Halmos - How to write mathematics - Executive Summary

Below is a summary of Halmos' classic essay that has been tailored for myself. I hope it is of use to others. I have left out all detail and thus the list below is in no way a substitute for the article itself.

1. Preface.
   1. What constitutes good mathematics exposition is subjective.
2. There is no recipe and what it is.
   1. Even if the ability to communicate clearly is innate, this essay may serve to "remind" (in the sense of Plato) the reader.
   2. The recipe for good exposition in mathematics is:
      1. Have something to say,
      2. Have someone to say it to,
      3. Organise what you want to say,
      4. Arrange it in the order that you want to say it,
      5. Write it, re-write it and re-re-write it, and,
      6. Think about and work hard on mechanical details.
3. Say something.
   1. There are to ways to fail to say something: no ideas or too many ideas.
   2. A good idea can shine even if buried in bad exposition, but don't rely on this.
4. Speak to someone.
   1. If you wish to reach your audience you must write as if writing for the ages.
   2. In addition to having a generally defined audience ensure that you focus your writing as if writing for a specific member of that audience.
      1. Avoid the temptation to include side remarks, polemic or in-jokes that comes with close identification with your audience.
      2. In writing for that specific person anticipate and avoid the problems you expect them to have.
      3. You may aim for an audience and you may miss. The chance, however, of hitting some audience is much higher than if you had not aimed at all.
5. Think about the alphabet.
   1. Invest an hour or two of thought in the alphabet; this'll save you a headache.
   2. Be consistent in your use of the alphabet.
   3. Bad notation can make otherwise good exposition bad.
   4. Good notation avoids dissonance.
   5. Avoid "freezing" notation or awkward or unhelpful notation.
   6. Do not increase the rigid frigidity of notation.
6. Write in spirals.
   1. Write your sections in this order: 1, 2, 1, 2, 3, 1, 2, 3, 4... This also applies to other divisions of writing, e.g. chapters, paragraphs, sentences, words...
   2. Write, re-write and re-write your work
   3. "Once you have a first draft in hand, spiral-written, based on a subject, aimed at an audience, and backed by as detailed an outline as you could scrap together, then your book is more than half done."
   4. On a first draft just write: ignore the rules.
   5. This is not literal advice. In a 10 chapter book the first chapter should not require 10 re-writes. I do, however, think 3 or 4 could be needed.
   6. After a spiral written first draft is completed I will usually re-write the whole book, adding mechanical reading aids (contents, index...). I then re-write the book again. It is this third draft that I give to others for review. Their comments are incorporated and this version is read, re-read, proof read and re-proof read. The result is the version that is sent to the publisher.
   7. Write a small part each day, no exceptions, no holidays.
   8. "Prime the pump for the next day"
7. Organise always.
   1. Organising material does not stop when writing starts: it continues throughout writing and editing.
   2. Enact spiral-organisation: write section 1, then section 2, then review section 1 in the light of section 2, and so on...
   3. The reader is supported by firm organisational scaffolding: even if he doesn't see it.
   4. Use sub-plots and leave clues of what is to come.
   5. Ensure that you have a logical structure.
8. Write good English.
   1. Use spelling and grammar that doesn't cause distractions.
   2. The style should be completely unobtrusive.
   3. Mathematical books should be written in good English style: that is with "correct" style according to current and commonly accepted public standards.
   4. Use common sense in your choice of words, punctuation and grammar.
   5. Repetitive use has a cumulative abrasive effect.
   6. The English language is a tool. Train yourself to use it.
9. Honesty is the best policy.
   1. A good style should smooth the reader's way. Anticipate their difficulties and forestall them: clarity not pedantry; understanding not fuss.
   2. Do not lie! Sometimes messy computations are needed.
   3. A statement like, "Note that p does not imply q" requires explanation. No explanation is equivalent to not revealing the whole truth.
