The mathematics of physics is just an infinitesimal part of all of mathematics.
Physics could at some point be completely solved, which is to say that at some point, there would be no further knowledge that would ever allow us to do anything new, to make any better a prediction, to do anything more efficiently, etc.
There is no such limit to mathematics though, because mathematics, unlike physics, is not constrained by reality. It only needs to be self consistent (under perhaps limitless different conceptions of consistency) given a particular starting point, and there is no limit to the number of starting points or perspectives upon which can be built new systems. And there are concepts analogous to quantity, transformation, shape, etc., that that have not yet been invented but will be fertile ground for new branches of mathematics someday.
I think it’s possible that all useful mathematics could someday be discovered (if you consider all art useless), but that would still be just an infinitesimal part of all of mathematics. To say that mathematics could be exhausted is to say that all stories may one day have been told, and fiction may be exhausted. It just can’t happen.
The mathematics of physics is just an infinitesimal part of all of mathematics.
Physics could at some point be completely solved, which is to say that at some point, there would be no further knowledge that would ever allow us to do anything new, to make any better a prediction, to do anything more efficiently, etc.
There is no such limit to mathematics though, because mathematics, unlike physics, is not constrained by reality. It only needs to be self consistent (under perhaps limitless different conceptions of consistency) given a particular starting point, and there is no limit to the number of starting points or perspectives upon which can be built new systems. And there are concepts analogous to quantity, transformation, shape, etc., that that have not yet been invented but will be fertile ground for new branches of mathematics someday.
I think it’s possible that all useful mathematics could someday be discovered (if you consider all art useless), but that would still be just an infinitesimal part of all of mathematics. To say that mathematics could be exhausted is to say that all stories may one day have been told, and fiction may be exhausted. It just can’t happen.