Badiou’s Number and Numbers: Part {0}

Usually during winter break I try to get a scholarly project started. I tend to flounder until the last few days of the break, when the ideas seem miraculously to come together. Then the semester begins. Any project begun during break quickly dissipates, and if I’m lucky, I can pick up the pieces and turn it into something else a little later on. But during this break, I went in a slightly different direction. I simply spent much of my off time rereading Alain Badiou’s Number and Numbers.


I had read and cited Number and Numbers before, but always had the nagging sense that I had under-read it. The first time through, I read it as the 240 page book it appeared to be. This time, I read it as the 900 page tome it really is. Number and Numbers is one of Badiou’s least celebrated books, but it’s my favorite. It is to the Being and Event books what the Prolegomena is to Critique of Pure Reason. It’s a stripped down book of pure ideas without much meditation on their implications. But like the surreal numbers Badiou champions, it’s a dense, dense fabric. It’s the thought process of a genius on full display, but it also gives us a glimpse of Badiou the teacher. What he gives the reader is difficult, but he’s gentle and generous in the delivery. Credit here must also go to Robin Mackay for the beautiful translation.

Writing as I am on WordPress, I don’t have the characters to get into the nitty gritty of Badiou’s logic in this post. Nor, frankly, do I have the confidence to do so at this stage. What follows is pretty surface level stuff upon which I will expand in an upcoming book project, which I am tentatively calling Creole Mathematics. I’ll focus here on some of the virtues and criticisms of the book…and I’ll probably wonder off on my own a bit as well.

Despite the heavy lifting it takes to get through the book with a modicum of understanding, Badiou’s primary aim is actually quite modest. There’s no articulation here of a complete philosophical system like you’ll find in the Being and Event volumes. He simply wants to demystify the concept of number. OK, that’s actually pretty lofty–but the book is a manageable attempt at it.

Of course, by putting number at the heart of his ontology, Badiou could–not unjustly–be accused of performing his own mystification. And there’s no point in pretending that the construction of surreal numbers is as easy and accessible as he prefaces, even if you do have some background in set theory. But here’s the real argument: the democratization of math begins with de-operationalizing number. Badiou, in other words, wishes to bring number back into the kingdom of ends. He speaks in generalities about “number’s despotism” in the reign of Capital (1), arguing that “The bureacratisation of knowledge is above all an infinite excrescence of numbering” (2). But it strikes a particular note in the America of 2017, with its slavish devotion to the STEM messiah. The idea that polynomial fluency will protect young people from the ravages of financial capitalism is just about as ludicrous as the notion that a strong leader will somehow bring manufacturing jobs back to the middle aged. Insofar as math is a skill and numbers are instruments, mathematics remains the only area of knowledge untouched by politics. It is, if you like, the Teflon discipline. Here, I’m reminded of Andrew Hacker’s interesting (if somewhat flawed) book, The Math Myth: And Other STEM Delusions. One of the things Hacker points out is that, globally, students with the highest math achievement scores often come from some of the most politically repressed nations in the world (China, Russia, Iran, etc.). Perhaps Hacker lifts causality a little too easily from correlation, but the point about math’s innoxious relation to politics is well-taken, and it leaves those of us in the humanities with a stark choice: either play down the importance of math and hope someone listens or return it to its proper place in philosophy. Badiou encourages us to choose the latter. I’m inclined to agree, even if I’m not prepared to accept mathematics as a general ontology. And, excluding the Alan Sokal wing, I don’t think that even Badiou’s most persistent critics would disagree with that either.

Criticisms of Badiou come in two types. The first criticism–that of the Sokal wing–states that mathematics, like science, is a serious business for grown ups, and that French philosophers shouldn’t be nosing around in it with their anachronistic notions of subjectivity and poetics. This is anti-intellectual dogshit. Obviously. The other type of criticism comes from the philosophical circles with which I tend to identify. Among these critics, there is no question of taking Badiou seriously, but it is supposed that he commits a category mistake by conflating existential freedom with the inconsistent multiplicity of ordinals, and by dissolving real objects into sets. For example, Nirenberg and Nirenberg, who authored a very thoughtful response to Number and Numbers, argue that Badiou pays an acknowledged price by his generalization of transitivity as stipulated in the Zermelo-Fraenkel axioms:

One of the ZF axioms is that for any two sets x and y there is a third set x∪y such that z∈x or  z∈y if and only if z∈x∪y. For the union to exist as stipulated, the elements of and the elements of y must be such that they don’t interact, meaning nothing happens (apathés) to the elements of x or the elements of y when we bring them together: no change in identity (607).

This is easily seen in Badiou’s historiographical meditations in other works, where he enumerates the elements of an event, such as the sans-culottes, printing presses, the Bastille, etc. in the French Revolution. On the surface, it looks like a classic Latour litany (to borrow from Ian Bogost), albeit with a particular historical theme. But for Badiou, it is an inconsistent multiplicity, like an infinite set. The significance of which is that the elements of the set (which are themselves sets) are not deducible from a given situation, and thus open up the possibility for a novel event. By contrast, when Latour talks about Pasteur, microbes, Petri dishes, wastepaper baskets, etc., he is describing actants, which are pathic elements, capable of effecting novelty in one another. The inability of Badiou’s system to conceive of novelty or emergence in these terms is indeed a high price to pay.

So then what, besides the non-deductibility of the event does his system purchase? That’s what I’ll be getting up to in the next post.

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