James Hartle and Mark Srednicki have a paper in Physical Review A arguing against what they call typicality assumptions (arXiv preprint here). A typical argument employing typicality assumptions goes as follows:
1. If theory T is true, then humans would be extremely atypical beings in the universe.
2. Any true theory must imply that humans are typical beings.
3. Therefore theory T is not true.
2 is the typicality assumption. It has been used in several papers by prominent physicists, notably Don Page, who uses it to argue against models of the universe that predict vast numbers of Boltzmann Brains (relevant papers include arXiv:hep-th/0610079v1, hep-th/0611158v1, hep-th/0612137v1, hep-th/0612194v1; J. Cosmol. Astropart. Phys 01 (2007) 004.). A Boltzmann Brain is essentially a vacuum fluctuation that happens to have the right configuration of atoms to form a (momentarily) conscious brain. Any universe that persists for an infinite amount of time would contain vacuum fluctuations forming an infinite number of Boltzmann Brains (BBs from now on). Most BBs would have nothing more than a series of disordered, disjointed experiences — they persist too fleetingly to have the continuous, coherent experiences that we have. So we can safely say that we are not BBs. In a universe with an infinite lifetime, then, we would be atypical observers, outnumbered by BBs. Therefore, if we accept the typicality assumption, then we should rule out theories that predict that our universe has an infinite lifetime.
First, let me say that I don’t even see why an infinite number of Boltzmann Brains need imply that our experiences are atypical. I have the impression of continuous, coherent experience, but surely this could just as well be due to my brain being a BB that just happens to have the right arrangement of neurons and electrochemical signals to give me the impression of having coherent memories of past experiences. If we have an infinite number of Boltzmann Brains, then surely we have an infinite number of BBs that are observers with coherent experiences, and it’s not obvious to me that one infinity is larger than the other in the strict mathematical sense. Page takes the fact that it is ambiguous how we take the ratio of ordinary observers (OOs) and BBs in this case as a reason to reject theories that contain infinities of both OOs and BBs (arXiv:hep-th/0612137v1). But if we take a step back and think about it, surely it is absurd to reject a possible reality just because that reality would not be amenable to us conducting certain mathematical operations on it. Why should any reality be such that we can unambiguously determine a ratio of OOs to BBs?
Another problem is that typicality has to be defined relative to some class. Since Page considers BBs to be in that class, we can make a reasonable guess that he means typicality of experience relative to all other conscious objects with experiences. But this class of objects seems to be quite arbitrary. How in the world are we supposed to decide what qualifies as a minimal experience? Would we include a Boltzmann Beetle Brain in that class?
Those are just some of my own objections. Hartle and Srednicki raise others. The centerpiece of their paper is a Bayesian analysis of candidate theories about a model cyclic universe. Hartle and Srednicki describe it as follows:
Consider a model universe which has N cycles in time, k = 1, . . . ,N. In each cycle the universe may have one of two global properties: red (R) or blue (B), which could be thought of as (for example) two different possible values of the CMB temperature. To further simplify the discussion, the only relevant observables are assumed to be (1) the value of the property and (2) the existence of an observing system that is able to determine this value. In each cycle, the probability for such an observing system to exist is taken to be pE; this probability is assumed to be independent of whether the universe is red or blue in that cycle. Furthermore, observations are assumed to be perfectly accurate, so that if red is observed in any cycle, then the universe is red in that cycle, and conversely.
Two competing theories are considered: one that says the cycles of the universe are all red, another that says that only some are red. Via a Bayesian analysis, Hartle and Srednicki show that the probability of observing a red universe given a theory that stipulates how many red cycles there are, depends on the sum over all possible values for the number of observing systems not observing a red universe. So the actual number of observing systems not observing a red universe doesn’t matter. The implication is that the probability of our observing a certain set of data is independent of how many other observers there are with different sets of data — only our set matters. We can ignore the observations of hypothetical BBs.
While that disposes of Page’s dismissals of theories that imply that our observations are not typical, a deeper problem with Page’s reasoning, I think, is that the probability of observing what we observe given that a theory is true is not the quantity we should be looking at to gauge if a theory is acceptable. Instead, we should be looking at the probability that the theory is true given a certain data set; that is, P(theory | observation) instead of P(observation | theory). Since Page is arguing against accepting theories with an infinite number of Boltzmann Brains, surely what we’re concerned with is the probability of a given theory being true given a set of data, not the probability of a set of data being true given a certain theory. With P(theory | observation), we would not need the problematic typicality assumption. In fact, if we look at P(theory | observation), we can see where the mistaken intuition that typicality matters comes from. Hartle and Srednicki show that in their toy universe, the probability that the all-red-cycles theory is true given that we observe a red universe, P(all red | our cycle is red), is much larger than P(a few red cycles | our cycle is red),* the probability that a given theory with a small number of red cycles is true given that we observe a red universe. In other words, the theory in which our observations are more typical is favoured. This, however, does not justify the typicality assumption, at least with respect to Boltzmann Brains. A crucial assumption in Hartle and Srednicki’s toy cyclic universe is that the probability of an observing system to exist in any one cycle is independent of the property of the cycle under scrutiny, that is its colour. For BBs, however, it is not true that the probability of an observing system to exist in a given ‘bubble’ of inflation in the universe (say) is independent of the ratio of BBs to OOs in the entire universe. So one cannot apply Hartle/Srednicki’s Bayesian analysis to the case of hypothetical BBs, and the typicality assumption is not justified.
*Note that in this context, “few” or “many” BBs/OOs is relative to the numbers of the other kind of brain; i.e. there are “few” BBs iff the BB:OO ratio is small and there are “few” OOs iff the BB:OO ratio is large, regardless of the absolute numbers of each kind.