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.
94 comments
Comments feed for this article
26 October, 2015 at 9:34 am
Right to Learn, Part 3: Challenge-Deficit | Minds on Fire
[…] See, for example, the essay Work Hard by Terry […]
29 October, 2015 at 3:29 am
Anonymous
dear prof tao how can you became so smart?
10 May, 2016 at 12:54 am
Not “too silly”, not “too girlish” for maths | Why It Rained Today
[…] and https://terrytao.wordpress.com/career-advice/work-hard/ by Terrence […]
2 June, 2016 at 9:04 pm
Terrence Tao’s Advice – ashadianand
[…] institution Y in only Z years, or on scoring A on test B at age C..Of course, one should still work hard, and participate in competitions if one wishes; but competitions and academic achievements should […]
13 June, 2016 at 8:01 am
diksha
Thankyou sir …I will try to opt it in my life
13 June, 2016 at 8:08 am
diksha
Dear terense tao,
it is really reflected in ur personality that immense love towards mathematics and dedication and hardwork brings a lot of commendable and eye opening results …Sir I really love to learn more and more mathematics as much as I can …really got inspired whenever I read your quotes and blogs …every time I get boost up and thankyou very much sir I hope u will keep motivating us and guide us and we r very thankful to god for giving such a brilliant mathematician ..god bless u sir
22 June, 2016 at 1:10 am
Smart Imposter Syndrome – G-Notes
[…] 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 fieldsand tools, ask questions, talk to other […]
10 July, 2016 at 10:24 am
Not “too silly”, not “too girlish” for maths – paularowinska
[…] and https://terrytao.wordpress.com/career-advice/work-hard/ by Terrence […]
18 July, 2016 at 8:29 am
Solving mathematical problems | nguyen Huynh Huy's Blog
[…] are mainly solved by knowledge (ofyour own field and of other fields), experience, patience and hard work; but for the type of problems one sees in school, college or in mathematics competitions one needs […]
31 August, 2016 at 1:54 pm
Kumar
Hi Prof Tao,
I am a computer programmer and have some research experience in practical computer science, with almost zero knowledge base in advanced mathematics (ie, beyond high school). I am trying to explore more on mathematics research, and have come to know that mathematicians are in search of some kind of truth (is it true?) Although till high school, I was good at it, but my current understanding of mathematics research suggests that merely being able to do small manipulations and memorizations (as you’ve mentioned in one of your articles) doesn’t mean that I’ll necessarily be good at mathematics research. As it’s all about proofs, and I have had minimal exposure to such mathematics, currently I am finding it extremely difficult to keep up with the material I am trying to learn (Abstract Algebra, Computational Complexity Theory). Also, I am 23 now and am just above average intelligence. Do you think starting at this age I can (given that I want to) learn (and ultimately “contribute to”) pure mathematics?
3 September, 2016 at 5:09 pm
Anonymous
Hello Prof. Tao,
Just want to let you know that I’m a big fan. Cheers!
24 September, 2016 at 3:07 pm
yuvallevental
“Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work.” That was me after I discovered an easy way to mitigate my autism symptoms.
13 January, 2017 at 1:33 am
Anonymous
Thanks for this … it really helped me out of a rather rough patch of disappointment and self loathing
14 April, 2017 at 7:32 pm
Naman Kumar
Dear Terry,
I am a 15 year old from India, currently in the 10th grade. I want to participate in the IMO. From now, I have about 5 months to prepare. The question is, is that possible? Can I really prepare myself from the beginner level to the level of the IMO in 5 months? How did you go about it?
