Writing Mathematics

• Steenrod distinguishes formal structure in mathematical writing from informal structure and asserts "This division of the material should be conspicuously maintained in any mathematical presentation" because it makes the logical structure clear.
  • Formal structure: definitions, theorems, proofs
  • Informal structure: motivations, analogies, examples, "metamathematical explanations"
• The primary purpose of a book is to present the formal structure.
• A secondary purpose (almost as important) is to offer the reader a method whereby they can fit the new structure into what they already know. It is here, in this secondary purpose, that authors exhibit greatest variations in skill and art.
  • What are the basic questions that will be answered?
  • What are the crucial examples that motivate development?
  • What are vaguely formulated principles from which the entire theory seems to unroll effortlessly?
• Huge danger in writing about mathematics instead of doing it

This is probably the most famous entry in the book, and can be found if you google for it. The essay spans pages 19–48 (so, roughly 29 of the 64 pages of the book, nearly half the book is Halmos's essay!). The essay consists of 21 sections, beginning with section 0 "Preface".

1. Have something to say
   • The problem is to communicate an idea. To do so, and to do it clearly, you must:
     • have something to say,
     • have someone to say it to,
     • organize what you want to say,
     • arrange it in the order you want it said in,
     • write it, rewrite it, re-rewrite it several times,
     • be willing to think hard about it and work hard on the mechanical details (e.g., diction, notation, punctuation, etc.).
   • That's basically it.
2. Say something
   • Problem 1: you have nothing to say.
   • Problem 2: you have too much to say.
     • Calculus textbooks have this problem because "calculus" isn't a single subject, it's an alloy: a mixture of many interesting subjects.
   • "To have something to say is by far the most important ingredient of good exposition".
3. Speak to someone
   • This determines:
     • the amount of motivation you need to put in,
     • the extent of informality you may allow yourself,
     • the fussiness of the detail that is necessary,
     • the number of times things have to be repeated.
   • All writing is influenced by the audience. But given the audience, an author's problem is then to communicate with it as best they can.
   • Anticipate and avoid the reader's difficulties.
4. Organize first
   • Organize and arrange the material so as to (a) minimize the resistance and maximize the insight of the reader, (b) keep the reader on track with no unintended distractions.
   • Why prefer a book over a stack of papers?
     • Efficient and pleasant arrangement
     • Emphasis where emphasis is needed
     • Indication of interconnections
     • Description of examples and counter-examples on which the theory is based.
   • These are precisely "organization".
   • Once you know what you want to say (and to whom you want to say it), the next step is to make an outline.
   • To organize a subject means: to decide what the main headings and subheadings are, what goes under each, and what are the connections among them. (A diagram of the organization is a graph, probably a tree, but not a linear chain.) There are many ways to organize a subject.
     • We then arrange the results of the method of organization in a linear fashion.
     • Organization > Arrangement.
   • The preparation of an outline can take years (or, at the very least, many weeks).
   • Halmos remarks that when he does his daily work, he daydreams about the project, and as ideas occur he jots them down on loose slips of paper then places them "helter-skelter" in a folder. Ideas could be what fields of math to include, it could be a proof, it may be a witticism. Then when Halmos writes, he "plays solitaire" with the slips in the folder. (Page 25, first full paragraph of the page.)
   • "What to put in?" is as important as what to leave out.
5. Think about the alphabet
   • Think hard about the symbols you use, the notation used, because if it's adopted…then it's impossible to change.
   • Halmos says he writes tables of notation before writing a book, with different letters in different fonts and what they correspond to.
   • Personally, I'm not as pessimistic as Halmos. When he says something like "Who would dare write 'Let 6 be a group.'?", well finite group theorists do this! So perhaps we should not be quite as paranoid as Halmos when it comes to notation, but we should abstract away notational choices as LaTeX macros.
6. Write in spirals
   • It's impossible to know what to write until after it's done. You know what to put in chapter 1 (or how it should've been written) only once you've completed your draft and have moved on to chapter 2. But once you've finished chapter 2, you realize how much better it could have been if you had only written Chapter 1 slightly differently.
   • Halmos says this is fine and natural, and that the correct order for writing a book is: 1, 2; 1, 2, 3; 1, 2, 3, 4; etc.
   • But the first step in writing, rewriting, re-rewriting, etc., is writing.
   • "There is no better incentive for writing a good book than a bad book."
   • In the first draft of each chapter, Halmos recommends that "you spill your heart, write quickly, violate all rules, write with hate or with pride, be snide, be confused, be 'funny' if you must, be unclear, be ungrammatical—just keep on writing."
   • Do not edit, but rewrite.
7. Organize always
   • Spiral writing goes hand-in-hand with spiral organizing
   • Review the material introduced in section 1 in future sections (as examples, limitations, generalizations, etc.). The material in sections 1 and 2 should be reviewed in section 3 (as examples, counter-examples, epitomizing various properties, etc.), and so on.
   • But in this process, you may realize there are important examples which should have been given in section 1 for use later on
   • This seems to be the gist of "spiral organizing"
8. Write good English
9. Honesty is the best policy
10. Down with the irrelevant and the trivial
11. Do and do not repeat
12. The editorial we is not all bad
13. Use words correctly
14. Use technical terms correctly
15. Resist symbols
16. Use symbols correctly
17. All communication is exposition
18. Defend your style
19. Stop
20. The last word

