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Your claim we could reduce mathematics research misses one big step -- that we can tell, in advance, which mathematicians (and researchers in general) are doing

Your claim we could reduce mathematics research misses one big step -- that we can tell, in advance, which mathematicians (and researchers in general) are doing the "useful" research. In my experience that is very hard to predict. Also often researchers are "standing on the shoulders of giants", so the people who look most real world useful are extending earlier, "not useful" research.

JadeNB 8y ago
In perhaps pithier words, accepting abnry's figures > It may be true that 80% of the mathematical research that is valuable to society is done by 20% of mathematicians. In this case, not much can be lost by reducing research funding. , you can probably get 80% of the return by cutting the right 80% of research; but, if you cut the wrong 80%, then you might be left with just the 20% return on the remaining 20% of work, or 4%. (Also, there're lots of ways to cut the wrong 80%, and only one way to cut the right 80%.)
- abnry 8y ago
  Meritocracy. The mathematicians at Princeton University are going to be producing potentially more valuable pure math research than a professor at name state university 145. I don't know what the exact figure is, but it wouldn't surprise me if it was less than 25% of math PhDs who go on to get a research job in math. How much are we funding these students here? I was on the receiving end of some NSF money for a semester. Was it worth it for the NSF? I barely contributed much. Granted paying a grad student is relatively cheap. But I wouldn't hold it against the NSF if they were more stingy. It may be that we really are funding the right amount and the benefits to the whole ecosystem are great. But I want someone to give a cool-headed discussion of the numbers, not some vague persuasiveness motivated by job security.
  - JadeNB 8y ago
    > Meritocracy. The mathematicians at Princeton University are going to be producing potentially more valuable pure math research than a professor at name state university 145. On average, maybe … but, if we just axe those at NSU 145, then we're definitely not going to be funding the proof of the bounded-gaps conjecture. Now, Zhang managed to prove it anyway (https://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html https://golem.ph.utexas.edu/category/2013/05/bounded_gaps_be...), but who knows how many people at small universities have a big proof in them, if they could only get the funding to have time to explore it? (I would also argue that this is dangerously close to the point of view that big companies obviously know something about doing business successfully, so the best way to save government money spent on business is to cut out small-business loans.) > I don't know what the exact figure is, but it wouldn't surprise me if it was less than 25% of math PhDs who don't go on to get a research job in math. Did you flip a 'not' there? I suspect that it's the other way around, that less than 25% of math Ph.D.s do get a research job in math, or perhaps even worse. (At least, that's if by "research job in math" you mean "academic job in math with research expectations"; if you count industrial research, then maybe I believe it.)
    - abnry 8y ago
      Zhang is an outlier. From what I followed when his proof came out, there were actually very similar ideas being developed by Terry Tao at UCLA and by a PhD student at Oxford. The sort of places you'd expect. And IIRC, Zhang actually come from the top math department in China. He just had trouble getting a good job in the U.S. It is also fair to wonder just where this twin prime conjecture is leading us in terms of usefulness. I know it's hard, but we should be able to draw direct comparisons to how something like RSA panned out. Though RSA is pretty easy to understand. Maybe a better example is elliptic curve cryptography, though I know nothing about that. Can we at least provide a road map for how understanding the twin prime conjecture will lead to useful, practical techniques? (And yeah, I accidentally added a "not" there. Will edit.)
      - JadeNB 8y ago
        > Zhang is an outlier. … And IIRC, Zhang actually come from the top math department in China. Well, sure, and that's my point; there will be outliers. They'll probably have some indicators, like coming from good schools, or prior good work, even if they are currently in lower-ranked places. Every time an outlier comes along, one can certainly retroactively find something that reveals all along that he or she was going to excel; the challenge is finding the people with this potential in advance. (You don't want to fund only the people who have done good work; eventually you'll just get an unduly privileged class of people who did a lot of good math now and no longer can.) So we should have some way of finding these outliers by evaluating their academic history and apparent future potential … and that's a grant committee. (Hey, I hate to find myself defending them! I'm an academic and grant writing is low down on my list of favourite things to do; but it's better than being told that, since I'm not at Princeton, I won't even get a chance to seek funding.) > From what I followed when his proof came out, there were actually very similar ideas being developed by Terry Tao at UCLA and by a PhD student at Oxford. I don't know about the Ph.D. student at Oxford, but (although I can't find it now) I am pretty sure I remember reading a post on Terry Tao's blog in which he was much more charitable than this; essentially, his view (I believe, though I can't find it) was that commonalities could be found between his work and Zhang's, as there always can between even the most revolutionary work and its predecessors, but that Zhang's work represented a genuinely new idea and huge step forward. > It is also fair to wonder just where this twin prime conjecture is leading us in terms of usefulness. I know it's hard, but we should be able to draw direct comparisons to how something like RSA panned out. RSA took essentially from the dawn of recorded mathematics (Euclid) to about 70 years ago; it's now the prototypical example (second, perhaps, to Riemannian geometry) of apparently "purely pure" mathematics that turned out to have applications. I think that's an excellent argument for taking the long view.
thecleaner 8y ago
I suppose a derivative of serendipity is at work here. That is we accidentally find something unused and is suddenly very useful (backprop although not a perfect example). But I think there was a paper sometime back that showed if you want to take advantage of seredipity you may want to divide your resources equally.
WkndTriathlete 8y ago
You're not alone. I think it was Martin Gardner that observed that discoveries in mathematics predated their use in physics by approximately 100 years.
abnry 8y ago
I think we have 50 years worth of data, at the very, in the modern era to give at least some sort of evaluation of the usefulness of the money we are spending on math research. Yeah, it's hard to quantify the value of research. I want research to be funded too. But I prefer some quantification over none at all.

