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.

Right, the notation of derivatives is also totally confusing, especially for things like the chain rule. The sloppy way to write it seems very intuitive but isn't precise. To write that in a precise way, you have to use the vertical-bar-on-the-right (e.g. d/dx f(x) | x=0, but typeset properly), variable names etc. In high school I rewrote a lot of the textbook stuff in a super explicit version for myself like this and it got very verbose of course, but gave deeper understanding.

Same in college when learning the Fourier transform, a stumbling block was that the prof didn't properly explain that it takes a function as a whole and gives a whole new function as output. When you first learn this concept, it's a bit of time to wrap your head around, but when it clicks, everything makes more sense. But just writing F{sin(x)} = ... seems like F acts on a concrete value. A more explicit way would be F{x->sin(x)}={x->...}

Of course once you already know these fundamentals and they are baked into your brain and take them for granted, it's hard to see where beginners get confused, and writing in short hand is so much easier so you get sloppy while still unambiguous to experienced people.

This is why I always preferred to see coded-up demos and implementations as opposed to formulas on blackboards and slides. If you have to implement it, you can't handwave away things as pedantry. It forces precision by default.

Fully agree with this thought. Sloppy notation has been a hindrance to my personal understanding many times. Math made more sense for me when I could code it up and see the function doing things. Everything explicit vs inconsistent shorthand that saved the author a few pen strokes.

Which is why I am so favorable of Jupyter notebook-like teaching environments. Embed the (guaranteed to execute!!! no illegal shorthand) code so that learners can get a true representation that can be manipulated. Although, I think they are still unlikely to reshape education - now you require some coding fluency + the niche math topic.

> abusive language/notation basically everywhere in math

In most cases it is not as much abusing notation as overloading it. If you think of the context of a formula (say, adjacent paragraphs) as its implicit arguments (think lambda captures in c++), then it is natural that curly braces can denote both a set and a sequence, depending on this implicit input.

Such context dependent use of symbols is actually rather convenient with a little practice.

"it is natural that curly braces can denote both a set and a sequence, depending on this implicit input."

?

I don't even know where to begin. Overloading symbols in mathematics occurs all over the place. There's nothing wrong with that. The difference between overloading a symbol and abusing it is whether there is an agreed upon definition/convention regarding its use and to what extent its use conforms to that definition/convention. What I'm saying in my original post is that the statement "{x_n} is a sequence" disagrees with the formal idea of what a sequence is and that most writers don't bother to explain their own notational use.

If you wish to re-define the curly braces to have a context-dependent meaning, knock yourself out. But, I would imagine that that practice would confuse a lot of people. Math is a human activity. It's not a programming language.

In programming you would put function as a parameter. So, maybe something like this

F{sin}(x) = ...

is just as short and clearer?

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.

I think there's also a bit of sneakiness in that by not requiring actual understanding, it's possible to memorize equations and pretend to understand, and then pretend that students got more understanding than in reality.

Yes, but often the formula is more complicated like (sin(x) + 5)^2.

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.

> \Sum_{e \in S} e_i

I'm tempted to call that notation simply wrong instead of abusive. Generally "abusive" notation, while technically wrong, has some redeeming feature in intuition or consicebess.

In this case, the alternative notation would be to simply drop the index and write "\Sum_{e \in S} e", which seems to be all around better.

From having spent way too much time doing technical writing; I'm tempted to say the notation you are recalling really was a mistake. They probably started out with "\Sum_{e \in S} e", then decided to make all summations be index based instead of set based. Unless you spend a lot of time proofreading, that type of style change can easily lead to half translated expressions like what you recall.