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How to write math papers clearly

Writing a mathematical paper is both an act of recording mathematical content and a means of communication of one’s work.  In contrast with other types of writing, the style of math papers is incredibly rigid and resistant to even modest innovation.  As a result, both goals suffer, sometimes immeasurably.  The clarity suffers the most, which affects everyone in the field.

Over the years, I have been giving advice to my students and postdocs on how to write clearly.  I collected them all in these notes.  Please consider reading them and passing them to your students and colleagues.

Below I include one subsection dealing with different reference styles and what each version really means.  This is somewhat subjective, of course. Enjoy!

4.2. How to cite a single paper. The citation rules are almost as complicated as Chinese honorifics, with an added disadvantage of never being discussed anywhere. Below we go through the (incomplete) list of possible ways in the decreasing level of citation importance and/or proof reliability.

(1) “Roth proved Murakami’s conjecture in [Roth].” Clear.

(2) “Roth proved Murakami’s conjecture [Roth].” Roth proved the conjecture, possibly in a different paper, but this is likely a definitive version of the proof.

(3) “Roth proved Murakami’s conjecture, see [Roth].” Roth proved the conjecture, but [Roth] can be anything from the original paper to the followup, to some kind of survey Roth wrote. Very occasionally you have “see [Melville]”, but that usually means that Roth’s proof is unpublished or otherwise unavailable (say, it was given at a lecture, and Roth can’t be bothered to write it up), and Melville was the first to publish Roth’s proof, possibly without permission, but with attribution and perhaps filling some minor gaps.

(4) “Roth proved Murakami’s conjecture [Roth], see also [Woolf].” Apparently Woolf also made an important contribution, perhaps extending it to greater generality, or fixing some major gaps or errors in [Roth].

(5) “Roth proved Murakami’s conjecture in [Roth] (see also [Woolf]).” Looks like [Woolf] has a complete proof of Roth, possibly fixing some minor errors in [Roth].

(6) “Roth proved Murakami’s conjecture (see [Woolf]).” Here [Woolf] is a definitive version of the proof, e.g. the standard monograph on the subject.

(7) “Roth proved Murakami’s conjecture, see e.g. [Faulkner, Fitzgerald, Frost].” The result is important enough to be cited and its validity confirmed in several books/surveys. If there ever was a controversy whether Roth’s argument is an actual proof, it was resolved in Roth’s favor. Still, the original proof may have been too long, incomplete or simply presented in an old fashioned way, or published in an inaccessible conference proceedings, so here are sources with a better or more recent exposition. Or, more likely, the author was too lazy to look for the right reference, so overcompensated with three random textbooks on the subject.

(8) “Roth proved Murakami’s conjecture (see e.g. [Faulkner, Fitzgerald, Frost]).” The result is probably classical or at least very well known. Here are books/surveys which all probably have statements and/or proofs. Neither the author nor the reader will ever bother to check.

(9) “Roth proved Murakami’s conjecture.7 Footnote 7: See [Mailer].” Most likely, the author never actually read [Mailer], nor has access to that paper. Or, perhaps, [Mailer] states that Roth proved the conjecture, but includes neither a proof nor a reference. The author cannot
verify the claim independently and is visibly annoyed by the ambiguity, but felt obliged to credit Roth for the benefit of the reader, or to avoid the wrath of Roth.

(10) “Roth proved Murakami’s conjecture.7 Footnote 7: Love letter from H. Fielding to J. Austen, dated December 16, 1975.” This means that the letter likely exists and contains the whole proof or at least an outline of the proof. The author may or may not have seen it. Googling will probably either turn up the letter or a public discussion about what’s in it, and why it is not available.

(11) “Roth proved Murakami’s conjecture.7 Footnote 7: Personal communication.” This means Roth has sent the author an email (or said over beer), claiming to have a proof. Or perhaps Roth’s student accidentally mentioned this while answering a question after the talk. The proof
may or may not be correct and the paper may or may not be forthcoming.

(12) “Roth claims to have proved Murakami’s conjecture in [Roth].” Paper [Roth] has a well known gap which was never fixed even though Roth insists on it to be fixable; the author would rather avoid going on record about this, but anything is possible after some wine at a banquet. Another possibility is that [Roth] is completely erroneous as explained elsewhere, but Roth’s
work is too famous not to be mentioned; in that case there is often a followup sentence clarifying the matter, sometimes in parentheses as in “(see, however, [Atwood])”. Or, perhaps, [Roth] is a 3 page note published in Doklady Acad. Sci. USSR back in the 1970s, containing a very brief outline of the proof, and despite considerable effort nobody has yet to give a complete proof of its Lemma 2; there wouldn’t be any followup to this sentence then, but the author would be happy to clarify things by email.

