Among mathematical writers, Paul Halmos was unique because of his clear exposition and introduction of clever notations. Halmos, known for his famous autobiography, also wrote an insightful essay on how to write mathematics. Halmos claims in the beginning that the essay should be titled ‘How I write mathematics’. However, he is simply showing humility and we can get some important lessons from his essay. Here are some of his tips:
- Say something (Don’t communicate something non-substantive or with too many ideas)
- Speak to some one (Have the target audience in mind)
- Organize first (to maximize understanding and logical flow)
- Write in spirals (re-writing previous sections is required as the big picture emerges)
- Organize always (organization is required even while writing and after it)
- Write good English (correct and unobtrusive English will not distract the reader)
- Honesty is the best policy (clarity is better than pedantry)
- Down with the irrelevant and the trivial
- Do and do not repeat
- The editorial `we` is not all bad (Using a neutral voice is good but use of ‘we’ is preferable at times)
- Use words correctly (words of mathematics and logic should be used be used in an unambiguous way)
- Use technical terms correctly
- Resist symbols (minimal symbols makes the writing clearer)
- Use symbols correctly (correct and consistent usage of symbols is desirable)
- All communication is exposition (lectures, letters, articles and books should all be treated seriously)
- Defend your style (against editorial assistants who may mess things by mechanical implementation of style rules)
- Stop (one can’t keep improving the writing indefinitely)
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