I worry this might sound really vulgar and naive, but what if we were to raise certain questions about Badiou’s ontology in relation to the Barber of Seville paradox. Among Badiou’s most famous claims is the thesis that ontology belongs not to philosophy, but rather mathematics. Maths, and, in particular, Cantorian set theory, articulates all that can be said of being qua being. I remember the excitement and pleasure I took in this thesis when I first encountered it in graduate school. Not only did I already have a deep and abiding love of mathematics, but there was also something marvelously perverse in a Continental philosopher championing mathematics. Who can forget the title of Heidegger’s lecture course The Metaphysical Foundations of Logic, his claim that maths doesn’t think, or the generalized hostility towards maths one finds everywhere in Continental philosophy with the notable exception of Deleuze. What could be more contrary to Heidegger’s thesis than the Mathematical Foundations of Ontology? Moreover, in a field of philosophical alternatives dominated by obsessive meditations on the human, the body, language, and power, few things could be as “other-worldly” and inhuman as the elevation of humble mathematics… A humility that paradoxically is coupled with the most acrobatic conceptual innovation, daring to think spatial configurations, multiplicities, topologies, and all the rest remote from anything like the “everydayness” we experience in our intuitive relations to the world.
I suppose you could say that I took an impish pleasure in how Badiou must stick in the craw of my fellow Continentalists. I will never forget having coffee with a very well known Continentalist in his own right, my face, words, and gestures animated by my enthusiasm for Badiou like a child having at it with a new toy, only to hear him despairingly say “it’s kinda like analytic philosophy, though.” Kinda, but not quite. Badiou had really hit a symptom at the heart of contemporary Continental thought. Where Derrida and the others were endlessly talking about free play and dissemination, Badiou put his finger on the remarkable univocity of mathematical prescription. But this is not all. Where everyone was endlessly talking about difference, Badiou took this one step further, developing a radical articulation of difference. Many of us had become accustomed, through Heidegger, to thinking of maths as the most extreme form of enframing and identity thinking. What Badiou showed, through his deployment of set theory, was that far from the valorization of identity, maths give us the resources to think multiplicities qua multiplicities without one, or absolute difference and dissemination. Similarly, where many were celebrating the accomplishment of Derrida’s thought and the aporetic undecidables it acquaints us with in every domain, Badiou dared to declare that we must decide the undecidable, and articulated a rigorous account for doing so through his discussions of forcing and the generic with respect to truth-procedures. Indeed, the very fact that he said truth at all, and in such an interesting way, was a shock to the system within that intellectual context.
Yes, Badiou had hit a symptom. For those of us who had cut our teeth on the intricacies of Lacan and thinkers such as Laclau, reading these figures in the happy days following the advent of the beautiful work of Zizek and Fink where Lacanian thought had been freed from the endless rut of the imaginary and cinematic accounts of suture, where the late Lacan was finally, slowly, so slowly, becoming readily available, and who were already acquainted with the intellectual atheleticism required by set theory, topology, and all the rest, Badiou arrived at just the right time and just the right moment. Badiou arrived as the philosopher of these formalisms. Those of us intoxicated by Lacan and Zizek, worried, as philosophers, at how we might escape the rut of literary and cultural criticism. The question that haunted the time was that of how psychoanalysis might be put to use philosophically. Badiou provided precisely the answer to this question, not by virtue of being a psychoanalytic thinker, but by mobilizing all of these set theoretical and topological structures we had been exploring in Lacan but with respect to questions of ontology, ethics, and politics. And above all, Badiou arrived at a moment where interpretation, philosophy as interpretation, had largely exhausted its potency, becoming a dreary and oppressive activity, appearing daily to be more a way of insuring that everything remain in place and that the tradition be preserved against any and all change. The potency of Badiou lay not so much in his explicit declarations and theses, as in his invitation to think again and his reminder of what philosophy ought to be. Indeed, he went so far as to denounce doxa. Who, in our Protagorean age, in our age dominated by simulacra of Gorgias, in our age where rhetoric had come to trump philosophy (i.e., the triumph and primacy of ethos or local custom over logos), had dared denounce doxa? Badiou did.
read on!
