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 further advice on mathematical exposition:

- Michèle Audin’s “Conseils aux auteurs de textes mathématiques“
- Henry Cohn’s “Advice for amateur mathematicians on writing and publishing papers“.
- 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.
- Ashley Reiter’s “Writing a research paper in mathematics“
- Jean-Pierre Serre’s “How to write mathematics badly“

## 117 comments

Comments feed for this article

22 July, 2007 at 5:06 am

math_dreamer_liuHello:

May I ask you a simple question?

I am already 26. I found I was interest in Mathematics. I decide to begin from Math Analysis.

Some friends thought I was crazy.

Was it too late for me to study Math on such a 26 age?

22 July, 2007 at 12:40 pm

AnonymousAssuming you’ll be attending a University:

As someone who went back to study Math/Physics at around the same age, no it is certainly not too late. The main problem I encountered is that I forget just about everything I learned in high-school (hell, I forgot about cross multiplication). So, the first about 2 months of study was me sitting in the Physics student room studying and asking questions for most of the time I spent there (about 9am – 10/11 pm 5 to 6 day a week) re-learning high-school math in parallel with Intro Calc, etc. After that, the time I needed to spend studying was about the same or less than the other students.

I must say that the only linchpin in what I experienced was people not getting frustrated with my constant questions. I was fine because people realized that I only needed conceptual help, but that isn’t to say that you’ll run into similar people.

But, to increase your chances I’d suggest looking into any prep courses that your local university may offer. If it does have something like that, then you’d avoid a lot of the pain that I had to endure (assuming you’re in a similar situation to me of course).

If you think that you’ve maintained a lot of your high-school knowledge, I’d recommend verifying that by working through an intro calc text (e.g. Calculus: A Complete Course by Robert A. Adams). At the very least, this will fill in the holes that you don’t know aren’t filled.

Assuming self-study:

The term Analysis brings books like Rudin to mind. If this is the case, then yes, you’re crazy. Especially, after a hiatus from mathematics. I seriously hope that you mean the typical “pedagogically correct” way of learning calculus (i.e. starting from a book like Adams or Stewart). Because if you’re not, you’re setting yourself up for failure.

If you are doing the self study thing, I’d recommend getting a book on Naive Set Theory (don’t let the naive fool you, it can be quite difficult). This will give you a (relatively) gentle introduction to the way you have to think to do mathematics. After that, there’s things like Elementary Number Theory (Elementary refers to Number Theory of the Integers _not_ the difficulty), and the mentioned books for introductory calculus.

But, all this is for not if you don’t get someone to mark your work. I imagine that contacting your local math department, you could get a senior student to do this for a small fee (e.g. ~$10/hr). That way, you’ll know if you’re on the right track.

Hope that helps!

23 July, 2007 at 5:02 pm

math_dreamer_liuDear anonymous:

Very thanks for you to write guide for me. It is very useful for me.

Now I am work at day time and study in a adult college at night. I like Math as some people love drag.(I never take drag)

Maybe I will get some pain at beginning. But Math just like a pretty girl in front of me.

Thanks again

29 July, 2007 at 11:00 am

Terry - not Tao :)Dear Prof. Tao,

I have a question about proofs in mathematics research papers. Maybe you (or someone with good experience in mathematical writing) can help me here.

Certainly we must provide a proof for any new results claimed in our paper; that goes without saying. But I’ve seen (not too often) papers where the authors have included proofs for “well-known” results. I’ve always been under the impression that we should not do that; rather we should just refer the reader to a reference containing a proof of the known theorem, unless of course our proof is new or that the inclusion of the old proof is necessary to give valuable insight to the understanding of the present paper. In either case, a comment should be added to explain just that. When I see a proof in a paper, unless otherwise noted, I’m assuming that it has not been published elsewhere. Otherwise, why take up valuable journal space?

Is this practice acceptable to the mathematical community? If so, why?

Thanks in advance for any insight you (or your readers) can give me on this topic!

29 July, 2007 at 11:12 am

Terry - not Tao :)Please note that in the comment above, I was talking about original research papers only, not survey articles, textbooks, lecture notes, etc.

30 July, 2007 at 8:44 pm

Terence TaoDear Terry,

I discuss this in the subpage

https://terrytao.wordpress.com/advice-on-writing-papers/give-appropriate-amounts-of-detail/

2 August, 2007 at 10:06 am

Eno, IdGreetings, I have stumbled upon something new in the field of mathematics and it has increased my interest for mathematics ever since I made the discovery. It is not major, it is a new way of finding the square of two digit-numbers, I just finished a Bsc. programme in Electrical\Electronics Engineering. I am working on other papers in the field of mathematics.

