Saturday, March 19, 2005

Modern Mathematics and Academia.

This originally began as a reply to something a friend of mine wrote in her LiveJournal. If you're curious and I rememeber rightly, as an undergrad she survived a double major in English Literature and Physics.

Brave woman. \:

from First Post: Does anyone else find grad school to be lessons in how to grow an inferiority complex?

from Second Post: Oughtn't there to be some better way to do this?

My response is below, and full of my own opinions. Please be mericful.




You've got me. I can speak a little, but only about mathematics, and let me start with the present state of affairs.

Formalism. Walking into this subject is a little hazardous, because since the early 1900's mathematicians have sought (with some success) a general-use formalism with which to express their ideas. Modern mathematical arguments are embedded in logic and precise definitions of appropriate concepts to the argument at hand.

It's now what everyone expects and celebrates as a sort of "crown jewel," and serves as a basis for the rigor and 'infallibility' of mathematics. We know exactly why certain facts are true and why certain (theoretical) phenomena occur, because we are very specific about what causes them, what hypotheses we make, and what assumptions we use.

For example, are we in a space where interior angles of triangles add up to 180o? If not, some facts are no longer true (think of a surface of a globe, and draw arcs joining two points of the equator and the north pole: that gives you at least 180 degrees of angles). In such a circumstance, for instance, you can't move vectors the same way and expect to point in the same direction as before (i.e. the problem of parallel transport, from physics).

Comparison. With this in mind, I have to say that it's a lot harder to be a modern mathematician than an 'ancient' one (up to, say, the 1890's). Thinking rigorously is exceptionally hard and takes years of practice, and it's a wonder how newcomers can breach this obstacle and do "good" modern mathematics.

The precision of ideas in the 19th Century wasn't up to par with current logic, but they were extremely intuitive and serve as a basis for modern problems. One example is Riemann's Hypothesis concerning distributions of primes (which are related to roots of a function of 2 variables). Another is an idea of growing popularity: the concept of string theory sponsored by the successors of Einstein, Heisenberg, and the Bohrs. The essential problem lies in how to make the right definitions so that the modern mathematics work.

One key element to observe is that until 1904 or so, most mathematicians were universalists, in that they knew a little something about every area of maths (some say that Henri Poincare was the last universalist) and moreover, had interests in chemistry, mechanics and physics, and even small-scale economic models. In a way their ideas had to be intuitive, because they were being stimulated by so much of Nature.

In short, going into graduate school in mathematics is like learning a conceptual language to extreme fluency, yet preserving the curiosity which is present only in "newborns," who have no such language.

The best problems are the most intuitive ones, and many current research problems can't be understood by the layperson and survive as an artifact of the edifice of academia. They're interesting, but to a select few people who can understand them.

If I may consider myself a mathematician, the odd thing is that we mathematicians still think and work intuitively. We believe that a certain solution may work because of some heuristic (which is really a convenient rhetoric) but we won't know until we prove it: that we demonstrate its validity in a sufficiently rigorous argument, paying special attention to assumptions and smaller arguments which are individually 'infallible.'

That is one problem of modern mathematics, graduate school, and academia: learning how to argue with rigor and precision, and learning how to find good ideas which make good problems. Along the way there are hoops to jump and hurdles to pass, in the forms of exams and course-work, but I believe this is the crux of matters.

12 comments:

Anonymous said...

My answers would be shorter:

1. Yes. Unfortunately, many of them remain in grad school for too long.

2. No. Instead of better ways, you may be able to find better things to do.

I won't comment on your response since you asked to be merciful.

janus said...

Thanks. I appreciate it, because I'm sure that I left myself wide open for severe criticism in more than a few parts of that post.

What can I say? I have a habit of ranting purposelessly.

Anyways, back to work.

~ vyv said...

You remember rightly, except that it wasn't just a double major, it was a dual degree. Oi.

Re: the actual *content* of your post, and keeping the injunction to mercy in mind, I'm gonna disagree with the position (apparently) taken by the anonymous poster, because I still believe that there is something worthwhile in grad school. Maybe I have not been here long enough to get the idealism hammered out of me.

