Chapter 9

General Presentation of the IEML Semantic Sphere

This chapter is devoted to the general properties of the semantic sphere, which serves as a system of coordinates for the IEML model of the mind. As shown in Figure 9.1, the IEML semantic sphere is essentially made up of three concentric interdependent “layers”.

At the nucleus, an automaton, the semantic machine, generates, transforms and measures the giant hypercomplex graph of the semantic sphere.

In the layer that envelops the machine, the nodes and links of the semantic sphere become the texts (USLs) translated into natural languages from a bridge metalanguage. Each distinct USL encodes a distinct concept, and the connections between USLs indicate their semantic relationships.

In the outer layer, the semantic sphere ensures the interoperability of a global hermeneutic memory. This memory consists of a world of collective interpretation games from which an ecosystem of ideas emerges.

The general properties of the semantic sphere are related to these three layers: it is a calculable (machine) network of concepts (metalanguage) that can be used as an addressing system for ideas (hermeneutic memory). This is why, before describing the semantic sphere further, I will discuss the ideas and concepts in the IEML model of the mind. Once I have done this, I will examine the properties of the semantic sphere: unity, calculability, symmetry, internal coherence and inexhaustible complexity.

9.1. Ideas

9.1.1. Internal structure

Symbolic cognition is in an interdependent relationship with physical reality in all kinds of ways, since mind and matter are complementary spheres — actual and virtual — of the same communication nature. This point has been amply covered in Part One of this book and reviewed at the beginning of Part Two. Here I am concerned with the content specific to the mind. Let us consider symbolic cognition and ask ourselves what it contains. It is clear that it does not contain material objects or waves of energy, since all that belongs to physical nature. In the IEML model, the fundamental content of the mind consists of ideas. As illustrated in Figure 9.2, an idea is necessarily made up of a concept S (a general category), a percept T (a complex sensory-motor image) and an affect B (an energy of attraction or repulsion). Let us now examine these three components.

9.1.1.1. Percepts

As our daily experience shows, the mind contains images. These images may be visual, auditory, olfactory, tactile, gustatory or proprioceptive (felt by the body). Most often, sensory images combine these different types of sensory data. To avoid confusion with purely visual images, these multimodal sensory patterns are called percepts here. Percepts are associated with the “thing” (T) pole of ideas.

In the IEML model, percepts are represented by URLs, i.e. multimedia and multimodal data of all kinds found on the Web. From the point of view of the requirement of calculability of our model, the production and automated transformation of sensory or “multimedia” images present no basic problems. Methods for the automated synthesis of images, sounds and haptic data (measurement of pressure, force feedback) have been available for decades. These methods have been implemented on a large scale in scientific research and the design, illustration, music, entertainment and gaming industries. Countless Internet users (individuals, groups and institutions) produce, transform and connect the multimedia data that flood the digital medium. In addition, the means of physically addressing these data is universal and well established: URLs. While URLs are semantically opaque¹, their relationships are processed well using statistical methods (as Google does) or logical methods inherited from traditional artificial intelligence (as the web of data does).

9.1.1.2. Affects

In addition to sensory-motor images, the mind also contains emotions or affects. These affects distribute their polarities (positive, negative or neutral) and intensities to concepts and percepts. Emotions may be conscious or unconscious, simple and raw or subtle and nuanced.

With respect to the modeling of affects or the symbolic energy of ideas, I played with specific functions of distribution and calculation of the intensity of a current in networks of ideas for a long time before deciding that a general framework of modeling did not need to establish these functions. It was sufficient that the affective dimension or the “value” of ideas be represented by a semantic current describable with two numbers. One of the two numbers represents (a) the intensity of the current and the other number represents (b) its quality or its polarity on a negative–positive scale. From there, all kinds of functions can be imagined, tested and adjusted to specific goals. Since the numbers are obviously calculable, the functional modeling of affects in the IEML semantic sphere is ensured.

9.1.1.3. Concepts

After percepts and affects, the mind contains concepts, since it uses language and explicitly manipulates abstract classes or categories.

The capacity exists to provide a functional, calculable description of affects and percepts in the digital medium. The biggest modeling problem I had to solve was that of the production and functional transformation of concepts. The solution to this problem has been outlined in section 7.4.5. I will return to it in more detail in Chapter 11, after discussing the specifically linguistic dimension of IEML².

9.1.1.4. Internal unity

Concepts, percepts and affects do not emerge in the mind separately from each other, but are brought together in the unity of an idea. An idea combines a concept, a percept and an affect. The simplicity and clarity of this proposition should not obscure the fact that each of the three components of an idea can be vague, dynamic (evolving), more or less conscious and, above all, complex (i.e. it can envelop a multiplicity). I should note in passing that what in everyday vocabulary is called an emotion corresponds in the IEML model to an idea. In fact, emotion in common language actually includes not only an intensity and affective polarity, but also a semantic categorization (a concept) that can be very complex, as well as physical sensations and images (a percept).

The three aspects of an idea can only be distinguished logically. In the reality of the idea, none of its components can exist without the others. The concept may be seen as the “knowledge” of the idea, a knowledge that categorizes it and situates it in a network of other categories. The affect may be considered the dynamic “will” of the idea, the force and direction (attraction or repulsion) of its tropism. Finally, the percept may be associated with the “power” of the idea. Without the sensory medium of percepts, affects and concepts would vanish.

No concept arises in the mind without emotional and sensory dimensions. Similarly, affects do not arise without sensory-motor images or concepts (as unclear as they may be). Finally, although the percept is a product of sensory-motor functions, it would have no meaning — and would even be “imperceptible” by the mind — without categorization or emotional charge. To use a physical metaphor, physical objects in movement necessarily have a spatial location, speed and mass, which can only be distinguished from each other logically. Similarly, ideas can be broken down into concept, affect and percept, although their reality and their efficacy necessarily imply the concrete interdependence of these three variables.

9.1.2. Production of ideas

As we have seen in the preceding chapters³, the ideas we are concerned with here are not fixed and eternal like those of Plato, but they interact with each other through the cognitive systems that manipulate them, and evolve within ecosystems of ideas. These ideas are all the less fixed and eternal because they are indissociable from the hermeneutic functions of perception and thought that produce them.

The hermeneutic functions have been described in section 7.4.7.1. I will review them here as a reminder. As Figure 7.5 shows, ideas are made up of concepts (S: sign), affects (B: being) and percepts (T: thing) and are produced by two types of functions:

– functions of perception (A: actual) receive inputs of URLs (standing for percepts) and categorize and evaluate them to produce outputs of phenomenal ideas; and

– functions of thought (U: virtual) receive inputs of ideas and produce outputs of noumenal ideas.

