In a footnote to his Berlin paper Meillassoux writes:
I distinguish the indefinite from the infinite according to the classical difference between the potential infinite and the actual infinite. The indefinite is the endless augmentation of the finite (1, 2, 3, etc.); it is the indispensable, but not sufficient, preliminary to the thinkability of the actual infinity whose existence is guaranteed by a set theoretical axiom. I cannot explain here how I derive actual infinity factially, on the basis of the derivation of the indefinite.
I’ve been thinking about this enigmatic note recently, in light of the claim by Heller and Cogburn that Meillassoux requires the Domain Principle in order to break out of the correlationist circle, but has no principled way of giving it up after that — which he must do, they say, in order to avoid commitment to absolute infinity, which here means in order to avoid totalizing the possible. For Meillassoux, the fact that possibility cannot be totalized is essential to the distinction he wishes to make between probabilistic chance and genuine contingency. A useful way of thinking about this is in terms of emergence: for genuine novelty to be possible, possibilities must not be taken as germinally contained in actual being. Meillassoux is claiming that for us to properly grasp the nature of emergence and of the open future, Cantor’s transfinite paradise is required. (Cf. the discussion in ‘Potentiality and Virtuality’, p. 72, where new sets of possibles are said to arise, without any preformation or germinal antecedent, from nothing.)
Heller and Cogburn urge Meillassoux to adopt the Domain Principle because of the fortifying role it plays in their reconstruction of his argument in After Finitude. So why doesn’t he? At no time does Meillassoux mention or appeal to the Domain Principle, even though it originates in Cantor’s work and was explicitly used to argue that potential infinity depends upon actual infinity. Looking at the quoted passage above, we can see that Meillassoux claims to derive this dependence starting from factiality and proceeding through an intermediate proof of the indefinite, i.e. of potential infinity. The Domain Principle would clearly help with this task. However, there is a methodological issue here that renders any use of it potentially problematic for Meillassoux. This issue is noted but not confronted directly by Heller and Cogburn, and can be put crudely this like: why is Meillassoux trying to derive or prove things at all? Why not simply proceed axiomatically or speculatively? As they wonder at the end of their paper: does Meillassoux the philosopher of contingency really need the anhypothetical argument for contingency?
This methodology is clearly important to Meillassoux, however, and it seems to exclude using the Domain Principle in this way (unless of course it too could be derived from factiality). Putting this difficult methodological issue aside, it is at least clear enough why Meillassoux would reject the original motivation for the Domain Principle. As shown in Hallett’s fascinating book on Cantor, the original version of the Domain Principle was justified in theological terms, insofar as any potential infinite maps onto an idea in the mind of God, which must therefore contain an actual infinite. This would be completely unpersuasive to Meillassoux. Unlike Cantor, Meillassoux rejects the absolute infinite not simply as an object of mathematics but as any sort of object at all. So God cannot be a motivation for the Domain Principle for him.
Why though does Meillassoux reject the Priestian version of the Domain Principle? To be clear, I actually agree with Heller and Cogburn that the Domain Principle does lead all the way up to absolute infinity — or at least that it forces us to abandon classical logic in order to avoid this outcome. This claim is taken over from Graham Priest, whose general account of paradox in Beyond the Limits of Thought is very compelling. However, giving up classical logic is something Meillassoux evidently does not want to do.* Nevertheless, when Heller and Cogburn note that paradox can also be avoided by retaining the Domain Principle but adopting a weaker-than-classical logic such as the one developed by Hartry Field in Saving Truth From Paradox, they effectively emphasise the question: why does Meillassoux (seem to) reject the Domain Principle?
*This may be where the real answer lies, insofar as Meillassoux arguably has an theoretically unstable attitude towards classical logic, which I believe stands in a complicated relationship to his deep commitment to ontological emergence — I point will save for another time.
Priest’s view (heavily simplified) is that absolute infinity is inherently paradoxical and that there is no principled way to avoid absolute infinity once the transfinite has been accepted. Actually it is stronger than that, since as he notes on multiple occasions, even the smallest infinity (i.e. the countable infinity of the natural numbers) leads to paradox, namely the Berry Paradox. Priest relies on what he calls the Principle of Uniform Solution in order to create trouble here. This principle says that the paradoxes must be given a uniform or general solution. What this means can be made clear by noting the following: although absolute infinity is inherently paradoxical for Priest, we cannot avoid paradox by simply denying the existence of absolute infinity. The reason is that this will not solve every paradox, in particular it will not solve the Berry Paradox since the latter does not depend on the existence of the absolutely infinite.
I find the Principle of Uniform Solution to be persuasive and am prepared to attempt to operate within its constraint. But I share Meillassoux’s rejection of the absolute infinite as well as his belief, which I think is an extremely important insight, that the correct understanding of contingency depends upon this rejection and the attendant notion of incompleteness. So what is to be done here? This is the problem I am currently wrestling with.
Let me reformulate the constraint I take over from Priest in such a way that suggests one way forward: the full power of the Inclosure Schema and the Principle of Uniform Solution is that countable infinities generate paradoxes at the limits of (classical) thought, but cannot penetrate into the austerity of the finite. Everyone therefore has to make a decision at this point. We may join Priest in accepting true contradictions at the limits of thought. We may alternatively opt for a non-classical logic that does not involve truth gluts — namely one based on the rejection of the Law of Excluded Middle. This would take us in the direction of Field, perhaps, or into the unstable terrain of intuitionism in the style of Dummett. It is an interesting open question right now what the ultimate status of the indefinite is in this environment — a question for another time.
The classical alternative, as far as I can see, is strict finitism, the view that there are no actual infinites of any kind, countable or uncountable, transfinite or absolutely infinite. We already know, incidentally, that the Liar cannot arise in a finite truth theory, and it seems likely that strict finitism is both completely consistent and compatible with the Principle of Uniform Solution. So here is where we are: suppose we want to solve the paradoxes uniformly (in Priest’s sense) and classically (i.e. without giving up PNC or LEM). Now ask: is rejection of the infinite not merely sufficient but also necessary for classically and uniformly avoiding paradox? Consider how this rejection might take specific shape. Historically, at least, the rejection of the Domain Principle would seem to do the trick (of uniformly undercutting paradox whilst entailing finitism), since it was the bridge Cantor used to first enter the transfinite, by licensing the move from potential to actual infinity. We just need to hold on to the view that we cannot be otherwise compelled to accept the existence of actual infinity.
Let me try to give one last summary statement. Taking up what I said about possibility and incompleteness earlier, it is good to say that possibility — modal space — is incomplete or incapable of being totalised because it is uncountably infinite (and because orthodoxy has retained the view that nature at least abhors the absolute infinite). It is good because this already suggests a need to revise our modal semantics. But it is perhaps even better to say that possibility is incomplete because it is finite, and because the finite is intrinsically contingent. On this alternative view, there does not need to be an infinite semantics underlying this, as in classical modal logic. We have, rather, a finite means of conceiving alternative possibilities, just as we have a finite ability of conceiving successively larger numbers on this view. Yet we naturally believe there could have been more numbers than there are, and so believe the same in the case of possibility. The goal of developing such a picture is to have possibility be understood as incomplete and unpredictable, as a phenomenon of logical emergence. Whether or to what extent this is viable remains an open question.