Why worry about commutativity but not associativity in The Fundamental Theorem of Arithmetic? A common statement of The Fundamental Theorem of Arithmetic goes:

Every integer greater than $1$ can be expressed as a product of powers of distinct prime numbers uniquely up to a reordering of the factors.

Now the statement makes a point of mentioning that factorization is unique up to reordering of the factors, saying basically that we don't have to worry about it because multiplication in the integers is commutative. But why not specify that it's also unique up to the choice in which order we multiply the factors? I.e, that we don't have to worry about it because multiplication in the integers is associative too? If we insist on multiplication being a binary operation, then we need to define some grouping when we have a product of more than two integers. Shouldn't there be a clause in the Fundamental Theorem that indicates, for example, that $30 = (2\times (3 \times 5))$ and $30 = ((2\times 3) \times 5)$ are not distinct factorizations? ​

It should be noted that some answers to this question were merged from another question, so they may not be completely consistent with this question exactly as it's stated.

While being sloppy, mathematicians create an intuitive and simple language that allows deeper and deeper investigations in an extraordinary creative activity. Keeping all parenthesis and avoid all informal constructions would make mathematics more static.
Of course one can define a normal form which cooperates with the idea of the Fundamental Theorem of Arithmetic. 
