4 ms·

How much of this argument should apply to more theoretical areas of research like pure mathematics?

abnry 8y ago
I used to be in a math PhD program. Probably would still be if not for some life circumstance. I think it is worth taking a hard look at the value all of this mathematical research actually produces. I understand how number theory has been useful to cryptography. I understand how branches of pure math can have a surprising influence. But when these examples are given by pure mathematicians, it often strikes me as anecdotal and motivated reasoning. Where are the hard numbers? Where is the cool-headed evaluation? They very much want the NSF to continue giving them grants so they can keep funding their mathematical interests. Because it personally and immediately benefits them. 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. This is how I look at it: Funding mathematical research means your society is wealthy. When the vast majority aren't worried about putting food on the table, it is a privilege when you can get paid by them to pursue your mathematical hobby. A hobby that has some relatively low chance of impacting society.
- thecleaner 8y ago
  I can't argue for pure math. But atleast in applied areas such as pde a lot of the problems are motivated by real physical problems. To solve many of these would require theoretical breakthroughs I guess (cant be sure I dont have a phd). So perhaps funding pure math may not be such a bad idea.
  - ginnungagap 8y ago
    I always find the example of PDEs interesting, depending on which mathematician you talk to PDEs could be anywhere from "as pure as it gets" to "extremely applied"
    - thecleaner 8y ago
      I believe this demonstrates the spectrum of pde research. You have people from both ends contributing. Since functional analysis gets used a lot in pdes especially when dealing with weak solutions it would make sense that it appeals to be pure math folks. But once you find ways to construct viable test functions it becomes the basis for writing numerical schemes. So that appeals to the applief guys and of course once you have the numerical solution this can be used for engineering. This demonstrates a good pipeline for "consuming" science (of course assuming this pipeline is indeed correct). However take any component out and the value creation wont be that high. Perhaps finding more such pipelines for pure math woulf help evaluate its valur.
- CJefferson 8y ago
  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.)
- leereeves 8y ago
  Cryptography is a trivial example. The entire field of computing was invented by pure mathematicians like Turing and Church working in the 1930s. At the time, no one had any idea what the applications, if any, would be. It's like any science. Some work has immediate applications. Other work is pure exploration of the unknown. And it is pure exploration of the unknown that leads to the truly revolutionary discoveries. You can't set out to discover penicillin when you don't even know it exists.
  - maxxxxx 8y ago
    Exactly. A lot of discoveries can be viewed as useless for quite a while before someone figures out a practical application.
  - abnry 8y ago
    Something like the Fast Fourier Transform has had a much significant impact on society than a proof of the incomputability of the Busy Beaver function. Saying that Turing and Church "invented computing" is too vague of a justification. I want numbers and details. Folks like Babbage were already thinking procedurally in practical enough terms that once the technological capability (not the theoretical capabilities) caught up, algorithms like the FFT could be discovered and put to use.
    - leereeves 8y ago
