#### 2.1. Introduction to Structural Analysis

For a period, Claude Lévi-Strauss’ structural analysis had a powerful influence on both British and American anthropologists, offering a way out of functionalism and empiricism [

14,

15]. Although his reception in the Anglo-American world was always mixed [

16,

17], there was also support and admiration for Lévi-Strauss even late in life [

18]. More importantly, he had remained a source of fascination and inspiration for a younger generation of knowledge-based anthropologists. Over the years, the coming back of structural methodology as “the only way out of the postmodernist perplexing difficulties” is also heard from younger colleagues in international anthropological meetings. (According to Pierre Maranda in an essay discussing the impact of structural analysis presented at the University of British Columbia, quoted in [

19]).

Much understanding of the structural method emanated from Lévi-Strauss’ early manifesto-like pronouncements and the reinterpretations or misinterpretations offered by poststructuralists, notoriously Derrida and his imposturous critique of Lévi-Strauss on the place of writing [

20,

21]. Recently, several scholars have argued that structuralism, as it is popularly imagined, is a retrospective invention by poststructuralists, which has come to be substituted for the real thing [

18,

22,

23]. Such caricatured reconceptualizations of structure and structuralism cannot and should not obscure Lévi-Strauss’ more decisive notions of differential imbalance and transformation. Actually, some of Lévi-Strauss’ achievements could lay strong claim to having mapped within anthropology and the philosophical parameters of an increasing preoccupation with issues of contextualization and reflexivity in the face of the declining coherence of meta-narrative and grand theory, as well as with issues of political concern and engagement in the post-colonial era [

24]. We may be correct in asserting that Lévi-Strauss used structural arguments coherently and correctly to analyze the cultural order and its transient character by means of entropy and irreversibility, as well as, unsurprisingly, to examine deconstruction, or rather “dissolution” (in his own words), and self-reflexivity.

Lévi-Strauss’ contribution is also far-reaching and represents the most compelling challenge to the future of anthropology in the twenty-first century [

22,

25]. While establishing the theoretical and methodological foundations of a scientific revolution in anthropology, his ambition was to provide a new epistemology, involving a set of novel assumptions and procedures for the acquisition of knowledge, a new approach to methodology, and a new global awareness. For Lévi-Strauss, since human brains are themselves natural objects, and since they are substantially the same throughout the species homo sapiens, we must suppose that when cultural products are generated the process must impart to them certain universal (natural) characteristics of the brain itself [

17]. Thus, in investigating the elementary structures of cultural products and social institutions like myth or kinship, we are also making discoveries about the nature of humankind. Verbal categories found in myths and other narratives provide the mechanism through which universal structural characteristics of human brains are transformed into universal structural characteristics of human culture. In this way, category formation in human beings follows universal natural paths. It is not that it must always happen the same way everywhere, but that the human brain is constructed in such a way that it is predisposed to develop categories of a particular kind in a particular way.

The primary modes of thought in human mentality and the foundations of human social life conceal in themselves an infinite intellectual virtuosity by means of natural classifications that show logical possibilities unknown by classical logic. Herein lies the real source of that “science of the concrete”, which creates abstract classifications by means of contrasting similiarities and oppositions, not only to think and modify the concrete things, but also to construct mental structures that facilitate and aim at understanding the world. This speculative interest forms a total well-articulated system that remains independent of scientific logical systems, a concrete way of thinking that suggests that it is not our thinking that shapes the world, but that the world around us contains in itself a power to format our thinking.

Totemic beliefs and practices do not reflect the “primitive” ignorance of indigenous people or an important commodity for functionalist survival, but the concrete embodiment of the abstract means by which these people expressed fundamental ideas about their relations to each other and to their environment. As Lévi-Strauss stated, “the animals in totemism cease to be solely or principally creatures which are feared, admired or envied: their perceptible reality permits the embodiment of ideas and relations conceived by speculative thought on the basis of empirical observations” [

26]. We can understand, then, that natural species are chosen not because they are “good to eat”, as the functionalists held following Malinowski and Radcliffe-Brown who believed that the social lives of indigenous peoples were determined by basic needs like sex and hunger, but because they are “good to think” [

26]. This concrete way of thinking is a universal mode of thought, which Lévi-Strauss qualified as a “savage thought”. It is savage and undomesticated, just like the wild pansy that he put on the cover of The Savage Mind [

