One of the most attractive, problematic, and astonishing features of Badiou’s ontology is his strictly extensional understanding of sets or multiplicities. A set is not defined by its members sharing a common predicate or quality, nor by the relations among members of the set. Rather, a set is defined strictly by its extension or the members that belong to that set. From the standpoint of 20th Century French and German Continental philosophy, this thesis cannot but be a heresy, for the predominant trend in Continental thought has been a relational conception of entities. Whether we are speaking of language as a diacritical set of negative oppositions as defended by the structuralists and the post-structuralists, or Heidegger’s being-in-the-world where entities, the ready-to-hand, are defined by the relational networks to which they belong, the predominant trend has been to treat beings as bundles of relations such that the entity is nothing apart from its relations. In a spirit similar to Deleuze’s declaration that relations are always external to their terms, Badiou will have none of this. For Badiou entities are not defined by their relations and there are no intrinsic or internal relations that define the being of the entity. Rather, they are simply defined by their relations.
From the standpoint of both Heidegger’s being-in-the-world where each entity is thought as a “being-in” belonging to the worldhood of the world defined by an ensemble of relations defining meaning, or from the standpoint of structuralist and post-structuralist thought where the entity is an ensemble of internal relations from which it cannot be detached, or from the standpoint of Hegelianism where, as Hegel painstakingly shows in the Doctrine of Essence in the Science of Logic, where the entity simply is its relations or mediations, this move cannot but appear stunning. For what this extensionalist conception of sets authorizes is combinations of subsets in whatever order we might like. This, in short, is what the axiom of union tells us. What the axiom of union allows– if I understand it correctly (I’m sure Dominic will educate me if I don’t, thankfully) –is the construction of whatever sets we might like based on those elements belonging to our initial set. Thus, if I have a set composed of an umbrella, an apple, and the moon ({umbrella, apple, moon}), I certainly have a set composed of the apple and the moon ({apple, moon}), or a set composed simply of the apple ({apple}).
Now all of this sounds silly and unremarkable so long as we don’t contrast Badiou’s extensional notion of sets with the relational ontologies that have predominated during the 19th and the 20th century. If to be an entity is to be a bundle of internal relations, it follows that entities cannot be grouped in any way we might like. Rather, a model of the world based on internal relations dictates that each entity necessarily has a place within an Order and that the entity is nothing apart from this order. Thus the phoneme {c} is nothing apart from other phonemes such as {p}, {b}, {f}, etc., by virtue of the differentiality that allows it produce different senses at the level of the signifier: cat, pat, bat, fat. Insofar as these phonemes take on their value (in the linguistic sense of “value”) differentially in relation to one another, they are nothing independent of their relations to one another. This is what it means to say that each entity takes on a place within an Order. The Order is the totality of internal relations defining a system or structure, whereas the places are locations within that Order relative to the other terms. Because the relations are internal to the various beings in the Order, there is thus a Law that governs these beings and exhausts their being, legislating how they can and cannot act.
In proposing that sets are defined purely by their extension or their membership, Badiou undermines the thesis that to be is to be a bundle of internal relations. At the level of ontology, there is thus no intrinsic Order that defines entities. Rather, in their stark independence, the elements that make up a set not only can be decomposed into infinite subsets (through a recursive process of taking the power set of each power set), but the elements of each set can be related in a variety of different was or simply taken as singletons, thereby abolishing the notion of intrinsic or internal relations as in the case of Hegel’s logic of essence.
read on!
Of course, while this might be the case ontologically or in terms of what can be said of being qua being independent of any particular being, worlds, the domain of the ontic, do not work in this way. In worlds it is not the case that things can be combined in any old way, but rather there are ordering relations among entities. However, it’s notable that Badiou’s thesis is not that entities cannot enter into relations, only that they are not defined by their relations. Thus, while there is a gap between the ontological (being qua being or the discourse of multiplicity qua multiplicity) and the ontic (ordered or related elements in a world), Badiou’s ontology nonetheless strongly suggests that entities are prior to their relations in the domain of the ontic as well. Here relations are external to their terms, such that entities are not defined by their relations, but rather enter into their relations.
