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Of course I would never compare myself to, and don't expect to be, one of the great mathematicians. However, I am curious as to how my work habits, dedication and passion differs from theirs.

How hard did Gauss, Euler, Einstein,..., and so on work? How many hours were they spending learning and or thinking about their studies? Do the "best" mathematicians (or scientists) share many similar work habits?

Is it just sheer intelligence that separates the greats from the norm? Are there any other characteristics shared among them, like a great memory or being extremely quick learners?

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closed as primarily opinion-based by Arthur Fischer Nov 29 at 5:39

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

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There's a very good reason it took over one hundred years to make calculus a rigorous subject in mathematics. Math is not easy, even for the world's brightest men. There are some astonishing math savants that are difficult to understand (Rammanujan and Connes are such men) but generally math comes from lots of hard work. Memory plays a big role but that comes with practice and experience... –  Cameron Williams Jul 22 at 0:25
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A side word of advice: impostor syndrome is a huge issue in academia. It's really easy to doubt oneself and wonder if one is really good enough for the future they want to give themself. The world's best mathematicians get things wrong plenty and there's no reason for you to cast dispersions upon yourself because you get things wrong or struggle sometimes. Math is hard but we're all in the same boat as you. –  Cameron Williams Jul 22 at 0:30
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If Paul Erdos is anything to go by, you should spend 20 hours a day turning coffee into theorems. –  lemon Jul 22 at 0:31
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@ozo or ffee into cotheorems. –  Cameron Williams Jul 22 at 0:31
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@CameronWilliams: you have to reverse the arrows. Comathematicians turn cotheorems into ffee. –  Grumpy Parsnip Jul 22 at 1:04

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up vote 27 down vote accepted

I often wonder the same thing, and for concerns and interests around these lines found myself reading "The Cambridge Handbook of Expertise and Expert Performance." (I would suggest reading this more if you are interested - it is a well-supported book and says lots of things like practicing well for a few decades along with a supportive home environment is vital.) To put your concerns to rest, I was going to quote the aforementioned tome, but I couldn't find the quote (it's 800 pages) I was looking for and thus can only paraphrase: they found little to no correlation between intelligence levels (as measured by IQ tests) and internationally recognized performance (nominated by peers) amongst people with PhD's. Put another way, amongst mathematicians, physicists, molecular biologists, etc. those with an ostensibly higher level of intelligence have little to no greater chance of being amongst the elite few who are nominated by their peers for/win international level awards.

Hence I would say that if you have the potential to earn a PhD and begin research in mathematics, then it is likely that other qualities will determine your success; and thus hard work will indeed influence your success greatly. Note that this view doesn't necessarily fly in the face of the fact that some people are indeed smarter than others; it simply acknowledges that having a slightly higher IQ relative to others with very high IQs is far less of an advantage than other, more "soft" qualities and of course luck. Terry Tao says:

Does one have to be a genius to do mathematics? The answer 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 mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities... Of course, even if one dismisses the notion of genius, it is still the case that at any given point in time, some mathematicians are faster, more experienced, more knowledgeable, more efficient, more careful, or more creative than others. This does not imply, though, that only the “best” mathematicians should do mathematics; this is the common error of mistaking absolute advantage for comparative advantage.

This all fits in with one's sort of intuitive picture of things (if one has read biographies and such): certainly Einstein was extremely clever, and quite gifted at mathematics, but when comparing his first-blush mental abilities him to someone like von Neumann, one thinks Einstein was quite dumb!!! Yet both contributed seminal ideas to their fields...

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Note, though, that TT is famously hard-working (as well as incredibly efficient in his work-flow/work-habits)! –  user160609 Jul 22 at 2:38
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Terry Tao has zero experience on not being a genius and doing mathematics :-) –  Georges Elencwajg Jul 23 at 9:26
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@GeorgesElencwajg I see what you mean, but for me that makes him more qualified to judge. He is unbiased and he clearly has worked with a greater number of mathematicians, surely not all of them geniuses. Hence he would have a close view on their situation, and thus would be able to give a candid account about what separates the best "non-genius" mathematicians from the rest, as he did in the quote I gave. Moreover, I wouldn't much value the opinion of a mathematician who proclaims himself to not be a genius, since any such person is likely quite bitter about their supposed disadvantage. –  Austin Stromme Jul 23 at 19:52

I do not know how many hours did "great" mathematicians spend on doing research but I don't think it is of any importance. The most important think IMHO is to not worry to much about the result of your work, not compare yourself with others (from the past as well as from the present). Find what is really interesting for You in mathematics and have fun !

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I recommend reading Allyn Jackson's short biography of Alexander Grothendieck. Here's a highlight to give you an idea of how much Grothendieck worked:

Even among mathematicians, who tend to be single-minded and highly devoted to their work, Grothendieck was an extreme case. "Grothendieck was working on the foundations of algebraic geometry seven days a week, twelve hours a day, for ten years," noted his IHES colleague David Ruelle.

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Well, I can't say how much more time a mathematician put in his work more than other scientist but I can tell you something. The father of one of my colleagues is a mathematician. He is now 91 years old and still working on math every day except Sunday for 5 hours. Maybe is just a passion involved. When you are determined to do something and it doesn't matter what science it is, you put in a lot of work and don't think how much time you do it. Passion is passion!

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I heard here that "it takes roughly ten thousand hours of practice to achieve mastery in a field."

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This is a general comment, and unrelated. He isnt asking how much work it takes to be a great mathematician, but how much great mathematicians worked. Very different question –  Asimov Jul 23 at 2:45
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On the other hand this is a new user who has 7 reputation (you start with 10) and you people have down voted him twice... The comment is beneficial for learning, the down-votes at this stage in his/her account is just discouraging. +1 –  Albert Renshaw Jul 23 at 4:36
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And further studies are disparaging this thought. –  Almo Jul 23 at 13:28

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