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“.)
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”.
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3 August, 2007 at 7:30 am
Anonymous Admirer
Dear 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 Tao
Dear 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
AT
I 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 Post
Reply 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 Post
The 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?