I’m a curious person who finds pure math interesting, but I still care a lot about application. You can definitely estimate the usefulness of math, in a way like how people estimate the importance of pure math.
However, my understanding is that basically none of the famous conjectures are important because the statement itself is useful. Rather, we expect that if we can answer the question we should also have a better developed theory that will then have various applications (or on the pure end, let us understand various things).
There’s a whole bunch of unknown containments in complexity theory, and (reportedly) the theory currently just doesn’t seem to have good approaches to tackle the problems. This suggests that if we can resolve whether P=NP, we should also have some much better tools that can resolve a whole lot more and allow us to build up more theory.
I put like 80% on P ≠ NP since the experts think it’s likely true. Extra weight is given to people like Scott Aaronson, who also believe similarly I think. This is how I deal with fields I don’t know enough to have my own opinions on.
Empirical mathematics is fine, but once you get to famous outstanding conjectures the track record isn’t so good. For most random statements I cook up, empirics will vastly narrow down the possibilities and agree with proofs. But there are famous examples of this failing for famous conjectures in number theory, which a priori seems more amenable to empirics than computational complexity or, say, algebraic geometry. This suggests that once you’ve weeded out the easy stuff, empirics are less useful for what’s left.
But what’s the empirical (as in, similar to checking a number theory conjecture on all the integers up to however much you can compute) evidence for P ≠ NP, besides humans not finding the necessary algorithms? It’s not like someone has an algorithm that seems to have asymptotics of O(f(n)) and where proving it does would resolve the conjecture (Levin’s universal search doesn’t count on account of being too ridiculously slow to get empirical evidence about).
I’m a curious person who finds pure math interesting, but I still care a lot about application. You can definitely estimate the usefulness of math, in a way like how people estimate the importance of pure math.
However, my understanding is that basically none of the famous conjectures are important because the statement itself is useful. Rather, we expect that if we can answer the question we should also have a better developed theory that will then have various applications (or on the pure end, let us understand various things).
There’s a whole bunch of unknown containments in complexity theory, and (reportedly) the theory currently just doesn’t seem to have good approaches to tackle the problems. This suggests that if we can resolve whether P=NP, we should also have some much better tools that can resolve a whole lot more and allow us to build up more theory.
I put like 80% on P ≠ NP since the experts think it’s likely true. Extra weight is given to people like Scott Aaronson, who also believe similarly I think. This is how I deal with fields I don’t know enough to have my own opinions on.
Empirical mathematics is fine, but once you get to famous outstanding conjectures the track record isn’t so good. For most random statements I cook up, empirics will vastly narrow down the possibilities and agree with proofs. But there are famous examples of this failing for famous conjectures in number theory, which a priori seems more amenable to empirics than computational complexity or, say, algebraic geometry. This suggests that once you’ve weeded out the easy stuff, empirics are less useful for what’s left.
But what’s the empirical (as in, similar to checking a number theory conjecture on all the integers up to however much you can compute) evidence for P ≠ NP, besides humans not finding the necessary algorithms? It’s not like someone has an algorithm that seems to have asymptotics of O(f(n)) and where proving it does would resolve the conjecture (Levin’s universal search doesn’t count on account of being too ridiculously slow to get empirical evidence about).