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> Is all the good math institutionalized now (e.g., you can't do good math without all the benefit of learning background within the institution conventions)?

> Is all the good math institutionalized now (e.g., you can't do good math without all the benefit of learning background within the institution conventions)? The short answer is yes. We haven't had a Ramanujan in decades, and they were more or less always extraordinary. Insofar as you can generalize all of its subfields together, it's reasonable to say mathematics is an extremely mature discipline now. What I mean by that is there isn't much low hanging fruit around. Even when Ramanujan was dazzling Hardy with his results, many of them were already known. That was over a century ago. There just isn't much fertile ground left where a single brilliant mind can make significant headway based on raw talent or intuition, without first working through significant education in prerequisites. It's more and more commonplace for authors to collaborate on their work, because compelling problems at the forefront of mathematics research are increasingly requiring cross-disciplinary knowledge to resolve. To focus on Ramanujan a bit in particular - one of the reasons Ramanujan was so talented was because he not only came up with novel results in complete isolation; rather, he also came up with original perspectives and theorems. That means he was capable of posing interesting questions, which is usually much more interesting and inspiring for further research. That's very inspiring, but the reason I point it out is because Ramanujan wasn't solving open problems in the West left and right so much as he was exercising ingenuity to pose - and resolve - open problems in fertile ground he could reach. But a century later, the amount of existing mathematics has increased so much in both breadth and depth that a burst of creativity is almost certainly not going to get an undergrad to find something that e.g. Rudin hasn't written down somewhere already. But more importantly, it's vanishingly unlikely they will resolve a famous open problem which is the subject of intense research interest. All the stories of untrained amateurs doing that are from the mid 20th century or earlier. To turn your question on its head a bit: consider how astonishingly brilliant an untrained individual would have to be find a nontrivial result that the rest of the mathematical community hasn't found. Either they're inventing their own definitions and discovering entirely new mathematics in isolation (wow!), or they've somehow found a way to use the tools accessible to them (university analysis and algebra, at best) in a way every other trained mathematician has not. It's rare even for senior math undergrads to publish nontrivial research on their own. It's significantly rarer for that research to improve progress towards an open problem. The only example I can think of off the top of my head where undergrads actually resolved a well known open problem is AKS, and even then it wasn't a solo author.

neilv 7y ago
Thank you, that makes sense. Rather than this reality being discouraging to all the aspiring mathematicians who've been learning and building on their own, I think it should be encouragement -- to find ways to get closer to the community, to learn more of what they already know, and perhaps eventually find mentoring or collaboration.
piyushahuja 7y ago
But when we are talking about unsolved problems, we are in the land of vanishing small probabilities in any case. Could it not be that the very fact that someone has been trained in conventional mathematics a hindrance to them resolving an unsolved question, and someone thinking in isolation has an advantage? Here's Grothendieck, for example: “In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.” I can certainly see this happening: someone away from the pressures of publishing, the drudgery of admin tasks, the frustration of applying for grants or department politics, or the fear of looking "bad" to their peers etc. devoting their time to mathematics out of pure joy, play and drive and coming up with novel definitions and assumptions that those in establishment mathematics, out of pure sociological and psychological reasons, have not even dared venture into.
- throwawaymath 7y ago
  I can see a case for a trained mathematician contributing a legitimately new perspective to the community by withdrawing and spending time looking at things in an unorthodox manner. That's closer to that Grothendieck was describing than what you're posing. In the early to mid 20th century, it was feasible for someone to pick up a book on number theory and apply relative genius to an open problem to quickly solve it. The bottom line is that prerequisites for doing so were often just basic calculus and high school algebra. An understanding of sequences and series went a long way. This isn't the case in the 21st century. We're firmly out of that territory. You can't offer a profound new perspective on a thing which you can't understand, and the barrier to understanding research mathematics continually rises. The forefront of modern mathematics is so far removed from even graduate level mathematics course material that it's not going to just be intuited through untrained brilliance. You have to actively learn it, which is (unfortunately) vanishingly unlikely outside of academia. If the tools available to you are basic real analysis and linear algebra, P/NP is beyond your reach - full stop. At this point we actually have proofs that a valid proof of P/NP (and similar problems) cannot be achieved through large swathes of elementary techniques. We don't live in a world like Good Will Hunting where an amateur can succeed by being a genius. That's useful in the long term but not enough on its own.

neilv 7y ago

Thank you, that makes sense. Rather than this reality being discouraging to all the aspiring mathematicians who've been learning and building on their own, I think it should be encouragement -- to find ways to get closer to the community, to learn more of what they already know, and perhaps eventually find mentoring or collaboration.

piyushahuja 7y ago

But when we are talking about unsolved problems, we are in the land of vanishing small probabilities in any case. Could it not be that the very fact that someone has been trained in conventional mathematics a hindrance to them resolving an unsolved question, and someone thinking in isolation has an advantage? Here's Grothendieck, for example: “In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.” I can certainly see this happening: someone away from the pressures of publishing, the drudgery of admin tasks, the frustration of applying for grants or department politics, or the fear of looking "bad" to their peers etc. devoting their time to mathematics out of pure joy, play and drive and coming up with novel definitions and assumptions that those in establishment mathematics, out of pure sociological and psychological reasons, have not even dared venture into.

throwawaymath 7y ago
I can see a case for a trained mathematician contributing a legitimately new perspective to the community by withdrawing and spending time looking at things in an unorthodox manner. That's closer to that Grothendieck was describing than what you're posing. In the early to mid 20th century, it was feasible for someone to pick up a book on number theory and apply relative genius to an open problem to quickly solve it. The bottom line is that prerequisites for doing so were often just basic calculus and high school algebra. An understanding of sequences and series went a long way. This isn't the case in the 21st century. We're firmly out of that territory. You can't offer a profound new perspective on a thing which you can't understand, and the barrier to understanding research mathematics continually rises. The forefront of modern mathematics is so far removed from even graduate level mathematics course material that it's not going to just be intuited through untrained brilliance. You have to actively learn it, which is (unfortunately) vanishingly unlikely outside of academia. If the tools available to you are basic real analysis and linear algebra, P/NP is beyond your reach - full stop. At this point we actually have proofs that a valid proof of P/NP (and similar problems) cannot be achieved through large swathes of elementary techniques. We don't live in a world like Good Will Hunting where an amateur can succeed by being a genius. That's useful in the long term but not enough on its own.