Nonetheless, despite my enthusiasm for Badiou, I had cloying doubts about the relationship between his ontology of pure multiples without one and the relationship of being to what he calls “consistent multiplicities” or structured situations. The world, as Badiou emphasizes in Logics of Worlds, never presents itself as pure multiplicity or dissemination, but always as structured entities. This simple observation, I think, is enough to raise the question of the degree to which maths and ontology can really be equated with one another. This point can be illustrated by reference to the Barber of Seville paradox. This paradox is formulated in the form of a question:
If the Barber of Seville cuts everyone’s hair except for those who cut their own hair, who cuts the Barber’s hair?
The answer, of course, is that the according to the criteria set out in the proposition, it is impossible for the Barber to ever have his hair cut. If the Barber cuts his own hair then he violates the rule whereby the Barber cuts everyone’s hair except those who cut their own hair. If, by contrast, the Barber has someone else cut his hair, then he violates the rule whereby he cuts everyone’s hair except those that cut their own hair. Consequently, it is logically impossible for the Barber’s hair to get cut.
However, this is the whole crux of the matter (and here I become vulgar and naive): This paradox only occurs at the level of formalization. At the level of reality, the Barber easily gets his hair cut, most likely having someone else cut his hair as it is difficult to cut one’s own hair.
I think this simple observation brings to the fore certain problems with the equation of ontology with mathematics. A number of Badiou’s more startling claims issue from meditations on formal mathematical deadlocks of the sort outlined in the Barber of Seville paradox. For example, Badiou declares that “the One is not”. This claim is aimed at the sense of the One as a totality or whole of the universe, but also at the sense of objects as one or unities. In this latter sense, Badiou contests Leibniz’s claim that “there is no entity that is not a entity”. Let us take the One in the sense of a totality or whole of the universe. Employing the power-set axiom, we can easily demonstrate that it is impossible to form a set of all sets (i.e., a uni-verse). The power-set axiom authorizes us to form the set of all subsets of an initial set. Thus, for example, if we have a set composed of {comb, knife, moon}, the power-set of this set would be as follows: {{comb}, {knife}, {moon}, {comb, knife}, {comb, moon}, {knife, moon}, {comb, knife, moon}}. As can be readily observed, the power-set of our initial set is greater in size than the set from whence we began. From this we can see why it is not possible to form a set of all sets, for the power-set of the set of all sets will always be in excess of that set. As a consequence, we are led to the conclusion that there cannot be a universe or a totality. This is what is known as “Cantor’s Paradox”. Badiou makes similar arguments when demonstrating that all objects are infinite.
The question, however, is what allows us to assert that these formal properties of mathematics belong to being itself? What is it that allows us to conclude that the universe itself has the properties uncovered in Cantor’s Paradox? It will be objected that I am conflating two distinct domains in raising this common sense question. Ontology, the argument runs, deals with being qua being, or pure inconsistent multiplicities. In asking how Badiou’s various claims apply to entities and the world, I am crossing circuits that cannot be crossed, conflating being with existence. If I want to know about beings, entities, objects, existence, I must turn away from ontology to ontology or the logic of appearance and existence Badiou develops in Logics of Worlds.
This is a somewhat compelling argument. When we think ontologically, we are not thinking any particular entity, but rather pure multiplicities. It is thus a mistake to take the claims of ontology as applying to existence and the world. The problem, however, is that Badiou himself continuously crosses these circuits or levels. When Badiou claims that the One is not, he is not simply making a formal claim about the set of all sets, he does not hesitate to declare that there is no world, universe, or totality. When Badiou claims that every consistent multiplicity or “count-as-one” is infinite, he does not hesitate to declare that this is true of objects or entities.
Yet what is it that authorizes Badiou to claim that these formal truths apply to the world and objects themselves? In the case of the Barber of Seville, we saw that there is nothing in the formal rules governing his behavior that restricts that behavior in reality. Why would this not be the case with respect to mathematical necessities? Let me be clear. In asking this question, I am not arguing that there is a universe or a totality, a One. Nor am I arguing against the thesis that objects are infinite or subject to infinite decomposition. I do not know one way or another. The question I am raising is whether or not we can legitimately reach these conclusions through the sort of formal reasoning practiced by Badiou. Nor am I arguing that nature is not mathematical. The issue here is not whether or not nature is mathematical, but rather the gap between those instances of nature that we successfully mathematize and the infinite domain of mathematical formalization where it is not at all clear that there are any entities structured in this way. Because of this gap between the infinite field of mathematical formalization and the structured organization of nature, it is not clear that we can reason a priori from formal properties of this or that mathematics to properties of objects in nature. One will object by pointing out that I am a Lacanian and as a Lacanian I make reference to these sorts of formal necessities and paradoxes drawn from set theory all the time! However, here the situation is entirely different. If set theoretical paradoxes such as the Barber of Seville paradox are relevant in the case of psychoanalysis, then this is because the unconscious is, according to Lacan, structured like a language. As a consequence, the unconscious, forming a system, is capable of knots and aporia around which the symptom suffered by the subject are organized. It is not clear that the same can be said of the world and objects.
