>You missed my point entirely. I said that you should not optimize for writing math, you should optimize for reading math.
My point is that these goals are nearly one in the same when you get to the high level. Many mathematical relationships are very complex, usually not the step-by-step procedures that is common in code. Thus being able to hold the entire relationship in your head at once is crucial. Single letter names for variables and functions are critical here (for the previously mentioned reason working-memory-decay).
I basically have a bachelors in math, and I could not imagine reading complex equations with full variable names. The hard part is understanding the whole, not remembering what x or i means. If you find an equation cryptic, that just means you don't have the requisite knowledge to really understand it.
Sorry, if all you have left is the (very typical) argument of
"if you think that way you just don't understand it", then I'll
consider this discussion over. I have no interest in hearing
endless appeals to tradition. I'm sure people made similar
arguments for GOTO back in the day. Good thing we moved on.
I would like to think my argument is far more nuanced than you're giving it credit. This isn't an appeal to tradition; I'm making an argument that explains why traditionally math has stayed with the "cryptic" notation rather than descriptive variable names.
You're completely ignoring the working memory argument I'm making. A large part of math is pattern matching: recognizing a common relationship between variables (a common pattern or "theme") and investigating that relationship further. This ability is critical to mathematical ability. Visual compactness is crucial here. Using descriptive names will completely bog down your visual system with reading words rather than identifying patterns.
Pattern matching is a critical skill when one becomes an expert in any field; in math its of utmost importance. Math is optimized for reading by other experts who have those same visual patterns committed to memory. This is the optimal approach for the work's target audience. Yes, it makes advanced math largely inaccessible to outsiders, but that's a part of the trade off.
>You're completely ignoring the working memory argument I'm making.
I'm not ignoring it. I said I don't buy it. I generally don't believe that
you need visual compactness to to point of being cryptic to make use of pattern
matching. I'm not saying this is necessarily easy. Neither is choosing good
variable names in programming, especially at high abstraction levels. But
that's no excuse for not doing it.
The other thing is that mathematicians will finally have to accept that
they aren't the only ones who need math. It's not, for the most part,
a "field for experts" (to paraphrase you) which only those who are willing
to become experts at it are allowed to understand. It's a beautiful science
with lots of appliance in almost every other field. But it's inaccessible
as hell in large parts due to mathematicians with your attitude.
I strongly believe that math can be made more accessible and more
"user-friendly" for people who aren't experts at it. It will be hard to do
that without sacrificing its tremendous power and flexibility (which I am
totally against in software, as well). But it must (and will) eventually be
done.
Take note, by the way, that the accessibility problem is less and less
important the more specialized the math you are doing becomes. I don't really
care if some hyper-abstract field of math that a few dozen people in the world
are even able to grasp the basics of is arcane and cryptic. That's something
only experts will care for and if they are fine with their niche field being
overly cryptic, so be it. What I'm concerned about is math in general.
I firmly agree with the position that math should not be taught as an
appliance, because that is missing the point. However, it's downright fatuous
to ignore the fact that math has appliances, and that therefore, a lot of
people will have to learn a subset of it. To excuse bad practices which have
survived by virtue of no one questioning them with nothing but "it's optimized
for edge cases (experts)" is a cop-out. It ignores the problem.
I also find it funny that I already met with such vehement opposition for just
proposing meaningful variable and function names. That was just an example.
There are a lot more problems to be solved. Stuff like "this is trivial" or
"left as an exercise" as a way to avoid writing out cumbersome, but nonetheless
important (for non-experts) parts, the general terseness of mathematical texts,
including the parts in plain English. I could probably find more.
Let me conclude with this: Math as a whole has accessibility problems, just
like a lot of CS and computer stuff. Flat out ignoring them or even asserting
"it must be this way" is incredibly ignorant. And we need dialogue to do this -
not condescending "there is no problem" rebuttals.
You make a lot of points that I agree with. I just don't think your solution is the right one. Math, when it comes to writing proofs and peer review, should absolutely be optimized for other experts. Whatever makes communicating ideas precisely among themselves is what's important here. I don't see the benefit of changing this process so those of us as outsiders can take a peek.
But as you said, there is a very large part of math that deals with applying these concepts and communicating them to non-mathematicians. The way we go about this is definitely in need of a do-over. Finding more intuitive ways to communicate these ideas is critical a critical part of this (possibly including more descriptive variable names). But, I think there will always be an inherent chasm between the math experts and those who are using the results they discover. An example is the difference between the calculus track of courses that every science major takes, and say, abstract algebra. The math that mathematicians do and the math that the rest of us learn will always be vastly different. It's just the nature of the beast.
My point is that these goals are nearly one in the same when you get to the high level. Many mathematical relationships are very complex, usually not the step-by-step procedures that is common in code. Thus being able to hold the entire relationship in your head at once is crucial. Single letter names for variables and functions are critical here (for the previously mentioned reason working-memory-decay).
I basically have a bachelors in math, and I could not imagine reading complex equations with full variable names. The hard part is understanding the whole, not remembering what x or i means. If you find an equation cryptic, that just means you don't have the requisite knowledge to really understand it.