The concept of form is central to the history of Western metaphysics, aesthetics, and critical philosophy. In Plato, it establishes the distinction between metaphysics and physics, differentiating the intelligible from the sensible. In Aristotle, it functions as the primary term in the distinction between form and matter and occupies an important place in the theory of the four causes. In Spinoza, it refers to the essence of individual things actualizing substance.1 In Kant, the categories of the understanding determine the form of objects and of experience in general, while beauty is a purely reflective judgment concerning satisfaction or dissatisfaction in the form of the object thus determined. In Hegel, it is the dialectically integral relation of form and content, displacing the transcendental separation of form and matter, that endows the determinations of thought with their infinite movement. In Marx, value must be understood as a form—as form-determined—insofar as it is predicated not upon the material scarcity or constitution of a commodity, but rather the socially necessary labor time requisite for its production. For Adorno, the historical dimension of the artwork is legible in its law of form, which separates it from what it is not while nevertheless bearing the sedimented content of the social relations from which its autonomy distinguishes it. And in Badiou’s materialist dialectic, form returns to its Platonic identity with the Idea, yet undergoes an inversion insofar as it is the singularity of embodied procedures that generate the eternity of the Idea they incarnate, in the temporal modality of the future anterior. Throughout the history of Western philosophy, the concept of “form” thus anchors a series of metaphysical, transcendental, and dialectical determinations of relationships between the intelligible and the sensible, traversing movements of opposition, contradiction, or deconstruction involving movement, matter, content, history, process, and finitude.
How can we locate the concept of “structure” with respect to the perennial centrality of the concept of “form” in the history of philosophy? A curiosity of the concept of structure is that it occupies a position of latency throughout most of this history. Despite the occasional usage of the term, and despite its prevalence in classical, Renaissance, and modern architectural theory, it does not undergo development as a systematically determined philosophical concept until the twentieth century. Neither Plato nor Aristotle systematically develop a concept of structure. The term “structura” does not appear in Spinoza’s Ethics, though the term “forma” and its cognates occur seventy-three times.2 The concept of structure is nowhere to be found in the glossary of the Cambridge edition of the Critique of Pure Reason, and the term is not used with any conceptual precision by either Hegel or Marx. Yet, we might see, from our own vantage point in intellectual history, how Aristotle could have deployed a concept of structure to explain the consistency of the four causes; or how Spinoza might have used such a concept to describe the redetermination of axioms, definitions, and propositions characterizing his geometrical method; or how Kant might have theorized the relational “structure” of the faculties in their formal determination of nature or of aesthetic judgments; we might think that the concept of structure could have played a methodological role for Hegel in his theory of dialectical discursivity, or for Marx in describing the integral relationality of his major categories in Das Kapital. Of course, it was precisely by retroactively drawing a theory of “structure” from Spinoza and then applying it to Marx—a theory of the immanence of relational causality in its effects—that Althusser developed the concept of “structural causality” (a concept that might itself be considered an addition to or intervention upon Aristotle’s theory of causation).
The systematic development of the concept of structure in the twentieth century—and the import of that concept across linguistics, anthropology, philosophy, psychoanalysis, and Marxist theory—has probably obscured the position of latency it previously occupied. It now comes naturally to deploy this term in discussions of ancient and modern thinkers, as though the concept always enjoyed the prominent role it has played over the past one hundred years, to the point that even those who supposedly disavowed it are characterized as “poststructuralists.” Consider, for example, the following passage from Bas van Frassen’s 2008 book Scientific Representation. Defending a structuralist epistemology of science, van Frassen turns to Aristotle to exemplify the inaugural philosophical importance of “structure” as what unifies the referential dimension of theories with their methodological coherence. He writes:
The insistence on this unity as a hallmark of science has been with us in philosophy since the beginning. In Aristotle we see a remarkable parallel in his views of drama and on physics. The Physics presents us with a view of the structure of nature and natural processes, and also, in conjunction with the Posterior Analytics, of the structure of the science that deals with nature. The Poetics presents the structure of the human condition and of human action as they are depicted in tragedies, but also of the structure of those tragedies, that dramatize human existence, themselves.3
For van Frassen, “structure” holds together what theories describe with how they describe it, and thus plays a crucial epistemological and methodological role in the history of philosophy. I agree with him, and I think that he more or less accurately describes what Aristotle is doing in the texts he mentions. Nevertheless, Aristotle did not have at his disposal a coherently or systematically articulated concept of structure that would have allowed him to describe his project in the same way. The difficulty thus raised is how to articulate the place of “structure” in the philosophical tradition without occluding its relative absence from that tradition. Lacan and Althusser discerned a latent concept of structure in the texts of Freud and Marx, and they reconstructed the work of those thinkers by systematically developing that concept within the field of their theoretical legacy. In what follows, I want to locate the implicit place of structure within Plato’s phi-losophy—at the very hinge of the intelligible and the sensible—with reference to a single text: the cosmology articulated in the Timaeus. In doing so, I hope to defamiliarize any easy application of this concept to Platonic philosophy, while also theorizing the systematic import of its latency therein.