I want to recall that in the IEML model, concepts are represented by USLs, percepts are represented by URLs and affects are represented by a semantic current.

9.1.3. Networks of ideas

Ideas themselves are not separate, but arise in the mind interconnected in networks of relationships. The functions of thought assemble ideas in circuits capable of conducting the affective energy modeled by the semantic current. I am making no particular hypotheses on the order, causes, reasons or nature of the relationships among ideas. They can be connected through their concepts, affects or percepts, or through any complex combination of these three types of variables.

9.2. Concepts

Concepts can be broken down into symbols, networks and categories. It is clear, to begin with, that the mind can only manipulate concepts through symbols or signifiers. These symbols, moreover, represent abstract classes or categories. A symbol can only represent a category because a symbolic system or a language situates these categories in paradigmatic and syntagmatic networks.

9.2.1. A concept reflects a category in a symbol

The very existence of concepts, i.e. of explicit general categories represented by signifiers, ensures the unique nature of human symbolic cognition. Indeed it is likely that other animals have ideas, since they obviously know percepts and they categorize and evaluate their perceptions. The difference between a human idea and an animal idea is that for the human idea, the category (thing: T) is reflected in a symbol (sign: S) through a symbolic system that situates this category in a semantic network (being: B). Concepts can be considered the key to specifically human thought. As it can be considered explicitly by the mind, the self-reflecting category of the concept makes the manipulation, organization and filtering of ideas possible, not only on the basis of their affects or their percepts, but also their concepts.

9.2.2. A concept interconnects concepts

A concept never exists in isolation, not only because it is always integrated with an idea, but also because it is always situated in a semantic network of other concepts. On the paradigmatic axis, that of langue, or language, the concept is in complex relationships of inclusion, opposition, complementarity or genealogy with other concepts. For example, white is a color, it is the opposite of black and it has the same root as whiten. The concept of white contains this network of relationships. On the syntagmatic axis, that of parole, or speech, the concept is in grammatical relationships with other concepts. For example, white plays the role of modifier of the subject in the sentence “A white cloud floats in the sky”. It thus becomes an integral part of the syntagmatic network of the sentence. If the concept is itself represented by a complex expression, it contains the syntagmatic network of that expression. The concept thus always includes a network — or a position in a network — of semantic relationships (syntagmatic and paradigmatic).

9.2.3. The IEML model of the concept

As shown in Figure 9.5, in the IEML model of the concept (S) its symbol is formalized by a USL, i.e. a text in the formal IEML writing system; its network (B) is formalized by a subset of the big network of the semantic sphere; and the category it represents (T) is expressed in natural languages. The category can only be reflected in the symbol through a semantic network. The symbolic system represents the virtual dimension (U) of conceptualization, while its expression (A) represents its actual dimension. The advantage of the IEML model is that the network (both that of the semantic sphere and that of the USL) is automatically calculable based on the symbol, and the meaning of the category in natural languages is automatically calculable based on the network.

9.2.4. Addressing of ideas by concepts

9.2.4.1. On the relationship of ideas and concepts

The human experience of a concept is always integrated with the experience of an idea. This means that concepts never arise in the mind separately or independently of affects and percepts.

In the material cosmos described by physical science, an object must be located somewhere in space-time (whether deterministically or probabilistically) and must have some mass or energy to exist. Similarly, to exist in the mind as described by the IEML model, a multimedia datum or a percept must occupy (probabilistically or deterministically) a semantic circuit, i.e. a subset of the semantic sphere, with some affective force.

It should be understood that a mere address in the semantic sphere — a semantic circuit — does not necessarily imply the existence of an idea. In addition to the concept that formalizes this circuit, the full existence of the idea also requires some emotional charge (formalized by a current) and sensory images or data (formalized by a URL). Some cognitive system has to have invested certain sites of the semantic sphere with percepts and affective forces for ideas to begin to develop and circulate meaning there. This is why the graph of concepts formalized by the semantic sphere is not identical to the world of ideas: rather, it represents its abstract container, or its system of coordinates, in the form of a topological grid that makes it possible to process the location of ideas in a functional and calculable way. In short, what exists in the mind is ideas. Concepts precede existence; they are virtualities of existence: places, sites or addresses in the form of circuits capable of accommodating flows of semantic current and categorizing data⁴.

9.2.4.2. Why is it the concept that addresses the idea?

Why, in the IEML model, is it the semantic sphere that provides the nature of the mind with its basic system of coordinates? This question can be broken down into two sub-questions. First, why must the address of an idea be part of that idea and, second, why must that part be its concept and not its affect or its percept?

First, the “place” of an idea in the mind cannot be something outside the idea itself. This means that the cognitive operations that produce ideas naturally and automatically include an operation of addressing the ideas produced. It is the hermeneutic functions themselves that determine the place of the idea. In other words, the identity of the idea must imply its location, which immediately eliminates any form of container or addressing that is extrinsic, arbitrary or “material”.

Second, ideas have only three aspects: their concept, their affect and their percept. The choice of concepts as the basic addresses of ideas in the mind can then be justified by a process of reasoning by elimination:

– Rather than its meaning, the affect represents the energy of the idea. This energy can be formalized by two quantities: the intensity and the polarity (positive or negative) of the affect. However, the quantities are in general too limited to provide the basis for the semantic addressing of ideas.

– Nor can the system of coordinates of the mind be based on percepts, since they represent the sensory phenomena implied in symbolic cognition, and not the specifically semantic dimension of ideas. The URL that formalizes the percept provides only a physical address. Before being categorized by concepts (USLs), percepts (URLs) are semantically opaque. One of the privileges of the human mind is its freedom to creatively classify sensory data and space-time addresses. My scientific undertaking aims to extend this privilege rather than to abolish it.

– Since we cannot base the semantic addressing of ideas either on affects (quantitative) or on percepts or sensory images (opaque), all that remains is to organize the system of coordinates of the mind around the third dimension of ideas: their concepts (qualitative). Concepts (formalized by USLs) will therefore be the semantic addresses of ideas. I would like to add that, in the IEML model, each concept symbolized by a USL automatically implies a local network that is exactly situated in the global network of the semantic sphere. All local networks corresponding to USLs act as the variables of a transformation group, so the semantic sphere is the ideal candidate for the semantic addressing of ideas. Finally, the network corresponding to the USL can be used not only to address the idea but also to channel its current (the affect).

