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. 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
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.
Some further advice on mathematical exposition:
- Michèle Audin’s “Conseils aux auteurs de textes mathématiques“.
- Clark Barwick’s “Notes on mathematical writing“.
- 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“
- Jean-Pierre Serre’s “How to write mathematics badly“
147 comments
Comments feed for this article
31 January, 2010 at 1:46 pm
Menulis « random notes
[…] yang menyegarkan adalah mereka yang gemar menulis. Seperti Terence Tao. Ia rajin sekali meng-update blog-nya, serajin menulis risalah ilmiah dalam teori bilangan. Dalam […]
27 March, 2010 at 3:58 am
Euler
Dear Mr.Tao
I just ask if you can help me for published a paper in mathematics in a journal,because this is my first paper?
I have preprint on arXiv but I can not put it on a journals with editorl boards.
I have 5 months to have a favorable opinion.
Thank you for help.
12 April, 2010 at 9:33 pm
researcher
Dear Prof. Tao,
is there a web page to learn ”impact factor of mathematical journals”?
Could you please explain what this number means and its importance?
thanks
21 April, 2010 at 10:59 pm
Write in your own voice « What’s new
[…] On writing […]
22 April, 2010 at 6:47 am
Career advice « 火柴人的星空
[…] Write down what you’ve done, and make your work available. In this regard, I have some advice on how to write and submit papers. […]
11 May, 2010 at 6:17 pm
Im Chae Won
Dear, Mr. Tao
My nationality is Korea
I am elementary school student
I want to be a mathematician when I grow up
Please tell me the secret method of being a good mathematician
7 June, 2010 at 3:18 am
porton
What sounds better, “f is a lower adjoint” or “f has an upper adjoint”? (about a Galois connection).
7 June, 2010 at 3:44 am
porton
Also, what sounds better, “f is a lower adjoint of g” or “g is an upper adjoint of f” where f is an important function and g is a not so important function (g appears only as an upper adjoint of f).
27 June, 2010 at 2:40 am
Chaofeng Zhang
Dear Tao, I am very happy to come across you on the website. I am a new maths teacher in a college of southwest China. I major in algebra. But I feel sorry to refer to my study,since I can’t collecet any meaningful paper. If you have some meaningful references concerning algebra,I ecpect your help by email. I want to do some research on mathematics. thank you.
7 August, 2010 at 5:15 am
James Clark
Organization in writing is a huge key. As time goes on, your writing style will develop and organization changes with that. You have a good informative blog.
21 September, 2010 at 10:43 am
bre
I am a 8th grade teacher in NC and came across your site while researching some information about writing techniques for my English class this year. I just wanted to thank you for the great information and articles about writing, and let you know about a site we are putting together to help teachers find trusted resources.
We would love it if you could write a few articles for us, but understand that you are probably busy. I have included a link to the site below in hopes that if you can’t write some resources for us that you can at least link to it, tweet it, or add it to your Facebook profile to help us spread trusted resources throughout the educational community.
http://www.thefreeresource.com
Thanks and keep the great resources coming :)
Bre Matthews
28 September, 2010 at 1:51 am
Daviduniverse
我是一名中国的大学生,想知道你的思维方式是如何的,怎么学数学呢?我是每天泡在图书馆里学,总感觉没进步,你的思维速度很快么?智商怎么样能得到最好的提升?我通过加快思维速度与准确性能达到150左右,但已经到了极限,能不能突破?思维速度重要么?
28 September, 2010 at 8:46 am
Terence Tao
See my career advice pages for my thoughts on these issues, particularly
https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/
https://terrytao.wordpress.com/career-advice/ask-yourself-dumb-questions-%E2%80%93-and-answer-them/
In the long run, intellectual speed and the type of traits measured by such statistics as IQ are not actually all that relevant to the ability to do mathematics, which relies instead more on getting the right level of understanding for the subject and on systematically and patiently attacking a problem or range of problems.
4 October, 2010 at 11:36 pm
Pwri
I have read your advise on writing and career and I would say that all of them were very interesting and smack of reality. I would liketo hear about teaching from the perspective of a Mathematician of your calibre.
