"There is no uniform probability distribution of the whole real line. Any distribution you use must give greater weight to earlier brains than later ones (in the limit). So there may be infinitely many Boltzman brains and yet your probability of being one of them could still be negligible."If there is no uniform probability distribution of the whole real line, then there surely isn't a skewed distribution that gives "greater weight to earlier brains than later ones", if we don't have evidence that we are more likely to be born (if you are an ordinary observer) or fluctuate into existence (if you are a Boltzmann brain) at one time rather than another we should assume a uniform distribution; he needs to provide evidence for his assertion that a skewed distribution is the default.
CORRECTION: I have since learned that you cannot have a uniform distribution over a countably infinite set and find this extremely disturbing. See my conversation with Barnaby Dawson over at "The Seed Of Reason".
And when we talk about the "real line" we shouldn't be thinking in terms of time in this universe. If our universe arose in a random quantum fluctuation we need to consider that there are doubtlessly other universes out there, and think in terms of whatever dimension distinguishes one universe from another in the multi-verse.
This is because we must consider not only that an infinite number of Boltzmann brains will arise in our universe if it goes into heat death forever, but an infinite number will have arisen in any universe (that continues forever in heat death), and that universe may have arisen enough eons before ours so that the number of Boltzmann brains in it has far overwhelmed the number of ordinary observer in both itself, and in our universe. However, if we look at it relativistically, from outside of our universe, everything has already happened since time is just another dimension that can be used merely to distinguish between things, hence the point may be mute, since the infinite number of Boltzmann brains resulting from a universe that continues forever after heat death are already "there" essentially in the same way that my table exists but I can't interact with it until I cover the distance between it and me. And with current scientific theories the default position will be to assume the universe expands forever, and lasts forever, all evidence points in that direction; although some interesting implications for Boltzmann brains my result from the possibility of "The Big Rip Theory" being true, but I do not have the knowledge of physics to evaluate them so I will pass over that in silence.
But even regardless of that, let us say we can assume that our universe is early. What then follows? How do I know that I am not one the infinitely many future Boltzmann brains? To say that my observations reflect reality, and that I am actually in the early universe I observe begs the question as to whether or not I am a Boltzmann brain.
Now for his post. He make several claims in his post.
He says that heat death is only inevitable:
1: If the universe is finite in space or matter.
2: And there is a lower bound to how much energy is necessary to perform a given calculation.
3: The second law of thermodynamics is not certain to hold.
These are all valid from a theoretical standpoint, but pragmatically they are, in my opinion, very far fetched. Even if the universe is infinite, usable energy is still going to be lost everywhere in that infinite universe. So the more usable energy the universe has, the faster that usable energy will be running out, but by random chance, in an infinite universe, there is always bound to be a part that has some neg-entropy. However, the likely hood that we (or any other civilization) is going to be able to find those random parts, and stay in them (as those parts are going to change) is ridiculously improbable.
As for energy required to perform a given calculation, it is certainly possible (as he argues) that there are an infinite number of elementary particles, getting smaller, and smaller as we can detect more, but it is not at all clear that it will be possible to redesign ourselves (or any other life) to make use of this. The current paradigm has the human race becoming more "advanced" and using MORE energy on a larger scale, not less on a smaller scale.
And finally for number three. The second law of thermodynamics is not certain to hold, in the same way that I am not sure if the sun will rise tomorrow, dark energy could decay, another inflationary Big Bang could wipe out our universe. But I assume the sun will rise tomorrow, because along with every other sane person, and scientist, I use the principle of induction. The default position is to assume that it will hold, unless we are given evidence to contrary. This is why we can have theories about the end of the universe, because we assume that the laws of physics will hold. What Barnaby is doing is misplacing the Burden of proof; no scientist needs to prove that any theory he makes will hold forever into the future. If the theory fits current observations, and is tested, it is accepted, and no one complains about human uncertainty, because that is what science is all about.
And most importantly, if Barnaby wants to doubt that one of the most well established laws of physics will hold by pointing out other failings in science, he must also doubt the continued progress of civilization that he so passionately believes in. We establish that civilization will progress in the same way that we establish that physical laws will hold: through induction. It doesn't matter that civilization hasn't always progressed, or that there have been some failings in scientific history; we can still latch onto the overall trend, and make an assumption about where things are going; in this case, I think it is that civilization will continue to progress (if we don't kill ourselves first), but that the second law of thermodynamics will end that progress eventually, along with all other progress, and life in our universe.
And finally I have a question for Barnaby. How exactly does he judge "progress", and what does it mean to him? Why isn't an empty universe in heat death, with no civilization, no life, and no suffering considered progress? There are many zen masters who would disagree with Barnaby's views. This goes back to the old is-ought problem; what Barnaby means by progress has to be physical and based on our own subjective view of complexity and intelligence. Whatever it is, it can't be moral progress, and hence, no one but a bunch of sci-fi geeks is going to give a rat's rear.