9.2.4.3. The nature of semantic addressing

Once again, when I say that an idea is “in” the mind, I am not talking about location as position in a tri- or quadri- or n-dimensional system of geometric coordinates. I mean that the concept of the idea is explicit or codified and that the code of the concept can be precisely designated as a variable of a symmetric transformation group. The IEML system of coordinates of the mind — the semantic sphere — is certainly “mathematical”, but it meets conditions other than those of Euclidean geometry, its variants or its extensions. The concept is formalized as a semantic circuit in interconnection with a huge variety of other circuits within a system of coordinates woven by the semantic sphere. In the IEML model, the universal container of the mind (its fundamental place) is thus a graph made up of all possible semantic addresses of ideas, including the hypercomplex net of semantic relationships interconnecting these addresses. By formalizing the universe of concepts, the IEML semantic sphere makes it possible to precisely locate ideas and describe their relationships and the transformations of these relationships by means of calculable functions.

9.3. Unity and calculability

Having described ideas, concepts and their relationships, I can now introduce the main topic of this chapter: the general properties of the IEML semantic sphere, which is to the mind what geometric space is to matter.

9.3.1. Functional calculability

At the highest level of abstraction, the science of matter is based essentially on a calculable, or functional, codification of the interconnections among sensory phenomena. In the same way, a science of the mind must be based on a model that makes the calculability of the interconnections among processes of symbolic cognition possible. That is why the IEML model imposes the same requirements on knowledge of the mind as the contemporary scientific community imposes on knowledge of physical nature.

To begin with, the IEML model makes it possible to describe processes that occur in the mind using calculable mathematical functions. I adopt as my own the fundamental hypothesis of the contemporary cognitive sciences that cognitive processes must be able to be modeled by explicit mechanisms⁵. Far from being ineffable entities that are impossible to process rigorously, meaning, thought and the mind belong to nature, and it must therefore be possible to describe them scientifically using calculable methods. The requirement of calculability is selfevident. It is also justified by reasons of practicality, because it allows us to exploit the calculating power of the digital medium. There is no need to comment at length here on what has been an epistemological requirement of scientific knowledge at least since Galileo.

9.3.2. The unity of the mind

The originality of my approach is not based on the requirement of calculability and full explication, since this requirement is shared by the majority of researchers in the field. The uniqueness of the theoretical framework I am proposing is due to the fact that it allows us to grasp human symbolic cognition as the coherent unity of a nature. We know that, according to contemporary physics, bodies and all physical processes are interconnected (actually) in the same material nature. Likewise, according to the IEML model, all processes of symbolic cognition are interconnected (virtually) in the same nature of the mind.

Why require that the calculable functions that describe human symbolic cognition be coordinated within a common, coherent nature? The first reason is a practical one, because this type of modeling obviously favors semantic interoperability for collaborative knowledge management. But this requirement is also justified for a fundamental theoretical reason: the human symbolic faculty is unique. As the term indicates, human symbolic cognition cannot explicitly manipulate abstract classes without using some kind of language or system of symbols. In factual reality, there is no discursive thought, whether conscious or unconscious, that is not based on symbolic systems. It is, however, necessary conceptually to distinguish between the innate universal symbolic faculty⁶ that is unique to humanity — the capacity to explicitly manipulate classes, general categories and abstract intellectual essences — and the expression of this faculty in specific times and places. This expression is actualized through many symbolic systems that are by definition conventional or cultural.

I would now like to consider not the many conventional symbolic systems, but the universal symbolic faculty. Since there is a symbolic faculty that is common to our species, we can assume the existence of an objective counterpart to that faculty: the set of variables that faculty produces and transforms, i.e. a universe of selfreflexive concepts capable of categorizing sensory data and emotions. I believe that we will be able to observe and study the collective human intelligence that is expressed in digital data only if we possess a mathematical formalization of this universe of concepts, including their relationships and their transformations⁷. This is precisely why I constructed the IEML semantic sphere.

9.3.3. Requirements of calculability for a system of semantic coordinates

In the previous section, I made a distinction between ideas (living thoughts full of emotions and sensory images) and concepts (the semantic addresses of ideas). We saw that the IEML semantic sphere — the system of coordinates of the mind — formalizes the universe of concepts as a giant semantic graph. One of the major technical problems I faced for years was how to produce the semantic sphere automatically – because there was obviously no question of building it by hand. If the semantic sphere was to serve as a system of coordinates for addressing the nature of symbolic cognition, it had to be huge, if not infinite. Hence the need for a machine to trace, navigate and measure it. In addition to this, one of the goals of my project was to make maximum use of the ubiquitous calculating power now available in the digital medium. One of the reasons the semantic sphere was not constructed by previous generations is the lack of automated calculating power before the 21st century. To construct this sphere, I therefore designed an abstract machine capable of fully exploiting the computational resources of the digital medium. What made it particularly difficult to design this machine for weaving the semantic sphere was the fact that the nodes and links of this net for surveying the nature of the mind were not like the points and lines of the geometry of ordinary three-dimensional space, which differ only in their positions. The vertices and edges of this gigantic semantic graph had to be qualitatively unique, distinct texts expressing distinct categories, connected by distinct relationships of meaning. I had to construct a machine that was abstract — but that could be implemented by computer programs — and that was capable of weaving the huge fractaloid hypertext network of a system of semantic coordinates and translating this network of texts into natural languages without sacrificing the deducibility, precision and algorithmic manipulability of algebra. The way I proposed to use it (to model and observe cognitive ecosystems), meant that this mathematical-linguistic “container” of the nature of the mind had to meet many requirements with regard to automation. Its circuits and the paths in its circuits had to be capable of being drawn and transformed automatically. The identification of variations by invariances, i.e. symmetries and dissymmetries on its basic grid, also had to be automated. Finally, the semantic distances and analogies between circuits also had to be able to be measured using mechanisms (i.e. programs).

I gradually discovered that the interdependent requirements of calculability I have listed converged toward the concept of a transformation group⁸. If the USLs (symbolizing concepts) and their graphs of relationships were constructed as the variables of a system of symmetric transformations, their algebraic processing and the automation of this processing became not only possible but fully satisfactory in scientific terms. Symmetry, in fact, precluded giving undue privileges to any variable: the variables of a symmetrical system are distinct but “equal” in relation to the operations that mutually transform them. That is why I arrived at a solution in which both the USLs, which are valid IEML expressions or “texts”, and the circuits of the semantic sphere, whose vertices and edges are labeled by USLs translated into natural languages, are variables of symmetric transformation groups. In addition, these two symmetric transformation systems, (i) the regular IEML metalanguage and (ii) the semantic sphere, are themselves in a symmetrical relationship of reciprocal transformation (see Figure 11.3). As it is deeply integrated into the basic design of the semantic sphere and the machine for tracing and surveying it, I would now like to explain the importance of this concept of the symmetric transformation group.

