actual and absolute infinity

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.

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hegel and nothing

Pure being is so indeterminate that it is not even being and so is nothing. This is one gloss on the basic Hegelian idea at the beginning of the Science of Logic. But what does it mean?

To achieve a correct understanding of this equation of pure being and nothing, we need to consider if it is required that we screen out properties that are indeterminate in the sense of not distinguishing between determinate beings. If everything were necessarily spatiotemporally located, for example, then being spatiotemporally located would be an indeterminate property in this sense. Yet something spatiotemporally located is not nothing, so we must be careful if we are going to suppose that the equation of pure being and nothingness rests on indeterminacy of this sort.

Why would we want to avoid this supposition? Well, insofar as we are capable of contrasting relative and absolute nothingness, it seems clear that the latter is incompatible with this sort of indeterminacy. So we have to get rid of this indeterminacy if we want to reach all the way through to absolute nothingness. And, if we take away all the indeterminate properties of pure being, as well as all the determinate ones, then we arguably don’t have a subject of predication at all, and that would seem equivalent to nothing in this absolute sense; or at least it would be if there were a workable notion of negation floating free of any predication.

Again, if we still have some indeterminate properties, e.g. self-identity, then we have probably not done enough to purify the concept of pure being so that it is capable of validating the equation with absolute nothingness. (That is, we have not done enough to destroy any means of distinguishing it from the latter.)

What prevents us from succeeding along these lines, then? The only barrier would be the inconceivability of absolute nothingness itself, which arises from the apparent inability to express any sort of negation without having that negation mean that something (and hence not nothing) is the case.

On the other hand, how do things stand if we adhere closely to the view that absolute nothingness is inconceivable? Then any negation of pure being would at least require that ‘being’ be taken equivocally, i.e. as something that could be wholly replaced by something else (and not by absolutely nothing). Yet, in this case, the being in question would clearly not be pure, because it would be determined against the being that negates it. Basically, as long as we take being to be an individual, it’s not going to be possible to completely purify it, even if we give up the goal of removing all its indeterminate properties (so as to equate it with absolute nothingness). Such a being would still be determined against every determinate thing, and thus be itself determinate (and so impure).

So to summarise: the purest being is impure and the most absolute nothing is relative.

 

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follow up on paradox and totality

Meillassoux deploys a stock Cantorian argument to conclude that there cannot be a totality of possibilities, i.e. a universal set of all possible ways the world could be. This takes place in the context of his attempt to disqualify an updated rendition of the old Kantian rejoinder to Hume (“If cinnabar were now red…”), according to which, on the supposition that contingency is the only necessity, we should expect to see frequent/constant variation of the so-called ‘laws of nature’. In their discussion of this problematic, Joshua Heller and Jon Cogburn suggest that Meillassoux should follow Graham Priest in accepting as true the various contradictions that arise at the limits of thought. They assert moreover that there is no middle path between an accessible paraconsistent totality and consistent plurality without totality. This is the dilemma they construct for Meillassoux, and that I wish to follow him in resisting.

Of course I readily agree that neither wilful ignorance nor complacent dismissal are acceptable responses to the paradoxes of totality. In fact it is basically axiomatic for me that dogmatic quietism is off the table: something significant, something revisionary, must be done. I take it to be true, furthermore, that there are no straight solutions to these paradoxes; only sceptical solutions, solutions that cede something significant to the ‘sceptic’ (i.e. the imagined propagator of the paradoxes). Having said this, there is an important sense in which existing sceptical solutions cede too much. In their eagerness to acknowledge the inexistence of problematic totalities like the universal set, they cede the ability to think in absolutely general terms at all, or to quantify universally and unrestrictedly. This is a mistake: to wed the generality of thought itself to the existence of paradoxical totalities is an unnecessary concession.

Thus my conclusion is that we need to find a place for pluralism somewhere within our response to the paradoxes, but without completely abrogating the viewpoint of absolute generality.

We can call those who deny the ability to think in absolutely general terms generality relativists. This leads to a preliminary puzzle. For how is it that generality relativists manage to coherently express their viewpoint? Their view, taken straightforwardly, seems to require that, for any X, X is not an absolutely general thought. This however involves universal quantification over every X. The following, similar line of thought — taken from an interesting published correspondence between Patrick Grim and Alvin Plantinga — expresses the difficulty here clearly:

Were there a sound Cantorian argument with the conclusion that there can be no universal propositions — so the argument goes — it would require at least one universal proposition as a premise. But if sound, its conclusion would be true, and thus there could be no such proposition. If sound its premises would not all be true, and thus it would not be sound. There can then be no sound Cantorian argument with the conclusion that there can be no universal propositions.

