That's natural, since mathematical notation is heavily slanted towards writing on paper and symbol manipulation. Use of one-letter variable names and highly context-sensitive notation reduces the number of pen strokes, but doesn't help the reader.
Modern programming languages and conventions are quite a bit better in that respect, since people design them to be maintainable. A program using one-letter variable names for everything and a set of similar macros meaning entirely different things in different files would be considered unreadable by pretty much any programmer.
IMO, the world would immensely benefit from another version of mathematical notation designed specifically for explaining things, rather than doing calculations on paper.
The problem is that the mathematical notation, while not perfect, works really well for those adept with its use. It's an efficient shorthand. For explaining things in depth we have pictures and and natural language. I fail to see how creating a new set of symbols would do anything but hinder communication within the math community. If you have any deeper argument to support your position, I'd be interested to hear it.
>I fail to see how creating a new set of symbols would do anything but hinder communication within the math community.
I don't think this is just about the math community. The problem - at least as I see it, as a student myself - is that Math is a required part of many fields of studies. Fields which not necessarily have much to do with Math. And that's a problem for many of the students. I won't say that Math is entirely unneeded and should be completely abolished, but what I am saying is that many of these students (including myself) don't want to be part of said "math community".
We don't want to have to learn the arcane notations and "math language" just to be able to use some of the tools we might indeed need. Yes, I might sound like a whiny student who's butthurt over having to actually study for something. I may be. However, seeing as /so many/ other students (who, for the most part, are competent in their own chosen field) are in danger of failing their studies or struggle heavily with their math lectures simply because they are /not/ competent at getting into the "math community" is ridiculous to say the least.
For the record, I study Informatics at a university in Germany, so the situation might be different elsewhere, but I've read about similar issues at American colleges and universities, and every now and then, related articles pop up here at HN.
As a math major, my view is probably biased, but I feel like one of the reasons mathematics is so widely used and applied in other fields such as physics, CS, and even biology these days is that they express or model, to a high degree of accuracy, patterns that recur in all these fields. The mathematical notation is the most general and abstract form of expressing these patterns. My feeling is that to not learn the language of math dooms one to reinvent the wheel because they don't realize that the problem she is working on has been solved before in a more general case by some one else. In those cases where the problem hasn't been solved before, the solution leads to new areas or notation which can show up in unexpected ways in different branches of science.
Without this common language, we would have a specialized language for every field, allowing little "cross-pollination". As an example, in evolutionary biology, take the NK model of epistatic interactions which models gene interactions. It just so happens that this model is extremely similar to a thing in statistical physics called a spin-glass. Without mathematics the biologist would have to sit and work out all the details of these epistatic interactions before getting to actually do biology. So in short, mathematics saves you time by expressing common patterns or ideas.
There was an example not so long ago of a paper published in, I think, biology, where the author had invented a wonderful new technique for more accurate estimates of the area under a curve.
It was Simpson's rule.
CORRECTION:
It was worse than that. It was the trapezoidal rule:
>So in short, mathematics saves you time by expressing common patterns or ideas.
I wasn't arguing that. In fact, I said that math is lending me and students of other fields tools to use and apply. The problem lies in the expression of said tools (which is exactly what I mean with "arcance notations") and the inability of many mathematicians - no offense - to separate those two concepts (tools/mechanics and expression) from another. They assume that they are completely inseparable, which is not only part, but a major source of the problem. Any attempts to somehow make math less obscure (or even just suggest it) is met, in great parts, with conservative hostility. I've seen mathematicans claim "that I was telling them that their profession is pointless and should be abolished" when all I was saying that a majority of the expression of math is incredibly obscure/arcance to most people which are not mathematicians and mostly not suited for practical appliance (as in, appliance in fields of science other than math).
Also, it's a valid point to say that we shouldn't have special syntax/language for every appliance of math imaginable. That would indeed not only be stupid, but actually /increase/ obscurity. We should, however, stop excusing the obscurity of math with "it's shorter to write". Readability and - most importantly - comprehensibility should /always/ come before convenience. Even in math.
You speak about "math community" as if is some exclusive, elite club. In reality, anyone who needs to read a formula or some proof is part of that group.
Also, a formula can be written once and read hundreds of thousands of times. Because of this, I consider ease of reading math notation far more important than efficiency of writing in it.
I disagree. By "math community" I refer specifically to those people who create new mathematics, i.e. mathematicians. That is not at all as inclusive as you argue. Math notation comes from mathematicians and is designed to facilitate communication between people of that profession. The notation is very functional and efficient for its designed task.
To create a new set of symbols and notation designed specifically for those who are not mathematicians is a waste of resources. Professors, whom would be responsible for teaching such a thing, would have to translate all of their concepts from "professional notation" to "novice notation" when lecturing. This involves them learning a totally new set of notation designed for no reason other than teaching. Furthermore, anyone who wanted to pursue mathematics professionally would have to then learn "professional notation".
Aside from these logistical difficulties, I fail to see how a different set of notation would be any clearer or easier to understand than the current notation, which is already widely accepted and considered useful. Essentially, you are arguing that math notation is too difficult and so it should be simplified. That is like saying Faulkner or Joyce is too difficult to read so should be rewritten for those without the ability to comprehend the source document: it sort of misses the point.
Modern programming languages and conventions are quite a bit better in that respect, since people design them to be maintainable. A program using one-letter variable names for everything and a set of similar macros meaning entirely different things in different files would be considered unreadable by pretty much any programmer.
IMO, the world would immensely benefit from another version of mathematical notation designed specifically for explaining things, rather than doing calculations on paper.