ONE, TWO, THREE …
Plato’s cosmological dialogue famously opens with a scene of enumeration. Socrates counts the three interlocutors who are present, and then questions the absence from his count of a missing fourth. In Jowett’s venerable translation:
SOCRATES: One, two, three; but where, my dear Timaeus, is the fourth of those who were yesterday my guests and are to be my entertainers today?
TIMAEUS: He has been taken ill, Socrates; for he would not willingly have been absent from this gathering.4
The dialogue thus opens with an incomplete count and a missing number—the presence of one, two, and three, but the absence of the number four from the scene of the dialogue. “Then, if he is not coming,” replies Timaeus, “you and the two others must supply his place.” “Certainly,” Timaeus assures him, “and we will do all that we can.”5 This is a classic instance of Plato’s suggestive scenography, and as is sometimes the case, it is what is absent from the frame of philosophical articulation, rather than what is present, to which our attention is drawn.6 The dialogue begins in incompletion, with the specification of a structural lack, and with a promise to do whatever possible to compensate for an absence. Since four is missing, three will have to supplement its lack.
In his exquisitely attentive reading of the Timaeus, titled Chorology, John Sallis draws attention both to the pertinence of the number three in Plato’s dialogue and to the absent fourth with which it begins. Noting that the dialogue opens by putting an absence in question, he suggests that this question of a missing place implicitly turns the dialogue toward the chora, the theory of place, from its outset.7 Yet, as he also knows, the chora occupies the place of the number three in Plato’s text, and, indeed, this is the number with which Sallis will be most concerned. He argues that “the first three words of the Timaeus”—one, two, three—“bespeak the dialogue as a whole,”8 and he says of the number three that “it is this number and the counting numbered by it that will be decisively repeated throughout the Timaeus.”9 Here he is certainly correct: the relationship between the discourses of metaphysics, physics, and that which lies uneasily between them will be consistently articulated as a tripartite structure in Plato’s text, one that traverses ontological, epistemological, and methodological levels of its theoretical development. What I want to argue is that what is missing from this tripartite structure is precisely the place of structure itself, of a concept of structure that might play the role of a fourth term, holding together the uneasy relational placement of the other three. In this vein, Sallis notes a reference in Proclus’s commentary on the Timaeus to the suggestion of Iamblicus that the absent fourth bears upon the orientation of the dialogue toward the sensible, natural cosmos—its treatment of physis—and therefore the absence of someone fit to discuss the realm of the intelligible, the metaphysical. But while it is true that metaphysics is not the focus of the dialogue, the place of the intelligible and the metaphysical is nevertheless clearly designated within the three part methodological scheme articulated by Timaeus, and this is precisely the place of the concept of Form.10 His discourse begins with a classic Platonic distinction between the intelligible and the sensible, metaphysics and physics, and a great deal of attention has been paid to the manner in which the theory of the chora—of space or what Plato calls the Receptacle—comes to mediate this two part structure through the introduction of a third term. What I think has been missing is a clear account of where the number four intervenes in this dialectical development, precisely as a concept of structure that is implied in the dialogue, but that is not explicitly included in its conceptual scheme, nor in that of Plato’s larger philosophical project. As we will see, the place of structure implicitly opens at the moment when Timaeus turns from the theory of the chora to his geometrical description of the four elements—fire, earth, air, and water. What he describes is the structure of these four elements. And although a consistent terminology to designate this term is lacking, the place that it would occupy were it more precisely articulated does a great deal of work in Plato’s cosmological system.
