What's a mathematician to do? (2010)

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.

I had the privilege of discussing similar concerns with regard to theoretical physics with the late Richard Feynman. He told me the following, which has always served me in good stead: "You keep on learning and learning, and pretty soon you learn something no one has learned before." That was his "advice"; my advice? Go for it!

You don't have to be Michael Jordan to play basketball.

I appreciate the OP's question, but it seems to me that

One can rewrite their books in modern language and notation or guide others to learn it too but I never believed this was the significant part of a mathematician work; which would be the creation of original mathematics

is a widely-held fallacy. Perhaps 90% of the work of mathematicians is not brilliantly original creation, but the challenge of digesting and reworking over the cumulative insights and re-presenting them in a useful way for others (this generation and the next). This is a huge task, and it's often a very satisfyingly creative task which will well reward one with new insights, understanding of consequences of the major insights, and understanding their underpinnings.

And part of this is: teach and write well. Good exposition is perhaps the most important contribution that "lesser mortals" can make (and perhaps our collective livelihood depends on it -- Rota has some perceptive things to say about that in his Indiscrete Thoughts). You can get started on this even as an undergraduate: once you have understood something, share it with others. Give a good talk in your local math club. Pay it forward!

Terry Tao wrote on this subject; I think you will be happy with his conclusions. An excerpt:

The number of interesting mathematical research areas and problems to work on is vast – far more than can be covered in detail just by the “best” mathematicians, and sometimes the set of tools or ideas that you have will find something that other good mathematicians have overlooked, especially given that even the greatest mathematicians still have weaknesses in some aspects of mathematical research. As long as you have education, interest, and a reasonable amount of talent, there will be some part of mathematics where you can make a solid and useful contribution.

Let met add two quotations:

1. Fermat's motto was "Multi pertransibunt et augebitur scientia" (many will pass through and knowledge will be increased). At another occasion he wrote about "passing the torch to the next generation", which I find particularly nice.

2. "When kings are building, carters have work to do". Kronecker quoted this, in his letter to Cantor of September 1891.

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.

I'm not really qualified to answer this...or perhaps I am, since I'm no Terry Tao! I think what we bring to mathematics is the unique perspective afforded by our own individual experience. The more of us who are trying to do mathematics, the better, because this increases the diversity of perspectives in approaching the myriad problems we face. Also, as is popular to say, there is far more mathematics to do than any handful of powerful mathematicians can do. This is a "big tent".

Another thing that I've read someplace...someone was describing the difference between Kolmogorov and Israel Gelfand, and this person wrote/said the following: "when Kolmogorov went into a new mathematical landscape he immediately looked for the tallest mountain and climbed it, when Gelfand entered the same landscape, he immediately began building roads." (Someone please fill muad in on the proper location of this quote...I think it was the Notices...)

Others have said that most of what we do as mathematicians is organize and clean things up to clear the way for a future polymath, like a von Neumann, to really make some progress, and some road builders have built some unbelievable roads. (Didn't Serre say something like he spent most of his career rewriting other people's work?)

I don't know about you, but I'm fine with this! It's much better than having my name on the location of a transistor on some unknown circuitboard on the space shuttle...not that there's anything wrong with that!

Anyhow, don't worry about having things to do...there are plenty!

One possible way to think about this midnight question might be to ask what smaller results have been important or interesting to you, and then to appreciate the existence of the mathematicians who have discovered/invented it. My guess would be that everyone can compile a list of such results by players who are not in the league of Gauss and Euler. My own list would be quite long, and some of the results are recent enough to have been discovered by users of MO.

I like the cannon fodder analogy -- made me laugh! I think of myself that way sometimes. Also, though, deep down I like to think I might be smarter than I realize. If I keep at it, maybe I will tap into some hidden reserves of insight. Also, remember the line from Ecclesiastes:

"I returned, and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill, nor yet theorems to mathematicians of brilliance; but time and chance happeneth to them all."

OK maybe the part about theorems is not in the original, but you get my point :-)

If you are an undergraduate, you don't yet know what you can contribute to mathematics, and in particular you don't yet know whether, or at what level, you can create original mathematics. Fortunately, you don't have to work this out entirely for yourself; if you go on to graduate work in mathematics, you will have an advisor, whose job is to help you get the most out of your potential. Even after a student finishes a PhD, she's not alone; much of the best mathematical work today is collaborative, and whatever weaknesses you may think you have can be compensated for by the strengths of your collaborators (while you compensate for their weaknesses with your strengths).

