Meditation 15 of Badiou’s Being and Event is one of the most forbidding encounters of that book, pitching the Hegelian doctrine – or, more precisely, doctrines – of the infinite against Badiou’s own Cantorian conception. Keeping both these competing views in focus while attempting to grasp their precise differences and perhaps even adjudicate between them….. I remember someone stating that reading Difference and Repetition is intellectually akin to running into the ocean – splashing out those exhilarating first few steps before you abruptly come to a halt and tumble into the deep water of “sink or swim” – but I’ve read this meditation a few times before, and each time it feels more akin to running into a brick wall. And so, previously, I’ve settled for the bland “Badiou thinks Hegel cannot do justice to the mathematical infinite” conclusion. Which is clearly unsatisfactory in the long run, so this post is a first effort at dismantling the “wall” aspect of those nine pages.
Some of the obvious – too obvious – disagreements between Badiou and Hegel can be stated at the outset, simply to get a preliminary grasp of the battlefield:
Hegel thinks that infinity is 1) generated, i.e., immanently derived from the dialectical process, and 2) ‘good’, i.e., qualitative, so it is the qualitative essence of quantity that constitutes its true infinity.
Badiou, on the other hand, thinks that infinity is 1) postulated, i.e., that the existence of an infinite set must be axiomatically decided upon, and 2) ‘bad’, i.e., quantitative, since Cantorian infinity allows for “the very proliferation that Hegel imagined one could reduce” (B&E, 170).
Both these disagreements can be found more generally in Badiou’s engagement with the philosophical and mathematical tradition. For instance, Badiou likewise criticizes Dedkind’s attempted deduction of the existence of the infinite, in Number and Numbers, ch. 4. (“Now, just like the empty set, or zero, the infinite will not be deduced: we have to decide its existence axiomatically” (N&N, 44). And the qualitative conception of infinity is something Badiou takes the philosophical tradition in general to task for, seeing in it a latent religiosity.
We must immediately complicate the above assertions, however, for they do not adequately represent Badiou’s particular line of argumentation in meditation 15. It might, from the above, seem as if Badiou gainsays any deduction of the infinite, and that he opposes the proper, quantitative infinite to the spurious qualitative infinite. But in fact, Badiou does not object to Hegel’s deduction of the good qualitative infinite. And in fact, Badiou recognizes that there is not, in Hegel, a simple dichotomy between qualitative and quantitative infinity, but rather a fourfold of infinities: the bad qualitative infinity, the good qualitative infinity, the bad quantitative infinity, and the good quantitative infinity.
What Badiou seems to argue is 1) that Hegel cannot properly ground the third infinity, the bad quantitative infinity: “One must recognize that the repetition of the One in number cannot arise from the interiority of the negative” (B&E, 169). 2) That in any case, what Hegel puts forward as the fourth infinity, the good quantitative infinity, cannot properly be called “infinity” at all: “I have no quarrel with there being a qualitative essence of quantity, but why name it ‘infinity’?” (B&E, 169).
I must therefore demur from Hallward when he writes, in Badiou: A Subject to Truth, that “The crux of Badiou’s argument is that without tacit recognition of the disjunctive decision to affirm the infinite, the good, ‘qualitative’ infinity cannot join up with a properly quantitative notion of infinity at all.” (173). The problem is not that there are two different infinities, and they cannot be joined. Rather, the problem is as follows: there are two dialectics, quality and quantity, which Hegel connects by means of a “fragile verbal footbridge thrown from one side to the other: ‘infinity’.” (B&E, 170) But this footbridge is fragile indeed: It is in fact merely verbal, since on one side it is anchored to an ungrounded hallucination – the Hegelian ‘good quantitative infinity’.
I’ll try, next, to investigate more in detail Badiou’s analysis of Hegel, and perhaps even delve properly into the Science of Logic to see whether his objections can be met from a Hegelian standpoint.