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.
Transcissions 
Interesting post. I’m sure most of us (I know I am) would be happy to include maths among the domain of objects. Object is not a synonym for “physical thing”. The question would be just where to locate the objects of mathematics? Are numbers and shapes, for example, mathematical objects? I’m inclined to say no. Rather I would argue that a particular number and a particular shape is a local manifestion of a mathematical object. Each branch of mathematics would be a particular mathematical object. Thus Z-F set theory is one object, calculus another, topology theory another, Euclidean geometry another, and so on. This allows us to see how maths has submergent qualities and emergent properties: each object contains theorems not currently known or explored (submergent qualities), and each object produces theorems (emergent properties). Maths are among the best examples of objects in ooo’s sense, precisely because they are independent of their relations to other things and because they have infinite fecundity. For example, for Euclidean geometry a right triangle is a right triangle regardless of whether it’s written in chalk, marker, sky writing, or embodied in wood or steal. Triangleness is independent of its relations to other objects. When comparing Badiou and ooo it’s important to attend to our respective theoretical universes. Badiou is not using the term “object” in the way ooo is. As such his use of the term is ammere homonym of ours. Consequently, Badiou suggesting maths is without an object really is neither here nor there. All he means is that maths is not about physical things. But physical things are not the only type of objects that exist for ooo.
You might want to be careful here, as Graham has definitely committed himself to the idea that numbers (and thus perhaps other mathematical objects) are objects in his sense. Precisely what parts such objects are made out of, and how they enter into relations are another matter. Though obviously, given Levi’s comment, there’s some space for disagreement here within OOO.
Also, I’m no Badiou expert, but I’m pretty sure that Badiou’s claim that mathematics does not talk about objects is not merely a claim that mathematics does not talk about *physical objects*, insofar as it’s meant to be a response to mainstream debates in the philosophy of mathematics regarding what mathematical objects are (and pretty much no one holds that they are physical). The claim is that they are not *beings*, a category which for Badiou includes things other than physical objects.
Larvalsubjects: Thanks for the reply! That’s really interesting, the suggestion that what one would commonsensically call “mathematical objects”, i.e., things like numbers and figures, are in fact local manifestations of the real mathematical objects which are the branches of mathematics. I’m aware that OOO countenances a lot more than “physical things” as objects (though I disagree that Badiou merely means that maths is not about physical things when he says that it has no objects), my observation was that it does seems to require a part/whole structure, and the empty set does not, according to set theory, have such a structure. But taking the branches of mathematics to be the objects, rather than the structures of those branches, might obviate my worry. So thanks again for the response, interesting and not what I’d expected! I’ll have to think about it more to know what to say.
I was under the impression that every set, for Badiou, is necessarily infinite. Is that a mistake in the case of the empty set? At any rate, as strange as it sounds, I’m more than happy to treat the empty set as an object. Perhaps it could be argued that the empty set is that object that opens the set theory as an object (insofar as all the sets of set theory are built out of the empty set).
Pete: Yes, that’s what I would say about Badiou as well, going by his article in the Theoretical Writings about his own particular kind of Platonism, for instance. In fact, if pressed I would say that *beings* for Badiou is a wider category even than objects for OOO (in including much that OOO would label pseudo-objects).
Well, if it turns out Levi believes that set theory, topology, etc., are the mathematical objects, whereas Graham believes that numbers and figures are (also) objects, then that’s a pretty interesting difference! I guess the worry of my main post is more properly directed towards Graham, then…
Larvalsubjects: I am fairly certain that not every set is infinite for Badiou. What he might say is that every non-ontological situation is infinite in its being, or something like that…… But all the finite numbers, for instance, are finite sets. Including the number zero, which he equates with the empty set. (?? I’ll have to check Number and Numbers, but I think he does equate them)
Ah, yes I do find it strange that you consider the empty set to be an object! Since according to set theory, the empty set has no elements, thus no parts. But that might be because I am confusing Graham’s criteria for what an object is with yours, and if so I’m sorry. Do you not believe that an object must have parts? Or do you believe that the empty set has parts?
Oh, and feel free to just point me to prior writings if you want. I’m quite curious about how each of you object-oriented ontologists see mathematics, and haven’t seen much written about it, but I might just have missed it.
I was under the impression that every set included an infinite number of elements, while it’s not the case that all sets have parts. This, I thought, was how you can generate additional sets from the empty set. For OOO, at any rate, the being of an object is not defined by extensionality in a set. In my own framework, for example, objects like a society do not have humans as parts. Humans or people are strictly outside of society (I’m following Luhmann here). Likewise, cells aren’t “parts” of the body. This is the strange mereology I’m going on about. In this regard, a set without parts is not really a problem for me. As long as it’s a unity that’s enough for me. There might be a debate to be had between Graham and I regarding maths. My initial hunch is to say that a triangle is not an object at all, but rather a local manifestation of topology. But here I have to be careful, because my intuition is to say that Euclidean geometry isn’t an object because it can be subsumed under topology (topology would be the real object). I think similarly about language. On most days I’m inclined to say that words aren’t objects but are local manifestations of language, which is the real object. Likewise, a class wouldn’t be an object, but would rather be a local manifestation of the hyperobject, capitalism. These positions are soft, however, and I go back and forth, ie, I’m still working through them. All of this is a way of saying that I wouldn’t want to exclude maths from the domain of being. Rather, it’s a question of working out the appropriate regional ontology of maths.
In fact, it seems (I had to check this now myself and it wasn’t like I remembered… it is in meditation 7 of Being and Event) that while the empty set has no elements, it does have a part (i.e. subset), namely itself. And so one generates an additional set by using the power-set axiom to take the set of the empty set’s parts, which is {ø} rather than ø itself. And more sets by repeating that process….
In any case, I look forward to more about this kind of ‘strange mereology’! (Democracy of Objects, right?) I guess I have hitherto assumed that the point was merely that the whole could not be reduced to the parts, e.g., that the body could not be reduced to the cells and so on. So that even though the whole cannot be reduced to its parts, the parts still have to be there. That’s sort of a “normal mereology + emergence” picture. But you’re saying that it’s stranger than that, that the cells are not strictly speaking the parts of the body at all, that an object need not have “parts” in the normal mereological sense at all, right? Interesting.
The objects you mention here (language, capitalism, topology), and the way in which you mention them, remind me of Deleuze’s Ideas in chapter 4 of Difference and Repetition…. a chapter I love by the way. I’m intrigued, so let me just ask one question: If words are local manifestations of language, and classes local manifestations of capitalism, what stops, for example, individual biological creatures from being local manifestations of “the organism” (as per Deleuze’s second example of Ideas in chapter 4)? Can you have that kind of “object-oriented structuralism” (for want of a better expression) in certain regions of ontology without having it in all regions?
A lot of my inspiration for the internal structure of objects comes from chapter four of DR (chapter three of TDO is devoted to this). I’ve never liked the thesis of a biological Idea in that I believe biological entities are discrete units in their own right. In my view, they proceed by iteration and repetition (copies with a difference) not as local manifestations of one and the same object. This is the only way I can make sense of the Darwinian orientation.