It ain't easy, but it can be done.

On this issue I find a deal of comfort in the concluding paragraph of G. H. Hardy's A Mathematician's Apology:

The case for my life, then, or for that of any one else who has been a mathematician in the same sense which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them.

Mathematics needs people to create, to explain, to synthesize, to apply, to teach, to learn, even to proselytize (in socially acceptable ways). If you want fame bordering on immortality, solve a very hard problem or create something that both solves and poses hard problems. If you want instead to be a great contributor to mathematics, do as much of the above as your heart and mind allow.

What would you like to do for mathematics?

Gerhard "Ask Me About System Design" Paseman, 2010.10.27

Muad you raised not only a very important question but also a fairly enigmatic one. I completed my Maters in Mathematics 13 years ago and left mathematics then and there, what I realized during these years was that I could not sleep without thinking something new to discover or invent in pure mathematics. The numbers kept on haunting me. I was fascinated by Relativistic Mathematics as well had a great passion for Number Theory.I tried to find a suitable topic to do some original research work but left that to enter a totally different field. I opened a restaurant named e=mc^2. But still couldn't sleep well. So after running a successful talk-of-the-town restaurant for almost 4 years, I decided to turn to Computer Graphics, another of my passions as I was very involved in fine arts and poetry as well. Having graduated from Vancouver Film School, I am now a Computer Graphics Technical Director at a Visual Effects Studio, where I deal with mathematics in a more intimate way on a day to day basis. The point of all this autobiographical ramblings of mine is this : If you have the right motivation, and if you are curious enough, you have something to discover. My nights are still sleepless as I am yet to find something to discover !!!. So I think mathematics is inside you waiting for it to be discovered, just be passionate enough !!

I suspect that most interesting mathematical results raise more questions than they settle, so in this case you would not have anything to worry about.

@muad: You say you are not strong in mathematics. I don't know whether this is really such. But let's assume that you cannot do math discoveries for the purpose of this discussion.

You may do something else.

You may consider to write software for mathematics. One such project is TeXmacs. I would be grateful if you would manage to learn Scheme programming language and program proper creation of LaTeX macroses when exporting from TeXmacs to LaTeX.

Do you like my idea of what you can do?

I consider to take time to do this myself, but this would steal from my valuable time to do mathematics.

It's a good question. Sometimes you need to listen to your gut instinct and not to what people tell you. Generally, if you ask questions like these to mathematicians, they will offer reasons why you should stay in mathematics, why it's worthwhile, why your contributions matter, etc. But put the same question to a non-mathematician, and you are likely to get a totally different response. In my estimation, most non-mathematicians consider the majority of contemporary mathematical research to be an utter waste of time. Myself, I have gotten very tired of "selling" my own field of research (which is approximately number theory) to the lay public -- it's always the same handful of "applications" that are trotted out. Maybe the problem is that mathematics is perceived/presented as something that is worth pursuing for its own sake, yet in practice this pursuit is tied up with all kinds of incentives of a (much) less lofty nature. Money, power, reputation, you name it. I think this is a very bad combination. It's probably much better for the health of your soul to pursue things of value in a way that is completely separated from these lower incentives.

With a mathematics degree, we are uniquely able to translate the works of an Euler or a Gauss or Wiles or Andrew Ng or Dijkstra or Paul Erdős or Robert Tarjan and talk to a classroom or a layperson about what mathematics really is. What they think and have thought all their life is that it's simply adding numbers and arithmetic. So when you say you do mathematicians and they respond with "I hate math", they don't hate math research, they hate the arithmetic they were taught math was. But we have the ability to show that math is more than that.

Second, I agree with the Richard Feynman quote. Our knowledge consists of what we learn. What we learn consists is what interests us. What interest you may be different than that interested me. But if you learned some aspects of topology last semester while I was learning theoretical computer science, it doesn't make either class bad. But if we're both tackling the same problem, were going to have a different arsenal - a different thought process.

My final thoughts piggy back on my initial ones but more tractable. Math is about proofs. There is no getting around that. However, I would also say that learning to code is a valuable asset that can assist you with mathematics. You'll be able to test small cases and the intuition behind your proof. There are also more formal things like Lean and Coq to help verify formal proofs. This is also valuable in "translating" people like Dijkstra and Tarjan's work.