Any particular theorem? This links to hard/impossible to access materials, but does any of them actually contain a theorem which proof is not accessible in some form?
Further is it any relevant theorem, which has been cited many times?
John Beck's Monadicity Theorem is cited in "many sources" but a manuscript has been found and "The proof is reproduced for instance in (MacLane, p. 147-150, Riehl 2017, 5/5/)".
Fred Linton cites Michael Barr for a "Universal property of the Kleisli construction" and no copy has been found.
I don't know enough about the subject to say how difficult these proofs would be to recreate or whether they exist elsewhere, but these definitely seem to be specific theorems.
I'd like to see a specific theorem in any field that was once proven and accepted but all existing proofs have been lost. That would be extremely tantalizing
I don't think this is convicing at all. None of this shows that there are generally accepted theorems without proofs.
The first one isn't lost at all, the second is just a single lost citation.
All of this is very far away from mathematicians building upon thr shaky grounds of mythical theorems. Also, in all my mathematical experience (going up to recent research in analysis) I never encountered something like this. In all textbooks I ever read either everything was proven or was cited to other works, which every time I checked included the proof.
"Everything was proven" ... everything to satisfy a mathematician, not a logician. Formal proofs are generally very unwieldy, because informal proofs treat all the most obvious parts as obvious. In some cases these parts are not even trivial to prove formally. Last time I checked there was still no formal version of Cantor's diagonalization argument.
>because informal proofs treat all the most obvious parts as obvious.
No, they treat the most obvious parts as previously proven (which they are). That is an enormous difference.
I really have no idea what the difference to a "fully formal" and the standard proof would be, except the later argument includes verbose commentary, instead of logical symbols. But saying "since a is true and we formerly proved a implies b, we know b is true" instead of "(a and (a => b)) => b", does only seem to be a pedantic difference.
>Last time I checked there was still no formal version of Cantor's diagonalization argument.
AFAIK it is part of the standard libraries of multiple proof assistants, which means it is as fully formal as it can ever be.
https://ncatlab.org/nlab/show/list+of+lost+manuscripts+in+ca...