Priority of operations: Necessary or arbitrary?

We reflect here on the priority of operations, where an expression such as 3 + 4 × 5 is taken to be equal to 3 + 20 = 23, rather than to 7 × 5 = 35. In this example, the multiplication takes priority over the addition, meaning that it is carried out first, even though, when reading left to right, the addition operation is encountered first. In order to teach the priority of operations, we see it as important to establish whether it is ‘merely’ a common convention – something that could be otherwise – or whether it follows mathematically and ‘can be proven’. We were surprised to discover that the mathematics education literature appears to be unclear on this point. [...]