
What's a Mathematician to do? - StandardFuture
http://mathoverflow.net/a/44213
======
kasbah
Tangentially related: a humorous response to a similar question on Quora.

[http://www.quora.com/What-do-grad-students-in-math-do-all-
da...](http://www.quora.com/What-do-grad-students-in-math-do-all-day)

------
otakucode
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.

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

------
quarterwave
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.

~~~
acadien
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.

~~~
graycat
Long one reason I have wanted to be successful in business has been so that I
could afford to _retire_ and pursue theoretical physics!

------
bambax
> _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
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
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.

~~~
kaitai
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).

~~~
bambax
> _with his writing_

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.

~~~
kaitai
With all due respect, I find declarations like

> 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?

~~~
bambax
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.

------
wwweston
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."

------
dahart
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.

------
zak_mc_kracken
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."

------
graycat
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?

~~~
Energy1
I just looked up Luenberger's book. Seems like a nice book.

Can you, please, post all the books(and maybe papers) you think are
mathematical masterpieces? Subject doesn't matter, only the exposition.

~~~
graycat
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.

~~~
Energy1
Very nice!

Thank you very much!

------
iconjack
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
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
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.

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

~~~
jjoonathan
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 :/

~~~
vertex-four
> 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
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
Discourse is supposed to solve this problem, and Jeff Atwood is involved in
both.

~~~
jjoonathan
Does it solve the problem of spontaneous seeding & seamless organic discovery
of these conversations?

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

------
javert
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.

