[ithinkso]

> Moreover, mathematicians rarely prove anything from axioms, they start from other statements which are considered more trivial.

Not more trivial but from the ones already proved and those closer to the thing you are proving. There is no need to go back to the axioms if you know, and can reference, proof of step N-1, just go from there.

There is also this 'misconception' that mathematical theorems follow from the axioms. They do, of course, but the axioms were choosen just right to make things that were working to still work, with some weird consequences like axiom of choice

[cubefox]

There are are a lot of "folk theorems" in mathematics, and things which are simply considered obvious or common knowledge relative to a conjecture in question, without anyone being able to actually cite a proof for that. Mathematical proof is really just for convincing other mathematicians, there is no need to prove things which are considered obvious.

[ithinkso]

None of what you said is true except for

> Mathematical proof is really just for convincing other mathematicians

which is precisely the point of a proof, but 'convince' means something different than you seem to think it means (in the field of mathematics)

[constantcrying]

>There are are a lot of "folk theorems" in mathematics

Can you name some? Preferably in analysis.

[rdlw]

I found this, unfortunately it is about category theory: "manuscripts [...] that are cited but never became widely available"

https://ncatlab.org/nlab/show/list+of+lost+manuscripts+in+ca...

[constantcrying]

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?

[rdlw]

From the link I posted:

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

[constantcrying]

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.