I think you need an additional primer on the topic of: why I should care about "casual emergence". What's the point? How can it be put to use? Started reading your article but couldn't get into it. Is it just a pointless philosophical notion?
I think the third paragraph suggests the point is that it is allegedly a measure for whether a purely reductionist approach is a useful way to explain a given thing or phenomenon. For example, while I have no doubt that the history of life on earth could, in principle, be explained in terms of physics, I find that a Darwinian theory of evolution is a more useful way of looking at it.
Aaronson's argument seems to be that the resulting measure is an unsurprising outcome. I am not sure whether that proves it is irrelevant, but I think the onus is on its proponents to show that this measure goes beyond being trivially true, and is actually useful.
In the conclusion of the article, Hoel seems to be making a stronger claim: "The theory does imply that universal reductionism is false when it comes to thinking about causation, and that sometimes higher scales really do have more causal influence (or information) than whatever underlies them." There seems to be a hint of the motte-and-bailey strategy here, with Hoel defending the narrower claim but expecting us to accept the broader one.
I have not figured out how this measure is actually calculated without having a reductionist measure of the system's information content, so I wonder if it has been created to be used as a pawn in some philosophical argument, perhaps such as the nature of consciousness.
About half way down is a section that starts "But of course this still leaves the question: what is in the mathematical part of Hoel’s Entropy paper? What exactly is it that the advocates of causal emergence claim provides a new argument against reductionism?"
...but basically, I wouldn't bother. I regret the time I wasted deciphering what it was on about already.
