According to the spiral plan the chapters get written and re-written in the
order 1, 2, 1, 2, 3, 1, 2, 3, 4, etc. You think you know how to write Chapter 1, but
after you’ve done it and gone on to Chapter 2, you’ll realize that you could have
done a better job on Chapter 2 if you had done Chapter 1 differently
I guess this is one of the best tips in terms of general structured writing.
You write chapter 1. This is math, you've laid out several definitions, theorems, propositions. You write chapters 2-5. You get to 5 and you realize your choice of terms was poor. Or that you only define some term in chapter 5.
You realize that definition should be before chapter 5, but in which chapter? 1, 2, 3, or 4?
You have to review each chapter (probably in reverse order) to determine the latest position it should be in.
Additionally, as you add that definition you find out that there's even more that needs to be shifted. Do you reorganize the whole thing, or leave it as is? You'll end up spending as much time reworking the text as you did writing the original.
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Really, this is a philosophical thing. In the spiral approach you end up reviewing things early and realizing what's missing and where they should be moved. In your approach, you don't get to do the review until late. Is it wrong? No. But it can create a less flexible document that requires more effort (and not less) to correct.
This is very similar to the argument in lean software development of keeping batch sizes low (small number of changes at a time) but running them more frequently. Versus the waterfall approach which has large batch sizes and takes much longer to do it all, with a big review near the end.
Paul Halmos is known as one of the greatest and most enjoyable expositers of mathematics of the twentieth century. I've certainly learnt a lot about writing in general (not just mathematical) from reading his stuff, and his book "Finite Dimensional Vector Spaces" was an aid to me when I was learning Linear Algebra ten years ago.
If the linked text is too long, you might enjoy his shorter piece "Mathematics as Creative Art" [0].
This was part of a collection of four essays on mathematical writing, commissioned by the American Mathematical Society:
The committee was authorized by the Council of the American Mathematical Society in August 1968; the last appointment to it was made by Oscar Zariski, then president, in March 1969. The charge was to prepare "a pamphlet on expository writing of books and papers at the research level and at the level of graduate texts."
In May 1969, two months after the committee was completed, one of its members resigned. He said he thought the project was too interesting to leave to a committee, which would never get it done properly, and he said he wanted to be free to write and publish his version independently. Norman Steenrod (the chairman) declined to accept the resignation, preferring to allow the member the freedom he sought. This left the exact membership of the committee up in the air.
The work of the committee proceeded mainly on Steenrod's steam; he wrote to the other members (in triplicate), and occasionally they would write an answer (to him alone). The committee met only once (for an hour, at the Eugene meeting in August 1969, with three present). The result of the correspondence and the meeting was the decision to present to the Council, as the product of the committee, four separate essays, one by each of the four members, with the recommendation that the Society publish them, together, as this book. > > A year later (in August 1970) Steenrod had at hand only one essay. A year and six months later (in March 1971) that essay was published. (L'Enseignement Mathématique, 16 (1970), 123-152.) Even so, Steenrod was still hoping; he set August 300, 1971 as a target date for the receipt of all the essays. The solution he proposed for the problem created by the already published essay was to reprint it as is, as part of the AMS publication, provided the editors and publishers of L'Enseignement Mathématique agreed. They did.
Steenrod died in October 1971, before quite completing his own essay. Before he died he asked, through his wife, that his nearly finished work be prepared for submission to the council and presented together with the others. That was done.
Respectfully submitted,
J. A. Dieudonné
P. R. Halmos
M. M. Schiffer
The other three essays are excellent as well and I recommend that you check out all of them. All of the authors are excellent mathematical expositors, though perhaps not as well-known as Halmos outside of the academic mathematics community. Dieudonné in particular had quite an illustrious writing career as well, having been part of the Bourbaki group as well as Grothendieck's EGA project.
“What we call calculus nowadays is the union of a dab of logic and set theory, some axiomatic the- ory of complete ordered fields, analytic geometry and topology, the latter in both the “general” sense (limits and continuous functions) and the algebraic sense (ori- entation), real-variable theory properly so called (differentiation), the combinatoric symbol manipulation called formal integration, the first steps of low-dimensional measure theory, some differential geometry, the first steps of the classical analysis of the trigonometric, exponential, and logarithmic functions, and, depending on the space available and the personal inclinations of the author, some cook-book differential equations, elementary mechanics, and a small assortment of applied mathematics. Any one of these is hard to write a good book on; the mixture is impossible.
Really. And he’s gonna teach us to write about math?
Seems like his point is that "what we call calculus nowadays" is ridiculous--the form of his sentence propels the meaning he's trying to express as well as the words do.
I guess this is one of the best tips in terms of general structured writing.