The crucial notion of singularity, taken literally,
seems to belong to all the domains in which there is structure.
Gilles Deleuze

1 | Introduction

1In his early, programmatic essay “How do we recognize structuralism?” (1967) Deleuze presents a confrontation between structuralism and transcendental philosophy with the aim of demonstrating how structuralism transforms transcendental philosophy, and how a new transcendental structuralism becomes compatible with contemporary ideas in topology and dynamical systems theory concerning self-organization, singularities, structural stability, complexity, etc. Concerning the first point, the key novelty of structuralism is the idea that the constitution of sense and meaning in language does not presuppose an ideal, transcendental subject. Instead, language becomes the transcendental condition of possibility for both meaning and subjectivity. Unlike Kant and Husserl, who argued language presupposes a transcendental subject or ego, structuralists conceived language as a structural, symbolic/ideal realm that does not belong to the order of consciousness or subjectivity but that instead contains its conditions of possibility. Linguistic structure was therefore defined as a transcendental field without a subject that generates both meaning and subjectivity.

2Deleuze’s second key idea in this essay is that the foundations of structuralism are not only transcendental but also topological, and not logical.1 Indeed, Deleuze speaks of a “transcendental topology,” which grounds empirical psychology and conceives the psychic element of symbolic structure in terms of a topological, spatial order of proximities defined by differential relations of emergence. As he writes, “empirical psychology is not only founded, but determined by a transcendental topology.” (Deleuze 2004, 174) Psychic structure would emerge in a topological space, divided into regions (places) by a system of energetic differences and singularities. These intensive differences between the elements of a structure are not external relations between pre-existing identities, but rather generative relations of reciprocal determination. Generatively, psychic structures become organized by the unfolding of singularities, which differentiate and organize it into a properly structural space.

3The aim of this paper is to evaluate these two goals of Deleuze’s transcendental structuralism: how does structuralism transform transcendental philosophy, and how should the foundations of structuralism be modeled topologically and dynamically? To address the first question, we will consider some key convergences and divergences between structuralism and transcendental philosophy concerning the autonomy of cognition, the constituency of structure, the principle of reciprocal dependency relations, and the relation between language and subjectivity. To address the second question, we will turn to Jean Petitot’s dynamical structuralism, which has meticulously developed Deleuze’s proposal of a topological foundation of structuralism. Dynamical structuralism proposes an alternative to both traditional structuralism and classical computationalism, which both entail a strict dualism between the symbolic and physical levels, the relation between both being a matter of mere implementation. Instead of simply positing symbolic cognitive architectures in which cognition takes place, dynamical structuralism proposes to conceive symbolic structures as emerging from their material substrate base. Insofar as every cognitive structure is a system of reciprocal dependency relations, which links up different parts within a whole, every structure is modeled as a system of qualitative discontinuities emerging from the underlying substrate. But dynamical structuralism is nonetheless still considered a true functionalism because it considers emergent structures to have properties, which are to a large extent autonomous from the specific physical properties of their underlying substrata. Also, it fully endorses the important computationalist claim that high-level cognitive processes have properties (generativity, compositionality, systematicity, inferential coherence), which can only be modeled by constituent syntactic and semantic structures.2

4Our purpose in staging this confrontation between Deleuze and Petitot is fundamentally systematic-theoretical, rather than historical-exegetical. Beyond reconstructing a remarkable historical exegesis, we are fundamentally concerned here with the question as to what exactly this topological, dynamical foundation means for the future of structuralism and transcendental philosophy. In this regard, the key issue becomes the naturalization of constituent structure: either one conceives the foundations of structuralism and functionalism in a purely logicist manner, thereby adopting a resolutely dualist stance and leaving the naturalization of structure wanting, or one conceives these foundations in a dynamical and topological manner, thereby naturalizing constituent structure in physical, morphological and ultimately symbolic terms. To see what such a dynamical functionalism entails, we will consider two of its key applications to categorical (speech) perception and visual perception. In this light, the key theoretical innovation of Deleuze’s proposal for a transcendental structuralism will appear as a morphodynamic approach to psychic structure, which transforms the central structuralist principle of reciprocal determination into a morphogenetic process of psychic individuation, which generates ideal structure. Indeed, in Deleuze and Petitot, the transcendental itself becomes a morphodynamic field of ideal structure, which precedes the emergence of subjectivity.

2 | Transcendental Philosophy and Structuralism

2.1. The Autonomy of Cognition

5Structuralism shares with transcendental philosophy an exclusive concern for the autonomy of cognition, language and meaning. Both transcendental philosophers and structuralists claim that the phenomena of meaning and language have a priori foundations over and above physiological, psychological and cultural-historical conditions. For both, the faculty of language is psychic not in the psychological but only in the transcendental sense: the speaking subject processes linguistic information according to an a priori universal, constitutive system of formal, syntactical constraints that condition the assignment of semantic meaning to phonetic sequences. From a transcendental perspective, syntax is a constituent structure, which is akin to the constitutive function of the logical forms of judgment in constraining thought. Just as transcendental philosophy conceives the realm of meaning in terms of its own logical laws that are irreducible to physical reality and subjective psychological acts, so structuralists analyze our psychological linguistic experience as a sum of sense effects from the structural realm of language onto the psychological realm of experience.3

6In cognitive science, the structuralist principle of constituent structure in language has been extended by functionalist computationalism to the whole of cognition. What computationalists such as Jerry Fodor and Zenon Pylyshyn call the “systematic” and “compositional” structure of thought exactly recapitulates the traditional structuralist argument for constituent structure in sentences (cf. Fodor & Pylyshyn 1988). For them, the constituent structure underlying our faculty of language is a paradigm of systematic/compositional cognition, but this constituent structure is not limited to language alone and rather appears as a thoroughly pervasive feature of human and infrahuman mentation. Classical computationalism postulates that all cognition relies on symbolic constituent structure, which belongs to a so-called “language of thought” that has the structure of a formal language (with symbols, expressions, inferential rules, etc.).

2.2. The Transcendental Ideality of Constituent Structure

7Like transcendental philosophy, structuralism rejects empiricist reductionism and posits the ontological primacy of ideal structures, which are constitutive of meaning. Structuralism begins when one assumes the constitutive existence of structures that are purely ideal and not material (or imaginary). As such, linguistic structures are abstract forms of psychic organization that are irreducible to certain interacting material components. The founding idea of structuralism is that structure is the ideal form of organization of a material substance, and as such structure is not a sensible phenomenon. It is essentially invisible although its substantial realization and its meaning effects are observable and may be the object of certain rigorous empirical experiments. As such, structure is a theoretical object and not a fact. In Deleuze’s famous phrasing, structure is at once real and ideal: “real without being actual, ideal without being abstract,” a “pure virtuality of coexistence that pre-exists beings” (Deleuze 2004, 179). As an organizational principle, structure “incarnates” itself in its material substrate, it “expresses” itself in it, but it never actualizes itself as such. Instead, the sensible expression of a structure is always essentially a negation of its ideal, formal being.

