Subset of Theoretical Practice

The two-part presentation based on the text 'The Hustle of Struggle' was one of the richest and densest ones yet. It brought together a detailed analysis of a concrete political sequence in recent Brazilian history – the courier brakes –, a great summary and application of many of STP's ideas and a bunch of new developments as well. In this written commentary on the first of the two meetings, I would like to focus on a particular but very meaningful contribution, namely, the extension of the theory of "political bodies" through the use of an alternative definition of topological spaces.

Since a previous two-for-one presentation, from about nine months ago, people in STP have been exploring a way to think about the interactional aspect of our "organizational trinitarianism" by focusing on the idea of boundaries. The basic idea is very elegant, and it involves formally defining what a space is through Kuratowski closure axioms rather than the usual definition through open sets and, in this way, highlighting formal properties that are not necessarily conceptualized in Badiou's own presentation. Through the closure axioms, which allow for the distinction between interior, boundary and exterior of some space, it seems possible to locate and conceptualize what political struggle between forces would look like from a Badiouian standpoint.

But before we understand what these alternative axioms for topological spaces help us do, let us first understand the role of topological spaces in Badiou's own work. And this is perhaps most clearly articulated if we consider his work together with that of a close comrade of his, Sylvain Lazarus. One crucial shared idea between Lazarus and Badiou, two comrades from the now extinct L'Organization Politique, is that of thinking politics "in interiority". To think in interiority is to accept that one can only locate what is truly being thought in a political process by learning to recognize the inner language of that movement. Lazarus greatest quip about this was to say that political thinking was "a relation in the real" (rather than "to" the real), which Badiou later interpreted as a practice of immanent modeling – the possibility that, through our collective engagement with transforming the world we might also produce a better picture of the world as it is. For Lazarus, the point of view of interiority serves to oppose both those doctrines that suggest that when people revolt they are not thinking, merely reacting to social forces, as well as those that claim that political thinking is only present insofar as people align with theoretical or historical templates that we already recognize as politically creative or effective. With Badiou, we get to explore the structure of this interiority – rather than simply relegate it to some empty concept: as always, Badiou does not attempt to say what political strategy should be like, but he does give us some crucial conceptual markers on how to rationally conceive of a political process as something whose determinations qua political come from its intrinsic properties. Though these insights correspond, ultimately, to the whole theory of the subject and truth proposed in the three volumes of Being and Event, we can focus, for the purposes of this short note, on the specific contribution of Logics of Worlds, where the issue of interiority is most explicitly dealt with.

Badiou begins Book VI of Logics of Worlds with a definition of a "faithful subject" as "the form of a body whose organs treat a worldly situation point by point" (LoW, 399). Let us unpack this: a subject is (1) a form – meaning, it has distinguishable properties – of (2) a body – meaning, these properties are those of a material extension, of how it is put together – with (3) organs – that is, individuated parts that maintain some relation between them – that (4) treat – i.e. this organized body acts on something – (5) a worldly situation – recall that "world" means a logical space supported in material reality – (6) point by point – that is, which interact with a world in a localized and partial fashion. This already clarifies a bit better the ordering of these terms: since the form of the body is derived from the logic of its organs, which are themselves defined by the capacity to treat local parts of the world, the order here is inverted – points determine organs which determine bodies, which is in fact the order in which the concepts will be introduced in the book.

The theory of points asks, thus, how to treat a part of the world in such a way that we might produce a minimal consistency between something locally new – a "strong singularity" as Badiou calls in in Book V – and some already established thing, action or region of the situation. How to follow up on a love encounter –when to schedule a next date? How to turn an out of place musical sample into the basis for a new beat? How to produce, out of the mismatch between expectations and experimental results, a new testable hypothesis, or at least some useful information? Or – what will interest us here – how to turn a political meeting into a durable engagement, or a punctual protest into lasting collective relations?

It is precisely here that the two doctrines that Lazarus opposes would come in with ready-made answers: the first would claim that if such duration or consistency was found it is because people stopped thinking and actually are adhering to herd mentality, fanaticism or following some automatic social dynamic, while the second would lay down the general principles of organization that could make this happen, either according to philosophy or political history. What Badiou wants to demonstrate is that we do not need to place a political process within society or History to find justification for how people act politically – resorting, in the first case, to an established theory of motivations, in the second, to a theory of an underlying common impetus for why we break with these social structures. We do not need to place politics in a larger context because acting politically is to create a place, that is, a minimal way of localizing and referring to other parts of this shared world. Note that this does not mean we can simply disregard the social context of a political process – on the contrary: it means we will learn more about that context and the actual ways we rely and are determined by it if we look at it from the standpoint of the political actions taking place, observing how they change the situation, how the situation constraints this action, etc.