I think your focus on formalism is telling, and important - in English, we tend to phrase a goal for grad students as "entering the conversation." The idea is that as we write essays and articles and work on our dissertations, we will become familiar with the types of discussions going on in our area of interest, and become capable of generating our own research which will contribute substantially to those discussions. To do that, yeah, we need to know the language - the major theorists, some minor theorists if they're important, the general trends of what's being written now, what's still being cited even though it was written 30/50/100 years ago. Now, here we can go into a long wankish argument about whether English theory is useful, what sort of theories are more useful than others, how much it occludes some lovely imaginary "real" and how much it explicates what's already there, but that's not actually my point, believe it or not. I consider myself to have a reasonable skepticism of theory, at least one foot firmly attached to the ground at all times, and a willingness to go along with a good argument even if I don't initially see where it's going.

So. My problem is not *so much* with the idea that we have to learn a new language, or refine our existing language to meet new standards of rigor, to succeed in graduate school and the academic community. I'm a little more iffy on the *total* lack of generalization, or comparison, as you seem to be calling it - in my own experience, having a broader field of knowledge to draw on can be invaluable. I worry that if we are fitting ourselves to have a certain type of academic conversation, then that type of conversation is the only type that we will be able to have. And that seems rather restrictive. It's possible to argue that math, or English, is best served by having only one type of conversation with one standard of rigor and not much room for intuition, but frankly, I don't like that argument. (Which is why I tend to write on science in fiction, or other simiarly cross-disciplinary topics. That shit is *out* there, and an English major who has never taken a biology class is going to be at a great disadvantage when it comes to recognizing it. Possibly you will be able to come up with a similar example for math, possibly not.)

That's one worry. The valuing of one type of knowledge over others always seems an iffy proposition. But there are others. In my second post, I asked if there wasn't some better way to do this, and by "this," I want to broaden my scope - a *lot*.

Graduate school is an incredibly *weird* place. First, there's the academics, which I've kind of addressed (though not in the depth I could, and be grateful for my restraint); second, there's the social side; third, or maybe first, there's economics. We seem, to me, to be occupying some strange interstitial space where we are treated as adults and yet cannot make a living, where we spend our days (and nights, and weekends) in the pursuit of ever more rarefied knowledge, and in teaching it to other students, many of whom are only after a degree because it will make them more hireable, and this knowledge is only rarely rewarded in any tangible way (money, praise, respect, security, what have you). We have taken out a loan on our futures, believing that the four or five or seven years we spend pursuing a degree will, after a possible post-doc or years spent as an assistant professor, allow us to *begin* to recoup our debts on our selves and begin to be recognized for our acheivements in some meaningful manner. We socialize with other people who have taken out this same loan; we teach undergraduates who, when we give their - attempts - a low grade, will instantly go over our heads to the professor, because they perceive us as having no actual power, no matter how much knowledge we have attained. (I admit, some of this is on the bitter side, but on the other hand, I imagine anyone who's taught a few semesters as a TA has had the same experience).

To wrap up this overly-long comment, graduate school is more than the *learning* we do in it. It's more than our fellowships, and it's more than our teaching, and it's more than our rent payments. It's more than the conversations that we have in print with other scholars, and it's more than the ways we struggle to *have* those conversations, despite the difficulty of getting into print. It's more than the constant pressure to out-perform some imaginary ideal grad student. It's this whole complex of things that has accreted like particularly clingy barnacles to the academic hull over the years, and I'm not convinced that all of the time-tested traditions are worth having around.

I think that almost everyone who goes to grad school sincerely wants to *learn*. And I see so many people getting frustrated with the process, and I don't think it's because they've lost that drive for knowledge. I think it's something in the make-up of grad school itself. And *that's* why I ask - isn't there some better way to do this?

Anonymous said...

Whoa, Jasun's habit is contagious. I must be careful. :)

1) I agree that cross-disciplinary research should be encouraged by grad schools, and in particular by math departments.

2) I think that grad school is supposed to perform some sort of selection. And that is always going to be frustrating for some.

3) You reminded me of a comic: http://matik.bohemiaweb.cz/dilbert.gif
Originally it had "English" in place of "mathematics". :)

fragments of angry candy said...

Jasun, you give an appealing description of math: rigor and intuition, in concert.

- - -

Who's the Bohr that's not Neils and what did he/she do?

