It's formal compared to colloquial english, but completely vague compared to computer programs.
There are automatic theorem provers, but it isn't Godel's undecidability theorem that limits them. It's the difficulty of formally writing down our proofs in machine readable format that limits them for practical purposes.
Mathematical proofs are completely precise, that is to say, not vague in the least. They are, however, in an extremely high-level language; it's left to the reader to expand the notation enough to convince themselves of the validity.
(And, of course, there can be bugs - that is, mistakes - but there is no ambiguity.)
It's formal compared to colloquial english, but completely vague compared to computer programs.
There are automatic theorem provers, but it isn't Godel's undecidability theorem that limits them. It's the difficulty of formally writing down our proofs in machine readable format that limits them for practical purposes.