Integrate concise and effect explanations into the relevant Wikipedia articles and you at least give future generations a good head start on understanding these things.
Most of the Wikipedia articles on technical subjects, and especially on mathematical topics, are terrible as introductory exposition. They are jargony, highly technical, and self referential. They usually contain much that is irrelevant, and they almost never properly explain the context for an idea.
The main problem is that Wikipedia articles are tiny and atomic, so it’s difficult to synthesize and organize ideas into a coherent story. The culture of Wikipedia frowns on the kind of exposition found in textbooks or lectures. And perhaps most importantly, no one is responsible for either individual articles or sets of related articles in a field. Working within those confines is not the best way to spend your time if the goal is to give future generations a leg up, in my opinion.
If you want to learn about mathematics, even a mediocre textbook is nearly always better than the relevant Wikipedia pages. The Wikipedia pages are then useful later, as a reference, for people who already understand their content.
If I (still) knew it, I wouldn't need reference material. But I agree, if I never knew it, then reference material is... not quite useless, but quite useless to me at the moment.
I think language and symbology is at the core of why they are so impenetrable.
One major sin is taking new concepts and ideas and putting the primary discoverer's name on them. Such names yield no clue as to the interpretation or application of the idea itself.
Another problem is the symbols used in certain mathematical texts. Everyone who uses them treats them like they're universally understood, but in reality the syntax and meaning of the symbols can and frequently are recycled and reused across disciplines and even theories in the same discipline. You have to be close to the 'in-group'. Like reading other people's code where operators have been overloaded, it's like learning a new language every time you want to dig into a cool new maths paper.
I don't actually have any good solutions to these problems. I would guess there are lots of lessons to be learned from the history of Chinese characters, though. They have thousands of unambiguous symbols which _can_ be learned by non-natives and which _do_ give an appreciable degree of cross-lingual intelligibility among languages that use them.
I was extremely frustrated with my machine learning class because of this. The notation used was hazy. Also if you ask me, probability theory should just be squashed and the ideas reformulated with new more consistent syntax / symbols, because the entire thing is so damn inconsistent and disorganized at present.
>Also if you ask me, probability theory should just be squashed and the ideas reformulated with new more consistent syntax / symbols, because the entire thing is so damn inconsistent and disorganized at present.
How do you mean? Probability doesn't even have that much complicated symbolism... although I do wish we would teach in probability courses how to translate between random-variable "distributed according to" notation and actual density functions. As in, I wish I knew how to do that.
Wikipedia is a horrible way to learn math. At most it works as a way to get initial pointers for literature. In most specialized fields, don't expect wikipedia to provide any understanding of mathematics beyond a summary of formulas without much explanation of what they are.
Wikipedia has the same definitions that are in your textbook, but often provides much more readable proofs as compared to the very short and "elegant" ones or unwieldy monstrosities in your usual textbooks. At least that's true for textbooks at the level of, say, Dummit&Foot's Abstract Algebra or Baby Rudin.
