komponisto:

In my view, if someone's numbers are wrong, that should be dealt with on the object level (e.g. "0.001 is too low", with arguments for why), rather than retreating to the meta level of "using numbers caused you to err".

Trouble is, sometimes numbers can be not even wrong, with their very definition lacking logical consistency or any defensible link with reality. It is that category that I am most concerned with, and I believe that it sadly occurs very often in practice, with entire fields of inquiry sometimes degenerating into meaningless games with such numbers. My honest impression is that in our day and age, such numerological fallacies have been responsible for much greater intellectual sins than the opposite fallacy of avoiding scrutiny by excessive vagueness, although the latter phenomenon is not negligible either.

You also do admit in the end that fear of poor calibration is what is underlying your discomfort with numerical probabilities:

Here we seem to be clashing about terminology. I think that "poor calibration" is too much of a euphemism for the situations I have in mind, namely those where sensible calibration is altogether impossible. I would instead use some stronger expression clarifying that the supposed "calibration" is done without any valid basis, not that the result is poor because some unfortunate circumstance occurred in the course of an otherwise sensible procedure.

There is such a thing as a poorly-calibrated Bayesian; it's a perfectly coherent concept. The Bayesian view of probabilities is that they refer specifically to degrees of belief, and not anything else.

As I explained in the above lengthy comment, I simply don't find numbers that "refer specifically to degrees of belief, and not anything else" a coherent concept. We seem to be working with fundamentally different philosophical premises here.

Can these numerical "degrees of belief" somehow be linked to observable reality according to the criteria I defined in my reply to the points (1)-(2) above? If not, I don't see how admitting such concepts can be of any use.

If my internal "Bayesian calculator" believes P(X) = 0.001, and X turns out to be true, I'm not made less wrong by having concealed the number, saying "I don't think X is true" instead. Less embarrassed, perhaps, but not less wrong.

But if you do this 10,000 times, and the number of times X turns out to be true is small but nowhere close to 10, you are much more wrong than if you had just been saying "X is highly unlikely" all along.

On the other hand, if we're observing X as a single event in isolation, I don't see how this tests your probability estimate in any way. But I suspect we have some additional philosophical differences here.