Larval Subjects . 
June 17, 2009 at 3:54 am
You are heading almost exactly where I’m heading with this. The sorts of self-referential paradoxes that we find in every formal system are not, generally speaking, found in the “real” (empirical?) world. Achilles passes the tortoise, the arrow flies to its target.
There are a couple of features of the human symbolic ability that seem to lead to these paradoxes. One is self-reference. The other (which I’m blogging about now) is what I call the “imperative split” (the inability to define without reference to something else).
I haven’t seen a real, heavy-duty inquiry into why formal systems always have these characteristics, but if anyone has reading suggestions, I’d love to learn more about it.
June 17, 2009 at 5:55 am
Asher,
I can’t recommend Graham Priest’s “Beyond the Limits of Thought” highly enough. There’s some heavy duty logic in there, but 95% of it doesn’t presuppose that.
He tells an entire history of philosophy in terms of various attempts to deal with limits of expression, iteration, cognition, and conception. For Priest each such proof is structurally similar to the proof of Russell’s Paradox (which is similar to the Barber Paradox except you can’t solve it by just saying the Russell set is empty). These proofs canonically predicate some absolute limit (closure) and then go beyond that limit (transcendence).
Anyhow, my brain is too tired to to respond intelligibly to the above interesting substantive points. I’m going to read them again tomorrow.
But please read Priest. He’s dynamite and his dialetheism is far more plausible a solution to these issues than one might think. I think for him the paradoxes show that reality really is inconsistent, and that para-consistent logics are necessary to be able to describe it.
I want to re-check that with reference to you guys’ points.
Jon
June 17, 2009 at 6:10 am
hi LS,
I’m rusty on Badiou, but for what it’s worth….
“If the Barber of Seville cuts everyone’s hair except for those who cut their own hair, who cuts the Barber’s hair?”
I don’t think you’re right to say that the problem with this statement only occurs with formalization and that in reality this is no problelm. I think we can leave aside formalization and still have the statement have the odd results you point to. It seems to me there are only two possible answers to this question. One is “no one, the Barber doesn’t get hair cuts”, which is contradiction free. The other is to qualify the part about whose hair the barber cuts to note that the barber cuts the hair of at least one person who cuts their own hair (namely, the barber).
With regard to mathematical ontology, I remember thinking Badiou quite explicit that he was not ascribing mathematical properties to being, but that he was saying something like “insofar as we can say things about being qua being, we can say only what mathematics can say.” That doesn’t mean being has mathematical properties.
Personally, I like the ontology as mathematics thing quite a bit, but for reasons Badiou may not – I take it as a good deflationary argument against most uses of ontology (or at the least, against Deleuzian/Negrian ones), saying basically “if you want to talk about ontology, go do math.” I think it’s got a fair bit in common with Hegel in one of the logics – being has so little content it’s hard to tell apart from nonbeing. As such, for any X, X qua being is really rather dull. The interesting stuff is X qua something aside from being … well, being. Riots and books and paintings and meals (etc) qua being misses out on what makes those things interesting.
cheers,
Nate
ps- sorry if I’m being a Minotaur/Icarus!
June 17, 2009 at 7:08 am
I think the view that these paradoxes only concern representational machinery is perilously close to being a linguistified (where the representing medium being language rather than phenomena) version of Kant.
In the Dialectic Kant rubs up against himself stating Russell’s Paradox. His avoidance of them isn’t just positing the distinction between noumenal and phenomenal, but also limiting what can be said about the phenomenal as well (he has to take the second step of insisting that infinity needs to be understood in terms of potentially getting to places in larger finite series, as opposed to a completed totality within phenomenal experience (think paradoxes concerning infinite divisibility), for example).
I worry that the “it’s only a problem for the medium” is equal to the initial positing of a distinction between phenomenal and noumenal.
I realize that to say that something is similar to Kant is not to refute it, but maybe it is cause for worry.
