Boltzman brains, progress and the second law of thermodynamics!

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 Dwielz Camauf Descartes over at The Meatyard.

It started with this post concerning Boltzmann brains.  A Boltzmann brain is a hypothetical observer who appears as a result of a quantum fluctuation (this is incredibly unlikely but not impossible according to some quantum theorists).  The argument about Boltzmann brains which is obliquely referred to is the following:

A random quantum fluctuation only creating a single observer is more likely than one creating a multitude of galaxies in which an multiple observer could evolve.  The universe will suffer a heat death followed by an infinite expanse of time afterwards (this infinite expanse of time is an assumption the argument is meant to be criticizing).  A vast expanse of time occurs after the heat death and though individually unlikely there will be infinitely many Boltzmann brains popping into existence during this time period.  So, you should expect to be a Boltzmann brain as there are infinitely many such observers and only finitely many occurring before the heat death.

My response follows:

"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 missed Dwielz Camauf Descarte's assumption "if 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!

Comments

[Dwielz Camauf Descartes]

Barnaby, I think we are using uniform distribution in two different ways. You seem to be saying that the probability that you are born (or fluctuate into existence) is evenly distributed throughout time (continuous distribution). However this is not true for two reasons:

1 As space expands there will be more room for Boltzmann brains to fluctuate into existence.
2 There may be more observers per amount of time, before heat death due to the fact that evolution may contribute ordinary observers during that period of time.

I at first thought the uniform distribution was refering to the fact that you are equally likely to be any given observer in any universe (throughout time) (a discrete distribution). I now see that a discrete uniform distribution is defined as having a finite number of members, hence what I am refering to can not be described with one, but I don't see any reason why it can't be described at all. (you can't control where you are born or where you fluctuate into existence, hence your probability of coming into existence as any observer is evenly distributed over all possible observers, and there are an infinite number of observers in a universe that lasts forever because of Boltzmann brains. This is rather trivial, and follows from the principle of mediocrity.

If we can't map that out as uniform distribution.... well it still happens, I don't see how it couldn't, if I reach into a vat with an infinite number of marbles, the probability that I draw any individual marble is infinity small but I still can (theoretically) draw a marble.

I'm studying to be an engineer so I didn't get into much of the math theory (I memorized a bunch of formulas and applied them) but can't you get around this problem of 1 not being equal to "q+q+q.." by using limits? I mean if that doesn't work then how on earth does calculus work then? Lets say you have an infinite number of slices underneath a curve all of which have infinitely small areas. But they all add up to the area underneath the curve (which could be 1). So yes 1 can equal q+q+q.... if you define q as 1/n where n is the number of q's and you take the limit as n approaches infinity. And that can be proven using mathematical induction.

I have more to say but I'll have to post it later on my blog. Nice conversing with you.
-Dwielz

[Dwielz Camauf Descartes]

Oh, one more pertinent thing. When I said "if we don't have information about it we generally assume a uniform distribution" this seems to imply the Principle of Indifference, which leads to contradictions.

I should have said "if we don't have evidence that we are more likely to be born (if we are an ordinary observer) or fluctuate into existence (if we are a boltzmann brain) at one time rather than another, we should assume a uniform distribution"
My reasoning is once again coming from the Principle of Mediocrity that we shouldn't consider ourselves special, and that material interactions can replicate our consciousness.

I have since corrected this on my blog. Although, it seems that uniform distributions cannot have infinite members, so what I've said about that is false. Like I said, I'm an engineer, not a mathematician.

It kind of ticks me off though, I see no reason why you can't have a uniform distribution with infinite members (the only difference is that the mean and mode are undefined), but as I've said before, you can use limits to make 1=q+q+q....

And if you don't like infinite uniform distributions, we could just take a number like 1000^1000^1000^1000^1000 and use that in place of infinity, it would basically be the same thing.

-DCD

[Dwielz Camauf Descartes]

Another correction, when I said:
"(the only difference is that the mean and mode are undefined)"

I should say that the mean and mode would be infinitely large for a distribution of the whole real line. (if one could exist)

[Barnaby Dawson]

Your point about the increasing size of the universe over time makes no difference because it just means spreading the proability distribution over a 4D cone rather than a line. That makes no difference to the mathematics. The point is that the universe is infinite in time or space.

The inconsistency of a uniform distribution can't just be ignored. It explains the apparent paradox raised by Boltzmann brains.

If your intuition says that you can have a uniform distribution over an infinite space or an infinite collection of objects then its almost certainly your intuition thats wrong.

[Dwielz Camauf Descartes]

Ok, fine we can't have an infinite distribution; then let us take the real line from 0 to 10000^10000^10000^10000^10000.
Boltzmann brains will surely dominate if we use that number as an upper bound. What say you now? And the greater the number the better our distribution will be, because if will encompass more of the observers that are out there.

And I'm not just using my intuition. Define q as 1/n, define n as the number of q's, and then

n*q=q1+q2+... qn

and n*q=n*1/n=1

Therefore 1=q+q+q...for all n>0

Could you tell me why that is wrong? Thanks.
-DCD

[Dwielz Camauf Descartes]

Oh and let the big number in my last comment be refering to trillions of years :)
-DCD

[Dwielz Camauf Descartes]

And of course for 1=q+q+q.. you would take the limit as n approaches infinity; why can't you do that?

[Barnaby Dawson]

The problem is you just picked an arbitary upper bound. Why not choose 10^11? Then evolved brains would dominate.

You could get around that by picking an upper bound at random. But then you couldn't do that uniformly. You'd be back to precisely the same problem. Whatever you do your distribution contains an arbitrary choice of scale. Furthermore that choice cannot be made probabilistically as that would involve a further arbitrary choice.

This means the probability cannot be calculated objectively. Differing choices of distribution will give you different answers.

Your limit argument doesn't work because in the limit q=0. But the equation 1=0+0+0+0... is incorrect.

[Dwielz Camauf Descartes]

Hmmm, Yeah there's definitely a lot more I need to learn about math theory (the problem with choosing engineering as a major). I've talked to some of my professors about this, and I plan on researching this further, as I find this problem with infinite uniform distributions fascinating. I'll let you know what else I find; I am convinced there must be some way around this. Thank you for taking the time to discuss this further.

I agree that I was wrong on the math part, but I do have some more things to say about the philosophical arguments I have discussed with you that I'll be sure to post later. Thanks again for taking the time to discuss this.

-DCD