Penelope Maddy — Defending the Axioms

Infinity is a puzzling concept in itself and also in the way it has shaped the development of mathematics. Take for example the number line, the infinity of the infinity of the integers and the larger infinity of the real numbers. Cantor asked: is there also an infinity of intermediate size? This is called the Continuum Problem. Since it is independent of ZFC, there is a worry that it does not have a determinate solution. This raises a difficult question that I’ve been wrestling with for some time now: What is the provenance of this concern? Mathematicians worry about it, certainly. But is the worry itself mathematical? Or is it philosophical or perhaps a question ultimately of cognitive science or physics? And anyway, who has the right to pose questions in this fashion?

These are hard questions, so before I go further let me put two more examples on the table:

  1. The development of Non-Euclidean geometry from the work of Bolyai and Lobachevsky in the first half of the 19th century. They now co-exist side by side but neither can lay claim to the old mantle of synthetic a priori insight into the geometry of physical space.
  2. Intuitionism. Brouwer declared that the Law of Excluded Middle has the same status as the view that the Earth is the centre of the Universe. For him, the Copernican Revolution in mathematics was wedded to the view that only denumerable infinites exist. And his opinion was taken seriously. But unlike the case of Ptolemaic astronomy, it did not yield revisions in the practice of mathematics — or rather, its revisions were additions rather than the sort of thing Hilbert had in mind when he referred to Kronecker as the Verbotsdiktator. The answer everyone could agree on turned out to be ecumenical: classical and intuitionist mathematicians are free to proceed as they please. Let a thousand flowers bloom.

Question: when a set of axioms is demonstrably independent of some further hypothesis, is there a determinate answer regarding the truth or falsity of that hypothesis?

maddy cover.jpg

This question is explored by Penelope Maddy in her book Defending the Axioms. Her main focus is the addition of new axioms to Zermelo-Fraenkel set theory in order to settle the question of whether Cantor’s Continuum Hypothesis is true or false. She quotes Gödel on this (2011: 56): set theory “…must describe some well-determined reality, in which Cantor’s conjecture must be either true or false. Hence its undecidability from the axioms being assumed today can only mean that these axioms do not contain a complete description of that reality.” The key assumption here is that the reality in question is determinately there to be described.

What sort of process is involved in deciding the undecidable? Shapiro describes it as follows in his book Thinking About Mathematics (2000: 208): a proposition or formula may be undecidable in arithmetic or in some framework, so we decide it by embedding it into another more expressive framework. As we just saw, Gödel takes this process — undecidability + embedding in a richer framework — to constitute an argument for Platonism. But given that more than one embedding is available, what rule prescribes that we go in one direction rather than the other? Hilbert eventually declared Cantor’s conjecture to be meaningless on this basis. Interestingly, Shapiro also echoes this sentiment in his paper in the collection Meaning in Mathematics (p. 104): disagreements over set-theoretical axioms, he says, are in a certain sense not real disagreements but just talking-past-one-another phenomena. What is needed is not fruitless argument but what he calls an “Euclidean rescue,” wherein we declare the competitors equally valid in their own domains, which are then strictly segregated. (At the level of logic he repeats this gesture.)

All of this brings us back to the questions raised above: when mathematicians worry about Cantor’s Paradise, is the worry itself mathematical? Maddy mentions Dedekind’s belief, that the natural numbers are a free creation of the human mind, in order to illustrate the class of irrelevant conceptual accretions attendant upon but irrelevant to mathematical practice (2011: 53). On this point, I can’t resist sharing the following humorous story recounted by Dirk van Dalen in his biography of Brouwer (2013: 92). Brouwer was trying to assuage the doubts of his advisor Korteweg about the contents of his dissertation thesis. He wrote about Russell: “To put something higher than mathematics, you must feel a non-mathematical intuition; Russell has none of that, and yet he starts to bullshit.” Korteweg’s response included the following: “A kind of pessimistic and mystic philosophy of life has been woven into [the dissertation] that is no longer mathematics, and has nothing to do with the foundations of mathematics. It may here and there have coalesced in your mind with mathematics, but that is wholly subjective.”