If entities are independent or prior to their relations, it follows that we should abandon the concept of structure and instead shift to a network or assemblage based model of relations among entities. The problem with the concept of structure is that it treats relations as internal relations, such that the elements belonging to the structure have no existence independent of its relations. In his marvelous popularization of network science, the Columbia sociologist Duncan Watts admirable puts his finger on this problem, writing that,
The crux of the matter is that in the past, networks have been viewed as objects of pure structure whose properties are fixed in time. Neither of these assumptions could be further from the truth. First, real networks represent populations of individual components that are actually doing something— generating power, sending data, or even making decisions. Although the structure of the relationships between a network’s components is interesting, it is important principally because it affects either their individual behavior or the behavior of the system as a whole. Second, networks are dynamic objects not just because things happen in networked systems, but because the networks themselves are evolving and changing in time, driven by activities or decisions of those very components. In the connected age, therefore, what happens and how it happens depend on the network. And the network in turn depends on what has happened previously. It is this view of a network as an integral part of a continuously evolving and self-constituting system– that is truly new about the science of networks. (Six Degrees, 28 – 29)
Structuralism has been valuable in drawing our attention to the importance of relations. However, the cardinal sin of structuralist thought– and these assumptions still remain pervasive today –lies in its tendency to, as Bateson would put it, confuse the map with the territory. That is, the structuralist makes a map of relations among nodes in a network, but then treats this map as if it were itself a real and abiding thing such that these relations are abiding and eternal. The entities inhabiting the network then get treated as epiphenomena of this map of relations, such that it is the structure that is real and the entities populating the network that are illusions. This is a specifically Platonic tendency within structuralist thought.
In ontologizing structure in this way, the dynamics of structure through which structure is both produced and reproduced in time. In short, what is missed is the manner in which nodes in a network must be related. That is, the links among elements of a network must be forged for the network to function. Part of the great value of structuralism has been to draw our attention to the manner in which there are emergent properties of networks that exceed the intentions of any of those participating in the network (for example, patterns of wealth distribution). However, by ignoring the dynamics of networks and the fact that they have to be built, structuralists have drawn the wrong conclusion. Thus, for example, Althusser drew the conclusion that humanism must be mistaken as these networks function anonymously and not according to the intentions of those participating in the structure. The individual person thus becomes, under this reading, a sort of illusion and nothing more than its place in the social structure.
Althusser, however, is wrong on both counts. On the one hand, insofar as entities are prior to their relations, they are not simply illusions (though their effect might be negligible from the standpoint of the functioning of the network). Moreover, without the interactions among these individuals, the network could not exist at all. Thus, while a network cannot be reduced to the action of these individuals, it also can’t exist without the actions of these individuals forging links, making decision, becoming hubs, and therefore attracting more relations that then come to preside over the future course of the networks development. This last point is especially important. One thing network research has discovered is that those nodes in a network that possess more relations to other nodes within a network also attract more relations as the network evolves and develops. For example, wealth tends to attract more wealth such that it comes to be localized in one segment of the population. Thus, during the early stages of network development, the relations that can be forged among entities are relatively open. But, networks are defined by times arrow such that the forging of relations introduces elements of accretion that limit the direction in which the network can develop in the future.
From the standpoint of political theory, this simple observation is of tremendous importance. First, it underlines a point of strategy for targeting oppressive social systems in that a network will be weakest at those points where relations to a particular node or set of nodes are most extensive. Take out that node and the rest begins to fall (as we have learned from the California power outages). Second, it also underlines the importance of developing group relations in engaging a network and changing it. In other words, it is of vital importance to generate networked relations that will attract more relations to other nodes if the overall evolution or development of a network is to be affected in a significant way. This shows, for example, why forms of political theory written in such a way to interrupt discourse and communication so as to fight the metaphysico-politico structure in language itself are so misguided. By adopting this rhetorical strategy they limit the ability for links among nodes to be formed, thereby preventing the accumulation of relations within networks that are the best chance for shifting the organization of the network as a whole.
