8 ms·

What's a Mathematician to do?

140 points by StandardFuture 12y ago

dinglefuk 12y ago
Good luck having a discussion about this here. Math is for zany, nerdy hermits! Math is opposed to humanities -- right HN?
- steego 12y ago
  derp
bkcooper 12y ago
That's an excellent response. In one of the comments on it, there is a link to a paper[1] by Thurston that is well worth your time. I found it an inspiring read. [1] http://arxiv.org/abs/math/9404236 http://arxiv.org/abs/math/9404236
- poindontcare 12y ago
  +1 : Will Thurston's : "On Proof and Progress in Mathematics" is a highly highly recommended read. Very inspirational stuff.
ryan-allen 12y ago
Interesting read, and the whole thread was shut down by moderators... as usual!
- jjoonathan 12y ago
  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 a vaguely similar post that usually differs in at least one subtle but critical or aggravatingly inconvenient way. Then you're left to hope that a generous contributor has managed to slip an answer in before the hammer fell, because despite the "assurances" from all the "google it" posters, the other google hits always seem to be composed primarily of completely irrelevant SE posts, out of date forum discussions from 2003, Expert Exchange paywalls, and answers.com galleryspam. At least with the spam sites I can recognize that they will be useless from the URL alone :/
  - ryan-allen 12y ago
    I'm astonished at the number of top ranking SE questions I come across that are closed for irrelevance or some other questionable reason. If the answer has a lot of upvotes, how is it irrelevant in the greater scheme of things? If the answer has a high search ranking and you're using it to drive traffic to SE, and it's closed, is that even right? I think they should back their own horse and remove closed content from their Google search index. If anything this mod policy and closed tickets make it obvious there's a market need that's not being fulfilled.
  - vertex-four 12y ago
    > 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.
    - CamperBob2 12y ago
      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?
      - rlpb 12y ago
        Discourse is supposed to solve this problem, and Jeff Atwood is involved in both.
        jjoonathan 12y ago
        Does it solve the problem of spontaneous seeding & seamless organic discovery of these conversations?
deleted 12y ago
[deleted]
wwweston 12y ago
This is great. And this bit: "mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new" generalizes pretty well into: "Understanding only exists in a living community of thinker/practitioners refining and breathing life into ideas old and new."
iconjack 12y ago
A classic mathoverflow moderator debacle. Good question, marked community wiki, with good answers including the top-ranked one from Bill Thurston, an accomplished and beloved mathematician who died just a couple of years ago. Despite the amount of work put into the answers and the number of upvotes, the question is closed. Comments on the top answer deleted, even though other comments refer to them, leaving new visitors to the question momentarily confused. Luckily [closed] doesn't mean [deleted]. I guess the best place to post these kinds of questions is Quora? There seems to be quite a few high-caliber posters there now, including many from mathoverflow. I really like mathoverflow, it's just my personal opinion that the mods are too anal (read: conservative) about enforcing the narrow scope of the site.
- moultano 12y ago
  I don't get it. This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. How is that even remotely true? This is probably the most widely applicable question on the site?
  - techdragon 12y ago
    That's the generic label on the post, the real reason is usually it's "off topic". The bigger problem is that moderators close questions that re ask 5 year old questions, irrespective of how much programming has changed. They built mechanisms to deal with this but the karma limits to use them are so high that no new user can so stack exchange are now suffering from the same kind of reverse feedback mechanism Wikipedia has. New user arrives, asks simple question/fixes a typo, mods shut question down as dupe/bot reverts edit, user comments saying old question isn't accurate anymore because language X has change in way Y and he wants an answer for how it works in language X now/user fixes it again, mods don't care having moved on to other posts/bot reverts it again, user leaves in disgust. It's unfortunate and very difficult, but it's worse for stack exchange because they want the selective pressure against "bad" questions but the community are currently suffocating the churn required to keep the questions relevant to current.
javert 12y ago
Funny how everything after the first paragraph is living proof of just how wrong the first paragraph is. Turns out you can use reason and intellect to understand the world and justify values.
bambax 12y ago
> 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, because it lets you express something about you and provides you with a deep sense of satisfaction. Because it helps you live, not because it helps "the world". The world is big and doesn't care about you; there is no way to know what helps the world and what doesn't, anyway. If what fulfills you is healing people, or getting them out of poverty, then by all means go do that. But don't pollute mathematics, or art, with morals.
