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.
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12 comments:
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.
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.
Ah, so familiar. So true.
Please keep up the good work.
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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.
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
@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.
I think you will be interested to read this essay by William Thurston on the communications problems that mathematicians have.
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.
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.
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.
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