June 17, 2009 at 3:23 pm
It is always possible to nitpick anything that anyone ever says about anything. And Nate will always be right here, finding a way to do it.
June 17, 2009 at 3:36 pm
‘the generalized hostility towards maths one finds everywhere in Continental philosophy’
sorry who is this ‘continental philosophy’ that is negative towards math – bachelard, canguilheim, serres – i could go on…
interesting post however.
June 17, 2009 at 5:33 pm
I’m no cosmologist, but the ones I can think of would not be using set theory to prove that there’s a multiverse…in fact there’s scant evidence for it as of yet…
June 17, 2009 at 6:08 pm
Quite right, anodynelite.
Additionally, I think one of the central problems with the use of set theory for these sorts of ontological aims is that there are no ordering relations in elements of a set. Thus, for example, a set consisting of elements p, q, and r or {p, q, r} is identical to a set written {p, r, q} or a set written {q, r, p}, etc., because sets are defined purely in terms of their extension. As a consequence, a set theoretical ontology invites us to ignore ordering relations among entities. It will be objected that we have all sorts of logical operators that allow us to introduce ordering relations among elements. But this is not the point. The point is that whereas sets are indifferent to ordering relations, the world is not. Of course, it is precisely this issue that Badiou grapples with in Logics of Worlds, and for that I am grateful. What I find peculiar, however, is the thesis that being qua being is indifferent to ordering relations. Can we really claim this?
Zubiri’s On Essence has really gotten me thinking about these issues. One of Zubiri’s central claims is that the first term in philosophy is not being but rather reality. Reality, for Zubiri, precedes being. If that’s the case, ontology cannot be treated as a discourse on being qua being where ordering relations are irrelevant. Rather, ontology is an investigation of being qua reality, where ordering relations are of the utmost importance.
June 17, 2009 at 6:42 pm
I’m not all that interested in defending Badiou, ultimately, but it seems to me that you are at least coming close to getting the argument in the wrong order in this post.
that is, I take it that Badiou’s first claim is that the One is not, and it is from this claim that he moves to set theory as a logos of this inconsistent being. in other words, looking at the first few meditations of Being and Event, it seems clear to me that the claim that the One is not it NOT based on set theory, but serves to ground the claim that set theory is ontology.
otherwise, things simply will not work. if every situtation is structured by a count-as-one, and ontology is a situation, what prevents us from concluding that being (in-itself) is consistent?
it seems to me that an answer to this question is only possible if the decision that the One is not comes first and the move to set theory second. this is also what allows him to say that mathematics provides a logos of being, while insisting that he is not claiming that being itself is mathematical.
June 17, 2009 at 7:37 pm
Jon – thanks! Priest’s book is just the sort of thing I was looking for.
I am skeptical about dialetheism but interested and open-minded. Here is a very funny quote from the Wikipedia article that I think sums up my skepticism:
The argument itself isn’t what makes me skeptical. It’s the issue of whether para-consistency is going to end up meaning using a scalpel to slice off the parts of logic that aren’t working or whether it means something new about consistency.
Can’t wait to find out!
June 17, 2009 at 7:53 pm
“It is always possible to nitpick anything that anyone ever says about anything. And Nate will always be right here, finding a way to do it.”
Wow, ouch. Sad but true, I suppose.
June 17, 2009 at 10:31 pm
I take your point, Caemeron, but this analysis nonetheless seems problematic to me. While Badiou certainly proceeds textually in the way you describe, the interpretive conundrum that comes up under your analysis is that we would here have an ontology before maths, undermining the thesis that ontology belongs to maths. This would ruin the “deflationary” account of ontology that Nate referred to earlier in the thread. It seems to me that the core of his thesis is that set theory renders pure multiples or multiples qua multiples. In this respect, it would follow that the first few meditations of BE are an informal presentation of something that can only be demonstrated formally. In other words, what you get in the opening meditations is the thesis and then the subsequent demonstrations. I agree that for Badiou the thesis isn’t that being is mathematical, but rather that his thesis is a thesis about discourse or what can be said of being qua being. However, he does not hesitate to treat existence in terms of his claims about being, raising the question of what authorizes him to do so.
June 17, 2009 at 10:40 pm
Nate, I was joking of course. It’s a citation.
