This post has been a part of my thinking ever since I’ve read it, and is an important part of how I think about math!
Another lens: (good) definitions are True Names of important concepts.
Lastly, you say:
You may be thinking about some famous conjectures whose proofs would be the ultimate high-status accomplishment, such as the Riemann hypothesis or P = NP
My understanding is that for many of these, experts are interested in part because they believe that answering the question would require making a bunch of conceptual advances—that is, that the conceptual advances they expect are necessary are the real point of trying to prove e.g. the Riemann hypothesis. Possibly this is less true for P=NP (in that one might hope for general ways to do NP problems in polynomial time that can be made practical enough to matter for applications). Possibly this is just me not understanding the other Millenium problems enough to intrinsically care about them (though for the Riemann Hypothesis, I am ‘told’ that you get direct info about the distribution of the primes).
This post has been a part of my thinking ever since I’ve read it, and is an important part of how I think about math!
Another lens: (good) definitions are True Names of important concepts.
Lastly, you say:
My understanding is that for many of these, experts are interested in part because they believe that answering the question would require making a bunch of conceptual advances—that is, that the conceptual advances they expect are necessary are the real point of trying to prove e.g. the Riemann hypothesis. Possibly this is less true for P=NP (in that one might hope for general ways to do NP problems in polynomial time that can be made practical enough to matter for applications). Possibly this is just me not understanding the other Millenium problems enough to intrinsically care about them (though for the Riemann Hypothesis, I am ‘told’ that you get direct info about the distribution of the primes).