- vlasev 12y ago
  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 world.
  - bambax 12y ago
    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.
kasbah 12y ago
Tangentially related: a humorous response to a similar question on Quora. http://www.quora.com/What-do-grad-students-in-math-do-all-day http://www.quora.com/What-do-grad-students-in-math-do-all-da...
graycat 12y ago
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?
zak_mc_kracken 12y ago
I like the Feynman quote a few comments down: "You keep on learning and learning, and pretty soon you learn something no one has learned before."
otakucode 12y ago
Very insightful comment. My only quibble with it is that some of its points could be better fleshed out. For instance, when they say that it is difficult to intellectually know the impact of our actions, that can be made precise. Human society is a complex system which exhibits chaotic behavior and it is provably impossible to predict the consequences of actions. Turning to 'passion' and intuition because reason can lead one astray, though, seems slightly ludicrous. Passion and intuition are the outgrowth of brain processes and we know that these processes are very flawed in often predictable ways. It is true that reason can not guide you with precision, but also true that intuition will misguide you. We're left in the difficult position of only having 2 poor choices. Reason, at least, can recognize its own limitations and admits for course corrections if negative consequences are seen.
quarterwave 12y ago
In my humble opinion, the key is to judge oneself accurately. Or get someone to do so. Ramanujan wrote to Hardy: "I am already a half starving man. To preserve my brains I want food and this is my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the university of from the government." Returning to our world, a tenured university position could provide stability, while a contractual appointment to teach Linear Algebra I & II might end next year - and a journey by stagecoach to the next temporary appointment. In such a situation it may be simpler to write code for a living, while one 'dreamed a dream' on weekends. The way it normally works, one figures out these answers during the M.A level itself. The trouble starts if one does not & keeps plodding on. I apologize if this sounds dull & dreary, but I recently gave this advice to a nephew who wanted to study maths after an undergrad degree - he is now safely writing code for a salary & his parents are relieved. Have I stymied the next Ramanujan, perhaps out of a sense of sour grapes at my own inability to prove the Riemann Hypothesis? Both are possible, but unlikely.
- Energy1 12y ago
  You know you're being idealistic as well because ideally it's better to have money than not.
- acadien 12y ago
  I need to comment about going into academia. The end goal being to contribute to math/science in some meaningful way by working on theoretical research full time. I'm in my 6th year of my PhD as a theoretical physicist. I have been rejected for ever prestigious fellowship and tenure track position I have applied for, my academic career is dead. As I finish my PhD I'll start applying for national lab and industry positions and likely end up right next to your nephew writing code somewhere. I will never contribute in any significant way to the world of theoretical physics. Do I regret it? Hell no! At least I tried! I loved so much of my graduate school experience and I got to try my hand at real research, teach huge classes of undergrads about my favorite topics and travel all over the country giving talks at conferences. I imagine going to graduate school is a lot like starting up your own business, its a high risk/reward situation. And if you do succeed you get huge grants, a tenure track position, and wonderful students to work with. Failure means going back to industry with the everyone else (is that so bad?). If I had skipped grad school and moved straight to industry (I've had multiple internships in the valley) I would certainly be much further ahead, both financially and in terms of job security. But I would have missed out on an important life experience and more importantly I never would have made my attempt to contribute to math/science.
dahart 12y ago
Out of all the really excellent reasons given relating to how someone can still discover new things and/or contribute to society, nobody actually challenged the premise with the actual reason most people do math, or anything else: for themselves, because they enjoy it. Thinking that the only reason to do something is if you have a chance of doing someone nobody else has done, or to give a lasting and historical contribution to society, is to deprive yourself of the ability to enjoy what you do, which in turn is demotivating and ironically reduces the probability of achieving those goals. Math is worth doing because its fun to do, regardless of whether you are traveling on well-worn roads or exploring unknown places.
theodoree 12y ago
To claim that Ted Kaczynski was led astray because of "bare reason" is preposterous at best.

