Work hard

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.