Advice for writing good mathematics?

Solution 1:

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

Solution 2:

I sympathize with some of the comments of Dylan Frank but I would argue to the OP that deciding who the audience is and the purpose of the book (point 1) affects everything to the point of somewhat invalidating some of the other complaints in Dylan Frank's answer: e.g. some books are supposed to be calculation and symbol heavy so that they act as efficient-if-somewhat-dry references for the people who already understand all the motivation and context but don't want to waste time rederiving things that are standard. Or some books are known for their myriad and challenging exercises and examples, amassed from thousands of pages of papers and other books. Sometimes, the advanced reader is thankful for a few `clearly' s because he/she knows all the standard arguments and just needs to skim a proof for one or two ideas. Obviously, for all of these things some books do them better than others, but in themselves I would not say that they are undesirable.

Basically, I think for the majority of mathematicians, the majority of textbooks fall short of what is being sought. However, this same fact means that for the majority of mathematicians there is at least one book which is more or less exactly what is being sought. So the diversity one sees in books of too wordy/too symbol-heavy or too dense/too drawn-out or too many examples and questions/no examples, just abstract theorems is a good thing; the literature is richer for it.

Having said that, obviously some books do manage to appeal to larger numbers than others and some are widely-regarded as excellent, but this is appropriately rare!