27], which established that human thinking and social life are not emotional, instinctive or mystical. In their unprecedented symbolic drive and their capacity for infinite extension, they are inherently intellectual, rational and logical. Lévi-Strauss witnessed these concrete ways of thinking and this intellectual quest for order in everything among indigenous people, from their face-painting to the layout of their camps, and mostly in their myths, which they pieced together with borrowed scraps of older ones, in the same way that a computer programmer might patch together different codes. He found out that indigenous people are every bit as scientifically minded as the ethnographers who investigate them. The only major difference, Lévi-Strauss claimed, is the “totalitarian ambition of the savage mind”, which operates on the assumption that if you could not explain everything, you had not explained anything. Arguably, this kind of totalitarian ambition of the savage mind becomes the ethnographer’s own ambition for the objective conditions of scientific validity in anthropology.

In Lévi-Strauss’ work, the properties of the structuration “frames” of myths open onto a second level of structuration of the transformations of myths, which implied the “mythical thought in a savage state” where the procedures of information processing of the brain had not yet been reformatted by the intellectual technologies of writing, computing, or scientific experimentation. Lévi-Strauss used these cognitive structures in his analyses of the convertibility and mediation of codes and axes, or in his analysis of the combinatorial permutations by inversion or symmetrization of mythical structures (armatures). His famous statement about the myths that “think each other (se pensent) in people’s minds without their being aware of the fact” [

8], is quoted at length by his critics and is often interpreted obtusely as being literal. For some, “Lévi-Strauss’ formulas are superb and provocative, but they, literally, make no sense. Myths are thought expressed in different languages, but they are not thinking subjects. They can neither operate (penser) through men nor reflect (penser) upon themselves. A physicist may as well say that the waves and particles that make up light are operating (pensent) in his/her mind and reflecting (se pensent) upon themselves there.” [

28].

Acute and subtle thinkers among Lévi-Strauss’ critics must know that the algorithmic formality of myths demonstrated by Lévi-Strauss is actually something existing independently of individual human minds. In addition, a simple algorithm can produce complex autonomous systems that can reproduce human conditions, and one can read Lévi-Strauss’ Mythologiques as a sort of Turing test of the corpus of Amerindian mythology [

29]. Critics might also be aware that in quantum mechanics, waves and particles operate with a “spooky action at a distance”, as contemporary physicists put it (after Einstein [

30]), and they parallel mythical structures in Lévi-Strauss’ sense, by actually reflecting upon each other, not in the anthropologist’s or physicist’s mind, but like myths, in their own quantum entanglement [

31,

32]. Far from a pseudo-mathematical mystification, as many Anglo-American anthropologists learnt it [

33,

34,

35], Lévi-Strauss’ structural method originated in mathematics and in applied group theory, dealing primarily with the algebraic models of kinship structures [

36,

37,

38,

39]. Afterwards, it has gone on to be well received by modern scholars seeking to study culture and society by formal means [

40,

41,

42,

43,

44]. The majority of commentators, either admirers or critics, have retained from the structural analysis of myth only its capacity to disclose stable, common, and probably universal frameworks. Fundamentally, however, Lévi-Strauss preferred to look for rules that would ideally make it possible to generate, starting from an unspecified myth of reference, the finite or infinite whole of all other real or possible myths. In the structural study of myths, Lévi-Strauss demonstrated the transformational morphodynamics of mythical networks. One of the more powerful of Lévi-Strauss’ ideas is his description of the generative engine of myths on the basis of the set of their own transformations.

In [

1], Claude Lévi-Strauss stated: “As a myth is spread and retold in different contexts, new variations occur”.

While the basic mythical structures (armatures) remain unchanged, characters may change, roles may be inverted based on symmetry, elements of the myth may be lost or inserted, oppositions may become weaker, etc. In short, a number of what Lévi-Strauss calls transformations of the myth will occur. Since they are all related to one another, they form a group of transformations, where each variant is a symmetric transformation of the others and none of them has any preeminence in logic, analysis, or history over the others.

