According to my tour guide at the Bodleian Library at Oxford University, preservation of books back in the olden days was all about reducing humidity. And the best way to achieve this before proper isolation were invented was to bring them up as high as possible. Rich people who collected books therefore always kept their books upstairs.
Also the curators of such collections are also often scholars. Old men, going up stairs to read their old books.
An hypothesis on my part of course, but to me I can se how this over generations gets morphed into "wizards in towers".
I agree that fuzzy logic wouldn't work for that purpose. But it addresses a formalism around the foundation of what probabilities are, which to what I could see was something you guys were doing as well. Just a thought.
As for actually addressing logical uncertainty and asymptotic convergence, I think subjective Bayesianism can be used in both cases. For example you write "the axioms of probability theory force you to put probability either 0 or 1 on those statements", which I think is simply not true. If I as an "expert" claimed that "in my experience there is a 70% chance of conjecture being correct", I can set "Prior(conjecture)=0.7".
Secondly, is there a reason why subjective Bayesian theory isn't mentioned? To me it seems obvious that expert elicitation and assigning probabilities to logical uncertainty is perfectly fine.