Author’s note: I wanted to wait before doing another post on anthropic reasoning, but this topic was just too good to pass up just after Halloween .
1: Are You A Disembodied Brain?
Our story begins with Ludwig Boltzmann’s thermodynamic solution to the arrow-of-time problem. The problem is to explain why the laws of physics at the microscopic scale appear to be reversible, but the laws as seen by us seem to follow a particular direction from past to future. Boltzmann argued that, provided the universe started in a low entropy state, the continual increase in entropy due to the second law of thermodynamics would explain the observed directionality of time. He thereby reduced the task to the lesser problem of explaining why the universe started in a low entropy state in the first place (incidentally, that is pretty much where things stand today, with some extra caveats). Boltzmann had his own explanation for this, too: he argued that if the universe were big enough, then even though it might be in a maximum entropy state, there would have to be random fluctuations in parts of the universe that would lead to local low-entropy states. Since human beings could only have come to exist in a low-entropy environment, we should not be surprised that our part of the universe started with low entropy, even though this is extremely unlikely within the overall model. Thus, one can use an observer-selection effect to explain the arrow of time.
Sadly, there is a crucial flaw in Boltzmann’s argument. Namely, that it doesn’t explain why we find ourselves in such a large region of low entropy as the observable universe. The conditions for conscious observers to exist could have occurred in a spontaneous fluctuation much smaller than the observable universe – so if the total universe was indeed very large and in thermal equilibrium, we should expect to find ourselves in just a small bubble of orderly space, outside of which is just featureless radiation, instead of the stars and planets that we actually do see. In fact, the overwhelming number of conscious observers in such a universe would just be disembodied brains that fluctuated into existence by pure chance. It is extremely unlikely that any of these brains would share the same experiences and memories as real people born and raised on Earth within a low-entropy patch of the universe, so Boltzmann’s argument seems to be unable to account for the fact that we do have experiences consistent with this scenario, rather than with the hypothesis that we are disembodied brains surrounded by thermal radiation.
Matters, as always, are not quite so simple. It is possible to rescue Boltzmann’s argument by the following rationale. Suppose I believe it possible that I could be a Boltzmann Brain. Clearly, my past experiences exhibit a level of coherence and order that is not typical of your average Boltzmann Brain. However, there is still some subset of Boltzmann Brains which, by pure chance, fluctuated into existence with an identical set of memories and past experiences encoded into their neurons so as to make their subjective experiences identical to mine. Even though they are a tiny fraction of all Boltzmann Brains, there are still vastly more of them than there are `really human’ versions of me that actually evolved within a large low-entropy sub-universe. Hence, conditional on my subjective experience thus far, I am still forced to conclude that I am overwhelmingly more likely to be a Boltzmann Brain, according to this theory.
2: Drama ensues
Quite recently, Sean Carroll wrote a paper (publicized on his blog) in which he and co-author Kim Boddy use the Higgs mechanism to “solve the problem of Boltzmann Brains” in cosmology. The setting is the ΛCDM model of cosmology, which is a little different to Boltzmann’s model of the universe, but suffers a similar problem: in the case of ΛCDM, the universe keeps expanding forever, eventually reaching a maximum entropy state (aka “heat death”) and after which Boltzmann Brains have as much time as they need to fluctuate randomly out of the thermal noise. Such a model, argues Carroll, would imply that it is overwhelmingly likely that we are Boltzmann Brains.
Why is this a problem? Carroll puts it down to what he calls “cognitive instability”. Basically, the argument goes like this. Suppose you believe in a model of cosmology that has Boltzmann Brains. Then you should believe that you are most likely to be one of them. But this means that your reasons for believing in the model in the first place cannot be trusted, since they are not based on actual scientific evidence, but instead simply fluctuated into your brain at random. In essence, you are saying `based on the evidence, I believe that I am an entity that cannot believe in anything based on what it thinks is evidence’. A cognitively unstable theory therefore cannot both be true and be justified by observed evidence. Carroll’s solution to this problem is to reject the model in favor of one that doesn’t allow for the future existence of Boltzmann Brains.
Poor Carroll has taken a beating over at other blogs. Luboš Motl provided a lengthy response filled with the usual ad-hominems:
`…I really think that Carroll’s totally wrong reasoning is tightly linked to an ideology that blinds his eyes. As a hardcore leftist […] he believes in various forms of egalitarianism. Every “object” has the same probability.’
