represents the anti-king, a character whose expulsion makes it possible to bring the myth full circle by enabling the golden age law-king to return. But the comparison cannot be pushed any further. For in Mezzadri's interpretation, y designates the generalized hubris that engenders the anti-king, while in the Marandas's y designates the good generated by the elimination of the antihero. Sceptics will undoubtedly deduce that, by cancelling each other out, these two interpretations demonstrate once more the futility of any speculation on the canonical formula. However, the text by Vernant that we have added to the discussion suggests another solution. For if the return of the king requires killing (or expulsion) of the anti-king (or of the pharmacos), th two rival interpretations then become not only compatible but complementary. They would actually describe two successive stages of the same overall process: the choosing of a scapegoat by the society threatened by hubris (Mezzadri) and the killing (or expulsion) of this scapegoat (Maranda). This complementarity would imply that, in spite of appearances, the authors of those interpretations both endow the canonical formula with the capacity to model one and the same cyclical process, and the only point at issue would be the capacity of the formula to express simultaneously (at least in its original form) all the aspects of this process. We should also note that

154 Lucien Scubla Burkert, although very cautious as far as the canonical relation is concerned (1979:10-14), cannot help remarking that he finds it particularly well suited to the ritual of the scapegoat. The process can be brought/ he writes, 'into a nearly perfect Levi-Straussian formula, the scapegoat being the mediator who brings about the reversal from common danger to common salvation: the situation "community endangered" versus "individual distinguished" is turned into "individual doomed" versus "community saved," f x (a): f y (b)-»f x 2(b):f x -i(a)'(ibid.:67). 21 While for us the most significant aspect of Mezzadri's article seems to be the hypothetico-deductive nature of his method, he himself tends to gloss it over. He in fact endows hubris with the capacity to produce anti-kings (1988:53) before turning to the canonical formula (ibid.: 54) as if he had obtained this property by drawing 'all the consequences' of Vernant's analyses (53) and realized only afterwards that it achieves the closure of the myth 'at the price of a double twist recalling rather precisely the two characteristics' of Levi-Strauss's formula (54). But after reading Hesiod's text and Vernant's commentaries, there can be no more doubt concerning this issue: it is precisely the application of the canonical formula to the myth of the races - made possible, but not in any way imposed, by Vernant's work that enables Mezzadri to go beyond the results obtained through the textual analysis to which the great historian scrupulously confines himself.

References Burket, W. 1979. Structure and History in Greek Mythology and Ritual. Berkely, Los Angeles, London: University of California Press. Casajus, D. 1990. 'A propos de la formalisation en anthropologie.' Lettre d'information, Association pour la recherche en anthropologie sociale (APRAS), no. 2:1-3. Culioli, A. 1968. 'La formalisation en linguistique.' Cahiers pour I'Analyse 9: 106-17. Delcourt, M. 1944. Oedipe ou la legende du concjuerant. Liege: Bibliotheque de la Faculte de Philosophic et Lettres de 1'Universite de Liege. Hocart, A.M. 1927. Kingship. London: Oxford University Press. - 1954. Social Origins. London: Watts. Le Herisse, A. 1911. L'ancien royaume du Dahomey: moeurs, religion, histoire. Paris: Larose. Levi-Strauss, C. 1955. The Structural Study of Myth.' Journal of American Folklore 78 (270): 428-44.

Hesiod, the Three Functions, and the Canonical Formula of Myth 155 - 1958. Anthropologie structurale. Paris: Plon. - 1962. La pensee sauvage. Paris: Plon. - 1964. Mythologiques I, Le cru et le cuit. Paris: Plon. - 1971. Mythologiques IV, L'homme nu. Paris: Plon. - 1973. Anthropologie structurale deux. Paris: Plon. - 1984. Paroles donnees. Paris: Plon. - 1985. La potiere jalouse. Paris: Plon. - 1991. Histoire de Lynx. Paris: Plon. Maranda, E.K., and P. Maranda. 1971. Structural Models in Folklore and Transformational Essays. The Hague: Mouton. Mezzadri, B. 1988. 'Structure du mythe et races d'Hesiode.' L'Homme 106-7: 51-7. Petitot, J. 1985. Morphogenese du sens: pour un schematisme de la structure. Paris: Presses Universitaires de France. - 1988. 'Approche morphodynamique de la formule canonique du mythe.' L'Homme 106-7: 24-50. - 1992. Physique du sens: de la theorie des singularities aux structures semionarratives. Paris: Editions du Centre National de la Recherche Scientifique. Scubla, L. 1992. 'Sciences cognitives, materialisme et anthropologie.' In Daniel Andler, ed., Introduction aux sciences cognitives, 421-46. Paris: Gallimard. - 1995. 'A propos de la formule canonique, du mythe, et du rite.' L'Homme 135: 51-60. - 1996. 'Histoire de la formule canonique du mythe et de se modifications.' Thesis, Ecole des hautes etudes en sciences sociales, Paris. - 1998. Lire Levi-Strauss: le deploiement d'une intuition. Paris: Odile Jacob. Sperber, D. 1985. On Anthropological Knowledge. Cambridge and Paris: Cambridge University Press and Maison des Sciences de 1'Homme. Thorn, R. 1980. Modeles mathematiques de la morphogenese. Paris: Christian Bourgois. Vernant, J.-P. 1960. 'Le mythe hesiodique des races: essai d'analyse structurale.' Revue de I'Histoire des Religions : 21-54 (reprinted in Mythe et pensee chez les Grecs, I [Paris: Francois Maspero, 1971], 13-41). - 1966. 'Le mythe hesiodique des races: sur un essai de mise au point.' Revue de Philologie: 247-76 (reprinted in Mythe et pensee chez les Grecs, I [Paris: Francois Maspero, 1971], 42-79). - 1970. 'Ambiguite et renversement: sur la structure enigmatique d'Oediperoi.' In J. Pouillon and P. Maranda, eds, Echanges et communications - Melang offerts a Claude Levi-Strauss, vol. 2:1253-76. Paris, the Hague: Mouton. - 1971. See I960,1966 above.

6 Classical Myths and Transformation: Computer Observation of the Levi-Strauss Formula at Work Sandor Daranyi

My study of the texts related to Attis goes back to a short treatise about this Phrygian deity from Asia Minor whose cult was imported to Greece and Rome (Trencsenyi-Waldapfel 1981). In an earlier article, I suggested that these myths are but one subset of otherwise typical stories about flower heroes such as Hyakinthos, Narkissos, Kyparissos, or Adonis, who characteristically are wounded and turned into plants at the story's climax. Following this repetitive action, I also offered a Proppian segmentation of the respective myth variants (Daranyi 1986). My aim with this study is to suggest the applicability of multivariate classification methods for structural investigations. The working hypothesis was that if the methods applied do not distort structural relations pertaining to the material, neither will they introduce foreign effects into their visual representations. The fact that logical oppositions pertinent to a body of myths could be visualized by computer classification indicated that this indeed was the case. 1. Methodological Considerations Before formalizing the corpus for computer classification, I postulated that the structural study of myth as suggested by Claude Levi-Strauss (1958) contains two paradoxical preconditions. The first of these suggests that a myth consists of all of its variants; therefore myths last as long as the process of mediation creating them does. In this case, however, the borders of a set will be impossible to guess, the definition anni-

Classical Myths and Transformation 157

hilating the entity it aspired to define. No wonder that traditional classification eager to grasp solid types runs into trouble when it is trying to apply structural means to a fluid material. With structuralism, both classification based on many content elements instead of some and fuzzy categories are a must; otherwise, the result will be disappointing. A more serious problem is this: myth variants, the rather momentary text states of a coherent mythology, are supposed to come into being by linguistic or analogous transformations of synchronic texts. But transformations embody change, change involves time, and thus any structural attempt dealing with more than one text variant will be as diachronic as those of the non-structural schools. Therefore, the problem we have to address is a kind of diachrony that depends on a sequence of otherwise synchronic variants, the result being an evolutionary process characterized by its temporal cross-sections. Its motor and catalyst of conceptual evolution in the case of myths is text variation, which keeps on producing variants as long as a particular belief system is working. Text variation leading to variants with a modified content can be modelled by sets of specific narrative elements underlying types that undergo transformations, that is, exchanges of elements among sets. Here, a transformation can correspond either to a linguistic rewriting rule or to a mathematical function. To connect this model to multivariate statistics, I suggest that classification techniques are able to extract such specific groups of variables accounting for the relevant phenomena. 2. Structural Invariances in the Attis Myth In the myth of Attis, there exist at least four kinds of structural invariance paired with variability on content level, as follows: (a) specificity and a set sequence of Proppian functions applied to myth (Daranyi 1994: 43), (b) specific and stable correlations among narrative elements and Proppian functions, acting as marker traits, the occurrence of one indicating or predicting the presence or absence of the other (ibid.), (c) a helicoid primary structure, as in riddles, because these myths belong to the problem-solving kind, in accord with Levi-Strauss (Maranda and Maranda 1971: 28), and

158 Sandor Daranyi

(d) in the variants treated as a transformation series, a second-order helix (ibid.: 29). My primary concern here is to show how such invariances can find a visual expression by means of statistical analysis. 3. From Myth Variants to the Input Matrix Multivariate statistical analysis needs input in matrix form, in which the column vectors of a matrix correspond, for example, to text or interpretation variants of a corpus and its row vectors to their distinctive features or aspects of comparison. The input matrix is processed by methods of increasing complexity that look for interrelated groups in the input data. I will confine myself to the use of cluster, factor, and principal component analysis. The abstract Urvariant of the Attis corpus establishes the story pattern characteristic of flower heroes. As stated in my earlier analysis, according to their formalism these myths tell about the supremacy of androgyny as compared to man and woman divided (Daranyi 1986). Following the miraculous birth of a bisexual being, in the course of conflicts he offers his virility to the Mother Goddess and, although dying in her honour, is restored again and again. Due to the circularity of the plot, different narrative elements, functionally equivalent with one another, appear at specific loci of the structure. For example, a deadly stroke can come from a spear, a discus, a knife, or a sickle, and the deadly wound is concentrated on, although not limited to, the lap or thigh of the hero. Also, the plant into which he is turned varies from violets to anemona and pines. We have good reason to believe that according to other myth variants, such interchangeable plants had a fertilizing charm, the substitutions generating new versions of the same message.1 I distinguished among four types of mutilation of the hero as encoded in myths about Attis, Agdistis, Uranos, Adonis, Hyakinthos, Kyparissos, and Narkissos. I also assumed that these variants result from the coincidence of two logical oppositions, 'murder is to suicide what erotics is to homoerotics' and 'other is to self what fertile is to barren.' We can rearrange these binary oppositions in the following matrix form: direct not-direct

self

not-self

Classical Myths and Transformation 159 Consequently, direct self-mutilation [DS] will correspond to emasculation (Attis, Agdistis), direct not-self-mutilation [DNS] to castration (Uranos, Adonis), not-direct not-self-mutilation [NDNS] to killing by accident (Hyakinthos, Kyparissos), and not-direct self-mutilation [NDS] to dying by passion (Narkissos). Due to the homoerotic content of the stories of Hyakinthos, Kyparissos, and Narkissos (in the case of the latter superseded by suicide instead of the normal killing of the hero), I postulated that the direction of transformation is from androgyny to homosexuality, in accord with the relatively late appearence of the latter in Greek mythology. According to this postulated trend, bisexual beings emasculating themselves were replaced by male heroes being castrated. By the end of the process, instead of loving and killing another man, Narkissos loves and kills himself. If we read the above matrix from variants A to C through D, the predicted second-order helix appears (Maranda and Maranda 1971: 29). We can easily convert the above arrangement of logical oppositions in an input matrix where 1 will indicate the presence, and 0 the absence, of a trait: direct mutilation not-direct mutilation self-mutilation not-self-mutilation

A 1 0 1 0

B 1 0 0 1

C 0 1 1 0

D 0 1 0 1

In this representation scheme, type A will correspond to direct selfmutilation, type B to direct not-self-mutilation, type D to not-direct not-self-mutilation, and type C to not-direct self-mutilation. Now we can start analysing the myth variants about Attis. They were collected by Hugo Hepding (1903). From 103 Greek and Latin sources, I chose the following ten: a b c d e f g h

= = = = = = = =

Hermesianax in Pausanias VII, 17, 9 Herodotus I, 34^5 Lucianus, De dea Syria c. 15 Pausanias VII, 17 / Arnobius adversus V, 5 Diodorus, Biblioth. histor. Ill, 58 Ovidius, Fasti IV, 221 Minucius Felix, Octav. 22 / Fulgentius, Mitol. Ill, 5 Servius, Schol. ad Aen. IX, 115

160 Sandor Daranyi Not-direct not-self: killing by accident

Not-direct not-self: celestial love

Not-direct self: death by yearning

Direct not-self: murder

Direct self: suicide"

Not-direct self: ???

Direct not-self: earthly love

Direct self: homoerotics

Figure 1 The cube of oppositions and distribution of narrative elements in factor space

i = Arnobius' Source 2 (Valerius pontifex in Hepding 1903:118) j = lulianus, Orat. V. / Sallustius, De diis et mundo c. 4 Before proceeding, I mention that according to Maranda and Maranda, opposites like male : female, old : young, or rich : poor that structure the European Marchen can be displayed as angles of a cube (Maranda and Maranda 1971: 23). Figure 1 shows the double opposition 'murder : suicide :: erotics (earthly love): homoerotics :: other (not self) : self as a similar shape, its angles standing for these opposites. This cube will be of substantial help when we arrive at the interpretation of factor analytical results. These texts will now be compared according to the list of narrative elements listed below as labels of the row vectors. The binary input matrix appears on the right side. Note that the binary oppositions from type A to C through D and leading to the second-order helix are included in the following input matrix (rows 21-4), acting as marker traits to track inversion of content.

Classical Myths and Transformation 161

1. 2. 3. 4. 5. 6 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

abcdefghij 1000000000 Sex absent due to birth 0011011111 due to mutilation not relevant 0100100000 0100000010 Mutilation by accident as punishment 1011111101 of adultery 1001110001 0011011011 against goddess against god /king 1000100100 of losing virginity 0000110011 because of jealousy 1001001100 hero being worthy 1100001110 goddess being unworthy 0000101110 0010001010 Wound by goddess god /king 1000000100 boar 1100000000 0100000000 'companion' 0000100000 parents himself 0001010011 0100000110 each other 1011101101 Wound deadly 0010100111 Direct self-mutilation [DS] ('A') 0010101000 Direct not-self -mutilation [DNS] ('B') 1100000000 Not-direct not-self-mutilation [NDNS] ('D') 0000000100 Not-direct self -mutilation [NDS] ('C')

4. Classification Results Because of the compound nature of the results, I have to emphasize what I consider to be of importance. First, the distribution of text variants in classification space, based on their factor loadings, was a good approximation of the theoretical possibility of a perfect dispersion. Here, I succeeded in naming the first and third principal component. Second, the agglomeration of factor score values fairly well reproduced the theoretical cube form in figure 1, and the geometrical shape and structure of this distribution compares well with the geometry of the Levi-Strauss formula. 4.1 Material and method I used the BMDP statistical program package for grouping the afore-

162 Sandor Daranyi

mentioned corpus of 10 texts, compared for the presence and absence of 24 distinctive content features (narrative elements) (Dixon 1990). After preparing the input matrix,21 carried out three versions of hierarchical cluster analysis, followed by block clustering and, finally, principal component analysis, maximum likelihood factor analysis, and Kaiser's factor analysis. In this combination of grouping methods, hierarchical clustering produced robust sets of texts with slightly irregular intragroup connections displayed as a dendogram, and block clustering assigned text and feature clusters to each other, suggesting homogenous dispersions of narrative elements over sets of texts. These dispersions corresponded to those content features that, as underlying complexes or clumps, determined types of texts expressed as clusters. Finally, versions of factor analysis were included in the analysis to suggest the underlying reasons of a given classification. For instance, if the classification was shown as a dendogram, the principal components or factors corresponding to the axes of a multidimensional discourse universe reduced to three dimensions were found to correspond to the causes of that particular classification. Generally, the clusters represented as branches of a dendogram appear as clouds of dots in such a discourse space. 4.2 Hierarchical cluster analysis

According to our input matrix, text type A (direct self-mutilation, DS) was represented by Pausanias, Ovidius, Arnobius' Source 2, and lulianus (texts 4, 6, 9, and 10); type B (direct not-self-mutilation, DNS) by Lucianus, Diodorus, and Minucius Felix (texts 3, 5, and 7); type C (not-direct self-mutilation, NDS) by Servius (text 8); and type D (notdirect not-self mutilation, NDNS) by Hermesianax and Herodotus (texts 1 and 2). The results of different cluster analyses - single, complete, and average - were fairly robust, the change of linkage methods leading only to minor intergroup modifications. As shown in figure 2, the computer reproduced types A (texts 4, 6, 9,10) and, partly, B (3 and 7, 5 missing), plus single and complete linkage put aside type C (text 8). For the overall recognition of types C and D, the small training set, altogether three texts, was not enough.

Classical Myths and Transformation 163

Figure 2 Classification of myth variants by hierarchical cluster analysis

164 Sandor Daranyi 4.3 Block clustering

The result of block clustering was already referring to the features' agglomerations later described by factor score values in factor analysis. The blocks of traits, displayed as ones in the context of zeros and homogeneous for a block (group, cluster) of texts, displayed a stratified structure of features as shown on the borderline, for example, between types A and B, underlying different text types. We can see the agglomerations of features constituting clusters of texts in figure 3. This figure also suggests an explanation for the strange phenomenon of discrete variables behaving as continuous ones both here and in factor analysis, the discourse universe of comparison corresponding to a conceptual continuum that is spatially structured by oppositions. The dimensions of comparison, continuous although their aspects represent discrete features, are embedding artificial entities, that is, groups of texts. As distributions the latter become per se discrete, yet they remain continuous at the same time according to their relations to other such artificial entities. 4.4 Factor analysis

4.4.1 Factor loadings reveal logical oppositions To learn about the statistical robustness of the results, I carried out three versions of factor analysis on the input matrix: principal component analysis (PCA), maximum likelihood factor analysis (MLFA), and Kaiser's factor analysis. Although PCA is designed for the causal grouping of variables (column vectors) and FA for that of cases (row vectors), both rely on matrix decomposition, where the resulting matrices correspond to factor loadings and factor scores. I have arbitrarily limited PCA to the extraction of three principal components in order to compare the numerical values of loadings and scores to three-dimensional FA. When they are displayed as coordinates of individual texts (variables) and features (cases), both factor loadings of the variables and factor scores of the cases showed quite similar distributions for all three methods (figures 4-6). In my interpretation, this is clear evidence for the robustness of the classification and its reasons, the factors or principal components providing us with a reliable base for the consequent structural interpretation.

Classical Myths and Transformation 165 X(5) Diodorus X(2) Herodotus X(l) Hermesianax X(8) Servius X(9) Arnobius X(4) Pausanias VII X(6) Ovidius X(10) lulianus c.4 X(3) Lucianus X(7) Minucius Felix

3 13 1 4 16 24 10 14 20 9 18 21 7

1100000000 0000110001 0010000000 0100100000 0100000000 0001000000 0011010001 0011000000 1111111000 1000101100 0000111100 0000111100 0000111111

3 13 1 4 16 24 10 14 20 9 18 21 7

2 5 6 17 22 8

0001111111 1011011111 1010011100 1000000000 1000000011 1011000000

2 5 6 17 22 8

12 19 11 15 23

1001100001 0101100000 0111100001 0110000000 0110000000

12 19 11 15 23

Sex absent not relevant Wound by goddess Sex absent due to birth Mutilation by accident Wound by 'companion' Not-direct self-mutilation [NDS] (type C) Mutilation because of jealousy Wound by god/king Wound deadly Mutilation as punishment for losing virginity Wound by himself Direct self-mutilation [DS] (type A) Mutilation as punishment for adultery against goddess Sex absent due to mutilation Mutilation as punishment Mutilation as punishment for adultery Wound by parents Direct not-self-mutilation [DNS] (type B) Mutilation as punishment for adultery against god/king Mutilation because of jealousy, goddess unworthy Wound by each other Mutilation because of jealousy, hero worthy Wound by boar Not-direct not-self-mutilation [NDNS] (type D)

Figure 3 Classification of myth variants and narrative elements by block clustering

166 Sandor Daranyi

Figure 4a, b Classification of myth variants by principal component analysis

Classical Myths and Transformation 167

Figure 5 Classification of myth variants by maximum likelihood factor analysis

Remembering that types A (4, 6, 9,10), B (3, 5, 7), C (8), and D (1, 2) were supposed to appear in the four opposing quarters of the discourse universe, we can state both that this is more or less the case and that the types constitute an A—» B —» D —>C circle as suggested by the helix interpretation of the Levi-Strauss formula. Further, marking the quarters by the members of the DS : DNS :: NDNS : NDS pair of oppositions - for example, in principal component analysis (figure 4b) - type A = direct self-mutilation (emasculation, DS) will belong to the upper-right quarter, type B = direct not-self-mutilation (castration, DNS) to the lowerright quarter, type C = not-direct self-mutilation (suicide by passion, NDS) probably to the upper-left, and type D = not-direct not-selfmutilation (killing by accident, NDNS) quite likely to the lower-left quarter of the discourse universe. This helix clearly reveals that the name of the first factor, that is, the reason of this grouping, is 'direct: not-direct [effect]/ Taking another example shown in figure 4b, the naming of the third principal component yields that the difference between the upper and lower hemispheres can be expressed as 'self : not-self (in the sense of reflexive versus non-reflexive).

168 Sandor Daranyi

Figure 6 Classification of myth variants by Kaiser's factor analysis

4.4.2 Factor scores corresponding to Levi-Strauss's formula Turning now to the universe of factor scores - in this analysis, the universe of feature dispersions distributing cases along causal dimensions called factors - the cube form of the dispersions is evident, although it is best expressed in the case of MLFA. Because I assumed that texts are grouped into clusters on the basis of feature agglomeration, it is logically consistent to expect that the oppositional structure of texts should rely on the oppositional structure of features. As I will show, this indeed was the case. Marking the angles of the observed cube-like dispersion of factor score values in figure 7 with the symbols cc, 6, y, 5, e, (|), K, and A,, the angles will correspond to the following narrative elements (comparison criteria - see also figure 9): a = = fi = y = = =

2. 7. 5. 9. 18. 21.

Sex absent due to birth Mutilation as punishment for adultery against goddess Mutilation as punishment Mutilation as punishment for losing virginity Wound by himself Direct self-mutilation [DS] (type A)

Classical Myths and Transformation

169

Figure 7 Distribution of narrative elements: principal components analysis

8 = 6 . Mutilation as punishment for adultery = 20. Wound deadly e

= 12. Mutilation because of jealousy, goddess unworthy

= = (|) = = K = = = = = = X = = = = =

13. 22. 10. 11. 3. 4. 16. 17. 19. 24. 1. 8. 14. 15. 23.

Wound by goddess Direct not-self-mutilation [DNS] (type B) Mutilation because of jealousy Mutilation because of jealousy, hero worthy Sex absent not relevant Mutilation by accident Wound by 'companion' Wound by parents Wound by each other Not-direct self-mutilation [NDS] (type C) Sex absent due to birth Mutilation as punishment for adultery against god/king Wound by god/king Wound by boar Not-direct not-self-mutilation [NDNS] (type D)

170 Sandor Daranyl

It is obvious from the above that the groups of narrative elements (comparison criteria) describe well-interpretable, homogeneous stories. The only major misclassification is the exchange of elements 23 and 24 between the groups K and X, indirect emasculation (criterium 24) being characteristic for none of the narrative elements of the K group. The reason for the misclassification may be that the concept of indirect, that is, sublime emasculation goes back to the myth of Narkissos, himself not present in this comparison. Its counterpart in this analysis is Servius' account, according to whom Attis and the king, before the latter could have raped him, castrated each other (Hepding 1903: 117). Although this motif is indirect suicide ('dying by passion/ notdirect self-mutilation = NDS), the killer and the victim are not one and the same person. Therefore, the motif was exchanged by the computer with that of indirect killing ('killing by accident/ not-direct notself-mutilation = NDNS), since for the latter two persons are required too. In the following, we take the angles y, e, K, and A, of the cube as marked by the four oppositions and interpret the pyramid-like structure (figure 9). The argument will run parallel to the geometrical interpretation of the Levi-Strauss formula (Levi-Strauss 1963: 228). Having postulated that, in the chosen set of myths, a transformation from type A to C took place in the sequence of A—» B —» D —»C (figure 8), this would correspond to the transformation DS -» DNS —» NDNS -> NDS as described by the formula fx(a) :/y(b): :fx(b) :/«_i(y). Here, the transformation NDNS -> NDS, that is, not-direct not-self-mutilation ('killing by accident') subliming into not-direct self-mutilation ('dying by passion') would equal the a"1 member of the formula. From this postulate, a geometrical interpretation of the Levi-Strauss formula emerges as follows (figure 10). Let us distinguish between function and term values in the formula in such a way that the functions fx andfy will be represented by two

surfaces, but the terms a and b by two lines. The/x surface is generated by/x(a) and/r(b), the/y surface by/y(a) and/y(b). It is an implication of this geometry that, although the latter expression is not used in the canonical formula,/^(a) =/y(a). The function and term values, assigned to the Levi-Strauss formula (see below), are identical with the names of the relations connecting the angles of the pyramidal shape, that is, the connector between DS and DNS can be identified as their joint intension, 'direct'; between

Classical Myths and Transformation Homoerotics

171

Androgyny III Self

Not-direct self, type 'C (suicide)

Direct self, type 'A' (emasculation)

Not dirre

Not-direct not-self, type 'D' (killing by accident)

Direct

I

Direct not-self, type 'B' (castration) Not-self

Figure 8 A helix interpretation of Levi-Strauss's formula

DNS and NDNS, 'not-self; between DS and NDS, 'self; and between NDS and NDNS, 'not-direct/ respectively. The sequence of transformation is indicated on the right side of the figure. The DS-NDS-NDNS triangle, shaded grey, is generated by the inverted, /a_i(y) member of the formula, as the area encapsulated by the/y surface, and line a. Further, one must keep in mind that a hypothetical fa(y) expression would have mirrored the pyramidal shape into the volume of the/.,. */y surfaces. Note that the fifth and sixth connectors of the shape, between DNS and NDS, and DS and NDNS, were not generated by the Levi-Strauss formula, because there are no logical connections between these oppo-

172 Sandor Daranyi

sitions, respectively. The same holds for the DNS-NDS-NDNS triangle. Hypothetically, however, the latter area could be defined as/fl_i(x), and its mirror image inside the/j y *fsurfaces' volume We can observe that there is a perfect overlapping between the distribution of marker traits, based on their factor scores in the cube of opposition (figure 9), and the above geometric interpretation (figure 10), because the same pyramidal shape, with the same oppositions in the same distribution, is part of both. Further, now we show that the transformation sequence coded in the geometric interpretation is the same as the one of the canonical formula. To this end, we can repeat the primary helicoid structure of Maranda and Maranda by choosing the following distribution of function and term values (Maranda and Maranda 1971: 28).3 Let/x = not-self, fy = self, (a tion, and (b) = direct mutilation. Then,

direct self

direct not-self

(3) NDNS

(DDS

(2) DNS

type D

type A

typeB

not-direct not-self

not-direct self (4) NDS

typeC

Let me reiterate the core of my argument. The transformation sequence suggested in the third and fourth lines above was the one observed by principal component analysis (figures 4,8). This sequence, the secondary helix, was depending on the underlying factor-score distribution observed by three statistical methods. The factor-score cube, on the one hand, was congruent in terms of shape with the one resulting from the geometrical interpretation of the Levi-Strauss formula; given the congruency of postulated and observed geometrical shape, statistical classification did not introduce distortions in the analysis and preserved the initial structural relations intact. QED. On the other hand, logical oppositions extracted from the discourse universe as term and function values could be distributed in such a way that the formula reconstructed a known type of primary structure as well. Note, however, that other primary structures also seem to exist, and

Classical Myths and Transformation 173

Figure 9 Distribution of marker traits, based on their factor scores in the cube of opposition

that the factor-score dispersions underlying oppositions were computed from a set of myth variants. Therefore, the displacement of term and function values in the formula depends on the range and consistency of the set of variants. 5. Discussion of the Results Basically, I see three major problems with these results. The first is that, with the use of presence/absence data, Boolean factor analysis may be better suited for the analysis of a binary matrix. Second, the real issue is how one should design a set of comparison criteria in order to arrive at the visual presentation of structural commonplaces such as oppositions opposing one another, as was the case in this study. Third, the challenge is that these results are highly interpretable, although they should not be. For example, MLFA is a method that

174 Sandor Daranyi

Figure 10 Geometrical interpretation of the Levi-Strauss formula

relies on the concept of a maximized likelihood, itself based on a normal distribution of the values of variables. It is simply not probable that, in a small database like this input matrix, the variables would have such a distribution. Yet MLFA led to the most precise representation of the cube of oppositions, questioning the standard statistical interpretation of these methods. The other possibility is that the cube as computed was a matter of accident. PC A and Kaiser's factor analysis resulted in similar though less accurate cubes, however, which again raises the question: if in terms of mythological text analysis the orthodox application of statistical techniques is leading to hardly interpretable results and the unorthodox one to highly interpretable ones, which way shall we go from here? All I want to suggest is that we will have to carry on the investigation of such questions, whatever the implications may be. Notes This article was based on a paper written in collaboration with Andor Abranyi, originally read at the 'Working Symposium on the Sacred Narrative/ Budapest (Daranyi and Abranyi 1988). I am most grateful to Otto Hajdu, University of Economics, Budapest, for his critical comments on the nature of multivariate methods and the limits of my approach. The original data were visualized in three dimensions. 1 For a related treatment of this material, see Daranyi 1996.

Classical Myths and Transformation 175 2 I must make the reader aware of the unorthodox way I am using multivariate methods. Regularly, the rows of an input matrix equal cases or observations (entities), whereas its columns stand for the observed variables (features). For practical reasons, I filled in this matrix vice versa, the text variants corresponding to column vectors. In the case of principal component and factor analysis, this transposition leads to interpretation problems, for within the classical frame of thought, if texts are variables and features equal cases, then features consist of texts, which is nonsense. This top-down argument is based on the concept of entities consisting of their features or, in the bottom-up way, features constituting entities as their specific agglomerations. Since different patterns of features underlie different entities, in my approach factor analytical methods can replace one another in spite of their opposite statistical design. This approach is legitimate because factor loadings and factor scores will under all circumstances reproduce the causal dependence of entities on their traits. These and some other problems of interpretation will, however, have to be addressed in a future paper. 3 Note that other distributions lead to variants of the helix. E.g., itfx = /„ = not-self, (a) = direct mutilation, and (b) = not-direct mutilation, then it runs opposite to the originally mediated process.

References Daranyi, S. 1986. 'Meghalo istenek a virag-mitoszokban ["Dying Gods" in Flower Aetiology Myths].' Ethnographia 97 (1): 25-44. - 1994. 'Valtozo es valtozatlan. A termeszetes hiedelmi rendszerektol a kognitiv konyvtarig [Variant and Invariant: From Natural Belief Systems to the Cognitive Library Model].' PhD thesis, Budapest. - 1996. 'Formal Aspects of Natural Belief Systems, Their Evolution and Mapping: A Semiotic Analysis.' Semiotica 108 (1/2): 45-63. Daranyi, S., and A. Abranyi. 1988. 'Attis, Propp, Levi-Strauss: Modelling Aspects of Greek Mythology.' Working Symposium on the Sacred Narrative, Budapest. Dixon, W.J., editor. 1990. BMDP Statistical Software Manual. Berkeley: University of California Press. Hepding, H. 1903. Attis, seine Mythen und sein Kult. Giessen: Topelmann. Maranda, E.K., and P. Maranda. 1971. Structural Models in Folklore and Tra mational Essays. The Hague: Mouton. Levi-Strauss, C. 1958. The Structural Study of Myth.' In Sebeok 1958: 50-66.

