Things between Relations

This post will be concerned with Structural Realism. More precisely, with an article by John Stachel, one of the fathers of the modern version of the Hole Argument, which was first considered by Einstein. For an introduction to the Hole Argument read the Stanford Encyclopedia entry here:
http://plato.stanford.edu/entries/spacetime-holearg/

In "The Relations between Things" versus "The Things between Relations": The Deeper Meaning of the Hole Argument (published in Reading Natural Philosophy, edited by D. B. Malament) Stachel considers a generalization of the argument which not only covers diffeomorphisms on manifolds, but permutations in general, including those arising in quantum mechanics due to the indiscernability of elementary particles. His ideas are highly controversial, but I am quite sympathetic about his views. This article also gave me the occasion to deal with the theory of reference of Hilary Putnam, which I will discuss in a separate post.

Let me briefly outline the general setting first: the so-called manifold-substantialist, who believes that space-time points are individuated separately from the fields that they "carry", is considered not to escape the hole argument. He is oblieged to the view that a hole diffeomorphism yields a physically distinct solution to the field equations. But since the fields outside the hole (where the diffeomorphism reduces to the identity) do not determine the solution within the hole uniquely, indeterminism arises. In contrast, the relationalist is not commited to such a view, he does not run into the hole argument since he does not think of space-time points as independently individuated entities. To him, all the solutions yielded by diffeomorphisms are the same, and no indeterminism arises.

Note that the indeterminism is destructive because the indeterminism is not a matter open to empiricical evidence, it is a priori implemented in the theory. But this is considered to be highly implausible. Of course, this kind of indeterminism has no observational consequences, and hence this conclusion was challenged (e. g. Liu, 1997).
However, for the sake of the argument made by Stachel, let us consider his view to be largely correct. Is it now possible to extend the hole argument in a way to apply to quantum physics as well? Is there a common feature concerning the underdetermination theories which will
And if so, can the structural content be isolated? These suggestions are absolutely mind-thrilling to me, and if the arguments made by Stachel turned out to be correct, it should have some impact on physical science as well, for certain aspects on the individuality of particles are still not well understood.

Stachel proposes to forget about the differentiable properties and consider just o set of points, then covariance of the field equations becomes permutablility (invariance under symmetric group)

Usually, one first introduces a set S of entities and then defines the relations R=(R1..R_M) on top, which form a relational structure (S,R). Stachel now also considers reflexive definitions of relations. More precisely, he offers the two alternatives:
(1) relations between things: things are independently
(2) things between relations: things are reflexively defined
Consider as an example for the second type simply the notion of king and subject: of course, there need to be people who can bear these roles, but this is not essential for this relation. There is no subject without a king, and no king without subjects: the relation is established at once, the people become subjects in the moment the king is crowned. Of course, the person who (accidentely) happens to be the king can be individuated otherwise, since he (the person) is an independent being.

Stachel then argues that in in a generally covariant theory, the points of space-time are defined reflexively. He then introduces the notion of general permutability, which simpliy means that any permutation of the entities in S yields the same possible world, i.e. all relations are defined reflexively (the "things" have no individuality, and its individuation depends entirely on the place it occupies in the relational structure). In this picture, the relationalist holds that space-time points are reflexively defined (spacetime-points are indistinguisable like electrons in an atom), whereas the manifold substantialist holdsthe view that space-time-points are independently defined (like cards in a card house), and hence the generalized hole argument arises: we can permute the entities which are not fixed by the relations, yielding a physically distinct wolrd but same observational consequences.

Pooley has challenged this analogy, and he .. haeccist and anti-haeccist....
[to be continued...]