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.

As a non-native speaker, another tip I remember reading somewhere and then finding it very useful: when you read a packed sentence that doesn't seem super clear and the next sentence starts with "Indeed, ..." that means you should read further as it will be a kind of explanation or elaboration. I mostly filtered out such words as "filler" or generic "emphasis words", but they guide the understanding also.

Good mathematical writing has this kind of cadence and pattern to it, and that's not a problem. For good writers some personal charm and flavor can still shine through, but it helps the reader to use the familiar trope structures. Unfortunately, this kind of "meta" is not taught much, so many students don't quite understand how to read math books, get frustrated when they progress slow, expecting to read it at the same speed as a history book or a novel. In a math book it's normal to re-read sentences or jump back half a page, to flip the pages back and forth, to put down the book for a moment and think etc. and it may take an afternoon to just digest one or two pages.

I vaguely remember that Prof Thomas Körner in Cambridge used to have some good notes on mathematical writing. I can't find them now but here is another lecturer's notes that look quite good: https://www.dpmms.cam.ac.uk/~grw46/Writing_Guide.pdf

Paul Halmos and Donald Knuth are good examples of people that have a very nice mathematical style imo.

> Note the difference between "where" and "such that". "Where" is used when the preceding notation is being defined; "such that" is used when it is already defined and its value is being restricted.

I found this interesting. SQL and Swift use “where” for restrictions. Any other examples in programming languages?

If a programming language wanted to use a keyword for restrictions that isn’t “where” (and still is a single word, hence “such that” doesn’t qualify), what word would be suitable instead? “With”? “Having”?

    >>> [(x, y) for x in range(10) for y in range(2, x) if x % y == 0]
    [(4, 2), (6, 2), (6, 3), (8, 2), (8, 4), (9, 3)]

    * (loop for x from 1 to 100 when (= x (expt (floor (sqrt x)) 2)) collect x)
    (1 4 9 16 25 36 49 64 81 100)

    DB<4> for (1..10) { print "$_ " unless $_ == int($_**.5)**2 }
    2 3 5 6 7 8 10

    struct S
    {
        x : num;
        y : num;
        where x < y
    }

    struct S
    {
        x : num;
        y : num;
    }

    let s : S where s.x < s.y;

   f x = y + 1
     where y = 2 * x

    f x = let y = 2 * x
      in y + 1

  [ x for x in range(100) if x % 2 == 0 ]

It's a process. There is little utility to be gained from specifying mathematics, the title and abstract should explain why a general audience might care, but the meat is often for your advisor. If you think that more people will care then expand it out, use your school standards, even more? sure you can check out some knuth books on mathematical writing or use this grammar, but have you considered programming? It's by far the most inclusive and effective way to tell other people what you actually mean. Maybe popular languages aren't optimal for this try a specification language.

I'm in no way saying you should DO mathematics in a programming language, rather translate it for a wider audience. At a sufficiently high level even little kids could understand your argument before drilling down into specifics, and since the spec never actually has to run it can be as high level as you want limited only by verification time

Very nice. I wish people put this kind of careful thought in academic manuscripts.

The whole point of introducing a math equation in a paper is to serve as a completely unambiguous formalism, devoid of the ambiguities of the spoken word.

And yet, it is all too common to read something many times and not make sense of it, until it hits me the author means something completely different than what the symbols would imply in principle, and what looked like a formalism is basically a sloppy direct translation of words as math symbols, combined with abuse of notation, idiomatic but undefined uses of established notation, or outright nonsense.