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Thank you!

I am 100% certain that had authors of a lot of papers that I have read included some examples and perhaps even calculation results, I would have understood and perhaps even been able to reason about some of their publications a whole lot better than I currently can.

Just the mere inclusion of a few examples makes a world of difference. The idea that X is clear to everyone and therefor we do not need to exemplify it for the reader is really back-breaking. Especially when you are not coming form a background that includes years of expertise within said domain!

Sure, but math papers are not written for general audience; they're written for mathematicians.

Elucidation that would be useful for the general public is very difficult or impossible to write in such a way that it wouldn't be extraneous distracting fluff for mathematicians. It would also make most papers dozens of times longer.

I'm not exaggerating here: I think you could probably state and prove the results from basic one-variable calculus (assuming as axioms some basic properties of the field of real numbers) in 10-20 pages in such a way that mathematicians would understand it, whereas beginner calculus textbooks are regularly over 1000 pages long. A hypothetical research mathematician who didn't know calculus would find the 20-page version way easier to understand than a slog through Stewart's 1300-page book.

What you want is undergraduate-level textbooks, not research papers. Good textbooks do contain all the stuff you want: examples, calculations, motivation, etc. I can recommend some if you are interested in a particular field.

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.

> Part of the elitism within the field is that, at times, the community places pride on being as incomprehensible as possible because we should be smart enough to fill in the "trivial" details. That's morally wrong and it turns people away from the field. Further, it locks out a huge number of people who are not specialists from accessing the material.

Saying something is trivial when its not is insecurity. See it for what it is. Has nothing to do with elitism. This is no different from a driver who claims he doesn't need a seatbelt because he is such a good driver.

But even recent PhDs have to understand that our old-ish professors simply deem things trivial that are very difficult for us to understand. They've just gotten used to the material. The mathematician I admire the most, from whom I've initially learned algebraic topology, would react to questions to the material with such sweeping, elegant, simplistic arguments that you would either be mesmerized by the beauty or call it arrogance/elitism. But that's just the level he operates on.

Pretend there was an alien arriving on earth asking "stupid" questions like "whats 2+2", expecting you to explain in detail how you arrived at the answer. If you can't "dumb down" enough to get to where you lose him, are you an elitist because you "pretend" the calculation is "trivial"?

I don't think we need to postulate about aliens to come to a similar scenario. If someone in primary school, an undergraduate, or a practitioner asks me a question and I continue to lose them because I am unwilling to revise my explanation to something suitable for them, then, yes, I am elitist. That's not to say that I can't write something incredibly high level that only specialists in my field will readily understand. Frankly, I don't start my papers by telling the readers that we will be using Zermelo–Fraenkel set theory before going off and proving something about differential equations even though I likely depend on these results in order to get things like the real numbers. However, I don't think that's where the frustration is. I believe the frustration is that there is a dismissive attitude toward others when they don't understand the results. This is elitism. I also think there's a general unwillingness to self-reflect and consider that maybe the problem isn't that everyone else isn't educated enough to understand the results and maybe I'm just a terrible writer.

For me, I'm about a decade out from my PhD and I've been working in my chosen field for almost two decades. As I've matured in my understanding of the material I actually write quite a bit more because I realize more closely that there are lots of little details that are really important and it's highly unlikely that only outside of a select few will pick up on that. Now, that is my choice and others can choose differently, but just because someone masters a field doesn't automatically mean that they can just skip the details.

As an aside, 2+2 isn't that hard to explain. "2" is just our name for 1 + 1. So 2 + 2 is 2 + (1 + 1). By associativity of addition, this is (2 + 1) + 1 or ((1 + 1) + 1) + 1. And "4" is just our name for that last value.

what if its brain is binary and it cant comprehend our way of expressing numbers.

Says you. It took me weeks when I first encountered it to understand the Peano axioms, how the natural numbers were constructed, and how addition is defined recursively given that definition.

>Awhile back, I had a coauthor on a paper screaming at me in his office that it was condescending for me to include a full set of calculations for a proof that I authored. We had the room the paper for the result. I, as the author of the proof, couldn't follow the proof without the full set of calculations. We actually walked around the office and most people agreed with my coauthor.

This is not a math-specific problem. I had the same issue with physicist and engineering department's coauthors.

Do you have any particularly memorable example of a complicated proof which was lacking complicated "trivial" details? My experience (I work in interactive theorem proving) is that many such proofs are subtly wrong.

Off the top of my head, I'd probably say integration by parts in multiple dimensions. Generally, I see people given an outline of a proof in two dimensions for Green's theorem. I'd say it's an outline because the typical qualification is that the proof is for simple regions and that all we have to do is patch these regions together for more complicated areas. However, the details of how that works, in my opinion, are complex. The only full proof that I've ever seen for this result comes from Daniel Stroock's book, "A concise introduction to the theory of integration." He takes a little over 100 pages of background work to get there. This actually leads to my math confession, "I have a PhD in math and can't prove integration by parts in multiple dimensions."

By the way, if anyone has another reference or good proof for proving integration by parts in multiple dimensions, please let me know. I'd love to see it.

As another aside, I'd love to start using interactive theorem proving tools in my field, which tends to be related to things like optimization and differential equations. That said, I've never seen examples for how to prove results for stuff in calculus using interact theory proving tools, so do you know of any? Really, things like the mean value theorem or Taylor series would be great to see.