METAPHYSICS AND PHYSICS
While Aristotle devotes distinct courses to Metaphysics and to Physics, it is at the conjunction of these two discourses that Plato’s Timaeus unfolds. As Alfred North Whitehead points out, this hybridity is proper to the philosophical discourse of cosmology. The conceptual tension between metaphysics and physics determines the system of distinctions with which Timaeus introduces his speech:
First then, in my judgment, we must make a distinction and ask, What is that which always is and has no becoming; and what is that which is always becoming and never is? That which is apprehended by intelligence and reason is always in the same state; but that which is conceived by opinion with the help of sensation and without reason, is always in a process of becoming and perishing and never really is.11
Being is distinguished from becoming; intellectual understanding is distinguished from opinion; and reason is distinguished from sensation. These binary distinctions, dividing the double subject matter of the text along the axis of the rational and the empirical, control the development of Timaeus’s discourse through its opening section (28a–48e). The classic question of what mediates the relation between being and existence, intelligible form and sensible likeness (the problem of participation) is exacerbated in this text by the assignment of each to a distinct register of philosophical discourse. Timaeus states that
in speaking of the copy and the original we may assume that words (logos) are akin to the matter which they describe; when they relate to the lasting and permanent and intelligible, they ought to be lasting and unalterable, and, as far as their nature allows, irrefutable and immovable—nothing less. But when they express only the copy or likeness and not the eternal things themselves, they need only be likely and analogous to the real words (logos). As being is to becoming, so is truth (aletheia) to belief (pistis).12
The logos of Being is a discourse of immutable rational certainty, whereas becoming is the subject of an eikos logos, a reasoned account that is only likely rather than certain. Several lines later, eikos logos will be terminologically transformed into eikos mythos, a likely story.13 Thus logos and mythos are each tied, in turn, to the merely likely status of speech about becoming. In this application of the term eikos to both the logos and mythos of becoming, we can perceive the asymmetry of a potentially tripartite division of philosophical discourse, with logos assigned to the discourse on being, while the discourse on becoming is split between and synthesized by the categories of eikos logos and eikos mythos, such that eikos, (analogy), mediates the relation between mythos and logos. Being is to becoming as truth is to belief, so mythos is like logos insofar as it is sufficiently convincing to approximate truth. An eikos mythos will be an eikos logos insofar as it is a convincing copy of the rational and the true.
Hegel teaches us to be attentive to the four-part structure of distinctions between three terms. There is an opposition between Being and Nothing. Pushed to the point of contradiction, this opposition produces the movement of Becoming. Becoming is the negation of the negation Nothing understood as the negation of Being. Thus we have a double structure of distinction: Being is opposed to Nothing, and Becoming—as the negation of the negation, is opposed to Nothing. In the methodological dialectic of the Timaeus, the structure is the same but the logic is different. Logos is differentiated from eikos logos, whereas eikos mythos is not differentiated from eikos logos. Logos is reason, eikos logos is reasonable, and eikos mythos is convincing insofar as it is reasonable, and thus akin to reason. Whereas the logic of Hegel’s dialectic is negation, the logic of Timaeus’s dialectic is analogy. In both cases, however, the explicit introduction of three terms implicitly entails a four part logical structure. One, two, three—where’s number four? That is the question we always have to ask when we think dialectically.
As we move from methodological and epistemological problems toward Timaeus’s story of the physical construction of the cosmos by the Demiurge, the structural role of the number four continues to be important. In order to compose the world soul the Demiurge mixes together three different essences: Being, the Same, and the Different.14 But each of these, in turn, is made of two components: the divisible and the indivisible. The first step is thus to compose three separate mixtures of that which is divisible and indivisible within Being, the Same, and the Different before combining the result:
1. divisible and indivisible components of Being are combined into an intermediate essence of Being
2. divisible and indivisible components of the Same are combined into an intermediate essence of the Same
3. divisible and indivisible components of the Different are combined into an intermediate essence of the Different
The Demiurge then takes the mixture of the Different and combines it with the mixture of the Same, before combining the result with the mixture of Being in order to produce a single new compound, a fourth essence, the Soul. So, each of three essences is composed of two aspects, which are combined into three mixtures before these are combined to produce a fourth term.
