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.

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%.)

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.

> 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_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.)

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.)

> 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.

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.

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.

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.

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.)

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.

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.

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.

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"

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.

> A hobby that has some relatively low chance of impacting society.

As opposed to more important pursuits, such as advertising.

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.

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.

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

2: https://www.quora.com/What-did-Alan-Kay-mean-by-Lisp-is-the-...

3: http://worrydream.com/EarlyHistoryOfSmalltalk/

4: https://en.wikipedia.org/wiki/Fast_Fourier_transform#History

Exactly. A lot of discoveries can be viewed as useless for quite a while before someone figures out a practical application.

but isn't pure math research very cheap (compared to the cost of other scientific research) and therefore could still be cost-effective?

It very well could be. I just dislike vague justifications that it actually is worth the cost. Give me numbers and details.