Michael Spivak's books have a unique style of presenting mathematics that makes the subject very enjoyable to read. It is something that I have tried to imitate myself during my mathematical noding here on e2. It's not something that he's ever so explicitly stated as I am about to. Spivak's philosophy can be extracted from his mathematical texts, because he has never published it in plain language. He is a real mathematician who does mathematics, and as G. H. Hardy mentions in his A Mathematician's Apology, it is a profound tragedy for a mathematician to be writing about mathematics rather than doing mathematics. In this spirit, perhaps, Spivak himself writes mathematics and lets us deduce from his presentation general guidelines of how mathematics is best written, but doesn't write about mathematics himself.
Opinions may differ on this matter, but I am fairly certain that Spivak would agree very much with the broad recommendations I give below for writing good expositions of mathematics for modern audiences.
The Spivak Style
- Embrace prose and avoid terse writing.
- Computations are just as important as concepts.
- Rigour is a means and an end of mathematics, not a nuisance.
- Mathematics is beautiful. Write it as such.
- Don't forget history and the human element.
- Have fun but keep a straight face.
Allow me to elaborate.
Embrace prose and avoid terse writing.
This piece of advice is a reaction against the Bourbaki style. Bourbaki is famous for explaining as little as possible in order to give the tightest presentation possible. While this is indeed, strictly speaking, correct mathematics according to our modern standards, it is also very hard to read and an altogether unpleasant experience unless you already know the subject matter and just want to review, not really learn a new subject.
It is better to write more than less. Explain ideas fully and clearly. You are writing in English, remember. Avoid unecessary proliferation of abstruse notation if possible. Mathematics has its mathematical language, but it's mostly for automatons to read, not humans. Have a clear structure of your paragraphs. Topic sentences. Do not hesitate to give your opinions and impressions of the subject. If you have a personal opinion of how something should be done, give that opinion and justify it. You are writing for people to read, not computers, so do not shy from writing more in order to explain more.
Computations are just as important as concepts.
This is a recommendation against that oh-so abundant "this exercise is left to the reader" clause when the author just doesn't feel like doing a little bit of hard work during a mathematical presentation. Don't give that clause. If there is a moment in your work whem you need to perform a long and boring computation in order to prove a point, then do the hard work! Your readers will be thankful for it. Complain if you want in your prose about how boring and hard the computation is, but do it anyways.
After all, this is how the masters of yore did mathematics, and something we have lost a little recently thanks to the laziness that computers and calculators afford us. Examine any of the old mathematical classics. You'll see that there are lots of computations in there embedded into the ideas that the authors are presenting. There is a computational and boring aspect of mathematics, and doing it peoperly is necessary in order to be able to step back and have a broader view of the topic.
Rigour is a means and an end of mathematics, not a nuisance.
Just because we recommend that your prose should be chatty, doesn't mean that it has to be messy or sloppy. There is a way to be rigourous and clear at the same time. Bourbaki-like rigour is possible without being boring and hard to read. Embrace the mathematical tradition that the logicians of the twentieth century have given us along with the analysts of the nineteenth century. Prove everything you use. Make your exposition as self-contained as possible. This is the way mathematics should be.
The detractors of this piece of advice often retort something along the lines that "too much rigour becomes rigor mortis". There is some truth in that, and it can be witnessed in the Bourbaki style. The issue here is the ability to separate the clear logical ideas from the presentation. I reiterate: just because logic is economical and austere doesn't mean that the prose has to be as well. Adhere strictly to the rules of logic in order to show how the subject you are presenting has internal coherence and fits in together with the larger picture.
Rigour is not something to be shied away from — it's the natural setting for mathematics to occur. It can be done properly, and it can even be fun.
Mathematics is beautiful. Write it as such.
Spivak didn't write The Joy of TeX and the AMS-LaTeX package for nothing. It is true that the peculiar mathematical notation necessitates some rather complicated typesetting, and in the past this has always been hard to do. I won't suggest to use LaTeX all the time for all mathematical typesetting (there are a few alternatives out there, though none quite as beautiful in my opinion), but do consider using LaTeX for all of your mathematical typesetting needs if it's possible.
Beautiful writing of mathematics isn't a mere matter of typesetting, though. Even with ugly typesetting there are other ways of bringing visual beauty into the presentation. Proper capitalisation, punctuation, and grammar almost go without saying, but you can also expand upon them by using descriptive language for your prose. You needn't write all of your mathematics as if you were writing on a strict template. Exercise some freedom with words from the dictionary. If you think a particular formula is attractive, say it exactly like that: "this formula is attractive" (assuming, of course, that "attractive" isn't already reserved in your prose for a mathematical/physical concept, such as an attractive force). Use techniques of classical storytelling if they help to push your exposition along. Build up suspense. Foreshadow a remarkable and surprising result that will come later. Express your admiration for a particular theorem if you feel like it. If you had to write down a particularly ugly computation because you had no alternative, then please apologise for displaying ugly but necessary mathematics.
Aesthetics. Always keep aesthetics in mind. Mathematics deserves them.
Don't forget history and the human element.
Mathematical history and folklore is rich. At least in Western tradition, to which most of the world now adheres mathematically, our history goes back thousands of years to the Ancient Greeks. Modern ideas didn't come out of nowhere, and almost always it is clearer for the modern reader to understand mathematics if there is at least a little bit of background ideas to support the current presentation at hand. Give those background ideas. Remember that mathematics is a supremely human activity, and humans have a history. Show us that history.
This is not to say that every mathematical presentation has to be a history text. You needn't give dates, places, particular events, nor even names of people who came up with the ideas. Instead, what you should do if it helps your presentation is to give a development of the ideas leading up to the current one. Why, for example, is the definition of a limit so complicated? Why should we care that numbers such as e and &pi are transcendental? These ideas are clarified by a context of other ideas, and we want to see that context.
Of course, not all ideas from the past are all that helpful. Humans make mistakes all the time. Do not include every little detail of history, like its burps, hiccups, and the times mathematics has stumbled. That would only muddle the real purpose of your presentation, which is to expose a piece of mathematics in the clearest and most enjoyable way possible. Hiding the embarrassments of the past is perfectly acceptable if you feel that they do not elucidate anything to us today, and this kind of latent revisionism is justifiable on the grounds that you are writing a text on mathematics, not history.
Have fun but keep a straight face.
Mathematics can and should be fun. Something has to counterbalance those boring and unavoidable calculations. We are not machines. Make a small joke now and then, very small, barely noticeable. Place a few puns, where appropriate. They should not distract from the real content. Have a sense of humour. Make a reference to popular culture, to literature, to history, if it seems in place. Michael Spivak likes to hide instances of his beloved Yellow Pigs and the number 17 in his books, hide something similar in your text if you want to. Enjoy, so that your readers notice your enjoyment and can share it too.
As a mean between extremes, do avoid being a complete buffoon. You're here to write mathematics, not to get cheap laughs from your audience. It's a delicate balance to strike between being a class clown and being a mathematician.
Mostly, the suggestion is to let your enthusiasm for mathematics show. Makes for a much more pleasant reading.