Everett and Elga’s Principle of Indifference

August 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)?



  1. That’s nice! I think out of the principles you mention, indifference is clearly the weakest link (because of Bertrand’s paradox and all that). So if it really is that one of the three must go, then that’s the one.

    However, I think we have to be really careful with probability in the Everett interpretation, as you note. I think we also need to be really careful with counting centers in the Everett interpretation. Are there really only two centers, one each in Rooms A and B? Or are there uncountably many in A and uncountably many in B? I guess it depends on whether we think of decoherence as involving a single center splitting into two, or whether we think of it as two already distinct (but qualitatively identical) centers becoming different enough that they stop interacting. On the latter picture, all the infinitely many future splittings already mean that there are uncountably many centers on each side. And thus, indifference can’t tell us what to do unless we impose some sort of measure on things. And the Born rule seems like a natural way to do so, though we still have to answer the standard question of why that’s the case.

  2. Hi Kenny,

    Thanks for the comment! Yeah, I was originally going to write it as a problem for indifference, but then I got more cautious. After all, people tend to find the Everett stuff unintuitive anyway.

    You mentioned Bertrand’s paradox as a problem for indifference. Did you mean the paradox with the three boxes and the gold and silver coins? I can see how that might be a problem for the unrestricted principle of indifference, but I can’t see how it might effect Elga’s principle? The unrestricted principle is pretty much doomed anyway – e.g. it is inconsistent with the Principal Principle, even without Everett, and entails skepticism.

    I hadn’t thought about how to count centres. I guess in the back of my mind, I had the Lewis worm view in mind – but, like you said, that gives us lots of centres (I guess you get the same problem if you’re an endurantist, with lots of colocation.) I have two thoughts on this.

    Obviously you can go counterpart theoretic and restore the right count that way. But if we stick to worms, the centred worlds looks like its going to need tweaking anyway. You don’t even need Everett for the problem to arise, it would be a problem if I undergo ordinary fission in my future. I don’t think that telling me I’m going to undergo fission in ten years time should make me half my credence in my current predicament.

    Probably the best way to do this would be to instead consider world/time/equivalence class triples, where the last coordinate is just the equivalence class of all individuals colocated with an individual at that time. This captures the intuition that the predicament the centre represents only includes what the individual is doing at that time. Without this tweak, there are continuum many centred worlds similar to my current one (provided some subtree splits unboundedly often along every branch) – but that shouldn’t mean I should have zero credence that I’m in the predicament that I’m actually in.

    Secondly, I was hoping that it might be physically possible that there is only 1 branching *ever*. I guess this depends on how you cut the worlds out of the universal quantum state, but I got the feeling it depended on the observers. Couldn’t the observers just disappear after the first branching?

    • You Sir/Madam are the enemy of confusion evweheyrre!

  3. Interesting stuff. I’d like to echo Kenny; the ‘naive Everettian’ viewpoint you sketch already has big problems with probability, and needs to be altered in a fairly substantial way. Many of the extant suggestions for doing this (eg the Saunders-Wallace style multiple-utterance view Kenny was alluding to) will have consequences for your puzzle.

    Although I haven’t thought it through in detail, I think the ‘caring measure’ version of EQM (cf Greaves) might have the most trouble accommodating the puzzle. But that view already involves modifying our primitive rationality principles, so its proponents probably wouldn’t mind having to drop or modify indifference.

    By the way, you say ‘the individuals in each branch are worldmates by anybodies standards’ – they’re not by my standards! I think the best way to think about modality and Everett is a modified version of modal realism that uses a alternative criterion of world-demarcation from Lewis’ criterion spatio-temporal isolation. This view allows you to maintain that inhabitants of different branches are parts of different worlds. My defence of this position goes via knowledge-maximizing interpretation of our ordinary modal talk.

  4. Hi Alastair,

    I haven’t actually read about this multiple utterance view – can you point me to any literature?

    I guess one way of dealing with the incoherence problem (making sense of any kind of uncertainty in branching time) is to treat the uncertainty as self-locating uncertainty. You may know all the facts about the world, but still not know where you are in it if you count people the Lewisian way: as linear non branching worms. Before I split there are two indiscernable colocated people, lefty and righty, and I don’t know which one is me.

    The problem with this answer to the incoherence problem is that it stumbles on the ‘quantitative problem’: not just accounting for uncertainty, but getting it to accord with the born rule. Before the split there are two colocated people who have identical mental states and are thus subjectively indistinguishable. Secondly, they are members of the same world (even by your standards, if I’ve understood you) because they overlap mereologically. Thus they are similar centred worlds and should get the same credence by Elga’s indifference principle.