June 18, 2009 at 1:41 am
I think cameron is right here. If I remember correctly (as I don’t have my copy of BE with me), Badiou actually characterizes his statement that “the One is not” as a decision, a qualification that implies a lot vis-a-vis his theory of the subject. So it’s a sort of hypothesis of the nature “IF, as much of the last century’s philosophy has asserted, the One is not, THEN we must overturn the classic one/multiple opposition in favor of the multiple. As a consequence, being, the multiple-without-one, requires a language that can think multiplicity as such, which just so happens to be mathematics.”
Sorry if I’m repeating things you’re aware of, but I think the point is crucial. The proofs you circumscribe in your post are actually put to use by Badiou in different circumstances than demonstrations the non-being of the One. In addition, you appear to lump together two separate proofs in order to make your point more quickly. I don’t mean to come off as didactic or pedantic or anything, so I’ll sketch what I mean and, if you want, lay out the details in another post.
–Russel’s paradox: assumption of set of all sets produces contradiction.
–Employed by Badiou in discussion of axiom of separation (BE) and the declaration of the inexistence of the whole, or single overarching World (LW).
–Cantor’s theorem: the powerset of a given set is always greater in cardinality (consequence: there is no set of greatest cardinality, i.e. Cantor’s paradox).
–Employed by Badiou in discussion of ontology’s problem of excess and the claim that ‘God does not exist’, which leads to the discussion of errancy (continuum hypothesis)(BE). Also referenced in the declaration that worlds are closed under the powerset operation, hence their size being that of an inaccessible infinity (LW).
Neither of these are directly brought up in order to argue against the One, but certainly seem to validate his point. That said, I agree entirely that his use of these results, (and set theory in general in LW) in his accounts of both being and appearance seems to blur the supposedly clear line between mathematics and logic. There is, however, a delicate interplay between the two that is so complex that I can’t really tell whether he’s making huge violations of his claims or not.
June 18, 2009 at 2:07 am
Dana,
I’m a little unclear here as to what you’re claiming. I was moving rather quickly in the post as the aim wasn’t to give a didactic presentation of Badiou’s position or the various implications he draws from the non-existence of the One. I chose to discuss Cantor’s Paradox as it is relatively straight forward and easy for readers not acquainted to understand so as to get a sense of the flavor of his arguments (such readers would become quickly bewildered, I think, were I to instead discuss the Theorem of the Point of Excess). The examples you give from BE and LW actually seem to validate my reading. Wholes and totalities are instances of the One, as is the God of traditional theology. In each instance you present, we have set theory being mobilized to undermine a particular ontological claim (I italicize the “logical” portion of “onto-logical” to emphasize that I’m referring to issues from LW, not BE). The consequence then is that you get Badiou wanting to have it both ways. On the one hand he wants to say, that he’s just making claims about what can be said of being qua being, not existence or what being itself is. Fine. On the other hand, however, he perpetually uses these claims to legislate over existence, making existential claims about what exists and does not exist, and the nature of its existence (his claims about God, wholes, World, etc).
An additional point worth observing relates to those who have wished to treat Badiou as a realist rather than as a new variety of idealist. If Badiou’s ontology is decisional as you point out, in what sense is it possible to treat him as a realist? If the thesis that the One is not is the result of a decision and not a demonstration, then it strikes me as impossible for Badiou to be a realist because this entails that his core ontological claim is necessarily sutured to the one making the decision, rather than being a claim about the world itself that would hold regardless of whether any human were about to make this ontological decision.
June 18, 2009 at 2:15 am
I guess my point is as follows: Just because I can conceive a particular number, it does not follow that entities corresponding to this number exist in nature. Likewise, just because I can conceive a particular relation, it does not follow that that relation is exemplified anywhere in nature or the world. Given this gap between our ability to conceive mathematical relations and mathematical relations exemplified in nature, we get the issue of what entitled Badiou to claim that the sorts of mathematical formalisms are exemplified by world(s), God, ones, entities, etc.,
June 18, 2009 at 2:19 am
SDV,
Clearly any generalization is a claim about what is prominent in a population. Certainly you would agree that, as admirable as the thinkers you mention are, the thinkers you list are also rather marginal figures in Continental thought (precisely, I suspect, for the sorts of reasons I outline). Continental thought grew predominantly out of romanticism, and romanticism was characterized by a good deal of hostility towards Enlightenment mathematism. At any rate, I think I fairly clearly indicated that I wasn’t making claims of the sort “All whales are mammals” or “All Continentalists are hostile to math” insofar as I wrote with qualifications like “with few exceptions…”
June 18, 2009 at 3:11 am
“I think the view that these paradoxes only concern representational machinery is perilously close to being a linguistified (where the representing medium being language rather than phenomena) version of Kant.”
