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What's a Mathematician to do?

mathoverflow.net

21–30 of 49 posts

Re: What's a Mathematician to do?

#21
post #19
post #15

> how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics I cannot begin to express how much I disagree with this. Mathematics should fight child abuse? And artists enlist in the military to fight despots overseas, by way of art?? Mathematics should be an end in itself. You should be a mathematician, or a stand-up comic, or a painter because it fulfills you , bec…

IMO Bill Thurston is saying people are people first and mathematicians second. People often feel the need to contribute to the world and mathematicians often wonder how they can do it through mathematics. And of course he says it's not easy to answer that. There's nothing wrong in trying to pursue mathematics to improve some aspect of the world. There are millions of way in which pursuing mathematics can improve the…

My above comment has started its slow descent into negative karma, and there's probably nothing I can do about it -- what's the point of karma if not to spend it from time to time anyway.

I'm currently reading "How Not to Be Wrong" by mathematician Jordan Ellenberg; here's what he has to say about his calling:

"Pure mathematics can be a kind of convent, a quiet place safely cut off from the pernicious influences of the world's messiness and inconsistency. I grew up inside those walls. Other math kids I knew were tempted by applications to physics, or genomics, or the black art of hedge fund management, but I wanted no such rumspringa. As a graduate student, I dedicated myself to number theory, what Gauss called "the queen of mathematics," the purest of the pure subjects, the sealed garden at the center of the convent, where we contemplated the same questions about numbers and equations that troubled the Greeks and have gotten hardly less vexing in the twenty-five hundred years since.

"At first I worked on number theory with a classical flavor, proving facts about sums of fourth powers of whole numbers that I could, if pressed, explain to my family at Thanksgiving, even if I couldn't explain how I proved what I proved. But before long I got enticed into even more abstract realms, investigating problems where the basic actors— "residually modular Galois representations," "cohomology of moduli schemes," "dynamical systems on homogeneous spaces," things like that—were impossible to talk about outside the archipelago of seminar halls and faculty lounges that stretches from Oxford to Princeton to Kyoto to Paris to Madison, Wisconsin, where I'm a professor now. When I tell you this stuff is thrilling, and meaningful, and beautiful, and that I'll never get tired of thinking about it, you may just have to believe me, because it takes a long education just to get to the point where the objects of study rear into view."

Maths appeal rests in it being a "sealed garden at the center of the convent". One should do math because this sealed garden brings you peace, because you belong there.

Re: What's a Mathematician to do?

#22
post #21
post #19

Earlier quoted context omitted.

IMO Bill Thurston is saying people are people first and mathematicians second. People often feel the need to contribute to the world and mathematicians often wonder how they can do it through mathematics. And of course he says it's not easy to answer that. There's nothing wrong in trying to pursue mathematics to improve some aspect of the world. There are millions of way in which pursuing mathematics can improve the…

My above comment has started its slow descent into negative karma, and there's probably nothing I can do about it -- what's the point of karma if not to spend it from time to time anyway. I'm currently reading "How Not to Be Wrong" by mathematician Jordan Ellenberg; here's what he has to say about his calling: "Pure mathematics can be a kind of convent, a quiet place safely cut off from the pernicious influences of t…

The existence of sociopaths demonstrates that his advice isn't universal. Nevertheless, it's widely applicable. And math is too often painted as a hyperindividualistic endeavor, like shopping is.

Some wish to spend all their days in a garden. They exist too; good for them. But math is not just for those who rate high on a hermit spectrum.

Re: What's a Mathematician to do?

#24

Earlier quoted context omitted.

> Organic discussion following a tangent that many readers and contributors found interesting and valuable as evidenced by their posts and upvotes? KILL THAT FILTH BEFORE IT SPREADS! The issue is, the StackOverflow boards aren't discussion boards. It's a Q&A system. If your question doesn't have one (or a few) unambiguously correct answer(s), it's not a fit for the site.

You know what? Maybe they should be discussion boards. If a large number of users seem to use your site's resources in a nonstandard way that a large number of other users find helpful, that might be a hint that you're not in the business you think you're in. In many areas of business, it also might be a hint that someone is about to disrupt you. If Stack Exchange is the Myspace of Q&A sites, who will be the Facebook…

Discourse is supposed to solve this problem, and Jeff Atwood is involved in both.

