Just a little more to add to my previous post.
Consider the following argument:
- Set theoretical paradoxes such as Russell’s paradox show that we can’t consistently quantify over absolutely everything.
- If we had a consistent concept of absolutely everything, then we could consistently quantify over absolutely everything.
- We don’t have a consistent concept of absolutely everything (from 1 and 2).
- We can’t consistently understand nothingness as the negation of absolutely everything if we don’t have a consistent concept of absolutely everything.
- 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.