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’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”.)