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.
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.
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.
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.
> 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.
I'm surprised to see a lot of resistance to your post here on Hacker News. I'm not a mathematician by trade, but I was involved fairly deeply in that world as an undergraduate and I certainly feel that many mathematicians would agree that mathematics is an end in and of itself. Hardy was actually quite adamant about favoring those areas that are least likely to yield real world applications (though as we know from history this back-fired on him quite spectacularly, through no fault of his own).
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.
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.
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.
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).
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.
> 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?
It seems we completely agree, but you have a peculiar way of putting things. I'm not the one accusing you of writing papers "just for jobs and tenure"... you are! In the same post! ;-)
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.
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.
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.
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.
My list (necessarily limited to what I know about,
have on my bookshelf, and have studied at least
significantly) of mathematical masterpieces? Sure:
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.
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.
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?
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.
By design, Quora I suppose does seem like a better place to post a question like this. But I really don't think Quora's community is anywhere near the caliber of that of Mathoverflow. I guess I haven't been active on MO in a few years so maybe things have changed, but when I last actively read it the site was frequented by quite a few very illustrious names and countless young prodigies and miscellaneous professors.
Sadly this seems to be the state of the higher level discussions on the various Stack Overflow sites that get shared through Y Combinator. Fundamental questions are closed off because they're out of scope of the site they're asked on.
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 :/
> 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.
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.
http://www.quora.com/What-do-grad-students-in-math-do-all-da...