Especially if they are using public money, because public monies need to be accountable.

Would the same apply to people who further explore questions like P = NP and what happens if it is true or false or what happens if the Riemann hypothesis is true? Sometimes digging more into something that doesn't seem testable results in a finding that is testable or into knowledge that helps advance a related field.

I think the deeper question is, how do we make sure that public money is spent in a justifiable fashion. And when it comes to science that is a hard question to answer, especially if one considers that science works a bit differently than the way the average voter thinks about things when it comes to notion such as proven or disproven.

In general all of theoretical physics is mathematics that takes inspiration from experiments, and takes shortcuts when proving theorems:).

Our current society is way to focused on short-term gains, mostly because of profit imperatives. Yet it is becoming increasingly obvious that being short-sighted will eventually doom us all. Coronaviruses were an obscure family of viruses no one but a handful of niche researchers were interested about until it became the most important thing in the world. As for climate change...