UPDATE 1. (Nov 1, 2017): There is now a video of the MSRI talk I gave based on the article.

UPDATE 2. (Mar 13, 2018): The paper was published in the Journal of Humanistic Mathematics. Apparently it’s now number 5 on “Most Popular Papers” list. Number 1 is “My Sets and Sexuality”, of course.

UPDATE 3. (March 4, 2021):  I wrote a followup paper and a blog post titled “How to tell a good mathematical story“, with a somewhat different emphasis.

  1. July 12, 2017 at 2:59 am

    I realize this is mostly humor, but I would disagree in several respects.

    First, I think it’s generally bad form to use citations as objects of prepositions, so I would rarely, if ever, write your example #1, preferring greatly to write #2 instead in nearly all circumstances.

    Second, your examples #3, #4, #7 are classic examples of “run-on sentences”, which both Wikipedia and I agree should be avoided. It is a common stylistic error.

    Third, style guides such as the Chicago Manual will tell you that you always need a comma after “e.g.”.

  2. July 12, 2017 at 3:11 am

    Thanks, Jeffrey. You are correct on the substance in all three instances. However, this is no humor and I personally dislike all three style rules which you mention, as I believe that they diminish clarity of the math writing for both native and (especially) non-native English speakers. In fact, given the precise manner in which we in math sciences need to communicate references, having more choice is good! So I don’t feel bad at all about abandoning century old rules which were designed for different types of communication. I explain it all in the notes I linked in the post.

  3. July 22, 2017 at 6:10 am

    @quasihuman (timestamps got reordered for some reason)

    About 1) – there is nothing to fix, in my opinion. The clauses and connections between them are clear. Also, I am against semicolons (this is explained in the writeup).

    About 2) – Right. I am perfectly comfortable if copy editors correct me. It’s their job. But my job is to communicate as clearly as possible in the version that’s free on the web. As I explain in the writeup, clarity is not a tradeoff, it’s *everything*. We have reached the point when over 90% of the world mathematicians are not native English speakers, yet English is the language we all use to communicate math. If some pedants are annoyed by the simplified and substandard writing style while still understand it clearly, it’s fine, really (cf. Simple English Wikipedia). This is because the clarity of understanding of people with poor English improves even by just a little then, which is all I want.

    In fact, if I thought that writing “ze rezults of zis teorem R enaf 2 show…” would improve the clarity of my math papers, I totally would use this style. I don’t believe it does though, not on any level. One reason is the modern technology – Google Translate wouldn’t get that, and I know many are using it to help with translation. I only recommend breaking standard rules about commas in a constructive way to signify the clauses. I feel this is a very minor grammar violation for good purpose.

    About completely restructuring sentences. I am not a fan. Sometimes this works well. But sometimes this results in a lengthy cumbersome construction communicating exactly the same thing, but whose only advantage is compliance with grammar rules. The result is less clarity. Remember, it’s people who are reading your paper, not some kind of compilers. You brain won’t stop working just because of some missing comma or a period.

  4. July 22, 2017 at 7:02 am

    1) Numbers 3, 4, and 7 can all be fixed easily; put in a semicolon in place of the comma.

    2) I have to disagree with the amount of encouragement you give for breaking grammar rules. The minor reason for the disagreement is that you annoy copy editors as well as grammar pedants like myself. More importantly, almost every time I am tempted to break a grammar rule for the sake of clarity, I can make the writing even more clear by completely restructuring the sentence or paragraph in the right way. Breaking a grammar rule should be the last resort when, after much effort, you find no other way of making the writing clear.

  5. October 1, 2017 at 11:32 pm

    There are citation typesetting styles that distinguish between textual (\citet) and parenthetical (\citep) references. For example, one could say, “Roth [1976] proved Murakami’s conjecture.” Being in the 21st century, the “Roth [1976]” would be a hyperlink.

    I very much enjoy this grammar book http://www.lel.ed.ac.uk/~gpullum/SIEG/
    You may too. Its point of view is that grammar shouldn’t be prescriptive but descriptive. That is, it should observe empirically what facilitates communication, and then describe it. (And they basically say that Strunk is an idiot.)

  6. October 1, 2017 at 11:40 pm

    @rgrig — Cool. Thanks for the link!

  1. July 25, 2017 at 1:35 pm
  2. October 26, 2022 at 10:23 pm

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