140 points by StandardFuture 12y ago

Good luck having a discussion about this here. Math is for zany, nerdy hermits! Math is opposed to humanities -- right HN?

steego 12y ago
derp

bkcooper 12y ago

That's an excellent response. In one of the comments on it, there is a link to a paper[1] by Thurston that is well worth your time. I found it an inspiring read. [1] http://arxiv.org/abs/math/9404236 http://arxiv.org/abs/math/9404236

poindontcare 12y ago
+1 : Will Thurston's : "On Proof and Progress in Mathematics" is a highly highly recommended read. Very inspirational stuff.

ryan-allen 12y ago

Interesting read, and the whole thread was shut down by moderators... as usual!

jjoonathan 12y ago
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 a vaguely similar post that usually differs in at least one subtle but critical or aggravatingly inconvenient way. Then you're left to hope that a generous contributor has managed to slip an answer in before the hammer fell, because despite the "assurances" from all the "google it" posters, the other google hits always seem to be composed primarily of completely irrelevant SE posts, out of date forum discussions from 2003, Expert Exchange paywalls, and answers.com galleryspam. At least with the spam sites I can recognize that they will be useless from the URL alone :/
- ryan-allen 12y ago
  I'm astonished at the number of top ranking SE questions I come across that are closed for irrelevance or some other questionable reason. If the answer has a lot of upvotes, how is it irrelevant in the greater scheme of things? If the answer has a high search ranking and you're using it to drive traffic to SE, and it's closed, is that even right? I think they should back their own horse and remove closed content from their Google search index. If anything this mod policy and closed tickets make it obvious there's a market need that's not being fulfilled.
- vertex-four 12y ago
  > 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.
  - CamperBob2 12y ago
    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?
    - rlpb 12y ago
      Discourse is supposed to solve this problem, and Jeff Atwood is involved in both.
      - jjoonathan 12y ago
        Does it solve the problem of spontaneous seeding & seamless organic discovery of these conversations?

deleted 12y ago

[deleted]

wwweston 12y ago

This is great. And this bit: "mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new" generalizes pretty well into: "Understanding only exists in a living community of thinker/practitioners refining and breathing life into ideas old and new."

iconjack 12y ago

A classic mathoverflow moderator debacle. Good question, marked community wiki, with good answers including the top-ranked one from Bill Thurston, an accomplished and beloved mathematician who died just a couple of years ago. Despite the amount of work put into the answers and the number of upvotes, the question is closed. Comments on the top answer deleted, even though other comments refer to them, leaving new visitors to the question momentarily confused. Luckily [closed] doesn't mean [deleted]. I guess the best place to post these kinds of questions is Quora? There seems to be quite a few high-caliber posters there now, including many from mathoverflow. I really like mathoverflow, it's just my personal opinion that the mods are too anal (read: conservative) about enforcing the narrow scope of the site.

moultano 12y ago
I don't get it. This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. How is that even remotely true? This is probably the most widely applicable question on the site?
- techdragon 12y ago
  That's the generic label on the post, the real reason is usually it's "off topic". The bigger problem is that moderators close questions that re ask 5 year old questions, irrespective of how much programming has changed. They built mechanisms to deal with this but the karma limits to use them are so high that no new user can so stack exchange are now suffering from the same kind of reverse feedback mechanism Wikipedia has. New user arrives, asks simple question/fixes a typo, mods shut question down as dupe/bot reverts edit, user comments saying old question isn't accurate anymore because language X has change in way Y and he wants an answer for how it works in language X now/user fixes it again, mods don't care having moved on to other posts/bot reverts it again, user leaves in disgust. It's unfortunate and very difficult, but it's worse for stack exchange because they want the selective pressure against "bad" questions but the community are currently suffocating the churn required to keep the questions relevant to current.

javert 12y ago

Funny how everything after the first paragraph is living proof of just how wrong the first paragraph is. Turns out you can use reason and intellect to understand the world and justify values.