   4. Using the word, "obvious" is ok if the result genuinely is obvious to people who have not written the paper.
   5. Do not hide the status of your statements or your attitude to them.
10. Down with the irrelevant and the trivial.
    1. Don't include unnecessary information, e.g. irrelevant assumptions.
    2. Insistence on legalistically correct but insufficiently explicit explanations is misleading, bad exposition and bad psychology: almost bad mathematics.
    3. State theorems up front, do not surprise the reader with, "Thus we have proven...".
    4. Do not include "chit chat" in theorems, e.g. "Moreover..." or "Without loss of generality we may assume...".
    5. Ideally a theorem is a single short sentence. Page long statements indicate a lack of clarity in thinking.
11. Do and do not repeat.
    1. Good repetition: word for word repetition of a phrase, or even many phrases, with the purpose of emphasising a slight change in a neighbouring phrase. Announce these differences clearly.
    2. Bad repetition:
       1. repetition with no reference to previous use.
       2. a proof or reference to a technique used in a proof. In such situations a lemma should be used.
12. The editorial we is not all bad.
    1. The best expository style is the least obtrusive one.
    2. Avoid the use of first person pronouns: simple declarative sentences are the best for communicating facts.
    3. Using the imperative will save time.
    4. Let "we" mean the author and the audience.
    5. Do not use "we" when you mean a singular author.
    6. Do not over use "we".
    7. "I", especially in overuse, can have a repellent arrogance: be careful.
13. Use words correctly.
    1. Think about and use with care the small words of common sense, intuitive logic and specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning.
    2. "Any" can mean either existence or be a universal quantifier. Avoid it. Use "each" and "every" instead.
    3. Formal logic is not good for communicating ideas. It is a code that the writer must encode and the reader must decode.
    4. Introduce assumptions first.
    5. "Equivalent" for theorems does not make sense. It usually means that the theorems imply each other: state this.
    6. Avoid statements logically correct by stylistically problematic statements like, "if p then if q then r".
14. Use technical terms correctly.
    1. "To illustrate the possibilities of the unobtrusive use of precise language in the everyday sense of the working mathematician I provide three examples:"
       1. A function and its values are different. So $latex z^2+1$ is even is bad usage while $latex f(z)=z^2+1$ is good usage.
       2. A sequence means a "function whose domain is the set of natural numbers". Hence "the union of a sequence of measurable sets is measurable" is bad usage while "the union of a countable set of measurable sets is measurable" is good usage.
       3. "contain" and "include" are almost always used as synonyms but $latex \in$ and $latex \subset$ have very different meaning.
    2. Bad use of specific technical terms distracts and postpones understanding.
    3. Failure of consistency can cause mild irritation through to serve misinformation.
15. Resist symbols.
    1. Everything said about words applies mutatis mutandis to even smaller units of mathematical writing: mathematical symbols.
    2. The best notation is no notation.
    3. Is the symbol needed for this sentence?
    4. Use no superfluous letters, use no letter only use, leave no variable free.
    5. "A rule can be bent but do not shatter it".
16. Use symbols correctly.
    1. Ensure that symbols have a consistent verbal translation.
    2. Do not overwork commas and fullstops. They are small and can easily be missed.
    3. Never start a sentence with a symbol.
    4. If you use an "if", then always add a "then".
    5. Observe the impact of each page of text: aim for an inviting picture.
17. All communication is exposition.
    1. "The differences between books, articles, lectures and letters (and whatever other means of communication you can think of) are smaller than their similarities."
    2. "Content, aim and organisation, plus the vitally important details of grammer, diction, and notation - they, not showmanship, are the essential ingredients of good lectures, as well as good books."
18. Defend your style.
    1. "Smooth, consistent, effective communication has enemies; they are called editorial assistants and copyreaders."
19. Stop.
    1. It is hard to stop. Take pleasure in allowing a manuscript to "ripen" by putting it aside for a few months before reviewing.
20. The last word.
    1. "Do, please, as I say, and not as I do, and you'll be better. Then re-write this essay and tell the next generation how to do better still.