The component parts of Soul are then divided into spheres whose movements are proportionate to each other, and these constitute the “symphony of proportion” which is the corporeal world.15 The unity of the Soul and the Body of the cosmos is a fabric of these proportionate relationships, the Soul enveloping and woven together with the body of the cosmos in all directions.16 This interweaving of the soul and the body has epistemological consequences: through the mixtures of the Same and the Different with which it is woven throughout the cosmos, the Soul is capable of sensing similarity or difference—that is, like-ness—whenever it comes into contact within anything belonging to either Being or Becoming. True opinions and true convictions come about through the resonance of the Different with statements about the perceptible, while understanding and knowledge derive from the resonance of the Same with statements about the intelligible. The Soul, as a fourth term composed of divisible and indivisible aspects of the Same, the Different, and Being, weaves these throughout the body of the cosmos to render it epistemically resonant, making truth relevant as it applies to both being and becoming. As a cosmological concept, the Soul the functions as a principle of unity drawing together metaphysics, physics, and epistemology. But it is in the cosmological role accorded to the four elements, and in the theory of their physical composition, that the place of structure is at issue most centrally, if only implicitly, in the Timaeus.
Timaeus first explains why there are four elements—fire, earth, air, and water—that compose the physical body of the universe. The primary two elements are fire and earth, which respectively account for the visibility and the solidity of the cosmos, and these are joined together through the mediation of air and water. Timaeus explains that “two things cannot be rightly put together without a third; there must be some bond of union between them.”17 Yet in the case of the elemental composition of the physical world, we require two mediating elements between fire and earth in order to uphold the three-dimensionality of the cosmos. According to Timaeus:
If the universal frame had been created a surface only and having no depth, a single mean would have sufficed to bind together itself and the other terms; but now, as the world must be solid, and solid bodies are always compacted not by one mean but by two, God placed water and air in the mean between fire and earth, and made them to have the same proportion so far as was possible (as fire is to air so is air to water, and as air is to water so is water to earth); and thus he bound and put together a visible and tangible heaven. And for these reasons, and out of such elements which are in number four, the body of the world was created, and it was harmonised by proportion, and therefore has the spirit of friendship; and having been reconciled to itself, it was indissoluble by the hand of any other than the framer.18
The physical solidity, consistency, proportionality, and indissolubility of the cosmos depend upon its quadripartite elemental composition, which endows it with “the spirit of friendship.” Again, four is the number of compositional cohesion in the dialogue.
It is following his enumeration of the elements and his account of the genesis of the world soul that Timaeus inaugurates a “second beginning” of his story, and here he develops his theory of the chora as a third kind mediating between Being and Becoming.19 In addition to intelligible model and sensible copy, there is a third kind that he refers to first as hypodoche (Receptacle) and then as chora (space). Moreover, while Being is grasped by reason, and Becoming is perceived by the senses, the chora must be apprehended by a kind of “bastard reasoning” (logos nothos).20 The subject of influential commentaries by Kristeva, Derrida, Sallis, and most recently Emanuela Bianchi, in her outstanding book The Feminine Symptom, Plato’s chorology will not be my focus.21 But here I would question Sallis’s judgment that “the first three words of the Timaeus”—one, two, three—“bespeak the dialogue as a whole.” It is indeed the case that, amid his account of the chora, Timaeus summarizes his discourse thus far by affirming his “verdict” that “there are being, space, and becoming, three distinct things that existed even before the heavens came to be.”22 The chora, space, adds a mediating term to the split between metaphysics and physics from which the cosmology initially sets out. But no sooner does Timaeus thus arrive, again, at the number three than he proceeds, again, to the number four: to the genesis and structure of the four elements.
STRUCTURE AND GENESIS
This is Plato’s famous account of the structure of the elements—and also of the cosmos as a whole—as modeled on the geometry of what have come to be called the Platonic solids.23 Fire has the geometry of the tetrahedron, Earth of the cube, Air of the Octahedron, and Water of the Icosahedron, while the totality of the cosmos is said to have the geometry of the Dodecahedron. This fifth geometrical figure is curiously mentioned as an afterthought, while Timaeus focuses his account on the geometry of the four elements. Each of these can be constructed on the primary basis of two elementary right-angled triangles—the isosceles half-equilateral and the scalene half-square—both of which in turn obey the structural principle of the Pythagorean theorem. We might say that the theorem is the geometrical Idea or Form grounding the composition of the triangles, while the triangles themselves combine to form the three-dimensional structure of the elements, from which the physical matter of the world is composed.
