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.
Best Answer
I've certainly not written a text, but I've read a few nasty ones. The best advice I would give an up-and-coming text writer:
1) Before you start writing, think long and hard about who you intend the book for -- Is it for teens? Undergraduates? Graduate Students? PhD wielders? This is something that really matters because of the "mathematical maturity" issue. Somebody that's been studying the subject for 20+ years will be better able to fill in any gaps you might leave (intentionally or unintentionally) in your treatment of the subject matter.
2) Super "symbol-heavy" books are not conducive to getting your message to your audience, although it might feed your ego. I think it is tempting for somebody who'd become so familiar with a subject that they can write a book about it to get carried away with showing off everything they know. If I had a quarter for every time I was reading something convoluted and came across the words 'trivially', 'obviously', 'clearly', etc... I would be a millionaire. Well, maybe not that rich, but I'd have at least $50.00.
3) It seems like mathematics texts come in one of two types: a) overly symbolic, cryptic, and dense; b) overly pedantic, example-ridden, boring to read.
4) Many mathematics texts seem to be very "method-driven" and not present the intrinsic subject matter they deal with. For instance, I'm currently reading a book on differential geometry that is extremely dense symbolically (granted, anything with tensors necessitates this to some degree) and the author hasn't bothered to tell my anything about why the hell what he is writing should matter to me: "Ok, I can do this curvature thing, and it's intrinsic, that's nice. So what do I do with it? What motivated the discussion in the first place?"
The above list is by no means complete, but I hope it helps a little. I encourage others to add to my list!
Disclaimer: I am partial to a more conversational reading style, as that is how I write. Don't be too formal in your writing style -- it's already mathematics for heaven's sake!
Best,
Dylan