Re: What's a Mathematician to do?

#25
post #21
post #19

Earlier quoted context omitted.

IMO Bill Thurston is saying people are people first and mathematicians second. People often feel the need to contribute to the world and mathematicians often wonder how they can do it through mathematics. And of course he says it's not easy to answer that. There's nothing wrong in trying to pursue mathematics to improve some aspect of the world. There are millions of way in which pursuing mathematics can improve the…

My above comment has started its slow descent into negative karma, and there's probably nothing I can do about it -- what's the point of karma if not to spend it from time to time anyway. I'm currently reading "How Not to Be Wrong" by mathematician Jordan Ellenberg; here's what he has to say about his calling: "Pure mathematics can be a kind of convent, a quiet place safely cut off from the pernicious influences of t…

Being in a "sealed garden" is not nearly the only reason to do mathematics.

Re: What's a Mathematician to do?

#26

Earlier quoted context omitted.

The SE mods truly baffle me sometimes. Organic discussion following a tangent that many readers and contributors found interesting and valuable as evidenced by their posts and upvotes? KILL THAT FILTH BEFORE IT SPREADS! There's nothing so frustrating as getting an exact hit to your question only to find SE filled with "just google it" responses and "closed because duplicate" moderator actions, complete with a link to…

> Organic discussion following a tangent that many readers and contributors found interesting and valuable as evidenced by their posts and upvotes? KILL THAT FILTH BEFORE IT SPREADS! The issue is, the StackOverflow boards aren't discussion boards. It's a Q&A system. If your question doesn't have one (or a few) unambiguously correct answer(s), it's not a fit for the site.

You must be a SE mod or something!

Re: What's a Mathematician to do?

#27

Earlier quoted context omitted.

> Organic discussion following a tangent that many readers and contributors found interesting and valuable as evidenced by their posts and upvotes? KILL THAT FILTH BEFORE IT SPREADS! The issue is, the StackOverflow boards aren't discussion boards. It's a Q&A system. If your question doesn't have one (or a few) unambiguously correct answer(s), it's not a fit for the site.

You know what? Maybe they should be discussion boards. If a large number of users seem to use your site's resources in a nonstandard way that a large number of other users find helpful, that might be a hint that you're not in the business you think you're in. In many areas of business, it also might be a hint that someone is about to disrupt you. If Stack Exchange is the Myspace of Q&A sites, who will be the Facebook…

Jeff Atwood hates "forum style" interfaces and discussions for Q&A sites. I've always disagreed with this. Forum style discussions not only solve problems, but also branch out into brilliant discussions. I myself have been part of a few math forums before.

Re: What's a Mathematician to do?

#28
It appears that in part the OP wants to know how to do "original" mathematics. Well, to raise the bar a little, the usual criteria for publication are new, correct, and significant. So, the OP was asking about "new", but likely he will also want to know about "correct" and "significant". Okay.

My suggestion is to do applied math. How? There is a famous recipe for rabbit stew that starts out, "First catch a rabbit". Well, for applied math, we could have a recipe that starts "First find an application" or at least find a real problem that needs a solution.

There is a broad range of how to use this recipe -- be in an applied department, e.g., business, engineering, agriculture, medicine, and there get the real problem to start. E.g., R. Bellman was in engineering and medicine. Or, maybe better yet, be in some such field in the real world, find your problem, get at least at first-cut solution, then go to graduate school in such an applied department and use your work as your Ph.D. dissertation. I did that.

Next, it turns out, if take what looks, first glance, like a serious applied problem, or, really, nearly any fairly new applied problem, and consider this problem fairly carefully, say, try to find the first good or a much better solution, then likely can find a doable math problem to attack.

Automatic Presto! If for that serious/new applied problem get a math result that is new and correct, then the math can be viewed as "significant" due partly to its being significant for the applied problem.

"Correct"? Sure, as a means of being correct, do the work with theorems and proofs. Compared with what is available in other fields for being correct, the theorems and proofs of math are a great advantage.