8To understand this abstract idea, one may compare it to the Kantian conception of a pure intuition of space. On this view, our sense of three-dimensional space as a framework within which things are perceived is not derived from material input of the senses, but rather produced by the (unconscious) mind as an a priori framework within which sensorial input can become organized. Space as pure intuition is an ideal or virtual structure, which the mind projects onto the physical world, and which is ‘filled’ and ‘realized’ by means of actual sensorial input. In Kant’s terms, space is an a priori “form of intuition” through which raw sense data are first “given,” and can subsequently be processed or “synthesized.” But this ideal form is not itself derived from sensory experience: spatial cognition is not an abstraction from the relations between objects or stimulus configurations, as empiricists would hold. On the contrary, cognitive processing or “synthesis” of sensory information is only possible based on the mental generation of a pre-existing spatial framework.

9Like spatial cognition, linguistic structure can be understood as a virtual frame that we project onto the physical world (i.e. the audio-acoustic flux of spoken sound). Deleuze gives the example of the phoneme. Phonemes play an essential role in the perception of sound sequences as linguistic sound sequences. They allow us to perceive the continuous audio-acoustic flux of sound sequences as a sound structure that consists of discrete interconnected parts, or units. The central idea is that speech perception is especially sensitive to qualitative discontinuities in the flux of sound sequences, which serve as cues of differences in meaning. For example, upon comparing “bat” and “pat,” a qualitative discontinuity is perceived by virtue of the phonemes /p/ and /b/. Here, the phonemes p and b function as cues, which highlight an elementary categorical discontinuity, or what Deleuze calls a singularity. Thus, the phonemes are not perceived along a continuum but rather as discrete categories. As such, the perception of phonemes transforms the continuum of audio-acoustic sounds into a discrete, categorized structure.

10But in what sense should the phoneme be a pure virtuality and not simply a well-defined speech sound? Here, we have to see that a phoneme can never be identified with one physical constant. Rather, there is a physical continuum of variations, which are all identified in perception as belonging to the discrete phoneme. There is, in short, a marked lack of correspondence between sound and perceived phoneme. The phoneme is an abstract type that is instantiated in actual speech by concrete tokens. For this reason, many structuralist linguists such as Roman Jakobson and Nikolaj Trubetzkoy considered the phoneme not as a physical, audio-acoustic property but rather as a psychological or mental reality (X). As Petitot emphasized in his early work, this leads to a “foundational aporia” underlying linguistics: how can an acoustic, phonetic flux of a physical nature become the perceptual basis of a phonological code of a strictly linguistic nature, governed by differential relations? (cf. Petitot 1985, 20; 2004, 93) This aporia concerns the relation between the physical-phonetic and the symbolic-structural dimensions of language.

11As abstract linguistic types, phonemes are not defined by intrinsic physical properties but by a network of differences and reciprocal dependency relations. Since Saussure and Jakobson, phonemes have been understood as purely relational entities, rather than substantial ones. As Petitot puts it, they are eidetic “Gestalten” depending upon a purely formal level of reality (Petitot 1988, 36). In this sense, the phoneme is essentially “double-faced”:

1. On the phonological side, it is a structural, linguistic category that is traditionally believed to be of an algebraic-combinatorial, computational kind: identity of position, relational unity, difference, discrimination, reciprocal determination, etc.
2. On the phonetic side, it is a psychophysical or neuropsychological, audio-acoustic category: spectral forms, deformation of these forms, control by acoustic cues, invariance/variability, categorization, boundaries, etc.

12Structuralists such as Saussure and Hjelmslev emphasized the ontological autonomy of the phoneme as an ideal “form of expression” that is “embodied” in the material “substance of expression” but that is not reducible to it. As such, there is a reciprocal dependence between the phonetic form and the phonetic substance. In the Kantian sense, phonemes function like categories that determine our phonetic perception. Thus, if Deleuze calls the phoneme an exemplary case of a “pure virtuality,” this must be understood in the sense that it is an abstract structure or organizational form that has a psychic reality and becomes attached or encoded in the physical audio-acoustic flux.

13The central principle of structuralist linguistics asserts that only the form of expression is significant, and not the substance of expression. Signs, i.e. linguistic functions, relate forms of expression to forms of content. As such, the effects produced by meaningful substances can be studied only by abstracting forms from them. Now, to see how form or structure itself can be meaningful, we must turn to a third concept marking a convergence between structuralism and transcendental philosophy: the concept of reciprocal dependency relations, which constitute meaningful form in an autonomous dimension of sense and meaning.

2.3. Reciprocal Dependency Relations

14One of the most fundamental concepts of structuralism is that of a relation of reciprocal determination. This category informs the very idea, proposed by Saussure, of a primacy of relations to their terms. Saussure’s main contribution to linguistics lies in conceiving linguistic units not as substantial ones but as purely relational ones. In a linguistic structure, the value or identity of a particular linguistic unit, such as a phoneme, is purely “positional” (Deleuze 2004, 174). The sounds of language become phonemes when they are reciprocally opposed to one another. This means a linguistic structure is not a system of relations between predefined terms. As regarding their value, the terms of the structure do not have any autonomous existence. They can be defined only by their reciprocal determination within a self-organizing whole. Similarly, words in natural language never possess meaning by themselves but only insofar as they are opposed to other words within the lexical structure of a language. Ever since Saussure, the category of reciprocal determination has been fundamental to structuralism, and as we will see, it plays a central role in Deleuze’s structuralism as well.4 Indeed, the category of reciprocal determination entails the structuralist principle according to which difference is prior to identity, so important to Deleuze.

15Transcendental philosophy easily aligns with this core feature of structuralism, which is the idea of relations without substantial relata. One might even say one of its first substantial philosophical developments appeared in Kant’s treatment of biological organization in terms of reciprocal dependency relations in the Third Critique. Likewise, Husserl’s Third Logical Investigation (“On the theory of wholes and parts”) is one of the key philosophical developments of the mereological relations between a whole and its parts. In his analysis of structural relations of dependencies, Husserl argued that such relations should be conceived as metaphysical connections that are not merely applicable to linguistics (i.e. to linguistic contents and their expression) but rather applicable a priori to many spheres of objects. His paradigmatic example is that of the dependency relation between qualities (e.g. the color red) and extension: the redness of a particular thing cannot exist independently from spatial extension and, vice versa, extension itself is also necessarily colored (cf. Husserl 2001, 7-8). The same holds for the relation between intensity and quality: both are reciprocally dependent upon one another. Eliminate quality and one eliminates intensity, and vice versa. For Husserl, these kinds of reciprocal dependence relations are not merely empirical – instead, they have an a priori necessity, grounded in pure essence. That is to say, the necessity of dependency relations “stands for an ideal or a priori necessity rooted in the essences of things” (Husserl 2001, 12).

