Every mathematician worthy of the name has experienced … the state of lucid exaltation in which one thought succeeds another as if miraculously… this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work…(André Weil, “The Apprenticeship of a Mathematician”)

Relying on intelligence alone to pull things off at the last minute may work for a while, but, generally speaking, at the graduate level or higher it doesn’t.

One needs to do a serious amount of reading and writing, and not just thinking, in order to get anywhere serious in mathematics; contrary to public opinion, mathematical breakthroughs are not powered solely (or even primarily) by “Eureka” moments of genius, but are in fact largely a product of hard work, directed of course by experience and intuition. (See also “the cult of genius“.)

The devil is often in the details; if you think you understand a piece of mathematics, you should be able to back that up by having read all the relevant literature and having written down at least a sketch of how that piece of mathematics goes, and then ultimately writing up a complete and detailed treatment of the topic. (See also “learn and relearn your field“.) It would be very pleasant if one could just dream up the grand ideas and let some “lesser mortals” fill in the details, but, trust me, it doesn’t work like that at all in mathematics; past experience has shown that it is only worth paying one’s time and attention to papers in which a substantial amount of detail and other supporting evidence (or at least a “proof-of-concept”) has already been carefully gathered to support one’s “grand idea”. If the originator of the idea is unwilling to do this, chances are that no-one else will do so either.

In short, there is no royal road to mathematics; to get to the “post-rigorous” stage in which your intuition matches well with what one can establish rigorously, one has to first invest real effort in learning and relearning the field, learning the strengths and weaknesses of tools, learning what else is going on in mathematics, learning how to solve problems rigorously, and answering lots of dumb questions, and so forth. This all requires hard work.

Of course, to work hard, it really helps if you enjoy your work. It is also important to direct your effort in a fruitful direction rather than a fruitless one; in particular, spend some time thinking ahead, and don’t obsess on a single “big problem” or “big theory”.

There will of course be times when one is too frustrated, fatigued, or otherwise not motivated to work on one’s current project. This is perfectly normal, and trying to force oneself to keep at that project can become counterproductive after a while. I find that it helps to have a number of smaller projects (or perhaps some non-mathematical errands) to have at hand when I am unwilling for whatever reason to work on my major projects; conversely, if I get bored with these smaller tasks, I can often convince myself to then tackle one of my bigger ones. See also my thoughts on time management.

There are also times when one realises that a project is simply too much to handle at the present time, and it then makes sense to modify one’s goals for the project, or shelve it and work on another project instead: see “be flexible” and “use the wastebasket“.

One final note: there is an important distinction between “working hard” and “maximising the number of hours during which one works”. In particular, forcing oneself to work even when one is tired, unmotivated, unprepared, or distracted with other tasks can end up being counterproductive to one’s long-term work productivity, and there is a saturation point beyond which pushing oneself to work even longer will actually *reduce* the total amount of work you get done in the long run (due to the additional fatigue, loss of motivation, or increasingly urgent need to attend to non-work tasks that this can cause). Generally speaking, it is better to try to arrange a few hours of high-quality working time, when one is motivated, energetic, prepared, and free from distraction, than to try to cram into one’s schedule a large number of hours of low-quality working time when one or more of the above four factors are not present.

See also this quote of Ira Glass on the hard work needed to bridge the gap between low-quality and high-quality creative output.

## 48 comments

Comments feed for this article

3 August, 2007 at 7:30 am

Anonymous AdmirerDear Terry Tao,

Thank you so much for your career advice page. I remember reading a comment some time ago from a person asking you about advice for time management. It seems to me also that you have an unlimited supply of energy in writing papers, books, teaching, maintaining your blog, giving talks, etc. etc., on top of your personal life. How do you do it???

