Very good question. No worries, I am not going to try and explain the end product of what I do. But how is it done? What does a mathematician do all day, or what do mathematicians do all day when they come together?
Well, first of all, all day is a strictly relative term. A famous mathematician once explained that he couldn’t do mathematics for more than six hours a day, and most of the time, this is true for most of us, at least while working alone. The rest of the time we while away reading, or running, or making music, or watching TV, or blogging - while somehow part of the brain keeps working, which can lead to a certain absentmindedness. Only when I am hot on a trail (sadly, a rare occurrence) will I completely concentrate on work for long periods of time - or when the work is strictly routine, like preparing classes, grading papers (an event all too common in my life), refereeing articles for academic journals or any of the other administrative and community duties that make up a large part of any academic’s day for at least nine months of the year.
Well, but what about the times I do work? Mostly, I sit at my desk for some time. I stare at a blank piece of paper. When I get tired of that, I might stare out of the window, or at the wall if in a basement office. After some time - maybe half an hour - I can’t stand it anymore. Now I have several choices: distract a friend and colleague from work and drag them to get some coffee; take a stroll to get the circulation going; or write something on that piece of paper. What I write on that paper will usually be only a slight variation of whatever I wrote on a different piece the day before - a piece now safely filed away in the “circular file”, or else buried somewhere in my office, or at home. The hope is that sooner or later this slight variation will lead me to an insight.
Then, some day, out of nowhere, I may have an idea. This could work! It’s a very good feeling, an idea. Can be an extraordinary high, in fact. Better not to think about it again for a day to preserve the good feeling (I know from experience that the idea is likely to be wrong, or useless). After a suitable time of enjoying the feeling of my idea, I’ll go back and check it out. Does it even make sense? Care is needed - questions may appear easier than they are (especially when under the influence - even a little alcohol makes everything seem clear). What does it imply? More often than not, it turns out the idea is correct, but only led me in a circle - it looks long and interesting travelled one way, but in the end I’m back where I started.
If the idea holds up for a couple days and seems worthwhile, I’ll write to a collaborator (I much prefer joint projects - which isn’t quite compatible with the requirements for such trivialities as tenure) and explain my idea. In most cases, I’ll hear back soon with a mistake I made (it’s too easy to get carried away by your own ideas), often dooming the whole train of thought. Only rarely does everything work out, and then it’s hard concentration until the possibilities are (for the time being) exhausted.
Once a problem is understood its solution has to be brought into a form other mathematicians will want to read, and be able to understand. Solving the biggest problem in mathematics won’t be of any import if you can’t explain the solution to your colleagues (although once you have a considerable track record of being right - something I’m a long way from - those colleagues will make more of an effort, and will also be more prepared to simply trust you on the details). Writing about mathematics is something that’s difficult to do well. Large numbers of published works are error-ridden, (though rarely are the errors central to the argument), and include convoluted, overly-technical, and notationally insane language: “Let P be point Q, which we’ll call R” is an old but accurate joke. It isn’t only literature professors whose writing makes you go “huh??”. Many mathematicians also don’t like spending time writing down things they, after all, have already understood. This problem seems to have been compounded by the advent of the world wide web making instant dissemination of unfinished manuscripts possible.
Discussing mathematics with others is what I find most enjoyable. I am not very discerning - I’ll happily spend hours explaining calculus to my undergraduates - but certainly discussing research with a colleague, or graduate student, is most satisfying. Thinking along with each other, seeing your vague intuition concretized and formalized - or taken apart - by another person, is an exhilarating experience. When working together ideas can be thrown out and discarded at high speed. Because no two mathematicians think alike, I have never come away from a meeting with a colleague without having learned something - even if no new result was produced.
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