On Thu, Apr 08, 1999 at 08:51:01PM +0000, Nick Bostrom wrote:
> The idea is simple, but it is not so obvious that it is right. I've
> been thinking about the possibility of using nonstandard analysis to
> deal with these cases. One would assign an infinitessimal prior
> probability to each of the hypothesis "I am the n:th human".
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?
> But think about what "a preferred point" means in this context. It
> means that you are more likely to find yourself near spatial point A
> than B, even though there may be somewhat more people around B.
> Presumably the preferred point is the same for everybody, since you
> say they are like physical constants. So everybody should think they
> are probably around B. But if everybody follows this recommendation,
> there will be more people who are wrong than if they just use the
> ordinary self-sampling assumption and think they are more likely to
> be around A. And all this would be known to everybody. What
> justtification, then, is there for following your rule?
If nonstandard analysis works, there is no justification, otherwise I would say the justification is that there is no alternative.
> 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.)
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. If the universe is spatially but not temporally homogeneous, a combination of nonstandard analysis and preferred time may be sufficient.