The programming world is like this too, only worse. Programming techniques transfer between languages, but nobody seems to realize that. Instead, you see each new language community "discover" something every other language community had known for years. "Those languages suck, so we ignore everything they do."
It is depressing, and I have nothing more to say about this.
In programming, the question of whether a C++ programmer can walk in and do Java is still open. People may claim it's not possible or they may give the person a chance. In mathematics, as far as I understand the current situation, there is almost never any debate. Once you are an algebraic geometer or whatever, you aren't going to walk into another subfield and start working except in very exceptional cases. The relative openness of programming, in fact, is why I switched from math to programming.
Unless you're dealing with business executive types (that wouldn't notice if you were lying anyway), most of the time, to most of the people, a Java programmer and a C programmer and a Ruby programmer are all really just programmers.
They may have different ways of dealing with problems (the C programmer will lie awake trying to remember if he dropped a free(), while the Ruby programmer will stay up nights trying to figure out how to stop copying that list O(n) times), but the point is that they're still dealing with the same problems (it's always memory, isn't it?). The techniques may be different, but the questions and basic concepts never change.
In math, this is far from the case. In geometry, you never have to deal with infinity the same way you do in set theory. In fact, you don't even have to understand the idea of infinity the same way as a set theorist, and because of this, you _can't_ become a set theorist (unless you want to go back to undergrad and disappoint your parents _again_). It's like saying that a Java programmer doesn't need to understand that memory exists. Maybe they don't have to directly allocate and free it, but they still need to know how much they have and what happens when they use too much of it, and they can certainly recognize the same issues in any other language, even if they don't know how to fix them.
Mathematicians just don't have the same amount of common ground, and it's not even that they can't (because there's too much or something), it's that they don't really even want to (because they see it as boring or a waste of time (and in the case of analysis, they'd be right ;-)).
The programming world is like this too, only worse
Much worse. I think the problem is magnified by our obsession with languages - overlapping subsets of syntax features that have highly intricate relationships with programming techniques (making certain techniques easier to implement, others-harder, regardless of problem domain).
There is also the language == word on your resume' issue. If something new and cool comes out, people will resist it because they will have to give up their "10 years of Foo experience" for "1 year of Bar experience". Sad but true. (I just say on my resume, "X years programming experience". Except I don't really know how to pick X, because I have been programming since I was 5 and writing useful programs since high school. Slightly different than going to a Java training class and showing up to work everyday for a few years...)
There's some overlap, as well as actor/message-passing concurrency (Erlang), dataflow (Prolog, Oz, etc.), constraint, vector-oriented (APL), etc. Probably a dozen more, depending on where you draw some fine lines. (See CTM for a good overview.) FWIW, Prolog is just as homoiconic as Lisp, and has compile and runtime macros.
Still, I was more wondering about teaching methodology, not enumerating paradigms themselves. What could be done to counteract the "standing on the toes of giants" effect?
If I understand correctly, I'd imagine teaching everyone to implement languages using a system like Ian Piumarta's COLA might do the trick. The point is to break open these black boxes of abstraction (even though black boxes are good sometimes).
This is essentially a critique of specialization, applied to the field of mathematics. I think that this trend might be inevitable, not just in the field of mathematics. The total amount of knowledge that humanity has accumulated continues to grow. If the amount of science that a single person can understand is finite, then individual scientists will understand an ever smaller percentage of it. In some ways that makes me sad.
This trend is inevitable if all of the "specialized" branches and sub-branches really have independent meaning. It's hard to talk concretely about this, but my intuition says this is highly unlikely: chances are that connections between seemingly unrelated subfields should allow us to unify concepts and discover more meaningful mathematical truths.
Maybe that's the reason. Or is it that the modern academic job scene pushes people to be big fish in small ponds rather than medium-sized fish in bigger ponds?
I don't see why two specialists couldn't link their specific knowledge when necessary by explaining it in the context of the fundamental knowledge they should all share. That is to say, as a person specializes more and more in mathematics, there's no reason to think they should forget the earlier lessons they've learned.
I visualize the situation like a search tree. As you travel down the tree away from the root, you become increasingly isolated from other branches of the tree, and you can't even talk to distant leaf nodes because you lack the same vocabulary and shared set of theorems.
Yes, we share some parent way up the tree, but we can't go all the way back to that as a starting point because it'd simply take too long to progress from there back down to the interesting leaf node.
This vision makes me wonder when mathematics will reach a point where mastering the material required to understand a leaf node will take greater than the average life span.
I think a critique that could try to control against expanded knowledge/specialization would be to try to look at the big problems in the field. Is specialization helping people tackle them with their deep knowledge or avoid them to work on things that are irrelevant?
Doron Zeilberger is a pretty clever mathematician, so at the very least, any of his opinions are certainly worth thinking about. I think as a comment about the state of modern research culture, he also recently had one of his paper's published in Rejecta Mathematica
From the article: "Sure enough, the best invited talk was Michael
Kiessling's talk that used the ancient technology of overhead
projector, and it would have been even better if he only used the
blackboard, and it would have been better still if he didn't use
anything, just told us a story."
