Friday, October 13, 2006

There are sets . . . (not really!)

I’ve been thinking a lot recently about relations of fundamentality. There are, I think, three relations here: a relation of ontological dependence that holds between entities, a relation of grounding/truth-in-virtue-of that holds between propositions, and the truth-making relation, that holds between an entity and a proposition.

One thing I am interested in is the connection between the relations. One potential connection is the following: if A makes p true and if p grounds q (i.e. if q is true in virtue of p) then A makes q true. This seems pretty plausible. If p grounds q then it doesn’t take anything more for the world to be a q-world than for it to be a p-world: so to make the world a p-world is to make it a q-world.

I’m currently intrigued by the potential of this to allow us to make sense of Fine’s distinction between what there is and what there really is. It would be nice to be allowed to make such a distinction. In particular, I’d like to be able to say that there are abstracta, but that there aren’t really any abstracta. I’d like to say that there are sets, for example, because it’s really useful to be able to talk about such things; but I’d like to deny that there are really any sets because an ontology without sets is, other things’ being equal, preferable to one with.

Now, it’s natural to think that a set is ontologically dependent on its members. Socrates’ singleton depends for its existence on Socrates, and not vice-versa. You might be tempted as well (perhaps as a consequence) to the claim that the proposition the singleton of Socrates exists is true in virtue of Socrates exists. Since Socrates is the truthmaker for Socrates exists the above principle will then imply that Socrates is the truthmaker for the singleton of Socrates exists.

First thought then: we don’t need there to actually be a singleton of Socrates. We only need Socrates, and he makes true all the truths talking about his singleton. Generalising, all we need are the ordinary concrete objects, and we get all the truths about sets for free. (Pure sets will be a bit trickier – but there are any number of stories we might tell here.) So we can secure all the truths we want – we get the benefit of talking about sets – without admitting sets into our ontology.

But that can’t be quite right. We can’t deny the existence of sets and affirm the truth of the proposition the singleton of Socrates exists. But what we can do is accept that there are sets and deny that there are really any sets. The thought is that the singleton of Socrates exists is true (and hence there are sets), and is made true by Socrates; but the singleton of Socrates really exists is not made true by Socrates; in fact, it’s not made true by anything, and so it’s false.

Armstrong says that a exists is always made true by a. I am denying that: I claim that the singleton of Socrates exists is made true not by the singleton of Socrates (since the truthmakers are what there really is, and there aren’t really any sets) but by Socrates. But I can accept a variant of the Armstrong position: that a really exists is always made true by a, which is a fundamental being.

So the thought is that we have a bunch of fundamental entities that do all our truthmaking. Some of the things they make true is that they really exist. Other things that they make true is that some non-fundamental entities exist (but not that they really exist – they don’t!). This is meant to secure all the benefits without the cost. We get the benefit of talking about sets, since all we need to secure that is that we can presuppose that sets exist – we couldn’t care less whether or not they really exist. And we secure a parsimonious ontology: since what we care about here is what there really is – what exists in reality – and sets don’t really exist.

I have no idea whether that is in line with Fine’s thinking on the distinction, but it seems to me not wholly crazy, and worthy of pursuit.

On other news, I’ve been invited to respond to Jonathan Schaffer (UMass) at the Pacific APA in San Francisco next April. I’ve never been to SF, and am looking forward to it. Schaffer is defending the view that, necessarily, every true proposition is made true by the world, which is the only fundamental entity that there is. We’ve already had some correspondence over his paper, after meeting at Bellingham, and it’s been lots of fun – so this should be a good time. San Francisco was the home of the one and only Emperor of the United States, which is reason enough to visit.

6 comments:

Ocham said...

A subject dear to my heart.

1. Why the distinction between what there is and what there really is? You say that sets are useful, and so we would like to say there are such things, so you arrive at a halfway house of, they do exist, but not really so. I don't follow. Either sets are required in the sense that the 'real' existence of certain things depends on the existence of sets, in which case there are 'really' sets as well. In the same way that if certain sub-atomic particles or fields are required to explain things that really exist like tables, chairs &c, then the partices, fields, really exist. Or sets are only required in the sense that, to make computations &c easier it is useful to pretend they exist, thought they don’t. In which case, we pretend that they really exist, but they don't. Either way, there is no need for the halfway house.

2. How can we talk about sets without 'admitting sets into our ontology'? Surely we can only talk about things we have 'admitted into our ontology'?

Ross Cameron said...

Hi Ocham,
I just replied to this, but blogger appears to have destroyed my reply. Sometimes I really hate technology.

So this might be briefer.

On (1). I don’t think your ‘either-or’ exhausts the options. Sets aren’t a pretence for me – there are sets, and that is literally true. But nor does “the 'real' existence of certain things depend on the existence of sets”. I agree that if the real existence of a depended on the existence of sets then sets would really exist, but I do not assert the antecedent. I assert that the (unreal) existence of sets depends on the real existence of non-sets.

The case of composition might provide a helpful analogy. The analogous view would be that only mereological simples really exist, but that propositions such as [there are tables] are true, and made true by those simples. Our theory quantifies over tables, but is not ontologically committed to tables, because it is only ontologically committed to what there really is (see discussion of point (2)). There is nothing whose real existence depends on the existence of tables. That would indeed require the real existence of tables. But nothing is ultimately dependent on the existence of tables. All the complex objects are ultimately dependent on the simples, and so the simples are all that we need to really exist: they are the truthmakers for any claims concerning complex objects.

On (2). I think your objection assumes a Quinean account of ontological commitment which the truthmaker theorist will reject. Of course if my theory says that there are sets then it quantifies over sets, and so is committed to them if Quine is right. But the TM theorist rejects this thought: the ontological commitments are just those things that must exist to make true the sentences of the theory. So if only follows that the Xs are an ontological commitment of a theory that quantifies over the Xs if the Xs are required to make it true that there are Xs. But this is exactly what I am rejecting. The ontological commitments of the theory are the truthmakers: so if [there are sets] is made true by something other than sets then the theory can say that there are sets without bringing an ontological commitment to sets. It is in this sense that we can talk about sets without admitting them into our ontology.

Ocham said...

On losing things, I always put any extended reply into Word or the draft section of Outlook (and save, given my temperamental computer).

I'm afraid you lost me completely on the distinction between something merely existing, and it 'really existing'. Can you make this any clearer?

Ross Cameron said...

I wrote this up into a short paper, and I've posted it on my webpage:

http://arche-wiki.st-andrews.ac.uk/~ahwiki/pub/Main/RossCameron/setsandrealexistence.doc

Maybe that will make things clearer. Maybe not :-)

Ocham said...

I have a theory that all this web stuff (blogs, usenet, &c) is massive displacement activity by people who have a block about writing papers and stuff.

This seems to be an exception! I look forward to reading the paper and will revert shortly.

Ross Cameron said...

An exception to *that* form of the rule, perhaps - but not, I suspect, to the general motivation. I just end up writing silly papers to avoid doing my admin!

:-)