JadeNB 8y ago

In perhaps pithier words, accepting abnry's figures > It may be true that 80% of the mathematical research that is valuable to society is done by 20% of mathematicians. In this case, not much can be lost by reducing research funding. , you can probably get 80% of the return by cutting the right 80% of research; but, if you cut the wrong 80%, then you might be left with just the 20% return on the remaining 20% of work, or 4%. (Also, there're lots of ways to cut the wrong 80%, and only one way to cut the right 80%.)

abnry 8y ago
Meritocracy. The mathematicians at Princeton University are going to be producing potentially more valuable pure math research than a professor at name state university 145. I don't know what the exact figure is, but it wouldn't surprise me if it was less than 25% of math PhDs who go on to get a research job in math. How much are we funding these students here? I was on the receiving end of some NSF money for a semester. Was it worth it for the NSF? I barely contributed much. Granted paying a grad student is relatively cheap. But I wouldn't hold it against the NSF if they were more stingy. It may be that we really are funding the right amount and the benefits to the whole ecosystem are great. But I want someone to give a cool-headed discussion of the numbers, not some vague persuasiveness motivated by job security.
- JadeNB 8y ago
  > Meritocracy. The mathematicians at Princeton University are going to be producing potentially more valuable pure math research than a professor at name state university 145. On average, maybe … but, if we just axe those at NSU 145, then we're definitely not going to be funding the proof of the bounded-gaps conjecture. Now, Zhang managed to prove it anyway (https://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html https://golem.ph.utexas.edu/category/2013/05/bounded_gaps_be...), but who knows how many people at small universities have a big proof in them, if they could only get the funding to have time to explore it? (I would also argue that this is dangerously close to the point of view that big companies obviously know something about doing business successfully, so the best way to save government money spent on business is to cut out small-business loans.) > I don't know what the exact figure is, but it wouldn't surprise me if it was less than 25% of math PhDs who don't go on to get a research job in math. Did you flip a 'not' there? I suspect that it's the other way around, that less than 25% of math Ph.D.s do get a research job in math, or perhaps even worse. (At least, that's if by "research job in math" you mean "academic job in math with research expectations"; if you count industrial research, then maybe I believe it.)
  - abnry 8y ago
    Zhang is an outlier. From what I followed when his proof came out, there were actually very similar ideas being developed by Terry Tao at UCLA and by a PhD student at Oxford. The sort of places you'd expect. And IIRC, Zhang actually come from the top math department in China. He just had trouble getting a good job in the U.S. It is also fair to wonder just where this twin prime conjecture is leading us in terms of usefulness. I know it's hard, but we should be able to draw direct comparisons to how something like RSA panned out. Though RSA is pretty easy to understand. Maybe a better example is elliptic curve cryptography, though I know nothing about that. Can we at least provide a road map for how understanding the twin prime conjecture will lead to useful, practical techniques? (And yeah, I accidentally added a "not" there. Will edit.)
    - JadeNB 8y ago
      > Zhang is an outlier. … And IIRC, Zhang actually come from the top math department in China. Well, sure, and that's my point; there will be outliers. They'll probably have some indicators, like coming from good schools, or prior good work, even if they are currently in lower-ranked places. Every time an outlier comes along, one can certainly retroactively find something that reveals all along that he or she was going to excel; the challenge is finding the people with this potential in advance. (You don't want to fund only the people who have done good work; eventually you'll just get an unduly privileged class of people who did a lot of good math now and no longer can.) So we should have some way of finding these outliers by evaluating their academic history and apparent future potential … and that's a grant committee. (Hey, I hate to find myself defending them! I'm an academic and grant writing is low down on my list of favourite things to do; but it's better than being told that, since I'm not at Princeton, I won't even get a chance to seek funding.) > From what I followed when his proof came out, there were actually very similar ideas being developed by Terry Tao at UCLA and by a PhD student at Oxford. I don't know about the Ph.D. student at Oxford, but (although I can't find it now) I am pretty sure I remember reading a post on Terry Tao's blog in which he was much more charitable than this; essentially, his view (I believe, though I can't find it) was that commonalities could be found between his work and Zhang's, as there always can between even the most revolutionary work and its predecessors, but that Zhang's work represented a genuinely new idea and huge step forward. > It is also fair to wonder just where this twin prime conjecture is leading us in terms of usefulness. I know it's hard, but we should be able to draw direct comparisons to how something like RSA panned out. RSA took essentially from the dawn of recorded mathematics (Euclid) to about 70 years ago; it's now the prototypical example (second, perhaps, to Riemannian geometry) of apparently "purely pure" mathematics that turned out to have applications. I think that's an excellent argument for taking the long view.

thecleaner 8y ago

I suppose a derivative of serendipity is at work here. That is we accidentally find something unused and is suddenly very useful (backprop although not a perfect example). But I think there was a paper sometime back that showed if you want to take advantage of seredipity you may want to divide your resources equally.

WkndTriathlete 8y ago

You're not alone. I think it was Martin Gardner that observed that discoveries in mathematics predated their use in physics by approximately 100 years.

abnry 8y ago

I think we have 50 years worth of data, at the very, in the modern era to give at least some sort of evaluation of the usefulness of the money we are spending on math research. Yeah, it's hard to quantify the value of research. I want research to be funded too. But I prefer some quantification over none at all.