From browsing your blog chronologically, it also seems like everyday you’re working on something different, or explaining something you learned just a day or two ago. Incredible! It boggles me how someone can just sit down and “work hard” to accomplish all that you have done…. Could you please shed some light on your intellectual wizardry? :-)

3 August, 2007 at 9:59 am

Terence TaoDear anonymous,

I am slowly collecting my thoughts on time management, but it will take a while to put it all together coherently. (The thing about time management is that it does not allow you to do every task instantly; but it does let you match one’s high-productivity periods of time to those “high-intensity” tasks which could really use one’s full faculties, and one’s low-productivity periods of time to those “low-intensity” tasks which don’t need one’s “premium work time”. Blogging about time management falls into the “low-intensity” category.)

As regarding learning different areas of mathematics, this is pretty much a continuous process; the need to keep on learning doesn’t stop once one exits graduate school. But it gets easier with time, because you start seeing more analogies and connections with the mathematics you already know, and also you get a sense of what parts of a theory are the key ones to focus on, and which ones are the routine computations which one can skim over; you start “seeing the plot”, so to speak. Also, the compounding power of experience is quite remarkable; if every year you manage to expand your mathematical horizons by (say) 10%, then by ten years or by thirty, you can handle a range of mathematics which may seem quite amazing to a less experienced mathematician.

27 May, 2008 at 10:35 am

Advice to the Bright and Young « Essays by Danielle Fong[...] big problem or theory. His advice? Don’t. Try instead to be patient, and flexible. Work hard. And above all, enjoy [...]

14 June, 2008 at 11:31 am

这等牛人也在wordpress上写blog！ « Just For Fun[...] is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other [...]

22 July, 2008 at 11:38 am

ATI apologize for the following longish comment, but I find that while “Work Hard” is, prima facie, good advice, it is also too simplistic to be helpful.

When I was in graduate school, my advisor told me the same thing: Work hard, it is the only way to get anywhere. It would have been good advice, except at first, I didn’t know what I should spend my time on: Should I spend most of my time sitting around thinking about a problem and hope I would find a solution eventually, or should I mainly read papers, and if so, how should I read papers, in what detail, etc.? Since I didn’t like reading papers, I mostly sat around hoping to come up with an idea that would solve my thesis problem. Needless to say, it didn’t work very well. It would have gone all wrong if it wasn’t that I eventually realized that there is a parallel between elite sports and mathematics, and that I could use my experience as a tennis player as a template for how to become a working mathematician.

My other great passion in life is tennis. But if you want to become a good tennis player it is not enough to go out and play a lot of matches. You have to spend time “off the court” in the gym and on the running track to get strength and endurance. You have to do lots of drills working with a coach to perfect your strokes and technique. And you have to relentlessly work on your weaknesses.

Similarly, to be a “professional” mathematician, you need to not only work on your research problem(s), but you should also constantly be working on learning new proofs and techniques, going over important proofs and papers time and again until you’ve mastered them. Don’t stay in your mathematical comfort zone, but expand your horizon by also reading (relevant) papers that are not at the heart of your own field. You should go to seminars to stay current and to challenge yourself to understand math in real time. And so on. All of these elements have to find their way into your daily work routine, because if you neglect any of them it will ultimately affect your research output negatively.

I realize everyone is different and we all work in different ways, but I wonder if Terrence Tao could comment on how he works, and how he balances his days between thinking about a problem, or reading papers, or…

22 July, 2008 at 4:57 pm

Jonathan Vos PostReply to “When I was in graduate school, my advisor told me the same thing: Work hard, it is the only way to get anywhere. It would have been good advice, except…”

The world is filled with people who work hard but have little or nothing to show for it.

The issues become: (0) what are the goals of the work; (1) work smart; (2) Have a mixed portfolio of work in progress; (3) have a mixed portfolio of collaborators; (4) Work long enough; (5) assess whether the work is taking you towards the goals; (6) return to (0), lather, rinse, repeat.

I presume that, simultaneously, the implementation of these rules vary considerably from one person to another; and that there are general laws that apply to the process integrated over all people.