One could devote many pages to the details of Plato’s geometrical construction of the elements and their interrelations, as do A. E. Taylor and Francis Cornford in their important commentaries.24 But here I am interested not in summarizing the geometrical details but in the philosophical concept that should be assigned to the place of this geometry in Plato’s cosmology. As I have suggested, I think this is the place of the concept of “structure” in the dialogue. And I think we can add this term to the other three articulated by Timaeus as the fundamental concepts of his cosmology: in addition to Being, Becoming, and Space, there is Structure, and this is the answer to the question—where’s number four—with which the dialogue opens. What is the significance of this answer for Plato’s philosophy?
When Timaeus turns from the chora to his geometrical description of the four elements he says, in Jowett’s translation: “And now I will endeavour to show you the disposition and generation of them by an unaccustomed argument.”25 The key terms here, translated as “disposition” and “generation” are diataxis (διάταξιν) and genesis (γένεσνν). For diataxis, Liddell and Scott’s lexicon also has “disposition” or “arrangement.”26 Cornford translates the term as “formation.” Both Desmond Lee and Donald Zeyl translate it as “structure.”27 Several paragraphs later, as he turns from elementary triangles to their combinations in elementary bodies, Timaeus tells his interlocutors, “I have now to speak of their several kinds, and show out of what combinations of numbers each of them was formed.”28 He then deploys a different term in the same context as he had earlier used diataxis. In Jowett’s translation, he says, “The first will be the simplest and smallest construction,”29 where “construction” translates the term sunistamenon (συνιστάμενον), which Liddell and Scott translate as a verb: “to set together, combine, associate, unite, band together.” Cornford has “First will come the construction of the simplest and smallest figure;”30 Lee follows Cornford’s translation, whereas Zeyl gives us “Leading the way will be the primary form, the tiniest structure.” Zeyl thus translates both diataxis and sunistamenon as “structure,” designating both the act of constructing the figures and the geometrical configuration in which they are constructed. This is an apt choice, but it obscures the discrepancy of Plato’s conceptual terminology, a discrepancy due to the fact that he does not have a consistent concept for the consistent referent of his discourse.
Interestingly, Zeyl deploys the term “form” in the second passage quoted above as an equivalent term of “structure,” and this indeed Plato’s usage. The Greek is proton eidos kai smikrotaton sunistamenon (πρώτον είδος και σμικρότατον συνιστάμενον), which could be directly translated, hewing as closely as possible to Liddell and Scott, as “primary form and smallest assemblage.” So form and assemblage, or form and construction, or form and structure are here deployed as concepts with the same referent.
The problem with this usage, in a Platonic context, is that it is philosophically equivocal. The term eidos, which occurs a number of times in the account of elemental geometry, is either used casually or should be reserved for pure intelligible forms of Being that are not subject to Becoming or transformation—and it is difficult to tell which is the case. But the account of the geometry of the elements is clearly a story of their genesis, of their formation by the Demiurge as physical, rather than metaphysical entities. Indeed, the geometrical foundation of the elemental structures, the shared basis of the two primary triangles, enables several of them (with the exception of the cube) to transform into one another. The three dimensional forms can fragment into their triangles and recombine into different bodies. Since these are bodies, they are not purely ideal Forms. Significantly, though, these are also bodies that are so small (smikrotaton sunistamenon) that they are invisible, which violates the epistemological criterion of visibility that is normally applied by Plato to physical things. These are physical entities that are not sensible entities; they are insensible entities that are not metaphysical forms. In apparent violation of Plato’s carefully articulated relationship between metaphysics, physics, epistemology, and methodology, these are assembled forms, both eidos and sunistamenon, insensible forms that are susceptible to processes of generation and transformation that should be reserved for the realm of the sensible.
Both the terms diataxis and sunistamenon encode a suggestion of this ambiguity in their function as either nouns or verbs: a structure is structured, an assemblage is assembled. In the Timaeus, the four elements are liminal beings. Before they were put in order by the Demiurge, there were traces of them scattered throughout the Receptacle, which its movement sorts into regions of likeness and difference according to their inchoate qualities. At this point, “before they were arranged so as to form the universe,” the elements were “without reason and measure”31 (alogos kai ametros) (άλόγως καί άμέτρως). The four elements are thus both prior to and products of creation, of genesis. They exist initially as inchoate qualities, and then as quantitatively determined bodies, once the Demiurge has “fashioned them by form and number”32 (eidesi te kai arithmois) (εϊδεσί τε καί άρνθμοΐς).
