## 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!

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**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 Very occasionally you have “*see* [Melville]”, but that usually means that Roth’s proof is unpublished or otherwise unavailable (say, it was given at a lecture, and Roth can’t be bothered to write it up), and Melville was the first to publish Roth’s proof,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.

## You say goodbye and I say hello

I’ve been meaning to write a few posts for a while now, but never could find the time. It really takes special effort to clean your thoughts and then put them in order. However, the following story just fell into my mailbox. It tells you how to save time by skipping on the greetings/salutations. I am removing all math matters and leaving it undedited otherwise. To protect the anonymity of my correspondent, I will call him “Kiran” throughout the email exhange. Enjoy! — IP

(1) [Math] Thanks! — Kiran

(2) Dear Kiran,

[Math] Best, — Igor

P.S. In the future, please address me as “Igor”, which is my first

name. It’s best to begin your email with customary “Dear Igor”.

Thank you.

(3) I’ve been writing emails for 25 years, so I’m not about to start taking advice on how to start them; but if you start a thread to me with “Dear Kiran”, I can be safely counted on to respond in kind for the *first* email in the thread. If it happens enough times, I might even remember to initiate same way. For instance, this is what happens when I exchange emails with Serre; but neither of us uses the salutation on replies after the first within a thread.

[Math] Best, — Kiran

(4) Dear Kiran,

[Math] Best, — Igor

P.S. With all due respect, I am going to continue using salutations

and expecting the same in every email irrespectively on the person or

the count in the thread. Neither the “25 years” nor argumentum ad

verecundiam seem convincing — I have been using email for just as

long and in similar circumstances. The 8 letters of “Dear Igor” is

really not too much to ask.

(5) Thanks, I think this last reference does exactly what I was looking for!

Best, — Kiran

P.S. I have something more to say on the subject of salutations, but since that is a low-priority discussion for me, I will have to put it off until I am more current on my email.

(6) “Dear” Igor,

I promised one more piece of information regarding salutations, so here goes. (Don’t bother replying to this email; I promise to delete the response without reading it!)

I recently had some email exchanges with Shinichi Mochizuki, and was a bit surprised by the fact that despite the fact that I met him more than 20 years ago, he began his email with “Dear Professor [redacted]” (and persisted with this in subsequent replies within the thread). However, when I asked about this, he made it clear that on one hand, he has a policy of using the same format of salutation no matter the recipient (to avoid having to worry about the level of formality, figuring it is safe to err on the side of being too formal sometimes), he has absolutely no expectations about how anyone will address his in response.

My point is that you misuse a certain term here and it’s not the gratituous Latinate rhetorical terminology; it’s the word “respect”. It is a fact that reasonable people can draw different conclusions about such matters as how it is appropriate to start an email. You are free to choose how you address me, but how I choose to structure my correspondence is my decision alone. What you think is “not too much to ask” is for me to keep you in mind as a special case when I don’t even have very much correspondence with you anyway; that’s a waste of mental real estate that I can little afford.

— Kiran