Sleeping Beauty and the Mathematical Alarm-Clock

In the chancy Sleeping Beauty case, popularized by Elga [2000], Beauty goes to sleep on Sunday night knowing that she will be awakened on Monday morning and then put back to sleep on Monday evening. A fair coin is tossed: if the coin lands tails, Beauty is awakened again on Tuesday with her memory of the Monday awakening erased, and if the coin lands heads, she sleeps through to Wednesday. Beauty knows all this. The puzzle is to say what credence Beauty should have on Monday in the proposition that the coin lands heads (call this proposition Heads).

In the mathematical Sleeping Beauty case, uncertainty about the result of a fair coin toss is replaced by uncertainty about the truth of a mathematical proposition. On Sunday night Beauty has credence 1/2 that Fermat’s Last Theorem is true. She will be awakened on Monday if the theorem is true, and on both Monday and Tuesday (again with her memories from Monday erased) if the theorem is false. Beauty knows all this. The puzzle is to say what credence Beauty should have on Monday in the proposition that Fermat’s Last Theorem is true (call this proposition True.)

The setup of the case entails that Beauty cannot be a fully rational Bayesian agent, since such agents are required to be logically omniscient.  Is there then any sense in asking what her credences ought to be? – after all, in one obvious sense, her credence in True ought to be 1. I think so: it is common to distinguish diachronic and synchronic constraints on credences, and evaluate an agent’s performance with respect to the diachronic constraints independently of her performance with respect to certain synchronic constraints (such as logical omniscience). Epistemologists had better hope that something like this is possible: we’re not perfect Bayesians, after all, and agents with non-trivial degrees of belief in mathematical propositions need to be modelled by any candidate epistemology for mathematics.

The difference between the two cases may seem somewhat incidental. General considerations relating to topic-neutrality perhaps make it natural to assume that the two puzzles have the same solution. In any case, discussions of Sleeping Beauty have usually paid little attention to the role that chance plays in the story. For example, in a discussion of the chancy Sleeping Beauty case Chris Meacham remarks that “the chanciness of the coin toss only plays a superficial role in the argument….  the argument goes through just as well if heads and tails are replaced by two different hypotheses we have other reasons for having ½ / ½ credences in.” (Meacham [2008]). Sleeping Beauty is usually described as a puzzle about self-location; and our two cases seem to involve structurally similar self-locating uncertainty. In both the chancy case and the mathematical case, when she awakens on Monday Beauty is unsure whether it is Monday or Tuesday.

In this post I will give some reasons for suspecting that the two puzzles may in fact have different solutions. Two of the most powerful arguments for the answer 1/3 in the chancy case (both of which appear in Elga’s original article) are inapplicable in the mathematical case. Moreover, an analogy between Sleeping Beauty and confirmation in Everettian quantum mechanics gives us reason to prefer the answer 1/2 in the mathematical case.

Consider first the long-run frequency argument for the answer 1/3. Beauty knows that if the chancy case is repeated infinitely many times, the ratio of Tails-awakenings to Heads-awakenings will (with chance 1) tend to 2:1. Since any awakening is indistinguishable from any other, her credence that a randomly-chosen awakening is a Heads-awakening should be 1/3.

This argument lapses in the mathematical case. The truth-value of mathematical propositions cannot vary between awakenings, so Beauty knows that in the long-run either all the awakenings will be True-awakenings or all the awakenings will be False-awakenings.

A similar case involves different mathematical questions on each run of the experiment. This variable-question mathematical case will probably pattern with the chancy case rather than with the fixed-question mathematical case, depending on the procedure by which the questions are selected. Be that as it may, here I will only address the fixed-question mathematical case.