Diataxis thus inhabits the exact point of coincidence between structure and genesis, the fashioning according to form of the primary constituents of physical bodies. The concept of diataxis either binds together or falls into the rift between metaphysics and physics. In this it is similar to the Receptacle, or chora, which also evades the binary of Being and Becoming, yet, unlike the chora, the formation of the elements is geometrical, ordered, quantitatively determinate. Moreover, the qualities of the elements, and thus the qualities of the entire physical world they compose, are also determined by their quantitative determinations. Fire is hot because the tetrahedron is the sharpest geometrical structure of the Platonic solids; Water is smooth because the icosahedron is the roundest; Earth is solid because the cube is stable. Diataxis or sunistamenon—structure, or disposition, or arrangement—entails a synthesis of quantity and quality, which is precisely how Hegel defines measure. The geometry of the elements is the primary measure of the physical world, and indeed the genesis of their structure is the genesis of measure itself.
THE ARCHI-THEATER OF WRITING
My argument is that Plato’s terminology is embroiled in ambiguity because he does not have a concept of structure—a consistently determined sense of such a term that runs throughout his philosophy—and thus he uses a number of different terms to designate its place. But this place is crucial, because both the content of his discourse on the significance of elemental geometry and the ambivalence of his conceptual terminology (shifting between “form” and “structure” or “assemblage”) suggest that structure is the place of participation in the Timaeus. The genesis of the structure of the elements is the genesis of structure itself, and thus of a relationship of likeness between metaphysical forms and physical bodies, between Being and Becoming. The chora, Space, is the medium of this genesis, the medium of participation. But to understand how it is the medium of participation, we need to move from one, to two, to three, to four: from Being, to Becoming, to Space, to Structure. As Timaeus says, “for these reasons, and out of such elements which are in number four, the body of the world was created, and it was harmonised by proportion, and therefore has the spirit of friendship.”33
The spirit of friendship alluded to here is not only the cohesion of physical bodies among one another, but also the participation of their particularity in the universality of the forms, and this spirit is embodied as structure, understood as the unity of form, quantity, and quality. Structure is the minimal physical correlate of metaphysical form.
But structure is also a concept with methodological implications. The philosophical discourse of cosmology unfolds at the intersection of the rational and the empirical, and structure is the site at which the rationalist and the empiricist dimensions of science are conjoined as the synthetic coherence of the intelligible and the sensible. This synthetic coherence is epistemologically secured, and methodologically tested, by the mathematical formalization of physical organization. If this was incipiently the case for Plato, as the eikos logos of the Timaeus makes clear, it is entirely the case for modern and contemporary science, in which the consistent relationality of equations is tested by experiment, and in which the empirical findings of the laboratory are vetted by criteria of mathematical consistency within the formalized field of physical theory. It is this relation of dialectical coordination and reciprocal critique between the rational and the empirical, through the mediation of formalization, that was the epistemological framework of the incipiently structuralist theory of science developed by Gaston Bachelard. In The New Scientific Spirit, he argues that we move beyond the primacy of the Cartesian subject in the theory and practice of science by learning “to measure precisely the limits of our thoughts, and to match, in a strict and rigorous manner, thought to experiment, noumenon to phenomenon, rather than allow ourselves to be misled by the deceptive appearance of substances both subjective and objective.”34 We now match thought to experiment through the mediation not only of mathematical formalism but also complex instrumentation, both of which not only filter the immediacy of sensible intuitions but also constrain the generalizations of preexisting mathematical determinations.
