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The third scenario here is equivalent to the "God tells us we're in one of two universes" experiment. The only difference between the two scenarios is that, in

The third scenario here is equivalent to the "God tells us we're in one of two universes" experiment. The only difference between the two scenarios is that, in the latter, the two universes "exist", but if God had said "there were two universes and I destroyed one; you inhabit the one which remains; which is it?" our answer wouldn't change. The fourth scenario is more interesting because we've admitted the distribution of p is unknown to us. You note that if p is uniform, all intelligent lifeforms should conclude they live in a universe where p is high, but I'm not certain of that. It seems to me that this logic only works if you conclude there is _one_ universe for each possible value of p. But if, for each possible value of p, the multiverse contains universes for each possible outcome of a Bernoulli trial with probability p, then we can't conclude anything at all. To see this, notice that the number of Lucky universes (where p is high) is uncountably infinite, just as the number of Unlucky-Lucky universes (where p is low but a miracle has occurred) is uncountably infinite. I haven't thought this next part through as deeply, but I believe this problem compounds given the unknown distribution of p. This is because subjectivist models of probability (e.g. Bayes' Theorem) are heavily influenced by the distribution of the prior when we have only observed one trial. And in cases like this one, where the only trials that _could_ be observed are successes, I believe it's all but entirely driven by the distribution of the prior. (You could imagine a hyperprior - a distribution of distributions of p, but I don't think we have any good reason for deciding what that would be either). Anyway, thanks for this. I love geeking out about epistemology.

inimino 5y ago
I agree that scenarios two and three are equivalent. But then in three, the fact that we exist is allowed to influence our belief in what the value of a physical constant is. From this contrived example, can't we get to the argument being made in the original post? The fourth scenario, to me, is the closest to our actual situation. The many universes correspond to our uncertainty about p. It doesn't matter whether they are actual universes and we are in one, or whether they are possible universes and we are in one. I wasn't sure why you had the number of universes uncountably infinite. Is it because p is a real number? To make this argument, wouldn't you need to show that not only are they both uncountably infinite, but that they have equal measure? We can construct two sets that are both uncountably infinite, but one has measure 1 and the other measure 0. If the Lucky and Unlucky-Lucky universes have equal measure, there's still more people in the Lucky ones. Only if the measure of Unlucky ones is much higher, then there would be more people in the Unlucky-Lucky universes. We said we don't know anything about the distribution. But! In my opinion, we do have some idea about the distribution. It's just our prior, from whatever we know of chemistry, biology, etc, about p. We have some reasons to believe it's low. (The arguments the intelligent design folks make.) Then we update that with the observation that we exist. In other words, we start with a distribution over possible universes based on what we know about physics, and the complexity that's necessary for life, and we say that it's remarkable that we are here. But then we update by saying that, of all beings capable of reasoning through this, more of them would be in universes where it's less remarkable. So it doesn't tell us what p is, but it should shift our distribution over values of p to the right. Eventually, we may get such a clear understanding of how life must have formed, that we could say exactly what steps had to happen and in what order, and how probable they all were, and could determine p exactly, and even simulate everything. Then this argument wouldn't buy us anything. Because the physics is what it is, and p is what it is. It's only because our prior over p is currently so vague, that the argument is interesting. Anyway, it's an interesting discussion and you may be completely right. I'm certainly not certain of anything here :-)

inimino 5y ago

I agree that scenarios two and three are equivalent. But then in three, the fact that we exist is allowed to influence our belief in what the value of a physical constant is. From this contrived example, can't we get to the argument being made in the original post? The fourth scenario, to me, is the closest to our actual situation. The many universes correspond to our uncertainty about p. It doesn't matter whether they are actual universes and we are in one, or whether they are possible universes and we are in one. I wasn't sure why you had the number of universes uncountably infinite. Is it because p is a real number? To make this argument, wouldn't you need to show that not only are they both uncountably infinite, but that they have equal measure? We can construct two sets that are both uncountably infinite, but one has measure 1 and the other measure 0. If the Lucky and Unlucky-Lucky universes have equal measure, there's still more people in the Lucky ones. Only if the measure of Unlucky ones is much higher, then there would be more people in the Unlucky-Lucky universes. We said we don't know anything about the distribution. But! In my opinion, we do have some idea about the distribution. It's just our prior, from whatever we know of chemistry, biology, etc, about p. We have some reasons to believe it's low. (The arguments the intelligent design folks make.) Then we update that with the observation that we exist. In other words, we start with a distribution over possible universes based on what we know about physics, and the complexity that's necessary for life, and we say that it's remarkable that we are here. But then we update by saying that, of all beings capable of reasoning through this, more of them would be in universes where it's less remarkable. So it doesn't tell us what p is, but it should shift our distribution over values of p to the right. Eventually, we may get such a clear understanding of how life must have formed, that we could say exactly what steps had to happen and in what order, and how probable they all were, and could determine p exactly, and even simulate everything. Then this argument wouldn't buy us anything. Because the physics is what it is, and p is what it is. It's only because our prior over p is currently so vague, that the argument is interesting. Anyway, it's an interesting discussion and you may be completely right. I'm certainly not certain of anything here :-)