Further indeterminate fallout
Sun 12 Apr 2009 09:37 PM
Thinking more about indistinguishable spacetimes has led me to think about the contrast between underdetermination and indeterminacy. Somehow, I wrote a dissertation on the former without clearly thinking through the latter.
In a discussion note that he wrote for the workshop but did not present, John Norton suggests that "the indistinguishability does not pertain to theory. We are not presented, for example, with general relativity and some competitor theory, indistinguishable from it. Rather, what we cannot distinguish is whether this spacetime is the one of our observations or whether it is that one" (p. 4). Yet this requires drawing a line between a theory and a detail filled out within a theory.
Suppose we accept a statement conception of theories. General relativity is a set of axioms and their consequences. Add some claim about spacetime (e.g. that it is hole-free), and it's one more axiom. The specification is still a theory and can be pitted against a rival specification.
One might still say: A theory is more than just any old colection of sentences. A theory is a system of laws. General relativity is a theory because it's got the general laws, but specifications just add local detail. This may separate the sheep from the population ecology, but it won't cut the mustard here. The specifications of spacetimes are definitely not local detail. Spacetime being hole-free is a fact about the global topology of spacetime. It's plausibly even a law. (Whether or not it is ultimately a law depends on what we think "law" means.)
Suppose we accept the semantic conception of theories instead. General relativity is a set of possible models. Models with hole-free spacetimes are a subset which can itself be treated as a theory.
I'm a pluralist about theory concepts, so I can't insist that there is no possible understanding of theory such that general relativity is a theory and the specifications are not. I just don't see what it would be.
So I don't think that Norton's attempt to make this not about theory succeeds. But I think he's right to think there's a difference between underdetermination between rival theories and indeterminacy within a theory.
Underdetermination obtains when we can't responsibly decide between rival theories. If this inability only holds for a narrow range of circumstances, then there isn't anything of especially philosophical interest: We begin in ignorance, do some research, and discover something. Underdetermination of the sort that typical concerns philosophers holds for a broad range of circumstances, possible every circumstance we could ever hope to be in. (In my work, I call this range of circumstances the 'scope' of the underdetermination.)
Indeterminacy obtains when a theory can be specified in different and significantly incompatible ways. For example, the number of particles in the universe is indeterminate in classical mechanics. If you specify the number of particles, then you can put the machinery of the theory to work - but the theory won't tell you how many particles you should consider. A different way of putting the point is that classical mechanics has models with any number of particles.
To get a feel for how underdetermination and indeterminacy interact, consider some examples:
A. The gravitational constant simply appears as a parameter in Newton's theory of gravity. The theory does not tell us what the constant must be. It's indeterminate. Yet, since there is no problem is supposing that there is a precise value for the constant in the world; we can try to formulate the specified theory that includes both the law of gravity and the correct value for the constant. We can determine the constant experimentally (within error bars) and so its value is not underdetermined.
B. It is possible to describe arrangements in classical mechanics such that two outcomes are equally compatible with the theory; e.g. Norton's Dome. This is clearly indeterminacy. We could specify which of the outcomes will occur, effectively forging a more specific theory that is not indeterminate in this way. Yet there is no principled reason to specify one outcome rather than another. Since we can't construct a Norton Dome, there is no way to resolve the matter experimentally. Even supposing we were justified in accepting classical mechanics, we would not be justified in accepting a more specific theory which avoided the indeterminacy. So the choice between specific theories would be underdetermined. Yet the Dome indeterminacy would not have conjured a rival to classical mechanics. Our acceptance of the more general theory might withstand underdetermination, even if it arises for the specifications.
C. In Quantum Mechanics, as it's usually construed, particles typically don't have determinate positions and momenta. Yet we can't just freely imagine precise quantities out in the world. The Kochen-Specker Theorem puts constraints on there being precise values, precluding certain kinds of hidden variable theories. Presuming that QM were true, this limitation would not be an epistemic problem at all - there would be indeterminate quantities in the world. We could (in principle) know them as precisely as the world defines them, so it wouldn't be a matter of underdetermination.
D. In General Relativity (to return to the Malament/Manchak proof) the theory is compatible with spacetime having different global features. So we have indeterminacy. If some of those features obtain, then neither observation nor theoretical considerations would justify our thinking that they obtained. (See my previous post for a brief explanation as to why.) So we'd face underdetermination. Yet this underdetermination does not show that our acceptance of GR is underdetermined. We cannot responsibly decide between GR&H and GR¬-H, say, but that does not show that GR itself has any serious rivals.
Notice also that we might fret over questions of indeterminacy within theories that we think are simply false. We can no longer responsibly believe classical mechanics, for example, so we are not in scope of any interesting underdetermination for it against any rivals - but it can be philosophically rewarding to consider indeterminacies like the Norton Dome. I think that this is because indeterminacy is fundamentally a logical or metaphysical question, and so it may be rewardingly chased even into the den of otherworldly counterfactuals. Underdetermination is primarily an epistemic or epistemological question, and so it matters primarily insofar as it tells us something about limits on our ability to actually know things.
Mon 13 Apr 2009 08:14 AM
Hi P.D. --
I'm very sympathetic with the basic thrust of this helpful post, but I do have one nit to pick, re: separating sheep from population ecology. You appear to be saying that the contrast class for laws is "local detail". In my idiolect, the right contrast class is initial conditions. I don't have a perfect grasp on what physicists mean by a 'local' quantity, but I think a complete specification of initial conditions would not qualify as local. In short: I think local v. global does not map neatly onto initial conditions v. laws.
The weird thing is that a spacetime's having the hole-free property doesn't (to my mind) fall clearly into the 'law' box or the 'initial conditions' box.
Another route to your conclusion (which you probably already know about): it is not clear that there is a genuine distinction between laws and initial conditions -- the 'past hypothesis' in thermodynamics (the universe started out in a law entropy state) is the paradigm problem case here.
Mon 13 Apr 2009 08:46 PM
Greg: I aagree that this gets somewhat murky. It will suffice for my purposes here that hole-freeness is not obviously not a law, because then theory-means-law can't be used to draw the line between theory and specification in the way suggested.