This week we discussed Cian Dorr’s ‘There are no abstract objects’, which isn’t currently available online, but is in ‘Contemporary Debates in Metaphysics’. Here’s the handout instead.
As we had Cian on the spot for this meeting, the discussion mostly took a question-and-answer format. So here are what I recorded of some questions and some answers, with a few that I didn’t get time to ask thrown in at the end.
Q: What about people who would resist the paraphrase strategy (p.37) because they think that counterpossibles are all vacuously true (Williamson takes this line in The Philosophy of Philosophy).
A: Nominalism/anti-nominalism are both contingent theses. But even if you think that nominalism is necessary if true, there will be certain kinds of truths like ‘there are possibly some things with a number-like structure’ which can be used to ground the relevant counterfactuals, along the lines of modal structuralism.
Q: Why require systematicity in our paraphrases?
A: Because we want to make straightforward sense of statements which mix different cases, such as ‘there are various different kinds of abstract objects’ – numbers, properties, sets, etc’
Q: Aren’t tables and numbers in the same boat according to the strategy applied here? What does this do for the nominalist intuition that tables are better known than numbers? How would the world look different if either existed
A: Bite the bullet – chairs do not differ from numbers in any interesting way. If they did exist, things would look just the same, but there’s still no good reason for us to suppose they do.
Q – Is the regress vicious (p.44)?
A – Yes, though it’s a little bit unclear why. One possible reason is that if denial of brute necessities is to get us anywhere, we’d better not have circular or regressive accounts of metaphysical primitives. For example, the following analysis does not get rid of brute necessities at any stage:
The necessity of ‘All fs are gs’ is explained by ‘to be an f is to be an f1 that is g.’ But why are all f1s gs? This is explained by ‘To be an f1 is to be an f2 that is g’… and so on.
Q: If we take property essences seriously, Kit Fine style, can we posit an essence of the instantiation relation that rules out pathological cases of instantiation?
A: The problematic cases can still be described in purely structural terms, so the problem has not completely gone away.
Q: Is this fictionalism?
A: Not if fictionalism is characterised by the literal falsity of claims about the domain one is fictionalist about.
Q: Why think that 6a or 7a in the fundamental sense do not analytically entail 6b and 7b in the fundamental sense? Isn’t it just the neo-Fregean project to defend these analytic entailments? What conception of analyticity makes it obvious that the entailment fails? Admittedly, a conception of analyticity which supports the entailment would have to have it analytically false that nothing exists.
Q: ‘Fundamental way’ – ‘ultimate furniture’ – ‘final analysis’ (p.34) – it seems like there are two ideas mixed up in the motivating metaphors. One is about the ideal science at the end of enquiry – but it’s not at all obvious (to me) that numbers won’t feature in this ideal science, if it’s formulated in the most natural way. Another involves ‘metaphysical explanatoriness’ – in this view, fundamental means something like ‘best trade off between simplicity and metaphysical explanatoriness’.
Q: Does it make a difference to the Alien Particulars Objection (p.49) whether we are committed to haecceitism? It looks like an anti-haecceitist can’t even express the objection.