Much more likely the Boltzmann Brain appeared right now this instant and you, the reader of this, are just hallucinating all of your experiences and will disappear in another nano-second.
That's what makes this something of a paradox. It's far more likely that we are experiencing an illusion in the heat death of the universe than we are experiencing the "original" universe.
Boltzmann's fluctuations are pretty easily defeated by Feynman's argument: small fluctuations are exponentially more likely than large fluctuations, so if we make a new observation (e.g. opening a door) the fluctuation hypothesis predicts that we'll see random noise; yet we don't, we see more ordered structure. The hypothesis is refuted.
What I find more interesting is if we're a random fluctuation on a Turing machine tape. "Randomness" takes a lot of space to encode in a program, so smaller (more likely) programs lead to less random, more structured outputs. This agrees with our observations.
> Boltzmann's fluctuations are pretty easily defeated by Feynman's argument: small fluctuations are exponentially more likely than large fluctuations, so if we make a new observation (e.g. opening a door) the fluctuation hypothesis predicts that we'll see random noise; yet we don't, we see more ordered structure. The hypothesis is refuted.
You've missed the point of Boltzmann brains. No one who studies this stuff thinks we're a Boltzmann brain. The question is whether we can rule out theories that predict Boltzmann brains as the numerically dominant observer with our observations. Indeed, many very popular theories of cosmology (e.g., eternal inflation) are in apparent conflict here.
Maybe. It's highly contested. One popular argument against that claim (which I agree sounds reasonable on its face) is that it turns out to allow us to draw some strong conclusions about the universe which, intuitively, should not be possible to conclude from our armchair (the "presumptuous philosopher").
In the philosophy literature, this often is framed in terms of a choice between the self-indication assumption or the self-sampling assumption.
In a cosmology with Boltzmann brains, your memories of opening doors and not seeing random fluctuations are more likely to be part of a random fluctuation than to be caused by having actually experienced that.
Heck, your moment-to-moment experience of being in a room is extremely unlikely compared to just a temporary Boltzmann brain hallucination upon arrival. No need to go find a door, just wait 10 milliseconds for the "Oh wait actually I'm in space and about to die" sense information to start pouring in. Oh it didn't happen? That's just because you only arrived a millisecond ago and have fake memories of it not happening and are still in the hallucination phase... (repeat indefinitely).
This moment-by-moment thing is what makes Boltzmann brains such a paradox in terms of epistemology. In a cosmology with Boltzmann brains, most of the entities who've just made some conclusion about Boltzmann brains are themselves Boltzmann brains.
Recommended, it's a very interesting reading.