This seems a deep insight simple enough to explain in a blog post (and so I’m probably not the first to see it): the self-indication approach to indexical uncertainty solves the time-asymmetry question in physics! To explain this, I must first explain time-asymmetry and indexical uncertainty.
A deep question in physics is time asymmetry – why doesn’t stuff happen as often "backwards" in time? We have no idea about the tiny CP-violation in particle physics, but all the other time asymmetries are thought to arise from a very-low early-universe entropy. The most popular explanation for this is inflation, especially eternal inflation, which says that any small space-time region satisfying certain conditions is connected to infinitely many large time-asymmetric regions much like what we see around us. Alas, the chance that any small region satisfies these inflation conditions is extremely small. As a recent paper puts it:
Initial conditions which give the big bang a thermodynamic arrow of time must necessarily be low entropy and therefore "rare." There is no way the initial conditions can be typical, or there would be no arrow of time, and this fact must apply to inflation and prevent it from representing "completely generic" initial conditions. … If you can regard the big bang as a fluctuation in a larger system it must be an exceedingly rare one to account for the observed thermodynamic arrow of time.
So the question of time-asymmetry reduces to this: why does the universe have enough independently variable small regions that at least one of them gives eternal inflation? That is: why is the universe so big?
Indexical reasoning is about where we are in space-time. Even if we knew everything about what will happen where and when in the universe, we could still be uncertain about where/when we are in that universe. To reason about this form of uncertainty we need something equivalent to a prior which says where/when we should expect to find ourselves, if we knew the least possible about that topic. This indexical prior over locations in a universe must be combined with a prior over possible universes to give a total prior over where we might find ourselves.
For a particular universe, if we would have been equally likely to find ourselves as any of the observer-moments (e.g., "me now") in that universe, then we should expect to find ourselves in the parts of that universe with the most such moments. Among humans on Earth, for example, you should expect to find yourself in the eras and nations containing the most people. The self-indication approach to indexical uncertainty says that this same reasoning also applies to possible universes: not only should you expect more to find yourself in universes that are more likely to exist, but you should in addition expect more to find yourself in universes that have many slots for creatures like you. For example, you should expect to be in a populous era in a big nation in a big long-lasting universe. (This approach favors thirders in the sleeping beauty problem.)
The self-indication approach says that if very large universes are at all possible, you should think yourself very likely to be in one. In the limit, if there is a non-zero probability of situations where the universe has infinitely many slots for creatures like you, you should be almost certain you are in that sort of situation. This may seem arrogant, and Nick Bostrom argues it is unreasonably presumptuous. Nevertheless, it does handily answer the key time-asymmetry question: why is the universe so big? (And it is more presumptuous than being sure to be in an infinite, vs. finite, part of an infinite universe?)
That is, given self-indication we should expect to be in a finite-probability universe with nearly the max possible number of observer-moment slots. Such universes seem large enough to have at least one inflation origin, which then implies at least one (and perhaps infinitely many) large regions of time-asymmetry like what we see around us. And if, as it seems, most observer-moments in such universes are in such regions, then we have explained why we see what we see.
I found the self-indication approach pretty persuasive on theoretical grounds before, and now I’m all the more sold. If that is presumptuous, so be it – I bite bullets. But do I contradict myself?