Its been a while since I've posted. Apologies for that. I have been busy and normal posts will resume soon. In the mean time I thought I'd give you a discussion that I've been having with"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 Boltzmann brains and yet your probability of being one of them could still be negligible.
Also can I recommend you read my post on why the heat death isn't so obvious."
Now I made a mistake here. I missedif they believed that the universe arose in a random fluctuation. . This does make things different. As I'm not committed to that assumption though it doesn't effect my argument against the above version of the Boltzmann brain problem. It also doesn't effect the discussion regarding scientific progress and the second law of thermodynamics that follow.
Never the less the resulting discussion was interesting. Dwielz Camauf Descartes responded with this post.
My responses follow:
Firstly you say:
"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 information about it we generally assume a uniform distribution; he needs to provide evidence for his assertion that a skewed distribution is the default."
No there are skewed distributions just no uniform ones. I can prove it rigorously as follows (I am a mathematician so it would be embarrassing to be wrong on this!):
Probability distributions integrate to 1. A uniform distribution on the positive real line has the same integral over any unit interval. So the probability of being in the interval n to n+1 is the same as being in the interval m to m+1. Call that probability 'q'. Then we get the following infinite sum:
1= P(x between 0 and 1)+P(x between 1 and 2)+P(x between 2 and 3)+...
which simplifies to
1 = q+q+q+q...
But this equation cannot be defined for any value of q. Therefore there is no uniform distribution on the positive real line.
Indeed it is clear that the limit as n tends to infinity of P(x between n and n+1) must be 0 because the equation could not be satisfied otherwise.
This proves that a probability distribution on the positive integers will always be skewed to the origin (at some scale). Of course one does not know the appropriate scale but this does not help your argument I'm afraid.
Again regarding my probabilistic argument you suggest that it wouldn't work in a metaverse.
If the metaverse is expressible as a subset of some finite dimensional space then the same reasoning applies (with unit blocks of the appropriate dimension taking the place of unit intervals). If not then you would need to specify some meaning to the term "probability measure" and then make your argument. I very much doubt though that this would save the argument.
Finally regarding your response to my arguments regarding the heat death:
The problem with using induction here is that we just don't have enough information to even begin guessing. We have less than a millennium (10^3) of proper scientific investigation of physics. But in predicting a heat death we are making predictions about events 10^40 years in the future.
Using induction here would be like picking up a single grain of sand at a beach and concluding that the entire solar system was made of silicon dioxide!
Don't get me wrong I'm all for induction but I'm just pointing out that we're effectively guessing in the dark when it comes to questions of this scale.
Finally (oops I thought I was finished) you accuse me of not trusting in scientific progress and ask what I mean by progress.
But what I've said isn't really a criticism of science. I'm only saying science cannot yet answer certain questions including the plausibility of a heat death. Given the scale of the question that's no real surprise.
I provisionally define progress to include (but not be limited to) increasing knowledge of strategies that allow multiple parties to achieve mutual benefit from non-zero sum games. That is part of what moral progress means in my view. This form of progress is certainly worth having and I think most people would have a great interest in it occurring.
It occurs to me that it would have been better to post these comments together as a blog post on my site. I shall do that and I suggest you delete the comments here for clarity. I'll post a link to the page once its up.
Cheers for an interesting discussion!