## Two Tough Cases of Underdetermination

John Earman once pointed out two tough cases of underdetermination, neither of which arises in a silly algorithmic way. Today, I’d like to argue for **an important difference between these two examples**. Ultimately, I suspect the first is more deeply intractable than the second.

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
- EFE + EC
- EFE + CC
- EFE + EC + CC

all of which correspond to different classes of models. So, when one argues, *model M and model N are observationally indistinguishable models of our world*, this underdetermination might be avoided by moving to a more restricted formulation of General Relativity. And, unfortunately, the issue of which formulation of general relativity is the correct one is deeply contentious.

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.

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Intuitively, I think I see the distinction you are trying to draw, but I’m not sure how you can draw it in a principled manner once you make clear what notion of theory you are working with. If, following the semantic view, you take a theory to be a set of models, then I don’t see how you can distinguish between the two kinds of underdetermination (what you call different formulations of GRT would simply be different theories). If you don’t and you think that theories are distinct from their models, then what is a theory? I think the most plausible answer would be a set of propositions but then again the different formulations of GRT would seem to count as different theories. However, and probably here is the difference the different formulations of GRT are such that one is a proper set of the other, while the different formulations of Newtonian Mechanics are not.

Does this capture the difference between the two cases. If not, how would you characterize it?

That’s very interesting. You may be more of an expert on semantic accounts than I am. A semantic theorist takes a set of models stand in for a ‘theory.’ But it seems that he would still make a principled distinction between the two kinds of underdetermination like this:

1. You have two sets of models, with distinct ontologies (the models of one set might be curved spacetimes, and the models of the other set might be flat). But the two sets are pairwise-indistinguishable — that is, each model in one of the sets has an empirically indistinguishable counterpart in the other set.2. You have just one set of models, with one ontology (say, curved spacetimes). But there are pairs of distinct models in the set that are empirically indistinguishable.Perhaps the ‘subset’ relation could also be used to make the distinction. I’m not sure, but it’s an interesting idea to try. Thanks for the thought!

Here’s how I see the two kinds of underdetermination:

For any theory, there are some parts of that theory that are considered to be the “fixed” parts of the theory, and other parts which are the “results” of that theory. Those parts that are considered to be fixed can turn into the results, and vice verso, according to our scientific whim/ experience, but at any given point these two parts always exist.

Secondly, insofar as all theories have these two parts, we can have distinct underdetermination claims about both parts within a single theory.