Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Subscribe to feed There are three rules for writing the novel. Unfortunately, no one knows what they are. ( W. Somerset Maugham ) Everyone has to develop their own writing style , based on their own strengths and weaknesses, on the subject matter, on the target audience, and sometimes on the target medium. As such, it is virtually impossible to prescribe rigid rules for writing that encompass all conceivable situations and styles. Nevertheless, I do have some general advice on these topics: Writing a paper Use the introduction to “sell” the key points of your paper ; the results should be described accurately . One should also invest some effort in both organising and motivating the paper, and in particular in selecting good notation and giving appropriate amounts of detail . But one should not over-optimise the paper. It also assists readability if you factor the paper into smaller pieces, for instance by making plenty of lemmas . To reduce the time needed to write and organise a paper, I recommend writing a rapid prototype first . For first time authors especially, it is important to try to write professionally , and in one’s own voice ; it can also help to make your past self your target audience . One should take advantage of the English language , and not just rely purely on mathematical symbols. The ratio between results and effort in one’s paper should be at a local maximum . Use the introduction to “sell” the key points of your paper ; the results should be described accurately . One should also invest some effort in both organising and motivating the paper, and in particular in selecting good notation and giving appropriate amounts of detail . But one should not over-optimise the paper. It also assists readability if you factor the paper into smaller pieces, for instance by making plenty of lemmas . To reduce the time needed to write and organise a paper, I recommend writing a rapid prototype first . For first time authors especially, it is important to try to write professionally , and in one’s own voice ; it can also help to make your past self your target audience . One should take advantage of the English language , and not just rely purely on mathematical symbols. The ratio between results and effort in one’s paper should be at a local maximum . Submitting a paper Proofread and double-check your article before submission ; you should be submitting a final draft, not a first draft . Submit to an appropriate journal . Proofread and double-check your article before submission ; you should be submitting a final draft, not a first draft . Submit to an appropriate journal . I should point out, of course, that my own writing style is not perfect, and I myself don’t always adhere to the above rules, often to my own detriment. If some of these suggestions seem too unsuitable for your particular paper, use common sense. Dual to the art of writing a paper well, is the art of reading a paper well. Here is some commentary of mine on this topic: On “compilation errors” in mathematical reading, and how to resolve them. On the use of implicit mathematical notational conventions to provide contextual clues when reading . On key “jumps in difficulty” in a mathematical argument, and how finding and understanding them is often key to understanding the argument as a whole . On “local” and “global” errors in mathematical papers, and how to detect them . Some general principles (with a worked example) on how to justify a complicated looking step in an argument, for instance by strategically guessing and introducing temporary notation . ( Here is a second worked example .) Some further advice on mathematical exposition: Michèle Audin’s “ Conseils aux auteurs de textes mathématiques “. Clark Barwick’s “ Notes on mathematical writing “. Andrea Bertozzi’s “ Advice for Writing Research Papers and Dissertations “ Henry Cohn’s “ Advice for amateur mathematicians on writing and publishing papers “. Keith Conrad’s “ Advice on Mathematical Writing “. Oded Goldreich’s “ How to write a paper “. David Goss’ “ Some hints on mathematical style “ Timothy Gowers on “ writing examples first! ” (see also this followup post ) Paul Halmos’ “ How to write mathematics ” (the book also contains similar pieces by Dieudonné, Schiffer, and Steenrod); the article can be found here . “ Mathematical Writing ” – notes from a lecture course by Don Knuth, Tracy Larrabee, and Paul Roberts. Dick Lipton on an analogy between paper writing and city planning . James Milne’s (sardonic) “ Tips for Authors “. Igor Pak’s, “ How to write math papers clearly “. Ashley Reiter’s “ Writing a research paper in mathematics “ Petra Schwer’s “ How to read Mathematics? A study guide “ Jean-Pierre Serre’s “ How to write mathematics badly “ My page on views on AI also contains some relevant advice on these topics. Recent Comments A Severe Misalignment of AI in Mathematics "Deep theorems were scarce and difficult and so became an effective mechanism to identify deep thought. AI has broken this system." 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Polymath 13 – a success! Non-transitive Dice over Gowers’s Blog Comments feed for this article 13 March, 2015 at 9:01 am I am writing a research monograph. What’s about this experiment: In my book write instead of ? 13 March, 2015 at 9:04 am Note that I don’t use $\frac{}{}$ construct to express division in my book (I use no division at all, as far as I remember), so it is free to be used for an other meaning. 18 March, 2015 at 6:37 am What also about a set definition laying on its side? . This may greatly reduce formula width (and this is a great problem for me). 