Next, often you will find that existing math still isn't quite or at all what you really need for the applied problem and, thus, have to create some new math, that, yes, will be guided by the applied problem and get some of the significance of your work from that problem. And at time can discover mathematical questions that, maybe, are not important for the applied problem but are interesting as mathematics or have some promise for other applications; so, starting with the applied problem can provide an injection of secret stimulant that can make clear several new directions to pursue.

Also, since nearly everyone in academic STEM fields or nearly everyone in academics has physics envy, in particular, mathematical physics envy, in nearly every field the work that is most respected is that which mathematizes the field. So, do some of that.

So, first get an application!

For tools, I would suggest a good background in analysis, say, through Royden's Real Analysis and/or the first half of Rudin's Real and Complex Analysis. Also get a good background in optimization. For a desert buffet of math, especially the Hahn-Banach theorem, useful for applications, take a pass through Luenberger, Optimization by Vector Space Techniques. Then, get a good background in stochastic processes. And have some more, really, essentially anything and everything in an ugrad math catalog.

You will find that in applied fields such as I mentioned, nearly all the workers struggle terribly with their math background -- they know that they need much more math than they have, and you will have a big advantage. Use it and, thus, do the original math you want.

Do this in a department of pure mathematics? Maybe some such department would like to have some applications from outside the department and be interested, but mostly not. But work in other fields, especially some parts of engineering, has long shown some really interesting and valuable questions and results. E.g., the question P versus NP is now taken as quite serious mathematics, but heavily the question started with integer linear programming in operations research. Some of the work in linear programming and special cases of linear programming on networks resulted in some darned interesting questions with some nicely non-obvious, original answers; in principle many of the questions could have been pursued directly from now classic work in linear algebra but were not and apparently because the motivation was not available or some significance was not clear. So, there was interesting work by W. Cunningham, K. Borgward, V. Klee and G. Minty, R. Bland, D. Bertsekas, and more, and some of this work was done outside departments of pure mathematics. Net, for over 50 years there has been a big theme: Not all the interesting, powerful, valuable, important research in mathematics is done in departments of pure mathematics.

Questions?

Re: What's a Mathematician to do?

#29

Earlier quoted context omitted.

> Organic discussion following a tangent that many readers and contributors found interesting and valuable as evidenced by their posts and upvotes? KILL THAT FILTH BEFORE IT SPREADS! The issue is, the StackOverflow boards aren't discussion boards. It's a Q&A system. If your question doesn't have one (or a few) unambiguously correct answer(s), it's not a fit for the site.

You know what? Maybe they should be discussion boards. If a large number of users seem to use your site's resources in a nonstandard way that a large number of other users find helpful, that might be a hint that you're not in the business you think you're in. In many areas of business, it also might be a hint that someone is about to disrupt you. If Stack Exchange is the Myspace of Q&A sites, who will be the Facebook…

It could also be showing that an additional product is required, rather than a replacement or dilution of the current one. The world doesn't require the destruction of one product to create another.

Stack Exchange's creator is looking at how to improve forums with Discourse, so that might very well happen.

Re: What's a Mathematician to do?

#30
post #21
post #19

Earlier quoted context omitted.

IMO Bill Thurston is saying people are people first and mathematicians second. People often feel the need to contribute to the world and mathematicians often wonder how they can do it through mathematics. And of course he says it's not easy to answer that. There's nothing wrong in trying to pursue mathematics to improve some aspect of the world. There are millions of way in which pursuing mathematics can improve the…

My above comment has started its slow descent into negative karma, and there's probably nothing I can do about it -- what's the point of karma if not to spend it from time to time anyway. I'm currently reading "How Not to Be Wrong" by mathematician Jordan Ellenberg; here's what he has to say about his calling: "Pure mathematics can be a kind of convent, a quiet place safely cut off from the pernicious influences of t…

And now Ellenberg is trying to educate and contribute to the world with his writing, because he has some semi-moral push to do something more than just appreciate that sealed garden.

I find cohomology of moduli schemes thrilling too, but it is really cool when you can use a little math to solve someone's real problem, or more often, use a little math to try to dismantle something bad (see mathbabe.org for some economic applications).

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