2.4. A Transcendentalism Without a Subject

16That being said, we should now consider a significant divergence between structuralism and transcendental philosophy. The greatest difficulties for the concept of a transcendental subjectivity come from the study of language. One of the fundamental problems, which animated Husserl’s phenomenology, concerns the relation between insight and expression. Husserl considers insight to be purely ideal and to be identically repeatable within consciousness. In order to share insight, it must be dressed in the physical form of an expression (e.g., the spoken word or the written sign). An expression then allows a reader or listener to ‘see’ a meaning, which as an ideal entity is itself irreducible to the expression (and to any psychological event grasping the meaning).5

17In The Origin of Geometry (1936), Husserl sketches this relation between insight and expression as follows. Imagining how a highly abstract discipline such as geometry may have emerged, he believes this must have taken place in an originary visionary intuition within the consciousness of a particular individual, which might be seen as the originary geometer. He or she ‘saw’ the geometric truth as an ideal given that was immediately present (selbst-da) and self-evident. Although this moment of discovery was psychologically localizable within the mental space of this proto-geometer, the geometric insight as such is not psychological in nature. Indeed, geometric existence is not psychic existence. For example, the Pythagorean theorem does not exist as something personal within the personal sphere of consciousness but rather it exists as an “ideal objectivity” (ideale Gegenständlichkeit) that is there for everyone to ‘see’ (Husserl 1978, 160). For Husserl, geometric insight has not a real, psychological but an irreal, purely ideal status. This means it is a) supra-temporal, b) infinitely repeatable and c) valid a priori for any other rational subject (Husserl 1978, 160). In order to transcend the intra-personal psychic origin and to attain an intersubjective objectivity, the insight must then be linguistically expressed. Important in this regard is Husserl’s view that insight in and of itself already has an ideal objectivity to it, prior to and independent of all expression. Expression is rather a secondary and even accidental matter. It merely transmits what will forever remain identically the same – in the original language of Euclid and in all its translations. As Husserl emphasizes, “the idealities of geometrical words, sentences, theories – considered purely as linguistic structures – are not the idealities that make up what is expressed and brought to validity as truth in geometry; the latter are ideal geometrical objects, states of affairs, etc.” (Husserl 1978, 161).

18It is these assumptions on the relation between insight and linguistic expression, which have been questioned by structuralism: can we posit a subjective sphere of pure insight and pure interiority, which would precede language, or is such subjectivity itself constituted by language? In the Logical Investigations, Husserl himself already pointed out that the ego is not only an instance of thought or of consciousness, but also of speaking. In speaking, the ego signifies itself, which leads to a remarkable doubling between the signifying ego and the signified ego. In this context, the Libanese linguist Emile Benveniste explicitly argued that the ego emerges within the signifying act. As such, language would be a transcendental condition of possibility for subjectivity: “It is only in and through language that the human being constitutes itself as a subject; because only language grounds in reality, in its reality, the concept of the ‘ego’” (Benveniste 1966, 259: my translation)6 Around the same time, the transcendental was conceived by Jean Hyppolite – Deleuze’s first philosophical master – as a transcendental field without a subject, a field that is neither of the order of consciousness or of the ego, but instead would contain the conditions of possibility of subjectivity, meaning and consciousness.

19Along these lines, Paul Ricoeur famously called structuralism a “transcendentalism without a subject” or a “Kantianism without a transcendental subject” (Ricoeur 2000, 52; 53) As Ricoeur points out, structuralism is concerned with “a categorial system without any reference to a thinking subject” (Ricoeur 2000, 33). It posits a pre-conscious categorizing activity that concerns our unconscious competences in understanding and expressing language. The object of structuralist linguistics is the preconscious rules and categories of language, i.e. the laws, which the competent linguistic subject has an implicit, unreflective knowledge of. As Ricoeur puts it, structuralist linguistics invokes therefore a “Kantian unconscious” rather than a Freudian one, since it is concerned not with the energetics of instincts, drives and desire, but rather with a preconscious categorization of auditory sound fluxes into phonemes, words, and sentences according to the structural laws of language. (Ricoeur 2000, 33)

3 | Deleuze’s Transcendental Structuralism: Key Principles

20Traditionally, the origin of structure was left open and its naturalistic knowledge was often considered to be illusory. Consequently, structure was considered to be a transcendent instance within which the differential reciprocal relations are established, which determine the meaning of signs and which principally exclude the question of the ‘origin’ of meaning.

21Moreover, given their ideal, non-phenomenal status, structures are ontologically ambiguous. Although it is an ideal eidos, a structure cannot be detached from the substance in which it actualizes itself. It is therefore both a structuring principle and a structured substance. In this regard, two different ontological approaches are possible: either one adopts a realist point of view, and one takes ideal structures to be given, or one takes a nominalist point of view, and one considers them as merely theoretically posited. The “great” structuralists of the 20th century, Deleuze and Petitot included, were realists in this regard: Saussure, Jakobson, Tesnière, Hjelmslev, Piaget, Levi-Straus, Chomsky, Greimas, Deleuze, Thom and Petitot.

22The main principle of structuralist linguistics asserts that all meaning is generated exclusively by reciprocal determinations in the autonomous, transcendent dimension of structure. Of course, meaning needs always to be realized concretely in the psychological immanence of speech acts. But nonetheless, the key idea of structuralism is that the reciprocal relations between linguistic elements such as phonemes, letters, words, etc. cannot be reduced to the immanence of psychological processes and pertains exclusively to disembodied, purely ideal signifying relations in the autonomous, ideal domain of structure. It asserts that for the order of meaning, causality is structural and formal before being physical and material.

23Although Deleuze is a realist in the sense just described, his proposal for a transcendental structuralism is one of the first critiques of the traditional conception of meaning in structuralism. For Deleuze, meaning cannot be seen as completely detached from its psychic and physical realization; by contrast, meaningful forms and structures emerge from dynamical processes of matter. His proposal for a transcendental structuralism claimed that the structural concept of form must be replaced by a genetic concept of structure or form, as an emergent self-organization or individuation. In structuralism, the symbolic order of structure is considered to be a transcendent instance within which the differential relations are established, which determine the meaning of signs and which principally exclude the question of the ‘origin’ of meaning. By contrast, Deleuze proposed that symbolic structures should be modeled topologically (i.e. geometrically) and that the mathematical concept of singularities could be used to explain the emergence of the structural values in symbolic structures. As he put it:

The crucial notion of singularity, taken literally, seems to belong to all the domains in which there is structure. (…) Every structure presents the following two aspects: a system of differential relations according to which the symbolic elements determine themselves reciprocally, and a system of singularities corresponding to these relations and tracing the [topological] space of the structure. (…) It is not a matter of mathematical metaphors. In each domain, one must find elements, relationships and [singular] points. (Deleuze 2004, 177)

24In this regard, it is possible to see in Deleuze’s proposal a key forerunner to the more recent debate in the cognitive sciences between dynamical systems approaches to cognition and classical functionalist approaches. Traditional functionalism, e.g. in Fodor and Pylyshyn’s foundational work, entails a strict separation between the symbolic (cognitive) and physical levels – the relation between the physical and the symbolic being one of mere implementation or realization, as in classical structuralism. Deleuze’s topological approach to structuralism, however, sought to overcome this dualism by providing an emergentist approach (cf. Petitot 2004, 11). On this view, every symbolic structure is epistemically reducible to a system of singularities or qualitative discontinuities emerging from an underlying physical (e.g. neural) substrate of morphodynamical, generative mechanisms. As Petitot has argued, “structures are essentially dependent on critical phenomena, i.e., on phenomena of symmetry breaking which induce qualitative discontinuities (heterogeneities) in the substrates (…). Discrete structures emerge via qualitative discontinuities” (Petitot 1995, 231). In his seminal Morphogenesis of Meaning (1985), Petitot pointed to Deleuze’s early proposal for a transcendental structuralism as one of the key philosophical inspirations underlying his application of René Thom’s Catastrophe Theory (proposed in the 1970s) to cognitive semiolinguistics.