9.4. Symmetry

The system of conceptual addressing of ideas, i.e. the fundamental system of coordinates that is the IEML semantic sphere, is symmetrical. Just as the space containing matter is symmetrical (it is a transformation group: for example, the movements of rotation and translation are reversible), the semantic sphere containing ideas has to process all the semantic addresses formalizing concepts as interchangeable variables that can be transformed into each other symmetrically. In other words, no semantic address, no conceptual “point of view” may be favored over another; changes of address must be reversible; and these changes (formalized by algebraic operations) must be rationally constituted. This requirement responds to the general intuition that nature (whether the nature of the mind or material nature) potentially places individuals belonging to the same layer of complexity in relationships of symmetrical interconnection.

9.4.1. Unity and symmetry

In the course of my work, I realized that there was a profound relationship between the calculability of cognitive processes, which was one of the basic hypotheses of my research⁹, and another of my basic hypotheses, that of the unity of the mind.

Let us think this through using an analogy with material nature. How is the unity of physical nature, which is one of the great discoveries of modern science¹⁰, represented mathematically? My answer is as follows: since the Newtonian revolution (for which Copernicus, Kepler, Galileo and Descartes paved the way), everything contained in material nature has been situated or addressed, by the sciences that study it, in a single space-time continuum. This fundamental container is formalized as a system of geometric coordinates or, in contemporary physics, a system of symmetrical relationships between systems of coordinates. One of the main properties of geometric space is that it is a transformation group. Rotations, translations and mirror symmetries can be carried out on geometric figures, and these operations can be combined and reversed at will. The very structure of space is generated by these reversible and recombinable operations. It is the symmetrical properties of geometric space (the fact that it is a transformation group) that make it a scientific system of coordinates. If the system of space-time coordinates was not a transformation group, it would not be possible, using calculable functions, to coherently describe the trajectories of material objects, or any kind of local or temporal transformation. The transformation group ensures the “rationality” of the changes described within it. All material nature is contained in the same geometric space, whatever the number of dimensions — depending on the model — of this space. The unity of nature does not come from its being contained in the same space, as if by a bag holding a chaotic, heterogeneous multitude of things. Rather, the unity ensured by the system of geometric coordinates comes from the internal symmetry of the operations that can be carried out on its addresses. It is therefore an intrinsic unity, inherent in nature as represented by our scientific models. The system of geometric coordinates simultaneously establishes the unity and calculability of the nature it makes it possible to model. It is an abstract machine, a coherent, symmetrical structure of relationships among variables through a few recombinable operations. In a sense, geometric space is generated by this abstract machine.

There are transformation groups other than those of geometry. The operations of addition and multiplication form a transformation group on rational numbers. Another example: the operations of intersection and symmetric difference form a transformation group on the subsets of a set. In the three examples given (geometric space, numbers and subsets), there are relationships of symmetry among the variables and among the operations, and the operations can be combined, recombined and reversed at will. However, their relationships and identities are given as soon as the algebraic structure that generates them is defined. Geometric space, rational numbers and subsets of a set respectively form intrinsic units, because the operations of a group create symmetrical relationships of reciprocal generation or transformation among their variables. Group structures are so fundamental in mathematics that, despite their abstractness, they are taught in secondary school¹¹.

To return to my problem: how can we scientifically represent the unity of the human mind while modeling its changing phenomena as calculable functions? As suggested above, a system of coordinates could be used to represent the unity of the mind in the form of an algebraic transformation group. With such a system, the changing phenomena of symbolic cognition can be modeled by calculable symmetric transformations on the variables of a single algebraic structure.

9.4.2. Graph theory and the human sciences

My emphasis on transformation groups may seem to conflict with the contemporary view that the “right” mathematical theory for modeling the cognitive sciences and human sciences is graph theory. This is not the case. As I will show, the model I am proposing combines both graph theory and group theory, because the semantic sphere is a set of graphs on which a group structure can be defined.

The idea of basing the human and social sciences — or the sciences of the mind — on graph theory, i.e. the mathematical theory of networks, is not new. It is one of the key ideas of the current in sociology that is interested in social capital¹². It is also one of the themes developed by creative contemporary sociologists such as Manuel Castells¹³, Barry Wellman¹⁴, Bruno Latour and, with him, the actor-network school of sociology¹⁵. Moreover, the cognitive sciences and artificial intelligence have for a long time modeled cognitive phenomena using semantic networks and graphs in general¹⁶. In the same vein, Albert-Laszlo Barabasi has argued eloquently in favor of an interdisciplinary “science of networks”¹⁷. I am in agreement with these writers and theoretical approaches, and I fully endorse the general research program that aims to use graph theory as much as possible to study cognitive, cultural and social phenomena. The vertices — or nodes — of graphs can be used to model actors (human or non-human), and their edges — the links or connections — to model relationships among the actors. In addition, networks can be considered circuits channeling all kinds of magnitudes (information, value, prestige, etc.) according to various economic, sociological, psychological and other models. However — and this is the key point of my argument — in order for graphs to be useful in advancing knowledge, their vertices and edges must be categorized. Consequently, the human sciences have every interest in possessing a system of semantic coordinates that encodes categories or concepts so that the vertices and edges of graphs, and finally graphs themselves, can be processed as variables of a transformation group. This is why I am proposing the adoption of a scientific model of symbolic cognition — symbolic cognition being the common ground of the human sciences — that integrates graph theory and group theory¹⁸.

9.4.3. Group theory and the human sciences

The importance of transformation groups for the scientific study of human society and the human mind has already been pointed out by major thinkers such as Jean Piaget (1896-1980)¹⁹ and Claude Lévi-Strauss (1908-2009)²⁰. Piaget showed, for example, that an “object” was an abstract cognitive construct, with the stable object emerging in the mind as the group of transformations of its appearances, i.e. as the structure that remains invariant through the variations presented by its successive aspects. Additional levels of abstraction in learning and thought are reached when different objects or areas of activity constituted as transformation groups are shown to themselves be variations of a single basic structure through morphisms that change them into one another while preserving certain of their properties. The mathematical theory that studies these transformation meta-groups — category theory — is in my view one of the areas of mathematics that has the most affinity with philosophy, epistemology and the cognitive sciences²¹.