bambax 12y ago

> 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, because it lets you express something about you and provides you with a deep sense of satisfaction. Because it helps you live, not because it helps "the world". The world is big and doesn't care about you; there is no way to know what helps the world and what doesn't, anyway. If what fulfills you is healing people, or getting them out of poverty, then by all means go do that. But don't pollute mathematics, or art, with morals.

vlasev 12y ago
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 world.
- bambax 12y ago
  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.

kasbah 12y ago

Tangentially related: a humorous response to a similar question on Quora. http://www.quora.com/What-do-grad-students-in-math-do-all-day http://www.quora.com/What-do-grad-students-in-math-do-all-da...

graycat 12y ago

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?

zak_mc_kracken 12y ago

I like the Feynman quote a few comments down: "You keep on learning and learning, and pretty soon you learn something no one has learned before."

otakucode 12y ago

Very insightful comment. My only quibble with it is that some of its points could be better fleshed out. For instance, when they say that it is difficult to intellectually know the impact of our actions, that can be made precise. Human society is a complex system which exhibits chaotic behavior and it is provably impossible to predict the consequences of actions. Turning to 'passion' and intuition because reason can lead one astray, though, seems slightly ludicrous. Passion and intuition are the outgrowth of brain processes and we know that these processes are very flawed in often predictable ways. It is true that reason can not guide you with precision, but also true that intuition will misguide you. We're left in the difficult position of only having 2 poor choices. Reason, at least, can recognize its own limitations and admits for course corrections if negative consequences are seen.

quarterwave 12y ago

In my humble opinion, the key is to judge oneself accurately. Or get someone to do so. Ramanujan wrote to Hardy: "I am already a half starving man. To preserve my brains I want food and this is my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the university of from the government." Returning to our world, a tenured university position could provide stability, while a contractual appointment to teach Linear Algebra I & II might end next year - and a journey by stagecoach to the next temporary appointment. In such a situation it may be simpler to write code for a living, while one 'dreamed a dream' on weekends. The way it normally works, one figures out these answers during the M.A level itself. The trouble starts if one does not & keeps plodding on. I apologize if this sounds dull & dreary, but I recently gave this advice to a nephew who wanted to study maths after an undergrad degree - he is now safely writing code for a salary & his parents are relieved. Have I stymied the next Ramanujan, perhaps out of a sense of sour grapes at my own inability to prove the Riemann Hypothesis? Both are possible, but unlikely.

Energy1 12y ago
You know you're being idealistic as well because ideally it's better to have money than not.
acadien 12y ago
I need to comment about going into academia. The end goal being to contribute to math/science in some meaningful way by working on theoretical research full time. I'm in my 6th year of my PhD as a theoretical physicist. I have been rejected for ever prestigious fellowship and tenure track position I have applied for, my academic career is dead. As I finish my PhD I'll start applying for national lab and industry positions and likely end up right next to your nephew writing code somewhere. I will never contribute in any significant way to the world of theoretical physics. Do I regret it? Hell no! At least I tried! I loved so much of my graduate school experience and I got to try my hand at real research, teach huge classes of undergrads about my favorite topics and travel all over the country giving talks at conferences. I imagine going to graduate school is a lot like starting up your own business, its a high risk/reward situation. And if you do succeed you get huge grants, a tenure track position, and wonderful students to work with. Failure means going back to industry with the everyone else (is that so bad?). If I had skipped grad school and moved straight to industry (I've had multiple internships in the valley) I would certainly be much further ahead, both financially and in terms of job security. But I would have missed out on an important life experience and more importantly I never would have made my attempt to contribute to math/science.

dahart 12y ago

Out of all the really excellent reasons given relating to how someone can still discover new things and/or contribute to society, nobody actually challenged the premise with the actual reason most people do math, or anything else: for themselves, because they enjoy it. Thinking that the only reason to do something is if you have a chance of doing someone nobody else has done, or to give a lasting and historical contribution to society, is to deprive yourself of the ability to enjoy what you do, which in turn is demotivating and ironically reduces the probability of achieving those goals. Math is worth doing because its fun to do, regardless of whether you are traveling on well-worn roads or exploring unknown places.

theodoree 12y ago

To claim that Ted Kaczynski was led astray because of "bare reason" is preposterous at best.