Jesse,
>Stathis Papaioannou wrote:
>
>>Now, look at p(n) again. This time, let's say it is not k, but a random
>>real number greater than zero, smaller than 1, with k being the mean of
>>the distribution. At first glance, it may appear that not much has
>>changed, since the probabilities will "on average" be the same, over a
>>long time period. However, this is not correct. In the above product, p(n)
>>can go arbitrarily close to 1 for an arbitrarily long run of n, thus
>>reducing the product value arbitrarily close to zero up to that point,
>>which cannot subsequently be "made up" by a compensating fall of p(n)
>>close to zero, since the factor 1-p(n)^(2^n) can never be greater than 1.
>>(Sorry I haven't put this very elegantly.)
>
>p(n) *can* go arbitrarily close to 1 for an arbitrarily long period of
>time, but you're not taking into the account the fact that the larger the
>population already is, the more arbitrarily close to 1 p(n) would have to
>get to wipe out the population completely--and the more arbitrarily close a
>value to 1 you pick, the less probable it is that p(n) will be greater than
>or equal to this value in a given generation. So it's still true that the
>probability of the population being wiped out is continually decreasing as
>the population gets larger, which means it's still plausible there could be
>a nonzero probability the population would never be wiped out--you'd have
>to do the math to test this (and you might get different answers depending
>on what probability distribution you pick for p(n)).
>
>It also seems unrealistic to say that in a given generation, all 2^n
>members will have the *same* probability p(n) of being erased--if you're
>going to have random variations in p(n), wouldn't it make more sense for
>each individual to independently pick a value of p(n) from the probability
>distribution you're using? And if you do that, then the larger the
>population is, the smaller the average deviation from the expected mean
>value of p(n) given by that distribution.
>
>>The conclusion is therefore that if p(n) is allowed to vary randomly, Real
>>Death becomes a certainty over time, even with continuous exponential
>>growth forever.
>
>I think you have any basis for being sure that "Real Death becomes a
>certainty over time" in the model you suggest (or the modified version I
>suggested above), not unless you've actually done the math, which would
>likely be pretty hairy.
>
>Jesse
>
Jesse,
It would be stubborn of me not to admit at this point that you have defended
your position better than I have mine. I'm still not quite convinced that
what I have called p(n) won't ultimately ruin the model you have proposed,
and I'm still not quite convinced that, even if it works, this model will
not constitute a smaller and smaller proportion of worlds where you remain
alive, over time; but as you say, I would have to do the maths before making
such claims. I may try out some of these ideas with Mathematica, but I
expect that the maths is beyond me. Anyway, thank-you for a most interesting
and edifying discussion!
--Stathis Papaioannou
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Received on Sat Apr 23 2005 - 00:38:23 PDT