Lévi-Strauss, considered the leader of the structuralist school, devoted the major part of his work to studying operations of transformation on symbolic structures such as kinship rules, myths, rituals, aesthetic forms, social forms, etc. For example, for him a myth may be defined as all its versions in space and time. Each of the versions is only one specific variant of a single underlying structure (often a graph of relationships) and can be obtained by the transformation of another version, such as the replacement of one vertex with another or a change of the direction of a relationship. Chapter 3 of one of his most famous books, The Savage Mind, is in fact entitled “Systems of transformations”²². At the beginning of the next chapter²³, after providing large number of examples of inversions, substitutions and transformations on the symbolic systems of various cultures, Lévi-Strauss presents the key hypothesis of his research program: “And it is groups in this sense, and not arbitrarily isolated transformations, which are the proper subject of the sciences of man”. In spite of this emphasis on the concept of group, the research program of structuralism failed to fulfill the requirement I have formulated here, namely that to meet the requirements of the scientific method, the concepts or categories manipulated by human cognition should be able to be processed as variables of a universal system of calculable symmetric transformations. I contend that without this system of mathematical coordinates, without this topological net that makes the universe of concepts the fundamental place of the mind — an abstract, infinite place — it is impossible to model culture (i.e. collective human intelligence) as a scientifically knowable cosmos.

First, except in The Elementary Structures of Kinship²⁴, during the writing of which he was working with mathematician André Weil (1906-98), we do not find in Lévi-Strauss any formal definition of symbolic sets or operations on the elements of these sets. We do not find any precise mathematical characterization of the groups (are they monoids, rings, Lie groups, etc.?). The master of structuralism does talk about transformation groups on signifieds or concepts, but in spite of the number and precision of the quasi-algebraic studies he proposes in his work, the concept of the transformation group is still generally just a metaphor or a suggestive image²⁵.

Second, although Lévi-Strauss considers each version of a symbolic system (myth, ritual, kinship structure, etc.) as a transformation within a group, he only rarely — and allusively or ambiguously — mentions the concept of a universal semantic transformation group specific to the human species. In other words, he refuses to posit, as I do here, the existence of a coherent universe of concepts assembled by a system of calculable symmetric transformations.

We have the impression when reading his work that particular cultural areas or subsets may very well be called transformation groups, but that the hypothesis of a universal transformation group capable in principle of accommodating, translating or modeling the structures of signifieds of the whole set of symbolic systems is taboo. For example, after quoting Balzac (“Ideas are a complete system within us, resembling a natural kingdom, a sort of flora, of which the iconography will one day be outlined by some man who will perhaps be accounted a madman”)²⁶ he declares: “But more madness than genius would be required for such an enterprise”²⁷. In disregarding of his warning – although extending the path he traced – however, the transformation system on concepts modeled by the IEML semantic sphere does indeed provide an algebra (in the formal sense of the term) capable of mapping and manipulating the huge set of conceivable signifieds in a regular fashion.

9.5. Internal coherence

Like the symmetrical system of coordinates of matter, the symmetrical system of coordinates of the mind, the IEML semantic sphere, obeys a strict requirement of internal coherence that does not change with the specific characteristics of any experience. To illustrate this point: there is no “high” and “low” in the geometric system of coordinates of material nature, although all human experience attests to the importance of the distinction between high and low in daily life. Similarly, the requirement of internal coherence for the IEML semantic sphere takes precedence over all specific practical considerations. In fact, it is precisely because of the symmetry and internal coherence of the system of coordinates supplied by IEML that the scientific study of the mind is able to identify regularities and irregularities, symmetries and dissymmetries among cognitive processes.

9.5.1. The mathematical formalization of concepts is a methodological necessity

As the foundation of a scientifically explorable nature of the mind, I am proposing a practically infinite graph of encoded concepts that meets the requirement of symmetry of a transformation group. I am well aware — and Lévi-Strauss’s words remind me — that this position will give rise to many philosophical objections. I will perhaps be suspected of excessive formalism or Platonic idealism. Some will also wonder why I do not start from empirical data on the neural, linguistic or social dimensions of symbolic cognition. With respect to Platonic idealism, I would like to point out, first of all, that my model describes a nature of the mind in which ideas are not fixed or eternal but, on the contrary, living, dynamic, evolving and interacting within cognitive ecosystems. It is only at the level of concepts, i.e. the semantic addresses of ideas, that I am proposing a system of mathematical/linguistic coordinates that serves as a fixed reference. It should also be understood that this fixed quality is quite relative because, as we will see in more detail below, the machine that weaves the semantic sphere is programmable. Second, this machine and the system of coordinates that it traces originate not from a transcendent eternity but from a formal requirement: that of making cognition describable using calculable functions within a symmetrical, coherent universe. I ask the reader to remember that this methodological approach has proven itself in the study of material nature. One of the most insightful historians of science, Alexandre Koyré²⁸, drew a connection between the revolution in the physical sciences in the modern period and the convergence of two currents of thought: (i) a certain “return to Plato”, as indicated by the importance accorded to mathematical idealities; and (ii) the unification of an infinite cosmos, as opposed to the closed, fragmented world of medieval Aristotelianism. There is, however, a caveat! There is no question here of imitating physics, but on the contrary, of thinking scientifically — and therefore mathematically — about the inherent nature of symbolic cognition, using a system of coordinates specifically suited to this purpose, i.e. designed from the outset to capture the interconnections of meaning.

With respect to the relation of my model to empirical data (neural, linguistic, social, etc.), the system of coordinates provided by the IEML semantic sphere is explicitly presented as a useful scientific convention rather than a natural given. The role of this convention is precisely to scientifically organize empirical data, i.e. to make the best possible use of them, and not to deny them or substitute some preconceived conception for them. The objective of the system of semantic coordinates is to inscribe empirical data on symbolic cognition within a framework that makes them calculable, interoperable, comparable and meaningful. The IEML semantic sphere should as far as possible permit relevant description of empirical data. That is why this organizing grid should not only have certain mathematical properties, but should also include the main characteristics of natural languages (which we will look at in Chapter 10). We know from experience that languages are suited to the description of phenomenal data from the human perspective. This is why the IEML semantic sphere encodes concepts simultaneously as mathematical variables (belonging to a transformation group) and as metalinguistic texts (automatically translatable into natural languages).

I will now show that if the formal identities of concepts were extrinsically determined, either by what they represent or by the physical/biological or social mechanisms of cognition, it would be impossible to define a symmetric transformation group on the set of them, and therefore to unify the mind in a way that would permit scientifically relevant mathematical modeling. My argument is essentially as follows:

– we have seen that a system of semantic coordinates of the mind had to take the form of a system of calculable symmetric transformations;

– the only way to obtain this result is to construct an autonomous, coherent system of relationships among concepts;

– rather than adapting to some state of data or to the biological and social mechanisms that support cognition, the semantic sphere must thus determine a strict interdefinition of concepts.