The description of these transformations relies on the concepts of terms and functions. When analyzing myths, Lévi-Strauss often talks of terms that are qualified with different functions. Terms can be persons (in the forms of humans, animals, divinities etc.), or things which have the ability to take up roles. Functions are the different roles carried by these terms. In myths, we encounter a number of characters (terms), all of which have a great number of possible roles (functions).

A myth, understood as a group of transformations occurring from one to another of its variants, is further decomposed into a set of basic elements called mythems, which are also characterized by the notions of terms and functions. Lévi-Strauss distinguished a set of basic operations of homology, inversion, opposition and symmetrization, between a number of characters or terms of myths and their large number of possible roles or functions. In mythical thought, these basic operations account for the convertibility, the mediation and the combinatorial permutations of codes and axes between terms and functions. They can be controlled by means of a special relationship that Lévi-Strauss formulated in a canonical way through various mathematical expressions, which demonstrate how the transformations of the myths can be captured and how new myths are generated from any specified myth of reference. Lévi-Strauss’ concept of canonical formulation that articulates the transformational dynamics of mythical networks transcends a simple analogical relation to a quadratic, as shown in Equation (

1),

This equation articulates a dynamic homology between meaningful elements and their propositional functions. This formulation made it possible for Lévi-Strauss to detect a sort of genuine logical machine generative of open-ended meaning within specified mythical networks. In a quadratic equation of this kind, the generative virtues of the so-called “double twist” of the canonical transformation in the structural study of myth imply two conditions internal to canonical formalization. According to Lévi-Strauss, a formulation of this type reflects a group of transformations in which it is assumed that a relation of equivalence exists between two situations defined respectively by an inversion of terms and relations, provided that (i) one of the terms is replaced by its opposite and that (ii) a correlative inversion is made between the function value and the term value of two elements [

1]. We may remark that Claude Lévi-Strauss used three or more mathematical expressions of the same quadratic Equation (

1) [

8,

9,

10,

11,

45,

46].

After the method for the structural study of myth was introduced in 1955, the generative virtues of the so-called “double twist” of canonical transformation have remained not understood for a long time. Lévi-Strauss almost never mentioned explicitly his formulation of transformational dynamics, even though this was implicit in the massive work of his Mythologiques series published between 1964 and 1971 [

8,

9,

10,

11]. The morphodynamic principles of canonical transformations were explicitly operationalized only in his more recent inquiries published between 1985 and 1991 [

45,

46]. This does not mean, however, that Lévi-Strauss did not understand from the start his own theories and that only advanced mathematicians are up to that task. Instead, we might wonder where that might bring anthropologists, were we to realize that we had to wait, starting from 1972 [

47,

48], for the knowledge progress in qualitative mathematics become sufficiently advanced for us to understand Lévi-Strauss’ theories. Indeed, they were made comprehensible afterwards as a complex variety of analogies, a torus, a Moebius strip, a Klein group, or aptly as an anticipated formalization of catastrophe models in new mathematics and morphodynamics [

49,

50,

51,

52,

53,

54,

55,

56,

57,

58]. What is more important, for a catastrophist operation of this kind to take place, is that the very idea of canonical relation requires a third operating condition as a boundary condition external to canonical formalization. In all cases, boundary condition refers to the empirical evidence from outside the realm of the myths being analyzed, which Lévi-Strauss carefully identified in each case as the necessity of the crossing of a spatiotemporal boundary, defined in territorial, ecological, linguistic, cultural, social, or whatever other terms. Boundary condition is also a formal mathematical concept, required to be satisfied at the boundary of a topological domain in which a set of differential equations is to be solved. A boundary condition of this kind is claimed by Lévi-Strauss to be important in determining the mathematical solutions to various mythical problems. In his further canonical formalisation, boundary condition is used to account for the morphogenetic and morphodynamic transformation of myths across the boundaries existing between one people and another. Namely, a series of variations inherent in the myths of a given people cannot be fully understood without going through myths belonging to another people, which are in a relation of inverse transformation with the formers.

#### 2.2. Understanding Claude Lévi-Strauss Method of Myth Analysis

Following Claude Lévi-Strauss’ methodology, we specify a representation for a myth in terms of the basic elements of the narrative, the mythems involving a term and a function. We can give an example corresponding to an excerpt of a basic folktale of Corsican oral literature. The beginning of the story can be described through the following mythems:

The ogre (named Orcu) knows the secret of the fabrication of Corsican cheese using milk.