For me, it’s only as much as to say that problems that arise in thinking about reality don’t necessarily arise for reality itself.
There’s a relationship between the two, and that’s been a philosophical problem of sorts, but I don’t see it as a huge one. When I distinguish the phenomenal world (where “world” is probably saying too much) and the noumenal one, I’m keeping in mind that the phenomenal world is embedded in the noumenal one, and that there’s no universality or necessity to correlations between them. Making the initial distinction seems wise to me, as a way of keeping clear on the fact that what’s represented in our heads isn’t the thing being represented, but apart from that, the phenomenal is a small and bizarre eddy in the noumenal, not something separate in any way.
But I have a feeling I’m not understanding you correctly.
June 18, 2009 at 4:56 am
Levi,
I think this gets to the crux of the issue:
“On the one hand he wants to say, that he’s just making claims about what can be said of being qua being, not existence or what being itself is. Fine. On the other hand, however, he perpetually uses these claims to legislate over existence, making existential claims about what exists and does not exist, and the nature of its existence (his claims about God, wholes, World, etc).”
two things here. 1) if the claim is that mathematics is ontology, but that being is not itself mathematical, this does not mean that Badiou wants to restrict himself to what can be said of being as if there were some ineffable excess. rather, ontology describes being as it is, as a multiple of multiples. it isn’t mathematical in itself but set theory describes it as it is (not). it is itself the void/excess of what exists.
what brings me to 2). I think it is very dangerous to conflate being and existence in Badiou, though I don’t think he is always terribly clear about this (and I have yet to read Logic of Worlds). as I understand him, existence always pertains to being in a situation, which means that it is structured by a count-as-one, whereas being is not one. in other words, things present themselves as one (say, I have a ball of wax) but they are not really one. ontologically, they are inconsistent (my ball of wax fell apart). but here the point would be that it is only because they are inconsistent that they can change.
and this is what fascinates me about Badiou, because it seems that it is not even appropriate to say that there is an “it” that changes, or, better, that it which changes is an inconsistent multiplicity.
I think Badiou is basically Descartes without God. instead of keeping with the tradition, which says that “what presents itself is multiple, what presents itself is one” Badiou decides for presentation of presentation, multiple of multiple. I judge that it is the same ball of wax, but nothing justifies my judgment.. in itself, the wax is inconsistent.
if existence means something like standing-forth, as I think it should in talking about Badiou, then it is obvious that God, e.g, doesn’t exist (read, is not a sensible given). the theistic tradition always tried to claim nonetheless that God is, has Being, stands behind what presents itself as its ground of foundation. Badiou removes the ground – thus the Void. the being of an object, underneath its presentation, does not unify it as a substance through time, but is rather the principle of its disunity, its death.
what unifies the object (who’s counting) is another question, but I am trying to work out an interpretation of this which would avoid the sort of anthropocentrism you have accused Badiou of elsewhere, though this may not be possible…
June 18, 2009 at 5:45 am
Hi Caemeron,
You write:
Yes, I agree. The issue here isn’t one of how Badiou is to be interpreted. He’s very clear in asserting that being and existence are not the same. BE deals with being whereas Logics of Worlds deals with existence. None of that is in question. The point is that he himself constantly crosses these different levels in ways that are philosophically questionable.
I don’t think this can possibly be correct as a characterization of Badiou’s position:
In your emphasis on what is sensible, you are consigning Badiou to Kantianism (the thesis that knowledge requires an element of sensibility) and finitude. Badiou’s demonstration that God does not exists is not premised on the vulgar thesis that God can’t be sensed, but rather is a formal demonstration proceeding from formal properties of sets and the sort of set that God would have to be in order to exist. Here again we have an example of something that ought to remain in the domain of ontology (without making any claims about existence or non-existence) legislating existence.
June 18, 2009 at 6:10 am
what I meant to say with the God bit is that, if we accept a distinction between being and existence, a theist can accept that God does not exist, but instead subsists, is Being and not a being, however one might want to put this.