20 January, 2011 at 8:01 am
我如何安排时间(译自陶哲轩博客)[转载自刘小川WordPress] | 宇宙海洋™
[…] 受到一些评论的鼓励,我最终决定在这里写一些关于如何安排时间的建议。其实,我有这个打算已经一段时间了,可是就我自己的情况而言,这方面也还在做 着探索(读者应该看看我等着写的论文排了多少!)而且很多想法未必成熟。(已经有一些经验写在advice on writing papers,比如page on rapid prototyping)而且,我的一些个人经验恐怕也不能对所有人通通适用,因为每个人都有不同的性格类型以及工作状态。 欢迎大家把自己的想法啊,经验啊,或者建议在评论中写出来。(其实,即使我自己的经验,我有时候也不能严格的遵照,挺遗憾的。) […]
1 February, 2011 at 9:09 pm
Zeeshan Mahmud
Professor Tao,
Thank you for taking your valuable time to open up a space for hive minds. It reflects great humility for a genius of your caliber to pave opportunities for interaction.
You mentioned:
“General note: Papers that are outside the scope of a mathematical research journal (e.g. a paper primarily concerned with physics, metaphysics, philosophy of mathematics, history of mathematics, mathematical education, or mathematical criticism) will almost certainly be rejected by any one of the journals above.”
However, in the Mathematical Subject Classification in JAMS, there is a section for history of mathematics, logic and foundations, and others.
I was wondering if you could clarify on the restrictions as far as philosophy of mathematics is concerned as that section would include logic and foundation.
Thank you once again your contribution in your blog and helping others.
15 March, 2011 at 6:56 am
liuxuewudi
[…] 受到一些评论的鼓励,我最终决定在这里写一些关于如何安排时间的建议。其实,我有这个打算已经一段时间了,可是就我自己的情况而言,这方面也还在做着探索(读者应该看看我等着写的论文排了多少!)而且很多想法未必成熟。(已经有一些经验写在advice on writing papers,比如page on rapid prototyping)而且,我的一些个人经验恐怕也不能对所有人通通适用,因为每个人都有不同的性格类型以及工作状态。欢迎大家把自己的想法啊,经验啊,或者建议在评论中写出来。(其实,即使我自己的经验,我有时候也不能严格的遵照,挺遗憾的。) […]
18 March, 2011 at 10:51 am
Mustapha Abboud
Dear Terry,
Do you usually go through your mail before you decide which one to read or someone else goes through your mail first before you receive it?
I sent you a letter last year and did not receive a response from you. I was wondering if you had actually read it.
Mustapha
3 April, 2011 at 5:48 am
我如何安排时间(译自陶哲轩博客)- (转载自Liu,Xiaochuan的博客) « What's new
[…] 受到一些评论的鼓励,我最终决定在这里写一些关于如何安排时间的建议。其实,我有这个打算已经一段时间了,可是就我自己的情况而言,这方面也还在做着探索(读者应该看看我等着写的论文排了多少!)而且很多想法未必成熟。(已经有一些经验写在advice on writing papers,比如page on rapid prototyping)而且,我的一些个人经验恐怕也不能对所有人通通适用,因为每个人都有不同的性格类型以及工作状态。欢迎大家把自己的想法啊,经验啊,或者建议在评论中写出来。(其实,即使我自己的经验,我有时候也不能严格的遵照,挺遗憾的。) […]
18 April, 2011 at 2:54 am
Len Shantz
I am extremely impressed with your writing skills and also with the layout on your weblog. Is this a paid theme or did you modify it yourself? Anyway keep up the excellent quality writing, it’s rare to see a nice blog like this one these days..
6 May, 2011 at 5:56 pm
Flavor: Working through a proof. « Flavors and Seasons
[…] blog post by Terence Tao has links to several comments on how to read […]
20 May, 2011 at 12:01 am
Season: Pulling together and writing up. « Flavors and Seasons
[…] Tao has advice on writing […]
20 May, 2011 at 12:24 am
science and math
@len shantz
Yeah The writing style of prof. Tao is great.
He writes with simplicity yet with the complex meanings given by the words.
31 May, 2011 at 11:59 pm
Advice on writing from Terence Tao | Science Library
[…] here to view the advice on writing from Terence […]
23 August, 2011 at 10:43 am
Descargar
Hey, do you have a facebook fanpage?
