Thanks for writing this. I don't have a solution but I'm just registering that I would expect plenty of rejected applicants to feel alienated from the EA community despite this post.
It's just an informal way to say that we're probably typical observers. It's named after Copernicus because he found that the Earth isn't as special as people thought.
Very nice list!
Hmmm isn't the argument still pretty broadly applicable and useful despite the exceptions?
If you want a single source, I find the 80000 hours key ideas page and everything it links to quite comprehensive and well written.
Like most commenters, I broadly agree with the empirical info here. It's sort of obvious, but telling others things like "don't go out of your way to use less plastic" or even just creating unnecessary waste in a social situation can be inconsiderate towards people's sensibilities. Of course, this post advocates no such thing but I want to be sure nobody goes away thinking these things are necessarily OK.
(I was recently reminded of a CEA research article about how considerateness is even more important than most people think, and EAs should be especially careful because their behavior reflects on the whole community.)
On second thoughts, I think it's worth clarifying that my claim is still true even though yours is important in its own right. On Gott's reasoning, P(high influence | world has 2^N times the # of people who've already lived) is still just 2^-N (that's 2^-(N-1) if summed over all k>=N). As you said, these tiny probabilities are balanced out by asymptotically infinite impact.
I'll write up a separate objection to that claim but first a clarifying question: Why do you call Gott's conditional probability a prior? Isn't it more of a likelihood? In my model it should be combined with a prior P(number of people the world has). The resulting posterior is then the prior for further enquiries.
The diverging series seems to be a version of the St Petersburg paradox, which has fooled me before. In the original version, you have a 2^-k chance of winning 2^k for every positive integer k, which leads to infinite expected payoff. One way in which it's brittle is that, as you say, the payoff is quite limited if we have some upper bound on the size of the population. Two other mathematical ways are 1) if the payoff is just 1.99^k or 2) if it is 2^0.99k.
If you're just presenting a prior I agree that you've not conditioned on an observation "we're very early". But to the extent that your reasoning says there's a non-trivial probability of [we have extremely high influence over a big future], you do condition on some observation of that kind. In fact, it would seem weird if any Copernican prior could give non-trivial mass to that proposition without an additional observation.
I continue my response here because the rest is more suitable as a higher-level comment.