Sincerely,
Naman Kumar
2 October, 2017 at 3:00 pm
The Flash
I am seriously failing at math. I study or try to study but i dont see any results. If you would be kind enough to help and guide me i will be grateful and so will my grades
PS – Huge admirer. You are awesome
10 December, 2017 at 12:38 pm
Training for the NYC marathon: an academician’s perspective | Forty-two
[…] The way a researcher is evaluated is different from most jobs. A surgeon or a clerk are expected to perform well every single day at work, since every mistake is extremely costly. A researcher publishes a bunch of papers every year, and in each of them there are a handful of ideas (sometimes by one of the coauthors). He is therefore expected to perform extremely well few times, and do whatever the rest of the time. This is why many people think of research as a product of pure creativity1. And creativity indeed matters, but, as Terence Tao – arguably the most famous mathematician of our times – puts it: […]
11 January, 2018 at 9:30 am
tornado92
Re “[you should] 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,” are these writeups supposed to be from memory?
2 November, 2018 at 4:49 pm
ncoppedge
As far as work, here is an alternate philosophy:
*CONCLUSIVE EVIDENCE OF PERPETUAL MOTION* (https://youtu.be/Xoou4dt2qk0)
2 November, 2018 at 5:25 pm
ncoppedge
Perhaps if a person is a walking golden age it is a literal genius, then they might desire the secrets of the walking golden age.
Perpetual Motion: Maybe if it’s not an idea I couldn’t be smart about it, as far as that goes it is a material idea with more meaning than most give me credit for.
27 November, 2018 at 11:05 pm
asahay22
The link in the word “post-rigorous” is broken (there’s an apostrophe where there shouldn’t be). You probably meant to link it here:
https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/
[Corrected, thanks – T.]
12 July, 2019 at 11:22 pm
Math grad student
Hello Prof. Tao,
Did you experience what is described in the Ira Glass quote as a graduate student? I feel like it perfectly describes the state of my lousy research right now :(
6 April, 2020 at 9:57 pm
Anonymous
Hi Professor Tao,
I hope you are doing well and safe.
I am Math PhD student and I have a question regarding woking hard. Every now and then I have days where I feel really tired and I can not focus for an extended period of time. It is not like I do not want to work in those days. In fact, it is the opposite, I am usually very eager and motivated to work. It is just I get tired very quickly.
I am wondering whether you or any of your blog readers have ever experienced something like this, and how did you mange to overcome it. Thank you!
26 July, 2020 at 4:37 am
Anonymous
Hi Prof. Tao,
Thank you so much for inspiring me. I’m currently a high school student in Australia, and everytime I get a burnout or recurring headaches, your existence motivates me to work harder (much apologies for the corny feels). Kind of the truth though teehee :). I get reminded that there are people who have it harder than me, for example those in poverty around the world, the oppressed citizens of corrupted countries, and you, who had accomplished much more than me and many others which I’m sure, took a billion times of hard work that I can’t ever imagine. I’m sorry about the spontaneity of this clichèd remark, but after seeing how much effort and time you placed into creating tips on things that seem undermined and easily forgotten (e.g. career advice and workload) along with your workload as a professor in UCLA, I’d just like to say how immensely grateful I am, and I’m sure many others are as well, for your beautiful maths proofs, enjoyable lectures (there’s some on Youtube), your humility (that sometimes is seen as ‘self-depracating’, quoted by nytimes o(≧▽≦)o ) and tenacity that proves that it is possible to achieve what seems impossible.
I guess my general message is that we should appreciate more of the many things we take for granted. I also guess I thought of this because of a book I’m studying in my English by Kazuo Ishiguro that comments on human passivity- and I want to break from my passivity for a bit my expressing my appreciation before it’s too late for me to change my mind and procrastinate and eat dinner and do something other that’s completely irrelevant and random.
I sound like an old man that’s dying, don’t I haha. An old man who’s speaking really embarrassing things you probably have heard for so many times in your life (if you have the time to see this long-winded, boring comment of course). Feel free to ignore everything this old man has ranted about if that is so. Finally, I didn’t really know if this was your thing- receiving ‘fan’ letters or having a public email account, so sorry if I kind of sacrilegiously spammed the comments (reply(?)) section. So, really, REALLY thank you. From the bottom of my heart. Best wishes on what life brings you.