The shepherds are jealous of the Orcu because of this secret.

The ogre is captured because of an ingenious trap.

The shepherds ask for the secret of the fabrication of the cheese.

We give below for each mythem the value of the variables a and x:

1st mythem: $a=orcu$; $x=secret$

2nd mythem: $a=shepherd$; $x=jealous$

3rd mythem: $a=orcu$; $x=trapped$

4th mythem: $a=shepherd$; $x=secret$

The structural analysis defined by Claude Lévi-Strauss involved in structural analysis consists in showing how a set of mythical transformations are generated from an initial reference myth. We can list several types of transformation of a given myth that are encountered in his books: (i) four basic kinds of transformations (homology, inversion, opposition, symmetry) (ii) the canonical formula; (iii) two additional transformations described as addition and suppression of a mythem. Each type of these transformations can be understood as follows:

A homology between two terms a and b. The result of a homology between a and b consists in replacing term a by term b in each mythem in order to obtain a new generated mythical transformation. For example, considering the four mythems previously introduced, we are able to generate mythems of a new myth using a homology between the ogre and the Sybille. The result consists in replacing the term ogre by the term Sybille in all the mythems containing the ogre term: first and third mythems.

Inversion. The result of an inversion of a term a consists in replacing the term a by term $1/a$ in each mythem of the given myth in order to obtain a new generated mythical transformation. For example, in the Corsican culture, the devil is often considered as having an inverse character to the ogre. So, we are able to generate a mythem of a new myth using an inversion of the terms ogre and devil. The result consists of replacing the term ogre by the term devil in all the mythems containing the ogre term: first and third mythems.

Opposition. The result of an opposition of a term a consists in replacing term a by term ${a}^{-1}$ in each mythem of the given myth in order to obtain a new generated mythical transformation. For example, we can point to usual oppositions introduced by Claude Lévi-Strauss in the Mythologiques Series: fire = $wate{r}^{-1}$, Jaguar = $anteate{r}^{-1}$, vegetable = $anima{l}^{-1}$.

Symmetry. The result of a symmetry of a term consists in replacing term a by term $-a$ in each mythem in order to obtain a new generated mythical transformation. For example, in some myths studied by Claude Lévi-Strauss, a menstruating woman is sterile. She becomes a non-woman (symmetry operation), but without being a man (opposition operation).

The canonical formula transformation (also double twist transformation) leans on the algebraic expression given in Equation (

1).

Equation (

1) can be understood as follows:

a /

b is a qualitative opposition of terms,

x /

y a qualitative opposition of functions, where

${F}_{f}\left(t\right)$ means that the term t has the function f and where

${F}_{a-1}\left(y\right)$ means: (i) that there has been an inversion of the value of term

a into an inverse value

${a}^{-1}$ and (ii) that there has been an exchange between a term value and a function value. These two kinds of exchanges have been called the double twist.

In order for the reader to better grasp the meaning of such a formula, we can take an example from The Jealous Potter [

45]. In the canonical formula, if one replaces term

a by term

n (nightjar) and term

b by term

w (woman),

x by

j (jealousy) and

y by

p (potter), then this gives:

${F}_{j}\left(n\right):{F}_{p}\left(w\right):{F}_{j}\left(w\right):{F}_{n-1}\left(p\right)$. This means: the

$jealousy$ function of the

$nightjar$ is to the

$potter$ function of the woman, what the

$jealousy$ function of the woman is to the inverted

$nightjar$ function of the potter. This last relationship (the inverted

$nightjar$ function of the potter) is equivalent to the function

$ovenbird$ of the potter (because

$nightja{r}^{-1}$ =

$ovenbird$). Claude Lévi-Strauss highlights that the ovenbird is a specialist in terms of pottery. This relationship introduces a new mythem: “the potter is good as an ovenbird for pottery”. Finally, Claude Lévi-Strauss points out that this new mythem belongs to a set of other myths of South America, so that he demonstrates that new myths can be obtained using the canonical formula. The generative engine of myths is perfectly illustrated through this example.

Adding a mythem: this is a very simple transformation which consists in adding a mythem to a given myth. The new mythem is built with a term a and a function x.

Removing a mythem: once again a very simple transformation which consists in removing a given mythem from a given myth.

