Terry Tao on some desirable properties of mathematical notation

Is it just me, or does probability theory in general have fairly terrible notation? Ambiguity between random variables and their distributions because of them simply being distinguished by being upper-case or lower-case, writing likelihood functions alternatively with an L() or p(), and using p() (with different arguments) to refer to different probability distributions. Perhaps I'm just having such a difficult time grokking probability theory because it's just difficult stuff, but I often find myself immensely frustrated with the notation.

A lot of people say this, but if I were to sit next to you and tell you what the notation means whenever you asked, would you instantly understand the concept? I don't think so. You still need to do work on your own with the concept before you can get to any understanding.

Say, you come across this symbol: ζ(s)

And you asked me, what does it mean?

Oh, that's probably the Riemann zeta function (it could be other things, but that's probably its most famous meaning).

Would you be like, ah, now I understand it all!

I sincerely doubt just knowing the English words for the symbols (and all mathematical notation is just shorthand for English, or whatever other natural language; it can all be read out loud) would get you that much further along. I could give you one of several possible equivalent definitions, but you still wouldn't really know much of anything. I could start explaining some things about primes and how they relate to the Riemann zeta function, and you'd still be in the dark. I'd have to talk about the prime number theorem and estimates and complex analysis and a whole bunch of other things before the Riemann hypothesis even started making sense. In fact, people have written entire books just trying to explain in the most accessible way possible what that one symbol is:

https://www.wstein.org/rh/

I believe what's going on is that a lot of people who find notation to be the first stumbling block are trying to learn mathematics in isolation, without the benefit of a mathematical community to be able to talk to. You pick up a book, see a bunch of weird symbols, blame the notation, you put it down. But people who train for mathematics specifically rarely learn that way. Mathematics is a social activity, you talk to other people, you read the symbols out loud to each other on chalkboards or side-by-side as you scribble notes on paper or on a napkin. You talk to each other a lot, pronouncing the symbols as much as you write them down.

The notation is a very superficial obstacle. Notation is very incidental, malleable, almost irrelevant (or as mathematicians might say, non-canonical or not natural). If notation is your biggest obstacle, there's a lot more underneath that you probably don't have enough experience with.

I don't really know, I'm not an expert in the field. What I do know, is that I can usually get almost anything written in English in a statistics text, but when it goes mathematical notation I really struggle even with the simplest of concepts.

Type theory is a little like that, at least from some authors that - although the are is well suited for symbols - kind of goes a little off the rails using seven different kind of arrows and all the symbols in at least three different alphabets, instead of maybe just write covariant, or even an abbreviated form adjecent to the arrow?

Almost ten years ago I was working on an interesting Linux multiboot system, googling to find far-from-default grub operation hints.

I think it can be agreed the present grub documentation is very broad, with only a few undocumented features, but still not very deep even on the default operation. Grub was also undergoing more rapid change at the time.

Came across a message where Tao had explained a concept like no other, and after what I had seen it was clear to me he understood it like no other, so it was purely logical. He knew more than the documentation. I didn't know he was a widely recognized mathematician or anything, I just thought he was a very bright computer scientist on a message board.

There were only a couple sentences that nearly applied to my system, within a couple paragraphs on his solution to a different problem.

There was no useful answer for me yet so I moved on.

Googled to exhaustion that session with no code changes to make, but I looked at it again and it was the only tab I kept open, even though Tao did not show the direct way forward for me at all, everything else was actually useless.

Next day I read it again, scrutinizing it over & over for an action I could follow through with, wishing someone had posted equally straightforward advice for my particular situation.

No such luck, but it inspired me to go forward in a similar fashion.

Mostly use both grub and syslinux as separate alternatives to boot my distributions ever since.

For years I've felt like I couldn't have done it otherwise.

And with Windows, grub or not, I ended up bringing more reliability to my employer.

Fairly recently I found out Tao had started out as a mathematical child.

That was incredibly helpful.

He actually communicated the unique abstract concept I needed without even knowing the problem and without intentionally trying.

I imagine he made up his own notations quite a bit before he carefully adopted the various professional terminologies.

We have symbolic expression in programming languages already.

It helps somewhat, but it isn’t a panacea.

We still have to comment our code and add context.