Linguists distinguish between "descriptive" and "prescriptive" treatments of language. A descriptive treatment is intended to describe the language as it is used in fact, whereas a prescriptive treatment provides rules for how the author thinks it should be used.
This paragraph is mostly descriptive. It is an attempt to describe accurately the language used by mathematicians in communicating mathematical reasoning as well as in other aspects of communicating mathematics, rather than some ideal form of the language that they should use. Occasionally I give opinions about usage; they are carefully marked as such.