Here is the point. You may agree that Brouwer operated on the basis of a fair amount of irrelevant philosophy when working out his alternative to classical mathematics. And although I am leery of this attitude, few would deny that there is some distinction to be had here between what is merely subjective and what Maddy describes as “shared convictions that actually drive the practice” of mathematics. The question is whether and to what extent this distinction supports the weight Maddy wants to place on it. To give just one example, it seems to rule out by fiat the entire project of cognitive science of mathematics proposed by Lakoff and Núñez in Where Mathematics Comes From. And that is a strange thing for an avowed naturalist to want to do.

Here I just want to flag the issue and mention what Mary Tiles says about this aspect of Maddy’s position in her review of Defending the Axioms. She leans on Bourdieu to make the point that there is a danger here of telling an epistemological or methodological story that simply “reproduces the dominant discourse of a sector of pure mathematics about itself while denying legitimate access to non-dominant discourses. While not denying that set theorists have internalised a sense of objectivity, there are grounds for thinking that this sense cannot be the sole arbiter of reliability when it comes to non-mathematical uses of their products.”

Of course, there are more and less extreme ways of fleshing out the complaint about “legitimate access to non-dominant discourses.” For example, it is clear that whatever exactly it is, the debate about Woodin’s program in contemporary set theory is not a debate about whether physics tells us that set theory has to be one way or another. The debate over whether we should we force mathematics to pass through the eye of the physicalist needle — is conducted on the basis of relatively lesser common ground. And this may be seen to count against it to some degree.

At any rate, there is no narrowly mathematical procedure for answering this latter question — except in the sense of a very narrow internal-to-mathematics perspective that makes the answer trivially affirmative — so it makes sense to view it as a question imposed on mathematics from the outside. Given this, are we right to explore such a question, or is this just wrong-headed and confused? Are we allowed to ask?

The distinction between internal and external questions is often wielded with deflationary intent — this is how Maddy deploys it. Set theorists have the wherewithal to function independently of external oversight, relying on an institutionally shared sense of mathematical depth and objectivity that does not even require that sets exist. A familiar way to make sense of the significance of this is as follows: we can view as ipso facto external to mathematics any global question — such as whether abstract objects exist as such — so long as none of the possible answers to this question are conceived of as capable of affecting mathematical practice. This may be part of the reason why it is felt reasonable to dismiss such external questions — they define themselves into irrelevance by requiring that mathematics be potentially based on an error undetectable not only to mathematicians themselves, but to us looking in from the outside.

If this is what the strategy is, then it’s a nice trick. External criticisms that propose to modify mathematical practice are dismissed on that basis, whereas criticisms that leave practice as it is are declared irrelevant. As for internal criticisms, one simply parameterises the disputants and returns to work. I have to admit that, in my more cynical moments, this strikes me as a Potemkin imitation of objectivity!

Regardless, there is a more specific issue here pertaining to the foundational aspirations of set theory in particular. To see what this is, begin by noting that there is a class of beliefs pertaining to what is invariant across any possible mathematics or logic, a class whose boundary shifts with time. The belief that classical mathematics is the only legitimate mathematics, and more generally the belief that a particular mathematical theory is in the intersection of ‘universal’ mathematics (in the sense, for example, that there is a universal geometry that is equivalent to Euclidean geometry minus the parallel postulate), illustrate what I am referring to here. When a hypothesis is formulated that is independent of a given set of axioms, and we relativise it to two different expanded sets of axioms, one yields that hypothesis as a formula and the other yields its negation. My point: there seems to be no universe, then, in which either the formula or its negation are mathematically necessary. And so the question becomes: what is mathematical necessity, ultimately speaking? What, for that matter, is mathematically universal? The idea of providing a foundation for all of mathematics seemed to require accounting for one or both of these notions (necessity, universality), and doing so in way that was internal to mathematical practice. But it seems that this cannot be done. So now we ask: are beliefs about mathematical necessity or universality ultimately mathematical or philosophical (or within the purview of empirical science)?

Being that metamathematics exists, I have to say I am wary of Maddy’s attempt to cast objections to set-theoretical methodology as both external to mathematical practice and inherently global (and by implication irrelevant). Regarding the latter, external questions are not always or even predominantly global in nature. Schematically, if I give you an argument for why there are no sets, but there are zets, where a ‘zet’ is defined such that at least some of the things you took to be existing sets do in fact exist, then I have not given you a global anti-realist argument against sets. (Compare ‘set’ here with ‘electron’.)

