I was watching this YouTube video and at around 40:40 the speaker himself states that he does not know why we have the order of operations we have today. This got me thinking and I realize that I couldn't prove it or even think of a reason for any of it, except for maybe parenthesis because of how it looks when you read it, but that, I think, is based on syntax and semantics and not mathematics.
I tried finding something on the internet but all I get is how it is supposed to be, but not why.
So, is it possible to provide proof for the order of operations in arithmetic, and if so, what is it?
P.S: I can't find any suitable tags for this question so I'm going with arithmetic and math-history.