system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: How did we decide in PEMDAS as the proper order of operations?
gpt
Parenthesis go first because you need a way to specify an order when the default order is incorrect (or you just want to be explicit).

From there it's entirely convention, but the convention isn't without reason.  One thing that winds up showing up a lot in mathematics is a polynomial, such as x^2 - 4x + 3.  Ideally you want to be able to write this without parenthesis, which requires that exponentiation goes first, followed by multiplication and division (at the same priority) and finally addition and subtraction (at the same priority).

The decision to make addition and subtraction the same priority (and therefore executed left to right) as well as multiplication and division because ultimately these are just the same operation with a slight twist.  If you have A - B then you really have A + (-B), just as if you have A / B you have A \* (1/B).