28 December, 2011 at 7:48 pm
Make your work available (By Terence Tao) « Nguyễn Đình Công
[…] also “Write down what you’ve done“, as well as my advice on writing and submitting papers. Share this:TwitterFacebook Hướng dẫn viết đề cương, khóa luận, chuyên […]
3 March, 2012 at 7:31 am
porton
Some more advice on writing math papers:
Click to access writing.pdf
These advices contain among other some very concrete suggestions, such as to display a formula longer than 3/4 of the width of the text, as well as more abstract suggestions.
16 April, 2012 at 3:30 pm
Caibin Zeng
Dear Prof. Tao,
Noting that we can get the Laplace transform of frequency differentiation $t^n*f(t)$ as $(-1)^n F^(n)s$. What about the case that $n$ is not an integer, such as 1/2 or other rational number? Do you find any results concerning to this situation?
Caibin
20 June, 2012 at 5:47 pm
KY
Dear Professor Terence Tao:
I have a question on young mathematicians writing reviews for AMS Mathematical Reviews. Even prematurely as a mathematician, when one starts to publish papers in journals, he/she starts being requested to write reviews for some articles and even books. On one hand, the young mathematician is motivated to try hard to write good reviews and hence forced to learn a new sub-field even if it is not completely the field with which he/she is familiar. On the other hand, perhaps he/she should avoid spending time on these things and rather focus on his/her own research which he/she has a lot more to explore? Any feedback will be appreciated.
Thank you.
22 October, 2012 at 11:03 am
porton
Should a. textbook; b. research monograph have a terminology index near the book end? Or is index a waste of space?
22 October, 2012 at 5:30 pm
quasihumanist
One of the most useful parts of Hartshorne’s algebraic geometry textbook is its excellent index!
Unfortunately, indices in most textbooks are not nearly so helpful.
3 November, 2012 at 8:03 am
Some advices to young mathematicians « Research and Lecture notes
[…] think, can specially be useful. The blog of Terence Tao is also an excellent source for advices on writing mathematics and academic careers in this […]
21 November, 2012 at 4:58 pm
thichchaytron
The blog of Tenrecen Tao is very good……
30 November, 2012 at 5:25 pm
J. T. Salamanca
VIC …Maestro Tao ¿viene a Spain para poder conocerle?
15 December, 2012 at 8:31 am
Zeraoulia
I very confused about the behavior of mathematical community. I have worked 15 years in chaos theory and I have published more than 80 papers and 8 books in several international publishers and in each time I seek comments from other specialist, they answred my positively and they never refuse to read may papers. For this time I send a request for comments (for my paper: http://vixra.org/pdf/1210.0176v6.pdf) to more than 800 mathematician arround the world and I receive only 7 replies. The main objective is to see the opinions of experts before sending the paper to a journal. Is the mathematical community behaves like the logistic equation (chaotic). More than this, some of them attack me personaly with very bold words in despite they do not know me.
29 December, 2012 at 7:31 am
Elias Rios
Hi, my name is Elias Rios, i’m professor the University National Technology, i can´t compile the paper when using latex, miktex. I using word WS.
I want to know if anyone can recommend a program to write and compile a latex and ArXiv?
15 April, 2013 at 12:12 am
我的动机们 | Shrinklemma
[…] 写作业的动机:为学习课程服务,如果只是为了完成任务,不如不写。写作业是一个锻炼writing的好机会,可以借此机会联系数学的表达和latex的使用,更可参见陶的博客On writing。 […]
3 May, 2013 at 10:09 pm
十一城elevencitys.com » 我如何安排时间(译自陶哲轩博客)
[…] 受到一些评论的鼓励,我最终决定在这里写一些关于如何安排时间的建议。其实,我有这个打算已经一段时间了,可是就我自己的情况而言,这方面也还在做着探索(读者应该看看我等着写的论文排了多少!)而且很多想法未必成熟。(已经有一些经验写在advice on writing papers,比如page on rapid prototyping)而且,我的一些个人经验恐怕也不能对所有人通通适用,因为每个人都有不同的性格类型以及工作状态。欢迎大家把自己的想法啊,经验啊,或者建议在评论中写出来。(其实,即使我自己的经验,我有时候也不能严格的遵照,挺遗憾的。) […]
5 June, 2013 at 1:21 am
Dr. Sunil Kumar Kashyap
Dear Professor Tao,
You can become next Newton, Gauss or Archimedes. Or you became.
