Some of the point that I was trying to make is that a large number of papers are incomprehensible for people in the field as well and often not for any good reason. Yes, sometimes it's hard to come up with good illustrations or examples, but, frankly, they help everyone. Let me give an example of a good way to do this. To me, Mark Embree is a really good mathematician. He also works in a somewhat esoteric area called pseudospectra, which helps us understand non-normal operators and, ultimately, how to effectively solve differential equations. Look at the following page:

http://www.math.vt.edu/people/embree/EF.html

Mark isn't dumbing down his material, but he is giving context for why it matters as well as graphical descriptions of how some of these ideas work. This is some of why I admire Mark's work. It's not just the technical results, but working to come up with better ways to relate the material.

Right, but we are talking about two different things. I agree with you 100% that mathematicians dickwaving about how hard they can make things for other mathematicians to understand is stupid, and I think quality of exposition for a mathematical audience is very important in math papers.

I strongly suspect the person who originally started this discussion doesn't understand "we are interested in the spectral properties of quasicrystals, as modeled by discrete Schrödinger operators with aperiodic potentials", at all.

(I'm not saying that to be an elitist asshole -- I have an undergraduate degree in math and I don't understand most of that jargon.)

So Mark isn't doing what OP is asking for, and so he isn't a good example to show that what OP is asking for is possible or desirable for mathematicians.

Well, there's a lot going on in the thread, so I'll try to be more specific. The context for much of what I've been writing here, other than my own frustrations, is the OP that stated:

Mathematics is not a subject reserved for a select set of chosen elite. It is a very necessary tool for almost every scientific endeavour. And this is a huge problem because people in those fields aren't just studying maths. Case in point, I'm a mechanical engineer. I have to study thermodynamics, materials, structural analysis, mechanics and on and on. In the midst of all this, I needed to learn advanced numerical methods, but was flummoxed because it was taught by a senior professor, to a class of mostly mathematicians, with an inordinately huge amount of effort placed towards proving things etc. and not much thought spared for us non-applied mathematicians.

This is something that I'm sensitive to and passionate about. Why I think Mark provides a good example for this is that an physicist or engineer may come looking for new techniques to analyze or solve the Schrödinger equation and the linked page provides context for why this may or may not be important. It also provides a visual description to help this explanation. For the technical results, this wasn't necessary. If Mark was just writing for mathematicians this isn't strictly necessary, but, to be clear, it helps someone like me dramatically.

What I'm interpreting from the overall thread, perhaps falsely, is that there's a sense of frustration that results are being made overly opaque. I agree with this and I don't think it's necessary.

I think its necessary to preserve a certain opacity.

I believe that there should probably some effort made to write tutorials on top of mathematical literature that explain how to use and apply at least the important results of mathematics to make it more accessible. But core literature should not be diluted for the purpose of making it more accessible.

That's a perfectly valid point of view. I believe that's not necessary and doesn't dilute the material. That said, I believe what you propose does exclude a sizable group of people from the field, which is elitism. It's a consciousness of belonging to a group of select few that have ground through the material.