Given that addition and multiplication are both commutative and associative, Expressions containing only addition or multiplication can be evaluated in a any order. However, when the operation changes from addition to multiplication or from multiplication to division it is necessary to specify which operation to perform first because multiplication distributes over addition.
This seems somewhat unsatisfactory to me because it relies on the assumption that there is no actual preferred order of evaluation built into the real numbers. I was wondering if I might have some help fleshing this out in a way that explicitly addresses the putative assumption that there is no explicit order of operations on the real numbers.
Starting with the field axioms, can one prove that there is a single interpretation for unbracketed real-number expressions?