## 13 September 2009

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

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

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)

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.

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

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

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

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.

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

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