12.6 The Regress of Relations

Let us turn to the second objection. This was to the effect that if anything is empty, some things cannot be empty. The most obvious candidates are the relations in virtue of which things are related. Note, in particular, that in the (“expanded”) structural trees of Section 11.4, the relations (arrows) of the tree are not analysed away in the same way in which the nodes are: they stay unanalysed. That some things are empty therefore presupposes that others are not.

There is a simple answer to this, however. The relations involved are just as empty as anything else. They too have their quiddity by relating to other things. They are therefore at the roots of their own trees. Indeed, there was nothing in the arboreal discussion to suggest that the object at the root of our original tree could not have been a relation. (Relations are objects too.)14 Of course, this will raise a question about the relations involved in those trees. But the response is exactly the same.Those relations are empty, and have their own trees. Obviously we have a regress. But there is no more reason to think this regress vicious than to think our original regress so.15

This raises another worry, though. Understanding the quiddity of objects and relations as I have done shows that they are not “free-standing”. But the understanding suggests another candidate for something which is—not the objects and relations themselves, but the very structure in which they are all embedded. Here is one way to see the worry. The account I have given is obviously some kind of structuralism. As we have already noted, structuralism is also a view held by some people in the philosophy of mathematics.Numbers are not platonic (freestanding) objects, but simply places in structures.Thus, the number 0 is just the marker for the first place in any ω-sequence. But what is a structure? One view is that these structures themselves are “ante rem”.That is, they are platonic structures that lie behind things like numbers.16 In such a view, then, we still have free-standing things: the structures. In the same way, I have analysed the quiddities of relations and (other) objects in terms of certain structure. But do we not, then, have to understand this structure as free-standing? After all, it is the very provider of loci, not itself a locus.

To assuage this worry, we will need to lookmore closely at something I have so far taken very much for granted: the notion of a locus. A suitable analysis will show that structure is as empty as anything else. The analysis I will give is somewhat

14 There may even be relations in a tree of which they, themselves, are nodes. The mathematical representation of this requires non-well-founded sets. More of this anon.

15 In reply to Bird (2007), Barker (2009) notes one regress of this kind, and simply claims that it is vicious. It is not.

16 See, for example, Shapiro (2000).

One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness by Graham