My question is, which is the most appropriate journal to publish what I have discovered(I have tons of documents on how to write a scientific paper)? Once you help me with this I’ll be able to continue with the others I have discovered(e.g a new formula for Pythagoras).

This is a wonderful forum!

2 August, 2007 at 7:37 pm

Xiaodong XuDear Eno, Id

I guess your can publish your paper in the journal of Zeilberger! I guess Terry will give different answers (if any).

Good luck!

3 August, 2007 at 9:49 am

Terence TaoDear Eno, Id,

I have a small amount of advice on this in the subpage

https://terrytao.wordpress.com/advice-on-writing-papers/submit-to-an-appropriate-journal/

Judging from your description, it seems that journals in undergraduate or recreational mathematics may be appropriate. Many mathematical organisations also have newsletters or other less technical publications which have space for things like this.

11 August, 2007 at 9:46 pm

Grétar AmazeenDear prof. Tao.

What application do you use to write LaTeX? I´m looking for a good one to maximize writing speed.

12 August, 2007 at 8:29 pm

Terence TaoDear Grétar,

I currently use TeXnicCenter. I’m probably not exploiting its full potential, but I do like its integration with LaTeX (e.g. the single-click build-and-view button, and the ability to cycle through the locations of the LaTeX errors), as well as the colorisation of math mode, etc. I used to use EditPad Pro, which was also quite good, though with less LaTeX integration.

22 September, 2007 at 1:10 pm

edwin lubangadear prof.

sorry am using this page to ask for help! i am an undergraduate student at politecnico di torino in italy and we just started but in my country(kenya) we never studied mathematics annalysis so i am new to this is there anyway you can help go through this topic easily and quickly.

28 November, 2007 at 10:09 am

Rebecca BooneI am looking forward to meeting Terrence Tao in San Diego. Loved the comment on the hazards of “self study.” That’s me all over but at least, for decades, I’ve realized how “crazy” I am but also how truly insane scholastic pedagogy can become. Words and symbols can never “prove” what is religiously or numerically eternally “true.” Only the flesh and blood progession of two “numbers” or “quanta” — I and You, really, and I surmise best analyzed, numerically speaking, as mother and offspring — can “prove” a creature’s functional which is to say true reality. I’m really into analysis of prime numbers and cannot for the life of me see how “professional” numerical buffs can consider any number other than the “given” No.1 as prime, as the nomenclature indicates. I’ve listened twice to Tao’s podcast and will try to read the books he recommends before the SD meetings. The problematic “gap” of primes seems to me clearly the “generation gap” between mothers and offspring. Dirk Struik’s History of Math and Hans Jahnke” History of Analysis are my bibles.

I am 86 years old. I call my stuff rebel math. I have grandchildren at Yale and Harvard (post graduate math majors, one in nuclear physics and one in epidemiology) who vet my logic and egg me on. Best wishes to all, Rebecca Boone

7 December, 2007 at 3:32 pm

BeansDear Professor Tao,

I thought I would just let you know that the link to Jean-Pierre Serre’s lecture is no longer available. However, the video can be found at this site: http://modular.fas.harvard.edu/edu/basic/serre/

Beans

7 December, 2007 at 5:01 pm

Terence TaoThanks for the update!

15 January, 2008 at 6:51 am

ajay rawatsir,

as a student of physics i always believed that there is very deep conection betwen geometry and physics………it really mesmerises me……….

25 January, 2008 at 3:33 pm

Ryan HarrisHey Terry, I like your site very much. You are writing on topics that aren’t covered very often in personal blogs. I will link to your blog from mine, as you provide some very good tips and common sense. Thank you. I’ll subscribe to your feed as well. Please keep up the good work!

26 February, 2008 at 6:56 am

math LoverWriting while studying,

Dear Prof Tao,

How do you write when you “study” maths, i.e when reading a book about a new topic and solving exercises therein? Do you use Latex or just regular pencil and paper to do the “rough” calculations while studying

Thanks.

26 February, 2008 at 10:28 am

dsilvestreDear Dr Tao,

Wich book would you recommend for a Calculus of Several Variables class, it hasn’t need to ve very very rigurous but I would like it to have most of the proofs of the theorems, and be the most intuitive as possible (with lots of graphics, etc, that clarify the subject).

