Is there a way to explain or prove that there is not a single order of operations on the real numbers? 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?
 A: Suppose we restrict ourselves to just multiplication and addition. The distributive law usually reads:
$$a(b+c) = ab+ac$$
Which we understand to mean:
$a(b+c)=(ab)+(ac)$, not $a(b+a)c$ or $(a(b+c))a$. This is as close as I can find to an implied order of operations from the field axioms. Personally I'd argue this is the writers abusing a standard, not intended as an actual definition for order of operations on the reals. Nobody would object to field axioms in post script reading: $$\cdot (a, + (b, c)) = + (\cdot (a ,b), \cdot (a, c)).$$ (parenthesis added for notational prettiness, they aren't required for this to be unambiguous)
A: We could simply say as convention that all operations of more than two operands must absolutely have parenthesis around every pair and that if the parenthesis are missing they expression is meaningless.
So $a + b*c + d*e +f$ (as we know it) must be written as $(((a + (b*c)) + (d*e))+f)$  This is, of course, tedious.
We could have as a convention that we always go from left to right, always.  So the expression $a + b*c + d*e + f$ would now mean $((a+b)*c + d)*e + f$ and to write the expression we meant we'd have to writh $a + (b*c) + (d*e) + f$.
This is perfectly acceptable.
However given that we have the distributive law that $a(b+c) = (ab) + (ac)$ there is an incentive to view $a*c + b*c$ so $(a*c) + (b*c) = (a+b)*c$ rather than as $(a*c + b)*c = a*c^2 + (b*c)$.  Our convention is consistent, less capricious, and usefl and insightful.
