Earman's first example is this.
I claim that there do exist examples of rival empirically indistinguishable theories that posit interestingly different theoretical structures. For instance, TN [Newtonian Theory] (sans absolute space) can be opposed by a theory which eschews gravitational force in favor of a non-flat affine connection and which predicts exactly the same particle orbits as TN for gravitationally interacting particles (Earman 1993, 31).In short: Newtonian gravitation can be described as particles interacting via forces on a flat spacetime -- or as particles in freefall in curved spacetime, as in Cartan's 'geometricized' Newtonian gravitation. (See Malament (1986) for a discussion of the latter. This underdetermination has been beautifully and rigorously argued for by Jonathan Bain -- PDF.)
Earman's second example is the existence of observationally indistinguishable spacetimes:
Relativity theory tells us that the data available to an observer through such interactions [as observation] are restricted to events that are swept out by the observer's past light cone.... As a result, even idealized observers who live forever may be unable to empirically distinguish hypotheses about global topological features of some of the cosmological models allowed by Einstein's field equations for gravitation (ibid).Indeed, my distinguished colleague John Manchak has greatly generalized this result, and shown that all spacetimes have an indistinguishable counterpart that preserves all the local properties of spacetime (Philsci-Archive).
What's the difference between Earman's two examples? In short, it is that the first is about underdetermination of theories, while the second is about underdetermination of models of a particular theory, by available empirical evidence.
What's the significance of that? The first example strikes me as a completely convincing case of underdetermination (especially on Bain's treatment). But note: whether or not the second kind of underdetermination obtains depends on how one formulates a theory. For that reason, this underdetermination might often be avoided by a well-motivated reformulation.
For example: there is a particular class of models of Einstein's Field Equations (EFE) that can be described. There is a smaller class of models when one also requires that the Energy Conditions (EC) be satisfied. Any causal condition (CC) will place further restrictions on the class of models. So, we have multiple formulations of general relativity:
- EFE + EC
- EFE + CC
- EFE + EC + CC
Of course, there are some cases of observational indistinguishability which are can obtain even in EFE + EC + CC -- for example, David Malament (PDF) describes observationally indistinguishable counterparts to de Sitter spacetime. But one wonders if this might not lead us to a suspicous game: you show me an example of observational indistinguishability, and I try to rule it out by further restricting the class of models of General Relativity.
At any rate, a claim about underdetermination at the level of theories (such as in Earman's first example) does not seem to suffer from this kind of regress.