Everett and Elga’s Principle of IndifferenceAugust 20, 2008
I’m a bit out of my depth on this one, knowing nothing about physics, so if this is all nonsense hopefully some one out there will set me straight. This is a blog after all!
I’ve been thinking about Adam Elga’s restricted principle of indifference (see here) recently. The principle plays a central role in his argument for thirding, in the Sleeping Beauty experiment, and seems to be plausible on independent grounds.
To state the principle we need to assume that the objects of our credential attitudes are sets of centred worlds, or something of this sort. The basic idea is that we can be uncertain about our place in the world, just as we can be uncertain about the way the world is (as demonstrated by Lewis’s twin gods thought experiment.) A centred world is just a world/time/individual triple, where the time exists at the world, and the individual exists at that time at that world. As Elga puts it, a centred world is a maximally specific predicament for a person to be in. You’re in the predicament iff you’re the individual, at that time at that world.
Say that two centred worlds are similar iff (1) their world coordinates are the same, (2) they are subjectively indistinguishable, in the sense that the individual coordinates both have the same beliefs, are having the same experiences, have the same memories, et cetera. The principle of indifference states that you should divide your credences equally between similar centred worlds. In particular if you know you are in one of two similar centred worlds, then you should assign each credence of a half. Even if you have strong evidence that you’re in one of the scenarios and not the other, you know your counterpart will have will have received equally strong evidence, and that at least one of you must be mistaken. (For a proper defence of the principle, I recommend you read the paper, if you haven’t already. It’s a good read.)
Now this is where I become slightly less certain, but it seems to me that if the Everett ‘many worlds’ interpretation of quantum mechanics is correct, and assuming the Principal Principle, we’re going to have to admit exceptions to this principle (and they’re going be widespread.) To put it more carefully, if the Everett interpretation can assign events chances which accord with the Born rule, then there will be exceptions to the Principle of Indifference (perhaps the Everettian can just stipulate that chances go by the Principle of Indifference, and not the Born rule, but then it is hard to see how the Everettian picture is being confirmed by past observation. For example, it needs to explain why is it likely that particles in the double slit experiment make that wavy pattern on the wall over time.)
So here is the counterexample. Depending on the outcome of a quantum measurement on Sunday you will be put to sleep and moved into one of two rooms, room A or room B, and awoken on Monday. From the inside A and B look exactly the same. Now, we may assume that the measurement creates two branches, and that we can ignore any subsequent branching as irrelevant to the probabilities involved. Lastly, and most importantly, we assume that the branches are not equally likely. That is, that one outcome of the experiment has a higher objective chance than the other, and that you know what this chance is. To fix ideas, you know that the chance that you end up in room A is 1/3, and that the chance you end up in room B is 2/3 (I think that it is a coherent quantum mechanical scenario that exactly two branches can be created of unequal chances.)
Here’s the problem. When you wake up on Monday, you should have credence 1/3 that you’re in room A, since you know that it’s less likely that the quantum experiment resulted in your being moved to room A. This is just an application of the Principal Principle: you know the chance that you’re in room A is 1/3, so you should set your credences as such. However, both your worms wake up on Monday in a subjectively indistinguishable state. You both remember being put to sleep, are both seeing indiscernible rooms, and both know the chance that you end up in room A, and room B respectively. What is more, these scenarios are both part of the same possible world. They are both concrete, so they’re both actual, and there’s only one actual world. Or, if you’re a modal realist, they’re both spatiotemporally connected, and they aren’t causally isolated. Thus the individuals in each branch are world-mates by anybodies standards. So, we have two similar centred worlds – they agree on world coordinate, and are subjectively indistinguishable – so both your successors in each branch should assign them equal credence, by the principle of indifference. This means, you should have credence 1/2 that you are in room A, which contradicts our previous credence of 1/3.
I’m not yet sure what to make of this. Is it a problem for the Everett interpretation (after all, we know they have difficulties with probability anyway), or should we instead reject the probabilistic principles we appealed to (the principle of indifference, and the principal principle)?