That proof for the order of operations sure seems to rely a lot on our current order of operations
Doesn’t use order of operations at all. It only uses the definitions of the operators. i.e. 3x4=3+3+3+3 by definition. i.e. nothing to do with order of operations.
If I have 1 2l bottle of milk, and 4 3l bottles of milk, how many litres of milk do I have? It can be solved by simply adding them up - again, nothing to do with order of operations here, just simple addition. Now, write it out as a mathematical expression which uses multiplication, and tell me which order of operations gets you the right answer. Voila! Welcome to how we worked out what the order of operations rules had to be.
2+(4x3) gives the right answer, with addition coming before multiplication
If we rewrote all of Maths so that addition came before multiplication, then no, 2+3x4 would no longer mean what it does now (because + and x would have to mean something different to what they do now in order for the order to be swapped). The order of operations rules come directly from the definitions. You can’t just say “we’ll do addition first” without having defined what addition is now, nor multiplication. In a world where addition comes before multiplication, that means multiplication is no longer shorthand for addition (because that’s the very thing which means we have to do multiplication before addition, so it can’t be true anymore if now we’re doing addition first).
Let’s take an imaginary scenario where we now use x for add, and + for multiply. That would indeed mean that + has to be done before x, but note that + now means multiply. That means your “addition first” 2+(3x4) is what we currently mean by 2x(3+4) which is 14. Now take away the brackets (since I don’t use brackets when adding up the milk! Just 2+3x4). Your addition-first 2+3x4 is equivalent in our multiplication-first world to 2x3+4 which equals 10 - the wrong answer! So now you’ve created a world where we have to add brackets to things just to get the right answer! Why would you even want to do that when it works the way it is? The whole point to having order of operations rules is to not have to add brackets!
Doesn’t use order of operations at all. It only uses the definitions of the operators. i.e. 3x4=3+3+3+3 by definition. i.e. nothing to do with order of operations.
If I have 1 2l bottle of milk, and 4 3l bottles of milk, how many litres of milk do I have? It can be solved by simply adding them up - again, nothing to do with order of operations here, just simple addition. Now, write it out as a mathematical expression which uses multiplication, and tell me which order of operations gets you the right answer. Voila! Welcome to how we worked out what the order of operations rules had to be.
2+(4x3) gives the right answer, with addition coming before multiplication
If we rewrote all of Maths so that addition came before multiplication, then no, 2+3x4 would no longer mean what it does now (because + and x would have to mean something different to what they do now in order for the order to be swapped). The order of operations rules come directly from the definitions. You can’t just say “we’ll do addition first” without having defined what addition is now, nor multiplication. In a world where addition comes before multiplication, that means multiplication is no longer shorthand for addition (because that’s the very thing which means we have to do multiplication before addition, so it can’t be true anymore if now we’re doing addition first).
Let’s take an imaginary scenario where we now use x for add, and + for multiply. That would indeed mean that + has to be done before x, but note that + now means multiply. That means your “addition first” 2+(3x4) is what we currently mean by 2x(3+4) which is 14. Now take away the brackets (since I don’t use brackets when adding up the milk! Just 2+3x4). Your addition-first 2+3x4 is equivalent in our multiplication-first world to 2x3+4 which equals 10 - the wrong answer! So now you’ve created a world where we have to add brackets to things just to get the right answer! Why would you even want to do that when it works the way it is? The whole point to having order of operations rules is to not have to add brackets!