176 Sandor Daranyi - 1963. Structural Anthropology. New York: Basic Books. Sebeok, T.A., editor. 1958. Myth: A Symposium. Bloomington: Indiana University Press. Trencsenyi-Waldapfel, 1.1981. Valldstbrteneti tanulmdnyok, 314-17. Budapest: Akademiai Kiado.

7 Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic Christopher A. Gregory

The laws of logic which ultimately govern the world of the mind are, by their nature, essentially invariable; they are common not only to all periods and places but to all subjects of whatever kind, without distinction even between those that we call real and the chimerical; they are to be seen even in dreams. Comte, quoted in Levi-Strauss 1963: 67

In this article I want to consider a suggestion first put forward by some linguists1 that Claude Levi-Strauss's thought can be seen as a partial rehabilitation of the Rarnist tradition of thought that flourished in the sixteenth and early seventeenth century.2 This tradition covered grammar, rhetoric, and logic, but I restrict myself to the latter here. The Ramist logical tradition, which has its origins in Aristotle's Topics, was superseded by the rise of the what is now called 'formal' or 'mathematical' logic. The work of Descartes and Leibniz in the seventeenth century was the beginning of the end of Ramist logic; by the time of Kant and Boole in the nineteenth century, all traces of this tradition were gone completely. Boole's binary logic (1854), the basis of the present-day computer logic, is regarded as the most important development in logical thought since Aristotle.3 However, Levi-Strauss's 'binary logic' is Ramistic not Boolean, pre-Cartesian not post-Cartesian. This, at least, is what I shall argue. I have been unable to find any reference to Peter Ramus's logic in Levi-Strauss's work. By contrast, references to Kant abound, which

178 Christopher A. Gregory suggests Levi-Strauss saw himself as working within a post-Cartesian logical framework. If Levi-Strauss has rehabilitated Ramus, then it has been unconscious on his part. But the thesis seems plausible when it is recalled that the last great Ramist logician was John Milton,4 the author of Paradise Lost. This is one of the great mythical epics in the English language and like all of Milton's poetic work it has been constructed using the tools of Ramist commonplace logic.5 Could it be that Milton's commonplace logic of myth is related somehow to Levi-Strauss's mythlogicl The only surprise would be if it wasn't. Levi-Strauss's 'canonical formula of myth/ which a recent interpretator (Mosko 1991) had to change in order to decode, is logical from a Ramistic point of view. Locating the formula within this tradition enables us to see it for what it is, an important and original contribution to the theory of logical similitude. Two Types of Logic Ramist logic begins with a distinction between invention and disposition. Following Milton, this can be understood as the difference between commonplace** logic and axiomatic logic. Logic of the latter kind is based on axiomatic contradiction, and Boole's binary logic is classic example of this form of reasoning. Commonplace logic is based on c monplace opposition. What is at stake here is two different kinds of 'binary' logic and this is best illustrated by means of a simple example. From the point of view of axiomatic logic a 'not-snake' is the opposite of a 'snake.' This follows from the principle of axiomatic contradiction, which holds that nothing exists that is both a snake and a not-snake. The universe of discourse, then, consists of things that are either snakes or not-snakes. Logic of this kind is abstract and requires no appeal to history or anthropology to establish its validity. The principle of axiomatic contradiction is the supreme rule for judging if an inference is correct or not. From the point of view of commonplace logic, many things can be the opposite of snake. In this culture it may be a 'mongoose/ in that a 'greyhound/ and so on. Logic of this kind is inherently doubtful and only holds within clearly defined cultural parameters. In Indian mythology, for example, the opposite of a 'snake' is a 'mongoose.' The story of the mongoose and the snake forms part of the Panchatantra,7 the famous collection of stories believed to have been composed over 2000 years ago. In this story the wife of a Brahman mothered a single

Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 179

son and a pet mongoose. She cared for the mongoose like a son, giving him milk from her breast. However, she never fully trusted the mongoose. One day she went out to collect water and told her husband to look after the child. But, after a while, the husband went out to beg for food. A black snake crawled towards the baby's cradle, but the mongoose attacked it and tore it to pieces. Delighted with his deed, he went out to greet the mother with blood dripping from his mouth. When the mother saw him coming, she thought that he had killed her baby boy and she killed the mongoose by dropping the jar of water on him. When she found the baby safe, she was overwhelmed with sorrow because she had thoughtlessly killed the mongoose, the animal she raised as a son. When her husband came home from his begging tour with a bowl of gruel, she blamed him, saying, 'Greedy! Greedy! Because you did not do as I told you, you must now taste the bitterness of a son's death, the fruit of the tree of your own wickedness. Yes, this is what happens to those blinded by greed.' Levi-Strauss has stressed the importance of a knowledge of natural history when trying to understand tales of this kind. Here we find that the mongoose and the snake are natural enemies and that whenever one strays onto the territory of the other a deadly fight invariably ensures. The mongoose is usually the winner, but not always. The mongoose depends on speed and agility, darting upon the head of the snake and cracking the skull with a powerful bite, and is especially successful against cobras, who are slow in striking and apparently unable to strike at close quarters. They are less successful against a viper, such as a rattlesnake, which has a very rapid strike and return to the pre-strike defensive position. Interestingly, the authorities8 usually cite Kipling's tale of Rikki Tikki Tavi in his Jungle Book9 as the best description of the battle between mongoose and snake. 'If you read the old books of natural history,' writes Kipling, 'you will find they say that when the mongoose fights the snake and happens to get bitten, he runs off and eats some herb that cures him. That is not true. The victory is only a matter of quickness of eye and quickness of foot, - snake's blow against mongoose's jump - and as no eye can follow the motion of a snake's head when it strikes, that makes things much more wonderful than any magic herb.' The story has inspired not only Indian storytellers but Indian logicians10 as well. We find that the 'dynamic opposition of killer to killed' plays an important role in certain Indian logical traditions. One scholar who has investigated the wide distribution of the tale of

180 Christopher A. Gregory

the mongoose and the snake argues that 'it is possible to trace every step of the wanderings and to localise exactly the changes from mongoose to weasel to dog, from snake to wolf (in the final form of the Welsh tradition), and from Brahman to officer or knight or nobleman' (Emeneau 1940: 507). In thirteenth-century France it assumed the following form. One day a knight and his wife went out leaving the baby boy with their nurse. She, in turn, went out leaving the baby alone in a cradle with the pet greyhound. A huge serpent entered the house and was about to attack the baby when the greyhound saw it and chased it. He eventually killed it but was badly hurt in the process of saving the baby. The cradle was knocked over in the fight and blood was everywhere. When the nurse arrived home she thought the dog had killed the baby and let out a scream. Her screams attracted the mother who drew the same conclusion. So too did the knight when he arrived and drew out his sword and killed the dog. When they realised their error they threw the dog into a well in front of the manor door, covered it with stones, and planted trees in memory of the event.11

The idea informing Emeneau's analysis is that as a story spreads it acquires a local coloration and name that reflects the natural and cultural geography of the area in question. In thirteenth-century France, for example, dogs were generally disparaged, but the greyhound was an exception for it symbolized chivalric virtues such as faith, occupations such as hunting, and the male aristocratic way of life generally. Another exception was the little lapdog, symbol of conjugal fidelity and domestic virtue, which was associated with women. If the lapdog was the complementary opposite of the greyhound, then the snake was its antagonistic opposite. Snakes dwelt in the hollows of old walls and were unwelcome visitors on a lord's estate.12 Thus, the opposition of greyhound to snake is a figurative expression of cultural history, whereas the opposition of mongoose to cobra has its roots in natural history. The underlying idea is the same: two sensible bodies cannot be in the same place at the same time without a struggle. Milton (1982: 254), following Aristotle, formulates this 'law' of commonplace opposition as follows: Opposites cannot be attributed to one and the same thing by reason of the same thing, with reference to the same things and at the same time. From whence comes the dictum that 'opposites eliminate each other.'

Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 181

Commonplace logic, then, is a concrete logic of sensible beings (real or feigned) and not an abstract logic of axioms. A further difference is that whereas axiomatic logic is an All (1) or Nothing (0) logic with no middle terms, commonplace logic is full of 'middle' terms. Or, to put it another way, if axiomatic thought is a matter of black and white, then commonplace logic has every colour of the rainbow. Just as in the colour cube, a binary logic informs the logical classification of these concrete qualities. However, the binary logic that opposes red, yellow, and blue to green, violet, and orange is not the same as the 'off-on' binary logic of the light switch. Levi-Strauss's binary logic, I assert, is of the rainbow variety. For him the canonical formula is a logical tool for understanding mythical transformations of the mongoose-into-snake type, but this is by no means the only thing it is useful for. Moreover, it is not a tool that can be mechanically applied. Commonplace logic, the genus of which mythlogic is a species, is an art that, like painting, requires practice to master. It is one thing to know the logic of a colour cube, but quite another to use this logic to produce a masterpiece. Commonplace logic is a art of this kind, but it is one that, unlike painting, has fallen into desuetude in post-Cartesian Europe. As a result, the thought of the pre-Cartesian European is, perhaps, more alien to the anthropologist than the thought of the American Indian. What is Commonplace Logic? Commonplace logic consists of a series of arguments that can be drawn upon to establish belief in an argument that is in doubt. To take one of Cicero's examples (1960:420), if a woman was bequeathed silver by her husband, is she entitled to the coin he left? In other words, is coin silver? Today the answer would be 'No/ because no agreeing argument can be found to link the antecedent, coin, with the consequent, silver. However, in Cicero's day, an affirmative answer existed because coin was made from silver. Lawyers could argue that coin was silver because coin was a species of silver. In other words, the proposition 'coin is silver' was defensible because of the commonplace argument from species. Argumentation of this kind makes explicit what is implicit in a proposition. However, this type of reason appeals to the geographical, historical, and anthropological context of the utterance rather than to the self-evident axiomatic truth of an argument. Commonplaces, then, are like axioms in that they are 'obviously' true, but the former are

182 Christopher A. Gregory 'other-evident' rather than self-evident. Commonplaces establish linkages between antecedents and consequents, and figure 1 shows how the argument from species does so for this particular example.

Figure 1 More generally, commonplace logic enables us to say that if term x then term y because of commonplace argument 2. Arguments of this kind are more or less persuasive and never certain. In other words, they are always subject to doubt. To take another example from the theory of totemism, when an aboriginal informant says 'My kaga (mother's brother) is an eaglehawk,' the utterance can be thrown into doubt by asking: 'Is kaga an eaglehawk?' The structuralist anthropologist13 notes that kaga is the opposite of kuling (sister's son), that an eaglehawk is the opposite of a crow, and that the differences are similar. In other words, 'it is not the resemblances, but the differences which resemble each other' (LeviStrauss 1963:149). Commonplace arguments of this kind employ two arguments, opposition and similarity, to create a compound argument of the kind shown in figure 2.

simila

oppposite

eaglehaw

ccrro

opposite

kuling

kaga

nature

culturre

Figure 2

Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 183

Thus, it is the argument from similar that brings about the transition from nature to culture. Like the argument about coin and silver, this argument is problematic. It has to be grounded in the natural and cultural history of the Australian Aborigines before it is persuasive, and even then it remains subject to question. It suffices to note here that all the terms are similar because they are all meat eaters; they are different in that the eaglehawk is a hunter and the crow a scavenger, as with the relationship between a mother's brother and a sister's son. Levi-Strauss's constant appeal to the notion of 'correlation and opposition' is, I assert, a commonplace argument of this two-tiered kind. The canonical formula of myth is a complicated five-tiered variation of it. In order to demonstrate this it is necessary to introduce, very briefly, the tools of commonplace argumentation. The Tools of Commonplace Logic The Ramist logician saw himself as an artisan who employed certain tools to transform raw materials into an end that served some useful purpose. These tools consisted of some twenty or so commonplace arguments neatly arranged on a binary tree diagram for quick retrieval. These arguments contained four main genera: agreeing arguments, disagreeing arguments, comparative arguments, and derived arguments. These four genera were the logical bases of the four main tropes: metonymy, irony, metaphor, and synecdoche respectively (Adams 1983). Figure 3 summarizes the organization of the first book of Milton's logic, showing the link to the second book on axiomatic logic. Levi-Strauss's logic is concerned for the most part with disagreeing and comparative arguments. One could develop a critique of his notion of opposition by contrasting his use of this rather blunt instrument with the razor-sharp distinctions the Ramists draw between contradiction, privation, adverse, relative, and disparate. Such a critique is anticipated by Levi-Strauss himself, who has long been aware of the very loose meaning he gave to key terms such as inversion (Levi-Strauss 1995: 186). Of interest, too, is his observation that different modes of opposition belong to heterogenous categories and that they 'degenerate all the time in the course of mythical transformations. An opposition of contradictories becomes an opposition of contraries, which in turn turns into a mere difference in degree' (ibid.). Milton's logic captures such subtle variations with a series of discrete but finely graded distinctions that range from absolute sameness (efficient cause) to radical opposition

Figure 3 cause full agreeing simple

efficient material formal end

effect subject partial adjunct diverses

disagreeing

disparates opposites

primary

quantity comparative quality nominal commonplace arguments

secondary

logic axiomatic arguments

testimony

divine human

relatives adverses

negating

contradictories privatives

contraries equal

inherent

affirming

unequal • similar dissimilar

more less

conjugation notation distribution

cause and effect subject and adjunct

definition

perfect description

real

parts and integral wholes generals and specials

Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 185

(privation). However, as it would take a book to develop this critique, I will restrict myself here to an elucidation of just one of Milton's commonplace arguments, that of the similar. Milton on the Commonplace Similar 'Similar things,' notes Milton (1982: 284), 'are those whose quality is the same.' Quality here refers to logical quality and is to be distinguished from logical quantity, the basis of equal arguments. There is, he notes, a great affinity between equal things and similar things. The differ in this: 'equal things do not admit of superiority or subordination, while similar things do: for even the most highly similar things can be greater or less, while equal things cannot.' He divides similar arguments up into those that are contracted and those that are explicated. The explicated, in turn, are divided into the continuous and the disjuncted. These three type are distinguished by the number of terms involved in the argument: two, three, and four respectively. An example of contracted similarity is 'my father is a god.' This has the general form x : a,

where x is the antecedent 'my father' and a the consequent 'a god.' A continuous similarity occurs when the first term is to the second so the second is to the third, as in 'As laws govern the magistrates, so the magistrates govern the people.' This has the general form x : a :: a : b.

Milton calls the first part (x : a) the proposition and the latter (a : b) the reddition. An example of the disjunctive similarity is 'as the helmsman is to the storm, so is the wise man to fortune.' The proposition here has the form x : a and the reddition y : b, which gives x : a :: y :b.

These three forms of the similar argument are very familiar even if the language and the context is not. The logic involved is that of pro-

186 Christopher A. Gregory portion and, as Milton notes, the terms are subject to alternation and inversion of the type that occurs in mathematics. My argument is that the canonical formula is a new fourth form of the argument from the similar. What distinguishes it is that it has five terms, where the fifth term is the commonplace opposite of one of the others. It has the general form

(x : a): (y : b):: (x : b): (m : y), where m is the opposite of a and can be written, following Levi-Strauss, as14

m :: (1 : a). Substituting gives the canonical formula: (x:a):(y:b)::(x:b):((l:a):y). This is a complex commonplace argument that can be doubted by asking, 'Is the proposition (x: a): (y: b) similar to the reddition (x:b):((l:a):y)?' This doubt can be removed by treating the problem as a mathematical one or a logical one. As a mathematical problem it is the simple one of musical arithmetic. If x = 1, b = 2, a = 1.33, and y = 1.5, then the proposition reduces to 0.75/0.75 = 1. Substituting into the reddition gives 0.5/0.5 =1, which is similar to the proposition. Thus we must conclude, pace some mathematically inclined critics,15 that the formula is mathematically sound. Of more interest is the logical and anthropological implications of the formula. From a commonplace logical point of view the argument has the form shown in figure 4. The fourth argument is opposition, which means that an additional four commonplace arguments have to be found so that the proposition is similar to the reddition. This is the task that Levi-Strauss sets himself in The Jealous Potter. The Commonplace Logic of the Canonical Formula of Myth In The Jealous Potter Levi-Strauss postulates the existence of a transformation in five stages:

Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 187

Figure 4

woman (b) —> jealousy (x) —» pottery (y) —> goatsucker (a) —» ovenbird (m) Having established his five terms from an analysis of the ethnography and mythology, he argues that 'the jealous function of the goatsucker is to the "potter" function of the woman as the "jealous" function of the woman is to the "reversed Goatsucker" function of the potter' (1988: 56). The five commonplace arguments implicit here can be formulated as five questions: 1 2 3 4 5

Is the goatsucker (a) bird a jealous (x) bird? Are women (b) potters? (y) Are women (b) jealous? (x) Is the ovenbird (m) the opposite of the goatsucker (a) bird? Is the ovenbird (a) a potter (y)?

Levi-Strauss spends some fifty pages trying to establish these premises by means of an examination of the ethnographic, mythological, and zoological evidence. As my concern here is with the logical form of his argument rather than its anthropological content, I will summarize

188 Christopher A. Gregory this material very briefly and concentrate on the implicit commonplace arguments of his analysis. Levi-Strauss answers the first question by drawing on natural history. Scientific observation reveals that the goatsucker bird is a lonely bird of the night. It has a huge mouth that it uses for catching insects on the wing. This fact is reflected in the mythology, where the bird is invariably depicted as sad, lonely, jealous, or greedy. From the point of view of commonplace logic the goatsucker bird is to jealousy as commonplace subject is to commonplace adjunct. This is an agreeing argument that connects the antecedent 'jealous' with the consequent 'goatsucker' by means of the argument from adjunct. (One could, of course, have the goatsucker bird as the antecedent, in which case the argument from subject is the one that links the terms. This would require systematic reversal of the whole formula, something that Levi-Strauss does upon occasion.) The second premise comes from ethnographic observation: making pots is usually women's work in South America. As the makers of pots women are the efficient cause of pottery. Pottery, then, is an effect of women. This is an agreeing argument. These first two arguments set up an opposition between an agreeing argument (x : a) and an agreeing argument (y : b) to form a Miltonic proposition, (x : a) : (y : b), that opposes an argument about the emotional disposition of a bird to one about the occupation of women. Are women jealous? Levi-Strauss does not address this third question directly. The key myth in the book is about a jealous man; it poses the question of how a myth about a jealous man provides the basis of Levi-Strauss's argument about the jealous woman. The argument that Indian men are jealous is plausible because of the system of polygamy they practise. The weakness of this argument enables one to advance an anthropological critique of the argument in The Jealous Potter. However, as my concern here is with Levi-Strauss's logic I will save this critique for another paper. The commonplace argument that enables him to argue that women are jealous (x: b) is the adjunct. Note that this adjunct is transferred from the goatsucker bird to the woman as we move from the left-hand side of the formula to the right. The fourth premise is established with reference to the scientific record. The ovenbird is a very happy bird of the day that lives with its mate, while the goatsucker bird is a sad bird of the night that lives by itself. In terms of commonplace logic, there is a commonplace opposition between these two birds. The precise nature of this opposition is debat-

Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 189

Figure 5

able. Is it a relational, an adversarial, a contradictory, or a privative opposition? Levi-Strauss is satisfied with vague terms like 'reverse' and 'opposite/ but Ramistic logic demands, and can supply, a range of fine distinctions to choose from, as the summary diagram of Milton's logic above (figure 3) suggests. I leave open the question as to the precise nature of the opposition involved. The fifth premise requires that the ovenbird be a potter. Scientific observation once again supports this argument. The bird lives in a large nest of clay that looks like a clay pot. The ovenbird as maker of this clay nest is, mythopoeically speaking, a master potter. In terms of Ramist logic the ovenbird is the efficient cause of pottery, that is, an artisan that creates things from clay (the material cause). Note that the reddition transfers this occupation from the woman to a new bird by a process of 'double twisting.' The first twist, from the point of view of commonplace logic, transforms an effect into a cause. The second twist changes the order of realms being compared: nature is antecedent in the proposition but consequent in the reddition. Figure 5 summarizes the foregoing argument. This diagram shows the five levels of logical analysis on which the formula operates. At the highest level, the argument from the common-

190 Christopher A. Gregory place similar is employed. At the second level, the continuous similar is implicitly employed. Here the three terms are nature (the jealous goatsucker), culture (the jealous woman potter, la potiere jalouse), and nature' (the ovenbird potter). At the third level, there is a double use of the disjunctive similar. The proposition and reddition, considered by themselves, are arguments from similar that link two terms from nature with two terms from culture. At the fifth level, an argument from opposition links two terms from nature. Levi-Strauss's formula conflates all these levels into one and clothes it in a language of his own. However, I hope I have shown that his private language of his mythologic is fully translatable into the more publicly accessible, if archaic, language of commonplace logic. One doubt remains: How can we be sure that the proposition is similar to the reddition? How can one prove that Levi-Strauss's 'double twist' preserves the similarity? The argument is a commonplace and, as such, is obvious. However, commonplaces only become so when all reasonable doubt of a logical nature is removed. I have already given one analogy from musical arithmetic above. For the reader not persuaded by this illustration, let me provide another. Suppose that the first four terms are Sunday (x), Wednesday (a), Thursday (y), and next-Sunday (b). What is the fifth term (m)? LastThursday, because this is the commonplace opposite of Wednesday in the sense that Wednesday is four days (inclusive) after Sunday and lastThursday is four days before Sunday. In other words, Wednesday is the opposite of last-Thursday with respect to an invariant Sunday. The first part of the formula says 'Sunday is to Wednesday as Thursday is to next-Sunday.' This formulation contains an 'opposition and a correlation.' The opposition sets the first half of the week against the last and the correlation says that both parts of the opposition are similar because they both include four days. The second part of the formula says 'Sunday is to next Sunday as last-Thursday is to this Thursday.' The opposition here is the two qualitatively different types of week; the correlation is that both include eight days. The equivalence of the left-hand side of the formula with the right is brought about by the appearance of the fifth term, last-Thursday, and the elimination of its opposite, this-Wednesday. This substitution is legitimate because the resulting argument says that 'this four days is to that four days as this eight days is to that eight days.'

Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 191

Figure 6

Figure 6 illustrates the argument graphically. It is clear from figure 6 that the proposition (x: a :: y: b) is similar to the reddition (x: b :: (I/a): y) in the sense that the relation of this four days to that four days is similar to the relation between this eight days and that eight days. Or, in mathematical terms, (4 : 4) :: (8 : 8). The terms of Levi-Strauss's 'reversed Goatsucker' argument shown in figure 5 can be mapped directly onto those in figure 6. This example demonstrates that a rigorous, and novel, logic of similitude underlies Levi-Strauss's canonical formula. The canonical formula of myth, I therefore assert, is a commonplace of logical similitude, a tool of logical thought that needs to be added to the Ramistic toolbox. Conclusion The binary logic informing commonplace logic is not the binary logic of a

192 Christopher A. Gregory

Boole's axiomatic kind. Ramist logic does not deny the importance of axiomatic contradiction, but it does give it a secondary place (as the summary outline in figure 3 above suggests). Boole's logic, by way of contrast, has no place for commonplace contradiction: axiomatic contradiction is the sole principle upon which this logic rests. But why? Why do axiomatic logicians deny the existence of commonplace contradiction? It is one thing to dispute the primacy of commonplace contradiction, but quite another to deny it altogether. The two traditions of thought were seen as complementary for thousands of years. Why, then, did the post-Cartesians deny this complementarity?16 What is the logic of this position? What is the history of the rise of axiomatic contradiction and the fall of commonplace contradiction in the history of European logic? How has this affected anthropological thought? Could it be, for example, that Levi-Strauss's theory of 'two modes of thought' is an artefact of the axiomatician's denial thesis? If so, is he talking about two modes of thought or two modes of complementary binary logic? If Levi-Strauss has partially rehabilitated Ramist logic he has also partially not. But Ramist logic does not need to be fully rehabilitated because the task is to move ahead by looking backwards. The time has come for anthropologists to reflect critically on the logic of the history of European logic, because if Levi-Strauss's work has solved the problem of 'their' logic he has posed the problem of 'ours.' European axiomatic logic is logical, but its history is not. Anthropological thought has not escaped the implications of this paradox. Notes 1 Kukenheim (1962:18) suggests that Peter Ramus (1515-72) was a precursor to structuralism in that he found order in chaos. See also Robins 1967: 102. 2 See Ramus 1574 for an early English translation of his logic. Ong (1958, 1978-80) and Howell (1961) locate Ramus's work in the history of European thought. 3 C.S. Peirce argued that Boole's Laws of Thought was 'destined to mark a great epoch in logic; for it contains a conception which in point of fruitfulness will rival that of Aristotle's Organon' (Peirce 1984: 224). 4 He wrote his logic in Latin and published it in 1672. It was first translated into English in 1935 (see Milton 1935) and again in 1982 (Milton 1982).

Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 193 5 See, e.g., Adams (1983:18), who notes that 'in the work of a poet-logician like Milton, the schematic practice of developing tropes out of the topics of invention [i.e., commonplaces] must have resulted virtually from a mental habit.' 6 'We have moved far from the time when the term "commonplace" had a good rather than a depreciatory meaning. Through the ancient world, the medieval world, the Renaissance, and even later, it was thought honorable and indeed commendably enterprising to avail oneself of preprocessed material and modes of expression. Indeed, laying in a store of such material was regarded as in great part the aim of education. Only as the modern world took shape did men find it difficult to tolerate openly the seeming lack of originality which such a tradition fostered. Gradually the concept of "commonplace" was downgraded until it came to mean only something worn, used up, generally worthless' (see Ong's preface to Lechner 1962). 7 Ryder 1972:432-3. 8 See, e.g., Encyclopaedia 1980, in particular Micropaedia 3:934 and 7: 995. 9 See Kipling 1959:163ff.; the following quote comes from p. 174. 10 See Stcherbatsky 1962:407. 11 Schmitt 1983: 5. 12 Ibid.: 59-60. 13 See Levi-Strauss (1963:160-5) for a critique of Radcliffe-Brown on this matter. 14 Levi-Strauss writes m = a"1, which, mathematically speaking, is the same as m :: (1: a). Strictly speaking, one should write m :: (x : a), where x is the invariant term likened to the number 1 by the argument from equal. In other words, m is the opposite of a with respect to x. This is illustrated below, where Levi-Strauss argues that the ovenbird is the opposite of the goatsucker bird with respect to jealousy. 15 See Hage and Harary (1983:131), who argue that the cannonical formula is a tool of rhetoric not of analysis because it doesn't add up. 16 The denial is implicit in the following quote from Leibniz (1714:184), wherein a species of contradiction, the principle of axiomatic contradiction, is equated with the genus: 'Our reasonings are based on two great principles: the principle of contradiction, by virtue of which we judge to be false that which involves a contradiction, and true that which is opposed or contradictory to the false; and the principle of sufficient reason, by virtue of which we consider that no fact can be real or existing and no proposition can be true unless there is sufficient reason why it should be thus and not otherwise, even though in most cases these reasons cannot be known to us.' This argument became the unquestioned orthodoxy by the time of Boole.

194 Christopher A. Gregory References Adams, B.S. 1983. 'Milton Metaphor and Ramist "Invention": The Imagery of the Nativity Ode.' In J.D. Simmonds, eds, Milton Studies. Pittsburgh: Pittsburgh University Press. Boole, G. 1854. An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities. New York: Dover Publications Inc., 1951. Cicero, M.T. 1960. Topics. Trans. H.M. Hubbell. London: William Heinemann. Emeneau, M.B. 1940. 'A Classical Indian Folk-Tale as a Reported Modern Event: The Brahman and the Mongoose.' Proceedings of the American Philosophical Society 83:503-13. Encyclopaedia 1980. The New Encyclopaedia Britannica. London: Benton. Hage, P., and F. Harary. 1983. Structural Models in Anthropology. Cambridge: Cambridge University Press. Howell, W.S. 1961. Logic and Rhetoric in England, 1500-1700. New York: Russell & Russell. Kipling, R. 1959. The Jungle Book. London: Macmillan. Kukenheim, L. 1962. Esquisse historique de la linguistique fran$aise et de ses rapports avec la linguistique generale. Leiden: Universitaire Pers. Lechner, J.M. 1962. Renaissance Concepts of the Commonplaces. Westport, Conn.: Greenwood Press. Leibniz, G.W. 1714. 'Monadology.' In G.H.R. Parkinson, ed., Leibniz Philosophical Writings. London: J.M. Dent & Sons. Levi-Strauss, C. 1955. 'The Structural Study of Myth.' Journal of American Folklore 68: 428-44. - 1963. Totemism. Harmondsworth: Penguin. - 1988. The Jealous Potter. Chicago: University of Chicago Press. - 1995. The Story of Lynx. Chicago: University of Chicago Press. Milton, J. 1935. A Fuller Institution of the Art of Logic, Arranged after the Method of Peter Ramus (1672). Trans. A.H. Gilbert. New York: Columbia University Press. - 1982. A Fuller Course in the Art of Logic Conformed to the Method of Peter Ramus (1672). Trans. W.J. Ong and CJ. Ermatinger. New Haven: Yale University Press. Mosko, M. 1991. The Canotic Formula of Myth and Nonmyth.' American Ethnologist 18 (11): 126-51. Ong. W.J. 1958. Ramus, Method, and the Decay of Dialogue: From the Art of Discourse to the Art of Reason. Cambridge, Mass.: Harvard. - 1978-80. 'Logic and Rhetoric.' In W.J. Hunter, ed., A Milton Encyclopedia. Lewisburg, Penn.: Bucknell University Press.

Ramistic Commonplaces, Levi-Straussian Mythlogic, and Binary Logic 195 Peirce, C.S. 1984. Writings of Charles S. Peirce: A Chronological Edition. Vol. I: 1857-1866. Bloomington: Indiana University Press. Ramus, P. 1574. The Logike of the Moste Excellent Philosopher P. Ramus Martyr. Trans. M. Roll. New York: Da Capo Press, 1969. Robins, R.H. 1967. A Short History of Linguistics. London: Longmans. Ryder, A.W., editor. 1972. The Panchatantra. Chicago: University of Chicago Press. Schmitt, J.-C. 1983. The Holy Greyhound: Guinefort, Healer of Children since the Thirteenth Century. Cambridge: Cambridge University Press. Stcherbatsky, F.T. 1962. Buddhist Logic. New York: Dover Publications.

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PART THREE The Logico-Mathematical Status of the Canonical Formula

The chapters by Scubla, Daranyi, and Gregory bridge Parts One and Three. Now, with chapters 8 to 10 we move to more theoretical epistemological and mathematical - considerations of the CF. Cote explores the 48 possible expressions of the CF to narrow them down to Levi-Strauss's own applications. Quinn investigates the formula in terms of connectionist philosophy and cognitive grammars, and his contribution leads to Petitot's. The latter recasts the CF, a 'structural equation/ in the framework of Rene Thorn's morphogenetic theory as Petitot himself has developed it.

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8 The Set of Canonical Transformations Implied in the Canonical Formula for the Analysis of Myth Alain Cote

Claude Levi-Strauss's canonical formula (CF) for the analysis of myth has been the object of several scholarly papers (see Scubla 1998), but it still remains a mystery of sorts. Controversy persists on its interpretation. Should it be discarded as nonsense, as some would have it? Or is it still to be better understood? Actually, as one may learn from the history of mathematics, it remains imperative to be cautious when dealing with that kind of issue. Remember the controversies about irrational numbers. Pascal and Barrow saw them merely as symbols - as some do with Levi-Strauss's canonical formula. By contrast, Descartes and Fermat took them for what they were, but remained somewhat in a haze about their nature. On the other hand, Stevin and Wallis acknowledged them fully as numbers, but could not provide a logical foundation on which to ground them. Likewise, negative numbers were nonsensical for Chuquet and Stiff el, a view challenged by Arnauld. Cardan considered them as mere symbols. Actually, imaginary numbers had been used for two hundred years before Gauss and Hamilton modelled them appropriately (cf. Kline 1992). Consequently, and along the same epistemological lines, the problem of interpretation of the CF does not mean we should discard it without further ado. Furthermore, its employment, arcane as some may find the formula, remains useful methodologically. From an anthropological point of view, the CF may remain of questionable use. But no firm conclusion can be drawn on that score as long as no adequate interpretation has been provided. We must therefore try to explore the operations for the description of which Levi-Strauss devised his formula.