(0) what are the goals of the work? I suggest that you have this explicitly in writing, my mantra being “either you have a written plan for success, or you have an unwritten plan for failure.”

(1) work smart. Experts such as Terry Tao and John Baez give exceedingly good advice on what this means, online.

(2) Have a mixed portfolio of work in progress. Parameters vary from person to person, but the endpoints of the distribution include at the low end: work only on problems easy enough that you can almost always solve the problems encountered; and at the high end: work on one huge problem which might take you years, your whole life, or never be completed, but which, if solved, make you rich and famous, with Clay Institute megabuck or another Hilbert problem resolved, or Fields medal in hand, and a profile in The New Yorker. What is your perception of the distribution? For Game Theoretic reasons, I suggest a mixed strategy. I do not deprecate the low end: many of my 1,994 contributions in the OEIS are aimed at what for me is one sweet spot: problems which are original, elementary, nontrivial, and good to teach even high school students. 174 of my other OEIS entries link to arXiv papers in pure Math, mathematical Biology, or Mathematical Physics, because the best way to read Math (in my humble opinion) is to read actively, with pen in hand, or multiple windows open on your computer’s desktop. Commenting on the work of others, with complete credit given, is collegial, and perhaps useful.

(3) have a mixed portfolio of collaborators. In school, and at home, and in social groups, we learn how to do collaborative work. In university, we have a chance beyond study groups, namely in working with one or more mentor professors. There is a network effect, where successful collaboration links you into the community of Mathematicians, and leads to more collaborators. Erdos number, and all that.

(4) Work long enough. That’s estimated on another thread of this blog. My rule of thumb is that it takes 10,000 hours of serious work to become proficient at any adult activity. Your mileage may vary.

(5) assess whether the work is taking you towards the goals. Develop a model of the part of the Ideocosm (Zwicky’s name for the space of all possible ideas) where you are working. Learn its topology. Is it a metric space? If so, what is the distance between where you are and where you goal may be? Or is it only a semimetric space (i.e. is there a triangle inequality on distance between points in your part of the ideocosm)? Assessment, with the help of collaborators, and self-assessment seems essential.

(6) return to (0), lather, rinse, repeat. I believe that a Mathematician should do Math every day, just as a musician should play music every day, and a writer should write every day. Oh, sure, there are vacations. But for every day you skip, it may take more than one day to “get your chops back.” How many hours per day is related to (4) and (5), at least.

Technically, there’s a (7): if you have done work of any consequence, then someday a student of yours, or a collaborator, or a reader of what you published or spoke about at a conference, will take it to the next step, maybe or maybe not the step that you would have found if you’d only had more time.

The above is merely my opinion. But it is close to what I tell my students.

23 July, 2008 at 7:00 am

Jonathan Vos PostThe endpoints of ther distribution of “(3) have a mixed portfolio of collaborators” were discussed on the “Followup: working in secret” thread of 19 July 2008 of the Secret Blogging Seminar blog,

Posted by Ben Webster

Gil Kalai said:

“There are two extreme ways to practice math (with many altenatives in between.) One way is to work secretly on a big problem, to tell nobody or very few people about it, to discuss with nobody the techniques you are using, and then after many years to astonish the world with a preprint or a lecture) presenting the solution. The other extreme way is to work while at any time discussing your thoughts and ideas with everbody (perhaps also on blogs), write papers with partial progress and conjectures etc.”

“The advantage of the first avenue is not just the fear that somebody will use your ideas but also that it helps the researcher to stay concentrated, and avoid outside preasure and distractions of various types. A clear disadvantage of the first avenue is that feedbacks from others can be useful at intermediate stages of the process towards a mathematical discovery.”