Taken straightforwardly, generality relativism is inexpressible by its own lights. It is for this reason that sophisticated formulations of generality relativism do not straightforwardly deny absolute generality in this way. But then what exactly do they do? As far as I can see, they revert to a case by case treatment of individual concepts of totality, showing how each is paradoxical, but without ever ascending to the general conclusion that there is no such conception free of paradox.

Although we might raise further questions here, I am going to assume that the generality relativist has at least succeeded in formulating her disagreement in a way that constitutes a challenge from the viewpoint of the generality absolutist. After all, and in any case, we should want the latter to actually solve the paradoxes of totality, rather than sanguinely resting her entire case on the referential incoherence of denying totality. For this reason, I think that the dialectical subtleties surrounding generality relativism are a red herring: the generality absolutist should just focus on producing an understanding of totality that isn’t susceptible to refutation on the basis of familiar relativist strategies. At any rate, this is what I propose to do.

The paradoxes of totality are ontologically fecund: rather than locking us up in some cloistered epistemic space, the inexistence of higher totalities allows what is left behind to come clearly into focus; it enables absolute generality rather than foreclosing it — but only on the condition that what we are thus generalising about is conceived in a specific, and surprising, way.

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note on heller and cogburn’s meillassoux paper

Meillassoux’s insistence on both de-totalising the possible and simultaneously exploding correlationism is something that I am trying to preserve in my adaptation of his position. Of course I have since interpreted everything in my own way such that it is no longer fruitful (or easy) for me to articulate myself in terms of Meillassoux’s concepts and reference points. It is nevertheless great to come across work that makes real progress in doing the sort of thing I’d have to do if I were to attempt such an articulation. Thus I was happy to discover today a paper by Joshua Heller and Jon Cogburn entitled ‘Meillassoux’s Dilemma: Paradoxes of Totality After the Speculative Turn’. It is a very nice paper that seems to confirm my thinking and hunches on several points. I won’t try to cover all of the richness it includes, but I do want to comment briefly on their central thesis.

The tension between de-totalising the possible and exploding correlationism can be restated like this: we want there to be no totality of the possible, but we nevertheless want there to be a (consistent) totality of the actual. This turns out to be an extremely precarious balancing act; or, less optimistically, a dilemma. Cogburn and Heller quote the following salient passage from After Finitude:

This ignorance [of whether the possible can be totalized] suffices to expose the illegitimacy of extending aleatory reasoning beyond a totality that is already given in experience. Since we cannot decide a priori (i.e. through the use of logical-mathematical procedures alone) whether or not a totality of the possible exists, then we should restrict the claims of aleatory reasoning solely to objects of experience, rather than extending it – as Kant implicitly does in his objective deduction – to the very laws that govern our universe, as if we knew that the latter necessarily belongs to some greater Whole (Meillassoux 2008a, 105).

What is important to note here is that aleatory reasoning applies legitimately to experience for Meillassoux, which forms a totality, whereas possibility does not. Cogburn and Heller do not like this combination of attitudes: they see it as Meillassoux reverting to an oddly Kantian and correlationist attitude in his approach to modal space, and thus undermining his general critique of correlationism.

Here is a quick statement of their argument.

  1. They construe Meillassoux’s rejection of correlational finitude in terms of Graham Priest’s Domain Principle, which says that claims of the form ‘all sets are so and so’ only have determinate sense if there is a determinate totality over which the quantifier (‘all’) ranges. More explicitly: correlational finitude is defined by Cogburn and Heller as the position that we cannot coherently conceive (“self-subsistent”) totalities or absolutes. They claim that, given the Domain Principle, this view illicitly requires reference to the actual totality of spacetime, and so is self-defeating. They take the felicity of their reconstruction as evidence for concluding that Meillassoux is actually committed to the Domain Principle (or else is reasoning fallaciously in his refutation of correlationism).
  2. Commitment to the Domain Principle is, however, in tension with de-totalising possibility. Hence the dilemma. The tension comes from the fact that, given the Domain Principle, when we quantify over possibilities we therefore incur commitment to the totality of possibility. Meillassoux has to quantify over possibilities in order to formulate and express his metaphysics of contingency. Hence…

Perhaps the best feature of Cogburn and Heller’s paper is the way it tries to simplify the dialectic by reducing the number of live options on the table. Specifically, one must, in their view, choose either inconsistent totality or consistent plurality. The importance they assign to the Domain Principle also seems instructive to me. However, in trying to simplify the dialectic in this way, they have ipso facto had to lay a number of cards on the table. I am happy they have done this, and I hope the dogmatic nature of the rest of this post – where I simply list my disagreements more or less without defense – does not come off as negative or dismissive.

