[jiggawatts]

Most of it.

Or conversely, a lot of it can be explained without having to resort to category theory or similarly dense terminology.

I've found that Mathematicians like to come up with the most general, tersest definitions possible, where every symbol is overloaded with layer upon layer of meaning.

You end up with language that looks like:

    a = b c 

For some absurd percentage of the statements. Sometimes the equal sign is an arrow, and the various symbols have superscripts and subscripts on them, but all meaning is lost unless you read the text, which then becomes an exponentially expanding set of hyperlinks to definitions that you need to unpack the definitions.

This is never useful as a method of pedagogy, yet this is about the only type of content you will ever find online in places like Wikipedia or nLab.

Any attempt to clarify with an example is resisted, because it's not "general", or "not the definition" of the concept. In the end, all practicality is erased, leaving definitions so pure that they could be referring to almost anything.

[gizmo686]

At the extreme end of this you get category theory, with Mathematicians affectionately refer to as "abstract nonsense".

I checked the homepage of nLab and found this description [0]:

> This is a wiki for collaborative work on Mathematics, Physics, and Philosophy — especially, but far from exclusively, from the n-point of view: with a sympathy towards the tools and perspective of higher algebra, homotopy theory, type theory, category theory and higher category theory.

And following to the description of n-point of view:

> In particular, the nLab has a particular point of view, which we may call the nPOV, the higher algebraic, homotopical, or n- categorical point of view.

So, nLab seems to have made a very deliberate choice to cater to the style you are talking about.

I think your complaint is more valid for Wikipedia, which seems to want to be more general purpose. However it is not specific to math. Most articles on Wikipedia have a tendency to assume a high level of subject knowledge on the part of the reader.

The math community is well aware of the pedagogical concerns. You don't see category theory until grad school, where it is taught with the assumption that you are well versed in many concrete instances of categories.

You don't even see abstract algebra until the later mart of a undergrad math major. Again, it is taught with the assumption that you are already well versed in several concrete instances of the algebraic structures.

If you get a math textbook, you will find that they almost never take the most generic approach; and instead tend to take the most concrete approach that would cover the subject matter.

[0] https://ncatlab.org/nlab/show/HomePage

[galaxyLogic]

I would like to see math and physics presentations always having source-code which would do the calculations. Then the explanation for theory could refer directly to explaining that computer-program. You could understand it all simply by understanding the programming language used. You could debug through it and see how the calculations happen step by step.

[drran]

Equality is very important concept in mathematics, because it allows to manipulate formulas by rules. When formulas are manipulated, it's important to keep them as small as possible, because it's hard to manipulate a wall of text. Instead of short sequence of formulas, which you can write and verify quickly, you will have lengthy books, like written by ancient mathematics.

For example, command line tools, which are used often, also have cryptic one-symbol options, like `tar xzf`, which looks like formulas, for same reason: to be able write, manipulate, and check them quickly. However, most command tools are distributed with a built-in help or manual for these cryptic options, while most formulas are not. IMHO, we need something like «formulapedia», with manuals for most popular formulas.

[paulpauper]

Because to expand the whole expression, such as the Einstein field equations, would fills many lines or pages.

[teaearlgraycold]

As someone that also doesn't like the status quo - I would love for there to be a better mathematical notation. If we could get some kind of functional programming-like syntax it would be nice and consistent.

Also what's up with the acceptance of single letter variables in math?

[joppy]

Single-variable letters and suggestive notation make it easier to recognise patterns, and give a good level of abstraction at which thinking about the problem rather than muddling through notation can happen. For example one can rather easily check

a(b + c) + (b - d - a)(b + c) = (b - d)(b + c)

But it will be hard to simplify

add(mul(amins, add(boo, cold)), mul(…

You are not expected to understand what the notation means without further context, in the same way a programmer is not meant to see a function call to main(foo) and understand what will happen.

[db48x]

Find yourself a copy of The Structural Interpretation of Classical Mechanics. The primary author of this textbook on mechanics was the inventor of Scheme. It uses two notations throughout. The first will be very familiar to mathematicians, but with some changes to disambiguate things like derivatives, while the other notation is Scheme.

The Mechanics package for MIT-Scheme adds a whole computer algebra system so that it can compute the derivatives of ordinary Scheme functions, symbolically evaluate them, etc.

https://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/bo...

[jasonwatkinspdx]

Interesting bit of history that's relatively unknown: APL was originally an attempt by Iverson to create a uniform and systematic mathematical notation. It was only after he'd designed the notation that it was turned into a programming language. This helps explain APLs quirks, and perhaps the limit of popularity it hit.

[kirkules]

I think maybe you mean what's up with the dearth of multi letter variables, and if so IMO the answer is, satisfying or not, that juxtaposition as an operator is too hard to give up.

Edit: juxtaposition without some sort of brackets, of course

[thereIsazrzn]

When math “started” paper and writing utensils were harder to acquire. One had to transmit idea’s succinctly.

Still it would take many more books to capture it all in English than a simpler glyph set. One will still go further in math learning the meaning of the glyphs than reading it all explained in “human language”.

It’s not so hard when one accepts math is the four common operations (add, sub, div, mult) on a variety of data structures though. “Square root of” is a shorter coded language for “use numeric result of this sequence of operations against this given number” (the data structure).

Math isn’t hard. Government mandate we teach students in speaking traditions instead of educate them to measure for themselves has resulted in piss poor education in math.