Wei Dai wrote:
> I have to admit that I am not very familiar with nonstandard analysis
> (want to recommend an introductory text?), but I don't see how
> infinitessimal priors can help resolve the self-selection paradoxes.
> Won't you still end up with undefined expectations?
I agree that it doesn't by itself solve all the problems of self-selection. My point was that it is by no means obvious that the right solution is to declare all universes with infinitely many observers logically impossible.
You might want to check out "Learning the impossible" by Vann McGee, chapter 10 in Probability and Conditionals, eds. Eells and Skyrms, 1994. For a concise not-too-mathematical overview of non-standard analysis, I would recommend Appendix 4 in Brian Skyrms' Causal Necessity, 1980. (I haven't read that appendix yet, but Skyrms is good. I am currently working on a theory of chance, i.e. objective probability, that draws a lot of inspiration from chapter 1 of that book, as well as from David Lewis' theory.)
> If nonstandard analysis works, there is no justification, otherwise I
> would say the justification is that there is no alternative.
Couldn't we just say that the rational probabilities in such a case are indeterminate? No preferred position, nor the assumption that all positions should have the same probability (if there are infinitely many observers).
> > Because there is no center of gravity if the universe is spatially
> > infinite (and roughly homogeneous).
> Ok, I see. What I said earlier only makes sense for a universe that is
> spatially infinite but has finite mass. Although for a homogeneous
> universe a preferred position may not be needed for SIA-1 since averages
> do converge in such a universe, and we can define the measure of
> observer-instants as the average density of observers weighted only in the
> time dimension. (A preferred time is still needed because the universe is
> not homogeneous in the time dimension.)
I am still worried about the justificatio of this. I don't see why it should matter how much space there is between observers.
> So let me clarify my current position. I think a nonstandard analysis
> approach is promising, but it can't work by itself unless the universe is
> homogeneous in all dimensions. If the universe is not homogeneous, we
> need a preferred point.
Or we could say that the probabilities are indeterminate; that is the solution that I am leaning towards at the moment.
http://www.hedweb.com/nickb email@example.com Department of Philosophy, Logic and Scientific Method London School of Economics