My friends and I were arguing about the fundamentality of the order of operations (PEMDAS). They argue that PEMDAS is correct because we said it is, whereas I am inclined to think that regardless of how you represent values, you would be required to follow a congruent order.
Is the fundamentally correct answer of (3 * 2 + 4 / 2) = 8, or is it only 8 because we decided to use PEMDAS? Is, say 5, a valid answer if we had different rules all else equal? To answer this, I was hoping to find out if distinct number/mathematical systems throughout history would reach the same answer for a given expression (as they would express it). I feel that addressing the validity of PEMDAS may be out of the scope of this subreddit, so if you could just answer the latter question, that would be fine (math in history).
What you are asking about are the formalisms of a given mathematical system. These are just the rules you assume to use the system, and are no different than saying that instead of an 8 were are going to use a shape like a G.
In principle you can translate any formalism into any other formalism. In a PEMDAS system we have certain rules about how you express the order of operations; in others you would have others. 5 would not be a valid answer if the original was a PEMDAS system, anymore than saying that the shape for "4" is actually equal to 5 would be allowable (giving a totally different result). You would have to translate the formalism from one to another, just as you would a different language.
For example, Ancient Egyptian mathematics had a very different formal system than the one we are familiar (which is derived from Indian and Arabic mathematics). But you can translate them into our system (in which case it looks like regular math), but you'd necessarily have to translate the order of operations as well. Note that you cannot always translate back into another system, if your current system (which ours does) allows for a wider variety of operations than a previous ones.
Mathematics is a language for expressing logical operations. PEMDAS is valid if everyone agrees to it. The formalism does not address any abstract matter of truth; it is a convention only. (One can argue that mathematics gives us insights into fundamental logical truths, but the formalism does not. To use an analogy, you could argue that philosophy lets us grasp fundamental truths, but it doesn't matter if you use English, German, or Chinese while doing philosophy.)