Comment by degamad
Specifically, reverse math (a subset of metamathematics which looks at swapping axioms and theorems) allows us to show that some hard problems are equivalent to each other.
EDIT: I think this line is the most telling:
> But he cautioned that the reverse mathematics approach may be most useful for revealing new connections between theorems that researchers have already proved. "It doesn’t tell us much, as far as we can say, about the complexity of statements which we do not know how to prove."
So, at this point, it helps us understand more about problems we already understand a little about, but nothing yet about new problems.
> So, at this point, it helps us understand more about problems we already understand a little about, but nothing yet about new problems.
I don't think this caution makes any sense!
The more we learn about theorem/axiom equivalences (or more generally, the lattice of such connections) between existing proofs, the more insights we will gain into how unproved conjectures may relate to existing proofs or each other.
Only in the strictest possible sense does saying showing X tells us nothing about showing Y. Meaning a proof or identification of X is not a proof or identification of related thing Y. But that is an obviously pedantic statement.
Not to critique the person being quoted. I feel like an offhand remark may have got unduly elevated by being quoted in a "two-sides of a story" writer's dramatization reflex.