What's a Mathematician to do?
mathoverflow.net
mathoverflow.net
http://www.quora.com/What-do-grad-students-in-math-do-all-da...
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.
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.
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.
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.
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.
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).
That's right: not with his math. The math he talks about in his book has been around for at least 200 years.
I don't and will never pretend that when you're a mathematician you're forbidden to help others -- that would be meaningless and ridiculous. I react to the "feel good" sentiment that the motivation to do maths should be found in a desire to better the world.
Math is an end in itself; if you want to help the world that should be on your own time.
Ellenberg's book is great.
> Math is an end in itself; if you want to help the world that should be on your own time.
the marks of an idealist who doesn't do math "for a living." I am a mathematician. I've been doing it for years. You argue for a purity that is naive and counterproductive.
Mathematicians have all sorts of motivations to do mathematics. Intrinsic beauty is certainly primary, but in order to continue in this job that doesn't pay all that well and requires sacrifices our families don't understand, we've needed to come to some terms with our roles in the world. We've needed to justify our apparent uselessness, because some of us in conscience can't be useless people and can't morally continue to do pure math if it is indeed contributing nothing of value to the world. Doing math (or writing music, or making art) for the sole purity of thought, the simple beauty of it, is allowed only to people with a certain sort of psychological and financial privilege. I was not raised with that privilege.
The intrinsic beauty of math and the fact that it's a contribution to the world are not in contradiction. Bach wrote beautiful music that has changed the way we hear and the way we think, changed the path of human civilization. He did it for a paycheck. He did it for the audiences who would hear it then. He did it for the beauty. People who write programming languages because they want more beauty in programming do it for themselves and others. If Bach's music wasn't shared, if Ruby just sat hidden on a hard drive, neither of them would have made a lick of difference in the world and I would argue they'd have no value. Mathematics exists without and beyond us. Our discoveries, and the way they're shared, are what make them valuable to human life.
I do math because I desire to better the world: not by ending child abuse, but by discovering and then sharing the beauty of new mathematics. That's why we write papers, you know -- not just for jobs and tenure. Sharing has its own benefits, as in encountering the ideas of others we are sparked into new inspiration.
Move beyond the political and charitable in thinking about how one might better the world. Many software developers are interested in bettering the world and are doing it through their work, even if it's not an app for water in Africa. Can you so readily dismiss all of them? or is it ok because software development is a dirty business that contrasts with that pure garden of mathematics?
I think it's good that you're "sharing the beauty of new mathematics" with your peers -- that's what I've been talking about all along.
But I also think it's presumptuous to want to have a job that "betters the world"; most jobs don't make any difference in the state of the world; many worsen it; and of course a lot of people don't even have a job in the first place.
What's more, history shows that most or all of math will be useful, eventually; the way it's put in the OP, it sounds like math should turn into some kind of vocational school producing teachings that should be immediately applicable; don't be in such a hurry.
You don't know what the future will need anyway; you only know the needs of the present, which are a very bad predictor of the future. It can be argued that by thinking about the present less, one helps the world more.
I see parallels between the garden you are describing and the society of intellectuals presented in Hesse's The Glass Bead Game, a game of pure mathematical abstraction that may only be played by the trained elite of the day.
The result for the protagonist in that book raises serious questions about the purpose of the purely intellectual pursuits and their role in society and human fulfilment.
Most mathematicians that I know or knew contributed to the world and fulfilled that part of them through the rest of their duties as a professor. That means teaching, mentoring, and helping to run an academic department which trains mostly young students who will not be researchers in pure mathematics but rather engineers and so on. It means helping teach a new generation how to think rigorously and energetically and passionately. These are clear (and incredibly important) real-world contributions which are at the same time tangential to the professor's research.
1. Mathematics does fight child abuse. Statistics and analsysis provide insight and understanding into a problem and the outcomes of working with it. Look at the NHS' HSCIC for example which actually looks for this sort of thing: http://www.hscic.gov.uk/
2. Art does indeed fight despots overseas. Look at modern propaganda and the Eastern Bloc socialist and communist art for example. Then there's music and ceremonyl; both arts as well. Even typography foundries are enlisted.
It's all glued together at the core, inseparable from morals which emerges from psychology which is a science, ultimately constrained by mathematics.
Do it because you like it, but it's everything so you can't lose.
"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."
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.
"You keep on learning and learning, and pretty soon
you learn something no one has learned before."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?
Can you, please, post all the books(and maybe papers) you think are mathematical masterpieces? Subject doesn't matter, only the exposition.
Halmos, Finite Dimensional Vector Spaces
He wrote this in 1942 as an assistant to John von Neumann at the Institute for Advanced Study, and the book is baby Hilbert space. Maybe use as a second book on linear algebra, but, if you wish and want to try, a first book.
Rudin, Principles of Mathematical Analysis
AKA baby Rudin. Prove the theorems of calculus; see how such math is done; learn some more material important in the rest of mathematical analysis.
Spivak, Calculus on Manifolds
The three above were at one time the main references for Harvard's famous Math 55.
Royden, Real Analysis
Measure theory and a start on functional analysis. Elegant.
Rudin, Real and Complex Analysis
Rock solid, measure theory again, and more on functional analysis. Also von Neumann's cute proof of the Radon-Nikodym theorem. Nice treatment of Fourier theory. Some more nice material not easy to find elsewhere.
Neveu, Mathematical Foundations of the Calculus of Probability
A second or third book on probability. Succinct. Elegant. My candidate for the most carefully done, serious writing ever put on paper.
Earl A. Coddington, An Introduction to Ordinary Differential Equations
Rock solid mathematically, nice coverage for a first book, and also really nicely written. Read after, say, Halmos and baby Rudin.
Luenberger, Optimization by Vector Space Techniques
Or, fun and profit via, surprise, the Hahn-Banach theorem, Kalman filtering, high end Lagrange multipliers, deterministic optimal control, little things like those, solid mathematically, succinct, at times very applicable. I suspect that one of his theorems is the key to a high end approach to the usually mysterious principle of least action in physics, etc. Reading the Hahn-Banach theorem is just a nice evening in Royden or Rudin R&CA, but seeing the astounding consequences for a lot of applied math, e.g., in parts of engineering, is not trivial and is made easy by Luenberger. It's a lesson: Some of pure math can be much more powerful in applications than is easy to see at first.
John C. Oxtoby, 'Measure and Category: A Survey of the Analogies between Topological and Measure Spaces'
Elegant. Astounding. Some of what learn via the Baire category theorem can shake one's intuitive view of the real line and our 3-space. Definitely a masterpiece. Maybe it's profound.
Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis
When studying Rudin, Royden, etc., don't be without this one! And it's astounding and clears up a lot. Or, why didn't Rudin state the theorem this way? Because that way it's not true -- see Gelbaum and Olmstead!
There are no doubt many more masterpieces, but these are the ones I can recommend.
But, for a good background in pure and applied math and for doing research and making applications, more is needed. While I can list more good sources, I can't regard them as masterpieces. E.g., I don't know of a masterpiece in optimization, statistics, stochastic processes, differential geometry, partial differential equations, or abstract algebra. Useful texts? Yes. Maybe really good? Yes. Masterpieces? No.
Thank you very much!
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?
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.
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 :/
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.
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.
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?
Stack Exchange's creator is looking at how to improve forums with Discourse, so that might very well happen.
Turns out you can use reason and intellect to understand the world and justify values.