Practically everytime I read a theorem proof in a book, paper, slides, website, pdf, anywhere, the proof is written using natural language (e.g. sentences in English) and sometimes drawings (e.g. 2D drawings if it's a proof about Geometry, Graph Theory, etc.). Even if the author does her/his best at writing a very technical and formal proof, full of formal symbols, at one point or another they can't help but resort to natural language and/or drawings to get some point across. My question is: does there exist a formal language capable of expressing any mathematical proof without resorting to drawings or natural language sentences?
UPDATE: As @Noah Schweber suggested in the comments to his answer, I should make clear that a very important underlying reason for my interest in formal proofs has to do with the issue of confidence in natural language proofs. That is, if we are not using any formal language to prove most theorems, then how can we be so sure that we have proved anything at all? Because, what if we made some mistake at some point during our natural language reasoning? Since we don't have a full symbolic formal proof, how can we truthfully check that our proof is indeed correct? I think this is a very valid concern considering that natural languages are incredibly prone to ambiguities, which I guess most would agree is an undesirable property when we want to prove mathematical statements with crystal clear certainty.