First, it is not clear to me that Meillassoux really wants to use the Domain Principle as part of his refutation of correlationism. I suppose this is an easy point to make, since Cogburn and Heller are explicitly offering a reconstruction rather than a straightforward interpretation of what Meillassoux really wanted to argue. Nevertheless, this Priestian offering looks to me like a Trojan Horse that is best left outside the city.

In any case, even if Meillassoux does want to use the Domain Principle, we can still argue on behalf of actual totality without having to appeal to it. Maybe in so doing we lose the ability to meet the correlationist on her own terms, but then I think we had best not try to do that anyway; better to simply resolve the paradoxes of totality. To try to generate this solution (as it were) directly  from the conditions of correlationism itself, and thus preserve a sort of neo-Cartesian, anhypothetical purity – this strikes me as a bridge too far.

I think the Domain Principle is pretty dubious. This can be guessed, I suggest, from the uses Priest puts it to. In particular, he uses it to bolster his Hegelian view that potential infinity requires actual infinity. This is an exact inversion of my own approach that is intended to rest heavily on the viability of potential infinity (which I compare with Meillassoux’s virtuality rather that his potentiality) as part of a consistent response to the paradoxes of totality. On my model, the same thing allows us to both successfully balance the explosion of correlationism (taken in the deflated sense implied above) with the de-totalization of the possible, and articulate the metaphysics of contingency: namely the finitude of actual totality. This basic gesture then informs all of my other disagreements with Cogburn and Heller. In particular, I disagree with – or would seek to interpret extremely carefully – the view that talking about possibility requires quantifying over possibilities. And I agree with Markus Gabriel that possible worlds semantics are actually a hindrance in the way of our correctly understanding the nature of modality.

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more on waghorn

Just a little more to add to my previous post.

Consider the following argument:

  1. Set theoretical paradoxes such as Russell’s paradox show that we can’t consistently quantify over absolutely everything.
  2. If we had a consistent concept of absolutely everything, then we could consistently quantify over absolutely everything.
  3. We don’t have a consistent concept of absolutely everything (from 1 and 2).
  4. We can’t consistently understand nothingness as the negation of absolutely everything if we don’t have a consistent concept of absolutely everything.
  5. Hence, we can’t consistently understand nothingness as the negation of absolutely everything (from 3 and 4).

Waghorn presents this argument (though not quite in this way) in his book Nothingness and the Meaning of Life (p. 60 onwards)There are three independent premises: 1, 2, and 4. Premise 4 doesn’t strike me as worth arguing over, so I will focus on premises 1 and 2 instead. (For the rest of this post, I abbreviate ‘consistent concept of absolutely everything’ to ‘concept of totality’, except where otherwise noted.)

Premise 2: We can distinguish conceiving something and being able to quantify over it, but with regard to totality these two things usually go together. In particular, if we treat the existential quantifier as implying an ontology of discreet individuals that is at odds with the phenomena of vagueness and indeterminacy, then we can motivate drawing this distinction. This is gestured at by Gabriel in Fields of Sense (p. 104), and it is also a feature of Patrick Grim’s more recent work on totality, which is an attempt to escape from the straight-jacket he had created for himself in The Incomplete Universe. Nevertheless, I am going to assume this premise is true or irrelevantly false for present purposes.

Premise 1: In general, Waghorn thinks that many responses to Russell’s paradox are philosophically ad hoc because they turn the spade as soon as they succeed in removing the threat of paradox from the realm of mathematical activity. Thus one common approach mentioned by Waghorn, the von-Neumann-Bernays-Gödel axiomatisation of set theory, stipulates that collections beyond a certain size are not sets but ‘proper classes’, which by definition cannot be members of themselves (here the notion is formalised, unlike in ZF). Waghorn’s sentiment, which I am sympathetic towards, is that mere technical solutions like this are not (philosophically) illuminating, and thus inadequate. That is one reason why I appreciate Grim’s book so much: he shows how to amplify the felt cogency of the paradoxes of totality beyond attempts to quarantine it within set theory or mathematical thought more generally. The paradoxes of totality are rightly taken as paradoxes of thought as such, not merely mathematical or logical paradoxes.