Consider now the Principal Principle argument for the answer 1/3. It is a mere procedural irrelevance whether the coin is tossed on Sunday night or on Monday night – so let us assume that it is tossed on Monday night, and that Beauty knows this. Now consider Beauty’s credences on learning that it is Monday. She knows that whether Heads is true depends on a future toss of a fair coin, so the Principal Principle constrains her to have credence 1/2 in Heads. But learning that it is Monday ruled out some Tails possibilities without ruling out any Heads possibilities; so conditionalization requires that Beauty’s credences prior to learning that it is Monday must be less than 1/2. If we assume that on initially awakening Monday and Tuesday deserve equal credence conditional on Tails (perhaps justifying this assumption via a restricted principle of indifference, as Elga does) then Bayes’ theorem tells us that Beauty’s credence in Heads on initially awakening on Monday should be 1/3.

This argument too lapses in the mathematical case. Since mathematical propositions have objective chances of either zero or one, the constraint that the Principal Principle places on credence in mathematical propositions amounts to the requirement of logical omniscience, which is explicitly suspended in the mathematical Sleeping Beauty case. Accordingly, nothing in the setup of the mathematical case prohibits Beauty from adopting credence 1/3 in True on being informed that it is Monday, and hence nothing prohibits her from adopting credence 1/2 in True when she initially awakens on Monday.

The breakdown in the mathematical case of these influential arguments for the answer 1/3 provides reason to doubt whether our two cases have a uniform solution. Moreover, consideration of the analogy between the mathematical Sleeping Beauty case and confirmation in Everettian Quantum Mechanics (EQM) provides reason to think that the correct answer in the mathematical case is 1/2.

The analogy between SB and confirmation in EQM was first discussed in print by Lewis [2007]. Papineau and Durà-Vilà [2009] reply to Lewis, and an extended discussion is provided by Bradley [2010]. Lewis and Bradley both claim that applying ‘thirder’ reasoning to confirmation in EQM yields the result that evidence that a quantum measurement has taken place confirms EQM over a one-world stochastic theory, even if the result of the measurement is unknown. This ‘easy evidence’ for EQM is deeply problematic, not only for Everettians but for anyone who is not prepared to rule out EQM a priori. It is highly implausible to think that we get evidence for EQM simply by finding out that a measurement has taken place.

If the correct answer to the chancy Sleeping Beauty case diverges from the correct answer to the mathematical Sleeping Beauty case, then we may ask whether Everettian confirmation aligns with the chancy case or with the mathematical case. Since it is not a matter of chance whether Everettian quantum mechanics is correct, I suggest that confirmation in EQM ought to align with the mathematical case rather than with the chancy case. If Everettian confirmation scenarios are analogous to the mathematical Sleeping Beauty case, then a rejection of easy evidence for EQM generates the answer 1/2 in the mathematical case.

I conclude that there are important disanalogies between the mathematical case and the chancy case. Two influential arguments for the answer 1/3 in the chancy casehese arguments are inapplicable to the mathematical case. And the similarity between the mathematical case and confirmation in EQM provides an indirect argument for the answer 1/2 in the mathematical case.

These differences between the two Sleeping Beauty cases are of interest in their own right. But they also cast an interesting light on the chancy case. Sleeping Beauty is usually described as a puzzle about self-locating belief. But the preceding discussion suggests that the chanciness of the coin toss is also playing a crucial role in generating the answer 1/3. Sleeping Beauty may turn out to be as much a puzzle about chance as it is a puzzle about self-location.

References

Bradley, Darren [2011]. ‘Confirmation in a Branching World: The Everett Interpretation and Sleeping Beauty’. British Journal for Philosophy of Science 62: 323-342.

Elga, Adam [2000]. ‘Self-locating belief and the Sleeping Beauty problem’. Analysis 60: 143-47.

Lewis, Peter [2007]. ‘Quantum Sleeping Beauty’. Analysis 67: 59–65.

Meacham, Christopher [2008]. ‘Sleeping Beauty and the Dynamics of De Se Belief’, Philosophical Studies 138: 245-269.

Papineau, David and Durà-Vilà, Victor [2009]. ‘A Thirder and a Everettian: A Reply to Lewis’ ‘Quantum Sleeping Beauty’’, Analysis 69(1):78-85.

Sleeping Beauty and the Mathematical Alarm-Clock