200 Alain Cote

The most common approach to the CF is to scrutinize and to parse the formula's components in order to depict its inherent characteristics and to decipher its underlying idea. Actually, when Levi-Strauss wrote the paper in which he introduced the CF, he referred to it as one of some possible formulations. And again, thirty years later in La potiere jalouse (1985: 207), he reiterated the same view when pointing out that the formula can take different forms. We must then conclude that the CF belongs to a set of possible variations. Accordingly, the point of this paper is to explore the whole set of those variations. In the first section, I shall investigate the meaning of 'canonical formula': how can that term be defined? Then, in section 2,1 shall propose a different notation that will help in presenting the full set of 'canonical expressions' in section 3, that is, the class of statements that can fit the CF and meet its requirements. This will yield 144 formulations, of which I shall examine the canonical transformations (section 4) and the classes that encompass them (section 5). I shall then present graphically the structure of that set in section 6. Finally (section 7), I shall consider the implications of that exploration, taking into account LeviStrauss's comments on my endeavour. Most certainly, my analysis is nothing more than a modest contribution to the understanding of the CF. As such it shows that the issue of the CF is of outstanding importance. Indeed, far from being a mystification, the CF is really meaningful. At stake are all the basic propositions on the structural analysis of myth, as is a better understanding of the thought of Claude Levi-Strauss. My paper is relatively easy to read despite its somewhat abstract character. I have kept technical and formal points to a minimum and the discussion on an intuitive level, close to everyday language. And I shall use as many examples as necessary, proceeding a step at a time. 1. The Canonical Formula In his famous 1955 paper, Levi-Strauss introduced the CF as follows: Every myth (considered as the aggregate of all its variants) corresponds to a formula of the following type:

Here, with two terms, a and b, being given as well as two functions, x and y, of these terms, it is assumed that a relation of equivalence exists

Canonical Transformations Implied in the Canonical Formula 201 between two situations defined respectively by an inversion of terms and relations, under two conditions: (1) that one term be replaced by its opposite (in the above formula, a and a-1); (2) that an inversion be made between the function value and the term value of two elements (above, y and a). (1963: 228)

In that statement, I underscore 'a formula of the following type,' which implies that other forms of the CF can be acceptable. And LeviStrauss himself bears out such an interpretation in La potiere jalouse: The formula appears here under one of its transformations: /x(a) : /v(b) ~/x(x) :/fl_i(b). This use is legitimate provided that the initial conditions be met: that one if its terms be replaced by its opposite concomitantly with an inversion between a term value and a function value' (1985: 207 n. 1). Clearly the expression 'canonical formula' refers to a set of relations that Levi-Strauss calls 'transformations.' In order to avoid ambiguity, I call 'canonical expression' a relation of the type found in the 1955 version of the canonical/ormM/fl, a relation that must meet the two requirements stipulated in the above quotation. What is a canonical expression? From a formal point of view, the two left-hand-side members always look the same, namely, /x(a):/y(b). Its reading: 'the function x of the term (a) is to the function y of the term (b)' (see Levi-Strauss 1985: 79). This seems to refer to setting up two oppositions: an opposition between two functions (x and y) and another between two terms ((a) and (b)). From Levi-Strauss's own uses, it can be concluded that such oppositions always figure in the members of the left side of the formula. Hence, the left-hand side of a canonical expression cannot take the form of (i) / x (a):/ x (b),orof (ii) /x(a) :/y(a) and still less of (iii)/x(a):/x(a). Now let us look at the two requirements any canonical expression must meet. First, one of the terms in the left-hand side of the expression must be replaced by its opposite in the right-hand side of the expression. For instance, (a) becomes a"1. For any and all applications

202 Alain Cote TABLE 1 Explicit canonical expressions in the works of L6vi-Strauss Fx(a) .. Fy(b)

Fx(b) Fa-1(y)

Fx(a).. Fy(b)

Fx(b) Fb-1(y)

Fx(a) .. Fy(x) Fy(b)

Fa-1(b)

by Levi-Strauss of his formula (1963: 252; 1984: 146; 1985: 207), there is only one inverted term (see table 1). But nowhere does Levi-Strauss specify whether this condition is a strict or a minimal requirement. In other words, he does not tell us if one must read this condition as 'there must be at least and at most one term that is replaced by its opposite/ or as 'at least one term is replaced by its opposite.' Since there is no way to decide, I shall take this requirement under its latter form (see below, section 7, for developments). Consequently, and in keeping with my interpretation, a canonical expression must contain an inversion between a function value and a term value. But I must be more explicit on two points. As for the first one, one does not know if the inversion must be carried out from a term that has already been replaced by its opposite, as seems to be the case when one takes a look at the examples of canonical expressions in LeviStrauss's work (see table 1). And, with respect to the second point, one may ask if the opposite of a term always implies its transformation into a function. If such is not the case, then the right-hand side of a canonical x(Here induction teach expression could be, for instance, us nothing definite, but a paper by Solomon Marcus (1997) allows us to

postulate that a term inversion to its opposite entails a concomitant permutation from a term value to a function value. Concerning the first requirement, I shall adopt the same broad reading as I did for the second: that is, that there must be at least one inversion between a function value and a term value. Consequently, the formulation (iv)f

is not acceptable as a canonical expression because it fails to meet those two requirements. And it also means that the relation expressed by '::' is not reflexive, so that the 'equivalence' between the right-hand and left-hand sides of the CF is not an equivalence relation in the mathe-

Canonical Transformations Implied in the Canonical Formula 203

matical sense of the word. It also means that no single group structure - again in the mathematical sense - operates upon the set of transformations implied in the formula, because leaving out of the set of canonical expressions those given above ((i)-(iv)) implies the exclusion of the identity transformation associated with that set. Before moving on to the analysis itself, let us modify somewhat Levi-Strauss's notation. 2. Notation To begin with, let us say that a canonical expression consists of four ordered pairs. The first component of each ordered pair remains a function value and the second one a term value. Thus, in the ordered pairs [/x(a)] and [fa-\(y)],fx and/fl_i are function values, and (a) and (b)

term values. And, as we have just seen, only the inverse of a term, such as a - 1, can take up a function value - to be more explicit, there cannot be such a pair as [/fl(y)l, whereas [/fl-i(y)] is perfectly admissible. My modification of Levi-Strauss's notation is as follows. First, I eliminate the symbol T' and the parentheses. Second, I keep 'a' and 'b' exclusively for terms and V and 'y' exclusively for functions. When 'a' or 'b' occurs as a function (cf. a"1 in Levi-Strauss's notation), I use primes, a',b'; and the same with V and 'y' when they occur as terms. Accordingly, a constituent ordered pair of a canonical expression may look like x,a (instead of/x(a) in Levi-Strauss's notation) or like a',x but it can never be x',a'. The following table shows the notations changes.

Terms

Levi-Strauss

Cote

a b

a b x'

y'

Functions

X

X

y1

y

a-

a' b'

Let me add that there is no need to use a notation other than the

204 Alain Cote TABLE 2 Ordered pairs that may appear in a canonical expression

prime ( ' ) for the opposite of a term because such inversions automatically imply permutations from term to function values, permutations expressed by the primes in a' or b'. To sum up my notation: {x,y} stand for functions, {a,b} stand for terms, and, in a constituent ordered pair of a canonical expression, any of those symbols may take on either a term or a function status. Thus, for instance, fx(y)

becomes simplified as xy' and/fl_i(b) becomes a*b. It is

then easy to define the set of all possible ordered pairs that can form canonical expressions. Table 2 displays that set in four classes, A, B, C, andD. Table 2 contains the full set of ordered pairs that can constitute canonical expressions; it results from the product of a class of four functions {x,y,a',b'} by four terms {a,b,x',y'}. The classes A, B, C, and D form a natural cluster because they all meet Levi-Strauss's two requirements. In A, we have all 'terms' as terms and all 'functions' as functions; in B, 'terms' are both functions and terms; in C, 'functions' are both functions and terms, and in D, 'terms' are functions and 'functions' are terms. These four classes make it easy to define the whole set of canonical expressions, on the one hand, and the set of transformations associated with them on the other hand. 3. The Set of Canonical Expressions We have seen that the first two members of a canonical expression are always fx(a) :/y(b) or, in my notation, xa : yb. Thus, it is the two righthand-side members that account for the variation in canonical expressions. We can then take advantage of the constant - the two first members - to simplify computation, which means that we can use only the

Canonical Transformations Implied in the Canonical Formula 205

two last members to carry on the operation. Hereafter, I shall then write what I call the emblematic canonical expression xa : yb :: xb : a'y' as xb : a'y'.

Keeping that in mind, we see that it suffices to deal only with the two last members of the expression to show all its possible permutations. These two last members represent a relation between the elements of table 2, which contains 16, so that there are 16 x 16 = 256 different possible pairs. One would not think of considering singly each one of all those possible pairs to determine which ones meet Levi-Strauss's formula. Let us consider instead the relations between the classes A, B, C, and D, and we will see that only 16 types of relations meet the requirements. As I have said, the elements that form those classes are similar with respect to the requirements of the formula. So, if one pair of elements in table 2 qualifies (or does not qualify) as a component of the righthand side of a canonical expression, then all the elements from the same class(es) also qualify (or do not qualify). For example, the relation xa : yb does not qualify for membership in the right-hand side of a canonical expression: indeed, it does not meet any of the formula's requirements. Since xa and yb are elements of class A (table 2), all the relations between any pairs of class A cannot qualify as members of a right-hand side of a canonical expression. Let us label such a class of relations A,: Ay (in which i may = j). Then, to see which ones of the possible 256 possible pairs of elements in table 2 qualify as components of the right-hand side of a canonical expression, it is enough to examine one representative pair in each class of relations resulting from the Cartesian product of the set of the classes A, B, C, and D with itself, and to decide whether or not that pair meets both requirements of the formula. We then see that the number of acceptable expressions one has to examine reduces to 4 X 4 = 16 (see table 3). Now, what is the procedure? First, for each class A, B, C, and D, select a representative element in each class, an element that stands for the class from which it is taken: xa for A, a'a for B, xx' for C, and a'x for D. Such selected elements will be useful to see if a class of relations

206 Alain Cote TABLE 3 The set of relations between the classes of table 2 A

A

A

A

B

B

B

B

A

B

C

D

A

B

C

D

C

C

C

C

D

D

D

D

A

B

C

D

A

B

C

D

may qualify or not as a right-hand side of a canonical expression. Let us begin by examining A,: A;. Replacing A by its representative xa, we get the relation xa : xa. One sees at once that it contains neither a term inversion nor a permutation of a function value into a term value. We must then exclude A, : Ay from the possible candidates to right-handside membership in a canonical expression, and the same obtains for all the combinations of two elements of A. By the same process we see that neither class B, : By nor class C,: C; can contain elements that would qualify for membership in the righthand side of the expression. Relations in B do not change from function value to term value, and in C there are no term inversions. However, relations in class D,: D; meet both requirements of the formula. So, we are left with 12 classes. But of those, only 6 remain. Why? Because if the relations in a class - say, x : y' - qualify (or not) as right-hand-side members of the canonical expression, so do y : x. The only six relations that remain are A : B, A : C, A : D, B : C, B : D, and C : D.

Let us continue with A : B and substitute xa and a'a respectively for A and B. Then one can see that there is no change of function into term in xa : a'a, so that the relations in the class A : B as well as those in the class B : A cannot qualify as members of the right-hand side of any canonical expression. Let us now take up the case of A : D. Let xa and a'x' stand respectively for A and D: the relation xa : a'x' meets the requirements of the formula. Therefore, all the relations of the class A : D as well as those of the class D : A qualify as members of the right-hand side of a canonical expression. The reader may easily check that, out of the 16 classes of relations, only 9 classes contain relations that qualify as candidates to

Canonical Transformations Implied in the Canonical Formula 207

membership in the right-hand side of a canonical expression. As each one of those classes contains 16 relations, we then come up with 144 canonical expressions (see the appendix for the set of those expressions transcribed in Levi-Straussian notation). Needless to say, this is less than the 256 instances mentioned above, but much more than the three explicit instances known up to now! 4. Canonical Transformations At the outset of this paper, I pointed out that, for Levi-Strauss, the canonical formula is linked to the idea of transformation. In a certain way, each and every canonical expression describes a transformation, and, with respect to the two requirements stated by Levi-Strauss, the canonical formula refers to a set of such transformations. I shall then call 'canonical transformation' a transformation that can be built from a particular canonical expression. First, what are those transformations? A canonical expression meets two requirements that can be reformulated as follows: it includes at least one permutation of terms and one concomitant permutation of functions. Nothing is simpler, then, than to characterize a canonical transformation as a double permutation of term and function in the left-hand side of the expression to generate the members of its righthand side. Function permutations. More specifically, let us define permutations on the sets A, B, C, and D. To begin with, there are two basic permutations: the first one I call 'external' and the second 'internal.' When applied to any element of a class A, B, C, and D, an external permutation generates an element that is a member of a different class - thus its meaning is external; and it is the other way around with internal permutations. And since I am talking about a permutation of functions that is a one-to-one mapping of the set {x,y,a',b'} onto itself, I shall call it'{', and 'f can be defined as follows:

f:(xa')(yb'). It means that x changes into a' and a' into x and that y changes into b' and b' into y. For instance, if one applies f to xa - a member of the set A - one gets

(xa)f = a'a

208 Alain Cote TABLE 4 Basic internal and external permutations over the set of ordered pairs that may appear in a canonical expression

change yb into a'y' [(yb)f't = a'y')]. To generate all canonical transformations, I shall proceed as I did for the canonical expressions, that is, at the level of classes of transformations. 5. Classes of Canonical Transformations We know which classes of relations may appear in the right-hand side of a canonical expression. In order to circumscribe the set of canonical transformations, one needs only to examine which permutations generate the right classes of relations when those permutations are applied to the single and only relation forming the left-hand side of a canonical expression, namely, xa : yb. And since xa : yb is a member of the class A,: Aj, the operation amounts to seeing how one goes from A,: Ay to each one of the nine possible classes of the right-hand side of the expression. To do it, let us first look at the permutation that transforms a member of A into a member of the classes A,B,C,D. With the help of table 4, the reader may easily check that the permutations i, f, f, and ft' transform an element of A, into an element of A and that only these do so, among the 16 listed above. Actually, f and t' are what I have called internal transformations, as are also ft' and i. Let

Canonical Transformations Implied in the Canonical Formula 209

However, this is not the set of all possible permutations of terms and functions. Indeed, additional permutations appear through combinations of the first ones. There are 16 possible permutations of terms or functions over the set of the elements in table 2 and they form a group, {i,f,f/,r,t,t/,t",ft,ft',ft",f/t,f/t/,f/t",f"t,fY,f/r} (note that 'i', the identity element in the group, is a permutation that changes nothing). I am now in a position to show how to construct a canonical transformation from a canonical expression. The emblematic canonical expression xa / yb ~ xb / a'y' offers a good starting point. To generate the transformation associated with the expression, we only have to take the transition from the left to the right of the expression as a double permutation: a permutation that transforms the left-hand-side numerator into the one on the right-hand side, and another permutation that does the same with the left-hand-side denominator. I shall write such a transformation as an ordered pair, the components of which are the permutations concerning, respectively, the numerator and the denominator on the left-hand side. Thus, the canonical transformation associated with the emblematic canonical expression is written: (t'.f",t), for one needs t' to go from xa to xb [(xa)t' = xb] and f"t to change yb into ay [(yb)f"t = a'y')]. To generate all canonical transformations, I shall proceed as I did for the canonical expressions, that is, at the level of classes of transformations. 5. Classes of Canonical Transformations We know which classes of relations may appear in the right-hand side of a canonical expression. In order to circumscribe the set of canonical transformations, one needs only to examine which permutations generate the right classes of relations when those permutations are applied to the single and only relation forming the left-hand side of a canonical expression, namely, xa : yb. And since xa : yb is a member of the class A,: A;, the operation amounts to seeing how one goes from A,: A; to each one of the nine possible classes of the right-hand side of the expression. To do it, let us first look at the permutation that transforms a member of A into a member of the classes A,B,C,D. With the help of table 4, the reader may easily check that the permutations i, f, t', and ft' transform an element of A, into an element of A; and that only these do so, among the 16 listed above. Actually, f and t' are what I have called internal transformations, as are also ft' and i. Let us call this set 'i/ then i = (i,f ,t',f t'}.

210 Alain Cote

The permutations f, f, ft', ft' transform an element of A into one of B. We have already seen that in the case of f. As for f", it amounts to the same operation as doing f, then f'(f" = ff). As f transforms an element of A into one of B, and f is an internal permutation, so does f" an element of A into one of B. The same applies to ft', and since f = ff, it is obvious that f"t' = ff't' so that with f"t one goes from A to B. Let us call'((>' the set of permutations that change an element of A into one of Proceeding the same way as I just did, one can check that when the permutations of the set t = {t,f't,t",f't"} bear on the elements of A, the latter undergo a transformation into members of C. Finally, with the permutations of T = {ft,f"t,ft",f"t"} we get an element of D from one of A. It is now possible to circumscribe the set of canonical transformations. In order to do it, it suffices to examine which permutations change a relation of the class A,: A; into one of each class of canonical expressions. For instance, take the class A : D. One transforms a relation in A,: A, into one in A : D by applying a function of the set {i, f, t', ft'} to the numerator of the relation in A,: A; and a function of {ft, ft, ft", ft"} to its denominator. It is easy to see that the transformations that link A,: A; to A : D in that manner form a set corresponding to the following product {i,f ,t',f t'} x {ft,f"t,ft",f"t"}, that is a set of 16 canonical transformations. There is no need to go to great lengths if one expresses the kind of link between A,: Ay and A : D as an ordered pair, the components of which are the class of permutations that apply to the numerator of a relation in A, : A; and the class of relations that apply to the denominator of the relation. In the present case, this ordered pair is (i,x) and it represents the class of canonical transformations associated with the class of canonical expressions A : D. It follows that the class of canonical transformations associated with A: D is nothing but ((|)T,i). Another example may be useful. A relation of the class A, : A. is changed into a relation of the class B : C if one applies a function of the class $ to the numerator of A,: A; and one of the class T to its denominator. Thus, (T) and (tyi,fyi), that is, (J>i,i. One could ask, 'What in all that is the point for anthropology?' I shall shortly tackle the issue by showing the kind of questions this type of analysis may raise - questions about the canonical formula and also, to some extent, about the whole of structural analysis.

214 Alain Cote

If one bears in mind that canonical transformations concern first and foremost the universe of myth, and that every myth could belong to such transformations, then one can see that Levi-Strauss is right. Indeed, the result of the preceding analysis corroborates his view of the world of mythology as spherical. One can also see that its sphericity is, rather than that of a ball, that of a doughnut. Figure 1 shows that, from the standpoint of topology, it is a torus:

On the other hand, given the fact that TC is not a group, LeviStrauss's view can be questioned when - as sometimes in Mythologiques - he infers the existence of a group from the closure of a sequence of transformations. For it is not enough to show that a set of myths is connected, or even that there are closed paths over the set, to conclude that it has a group structure. However, the fact that TC does not have a group structure does not mean that it is not pertinent with respect to myth analysis; it only means that TC is not immediately related to canonical transformations and, therefore, to the canonical formula. This point bears on the structural analysis of myth because it calls for a new appraisal of the position, the function, and the meaning of transformation groups in that field. It also bears on the whole of the creations of the Savage Mind (Levi-Strauss 1962) and calls as well for a reconsideration of the canonical formula and of its position within structural analysis. It is true that the word 'group' takes on different meanings in the works of Levi-Strauss. He points this out himself. Sometimes he uses it in the mathematical meaning, at other times in terms of 'system/ or as a reference to a whole, and even to mean merely 'connectedness/ However that may be, and despite those variations, it remains that we still have to take a closer look at the function and meaning of mathematical groups in the structural analysis of myth. Another point stemming from the structure of TC is that, insofar as canonical transformations are linked to the problem of transforming one myth into another, the fact that a set of myths is connected to another does not entail, in terms of canonical transformations, an immediate transformation between the two of them - which would require that TC be strongly connected. The structure of TC shows that it is strongly connected only between some members of such sets. Maybe it would be interesting to specify, as regards the canonicity of

Canonical Transformations Implied in the Canonical Formula 215

the formula, the conditions of immediate transformations between myths. Before going on with the structure of the set of the classes of canonical transformations, let us take a look at a last question that figure 1 may suggest. There are only nine classes of canonical transformations, each including sixteen canonical transformations. Does it mean that there could be only 144 mythical schemata or, if one prefers, only 144 myth types? The answer depends on what Levi-Strauss means when he writes that 'every myth (considered as the aggregate of all its variants) corresponds to a formula of the following type' ... Does it imply that all myths can be reduced to a single type or, conversely, that a single myth can be amenable to different reductions? We could answer by saying that the CF develops a set of possibilities based on the very caput variationis contained in the left-hand side of a canonical expression. Then the question becomes, Is the left-hand side of a canonical expression unique for any given myth, or are there some other pairs of terms and functions that could fit it? If the latter, then what characterizes a particular myth would not be a unique reading but a structured set of unequivalent relations, that is, unequivalent pairs. 7. Test My analysis has rested on the two requirements that Levi-Strauss said the CF must meet, and on my interpretation of them (above, section 2). Do those requirements define too large a set as regards the kinds of transformations Levi-Strauss has in mind? Only the author of the CF can answer that question, so I have addressed it to him. I have phrased it in the form of a list of the classes of the canonical expressions that meet the requirements (see appendix), asking Levi-Strauss to peruse it and to point out the classes that he spontaneously considers should be excluded from the list because they would not fit the CF. His answer: You are asking me for a spontaneous reaction to your three-page appendix. Intuitively, I would be inclined to exclude D/A and D/C. On the other hand, D/D and B/D raise a question because, as you correctly point out [above section 2], my examples of canonical expressions invert only one term. But, as a matter of fact, I sometimes find two of them and then either I keep only one in order to be able to use my formula, or I drop the formula and fall back on a description. That shows clearly in Histoire de Lynx, p. 146 ('triple contrast') and pp. 188-9, but one would find other

216 Alain Cote examples throughout Mythologiques. As a matter of fact, I have not been able to design a convenient scheme to represent such occurrences or, when I tried, the scheme seemed too complicated to provide the visual, telling image that I sought. Could your D/D and B/D examples serve that end? I do not know.

From the fact that we have to leave out some of the classes of canonical expressions, we draw the conclusion that Levi-Strauss's requirements do not properly characterize the canonical formula. Therefore, further requirements must be added, and Levi-Strauss's answer offers a good starting point to do so. First, let us leave out the classes D/D and B/D, which include expressions containing two term inversions, and let us do the same with D/B for the same reason. Then, let us consider the classes D/A and D/C: we see that they are in an inversion relation with D/A and C/D. In other words, the relation of inversion between them consists of an interchange of relative positions in the third and fourth pairs of canonical expressions. Hence, the requirements to be added should concern that interchange. Why would it be that such an interchange could yield something that is not canonical? What comes first to mind is that the ordered pair that Levi-Strauss calls the 'double twist' keeps a constant position in such an expression. Still more interesting is the fact that Levi-Strauss has already attributed a fixed position to that ordered pair as a condition related to the CF. He did it, so to speak, in passing, and not explicitly. When referring to the CF in the Histoire de Lynx, he writes: '[TJhat way we would verify once again that the last state of a mythical transformation implies a double twist' (1991: 179; see also La potiere jalouse, 1985: 206). Thus, it is clear that the last ordered pair of the CF, the 'double twist/ consists of a term inversion as well as a change of term value into a function value. Even if one finds equivalent statements throughout Levi-Strauss's works, he never made that point explicitly in terms of the structure of the CF - a point I deem necessary to state clearly. Up to now I have not considered the classes C/D and B/C. LeviStrauss did not mention them in his response. As for their canonicity, we can confirm it because the class C/B includes, in the form/Jx) : /fl-i(b), the explicit canonical expression found in La potiere jalouse (1985: 205). On the other hand, and as regards the third requirement of the CF - the one I just added with respect to the final ordered pair - we must discard the class B/C.

Canonical Transformations Implied in the Canonical Formula 217

Even though Levi-Strauss did not exclude right away the classes D/D, B/D, and D/B, I am inclined to leave them out at this juncture for two reasons. First, the expressions that they contain do not meet the third requirement of the CF; and, second and more importantly, those expressions represent co-occurrences of mythical transformations. In my view, it is better to keep here the focus on the set of simple canonical expressions than to try to grapple with the problem of representing with one expression the co-occurrences of mythical transformations. In the last analysis, we are left with only three classes of canonical expressions, namely, A/D, C/B, and C/D. They yield a set of 48 possible canonical expressions. Although the cardinality of the set of canonical expressions has thus been appreciably reduced, their number remains considerably higher than the three that appear explicitly in Levi-Strauss's works. How can we explain such a reduction of the possible to the implemented? As I have stated it elsewhere (Cote 1995), a partial answer to that question may come from the epistemological motivation underlying Levi-Strauss's analysis of myths. In his response, part of which I quoted above, he wrote I wonder whether the transformations of the formula [and here LeviStrauss refers to FffKb)] are not easy ways out that I took; would not a better analysis of the myths have brought them back to the initial formula? It could also be that another formula, better designed than mine, could encompass all the cases encountered.

We can conclude from that statement that Levi-Strauss sees his original 1955 canonical formula, designed to represent a particular transformation, as emblematic. At first sight, it does not seem impossible to reduce all canonical expressions to the original 1955 formula. However, if so, the different sets of canonical expressions I have displayed in this paper would represent shifts of points of view on a single phenomenon. Such shifts could be compared to looking at a circle from different angles: directly from above, it looks perfectly round, but move to one side or another and it looks like an ellipse. Then, the main idea that we see as fundamental in the CF would be that of the 'double twist,' that is, of the crossing of boundaries. After all, maybe the multiplicity of canonical expressions only points out different ways, corresponding to different viewpoints, to describe the same set of relationships.

218 Alain Cote

Finally, what can we say about the kind of structure that operates on the set of the three classes of acceptable canonical expressions we end up with? The most important point is that those classes are not canonically connected, nor is there any such connection among the canonical expressions within any of those classes. Thus, if there is a transformation from a canonical expression into another, the transformation cannot itself be a canonical one. To really understand the implications of that fact, we would have to establish clearly what the canonicity of the CF consists in - a task beyond the reach of my paper. Conclusion Bringing this paper to an end, I should like to lay emphasis on the following three points. First, the presentation I made of the set of canonical transformations has provided a background against which to read the CF. In so doing, my point was to bring the reader's attention to some of Levi-Strauss's basic ideas related not only to the structure of myth but also, as Marcus has put it (1997), to the whole of structuralism. On that score, it is of particular importance to define precisely the relation between mediation and the double twist [Editor: see Petitot, this volume]. Second, let us remember that the CF has something to do with the crossing of boundaries, be they linguistic, cultural, ecological ... So it may be interesting to explore the way canonical expressions relate to that phenomenon. In that connection, one should consider what I have called the third requirement that the formula must meet. Finally, I wish to stress the methodological and heuristic pertinence of Levi-Strauss's canonical formula for the analysis of myths. Far from being vacuous, the CF calls on our intellectual responsibility to understand better the processes it aims at modelling and mapping out, and to expand its methodological and heuristic potential. Any step in that direction would be important to structuralism, which, far from being dead as some hope, deserves further developments.

Canonical Transformations Implied in the Canonical Formula 219 Appendix: The Set of Canonical Expressions Note: The left-hand side of canonical expressions is written at the left. Each little table is made up of right-hand sides of canonical expressions that form an equivalence class. The name of these classes is written at the right of each table. Fx(a) Fx(a) Fy(b)" Fa-l(x)

Fx(a) Fa-l(y)

Fx(a) Fb-l(x)

Fx(a) Fb-l(y)

Fx(b) Fa-l(x)

Fx(b) Fa-l(y)

Fx(b) Fb-l(x)

Fx(b) Fb-l(y)

Fy(a) Fa-l(x)

Fy(a) Fa-l(y)

Fy(a) Fb-l(x)

Fy(a) Fb-l(y)

Fy(b) Fa-l(x)

Fy(b) Fa-l(y)

Fy(b) Fb-l(x)

Fy(b) Fb-l(y)

Fa-l(x) Fx(a)

Fa-l(x) Fx(b)

Fa-l(x) Fy(a)

Fa-l(x) Fy(b)

Fa-l(y) Fx(a)

Fa-l(y) Fx(b)

Fa-l(y) Fy(a)

Fa-l(y) Fy(b)

Fb-l(x) Fx(a)

Fb-l(x) Fx(b)

Fb-l(x) Fy(a)

Fb-l(x) Fy(b)

Fb-l(y) Fx(a)

Fb-l(y) Fx(b)

Fb-l(x) Fy(a)

Fb-l(y) Fy(b)

Fa-l(x) Fa-l(x) Fa-l(y) Fa-l(x)

Fa-l(x) Fa-l(y) Fa-l(y) Fa-l(y)

Fa-l(x) Fb-l(x) Fa-l(y) Fb-l(x)

Fa-l(x) Fb-l(y) Fa-l(y) Fb-l(y)

Fb-l(x) Fa-l(x)

Fb-l(x) Fa-l(y)

Fb-l(x) Fb-l(x)

Fb-l(x) Fb-l(y)

Fb-l(y) Fa-l(x)

Fb-l(y) Fb-l(y)

Fb-l(x) Fb-l(x)

Fb-l(y) Fb-l(y)

Fa-l(a) Fa-l(x)

Fa-l(a) Fa-l(y)

Fa-l(a) Fb-l(x)

Fa-l(a) Fb-l(y)

Fa-l(b) Fa-l(x)

Fa-l(b) Fa-l(y)

Fa-l(b) Fb-l(x)

Fa-l(b) Fb-l(y)

F» Fy(b)--

A D

D A

D D

B D

220 Alain Cote

F».. Fy(b)

Fb-l(a) Fa-l(x)

Fb-l(a) Fa-l(y)

Fb-l(a) Fb-l(x)

Fb-l(a) Fb-l(y)

Fb-l(b) Fa-l(x)

Fb-l(b) Fa-l(y)

Fb-l(b) Fb-l(x)

Fb-l(b) Fb-l(y)

Fx(x) Fa-l(a)

Fx(x) Fa-l(b)

Fx(x) Fb-l(a)

Fx(x) Fb-l(b)

Fx(y) Fa-l(a)

Fx(y) Fa-l(b)

Fx(y) Fb-l(a)

Fx(y) Fb-l(b)

Fy(x) Fa-l(a)

Fy(x) Fa-l(b)

Fy(x) Fb-l(a)

Fy(x) Fb-l(b)

Fy(y) Fa-l(a)

Fy(y) Fa-l(b)

Fy(y) Fb-l(a)

Fy(y) Fb-l(b)

Fa-l(x) Fx(x)

Fa-l(x) Fx(y)

Fa-l(x) Fy(x)

Fa-l(x) Fy(y)

Fa-l(y) Fx(x)

Fa-l(y) Fx(y)

Fa-l(y) Fy(x)

Fa-l(y)

Fb-l(x) Fx(x)

Fb-l(x) Fx(y)

Fb-l(x) Fy(x)

Fb-l(x) Fy(y)

Fb-l(y) Fx(x)

Fb-l(y) Fx(y)

Fb-l(x) Fy(x)

Fb-l(y) Fy(y)

Fa-l(x) Fa-l(a)

Fa-l(x) Fa-l(b)

Fa-l(x) Fb-l(a)

Fa-l(x) Fb-l(b)

Fa-l(y) Fa-l(a)

Fa-l(y) Fa-l(b)

Fa-l(y) Fb-l(a)

Fa-l(y) Fb-l(b)

Fb-l(x) Fa-l(a)

Fb-l(x) Fa-l(b)

Fb-l(x) Fb-l(a)

Fb-l(x) Fb-l(b)

Fb-l(y) Fa-l(a)

Fb-l(y) Fa-l(b)

Fb-l(y) Fb-l(a)

Fb-l(y) Fb-l(b)

Fx(x) Fa-l(x)

Fx(x) Fa-l(y)

Fx(x) Fb-l(x)

Fx(x) Fb-l(y)

Fx(y) Fa-l(x)

Fx(y) Fa-l(y)

Fx(y) Fb-l(x)

Fx(y) Fb-l(y)

C B

D C

Fy(y)

_D B

C_ D

Canonical Transformations Implied in the Canonical Formula 221

Fy(x) Fa-l(x)

Fy(x)

Fy(x)

Fy(x)

Fa-l(y)

Fb-l(x)

Fb-l(y)

Fy(y)

Fy(y)

Fy(y)

Fa-l(x)

Fa-l(y)

Fb-l(x)

Fy(y)

Fb-l(y)

Fa-l(a)

Fa-l(a)

Fa-l(a)

Fa-l(a)

Fx(x)

Fx(y)

Fy(x)

Fy(y)

Fa-l(b)

Fa-l(b)

Fa-l(b)

Fa-l(b)

Fy(x)

Fy(y)

Fx(x)

Fx(y)

Fb-l(a)

Fb-l(a)

Fb-l(a)

Fb-l(a)

Fx(x)

Fx(y)

Fy(x)

Fy(y)

Fb-l(b)

Fb-l(b)

Fb-l(b)

Fb-l(b)

Fx(x)

Fx(y)

Fy(x)

Fy(y)

B C

Note It is a pleasure to acknowledge my indebtedness to Professor Claude LeviStrauss, who read a first draft of this paper (in French) and assisted me with helpful comments. I must also thank Professor Solomon Marcus for critical and helpful comments. Finally, I wish to thank the Social Sciences and Humanities Research Council of Canada for a two-year postdoctoral fellowship.

