The existence axiom at the basis of ZF set theory postulates the existence of the empty set, the set without elements. But being without elements, thus also without parts, the empty set qua mathematical cannot be an object according to OOO (Object-Oriented Ontology), since an object is defined on the basis of specific relations (emergence, submergence) between the whole and its parts. It cannot even be a pseudo-object, since a pseudo-object is also defined on the basis of specific relations (lack of emergence, etc) between the whole and its parts. (See this post for a summary of Graham Harman’s views on objects and pseudo-objects).
Insofar as one can construct basically the whole of mathematics out of ZF set theory, it might seem as if mathematical structures in general are neither objects nor pseudo-objects. So what are they? What is mathematics, according to OOO?
In itself, classifying mathematics as non-objective need not be a problem. For instance, the point that mathematics is not “objective” and has no objects is central to Badiou’s account of mathematics. In fact, it is a precondition for his central claim that mathematics is ontology. Mathematical structures are not objects, they are the being of objects. For object-oriented ontology, on the other hand, it must mean that mathematics has no place within ontology. Of course, there are many ontologies which do not include mathematics and mathematical objects. But object-oriented ontology includes societies, ideas, phonemes, organizations, and so much else (not by coincidence can it be referred to as promiscuous ontology), so it is fairly surprising that it does not seem to include mathematics!
I do not mean, of course, that it does not include mathematics at all. It obviously does. Mathematics qua signs, for instance the name of the empty set, ø, is presumably an object. So is mathematics qua scientific practice, mathematics qua mental and physical ideas and constructions, and so on. But mathematics qua mathematics is, I suspect, not. If this is correct, then OOO is reductionistic with respect to mathematics, insofar as the existence of the subject matter of mathematics cannot be affirmed as such, but only as something else, whether signs, ideas, physical patterns, practices, or something else.
I’m fully aware that I’ve drastically simplified things here, basically reducing the mathematics-OOO interface to the encounter between the statements “the empty set exists” and “an object has parts”. My primary wish is to see the partisans of object-oriented ontology engage with mathematics, and I would be perfectly happy to stand corrected. As per now, whereas I am not convinced by either Badiou’s or Meillassoux’ account of mathematics (though seeing as Meillassoux has not yet published his thoughts on mathematics I can hardly expect to, in his case), when it comes to OOO I have only the vaguest idea of what their account would even be, as this post no doubt shows.