Fabio over at Hypertiling has put up a translation of part of one Meillassoux’s papers (here), and it is most interesting. The aim of the section Fabio has translated is to sketch out a strategy for demonstrating that mathematical thought can grasp absolute contingency, which for Meillassoux is the Real itself. The way he goes about this is fascinating, but, I think, potentially flawed. I won’t go over the piece in too much detail, but explain just enough to show where I think it goes wrong.
Meillassoux’s basic idea is that the condition under which anything like mathematical thought functions is the ability to grasp and deploy empty signs (such as the letters (P, Q, R, etc.) traditionally used to denote propositions in propositional calculus, or the letters (a, b, c, etc.) traditionally used to denote sets in set theory), and that our grasp of such empty signs consists in nothing but our grasp of them qua sign, as opposed to our grasp of ordinary signs, in which our grasp of what the sign stands for obscures this pure signifying character. Now, he thinks that he can show that mathematical thought grasps absolute contingency insofar as this grasp of a sign qua sign upon which it is founded itself consists in a grasp of pure contingency. This is an interesting argument, and I can certainly see where he is going.
The important structural feature of the sign is that precisely what the mark that instantiates it is does not matter, it could be a particular sound, or shape, or gesture, or whatever, there is no good reason why this given mark is needed to signify the particular thing it stands for, rather than some other mark. Indeed, we should be able to systematically substitute it for some other unrelated mark, without losing anything of what it signifies. Meillassoux takes this absence of reason to be constitutive of signs qua signs, and indeed, the only thing (or at least the essential thing) constitutive of empty signs. The real question is why this absence of reason should be identified with contingency. To quote a bit from the translation (with the relevant section emboldened):-
Now, going back to the vision of the mark-one [marque-une] as the occurrence-one [occurrence-une] of a sign-type [signe-type]. What do I do precisely, when I see a sign as a sign: when I stop considering a mark as a thing, in order to consider it as a sign? Well, I am making of this mark an essentially arbitrary entity, i.e., contingent in its being a sign. That is, I can not thematize the idea of a sign—cannot think the sign as a sign—without letting the contingency of its determinations come to the fore. What does this mean? As a thing, the mark can be thought as the necessary effect of a certain number of causes: possibly related to erosion, to a shock, to a constrained human action, etc. Even if this necessitarianism is illusory, it shows that the mark-thing [marque-chose] doesn’t require that its contingency be thematised to be grasped. Therefore even if I am a Spinozist, the same mark, now become sign, must be necessarily posited as arbitrary, since a sign has the characteristic of not having in itself any necessary determinations. Certainly there are structural constraints in a language (the signs for distinct things must be separate), but the characteristic of a sign, or of a system of signs must be capable of being encoded—transcribed—into another, structurally identical, system of signs. A sign therefore exhibits its contingency ‘on its front line’, so to speak—at least when I grasp it as a sign, one that I thematize as such.
In essence, Meillassoux is claiming that if we view the mark as a mark, we can find reasons for why it has the particular empirical determinations it does (why it is this or that sound or what not), but seeing it as a sign involves putting these to one side and focusing upon its arbitrariness. The problem I have with this is that arbitrariness and contingency aren’t necessarily the same thing. This is because, although both of them can be understood as an absence of reasons, the kind of reasons involved in each case are distinct. Contingency is matter of an absence of reasons in the sense of an absence of causes, whereas arbitrariness is a matter of an absence of reasons for action. When we look at the sign, it is indeed true that we abstract from the various contingent factors that have caused us to use this mark. However, what we focus upon in doing so is the lack of a good reason for someone to choose this mark as opposed to another.
It seems to me that because of this difference, it doesn’t necessarily follow that what we’re grasping in grasping the sign qua sign is the kind of absolute contingency Meillassoux discussed at length in After Finitude. If nothing else, Meillassoux needs to say something about how all of this links up with the notion of action or choice. My sneaking suspicion is that this conflation of arbitrariness and contingency is actually based on a deeper conflation of deontic modality (obligation and permission) with alethic modality (necessity and possibility). If you want to get a handle on the distinction, think about the ‘possible’ moves one can make at any given point in a chess game. These are not really the only possible things that one could do, or that could happen (e.g., one could topple all the pieces, or beat one’s opponent to death with the board). They are really what one is allowed to do in the context of the game, as determined by the rules (or norms) of chess. It is a similar sense in which one is allowed to choose almost any mark as a sign for something.
I might be wrong about this, and Meillassoux might be able to tell an illuminating story about the relationship between deontic and alethic modality (Brandom already has quite an interesting one). This would be particularly perspicuous given that this argument is intended to be the centrepiece of an account of how formal systems can grasp the Real. After all, such systems just are sets of rules legislating what count as well formed formulas and, more importantly, the kind of operations one can perform on these formulas. There could thus be a good sense in which a grasp of what we can do, indirectly underlies our grasp of the absolute structure of what can be. However, demonstrating this requires that one nonetheless keep these two forms of modality properly separate.
At some point I’d like to go back and have another look at After Finitude, as I think there are deeper issues to do with the conflation of different forms of modality lurking in the core of the argument. Beyond alethic and deontic modality there is also epistemic modality. The best way of demonstrating this kind of modality is to consider statements like “192,335,766,081 could be a prime number”. Most of us would recognise that if that number is indeed a prime number, then it is necessarily so in the alethic sense. The point here is that we don’t yet know whether this is the case. A better example might be “Golbach’s conjecture could be true”, as this is something that our best mathematicians don’t yet know. I have a feeling that this difference between alethic and epistemic necessity is potentially conflated at some points in After Finitude, but I need to do some more serious work on it to be sure.