Frequentist here. This paper makes me hate Bayesians a little less. The reason is because a general thrust of the paper (since I have only had time to give it a once-over) seems to be that just because you are a Bayesian it doesn't mean that you have to get rid of model adequacy checks. Not having model adequacy checks is why I think Bayesians run around with a magic wand saying, "poof! there's an optimal model." After proving a theoretical optimality they never check to see if the real world data supports their arguments. So I'm glad to see a prominent Bayesian saying that you don't have to throw model checking out the window.
On a secondary note, I have to lament the use of philosopy in a math paper. I realize that many prominent mathematicians are/were also philosophers and that the two subjects are somehow linked at some level. But really I think that putting philosophy in a math paper is an excuse to use more big words and sound smart. Most of us would like to have a set of formulas to apply and not worry about-- forgive me if I for what I'm about to say-- fuzzy non-science like philosophy and the implications that it might have on our cold hard numbers.
Can each 31 page paper that combines math with philosophy come with a 5 page companion paper that leaves out the philosopy and just has the applicable math stuff?