      You're only thinking of the immediate practical applications of the mathematics, not the importance of that work for future work. Turing's work was so important that he, not Babbage, is generally recognized as the founder of computer science. One example of the importance of Church's work is Lisp. Church's lambda calculus is the foundation of Lisp, which pioneered nearly all the features of modern programming languages [1]. The designer of Smalltalk, Alan Kay, called it "the greatest single programming language ever designed" [2] and talked about its influence on Smalltalk quite a bit [3]. And your example, the Fourier transform, was itself a mathematical tool long before the first computer was built, and the first published FFT algorithm also dates from the 1930s. [4] 1: http://www.paulgraham.com/diff.html http://www.paulgraham.com/diff.html 2: https://www.quora.com/What-did-Alan-Kay-mean-by-Lisp-is-the-greatest-single-programming-language-ever-designed/answer/Alan-Kay-11?share=1 https://www.quora.com/What-did-Alan-Kay-mean-by-Lisp-is-the-... 3: http://worrydream.com/EarlyHistoryOfSmalltalk/ http://worrydream.com/EarlyHistoryOfSmalltalk/ 4: https://en.wikipedia.org/wiki/Fast_Fourier_transform#History https://en.wikipedia.org/wiki/Fast_Fourier_transform#History
- mattkrause 8y ago
  Number theory is an excellent example since it was an academic curiosity until it suddenly became very very useful. While cryptography is built on number theory, that earlier research was not done with the explicit goal of developing cryptography. Instead, people took existing math and built new applications around it. Obviously, it would be nice to fund only things that are eventually useful, but this is virtually impossible to predict in advance so....we fund a bunch of things and see what works. (Also, math is shockingly cheap compared to lab sciences so it makes even more sense to spread the bets widely.)
- ipioxu15 8y ago
  > A hobby that has some relatively low chance of impacting society. As opposed to more important pursuits, such as advertising.
- LeanderK 8y ago
  but isn't pure math research very cheap (compared to the cost of other scientific research) and therefore could still be cost-effective?
  - abnry 8y ago
    It very well could be. I just dislike vague justifications that it actually is worth the cost. Give me numbers and details.
- chess19 8y ago
  I imagine that a lot of people responding to this do not realize how abstract modern pure mathematics is. The number theory that makes up the basis of cryptography was established in the 1700s. For example, Euler's theorem is the basis of RSA and was proven in 1763. The theorem is a small generalization of Fermat's little theorem which was known (but not proven) in 1640. These theorems are really just simple facts about groups and other cryptosystems, such as elliptic curve cryptosystems, are essentially the same facts except the multiplicative group of integers is replaced with an elliptic curve group. These concepts could be taught to advanced high school students with no formal pure mathematical training. The "hot" areas in modern mathematics require not only an additional 4 years of undergraduate mathematics but usually ~2 years of a PhD program to begin to understand the current papers. This is extremely different from other fields such as theoretical computer science which seems to have applications almost immediately. Even professional mathematicians likely do not research in hopes of applications hundreds of years later. I will not claim that modern mathematics cannot possibly have applications. I will, however, claim that pure mathematics is an extremely poor way to allocate funds if you are simply looking for a return on investment in terms of "useful theorems proved per dollar". Mathematics research should be justified by stating that people trained in pure mathematics can be useful in industry, other applied fields or to teach mathematics.
  - abnry 8y ago
    Exactly. Modern pure math is extremely abstract. I want careful, level-headed arguments justifying research in swath of pure math fields. Give me numbers, give me details. Not just vague anecdotes. It may be worth the cost, but I don't want that taken for granted.
deleted 8y ago
[deleted]
stvswn 8y ago
While everyone is pointing out the practical, long-term benefits of pure mathematics, I disagree with the premise that there needs to be a practical pursuit associated with mathematics or any other theoretical fields (including the humanities). Sometimes we study things to pursue knowledge itself. If we're talking strictly about public funding, then sure, let's talk about practicality, but otherwise I'm happy that someone is getting funded by a university to devote their lives to something like the Riemann hypothesis. Expanding known human knowledge is justification itself I think. One shouldn't expect it to be easy, or to pay especially well -- it's its own privilege.

