Badiou on objects and relations. Set-theory is object-oriented, category theory is relation-oriented.
For Badiou, Plato is object-oriented, and “imposes as norm, as proper object of true thought the real being of ideas, whereas Kant organises the thought of knowledge on the basis of relations”. Badiou argues that “on an objectivity in some sense unknown, the knowing subject operates by relational projection so that a coherent phenomenal world appears. But this phenomenal world is coherent on the basis of the rationality of relations, since the being in itself of this world is unknown to us, is beyond us” (“nous échappe”, we could translate this as “the being in itself of this world withdraws from us”). It is clear that in Badiou’s categorisation Graham Harman is relation-oriented in effect, even if he is object-oriented in intent.
Applying Badiou’s categories, we can see that Harman’s so-called “object”-oriented ontology is in fact relation-oriented, as the real object is behind the veil of unknowing (or veil of withdrawal). This is why I disagree with the necessity in the first chapter of his book of Wolfendale’s excellent summary of Harman’s ideas about relations between real objects and with his taking Harman’s “fourfold” seriously. The real object withdraws, period. All this talk about “molten cores” and the “excess” of the object, that cannot be “exhausted” by its encounters, is not a valid part of Harman’s core philosophy. It is part of the ad hoc adjustments necessary to reconcile, in appearance only, his esoteric ontology of withdrawn real objects with his exoteric “naive” ontology of known or experienced objects.
Graham Harman’s “ontology” is in fact at least three ontologies. Firstly, an exoteric “naive” ontology of objects as we experience and know them, declared by Harman to be “pure sham”. This is the ontology picked up by the artists and architects, especially when it is combined with the third ontology. Secondly, an esoteric ontology of unexperienced and unknown real objects. Thirdly, a metaphoric ontology with “molten cores” and shadowy allusions, bridging the gap between the withdrawn, unknown, and unexperienced real and the manifest, experienced and known sensual realm.
“Between Plato and Kant we find again these two orientations: the priority of objectivity or the priority of the constitution of the known object by relations. It seems to me that the contemporary allegory of all that has also been the discussion on the foundations of mathematics. The creators of set theory seem to be in general philosophically oriented towards a realism of the object of thought. This was typically the case of Gödel. Category theory would orient us towards forms, that are disciplined and coherent, but nevertheless forms of relativism, because it proposes a geometry of relations subject to principles of variability, like elementary geometry itself in its different post-Euclidean possibilities. To help you understand this point, I think I will refer to the American mathematician Bell who has developped, in a polemical manner, in a book called TOPOSES AND LOCAL SET THEORIES. Local set theories, thus theoretical variability on the notion of set itself. Bell affirms at the same time the existence of several set theories, perhaps even of a virtual infinity of set theories, thus he affirms the indetermination of the concept of set itself, which radically de-objectifies, in a certain sense, the set theoretical universe”.
This is part of the later “allegorical” Badiou, who I find more inspiring than his objectivised technical set-theoretical avatar. For more on the allegorical Badiou, see my review of his CINEMA.