Let me try another tact and motivate arguendo the assertion that external questions are always global. Well, how many elves lived in Rivendell? Are we asking from within the perspective of Tolkien’s universe, or from an external perspective? Maddy’s complaint might be glossed like this: philosophers or physicists ruin the fun when they come along and ask which axiomatisation is ‘really’ correct, which abstract objects ‘really’ exist, and so on. The complainer might insist, in a Carnapian way, that this question should be taken only in an ‘internal’ manner, and that it is senseless to pose it externally, as to do so is to render it undecidable and theoretically inert. (But if it is undecidable why not transpose the mathematician’s own practice and embed it into another theory — this time perhaps that of cognitive science?) She might then observe, moreover, that the question of how many elves are in Rivendell is, necessarily, either an internal question to be answered as such, or senselessly narrow and specific. Narrow and specific in this sense: asking it from the external perspective is wholly unmotivated and weird — like a scientist who comes along claiming that although elves are complete myth and fantasy, the land where Rivendell was built was an actual geographical location in our planet’s prehistory. It is just weird. So how can it be less weird to say, for example, that although the rational numbers exist, the irrational numbers are just fictions that should never be reified into actual beings? This, perhaps, is the reasoning behind the conclusion that external questions are essentially global.

Yet, if this attitude is correct, why not apply it within mathematics as well? Let me try to articulate my discomfort here a little more clearly. Why not just dissolve foundational debates by allowing each otherwise competing set theory to bloom independently and freely? Each one will then have its own Rivendell, with as many or as few elves as you like. The existence or non-existence of a particular elf will be axiomatically decided one way or the other within these competing systems. If external critiques of mathematical ontology are senseless and undecidable, is it not the same way within mathematics if we frame things thus? “My own practice will not, after all, be affected by the correctness of whatever ontological revisions you wish to impose on me. So your objections are quite irrelevant! In fact, even if you could falsify my belief, I could just add a new axiom to my system to inoculate myself from your attack! Maybe, then, we should just get on with the business of developing our own systems, and not waste each other’s time…”

But from the viewpoint of mathematical necessity, this looks suspiciously like saying that we can project an indefinitely large number of incompatible patterns across the same data set, none superior to any other. The data set is random in this sense, and I find this hard to distinguish from the observation that a false belief ‘represents’ the empty set — an analogy that if accepted suggests that beliefs about mathematical necessity/universality are all false.

In general, the idea of arbitrarily adding axioms whenever we please should give us pause to consider what sort of constraint there is on set-theoretical activity. But this is where we seem to have ended up, with dismissing the validity of external questions making possible a corresponding internalising of the strategy underlying this dismissal, which in turn erodes the legitimacy of internal questions by rendering them all external in principle. In general, if a global challenge to a domain is simply the idea that nothing exists in that domain (but this fact is invisible from within), then any sub-domain could use this strategy to inoculate itself from any ‘outside’ criticism. But this is absurd. And this suggests that to avoid eroding the purchase of internal questions, we need to preserve the openness of mathematics to its outside, at least in principle — whether this be philosophy or science. (And this is the reason why I am leery of saying that e.g. Brouwer’s background philosophy is merely subjective/irrelevant to his mathematics.)

Set theorists do recognise a legitimate question about what unity, if any, underlies the plurality of competing axiom schemes — but for Maddy this question only has mathematical (i.e. legitimate) sense if taken as a question about which axiomatisation allows for the broadest and most complete reconstruction/accommodation of mathematical practice — not a question about whether mathematics as such models an ‘independent’ reality. Within the meta-axiomatic ‘game’ of choosing between rival set theories, selecting a different axiomatisation involves making a choice about what to accept as existing or not existing. A question of this sort is an ontological question, so there are in some sense ontological questions included in mathematical practice beyond those of axiomatic fiat (since there is no axiom system for making this choice). This is a step towards unity, but it does leave something to be desired, at least in my opinion. I want a mathematical problem to teach me something about reality — not simply about a ‘game’ we play or something which may turn out to be as illusory as Rivendell! (I’m also inclined to say that, to the extent that she relies on shared practice and perception of mathematical depth to guide axiom choice, Maddy ends up with a non-classical meta-theory, because facts about mathematical depth are presumably finite and physical.)

