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.