References Cote, A. 1995. 'Qu'est-ce que la formule canonique?' L'Homme 135: 35^41. Kline, M. 1992. Mathematics: The Loss of Certainty. New York: Oxford University Press. Levi-Strauss, C. 1962. La pensee sauvage. Paris: Plon. - 1963. Structural Anthropology. New York: Basic Books - 1984. Paroles donnees. Paris: Plon. - 1985. La potiere jalouse. Paris: Plon. - 1991. Histoire de Lynx. Paris: Plon. Marcus, S. 1997. The Logical and Semiotic Status of the Canonic Formula of Myth/ Semiotica 116 (2/4): 115-88. Scubla, L. 1998. Lire Levi-Strauss. Paris: Odile Jacob.

9 On Some Philosophical Dynamic and

Connectionist Implications of the Canonical Formula of Myth Seen as Space Categorization Andrew William Quinn

The specific categories and myths that are connected with them can also serve to organize space, and the classificatory system is then extended on a territorial and geographical basis. Levi-Strauss (1962: 218-19)

This chapter stems from a question formulated by Pierre Ouellet in his presentation of an issue of the periodical RS/SI, entitled 'Symbolic Representation and Spatial Organization' (1992), and proposes a series of reflections on Levi-Strauss's renowned canonical formula of myth involving certain cognitivist themes. The question was as follows: 'How does the topological or morphological structure of space give rise to different symbolic representations of a perceptual, conceptual or more broadly socio-cultural nature?' (Ouellet 1992:186). We shall address the question on a background of Kantian philosophy by first presenting the Connectionist (hereafter CN) Paradigm and its commitments along with the notion of semantic pregnancies (section 1), and its links to the canonical formula (section 2). That will lead us to the concepts of the structural and categorial unconscious and modal categorization (section 3). The morphological model (cf. R. Thorn's and J. Petitot's morphogenetic approach), connected to the anthropological semantic pregnancies (section 4), will take us to an investigation of the symbolic (section 5) and the sub-symbolic (section 6) that will loop back on the unconscious, this time in relation with topology, narrative, and semantic idealism (section 7). We will then

Dynamic and Connectionist Implications of the Canonical Formula 223

have to explore the problem of structural finality, in connection with the canonical formula (section 8) before returning to our point of departure, namely, the categorization of spatiality (section 9). The problem of categorization is indeed at the heart of Levi-Strauss's approach; and relying on cognitive semiotics, which derives from recent morphodynamic and CN paradigms, we will propose that a relationship exists between language and mythological categorization - that is, an equivalence that, in the strongest transcendental sense of the word, is a form of categorial constitution of social space(-time); hence there exists the possibility of a 'parallelism' between the semiotic mechanism of mind and the evolutionary process of Nature (Sebeok 1989). In addition, we will interpret the canonical formula of myth not as a logical formalism, but as a formalization of virtualities for framing relationships in space, or as a genuine 'modal framework' (Brandt 1992). Between social organization, which is spatial in nature, and the form of narratives there is a canonical configuration, and this formalism rests on a 'universal code' that was postulated by Levi-Strauss (1958: 71) as having a basis in the brain - a problem to which he has not returned since. He has, of course, strongly emphasized the importance of 'deep structures/ and the necessity of uncoupling nature/culture dualism (1983b: 1227), a preoccupation already figuring centrally in Les structures elementaires de la parente (1967) in relation to a treatment of the complex problem of the incest taboo (see Simonis 1980). He nevertheless attempted to uncover a logic of structures in a span of intellectual development ranging from the combinatorial rules that organize social systems to the entity in which such systems are ultimately founded: the structural unconscious. It is true that the quest for the formally invariant, the unveiling of a universal order of structuring, makes it possible to introduce something along the lines of a 'universal mathematical language' (Braudel 1969: 71) into the field of anthropology. The crucial issue remains, however, that of penetrating into the 'anthropological function of myths' (Petitot 1985: 49). As raised by Levi-Strauss, the problem intersects the theme that has been anticipated by Kant of a form of a deep semantic element, one that is 'quite different from the superficial, discursive and figurative variety of semantics which distributes and disseminates lexical semantics in superficial syntactical structures' (Petitot 1985: 49). And the question is equally crucial for the possibility of asserting a kind of logical coherence of myths; that is, an approach that is semantic rather than morphological-syntactical in nature.

224 Andrew William Quinn

In a manner that yet again presents analogies with the Kantian approach, Levi-Strauss has introduced 'into the theory of deep semionarrative structures the second fundamental aspect of the paradigmatic dimension which, namely, is no longer one of substitution but of categorization' (Petitot 1985: 50), that is, the complexity/diversity of systems converging toward the human brain (Levi-Strauss 1973: 3589). It is a matter, actually, of resolving the apparent antinomy between the unicity of the human condition and the seemingly inexhaustible diversity of cultures and forms, by resorting to a structural invariant. The solution being contemplated consists of setting forth the laws of order occurring in every register of the human mind. In short, the heterogeneity of contents is reducible to a 'dialectic of oppositions and correlations' (Descola 1989: 1521): identity is differential, constructed by combinations. But this approach would appear to fall within a logico-formalistic paradigm that is close to classical cognitivism (hereafter CC). 1. CN Commitments Before tackling the question of categorization and cognitive (nonlinguistic) universals, we will look at the Greimassian generative process, as revisited by morphodynamics. The latter deals with deep anthropological structures, connecting semantic and syntactic universals, as does Thorn's concept of pregnancies, which aims at raising a number of data to a formally structured qualitative plane through internal dynamics. Such data consist of events, paths, states of things perceptibly and linguistically describable and more deeply categorizable, data linked to categorial perception (Petitot 1992:1-2; Bechtel and Abrahamsen 1993: 106f.) by conjoining cognitive linguistic and CN modelization. On that level, we will position the canonical formula as a kind of Kantian schematization that works between the subsymbolic level and constituted emergent meaning. The operation will give the formula a status on the symbolic and subsymbolic levels, that is, confer to it a morphodynamical schematization (Thorn 1988; Petitot 1994: 534). In that connection we will consider summarily the status of the canonical formula of myth in the context of the debate between CL and CN cognitivism. This will lead us to crucial philosophical and semiotic issues. We will try to tackle this question 'at the very heart of dynamical structuralism' (Scubla 1995) principally by resorting to what Langacker calls

Dynamic and Connectionist Implications of the Canonical Formula 225

'connectionist philosophy' (1987: 9). Connectionist philosophy cannot be understood as a philosophy per se, but more accurately as the conjunction of morphodynamic theories (Thorn, Petitot), cognitive grammar (Langacker, Talmy, Lakoff), and what is called CN (Smolensky, Lakoff, etc.). We will then be in a position to characterize CN differently, and occasionally in opposition to CC. Actually, our characterization will position it deeper than a mere reduction to a sophisticated technical implement. It will indeed provide a theoretical framework that opens up convincing philosophical statements on the mind. Along such lines we can already sketch a philosophical consequence that leads to a naturalistic orientation by which the phenomena of meaning at the heart of mythical categorization are natural phenomena in the same sense as biological phenomena, although they require a more complex modelization. This naturalness must be converging toward a non-standard physicalism;1 here we must emphasise a certain autonomy in the realm of meaning according to emergential theory. The latter constitutes the nucleus-princep of the morphodynamic theories already outlined in the Ubergang in the Kritic der Urteilskraftt to Kant's Opus Postumum. Those philosophical and semiotic issues, by resorting to morphodynamics and CN, are among the most significant developments in the last twenty years, as much by their ontological as by their epistemic reach (Lakoff 1990a and b). That is so particularly as regards the connection between what Langacker called 'faculte de pensee' (1983: 6-7) and the morphodynamic program first established by Thorn's catastrophe theory of cognition (Thorn 1988, 1990), and afterwards deeply elaborated by Petitot's theory of space cognition and especially of emergent meaning (1994, 1995). Consequently, philosophy will now have to cope with morphodynamic and CN architecture, cognitive space and its hermeneutics, and human temporality hand in hand with highly complex geometrical models. With regard to mythical thought, we will stress the complex ability of the human mind to create continua in its natural need to order the world.2 To do so, it uses categorization, semantic classes, and permutations between semantic fields, where metaphorizing processes operate at a high level of rationality providing a logical framework. In this way, mythical categorization needs a complex formalism3 that is provided by the morphodynamic framework - used particularly by Petitot in his writings on the canonical formula of myth (1988, 1992, 1995a), to which we will refer for mathematical analysis. Such analysis necessarily implies a cognitive apparatus for cognition refers primarily to world-human

226 Andrew William Quinn

relationships and, as we will see, cannot be separated from the anthropological structures of the imaginary at the very heart of LeviStraussian semiotics. We argue here that Thomian morphodynamic space-time theory complements, and surely improves CN models with more than a complex geometry: it adds to CN models biological explanations to the effect that humans should ensure stability in a space. In reference to the canonical formula of myth and cognitive ability, cognition is not a mechanistic activity but, as in Kant's Third Critics, a teleological causality corresponding to what we call self-organization. We would surmise that our problematic has more to do with the new paradigms that have arisen from what a CN allows us to call 'cognitive semiotics.' Within this perspective, the symbolic is understood as being a structure of a topological nature that contains a productive dynamic process. Although both CC and CN share a principle concerning the predetermination of the world to which common sense provides access, they are nevertheless categorically opposed to one another in that, for CC, cognitive categorization is conducted on the basis of rules that govern the manipulation of symbols which are given to an 'intellect' and which constitute meaning-laden entities. Both CC and CN concern the very nature of representations and the commitment to learning, that is, relations between the forms of representations and learning. But, as we said, we will refuse to reduce CN to technical modelling. For CN, a relationship exists between simple constituents (micro level) that are undissociable from a principle of locality raised to the status of finality, from which global properties (macro level), such as those on the symbolic level, may emerge. Accordingly, meaning-laden entities derive from a network of complex configurations of activities, with the latter comprising finer units than symbols. As we will see, symbolization results from complex activities founded in the complex relation of mind-brain and world (see Deane 1992: 95). The CN learning rules provide a specific measure of the stimulus itself, as represented by the function (i.e., hyperplan) computed in the feature space. 'Learning-acts' of this internal representation refer directly to the properties of the external world (Langacker 1991: 69, 298). To agree with Andler (1992) that cognitive sciences are the natural sciences of mind and meaning, and to thus bridge the gap between Geisteswissenchaften and Naturwissenschaften, as Levi-Strauss has stressed, we need a deep and strong theory. The needed theory is, in fact, the morphodynamic paradigm associated to CN, this latter which, as emphasized by Lan-

Dynamic and Connectionist Implications of the Canonical Formula 227

gacker (1988a and b, 1991), denies epistemic autonomy to the merely formal level. In fact, if CC mental representations concern images of well-formed formulas, of an internal language, the CN commitments are intrinsically ontological and for them recognition of forms is a crucial point, which contributes to a qualitative ontology and to the theoretical framework of self-organization theory. This also means that morphological descriptions are significant manifestations for they take account of constraints that are inherent in structural stability. Indeed, morphological descriptions analyse observed singularities of the morphodynamic process and its functions, and the bifurcations of attractors that describe underlying dynamics (Petitot 1992,1995b). CN, then, deals with two levels: micro (substrata) and macro, where the latter is causally reducible to the former and nevertheless the one whose structural characteristics are independent of the nature of substrata. The legacy of forms is mathematically expressible and linguistically describable. Accordingly, we can say that the qualitative and physical ontology of morphological structures possesses more than a material and mechanistic causality, namely, a teleological causality, which is a natural process carrying information. Briefly, this qualitative ontology permits the spatializing of non-space entities and of meanings, and is of a great import in the field of hermeneutics. Thus, the problem of categorizing space, which falls, like every semio-linguistic problem, within the field of the perception/language cognitive complex (Talmy 1988: 195; Lakoff 1989; Langacker 1988c, 1991), stands out from logicism and is related to the way in which a process of spatialization involving non-spatial symbolic structures is organized. Accordingly, the relationship to the world entails a kind of isomorphism between formal relationships and the sensory world by means of a dynamics of forms that gives rise to the quasi-infinite multiplicity of investments of pregnancies (the localist hypothesis, Petitot 1989). And all investments of pregnancies presuppose an intrinsic relationship between cognitive structures and spatiality, inasmuch as, semiotically speaking, discursive/figurative-level classes symbolize those pregnancies. Moreover, Talmy makes substantially the same point when saying: 'In particular, perhaps the principal overarching function of the structuring common across cognitive domains is that of providing conceptual coherence, that is, acting as a means for integrating and unifying a body of otherwise disparate conceptual material. In language and ... in vision, this fundamental function has three main global forms of realization: coherence across a conceptual inventory, coher-

228 Andrew William Quinn

ence with a scene, and coherence through time ... to be able to cohere in any sensible way and hence to be amenable to simultaneous cognizing as a Gestalt' (Talmy 1988:195-6). Agreeing with Levi-Strauss's postulate of a universal brain, we reject CC mentalistic ontology. Levi-Strauss conceives the mind as a categorizing cluster of non-semantic universals that can generate semantic categories. This principle of world-determination implies a deep semantics that operates as a universal grammar subjacent to myths. The essential point of our contribution is to inscribe the mythicalcategorization process into the frame of Petitot's revision of Greimas's semio-narrative structures (1992). Those structures put into play deep categories made of pregnancies. In this way, we will be able to attribute a deep meaning to narratives and their truly anthropological function; and we will be able to deal with the syntactic and semantic forms of myth. We think that it is possible in terms of space categorization (space is here considered in a very wide sense, more abstract than Euclidian) to better understand the mythical imaginary and to make the 'linkage' between a variety of myths in terms of semantic classes (e.g., Petitot 1995,1994). What are pregnancies? To answer the question, we have first to define saliencies. Saliencies are discontinuities that stand out of a physical continuum as informational sensorial stimuli. Pregnancies are salient forms that are biologically and psychologically significant: necessary to survival in a natural search for equilibrium. As invested salient forms, pregnant forms are those the perception of which activates irresistible physiological and psychological reactions in subjects. In the perspective of the anthropological structures of the imaginary, we are faced with a metapsychological semiotics that links subjects and their intentionality to value-objects: 'The perception of meaning results from a deep process in which semantic pregnancies are grasped by subjective awareness through an actantial syntax, through the destinies of values into objects ... Semes of fundamental semantics ... are pulsional and unconscious thymical contents, which are both interoceptivical and proprioceptivical... To distinguish them from the semes, we could qualify them as semantic pregnancies ... [that] constitute the substantial face of the virtual and abstract values' (Petitot 1986:175). In Greimassian semiotics, as in classical structuralism, the seme is defined as a minimal unit opposed to other semes through distinctive features. Its nature is thus non-substantial and relational. We refer here to abstract semes as apices of contents that are without a direct refer-

Dynamic and Connectionist Implications of the Canonical Formula 229

ence to the external world, but that are essential for its categorization (Greimas and Courtes 1979: 333-4). Thymic semes (proprioceptiveness) connote semic systems according to euphoria/dysphoria categories. They are, then, axiological values governing the production of meaning, but which remain inexpressible without referring to the event-generated logic of action into which the actantial syntax converts paradigmatic values. We are then led to fundamental pregnancies basically linked to a deep semantics that is the Levi-Straussian code system converted into actantial interactions (cf. Propp's narrative functions).4 Semantic pregnancies are thus proprioceptivity, thymicity, intentionality, and desire (Petitot 1987). We call here for Thomian catastrophist schematism, in which an utterance describes a space-time process. Its syntactic structure expresses topological interaction (contact, capture, etc.) between topological cells that are actants in process that express a space-time conflict between units reducible to their location; thus a positive value is instantiated. In that connection, Thorn may rightly assert that 'one would do well to posit as principle that every signifier is generated by its signified' (1980b: 197). This principle, as we shall see, may correspond to Levi-Strauss's position, but this is not so easily shown. Thus, in the localist hypothesis, 'the spatial and temporal uses of prepositions' may be inferred from the perception of space in which human beings are located. In effect, 'spatial expressions appear before temporal expressions do; each term which may be used in a spatial and temporal meaning ought to be acquired with its spatial signification' (Descles 1991: 85). If the primacy of the space (time) is problematic, it remains important, however, to ground 'certain schemata of predication in schemata which are constructed by visual perception. The predicational schemata of languages thus appear as the extensions of morpho-syntactic schemata which encode the universalized representations of the position of an object in space ... The cognitive sciences invite us to conceive of these problems via representations which are not autonomous, but which are instead inserted in architectures which articulate different levels and types of representations' (ibid.). With respect to the 'architectures' mentioned by Descles, we think that the canonical formula takes on a major interest for the cognitive sciences and their related disciplines, particularly for the more specialized field of non-classical paradigms associated with the development of dynamic structuralism and CN models. If we allow that social space

230 Andrew William Quinn

is intrinsically determined by a dynamics of forms comparable to a geometry capable of accounting for the production of the symbols emerging at its surface, we are led to recognize the existence of a primitive cognitive level of representation, a deep conceptual structure that, contrary to semantic structure, is nonlinguistic (Langacker 1991: 108-9; see also Jackendoff 1994, Lakoff 1990a), and that assures a compatibility of morphosemantic type between perception, language, and action: 'at least basic conceptual ability is universal without entailing that semantic structure also is' (Langacker 1991: 109). In other words, universals are of a cognitive kind. In local domains where space-time entities coexist, positional relationships serve as archetypes for syntactical structuring. The grounding of language in perception calls for a real perceptual foreshadowing of elementary syntactic structures. The localist hypothesis posits the strong requirement of a cognitive-type linkage involving syntax, perception, and action. In the anthropological perspective of Levi-Strauss, every phenomenon involving the production of meaning is linked to dynamic unfolding throughout a territory, as has been attested by Maranda and Maranda (1971). For that reason, the symbolic meaning of narrative remains inseparable from the spatial structuring (see Deane 1992: 47) of the societies wherein they dwell. If, from a standpoint working from bottom up, social space appears as a dynamic morphology that intrinsically structures emergent-level 'symbolic organizations' in terms of their diversity, it is equally conceivable that these organizations categorially operate upon this space (Langacker 1993: 3)5 in the form of feedback. The underlying syntactic structures are themselves governed by 'the topological structure of [space-time] locating processes' (Petitot 1989: 69), which imposes asymmetry in 'locating entities' (Svourou 1994: 8-9, 213n6). Archetypes thus appear as deriving from elementary space-time interactions between equally space-time actants. It is for this reason that the canonical formula does not represent a loop that closes back on itself; this is also why it is possible to approach the question of myths as natural semiotics of the natural world; that is, a semiotics in which the iconicity of natural language, which Peirce has already dwelled upon, constitutes a fundamental point. Within this perspective, then, it becomes plausible to assert that the categorization operated by myths linked up with an a priori space temporality presents a parallelism with the spatial organization constitutive of the social order. Following Petitot, two levels of morphological structuring may be introduced: the pheno-cognitive and the pheno-physical. Those levels

Dynamic and Connectionist Implications of the Canonical Formula 231

permits us to avoid the trap of world/language dualism (CC), and give access to the anthropological structures of the mind. It is important to say that the human mind naturally (our common sense) has the property of being spatially invested by the dissemination of a pregnant form over or upon a salient form (Thorn). Thus, there is no antinomy between continuum and discrete terms that articulate possibilities of domains (Langacker 1991: 69) without falling into an open-ended structure. This is a strong commitment of Petitot's emergence dynamics of discreteness from continua (see also Langacker 1993: 4; 1988b: 53). In this way, categorization appears to be among the most significant manifestations of the human mind. To affirm that the anthropological structures of the imaginary are invested in a space by means of categorizing implies that the spatial order is of a differentiating kind and that it reveals the traces of this imaginary. This also implies, however, that the symbolic field connects meanings on a topological basis. In this case, meaning is of a positional kind, spatially invested by a categorial unfolding of space-time axes, as well as situating actants and their agential virtualities, be these of a narrative nature or other (go up, come down; good, evil; etc.). In short, the phenomenon of categorization comes within the scope of cognitive dynamics, owing to the fact that the narrative path (Greimas 1970; Greimas and Courtes 1979) is situated, set on a scene. Actants are always internal to the constitutive dynamics (principle of immanence), and they unfold positionally. All specific semiotics constitute a relational network through which elementary structures are worked out; indeed, this system of relations constitutes a network of semantic categorization. Inasmuch as the intelligibility of a path does not imply resorting to any 'arbitrary principle/ it is thinkable that both universal processes of interaction between human beings and their environments and of the phenomena of categorization should be conceived as being independent of linguistic classes. Meaning, of the kind that Thorn refers to, is of a piece with the world, in its pheno-physical, phenocognitive, phenomenological aspects, as well as in the way it summons forth a formalistic-objective hermeneutics (Salanskis 1990). Categorization reveals space-time at work in cognition, as related to Thorn's crucial structural stability principle (1972), a dynamic system through which the symbolic level emerges from a sub-symbolic level. This is a very fundamental reference to the common-sense understanding of space-time at the basis of natural hermeneutics (Quinn 2000,1998). In fact, natural geometry constitutes this hermeneutic configuring of

232 Andrew William Quinn space-time events. The symbolic-emergent level is deeply linked to a bottom level of categorization in specific, contextual space-time terms and relations (Langacker 1991: 115, 265; 1988c: 131). This leads us to a realistic and qualitative ontology associated with a cognitive semiotics that stratifies forms of meaning from a micro-substrate level to a macro-emergent one. 2. The Canonical Formula of Myth Levi-Strauss has always been guided by this formula, beginning with Anthropologie structurale, continuing thereafter with the four volumes of the Mythologiques, and finally with the more recent investigations of La potiere jalouse and Histoire de Lynx. There is no way of denying the fundamental importance of his work 'beyond the mythological universe' (Marcus 1993), given the relevance it has demonstrated in the field of semio-narrative structures. The canonical formula is related to a paradigmatic reading of a whole class of related myths' (ibid.: 1). On convergent lines, Marcus writes also that La pensee sauvage has a 'high degree of structural and systemic virtues' (ibid.: 4). And he states that the canonical formula expresses a fundamental aspect of Levi-Strauss's theory of the mind with regard to the 'interpretation of myths as antinomic processes' (ibid.: 1). Here, it may be useful to summarize briefly the difference between classical analogy and the canonical formula. With regard to this question of analogy, we partially subscribe to the view that sees the canonical formula as 'the figuration of a mediating process where some dynamic roles are expressed more accurately than in a simple analogy model' (Maranda and Maranda 1971: 26). In the canonical formula, '(b) is the mediator; (a) is the first term, which expresses, in connection with the socio-historical context, a dynamic element (specifying function der the impact of which the item unfolds. The other function, which is opposed to the first one, specifies (b) in its first occurrence. Thus, (b) is alternately specified by both functions, and thus can mediate opposites' (ibid.: 26). 'While analogy is specifically "linear," Levi-Strauss's formula is non-linear, i.e., it implies a permutation of roles or functions and of terms, since (a), which is given as term becomes, once inverted, a"1, a sign of function, and (y), which is given as a sign of a function, becomes (y), i.e., a term which is the final outcome of the process' (ibid.: 26). However, permutation or 'bringing the loop back around upon itself does not

Dynamic and Connectionist Implications of the Canonical Formula 233

constitute a conclusion (end-point); we are dealing here instead with a 'helicoidal step' (ibid.: 26) that brings about a new situation: 'If a given actor (a) is specified by a negative function (a villain), and anothe one (b) by a positive function (and thus becomes a hero), (b) is capable of assuming in turn also the negative function, which process leads to a victory so much more complete in that it proceeds from the ruin of the term (a) and thus definitely establishes the positive value (y) of the final outcome. This time as a term, (y) is specified by a function which is the inverse of the first term' (ibid.: 26-7). So, we obtain a gain:/fl_i (y) >/c(b). As Maranda and Maranda have also written, the formula's 'first three members, and express a dynamic process whose final outcome, expressed by the last member, r\ (y), is the result of a state, i.e., the en mediation' (ibid.: 27). We are now faced with the crucial question of structural dynamism, which encompasses far more than analogism. To answer some questions raised by Pierre Maranda concerning this structure as well as the question of feedback and loop, we will first emphasize that we need an additional formalism. The canonical formula of myth does not frame a closed system: it ends up in a gain. The problem, therefore, is how to express this gain, that is, to show how a gain may emerge from a structure-formula where the number of relations and terms is fixed (two terms in relation with two relations). We have to heed the fact that analogical models cannot adequately account for this complexity. Along with Petitot and Scubla, we emphasize the dynamics of the formula, which has been underscored by the Marandas (1971: 30) as being teleological. Moreover, Levi-Strauss himself, in Mythologiques III (294-5) and II (211-12), points out the dynamic character of mythical thought. The helicoidal structure of the formula illustrates an abstract principle that sets out the complexity of dynamics that analogy cannot model properly. With overlooping, we reach qualitative dynamics: 'The deep structure of a myth is the solution to a problem of cognition ... It generates the myth in the codes available to the society' (Maranda 1971: 30). Indeed, mythical thought is further intelligible as a morphodynamic process (cf. Maranda and Maranda 1971: 29) whose operations are illustrated by topological diagrams or rhetorical images such as the double twist or the double chiasm. We get more intelligibility than the symbolic paradigm (closed to CC) gives us. The mathematical concept of group is inadequate to formalize this process: the canonical formula of myth is a real 'genetic model of myth (i.e., that engenders it while simultaneously giving it its struc-

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ture)' (Levi-Strauss in Marcus 1993: 5). Myth, like any society, does not consist of a closed system. Although the helicoidal structure leads us to a morphogenetic view that takes into consideration complex operations as we will see later, we lay stress on 'looping with a gain' (Maranda and Maranda 1971: 27), which is deeply theorized in double-cusp formalism (Petitot 1992), where there is a coupling of two semantic spaces that produces a third term, a~l. We then move beyond analogy to work along the lines of catastrophe theory (Scubla 1995:17) in order to be able to deal with categorization, in which non-linear and non-symmetric processes obtain. If, in CC, functions like/j are of a prepositional type, CN commitments ensure that those functions are not of analytical or prepositional type. To support the understanding of this level of categorization, let us recall that the last part of the formula does not in any way proceed from predicate calculus formalism. CN commitments go beyond this kind of formalism. Let us emphasize once more the fact that categorization does not only serve to resolve contradictions, but also allows for the setting in of actantial relations. And actantial schemas call for a canonical syntax in formal semantics that operates in pregnancies. Indeed, and as implied in the last member of the canonical formula (Maranda 1971: 30), Levi-Strauss states the essential idea that the structure of myths in general implies a teleological view, which, according to Petitot, serves to resolve contradictions (see also Maranda and Maranda 1971: 30). The deep structure of myth is a dynamics inherent in cognitive processes even though it remains unconscious. In Marcus's words, 'in the same way in which Levi-Strauss looks [at] and accounts for the unconscious structure of society, we think that the capacity to perceive the intuitive aspects of the mathematical concepts, models and theories and their relevance for social sciences belongs to unconscious structure' [of mind in Levi-Strauss's theory] (1993: 4). At a deeper level, the canonical formula of myth is an essential component of the formalistic framework that constrains the mythic productions of the human imaginary and constitutes the formalism of the latter. According to Petitot, the formula offers 'a crucial nexus for theoretical structuralism' of a formal and dynamic kind (Petitot 1988: 26): (1) 'it provides empirical diversity with categorial unity by restricting the inferences which are specific to every hypothetico-deductive approach' (ibid.: 25); (2) it makes possible 'explicit mastery of situations whose intuitive component has become inoperative through a latent and implicit complexity' (ibid.); and (3) it brings out 'unsuspected soli-

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darities between realms of experience which otherwise would appear to be foreign to one another' (ibid.). In that connection, La potiere jalouse posits a 'dual relationship linking, on the one hand ... the Goatsucker to marital jealousy ... and, on the other, pottery with the same jealousy' (ibid.). It means that the relationship of Goatsucker to jealousy transgresses mere logical analysis: '[O]ther myths are connected, in terms of transformations, to the myths of the Jivaro, wherein a bird diametrically opposed to the Goatsucker - the Ovenbird - enters the scene. Furthermore, by means of an "empirical deduction," one may observe an anthropomorphic interpretation of the bird's anatomy and observable habits' (ibid.). The Goatsucker signifies jealousy, whereas the Ovenbird refers to 'the function of harmonious conjugal relations and the making of pottery (it is a remarkable nest-builder, a "master potter").' And, Petitot carries on, 'as a term (... actant), the second thus inverts the semantic values of the first, [whence] a"1. However,... it is absent from Jivaro myths and thus cannot operate as a term. The hypothesis thus rests on the fact of an exchange of its value as term for a value as function: the "jealous" function of the Goatsucker is to the "potter" function of the woman as the "jealous" function of the woman is to the "(inverted) Goatsucker" function of the potter' (Petitot 1988; quotation from Levi-Strauss 1985). And then, as illustrated by Petitot (1992: 395-6), / x (a):/ y (b)-/ x (b) :j Uy) becomes ^.(e):/p(f)-/;.(f):/^(p), where/(= *) = 'jealousy/ p(= y) = 'pottery,' e(= a) = 'Goatsucker,'/(= b) = 'Woman.' Now, we wish to explain why we believe that the dynamic and CN framework is not only compatible with Levi-Strauss's anthropological theory of the imaginary, but also how it prevents the reduction of his theory to the level of a simple symbolic paradigm. Indeed, his theory points in the direction of the real morphodynamic approach: his taxonomy goes further than classical categories,6 not by its categories per se, but by the way it introduces a dynamic approach to the symbolic level. Thus, Levi-Strauss's theory converges implicitly with CN in that