25However, just like Petitot’s dynamical structuralism, Deleuze’s genetic structuralism is at the same time still a true functionalism, because emergent structures have properties, which are to a large extent independent of the specific physical properties of their underlying substrata. In this sense, transcendental structuralism and dynamical structuralism share with traditional functionalism the refusal of ontological reductionism or reductionist physicalism (defending only a ‘softer’ epistemological reductionism), and recognize the immaterial content of mental acts (Petitot 2004, 69). As such, both transcendental and dynamical structuralism can be seen as a variant of what Petitot calls a “dynamical functionalism” (Petitot 1995, 11).

26Simplifying a lot, we may summarize Deleuze’s proposal in the following 3 key points.

3.1. The Topological A Priori

27Deleuze’s proposal starts with a monist-naturalist assumption, which is to say that both mind and life share fundamental organizational principles, and the organizational principles distinctive of mind are an enriched version of those fundamental to life. Like contemporary proponents of the enactive approach in cognitive science, Deleuze argues that the natural continuity between life and mind justifies approaching the emergence of structure and meaning in terms of morphogenesis (Deleuze 1994, 251).7 Structure and meaningful form emerge then like dynamical forms do in embryogenetic development, namely as morphodynamically self-organized and self-regulating structures or wholes.

28Now, insofar as every natural being has a form, one must imagine a space in which this form unfolds itself. Deleuze proposes to conceive this primary space as a continuous space that becomes categorized or discretized by a system of discontinuities or singularities. Structural space is divided into regions (places) by a system of differences, and for the elements in this space, the differences are not external relations between pre-existing identities but rather generative relations (Deleuze 2004, 174).

29Genetically, the generative relations between regions in structural space are obtained from the unfolding of singularities, whose unfolding organizes the structure of the space. These singularities are intensive discontinuities within the substratum space. This means ultimately that a meaningful form or structure must be defined as a system of intensive discontinuities or singularities that unfold within an underlying substratum space. However, as Petitot notes, the singularities themselves are not signifiable: “their inscription in the signifiable (expression and representation) reformulates them as lack, as lacuna” (Petitot 1977, 115).

30Petitot proposes a connection between this conception of singularity to Thom’s concept of “catastrophe,” which opens up a rigorous understanding of how the general and primitive phenomenon of discontinuity can give rise to structures. We will consider this application below (4.4).

3.2. Differential Structure

31Secondly, a structure becomes properly organized only by virtue of the principle of reciprocal determination. Deleuze proposes to conceive the principle of reciprocal determination in terms of Leibnizian differential calculus. Following the work of Jules Vuillemin and Hoëne Wronski, he proposes that differential calculus may function as a universal algebra of pure thought: a mathesis universalis of the Cogito. The general idea is that in all the different domains of thought – mathematics, literature, physics, philosophy, culture, etc. – the psychic individuation of new ideas involves one universal genetic mechanism taking place in structural, intensive space. The shift to differential calculus is meant to formalize this mechanism as an a priori principle of reciprocal determination, which is at work in the emergence of all new meaningful form or structure. In this sense, Deleuze’s transcendental structuralism extends the idea of constituent structure from language to the whole of thought – a classical functionalist move. Deleuze then defines three phases in the emergence of a structure:

1. First, there are undetermined differential elements (dx, dy), which corresponds to a principle of determinability;
2. Second, differential elements become determinable (dy/dx), which corresponds to the principle of reciprocal determination;
3. Finally, the determined (the values of dy/dx) corresponds to a principle of complete determination.

3.3. Virtual Multiplicity

32Deleuze maintains that the process of reciprocal and complete determination belongs to a differential unconscious of pure thought: an unconscious of pure reason. The primary model for this differential unconscious is Bernhard Riemann’s mathematical concept of multiplicity. In this context, Deleuze emphasizes that the concept of multiplicity may not be interpreted as a simple set or collection of elements that are united in some way. In the Riemannian sense, a multiplicity is a collection of elements, which are not merely united but also organized and connected in a continuous manner. A multiplicity is an organized connection that is defined by the relations between its elements rather than by those elements themselves. Deleuze then specifies the three conditions, which together allow us to define the process in which a structure emerges in the differential unconscious as a multiplicity:

1. Virtual indetermination: the elements of the multiplicity must have neither sensible form nor conceptual signification, nor, therefore, any assignable function. They are, however, inseparable from an “incorporeal” potential or a “virtuality”;
2. Reciprocal determination: the differential elements are determined through the process of reciprocal determination, which forms reciprocal relations that allow no independence of the elements to subsist. Such reciprocal relations are non-empirical, non-localizable ideal connections, which determine the ideal multiplicity globally;
3. Actualization/incarnation: the differential connections or relations of a multiplicity must become actualized in diverse spatio-temporal relations, at the same time as its ideal elements are actually incarnated in a manifold of terms and forms.

33Together, these elements allow Deleuze to define structure as an internal multiplicity: “a system of multiple, non-localizable connections between differential elements which is incarnated in real relations and actual terms” (Deleuze 1994, 183). But what exactly does the concept of multiplicity add to the concept of reciprocal determination? The important thing here is how we understand the organizational function of reciprocal determination. We already noted that a multiplicity can only be defined in terms of its connections between its elements, i.e. in terms of the relations established by reciprocal determination. Now, the concept of multiplicity brings with it the important idea of genesis or individuation: in a multiplicity, there are only elements with a particular function, value or signification insofar as the latter emerges from the connections between elements. Thus, the process of reciprocal determination is generative. It actualizes values by means of relations between elements that are in themselves merely undetermined and virtual. As such, the generative process of reciprocal determination is always a process of qualification.

34Once we understand that the emergence of a structure entails (1) a qualification of a virtual multiplicity, which is (2) driven by a genetic process of reciprocal determination, we should also see that the structure’s determination happens by means of a principle that is intrinsic to the system. The connections between its virtual elements are not external relations between pre-existing identities, but constitutive, generative relations of reciprocal determination within the system. Thus, the determination of a symbolic structure happens by means of a creative, organizational process that is immanent to its ideal-virtual structure. According to Deleuze, this is ultimately the fundamental property of structures: they dispose of a genetic, self-constructive principle of reciprocal determination that is characteristic only of ideal (also called symbolic or virtual) structures and not of the empirical real, the imaginary or even the conceptual.

35Thus, conceiving reciprocal determination as a process taking place in a virtual multiplicity allows Deleuze to develop its genetic, constructive, individuating nature: reciprocal determination is an ideal event taking place within a virtual structure that is in the process of being actualized and incarnated.

3.4. The Example of Linguistic Structure

36Let us consider at this point Deleuze’s example of the phoneme in linguistic structure (Deleuze 1994, 193; 203-6). In the discussed formal terminology, linguistic structure can be regarded as a virtual system of reciprocal connections between “phonemes” which is incarnated in the actual terms and relations of diverse languages. From a phonological point of view, a linguistic structure indeed has the mentioned properties of a differential structure:

1. The presence of differential ideal elements (phonemes) which are extracted from the continuous sonorous flux;
2. The existence of differential relations (distinctive features) which reciprocally determine these elements;
3. Finally, the ideal value or signification of singular points assumed by the phonemes in that reciprocal determination.

