Without a common understanding of what the symbols mean, communication isn't even possible.

A common understanding is a convention. Without conventions communication isn't even possible.

We agree on the meaning of things. We agree to all speak english here, if we didn't agree on that, communication would be quite hard. But we could also have chosen french or even esperanto as long as everyone speaks the same language.

Yet it's done all the time and is completely unambiguous and easy to understand with proper use of bracketing. (2x^2 + 3x - 9) / (ix^2 - 3) is unproblematic.

Is that so? "2x^2" means "(2x)^2" or "2(x^2)"? Of course I know it's the second one, we all know the convention perfectly well and therefore there's no ambiguity.

The problem with a line like "2x/2x" is that no one knows the convention. You say that a convention is unnecessary if we use brackets and I say that brackets are unnecessary with proper convention.

Couldn't we just agree that we need either one?

So BEDMAS is wrong? It's actually BEIDMAS, with the 'I' for 'implicit'?

So says wolfram alpha apparently. It doesn't mean I agree with it. For me every kind of multiplication has same level of priority as division. Without bracketing I go left to right, therefore 2x/2x = x².

If I write something like that just for me or for someone I know uses the same convention I use, there is no problem.

If I write this for someone else, I would be careful. I may use brackets because I know the person I write my line for uses the same convention for the meaning of brackets I use (almost everyone does ^^). But I may also write down my convention. In that very case it's stupid, but there are lots of cases where writing down the convention you use in a math exam is no waste of time.

That would take a while to explain, though there's nothing complicated (or controversial) about it:

http://en.wikipedia.org/wiki/First-order_logic#Syntax

I know how to construct first-order logic, but you're going too far above the question. I was just asking how you build the language you need to do elementary maths, nothing more than division. At that point you can assume logic.