# On mathematical writing

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This is maintained by Sang-hyun Kim

## References

1. [H] P. R. Halmos, How to Write Mathematics
2. [S] J. P. Serre, Writing Mathematics Badly
3. [K] S. L. Kleiman, Writing a Math Phase Two Paper
4. [BAMS] A manual for authors of mathematical papers, Bull. Amer. Math. Soc. 49 (1943), 1--16.

### Quotes

The sentences are rephrased by myself.

1. If there is if, write then. [H]
2. Don't start a sentence with a mathematical symbol. [H]
3. Don't ever use the word any for any circumstances. It's confusing even in the real world. [H]
4. Pick an expert of the field, and imagine you are writing a letter to that person. [H]
5. That is for a limitation, and which is for additional comments. [H]
6. When you say P does not imply Q, you should provide a counterexample or a citation. It doesn't mean I do not know how to prove Q from P. [H]
7. One (has thus obtained ...) doesn't make sense. Mathematics writing is a description of facts. Who are ones? [H]
8. Use we even the paper is single-authored. We means the reader and the author. But you should not write We have proved this in 2015 it is singled authored. We thank our wife for her help is illegal in many countries [H].
9. Use I only for something specific to yourself. Not in the content of the paper. [H]
10. If possible, avoid using 1st person reference. [H]
11. Given p, there is q doesn't make sense. It should be (If you are) Given p, (you) find q [H]
12. Repetition of sentences is not necessarily bad in mathematics. If there are five conditions appearing in chapter 1 and 2, and only the sixth condition is added, and write all five twice and emphasize the difference between chapters. [H]
13. Repetition of proofs are usually bad. Find a common lemma. It is annoying to see by the same technique (method/device/trick) as in the proof of Theorem 1 or see the proof of Theorem 1 [H]
14. For a pleural, you can use arbitrary to avoid any. [H]
15. Theorem must be short, in one short sentence. If you need eight conditions and five conclusions, you'll probably need a term for that. [H]
16. Don't use Where. $$a^n = 0$$ where n is... sounds, you forgot to put the condition at the beginning.[H]
17. Equivalent means the truth values coincide. What do you mean by two theorems being equivalent? [H]
18. Don't index equations that won't be used again. [H]
19. Like periods, the commas make reading difficult. Since $$p\ne0$$, $$p\in U$$ can be rephrased as Since $$p\ne0$$, it follows that $$p\in U$$. [H]
20. An example of semicolon: This is proved by Doe; see [Do]. C. Gordon told me this.
21. Don't name a thing that won't be used again. [H]
22. American English "Sounds of music.", British English "Sounds of music". I was personally told of this.
23. Authors are urged to make references specific by including page numbers. The extra space and trouble is fully justified by the saving to the reade. [BAMS]
24. A formula whose length is nearly half a line should be displayed. [BAMS]
25. Eschew the medieval practice of letting the title serve as an inflated advertisement. A title of more than ten or twelve words is likely to be miscopied, misquoted, distorted, and cursed. [BAMS]
26. Strive for proofs that are conceptual rather than computational. [BAMS]
27. It is good research practice to analyze an argument by breaking it into a succession of lemmas, each stated with maximum generality. It is usually bad practice to try to publish such an analysis, since it is likely to be long and uninteresting. The reader wants to see the path—not examine it with a microscope. A part of the argument is worth isolating as a lemma if it is used at least twice later on. [BAMS]