Um, I think we're agreeing. The expression is not arbitrary, it only has one correct answer. We agree on that. I'm saying that using PEMDAS is an arbitrary convention. If we all agreed to rewrite our equations in PEASMD, it would be ugly, you'd probably need more parentheses, but it would still work. People in this thread have used set theory to explain that PEMDAS makes more sense, and it totally does, but it doesn't strictly have to be that way.

I'm actually finding two different definitions of arbitrary on the Internet: 1. Based on individual discretion, 2. Random. I had the first in mind.