Kind regards,
Words of an unreliable high-schooler (who can’t even do a tenth of the maths you do)
27 August, 2020 at 6:15 am
Anonymous
A related funny quote from Feynman’s autobiography “Surely you’re joking Mr. Feynman!” (1985):
I still remember a guy sitting on a couch, thinking very hard, and another guy standing in front of him, saying, “and therefore such-and-such is true”.
“Why is that?” the guy on the couch asks.
“It’s trivial! It’s trivial!” the standing guy says, and he rapidly reels off a series of logical steps … The guy on the couch is struggling to understand all this stuff, which goes on at a high speed for about fifteen minutes!
Finally the standing guy comes out the other end, and the guy on the couch says “yeah, yeah. It’s trivial.”
We physicists were laughing, trying to figure them out. We decided that “trivial” means “proved”. So we joked with the mathematicians: “We have a new theorem – that mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial.”
11 October, 2020 at 6:15 am
Does one have to be a genius to do maths? – By Terence Tao – A2M1N
[…] 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 […]
26 August, 2021 at 11:43 am
Work Hard (By Terence Tao) - The web development company
[…] Article URL: https://terrytao.wordpress.com/career-advice/work-hard/ […]
26 August, 2021 at 11:43 am
Work Hard (By Terence Tao) – Cyber News Network
[…] Article URL: https://terrytao.wordpress.com/career-advice/work-hard/ […]
26 August, 2021 at 12:19 pm
CR Drost
The quote from Ira Glass was part of a larger interview on Currents.TV a long time back and all of the parts are worth a watch if you’re in any sort of creative work, YouTube playlist available here: https://www.youtube.com/playlist?list=PLuIu7YRdU32NIr2DysU3kxqnvs_flQyvs
29 August, 2021 at 3:22 am
HackerNews top stories #72 | Damian Mąsior
[…] https://terrytao.wordpress.com/career-advice/work-hard/ […]
30 August, 2021 at 11:59 pm
The 10+ Commandments of Undertaking Postgraduate Research – /* bitsilla ~ weaving code */
[…] – and they will be, don’t get too attached to your methods or results, network, work hard, etc.). But perhaps these will make up a future […]
19 September, 2021 at 9:24 am
Jacob Wakem
Work hard but beware of coffee.
11 February, 2022 at 9:52 pm
Akash Banerjee
Hello Professor,
This blog’s are very very helpful for me and many like me.
Thank you so much for making these invaluable thoughts available for us.
6 March, 2022 at 12:06 am
Dành cho những người thấy việc học là khó, nhưng không muốn từ bỏ … – Nippon Kiyoshi Blog
[…] Mượn ý từ Ira Glass, đăng trên trang của terrytao. […]
22 May, 2022 at 7:46 am
做數學一定要是天才嗎? – 嘎嘎酷
[…] 這個問題的回答是一個大寫的:NO!為了達到對數學有一個良好的,有意義的貢獻的目的,人們必須要刻苦努力;學好自己的領域,掌握一些其他領域的知識和工具;多問問題;多與其他數學工作者交流;要對數學有個宏觀的把握。當然,一定水平的才智,耐心的要求,以及心智上的成熟性是必須的。但是,數學工作者絕不需要什麼神奇的「天才」的基因,什麼天生的洞察能力;不需要什麼超自然的能力使自己總有靈感去出人意料的解決難題。 […]
21 September, 2022 at 3:04 am
Collatz conjunction
Dear terry tao
Every thing that you say it’s correct but mathematics isn’t modern or it needs to rewrite who can rewrite it if you say I will rewrite it. I am working on the reimann hypothesis and other problems like collatz conjunction . I Have done 50% after a few months a will report to clay mathematics institution I will see you soon professor terry tao. From Ethiopia mahammed Ahmed age 15.
In the future we will work together. My pleasure