Badiou, I think, goes a step further. it is not merely that God doesn’t exist, but that God doesn’t have being either, is ruled out by being as inconsistent multiplicity, because God would be what would make Being consistent. this is why Badiou equates the claim that “the One is not” with atheism.
I’ll defer about the extent to which he crosses the levels in his work, since you have read more than I have here, but part of the reason I mentioned the God stuff was that you used God as an example of such crossing. I just don’t see what is illegitimate here.
certainly ontology is a condition of possibility for existence, such that if the nature of being rules something out it cannot exist, no?
June 18, 2009 at 6:29 am
Caemeron,
I think this is the core of the issue:
What I was trying to get at in my example of the Barber of Seville was that restrictions on formalization often diverge markedly from restrictions on reality. At the level of the proposition, it is impossible for the Barber of Seville to get his hair cut. At the level of reality, however, the Barber has no trouble getting his hair cut. The formalization doesn’t impede the reality. I’m not sure who it was that said the proposition doesn’t prohibit him from getting his hair cut. But that’s not true. The Barber of Seville paradox is just a variant of Russell’s Paradox. There is no way he can get his hair cut within the constraints of the formalization.
I think the issue here is that there is a difference between mathematical necessity and natural necessity. The scope of mathematical necessity is far broader than that of mathematical necessity. That is, there are all sorts of mathematical necessities that don’t occur anywhere in nature. There’s clearly a relation between mathematical necessity and natural necessity, but whether nature has the properties of sets as described by Z-F set theory is an entirely different matter. First, nature has ordering relations whereas formal sets do not. Second, it is not clear that existence can be infinitely decomposed whereas sets can. Third, because nature does have ordering relations, it is not clear that the subsets of a natural situation are always in excess of the initial natural set because natural entities cannot be combined in any old way (there are subsets that cannot occur in nature) whereas set theory allows us to form whatever subsets we might like from an initial set. Examples could be multiplied indefinitely.
So what’s the cash-value of all this. Setting aside the choice of any particular ontology, we would like to say that a good ontology will give us some insight into what’s possible and what’s not possible. The problem with Badiou’s particular ontology is that the gap between the infinite domain of mathematics and the structured domain of nature doesn’t really allow us to do this.
June 19, 2009 at 11:59 am
Asher Kay, I do not know how much ethnography you might be willing to read, but you might find Roy Wagner’s notion of obviation interesting. See, for example, Symbols that Stand for Themselves, particularly the chapters on New Guinea. You might also want to look at Levi-Straus’s claim in Do Dual Organizations Exist? that the north-south axis in Bororo villages is there solely so the village can exist; my dissertation supervisor, the late J. Christopher Crocker, actually found that axis while working among the Bororo.
June 19, 2009 at 12:08 pm
I should add that Symbols that Stand for Themselves is a sort of users guide to another book entitled Lethal Speech.
June 19, 2009 at 8:40 pm
I’m not sure I follow all of this discussion (though this has convinced me to buy Logic of Worlds, I’d been waffling before) so I apologize if I’m missing something.
“If Badiou’s ontology is decisional as you point out, in what sense is it possible to treat him as a realist? If the thesis that the One is not is the result of a decision and not a demonstration, then it strikes me as impossible for Badiou to be a realist because this entails that his core ontological claim is necessarily sutured to the one making the decision, rather than being a claim about the world itself that would hold regardless of whether any human were about to make this ontological decision.”
I may have misses something, but I don’t see why this is so. I guess it depends in part of how much metaphilosophy you want. I get the impression you’re saying “Unless Badiou can demonstrate that it really is the case that the One is not then his claim ‘the One is not’ can not be a claim about reality.” Did I misunderstand you or is that in the neighborhood of what you mean? If I get you, then I think that doesn’t make sense. One can have beliefs about reality which are not exhaustively demonstrated and still be a realist.
It seems to me that someone who says “I have among my intuitions the intuition that there is a real world and it has demonstrable properties” can be called a realist, though that person may fail as someone offering arguments as to why others should be realists. (By ‘intuition’ I mean some belief which one can not yet offer exhaustive demonstration, apologies if that’s an idiosyncratic use of the term; I take Badiou’s ‘decision’ to be not far off from this.)
I think you have a fair point that Badiou doesn’t do much to help convince others of realism in some sense, but if that’s true it still doesn’t entail that Badiou is not a realist in the sense of holding to some sort of realist philosophical position (regardless of the rightness of that position), let alone that he’s an idealist.
cheers,
Nate