Peng Zhou

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====== Thurston says ======
(in reply to https://mathoverflow.net/questions/43690/whats-a-mathematician-to-do)


It's not mathematics that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding. We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.

The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.

I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction. Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining --- they depend very heavily on the community of mathematicians.


===== and Joe Silverman says =====
It's 12:40am New Year's Day, so maybe not the best time to be writing MO responses, but I loved reading everyone's encouraging answers to this question. (BTW: @Lubin: Taking algebra, with a dollop of algebraic number theory, from you was inspiring.) Anyway, my answer to the OP would be to learn lots and to work hard on problems that interest you. Don't worry so much about whether you're proving breakthrough theorems, just try as hard as you can to understand the parts of mathematics that interest you the most. (By "understand", of course, I mean get into the guts, figure out what's really going on, and prove as much as you can.) Also don't worry that you won't solve every problem (or even a majority of the problems) on which you work, and don't worry that you won't ever feel you fully understand everything about a problem; that's why there's always more to investigate. Then, after a decade or two, feel free to look back, and I think you'll find that you have made a contribution to humanity's knowledge of mathematics.

And even when you're doing research, it's hard (at least, I've found it hard) to decide on the significance of what you've done. I think the difficulty is that after working hard on a problem for a year or two and making enough progress to write a paper, one understands the problem so well that everything that one can prove seems trivial, while everything that's left undone seems hopeless. So maybe my saying "wait a decade or two" is a bit excessive, but it's definitely worth waiting a couple years before you decide on the quality of the work you've done.

Happy New Year to one and all at MO.