37From this point of view, linguistic meaning can be regarded as an ideal system of reciprocal connections between phonemes (1 & 2), which becomes incarnated (3) in the actual terms and relations of diverse natural languages. The process of reciprocal determination thus makes possible speech perception and actual speech as a faculty.

38How does this proposal differ from classical structuralism? In structuralist linguistics, the structural connections between phonemes, lexemes, words, etc. form the crux of syntax. These connections are immaterial, disembodied, or as Deleuze has it, “incorporeal.” Traditionally, structuralist linguistics understands the relational structure as a logical-combinatorial description of the relevant forms. However, in Deleuze’s topological approach, the reciprocal and complete determinations of a structure occupy particular places in a structure, which are defined by a dynamic process of reciprocal determination that is continuous and energetic, making use of relative forces linking the determinations. Deleuze introduced the mathematical concept of singularities to conceive the generating potentials, which underlie the reciprocal and complete determination of structure. In other words, for Deleuze structures become dynamical entities that are regulated and controlled in so-called possibility spaces or phase spaces.

4 | Naturalizing Constituent Structure: Dynamical Structuralism

39Jean Petitot’s dynamical structuralism is an attempt to ground cognitive phenomena such as language and perception – studied in semiotics, linguistics, phenomenology of perception, etc. – in the mathematics of René Thom’s Catastrophe Theory. One of its central theoretical ideas is that cognitive structures can be modeled at the macro-level by the topology of an “attractor” of an underlying micro-dynamics. Syntactic and semantic constituent structures should then be considered as emergent phenomena, analogous to the processes we find in physics under the name of critical phenomena and thermodynamic phenomena of phase transitions. This constitutes a key innovation compared to traditional linguistics: structures should be understood in terms of their emergence from underlying dynamical mechanisms.

40Secondly, a key idea underlying dynamical structuralism is that the syntactic and semantic constituent structures of language should be understood in relation to constituent structures of perception. Just like underlying dynamical mechanisms, so the structures of perception impose certain universal constraints on the grammatical structures of language (cf. Petitot 2011, 19). In line with Ray Jackendoff, Petitot proposes there is one single level of mental representation – “conceptual representation” – at which linguistic, sensory and motor information are integrated (Deleuze 1994, 22). Although this hypothesis remains compatible with the framework of computationalism, it introduces the idea that perceptual structures constrain semantic structures, and these in turn constrain syntax. Conceptual structure first transforms the physical world into a projected morphological world – the phenomenal, sensory world as qualitatively structured and phenomenologically organized – and into semantic structures, which in turn constrain the syntactic structures of natural languages. Thus, the perceptual level is more primitive than the semantic level: the latter is grounded on the former, not the other way around. This claim should be understood in an evolutionary sense: phylogenetically speaking, perceptual structures are deeper than linguistic structures. As a result of evolutionary adaptation, the perceptions and actions of an organism are fine-tuned to the qualitative structure of its environment (forms, qualities, etc.): this qualitative structure has become intrinsically meaningful to the organism. It is this perceptual / morphological structure, which in turn constrains linguistic structure. As Petitot puts it, “The idea is that the spatio-temporal a priori is deeper than the symbolic a priori: the human visual system is inherited from a very long natural evolution, while ideography and writing are extremely recent cultural acquisitions” (Deleuze 1994, 19-20).

4.1. Constituent Structure in Categorical Perception

41One of Petitot’s first proposals was a catastrophist reworking of Roman Jakobson’s structural phonology via the concept of “categorical perception.” Given Deleuze’s mentioned focus on the notion of the phoneme, we will briefly consider this case.

42Categorical perception is a remarkable phenomenon: although the acoustic-phonetic input of language is a variable continuum of sound, the perception of speech sounds is discrete, discontinuous and categorical. In speech, the acoustic-phonetic flux of individual sounds may vary substantially but listeners typically perceive them unambiguously as canonical sound types. For example, when presented with intermediate varieties between /b/ and /p/, listeners perceive each sound either as a distinct /b/ or /p/, and not an intermediate sound (Ishkhanyan et al. 2021). In other words, there is no ‘intra-categorical’ discrimination: listeners do not distinguish between two adjacent sounds unless they belong to two different categories. Contrary to what happens in continuous perception, categorical perception depends on an identification of categories, which turns the audio-acoustic continuum into a discrete structure that can be used to support a phonological, linguistic code. Indeed, by virtue of categorical perception, phonemes become encoded in the audio-acoustic flux (Petitot 2004, 83).8 And it is only the phonemes and their differences, which are linguistically pertinent – not the phonetic sounds as such.

43One of the key challenges of structuralist semio-linguistics is to understand the relation between the physical level of audio-acoustic sounds and the phonological level of linguistic forms (phonemes), which are encoded in the physical flux based on qualitative discontinuities and differential thresholds. How can one relate the essentially continuous nature of speech with the essentially discontinuous nature of speech perception and linguistic description? How can the audio-acoustic flux of sounds become encoded as instantiating the abstract, linguistically functional units or types, which are defined not by intrinsic (physical) properties but by a linguistic network of differential relations?

44Tackling this question, Petitot’s morphodynamic approach starts from two assumptions: a) categorical perception functions as an a priori form of perception, projecting its eidetic types onto the physical continuum, and b) this a priori form is essentially a sensitivity to discontinuity (Petitot 2004, 76; 94). The overarching idea is that the phoneme as a linguistic-relational form of expression emerges from the dynamical organization of the phonetic substance of expression. This implies assuming a morphodynamical mechanism “X” that is neurophysiologically implemented and generates this expressive, phonemic morphology. Instead of first defining this generating mechanism explicitly and then deriving the observable discontinuities from it, the morphodynamic approach first describes the observable discontinuities geometrically and then derives from them a minimally complex generating dynamics (Petitot 2004, 11). This minimal, explicit dynamics is then conceived as a simplification of the real, implicit dynamics (X).

45For Petitot, the morphogenesis of phonemes as abstract forms is structurally similar to the case of phase-transitions in physics: the boundaries between categorical units would correspond to the existence of critical points – for example, the end point for the liquid/gas interface for water (Petitot 2004, 100; 102-3). Categorical perception is modeled as an essentially non-linear process in relation to acoustic parameters. Although the physical stimulus changes gradually, categorical perception refers to the phenomenon where continuous variations in the stimulus are perceived as distinct categories with thresholds between them (e.g., r/l). This is the fundamental phenomenon that allows using models of bifurcation. Non-linear systems can exhibit bifurcations, where small, gradual changes in input lead to sudden, large changes in the system’s behaviour. In such systems, an attractor is a stable state toward which the system evolves. Applied to categorical perception, the brain/mind system can have multiple attractor states that correspond to different categories of perception, the attractors being phonemes. When input reaches a critical threshold, the system can settle into one of these attractor states, which corresponds to a discrete perceptual category.

46A crucial feature of the Catastrophist model of categorical perception is the concept of critical thresholds, which are points in the input space (e.g. acoustic properties of speech sounds) where the system undergoes a sudden discontinuous transition or “catastrophe” from one category/attractor to another. If, then, on the classical structuralist view, categorical perception functions as an a priori form of perception, projecting its categorical types onto the physical continuum, for Petitot, the transcendental ideality of linguistic form or structures emerges from the dynamical organisation of the phonetic continuum. This implies a morphodynamical mechanism that is neurophysiologically implemented and that generates this linguistic morphology.