On the other hand, if Althusser’s anti-humanism is mistaken it is because it treats the individual within a structure or network as a sort of illusion or effect of the structure. Althusser’s point is well taken. There are emergent properties of networks that can’t be reduced to the intentions of the individuals caught like a fly within these networks. However, this is very different than the conclusion that the individual is nothing but an effect of its place within a network. First, individuals are prior to their relations, and as such cannot be reduced to their relations. Indeed, in the world nothing ever functions as smoothly as our maps of networks suggest (Bourdieu analyzed this point to great effect in his critique of structuralist models of kinship relations in The Logic of Practice). Second, individuals move among different networks that are discontinuous to one another, thereby indicating that they are irreducible to their relations. This does not mean that the ongoing relations among elements in a network don’t play a tremendous constraining role on individuals participating (whether or not they know it) in a network; but to point this out is different than claiming that the individual is its place in a network.
Ontology matters. As Mark of K-Punk so beautifully put it recently,
My instinct would be to reverse this, i.e. it’s not that ontology is always constructed through a political battle, but that politics is always constructed through an ontological battle. Politics certainly presuppose ontology – to take a glaring example, the key slogans of Thatcherite capitalist realism, for instance (“There is no such thing as society, only individuals and their families” and “There is no alternative”) were explicitly ontological claims, claims about what sort of entities can be said to exist in the world. But that isn’t to say that all ontologies presuppose a politics.
This tendency to ignore the dynamics of networks, the fact that they must be produced and reproduced, the fact that relations among nodes in a network are perpetually shifting and changing, leads to the wrong sorts of questions and problems at the level of political theory. Thus, for example, Žižek has written thousands of pages arguing that we must conceive of a split subject, a subject that is a gap within structure, in order to account for how change is possible. However, this thesis only rings true if we begin from the premise that individuals are effects of structure without any independence of their own. All of this changes when we no longer ontologize structure as a Platonic form presiding over entities and reducing them to pure vehicles of structure. Rather, the question instead becomes that of how links between nodes are formed, what those links are, where the hubs are, and how it might become possible to form new networks. The question of change no longer looks so mysterious or intractable.
February 20, 2009 at 9:21 pm
Wrong axiom, I’m afraid – the powerset axiom is the one that gives you all the possible combinations of the elements of a set. The axiom of union is rather more interesting. It gives you all of the elements of all of the elements of a set – so U {umbrella, apple, moon} might be {dust, pips, craters, stalk, canvas, worm, stalks…}. You could see it as opening all of the black boxes inside the open box of your original set.
February 20, 2009 at 10:21 pm
Levi,
Great post. Which Duncan Watts book are you citing above?
Adam
February 20, 2009 at 10:24 pm
Um, I don’t mean to nitpick, but I’m pretty sure that set (or class) membership is in fact an extensional relation. To say that some a is a member of (or belongs to) some G simply is to say that there exists some predicate G, such that G(x) is the necessary and sufficient condition for membership in G for any arbitrary x.
In fact, extensionality is formally defined as follows: a is a member of {x:G(x)} iff G(a). Hence membership is defined in terms of predication, which is understood in extensional terms.
The appropriate distinction, then, isn’t between extensional and relational understandings of something, since extension just is a kind of relation, but between extensional and intensional ones. The difference is roughly the same as the one obtaining between denotation and connotation, or again between internal and external relations.
