"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.
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.
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.
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.
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.
Great to hear this view applied to maths as well, and curiously enough from someone like Terrance.
I wonder, if you were to ask him if he was born with "an abundance of raw talent" in maths, do you think he would say yes? And if so, I wonder if he has found it partially harmful to his development (as he mentions is a possibility for those with raw talent, and what has been found to be a potentiality by experts). And if so, I wonder if he feels he has had to develop various means (or put forth an unusual amount of effort) in circumventing it?
Alternatively, does he explain his situation as a case of having put forth an amazing amount of effort over the years?
Anyway, this is very much in alignment with recent studies from cognitive scientists and social/developmental psychologists in all sorts of fields, which I'm sure influenced Terrance's view and article here. Cool to hear.
The thing that has always puzzled me is if one's opinion on "talent vs hard work" is actually correlated to one's amount of talent. You can often hear very gifted people say things like "it's hard work that actually matters". At the same time, underperformers' claims that it's the talent that does the thing maybe treated as rationalizations ("I'm lazy, so, yeah, let's blame that on lack of talent, not work"). I don't see any easy way of studying this, though looking for such correlations might prove interesting.
The thing that has always puzzled me is if one's opinion on "talent vs hard work" is actually correlated to one's amount of talent.
There has been some research on this. Maybe I can open a thread sometime with some link to a good article on this issue. Meanwhile, I am reminded of pg's statement in his essay "What You'll Wish You'd Known"
"I'm not saying there's no such thing as genius. But if you're trying to choose between two theories and one gives you an excuse for being lazy, the other one is probably right."
I'm going to be a junior next year and I'm a math major. I've often wondered if math is even for me as an academic career as I don't see myself having any sort of significant insight to make it worthwhile. Perhaps I'm wrong. This is nice.
You definitely don't need to be a genius to do math. To everyone out there with small children, I would recommend this site - http://jumpmath1.org - they product a system that helps children learn the basics s.t. advanced features become self-explanatory.
What are you interested in? The theory of information? Ideas about space that go behind the flat plane of a piece of paper? How to communicate and co-operate securely when people might be eavesdropping and impersonating your allies? How to model various kinds of learning? How one studies the interacting strategies of military, economic, or biological adversaries? How complexity arises in simple systems? How sounds and images can be compressed?
These are just examples, but each one of these topics will select a certain 'diet' of pure mathematics that will keep you well fed with cool ideas as well as 'in's to different fields.
Although it does work for some people to just pick up a book on linear algebra and read it from cover to cover, I've never found that a very interesting or self-motivating way to learn. I think its better to hang one's knowledge on a tree of interlinked explorations.
I second that. My main side-project-esque interests involve AI and Computer Vision. Knowing that cleared up a lot of confusion as to what was 'best to learn'.
I graduated with Math 11 in highschool and never pursued it again to any extent until I was out of programming the first time and about 3 years into my culinary career.
I decided to pick up calculus and linear algebra citing a lack of intellectual stimulation. I used the dummies books and augmented them with schaum's outlines (not easy studying).
I got back into programming again, and although I didn't make it all the way through the first time, i've picked it up and have found the Demystifying series pretty good (better than the dummies books by a long shot in fact).
These are also bolstered by other books I find that fell off the back of the internet truck. There's lots of great website for study notes and tutorials. One I like is Paul's Online Math Notes (http://tutorial.math.lamar.edu/).
Probably not a bad place to start is just iTunes U. There is some pretty cool stuff curated there -- I'm just delving into Andrew Ng's machine learning course.
"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.