Ben Webster’s comments includes: “I’m curious: does anyone out there think that Gil’s ‘first avenue’ sounds like a good idea? It sounds crazy to me. Maybe I lack the self-confidence to think I would succeed at it (not something I’m regularly accused of), but it seems like asking for trouble, both in terms of actually getting the math done and in terms of one’s career. Obviously, there are dangers in revealing your ideas and results to other people. I think outright theft is relatively rare, but someone ‘eating your lunch,’ implementing something you had hoped to do before you have a chance, is a very serious concern.”

Or is it?

6 March, 2009 at 12:46 am

AnonymousDear Professor Tao,

Do you think it is good to study and do mathematics all the time? What do you like to do besides mathematics especially when you are a student, if you wish to answer, or may be you are never tired? Thanks.

7 March, 2009 at 5:24 am

tuDear , Prof Tao

I admire you so much and I’am trying to proceed a paper

5 June, 2009 at 4:54 am

Essential Career Lessons[...] 6. Breakthroughs come from hard work One needs to do a serious amount of reading and writing, and not just thinking, in order to get anywhere serious in mathematics; contrary to public opinion, mathematical breakthroughs are not powered solely (or even primarily) by “Eureka” moments of genius, but are in fact largely a product of hard work, directed of course by experience and intuition. (source) [...]

10 February, 2012 at 1:27 am

ian kennedyGoogle reports 10 sites with this wording. My existing rule of thumb is that >10 sites constitute “Common knowledge” and therefore does not require a statement of source. Should I change my rule of thumb to >=10 sites?

Here is my paraphrase which is an alternative solution to sweeping it under the “common knowledge” carpet: “The researcher needs to do a large amount of reading and writing, and not just thinking, to get anywhere in research. Contrary to the popular image, breakthroughs in research are not powered by a series of “Eureka” moments, but are the result of hard work, guided by experience and intuition. “

21 July, 2009 at 10:13 pm

AMS Graduate Student Blog » Blog Archive » Work Hard[...] Work Hard [via Terry Tao] [...]

22 July, 2009 at 12:34 am

Research tips: work hard « Entertaining Research[...] tips: work hard By Guru Terry Tao; link via AMS graduate student blog which summarises the post nicely: The two main problems are [...]

23 July, 2009 at 4:44 am

adamtrybusDear Terry,

I am a frequent reader of your posts. I can’t express how helpful it is for a (at times) confused and frustrated PhD student to read posts like the one above. It not only gives useful tips on how to do research but also demythologizes it and gives relief by making one’s own struggle with mathematical matter a part of common experience shared even by mathematicians of such a magnitude as yourself.

Thanks a lot!

31 July, 2009 at 11:48 am

AliDear Prof. Tao,

As a first year graduate student, I am finding it difficult to pick a subject to focus on in my graduate study. I find myself being drawn to so many different (and unrelated?) subjects, ranging from Probability to Algebraic Topology. How can one be sure that a certain subject is a better fit for them compared to another. I was hoping you could shed some light on your thought process when you chose to focus on Harmonic Analysis in your graduate study. Thank you so much!

Regards,

Ali

30 November, 2009 at 5:00 pm

30 days of power – jordan evolves[...] the years I’ve seen some great success with this method of study. However, as Terence Tao so eloquently points out in the opening lines of his post on graduate-level mathematics, there comes a point at which hard [...]

4 April, 2010 at 1:43 am

Learning new material « Rochester Probability Blog[...] As a recommendation for the productiveness of your learning, I suggest this blog: Ams Graduate Student and of course the reference of the blog is a must: Work Hard by Terry Tao. [...]

25 June, 2010 at 9:49 pm

Koushik GhoshHow much is Computer Knowledge essential for mordern mathematics research?

9 August, 2010 at 12:07 pm

arvosteluOn the other hand, elementary mathematics suffices for CS research:

http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf

25 June, 2010 at 10:02 pm

Koushik GhoshProf.Taos comments on hard work is truly good.I was told by my batchmates that I would not be able to reach the heights of some and that they do not do very much mathematics to to attain such high levels.But I have seen through my little efforts that it is the hard effort that actually pays off.While some of those so called naturally gifted ones have faded off.