47The idea is thus that categorical perception itself segments and categorizes the audio-acoustic continua by virtue of its sensitivity to qualitative discontinuities, thresholds or singularities. The internal cortical/perceptual state is controlled by the external audio-acoustic parameters. This neural/perceptual state remains stable (phonologically invariant) faced with much of the audio-acoustic flux, but it undergoes sudden variations (“bifurcations” or “catastrophes”) when the external continuum reaches certain critical values. For Petitot, the main import of Thom’s Catastrophist mathematical models concerns the analysis of these singularities and discontinuities, which emerge at the macro level from underlying micro-physical mechanisms. Their main purpose is to explain how observable morphologies, which dominate the phenomenologically experienced world (such as phonemes), can emerge from an underlying physics (in this case the neural substrate of speech perception). As such, the theoretical import of Catastrophe Theory is to introduce – by mathematical means – a new level of functional, cognitive architecture, which would function as an a priori condition of possibility for the implementation of syntactic processes into brain dynamics. The challenge for this approach is to show how it can also be synthesized with an effective computational theory, which captures other fundamental levels of cognitive architecture. However, dynamical functionalism was certainly not conceived as antithetical to computationalism.

48Moreover, the dynamical approach to categorical perception should also not be mistaken for an ontological reductionism or reductionist physicalism. What Petitot calls “morphodynamical functionalism” entails that “the relational form of expression is ontologically autonomous even if it emerges from the substratum matter” (Petitot 2004, 102). Phonemes are not treated as bundles of separately extracted phonetic features, but rather as integral eidetic Gestalt-structures, which serve as prototypes for a class of allophones. Categorical perception does not simply depend on a detection of acoustic physical cues, which are bundled into a unified percept, but rather the acoustic sound initiates the unfolding of a complex pattern of neural response that directly supports an eidetic, morphological, phenomenal experience (Petitot 2004, 105). Treating categorical perception morphodynamically entails thus two distinct levels of analysis: as a cortical/perceptual case of critical phenomena or singularities, dynamical structuralism models categorical perception as a psycho-physical process. But as a properly structural phenomenon, it treats categorical perception as a schematization of the phonological domain into structural categories with an eidetic content. As such, the morphological level is neither the conceptual and symbolical level, nor the purely neuronal one. As Petitot emphasizes, the differential thresholds and qualitative discontinuities, which discretize the continuum, engender a topology of symbolic/virtual places and reciprocal relations between positions, which are the veritable elements of ideal structures as envisioned by Deleuze (Petitot 2017; see also Petitot 1990, 2-3).

4.2. Constituent Structure in Visual Perception

49Turning to the phenomenology of visual perception, dynamical structuralism posits a morphological level of reality that is intermediate between the physical and the symbolical: “it is of a physical (emergent) origin but without being material, it is formal but without being symbolic; it is topologically and geometrically formal and not logically formal” (Petitot 1990, 145: my translation). Accordingly, the external world would contain a morphological, qualitative dimension of information, which has a physical origin but nonetheless is phenomenological in nature and, as such, is intrinsically significative. As noted, it is this morphological, intrinsically meaningful dimension, which serves as the basis for the properly symbolic information processing at higher levels of cognition. Its structural constituency derives from a spatio-temporal a priori that is phylogenetically more originary than the symbolic a priori of higher cognition. Indeed, one of Petitot’s key disagreements with classical computationalism concerns the idea that the morphological/phenomenological dimension cannot itself be derived from the syntactical-semantic: “this is clearly impossible, since the intrinsically spatio-temporal and dynamical dimensions of the morphological are not of the formal order in the sense of the logico-symbolical”(Petitot 1990). For Petitot, the classical computationalist approach is legitimate only in the context of a logical objectivism, a formal semantics and/or a phenomenological logic of essences. However, it is generally incompatible with naturalist approaches to cognition, since it offers no clue as to the existence of symbolic forms in nature. The morphodynamic approach, by contrast, assumes that all the constituent structures of mind, language and meaning are dynamical and geometric forms (Petitot 1990, 146). Of course, these must be symbolically translatable and treatable at superior levels of cognitive representation. But their type of objectivity can, for Petitot, not originally be that of symbolic objectivity. If the constituent structure or form of mind, language and meaning is to be natural, it cannot be symbolical. To formalize, describe and explain constituent structure, one must pass from the logical to the topological.

50How, then, does dynamical structuralism conceive of constituent structure at the spatio-temporal, morphological level? In this context, the paradigmatic case is the study of visual perception. Once again, the general problem is to understand how perceptual ‘macrostructures’ and their morphodynamics can emerge from an underlying neural ‘microlevel’. The approach is again functionalist, meaning it attempts to formulate the cognitive architecture (the constituent structure) that is implemented in neural networks considered as functional units.

51Taking his cue from David Marr’s pioneering work in vision research, Petitot starts from the question how vision can reconstruct 3-dimensional objects on the basis of 2-dimensional retinal images. Computationalists such as Marr and Fodor treat this question by distinguishing two types of information processing. On the one hand, there are peripheral, “modular” systems, which transform or “transduce” peripheral neuronal information into a propositional form that is adequate to symbolic cognitive processing. These processes function in a “bottom-up” manner, ascending from peripheries to the center of the brain. For Fodor, whose terminology we are adopting, phoneme or face recognition, and perception more generally, are typical instances of modular or vertical processes. Although these bottom-up processes already analyze sensory information, they are informationally ‘insensible’ to higher-order cognitive knowledge, belief, etc. of the subject.

52On the other hand, computationalism posits “central” or “horizontal” cognitive processes, which are non-modular, non-specific, descending “top-down” and interpretative (sensible to beliefs, memories, knowledge, etc.). Today, the wide popularity of top-down approaches to cognition – so called “predictive processing” and “predictive coding” approaches – consider information processing in vision to be essentially concerned with top-down predictions, which are compared with lower level representations to verify potential prediction errors (cf. Friston 2013). Marr and Petitot, however, place a firm emphasis on bottom-up processing, casting perceptual synthesis as the serial extraction of visual information from primary physical information. Such approaches lend themselves very well to natural (rather than artificial) vision, because unlike top-down processes, bottom-up processes are governed by universal rules that are largely ‘built’ into the brain at birth by biological evolution. They allow different species to extract particular key elements of images in the physical world and project them onto objects. Such extracted elements include contours, colors, intersections, and the crossings of lines and junctions, which animals use to discern objects, people, and facial gestures, to ascertain their placement in space (perspective), etc. For example, if the neurons of the brain of a hungry frog allow it to catch an earthworm by jumping up and capturing it, it is because its neurons are biologically wired, by evolution, to be sensitive to a particular configuration of visual stimuli, a particular form or Gestalt: longitudinal movement of any elongated subject (Berthoz 2000, 165-7). Similarly, the neurons of the brain of a cat allow it to catch a mouse by anticipating its future positions because the neurons are sensitive to the speed of the movement.

