Alain Badiou = Being and Event = Mathematics as Ontology

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Niall O&#8217-Connor has more in common with Robin Hanson than you would have thought one hour ago: They both love philosophy (i.e., blah blah blah).

Niall O&#8217-Connor:


On a broader point, you would be well served as a Frenchman to promote Badiou and his notion of “mathematics as ontology”. Moreover, you and your merry band of readers would all do well to put Badiou’s “Being and Event” on your holiday reading lists- for a little light vacation reading.


Introduction to Being and Event
Drawing from 8 March 2006 &#8220-Art&#8217-s Imperative&#8221- lecture

The major propositions of Badiou&#8217-s philosophy all find their basis in Being and Event, in which he continues his attempt (which he began in Theorie du sujet) to reconcile a notion of the subject with ontology, and in particular post-structuralist and constructivist ontologies.[3] A frequent criticism of post structuralist work is that it prohibits, through its fixation on semiotics and language, any notion of a subject. Badiou&#8217-s work is, by his own admission,[4] an attempt to break out of contemporary philosophy&#8217-s fixation upon language, which he sees almost as a straitjacket. This effort leads him, in Being and Event, to combine rigorous mathematical formulae with his readings of poets such as Mallarme and Holderlin and religious thinkers such as Pascal. His philosophy draws equally upon &#8216-analytical&#8217- and &#8216-continental&#8217- traditions. In Badiou&#8217-s own opinion, this combination places him awkwardly relative to his contemporaries, meaning that his work had been only slowly taken up.[5] Being and Event offers an example of this slow uptake, in fact: it was translated into English only in 2005, a full seventeen years after its French publication.

As is implied in the title of the book, two elements mark the thesis of Being and Event: the place of ontology, or &#8216-the science of being qua being&#8217- (being in itself), and the place of the event — which is seen as a rupture in ontology — through which the subject finds his or her realization and reconciliation with truth. This situation of being and the rupture which characterizes the event are thought in terms of set theory, and specifically Zermelo–Fraenkel set theory (with the axiom of choice), to which Badiou accords a fundamental role in a manner quite distinct from the majority of either mathematicians or philosophers.

Mathematics as ontology

For Badiou the problem which the Greek tradition of philosophy has faced and never satisfactorily dealt with is the problem that while beings themselves are plural, and thought in terms of multiplicity, being itself is thought to be singular- that is, it is thought in terms of the one. He proposes as the solution to this impasse the following declaration: that the one is not. This is why Badiou accords set theory (the axioms of which he refers to as the Ideas of the multiple) such stature, and refers to mathematics as the very place of ontology: Only set theory allows one to conceive a &#8216-pure doctrine of the multiple&#8217-. Set theory does not operate in terms of definite individual elements in groupings but only functions insofar as what belongs to a set is of the same relation as that set (that is, another set too). What separates sets out therefore is not an existential positive proposition, but other multiples whose properties validate its presentation- which is to say their structural relation. The structure of being thus secures the regime of the count-as-one. So if one is to think of a set — for instance, the set of people, or humanity — as counting as one the elements which belong to that set, it can then secure the multiple (the multiplicities of humans) as one consistent concept (humanity), but only in terms of what does not belong to that set. What is, in following, crucial for Badiou is that the structural form of the count-as-one, which makes multiplicities thinkable, implies that the proper name of being does not belong to an element as such (an original &#8216-one&#8217-), but rather the void set (written O), the set to which nothing (not even the void set itself) belongs. It may help to understand the concept &#8216-count-as-one&#8217- if it is associated with the concept of &#8216-terming&#8217-: a multiple is not one, but it is referred to with &#8216-multiple&#8217-: one word. To count a set as one is to mention that set. How the being of terms such as &#8216-multiple&#8217- does not contradict the non-being of the one can be understood by considering the multiple nature of terminology: for there to be a term without there also being a system of terminology, within which the difference between terms gives context and meaning to any one term, does not coincide with what is understood by &#8216-terminology&#8217-, which is precisely difference (thus multiplicity) conditioning meaning. Since the idea of conceiving of a term without meaning does not compute, the count-as-one is a structural effect or a situational operation and not an event of truth. Multiples which are &#8216-composed&#8217- or &#8216-consistent&#8217- are count-effects- inconsistent multiplicity is the presentation of presentation.

Badiou&#8217-s use of set theory in this manner is not just illustrative or heuristic. Badiou uses the axioms of Zermelo–Fraenkel set theory to identify the relationship of being to history, Nature, the State, and God. Most significantly this use means that (as with set theory) there is a strict prohibition on self-belonging- a set cannot contain or belong to itself. Russell&#8217-s paradox famously ruled that possibility out of formal logic. (This paradox can be thought through in terms of a &#8216-list of lists that do not contain themselves&#8217-: if such a list does not write itself on the list the property is incomplete, as there will be one missing- if it does, it is no longer a list that does not contain itself.) So too does the axiom of foundation — or to give an alternative name the axiom of regularity — enact such a prohibition (cf. p. 190 in Being and Event). (This axiom states that all sets contain an element for which only the void [empty] set names what is common to both the set and its element.) Badiou&#8217-s philosophy draws two major implications from this prohibition. Firstly, it secures the inexistence of the &#8216-one&#8217-: there cannot be a grand overarching set, and thus it is fallacious to conceive of a grand cosmos, a whole Nature, or a Being of God. Badiou is therefore — against Cantor, from whom he draws heavily — staunchly atheist. However, secondly, this prohibition prompts him to introduce the event. Because, according to Badiou, the axiom of foundation &#8216-founds&#8217- all sets in the void, it ties all being to the historico-social situation of the multiplicities of de-centred sets — thereby effacing the positivity of subjective action, or an entirely &#8216-new&#8217- occurrence. And whilst this is acceptable ontologically, it is unacceptable, Badiou holds, philosophically. Set theory mathematics has consequently &#8216-pragmatically abandoned&#8217- an area which philosophy cannot. And so, Badiou argues, there is therefore only one possibility remaining: that ontology can say nothing about the event.