Following PEbird’s suggestion, it may be worthwhile to conceive the decision between set theory and category theory as a sort of parallax at the heart of ontology. As Zizek describes it,

The standard definition of parallax is: the apparent displacement of an object (the shift of its position against a background), caused by a change in observational position that provides a new line of sight. The philosophical twist to be added, of course, is that the observed difference is not simply “subjective,” due to the fact that the same object which exists “out there” is seen from two different stances, or points of view. It is rather that, as Hegel would have put it, subject and object are inherently ‘mediated,’ so that an ‘epistemological’ shift in the subjects point of view always reflects an ‘ontological’ shift in the object itself. Or– to put it in Lacanese –the subject’s gaze is always-already inscribed into the perceived object itself, in the guise of its ‘blind spot,’ that which is ‘in the object more than the object itself,’ the point from which the object returns the gaze. (PV, 17)

A parallax, then, is not a subjective point of view, not a relativism to a subject or observer, but rather a displacement or split ontologically inscribed in the object itself. This split, moreover, is absolute in the sense that it doesn’t allow for a higher resolution or reconciliation, but is rather irreducible. As Luhmann aphoristically puts it in “The Cognitive Program of Constructivism and the Reality That Remains Unknown”, “Cognition deals with an external world that remains unknown and, as a result, has to come to see that it cannot see what it cannot see” (Theories of Distinction, 129). And a bit more forcefully later, “Reality is what one does not perceive when one perceives it” (italics mine, 145). Luhmann, of course, was not familiar with Zizek’s concept of the parallax as it had not yet been invented, but the two are remarkably close here. A parallax revolves around that blind spot, that which I do not perceive, my gaze inscribed in the thing itself, that allows beings to appear as they do. If I adopt this point of view on the world– or an Whithead’s language, if I am a “superject” of this particular blind spot –another field becomes invisible to me (incidentally, this calls for a radically different reading of Hegel or new dialectics that has not yet been formally worked through or elaborated).

Like Holbein’s Ambassadors above, I can either see the men depicted in the painting or the skull floating in the foreground, but I cannot see both the figures in the foreground and the skull flying in the foreground at one and the same time. A parallax thus has the structure of a moebius strip. Topologically a moebius strip has only one side and one edge. However, in order to discover this, I require time. It is only by traversing the surface of the strip that I discover that what originally appeared to be on the other side was really on the same side. Zizek’s suggestion seems to be that a parallax is such that “front” and “back” turn out to be on one and the same side, but only when unfolded in the order of time or through a series of dialectical operations. As Zizek argues with regard to sexual difference, feminine sexuation is not the Other of the logic of the One found in masculine sexuation, but rather the the parallax of masculine sexuation or the One.

I am still trying to get my mind about the idea of the parallax and how exactly one goes about constructing one or perceiving a parallax at work. Despite this, PEbird’s suggestion that the opposition between category theory and set theory forms a parallax strikes me as fertile and perhaps productive in staving off a number of debates likely to emerge around ontologies implicitly organized around this opposition.

The primitive operation in set theory is that of membership. On the one hand, set theory in its ZF formulation leaves the concept of set itself undefined. Initially this might appear to be a point of minor interest. However, this point is crucial for the thinking of difference for if we could give a univocal definition of “set”, we would have reintroduced the One back into ontology, such that the One would precede difference. In leaving “set” undefined– if we were being cute Lacanians we could aphoristically say “The Set does not exist” –set theory insures that pure multiplicity without One serves as the ground of being. Being can thus be conceived as pure multiplicity without One, such that any identities we find among beings are the result of operations, but are not themselves primitive. As Badiou puts it,

if an ontology is possible, that is, a presentation of presentation, then it is the situation of the pure multiple, of the multiple ‘in-itself’. To be more exact; ontology can be solely the theory of inconsistent multiplicities as such. ‘As such’ means that what is presented in the ontological situation is the multiple without any other predicate than its multiplicity. Ontology, insofar as it exists, must necessarily be the science of the multiple qua multiple. (28)

When we subtract all other predicates from being– for instance, Aristotle’s categories –we are left with the multiple alone. It is precisely set theory that is capable of thinking such pure multiplicities. If it proves necessary to separate being from the One so as to think it as pure multiples of multiples without One, then this is because the suture of being to the One leads to irresolvable paradoxes as can readily be seen in Plato’s dialogue Parmenides. Any unities we do find among beings are thus an effect or product of an operation, as the One is not and being consists only of multiples of multiples. Of course, this gives rise to a number of questions as to who or what operates these operations and how it is possible to move from pure multiplities to the various ones we find in the world about us.

