Here’s a good post from Graham on Badiou.

This is the weird thing about Badiou that I can’t get my head around. Knowing what I’m going to say next will probably provoke a ‘misreading’ of some sort. I’ll have to be brief and hence, woefully disingenuous (despite what I say, I have deep respect for Badiou’s work).

This is only something that has really made sense since I began to re-read the Badiouian corpus. For Badiou, there is nothing behind number, quite literally. This is the anti-Heideggerian move that Badiou puts forward in the beginning of B&E (although Graham’s right, he’s doesn’t have a dialogue with Heidegger as much as one would suppose); number and thus sets of number are thoroughly immanent, and thus cannot be founded on anything deeper than a formal structure of the void in a real presentation or representation.

Before ones delves into the math, Being is presentation, and thus consistent and inconsistent multiplicity of sets are derived from the same Being-ness in presentation. The outcomes of this has only made sense to me recently; for most people who have an interest in this, but don’t consider philosophy to be their primary field, one would have thought that for Badiou, Being is mathematical. In fact it isn’t, it’s more discursive than that. Set theory establishes what is expressible in presentation, and Badiou makes the case that immanent presentation is ‘all’ there is (much more so in N&N incidentally). Real sets are all there must be in their unbridled immanence. One could, if they were a betting man, wager that this is where Meillassoux gets his ‘creation from ex-nihilio’ trope from – another mathematical thinker of immanence. There is nothing deeper than what must be given – a Hegelian move if ever there was one.

This is one of the bigger gripes that I have with semantic-lead ontologies – especially Badiou who promotes an specific anti-constructivist ontology of number, with nothing behind it. It’s focused on a thought centred structure of meaning onto a flat immanent system of formal semantics, with the ‘real’ posited somewhere that doesn’t really fit or can’t be presented immanently, but not from something concealed from presentation itself – otherwise Heidegger would creep in through the back door.

Now I know Badiouian’s are fond of countering this with “No – the agent is not human, nor biological, nor founded on anything”. This is why Badiou favours the semantic led approach to a set theoretical ontology. Everything is founded on ‘nothing’. Praxis is delivered through an unfolding, unbounded contingent order, occasionally punctured by some real alibi, just to check it isn’t idealism every now and then.

Here’s the problem, (which at some point I will hopefully be able to explain in better detail than this); anti-constructivism distorts Badiou’s appropriation of set theory and Being. Apart from Badiou’s two early essays in Cashiers pour l’analyse (and these essays are vastly different from the Badiou most know; they are almost militantly anti-subject-Brassieresque), he never sufficiently deals with computer science’s own take on number and computational sets. The question we must ask is why?

My own take on this is that Badiou is not willing to surrender recursive procedures away from human concepts. When you are discussing any aspect of axiomatic set theory, you have to rely on recursion. In discussing the infinite set of natural numbers, or primes for instance, one must use a finite procedure recursively to derive what elements are in that set. Set theory is fundamentally built on recursion, which is why systems like Godel numbering achieved so much progress in the 40s.

What Badiou ignores in the history of number is the very fact that before the 40s, both Turing and Church independently discovered automated mechanisms which could automate identical recursive equivalences of mathematical reasoning. The very sort of recursion that humans thought they were particularly good at.

Bizarrely enough in Turing’s 1936 paper you even have a computational version of the Cantorian diagonal proof, which showed that in a list of computable algorithmic sets, there must always be an uncomputable ‘real’ number.

Even though Badiou never talks about this, one would have to assume given Badiou’s deep political system he’d have to account for it. If Badiou does, he faces two consequences, both of which aren’t particularly fun to accept.

1.) The first is that the ‘rare’ subject or the agent that performs the operation of the ‘count-for/as-one’ must be something other than human thought, i.e, a computer function or system. Although Badiouians are always careful not to conflate political activity and subjectification with ‘human’, tendencies, in my opinion, Badiou doesn’t give any explicit ground to suggest this magical ability is available to anything other than human thought. But one could envisage this criteria applying to an automated computational procedure just as much as a human one. No one has yet disproved it anyway.

2.) Computational procedures and their formal languages can only be interpreted as real sets just like any other set-like situation. Only subjects have a pivoted ‘true’ access to the real of a non-represented presentation.

Most Badiouians will probably plump for choice number two; and this second choice explains a lot, because during the 70s, early 80s Badiou’s ‘evental’ turn turned from eliminating the subject in favour of a mechanic a-subjective science, to favouring an objectless – subject born out of a truth procedure. No need to go into details here for now, suffice to say the relationship between Althusserian science and the Lacanian ‘Real’ changed Badiou’s thought dramatically.

Given that both positions still try to rescue a semantic viewpoint, they are rendered problematic by the fact that these equivalent procedures are constructed by syntactical recursive rules and strings. Here’s where Badiouians tend to come unstuck. By immediately subscribing to the Badiou school of anti-constructivist number, and rejecting subjective intuition (usually associated with mathematical constructivism) Badiouian scholars unnecessarily remove any question concerning independent procedures that are constructed by automated syntactical recursion. That formal systems, derived from sets, grammar and inference rules have the capacity to generate (or enumerate) phenomena one might never expect. And the crucial point is that these surprises emerge from the procedures themselves, not from ‘nothing’. If one accepts this (very hurried) notion, Badiou’s immanence goes out the window, precisely because formal rules of syntax actually ‘do things’ beyond the presentation of immanent sets.

Of course as Graham states, most Badiouian’s are so quick to bank on the politics that they accept certain problems concerning the ontological foundations. True, this mini post deals with more of the wider historical relevance in computer science than one usually associates, but nonetheless – specific troubling consequences emerge.