Jacques Distler wrote:
`…This is plainly nuts. How can a phase transition that may or may not take place, billions of years in the future, affect anything that we measure in the here-and-now? And, if it doesn’t affect anything in the present, why do I &#%@ care?’
Only Mark Srednicki seemed able to disagree with Carroll without taking the idea as a personal affront; his level-headed discussions with Carroll and Distler helped to clarify the issue significantly. Ultimately, Srednicki agrees with the conclusions of both Motl and Distler, but for slightly different reasons. The ensuing discussion can be summarized something like this:
Distler: A model of the universe does not allow you to make predictions by itself. You also need to supply a hypothesis about where we exist within the universe. And any hypothesis in which we are Boltzmann Brains is immediately refuted by empirical evidence, namely when we fail to evaporate into radiation in the next second.
Motl: Yep, I basically agree with Distler. Also, Boddy and Carroll are stupid Marxist idiots.
Srednicki: Hang on guys, it’s more subtle than that. Carroll is simply saying that, in light of the evidence, a cosmological model without Boltzmann Brains is better than one that has Boltzmann Brains in it. Whether this is true or not is a philosophical question, not something that is blindingly obvious.
Distler: Hmmpf! Well I think it is blindingly obvious that the presence or absence of Boltzmann Brains has no bearing on choosing between the two models. Your predictions for future events would be the same in both.
Srednicki: That’s only true if, within the Boltzmann Brain model, you choose a xerographic distribution that ensures you are a non-Boltzmann-brain. But the choice of xerographic distribution is a philosophical one.
Distler: I disagree – Bayesian theory says that you should choose the prior distribution that converges most quickly to the `correct’ distribution as defined by the model. In this case, it is the distribution that favours us not being Boltzmann Brains in the first place.
(Meanwhile, at Preposterous Universe…)
Carroll: I think that it is obvious that you should give an equal credence to yourself being any one of the observers in a model that have identical previous experiences as you. It follows that a model without Boltzmann Brains is better than a model with Boltzmann Brains due to cognitive instability.
Srednicki: Sorry Carroll – your claim is not at all obvious. It is a philosophical assumption that cannot be derived from any laws within the model. Under a different assumption, Boltzmann Brains aren’t a problem.
There is a potential flaw in the Distler/Motl argument: it rests on the premise that, if you are indeed a Boltzmann Brain, this can be taken as a highly falsifiable hypothesis which is falsified one second later when you fail to evaporate. But strictly speaking, that only rules out a subset of possible Boltzmann Brain hypotheses – there is still the hypothesis that you are a Boltzmann Brain whose experiences are indistinguishable from yours right up until the day you die, at which point they go `poof’, and the hypothesis that you are one of these brains is not falsifiable. Sure, there are vastly fewer Boltzmann Brains with this property, but in a sufficiently long-lived universe there are still vastly more of them than the `real you’. Thus, the real problem with Boltzmann Brains is not that they are immediately falsified by experience, but quite the opposite: they represent an unfalsifiable hypothesis. Of course, this also immediately resolves the problem: even if you were a Boltzmann Brain, your life will proceed as normal (by definition, you have restricted yourself to BB’s whose subjective experiences match those of a real person’s life), so this belief has no bearing on your decisions. In particular, your subjective experience gives you no reason to prefer one model to another just because the former contains Boltzmann Brains and the latter doesn’t. One thereby arrives at the same conclusion as Distler and Motl, but by a different route. However, this also means that Distler and Motl cannot claim that they are not Boltzmann Brains based on observed evidence, if one assumes a BB model. Either they think Boddy and Carroll’s proposed alternative is not a viable theory in it own right, or else they don’t think that a theory containing a vast number of unfalsifiable elements is any worse than a similar theory that doesn’t need such elements, which to me sounds absurd.
I think that Mark Srednicki basically has it right: the problem at hand is yet another example of anthropic reasoning, and the debate here is actually about how to choose an appropriate `reference class’ of observers. The reference class is basically the set of observers within the model that you think it is possible that you might have been. Do you think it is possible that you could have been born as somebody else? Do you think you might have been born at a different time in history? What about as an insect, or a bacterium? In a follow-up post, I’ll discuss an idea that lends support to Boddy and Carroll’s side of the argument.
 If you want to read about anthropic reasoning from somebody who actually knows what they are talking about, see Anthropic Bias by Nick Bostrom. Particularly relevant here is his discussion of `freak observers’.