Criterio vector 7, chacra 7

Interestingly, we can sometimes find in the cosmology literature explicit references to the Sleeping Beauty problem (and the thirder solution in particular) to justify the broader, ambigious formulation of typicality assumption according to which, as we just saw in

§ 4.3, we should consider that we are equally likely to be in the multiverse in any of the locations consistent with our data; that is, that we are equally likely to be any of the observers having experiences indistinguishable from our own.

If Beauty adopts this typicality assumption about her location, then, in the standard setting of the Sleeping Beauty problem, she will conclude that she is equally likely to wake up in one of the three possible locations: on Monday after aHeads toss, on Monday after aTails toss, or on Tuesday, necessarily after aTails toss. However, this claim rests on confusions about the Sleeping Beauty problem. Indeed, this claim entails the following interpretation of the Sleeping Beauty problem, which for convenience I will call

the Indifferent Sleeping Beauty problem:

1. before she is put to sleep, Beauty is told that there is a nonzero proba- bility that she will either be woken up once or twice,

2. she will have no memory of any possible awakening until the experiment is over, and

3. each of the possible awakenings is as likely as any other.

With this interpretation of the Sleeping Beauty problem the probability of Heads is consistent with the standard thirder answer to the Sleeping Beauty problem. However, unlike in the standard version of the problem (or the one I gave, which is equivalent), the probability of Heads is given to Beauty as a premise. In the Sleeping Beauty problem, Beauty’s equal credence in her awaking in one of the three possible predicaments (Monday after Heads, Monday after Tails, or Tuesday) is due to the fairness of the coin. In the

standard Sleeping Beauty problem, there’s no equivalent to point #3 in the Indifferent Sleeping Beauty problem.

Predictions in the multiverse that assume that we are typical observers based on the assumption that, based on an entirely epistemic principle of indifference, all scenarios generating our locations are equally likely are analogous to the Indifferent Sleeping Beauty problem. These predictions are akin to a version of the problem in which we tell Beauty that she should be indifferent about what determines her location when she wakes up. In such a version of the Sleeping Beauty problem, she doesn’t need to know what process will determine whether she will be awoken once or twice and can just assume that whatever that process, it will make her location typical. In terms of the betting framework seen earlier in § 4.2.2 (and in Appendix 4.A), it’s equivalent to asking one’s wager without telling them what the bet is about.

It’s easy to see how such predictions don’t follow from the standard version of the Sleeping Beauty problem. Suppose that, instead of a coin toss, it is the throw of a fair die that will decide how many times she will be awoken: once if the die comes up 1, twice otherwise. There is now only 1

6 chance that she will only be awoken once.

We can arrive at this result by correcting for Beauty’s questioner’s bias: she’s twice as likely to be asked a question after a number other than 1 was rolled, which is five times more likely to occur than the roll of a 1. We have P(2 to 5) = 10×P(1) and P(1) +P(2 to 5) = 1, and therefore P(1) = 1

11. This result is incompatible with point #3 in the Indifferent Sleeping Beauty problem.

Assuming that our place among observers in the multiverse is typical comes to as- suming that the initial conditions for each “pocket universe” (at least those that are life-permitting) are equally likely. With that assumption, making predictions in the mul- tiverse would rest solely on how favorable to life each “pocket universe” is; it wouldn’t depend anymore on the prior probabilities for their initial conditions. This indeed would amount to asserting, without any justification, what Hartle and Srednicki (2007) call a

4.3. Self-Locating Uncertainty and Typicality in Cosmology 87

“mere personal preference for theories in which we are typical of something.” This is closer to a form of strong anthropic principle than to usual scientific practice.13 _{We will} see in § 4.3.3 in greater detail what argument could be more rigorously based on the thirder solution to the Sleeping Beauty problem, but we will also see why it is unwar- ranted.

It should be noted that if one uses the work of Adam Elga—and in particular his defense of the principle of indifference in the case of self-locating uncertainty in (Elga, 2004)—to justify assumptions of typicality in the multiverse, one in fact makes “an absurd claim that [Elga] do[es]n’t endorse,” namely that all physical processes or hypotheses having the same observable consequences are equally likely.14 Elga distinguished this claim from another, uncontroversial claim that he endorses, namely that states differing only on indexicality deserve equal credence (which doesn’t mean that our credence should necessarily be evenly divided among all states differing only on indexicality). Sebens and Carroll (2015) discussed a roughly formulated principle of indifference for cases of self-locating uncertainty adapted from (Elga, 2004), according to which “an observer should give equal credence to any one of a discrete set of locations in the universe that are consistent with the data she has”; to avoid possible confusion about this claim, we should add the following proviso, implicit in (Elga, 2004): “provided all those locations are equally likely to exist.”15

13_{Bostrom (2002) has pointed out what he calls the problem of the Presumptuous Philosopher: if the} likeliest theory is,a priori, the one that predicts the largest number of observers, then any theory choice could be made by any presumptuous armchair philosopher.

14_{Or with Elga’s terminology: “centered worlds representing indistinguishable predicaments deserve} equal credence” (Elga, 2004, 387).

15_{To be sure, conditional on the multiverse scenario, all possible observers are guaranteed to exist. The} relative likelihood of their existence is determined in one region of the multiverse or another by the ratio of the occurrence of these regions. There, Sebens and Carroll are concerned with claims evoked earlier in fn. 2 according to which indifference in cases of self-locating uncertainty would amount to branch counting—and therefore have disastrous empirical consequences—in Everettian quantum mechanics. However, it is questionable whether considerations about self-locating uncertainty add anything new in this context, or to our usual treatment of predictions in physics more generally.