For readers who have already thought about the function of systems of coordinates in scientific knowledge, this “transcendent” self-positioning of the semantic sphere will not be surprising²⁹.

9.5.2. The identification code for concepts cannot be based directly on empirical data

9.5.2.1. Inadequacy of a neural basis

I will first show that the system of coordinates of the mind cannot be based on neurobiological data. The nervous system, the organic medium of animal cognition, emerged from biological evolution as the producer of phenomenal forms against the ground of memory. Neural circuits implement operations of categorization on looped flows of sensory-motor data. At the level of their simplest organic inscription, these operations are implemented by neurons that process electrochemical signals: activation through thresholds, amplifications, etc. At a higher level of complexity but still in the layer of neural encoding, categorization operations emerge from the self-organized dynamics of electrochemical states in “assemblies of neurons”³⁰. These dynamics inscribe relatively stable circuits of categorization in the neural material, which are shaped by our learning. When the same words and sentences meaning the same concepts are pronounced (or even only thought) by different people, they result in the activation of neural circuits that are not only physically distinct but are also dissimilar in their formal patterns³¹. Moreover, different sentences in different languages can refer to identical concepts (I drive my car/je conduis ma voiture). This is why I feel that there is no functional correspondence between concepts and the dynamics of neural states that would be of practical use from a perspective of large-scale semantic encoding.

9.5.2.2. Inadequacy of a sociotechnical basis

In addition to their neural mechanisms, operations of categorization are also implemented in culturally determined circuits that are indissociably semiotic (networks of signs and messages), social (networks of people) and technical (physical networks). Generally, complex categorization operations can be implemented in heterogeneous networks of interconnected artifacts, institutional operations, symbolic systems, etc. This is how institutions (such as families, schools, courts and political bodies) categorize the differences they produce with regard to themselves, their members and their environment. Can the system of coordinates of the mind be based on these sociotechnical mechanisms? I do not think so, since categorization operations involving the same categories (parent, graduate, guilty, elected, etc.) can obviously be actualized by completely distinct (actual) space–time mechanisms, among which it is very difficult to establish calculable transformation functions. In short, the formal, or abstract, identities of our categories must be distinguished from the concrete — neural or sociocultural — mechanisms that effectively implement categorization operations in the space–time continuum.

9.5.2.3. Inadequacy of a basis in natural languages

Finally, a system of coordinates of the mind in the form of a calculable transformation group cannot be based on a natural language. Concepts must be distinguished from the words or sentences that refer to them in natural languages. It is clear that the same signified, or concept, can be designated by expressions from different languages (dog, chien, kelb, etc.). Also, there is no reason for choosing one natural language rather than another to formally encode a concept. In addition, although we obviously use natural languages to think about and communicate abstract categories, the synonyms, homonyms, ambiguities and irregularities of natural languages make it difficult to use them as tools for the scientific identification of concepts. A natural language is not a calculable transformation group on concepts.

9.5.2.4. Conclusion

We want to be able to manipulate concepts automatically, transparently and symmetrically, and therefore to represent them as variables of calculable functions in a transformation group. To achieve this, we cannot base the identity, formal description or scientific encoding of concepts on (i) natural symbolic systems³², (ii) sociotechnical mechanisms or (iii) any kind of biological circuitry. Natural languages, sociotechnical systems and neural circuits are empirical — thus opaque to calculation, implicit, actual — mechanisms of manipulation of theoretical concepts, i.e. variables transparent to calculation: explicit, virtual, formal, symmetrical, conventionally assumed. The scientific identity of concepts cannot be based directly on empirical data. But then, how can concepts be encoded in a way that is transparent to calculation?

9.5.3. Concepts can only be distinguished through their mutual relationships

Since concepts must be encoded as variables of calculable symmetry operations and since they cannot be distinguished from each other using empirical data — the natural signs³³ designating them or the concrete mechanisms manipulating them — I had to develop a method for distinguishing them rigorously from each other on the basis of their mutual relationships. This is why I constructed the identification codes for concepts (USLs) differentially or relationally, in a network of symmetrical relationships, and not using references to a set of phenomenal data. Since concepts are only scientifically definable through their mutual relationships, and since the parsimony principle³⁴ requires that the code of a concept be the same as its scientific definition, the identification code of each concept is equivalent to the node of its relationships with other concepts. It is precisely this requirement that the IEML semantic sphere meets, since it makes it possible to automatically go from a USL to a semantic circuit translated into natural languages.

In the IEML model, a concept is thus presented a priori as a hypercomplex intersection of relationships with other concepts³⁵. In the semantic topology of IEML, a unique circuit corresponds to each USL (to the formalization of each unique concept). This circuit connects the USL of which it is the expression to other USLs, and the interconnected expressions of USLs delineate the semantic sphere. Like Plato’s ideas or Leibniz’s monads³⁶, the concepts of the semantic sphere are mutually defining. With the IEML semantic sphere, however, we move from philosophy to science, since the strict interdefinition of concepts uses a system of calculable symmetric transformations.

Symmetry and internal coherence obviously concern only the system of coordinates for addressing ideas and the circuits that connect them. On the symmetrical ground of the system of semantic coordinates, cognitive processes draw figures ranging from the most to the least symmetrical. To make an analogy with terrestrial coordinates, the fact that the meridians and parallels trace a perfectly symmetrical grid on the sphere does not mean that the continents, rivers or paths of cyclones drawn on maps are themselves symmetrical. The cognitive functions that will be able to be automated using the fundamental grid provided by the semantic sphere will be as unique and as complex as we wish.

9.6. Inexhaustible complexity

To end this chapter, I would like to highlight the inexhaustible complexity of the functions for producing circuits among concepts. The IEML semantic sphere lends itself to the automatable tracing of a practically infinite number of distinct semantic circuits and the programming of a practically infinite number of functions describing transformations among these circuits. One of the main issues in this section is to show that although the semantic sphere is mathematically finite for purposes of theoretical calculability, it is in practice inexhaustible.