For Plato, geometry organizes the rational field of application to the thinking of physical qualities through the mediation of plausible (eikos) structural determinations of matter. Consider Whitehead’s evaluation, in 1920, of Plato’s degree of success in attaining this plausibility:
In the Timaeus Plato asserts that nature is made of fire and earth with air and water as intermediate between them, so that ‘as fire is to air so is air to water, and as air is to water so is water to earth.’ He also suggests a molecular hypothesis for these four elements. In this hypothesis everything depends on the shape of the atoms; for earth it is cubical and for fire it is pyramidal. Today physicists are again discussing the structure of the atom, and its shape is no slight factor in that structure. Plato’s guesses read much more fantastically than does Aristotle’s systematic analysis; but in some ways they are more valuable. The main outline of his ideas is comparable with that of modern science.35
It is the emphasis of Platonic cosmological theory upon structure that renders it comparable, eikos, to twentieth century science. And the “likeliness” of physical theory—its aspiration to the status not just of opinion, but true opinion, if not absolute truth—is vetted by the plausible cohesion of geometrical principles with physical traits and relationships. The latent but not systematically developed concept of that cohesion—provided neither by the rational theory of forms nor the empirical immediacy of the sensible—is structure.
But what exactly is the status of these ideal geometrical formalizations which Plato speculatively coordinates with the phenomena of the physical world, as the elementary structure of matter? There is no story of the genesis of geometry itself in the Timaeus; its formalisms are already available to the Demiurge who structures the minimal constituents of material bodies after their model. In the framework of the dialogue, and indeed of Platonic theory more generally, these Forms are eternal. But this will not suffice for a rationalist empiricist approach to the philosophy of scientific knowledge. Geometry does indeed have a material genesis; it has a history. The problem of the genesis of this history, of the possibility of the historical transmission of geometrical idealities, and thus of the relation between structure and genesis within the field of geometry, is precisely the problem addressed by Husserl in his late text, “The Origin of Geometry” (1936) and by Derrida in his early introduction to Husserl’s essay (1961). Indeed, this introduction is Derrida’s most lucid and consequential encounter with the famous problem of “structure and genesis,” through which the histories of structuralism and deconstruction intersect.
In his text, Husserl emphasizes the constitutive importance of written signs in the transmission of geometrical knowledge. Without assigning the origin of geometry to the agency of writing, he does note that there is a passive genesis involved in the reception of written signs, whereby what is transmitted and legible as writing “is passively awakened [and] can be transformed back, so to speak, into the corresponding activity.” This “capacity for reactivation,” Husserl argues,
belongs originally to every human being as a speaking being. Accordingly, then, the writing-down effects a transformation of the original mode of being of the meaning-structure, within the geometrical sphere of self-evidence, of the geometrical structure which is put into words. It becomes sedimented, so to speak. But the reader can make it self-evident again, can reactivate the self-evidence.36
It is not just structure, but written structure, that constitutes the condition of possibility for the sensible transmission and intelligible reactivation of geometrical idealities.
In his “Introduction,” Derrida notes that “by absolutely virtualizing dialogue, writing creates a kind of autonomous transcendental field from which every subject can be absent.” He then devotes to Husserl’s remarks on written signs a declarative paragraph that inaugurates the field of grammatological research as an investigation of a transcendental field without a transcendental subject:
In connection with the general signification of the epoché, Jean Hyppolite invokes the possibility of a “subjectless transcendental field,” one in which “the conditions of subjectivity would appear and where the subject would be constituted starting from the transcendental field.” Writing, as the place of absolutely permanent ideal objectivities and therefore of absolute Objectivity, certainly constitutes such a transcendental field. And likewise, to be sure, transcendental subjectivity can be fully announced and appear on the basis of this field or its possibility. Thus a subjectless transcendental field is one of the “conditions” of transcendental subjectivity.37
The import of this paragraph is that it displaces the constitutive status of the transcendental subject, in formulating mathematical idealities, into the transcendental field of writing as a condition of possibility for the production and transmission of formalizing thought. We note that if this subjectless field is “transcendental,” insofar as it is a condition of possibility for transcendental subjectivity, it is not ideal. It is a material field of written traces. Therefore, what is at issue is a material genesis of the transcendental, which is exactly the amphiboly Kant’s framework of transcendental critique aims to block. If, for Derrida, writing constitutes a subjectless transcendental field, we can also say that it is a condition of possibility that deconstructs the transcendental, since it ungrounds the ideality of the transcendental through an account of its material genesis.
