This piece nicely spells out the Boltzmann-brain paradox:
if we are explaining our low-entropy universe by appealing to the anthropic criterion that it must be possible for intelligent life to exist, quite a strong prediction follows: we should find ourselves in the minimum possible entropy fluctuation consistent with life's existence.
And that minimum fluctuation would be "Boltzmann's Brain." Out of the background thermal equilibrium, a fluctuation randomly appears that collects some degrees of freedom into the form of a conscious brain, with just enough sensory apparatus to look around and say "Hey! I exist!", before dissolving back into the equilibrated ooze.
You might object that such a fluctuation is very rare, and indeed it is. But so would be a fluctuation into our whole universe - in fact, quite a bit more rare. The momentary decrease in entropy required to produce such a brain is fantastically less than that required to make our whole universe. Within the infinite ensemble envisioned by Boltzmann, the overwhelming majority of brains will find themselves disembodied and alone, not happily ensconsed in a warm and welcoming universe filled with other souls.
Since we don't find ourselves to be isolated, disembodied minds, the Boltzmann-brain scenario would presumably explain our memories of the past and our sense of embodiment as temporary illusions, rather than veridical sensory information. An interesting consequence of this, however, is that the Boltzmann-brain hypothesis becomes somewhat epistemologically self-defeating, in the following sense. If I am a Boltzmann brain, the probability is extremely low that my randomly determined illusory view of the way the universe works will happen to correspond with reality. Yet it's this view of the world (that consciousness arises from particular orientations of matter, that random quantum fluctuations occur, etc.) on which the Boltzmann-brain hypothesis is based.
We can think about this more formally. Take our sample space to be some set of possible worlds, and let T be the subset of worlds in which the underlying assumptions behind Boltzmann brains hold (e.g., the second law of thermodynamics, that configurations of matter give rise to consciousness, etc.). Let B be the subset of T in which I am a Boltzmann brain, and let U be the subset of T in which I am part of an ordinary complete universe of the type we normally envision. Let E be the subset of worlds in which I observe the evidence that led people to imagine Boltzmann brains in the first place (i.e., worlds in which I observe that the second law of thermodynamics holds, and so on). The point made above was that P(E | B) is quite low--it's the probability that a random fluctuation-created brain will imagine that it observes E, out of the vast space of possible imagined observations. On the other hand, P(E | U) is quite high, perhaps close to 1 depending on how exactly the set U is defined.
Nevertheless, Boltzmann brains needn't be defeated by this. From Bayes' theorem we have
P(B | E) / P(U | E) = [ P(E | B) / P(E | U) ] * [ P(B) / P(U) ].
The term P(E | B) / P(E | U) is, as we saw above, roughly the chance that a random brain will think it observes E. On the other hand, P(B) / P(U) is roughly the number of Boltzmann brains created randomly divided by the number of brains in normal universes created randomly. My impression is that, under several models, this ratio is far bigger than the inverse of P(E | B) / P(E | U), so that P(B | E) / P(U | E) remains far bigger than 1.
However, there are theories that predict a smaller value for P(B) / P(U). For instance, with inflation, whole universes can arise from small arrangements of matter, so that entire-universe-creating fluctuations may be more probable than brain-creating fluctuations. Also, in this paper, Andrei Linde shows that certain "probability measures for eternal inflation" avoid the Boltzmann-brain problem. I'm sure there's additional literature on this subject of which I'm unaware.
Are there implications for utilitarianism? One point is that if we are Boltzmann brains, there would seem to be little we could do help others (though perhaps this wouldn't be true if we were random fluctuations bigger than brains but smaller than entire universes). In view of this, we might adopt the principle (overly simplistic, I'm sure) that we ought to act as though we're not Boltzmann brains because only in this case will our actions make a difference. Even maintaining this assumption, we needn't believe that Boltzmann brains don't exist, but our probability distribution for the number of Boltzmann brains in existence conditional on our not being one of them should have fairly low expectation.
An interesting question is, What's the probability distribution over utility levels for Boltzmann brains? The mean of the distribution might be near zero, or perhaps quite negative if we weight disutility more heavily. This has implications for the ethics of creating lab universes, among other things. |