Badiou’s basic point, however, doesn’t seem to be about either extensionality or intensionality. Rather, he seems to be much more fascinated by the fact that sets don’t seem to presuppose anything (they offer a model for presuppositionless thinking). Literally speaking, they presuppose nothing – {} – and can be constructed through the reiteration of this nullity, which does not end up determining the nullity as ‘a something’ itself.
As far as I can tell, however, Badiou’s fixation on set theory and mathematics really is nothing more than analogical or perhaps metaphorical. You can read his work without ever once mentioning set theory or using its formalism and get exactly the same content (except maybe his book on number, which really is a mathy book).
February 20, 2009 at 11:36 pm
[…] Badiou, Extension, and Networks. Doing philosophy via a network ontology rather than the way Continental philosophy normally works. […]
February 20, 2009 at 11:55 pm
The question of change no longer looks so mysterious or intractable.
So unless you’re even more of a misunderstood genius than Dr. Harman, howcome nobody’s ever thought of that one and we’re stuck so long without change?
February 21, 2009 at 12:26 am
Hi Alexei,
I think Badiou’s conception of maths as ontology is more than metaphorical or analogical. He is genuinely committed to the thesis that maths is ontology and that it says all that can be said of being qua being. This would be the significance of the empty set, which speaks of no entity in particular and therefore (according to Badiou) saves ontology from presupposing the primacy of a particular sort of entity as the ground of ontology (i.e., he thinks in this way he’s able to sidestep Heidegger’s critique of metaphysics where a particular entity is raised to the status of the ground of all others). Extension is indeed a form of relation (membership), but the point is that the entities belonging to a set are not themselves defined or determined in terms of their relations to one another. No doubt Badiou finds this feature of sets attractive from a political standpoint insofar as it entails that the members of a set pertaining to the social do not have an intrinsic relational identity defining their place in the social order.
February 21, 2009 at 12:27 am
Thanks, Adam. The book is entitled Six Degrees… Very nice as popular science writing goes (short on all the biographical and historical narrative that often blemishes popular science books while heavy on the theory in a very accessible way).
February 21, 2009 at 12:29 am
Thanks for the correction, Dominic. I knew you’d catch me! I was under the impression that the powerset axiom only allowed you to take all possible subsets of a set. Are you saying that the powerset axiom allows you to take a specific subset from your initial set as well?
February 21, 2009 at 12:57 am
Alexei,
I would also add that characterizing Badiou’s fondness for set theory in terms of the epistemic issue of presuppositionlessness misses the whole point (and real force) of his ontological move. What interests Badiou about ZF set theory is not that it is presuppositionless, but that it sidesteps these sorts of questions about epistemology and foundations altogether. The axioms of ZF set theory are not self-evident truths, but are decisions from which set theory begins. As far as I can tell, this is an entirely new beginning point in philosophy. No other philosopher I can think of has thought to begin from a decision rather than some sort of question of knowledge. If Badiou is able to begin from the axioms of set theory without having to engage in something like Kant’s critical analysis or a Heideggerian hermeneutic analysis of everydayness, then this is because the decisional starting point subtracts itself from any and all epistemic questions of evidences, asking only what follows from the constraints of these decisions, not what authorizes these decisions/axioms in the first place. This simplifies a complex matter a bit too much, but hopefully draws attention to just what is important about Badiou’s move in the history of philosophy.
February 21, 2009 at 1:46 am
Could you give the specific to the Duncan Watts book? I’m interested in reading it but am unsure from which of his books it originates? Thanks.
February 21, 2009 at 1:59 am
Anonymous,
The book is entitled Six Degrees. Don’t let the title throw you, it discusses far more than the somewhat facile “there are six degrees of separation between all people” thesis. The title, I think, was a rather poor choice for the sorts of issues the text deals with.