53This selective response of the mammalian eye to information makes vision something profoundly different from simply recording the image of a scene or a person, like a camera would, pixel by pixel. The mammalian visual system selects and discards certain contours and lines with a specific orientation, which form the physiological building blocks for the neural elaboration of the image that will be consciously perceived. By means of this bottom-up processing, each human visual system extracts the same essential information from an environment. It is what enables animals to scan an open field and tree line for predators and prey, allows babies to recognize faces, and it is why, after not having seen a friend for many years, we may instantly identify her in the midst of an anonymous crowd at the single sight of the facial expression characteristic of her smile (cf. Kandel 2016, 22).

54Petitot considers Marr’s bottom-up approach to be a neo-ecological one, close to James Gibson’s ecological theory of perception. Gibson posited that in an animal’s Umwelt there exist qualitative and cognitively significant structures, which are objective without being strictly physical (they are morphological in Petitot’s sense).9 An animal’s visual system would choose, select and extract particular morphological invariants and based on these, it would construct its inferences, anticipations and interpretations.

55At the heart of Marr’s computational theory of perception, we find three distinct levels of information processing. Marr based these proposed mechanisms on what is known about mammalian, and in particular, primate, vision. He distinguishes the following three stages:

1. The first level is that of the so-called “2D primal sketch.” It makes explicit the local geometrical and morphological organization of the sensory signal. The first stage is concerned with detecting light intensity edges or what Marr calls “zero-crossings” of the visual scene in order to represent it in a “primal sketch”. A zero crossing is an intensive point where the value of a function changes its sign; it correspond to a sudden change in intensity in the image. The primary processing of such zero crossings allows for segmentation processes that will support the intermediate and final (cognitive and inferential) stages, such as interpretation, recognition, understanding, etc. As Petitot emphasizes, the primal sketch essentially involves a local analysis of the sensory signal in terms of qualitative discontinuities: edges, ridges, blobs with a particular elongation and orientation, bars, etc.
2. The second level is that of the so-called “2 ½ D” and is an intermediate level between the 2D and 3D levels. It is, for Marr, the essential level of the theory, calling it the “pivotal point” of “pure perception.” It is a viewer-centered and globally organized level, integrating several modular computations carried out on the primal sketch: contours of visible surfaces, textures, stereopsis, movement, shading effects, etc. It represents the external world as composed of visible surfaces filled with sensible qualities and moving in three dimensions. Petitot notes this level is neither purely sensory (because surfaces are distal and not viewer-centered), nor purely objective (because the appearances are still subjective). Rather, it is concerned with interpreting certain 2D qualitative continuities as apparent contours of 3D objects. As such, he considers this level, along with Ray Jackendoff, as the level of phenomenological appearance or phenomenal consciousness.10 In Petitot’s terms, the 2 ½ D level is “of a true morphological nature” (Petitot 2011, 113). This processing of contours is crucial, because it is what allows the transition form 2D to 3D perception in space-time.
3. The third level of “3D models,” finally, is the objective level of real things and material volumes with their real properties. It is from this level that superior cognitive constituents of “conceptual structure” (Jackendoff) operate, e.g. the hierarchical decomposition of shapes into parts, the constitution of prototypes, etc. Jackendoff contends that consciousness lies in the 2 ½ D sketch and not in the 3D model, which corresponds to a higher cognitive level of the hierarchy that remains outside of our consciousness. Like many of our memories, anticipations and understanding, the 3D model would remains outside of conscious awareness yet influences heavily the final product of which we are conscious. It should be noted, however, that these contentions are purely psychological (independent from the actual characteristics of the neuronal implementation) and, in fact, no one really knows where the consciousness determining neurons are located.

56Following Marr and Jackendoff, Petitot considers perception to be, primarily, a bottom-up process that ascends from 2D => 2 ½ D => 3D => Conceptual structure. This bottom-up process possesses top-down feedback mechanisms, which regulate it by means of anticipation, inferences, interpretations, etc. Top-down feedback moves from conceptual structure “down” to the morphological level of 2 ½ D. As such, the 2 ½ D level would be the final level of bottom-up perceptual processing.

57The distinction between three levels of perceptual information processing corresponds to the above-mentioned distinction between the physical, the morphological and the symbolical. In fact, Marr himself emphasized the importance of the 2 ½ D morphological level for higher levels of symbolic representation. He introduced the idea that the passage from the physical to the morphological involves primarily finding the light-intensity edges of the visual scene in order to represent these qualitative continuities in the primal sketch. The morphological structure of perception would derive primarily from a detection of zero-crossings, which analyzes physical stimuli in terms of qualitative discontinuities of a physical (intensive) and geometrical nature, and which subsequently are processed as perceptual edges or borders. With regard to this morphological information processing, Marr writes: “zero-crossing provides a natural way of moving from an analogue or continuous representation like the two-dimensional image intensity values I(x,y) to a discrete, symbolic representation” (Marr 2010, 67; cited in Petitot 1990, 157).

58The morphological structure of perception begins with raw intensity values, which are gradually transformed into a more symbolic, compact and robust representation of the world. But this conceptual structuration of vision is for Marr and Petitot not essentially a top-down process. On the contrary, it starts from a bottom-up processing of intensity variations, which become coded in primitive morphological signs or representations. This morphological processing, taking place in the 2D and 2 ½ D phases, does not have to be inferred (top-down) from a priori supplementary knowledge. It is largely constructed bottom-up, starting from what Marr calls “the physics of the situation.” As Marr writes:

The raw primal sketch is a very rich description of an image, since it contains virtually all the information in the zero-crossings from several channels. Its importance is that it is the first representation derived from an image whose primitives have a high probability of reflecting physical reality directly. (Marr 2010, 71)

59If the first level of the “primal sketch” is concerned with detecting local singularities or qualitative discontinuities, the second, “pivotal” level of “pure perception” is concerned with integrating the local morphological information obtained in 2D. Marr emphasizes this is an internal representation of objective physical reality that precedes the decomposition of the scene into ‘objects’. This integrated, morphological level of pure perception – being still pre-conceptual and pre-objectifying – provides the cornerstone of Petitot’s transcendental aesthetics.

60Petitot emphasizes that perceptual information processing becomes accessible to symbolic, higher-order cognitive processing only on the basis of this intermediate morphological structuration of physical stimuli. The essential idea is that morphological structure is situated at a cognitive level between the physical continuum and the symbolical discrete, and that all natural vision presupposes it. As Petitot writes:

For natural systems (where the symbolical discrete cannot exist immediately), qualitative morphological discontinuities provide, once they are made explicit, the condition of possibility for the constitution of a symbolical level. Insofar as they are objective singularities that are encoded in the physical signal, they carry information (Petitot 1990, 157).

Conclusion

61In the spirit of the rich work of Jean Petitot, we would like to conclude this article with an epistemological reflection.

62Transcendental and dynamical structuralism start from a naturalist, genetic project of accounting for the dynamical individuation or emergence of cognitive a priori structure, from the perceptual up to language and thought itself. Whereas classical symbolic cognitivism (CLC) develops:

1. an inneist conception of the a priori and,
2. a symbolical, computational conception of conceptual structure,

63On the contrary, dynamical structuralism develops:

1. a transcendental, anti-inneist conception of the a priori and,
2. a dynamical, connectionist conception of conceptual structure.