The world salute to you. All the best.
Sunil Kumar Kashyap
12 July, 2013 at 12:48 am
Katy Smith
Lots of learning stuff here. Thanks for people like you. After reading, I realize that a certain rule is not always applicable to everyone and that includes writing.
1 August, 2013 at 11:36 pm
neymetms
I proved hypothesis Legendre wanted to send in arxiv.org, but there
require Endorsement needed for math.NT, you would not have helped me?
Я доказал Гипотезу Лежандра , хотел послать в arxiv.org, но там
требуют Endorsement needed for math.NT, Вы бы мне не помогли ?
6 August, 2013 at 10:31 am
neymetms
By Chudakov is that if you take the odd $ x> P_ {n} $, where $ P_ {n} $ – simple and to consider
difference $ X-P_ {i} $
and taking into account the
$ \ beta (x) = O \ left (\ frac {x} {\ ln ^ M x} \ right) $ – even among those who can not be branded
as the sum of two prime numbers, it follows that one of the differences
$ X-P_ {i} = P_ {k} + P_ {j} $,
$ X = P_ {i} + P_ {k} + P_ {j} $.
We chose $ x $ and got Vinogradov theorem …
To answer this Tao?:
17 August, 2013 at 6:27 am
How to write mathematics badly « Research and Lecture notes
[…] is not always clear how to write a mathematics paper but it is much more clear how not to write it […]
19 October, 2013 at 12:15 pm
lm
SVP Monsieur j’aimerais savoir dans le problème de Goldbach binaire, si on estime l’arc mineur est-il simple de calculer l’arc majeur? merci .
26 June, 2014 at 2:03 am
Terry Tao: On Time Management | Fahad's Academy
[…] up) and I don’t yet have a coherent or definitive philosophy on this topic (other than my advice on writing papers, for instance my page on rapid prototyping). Also, I can only talk about my own personal […]
28 July, 2014 at 9:26 am
Anonymous
Dear Professor Tao,
Thank you very much for all the great advice! Unfortunately, since Google Buzz closed down, the following links are broken:
“The ratio between results and effort in one’s paper should be at a local maximum.”
“On the use of implicit mathematical notational conventions to provide contextual clues when reading.”
and
“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.”
I would really appreciate it if you would reprint these articles.
Thank you in advance!
29 July, 2014 at 7:02 am
Terence Tao
The articles are now reprinted on my blog:
9 October, 2014 at 2:54 am
Anonymous
What tools do you use for bibliographies? I find it very time consuming to do it by hand. I have heard that many people use “BibTeX” with some other program like “Mendeley”.
16 October, 2014 at 9:57 am
DP
Typo here: It is also assists readability if you factor the paper into smaller pieces, for instance by making plenty of lemmas.
[Corrected, thanks – T.]
26 November, 2014 at 2:42 am
Bek
Dear Prof Tao
I am fun of your math I am master student My scientific work about Discrete time dynamical System in simplex. Which university i can study PhD. Thank you.
12 January, 2015 at 7:07 pm
JoAnne
Dear Terry Tao —
Often I enjoy and learn from your blog. But today when I searched using the term “poetry” I came up with no matches. Perhaps you can fill that gap by noticing my blog “Intersections — Poetry with Mathematics” at http://poetrywithmathematics.blogspot.com.
Enjoy!
JoAnne Growney
29 January, 2015 at 2:57 pm
Xu Peng
Dear Prof. Tao, I have a question about trigonometric series: We know that the series \sum cos(2^nx)/\sqrt{n} is almost everywhere divergent in [0,2\pi] because it is a Lacunary series, But can we prove that the L1 norm of this series goes to infinity? Thank you very much for your help!!
29 January, 2015 at 3:17 pm
Xu Peng
Sorry I made a mistake, I mean, can we prove that the L1 norm of \sum cos(2^n x)/\sqrt{n} is not bounded? Thank you very much!
11 February, 2015 at 2:26 pm
On writing | Ichigo Ichie
[…] On writing. […]