February 21, 2009 at 3:17 am
Just a few follow up responses, Levi,
First off, I meant ‘presuppositionless’ in the sense you identified in your first reply to me (i.e. ZF+AC set-theory doesn’t take a particular entity as ontologically basic or originary, which is precisely the same sense in which Hegel’s philosophy is ‘presuppositionless’). And I agree with you that Badiou begins with a decision, rather than a skeptical analysis of knowledge claims. I just don’t think that beginning with an axiom, or some kind of decision is really so radical (Kant’s transcendental deductions begin in precisely the same way; Fichte’s whole account of ‘positing’ is a metaphysical account of what follows from a decision; Euclid’s elements do the same thing; ditto for most of the presocratics; same same for many of the Romantics [though for them it’s art, not set-theory, that’s the standard bearer of the metaphysics of the decision] and there’s some complex decisional thingie in Hegel’s development of the various shapes of consciousness too). I’m not trying to reduce our conversation to the level of ‘who did what first,’ I just want to point out that, whatever else is radical in Badiou, it’s certainly not the fact that he begins from the decision (for beginning with a decision is just a new fangled way of parsing what Hegel meant by the absolute beginning).
Second, now that I think of it, could you please define what you mean by ‘epistemic’ (as opposed to ‘epistemological’) and in relation to ‘ontological.’ For on at least one interpretation of the term ‘epistemic,’ an epistemology and an ontology are presupposed by an epistemic claim. IN fact, I take it that this insight inaugurates modern philosophy. Moreover, Realism (in its metaphysical and epsitemological sesnes) is typically understood as implying that we know the way the world is and that the world would be the same independently of us as knowers. Realism thus implies the identity of epistemic claims and ontic ones. But you keep using this word as if it were some kind of pejorative, and I’m really confused by your insistence on it — especially since you keep insisting that I’m reducing everything to the epistemic. I’m pretty sure I’m doing no such thing — especially since I have broadly anti-realist sympathies due precisely to my belief that you can’t reduce an ‘object’ to some set of justified — hopefully true — beliefs we hold concerning them. But I would say, however, that whatever we take to be ontological is going to have an effect on our epistemology (theory of justification, theory of knowledge), although the two need not be coextensive. Otherwise why would we give a fig about ontology in the first place? Really, why care about ontology if it has no impact on epistemology etc?
Third, I also agree with you about Badiou’s commitment to the claim that ontology = set-theory. The problem is that Badiou doesn’t actually engage set-theory (which is not to say he deosn’t understand or know set-theory, just that he isn’t doing set-theory when he is talking about it). He tends to use it as a prop to explore philosophical ideas, commitments. In fact, he says as much in the intro to Being and Event, if I remember correctly. So, to the extent that he’s not doing math, he is either he’s not engaging in ontology, which is false, or he’s using set-theory as an analogy to help develop his ontology. What’s more, he’s offering an interpretation of set-theory. so it’s not like he’s really escaping the hermeneutical analysis of everydayness, or the conditions for the possibility of experience. He’s just using a different model of everydayness or the conditions for the possibility of experience. Same Transcendentalism, different day.
February 21, 2009 at 3:26 am
Alexei,
Thanks for the clarification. Glad we’re on the same page vis a vis Badiou. I’m not sure I understand your question about the epistemic versus the epistemological. My use of one term rather than the other is, I think, simply a matter of grammar, not a distinction between two different things. What I do object to, however, is what I call, following Bhaskar, the reduction of ontological questions to epistemic questions. This would be summed up in the Kantian aphorism that the conditions for the possibility of experience are identical to the conditions for the possibility of the objects of experience. That is, if Kant is to be followed, the latter claim is making an ontological claim about what any object must be, suturing it to conditions for our knowledge of that object. I would argue that we can, in fact, (in some cases) have a knowledge of what a world would be independent of our knowledge of it or our existence. Indeed, many of the objects dealt with in the sciences are beyond the scope of anything humans can experience or intuit. The problem with correlationism, especially in its phenomenological variant, is that it sutures all talk of objects to what can be experienced by humans. Thus, when we talk of cosmic time and space scales, or quantum time scales, we’re necessarily led to the conclusion that these things are pure fictions as they’re thoroughly unintuitable for humans.
