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.
Best Answer
There is no such proof. The order of operations to which we are accustomed is really nothing more than a mathematical convention to which most adhere in order to help eliminate the alternative of ambiguity.
But it never hurts to use parentheses to designate operations to perform first (inner to outer), which is virtually universally understood, thus eliminating our reliance on convention in the hopes that others will know the convention!