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I quite disagree with this. Visionaries come along to unseat centuries of limitations, raising the ceiling forever. Those visionaries are rare, but there will b

by jb1991 4y ago

I quite disagree with this. Visionaries come along to unseat centuries of limitations, raising the ceiling forever. Those visionaries are rare, but there will be more.

vouaobrasil 4y ago
Maybe, but if you look at the list of famous unsolved problems in math, it has been dwindling for a long time. The list used to contain problems that undergrads could understand. Biology is still like this -- you can take quite a few interesting unsolved problems in biology and explain it to most people. Math has a few left like the Riemann hypothesis, but the vast majority of new work that has been coming out is more like refinements (I say this as a PhD in math who as actually read them). We look at history and expect it to repeat itself, but if you closely examine math history you can actually see a decline. Take the time to truly examine "visionary" things that have happened in math and those events are actually getting fewer and fewer. I seriously doubt there is much left to do that is of interest to more than 5-10 people in each subfield.
- galleywest200 4y ago
  Are you suggesting there are no more unknown problems in mathematics, that "the list" will not be added to?
  - vouaobrasil 4y ago
    Well, due to the nature of math there are technically hundreds of questions you can ask (specific behaviour of the infinite number of diophantine equations, improving constants in estimation, whether there exists a prime number that satisfies XYZ, etc), but those problems are becoming so hyperspecialized that very few people if any will care about them. There may be one more paradigm shift that is related to stuff that Sholze is doing and there is certainly some work to do in proof theory ala Voevodky's univalent foundations so we could see a couple more things but the vast majority of basic foundational work in math is done. So while we have a few more fundamental problems that most people have heard of like the Riemann hypothesis and the twin primes conjecture, don't expect to have any more of those. (Certainly the more recent ones are already so specialized that few people outside the immediate area would ever understand or even care about them). I'm also not sure why there's such a negative view towards the end of progress on this site---a lot of people seem to take it as an axiom that currently "useless" things will eventually become useful but I simply do not believe that is the case (and this is coming from someone who has published quite a few papers in pure math).

by jb1991 4y ago

I quite disagree with this. Visionaries come along to unseat centuries of limitations, raising the ceiling forever. Those visionaries are rare, but there will be more.

vouaobrasil 4y ago

Maybe, but if you look at the list of famous unsolved problems in math, it has been dwindling for a long time. The list used to contain problems that undergrads could understand. Biology is still like this -- you can take quite a few interesting unsolved problems in biology and explain it to most people. Math has a few left like the Riemann hypothesis, but the vast majority of new work that has been coming out is more like refinements (I say this as a PhD in math who as actually read them). We look at history and expect it to repeat itself, but if you closely examine math history you can actually see a decline. Take the time to truly examine "visionary" things that have happened in math and those events are actually getting fewer and fewer. I seriously doubt there is much left to do that is of interest to more than 5-10 people in each subfield.

galleywest200 4y ago
Are you suggesting there are no more unknown problems in mathematics, that "the list" will not be added to?
- vouaobrasil 4y ago
  Well, due to the nature of math there are technically hundreds of questions you can ask (specific behaviour of the infinite number of diophantine equations, improving constants in estimation, whether there exists a prime number that satisfies XYZ, etc), but those problems are becoming so hyperspecialized that very few people if any will care about them. There may be one more paradigm shift that is related to stuff that Sholze is doing and there is certainly some work to do in proof theory ala Voevodky's univalent foundations so we could see a couple more things but the vast majority of basic foundational work in math is done. So while we have a few more fundamental problems that most people have heard of like the Riemann hypothesis and the twin primes conjecture, don't expect to have any more of those. (Certainly the more recent ones are already so specialized that few people outside the immediate area would ever understand or even care about them). I'm also not sure why there's such a negative view towards the end of progress on this site---a lot of people seem to take it as an axiom that currently "useless" things will eventually become useful but I simply do not believe that is the case (and this is coming from someone who has published quite a few papers in pure math).