64In both Deleuze and Petitot, this morphodynamical turn leads to a return to empiricism, which is supposed to be compatible with transcendental philosophy. Indeed, just like connectionist (subsymbolic) cognitivism (CNC) and artificial neural networks, dynamical systems theory applied to the brain/mind is an empiricist theory that derives the principles of thought from the causal history of the organism-environment interaction. However, if “active synthesis” (causal inference, categorial synthesis, predicative judgment, etc.) is not distinct in kind from and reducible to “passive synthesis” (pre-predicative intuition, temporal syntheses, affective syntheses, association, etc.), psychologism results (cf. Lopes 2024 & 2025).11 If all mental content is built from experiential input, then empiricist theory of abstraction prevails, which has been refuted by Kant and Husserl.

65For this reason, Fodor and Pylyshyn rightly refuted the “associationism” inherent to connectionism (see Fodor & Pylyshyn 1988). They claimed the Parallel Distributed Processing of the 1980s was basically equivalent to the kind of associationism in Locke or Hume, albeit dressed up in the jargon of vectorial patterns of activation, matrices of weighted connections, and gradient descent learning. Aknowledging this pivotal criticism, Petitot wrote:

There are certainly higher processing levels of the cognitive system, which are of a symbolic nature. But this does not entail that there are no lower levels that are of a dynamical nature. Associationist processes are certainly not sufficient to explain the structure of cognition, but they can nevertheless be necessary. Logical-symbolical superstructures can possess associationist infrastructures. The question is not whether CNC should replace CLCL (or if the latter should excommunicate the former), but to find out whether the structural hypothesis can or cannot be already elaborated at the dynamical level of cognitive processes (Petitot 2011, 188).

66In his late 2008 book, LOT2: The Language of Thought Revisited, Fodor himself explored the idea that “what’s innate is the geometry of the attractor landscape” (Fodor 2008, 161) Here, he endorses attractor landscapes in relation to his Language of Thought (LOT) Hypothesis, and argues dynamical systems theory and LOT are in fact compatible, but necessarily distinct in kind, not in degree, and therefore non-reducible – on pain of psychologism, one might add (cf. Lopes 2025). On his view, morphodynamics can account only for low-level transcendental philosophy (passive synthesis, association, etc.), but it cannot serve as the foundation for higher-level transcendental philosophy (active synthesis). That is, if passive synthesis is directly related to active synthesis and they are continuous, the one would be reducible to the other, and that would be exactly what Hume envisioned, and logico-syntactic phenomena would be reducible to the causality of associations. But that is psychologism.

67By contrast, for Petitot, the morphodynamical viewpoint should stay ahead of classical – “merely” empiricist – connectionist theories, and be compatible with symbolical, computationalist approaches. However, the question of an integration between morphodynamical and computationalist approaches remains an open, important area of investigation. If dynamical structuralism can indeed be a proto-symbolic structuralism compatible with the symbolic level, it must be able to account for the emergence of a structural, symbolic macro-level from an underlying dynamical micro-level, without falling into plain empiricism and psychologism. The lesson of dynamical structuralism is that if this is to be the case, then this dynamical level, in order to be considered cognitive, should itself already be structural. That is to say, it must be shown that the symbolic form of the cognitive system does not exhaust its structure, and that this form operates on a dynamical functional architecture. In other words, it must be shown that in the case of syntactic structures, there are two authentically structural levels which correspond to the dynamical and symbolic levels respectively. Crucially, then, it must be shown how a dynamic functional architecture is not merely derivative from empiricist theories of learning by imitation, association, assimilation or induction, but also deeply innately, genetically constrained by a structural a priori (Petitot 2011, 190-191). This is precisely the objective of a topological theory of syntax within the framework of dynamical systems theory.

Notes

• 1 When discussing the mathematical origins of structuralism, Deleuze also points to the domain of differential calculus, “specifically in the interpretation which Weierstrass and Russell gave to it, a static and ordinal interpretation, which definitively liberates calculus from all reference to the infinitely small, and integrates it into a pure logic of relations.” See: Deleuze (2004, 176).
• 2 Computational functionalism rests on the claim that high-level cognition requires a highly specific cognitive architecture, namely one that gives cognitive states a combinatorial syntactic and semantic structure (which forms a “language of thought”). There are several good arguments for this claim, which rest primarily on four closely related properties of cognition: the systematicity, compositionality, and productivity of cognition, as well as its inferential coherence. Systematicity refers to the fact that the ability to produce/understand certain sentences is intrinsically connected to the capacity to produce/understand certain others. Cognitive capacities always exhibit certain symmetries, so that the ability to think a certain thought (“John loves the girl”) is tied to the ability to entertain thoughts with semantically related contents (“the girl loves John” but not “2+2=4”). Compositionality is closely related to this: insofar as language is systematic, a lexical term must make approximately the same semantic contribution to each expression in which it occurs, independent of its context. The productivity or generativity of thought entails that the representational capacities of cognitive systems are unbounded, and there are indefinitely many propositions, which the system can encode. In other words, they have unbounded expressive power, although this is achieved by finite means. Finally, inferential coherence entails that mental processes use syntactic structure to encode inferential roles and relations between certain representations. For a detailed discussion, see Fodor & Pylyshyn (1988)
• 3 Cf. Dario Compagno’s instructive introduction on the relations between transcendental philosophy and semiolinguistics (Compagno 2018).
• 4 In his Prolegomena to a Theory of Language, the Danish structuralist linguist Louis Hjelmslev equally emphasizes the essential methodological importance of this category of reciprocal dependence relations: “It soon becomes apparent that the important thing is not the division an object into parts, but to adapt the analysis in such a way that it conforms to reciprocal dependencies that exist between these parts and that allows us to give an adequate account for these dependencies.” See Hjelmslev (1963, 22: translation altered).
• 5 Cf. Husserl’s analysis of the expressive sign in the first Logical Investigation.
• 6 Cf. Hinzen & Schroeder (2015, 149): “We conclude that grammar and human-specific forms of selfhood form an inextricable unity, inviting renewed attention to language as a constitutive condition of personhood in at least some of its forms.”
• 7 In Mind in Life: Biology, Phenomenology, and the Science sof Mind, Evan Thompson draws on developments in dynamical systems theory to investigate how form and structure emerge in both life and mind, prolonging Petitot’s seminal idea that structures are essentially dependent on critical phenomena, i.e. on phenomena of symmetry breaking which induce qualitative discontinuities in their substrates. See Thompson (2007, 71).
• 8 Contrary to what was initially assumed, it was discovered that categorical perception is not specifically phonetic, i.e. linguistic, but also exists in the perception of musical timbres, musical intervals, colors, etc.
• 9 The reason why significant factors of a physical environment cannot be considered purely physical is that nothing physical is significant in and of itself, but only for a certain system. For this reason, classical computationalists such as Pylyshyn argue for a divide between the physical and the symbolical: what is informationally meaningful for a cognitive system cannot be analyzed purely physically.
• 10 Petitot considers Marr’s 2 1/2D sketch to correspond quite neatly to Husserl’s analysis of “adumbrative” perception in Thing and Space and Ideas I. Like Marr, Husserl examines how objects immersed in an external objective 3D space can be constituted from 2D “adumbrations” (Abschattungen). See Petitot (2011, 114).
• 11 I am thankful to Jesse Lopes for numerous discussions, which have greatly guided my thinking in this article, particularly in these conclusory remarks.