The Grammar According to West

You encounter abusive language/notation basically everywhere in math. Open up a calculus/real analysis textbook. A lot of the old ones write sequences in the curly brace/set {x_n} notation:

"let {x_n} be a sequence"

As the author points out, a sequence is a function. The statement {x_n} is the set of terms of the sequence, its range. A function and its range are two different things. And also sets have no ordering. It might seem like a minor thing, but I thought we were trying to be precise?

A second example: at the high school level, I'm pretty sure a lot of textbooks don't carefully distinguish between a function and the formula defining the function very well.

The author of this web page has a section on what he calls "double duty definitions". Personally, I don't find anything wrong with the language "let G=(V,E) be a graph". G is the graph and we're simultaneously defining/naming its structure. So, some of this is a matter of taste. And, to some extent, you just have to get used to the way mathematicians write.

I remember being very pleased when I first encountered what might be an abuse of notation, but is nonetheless super convenient: \Sum_{s \in S}, where s is an element of a set S. Or, even better, just: \Sum s_i. Of course, this is defensible as I was writing out 100 summations in a stats course, not writing teaching material.

My guess is that most mathematicians wouldn't consider that notation abusive. Why do you think it is? That notation is a convenience that allows us to represent a sum over a set where the elements aren't indexed by integers. So, in this particular case, I think there is a utility to the notation. And also: the definition of \sum_{e\in S} is ubiquitous.

I'm not a mathematician, but I went decently far into math and hardly ever encountered a summation over non-indexed elements, or really anything beyond the standard \Sum_{i=1}^{n}, even up to my final math courses.

I wasn't aware of its ubiquity! I may only think of it as "abusive" due to lack of familiarity. The way I've seen it used is: \Sum_{e \in S} e_i, where 'i' is never explicitly defined, and this still assumes elements indexed by integers. The only utility seems to be from the abbreviation, leaving out the range of indices being iterated over. Not saying that isn't useful, but the rigor of the math probably doesn't benefit from time-saving omissions.