Exactly this. The fact that all of the mathematicians I've been around
to date (except for one, but he's a ham, so he doesn't really count in
this statistic since he already counts as a hacker :) have treated their
overall field more like a science than an art has really disenchanted me
from it. Yes, mathematics is regarded as the queen of all sciences, but
I don't really buy entirely into that. It has applications there, and
that's probably as far as it goes as far as science is concerned. (Disclaimer: I studied applied
mathematics at a small state university at the undergraduate level.)
Math is beautiful not because it is full of intricate logical machinery
and full of useful computational tools and full of pretty pictures;
rather, math is beautiful because the intricate logical machinery can
take different forms (how many different proofs of the FTIC are there?
Pythagoras' theorem?) and because it's an imaginary world inside one's
head where there are arbitrary -- even infinite -- dimensions and
fungible axioms.
The point of these meetings is to inform (and even pique the interest of) your colleagues. If they have
no idea what's going on and don't even understand the fundamental
notions, what's the point? You're wasting your time, keystrokes,
breath, and energy, not to mention the money provided to you by some
grants.
but what about when these junior faculty etc become senior faculty? as time goes on, if you're not habitually trying to be broadly knowledgeable, how will you ever be?
I think the point of the article is to point out this issue and provoke thought, because it certainly doesn't provide a solution.
Younger mathematicians need to worry about tenure and can't waste their time on big picture things that are not likely to pay off
Uh, mathematics departments decide what the criteria for tenure are. As the article suggests, perhaps mathematics departments could make knowledge of the field in general a criterion.
I would go as far as to say that we need generalists who will take the time to chronicle and revive works from the past to make them relevant again.
Better still, we need the mathematics community at large to become more accustomed to seeing papers published in journals that do this so that a larger contingent can be made aware that these papers/ideas/theorems/corollaries/lemmata/definitions/people/pictures/etc. exist.
Note also that this doesn't necessarily preclude anyone in mathematics from adopting a specialization. In my eyes, the younger mathematician is in the same place as the startup founder: in the position to take a big risk and do something that sounds like a stupid idea (e.g., avoid publishing results anywhere but freely and openly on the WWW; see also: Daniel Bernstein) in order to have it pay off (e.g., earn tenure).
My general understanding is that the last person to be well-versed in basically all of mathematics was Euler. When he got through with it, the field was too broad for any mortal to obtain a working knowledge of most of it. It seems that now, given the above, hardly anybody tries to broaden their horizons after a certain point in their career. (It feels like every course I take these days, I end up making a connection between a topic and another branch of math or computer science, but my instructor doesn't know enough about the other field to appreciate what I bring up.)
While I don't think the problem is quite as severe as Zeilberger claims, I do feel like conference talks should strive to be more accessible to specialists of other fields and to students. Particularly for a conference like the Joint Mathematics Meetings, it would be cool if speakers prepared their talks to be more like TED talks: just technical enough to make sure that the audience can understand the really interesting aspect of the research.
One of my college professors put that suggestion like this: Write to an audience of intelligent 17 year olds. Ignorant of your topic and unwilling to put up with confusing writing, but curious and capable of understanding complex ideas.
When I followed that advice my writing was clearer and better balanced. I also found that the narrative in every piece (even technically orientated stuff) became stronger.
I don't write essay's any more, but it's still the standard by which I measure every email and comment I write.
I'm in the process of writing my thesis and one of my advisors asked me: "Who are you writing this for?". I had a few ideas, but I answered, "For him and our collaborators."
"Wrong! - You should be writing for the graduate student who will be picking up your work."
Any graduate student or even many professors will be on a similar level when exposed to something new. By explaining stuff at a relatively simple level, with enough intermediate steps to outline the method, most people can grasp how you did what you did.
Maybe not 17 year olds, but aimed at people well versed in mathematics, but not necessarily in that field.
One of the reasons that mathematics is so vast and powerful is because people have focused their efforts in specialized areas. If math weren't so broad that a single person could understand it all then it would be pretty unimpressive.
I disagree the characterization that mathematicians don't try to broaden their horizons. Like scholars of any subject, mathematicians follow their interests, which often take them to places never expected to be. To me, that fits the bill for "broadening horizons". And if they don't seem to move far outside of their focus of study, well, I say it has to be this way because mathematical material is often so dense and precise that it takes a very long time to understand what's going on in any one subject.
I've heard the last person to know all of mathematics to be Pascal (d. 1662), Euler (d. 1783), Poincare (d. 1912), or Hilbert (d. 1943). Most of the internet seems to believe that it was Poincare.
Some or most of this may be related to how complicated modern mathematics has become, I know at my college there are 6 undergraduate mathematics degrees to choose from (in regards to specialty) and only grows when you move into graduate school.
