Leithart
Frederiek Depoortere’s Badiou and Theology (Philosophy and Theology) 
What might those be? Depoortere’s book answers in several ways. Early on he summarizes a 1999 lecture of Badiou’s in which he reviews the sickness of contemporary philosophy, whether in its hermeneutic (Hedegger, Gadamer), its analytic (Wittgenstein and disciples), or its postmodern (you know who) guise. Despite their differences, these three trends share two flaws: all assume the end of metaphysics, which means the end of truth, and all assume that language is “the crucial site of thought.” Badiou finds both of these themes disastrous. Philosophy is dead unless it can “establish itself beyond the multiplicity of language games” (Depoortere’s phrasing – this is the Kantian agenda, as described by Hamann) and unless philosophy can affirm Truth, it has no way to stand against “the monetary uniformity imposed on us by global capitalism.”
True philosophy adheres to a “fixed point within discourse, a point of interruption,” an event to which one remains absolutely loyal. Hence Badiou’s interest in Paul, “a poet-thinker of the event.” Badiou’s own Damascus Road is less transcendent: It’s May 1968. A philosophy loyal to the event is characterized by revolt, logic, universality, and risk – all the features of genuine philosophy. As Depoortere puts it, “Without ‘the discontent of thinking in confrontation with the world as it is’ (revolt) and ‘a belief in the power of argument and reason’ (logic), true philosophy is seen as not being possible. True philosophy ‘addresses all human beings as thinking beings since it supposes that all humans think’ (universality’ and is, finally, always a decision which supports independent points of view’ (risks).”
That’s one way to put it. Another way is in Badiou’s own words: “mathematics is ontology.” This equation, Depoortere tells us, is “the basis of his entire philosophical system.”
Mathematics offers a way past the relativity of language, and also offers a way of getting into basic traditional questions of philosophy – one and many, for instance, infinity, the void. And for Badiou it also offers a way of conceiving a world with no more room for God.
We can start with Badiou’s concise description of the problematics of the one and many: “if being is one, then one must posit that what is not one, the multiple is not. But this is unacceptable for thought, because what is presented is multiple and one cannot see how there could be an access to being outside all presentation. If presentation is not, does it still make sense to designate what presents (itself) as being? On the other hand, if presentation is, the multiple necessarily is. It follows that being is no longer reciprocal with the one and thus it is no longer necessary to consider as one what presents itself, inasmuch it is. This conclusion is equally unacceptable to thought because presentation is only this multiple inasmuch as what it presents can be counted as one.” Badiou claims that the only escape from this dilemma is to decide that “the one is not” and to posit instead that “there is no one, only the count-as-one” (il n’y a pas d’un, il n’y a que le compte-pour-un).” One is always an operation, never a presentation.
Depoortere digresses into set theory to explain this non-being of the one. Set theory posits that no set can belong to itself; “the set of all cars is not itself a car.” As Bertrand Russell pointed out, this means that “there are collections which are not sets, or, formulated differently, there are multiples that cannot be counted as one.” Yet this creates a paradox. Suppose A is the collection of all sets that do no belong to themselves; is A itself a set? If it is, then by definition A does not belong to A; but not belonging to itself is the condition of being a set, so A does belong to A. But we can’t derive “A belongs to A” from “A doesn’t belong to A.” On the other hand, if we start from the premise that “A belongs to A” we end up conclusion that “A doesn’t belong to A” since A is the collection of all sets that don’t belong to themselves.
Rather than accepting this aporia, Badiou wants to find a way out of the impasse and does so by positing the axiom that paradoxical collections like A cannot be considered sets. For Badiou, A as the collection of all sets that do no include themselves is a description of U, the universe, which is “the collection of all possible sets.” This means, in turn, “that the universe is not a whole, not a one, but an infinite ‘multiplicity ‘made’ of nothing but multiples of multiples.” This gives Badiou a way of understanding infinity, a way that is resolutely antitheist.
“God is dead.” Badiou agrees, but recognizes that Nietzsche’s declaration is ambiguous. It could mean that the God of Christianity is dead – which he is since we can no longer encounter him (Badiou says); it could mean the god of metaphysics is dead – nonsense, since this god was never alive; or it could mean that the nostalgic poetic god of Heidegger is dead – and good riddance, Badiou says. And the mathematical pointer to God offered by Georg Cantor, the founder of transfinite set theory, is no longer plausible either, Badiou insists. Cantor was able to show that actual infinities do exist, and can be thought mathematically, but Cantor believed that beyond this “increasable actual infinite” that mathematics can describe there is an “unincreasable or Absolute actual infinite” that is God. Badiou saw this as Cantor’s way of dealing with the problem of paradoxical sets: “If a multiplicity cannot be counter-as-one in a coherent way, it is because it escapes from mathematics’ grasp. There, where the count-as-one fails, one humps into the Absolute, ‘the Infinite as supreme-being,’ or God.” For Badiou, Cantor is still a theologian: There is still one beyond the multiplicity.
Badiou finds what he thinks is a neater way to dispose of the paradox: “It suffices to axiomatically rule out these sets.” Following the work of set theorists after Cantor, Badiou argues for a “laicization or secularization of the infinite in which there is neither need nor place for God. Or, to put it more concisely: for Badiou, set theory demonstrates that ‘God is truly dead’ and enables a genuine atheism.” Set theory rules out an ultimate unity, and it secu
larizes and immanentizes infinity. Set theory disproves God.
But this, Depoortere says, only works because Badiou has excluded the possibility from the start. According to the “axiom of foundation” followed by Badiou (and other set theorists) paradoxical sets are simply ruled out axiomatically. Kenneth Reynhout (quoted by Depoortere) says that this is “a priori exclusion by fiat, and a fiat that serves no useful purpose other than making that specific exclusion. It would not be an exaggeration, therefore, to rephrase the axiom of foundation in this way: ‘there are not other infinities than the ones we construct.’” In that case, the argument that “eliminates the need for an extramathematical, theological infinite can rightly be regarded as begging the question.” Reynhout further raises the question of whether the axiom of foundation imposes “an unacceptably limitation of the scope of ontology from the very beginning.”
Cantor’s theological version of set theory (which I’ll briefly summarize in another post) is just as plausible as Badiou’s atheistic one.
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