This is an underappreciated part of Badiou's theory, but it is absolutely remarkable that the same formal and conceptual tools that account for actions – be these punctual or concatenated into tactical and strategic clusters – also account for models – be these ephemeral slogans shifting the value of some specific fact, shared narratives about why things are the way they are or even full-blown cognitive mappings of social reality: "where there is a choice, there is a place" (LoW, 401).

Before I continue, a quick word on Badiou's choice of words. The whole theory of points is discussed, exemplified and conceptualized in terms of "choices" and "decisions" and followed by philosophical commentaries on Sartre and Kirkegaard. All of this contributes to the general sense that we are talking about the drama of human subject making individual choices – which Badiou's timid remark that the operator of decision "has no need of a 'decider' in the psychological or anthropological sense" (LoW, 400) is not enough to dispel. However, a reading of this chapter in the context of the whole book, specially considering the formal discussions, does help to clarify that the theory of points is not a theory of how humans make free decisions (if anything, it is existential philosophy itself that borrows the individual stage to dramatize a particular form of the broader problematics of points), it is rather a theory of decisiveness. In a political movement we make thousands of decisions and choices all the time, both as individual militants as well as collective agents: choices about when to meet, how to behave and how to engage with different situations. But only some of these are decisive, only some of these will need to be recuperated if we try to explain the form of our victories and defeats, only some of them shed a light on our real capacities and the actual forces we are up against. The theory of points is a theory of what it means for an interaction to be decisive – to act in such a way that whatever is chosen, counts, and whatever counts, illuminates the situation. And it might even illuminate who is in fact an actor.

The greater part of Book VI is dedicated to defining – from the standpoint of a strong singularity – what it means to make a decisive choice and why the set of all such choices in a world has a power to localize other phenomena in this world, re-indexing them to this set in a consistent way. Being rather brief – so we can get back to discussing the STP meeting – and already filtering the reconstruction through clarifications we deem necessary, the core architectonics of Book VI are basically as follows:

(1) Let us define a functor** P**, called a point, that filters the differential logic of a world W in a binary way: given some part X of W, P(X) will give us either 1 or 0, either yes or no. [Note: Badiou restricts himself to Boolean classifiers, but part of our research into "multilayered social worlds" suggests this does not need to be the case, if the strict conditions presented next are maintained.]

The definition of point is somewhat counterintuitive if you assume that P should take a part of the world as input and output its transformation – an action we decide to take, for example. But the whole point of a general theory of decisiveness is to not predetermine what type of actions are decisions and which are not, so a point is not so much a transformation of the world, but the logical value that such modification can acquire. I can talk about politics with a stranger in many situations, but only in some of them will it be decisive, an action whose logical value attests to the existence of a political bond, rather than a passtime. In fact, it is not often that a choice appears as a choice, and this is not really connected to what is the content of the action and much more to it being located at a bifurcation between two types of commitments: if one goes down one path, then one is assuming such and such commitments as a consequence, if one takes the other path, there are other implications to what one is assuming to exist. So the best way to think of the point P is to think of it as asking a question to a bundle of actions in the world: do these actions – which might connect unconnected things or disconnect connected ones – imply that the strong singularity exists or not?

Of course, if we "aim" our functor P at trivial existences, we will get trivial results, but if you ask "does this protest of couriers rejecting unionization exist as a political group?" or "is the alliance between students and couriers an existing political force or is this just the effect of their contingent shared interests?", then the answer starts to become more decisive: some things follow from how you answer, and other things do not. If the couriers can organize without a union, then something has changed in the Brazilian labor movement and strategies that relied on the assumption the old paths are the best ones might no longer be relevant. If we recall that "parts of a world" can be pretty much anything – meetings, actions, technical objects, words, groups, etc – then you see that a point, in its minimal form, is a consequential and situated question about whether the existence of some interaction implies the existence of a strong singularity.

(2) P is not only a functor that maps onto 1 or 0, it is also a functor that respects properties of W. More precisely, it respects the properties of conjunction and envelope (meets and joints) of a world's logic. If P(cup) is yes, the cup exists, and P(table) is also yes, the table is there, then P(cup on table) is also true, given that the cup is on the table.

This is a crucial aspect of the construction, which binds the divisive aspect of decisions and their capacity to serve as new placement systems. The possibility of verifying, solely by looking at how morphisms compose together, that some of the structure of the domain of our mapping has been preserved is one of those amazing things about category theory, and Badiou mobilizes the fact that P is actually a "surjective homomorphism"to argue that even if the set of decisive answers totally transforms what counts as existing in the world – from the standpoint of a political process, all the different reasons we have not to join in become somewhat similar, some degree of distinction disappears there, while the most subtle differences of what to do as part of the movement might create big political scissions – that new "space of reasons" still includes a picture of the world in it. In other words: the set of new existential commitments we maintain when we politically organize – even if these are abstractions implied by the organizational form and not ourselves individually – might make the difference between two previous positions meaningless, but it does not make them false or irrational.