18 April, 2015 at 6:56 am I am an amateur mathematician without official scientific degrees. I am writing a breakthrough research monograph in abstract mathematics. When I finish it, what is better: to publish it traditionally or to put its LaTeX files into GitHub.com under a free copyleft license so that everyone could be able to edit my work. That it needs editing, is quite probable because this is a very new field of research and the book may require changes to make it better and more general. The main issue here is that after the decision there is no way back: If I publish it traditionally I may lose copyright and be not able to distribute my LaTeX files for free, and reversely if I put it online with a free license, this may be an obstacle for publishing it. I ask you the advice, what to choose? I am also afraid, that if I don’t publish it traditionally, math community may refuse to cite my work (and thus the world is not worth to receive my discoveries). This is even despite that publishing under copyleft is better for hunting errors, as in GiHub and similar free Git hosters there are error reporting zillas. Please help me to make the correct choice. 9 August, 2021 at 10:59 pm I’m not Dr.Tao but here’s a link that might help you: https://math.mit.edu/~cohn/Thoughts/advice.html 30 May, 2015 at 10:12 pm Stephen King to share writing tips in new short story collection | Ismael Olson […] I also came across with this article as well: https://terrytao.wordpress.com/advice-on-writing-papers/ […] 14 June, 2015 at 11:53 pm Hi Terry, Some time ago you were an advocate of publishing no further than arxiv. However, with their comments it is more of the same. Is there a time stamping technique you can recommend so that we can move on and publish at our websites? 13 September, 2015 at 10:25 am Mathematic Reading | futileinfo […] Terry Tao’s https://terrytao.wordpress.com/advice-on-writing-papers/ […] 23 October, 2015 at 4:20 am Dear Tao Terence, Just for my curiousity, 1+2+3+4+5+6 …. = -1/12 is it true ????? It is used to resolve the “Casimir effect” ! (see demonstration : In french but easy to understand: https://sciencetonnante.wordpress.com/2013/05/27/1234567-112/ Do you work on this topic or it is not a serious topic ??? 23 October, 2015 at 9:12 am Ok I found the answer here: The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation thanks to being awesome 26 December, 2015 at 11:42 am If a lattice has no least element, then the difference a\b is undefined for every a and b in this lattice. So, is it worth to explicitly say about the lattice “with least element” when mentioning a\b? Omitting it would decrease the length of theorem conditions. But without this condition conditions would be warrantenly false. Example: “for a distributive lattice with least element there exists no more than one difference a\b of elements a and b” vs “for a distributive lattice there exists no more than one difference a\b of elements a and b” I suspect that omitting existence of least element would hinder exposition, because in this case instead of saying “the least element” I need to say in the proof a little more subtle “an element of the set of least elements” (this set is always of one or zero elements). What is better for: a. research monograph; b. textbook? 17 August, 2016 at 7:21 am On writing | PRINCE THEMBA […] Source: On writing […] 27 December, 2016 at 8:55 am It seems that the link to On key “jumps in difficulty” in a mathematical argument, and how finding and understanding them is often key to understanding the argument as a whole gives something else? 1 January, 2017 at 8:16 pm bhaskar singhania Sir! I find interest in maths and equally in philosophy.I have been writing them linking them in the best possible way I can..I adopt equations and theories from math and duduce explainations for some deep philosophical logics.like I have been working on limits and curves along with behavioural aspect of maths.The fact that my work doesnt incorporate(for now) a substantial amount of equations and symbols pulls me from submitting my works..kindly guide me as to where, or which university or professor should I write to! 30 January, 2017 at 4:10 am Dear sir, I am a 11 grader. I wanted to know what would be the sum of all the real numbers . I will be grateful to you,if you provide the answer. 15 February, 2017 at 5:49 pm More links from math professors – Lucy's World of Technical Communication […] math professor has a whole section On Writing on his website. A couple of my […] 24 June, 2017 at 4:05 am Writing 101 | Snowdrop's Notebook […] Terence Tao’s “On Writing“ […] 15 July, 2017 at 11:24 pm On writing « Abdulafeez Abdulkareem […] Source: On writing […] 2 September, 2017 at 11:06 am himanshubajpaiblog Hello Sir, I am graduate student in Mathematics .I have some basic results related to Fibonacci sequence and partition but I am not able to deciding whether it is publishable or not.Tell me How can I proceed ? 1 December, 2017 at 1:40 am Hi Dear professor, I am a master’s student in math who has a master’s degree in accounting and declined my Ph.D. admission in accounting and finance because I am really interested in math. In spite of don’t have enough math background but I think I can pass all courses in master’s period with A. but, it doesn’t drench my thirsty. I want to publish papers in this field but I study in CSUN in LA which is not a researched-based university. My goal for PhD is Caltech or UCLA. I want to know how can I write a paper or work under the supervision of a professor? Please help me. best, Reza 30 March, 2018 at 9:05 am DEAR TERENCE TAO, I AM A HIGH SCHOOL STUDENT AND HAS A MATHEMATCAL FINDING WHIVH I WISH TO PUBLISH.SO COULS YOU PROVIDE THE FORMAT IN TERMS OF MANUSCRIPT TITLE WITH AUTHOR ETC SINCE I DONT UNDERSTAND THEM NICELY 6 April, 2018 at 12:40 pm Dan Christensen Dr. Tao, I have written some proof-checking software and want to use it to write a series of formal proofs constructing the real numbers (as Dedekind cuts or Cauchy sequences) and establishing the various “axioms” for the real numbers (the field axioms, etc.) Can you recommend some book(s) with rigorous proofs and lots of detail, that I might use as templates for formal proofs? Regards, Dan Christensen (not the math prof at UWO) 6 January, 2019 at 7:20 am David J. Marcus The link to Keith Conrad’s “Error’s in mathematical writing“ does not work. It produces a PDF that says, “This le is no longer available.” 