Badiou’s Number and Numbers: Part {0{0}}

I left off the last post asking what Badiou gets out of his reliance upon Zermelo-Fraenkel axiomatic transitivity, given its crippling exclusion of pathic interaction. His characterization of ontology in terms of the inconsistent multiplicity of ordinals is certainly capable of describing the possibility of existential novelty (or emergence), but performs poorly for novelty itself. For similar reasons, Levi Bryant claims that Badiou makes a category mistake by confusing essence with existence, Badiou’s set theoretical truths describing only the former. But, Bryant concedes, it is quite a powerful description of essence. Here I want to talk a bit about just why it is such a powerful description of essence.

The idea that reality is composed of matter and the Void, and that matter is governed by mathematical law, is of course nothing new. It’s Modernity in a nutshell. And while Badiou is certainly faithful to that epochal split of corpuscularianism from plenary scholasticism, his materialism is way more radical. It’s not just that matter interacts in ways that mathematics is uniquely qualified to describe. And it’s not even that mathematical laws constitute the self-same forms into which material bodies aggregate. For Badiou, matter itself is mathematical and vice-versa. Verily, matter as unitary bodies is anathema to Badiou.

Badiou can be described as a Platonist with (material) benefits. But actually, his understanding of matter sounds suspiciously Aristotelian, at least from the outset. The idea is that while it is basically impossible to think of general matter outside of particular forms, for matter to be matter, it must exceed any given form. So, for water to be the stuff of oceans, lakes, and rivers, water must be more than the oceans, lakes, and rivers of which it is the stuff. Otherwise, the oceans, lakes, and rivers would be pure forms. And you can’t dive into a form. You’ll break your crown. Similarly, the form of an ordinal (that which we recognize as a number) is exceeded by its ordinal matter. (Badiou gives a rigorous explication of ordinal matter, form, and residue as he approaches his discourse on surreal numbers, which finally serves as staging point for operationality). Strange as it may seem, the easiest way to picture this is to think of an infinite series: 1, 2, 3, 4…ω. The mark of the infinite (ω) is the limit ordinal. Now think of the last number before you get to ω. Call it n. What you get is not: 1, 2, 3, 4,…n, ω. There is always an n+1. This is pretty intuitive, because no matter how hard it is to think of infinity as a number, it’s probably even more of a stretch to think of what comes just before infinity. It’s similarly difficult to imagine something between the cardinality of an infinite set and its power set, even though the Continuum Hypothesis suggests this is the case. I remember, as a third grader, making what I thought was a pretty good argument for a gazillion-gazillion nine as the last number before infinity.But short of that, it has to be said that there is always ordinal matter between and ω. This works for all ordinals, and thus there is a convenient isomorphism between, on the one hand, the traditional matter which we cannot think of without particular forms, and ordinal matter on the other. This difference is that with set theory, we can think of ordinal matter inductively, and thus rigorously and publicly. So here we get a convergence of thought and being, perhaps even an indistinction. That, as far as Badiou is concerned, is value for your money. (Though, of course, one of the things he wishes to communicate is that numbers cannot and should not be reduced to value.)

And that gets to one of the wonderful things about mathematics, if you have the imagination to think of it in a non-operational or instrumental sense. Mathematikos (μαθηματικός) in the original Greek connoted something between learning and creating. It’s something you learn, but not as a series of facts. It has a way of bootstrapping that occurs somewhere between you and it. This is what Kant was thinking when he put something like 5+7=12 in the realm of the synthetic a priori. 5+7=12 is not a fact of causality that you experience and record (although most people sort of do learn it that way), but its also not something you totally create yourself, ex nihilo. Mathematics appears to think itself as you’re thinking it. And this sounds suspiciously like the Aristotelian version of God. It moves by thinking itself, causing others to move with it.

And this is a difference between a math-centered ontology and the bleak materialism which grew out of Modernity, and which some call “scientism.” In the latter, there is nothing thinking itself. There are only material bodies, which, in relation to one another, we think of as facts. To be sure, Badiou doesn’t hold out for an unmoved mover (irreducible multiplicity is enough for him), but his mathematical ontology, I believe, is a great intervention into the banality of contemporary materialism.


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