February 21, 2009 at 3:46 am
Hi Levi,
Very interesting post (and your response to Graham re: being, event & subject is also much appreciated). Just a small, but to my mind crucial point: As I read Badiou, “entities” are precisely what are decomposed at the level of ontology. As you rightly point out, ontology is defined as “what can be said of being qua being independent of any particular being.” “Entities” thus emerge only through the ontic ordering of worlds, and therefore cannot properly be said to be “prior to their relations in the domain of the ontic.” The ontological situation of inconsistent multiplicity is not composed of “entities.” So, in that case, entities are precisely defined by and constituted through relations, though being qua being is non-relational.
February 21, 2009 at 4:12 am
Thanks, Nathan. I like your way of putting it:
I share your reading. This is why questions of individuation or the count-as-one are at the heart of the whole ontological. What, precisely, are these mechanisms and where do they come from? I’m sympathetic to Badiou’s infinite decomposition but have reservations as well. I think it’s precisely these questions that tend to be ignored in the reception of Badiou. On the other hand, this is Badiou’s “Judo” move. Anyone who has ever practiced Judo knows that you use the force and momentum of your opponent against them. This is the beauty of Badiou’s move against the postmodern skeptics. Rather than claiming that difference or infinite decomposition is the ruin of metaphysics, he asserts that this is metaphysics. That is, against the sophists he claims that their very gesture reveals the essence of being, rather than spelling the ruin of being. That’s something worth preserving.
February 21, 2009 at 9:44 pm
I suppose I’m not terribly clear on the difference between ‘epistemic’ and ‘epistemological’ either, Levi. But I seem to recall that Mikhail once pointed out that these two terms seem to have the same kind of relationship as ‘ontic’ and ‘ontological’ do.
Should that be the case, any knowledge claim would be an epistemic claim, and any claim about the framework in which that knowledge claim is coherent would be an epistemological claim. Given that distinction, Kant avoids the epistemic fallacy, since he’s committed to saying only that any judgment (an epistemic claim) is possible on the grounds of the pure forms of intuition, the categories, the schematism, and the transcendental unity of apperception (a series of epistemological claims). And that says nothing about the thing in-itself or the speculative use of reason — nor does it say that there is no such thing as a thing in-itself, etc. it’s a rather deflationary claim, really. So when you say,
(which strikes me as a tautology actually — what else do we experience besides the objects of experience?)
is an epistemological claim that makes certain kinds of epistemic claims coherent. But it doesn’t say anything about ontology — we’re still talking of objects-of-experience, and not objects as such.
Viz science, just because I’m feeling nit-picky: regardless of how unintuitible the object itself might be, science always revolves around intuition in the form of indirect observation (hence Hacking’s ontological prinicple: if you can spray it, it’s real). So I don’t think anyone is really licensed to talk about objects independent of experience. The real issue — so it seems to me — is the priority given to experience.
February 21, 2009 at 11:34 pm
[…] cognition or perception. Consequently, Alexei (a correlationist), in response to one of my posts, writes: I suppose I’m not terribly clear on the difference between ‘epistemic’ and […]
February 27, 2009 at 12:25 pm
“The axioms of ZF set theory are not self-evident truths, but are decisions from which set theory begins. As far as I can tell, this is an entirely new beginning point in philosophy. No other philosopher I can think of has thought to begin from a decision rather than some sort of question of knowledge.”
Isn´t this a quite common view in analytic philosophy?
Isn´t it also common in analytic philosophy to take class-membership to be entirely extensional?
March 5, 2009 at 10:07 pm
And for a work-in-progress using the idea of tracing networks to rebuild a political project, see also http://yannickrumpala.wordpress.com/2008/09/16/knowledge-and-praxis-of-networks-as-a-political-project/
Could this be an idea to develop ?