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. 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 is 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. One should take advantage of the English language, and not just rely purely on mathematical symbols.
- 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.
Some further advice on mathematical exposition:
- Michèle Audin’s “Conseils aux auteurs de textes mathématiques“
- 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.
- Ashley Reiter’s “Writing a research paper in mathematics“
- Jean-Pierre Serre’s “How to write mathematics badly“
47 comments
Comments feed for this article
22 July, 2007 at 5:06 am
math_dreamer_liu
Hello:
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
Anonymous
Assuming 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_liu
Dear 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 Tao
Dear Terry,
I discuss this in the subpage
http://terrytao.wordpress.com/advice-on-writing-papers/give-appropriate-amounts-of-detail/
2 August, 2007 at 10:06 am
Eno, Id
Greetings, 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 Xu
Dear 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 Tao
Dear Eno, Id,
I have a small amount of advice on this in the subpage
http://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 Amazeen
Dear 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 Tao
Dear 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 lubanga
dear 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 Boone
I 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
Beans
Dear 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 Tao
Thanks for the update!
15 January, 2008 at 6:51 am
ajay rawat
sir,
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 Harris
Hey 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 Lover
Writing 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
dsilvestre
Dear 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
Anonymous
Dear 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
dsilvestre
Dear 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 Tao
Dear 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
dsilvestre
Dear 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
dsilvestre
For 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 Tao
Dear 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
Anonymous
Dear 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 ZEMAITIS
Hello 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
victor
Hi 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
satya
Dear 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
ouboub
Hello
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
DongPhD
Dear 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
drhwa
Thank 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 Porton
How 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 One
For 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 Porton
If 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 Post
Victor 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 Porton
Jonathan 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 Yuan
As 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 linh
Dear 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