TeX has "{n \choose k}".

(Replaced by "\binom{n}{k}" in LaTeX with amsmath.)

The fact that semantic commands are available in TeX doesn't change the fact that it is, unabashedly, a display rather than a semantic language. Knuth's goal was to create a language to facilitate beautiful typography, rather than one directly for expressing mathematical meaning.

Hence amsmath.

You should define your own commands and then use them, in preference to directly using markup. For example, at the start of my essay on non-standard analysis, I define some commands:

- "\near" as a synonym for "\simeq"

- \newcommand{\hyp}[1][\mathbb{R}]{\prescript{*}{}{#1}}

- \newcommand{\powerset}{\mathcal{P}}

This way you can change the notation easily and all-at-once if you need to, as well as making the semantics clear to the source-reader.

I think we do not need semantic representation in HTML case at all. For example, f(x+1) can be a multiplication, or a function, but should we write something like \function f (x+1) ? I think knowing the layout similarity with query is enough for math-ware search engine to identify similar math expressions. Adding too much in Math HTML standard is not helpful but redundant.

For sighted users it may be possible to differentiate, but whether "f of x plus 1" or "f times x plus 1" is read aloud is a significant difference.

What I really want to say is whether f(x+1) is function or multiplication does not matter that much in terms of both browser presenting and math-aware search, moreover, extracting semantics can be done by algorithm from context. Considering that few author want to annotate on their expression semantic, and adding semantic does not really help math-aware search, I argue for the necessity for bringing math semantic notation into WEB.

The same way a sighted person does?

Yes, you could try to build algorithms that turn that markup into semantic meaning but now we're almost talking about computer vision. I don't think a solution like that is likely to ever work reliably, and it's better to encode the semantic meaning so that it doesn't need to be re-derived.

I think they meant that a person needs to interpret “two numbers arranged vertically between parentheses” according to context, regardless of whether they’re sighted.

Having more semantic information in that scenario would help the sighted just as much as the visually impaired—imagine hovering over an unfamiliar notation with your mouse and seeing it explained.

The other reply made the same point I'm about to make, but I think it's worth clarifying.

Specifically, just like a sighted person has to determine from context whether "a(b)" is function application or variable multiplication, a blind person can be asked to determine from context whether "a left-parenthesis b right-parenthesis" is function application or variable multiplication.

No AI or CV is required for such a reading algorithm. It's unfortunate that this reading algorithm doesn't quite match what two mathematicians would say if they were conversing with each other, but there's at least one good reason to believe this is still a useful reading algorithm: it's exactly the system used by blind mathematician Abraham Nemeth with his readers, called "mathspeak":

    The speech generated by this protocol is not exactly what a 
    professor in class would use, but it is absolutely unambiguous 
    and results in a perfect Nemeth Code transcription. It avoids 
    largely unsuccessful attempts by a reader to describe the 
    notation he sees, accompanied by the shouting and gesturing that 
    such attempts at description engender.

http://www.nfbcal.org/s_e/list/0033.html

(Nemeth notably created the Nemeth Braille Code for Mathematics, which is part of Unified English Braille and is probably the most widely used Braille code for math. MathSpeak hasn't enjoyed the same level of adoption, but only because there's no standard, MathPlayer, VoiceOver etc all have their own ad hoc rules for how to read math.)

(Strictly speaking the example I gave was a verbose variant of the system described in the email: http://www.gh-mathspeak.com/examples/NemethBook/?rule=18 ) [2]:

That's interesting, thanks for the references. I wonder if this is applicable to screen readers as well as Braille?

Oh, I was talking about screen readers the whole time, except for an aside that the speech rules system I was discussing was created by Nemeth who is known for his Braille system. Other than that though, I wasn't talking about Braille at all.

The problem is that mathematical notations can vary from topic to topic, from university to university, even from office to office inside a university department, even within the same lecture of a single professor. (Been there: "Notation will be different in this chapter since I've copied it from another author's paper.")

I'd imagine any real solution would actually be akin to ruby markup, with the author of the content needing to supply it since there will be cases where it simply cannot be usefully deduced by a browser.