26 June, 2010 at 6:20 am

FengHi Tao, can you provide rss feed for some individual tags. I am not a mathematician (which means not very much interested in your mathematical post). But I’d like to read some of your topics, like “career-advice”. which are also very fruitful to general research areas. Thanks!

26 June, 2010 at 9:23 am

Terence TaoI believe RSS feeds for individual tags or categories are available, see

http://en.support.wordpress.com/feeds/

for details.

15 September, 2010 at 10:41 pm

Aritra KarYour pages in real analysis is really good.

19 September, 2010 at 12:30 pm

chandrasekharRespected Sir,

Is this advice “don’t obsess on a single “big problem” or “big theory”.] is really good? I mean if you weren’t obsessed with Green-Tao theorem then do you think, you could have proved it afterwards. Also, if Prof. Wiles wasn’t obsessed with FLT, probably we couldn’t have seen a proof of FLT.

“Obsession” only arises when one is not enjoying his work. If one is enjoying his work then i see no point in getting obsessed.

I hope you don’t take this in the wrong way. Have you never thought about a single problem for a long time? If you didn’t then you wouldn’t be rated as the best.

Chandrasekhar

19 September, 2010 at 2:17 pm

Terence TaoThis is discussed at the relevant article

http://terrytao.wordpress.com/career-advice/dont-prematurely-obsess-on-a-single-big-problem-or-big-theory/

(and note in particular the crucial adverb “prematurely”).

Regarding what is now known as the Green-Tao theorem, Ben and I in fact were not directly working on this problem to begin with, but on a much more modest problem (namely, trying to improve the best known bounds for Szemeredi’s theorem for arithmetic progressions of length four) when we stumbled upon a general strategy (namely, viewing sparse sets as dense subsets of pseudorandom sets, and then modeling the latter by dense sets of the entire integers) which Ben then realised had the potential to capture progressions in the primes. It was only then that we devoted our full efforts to this problem (which, after about six months, we eventually were able to solve). But until we had that observation, devoting significant amounts of time to the problem would indeed have been premature.

24 September, 2010 at 9:15 am

chandrasekharSir,

It’s always a pleasure having reply. Next your reply was very much interesting. By the way you say, it took about six months for you to solve, i would like to ask an another question. How did you know that you were making progress on the problem? Was it daily, that you made a sufficient amount of progess.

15 November, 2010 at 8:48 am

A TLucy Kellaway of the Financial Times (and Business Daily on the BBC World Service) makes a related point: http://bit.ly/aKdOdS .

10 December, 2010 at 7:06 am

StudentDear Prof Tao,

Sorry if this is an inappropriate question, I am an undergrad student that have recently found a great interest in mathematics. I am soon about to write my bachelor thesis, which I am both looking forward to and not.

The reason why I am not looking forward to it is that, when skimming through old students bachelor thesis’s, I realize that most of the students have reproduced an old result, or done a literature review, or equivalent. This is not strange to me, in the sense that most of the “cutting edge” research in mathematics have a quite high barrier of entry.

But, I do think, and I do hope, that are many interesting problems out there to answer, that are accessible even for undergrads . The problem for me is that my experience in mathematics is small, and hence some problems which I think are interesting, more often than not have already been solved.

To make my experience writing the bachelor thesis, as pleasant as possible, I humbly wonder If you have some general advice on this?

Regards

H

10 December, 2010 at 7:54 am

Terence TaoThis issue is discussed in the following MathOverflow thread:

http://mathoverflow.net/questions/45802/undergraduate-math-research

10 December, 2010 at 12:33 pm

StudentSpot on, thank you very much!

Regards

H

21 December, 2010 at 1:21 am

Chris LeeHere, the key is “Of course, to work hard, it really helps if you enjoy your work.”, I believe.

