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> Also, I think that making formulas smaller makes it easier for the eye to see the whole formula together. This makes it easier to quickly understand the meaning of the formula, at least once you've got used to the notation for that particular area of mathematics

Not only this, but the notation itself is often intentionally used as a particular abstraction to aid intuition, see bsilvereagle's example of Einstein notation below as a great example. This makes things easier to reason about with our human brains even when dealing with very complex ideas.

The abstraction is usually leaky, however, and some notation that aids intuition in one way often makes some other interpretation horribly verbose and unintuitive. Hence the need for more than one notation for the same situation.

All of which isn't to excuse the many historical accidents that haven't been winnowed out by our predecessors. That's often a cultural problem, however, and while some of it can be realistically fixed by teaching the same way we teach good grammar, some of it is as hopeless to rail against as asking to fix every irregular conjugation in the english language. Good luck with that, honestly.

Yes, that's the tradition part. I certainly don't object to using whatever notation you like when you're at a whiteboard or scribbling on paper. But math papers are carefully formatted and published using LaTEX, not by taking pictures of whiteboards. This isn't because it's easiest. It's supposedly to benefit the reader, and yet fails in practice.

It seems like coming up with a format that at least allows definitions and usages to be hyperlinked might be pretty nice, instead of a format whose primary benefit is looking pretty when you print it on paper.

FWIW internal links between equations and figures in LaTeX is pretty easy and should be supported by most readers.

Poor mathematical writing is pretty rampant, but that's partly due to the fact that it's not taught well and there's definitely some cultural biases towards jargon. I don't think that negates the fact that even within a typewritten document a human needs to read and understand what's going on, so often terseness is desired, as long as the reader understands the notation enabling it.

Go to definition (for symbols or notation) would certainly be helpful in reading a mathematical paper, but how is that related to choosing good notation? Do you have an example of mathematical notation that "fails in practice"? In my experience, mathematical notation becomes established because it works well enough in practice.

I don't have any specific example, but I've read that many (most?) math papers are incomprehensible to outsiders, even mathematicians in other fields. I also seem to remember an article about a proof that's gone unchecked for years.

So, from the outside, it seems like there are big problems with communication in the mathematics community, and it's simply accepted that it has to be that way because math is hard.

Good notation certainly aids understanding, but math is a vast endeavor where the details matter. In switching from one programming language to another, often a rough understanding of corresponding concepts is enough: how to print has different details in different languages but it's not that different, and the details are easy to look up. On the other hand, just switching from topology to symplectic geometry involves asking entirely different questions -- having different values and pursuing different goals. Maybe it's more like switching from web dev to scientific computing as well as switching languages.

Yes, a lot of mathematical papers are incomprehensible to others in the field. But it's not just because of notation. You wouldn't ask a content marketer who is great at A/B testing to deal with tweaking something in the linux kernel because it's all "computers".

This is like saying that the Windows codebase is incomprehensible and thus C / C++ must need better syntax.

I think papers are primarily aimed at other expert researchers, who actually do prefer brevity.

Going through Calc III for the first time. I can't imagine how many more trees I would destroy if I had to write out double integrals longhand. Not to mention, it would make it a lot harder to reason about everything going on...

When I first started in my adult adventure in math (never doing well with it academically, I started on pre-calc with Khan Academy about a year and a half ago; I'm now working through multivariable calculus on MIT OCW), I used scratch mode in emacs because hey, I'm right there on the computer.

I gave up and started using graph paper as my scratch pretty quickly :)

I can't imagine trying to actually _do_ math with something "descriptive" like LaTeX.

Math, from what I've come to realize, is less about describing how to compute something and more about describing something in such a way that it can be manipulated in order to reveal something new.

And that's why the obtuse iconography is popular in math, and why the obtuse iconography of APL never became truly mainstream in programming.

>Programming is different because you can use a text editor with copy, paste, autocomplete etc.

But we don't do this. We use abstractions. Why doesn't the mathematician? (And you'll say of course they do). In which case we go back to the original question of why dense notation?