TITLE: Advice for writing good mathematics?
QUESTION [27 upvotes]: It's been a (far-fetched, possibly) goal of mine to some day write a math Textbook. I've been thinking about writing this question for a while, but reading an exceedingly mediocre text on Mathematical Modeling has finally provoked me. In several paragraphs, I have already spotted numerous changes I would have made to the book (not the least of which is publishing somewhere besides Elsevier, haha); so it got me thinking about mathematical exposition.
At this point I'd really just like writing advice. I've seen some articles e.g. (A Guide to Writing Mathematics and seen Serre's "Writing Mathematics Badly," and I know some of the "basic" authors that are celebrated expositors (Halmos, Spivak, Rudin...) 
But I'm asking for you to share the benefits of your experience: Unforgivable mistakes you've seen made (repeatedly), great expositions you've read, or any other sort of advice you have (apart from 'just do it!' I am!). 
Certainly I'm lucky to have this great website to practice on and see a vast number of others' style.
Lastly: I am not entirely sure if this question is appropriate, or if it is too broad or vague.

REPLY [3 votes]: There are at least two cognitive processes happen during the information transmission:  

Illusion of transparency or curse of knowledge: the author always overestimates how the readers understand their work. In an experiment about guessing a song by listening to how it's tapped, the tappers guess that 50% of time the listeners will guess right, while in fact the number is 3%.  
Cognitive load: the readers read the paper to find solutions for their problems, not the author's problem. What is worse is that the desire to get them solved will cloud their minds and make what is written black and white in the paper be unavoidably exotic, misunderstood or, to an extent, distorted. Not getting what you are expecting is so frustrating, like you are in a rush and your car suddenly stops working. A related phenomenon is information overload.    

Of course it is impossible to write a smooth research paper, and the readers are expected to overcome it anyway, and I may exaggerate a bit, but overall it's still better to keep them in mind when writing anything. It's important to have a concise writing, but when the work is long, complicate and abstruse like in a math paper, only relying on conciseness may make it become too dense, and will backfire the intention.  
My advice are:  

Make a concrete analogy. A concrete analogy will intertwine to the text and allow room for the readers to project their background into it.  
Make the ideas constantly contradict each other. "Contradiction" here doesn't mean as a logical contradiction, but more about "a surprising, but still logical step of development". It introduces why the topic is important, and is the source of excitation, enlightenment, and satisfaction. Being able to solve contradictions is the reason why the ideas survive and are worth the attention.  
Notice where the flow emerges and dissipates. This will help overcome the jargon barrier without having to oversimplify them. Imagine the article is like a heatmap, and each jargon/theorem/proof is a heat source, then the writer's job is to locate them not too hot (too dense) or too cold (too uninformative).
Viewing the topic in as much perspective as possible. Each unit of the article (phrase, sentence, paragraph, section) should be thought as an unique, different perspective, so the readers can see the topic in fresh perspectives. This makes a profound, advanced topic more playful, imaginative and transformative. The simplest trick is to start a new sentence with a different subject than the preceding one. If the readers are about to leave and we have only one more sentence to keep their attention, what should it be?    

Though not about math paper, research about myth debunking and web usability (widely known for the F-shaped pattern) will provide experimental insights about what conciseness really means. For complicated and long writings, I also have an article for this, you can check it out: Making concrete analogies and big pictures.