    So the way I see it, the indifference principle hits this kind of view the hardest.

    I think the Greaves view comes away unscathed by Elga’s principle. However problems arise if you think the analogous principle for caring holds: you should care just as much about each individual from a collection of similar centred worlds.

  5. The best place to look is Saunders and Wallace’s forthcoming BJPS paper: http://users.ox.ac.uk/~lina0174/uncertainty.pdf . Their position is based on the self-locating uncertainty idea that you describe.

    This version of EQM only has problems accounting for the Born rule if we admit that the two branches in question are parts of the same world. And as I suggested above, Everettians should deny this. It’s true that something subtle has to be going on with the mereology to make the view work out; but I don’t think it’s impossible. Have you looked at the comments on this post of Robbie’s?


    BTW, I agree with your comments on the ‘caring measure’ view.

  6. Thanks for the reference!

    I saw Robbie’s post, but I didn’t read the linked pdf. I’ll have a look today.

    I’m slowly warming to your view on worlds, but I’m still not sure if it helps here. My worry with mereology wasn’t so much to do with different worlds overlapping – there are modal realists who accept overlap (Kris McDaniel I think holds this view.) It’s more that their temporal parts at a particular time before the split are identical, which makes me want to say that they are world mates at that time.

    For example – surely me and you are worldmates. However, assuming we are both particular linear worms, the chances are that at some point in the future our worms will part and take different world paths. So if you and me have to be *entirely* within the same world to be world mates, then chances are we’re not world mates after all.

    Of course, our temporal parts at this time are world mates. But if you want to keep to the Lewis way of counting people this is no consolation. Probably the best way to think of it is that the relation of ‘being a world mate’ is time dependent. For example, world-mates(Al, Andrew, now) is true, but world-mates(Al, Andrew, t) may be false. But on this view, before the quantum measurement, lefty and righty *are* world-mates, and so the principle of indifference can be applied.

  7. BTW I’m beginning to think the Greaves line might be the most attractive way out. The indifference principle for caring only seems to be plausible when you are uncertain about which predicament you are in. If you are in exactly one of them, without the uncertainty, you should only care about your actual predicament.

    My problem with the caring view is I can’t see how it gets us the statistical predictions that quantum mechanics makes. The relation between frequencies, chances and credences is clear, but I can’t see how they relate to caring measures.

    Another way of getting uncertainty in Everett might be to think of the logic of time as being that of linear time – but to supervaluate out the branches. Thus many statements about the future will be indeterminate. When we learn a sentence is indeterminate, our credences in it can evolve to non trivial probabilities – thus simple indeterminacy about the future allows us to be uncertain about the future. But, of course, you still have the quantitative problem to deal with on that view.

    (The analogy being that, if I learn that ‘Fred is bald’ is indeterminate (borderline, vague), I may still be uncertain about whether Fred is bald, even if I know *all* the de dicto facts.)

  8. The Saunders/Wallace way of dealing with your example is to say that the continuant that is you and the continuant that is me are worldmates timelessly. There’s no chance that our continuants can diverge. Of course, I may not be able to say in purely descriptive terms which continuant person I’m talking to – or indeed which continuant person I am – that would require knowing exactly which world is actual. Nonetheless I can refer to you demonstratively; the person that I am in fact in causal contact with.

    Making this work requires denying the straightforward gloss on the SW view, that a temporal part of me at time t is identical with the temporal part at t of the guy just like me who gets run over by a bus tomorrow. Instead, I think we have to say something like ‘world-bound temporal parts are ordered pairs of wavefunction segments and continuant branches’. Robbie and I thrashed this out a bit in the comments to his post.

    The problem I have with using the caring measure view as a way out of the indifference puzzle is that the cure seems worse than the disease. To recover the Born rule, we need to postulate a new primitive rationality principle involving the caring measure. That seems worse than just dropping indifference…

  9. […] and our uncertainty is about which one this is. (From the comments of my last post, it seems that Alastair holds the branches as worlds view – but I don’t know what he’d make of this […]

  10. The only point I was trying to make was that there are strong reasons to think that, pre-measurement, there are going to be violations of Elga’s principle of indifference – even by fairly stringent standards of world-matehood.

    However – if you take the line that the spacetimes diverge you may be able to get round this. I’m not entirely sure I understand it though – I’ll mull it over.

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