9.6.1. The inexhaustible complexity of the mind

I start from the hypothesis that the nature of the mind is infinite. This postulate of infinity should be considered a fundamental principle of openness, which is justified primarily by its practical conclusions. I simply wish to indicate by this that no knowledge of the mind, however scientific, comprehensive and precise it may be, will ever be complete or finished. Indeed, it is clear that scientific theories are perfectible human constructs and that our capacities of observation, measurement, memory and calculation are necessarily finite. Knowledge of the mind is in this respect, once again, exactly like the knowledge of material nature. Since all scientific knowledge of the mind depends on our theories and capacities of observation, measurement, etc., which are finite, and since the nature of the mind, by hypothesis, is infinite, it necessarily follows that scientific knowledge of the nature of the mind can only be approximate and incomplete. We can also reason as follows: since the nature of the mind is infinite, and finite human knowledge can only explore it gradually over an irreversible duration, then the sphere of symbolic cognition will always hold something unforeseen for us. Any capacity for prediction based on a finite memory of the known is structurally exceeded by a huge reserve of the unknown that will never be completely discovered. The word infinite has a precise meaning in mathematics and because, as we will see in this section, while the IEML semantic sphere (the system of coordinates of the mind) is huge it is not mathematically infinite in terms of calculability, I prefer to say that the IEML model of the mind permits the exploration of an inexhaustible complexity. The expression inexhaustible complexity suggests that missing knowledge cannot be reduced to a matter of decimal places or the best quantitative approximation: it implies the future discovery of new forms, new structures and new layers.

9.6.2. The unlimited variety of concepts and their transformations

My system of coordinates of the mind has to meet two apparently contradictory requirements. First, the topology of the semantic sphere has to meet the requirement of calculability. In other words, the functions of the machine that traces this semantic sphere must be able to be executed using finite algorithms in a finite time. Calculability assumes finitude. If the variables processed by my semantic machine were infinite in number, they would fall under the limit theorems of Gödel, Church and Turing³⁷. That is why the semantic sphere cannot be infinite in the strict mathematical meaning of the term. In addition to this, however, the semantic sphere must meet a constraint of unlimited openness, and quite rightly so. How could a finite model represent the potential playing field of the human mind? There can be no question of in any way closing the process of creation or discovery of new concepts. I was therefore confronted with the problem of representing in a finite way a reality that is in principle infinite. To solve this problem, I adopted a model that is finite but huge, i.e. of which the order of magnitude is beyond astronomical, and is therefore equivalent to infinity on the scale of human intellectual and technical possibilities.

Cosmologists estimate the maximum number of particles in the material universe at 10⁸⁰. In his article “Computational capacity of the universe”, physicist Seth Lloyd calculated that if each particle in the universe could be used as a part of a giant computer, employing the possible quantum states of the particles, that computer could only contain 10⁹⁰ bits³⁸. In comparison, the number of connections in the human brain is (only!) 10¹⁴. We may nevertheless consider that the thoughts emitted by a brain correspond to configurations of connections, i.e. to a space of possibles much greater than that of the connections themselves. We can call the numbers — which are huge and, while finite, forever beyond the possibility of being written exhaustively — cryptographic numbers. Indeed, universes of combinatory possibilities of this type are used in cryptography to prevent codes from being deciphered by brute calculating power.

The semantic sphere generated by the IEML machinery does provide a practical approximation of infinity, since an encoded recording of all its nodes exceeds the computational possibilities of the real physical universe as calculated by Seth Lloyd by many orders of magnitude. The model provided by the semantic sphere is “bigger” than the physical universe, in the sense that it is beyond the reach of physical recording or complete writing of the list (encoded in IEML) of the vertices of its circuits. The list of distinct calculable functions capable of describing all the paths in its circuits is greater still. Indeed, let us now consider the functions that transform the semantic variables (the circuits of USLs), functions that can be compared to conceptual trajectories. Since all the circuits are already given formally by the system of coordinates, the transformations among circuits can be translated into the production of networks of circuits or the tracing of paths among those circuits. The algebraic topology of the semantic sphere permits the creation of a practically unlimited variety of calculable functions describing conceptual trajectories among semantic circuits.

In short, the cryptographic immensity of the semantic sphere is quite simply beyond the reach of the finitude of the physical cosmos. The semantic sphere provides an acceptable approximation of infinity because its total hypertext, and even more so the paths through that hypertext, will forever remain indecipherable in its entirety. At the same time, it is finite and countable mathematically, thus avoiding the limit theorems of Turing, Church and Gödel. In practice, its regular, symmetrical structure makes it available for all kinds of automatable functions. The semantic sphere is therefore indefinitely explorable by finite automata, even though its total exploration is beyond reach.

9.6.3. The unlimited size of concepts

The semantic sphere is not only open in qualitative variety, since all USLs are distinct “texts”; it also accommodates as many degrees of complexity of concepts as we might want. We generally understand the terms category and concept to refer to the signified of a word or a short expression, e.g. “the equine species”, “justice”, “spring”, “laughter”, but there is no reason to limit the complexity of concepts to the signifieds of short expressions. The signified of a sentence is also a concept — a propositional concept — and can therefore be represented by a node of the semantic sphere. It should be noted that philosophers have generally focused on the referents of sentences (because the truth of the sentence depends on its referent), than on their signifieds or their meanings. The German philosopher and logician Gottlob Frege (1848–1925), whose ambition was to devise a “conceptual notation” and who is considered one of the founders of contemporary logic, distinguished between the referent (Bedeutung) of an expression and its meaning (Sinn). The classic example is “the evening star” and “the morning star”: the two expressions have different meanings but the same referent (the planet Venus). The truth of a proposition is determined by its relationship to its referent: if I indeed saw Venus, it would be just as true to say that I saw the evening star³⁹. However, here I am not talking about the referent, but about the meaning — the unique semantic quality — of a long linguistic expression. A paragraph, a book, an entire library, a discursive ensemble or a collection of documents can be counted as expressing “one” hypercomplex conceptual identity. It is difficult for beings whose short-term memory is as limited as ours to grasp concepts of such “size” in their unity, their internal variety and their interdependence with other concepts. Acting as an intellectual technology⁴⁰ that augments our cognitive capacities, the modeling of the mind coordinated by the IEML semantic sphere will make it possible to refine our understanding of such mega-concepts.

As we already know, concepts are modeled by USLs, affects are modeled by semantic currents and percepts are modeled by URLs. If all the possibilities of the hermeneutic functions that assemble concepts, affects and percepts and produce networks of ideas are combined, then it is clear that the semantic sphere permits the scientific modeling of a nature of the mind with inexhaustible complexity. This is the formal translation of what in more intuitive terms could be called the unlimited openness of symbolic cognition, or the natural freedom of the mind. Understanding the mind as an open and coherent natural totality meeting the requirements of calculability, unity, location, symmetry, internal coherence⁴¹ and inexhaustible complexity implies a profound change in our vision. It is only through such an intellectual change that we will be able to initiate a true scientific exploration of human cognition.

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1 See [BER 1996].

2 A complete mathematical formalization of its production and of the production of concepts will be presented in Volume 2 of this book.