Here we encounter a theme that also marks Badiou’s early work, during the period of his involvement with Cahiers pour l’Analyse. In “Mark and Lack,” he describes science as “the psychosis of no subject, and hence of all: universal by right, shared delirium, one has only to maintain oneself within it in order to be no-one, anonymously dispersed in the hierarchy of orders.”38 Here, science is the dispersion of the subject into an anonymous transcendental field that forecloses the unity of consciousness through its ramified stratification of discrepant orders of formalization. At the end of that essay he writes: “Science is the veritable archi-theatre of writing: traces, erased traces, traces of traces; the movement where we never risk encountering this detestable figure of Man: the sign of nothing.”39
What Derrida and Badiou share in the 1960s is an anti-humanist theory of scriptural formalization as the displacement of the subject of consciousness into the historical movement of rational inscription. At the crux of the structuralist project, and of its discrepant legacies, we thus find a theory of written formalization as the material condition of possibility for the production and reception of mathematical structures, and thus for the subjective consciousness of geometrical idealities. To a degree, this is in accordance with Platonic theory, since the existence of the Forms is certainly anterior to subjective consciousness of their universality. But a theory of the dependency of mathematical ideas on the historical transmission of written signs reinscribes Platonic theory within a materialism of the idea. Derrida displaces the Husserlian question of the origin of geometry, the problem of its singular point of genesis, into a plural genesis of signs through which the capacity for geometrical thought is constructed—as Badiou aptly puts it, an “archi-theatre of writing: traces, erased traces, traces of traces.”40 The place of structure, in Plato’s Timaeus, is the place at which the already-inscribed structure of geometrical idealities is inscribed at the intersection of the rational and the empirical, metaphysics and physics, and at which the capacity of such structures to unite quantity and quality within an eikos mythos, a likely story, is the very condition of plausible rationality: of an eikos logos that is neither relativistic nor absolute, but rigorously coherent and thus subject to revision. The development of a materialist scientific structuralism—which is how I would characterize the agenda of Bachelard’s epistemological writings—has its implicit genesis in Plato’s cosmology. If rationalism and empiricism are joined “by a strange bond, as strong as that which unites pleasure and pain,”41 perhaps it requires so strange a text as the Timaeus to indicate the structure of that bond, for which it makes an absent place.
One, two, three, four.
CODA: STRUCTURE AND FORM
Form may be considered as ideal, material, or phenomenal, designating mathematical or conceptual abstractions, physical shapes, or exterior appearances. Structure mediates relations between the ideal, the material, and the phenomenal. This concept of structure, drawn from my reading of the Timaeus, is implicit in the preceding studies of the metrological redefinition of the kilogram and of Nicolas Baier’s art practice.
The shape of Baier’s meteorite is abstracted from the object as a digital model, and it appears as a sculptural reproduction after the object’s disappearance. Its form may be considered in mineral, digital, or sculptural manifestations; structure passes through and mediates these transformations without quite conforming to any one of them. It pertains to the relation between chains of digital code, or molecular bonds, and the form of the object they sustain. Structure enables the reintegration of form and matter—the retention of the former across transformations of the latter—through these discrepant instantiations. As the mediating relation of ideal, material, and phenomenal form, structure never rests easily within these categories; it is the how of form’s passage between them.
The redefinition of the kilogram abstracts a quantity from the qualitative properties of an object constructed as the material instantiation of that quantity; it reinscribes that relation between quantity and quality by transforming it into a distributed apparatus, a procedural system regulating relationships between devices, numbers, and operations. Redefinition shifts the fundamental referent of a metrical unit from a material object to a number, from the International Prototype Kilogram to the Planck constant; it is the structure of the relation between quality and quantity that makes this transformation possible. Whereas form may be understood to designate the concept “kilogram” (from a Platonic perspective), the physical shape of “the kilogram,” or its phenomenal appearance, structure may be understood to hold these discrepant senses together and also to mediate between radically different relationships between ideal and material form: the difference, for example, between the kilogram understood as the mass of a liter of water or of a platinum cylinder. Structure is the relational coherence of transformations among material and ideal form, and of their relation to phenomenal appearance.
The earth, for example, appears as a form. Structure is what makes thinkable, also, the transformation of its form prior to manifestion: its formation. In this sense there is no opposition between “structure and genesis.” In the case of the formation of the earth, structure may be understood as those relations among physical materials, chemical events, temperature, gravitational force, and so on that are requisite for the form of the planet to come into being, to persist, and to change without ceasing to persist. Structure is ideal/material: it is the relational determination of the in-itself. As relation, it does not appear. If form can be thought, and if form can be sensed, structure is the mediation of the thinkable and the sensible. That is why we can conceive of structure as the mediation of the rational and the empirical.