I know Stewart’s book and Marsden Tromba’s one, and I think they are good, but maybe you could point me another good one.

Thanks!

28 February, 2008 at 9:55 am

AnonymousDear Prof Tao,

I would like you to recomend me a book for studing differential geometry. In particular what has been hard for me to understand is the covariant derivative. I don’t understand how the christoffel symbols are calculated from the derivatives of the metric tensor.

Thanks a lot for your advices!

4 March, 2008 at 7:54 pm

dsilvestreDear Prof Tao,

Haven’t you ever discover a new mathematical formula/theorem, and maybe years later you see that someone else has discovered it before you?

It happens to me very often. When I see it I don’t know whether to be happy or sad for that.

5 March, 2008 at 3:57 pm

Terence TaoDear dsilvestre,

If you are rediscovering known facts and theorems while trying to learn a subject, or when just playing around with it, it shows that you are asking interesting questions and are on the right track, so I wouldn’t be too concerned.

If instead you are proving theorems for a research project which one later finds to already be in the literature, this may be a sign that you have to do more library research (or talk to other experts in the area) before you are ready to publish a paper on the subject. But even if a result is partially duplicated in the literature, the new result or argument may have some original virtues (e.g. a simplified proof, or a strengthened conclusion) that make it worth keeping around for future use (though perhaps the result may be too slight to publish just by itself).

5 March, 2008 at 5:09 pm

dsilvestreDear Prof Tao,

Thanks a lot for your answer!

I wrote the original post yesterday after reading in wikipedia about Quantum Calculus (http://en.wikipedia.org/wiki/Quantum_calculus), and discovered the existence of this book: Victor Kac, Pokman Cheung, Quantum Calculus.

I for myself had discovered what is now called quantum calculus back in 1999 when taking my 1-dimensional calculus course, and called it “calculus base h” because it generalized the traditional calculus in the sense that it added an additional base variable “h” such that if we take the limit h->0 it gives the traditional notions of derivative and integral.

Now I know it is called h-calculus. And I learned there is even a q-calculus that is related to it but I didn’t knew anything about it.

How could I had realized back then that this was already “invented”? I think this book didn’t even exist then. I remember on that time searching the web and reading all the literature I could find about calculus, and I never found anything very much related to it until yesterday. I didn’t knew how to search, because there was nothing on “calculus base h” on the search engines.

Who could had been an expert on the area that could had helped me? I’m not a mathematician, and I don’t know many mathematicians. I talked about this subject to one mathematician long ago (by chat), but I think he didn’t gave too much importance to the subject.

But of course as you say, some of the things I wrote are somewhat different that what I found on the book, so there still is some original value in my writings (I think). I found some formulas I still hadn’t found on any other book. So maybe some day I will write everything down on latex and publish it on a website or something, but not yet, as there are much more things that I could add to it.

The question is: how can someone be sure that what she is researching is in fact something new, as maybe she doesn’t know how to name it in order to search/ask about it. I don’t remember how I found this wiki entrance, but I was not searching particularly for it, but found a link on another related article.

I hope this post makes some sense as I realize that what I make are “toy-researchs” as I can’t consider myself on beeing inside the mathematical comunity, as otherwise surely someone could had told me about this book/subject long ago.

Did you know about this quantum calculus subject? I’m sure you did.

Cya!

6 March, 2008 at 7:32 am

dsilvestreFor example this is a formula I found by myself, and I hadn’t seen yet on any book:

and

These integrals are calculated base h=1, so they are in fact sumations.

For example to calculate sin(1)+sin(2)+sin(3)+…+sin(100), you simply

calculate sumsin(100)-sumsin(1).

They work in radians and degrees, and if you change sin by sinh and cos by cosh the formulas are also valid.

Where can I find information about this kind of formulas?

7 March, 2008 at 9:21 am

Terence TaoDear dsilvestre,

There is no “quick fix” to gain the type of experience one needs to be fully familiar with what’s going on in a field of mathematics; generally speaking, one needs to spend three to five years in graduate school studying one subfield of mathematics intensively (while also talking to other mathematicians, attending talks and conferences, reading a lot of books and papers, and so forth), and then spend another three to six years as a postdoc branching out into other areas of mathematics (while continuing to talk to mathematicians, attend talks and conferences, etc.) before one really begins to have a solid grasp of one’s subject. (Even then, it is a continual learning process; for instance, I find myself having to learn new bits and pieces of mathematics constantly for my own research.)