3 See section 6.4, for example.

4 Alain de Libera has examined the logical precedence of essences (which I call concepts here) over existences in the medieval philosophical tradition and the sources of this precedence in Greek philosophy. It is Avicenna (Ibn Sina) who is first credited with stressing the independence of essence in relation to existence. In the metaphysics of Avicenna, God gives beings existence by choosing from among the intellectual essences that are formal virtualities of existence and are logically anterior to existence. See L’Art des Généralités. Théories de l’Abstraction [DEL 1999].

5 On this point, see Howard Gardner, The Mind’s New Science: A History of the Cognitive Revolution [GAR 1987], Jean-Pierre Dupuy, On the Origins of Cognitive Science [DUP 2005] and Margaret Boden, Mind as Machine: A History of Cognitive Science [BOD 2006].

6 Or intellective faculty; see Book III of On the Soul by Aristotle [ARI 2009c].

7 The intelligible universe is a classic and very old theme of philosophy that begin with Plato. It was extensively developed by the Neoplatonist schools of antiquity, was taken up by many medieval Aristotelian philosophers, was renewed by Leibniz in the Renaissance [LEI 1704]. It was then brought back in the contemporary period by Alfred North Whitehead [WHI 1925, [WHI 1929, WHI 1933], Karl Popper [POP 1972], Edgar Morin [MOR 1977-2004], etc. For further information, see section 6.4.

8 On symmetries, transformation groups and their role in the scientific process, see [BAC 2000, BUT 1991, LOC 1994, MIR 1995].

9 I want to reiterate that there is nothing original about this hypothesis. It is shared by most researchers in the cognitive sciences.

10 As opposed to the Heaven/Earth or sublunar world/celestial world fragmentation of medieval cosmology. In the finite, fragmented medieval cosmos largely inherited from Aristotle, only the celestial world could be described mathematically. Modern science unified physical nature, envisaging it as an infinite universe with no absolute center and all its parts able to be mathematically modeled. I refer once again to [KOY 1958].

11 I learned a lot about group structures and symmetry by reading Henri Bacry, La Symétrie dans tous ses États [BAC 2000] and R. Mirman, Group Theory, an Intuitive Approach [MIR 1995].

12 See [DEG 1994, FUK 1995, LIN 2001, PUT 2000].

13 See [CAS 1996, CAS 2009].

14 See [WEL 2001, WEL 2012].

15 See [CAL 1989, LAT 1987].

16 See the works of John Sowa [SOW 1984, SOW 2000].

17 See [BAR 2002].

18 To be complete, this model would also integrate the theory of computability and the theory of regular languages (see Volume Two).

19 See his book on structuralism [PIA 1970b]. This theme is found in most of his work on the modeling of human intelligence.

20 See The Savage Mind [LÉV 1966]; this is a recurring theme in all Lévi-Strauss’s works.

21 On this point, see the posthumous work by Jean Piaget, with his collaborators, Morphisms and Categories: Comparing and Transforming [PIA 1992]. On the relationships between category theory and philosophy, see Alberto Peruzzi, “The meaning of category theory for 21st Century philosophy” [PER 2006], and for a recent application of category theory to the study and automated processing of metaphors, see Yair Neuman and Ophir Nave, “Metaphorbased meaning excavation” [NEU 2009].

22 The Savage Mind, p. 75. As I have already pointed out, another great thinker in the human sciences, Jean Piaget, devoted a great deal of thought to the importance of transformation groups for the scientific modeling of the mind; see [PIA 1970b].

23 Op. cit., p. 109.

24 See Lévi-Strauss, The Elementary Structures of Kinship [LÉV 1969].

25 This opinion is shared by Klaus Hamberger: “If the road opened up by Lévi-Strauss is to one day lead to a real science of symbolic transformations — which implies the possibility of reproducing all the results of the transformational analysis in a series of explicit, comprehensible steps — it is clear that the formal basis of such a science will be group theory. Lévi-Strauss had already indicated this orientation by using, if only for purposes of illustration, the simplest symmetric transformation group that can be imagined beyond trivial groups (of first or second order), namely the so-called Klein group of fourth order created through two transformations of period 2 (in other words, two oppositions)” [translation]. This quotation is from Klaus Hamberger, “Le continent logique. À propos de Quadratura Americana d’Emmanuel Désveaux” [HAM 2004].

26 The quotation is from Honoré de Balzac, Louis Lambert, translated by Clara Bell and James Waring (New York: P.F. Collier, 1900).

27 The Savage Mind, p. 130 [LÉV 1966].

28 See his From the Closed World to the Infinite Universe [KOY 1958].

29 On systems of coordinates, see Peter Galison, Einstein’s Clocks, Poincaré’s Maps [GAL 2003]. On the importance of basic theoretical frameworks in scientific knowledge see Karl Popper, Objective Knowledge: An Evolutionary Approach [POP 1972]. It should be noted that Popper is taking up the concepts of the founders of contemporary physics, in particular Albert Einstein. On the pioneering and decisive role of theories and conceptual frameworks in the history of science, see also Thomas Kuhn, The Structure of Scientific Revolutions [KUH 1962].

30 I have borrowed the term “assembly of neurons” from Jean-Pierre Changeux [CHA 1985].

31 See Terrence Deacon, The Symbolic Species [DEA 1997].

32 It goes without saying that symbolic systems are always cultural, and therefore artificial. I am using the word natural in the same sense as it is used in “natural language”, however, a natural symbolic system as opposed to a symbolic system deliberately designed for scientific reasons and to meet scientific requirements.

33 See preceding note.

34 This principle, also known as Occam’s razor from the name of the medieval philosopher and theologian who formulated it most clearly, states that theoretical entities must not be multiplied unnecessarily. This is why the code for a concept must contain everything necessary for its scientific processing, with no need to add an additional definition.

35 If this is clear, there is obviously no reason — quite the contrary! — the nodes of the semantic sphere (representing concepts) could not be used to index phenomenal data.

36 See [LEI 1695, LEI 1704, LEI 1714a, LEI 1714b]. On Leibniz’s monadology, see also Michel Serres, Le Système de Leibniz et ses Modèles Mathématiques [SER 1968].

37 See Marvin Minsky, Finite and Infinite Machines [MIN 1967] and Barry Cooper, Computability Theory [COO 2004].

38 Lloyd Seth “Computational capacity of the universe” [SET 2002]; see also [SET 2000].

39 Gottlob Frege, Philosophical Writings of Gottlob Frege, edited by Peter Geach and Max Black [FRE 1952]. The original article distinguishing between meaning and denotation is from 1892.

40 On this point, see my book Les Technologies de l’Intelligence [LÉV 1990], as well as section 12.1, which refers to numerous sources.

41 The symmetry and internal coherence of its system of coordinates.