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it presupposes a cognitive categorization and a context-based semantics. Lakoff, opposing CC-CN paradigms, writes that 'a cognitive linguist ... expects categories to have one of the various kinds of prototype structures and to be organized in terms of basic-level, superordinate and subordinate levels' (1990a: 41). Opposing the logico-combinatorial fallacy, including Frege's theory of the truth conditions of the relation of symbols to physics (Langacker 1991: 69; Bechtel and Abrahamsen 1993: 123), Lakoff's cognitive and general commitments require us to 'take basic-level categorization and image-schemas seriously' (1990a: 42). These are inherently general cognitive processes grounding meaning in bodily and social experience, using neural foundations (see Langacker 1991:149). We recall that CC commitments shape closed-ended systems as elementary structures requiring a kind of autonomy in which it is impossible to link language, perception, and action. This is not the case in the Levi-Strauss's theory of mythology. In contrast with an analogical and combinatorial approach that radically erases cognitive and dynamic complexity, as discussed by the Marandas (1971), we have to hold to cognitive and dynamic complexity in order not to evacuate both dynamic syntax and semantics. We are thus seeking a higher level of general principles. If one agrees to consider mythical categorization as a metaphorical process, we may then be correct, according to Lakoff, in asserting that 'the metaphor involves understanding one domain of experience.' Such is the case in the Jivaro myth analysed by Levi-Strauss (1985), where 'jealousy' appears 'in terms indeed of a domain of experience': the Goatsucker bird. 'The metaphor can be understood as a mapping (in the mathematical sense) from a source domain (in this case, [Goatsucker]) to a target domain (in this case, [jealousy])' (Lakoff, 1990a). That is what happens in Levi-Strauss's deep analysis of the pottery. 'The mapping is tightly structured. There are ontological correspondences, according to which entities in a domain correspond systematically to entities in another domain' (Lakoff 1990a: 48). As Petitot wrote, 'the master of pottery is correlated to jealousy and represented by the Goatsucker, which is a solitary, greedy and dismal nocturnal bird which is also correlated in myths with conjugal discord' (1988: 26). Lakoff's theory of mapping includes epistemic correspondences in which knowledge about 'a domain is mapped onto knowledge about another.' But mostly, there are

Dynamic and Connectionist Implications of the Canonical Formula 237 ontological correspondences [that] map this scenario ... (a knowledge structure in the cognitive sciences) onto a corresponding [jealousy] scenario, in which the corresponding alternatives for actions are seen. Here is the corresponding [jealousy] scenario that results from applying the ontological correspondences to this knowledge structure. What constitutes the [jealousy-as-Goatsucker] metaphor is not any particular word or expression. It is the ontological and epistemic mapping across conceptual domains, from the source domains of [Goatsucker] to the target domain of [jealousy]. The metaphor is not just a matter of language, but of thought and reason. (Lakoff 1990a: 49)

Metaphorical categorization should help understand the multifarious capabilities of the human mind to extend knowledge and the imaginary indefinitely. A metaphorical correspondence, which is a high-level categorization, has to come from a conceptual system depending on a sub-conceptual structure (Langacker 1991). Agreeing with the idea that a metaphor is a real categorization, we are led to investigate 'the nature of categorization, both prototype and basiclevel' (Lakoff 1990a: 50). Thus, we take a stand against the fallacious Saussurian arbitrariness of sign because within cognitive linguistics, 'signs are motivated, and conceptual metaphor can be one of the things motivating an idiom' (Lakoff 1990a: 50). For example, pottery comes with a conventional image, that of an activity. In this view, 'insofar as pottery consists in transformation of a matter into form, particularly into a form of corporeal "container" from a "content," a woman's body is symbolically related in some myths to pottery' (Petitot 1988: 27). As we can see, 'metaphorical mappings preserve the cognitive topology ... of the source domain' (Lakoff 1990a: 54), and all domain inferences from cognitive topology will be preserved in the mapping: 'Abstract inferences are actually metaphorical versions of spatial inferences that are inherent in the topological structure of images-schemas. Thus, the generalization commitment leads us to the Invariance Hypothesis, which in turn makes a major controversial claim about the nature of abstract reason' (Lakoff 1990a: 54; see also Deane 1992: 306). This is also true about mythical narrativity, and 'if those abstract concepts are metaphorically understood, then their imagistic representations are the image-schemas that have been metaphorically projected from source domains of the metaphors' (Lakoff 1990a: 54). To emphasize this point, let us quote Lakoff once more:

238 Andrew William Quinn Mapping one image onto another can lead us to map knowledge about the first image onto knowledge about the second. Consider for example from the Navaho: My horse with a mane made of short rainbows ... The structure of a rainbow, its band of curved lines for example, is mapped onto an arc of curved hair, and many rainbows onto many such arcs on the horse's mane. Such image-mapping prompts us to map our evaluation of the source domain onto the target. What is preserved across the mapping is the causal structure, the aspectual structure, and the persistence of entities. We referred to this as generic-level structure. (1990a: 67, 69)

By this detour via Lakoff's theory of mapping, we underline the inter-code and inter-myth dimensions mentioned by Petitot and by Scubla (1995), which brings us to the categorization of space and to the complex mathematical formalism of the double cusp (Petitot 1992: 388f.) entailing that two semantic classes contain an emergent product. We emphasize here the gain at the end of the canonical formula, and also the complexity of the relations between terms of equal statuses, as well as the bearing of the overloop that leads to the final gain - a modelling operation impossible to formalize in analogistic formalism. Referring to catastrophe schematism makes it possible to conceive all the possibilities of articulation, on the one hand, between units and, on the other, the paths that connect one local content to another. Reference to two semantic classes prevents us from a paradoxical understanding of opposed terms in the formula by using a finite number of terms and functions. The explanation of the internal complexity by relational topology is achieved by means of the geometrical double cusp, where path-events in a dynamic structure with two semantic fields allow for a linkage. Petitot's double-cusp geometry (1992: 394f.) refers realistically to the theorization of actantial relations (conflict, etc.) necessitating a topological syntax. This modelization allows us to integrate space categorization by actantial relations and space domains without analysing categorization in the terms of logico-combinatorial systems (Klein's group, analogous, generative grammar,7 etc.). We are thus led to understand the emergence of meaning, of mythic narrativity, by the hypothesis that for naturalness philosophy, pregnancies invest salient forms that become objects-values, and this is true for metaphorical categorization. We refer to Langacker's predication theory by which domains profiling is topological and geometrically constituted (Lan-

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gacker 1991; Deane 1992: 120) with regard to prototypical substructures. At this point, we would like to bring forth a number of princeps and consequences of this modulation, leaving the further details of the modelization to mathematicians. 3. The Structural and Categorial Unconscious One of the essential objectives of structural anthropology is to account for the complexity of human cultures in terms of their diversity by attempting to bring this diversity back to universal/human forms (or architecture: Levi-Strauss 1985: 264) without voiding it. Moreover, such an attempt requires that philosophies of consciousness will not be resorted to: this is borne out in the celebrated 'Finale' of L'homme nu (1971: 562-3). 'Anthropologies of consciousness remain locked within the circuit of subjectivity.' Structural anthropology has thus entered another field of operations, and chosen to checkmate the ineffectualness of the social sciences in terms of their own practice, since consciousness is an unsatisfactory (to say the least) epistemological postulate (1971: 563). In the terms of classificatory logic to which Levi-Strauss has addressed himself many times (see in particular the question of totemism, 1962), the species functions as an operator of logic (Henaff 1991). However, the problem lies, precisely, in how to identify this logic, how to choose the paradigm required for understanding the type of categorization involved (CC or CN). A fortiori, it is necessary to remain attentive to the closeness of anthropology and language, especially since Levi-Strauss has often stated that 'social universes are universes of communication'; it must not be forgotten that we are dealing here with a method-oriented analogy, that is, a type of method that indeed expands the notion of communication: To derive from language a logical model which, being more accurate and better known, may aid us in understanding the structure of other forms of communication, is in no sense equivalent to treating the former as the origin of the latter' (Levi-Strauss 1958: 96). On the contrary, the confluence of disciplines is the critical point in the search for the deep level that underlies the syntactic and semantic form of narratives. Thus, the search for invariants does not reduce the narrative universe to a closed set, anymore than the attempt at reconstituting its physics implies a shapeless kind of universe, as the latter, indeed, appears to be dynamic. The social universe of myth is one in which framings and meaning circulate.8 It is by

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resorting to structural semantics, which is not to be associated with the semantics of linguistic theories which veiling canonical semantics, that we will relate the myth as canonical form to the narrative level. Modal categorization. We will now take up the idea that Per A. Brandt has advanced, according to which there is a 'genuine modal dynamics whose constraining effects determine not only meaning in general but all known semiotic registers as well' (1992: 11). It would appear that, immanent in narratives, there is a modal cognitive grammar that is undissociable from the 'perspective of the analysis of dynamic spaces.' We must take into account deep narrativity, which is a more abstract organization, a kind of narrative competence. It seems that there occurs in deep narrative structures the same dynamic organization as we find within the mental apparatus, which accounts for the welling forth of meaning occurring in, but not limited to, narrative performances. It is equally a question of searching for the stratum of deepmeaning forms in the multiplicity of forms assumed by discursive competence. Such a stratum would support discursivity and enunciation by actualizing the 'space of semiotic virtualities/ modality, and the dynamic circuit represented by narrative framing: Modality [is] the structuring principle at work on the level of deep narrativity; on the discursive level, where cognitivity predominates, the modal phenomenon founds the spatialisation of an event in a way making it possible to understand aspectuality somewhat better ...[;] at the sentence level, (we find a) decisive relationship ... between a syntax of cases and the underlying actantial semantics which, without involving sentence-level embedding structures, appears to imbue the modal in a way permitting the temporality specific to grammatical tenses to unfold;... at the enunciation level, we find the morphology of mode, which becomes analysable with the return to deep narrativity concerning the subject, with the latter appearing as both enunciator and actant in a context which is narratively structured, interactional. (Brandt 1992:11)

On the subject of the bird's-nester, Levi-Strauss has written that he 'establishes a relationship between the circulation of salmon and the circulation of women' (1991:41); a 'symmetrical myth exists which presents the adventures of the Coyote and his son/ but in a way that inverts the positive value of exogamy. But does the combinatorial principle reveal the 'modus operandi of mythic thought/ which codifies by means of thematic isotopies and discursive procedures that are so bril-

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liantly described by Levi-Strauss, and which relate elements drawn from the vegetable, animal, human, and supernatural worlds to one another? The transformation of these elements by myth has an equal impact on the order of things and the social order. At stake here is an intrinsic and teleological vector of myth that the canonical formula formalizes as open and dynamic. The impossibility for the loop to close completely back in on itself (ibid.: 47) is due partly to two facts. (1) The logic of a narrative involves contrary assertions that are held to be true, and (2) the kind of logic involved cannot attain completion on a narrative level alone; indeed, it brings the canonical schema to a close only on the level of social practice itself inasmuch as, implicitly, this closingoff or completion refers to its enunciation and to its locus. We emphasize that it will not suffice to state that myths are reflections of culture or space; instead, they must be seen as inscribed in creative dynamics specific to particular cultures. As expressed so well by Petitot, The form of the formula ... [is] not "ordinary"; its open-ended loop consists in the expression fa-\(y}' (1988: 41). That is why it is important to emphasize 'this supernumerary twist, which always appears in the final stages of mythical transformation/ Hence, 'the exchange in value between term and function is evidenced rhetorically on the narrative level by a metamorphosis. "A super-natural creature that used to be a Goatsucker only in name, that is, figuratively, actually turns into this bird when, in disappearing physically, she leaves mankind with the raw material used in pots, namely, clay, which, sub specie naturae, only her opposite, the Ovenbird, knows how to work'" (Petitot 1988: 27; quotation from Levi-Strauss 1985: 80). Petitot adds that this 'remarkable semiotic procedure, which is essential to the imaginary, evidences a formalistic operation in which an inversion of values is combined with communication between two ontologically heterogeneous worlds (natural/supernatural, or physical /meta-physical)' (ibid.: 27). If we speak of categorization, this is only to the degree that the dynamic form that unfolds throughout the quasi-infinite diversity of narratives rests on a common cognitive architecture. Such an architecture, in the theory of semio-narrative structures, has been described as an interconnection intrinsic in a complex of narrative variations having undergone transformations in space-time settings. And the 'rules of transformation' make it possible 'to move from one variant to another by operations similar to algebra' (Levi-Strauss, 1973: xii in Marcus 1993: 3). In accordance with a certain formalism, a form of content is

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syntactically articulated but often remains veiled by the concrete semanticism of content matter. It is a question of resolving the existing contradictions (anxiety, etc.) 'by playing off the transformational possibilities occurring between several mutually convertible codes. "Each code constitutes a grid for deciphering an empirical given," each myth always uses "several codes," but "retains only a few cells or blanks from each grid, which the myth combines with blanks from other grids. In actuality, a myth constitutes a meta-code, which is something altogether different from a metalanguage'" (Petitot 1988: 28-9; quotations from Levi-Strauss 1985: 228-9). Let us remember that one of the essential aspects of Langacker's cognitive grammar consists of refusing to reduce semantics to the grammatical level, and it offers the possibility of a metalanguage, as we said before. Somewhat in parallel with CN paradigms, Langacker holds that meaning derives from a cognitive operation that the narrative level places 'in perspective'; in the manner of a picture (Bild). Categorization is a mapping process or, at a more primal level, a scanning process (Langacker 1991: 69, 78f.) that originates in sub-symbolic networks and pursues the detection of contrasts and the sampling of qualitative discontinuities in the space-time field; it is an 'active cognitive functioning' (Langacker 1987:110). Inasmuch as meaning constitutes a formal order deriving from physical supports, it cannot be conceived within a dualistic perspective. Meaning is connected to the interfaces occurring between sensorial systems and the physical world. All deep categorial organization is the result of cognitive processes that are connected to physics. Linguistic units themselves derive from a geometric segmentation of local regions, beginning with the physical continuum as interpreted in terms of space-time regions. As for the regions themselves, they form the 'primal representational space' that constitutes the condition under which it is possible for forms of categorization to appear. Hence, before being symbolic, these forms are geometric (Petitot 1992). According to Langacker, the primal level at which language and cognition interconnect to form a complex determines the categories that are a kind of isomorphism with the forms appearing at the level of geometric spaces (Langacker 1990; see Thorn 1988: 211f. and Langacker 1991: 328 about iconicity). We should therefore go beyond the analogistic thesis mentioned above. In fact, the work of Levi-Strauss could lead us to the formalism of analogism close to the CC symbolic paradigm, and make it impossible to understand how, in terms of its infrastructure, the mind codifies.

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One must be careful to avoid the blind alley that inherently arises in conceptions of the mind presenting it as a processor of propositions, as thought-language like Fodor's mentalese; or in terms of world and categorization dualism as well as solipsism (even of a methodological variety). Instead we should privilege deep semantics that determines values on the basis of pregnancies invested in forms having tremendous diversity. And such determination indicates what is or what should be. There is, in fact, no dichotomy cutting off the practical from the mythical; there is, rather, a modal, space-time organization in which, beginning with the deep semiotic structure, it is possible to access the specifically figurative level, that is, the level of action. What must be opted for is a weaker isomorphism that situates mediation between world and language in a kind of phenomenological and morphodynamic opening. In that connection, strong isomorphism is quite 'insufficient/ on account of its remaining a 'simple isomorphism between semantic isotopies/ But this morphodynamic opening may be obtained only at the cost of strong constraints that only a linkage can provide. As emphasized by Petitot, 'in an isomorphism,9 an identical form of content is implanted upon two different substrata, whereas in a linkage, the form of content is enriched' (1988: 42). This linkage leads to a condensation of operations, already underlined by the double cusp that implies an overlap between two semantic spaces. Hence, in the Jivaro myth, as the woman dies, she 'undergoes metamorphosis into the Goatsucker.' Next, there is a change, a crossover, involving the Goatsucker (which is common to both codes), from the code of conjugality to the code of pottery. Finally, a change of space is translated by a 'semantic inversion of the Goatsucker into the Ovenbird, which is absent:10 'If, despite being absent from a given milieu, a species remains present in myth, it is projected into another world in which the semantic functions which myths have ascribed to it elsewhere as a real animal are systematically inverted' (Petitot 1988: 27). Although emphasis on the mediation process is of high interest (Maranda 1971), nevertheless it does not constitute a fundamental aspect of mythical thought. We have to take account of the dynamic emergence of differences, of the passage from continuum to singularities. Thus, the linkage maps two different semantic spaces; it proceeds to the categorization of semantic spaces in sub-domains (Langacker 1991: 64-5; Croft 1993; Svourou 1994: 34, 58-9), to that of values produced by a group of generating potentials. In this connection, we want to stress two points. (1) The cognitive process underlying the categori-

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zation, which (2) is ruled by a sub-level of a variational optimization processing (Smolensky 1992). In following these CN statements, we cannot agree that the canonical formula would be an elementary structure. There is (1) a linkage between two elementary structures against a formal isomorphism or a homomorphism (Petitot 1992), and (2) the value of opposites of term and function is syntactic and semantic. The one very deep and indispensable formalism is the inter-myth categorization.11 If an intra-code modelization deals with two half-axes in the same semantic class, then inter-code modelization deals with two axes as two semantic classes (see Scubla 1995). In short, the 'reciprocal metamorphosis of the Ovenbird (absent from the myth) into Woman transforms the latter into (male) potter' (Petitot 1988: 42). By means of an adequate schematization, one and the same actant can operate on two different codes, x and y, as follows and (not present), and into two semantic classes for mythical codes formalized by the double cusp. In order to grasp the identity of a linkage with the conversion worked out between value as term and value as function, and mostly in an inter-myth modelization, it is necessary to refer to the approach that catastrophist models take to semantic and syntax conversion. 4. The Morphodynamic (CN Dynamic) Model Formal aspects posed by mythology seem to be undissociable from aesthetic forms by which they are objectified. Then, these forms belong both to continuum and discontinuum, an antinomy that the catastrophe theory offers a new means of surmounting (Levi-Strauss and Eribon 1990:192). Actants operate as localisers of semantic pregnancies. As schemata of actantial interaction, actant catastrophes thus convert conflicts occurring between pregnancies. The consequence of this is that in an actantial model for example, the standard model of the cusp having an internal space of dimension 1 (variable x) and an external space of dimension 2 (variables u and v), semantics is brought into play in a threefold way: i) ... in the determination of the semantic content of internal variable x; ii) ... (conversion by duality) as the semantic contents of the threshold between both actants; iii) ... as the semantic interpretation of axes u and v of the external space in which unfolding occurs. (Petitot 1988:42)

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All semiotic structures contain a fundamental syntax and semantics. 'All structures are a morphology resulting from an internal dynamics; and all morphologies are a system of qualitative discontinuities that deploy themselves in a substratum space' (Desmarais 1992; also 1995). This morphodynamic principle means that once the narrative universe is formalized in terms of its virtualities by the canonical formula, it ought, via a dynamic and relational topology, to reconstitute either the description or, at least, the explanation (hermeneutic) of the relationships occurring between the symbolic structure constituted by the mythic narrative and the space in which this is embodied. Contrary to Henaff (1991), we would say that the Levi-Strauss's logical operator is itself the result of a cognitive architecture in which the operations are not of a logical kind, and as performed in reality, do not refer to a rational a priori, but do, instead, refer to common and primal cognitive activities in which categorization is to be considered as an adaptative and phylogenetic finality. Cognition may be conceived of as a dynamics that is constitutive of the real, with the latter being categorized in a naturally permanent way, not as an afterthought; but such a dynamics would also be driven by the constraints deriving from the complexity of the space that is inhabited (Desmarais 1992). Hence, at a deep level, there are some basic conceptual relations that articulate predication of a genuinely spatializing kind (Langacker's space grammar) in connection with the relations that link the language-cognition complex to the space-time world (a continuum) and perform sampling of identity and difference (the realm of the discontinuous). These categories are related to the situations at hand, and, in terms of their actualization, they display tremendous variability. Processes such as these do not depend on the intrinsic value of representation (CC), but do, instead, depend on their positions in space. Thus, the symbolic must be defined as the result of a space categorization process that institutes relations of a type/token variety and that produces configurations of doing and saying on the basis of this network of local and distributed representations. 5. The Symbolic By postulating that social systems are systems of communication, hence, symbolic systems that are 'never clearly perceived/ one is also recognizing that the symbolic is tied up with the very foundation of social existence. But how can we conceive the genesis of the symbolic (Levi-Strauss 1983b: 1229)? To undertake such a task, it is important to

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consider first of all that the emergence of symbolic thought is undissociable from that of the social-order process. In other words, one might say that, for Levi-Strauss, the social is not symbolic but that it is run through and through with the symbolic. Symbolic thought makes society not only possible but necessary. Thus stated, the very meaning of the structural unconscious is brought into play: the structural unconscious 'is reducible to a term by which we design a function, the symbolic function, which no doubt is specifically human, and which is carried out according to the same laws among all men' (Levi-Strauss 1958: 224). Hence, the unconscious is, by definition, the symbolic system, the set of laws that governs the very nature of humans and their productions (language, etc.) and, in short, the condition of meaning. There is no doubt, then, that it is related to the biological nature of man, to the very structure of his mind and his specifically structural modalities. However, in contrast with the psychoanalytic unconscious, the structural unconscious admits a formal framework (cognitive architecture) that Levi-Strauss both anticipated and called for. We should not imagine that unconscious is the ultimate haven of individual particularities (Levi-Strauss 1958: 224), any more than it is a Jungian unconscious 'replete with symbols and even symbolized things which provide it a kind of substratum' (1950: xxxii); it is uninhabited. Its status is more problematical than the Freudian unconscious, and, inasmuch as it presents itself as a regulator of the 'regions' under consideration (communication, etc.), it remains 'empty, ... as alien to mental images as the stomach is to the foods which pass through it. As the organ of a specific function, the unconscious merely imposes structural laws, which stretch its reality to the limit, upon inarticulated elements which originate elsewhere: impulse, emotions, representations, memories' (1958: 224). If we assert the pre-eminence of the symbolic and, what is more, the categorial character of the unconscious, the issue no longer involves constituting man as the Other of nature, quite the contrary. In fact, the symbolic sets out the widest possible differential for man, in a way that integrates the other dimensions; man is more a symbolic than a rational animal. However, the origin of this differential is to be found in the very 'architecture' of the brain as an integral part of the space-time reality. Of course, by remaining at the surface of narratives, or from LeviStrauss's formalist perspective, it is possible to claim that in a similar way to that by which language combines different phonemes, social systems combine elements that acquire meaning only by virtue of the dif-

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ferential gaps' that separate and oppose them: 'The practice-theoretical types of logic which govern the activity and thought of societies termed primitive are driven by the requirement of differential gaps (Levi-Straus 1962:100). And, 'the real contribution of cultures is not to be found in a list of their particular inventions, but consists, rather, in the differential gaps which they give rise to' (Levi-Strauss 1973: 417-18). Assertions such as these give the impression that Levi-Strauss has remained attached to the symbolic paradigm.12 In that light, we have to take into account that the apparent problem is that structural analysis makes use of reduction (1962:100-1) in an attempt to bring out the differential gaps that are constitutive of systems by referring gaps back to their symbolic nature. In our opinion, there is a way of exiting the symbolic paradigm to which Levi-Strauss's theory seems to be reducible. 6. The Sub-symbolic Option As systems of signs, Levi-Strauss's symbolic systems comprise meaning-laden gaps that are not to be reduced to constants, or sub-networks with a kind of creatorless finality. Occurring on the level of surface semantics, meanings are generated by the gaps that appear between these sub-networks, and meaning is worked out in terms of relationships: 'in short, the emergence of a semantic field can occur only in the presence of a gap, whether this be in form of symmetries, contradictions, etc.' (Levi-Strauss 1991: 249). Following Henaff, a general theory of the symbolic is possible if it is based on such an unconscious infrastructure (Henaff 1991: 157f.) undergoing metamorphosis into the 'organ of symbolic structuration.' However, while admitting that categorization constitutes a kind of application of a natural language to the world, or a 'conceptual segmentation' (space localization), as has been well shown in cognitive linguistics (Langacker 1988b, 1990, 1991; Talmy 1988), one has to consider that this categorization of a perpetually local space belongs to a dynamics that brings values into play and whose essence is topological (Deleuze 1973). As Maranda and Maranda have commented, "The mythical mind establishes a web of relationships between socio-historically given facts and it works on a symbolic level underlaid with the interplay of infra and super-structures. It thus performs to the pattern of conciliation of opposites,... through a mediator, which other students of symbolism consider as the constant dynamic core of mental processes' (1971: 24).

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However, it is not enough to merely speak of the dynamics of 'mental process/ One must be able to uncover 'the common, grammarfounded modal structure' of narratives (Brandt 1992: 13). If we take into consideration the stability of meaning forms, it follows that such stability cannot rest on a combinatorial type of logic. As Brandt has pointed out, these forms involve a 'modal framework which is always present in their internal morphology ... Insofar as thought is modal, it appears to be spatial and governed by constraints which reproduce external spatial dynamics (root). In that light, thinking amounts to inscribing a history within a potentiality for histories, so as to either transform it (deontic thought) or to understand it (epistemic thought); this potentiality is obtained by means of the continuous variations performed on the variables of the dynamics, until qualitatively distinct and discontinuous destinies (different histories) obtain' (ibid.: 11-12, 14). Catastrophe theory is engaged in a quest for intelligibility, that specific mode which connects 'continuous variations' and the 'discontinuity occurring between modal values': 'Continuous variations on the catastrophist form of a Thomian differentiable function cause crossings to occur over the thresholds separating the qualitatively distinct strata involved, thus accounting for the topological structure of the functional curve. In order to semiotize these catastrophes, one has only to interpret the minima as loci in a space (a scenario)' (ibid.: 15). In addition, writes Brandt, and this is where our contribution comes into play, one has to instal autonomous entities constrained by function, conferring on them existence and a condition admitting of their theorization. Such entities are endowed with principles allowing them to move locally and yet comply with the specified conditions. We call such entities dynamic. By postulating principles that regulate a dynamics-based type of behaviour, with the latter itself depending on an exchange of regulating factors between different dynamics, we arrive at a cognitive grammar that is 'a semiotic grammar of modalities/ The canonical formula might then be interpreted as the (1) modalization of a situation, as (2) structures of variation for subjects, or again as a (3) form of modal panorama (ibid.: 16). The 'narrative path' thus corresponds to a structure-generated unfolding that is already and always structured to produce 'meaning effects,' to borrow a term from Deleuze. Catastrophe theory attempts to understand the potential qualitative distortions: 'The thinkable world is fundamentally a world made up of

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constraints and conflicts between constraints' (ibid.: 69), that is, a world composed of conflict and its local resolutions. Thinking thus consists in 'over-modalizing an alethiological potentiality with an epistemic necessitude, thereby creating expectation at the level of this alethiology (of observation) and its corresponding necessitude' (ibid.). The grammaticality of the mythic and the diversity of codes give rise to mutual convertibility between codes that is not a real meta-code. The universal need involved in dream work consists of submitting terms that spring forth in a state of disorder to a grammatical discipline. And formalistic mental constraints rein in passions and drives; the architecture of cognition contains organizational schemata that articulate the form of content by means of rules that are bound to produce coherence and, for Levi-Strauss, such production constitutes the function of the intellect. Formalism acts as a framework for the mythical imaginary, which undergoes formalization in accordance with (1) the fundamental principle of structuralism, which holds that symbolic terms have only a positional meaning; and (2) the dialectic of the paradigmatic and the syntagmatic, of metaphor and metonymy, for symbolic thought establishes a pradigmatic relationship between homologous terms, with each one being subordinated to a particular syntagmatic relationship (Levi-Strauss in Petitot 1988: 29). The central issue here turns on the notion that, in terms of its universality, the canonical formula 'reduces a considerable number of different myths to their unity and that, implicitly, it includes a tremendous internal diversity' (Petitot 1988: 29), in addition to fulfilling our expectations of rationality, by virtue of the cognitive constraints inherent in the spatiality in which our imaginary takes shape. 'By favouring the catastrophist and spatial approach, we have found that semio-narrativity may be considered one such ab quo sub-structure, owing to [semionarrativities] being founded on the interplay of dynamics in controlled space-time, thus making possible the unfolding of a narrative universe which is objectively actantial and subjectively topological (with the "objective" aspect being defined by the catastrophist core of mental processes and the "subjective" aspect being defined by the correlative dynamic inter-play)' (Brandt 1992: 220). This interplay can be referred to the inter-myth model. In essence, to speak of dynamic interplay amounts to saying that narrative spacetime is a catastrophic derivation, insofar as it is on the enunciation level that stems from the semio-narrative level, because the latter moves scenes toward interactions. Questing by the subject is worked

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out through the modal grid of his deeds, which are functions of the interplay of the conflicting forces that surround the subject of the quest. Or, to put it differently, the ought-can cusp occurs in a duality of symmetry and asymmetry as it obtains on the semio-narrative deep level. The importance accorded to the theorization of semes and their relationship to actants consists in accounting for the semantization process of the actants themselves (and states of affairs) of which semantizations consist. Semes belong on the deep level of semiotic creation; moreover, some semes are laden with images (Lakoff 1990a) and thereby connect a 'knowledge form or a history to their actantial supports: cognitive networks that bind together a multiplicity of worlds and finite, distinct narrative universes become thinkable at that point' (Brandt 1992:17). Constitutive, natural semiotics must account for the relationship between the subject and the world. In that context, the entities through which meaning operates are no longer symbols. Accordingly, the syntax/semantics relationship inherent in the subject/world relationship implies an investment of semantic values in objects, and it also implies that syntactic terms support semantic functions. 'A subject S acquires a semantic value s by becoming conjoined with a value-object O in which the semantic value is invested. This fundamental metalinguistic equivalence S n O = s allows for a catastrophist schematization because values are identified with the thresholds differentiating subjects and objects from one another, thus giving rise to a duality between the schema and actants' (Petitot 1988: 40). Accordingly, form and meaning can no longer be separated. The emergence of the symbolic is governed by a complex dynamics that is no longer related to representational processing (CC). Instead the emergence of the symbolic comes under symbolic commitments to cognitive archetypes that are of sub-symbolic nature. And it is on that very high level that language, perception and action are linked, making possible thought-world relationships. 7. The Unconscious and Topology: The Narrative Aspect, or Semantic Idealism It can no longer be a question of defining the notion of the unconscious as 'the ineffable haven of individual particularities, the repository of a unique history' (Levi-Strauss 1958: 224). Instead, as Levi-Strauss brought home so emphatically in La potiere jalouse, the unconscious should be envisaged as a kind of collective, 'polymorphously social/

