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Here's a perspective I have some sympathy with: "It is a huge waste to have gifted problem solvers off in la-la land writing papers that start with the sentenc

Here's a perspective I have some sympathy with: "It is a huge waste to have gifted problem solvers off in la-la land writing papers that start with the sentence "Let G be a Grothendieck category of the first type." The world is a place churning with real problems that affect real people, and one of the most severe bottlenecks is that of smart and competent people." There are a bunch of counter-arguments one could give to this complaint, ranging from vague promises of long-term usefulness, through the idea that pure math is a intimate bed-follow of critical thinking and so its practice should be exalted, to the notion that theorems of higher mathematics are somehow art-forms, albiet esoteric ones, worthy of state support. Another point might be that pure math is often practiced by people who are deliberately avoiding the real world, who want a fantasy land to be clever in, and they'd be off doing something else rarefied if there weren't math departments. I don't believe that for a moment.

InclinedPlane 15y ago
I think there's a balance. While a lot of the work being done in advanced mathematics may never impact anyone's lives in any substantial way that's not always true. It took a lot of effort to reinvent mathematics and geometry from the ground up in the 18th and 19th centuries, and in many ways it could seem as if that work was purely abstract, but it has surely had a very profound impact on the entire modern world. The same goes for, say, Boolean algebra, Turing machines (which at one time were incredibly abstract and theoretical), and a lot of number theory. Yet today our computing systems and especially cryptography is based on what used to be very abstract mathematics. Do you think the wonks who worked on elliptic curve theory imagined that there would eventually be custom purpose micro-chips designed to perform elliptic curve math? Sometimes it's impossible to know what lines of research will prove practical in the next decades and centuries, so it's smart to balance between some focused practical application research as well as blue-sky research.
- taliesinb 15y ago
  That's true. It's said that Turing himself didn't immediately realize the importance of his machines -- he went straight on to considering oracles and super computation, as if Turing machines themselves weren't worthy of further study. Perhaps it is nit-picking, but I'm going to play devil's advocate and question exactly how important the theoretical notion of a Turing machine actually was for the practical development of computer algorithms. Probably a much more important development was the invention of the Von Neumann architecture and its use in the IAS machine and later the JOHHNIAC. And also lambda calculus for its spawning of functional programming languages. Von Neumann is the best example I can think of a blue-sky thinker whose work had profound practical impacts all over the place. Maybe a recent example of an ingenious pure mathematician who made a practical impact is Terence Tao's invention of compressed sensing. But on the whole, no-one would argue that the next cryptographic breakthrough is going to come from category theory. Some parts of pure mathematics will always be useless, and everyone knows it.
snikolov 15y ago
It would be great to have smart people working on real problems, but problem solving skills without a desire to work on those problems just won't work. I think a lot of mathematicians just like doing mathematics and would be really bored solving "real problems." You could also say it's a huge waste to have gifted problem solvers working on photo-sharing apps when they could be helping the developing world with clean water and energy. Could these problem solvers provide more benefit to the world? Sure. Will they, if they don't enjoy it? Not for long. The real question is how to inspire people with those skills to tackle important problems.
- taliesinb 15y ago
  Yes, that is exactly the problem. One interesting anecdotes I heard from Stephen Wolfram the other day is the inverse correlation he's noticed between an employee having a post-graduate degree and their general curiosity and knowledge about the wider intellectual world. It's far from a perfect correlation, he said, but it is noticeable. It's probably fair to say that the problem lies largely with universities and the way they structure their curricula and more generally the form of post-graduate studies. I have no idea how to improve matters. But the obsession with early specialization and standardized testing is probably a good place to start.

InclinedPlane 15y ago

I think there's a balance. While a lot of the work being done in advanced mathematics may never impact anyone's lives in any substantial way that's not always true. It took a lot of effort to reinvent mathematics and geometry from the ground up in the 18th and 19th centuries, and in many ways it could seem as if that work was purely abstract, but it has surely had a very profound impact on the entire modern world. The same goes for, say, Boolean algebra, Turing machines (which at one time were incredibly abstract and theoretical), and a lot of number theory. Yet today our computing systems and especially cryptography is based on what used to be very abstract mathematics. Do you think the wonks who worked on elliptic curve theory imagined that there would eventually be custom purpose micro-chips designed to perform elliptic curve math? Sometimes it's impossible to know what lines of research will prove practical in the next decades and centuries, so it's smart to balance between some focused practical application research as well as blue-sky research.

taliesinb 15y ago
That's true. It's said that Turing himself didn't immediately realize the importance of his machines -- he went straight on to considering oracles and super computation, as if Turing machines themselves weren't worthy of further study. Perhaps it is nit-picking, but I'm going to play devil's advocate and question exactly how important the theoretical notion of a Turing machine actually was for the practical development of computer algorithms. Probably a much more important development was the invention of the Von Neumann architecture and its use in the IAS machine and later the JOHHNIAC. And also lambda calculus for its spawning of functional programming languages. Von Neumann is the best example I can think of a blue-sky thinker whose work had profound practical impacts all over the place. Maybe a recent example of an ingenious pure mathematician who made a practical impact is Terence Tao's invention of compressed sensing. But on the whole, no-one would argue that the next cryptographic breakthrough is going to come from category theory. Some parts of pure mathematics will always be useless, and everyone knows it.

snikolov 15y ago

It would be great to have smart people working on real problems, but problem solving skills without a desire to work on those problems just won't work. I think a lot of mathematicians just like doing mathematics and would be really bored solving "real problems." You could also say it's a huge waste to have gifted problem solvers working on photo-sharing apps when they could be helping the developing world with clean water and energy. Could these problem solvers provide more benefit to the world? Sure. Will they, if they don't enjoy it? Not for long. The real question is how to inspire people with those skills to tackle important problems.

taliesinb 15y ago
Yes, that is exactly the problem. One interesting anecdotes I heard from Stephen Wolfram the other day is the inverse correlation he's noticed between an employee having a post-graduate degree and their general curiosity and knowledge about the wider intellectual world. It's far from a perfect correlation, he said, but it is noticeable. It's probably fair to say that the problem lies largely with universities and the way they structure their curricula and more generally the form of post-graduate studies. I have no idea how to improve matters. But the obsession with early specialization and standardized testing is probably a good place to start.