#### 2.3. Applying Claude Lévi-Strauss Method of Myth Analysis

The transformational analysis of myths according to the Claude Lévi-Strauss’ structural method is complex enough to be a nightmare when being performed by an anthropologist. Anthropologists are confused because they cannot understand it without the help of mathematics and computer science. For that reason, they are unable or reluctant to apply it to other domains, even though the validity of Lévi-Strauss’ theory and method is beyond discussion, as it is by now confirmed by mathematical validation [

49,

50,

51,

52,

53,

54,

55,

56,

57,

58]. This sub-section presents the specifications of the tasks that have to be realized using a software approach in order to provide an additional experimental validation of Lévi-Strauss’ theory and method by using an M+S approach based on discrete-event system specification dedicated to show how a set of folk narratives are generated based on symmetry and double twist transformations.

Existing works have already explored the links between computer science and structural anthropology, focused on the structural study of narratives [

59,

60], or on myth transformation modeling [

61,

62,

63], including attempts to formalize the analysis of myths and folktales developed by Claude Lévi-Strauss [

1] and Vladimir Propp [

59]. Another logical-mathematical methodology, based on Lévi-Strauss’ claim of an algebraic structure to human mind and cognition derived from his structural analysis of myths, is used to map sub-literal meanings of narrative and discourse analysis. The new method is validated by findings of research on a set of systemic numeric references found in oral narratives and forming mathematically constructive algebraic groups, such as the specifically commutative Abelian semi-group with identity [

64,

65,

66,

67]. Some approaches are also based on computer sciences in order to perform a systematic narrative transformations software [

68,

69]. Despite their good results in terms of narrative generation software, even these interesting attempts have not succeeded in this task, as they do not include myths analysis. In order to help an anthropologist to perform structural analysis, we propose a software approach which may guide them in the different steps of the research method: (i) selection of a “reference” myth (ii) definition of the transformations in order to generate a set of myth (iii) generation of the set of new myths according to the previously defined transformations (iv) overview of the different transformations based on a graph visualization. The software approach should be able to realize the following tasks:

To allow a user to define an initial myth of reference, defined the Myth M1 by Claude Lévi-Strauss.

To allow the modeling of a given myth through a set of mythems. The software will allow a user to define a myth as an interconnection of mythems which is represented using two variables: a term usually noted a and a function usually noted x.

To allow a user to generate a new myth from a given myth by performing a transformation selected through the following set of basic operations which have been presented and detailed previously (homology, inversion, symmetry, opposition, addition of a mythem, removal of a mythem) as well as the Canonical formula.

To allow a user to visualize the set of transformations already performed using a graph representation. The graph should present the transformation relations between the myths. The nodes of the graph will represent the set of myths labeled by their associated number.

In order to implement these tasks, we propose to develop a set of models based on the formalism of discrete event systems specification (DEVS), introduced by Zeigler [

2] in the late 1970s. It provides a way to specify any system as a mathematical object, which enables the modeling and computer simulation of complex discrete event systems by means of a formal method mathematically demonstrated [

2]. It is based on a formal representation of discrete-event systems and allows an explicit separation from the M+S part. This means that one can define a model of the behavior of a given system without having to take into account the simulation phase. This explicit separation between the modeling phase and the simulation phase [

4] allows DEVS formalism to be one of the best ways to perform efficient simulations of complex systems. Highly pertinent to our concerns at this time is the fact that we have recently extended DEVS formalism in its PythonDEVS kernel to be used within a DEVSimPy environment [

12] and software mechanism, which could now be used for the analysis of myths following the morphodynamics of Lévi-Strauss’ structural analysis of mythical thought. In practical terms, the software mechanism is based on the modeling of myths as discrete-event systems by means of a DEVS simulation that allows the generation of myths from an original myth based on structural analysis as defined by Lévi-Strauss [

6]. Such a recent elaboration of computer M+S concepts [

70] is further used to deal with dynamic variable structures applied in several approaches from social sciences and the humanities to object-oriented models for the analysis of narrative transformations [

7,

71].

The next section will present the DEVS formalism and the simulation software framework (the Python Based DEVSimPy Framework) that has been used in order to implement the DEVS models to perform the generation of a set of myths from a reference myth based on mainly symmetry and double twist transformations.