Before I get into the mathemes themselves, I’d like to lay out the four terms by which we will need to read each matheme.
In Lacan, the “$” can be read as the barred or split subject. As Bruce Fink points out, "The castrated subject is the barred subject, the subject under the bar: it is a product of every attempt and intent to signify to the other" (73) Since my goal is to move these mathemes into the realm of OOO, we will need to read the “$” as the split object of onticology. As Levi Bryant states in The Democracy of Objects, “…[S]plit-objects refers not to a physical split such as the idea that objects can always be broken in half or divided, but rather to the split between the virtual proper being of objects or their powers and their local manifestations or qualities” (70). There are a couple of points that need to be made about this split, though. First, every object (including humans) is split between a virtual proper being and local manifestations. This means that “$” is representative of all objects, including the split subject in Lacan. Second, this “$” is the object proper, meaning that “$” represents precisely this onticological idea of an object consisting of both a virtual proper being and local manifestations (whether experienced or not by another object).
$ = Split ObjectS1 = Local ManifestationS2 = Environmenta = Withdrawal or Virtual Proper Being
Next, we will need to read Lacan’s “S1” not as a master signifier, but as an object’s local manifestation. For Lacan, as Fink points out, “An S1 is often recognizable in analysis by the fact that the analysand repeatedly butts up against the term; it may be a term like ‘death,’ for instance, or any other term that seems opaque to the analysand and that always seems to put an end to associations instead of opening things up” (Fink 77). In other words, the master signifier is an actualization of a lack in the subject, but an actualization that is so strong or frequently repeated that it blocks further actualizations. Onticologically, however, the local manifestation can be seen as a quality or actualization of a power in the object’s virtual proper being, often when it comes into an external or exo-relation with another object. As Bryant notes, “As a function of the exo-relations objects enter into with other objects, the attractors defining the virtual space of a substance can be activated in a variety of different ways, actualizing objects in a variety of different ways at the level of local manifestations” (Bryant 114). But like the master signifier that could keep the analysand from opening up, in a sense defining him, there is also a danger to each local manifestation. Because local manifestations are actualizations of withdrawn virtual or potential powers in the object, each object runs the risk of reducing the entire split object into only its local manifestations. “It is for this reason,” Bryant argues, “that the confusion of objects with their actualization in local manifestations always spells theoretical disaster, for in doing so we foreclose the volcanic potentials harbored in the depths of objects” (114). Therefore, like Lacan’s master signifier, each local manifestation harbors the danger of eclipsing the split-object by reducing it to its local manifestations at the detriment of its virtual proper being.
We will also need to read Lacan’s S2 in a modified way, since for Lacan the S2 represented at times knowledge and “‘other’ signifiers” (75). Fink explains how for Lacan it was the S2 or the group of signifiers, the chain of potential signifiers that gave meaning to the master signifier. In other words, the S2 is always multiple, S2s. And, “If S1 is not in place, every S2 is somehow unbound. The S2s have relations amongst themselves; they may be strung together in perfectly ordinary ways by a psychotic [someone who has no master signifier], but they do not seem to affect him or her in any sense; they are somehow independent of him or her” (75). Therefore, the S2 is a chain or group of signifiers that retroactively brings about the meaning of an S1. In onticological terms, however, we need to read this S2 as representing the environment of the object. For Bryant, though, “[T]he environment is not a container of substances or systems that precedes the existence of substances or systems. There is no environment ‘as such’ existing out there in the world…Rather, we have as many environments as there are substances in the universe, without it being possible to claim that all of these systems are contained in a single environment” (146). Environments, or S2s, are again plural. And because of the ongoing and metonymical form of identification, environments can be seen as in-formational. As Bryant puts it, “While there is indeed an identity to the object, in the sense that it has a virtual endo-structure that persists across time, this identity is always manifesting itself in a variety of ways” (166). In this sense, objects take on new forms and actualize new local manifestations, so that “[i]n both allopoietic and autopoitic systems, information is an event that makes a difference by selecting a system-state” (166). In this way, the environment has an effect on the object retroactively, by producing the space available for the object’s local manifestations, so that the system or virtual proper being of the object is what constitutes the environment. As Bryant points out, “Although this distinction refers to two domains (system and environment), the distinction itself originates from one of these domains: the system” (144). S2, or the realm of other possible signifiers, can effectively be read as the object’s environment, since both operate as multiple arenas by which the S1 becomes actualized retroactively.
Finally, for Lacan the object cause of desire is represented by the small “a.” Fink describes object (a) as “the leftover of that process of constituting an object, the scrap that evades the grasp of symbolization. It is a remainder that there is something else, something perhaps lost, perhaps yet to be found” (94). In short, object (a) is that which is in excess of every relation between S1 and S2. It is, speaking onticologically, that which withdraws from any relation. It is the object’s virtual proper being. As Bryant argues, “objects are always in excess of any of their local manifestations, harboring hidden volcanic powers irreducible to any of their manifestations in the world” (70). So that “[w]ithin the framework of onticology, the claim that objects are withdrawn from other objects is the claim that 1)substances are independent of or are not constituted by their relations to other objects, and 2) that objects are not identical to any qualities they happen to locally manifest” (70). Much like Lacan’s object (a) which escapes (and is thus produced by) every signifying relation, the virtual proper being of an object withdraws (and is inferred) from any of the object’s relations and local manifestations.