I would like to find a way, then, to render it plausible and right to continue asking unifying ontological questions at broader and broader scopes without succumbing to the charge of regressing into objectionable metaphysics. So here it is: we noted that the viewpoint of a greater unity is invoked when adjudicating issues within foundational mathematics such as which set theory best captures the mathematical hierarchy. This immediately generates a new line of questions: who decides how unified mathematics is? Who decides that pure mathematics is one unity whereas applied math is another, or more to the point, that the unity of mathematics is not itself merely a part of a greater unity, over which mathematics cannot have any sort of legislative fiat or veto? Tiles makes this point interestingly in her review: pure and applied mathematics appear to be merging once again, but Maddy’s strategy requires them to remain firmly separated. I think we can see a way towards formulating an external but non-global challenge to mathematics by focusing on this dividing line between pure and applied mathematics. This line is what filters the purely abstract and brings it back down to Earth.

But hang on a minute! Imagine someone saying: “Mathematics methodologically differs from natural science; this is enough to account for the unity of each and their robust difference from one another. So that’s why it is illegitimate for physicists to pose external questions to mathematicians.” It is worthwhile to draw attention to this line of response. We can even add to it a bit: Gideon Rosen, for example, in his contribution to the book Meaning in Mathematics (p. 116) reasons as follows: mathematics is unassailable by all pertinent mathematical, scientific and common-sense standards — so to assail it necessarily requires dubious philosophical tools. In other words, isn’t mathematics the best method we have for evaluating the ontological status of mathematics? By contrast, armchair philosophers would only be bringing their own methodologically obscure techniques to bear in trying to adopt an external perspective.

In response, consider this: I would not want to be told that science cannot adjudicate the question of whether elves exist because Tolkien’s universe is unified by the methodology of creative writing and so can only be internally adjudicated. Elves exist in the story — but if we wish to say that they also exist tout court, then science gets to have an opinion on whether this is a wise thing to want to say, or not. Anyone — even a philosopher — can adopt a perspective methodologically internal to science in order to externally evaluate the claim of a particular discipline or field of inquiry. This is simply not the same as relying on dubious a priori methods, whatever these might be. And it means that people outside of mathematics get to have a say on the ontological import of mathematics — in particular, over what ‘exists’ means. My conclusion: it is subtly illegitimate to use the word ‘exist’ in speaking of mathematical objects unless one is prepared to allow this point to be adjudicated by a perspective external to mathematics itself.

Which brings me to my competing desire: suppose we want to develop a single or univocal ontological viewpoint. How should we proceed? One suggestion, which I would just like to float for now, is that everything must bend to the perspective of empirical science broadly conceived. This yields the following response to the charge of weirdness articulated above: the scientist is not going in willy nilly and arbitrarily declaring classes of similar objects either non-existent or existent without further ado. There is a demarcation corresponding, for example, to what could be applied in principle to the physical universe or some possible universe, and what cannot. The former are taken as potentially descriptive or representational, the latter as fictive, expressive or broadly epistemic.

Let me summarise briefly. For two supposedly rival hypotheses x and y, we can treat them in two different ways. First, we can treat them as belonging to the same domain of inquiry, in which case the question of which is correct is an internal question of that domain. Second, we can also treat them as belonging to different domains, such that y is external to x, and in this way the Carnapian manoeuvre seems to trivially solve any dispute or problem. This seems to be a reason for looking suspiciously upon that manoeuvre. I have suggested that we must pay due respect to this suspicion in order to properly follow the imperative of developing a unified — univocal — ontological perspective.

In this respect I am a little less sanguine than Maddy. Her strategy is to try to have internal realism and external anti-realism behave like two sides of a Möbius strip, such that it is indeterminate and undecidable whether mathematical objects exist or not. External evaluations of mathematics are like party-crashers that rip the cloth off of the dinner table so perfectly that nothing is disturbed or altered at all. But I’ve identified weaknesses in this approach. It is stronger when able to equate external challenges with global challenges — and correspondingly weaker at coping with non-global but still external challenges. The external perspective can, fairly obviously on reflection, be as fine-grained and as discriminating as you like; it can even, for example, choose one particular side in a debate otherwise internal to mathematics — e.g. the debate about what constitutes real as opposed to merely apparent mathematical depth, to use Maddy’s vocabulary — and attempt to buttress that side by embedding it in a broader ontological perspective.

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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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