(3) Let us write P(W) the set of points in a world. Now, for every part X of the world W, we have a series of points P(X) that were mapped onto 1 at that part, Badiou calls these points the "positivations of X" – all the decisive things that happen at X. Let us write these positivations as Px. Recall that, as per point (2), we know that the morphisms P preserve the ordering structure of W, even if they lose some information in the process. That means that selecting subsets of morphisms Px – points P that are indexed by a given part X – is an indirect way to select parts of W, but indexed on their decisiveness with regards to the existence of a strong singularity. Finally, the crucial theorem mobilized at this point states that the set of parts Px forms a topological space – which we write as Int(P(W)).

To prove this, Badiou lists four axioms that define a topological space and checks, one by one, whether the interrelations between positivized bundles Px really respect these. Before we list them, let us just clarify what is at stake: we are claiming that if we look at the decisive moments of action in a political process – moments in which doing something implies the existence of this singular new form – then not only will these moments preserve, even if only coarsely, the structure of the world we are intervening upon, but that these decisive moments constitute a shared common ground. You can not only move from one decision to another – connecting these moments or actions to each other – but you can also use them to localize and place everything else in the world – redescribing what matters and what doesn't, what counts and what does not, from the standpoint of this new internal ordering.

The axiomatization that Badiou chooses to employ at this point is canonical, and it is usually called the definition by "open sets", which verifies that, besides the empty set, any arbitrary union of (finite or infinite) parts of that space is also itself a part of that space and that the intersection of any two (finite) parts of that space is also part of the space. If we can prove that a collection of subsets satisfies these axioms, then we can say that it forms an interior – we know that there is a consistent operation of being "inside" the structure these subsets form.

Having proven, by the end of Book VI, that the set of of positivized subsets of the form Px have enough structure to have an interior, Badiou has shown, generally, that, under certain conditions, it is rational to talk about the set of transformations to some structure as being itself as a structure, a topological space, and, more specifically, that we can extract the most basic form of a political process from the set of situated interventions that compose it - provided, of course, that there are decisive interventions, that is, interventions that share the commitment to some common singular and situated form. The structure of these interventions – small actions composing larger tactical processes, connected across meetings, gatherings, different forms of political work, different material supports, etc – will both include a picture of the world being transformed (recall that points preserve envelopes and conjunctions in the world's transcendental) and a form that is immanent to their composition (recall that the topological space is formed only by the parts of the world that have confirmed connection to the militant process). This interiority "spaces out and conjoins the subjective and the objective" (LoW, 399): it remains as close to the situation as possible, while adding a bundle of new relations (positive connections to what is politically singular) to it. It is "a relation in the real", in Lazaraus' terms.

Now – after aaaaall this! – we are in position to understand in which sense the point of view of interiority could be supplemented by an alternative description that (1) emphasizes not so much the internal cohesion but the existence of a consistent frontier between political process and its environment and (2) which remains formally consistent with this whole construction.

The first difference between Badiou's construction, which takes interiority for a primitive concept (remember, the axiomatization used was based on open sets and their intersections and envelopes), and this alternative one is that to define the closure of a space is to divide a space into three, rather than two parts: instead of an interior, and a complement, which is the exterior, we get the closure as the sum of the interior and the frontier, and then the exterior as the complement of that sum. A simple way to consider this distinction is that the description we get when we start from the interior is one where the connectivity that interests us is, above all, the one that triangulates between the strong singularity and two or more positivized points: we want to show these decisive moments form a space insofar as they filter these interventions through the common light of what has taken place – the further we are from this interior, the less common it is to find positivized points, the less connected the space is, but our description of this process is just fuzzy. On the other hand, as we will see shortly a bit better, to start with the idea of closure is to start from these outskirts and ask what are the minimal conditions that would lead us to say that some arbitrary point is "close" to the movement, at its very frontier. The interior can be then defined as what remains when we remove both outside and frontier elements, just as we can also define the frontier itself as a sort of point of overlap between the inside and the outside – the point of dispute or struggle.

In order to clarify this, let us look at the closure axioms developed by Kuratowski and Monteiro. Let us define δ as a boundary operator, and M as a political movement:

(1) the boundary of the empty set is the empty set: δ(Ø)=Ø

(2) the boundary of the movement is the boundary of the world minus the movement itself: δ(M)=δ(W\M)

(3) the conjunction of two parts of the movement and the boundary of their conjunction is the conjunction of the two parts with the union of their respective boundaries: P1 ∩ P2 ∩ δ(P1 ∩ P2) = P1 ∩ P2 ∩ [δ(P1) ∪ δ(P2)]

(4) the boundary of a part minus its boundary is included in its boundary: δ(P\δ(P)) ⊆ δ(P)

