Terry Tao on some desirable properties of mathematical notation

Terry Tao on some desirable properties of mathematical notation
mathoverflow.net

Usually Tao's posts are so insightful, and crystallise some idea so perfectly that it feels like I was just on the cusp of discovering it myself—a rare talent, and hard to cultivate since it goes against the ego. In this case, though: I'm a professional mathematician, and as prone as anyone in my discipline to use mathematical language to describe not strictly mathematical things, but the pseudo-mathematisation here ("Notation^{-1}(C)", for example) seems more like wit than clarity. Not that there's anything wrong with wit, but in this case it seems to me that it's at the expense of, rather than a pleasant addition to, the central point.

I'd like to hear especially from anyone who isn't a professional mathematician: did you feel that this post improved your understanding of the purpose and function of good notation?

(EDIT: I was scared about making this post, since there's rightfully a lot of respect and appreciation for Tao—and I hope it's clear that I concur on both counts—and I wasn't sure how my reticence on his post would go over; but I'm super glad I asked. Thanks so much to everyone downthread; these are wonderful responses and I feel that it benefited me a lot to read them.)

No, but I did learn about new dialects and conceptualize mathematical notations better. I think for someone new to languages/notations the first set of detailed bullet points are useful. For more experienced practitioners I think it sums up to "use the right tool for the job".

What I found very interesting was the latter set of bullet points where he presents a sort of Rosetta Code (http://rosettacode.org/wiki/Rosetta_Code) of math. I am familiar with thousands of computer languages but not a mathematician and not familiar with what I'm sure are hundreds (thousands/more?) of Mathematical notations, so the few examples he lists here have me intrigued.

Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head.

Math notation sucks because none of it maps to things non-mathematicians know. Every time a new symbol is introduced, whether it be a greek letter or a operator, it's one more mapping your brain has to create to remember it. And on top of this, you also have to remember the English names too. Yes I said names - most math concepts have so many different names it's crazy. Even basic arithmetic can't escape this. There are two names for multiplication (multiply, product) and four common notations for representing it (*, x, ·, and whatever you call it when two variables are next to each other).