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structure-generated core of mental processes. All in all, the problem represented by history, language, and the symbolic rests on the limitations inherent in the cognitive apparatus of the human species and on the complexity of the contingent or necessary relationships involved. 'Narratives and ... myths are underlaid with deep semantic categorizations which articulate values, constitute unconscious codes (involving kinship, the natural world, etc ...) and which are projected on syntagmatic articulations' (Petitot 1985: 50). These 'semantic categories are universals of the imaginary.' They are 'unconscious,... subjectivable only by means of a logic of action' (ibid.: 50-1). They are open to subjectivization only within the articulation to a narrative syntax that displays (Darstellung) 'an event-based logic of action.' Hence, it is absurd to maintain that there are no general a priori laws. On the contrary, it is imperative to grasp the 'anthropological function' of narratives in terms of deep semantics, relying on the unconscious pregnancies of subjectivity in order to do so (ibid.: 51). In that connection, we are not far from an ontological conception of structure: the primacy of structural articulation is based upon a more than merely epistemological basis (ibid.: 25-6). Contrary to the symbolic formalist thesis of Henaff, the code refers to the articulation of deep semantics. As we have seen, the semes of such foundational semantics are anthropological universals of the human imaginary (life and death, etc.). Universals are not meanings in the usual sense of the word, that is, coming within the scope of denotative semantics (exteroceptive in semiotic metalanguage). They are pregnancies, or 'interoceptive signifieds' that have been 'phylogenetically inherited' from major ethological regulations and that, in general, belong to the realm of drives, psychodynamics and the unconscious (Petitot 1988: 29f.). As for the related semantic spaces, universals are axiologically polarised and thymically invested by forces of attraction and repulsion, all of which represent the semiotic reformulation of the unconscious code of LeviStrauss. Ontological primacy is granted to the structural and the relational (Levi-Strauss 1983a; Petitot 1985: 62f.; 112): The scientific ambition of structuralism is not quantitative, but topological and relational' (Deleuze 1973 in Petitot 1985: 66). It is not merely a question of establishing equivalences between different semantic spaces: a foundational semantics has to be introduced. 'Beyond the fact that one and the same form of contents may operate upon different content matters, it nevertheless remains that, under the influence of the twist in the canonical

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formula's loop and the rhetoric of metamorphoses, contents are concerned with the effects of interaction, crossover and linkages between codes' (Petitot, 1988: 31). 'As Levi-Strauss has, with rigour, put it, [the elements of a structure] have nothing more than a meaning: a meaning which is necessarily and uniquely one of "position"' (Deleuze 1973: 304 in Petitot 1985: 66). A symbolic system of differences (taxonomy, zoology), such as is to be found in the interpretation of totemism by Levi-Strauss, 'may serve to encode another symbolic system (social positionings)' (Petitot, 1985: 69), as occurs in 'the way by which these effects are related to an enigmatic phenomenon ... in which a foundational semantics is converted into an actantial syntax. The dialectic consisting of value as function and value as term brings into play a quite specific dialectic involving semantics and syntax. Actantial syntax "externalizes" in the form of the relations established between the actants "internal" relationships between semantic pregnancies of the imaginary' (Petitot 1988: 31-2). 8. The Problem of Structural Finality As regards myth, it is important to put the question of dynamic structuralism and its essential problem centring on finality (Petitot 1985: 208-9). To quote Levi-Strauss: According to many, to describe the diversity of institutions, customs and beliefs as the result of an equivalent range of choices, which are exercised by each society in a kind of ideal repertoire in which all potentialities were inscribed beforehand, is to indulge in an excess of language ..., in a series of arbitrary comparisons whose anthromorphism is irritating ... [In keeping with this view,] societies are not persons, and nothing justifies representing them as though they were individual customers who thumbed through the catalogue of heaven knows what metaphysical supplier, each one retaining for his particular use certain models of wares that in other societies are replaced by different models for identical uses. Nevertheless, there are cases in which this figure of thought lends itself to controlled experiment. This is notably the case with myths. (1983a: 215)

Organization is the schema involving first principles: behind the procedures of division and composition affecting entities having previously undergone segmentation there are operators of differentiation

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and integration at work to produce each culture as a solution to a 'general problem' (Chemla and Pahaut 1989: 957). This teleology is inherent in the construction of cultures. In his reinterpretation of a certain number of artefacts from exotic worlds, Thorn has shown the far-reaching consequences of this assertion: Tn effect, there is room in science for a kind of dynamic analysis which is perfectly independent of the nature of substrata; regardless of whether one is dealing with a material object or a mental object, one can expect that they will display behaviour which, in certain circumstances, is perfectly isomorphic' (Thorn 1980a: 12; see also Chemla and Pahaut 1989: 957). Variational procedures (Bechtel and Abrahamsen 1993: 34f.; Smolensky 1992; Clark 1989: 76) involving the transformation of a structure such as mythic codes, and formalized by the canonical formula, are raised to the status of finality, coming close to a principle of maximum cohesion. And that is in keeping with the conception of the intellect that has been so often expounded by Levi-Strauss. 'By means of the projection of the paradigmatic on the syntagmatic, the symbolic elements of a structure are organized into series, and a series always refers homologically to another series'13 (Petitot 1985). What is being proposed here is not something 'along the lines of a banal encoding process/ that is, 'a one-to-one correspondence.' The 'positional identities developed by the first series are inseparable from the shifts originating in the second series' (Deleuze 1973: 320 in Petitot 1985: 19). This is really a 'semantic shift' (Langacker 1991: 149). Thus linked to each other once more, forms and functions are intrinsic in social existence. What we are encountering here is the essentially Levi-Strauss's point of view as expressed in his remarks on how the versions of a myth in and of themselves make up a paradigmatic group or set which, when taken individually, may be inserted in other paradigmatic groups which intersect the first set at various points (1991: 201). Deleuze's thesis, which Petitot has since taken up, insists on the fact that every structure encompasses a paradoxical element: 'The relative position of terms within the structure depends on their absolute position in relation to that element' absent from the position befalling it (Petitot 1985: 72; see also Levi-Strauss, 1991: 197f.). The form of exchanges defines the structure as the 'order of positions subordinated to the variation of relationships' (Petitot 1985: 70); relationships enjoy ontological primacy over structures. Stated as such, this thesis represents the new version of transcendentalism deriving from Kant's theory of relations; it puts the question of

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meaning as follows: 'An apparently meaningless detail of the syntagmatic chain turns out to have relevance when placed within a paradigmatic set' (Levi-Strauss 1991: 187a; also 189n). This thesis is a defining aspect of what we would call differential topology. We have to take seriously this kind of Kantian schematization of the language and world relationship in the too-often recurring deletion of the a priori by analytic philosophy. We do so in the hope of formulating principles for an a priori spatialization in language-categorization (Quinn 1991; 2000). There is a pre-established spatiality of objects of the world that makes necessary a determination of language and thought relationships linked in space-time experience (perception and action). This is a crucial aspect of cognitive structure, a main point for categorization in a dynamic system in which the natural mind is much more than some sequential temporal process. Morphodynamic commitments deal with physics, a substrate space from which information coding at different levels emerges in a stratified ontology (Quinn 2000). As much as the internal dynamics of micro-physical substrate is unexplained, our understanding of the macro emergent structures of semio-physic and semio-cognition is in a large part limited. Dynamics are more than just postulated to scientifically demonstrate structural explanations applicable to human conceptual representations; qualitative discontinuities spread from one substrate to another on a sub-level underlying the essential problem of intentionality and meaning. What is significant for cognitive subjects are qualitative structures and dynamic forms. The significant relationship between cognitive subject and external world requires a nomological description that must characterize both the very nature of dynamics operating on substrates and substrate-vehicles by which strongly dynamic forms carry morphological information (Petitot 1992: 14; Smolensky 1988,1990b; Talmy 1988). We are thus seeking a deep cognitive level linked to the dynamics of the physical world,14 deeper than the CC symbolic paradigm. This is, in fact, a cognitive micro-structure (Clark 1989). Philosophy of mind and dynamic cognitive semiotics take up the issue of intra-cognitivity linked to the external world, in which the symbolic emerges from the sub-symbolic without uncoupling physical and cognitive levels. Instead, philosophy and semiotics have to use a hierarchy of levels that implies a stratified ontology by using dynamic concepts arising from sub-conceptual structures, a hierarchy that takes into account the specificity of cognitivity and significant forms. Mental symbolic representations are determined by the objective content of

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external physical information (Langacker 1991: 196, 209; Talmy 1988). Thus, we are led to take into account the constraints that frame the dynamic of the inner and outer space, and also of the inherent complexity of mental representations and mythical categorization. In that way, we are compelled to nomologize these relations and to recognize the necessity of their integration into the specific constraints of a cognitive domain made up of spatial relations (Svorou 1994: 32-3). In this way, we emphasize that the problem of intentionality can be partly understood by taking into account the representational theory of content, so as to void the fallacy of solipsism and eliminativism concerning representational narrow /broad content. The former (narrow content) implies specific constraints resulting from projected morphologically structured information (Jackendoff 1994) and a reorganization of this information; as shown by Langacker (1993: 3), there is a corresponding categorizing level controlled by specific innate structuring principles. The latter (broad content) implies a categorization level built on morphological properties of information in terms of perceptual categorization (Petitot 1994: 503; Langacker 1991: 59-61,149, 209) and the natural saliencies of correspondent categories. Narrow content comes from the investment of salient forms raised to a higher level by perceptual categorization in cognitive pregnancies (corresponding to Lakoff's cognitive commitments) that constitute specific structuring principles of a domain (Croft 1993). Through this investment of salient forms by cognitive pregnancies, an emergent high categorizing level of complex categories (Langacker 1991: 282) shows up in the pairing of new categories (Langacker 1988: 65; Thorn 1988; Talmy 1988: 194f.). Briefly, the categorization of abstract domains, such as in Lakoff's theory of metaphor, comes about by the penetration of cognitive pregnancies into object-values. The same holds true for the narrative model. Indeed, it is at this particular juncture that dynamic structuralism proposes a conception of intentionality governed by the unconscious. In myths, as Levi-Strauss has emphatically brought out in Histoire de Lynx, 'there are globally determinant paradigmatic projections which cover syntagmatic sequentiality' (Petitot 1985: 209-10). Actually, the syntagmatic substructure of narratives is raised to the status of finality, 'as though it w governed by a kind of intentionality' (ibid.: 209). And, since it is a question of moving from the manifest to the hidden, this intentionality will be defined as a ' "series" of "deep logico-semantic" operations of a paradigmatic and global nature ... As with the raw and the cooked in

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the case of the Bororo myth analysed by Levi-Strauss, we are dealing here with immanent semes which categorize and classify, and which belong to codes' (ibid.: 210; also Greimas 1970:117f.). In the depths of discourse, we find 'semiotic structures which possess a foundational syntax and semantics' (Greimas and Courtes in Petitot 1985: 211). The semantic component of discourse constitutes the most difficult aspect of myth analysis, inasmuch as it provides myth with its logical coherence. Levi-Strauss does not stumble into the symbolic paradigm close to CC when he refuses to reduce the paradigmatic aspect to the simple dimension of substitution 'reached by commutation tests' (Petitot 1985: 50), thus introducing a deep semantic dimension (Levi-Strauss 1973: 161f.). Accordingly, it is then necessary to resort to the morphodynamic epistemological and ontological commitments that hold up the CN paradigm, which are inherent in the localist hypothesis and make it possible to grasp categorization as belonging to deep semantics, of a kind occurring at a level that articulates not only values but the codes categorizing social and mental space as well. It is crucial, then, that this semantics be both contextual and global, and that it also be intrinsic in space-time. The canonical formula must bring this out without resorting to logicism. To be sure, a 'unitary perspective' (Descola 1989: 1521) must be developed in which difference is the prime feature. If approached in terms of their diversity, 'myths objectively form a group of transformations ... [which] derives its character [from] the fact that [this character] has, so to speak, been crystallized in a pre-organized semantic environment' (Levi-Strauss in Descola 1989: 1521). In a way, myth founds a 'heuristic principle,' or a classificatory type of logic that 'breaks with the hold exercised by the sensible that originated by introducing discontinuity' (Desveaux 1989: 1505), and that calls imperatively for a kind of reasoning that works 'in terms of relational gaps.' Thus, the coherence of myths extends far beyond words while yet falling far short of them, since it is possible 'to view narrativity as a "window" on the psycho-anthropological structure of the imaginary' (Petitot 1985: 212). In keeping with the Kantian concept of reason, the intellect in fact constitutes the required topos. In effect, the pertinent logic is relational, of a 'natural,' 'oriented' variety (Godelier 1971: 544), in which modality is the prime feature. That is why the formalism of the canonical formula cannot be of a logical variety, insofar as its constitutive units are not objects that are possibly subjected to the identity principle, but are, instead, positional and relational values. The semantics prescribed by

Dynamic and Connectionist Implications of the Canonical Formula 257 this type of formalism does not have to meet the requirements of logicism, that is, of being denotative; instead, it must be conceived as the articulation of a substance. As for its syntax, the formula does not refer to a denotational semantics either; instead, it provides evidence of the 'actantial conversion' of a semantic level that remains open, rendering inoperative any definitive closing of the 'loop' back on itself. To be sure, a myth 'is constructed as a closed system' (Godelier 1971: 550-1). But there again, 'everything occurs as though the struggle against entropy testified to a kind of nostalgia ... vis-a-vis the original (i.e., generative) continuum, whose self-destruction was necessary for the emergence of culture' (Descola 1989: 1523). Thus, the 'canonical narrative' operates like a machine for liquidating an initial absence (Petitot 1985: 259). Assuredly, the coherence of myths and narratives is to be found in a 'semantic approach' (ibid: 49). Even as it searches for constants and hypostasizes differences, structural analysis must nevertheless take on the 'purely' operative character of the human mind (Levi-Strauss 1991: 254-5). All the same, Levi-Strauss can never be reproached for underestimating the depth of the real; on the contrary, from his perspective, the 'real' overflows with meaning. This is what justifies the search for the elements necessary for going beyond symbolic paradigms close to the CC formalism, a search initiated by LeviStrauss himself. 9. Categorization of Spatiality 'On account of a void occurring at the centre,' meaning, 'far from having no greater dimension than that which is to be found in play' (Court 1989: 640), the transcendental field remains open to topological intentionality, the latter constituting a kind of semantic stratification. For Levi-Strauss, decentring intersects with a form of interplay of substitutions (Dupuy 1989: 256). Thus, he is opposed to a centred structuralism; instead, the non-centre, the interplay of substitutions, and absence, which are all marks of the impossibility of a metalanguage, serve to accentuate the axiological imperative to exit from centrisms and to set forth 'the relative anonymity of the longue duree/ the quasi-geological strata of historical time as first proposed by F. Braudel (Lenclud 1989: 1695). In our view, and following Kant's anthropology, man is a problematic epistemic unity. But is this not an instance, precisely, of Kantian symbolism, in which 'the freedom of the imagination' admits of conceptless schematization, thereby opening onto the symbolic in which

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thinking (Denken) is more far-reaching than knowing (Kennen), and in which forms of knowledge rooted in the imagination encompass pure reason? As Levi-Strauss writes, 'It seems to me, however, ... that in societies without writing, positive knowledge fell quite a ways short of the powers of imagination, and that the responsibility for filling this gap fell to myths' (1991:11). As a gap of a space-time variety, the distinction between the human and the natural ought to progressively diminish (Levi-Strauss 1973: 345). Morphodynamics offers an appropriate theoretical framework for making this process of diminishment thinkable16 insofar as entities with a semantics at the micro level appear as complex and global patterns for the activation of elementary local units that are connected with one another and whose operations parallel one another (Smolensky 1988; 1992; Bechtel and Abrahamsen 1993: 70-1). Admittedly, at the input level, there is already a finality to information that comes under an intrinsic, weak isomorphic phenomenology at the physical and cognitive levels. If the goal of anthropology is to arrive at a kind of logical and differential architecture beneath consciousness (Levi-Strauss 1958: 30-1), without also erasing the specific factors that are worked out at that level, the central problem can no longer lie with symbolic forms. Considering the contents of syntagmatic developments, the 'logical architecture' must offer the possibility of retracing these very constituents by means of reduction. Levi-Strauss's work demonstrates that the diversity of cultures is related to the actualization of the possibilities inhering in that architecture, with such actualization being itself derived from a kind of a priori (1983b: 1227): 'A thoroughly grounded systematics outlines the relational network by which the possible relationships between elements may be figured' (ibid.: 1229). On the basis of that statement, one may formulate a hypothetical postulate ultimately related to an underlying connectionist-type dynamics: the complexity of cultures, that is, the complexity to be found within cultures and in the domain of differential gaps, belongs to a dynamics that links categorization to the world. In this perspective, then, the whole will be ascribable to one and the same articulation, which is dynamic and occurring at an emergent level. Remember that deep semantic structuring brings into play cognitive representations that, besides being topological and dynamic, are compatible with primal structures of perception and are linked to categorization (Varela 1993). Structuralism, and more precisely modal analysis, through the interpretation of the canonical formula, serves to shed light

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on 'unexpected paths, on heretofore obscure rites and customs to which a fundamental meaning must nevertheless attach, since identical rites and customs may be found in areas of the world which are at the farthest possible remove from one another' (Levi-Strauss 1968: 331). On the subject of formalism, Histoire de Lynx states that structural analysis 'sheds light on how a mind functions when, by producing an empty discourse, [it] unveils and strips bare the mechanism of its operations' (Levi-Strauss 1991: 255). Anthropology, whose task is, precisely, to study symbolic systems and deep-level manifestations, must take as its point of departure a problem that has been posed in La potiere jalouse: 'A myth appears as a set of equations in which the symbols, never clearly perceived, are approximated by means of concrete values chosen to give the illusion that the underlying equations are solvable. Such choices are guided by an unconscious finality, but they are made among arbitrary and contingent elements, the products of history, so that the initial choice remains as impossible to explain as the choice of phonemes that come to make up a particular language' (Levi-Strauss 1985: 228). It would appear that behind the structural varieties of logic governing social systems, a genuine formalism is at work, which is accounted for by the canonical formula: 'Smooth running of the system depends on a dynamic disequilibrium' (Levi-Strauss 1991: 90). Wherever we turn our attention from systems to the structures of systems, and from structures to their foundations, we discover a so-called cognitive architecture. In the opposition of terms like 'heaven and earth, fire and water/ the mind 'strives to link [terms], but fails to establish parity between them, for it is this chain of differential gaps, as conceived of in mythic thought, which sets the machine of the universe in motion' (ibid.: 90-1). But it cannot be a question here of remaining satisfied with the thesis of binarity upon which a universal mythology would be based. It is instead a matter of taking these binary oppositions back to 'the smallest common denominator of the changing values assumed by comparison and analogy' (ibid.: 249; 316), that is, we must move onward to a 'different order ... which obeys distinctive constraints and rules' (ibid.: 10). And lest the meaning of reductionism remain unclear, it should be pointed out that the elements as such do not undergo reduction, which in itself would amount to ignoring the implications of relations and modality. On that score, and following Kant more than Descartes, structural analysis has not to comply with the Cartesian principle of 'split-

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ting the difficulty into as many parts' as are required to solve it. There is no real end to myth analysis, no secret unit to be grasped once the analytical breakdown has been completed. Every looping of the cycle or circuit inward does indeed constitute an opening. The relationship is what has prime importance. Themes unfold ad infinitum. No sooner have they been untangled and held separate from one another than they fuse together as a response to the tug of unforeseen affinities. Accordingly, 'the unity of a myth is only tentative and projective; it is never the reflection of a state of, or a moment in, the myth' (LeviStrauss 1964: 13). Reduction does not consist of merely observing elements, but of distinguishing the relations between elements along with the rules governing transformation and opposition (Levi-Strauss 1985: 77f.). It is only this kind of reduction that has made it possible to provide evidence of the circularity and the transformation of certain myths (Levi-Strauss 1991: 82; 198), evidence, that is, of the fluidity of mythic forms (ibid.: 250-1) that multiplies values and meaning, with the latter itself being space-time invested. It is only in this way that the aporia associated with the isomorphism occurring between mental space and social space may be resolved (see Smolensky 1994). This is attested by the morphodynamic and CN development of the canonical formula. Conclusion

The problem of narrative analysis involves the plausibility of the weak isomorphic relationship between man and nature in a way that delves deeper into the subject than is the case with the classic debates opposing culture and nature. The issue is not only the symbolic nature of man, but also, and more especially, the categorialization underlying it. We are dealing here with the ideal that runs through the whole of LeviStrauss's work: to work out the social sciences according to the model of the physical sciences (Levi-Strauss 1973: 29), an end to be attained by resorting to the realm of possibilities represented by invariants (ibid.: 323f.), wherein cognitive architecture corresponds to the structuring environment that itself may be event-determined. This isomorphism makes it possible to conceive a dynamics as the only way by which form may be imparted to content. The logic driving social systems is a true 'primal logic,' the direct expression of the structure of the mind, and underlying this, of the brain, and not the passive product of the action of an environment

Dynamic and Connectionist Implications of the Canonical Formula 261

upon a shapeless conscience. There is no mentalism here, superseded by a theory of morphological linkage. Essentially, all that anthropology does, according to Levi-Strauss, is to attest to a structural homology between the human mind at work and the object to which it is applied. The aptness of this assertion is borne out by the observation that, effectively, with a structural logic according to which it is the mind that explains social laws, we reach a categorial unconscious that could be the cognitive subject of the canonical formula and that constitutes the common denominator of cultures. Indeed, as a result of reduction, it is possible to state that the laws of the human mind operate universally. It is a matter of uncovering the 'system of common, fundamental constraints' (Levi-Strauss 1964: 19) to which collective modes of understanding belong. With this, the aim becomes one of identifying 'the mechanisms of objective thought' (ibid.: 21), and of showing that the diversity of cultures17 functions as a single bundle of categories. However, the type of formalism thus required is a matter of models that are neither common nor banal, and which involve a modal kind of logic. Structural analysis, which works with constructions, comparisons, correlations, empirical inferences, oppositions, transcendental deductions (Levi-Strauss 1958: 304f.), may be conceived as abductive as emphasized in Peirce's semiotics. The selection of that which specifically befalls to the real cannot be carried out deductively ... or inductively ..., but only abductively (whence the "generative" role filled by transcendental imagination in the problematic of constitution)' (Petitot 1985: 55). Notes The author gratefully acknowledges fruitful discussions with Pierre Maranda, and thanks Ronald Wullner and John St James for the revision of the text. 1 Standard physicalism asserts that there are no mental states over physical states, which implies reductionism and rarefied ontology. 2 This problematic goes beyond mythical thought; it concerns occidental metaphysics, sciences, religions, etc. 3 We use this term to signify a theoretical point running contrary to the 'formalistic fallacy' associated with logico-combinatorial formalism that implies mentalism, thought-language, or Fodor's mentalese, which refers to the CC.

262 Andrew William Quinn 4 As suggested to me by Pierre Maranda, we can say that dramatis personae are accretions of the interplay of functions, an actantial point of view leading to a teleological and dynamic theory of narrative functions. See Maranda 1971: lOf. 5 'Semantic space is the whole network of an individual's and a community's [structured] knowledge'; Croft 1993:337. 6 As Lakoff says about this passage from classical linguistic to the cognitive approach (1990a: 42f.), only cognitive CN commitments rid us of a CL syntactical autonomy. What makes this passage possible is the 'discovery of basic-level categories and image-schemas, [and] also conceptual metaphor' (ibid.: 42). We will insist on those points. 7 Any finitist grammar, including Chomskyan generativism, is a simple twodimensional and logico-combinatorial structure without any possibility of linking language to the world. 8 Its constitutive dynamics evokes the Peircean theory of semantics in which it is asserted that, in the temporal, physical continuum, the dynamic object exceeds the immediate object. For more on this subject, see Peirce 1960:4.536. 9 Contrary to correspondentalism, it must be pointed out that although language refers to the world realistically, it also derives from an imaginary that itself stems from the major pregnancies of the species. 10 On this point, see 'the place of the Sloth as forming a kind of gap within certain myths ... in which he does not appear. An animal which is absent from a new environment may nevertheless conserve a Metaphysical existence in the mythical imagination' (Petitot 1988: 26). Likewise, in other myths originating in North America, the Other is awaited, as though there were a gap to be filled. On the subject of the White, see Levi-Strauss 1991. 11 We find this formalism in the mathematical analysis of the double cusp for which we refer to Petitot (1988,1991,1995, and this volume). 12 Society has a symbolic origin, instead of the symbolic having a social origin (1950: xxii). 13 For further geometrical illustration, Petitot 1992:399^405. 14 'Human linguistic abilities depend upon the processing of linguistic information by brain structures whose primary function is the processing of spatial structure' (Deane 1992:47; also 61). 15 For more on these questions, see Clark 1989: 42 f., 188f. 16 As a permutation group (Levi-Strauss 1958:248), a myth is not constituted by isolated relations, but by bundles of relations (ibid.: 234) that are essential to define a myth and, as suggested to me by Pierre Maranda, as sets of embedded mediations (ibid.: 237f.) by which the categorization is articulated.

Dynamic and Connectionist Implications of the Canonical Formula 263 17 The reduction of this diversity is, obviously, far from constituting an objective for Levi-Strauss. References Andler, D. 1992. 'Calcul et representation.' In D. Andler, edv Introduction aux sciences cognitives. Paris: Gallimard. Bechtel, W., and A. Abrahamsen. 1993. Connectionism and the Mind. Great Britain: T.J. Press. Brandt, P.A. 1992. La charpente modale du sens. Philadelphia: John Benjamins Publishing. Braudel, F. 1969. Merits sur I'histoire. Paris: Flammarion. Burguiere, H. 1971. "Histoire et structure.' Annales, nos 3-4 (May-Aug.). Paris: A. Colin. Chemla, R., and S. Pahaut. 1989. 'Objets et artefacts, les sciences et la culture.' In Encydopedie Philosophique Universelle, 1:953-8. Paris: PUF. Clark, A. 1989. Microcognition. Cambridge, MA: MIT Press. - 1991. 'In Defense of Explicit Rules.' In W.M. Ramsey, S.P. Stich, and D.E. Rumelhart, eds, Philosophy and Connectionist Theory, 115-28. New Jersey: Lawrence Erlbaum Associates. Court, R. 1989. 'La faillite de la representation et 1'esprit de la musique moderne.' In Encydopedie Philosophique Universelle, I: 635-42. Paris: PUF. Courtes, J. 1973. Levi-Strauss et les contraintes de la pensee mythique. Paris: Mame. Croft, W. 1993. 'The Role of Domains in the Interpretation of Metaphors and Metonymies.' Cognitive Linguistics 4: 335-70. Deane, P.D. 1992. Grammar in Mind and Brain. Berlin, New York: Mouton de Gruyter. Deleuze, G. 1973. 'A quoi reconnait-on le structuralisme?' In Histoire de la philosophic, 8: 299-335. Paris: Hachette. Descles, J-P. 1990. Langages applicatifs, langues naturelles et cognition. Paris: Hermes. - 1991. 'La predication operee par les langues.' Langage 103: 83-96. Descola, P. 1989. 'Pensee mythique et theories de 1'identite dans le monde amazonien.' In Encydopedie Philosophique Universelle, 1:1521-4. Paris: PUF. Desmarais, G. 1992. 'Projection ou emergence: la structuration geographique de 1'etablissement bororo.' RS/SI, Symbolic Representation and Spatial Organisation, 189-216. Montreal: Canadian Semiotic Association. - La morphogenese de Paris. Paris, Ste-Foy: L'Harmattan, CELAT.

264 Andrew William Quinn Desveaux, E. 1989. 'Les pensees indigenes de 1'Amerique du Nord.' Encyclopedic Philosophique Universelle, 1:1504-7. Paris : PUR Dupuy, J-P. 1989. 'L'autonomie du social.' In Encyclopedic Philosophique Universelle, I: 254-5. Paris: PUR Greimas, A.J. 1970. Du sens. Paris: Seuil. - 1972. Essais de semiotique poetique. Paris: Larousse. - 1976. Semiotique et sciences saddles. Paris: Seuil. Greimas, A.J., and J. Courtes. 1979. Semiotique. Dictionnaire raisonne de la theorie du langage. Paris: Hachette. - 1986. Semiotique. Dictionnaire raisonne de la theorie du langage, II. Paris: Hachette. Godelier, M. 1971. 'Mythe et histoire' Annales, nos 3-4 (May-Aug.). Paris: A. Colin. Henaff, M. 1991. Claude Levi-Strauss. Paris: Belfond. Jackendoff, R. 1994. Patterns in Mind. New York: Basic Books. Kant, 1.1951. Critique of Judgement. Translated by J.H. Bernard. New York: Hafner Press. Lakoff, G. 1989. 'A Suggestion for a Linguistics with Connectionist Foundations.' In Proceedings of the 1988 Conference on Connectionist Models. San Francisco: Morgan Kaufmann Publishers. - The Invariance Hypothesis: Is Abstract Reason Based on Image-Schemas?' Cognitive Linguistics 1: 39-74. - Tensor Product Variable Binding and the Representation of Symbolic Structures in Connectionist Systems.' Artificial Intelligence 46:159-216. Langacker, R.W. 1972. Fundamentals of Linguistics Analysis. New York: Harcourt Brace Jovanovich. - 1983. Foundations of Cognitive Grammar I: Orientation. San Diego: Linguistic Agency, University of Trier. - 1987. Foundations of Cognitive Grammar I. Stanford: Stanford University Press. - 1988a 'An Overview of Cognitive Grammar.' Topics in Cognitive Linguistics, Current Issues in Linguistics Theory 50: 3^18. - 1988b. 'A View of Linguistics Semantics.' Topics in Cognitive Linguistics, Current Issues in Linguistics Theory 50: 49-90. - 1988c. 'A Usage-Based Model.' Topics in Cognitive Linguistics, Current Issues in Linguistics Theory 50:127-61. - 1990. 'Subjectification.' Cognitive Linguistics 1 (1): 5-38. - 1991. Concept, Image and Symbol: The Cognitive Basis of Grammar. New York, Berlin: Mouton de Gruyter. - 1993. 'Reference-point Constructions.' Cognitive Linguistics 4 (1): 1-38.

Dynamic and Connectionist Implications of the Canonical Formula 265 Lenclud, G. 1989. 'Anthropologie et histoire.' In Encyclopedic Philosophique Universelle, 1:1693-1701. Paris: PUR Levi-Strauss, C. 1958. Anthropologie structurale. Paris: Plon. - 1962. La Pensee sauvage. Paris: Plon. - 1964. Mythologiques I, Le cru et le cuit. Paris: Plon. - 1967. Les Structures elementaires de la parente. Paris: Mouton. - 1968. Mythologiques III, L'origine des manieres de table. Paris: Plon. - 1971. Mythologiques IV, L'homme nu. Paris: Plon. - 1973. Anthropologie structurale deux. Paris: Plon. - 1983a. Le regard eloigne. Paris: Plon. - 1983b. 'Histoire et ethnologic.' Annales, no. 6 (Nov.-Dec.). Paris: A. Colin. - 1985. La potiere jalouse. Paris: Plon. - 1991. Histoire de Lynx. Paris: Plon. Levi-Strauss, C., and D. Eribon. 1990. De pres et de loin. Entretien. Paris: Editions Odile Jacob. Maranda, E., and P. Maranda. 1971. Structural Models in Folklore and Transformational Essays. The Hague, Paris: Mouton. Maranda, P. 1971. 'Introduction.' In P. Maranda, ed., Soviet Structural Folkloristics, I. The Hague, Paris: Mouton. Marcus, S. 1993. The Logical and Semiotic Status of the Canonical Formula of Myth. Quebec: Universite Laval, Laboratoire de Recherches Anthropologiques. Ouellet, P. 1992. 'Presentation.' RS/SI, Symbolic Representation and Spatial Organization, 183-7. Montreal: Canadian Semiotic Association. Peirce, C.S. 1960. Collected Papers. Volumes 1-6. Edited by Charles Harshorne and Paul Weiss. Cambridge, MA: Belknap Press of Harvard University Press. - 1966. Collected Papers. Volumes 7 and 8. Edited by Arthur W. Burks. Cambridge, MA: Belknap Press of Harvard University Press. Petitot, J. 1985. Morphogenese du sens I. Paris: PUR - 1988. 'Approche morphodynamique de la formule cahonique du mythe.' L'Homme 28 (106): 24-50. - 1989. 'Hypothese localiste, modeles morphodynamiques et theories cognitives: remarques sur une note de 1975.' Semiotica 77 (1 /3): 65-119. - 1992. Physique du sens. Paris: CNRS. - 1994. 'La semiophysique.' In Passion des formes. Fontenay, St-Cloud: ENS Editions. - 1995. 'Note complementaire sur 1'approche morphodynamique de la formule canonique du mythe.' L'Homme 135:17-23. - 1996. Contribution in Greimas and Courtes 1986. Quinn, A.W. 1991. 'Sujet et representation. Essai su I'episteme structurale, son sens

266 Andrew William Quinn problematique, topologique et relationnel,' II. These de doctoral, Universite Laval. - 1998. 'Hermeneutique et cognition.' L'Interpretation, Protee 26 (1): 29-42. - 2000. 'Representation, perception et questions d'ontologies: anthropologie culturelle et cognitive.' Visio 5(1). Rumelhart, D.E. 1992. Towards a Microstructural Account of Human Reasoning.' In S. Davis, ed., Connectionism, 69-83. New York, Oxford: Oxford University Press. Salanskis, J.-M. 1990. L'Hermeneutique formelle. Paris: Editions du CNRS. Scubla, L. 1995. 'A propos de la formule canonique du mythe et du rite.' L'Homme 135:51-60. Sebeok, T. 1989. The Sign and Its Masters. Texas: Texas University Press. Simonis, Y. 1980. Claude Levi-Strauss ou La passion de I'inceste. Paris : Flammarion. Smolensky, P. 1988. 'On the Proper Treatment of Connectionism.' The Behavioral and Brain Sciences 11. - 1992. 'Principles for an Integral Connectionist/Symbolic Theory of Higher Cognition.' Technical report, Department of Computer Science, Colorado University. Svourou, S. 1994. The Grammar of Space. Philadelphia: John Benjamins Publishing. Talmy, L. 1983. 'How Language Structures Space.' In Spatial Orientation: Theory, Research and Application, 225-82. New York: Plenum Press. - 1988a. 'The Relation of Grammar to Cognition.' Topics in Cognitive Linguistics. Current Issues in Linguistic Theory 50:165-205. - 1988b. 'Force Dynamics in Language and Cognition.' Cognitive Science 12 (1): 49-100. - 1995. The Cognitive Culture System.' Monist 78: 80-114. - 1996. 'Fictive Motion in Language and "Ception."' In Bloom et al., eds, Language and Space, Cambridge, MA: MIT Press. Thorn, R. 1972. Stabilite structurelle et morphogenese. Paris: Ediscience. - 1980a. 'Artefacts et structures infravivantes.' Debat 3 (30): 11-33. - 1980b. 'L'espace et les signes.' Semiotica 29 (3/4): 193-208. - 1988. Esquisse d'une semiophysique. Paris: InterEditions. - 1990. Apologia du logos. Paris: Hachette. Varela, F. 1993. L'inscription corporelle de I'esprit. Paris: Seuil.