Luhmann begins from the other side of the issue. Sadly Luhmann’s work is almost completely unknown in the English speaking world. Indeed, whenever I evoke his name I always feel a little bit silly as no one seems to know what I’m talking about. Nonetheless, it is my view that Luhmann has developed the most robust account of social systems or organizations currently available, and one that is fully compatable with a number of other trends already out there in social thought. Where Badiou begins with pure multiplicities, Luhmann begins from the perspective of constituted organizations. Where Badiou begins from the perspective of set theoretical membership as the primitive operation in ontology, Luhmann begins from the perspective of relation. That is, Badiou’s elements are ultimately unrelated, whereas Luhmann’s elements always only exist in relation. As Luhmann puts it,

Whether the unity of an element should be explained as emergence ‘from below’ or as constitution ‘from above’ seems to be a matter of theoretical dispute. We opt decisively for the latter. Elements are elements only for the system that employs them as units and they are such only through this system. This is formulated in the concept of autopoiesis. One of the most important consequences is that systems of a higher (emergent) order can possess less complexity than systems of a lower order because they determine the unity and number of the elements that compose them; thus in their own complexity they are indepdendent of their material substratum. This also means that the complexity that is necessary or sufficient to a system is not predetermined ‘materially,’ but rather can be determined anew for every level of system formation with regard to the relevant environment. Thus emergence is not simply an accumulation of complexity, but rather an interruption and new beginning in the constitution of complexity. Accordingly, we take the unity of an action to be not a psychological, but a sociological fact; it does not emerge through the decomposition of consciousness into the smallest unities that cannot be dissolved further, but rather through the social process of attribution. (Social Systems, 22-23)

For Luhmann, then, we are unable to think elements independent of their relations and systems in which they are constituted. Elements are only elements for a system. Take Althusser’s critique of humanism. If I am not mistaken, Althusser’s point is that humans are always humans for the particular ideological state apparatus that subjectivizes them or constitutes them as beings. There is no humanity that underlies all historical change, but only the autopoiesis that occurs in each and every historical situation, constituting subjects as this type of subject at this particular point in time. The beauty of Luhmann is that he gives an abstract formulation of the mechanisms by which social systems constitute these identities or subjects.

Now Luhmann is a functionalist in his account of the mechanisms by which these operations take place. Here we must take great care not to confuse Luhmann’s functionalism with some sort of utilitarianism, where needs and desires are pre-established and the social is teleologically driven to certain ends. Luhmann’s functionalism is neither a teleology, nor a utilitarianism, but would better be thought in mathematical terms as akin to mathematical functions, or as what category theory refers to as “arrows” presiding over morphisms and transformations. That is, functions describe the processes by which elements are constituted.

So far it thus appears that there’s a remarkable proximity between Luhmann and Badiou. Both are in agreement that there are no primative elements, but rather that elements are products or effects of operations, out of what is essentially chaos. As Luhmann puts it,

Every observation– this holds for second-order observations as well –uses a distinction to mark one side (but not the other). No procedure can get around that. Even negations presuppose the prior distinction and indication of what one wants to negate. One cannot start from an immediately given nondetermination– an unmarked space, a primordial entropy or chaos, an empty canvas or a white sheet of paper –without distinguishing this state from what is being done to it. Even when moving toward fictionality, away from the real world in which we exist, we need this distinction in order to indicate the ‘whence?’ and ‘where?’ This is how we construct reality. (Art as a Social System, 55)

(It’s interesting to note that Deleuze, in Francis Bacon: The Logic of Sensation, categorically denies that there’s such a thing as a blank canvas, but that there’s always a virtual painting in the canvas already, populated by the history of painting). Luhmann’s point is that organization emerges from this chaos. The principle difference between Luhmann and Badiou would be that where the former believes that we can say nothing about this chaos, Badiou argues that it is precisely this chaos– inconsistent multiplicities –that ontology studies. Where Luhmann focuses on the operations by which elements are constituted, Badiou focuses on this chaos of inconsistent multiplicities revealed by set theory. Similarly, both Badiou and Luhmann would agree that there are only situations (or in Logiques des Mondes’ terminology, “worlds”), and no global whole in which these situations inhere. For Luhmann this would be a logical consequence of the fact that worlds only emerge through drawing a distinction in which one space is unmarked (there’s always a blind spot or, following Zizek’s Luhmanianism, a frame), whereas for Badiou it follows as a consequence of the thesis that the One is not. There is thus a sense in which the two positions are complimentary.

However, we might wonder whether Badiou’s formalism and tendency (until Logiques des Mondes) to ignore the structure of situations, doesn’t cause problems elsewhere. As Luhman remarks, systems do not simply constitute their elements, they also regulate their elements. “Out of the relation among elements emerges the centrally important systems-theoretical concept of conditioning. Systems are not merely relations (in the plural!) among elements. The connections among relations must also somehow be regulated. This regulation employs the basic form of conditioning. That is to say, a determinate relation among elements is realized only under the condition that something else is or is not the case” (Social Systems, 23). In Althusserian terms, this would be the reproduction of the conditions of production, where the social system must perpetually reproduce itself by excluding other forms of relating among elements. The great service that Badiou has done, the tremendous power of his thought, lies in showing how formal ontology brings us to the point where we can see how an event is possible, or how there can be something other than the structuration of a situation. However, in ignoring the mechanisms by which situations regulate themselves, one wonders whether Badiou’s truth-procedures might not be a bit utopian in not attending to the sort of action necessary to transform the self-regulating mechanisms of social systems.

Where Luhmann is blinded to the possibility of events by virtue of seeing too much closure in systems (nothing seems to escape the constituting force of systems), Badiou, in focusing on pure multiplicities and formalism, seems blinded to the regulatory functions of systems in constituting their elements. I’m not sure where these wandering musings lead, but hopefully they get a bit further in framing a question.