I've written a little (4-page) paper about diffeomorphism equivalence and permutation equivalence. It is on my university homepage, here. The argument, though a bit technical, is quite short. A diffeomorphism of a manifold M to itself is a permutation of the points that is an automorphism. A spacetime S is something of the form (M, g, phi_1, ..., phi_n). Suppose h : M -> M is a diffeomorphism of M (i.e., an automorphism). Then we can drag all other fields along by h, and obtain another spacetime, call it h[S]. By construction, S and h[S] are isomorphic. The "gauge equivalence" claim is that S and h[S] are "physically equivalent" solutions. However, a diffeomorphism h is just a special case of permuting the elements of the base set of the manifold. And one can in principle use any permutation one likes. Given a spacetime S and a permutation p, one can drag the topology along too, as well as the metric and other matter fields. Then one gets a spacetime p[S] isomorphic to the first, by construction. The equivalence claim now is that S and p[S] are "physically equivalent" solutions. If so, then perhaps the "gauge symmetry" of GR is not restricted to permutations of the points which are diffeomorphisms. Rather, it involves all permutations of the points: a kind of general permutation symmetry. (I relate this in the paper to Quine's idea that permutations of the interpretation of an interpreted language (using proxy functions) yields an interpreted language which is in some sense indistinguishable from the first.)
Anyway, the paper is short and comments by anyone who works on this kind of thing would be welcome.
UPDATE (May 18th).
After several days checking for any other related literature, I see that John Stachel seems to have made a somewhat similar suggestion in:
- Stachel, J. 2002. "‘The relations between things’ versus ‘the things between relations’: The deeper meaning of the hole argument." In D. Malament (ed.), Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics. Chicago and LaSalle, IL: Open Court.
I haven't read this paper yet. Stachel's proposal is discussed in a very interesting 2006 article by Oliver Pooley,
- Pooley, O. 2006. "Points, Particles and Structural Realism", in D. Rickles, S. French & J. Saatsi (eds), The Structural Foundations of Quantum Gravity, Oxford University Press.
This is available here
[Pooley's paper contains a nice discussion of the way to categorically axiomatize a structure M by, in effect, taking the logical conjunction of its diagram (set of atomic sentences true in M, in a language L_M with a name for each domain element), adding all inequation clauses saying all the names denote distinct things and that they exhaust the domain, and then existentially quantifying the result. For an infinite structure, clearly this is an infinitary formula.]