15 February, 2019 at 9:28 am Hi, I didn’t know if you were aware but Google+ is shutting down in April, which might mean that your posts on “compilation” errors and “local” and “global” errors might get lost. I’ve found those to be very helpful so was hoping you had found a back up, as it were. [Now saved to this blog – T.] 7 September, 2019 at 12:18 pm پیشنهادهایی برای دانشجویان تحصیلات تکمیلی سیستم‌های پیچیده - سیتپـــــور […] نوشتن مهمه. گاهی باید به استاد راهنماتون گزارش بدین، گاهی باید مقاله بنویسید و در نهایت پایان‌نامه خواهید داشت. جوری بنویسید که مردم احساس خوبی از نوشته شما پیدا کنند. […] 14 November, 2019 at 6:00 am Algunos consejos sobre Latex para principiantes | Caminos aleatorios […] a tener en cuenta sobre el estilo adecuado para escribir matemáticas. También puede ser útil esta entrada del blog del matemático Terry […] 18 February, 2020 at 6:02 pm Your link to Henry Cohn’s article seems to be dead. I think this is the new link: http://math.mit.edu/~cohn/Thoughts/advice.html [Corrected, thanks – T.] 27 February, 2020 at 7:57 am Prof Terence Tao , The strong Goldbach conjecture is a theorem, well I have proof. How i can send it ? 27 February, 2020 at 10:44 am Create a PDF file with your proof with the date and your name attached. Store this file securely. Use https://www.proofofexistence.com to store a crypto hash of your file. Now it is proved that you are the author and you can post your file publicly. But be advised that most proofs of strong Goldbach conjecture either just contain errors or are even a complete nonsense. Don’t wonder that others expect this from you. Expect that there is most probably an error yourself, too. But good luck. 27 February, 2020 at 8:24 am Prof Terence Tao , Euler and Goldbach are right when they announce this elegance guess, for years I have been fascinated by the strong conjecture of goldbach, but now I have come to a proof through pure mathematics. 28 February, 2020 at 12:38 pm Thank you, Mr. PORTEN, Thank you very much for your advice, the proof contains no errors, I am sure and certain. But also I expect nothing at all. 28 February, 2020 at 1:14 pm The description of the proof: Using pure mathematics and imagination, knowing that the Goldbach problem is a problem of pure mathematics: 1. I built a system 2n = prime (i) + composed (i) with i belongs to I 2n = prime (j) + composed (j) with j belongs to J 2.I have extracted two properties (A) and (B) from this system 3. By combining these two properties, we obtain prime (j) = composed (i). And finally: 2n = premium (i) + premium (j). 28 February, 2020 at 1:26 pm I will post the proof on arxiv.org thank you so much :) 10 March, 2020 at 2:11 pm On writing – michaelmorrowblog […] via On writing […] 14 August, 2020 at 3:55 pm Schreiben wie das Genie Terence Tao – Daniel Wieser […] und war Autor beziehungsweise Co-Autor vieler wissenschaftlicher Publikationen (Papers). In seinem Blog gibt er seine besten Tipps zum Schreiben solcher Arbeiten. Dennoch eröffnet er diese Liste mit […] 26 March, 2021 at 6:43 pm Vaibhav Kandari Hello Mr. Terrence Tao, I’m just 16 years old and I have two research papers. One is on a formula that I have made for ease in Statistics and the other research paper is on the useful patterns and observations on Collatz Conjecture. I have got them checked from my teachers. I want to publish those papers in nice journal, but I can’t find one, as they require me to belong some University… Therefore, I request u to name some nice journals, where I could publish my papers easily. Also, plz tell whether it is necessary to get those papers copyright… Any other extra advice is welcomed… 18 February, 2022 at 4:56 am jyotishraj thoudam Hi Terry, Thank you for posting all your thoughts in this wonderful platform. I also got inspired by your methods and your mathoverflow questions and answers with the late William Thurston on “Thinking & Explaining” 20 June, 2023 at 8:14 am 7 May, 2024 at 2:28 pm Why have you so far avoided the mathematics of war? 18 December, 2024 at 8:23 am gianmarcobrocchi The link in “make your past self your target audience” appears to be broken. I am curious to read the article which it refers to :) [Link added, thanks – T.] 5 April, 2025 at 7:23 am Dear Mr. Tao, I have bought Analysis I/II and I am going to read through them, write notes and comment here. I’d like to write a math book for my baby. Thank you for staying online. It’s hard for me to understand what you are blogging mathematically. But I can feel that you are warm hearted. 21 June, 2025 at 10:23 pm Recent articles on academic writing using AI – Arindam Sutradhar […] but then I saw the above one, and restrained myself from doing that (also, I have borrowed it from Terry Tao’s blog). But at least I should note my understanding about what to write in the age of AI, when AI itself […] 25 August, 2026 at 6:31 am 陶子, thank you for all the advice. This post and the blog is a treasure trove for me. 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