(5) the boundary of the boundary of a part is included in the boundary of a part: δ(δ(P)) ⊆ δ(P)

What is amazing is to realize that in order to adopt this alternative axiomatization we do not need to change anything about our theory of points – the definition of P as surjective homomorphisms mapping W into a "yes or no" can be preserved, as well as the definition of positivations Px – what we can do, however, is focus on what happens at the vicinity of points and use this to distinguish some further characteristics of a political process. One way of parsing out these regions of our space, based on these axioms, is to say:

(1) Whenever all the neighbors of a positivized point are also positivized, we are in the interior Int(M). (2) Whenever some of the neighbors of a point are also positive, but some others are not, we are at the frontier or boundary δ(M). (3) The exterior is then the interior of the world, minus the interior and its boundary:** Ext(M) = Int(W \δ(M) ∩ Int(M))**

Note that this helps us to categorize three types of regions:

(1) Those where all the main connections around some decisive action are also points: for example, the inner structure of a political organization, the existence of meetings, documents, people acting in certain ways, etc, all these actions rely on one another and all of them rely on some singular political process the organization is helping to unfold – none of those things would be happening if we were to sever the connection of these militants to the political situation they are in.

(2) Those where points rely both on other decisive actions and on regular parts of the world: for example, if a movement is manufacturing banners to use in a protest and has hired some design artist to do them, though creating banners to support a political cause might constitute a positive point, this part of the world relies a lot on another, of the same world, which remains quite regular – namely, the monetary relation connecting militants to the designer.

(3) Finally, there are those parts of the world that remain untouched by the movement – if it were to disappear, there are so few connections, so few things that exist there because the movement exists, that the world would remain in place. The region composed of all parts of the world such as this (and their connections) forms the exterior of the movement.

Though I will not go into this here, consider how complex the political space might look when we recall that it is, for us, in fact composed of three different logics, and that decisive actions at the level of community-forming might rely on regularities at the level of law and economics (or any combination of such layers)! I would venture that without this new way of approaching the inner structure of positivized points, we would not have the means to consider the dazzling ways in which multilayered social worlds constraint and challenge political organization. Though the fundamental statement proposed by Badiou and Lazarus remains essential – yes, we can construct, solely through the connections we establish in the name of some ephemeral new sense of justice, a rational operator of participation, of what it means to be included in a political space – the anatomy of political life in contemporary capitalism requires us to pay special attention to the hybrid places, where the advancement of the new is conditioned on the old, and the challenge of producing non-trivial connections at all social layers at once is not guaranteed by the fact that all layers ultimately coincide in some homogeneous logical space. In fact, without the notion of boundary, we could not include in our account of political practice the crucial contradiction and connection between the figure of the militant and the figure of the worker, so beautifully explored in 'The Hustle of the Struggle'. The primacy of the frontier – albeit not conflicting with the formation of an operator of political participation – highlights, in a way that perhaps a French citizen like Badiou would not deem necessary, that the conditions for political transformation often clash and interact with the conditions for social reproduction.

Now, in the two presentations on the text, the schema of the interior, boundary and exterior was used to directly model the reproduction, production and counter-production of a political process. The idea is quite intuitive: the interior is composed of the structures and actions that reproduce the movement's own political constraints, the boundary is where actions framed by these constraints meet with those that come from the exterior of the movement, producing struggle and a fight to see if the political process gets to expand or retreat, and finally the exterior is the place of the world's resistance to the movement, through which it imposes its regularities and logical structures.

Though I like this idea and think we should continue developing it, there is something about this interpretation that doesn't seem to be supported by the formalism, though it is quite subtle. It is not that we cannot associate this triadic structure to the reproduction, production and counter-production of a political process, but rather that perhaps what these terms mean might become counterintuitive and new. First of all, it seems to be that either we do not have the means to distinguish between hybrid dependencies (such as the one between worker-commitments and militant ones) and points of struggle (which do seem to have a particularly hybrid or limit quality to them) or we ended up defining, through the notion of boundary, the "hussle" more than the "struggle" – after all, points of struggle, if they do confirm the existence of our movement, could just be part of the interior (nothing requires the interior to be a non-conflictual part of the world), while the boundary points would display a sort of non-political interaction with the conservative forces of the world. Perhaps we do need something more here in order to define the difference between the boundary points between political work and social constraints (conservative dependencies) from boundary points between political work and reactionary resistance (points where, perhaps, to arrive at a disconnect or a defeat, implies larger sets of political points get deactivated as well – murder, political arrests, betrayal, defamation campaigns: these are more than the world moving along its way, these are actions that are meant to undo whole bundle of points through the defeat at a particular point at the border, I think). Though this would suggest that there is more work to be done to compatibilize these political insights with our own theoretical framework, it also makes me particularly enthusiastic at the possibility of having found a way to think through the idea of the hustle as this liminal border between truly living and merely surviving.