Monday, November 2, 2009

Why I left math

Reading how shocked Doron Zeilberger is at the state of modern mathematics reminded me of why I left the subject.

Math departments regularly have visiting mathematicians come and give talks. Or at least the one I was at did. For the visiting professors these talks were a confirmation of success, all of these people came to hear about their research. So they would talk about their research and get quite excited about what they were describing.

As a grad student I attended. I quickly noticed that most of the professors in the math department went out of politeness. However they knew they wouldn't understand the talk, so they brought other things to do. If I looked around about 15 minutes into the talk, I'd see people reading books, grading homework, and otherwise not paying attention. At the end of the talk the speaker would ask whether there were questions. Inevitably the mathematician who invited the speaker would have some. Occasionally a second mathematician would have some. But the rest of the room wouldn't.

This was supposed to be the high point of the life of a mathematician? That's when I decided that, no matter how much I loved mathematics, I wanted a different career. Unfortunately my wife was in grad school as well, and we were in such a small town that I didn't have any immediate employment options. Therefore I remained a rather unmotivated grad student. In the end my wife switched to medical school just before I would have finished the PhD. I'm mildly disappointed that I didn't finish, but it really has been no loss.

Why do mathematicians put up with this? I'll need to describe a mathematical culture a little first. These days mathematicians are divided into little cliques of perhaps a dozen people who work on the same stuff. All of the papers you write get peer reviewed by your clique. You then make a point of reading what your clique produces and writing papers that cite theirs. Nobody outside the clique is likely to pay much attention to, or be able to easily understand, work done within the clique. Over time people do move between cliques, but this social structure is ubiquitous. Anyone who can't accept it doesn't remain in mathematics.

It is important for budding academics to understand this and get into a good clique. This is because your future career and possible tenure is based on your research. But the mathematicians making those decisions are unable to read your papers to judge your work. Therefore they base their decisions on the quality of journals you get your papers into, and the quality of people you get writing recommendations for your work. But both of those come down to getting into a group that includes some influential mathematicians who can get your papers accepted in good journals, and that can write strong letters of recommendation.

In fact if, like me, you are someone who likes to dabble in lots of things, you will be warned (as I was by multiple professors) about the dangers of not focusing on one small group. You will be told plenty of cautionary tales of mathematicians who published a number of good papers, but who didn't publish enough in any specific area to get good mathematicians to stand behind them. And therefore the unlucky generalist was unable to get tenure despite doing good work.

For a complete contrast, look at the situation in biology. A motivated advanced biology undergrad is both capable of, and expected to read current research papers. When biologists go to a talk they both expect to understand the talk. And biologists have no trouble making tenure decisions about colleagues based on reading their papers.

I subscribe to the belief that the difference is mainly cultural. Biology is fully as diverse and complex as mathematics. Furthermore what I have read about the history of mathematics suggests that the structure of the mathematical community was substantially different before WW II. For example David Hilbert was known for stopping speakers and forcing them to define anything he found unclear. (Amusingly he once had to ask Von Neumann what a "Hilbert Space" was.) But after WW II an explosion of university math departments and a focus on solving concrete problems lead to a fragmentation of mathematics. And once mathematicians came to accept that they couldn't be expected to understand each other, there was nothing to prevent mathematics from splintering into fairly small cliques. Which has happened, and this is unlikely to ever be reversed.

PS I'm amused at the fact that a number of comments at Y-combinator thought that the situation with programming was worse than mathematics. Yes, there are divisions within programming. But they are nothing compared to the fragmentation in mathematics. I've done both and there is simply no comparison.

17 comments:

Peter Grey said...
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Peter Grey said...

I hadn't put my finger on it, but you described exactly the problem I have with academic math, and I would agree that the problem is worse than in computer science, though I think all academic disciplines suffer from it. It basically boils down to the fact that if you don't do work isolated in one particular field, it becomes much harder to get funding.

Dave Rodenbaugh said...

This was the exact same reason I couldn't continue with a degree in Physics. :) I changed directions after university and became a software nerd instead. People actually care about that work.

Artur Adib said...

Ah, so familiar. So true.

Please keep up the good work.

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cakesy said...

It is the same with swimmers and runner. Why do they just aim to win the 100m and 200m, why don't they also go for the 5km and the marathon?

Same question, and it is the same answer. You need to specialize, concentrate on one area, or you never understand anything really well.

The mathematicians who show up at the visiting lecture, and do other work are actually being polite. They don't understand the intricacies, but want to show support. They are also very busy.

As to Peter Grey, I would say the problem is much more understandable in maths. CS isn't that complex a field, you can pretty much understand a completely different part to the bit you study. Maths is a huge field, around a million times bigger than CS.

red dave said...

How complicated are maths proofs? The four colour proof was so complicated I believe no one yet fully understands it. How many people understand the proof of Fermats last theorem? And these are proofs almost everyone recognises as important.

If the problem is with maths being too concrete surely these pure maths areas would be better understood?

Great article

btilly said...

@cakesy: The process of intellectual knowledge needs both specialists and generalists. It is true that specialists tend to learn more about a specific subject and are more likely to push it forward. But acquiring facts is not enough, to be useful knowledge needs to be integrated, organized and arranged for a broader audience. Else the theorems proved today are forgotten tomorrow. Generalists tend to play this role.

It seems to me that mathematics has gone too far in the specialization direction. There needs to be a balance of activities. But currently that balance seems to be missing.

Incidentally, speaking personally, comprehension was not a challenge for me. I managed 2 publications and a math monthly problem in 3 different areas before finishing my masters. Nobody doubted that I would be able to go on to do research. What I was warned of is that I couldn't possibly be recognized for the research I had done unless it was focused in one area.