10 A Morphodynamical Schematization of the Canonical Formula for Myths Jean Petitot (Translated from the French by Mary Richardson)

To Claude Levi-Strauss

with deep admiration

This new reflection on the canonical formula for myths (CF) proposed in 1955 by Claude Levi-Strauss provides me with the opportunity to clarify the morphodynamical model I proposed in 1988 in the journal L'Homme (HI).1 During the exciting seminar organized by Professor Solomon Marcus at the College de France in Paris in the fall of 1993,1 clarified some of its epistemological difficulties. This work was published in L'Homme in 1995 (H2).2 I will attempt here to explain in further detail the so-called double cusp model presented in HI. My basis will be the pioneering work of Pierre Maranda3 and Lucien Scubla's outstanding dissertation.4 Since this paper must remain within reasonable limits, I cannot, of course, present either the history of the CF or the details of morphodynamical modelling. For the former, I refer the reader to Scubla's treatise, now the authoritative reference on the subject. As for the latter, the reader may refer to HI and to my book Physique du sens (1992), in which they will find the essentials of the mathematical, physical, and epistemological bases of Thomian morphodynamics. In order to ensure that this paper is somewhat self-contained, I will first present a few brief reminders. I begin with an epistemological foreword about the problems of the experimental method, and that of modelling. Then I consider the canonical formula as a 'structural equa-

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tion' before presenting my morphodynamical approach. In section 2,1 offer reminders concerning morphodynamical models - both semionarrative and morphodynamical ones - and I give the example of the 'cusp/ Section 3 deals with P. Maranda's mediation model, section 4 with addressers and external dynamics. In section 5,1 take up the paradoxes of mediation, the contradictions of which I move beyond in section 6. Section 7, on the internalization of external spaces, leads to the last section on the double cusp and the double twist. 1. Epistemological Foreword 1.1 The Problem of the Experimental Method

Many adversaries of the CF deny it any value whatsoever and even denounce it as absurd. They maintain that its applications project and superimpose a preconceived formal structure on mythical reality. This criticism is quite serious and justified to a large extent. It raises a fundamental problem: that of the experimental method in structural analysis. It is, indeed, impossible to come up with a direct experimental confirmation of formal structures of meaning, such as the CF: this is an aspect of the hermeneutic circle. Due to the 'resonant' character of meaning, such structures can always be recognized projectively in data. Applying a form of meaning to an empirical content always makes sense and therefore has an interpretative, not an objective, status. This justifies the objections regarding superimposition. But does it mean that structuralist approaches of meaning have to be abandoned? I do not think so. It is in fact possible to break the hermeneutical circle by conferring on structures a mathematical status rich enough to enable them to generate models that can be compared to empirical data in accordance with suitable experimental methods. In order to do so, we have to probe considerably deeper into the conception of structures and their models. For as long as the CF is considered a simple elementary algebraico-combinatorial structure, there is no way that it can generate models, and consequently we come up against antistructuralist criticism (see H2). 1.2 The Problem of Modelling

The crucial problem of modelling generally tends to be poorly understood in the humanities. It is rather naive to believe that a

A Morphodynamical Schematization of the Canonical Formula for Myths 269 mathematical model can be derived inductively from empirical observations and theoretical conceptualizations of phenomena. The history of formalized sciences (above all physics) demonstrates that such is not the case. Briefly summarized, what happens is instead the following. The scientific description of empirical data leads, through abstraction and categorization to basic concepts specific to a discipline. These are 'regional' or 'domain dependent' concepts, even if they are related to rational principles of universal significance (based on categories of formal ontology such as object, property, relation, whole/part, continuous/discontinuous, invariance, cause, etc.). At this level, each discipline uses a specific operational language imposed by its field. The first stage of mathematization then comes into play. It concerns the mathematical schematization of the regional concepts and consists of a mathematical interpretation of their theoretical content. Such an interpretation cannot be inductive; it is rather abductive (in Peirce's sense) and, consequently, not experimentally falsifiable. Of course, it makes use of available tools. Riemannian geometry and tensorial calculus, upon which general relativity is based, both existed long before Einstein and were elaborated by pure geometers (from Riemann to LeviCivita). Physicists, however, never spoke of superimposing mathematical structures on data, nor did they propose a purely physics-specific mathematics. Likewise, the geometrical concept of connection that underlies contemporary gauge theories (from Yang-Mills's theories to more recent works by Witten) was conceived of by Elie Cartan at the beginning of the century. Again, no physicist ever put mathematicians on trial for being too avant-garde. From the mathematical schematization of the theoretical concepts specific to a discipline, a variety of different accurate models can be elaborated and measured against experience. It is impossible to empirically falsify (in Popper's sense) a purely conceptual description (in fact, that is what enables the humanities to theorize without risk). Only explicit and effective models can be falsified. But this is only possible if theoretical contents are transformed into generative algorithms. Every concept is an unknown algorithm and the invaluable role of mathematics has always been to substitute algorithms reconstructing phenomena for purely conceptual contents. Thus, we see that in properly formalized sciences, the conceptual subsumption relationship of empirical diversity under a theoretical unity admits a converse relationship unfolding a constructed diversity that can

270 Jean Petitot

be measured against the given empirical diversity. The way in which this correspondence - that is, modelling - is related to the general theoretical concepts and principles guarantees its explanatory power. Modelling aims to reconstruct the phenomenal diversity of a domain of reality from its constitutive concepts. It solves the opposite problem to that of conceptual subsumption. In science, conceptual analysis must be convertible into computational synthesis. Obviously, this is only possible because phenomena are only phenomena and not ontological an sich realities. The purpose of morphodynamical models is to achieve such a program for semio-narrative structures.5 1.3 The CF as a 'Structural Equation' In H2 I reconsidered the central issue of the formal status of the CF as a constitutive element of what Claude Levi-Strauss has referred to as the 'grammatical discipline' of myths. The problem is, how can we do justice to the richness of the CF at the level of pure form? I showed that to interpret the CF as an elementary structure (Klein group, semiotic square, etc.) is insufficient for at least three reasons: (1) First of all, the CF expresses a coupling between two qualitative oppositions, which is much more than a simple Cartesian product (Klein group). (2) Second, the difference between what Levi-Strauss called the 'term value' and the 'function value' of the CF's constituents comes from that between syntax and semantics in structuralist models. It cannot be represented in an elementary structure. (3) Third, the CF deals with sets of mythological variants. It is not 'intra-,' but rather 'inter'-mythical; not 'local' but 'global.' And this property cannot be encoded in an elementary structure either. In order to overcome these difficulties and make the formula both theoretically compatible with the categories and principles of structuralism (which eliminates the risk of superimposition) and algorithmically with the generativity of models (which makes it possible to resort to the experimental method), I suggested that the CF be conceived of as a sort of basic 'equation' whose 'solutions' consist of different formal semio-narrative structures. The basic equations in physics (for instance, Newton's equation in

A Morphodynamical Schematization of the Canonical Formula for Myths 271

rational mechanics, Navier-Stokes's equation in hydrodynamics, or Feynman's path integrals in quantum field theory) express very general principles (principles of relativity, symmetry, least action, conservation, causality, etc.). Their solutions, however, are remarkably precise models of an amazing phenomenal variety. They encode in their very compact and universal mathematical form an unpredictable universe of empirical diversity and complexity. This astonishing ability - this 'miracle' - earned them the title of 'intelligent equations/ The CF, in my view, is an 'intelligent formula.' It, too, encodes in a compact algebraic form expressing general structuralist principles of conflict and stabilization an unpredictable universe of empirical diversity and complexity. I consider it, rather than a formula, a 'structural equation' of mythological syntax, an equation to which myriad actual myths offer so many empirical 'solutions.' This diversity explains the risk of superimposition. Indeed, by making the CF into an elementary structure, 'equation' and 'solution' become one and the same thing, with the result that the fundamental dialectic between subsumption and modelling is destroyed. The elementary structure is then uniformly 'superimposed' onto empirical diversity. Instead of modelling its complexity, it reduces it to a repetitive archetype. 1.4 Towards a Morphodynamical Approach In order to maintain the difference between the universal form of the CF 'equation' and the vast diversity of its 'solutions,' its constituent relations must be interpreted mathematically. To do so, an adequate mathematical universe must be chosen. To the extent that the CF intimately depends on the theoretical concepts and principles of structuralism, I considered it relevant to adopt the mathematical universe of dynamical structuralism and to model structures using morphogenetic models of differentiation, organization, and regulation. Other choices undoubtedly exist, but the morphodynamical option has the advantage of being based on powerful mathematics whereby the theoretical problems of structuralism can be adequately interpreted and structural categories properly schematized (see Morphogenese du sens and Physique du sens).6 The main hypothesis, as we have seen, is that the CF is more than the expression of a simple semantic analogy between two qualitative oppositions, namely, a coupling between two oppositions defined on different semantic dimensions. In the morphodynamical schematiza-

272 Jean Petitot

tion, the binary paradigm constituted by a simple qualitative opposition is modelled by a singularity called a 'cusp/ The analogy between two oppositions is therefore modelled by a simple isomorphism between two cusps. But a coupling between two qualitative oppositions is modelled by a singularity called a 'double cusp' in which both cusps interact. However, the double cusp is definitely more complex than the simple Cartesian product of two cusps. Out of its complexity a considerable number of different semio-narrative structures can be drawn, all of which are solutions to the problem of coupling ('modes' of coupling). It is this idea that I will attempt to develop below. 2. Reminders Concerning Morphodynamical Models 2.1 Semio-narrative Models

The models we use are semio-narrative ones. They are based on the hypothesis that there are: (1) terms, at the syntagmatic level, that is, 'actants' in the sense of an actantial syntax (to be distinguished from actors or characters that usually syncretize several actants and support thematic roles), (2) semantic functions, at the paradigmatic level, that depend on codes (in Levi-Strauss's sense) belonging to deep structures: they are values categorizing the continuous substratum of paradigms into discrete units. That an actant t takes on a semantic value v is written Fv(t) in the CF. The problem, which is an extremely difficult one, lies in holding together the paradigmatic (semantic 'codes') and the syntagmatic (actantial interactions) levels. A basic thesis of semio-narrative structuralism is that paradigmatic semantic relations can only be implemented through actantial syntagmatics. Semantic values are 'confined/ 'invested' in the actants, and circulate through their interactions. Three theoreticians have played a crucial role in elucidating these relationships: V. Propp, C. Levi-Strauss, and A.J. Greimas.7 With his grammar of functions, Propp overly dissociated the narratively dominant actantial syntax from the semantic content. All too often he reduced the latter to simple thematic roles. On the other hand, by focusing dually on the paradigmatic axis and its projection on the syn-

A Morphodynamical Schematization of the Canonical Formula for Myths 273

tagmatic axis, Claude Levi-Strauss somewhat underestimated the problem of actantial syntax. The synthesis was achieved by Greimassian theory, which showed in detail how actantial syntax could handle logico-combinatorial operations on paradigmatic values. We readily acknowledge that, with regard to Levi-Strauss's Mythologiques, the semio-narrative point of view may be somewhat biased in that there appear to be few constraints applicable to the actants' identity in myths. Metamorphoses systematically occur in myths that are very different from those found in fairy tales and, a fortiori, in more realistic stories. But these different types of identity concern the actors (characters) rather than the actants themselves. 2.2 Morphodynamical Models

Morphodynamical models rest on three basic hypotheses that make explicit the constitutive operations of the components Fv(t) in the CF.8 1. First of all, we hypothesize that the semantic axes (the continuous substrata that the values discretize) constitute spaces, referred to as internal spaces. Each dimension of such a space M usually will be the continuous substratum of a qualitative opposition (x/y) (of a distinctive feature +/ —), somewhat as in elementary geometry an axis of coordinates is divided into a positive half and a negative half. When there are several dimensions, several semantic axes are likely to be coupled. Coupling entails that the decomposition of M is not a simple combinatorics of + and - (the way two Cartesian axes split a plane into four quadrants ++, +-,-+, —). But the space M is still subdivided into domains (categories) whose centres represent values and the boundary relationships (usually of opposition) between values. It is, as we say, stratified, its stratification being more complicated than that brought about by hyperplanes of coordinates. 2. The syntactic confinement of values, their embodiment in an actant, is then expressed by a variational principle, referred to as the model's internal dynamics. That is the second basic hypothesis. We assume that there exists a potential function/(j) defined on the internal space M, and that one of its minima represents the actant a under consideration. If this minimum corresponds to a point on M belonging to the domain of the value x, it will be said that x 'invests' a or that a 'represents' or 'actantializes' x. If /has several minima, there are therefore several actants a, b, c... invested with different values x,

274 Jean Petitot

y, z ..., and the relationships between the minima become actantial relationships 'actantializing' relationships between values. This schematizes the paradigmatic organization of the semantic substrata, and the dialectic between the paradigmatic-semantic and the syntagmatic-actantial dimensions of semio-narrative structures. 3. To be able to move from static relations to dynamic interactions, the potential/must, moreover, be able to evolve over time. This means that the potential/must be temporally parametrized:/^) —>f t (x) = f(x,t). Furthermore, several parametrizations, which correspond to as many different scenarios of interaction, must be possible. This brings us to the third basic hypothesis: there is a second space W, referred to as external space, which parametrizes the internal dynamics /:/(*) and is in fact of the form//*). The temporal evolutions of/ are therefore temporal paths in this external space: f Such models, where the categorization of a continuous substratum space into sub-domains (values defined by reciprocal determination) is generated by a family of generating potentials, have become widespread in contemporary cognitive sciences. If the categorization process is implemented in a network of formal neurons, the generating potentials become true potentials in the physical sense of the term, that is, 'energy' functions whose minima9 determine the terms of the categorization. For example, in the case of a neural net R implementing a phonetic categorization, M will be the space of the instantaneous states of R, and W a space of acoustic cues (voicing, articulation point, etc.) controlling the phonetic percepts. The minima phonemic values as well as the relations of dominance between them. For each w, a phonemic value will be dominant and, therefore, actualized, the others remaining virtual. But for different w, a transition of dominant values will occur. Each value Aw will therefore have its own domain of domination WA within W. These domains geometrize the reciprocal determination relationships between values. They constitute a partition of the external space W that is an external categorization generated by the family of generating potentials fw(x). Their boundaries form a critical set K that stratifies W and materializes the categorization (any categorization can thus be identified with a set of boundaries in a substratum space). Therefore, M undergoes an internal categorization through the attraction basins of fw(x) and an external categorization through the catastrophic set K. This results in a subtle internal/external dialectic allowing for an externalization of the internal paradigmatic

A Morphodynamical Schematization of the Canonical Formula for Myths 275

relations between values. The stratification (W, K) unfolded by the generating family fw(x) externalizes the internal paradigm P. As we have seen, a syntagmation of the paradigm P involves introducing paths in the external space W. The crossings of K are therefore comparable to syntactic events that commute the values of P while at the same time sequentializing them. That is how the projection of the paradigmatic axis on the syntagmatic axis takes place in these dynamical models. 2.3 The Cusp Example

The best-known example of such a model is the cusp catastrophe unfolding a cusp singularity. The normal form of the family of potentials fw(x) is given by the formula

The internal space M is one-dimensional (x coordinate). It serves as a substratum to a qualitative opposition. The external space W is twodimensional (u, v coordinates). Depending on whether u > 0 or u < 0, f w(x) has either a single minimum or two conflicting minima separated by a maximum. For this reason the u parameter is referred to as the 'splitting factor.' It controls the formation or neutralization of a conflict between two determinations. As for the v parameter, in the case where u < 0, it controls the domination of one of the minima over the other. It is called the 'bias factor.' The cusp sheets correspond to the extrema of fw(x). For u < 0 there are two attracting sheets (minima) separated by a repelling intermediary sheet (maximum). The three sheets connect along a fold curve and fuse for u = 0, and for u > 0 just a single undifferentiated sheet remains. The (W, K) stratification is, therefore, essentially composed of a twodimensional space W made up of a neutral zone and a conflict zone. The latter is limited by bifurcating strata Kb along which one of the minima disappears by collapsing with the maximum and becomes captured by the other minimum. It is separated into two domains by an interface Kc (a threshold corresponding to a conflicting equilibrium between the two minima). The bifurcating strata Kb are the projections of the folds of the cusp surface (see figure 1). Figures 2 and 3 illustrate an example of a path in the external space W. This corresponds to a capture catastrophe.

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Figure 1 The cusp catastrophe, (a) The external space of the cusp and the bifurcation strata; (b) the cusp surface; (c) the cusp surface and its projection on the external space

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Figure 2 A path in the external space of the cusp (chosen so that the dominant minimum remains stationary), (a) The path crossing a bifurcation stratum; (b) the evolution of the internal potential along the path

As a model of structural relations, the cusp schematizes a qualitative opposition A/B. On one side of K the A value dominates. On the other side, it is the B value that dominates. A path that crosses K therefore corresponds to an A —> B type of syntagmation: the initial state where A dominated over B is replaced by one in which B dominates over A. In Physique du sens, I explain the way in which the cusp models the syntactic handling of qualitative oppositions both through SubjectAntisubject (S/S) actantial conflicts and through Subject-Object (S/O) conjunctions.

278 Jean Petitot Capture dynamic

Figure 3 The capture scenario associated with the external path of figure 2. (a) The capture of one place by another; (b) the actantial capture

3. Pierre Maranda's Model of Mediation 3.1 Mediation

In his dissertation, Lucien Scubla presented detailed comments on Pierre Maranda and Elli Kongas's model, which describes the CF as an

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algorithm mediating an initial contradiction. The CF is thus 'entirely deduced from the single idea that mythical operations aim to resolve a contradiction through a process of mediation and that the formula describes the passage from the initial state to the outcome of such a mythical transformation.'10 I think that a reconsideration of the mediation model allows us to improve significantly on the double-cusp model presented in HI. Let us begin with the standard form of the CF:

1. The first part of the formula F x (a): Fy(b) corresponds to a qualitative opposition between both the terms a/b and the functions x/y. This means that two opposed x/y values (for example, Nature/Culture) are actantialized by a conflict between two actors a/b (for example, Dragon/Hero). 2. The second step is a mediation effected by b. Having initially embodied the 'right' value y, the term b also actantializes the opposite 'evil' value x. This explains the third component in the formula, Fx(b), corresponding to the mediation. 3. Obviously, this mediation does not imply a simple swap, a switchover between a and b, a in turn mediating the opposition by actantializing symmetrically the 'right' value y. The appropriation of the initially negative value x by the positive term b is accompanied by an elimination of the term a and a sort of 'assumption' (an 'apotheosis' in Scubla's words) of the positive y value, a is eliminated as a term value and it is this operation itself which now becomes a value a"1. At the same time, the assumption of y is an 'embodiment' of it, which explains the inversion of the term value and the function value characteristic of the terminal twist. This generates the fourth component of the formula: /fl-i(i/). Hence the presentation of the formula as a structure that is both static and dynamic:

IS REP where A/B (i.e., A : B) is the initial oppositior resented by a/b), M is a process of mediation and indifferentiation, and

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X is an inversion of the mediation and a new process of differentiation generating a new qualitative opposition between A'/B' (looping twist). We will now develop the idea that a morphodynamical formalization of Maranda's model automatically leads to a structure embedded in the double-cusp catastrophe. 3.2 The Main Ideas of the Model We will develop the following main ideas. 1. With regard to the CF, the cusp models the dynamics of the conflict fx(a)/fy(b). It formalizes (i) the genesis of the conflict between a/b, (ii) the possible neutralization of the conflict, and (iii) the possibility for an actant (for example, b representing y) to wind around the organizing centre of the cusp and to represent the opposite function x (this corresponds to a transformation fy(b) —> fx(b)). This mediation can, obviously, be symmetrized (transformation fx(a) -»/y(fl)). In conformity with Maranda's idea, a mediation is therefore not a logical operation but rather the exploration of a conflict dynamics. 2. The inversion of the term and function values as well as the 'looping twist' characteristic of the CF are to be interpreted as an 'internalization' of the external parameter (that drives the mediation) into a new internal variable (in the sense of the duality between external and internal spaces). This automatically leads to the double-cusp model. 4. Addressers and External Dynamics Let us begin by clarifying certain constituent operations necessary to the proper functioning of Maranda's model. 1. First, a specific status must be attributed to the concept of 'representation' or 'actantialization' of a value v by an actantial term t (i.e., a Fv(t) component). We have seen that in morphodynamical models actantial relations are described by the relationships between minima of generating potentials. The relationship between an actant and a value is more Proppian-Greimassian than Levi-Straussian as it is built through value-objects. Values are invested in objects, and it is the syntactic relationships of junction/conjunction between the subjects and anti-subjects and such objects that make them represent values. But what drives the syntactic relationships actantializing the relation to values? Simple 'confinement' as described above is not sufficient. 2. The answer to this question lies in the fundamental actantial con-

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cept of Addresser.11 Actantially speaking, the Addresser is the actant that, on the one hand, provides and guarantees values and, on the other hand, modalizes the subjects (e.g., casts them in the 'will' mode) and controls their actantial trajectories. For example, in myths of the Hero/Princess/Dragon type, the Addresser is the royal figure of the father. But he does not necessarily have to be anthropomorphic. The purest narrative example of an Addresser that I ever encountered is the eagle's flight in Stendhal's La chartreuse de Parme that persuades Fabrice, in a flashing and imperious way, to leave Milano for Waterloo. When the Addresser is external to the Subject actant, he modalizes him deontically and casts him in the 'duty' modality (the Subject acts according to the program assigned to him by the Addresser). When the Addresser is internal to the Subject the modality becomes that of 'will' (the Subject intentionally controls his actantial trajectory, his 'will' acting as a self-destination). In my dissertation on actantial morphodynamics,12 I proposed that the connection between value, value-object, Subject, and Addresser be modelled as follows: (a) The value v is identified with the threshold separating the subject S from the object O and the capture of O by S is therefore ipso facto a capture of v by S. (b) Moreover, the value is also the semantic content of the internal axis on which the junction (disjunction/conjunction) S-O is defined. (c) The Addresser controls the external trajectory that leads from the disjunction S/O to the conjunction S-O. Per Aage Brandt showed in his reference work 'La Charpente Modale du Sens'13 that, generally speaking, Addressers can be described morphodynamically as control dynamics in the external spaces of models. External dynamics drive the actantial interactions (actants' 'programs' of action). When a Hero-Subject embodies a value, an Addresser assigns him the mission of outdoing antagonistic forces represented by an AntiAddresser. Subject/Anti-Subject conflicts in the internal spaces are therefore controlled by conflicts in the Addresser/Anti-Addresser dynamics in the external spaces (see figure 4). To say that a Hero t 'embodies' a value v means that he becomes his own Addresser and controls his external dynamics, causing v to triumph.

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Figure 4 The external dynamics of the Addresser and Anti-Addresser 3. There exists an essential connection between Addresssers and the axiological dimension of any conflict. To say that a value is axiologically positive means that it drives the modal intentionality of the external dynamic. In other words, axiology polarizes external spaces positively/ negatively and, through dynamical attraction/repulsion interplays, generates external dynamics. For example, in the cusp model, the halfaxis v > 0 drives the Addresser's dynamics (S is captured by S) while the half-axis v < 0 drives the Anti-Addresser's one (S is captured by S). 5. The Paradoxes of Mediation Now let us apply the simplest model, namely the cusp, to the problem of mediation. As we will see, it solves half of the problem, that is, the first twist in the CF, but leads to paradoxes that can only be solved by complexifying the model. 5.1 A Model of Mediation as Twist The cusp model makes it very easy to interpret the first twist, that is, of the cf and the moebiu the third component f fx(b). The characteristic of the cusp is the fact that an external path can

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Figure 5 A loop around the organizing centre of the cusp

wind around the organizing centre and, like a loop, come back to the starting point (see figure 5). Such a cycle is very particular. It is a 'marked' cycle generating what Thorn named an 'actant confusion' (see figure 6). The 'marked' cycle perfectly models the mediating component Having bypassed the organizing centre, the trajectory of b (which captured a) leads it from sheet y to sheet x, which corresponds exactly to Moreover, mediatttion taked placce by netaliszing the tthresold arating x and y: the single sheet beyond the organizing centre is neutral in relation to the x/y opposition. 5.2 Mimesis and Mediation

But in this cusp model, mediation proves paradoxical for at least two reasons. First, as we have seen, a's capture by b transforms a into a value-object for b. But, whereas b is a 'subject' actantializing the value y, a is more of an 'anti-subject' (enemy, adversary) than a value-object to be conquered. Furthermore, the actantial confusion involved in mediation implies that b has, in Thorn's terms, 'alienated itself in the role of a. In other

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Figure 6 The marked cycle associated with the loop in figure 5: the Thomian phenomenon of actant confusion

words, by virtue of mediation itself, the S/S opposition typifying the b/a opposition becomes somewhat analogous to a mimetic relationship in Girard's sense (see figure 7). This seems to be a deep and well-grounded anthropological phenomenon, essential to understand rituals. But it is nevertheless rather paradoxical. 6. Beyond the Mediating Contradiction As we can see, attempting to solve the contradiction means of a mediationf x (b) leads to an even more serious contradiction, logical this time, concerning identity. This new contradiction needs also to be solved. 6.1 Introducing Value-objects

The first possibility is to introduce a value-object that the subjects S (i.e., a) and S (i.e., b) are competing to conquer. This is the case in typical Hero/Princess/Dragon myths.

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Threshold neutralization Organizing centre Mediation

Conflict zone

Figure 7 Pierre Maranda's mediation as a marked cycle on the cusp surface; actant confusion implies that the subject is mimetically alienated in the antisubject

In Physique du sens I showed that such actantial scenarios are accurately described by the Thomian transfer catastrophe called 'butterfly/ which expresses the way in which a third actantial place (that of the value-object) Ov is transferred from an S/OV cusp to an S/OV cusp. If the external dynamics of the Addressers that control the subjects' intentional duty and will are introduced, the domination of S over S is expressed by the capture S-OV (which is a tearing away, a disjunction, S/0,). In much the same way as the cusp x4 - iix2 + vx is the universal unfolding of the x4 singularity, the butterfly catastrophe is the universal unfolding of the x6 singularity. Its external space is four-dimensional (coordinates t, u, v, w). It can be reduced to three dimensions since it is only for t < 0 that three-minima potentials occur. For t > 0, the butterfly is simply a cusp. Figure 8 shows the evolution of the plane sections (v, w) of the catastrophic set K for t = -1, when u varies from 0.3 to -0.3. An initial cusp 'emits' a 'swallowtail' along one of its branches, one cusp of which generates a new minimum, while the other cusp, called a dual cusp, generates a new maximum. The new

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Figure 8 The transfer catastrophe, (a) A (symmetrical) example of a planar section of the universal unfolding; (b) the evolution of the (v,w) planar sections for variable u showing the transfer phenomenon

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Figure 9 A few surfaces of the transfer catastrophe

cusp becomes progressively dominant while the initial cusp regresses. Finally, the latter collapses together with the dual cusp and we end up with a single cusp. Figures 9,10, and 11 show, respectively, the evolution of the butterfly sheets above the (v, w) plane when u varies, the evolution of the internal potential along a transfer path (the intermediary minimum transiting from one minimum to the other is clearly visible), and the three phases in the transfer: emission-transit-reception. One of the main interests of this model is that it obstructs the mimetic relationship S/S. In fact, it contains a value that is not actantialized by Ov (as it can be in a Subject actant S or S), but is simply 'invested' in O^ The object Ov does not itself represent the value, it only 'confines' it. It is true that the relationships and V and are organized singularities, and that these cusps allow for confusion between actants, but, insofar as the object Ov does not actantialize the value v, this 'fusion' essentially expresses the fact that the Subject acquires the value v by capturing the object Oj,. On the other hand, the relation S/S is a pure conflict, without any organizing cusp. Actant confusion S/S (mimesis) is thus avoided. Figure 12 shows a very simplified section of the

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Figure 10 Evolution of the internal potentials along a transfer path (the parameters were chosen so that the 'source' and 'goal' minima remains stationary); the intermediary minima transits from the source towards the goal

Figure 11 The three phases of a transfer: emission/transit/reception

external space of the transfer catastrophe containing not only the bifurcation strata but also the conflict strata. The stratum of pure conflict S/S is clearly visible. Another point of interest in this model is that S represents the value v by capturing an object Ov that 'confines' v. In other words, through the mediation of Ov,a component of the CF such asfy(b) is now inter-

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Figure 12 A simplified section of the external space of the transfer catastrophe showing the conflict strata; the pure conflict stratum S/S (written S/S* for graphics reasons) is clearly visible

preted as a trajectory in the external space (i.e., as a narrative program of conjunction S-Oy (with S = b)). This leads to another means, more apt in my view, of resolving the mediation contradiction. 6.2 Identity Reintegration and Value Incarnation

The origin of the mediation contradiction lies in the fact that actant

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Figure 13 The three dynamic cusp scenarios for the occupation of a sheet by an actant. (a) The actant occupies statically its natural sheet; (b) the actant occupies its natural sheet after having captured the anti-actant; (c) by a process of actant confusion, the actant occupies the sheet opposite its natural one.

confusion causes actant b to occupy the x value's sheet initially occupied by a. But b occupies this position as a result of a dynamical process that is fundamentally different than that which determined a. In the cusp model, there are essentially three dynamical scenarios leading to an actant's occupation of a sheet (i.e., to a relationship of the/ D (f) type): (1) The actant occupies statically its natural sheet: this corresponds for instance to the/x(0) or/v(b) components of the CF (see figure 13a). (2) The actant occupies its natural sheet, but after having captured the anti-actant (see figure 13b). (3) The actant occupies the sheet opposite to its natural one following a process of actant confusion (fx(b) in the CF) (see figure 13c). A first means of resolving the paradox of mediation would therefore be for actant b to reintegrate its natural sheet after stage (3) through a sort of 'tunnel effect' (see figure 14). Actant b thus reintegrates its identity having doubly eliminated a. In fact, it not only captured a, but it transcended the mimetic effect connected with this capture. It threw off the residue of a in which it was alienated and henceforth integrated the 'evil' value of a to the 'good' one y. This expulsion of a can be identified with the term a~l, and can be referred to as an 'assumption' of b and y. Actually, insofar as the provider and guarantor of values is the Addresser, b has shifted from the status of Subject to that of Addresser. The fourth component of the formula f could therefore be interpreted as a 'reprogramming' of identity that makes it possible for the alienation constitutive of mediation to be disalienated. Hence, the

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Figure 14 The 'tunnel effect' by which the actant reintegrates its identity after actant confusion

idea that actant confusion leads to a new differentiation, which is no longer a simple x/y static opposition but a complex dynamical process. If this idea of a 'tunnel effect' seems too sophisticated, consider that b simply continues its cycle and, after a sufficient delay, reintegrates its natural sheet by a standard catastrophic jump (see figure 15). This would give us an even more simple interpretation of the fourth component of the CF. Indeed, since the catastrophic jump takes place between the mediation and a return to the initial component fy(b), it could be said that actant b appropriates the entire x/y opposition in order to 'represent' it. Actant a is therefore completely eliminated, which is expressed by the term a"1. This gives us the global dynamical schema in figure 16. The stages in the CF's cusp model can therefore be summarized as follows: 1 fx(a) means that a occupies a natural sheet that makes it representative of the x value. 2 In the same way,/y(b) means that b occupies a natural sheet that makes it representative of the y value. 3 means that these two representations are in conflict: it corresponds to the double-sheeted conflict zone inside the cusp.

