Fallout from Pittsburgh 
A few weeks ago, I participated in a workshop on underdetermination at the Pittsburgh Center for Philosophy of Science. The conference was fabulous, both socially and intellectually. Here's a post growing out of that, specifically about John Manchak's work on global features of spacetime.

The post is somewhat rambling, so let me begin by summing up:
The underdetermination facing our theorizing about global features of spacetime is formally more like familiar illustrations of the problem of induction than it is like familiar examples of empirical equivalence. Yet (if Manchak is right) it is different than usual worries about induction because we could never have the right kind of background knowledge to justify the inductive generalization.

John Manchak discussed his work on observationally indistinguishable spacetimes: There are some properties which, if they held in our universe, we could not know to hold. Manchak gave the example of 'hole-freeness', the property that a spacetime has got if there aren't any holes in it. I'll stick with that example.

The proof (originally sketched in the 1970s by David Malament and recently proven by Manchak) models each spacetime point as a possible observer. Observers are then treated as knowing everything about the contents of their past light cones.* Now consider an (almost**) arbitrary hole-free spacetime ALPHA. We can cut up ALPHA and assemble a different spacetime BETA, such that there is a region of BETA corresponding to the past light cone of every observer in ALPHA plus an additional region that contains a hole. Observers who collected all of the observations afforded by ALPHA could not know whether they were in hole-free ALPHA or holey BETA.

Manchak argues that this is a result particularly about global spacetime structure and that it does not hint at any general kind of underdetermination plaguing scientific inference. Physicists attempt to overcome the underdetermination by putting restrictions on what counts as a physically plausible spacetime, but such restrictrions (argues Manchak) are ad hoc and ultimately unsuccessful. We don't ordinarily reckon with things like the structure of the entire universe, so our ordinary intuitions can't be relied on to constrain the space of possibilities.

All of that seems right to me, as far as it goes. Note that the underdetermination here is asymmetrical. If spacetime is not hole-free, then the hole must be in the causal past of some spacetime point and so an observer there could know about it. The underdetermination arises only if spacetime is hole-free, because all the hole-free observations can be embedded in a spacetime that includes holes elsewhere.

Familiar cases of (allegedly) empirically equivalent theories are not like this, but instead are symmetrical. Take the claim that we live in a physical world. Consider the rival Cartesian claim that we are immaterial things deceived by an evil demon into thinking that there is a material world. Assuming that the choice between these is underdetermined, it is underdetermined regardless of which of the two possibilities actually obtains.

Note, however, that common illustrations of the problem of induction do have an asymmetric structure like the spacetime case. Suppose we start with a world ALPHA in which all swans are white. We can construct a world BETA in which there are all the swans in ALPHA plus a black swan. If we only observe white swans, then it might or might not be the case that all swans are white. Yet if it's not the case that all swans are white, there is an observation that would show as much (an observation of one of the non-white swans).***

Or consider instead a world ALPHA in which all humans are mortal. Provided every observer has an open future, we can construct a world BETA in which all of the observations in ALPHA occur at some place and time but there are immortal people. This seems strictly analogous to Manchak's case, because the requirement of an open future is just the requirement that ALPHA not be causally bizarre.

If I am right about all of this, then there is a parallel between the inductive conclusions 'Spacetime is hole-free', 'All swans are white', and 'All men are mortal.' Yet I agree with Manchak that the first of these is importantly different.

Let's start with what John Norton calls a material theory of induction: Induction requires background knowledge about the domain of objects about which we are generalizing.

In the case of swans and whiteness, we know that natural species typically have variable colouration. So we conclude that swans are not the kind of thing that are likely to all be of the same colour. The observation of many white swans does not suffice to show that all swans are white, regardless of how many we observe.

In the case of humans and mortality, we know that human bodies are fragile things. People are apt to get injured or sick eventually. Moreover, bodies grow decrepit with age. So we are justified in concluding that all humans are mortal.

In the case of spactime and hole-freeness, as Manchak argues, we don't know anything which constrains the global structure of spacetime sufficiently to underwrite an induction. So we would never be in a position to conclude that spacetime is hole-free.


* John Norton raised the worry that there might be holes in our causal past which we wouldn't be able to notice. That kind of underdetermination is not at issue here. The question is just whether infallible observers of their hole-free pasts would ever be able justified in concluding that all of spacetime is hole-free.
** The proof excludes 'causally bizarre' spacetimes in which a single observer can survey all of an inextensible spacetime at once. This would require that some observers be able to see their own future; ie, time travel would be possible. Manchak retorts that if time travel were possible then indistinguishability would be the least of our problems.
*** To make it exact, add the constraint that one could never have observed all of the swans in the world. This is formally parallel to the assumption that spacetime is not causally bizarre. (Alternately, let the constraint be that one would never be justified in believing that one had observed every swan.)

John Manchak 
Couldn't have said it better myself!

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