1. Schiffer writes "more as a reader…than as an author". But there is no universal prescription for writing in a clear, informative, attractive manner.
2. When planning, the author must determine their audience and what amount (and type) of information the author wishes to transmit. Schiffer partitions the space of writing into four types of expositions: research paper, monograph, survey, textbooks.
3. Research Paper.
   • Generous quotation of sources, clear stating of facts used, precise definitions, complete proofs (if proofs are given at all).
   • It is very important to write a good introduction. You must help the reader determine if the paper is relevant to their research as quickly as possible.
4. Monographs.
   • Requires much more planning and attention than the research paper.
   • Since monographs have a longer lifetime than research papers, it's more important to plan and write well.
   • A monograph should include the background and motivation for basic concepts, the growth of ideas and methods, more detailed proofs should be provided, an extensive bibliography is a must.
   • A good introductory chapter should whet the appetite of the reader.
   • A historic and genetic approach may yield a good general guideline for organizing the monograph.
   • A monograph is not written to show how erudite and clever the author is, but to teach the reader some new material.
5. Surveys.
   • A survey is an invitation to a field of research, not an introduction.
   • Motivation, background, history, and problems should be displayed.
6. Textbooks. (This is the longest, or second longest, section of the piece)
   • Schiffer restricts discussion to writing advanced textbooks for senior or graduate level.
   • The purpose of a textbook is to take a student (with a specific amount of preparation) and introduce them to a new field of mathematical endeavor. Therefore the presupposed knowledge of the reader should be precisely realized, and the treatment of the textbook takes this carefully into consideration.
   • The textbook should rise from the known to the unknown in easy steps. It should start from the special and intuitive to the general and abstract.
   • Applications and examples should be generously given.
   • A real textbook should contain more material than could be covered in a single course. This allows teachers a certain amount of flexibility when using the textbook.
   • "The main desideratum would be that the solution to all significant problems in the exercise section should be given, or at least hinted at."
   • The textbook author should "aim for clear and interesting exposition rather than for completeness or novelty."
7. Now, instead of discussing "what book to write", Schiffer turns to the question "How to write it?"
   • Plans change and the production of any book is a process of successive approximation.
   • The worst moment in preparation of a book (or any writing) is the first moment staring at a blank sheet.
   • Schiffer's advice: start with a chapter which interests you the most, in which you think you can be most original and helpful.
   • Then ask yourself what material is needed to bridge from the presupposed knowledge of the reader to that chapter you've just written?
   • The auxiliary chapters are probably dry, unintersting, and consists of most of the text you've written. These chapters have to be fleshed out. "The important principle is that no chapter shall be entirely auxiliary and be the servant of some other chapter."
   • Schiffer contrasts this to how Polya writes his books. Polya uses a system which Polya calls "skeleton writing". Prepare the book according to your plan, but in a very sketch and incomplete way. Once the skeleton is ready, flesh it out and bring it to life.
   • Use the manuscript as the basis for a course to test (1) the clearness of exposition and (2) the logic of the arrangement.
   • Final advice: "Enjoy your writing and relax while doing so. Write in a natural style and leave the officialese and formal style to administrators and government departments."

• This is a short essay (a mere 3 pages!) disagreeing with everything Steenrod advocates.

There's frustratingly little of use in this lecture.

A lot of rules about Pascal programming in WEB. Perhaps the only gem in this lecture is: "We are trying to convey an intuition of what is going on, so a high-level account is much more helpful."

Each module/section/chunk of code should be manageable in size. Knuth thinks about a dozen lines of code is just about the right size.