- - -

I've always sensed similarities between math and literary scholarship. Only a narrow community understands the nuts and bolts of what you do, but you might comfort yourself with the faith that your pursuit has significance in the big wide world. For example, your results might lead to an application appreciated in other fields (although to describe "applied English," which I believe exists, entails going into the "wankish argument" mentioned above). Or maybe you're aiming at Meaning significant in itself, and you don't care if most of the big wide world knows this, as long as some do, which is the point of writing things down, and "entering the conversation."

- - -

I'm grateful for that weird interstitial space called graduate school, even though, yes, Inferiority Complex has moved into my mind and become like an uncool roommate I somehow manage to tolerate. Granted, I haven't had a chance to meet Bitterness From TAing, yet. But without teaching, mathematicians and lit scholars have a hard time justifying themselves economically.

I don't think grad school should be like a loan on one's future. There better be return on investment right now. Maybe there are ups and downs, but on average you ought to be doing what you enjoy. I used to work and put money into 401k, and I don't miss it; doing uninteresting work feels like a genuine waste of one's mind and one's life.

I'm not sure what are the "time-tested traditions" of grad school being critiqued in the conversation preceding this set of posts. Is there really something inherently wrong with grad school itself, or are there concrete aspects that can be fixed (teaching load, funding, etc.)? Or is it the life of a scholar that's naturally problematic? One can think of academia as an imperfect solution to the problem of how individuals can devote much of their lives to thinking and communicating abstractions whose use to society is far less immediate than that of sandwiches, religion, guns, or music.

--johanna

fragments of angry candy said...

sorry, this is bugging me. "wankish" should be "wonkish."

~ vyv said...

Anonymous person, yeah, it's contagious all right. I'm not usually this ranty. As you might have guessed, I've been feeling a little bit frustrated lately with the types of research encouraged by my area & department, with my students, with the whole process of academic one-upsmanship, and with the increasingly depressing economics of living on a tiny stipend. Any & all of this is probably coloring my response.

To Johanna (and I *love* the 'fragments of angry candy' name, btw), the "time-tested traditions" I was referring to *were* the concrete circumstances of the academic life: teaching, funding (and the cut-throat competition for funding), the pressure to publish, all of it. I wasn't trying to be abstract on that one. As I tried to say in my first post, I'm still idealistic enough to believe that there's good in academic study and that I am doing something worthwhile by devoting such a large segment of my life to reading texts. It just seems to me that there must be some ways by which that process could be made - easier? less damaging to health and sanity? If it's an imperfect solution, by what means could it be made more perfect? Sure, it's not always easy and any job is going to have its casualties, but the fact that of my statistically-questionable sampling of "friends in grad school" over 75% are *miserable* at any given time - and mostly due to money! or stretching themselves thin trying to publish and go to conferences and be scholars and still teach undergraduates who don't respect them! and also money! and the reading and classwork and search for a mentor! and the feeling that everyone else has got it more together than they do! and did I mention money? - leads me to conclude that something is not *right* with this system.

I came here because I wanted to do interesting work that engaged my intellect and stretched, if only a little bit, the boundaries of current thought. I still do. It seems, however, that I'm being held back from a full expression of my faculties by what I called the "barnacles," earlier. And that bothers the hell out of me, quite frankly. I'd rather be chasing some aesthetic theory of Oscar Wilde's across a wildnerness of fin-de-siecle French literature than wondering how I'm going to feed myself or grade 35 execrable midterms or finish my *own* papers for this term or revise the one from last semester that my professor graded poorly because the kinds of literature I like to analyze are apparently not canonical enough to merit attention.

As I said, I'm kind of bitter. I love my job despite its flaws. I just wish there weren't so many of them.

And finally, it's "wankish" when it's an abuse of the English language meant to refer to self-abuse, not a nerdy dedication to the chosen field.

Anonymous said...

You should not forget that poverty hurts only in the first few years of your academic career. After that, you'll get used to it.

janus said...

There have been many comments, and let me first thank the lot of you for doing so. I hadn't thought of most (if not all) of these issues and subtleties therein.

Let me reply to a less serious matter first, about why I wrote "Bohrs" and not "Bohr." You can read more about Niels Bohr (also known once as 'Nicholas Baker' when working at the Manhattan Project) here.