Although it isn’t completely clear what philosophical illumination actually requires in this case, to count as satisfactory a solution ought to have some independent motivation behind it. Now, the motivation behind my own solution is quite rhizomatic and multifarious, but can at least be hinted at with the following:

a) Nothingness is both conceivable and possible.

b) Being is finite.

c) There are no abstract objects.

These can be taken as mutually reinforcing but also as receiving independent inputs. (Note that b and c are the pillars of Goodman and Quine’s famous 1947 paper ‘Steps Towards a Constructive Nominalism’.) Listing all these inputs and connections would be time-consuming, but one thing we can see straightaway is this: Waghorn’s claim, that the paradoxes of totality preclude the conceivability of nothingness, arguably rests on the premise that there are abstract objects, since arguably these paradoxes require abstract objects to work. He is effectively claiming, then, that the existence of abstract objects is more probable than the ability to conceive totality. Actually, to be more precise, he is committed to the stronger premise that the existence of the subset of abstract objects necessary to express the paradoxes of totality is more probable than the truth of the proposition ‘we can conceive totality’. That however does not seem plausible to me.

To be fair, it is common to hold that the sort of nominalism demanded as minimally necessary to avoid the paradoxes of totality precludes the conceivability of totality. In other words, we must eschew all thought of totality in order to avoid paradox. Thus Geoffrey Hellman argues (in his paper ‘Against Absolutely Everything’) that the phenomenon of indefinite extensibility – which Waghorn rightly emphasises in his book – forces us away from a Platonist philosophy of mathematics and leads to the inconceivability of totality. He calls the latter, following standard usage, generality relativism. However, as Hellman notes, neither the indefinite extensibility of concepts like ‘ordinal’ and ‘set’, nor nominalism as such, imply generality relativism. Further arguments are needed. In general, it shouldn’t be that surprising to find that, after we prune the ontological bush there is nothing preventing us from quantifying over the totality that remains, so that premise 1 of the above argument is false. (Again, contrast this with the view that we must eschew all thought of totality in order to avoid paradox.)

Of course, I am sweeping under the rug for the time being the question of what happens to mathematics when we start gleefully pruning the ontological bush in this manner! Looking at the fate of Quine and Goodman’s early project, for example, it should be clear that my approach does require a degree of optimism.

 

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may update…

At present this blog is as acting as a sort of placeholder for future input, so for those of you who are still following, please bear with me! At the moment I am working a heavy teaching load, which doesn’t leave much time for independent study or writing; and in any case my ideas are — even without external interference — slow to develop.

Nevertheless, I recently had a chance to read through Nicholas Waghorn’s Nothingness and the Meaning of Life, which is probably the most exhaustive examination of the concept of nothingness that covers both analytic and continental philosophy. One could with profit read it as a companion text to Markus Gabriel’s Fields of Sense, since it develops a kindred line of thought, one that attempts to block the conceivability of both nothingness and totality (the one following from the other). The comparison is also instructive in the following way: Waghorn is acutely aware of the performative tension involved in his investigation, and has written a book with a truly tortured reflexivity to it (you have to read into it a bit before you fully realise this).

A recurrent theme in Waghorn’s book is that we don’t have a concept of (absolute) nothingness. The reasoning here is simple and yet mysterious to me: we can’t characterise nothingness, since it does not have any properties. To which we can immediately add: to assert that we can’t characterise nothingness is also to characterise it (p. 162), which illustrates in nuce why Waghorn’s approach contorts itself into an indefinite regress. Another way to put the basic claim is like this: to say that nothingness is not some X is necessarily to say that nothingness is a Y such that Y ≠ X. It is mysterious to me why, underneath every refusal to countenance the conceivability or possibility of nothingness, there appears finally this premise, like Plato’s ghost.

 

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two new papers…

My first paper, ‘Why not Nothing? Meillassoux’s Second Figure of Factiality and Metaphysical Nihilism’ has been published in the journal Speculations; it can be found here.

My second paper ‘Markus Gabriel Against the World’ has also been published in the journal Sophia; it can be found here.

In my next paper, I will try to push further into the philosophical and (broadly speaking) logical motivations for thinking of being as finite and contingent.

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