For your particular summations, for instance, if you take a course in Fourier analysis in graduate school (or as an advanced undergraduate class) you might recognise these formulae as being essentially the identities for the Dirichlet kernel,

http://en.wikipedia.org/wiki/Dirichlet_kernel

14 April, 2008 at 4:02 pm

AnonymousDear Dr. Tao,

I have been an adjunct faculty at a small community college for the past 4 years, and I have applied for a Ph.D. program and have been accepted to start in Fall of 09.

I am actually 33 and was wondering if it is too late for me to get in the 5 year run for my degree; is it too late to start? or should I not even ask this Question since I enjoy Mathematics!

Brian

2 June, 2008 at 5:54 pm

VINCENT ZEMAITISHello Dr. Tao,

“Several complex variables” is a field still in its adolescence. What opinions, if any, do you have concerning its current and/or future use as a working tool for physicists? Will it ever find itself within a large whole chapter of a standard textbook on mathematical methods for physicists? Will the edge-of-wedge theorem become popular?

Is the SCV textbook by Stephen Krantz the most workable, viable, and current one?

I would like to begin learning SCV but find the prerequisites have prerequisites.

Sincerely,

VZ

27 June, 2008 at 2:00 am

victorHi Dr. T

I started Math when I was quite young…and even while growing I never imagined the “ferocity”, for the sake of no greater word, by which math is applied in our daily lives.

Recently, I started work on design of cryptosystems, work that involves a very diverse amount of knowledge and mathematical ebbs….

Math’s great…

Dr. I understand that in research of perfect crypto-systems that the existence of a one-way function is hard, especially one based on probability…

Do you think it will be plausible to solve such functions, based on a private key, k, for example. and do you think a perfect cryptosystem can be created…I have started some hard work at it…mostly borrowing something from nature’s wonder!!!!

25 July, 2008 at 10:46 pm

satyaDear Prof Terence Tao,

I have done B-Tech degree in Electrical Engg and Masters degree in Software Systems. I have about 10 years work experience in the area of Embedded systems. I do not have any formal degree in Mathematics but have done some basic courses during the first two years of Engg. I am extremely fascinated and interested in the area of Compressed Sensing and would like to take it up as my topic of research for PhD which I have registered into as a part time student while doing my job.

What I could gather from the available literature that Compressed Sensing has lot of applications in many areas of Science and Engg and is almost like the next big wave.

Kindly advise me if I can take it up with my limited Mathematics background. If yes, what are the Mathematics tools which I need to get introduced to with the limited time available to me as a part time student.

Or else is it simply not my cup of tea and I should take up something else.

7 August, 2008 at 10:37 am

On time management « What’s new[…] On writing […]

8 August, 2008 at 4:27 am

我如何安排时间（译自陶哲轩博客） « Liuxiaochuan’s Weblog[…] 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想法。其实，我怀有这个想法已经有一段时间了，可是就我自己的经验而言，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。（除了有一些经验写在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 September, 2008 at 9:11 am

ouboubHello

I find your various comments on writing very useful. Could you

joint them together into a single pdf file? I think you did something

similar for the Navier Stokes problem.

thanks and regards

Uwe Brauer

21 September, 2008 at 12:19 pm

On Time Management « What’s New[…] writing 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 […]

17 November, 2008 at 10:26 pm

DongPhDDear Mr Tao,

I’m a student in Vietnam. Your page is useful to me. My friend said it was interesting when i told them what you wrote, of course in Vietnamese. Almost of them are poor in English. I intend to translate your series about career advice and on writing into Vietnamese. Would you please accept my idea? Wish your site more and more perfect.

I’m looking forward to hearing from you.

Thanks,

Dong

15 March, 2009 at 4:55 pm

Career advice from the Fields medalists and some other mathematicians « Academic Career Links[…] advice and writing tips from the blog of Terence […]

20 March, 2009 at 6:22 pm

drhwaThank you!

26 March, 2009 at 6:41 am

How to Maximize Citations « Academic Career Links[…] for the general advice on writing research papers, see excellent writing tips from the blog of Terence […]

23 April, 2009 at 7:26 am

Victor PortonHow is better to write in math texts “Exists exactly one” or “Exists unique”?