I used to be in a math PhD program. Probably would still be if not for some life circumstance. I think it is worth taking a hard look at the value all of this mathematical research actually produces. I understand how number theory has been useful to cryptography. I understand how branches of pure math can have a surprising influence. But when these examples are given by pure mathematicians, it often strikes me as anecdotal and motivated reasoning. Where are the hard numbers? Where is the cool-headed evaluation? They very much want the NSF to continue giving them grants so they can keep funding their mathematical interests. Because it personally and immediately benefits them. 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. This is how I look at it: Funding mathematical research means your society is wealthy. When the vast majority aren't worried about putting food on the table, it is a privilege when you can get paid by them to pursue your mathematical hobby. A hobby that has some relatively low chance of impacting society.

thecleaner 8y ago
I can't argue for pure math. But atleast in applied areas such as pde a lot of the problems are motivated by real physical problems. To solve many of these would require theoretical breakthroughs I guess (cant be sure I dont have a phd). So perhaps funding pure math may not be such a bad idea.
- ginnungagap 8y ago
  I always find the example of PDEs interesting, depending on which mathematician you talk to PDEs could be anywhere from "as pure as it gets" to "extremely applied"
  - thecleaner 8y ago
    I believe this demonstrates the spectrum of pde research. You have people from both ends contributing. Since functional analysis gets used a lot in pdes especially when dealing with weak solutions it would make sense that it appeals to be pure math folks. But once you find ways to construct viable test functions it becomes the basis for writing numerical schemes. So that appeals to the applief guys and of course once you have the numerical solution this can be used for engineering. This demonstrates a good pipeline for "consuming" science (of course assuming this pipeline is indeed correct). However take any component out and the value creation wont be that high. Perhaps finding more such pipelines for pure math woulf help evaluate its valur.
CJefferson 8y ago
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.)
leereeves 8y ago
Cryptography is a trivial example. The entire field of computing was invented by pure mathematicians like Turing and Church working in the 1930s. At the time, no one had any idea what the applications, if any, would be. It's like any science. Some work has immediate applications. Other work is pure exploration of the unknown. And it is pure exploration of the unknown that leads to the truly revolutionary discoveries. You can't set out to discover penicillin when you don't even know it exists.
- maxxxxx 8y ago
  Exactly. A lot of discoveries can be viewed as useless for quite a while before someone figures out a practical application.
- abnry 8y ago
  Something like the Fast Fourier Transform has had a much significant impact on society than a proof of the incomputability of the Busy Beaver function. Saying that Turing and Church "invented computing" is too vague of a justification. I want numbers and details. Folks like Babbage were already thinking procedurally in practical enough terms that once the technological capability (not the theoretical capabilities) caught up, algorithms like the FFT could be discovered and put to use.
  - leereeves 8y ago
    You're only thinking of the immediate practical applications of the mathematics, not the importance of that work for future work. Turing's work was so important that he, not Babbage, is generally recognized as the founder of computer science. One example of the importance of Church's work is Lisp. Church's lambda calculus is the foundation of Lisp, which pioneered nearly all the features of modern programming languages [1]. The designer of Smalltalk, Alan Kay, called it "the greatest single programming language ever designed" [2] and talked about its influence on Smalltalk quite a bit [3]. And your example, the Fourier transform, was itself a mathematical tool long before the first computer was built, and the first published FFT algorithm also dates from the 1930s. [4] 1: http://www.paulgraham.com/diff.html http://www.paulgraham.com/diff.html 2: https://www.quora.com/What-did-Alan-Kay-mean-by-Lisp-is-the-greatest-single-programming-language-ever-designed/answer/Alan-Kay-11?share=1 https://www.quora.com/What-did-Alan-Kay-mean-by-Lisp-is-the-... 3: http://worrydream.com/EarlyHistoryOfSmalltalk/ http://worrydream.com/EarlyHistoryOfSmalltalk/ 4: https://en.wikipedia.org/wiki/Fast_Fourier_transform#History https://en.wikipedia.org/wiki/Fast_Fourier_transform#History
mattkrause 8y ago
Number theory is an excellent example since it was an academic curiosity until it suddenly became very very useful. While cryptography is built on number theory, that earlier research was not done with the explicit goal of developing cryptography. Instead, people took existing math and built new applications around it. Obviously, it would be nice to fund only things that are eventually useful, but this is virtually impossible to predict in advance so....we fund a bunch of things and see what works. (Also, math is shockingly cheap compared to lab sciences so it makes even more sense to spread the bets widely.)
ipioxu15 8y ago
> A hobby that has some relatively low chance of impacting society. As opposed to more important pursuits, such as advertising.
LeanderK 8y ago
but isn't pure math research very cheap (compared to the cost of other scientific research) and therefore could still be cost-effective?
- abnry 8y ago
  It very well could be. I just dislike vague justifications that it actually is worth the cost. Give me numbers and details.
chess19 8y ago
I imagine that a lot of people responding to this do not realize how abstract modern pure mathematics is. The number theory that makes up the basis of cryptography was established in the 1700s. For example, Euler's theorem is the basis of RSA and was proven in 1763. The theorem is a small generalization of Fermat's little theorem which was known (but not proven) in 1640. These theorems are really just simple facts about groups and other cryptosystems, such as elliptic curve cryptosystems, are essentially the same facts except the multiplicative group of integers is replaced with an elliptic curve group. These concepts could be taught to advanced high school students with no formal pure mathematical training. The "hot" areas in modern mathematics require not only an additional 4 years of undergraduate mathematics but usually ~2 years of a PhD program to begin to understand the current papers. This is extremely different from other fields such as theoretical computer science which seems to have applications almost immediately. Even professional mathematicians likely do not research in hopes of applications hundreds of years later. I will not claim that modern mathematics cannot possibly have applications. I will, however, claim that pure mathematics is an extremely poor way to allocate funds if you are simply looking for a return on investment in terms of "useful theorems proved per dollar". Mathematics research should be justified by stating that people trained in pure mathematics can be useful in industry, other applied fields or to teach mathematics.
- abnry 8y ago
  Exactly. Modern pure math is extremely abstract. I want careful, level-headed arguments justifying research in swath of pure math fields. Give me numbers, give me details. Not just vague anecdotes. It may be worth the cost, but I don't want that taken for granted.

deleted 8y ago

[deleted]

stvswn 8y ago

While everyone is pointing out the practical, long-term benefits of pure mathematics, I disagree with the premise that there needs to be a practical pursuit associated with mathematics or any other theoretical fields (including the humanities). Sometimes we study things to pursue knowledge itself. If we're talking strictly about public funding, then sure, let's talk about practicality, but otherwise I'm happy that someone is getting funded by a university to devote their lives to something like the Riemann hypothesis. Expanding known human knowledge is justification itself I think. One shouldn't expect it to be easy, or to pay especially well -- it's its own privilege.