Regarding the first two types, I think that it’s practically never the case and one can always make progress—even if that progress is in work done on analogies or heuristically relevant techniques. The Riemann hypothesis is actually a great example of that; there are many paths currently pursued to help us understand it better, even if there aren’t any especially promising reductions (not sure if that’s the case). But I guess that your point here is that this are distinct markers for how easy is it to make progress.
What is the alternative strategy you are suggesting in those exceptions? Is it to work on problems that are weakly related and the connection is not clear but are more tractable?
If so, I think that two alternative strategies are to just try harder to find something more related or to move to a different project altogether. Of course, this all lies on a continuum so it’s a matter of degree.
Regarding the first two types, I think that it’s practically never the case and one can always make progress—even if that progress is in work done on analogies or heuristically relevant techniques. The Riemann hypothesis is actually a great example of that; there are many paths currently pursued to help us understand it better, even if there aren’t any especially promising reductions (not sure if that’s the case). But I guess that your point here is that this are distinct markers for how easy is it to make progress.
What is the alternative strategy you are suggesting in those exceptions? Is it to work on problems that are weakly related and the connection is not clear but are more tractable?
If so, I think that two alternative strategies are to just try harder to find something more related or to move to a different project altogether. Of course, this all lies on a continuum so it’s a matter of degree.