I imagine that there are three kinds of “unsolved” problems in mathematics: problems that are unsolved because people have tried and failed to solve them, problems that people haven’t yet tried to solve but aren’t particularly difficult to solve once attempted, and problems that both haven’t been tried but would likely result in failure anyway.
How much math does one have to study before one has a reasonable chance of encountering a problem of the second type—one that an “average” tenured mathematics professor at an “average” university of no special prestige is likely to be able to solve once the problem is brought to their attention? Do they even exist?
I’d say the average Mathematics PhD student that has a publication has already solved such a problem!
There’s usually a fair amount of low-hanging fruit in niche disciplines—for certain subdisciplines of mathematics, you really can count the number of people working on that discipline on one hand.
I would guess that there are many problems that nobody cares about. Producing new problems in math isn’t really that hard. Just add a new axiom to an existing theory and you have a bunch of new problems.
The problem is that nobody necessarily cares and so they won’t cite you.
It’s probably not hard to find a question of the second type at the bright undergraduate level or earlier if you drill down into a subdiscipline that’s both obscure and requires relatively few prerequisites (such subdisciplines do exist, e.g. probably some branches of combinatorics), but I don’t really see the point of doing this.
I’m not actually pursuing it right now. I’m trying to estimate its value (to self and others) for someone who’s below the “super-genius” level of math talent: how hard is it, really, to make useful progress in mathematics?
In my opinion, based on non-scientific examination over decades first as a graduate student and then as a professor, the primary value of the middle level of talent (where most of us are if we are lucky) is usually referred to as “teaching” and might more generally be referred to as “socializing.” An at least occasionally interesting teacher has the possibility of exciting interest in somebody of top talent, and has the near-certainty of exciting interest in many others who can become teachers.
Upon my graduation with a PhD from Caltech, I took a faculty position at the University of Rochester. I got funding, I plodded along. I had a crisis of confidence: the lesson I felt I had learned at Caltech was that “we” tolerate the bottom 99% because the results from the top 1% that our tolerance makes politically possible are more than worth it. As a prof. at UR, it was difficult for me to believe that I was anywhere near the top 1% of working research PhDs in the country. Years later, many of my graduate students hold VP and other leadership positions in industry: that is they are all apparently more productive than me. Considering the efforts they put in to getting me as an advisor and the efforts they put into getting my attention to talk through their projects once they were my students, it is reasonable to think that some part of their output is attributable to me.
Extremely hard, I think, but I may have a high bar for what constitutes useful progress (in general; FAI research might be an exception but you don’t need to pursue graduate study to get in on that).
My impression is the following. Most academic research seems more or less useless, and that’s no less true of mathematics than other fields. There are probably too many research mathematicians at the moment. The incentives are aligned pretty strongly towards research and away from other arguably more useful-at-the-margin activities like exposition, synthesis of previous research, meta-research, and so forth. Research mathematics also seems highly competitive relative to other areas where math-related talent has applications (e.g. programming, maybe finance).
I imagine that there are three kinds of “unsolved” problems in mathematics: problems that are unsolved because people have tried and failed to solve them, problems that people haven’t yet tried to solve but aren’t particularly difficult to solve once attempted, and problems that both haven’t been tried but would likely result in failure anyway.
How much math does one have to study before one has a reasonable chance of encountering a problem of the second type—one that an “average” tenured mathematics professor at an “average” university of no special prestige is likely to be able to solve once the problem is brought to their attention? Do they even exist?
I’d say the average Mathematics PhD student that has a publication has already solved such a problem!
There’s usually a fair amount of low-hanging fruit in niche disciplines—for certain subdisciplines of mathematics, you really can count the number of people working on that discipline on one hand.
I would guess that there are many problems that nobody cares about. Producing new problems in math isn’t really that hard. Just add a new axiom to an existing theory and you have a bunch of new problems.
The problem is that nobody necessarily cares and so they won’t cite you.
It’s probably not hard to find a question of the second type at the bright undergraduate level or earlier if you drill down into a subdiscipline that’s both obscure and requires relatively few prerequisites (such subdisciplines do exist, e.g. probably some branches of combinatorics), but I don’t really see the point of doing this.
1) PHD thesis topic.
2) Grinding out least publishable units in anticipation of a tenure application.
I’m trying to estimate the level of “talent” one needs to make graduate study in mathematics not entirely pointless.
From my understanding of grad students per year:new faculty hires per year ratios, you need to be in about the 95th percentile of math grad students.
Okay, but this approach sounds boring and painful. Why are you pursuing graduate study in mathematics?
I’m not actually pursuing it right now. I’m trying to estimate its value (to self and others) for someone who’s below the “super-genius” level of math talent: how hard is it, really, to make useful progress in mathematics?
In my opinion, based on non-scientific examination over decades first as a graduate student and then as a professor, the primary value of the middle level of talent (where most of us are if we are lucky) is usually referred to as “teaching” and might more generally be referred to as “socializing.” An at least occasionally interesting teacher has the possibility of exciting interest in somebody of top talent, and has the near-certainty of exciting interest in many others who can become teachers.
Upon my graduation with a PhD from Caltech, I took a faculty position at the University of Rochester. I got funding, I plodded along. I had a crisis of confidence: the lesson I felt I had learned at Caltech was that “we” tolerate the bottom 99% because the results from the top 1% that our tolerance makes politically possible are more than worth it. As a prof. at UR, it was difficult for me to believe that I was anywhere near the top 1% of working research PhDs in the country. Years later, many of my graduate students hold VP and other leadership positions in industry: that is they are all apparently more productive than me. Considering the efforts they put in to getting me as an advisor and the efforts they put into getting my attention to talk through their projects once they were my students, it is reasonable to think that some part of their output is attributable to me.
Extremely hard, I think, but I may have a high bar for what constitutes useful progress (in general; FAI research might be an exception but you don’t need to pursue graduate study to get in on that).
My impression is the following. Most academic research seems more or less useless, and that’s no less true of mathematics than other fields. There are probably too many research mathematicians at the moment. The incentives are aligned pretty strongly towards research and away from other arguably more useful-at-the-margin activities like exposition, synthesis of previous research, meta-research, and so forth. Research mathematics also seems highly competitive relative to other areas where math-related talent has applications (e.g. programming, maybe finance).
Maybe I should say “interesting progress” instead of “useful progress”, then?
(For example, literature is often interesting but rarely “useful”.)