(enjoy your work)s -> (work hard)s -> (lucid exaltation)s ->(R.I.P.)

“Was it daily, that you made a sufficient amount of progess.”

I imagine: continuously.

17 July, 2011 at 8:16 pm

Career Advice From a Mathematician « I, Geek[...] Work hard [...]

18 September, 2011 at 8:39 am

Be considerate of your audience[...] exposition is hard work – almost as hard as good research, sometimes – and one may feel that having proved the result, [...]

19 January, 2012 at 6:52 am

BISOGNA ESSERE UN GENIO PER FARE MATEMATICA? | fumagallierica[...] risposta è un NO enfatico. Per dare dei contributi buoni ed utili alla matematica, uno deve lavorare duramente, conoscere bene un settore, conoscere altri settori e altri strumenti, fare domande, parlare con [...]

30 March, 2012 at 8:12 am

JacobThis is helpful

14 April, 2012 at 5:56 pm

» Does one have to be a genius to do maths? [By Terence Tao] Jerry's Laborary[...] an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other [...]

21 May, 2012 at 7:50 pm

重学数学 | 10 years of endeavor[...] PS: 陶哲轩的博客上面也有许多有价值的文章，比如这个，这个，这个。以前觉着他的博客一定充满了看不懂的数学公式，今天翻了翻才觉着，当时太轻易就放过这个好资源了。 Like this:LikeBe the first to like this post. 2012/05/21 by meglory Categories: Blog, Reading | Leave a comment [...]

8 July, 2012 at 11:50 pm

O projektu MatemaTech | Matematech[...] is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other [...]

17 August, 2012 at 6:30 am

Nassim Nicholas Taleb on Role of Luck « Books I Read[...] you can do to sway the chances of luck helping you along the way. But that doesn’t mean hard work, confidence, and talent should be discounted. Share this:MoreEmailLike this:LikeBe the first to [...]

28 September, 2012 at 9:01 pm

The Art of Winging It :: The Daily Touch[...] Tao, a genius who has won the Field’s Medal said this about working in Mathematics, but it applies to every kind of study; “Relying on intelligence [...]

29 September, 2012 at 5:54 pm

Posts from Terry Tao | 10 years of endeavor[...] http://terrytao.wordpress.com/career-advice/work-hard/ [...]

9 December, 2012 at 2:26 am

[Skills] Làm việc chăm chỉ – GS Terrence Tao | Nguyen Hoai Tuong[...] (Nguồn: http://terrytao.wordpress.com/career-advice/work-hard/) [...]

21 May, 2013 at 3:42 am

Bisogna essere un genio per fare matematica? - Maddmaths[…] La risposta è un NO enfatico. Per dare dei contributi buoni ed utili alla matematica, uno deve lavorare duramente, conoscere bene un settore, conoscere altri settori e altri strumenti, fare domande, parlare con […]

19 July, 2013 at 8:57 am

Does one have to be a genius to do maths? | Singapore Maths Tuition[…] an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other […]

2 October, 2013 at 8:14 am

AnonymousThe journey to become a mathematician is long and difficult.

Ivan Alim

24 October, 2013 at 9:42 am

Terry Tao: On Hard Work | Fahad's Academy[…] TERRY TAO ON HARD WORK […]

29 October, 2013 at 5:14 am

Work hard | hrbatanu[…] Work hard. […]

14 May, 2014 at 10:36 pm

AnonymousDear Prof. Tao,

I have an idea and I would like share it here. I think if you keep a special section in your blog which collects the potential research problems for young researchers, it would be great. You could use the International Congress of Mathematician field list such as algebra, analysis, probability, topology, ….then senior professional mathematicians post some research problems which could be a dissertation problem or research paper. They could comment on the problem difficulty level or point out the relevant literature. Young early career mathematicians pick up some of them and start working on it, ask their questions or share their ideas with other people, collaborate and finally write a paper. It would be great for the young mathematicians.

Thanks