29 April, 2009 at 2:15 pm

How to Write a Really Good Research Paper « Successful Researcher[…] excellent advice on the subject (primarily for mathematicians) can be found at the blog of Terence Tao; see also […]

19 May, 2009 at 3:52 pm

Successful Researcher: How to Become OneFor V.P.: “there exists a unique”

5 June, 2009 at 12:29 pm

Hunting Down the Old References « Successful Researcher[…] Down the Old References While writing the research papers one quite often needs to get back to the full texts of old (pre-Internet or at least pre-arXiv) […]

5 June, 2009 at 8:20 pm

Hunting Down the Old References | Phasing[…] writing the research papers one quite often needs to get back to the full texts of old (pre-Internet or at least pre-arXiv) […]

20 June, 2009 at 8:53 am

Victor PortonIf a theorem consists of several items or of equivalence of several items, should one put first simpler proofs (simple to prove items or implications) or harder proofs (hard to prove items or implications)?

20 June, 2009 at 9:38 pm

Jonathan Vos PostVictor Porton: IMHO opinion, we’ve learned since Hilbert that it’s a slippery slope to seek a “simplest proof.”

21 June, 2009 at 1:46 am

Victor PortonJonathan Vos Post: I am not asking you how to search the simplest proof.

I ask for example if a theorem states that (1) is equivalent to (2) and (1)=>(2) implication is trivial while (2)=>(1) implication requires some lengthly proof, then which part of the proof put first, (1)=>(2) or (2)=>(1)?

21 June, 2009 at 6:50 pm

Qiaochu YuanAs a matter of style? I believe one generally puts the trivial implication first; it’s usually the one that motivates the theorem, i.e. “since this implication is trivial, is its converse true?”

26 June, 2009 at 11:24 pm

Pham Huong linhDear Mr.Tao

toi dang gap mot van de kho voi bai nay.Mong anh giai giup.De bai la:

Cho:

1<a<b+c<a+1 voi b<c

Chung minh rang:b<a

30 October, 2009 at 11:56 am

S.C. KavassalisThis is a very useful article.

It’s great to see people with such expertise sharing their knowledge in such an open format.

22 November, 2009 at 6:23 am

Advice on writing paper « Computer Vision[…] 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 […]

27 January, 2010 at 11:56 pm

Advices on writing papers « Hailongdao's Blog[…] Advices on writing papers By hailongdao Getting stuck on the intro of my paper. This looks helpful. And obviously, this. […]

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

EulerDear 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

researcherDear 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 WonDear, 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

portonWhat sounds better, “f is a lower adjoint” or “f has an upper adjoint”? (about a Galois connection).

7 June, 2010 at 3:44 am

portonAlso, 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 ZhangDear 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 ClarkOrganization 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

breI 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 TaoSee 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/does-one-have-to-be-a-genius-to-do-maths/

https://terrytao.wordpress.com/career-advice/ask-yourself-dumb-questions-%E2%80%93-and-answer-them/

https://terrytao.wordpress.com/career-advice/be-patient/

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

PwriI 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 MahmudProfessor 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 AbboudDear 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 ShantzI 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

DescargarHey, 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

portonSome more advice on writing math papers:

http://www.impan.pl/EN/PubHouse/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 ZengDear 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

KYDear 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

portonShould 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

quasihumanistOne 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

thichchaytronThe blog of Tenrecen Tao is very good……

30 November, 2012 at 5:25 pm

J. T. SalamancaVIC …Maestro Tao ¿viene a Spain para poder conocerle?

15 December, 2012 at 8:31 am

ZeraouliaI 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 RiosHi, 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 KashyapDear 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 SmithLots 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

neymetmsI 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

neymetmsBy 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

lmSVP 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

AnonymousDear 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 TaoThe articles are now reprinted on my blog:

https://terrytao.wordpress.com/advice-on-writing-papers/maximising-the-results-to-effort-ratio/

https://terrytao.wordpress.com/advice-on-writing-papers/implicit-notational-conventions/

https://terrytao.wordpress.com/advice-on-writing-papers/on-the-strength-of-theorems/

9 October, 2014 at 2:54 am

AnonymousWhat 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

DPTypo 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

BekDear 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

JoAnneDear 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 PengDear 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 PengSorry 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. […]

13 March, 2015 at 9:01 am

portonI am writing a research monograph. What’s about this experiment: In my book write instead of ?

13 March, 2015 at 9:04 am

portonNote 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

portonWhat 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

portonI 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.

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

observerHi 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

SalinasDear 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 ???

Thanks

Miguel SALINAS

23 October, 2015 at 9:12 am

SalinasOk I found the answer here:

https://terrytao.wordpress.com/2010/04/10/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

portonIf 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

AnonymousIt 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 wholegives something else?1 January, 2017 at 8:16 pm

bhaskar singhaniaSir!

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!