Bohr's son Aage also became a physicist and shared the Nobel prize for Physics in 1975. (This is just one instance of famous scientists in the same family. Others are the Van Vlecks as well as the Braggs and Madame Curie and her daughter Irene Joliot.)

janus said...

Here we go again; I'm choosing another quick-to-reply matter to attend to, and artificially increasing the number of comments on this post .. which conveniently makes me seem more popular than I really am. Fortune of fate, that.

I apologize if this is more mathematical than philosophical, but then again, this is a maths blog .. (;

Anonymous: I agree that cross-disciplinary research should be encouraged by grad schools, and in particular by math departments.

I don't think I explicitly stated that I support cross-disciplinary research in my original post. It's something I appreciate and approve of (though I believe it is not for everyone, due to style of thinking), but my main point was to suggest that some sort of intuition in mathematical ideas is useful and preferable. As humans, many parts of mathematics do not suffer from analogy or comparision with existing models of thought.

For example, it helps to perceive the notion of (analytic) capacity as the behavior of a condenser relative to electric current, even though the working mathematical definition has nothing to do with electromagnetism (though it is probably applicable to that end).

Then again, perhaps intuition and cross-disciplinary study cannot be separated as concepts. After all, how else would we acquire intuition (or even think of) certain mathematical concepts and constructions?

So short of intuition arising from direct computations of things, can anyone think of a circumstance where we gain mathematical intuition from non-natural sources?

fragments of angry candy said...

yet another post!

wank v wonk: aha, i get it, nice...

grad school misery: before going, i was forewarned right and left about how horrible it'd be, number one population for depression, etc. yeah, money's got to be a major cause, plus general terms of employment. there's a GSI strike planned here at UM, in just a few days, in fact. Hope we get good terms out of all of it.

angry candy: yay! someone gets the reference. for the record, ee cumming's "the Cambridge ladies who live in furnished souls" is a sexist poem whose sentiment I wholeheartedly disagree with! he's implying that university women (probably meaning wives, then) live sheltered, insipid lives. Bah! I say the women are the scholars, and have a huge house party of the mind, dance on the tables, and kick over the couch! OK... have been wanting to stick that commentary somewhere... and you're the first to comment on my blogname...

apologies to Jasun! absolutely no math in this post.

Anonymous said...

My 4 cents: 1. While it is true that certain experiences are universal, or nearly so, to graduate students, I think the treatment of "graduate school" as a monolithic entity is a hasty generalization. The main problems with graduate school are poverty, teaching and the research. In your case, the first one is purely a function of your location. If you were at say, Iowa or Kansas, this would be less of a problem. The teaching, imho, is also a function of your location. Having moved around quite a lot, I can say that experiences vary _greatly_ among universities. Graduate students are generally (cf. above comment (; ) assigned the least desirable courses. Use this experience to appreciate your future teaching assignments. (A loan against future "happy teaching".) Lastly, the research level there is one of the most intense. Because everyone has dreams/delusions of gradeur, everyone is playing academic "king of the mountain". (Ever play that as a kid? What was the inevitable outcome? Someone would get seriously hurt everytime, but it wouldnt stop the current or future games.....)
2. Because much of what is happening now is a function of your location, you should use this as a learning experience. Is this what you want your life to be? Do you want it 50% of this? 75%, etc? When applying for tenure-track, this will help you decide what your ideal situation is. Oddly, graduate school was hugely different for me, but since my postdocs were very different from graduate school, I got to experience what was and what could be.
3. For reasons not mentionable (you may guess why), I thought of grad school as the hazing period before joining a frat/sorority. And I mean the old-fashioned, now-illegal hazing in which the pledges were servants without dignity. Sadly, there are those who espouse such beliefs still (yet?) and extend it to postdoc positions. Anywhay, once through this period, the payoff was huge-you were a full-fledged brother/sister.
4. In light of the above, use this experience to learn about yourself, others, mathematics, and their levels of interaction. If you dont like something, now is the time to learn that, when there are no consequences. Imagine getting a tenure-track at a liberal arts school with 12 credits a semester teaching and only then realizing you hate teaching.....

We rejoice in our sufferings, because we know suffering produces perseverance, perseverance, character; and character, hope. And hope does not disappoint us. (Rom. 5:3-5)