In his funeral oration for Jean Hyppolite (collected in Pocket Pantheon), Badiou recalls that during his entrance examination for the Ecole Normale Supérieure, Hypollite, who was his examiner, asked him what the difference was between a thing (chose) and an object (objet):
I improvised an answer. And I have to say that, having worked in recent years, and with great difficulty, on the notion of an object, I still bear that warning in mind. I fear that, even today, I still confuse the two.
From Theory of the Subject, which opens with a consideration of the dialectic of force and placement, through the treatment in Being and Event of the excess of representation over presentation, to Logics of Worlds and the topic of “being-there”, the distinction between the thing-in-itself (chose) and the thing-in-its-place (objet) is a recurring concern of Badiou’s. To give it a simple, if slightly gnomic, formula: the thing in-sists, but the object ex-sists.
In Being and Event, a “thing” is simply a “being”, a multiple counted-as-one. The role of “object” or thing-in-its-place is filled by discursive objects, referents of encyclopedic predications. There are thus two ways in which a “situation” is structured: as a multiple of multiples, a thing composed of things, and as a system of referents classified and named through some predicative language which enables the discrimination of “parts” of the multiple. The power of the “event” is that it presents some chose as causa sui, and so initiates the composition of a new multiple (the “generic extension” woven from the elements of the situation) which has no place in the existing system of names. The event disarranges the familiar furniture of the world: it reveals that the contents of our “Latour litanies” of nameable parts of reality are provisional, that the true multiple-composition of a situation is neither dependent on language nor restricted to that which is able to be said.
This is not quite the same - nota bene - as saying that “objects” are arbitrarily carved (or autopoetically self-stabilised) out of some pre-objectal flux of becoming. The point here is that Badiou’s “mathematical ontology” gives us a picture of a “situation” that is not only intricately structured (being a multiple-of-multiples, to any degree of recursive depth you like*), but structured in such a way (or to such a degree) that the resources of its own language are unable to discriminate every part of it. No catalogue of objects capable of being compiled using the language of a situation can ever exhaust the multiple-being of the situation itself. But this does entail that an “object” be considered solely as the referent of an entry in such a catalogue.
The story changes significantly in Logics of Worlds, which I’ll come to next: there, the “being-there” of the object is decoupled from language, but attached instead to the “transcendental” which governs relations of identity and difference between objects. The object is no longer considered primarily as a referent pinpointed within a system of names, but it remains in a precarious relationship to the pure “thing” which still - through the mediation of the event - has the power to disturb its worldly co-ordinates.
* unless you like violating the axiom of foundation.