Mark Dominus said...

I think you will be interested to read this essay by William Thurston on the communications problems that mathematicians have.

zbyszek said...

This rings true to me. I was a theoretical particle physicist, and everywhere I worked one was expected to take an interest in the research of other particle physics theorists and experimentalists; attending their seminars was de riguer, for instance. Then I worked in a physics theory group that happened to be part of a mathematics department. The head of theory became the head of department, and was suprised to find the culture among the mathematicians very different. They would hardly ever go to the talk of a visiting speaker and often worked from home.
I suspect that the differences are that physics, or certainly the stuff I was doing, tends to be collaborative, but also that a knowledge of fundamental physics means that another area of physics might not be entirely opaque.
In contrast to Mr. Rodenbaugh, I find myself more isolated now that I write software for a living.

btilly said...

Thanks, Mark. I haven't read that paper in many years. It shaped a number of my opinions, including my belief that there is no point in having a professor lecturing about things that the class is not expected to understand.

I do have minor quibbles with some parts. For example I'd say that one of the prime reasons why verbal communication is faster than written is because of how much detail we wish to see. When we listen we're willing to accept that someone, somewhere, has worked out the details. In reading we wish to see the details.

But on the whole it is an excellent essay. Thank you for reintroducing me to it.

Wing said...

Thanks. That was a great post. I also left grad school for somewhat similar reasons. I really find no pleasure in communicating something I enjoy working on with maybe only a dozen other people in the world. I heard that it's also pretty bad in linguistics.

Christian R. Conrad said...

Hmm... So what's a field?

I mean, why is there a separate field of "Statistics" -- why isn't that part of "Mathematics"? Or should that be, of "Economics"? Or should that be a part of "Mathematics"? Or the other way around? And why, for example, shouldn't "Medicine" be part of "Biology", or vice versa?

That is, to quote: "These days mathematicians are divided into little cliques of perhaps a dozen people who work on the same stuff. All of the papers you write get peer reviewed by your clique. You then make a point of reading what your clique produces and writing papers that cite theirs." Yeah, like a bunch of guys write about how living organisms work, and a "field" crystallizes around them, and comes to be called "biology". And so on and so forth, and we get the fields that are reflected in the names of university faculties today. But perhaps the splintering you noticed just means that there should be several new separate fields/faculties, parts of the conglomerate that used to be called "mathematics"?

Then it would make perfect sense that some of you "knew they wouldn't understand the talk, so they brought other things to do", etc -- they're like history post-graduates at a paleontology lecture, or physics researchers at an engineering symposium. The "field" should be split up into several new ones.

Not saying it is necessarily that way, but it seems to be a possible explanation that you hadn't acknowledged.

It's like that old standby in political discussions: "That's just semantics!"

Yeah, sure... But one kind-of-valid reply to that is, "Everything is 'just' semantics."

John Baez said...

I don't experience the problem you're talking about, because I avoid going to talks I think will be boring, and I try hard to give talks that are fun and comprehensible. As a result, lots of people enjoy my talks. They say things like "that was a really great talk!", and I get that pleasantly inflated ego feeling that makes you want to keep getting better at something.

It takes endless thought and effort to give really good talks. Most academics don't think about the psychological principles involved, so they give really bad talks. They're too busy thinking about their subject matter to think about what the audience wants: a terrible mistake. But I bet they could improve if they bothered to try.

I have a page of advice on how to give good talks.

Nathaniel R H of Nauvoo AL said...
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Nathaniel R H of Nauvoo AL said...

What a pity.

I came here trying to understand the jumble of maths developed but unsolved.

Out of frustration, I googled why mathematicians make things so complicated.

It's not that they do, it's simply that they develop more math than they do language to describe their maths.

Best of wishes and I hope you are happy with your field.

David FernándezBretón said...

A lot of people seems to be echoing your feeling, so I thought I'd add my two cents since I feel that my experience is quite different. I am quite happy with what happens in talks, seminars and colloquia in my current department. People don't try to be polite: like John Baez, they don't attend a talk unless they feel that it's interesting enough to at least pay attention to it. As a result, the talks are almost never crowded but the four or five people who attend do try to at least understand the big picture and there's usually a good amount of questions (maybe not necessarily in the end, but people do interrupt the speaker when they want clarification). Of course, it's possible that someone gets lost at some point in what s/he thought would be an interesting talk, but overall the people attending the talk do try to follow as much as they can.

I mentioned before that people try to get "the big picture". Of course there's a lot of specialization in math, and you typically can't expect to understand all of the details on a paper that's not in your area unless you're willing to put some couple years (more or less, depending on how close or far away from your area) into it. But it is certainly possible to get an idea of the big picture, by making analogies with what you do know and so on. Although it is hard and it takes time to get used to getting the big picture of anything, because at the end of an undergraduate degree and at the beginning of a graduate one, you become convinced that the most important part of math is getting all the details right. But once you have to actually prove something on your own, you get pretty comfortable with "big pictures" (for example, if you want to mimic a certain technique from someone else's proof and apply it to your research problem, you can typically do with just a rough idea of the technique and without any details: you only work out the details for your problem in case the technique works --and if that's the case, you typically find that you can reconstruct pretty easily the details from the proof that you were adapting).

You mention biology, but I'm not sure to what extent what biologists do is just feel more comfortable with only getting the "big picture" of some paper they read or some talk they attend. If that's the case, then what they do wouldn't be so different than what I've seen in my Math department.

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