6 ms·

The Shocking State of Contemporary "Mathematics"

55 points by andreyf 17y ago

amichail 17y ago
For the good of future mathematics we need generalists and strategians who can see the big picture. Older mathematicians probably already see the big picture -- well enough anyway. Younger mathematicians need to worry about tenure and can't waste their time on big picture things that are not likely to pay off.
- carterschonwald 17y ago
  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.
- nrr 17y ago
  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).
- joe_the_user 17y ago
  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.
carterschonwald 17y ago
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
sramam 17y ago
It seems what the 'Kingdom of Mathematics' really needs, is a good toastmasters club!
- tome 17y ago
  Yes, very much so. Mathematical expository talks are, in my experience, of appallingly low quality.
fh 17y ago
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.
- thras 17y ago
  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?
- gregwebs 17y ago
  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?
- andreyf 17y ago
  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.
- baddox 17y ago
  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.
  - msluyter 17y ago
    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.
wtallis 17y ago
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.
- shib71 17y ago
  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.
  - maximilian 17y ago
    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.
  - 17y ago
tybris 17y ago
Math has always relied on a few geniuses to set things straight, the rest is just noise.
- eru 17y ago
  Sources?
buugs 17y ago
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.
deleted 17y ago
[deleted]
nrr 17y ago
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.
- dazmax 17y ago
  It would be nice if it were treated more as an art, but don't you think whoever is paying for the grants would rather the effort be put into digging deeper into specific research topics that may eventually be of use to science rather than finding beautiful proofs and relationships between things we already know to be true?
jrockway 17y ago
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.
- andreyf 17y ago
  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).
  - jrockway 17y ago
    Exactly right. 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...)
  - silentbicycle 17y ago
    Would you say that focusing on programming paradigms, rather than language implementations, would help to counteract this?
    - andreyf 17y ago
      I think so. The big ones I know of being: - dynamic object oriented (Smalltalk) - homoiconic for compile-time metaprogramming (Scheme, Lisp) - dynamic for runtime metaprogramming, like continuations (Scheme, Ruby, Smalltalk) - functional (Scheme, Haskell) - strongly typed (Haskell, Scala) - logical (Prolog) These are the ones I've come across, but there are more on wikipedia: http://en.wikipedia.org/wiki/Programming_paradigm http://en.wikipedia.org/wiki/Programming_paradigm
efaith 17y ago
An alternative way of measuring the degree to which mathematics has "fragmented" might be to look at the pattern of citations in the mathematical literature. Look at it over time to see how it develops. And compare it to citations in other fields to see whether the phenomenon is specific to mathematics or is part of a more general development toward greater fragmentation.

55 points by andreyf 17y ago

For the good of future mathematics we need generalists and strategians who can see the big picture. Older mathematicians probably already see the big picture -- well enough anyway. Younger mathematicians need to worry about tenure and can't waste their time on big picture things that are not likely to pay off.

carterschonwald 17y ago
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.
nrr 17y ago
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).
joe_the_user 17y ago
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.

carterschonwald 17y ago

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

sramam 17y ago

It seems what the 'Kingdom of Mathematics' really needs, is a good toastmasters club!

tome 17y ago
Yes, very much so. Mathematical expository talks are, in my experience, of appallingly low quality.

fh 17y ago

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.

thras 17y ago
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?
gregwebs 17y ago
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?
andreyf 17y ago
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.
baddox 17y ago
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.
- msluyter 17y ago
  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.

wtallis 17y ago

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.

shib71 17y ago
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.
- maximilian 17y ago
  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.
- 17y ago

tybris 17y ago

Math has always relied on a few geniuses to set things straight, the rest is just noise.

eru 17y ago
Sources?

buugs 17y ago

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.

deleted 17y ago

[deleted]

nrr 17y ago

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.

dazmax 17y ago
It would be nice if it were treated more as an art, but don't you think whoever is paying for the grants would rather the effort be put into digging deeper into specific research topics that may eventually be of use to science rather than finding beautiful proofs and relationships between things we already know to be true?

jrockway 17y ago

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.

andreyf 17y ago
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).
- jrockway 17y ago
  Exactly right. 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...)
- silentbicycle 17y ago
  Would you say that focusing on programming paradigms, rather than language implementations, would help to counteract this?
  - andreyf 17y ago
    I think so. The big ones I know of being: - dynamic object oriented (Smalltalk) - homoiconic for compile-time metaprogramming (Scheme, Lisp) - dynamic for runtime metaprogramming, like continuations (Scheme, Ruby, Smalltalk) - functional (Scheme, Haskell) - strongly typed (Haskell, Scala) - logical (Prolog) These are the ones I've come across, but there are more on wikipedia: http://en.wikipedia.org/wiki/Programming_paradigm http://en.wikipedia.org/wiki/Programming_paradigm

efaith 17y ago

An alternative way of measuring the degree to which mathematics has "fragmented" might be to look at the pattern of citations in the mathematical literature. Look at it over time to see how it develops. And compare it to citations in other fields to see whether the phenomenon is specific to mathematics or is part of a more general development toward greater fragmentation.