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Figure 15 The return catastrophe reprogramming the actant's identity

Figure 16 The Canonical Formula and the global dynamic schema of the cusp; summary of the cusp model

4 The mediation means two things: (i) that b actantially dominates a; and (ii) that it alienates itself in the value x represented by a. 5 The twist component means also two things: (i) that b reintegrates its identity through the 'catastrophic' expulsion of a (term a"1); (ii) the assumption of b as an Addresser 'incarnating' the y value.

A Morphodynamical Schematization of the Canonical Formula for Myths 293 6.3 The Problem of Internalization The dynamical scenario leading from the input f to the output fa-\(y) is rather subtle (mimetic mediation + identity reintegration). The external dynamics controlling b's actantial modalities and revolutions displays the Addresser's action. Hence, the issue of whether or not this dynamics can itself be represented statically on the condition that the internal spaces defining the relationships are complexified. The most natural idea consists in internalizing the external dynamics. This idea leads directly to the double-cusp model. 7. Internalizing External Spaces 7.1 The Problem Let us suppose that in the cusp example we wanted a more complicated internal dynamic to lead necessarily from an initial situation fx(a): fy(b) to an outcome where b has captured a. A simple way to do this is to consider a section of the cusp X4 + uX2 + vX with u < 0 (for example, u = -1), with v varying from negative to positive values, and to treat v as a new internal variable.14 More precisely, we want a value close to a VQ value of v located on the right outside the cusp to attract the trajectory y (see figure 17). When we try to put it into practice, this natural idea proves rather tricky technically because it is difficult to master intuitively the couplings of variables that come forth. We can, however, provide some fairly simple indications with the help of illustrations. 7.2 The Energy Landscape f(X,v) Let us begin by making v an internal variable in the cusp model by treating the potential f(X,v) = X4 - X2 + vX no longer as a family of potentials fv(X) with an internal variable X dependent on an external parameter v, but rather as a unique potential with two internal variables. If we look at the graph of/(X,u), we can see that it is composed of (1) a main Valley' (the principal minimum that slopes downward, and (2) a secondary valley (the secondary minimum ms offv(X)) that slopes upward and disappears through bifurcation by coalescing with the maximum (see figure 18).

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Figure 17 The idea of internalization

The bifurcation takes place for the value vcrit ofv>0, for which has a double root (inflection point where ms collapses with the maximum M). vcrit is the solution to the system of equations

Its numerical value is vcrit = 0.54. The valleys are generated by the minima of fv(X), that is, by the solutions to fv'(X) = 4X3-2X + v = 0. Figure 19 represents them in the plane with the ridge separating them. It clearly shows the descent of mp, the rise of ms, and its disappearance by collapse with the maximum M (bifurcation). 7.3 Making Capture Necessary

If we want the capture scenario driven by the external dynamics of the initial internal variable X to be driven by the internal dynamics

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Figure 18 The energy landscapef(X,v) and its two 'valleys': the main valley (sloping downward) and the secondary valley (sloping upward and bifurcating; x, v are the two internal axes, and the figure displays the graph of/)

of the new internal variable v, the energy landscape must be modified in two ways: (1) the main valley also must admit a minimum (in the neighbourhood of v0), but relative to the v dimension (and no longer to the X one); (2) the slope of the secondary valley must switch downward. To make the secondary valley slope downward, we can add a term -av (independent of X) to/p(X), whereby a is a coefficient accentuating the slope of the valley ms). For « = 1, we get figures 20 and 21, which show that the valley ms has indeed become downward-sloping. But in order for the main valley mp to have a minimum near v0, we need a little more. The simplest approach is to introduce a quadratic term in (v - v0)2 with v0 far enough removed from vcrit. This gives us the potential

For v0 = 1.5 (remember that vcrit = 0.5), we get figures 22 and 23. This simple quadratic transformation of the standard cusp model ensures that the ms valley slopes downward. This comes from the fact that we can add a constant term c(v) to the potentials fv(X) without altering their gradient dynamics. To make the ms valley slope down-

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Figure 19 The graph of extrema (relative to X) of the potential/(X,y): the mp and ms valleys and the M ridge. ward, the constant c(v) simply needs to decrease sufficiently until ms bifurcates (for vcrit ~ 0.5). This was the case for c(v) - —v. But it is also the case for c(v) = (v - v0)2, when v0 is sufficiently greater than vcrit. The minimum (Xmin, vmin) off(X,v) is then represented by the system of equations

In the case v0 = 1.5, we get vmin =>* 1.81. 7.4 Coupling Internal Variables The potential/(X,i?) = X4 - X2 + vX + (v - v0)2 can be interpreted as a coupling between two independent potentials f^X) = X4 - X2 and/2(p) = (v - vrespectively defined along the internal axes X and v.fj rules a 0)2,

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Figure 20 The energy landscape of figure 18 modified by the adjunction of a term -av

Figure 21 Both valleys now slope downwards

pure conflict (the x/y conflict on the X axis), and f a state of quadratic equilibrium that attracts v toward v0. The 'free' potentials/j and/2 are linked in/by a coupling (or 'interaction') termf . This multipli cative coupling is the simplest possible. It is, in a sense, 'minimal.'

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Figure 22 Internalization of the external capture dynamics

Figure 23 The main valley has a minimum

The minimal multiplicative coupling implies that v actually operates as a bias factor for/j15 and that, therefore, when it is attracted by v0, v drives the a/b conflict towards a capture catastrophe, that is, towards the domination of b over a on behalf of the y value. The idea of inter-

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Figure 24 Internalization defines a slow/fast system in which both time scales may vary

nalization is therefore to introduce a dynamics f2(v) for v and to couple it with the dynamics/?(X) in order to force the latter to evolve. 7.5 Slow and Fast Dynamics In this type of internalization of the external parameter v we do not take into account the fact that, in the standard model fv(X), the internal dynamics on X is a 'fast' dynamics that 'instantaneously' projects the representative point X onto a minimum of fv(X), that is, onto the surface of states, while the external dynamics on v is a 'slow' dynamics that makes these minima evolve on the surface. The system f(X,v) is what is called a slow/fast system, also referred to as 'adiabatic/ that is, a system with two very different time scales. If we wish to take this slow/fast difference into account in the internalization, we simply need to change the v scale (see figure 24). We will not take this problem into consideration here. 7.6 The Meaning of Internalization: Inverting a Term Value and a Function Value Let us clarify the structural meaning of the procedure of internalization. The/(X,z;) energy landscape now describes the capture of actant a by an actant b as a function of two internal parameters. The first, namely X, supports the opposition of values x/y. The second, namely v, on the contrary is the support for what used to be the external capture of a by b, that is, the domination of the y value over the x value. As we have seen, these external dynamics described how the Addressers modalize the subjects.

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Internalization therefore means (1) that the process of capture itself becomes a value, in other words, the 'triumph' itself is a value - whatever the content of the x/y values may be; (2) that actant b was enriched by the 'triumph' involuting (internalizing) the role of Addresser, b becomes its own Addresser during the end of the canonical process and its intentionality becomes involuted in the certainty of victory, instead of becoming externalized in a quest. In this sense, the y value is 'incarnated' by b. Such is the dynamical interpretation that we give of the inversion between term value and function value: a y value is transformed into a term (is actantialized) through the 'triumph' of the term b, which incarnates it; in other words, when the dynamical principle of its axiological domination is itself internalized into a value. This step provides new elements for interpreting the fourth component /fl-i(y) of the CF. It can be said that during the capture process, the elimination of a - that we will note/fl-i(y)° - is also reinterpreted as a value (the anti-subject a becoming a scapegoat). This is a 'precursor' of the final twist. 8. The Double Cusp and the Double Twist 8.1 Internalizing the Conflict of External Dynamics Internalizing the bias factor v of the cusp model X4 - X2 + vX thus accounts for a component of the/fc(y) type expressing b's 'triumph,' its capture of a, its 'rising' to the rank of Addresser, and the resulting inversion between b as term value and y as function value. But there also exists a sort of 'second degree' opposition between 0's and b's intentional dynamics, that is, between Addresser and AntiAddresser. This means that in the internalization of the bias factor v, the f2(v) dynamics controlling v possesses two opposing minima, vmin and -v In other words, the dynamics f2(v) must itself be a conflict dynamics. The simplest model is once again that of the cusp. We will therefore 2 take as a 'free' potential f the function whose minima are v = ±V0. For the sake of simplicity, we will use v0 = \ (which is sufficiently > v^. The minimal multiplicative coupling Xv therefore yields the potential

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Figure 25 Symmetrization of the situation in figure 23 by introducing a term

in v4 - v1

The analysis of this kind of potential, which expresses the simplest interaction between two pure conflicts, is already complicated. Its critical points are given by the system of equations

If we eliminate, we get an equation in X of degree 9:

But luckily, these nine critical points are not all relevant. If we maintain the special coordinates (X,i>), and if we follow the critical points (the two minima and the maximum) of fv(X), we get the swallowtail graph in figure 25. Figure 26 clearly shows the two antagonistic capture dynamics.

302 Jean Petitot

Figure 26 Internalization of two antagonistic external dynamics 8.2 The Double-cusp Model

The above potential/(X,#) is a partial unfolding of the singularity X4 + z;4, which is none other than the double cusp, a singularity that expresses the coupling of two cusps, that is, the interaction between two oppositions with different supports. This interaction explains why in Physique du sens and in HI, I proposed the double cusp as a global model (of which the CF is a submodel) for the internalization of external dynamics. At the time, it may have seemed somewhat speculative. I am pleased that this new opportunity to reflect on the CF has enabled me to clarify this point. The double-cusp model couples two independent oppositions. In the sub-model currently under study, the first, X-supported, is the initial axiological x/y opposition. The second, ^-supported, is the internalization of the opposition between the Addressers' dynamics (i.e., the actants' intentional dynamics). It expresses the conflict of the components fb(y) andfa(x) or, correlatively, of the components and/fl-i(y)° and 8.3 The Loop in the First Twist of the CF

How can actant confusion be taken into account in this internalized model? The simplest way is to internalize the bias factor v no longer as

A Morphodynamical Schematization of the Canonical Formula for Myths 303

Figure 27 Internalization of a loop winding around the organizing centre; the evolution of the dominant valley displays the switch of minimum (actant confusion).

a linear dimension, but as a cycle u2 + v2 = 1 winding around the organizing centre in the control plane (u,v}. If we introduce an angular variable 9 varying from 0 to 2n, we have u = -cos(0) and v = sin(6). For 0 = 0, the cycle takes off at (-1,0). In order for this cycle to correspond to a valley in the energy landscape f(X,v), a term that plays the role of -v and in the preceding models must be introduced. The simplest is to take - 8 . This yields the potential

The analysis of its graph very clearly indicates how b's valley lands in position a at the end of a cycle. This leads us back to the third component of the CF, Fx(b), that is, the first twist (see figure 27). If we wish, parameters can be slightly adapted. In this model, v = sin(0) and therefore | v \ = \ sin(9) | ^ 1. If we want to stretch the cycle in the v direction, we can add a multiplicative factor and take v = c sin(9), or else use an oval-shaped cycle and not a circle-shaped one, and so on. Qualitatively it does not change anything. Likewise, if the

304 Jean Petitot

downwards slope imposed by 0 seems a little too steep, 9 can be replaced by eQ. In fact, the first twist gives all of its meaning to the precursor of the second twist /fl-i(y)° (see section 7). Already the internalization of b's external dynamics could be interpreted as a fb(y) component whose effect on a was /fl-i(y)°. But this negation of a presupposes that the value x, initially actantialized by a, is in a sense 'recovered/ The component /fl-i(y) takes on its true meaning only after the actant confusion (first twist) 8.4 The Second Twist As we have seen in section 6.2, the second twist corresponds to b's reintegration of its identity, either through a sort of 'tunnel effect/ or through a catastrophe fx(b) -*fy(b) closing the cycle, and expressing the fourth component/^(y). In order to stabilize the internal dynamics on this final state while guaranteeing the symmetry between b and a, a term c04-rf02 must be added (the constants c and d serve to decrease the slope, since 2ji4 = 195). This yields the potential

If we want 0's minimum to be located, for example, at 27t+7i/4, we need c = 0.0004e and d ~ 0.04e. In the following figures, we used e = 5. The different stages of the process are clear. Figure 28 shows the evolution of the valley bottoms: b can be seen returning to its original sheet. Figures 29 and 30 show the loop taken by b from a variety of perspectives, as well as its different stages. This having been said, there is an aspect of the potential/(X,0) that is not very satifactory. It mixes algebraic and trigonometric terms together. To remedy this incongruency, we can come back to the universal unfolding of the double cusp. 8.5 Double Cusp and Generalized Couplings In chapter 7 of Physique du sens, I explained at length the genesis of elementary catastrophes leading to the double cusp. I also summarized the work of C. Zeeman and J. Callahan on the highly complex geometry of the double cusp (which the symbol X9 stands for in the list of

A Morphodynamical Schematization of the Canonical Formula for Myths 305

Figure 28 Evolution of the valleys for a full cycle (including the reprogramming of b's identity)

Figure 29 Three views of the double twist

306 Jean Petitot

Initial conflict Capture of a by b

Cycle winding around the organizing centre

Actant confusion Reprogramming b's identity Return of b alienated in a Figure 30 The full loop and the double twist (including the reprogramming of b's identity)

singularities). The double cusp X4 + Y4 is of a codimension 8 (in fact, reducible to 7). Its universal unfolding is given by the normal form

where w is the multiparameter (a,b,c,d,e,f,g,h) varying in the external space W. If desired, a weight coefficient for Y can be introduced and Y4 can be

A Morphodynamical Schematization of the Canonical Formula for Myths 307

replaced by Scaling Y in this way does not change anything qualitatively. A constant term p can also be introduced. The universal unfolding fw includes all the monomials in X and Y of degree excepting those in the ideal (called the Jacobian ideal) generated by the partial derivatives and of the organizing centre namely the m o n o m i a l s a n d This is due to the fact that in the functional space of functions, the universal unfolding of is a transverse section of the orbit of under the group action defining its qualitative type. The Jacobian ideal of f0 is the 'tangent' space to at and are therefore 'tangent' to the orbit, and have to be excluded from the universal unfolding. But obviously, nothing prevents a general (not universal) unfolding of/0 from having components tangent to the orbit. Let us go back to the potential For small 0 we can expand the trigonometric functions cos6 and sin0 and use the approximations This yields the potential:

that is, an infinitesimal path in the universal unfolding of the double 17 cusp with and These potentials are semi-local: 6 must remain small, whereas X can vary more freely. Furthermore, insofar as the potential is of degree 4 in we could extend the approximation of sinG to the order 3: and that of cosG to the order 4: The problem is then to globalize such a description to the cycle that 0 describes. This problem is rather tricky and can have several answers. The easiest is simply to rhoose as internal space not the R2 plane but rather a cylinder = unit circle) of coordinates The potential is then a potential on a part of the cyclinder. By using the analysis made for the small values of 0, we can also consider for a given value of an expansion around 00 and write 0 = with small. This yields the potential/wfe^(X,Y) of the double cusp that most closely approximates the potential at 19 This is quite an interesting idea. The internal variable X that defines the underlying semantic axis of the opposition and the variable 0 that internalizes an external parameter are clearly distinguished. The double cusp is then considered a generating singularity, that is, a singularity whose universal unfolding is so rich that many different scenarios can be embedded in it. That was the main idea presented in

308 Jean Petitot HI: to treat the double cusp as a 'classifying space' for the structures ruled by the CF. If we wish to internalize an external path of the cusp (u(Q),v(Q)) parametrized in a complex manner by 0 (like a cycle (u = cos0, v = sinO)), we can consider for each value of (u(Q),v(Q)) the potential fw(X,Y) of the double cusp that best approximates it. This defines a sort of infinitesimal internalization of the path (w(0), #(0)), the global internalization being a sort of 'integration.' This point of view is acceptable inasmuch as the condition of adiabaticity (see section 7.5) makes the external dynamics internalized by 0 a slow dynamics relative to that on X. If we apply this strategy to our example, we get the following results:

If we expand sin (0 - 00) and cos (0 - 00) around s = 0 - 00, it yields the potential:

For each value 00 of 0, the potential /(X,0) is approximated around 00 by the potential/w((? )(X,Y) of the double cusp defined by the following values of w(Q0):

where g' is the coefficient of a supplementary term in s3 and p a constant term. This is what I meant in HI when I said that the CF can be represented by a 'path' in the external space of the double cusp. I hope that this way of expressing it has now been sufficiently clarified. The path expresses the progressive internalization of the domination of the y value over the x value through the mediation of b, and then the 'triumph' of b (elimination of a).

A Morphodynamical Schematization of the Canonical Formula for Myths 309 8.6 The Double Cusp as a Classifying Space Once we have demonstrated that Pierre Maranda's model of mediation can be embedded in the double cusp, the geometric richness of this complex singularity can be used as an underlying complexity immanent to the CF conceived of as a 'structural equation.' Many other paths can be found, which constitute so many different solutions to the problem of coupling two oppositions defined along different semantic axes. Some will be modes of internalizing the dynamical evolutions of an actantial antagonism handling an initial semantic conflict. But the study of other paths may prove equally interesting. Acknowledgments I wish to thank those who played a specific role in motivating the formulation and elaboration of these comments. First of all, Claude Levi-Strauss, whose outstanding genius remains the brightest light in theoretical structuralism; then Rene Thorn, another genius, mathematician, structuralist, and philosopher, without whom this type of modelling would not exist at all. It has been an extraordinary privilege for me to develop my ideas in contact with the teachings of these two masters. I would also like to thank Lucien Scubla, who certainly has the most indepth understanding both of the theoretical problems of structural anthropology and of morphodynamical models, and with whom my many discussions have been essential. Pierre Maranda, whose work is at the origin of this morphodynamical schematization of the CF, Michel Perrin, who encouraged me to write the paper in L'Homme, Solomon Marcus and Charles Henri Pradelles de la Tour, whose seminars prompted me to clarify some technical points, and my friend Bernard Tessier of the Mathematical Centre of the Ecole Normale Superieure in Paris also deserve my heartfelt gratitude. Of course, it is also a great pleasure to thank Mary Richardson for her beautiful translation. Finally, as this paper will appear in a book generated in 'la belle province' of Quebec, I would like to mention that it was thanks to my friend Pierre Ouellet that I published my first article, Topologie du carre semiotique,' on morphodynamical models of semio-narrative structures (Etudes Litteraires, Laval University, 1977). Twenty years later, nothing seems more fitting than to come back to Quebec to tie up this adventure.

310 Jean Petitot Notes 1 2 3 4 5 6 7

Petitot 1988. Petitot 1995. Maranda and Maranda 1971. Scubla 1996. See Petitot 1992. Petitot 1985. Their works are masterpieces of structural epistemology and methodology. They relate to those of Saussure, the linguists of the Prague Circle, Mussel's third Logical Investigation, and Jakobson, Hjelmslev, and Br0ndal. 8 This section summarizes H2. 9 In physics, to minimize energy is the basic variational principle. 10 Scubla 1996: 210. 11 The French name 'Destinateur' would be more convenient, for it includes the connotation of destiny and fate. 12 Petitot 1982. 13 Brandt 1986. 14 The internal variable X is the support for the opposition between the x/y values that label the cusp sheets. 15 ifnt also operates on/2, but since/2 is a quadratic potential, this does not produce any dynamical effect. 16 For further precisions, cf. Petitot 1992. 17 Although they sometimes share the same symbol, the general coefficients of the double cuspfw(X,Y) must not be confused with the constants appearing in the potential/W(X,6). 18 This is the approximation obtained by taking the first terms in s of cos(s) and sin(s), that is, by reducing their Taylor expansion to order k>3 or 4. This is referred to as the path's k-jet.

References Brandt, PA. 1986. 'La charpente modale du sens.' Dissertation, Universite de Paris 3; published 1992, Philadelphia: John Benjamins. Maranda, E., and P. Maranda. 1971. Structural Models in Folklore and Transformational Essays. La Haye: Mouton. Petitot, J. 1982. 'Pour un schematisme de la structure.' Dissertation, Ecole des hautes etudes en sciences sociales, Paris. - 1985. Morphogenese du sens. Paris: Presses Universitaires de France.

A Morphodynamical Schematization of the Canonical Formula for Myths 311 - 1988. 'Approche morphodynamique de la formule canonique du mythe.' L'Homme 106-7:24-50. - 1992. Physique du sens. Paris: Editions du CNRS. - 1995. 'Note complementaire sur 1'approche morphodynamique de la formule canonique du mythe.' L'Homme 135:17-23. Scubla, L. 1996. 'Histoire de la formule canonique du mythe et de ses modelisations.' Dissertation, ficole des hautes etudes en sciences sociales, Paris.

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Conclusion Pierre Maranda

This conclusion draws from comments made on the manuscript by Claude Levi-Strauss, Lucien Scubla, Solomon Marcus, and Sandor Daranyi. Levi-Strauss's remarks bear on types and ranges of application of his CF and also on its capacity to deal with diachrony (le temps du recit I le temps historique). On convergent lines, Scubla offers a threefold and technical classification of static versus dynamic interpretations of the CF. Finally, Marcus's and Daranyi's points explore - the first, briefly, the second, somewhat more at length - the epistemological status of the formula. Actually, Daranyi happens to develop an aspect of Marcus's statement about the radical implications of the formula with respect to narratology. For Daranyi, the CF embodies a 'geometry and meaning-marked loci in that geometry/ which leads to a morphogenesis of meaning. Types and Ranges of Applications of the CF Levi-Strauss points out that Petitot and Maranda use the CF to ends different from the one for which he himself designed it. Indeed, he created the CF to 'represent every myth (considered as the aggregate of all its variants)' (emphasis supplied) (1958 [English translation]: 228). As for Petitot and Maranda, their applications remain within the confines of one narrative. Consequently, 'the CF is amenable to different applications' (Levi-Strauss, personal communication, 30 January 1993). On the same lines, Levi-Strauss (1994) gave the following enlightening response to questions addressed to him by Solomon Marcus in connec-

314 Pierre Maranda

tion with the latter's 1993 paper (Marcus 1997). The excerpt (my translation) also bears on the relation of the CF to diachrony. You distinguish two uses of the formula, one diachronic, the other synchronic. The first one can be illustrated by Maranda and Maranda [1971] who apply it to the time-dimension within narratives whereas I use the formula in order to sort out variants from a purely formal viewpoint. Nonetheless, even my use of it implies a diachronic aspect. The variant that comes last (the 4th member of the formula) stems from an event that occurred in time: overriding cultural or linguistic borders, borrowing by foreign audiences, etc. That is why, in response to a question raised by Cote [this volume], I have excluded those that set the twist at the beginning, from the 144 states he lays out... To summarize: one may conceive of diachrony in two ways: either as inscribed in the internal time span of a specific narrative (le temps du recit), or as the inscription of several related narratives in an external time span (le temps historique). Now I wonder whether, from a more general viewpoint, individual minds that consciously invent narratives, and collective minds that unconsciously generate series of mythical transformations, do not abide by the same constraints or do not resort to the same possibilities.

Static versus Dynamic Interpretations of the CF Scubla's (1998b) comments (my translation) bear on the 'static' and 'dynamic' interpretations of the CF: (i) There are two possible readings of x and y in the formula: one may see them either as the two poles of a binary opposition (x/y), or as two independent semantic axes x and y and, consequently, two distinct binary oppositions (x/x' and y/y')(ii) On the other hand, and more importantly, one can use the CF either from a static or from a dynamic standpoint. Hence three vast classes of possible interpretations: 1. The weaker interpretation consists of taking the x/y opposition and criss-crossing it with the a/b opposition also present in the formula. One then gets what Petitot calls 'an elementary structure': a Klein group (LeviStrauss) or variants of the classical analogy (Racine [this volume], Mosko [1991], Cote [this volume]).

Conclusion 315 2. In the strongest possible interpretation one takes two distinct opposites or semantic axes x and y and couples them in a double cusp as does Petitot in his masterly paper (1988) where he proposes to consider the CF as an 'intelligent equation.' 3. The middle way consists of taking a single opposition x/y as a point of departure but this time in a dynamic mode as did Greimas in the semiotic square and as I do it in developing your own work [Maranda and Maranda 1971] in my thesis [Scubla 1996]. On the formal plane, one then backtracks from the cusp to the dual butterfly in order to reconstitute the genesis of the semiotic square (as Petitot shows it in his Physique du sens [1992], or from the cusp to the double cusp in which to inscribe your model (as Petitot shows it in his paper) [this volume].

I should add here a comment by Levi-Strauss on Scubla's 'weaker interpretation/ He points out that his formula goes beyond a mere Klein group. Actually, one might say that Levi-Strauss inflects his own use - depending on contexts of application - in ways that can eventually fit within any of Scubla's 'three vast classes of interpretation.' Epistemological Aspects For Marcus (1997), the papers of this collective work fall into two different categories: those that use the CF as a 'metaphor' (Part One) and those that aim at a conceptual approach (Parts Two and Three). On the epistemological level, Marcus states that the formula questions radically 'the quasi totality of all fundamental concepts of narratology.' Indeed, the morphogenetic approach to narratology implied in the CF imposes a new problematics that leads to rethinking both the widespread and the traditional conceptions of narratives and even of general semantics. On those lines, Daranyi writes (1998): The richness and wide-spreading span of the ideas inspired by Claude Levi-Strauss, indeed the hermeneutics of the CF, mark a sub-field of research which, apparently, has not stopped emerging and has contact points with many other contemporary disciplines and areas of interest... Such a construct [as implied in the CF] is a metaphysical field as well, since it is in the intersection of two abstractions, a geometry and meaningmarked loci in that geometry. Semantics can be interpreted on such a field in terms of logical operations on spatial distributions of positions; the geometry is spatiotemporal, i.e., can evolve.

316 Pierre Maranda

Actually, in Scubla's words (1998a: 292), the important point is that 'there exists a reasonably good fit between the model that mathematicians have built a priori to give a respectable status to Levi-Strauss' formula and the one that anthropologists have obtained inductively from converging ethnographic data.' The double permutation - double twist - of positions of both 'terms' and 'functions' maps out a generative operation, that of the creation of complex meaning. The CF thus strives to untangle a basic dynamics that structures fundamental thought processes whenever human minds have to resort to tools more powerful than the simple capabilities of analogy and metaphor to cope with their own mission and fulfil their function as interprets of both the world and themselves. References Daranyi, S. 1998. Personal communication. Levi-Strauss, C. 1958. Anthropologie structurale. Paris: Plon. - 1993. Personal communication. - 1994. 'Lettre a Monsieur le Professeur Solomon Marcus.' Maranda, E.K., and P. Maranda. 1971. Structural Models in Folklore and Transformational Essays. Paris, The Hague: Mouton. Marcus, S. 1997. The Logical and Semiotic Status of the Canonic Formula of Myth.' Semiotica 116 (2/4): 115-88. Mosko, M. 1991. The Canonic Formula of Myth and Nonmyth.' American Ethnologist 18 (1): 126-51. Petitot, J. 1988. 'Approche morphodynamique de la formule canonique du mythe.' L'homme 106-7: 24-50. - 1992. Physique du sens: de la theorie des singularities aux structures semionarratives. Paris: Editions du CNRS. Scubla, L. 1996. 'Histoire de la formule canonique du mythe et de ses modelisations.' Doctoral thesis, Paris, Ecole des hautes etudes en sciences sociales. - 1998a. Lire Levi-Strauss. Paris: Editions Odile Jacob. - 1998b. Personal communication.

ANTHROPOLOGICAL HORIZONS Editor: Michael Lambek, University of Toronto This series, begun in 1991, focuses on theoretically ethnographic works addressing issues of mind and body, knowledge and power, equality and inequality, the individual and the collective. Interdisciplinary in its perspective, the series makes a unique contribution in several other academic disciplines: women's studies, history, philosophy, psychology, political science, and sociology. Published to date:

1 The Varieties of Sensory Experience: A Sourcebook in the Anthropology of the Senses Edited by David Howes

2 Arctic Homeland: Kinship, Community, and Development in Northwest Greenland Mark Nuttall

3 Knowledge and Practice in Mayotte: Local Discourses of Islam, Sorcery, and Spirit Possession Michael Lambek

4 Deathly Waters and Hungry Mountains: Agrarian Ritual and Class Formation in an Andean Town Peter Gose

5 Paradise: Class, Commuters, and Ethnicity in Rural Ontario Stanley R. Barrett 6 The Cultural World in Beowulf John M. Hill 7 Making It Their Own: Severn Ojibwe Communicative Practices Lisa Philips Valentine 8 Merchants and Shopkeepers: A Historical Anthropology of an Irish Market Town, 1200-1991 Philip Gulliver and Marilyn Silverman

9 Tournaments of Value: Sociability and Hierarchy in a Yemeni Town Ann Meneley 10 Mal'uocchiu: Ambiguity, Evil Eye, and the Language of Distress Sam Migliore 11 Between History and Histories: The Making of Silences and Commemorations Edited by Gerald Sider and Gavin Smith 12 Eh Paesan! Being Italian in Toronto Nicholas DeMaria Harney

13 Theorizing the Americanist Tradition Edited by Lisa Philips Valentine and Regna Darnell 14 Colonial 'Reformation' of the Highlands of Central Sulawesi, Indonesia, 1892-1995 Albert Schrauwers 15 The Rock Where We Stand: An Ethnography of Women's Activism in Newfoundland Glynis George 16 Being Alive Well: Health and the Politics ofCree Well-Being Naomi Adelson 17 Irish Travellers: Racism and the Politics of Culture Jane Helleiner 18 Writing and Colonialism in Northern Ghana: The Encounter between the LoDagaa and the 'World on Paper/1892-1991 Sean Hawkins 19 An Irish Working Class: Explorations in Political Economy and Hegemony, 1800-1950 Marilyn Silverman 20 The Double Twist: From Ethnography to Morphodynamics Edited by Pierre Maranda