In mathematics, they do this thing where they figure out what are called "proof-theoretic ordinals" (see this section of the SEP article on proof theory for background details) of theories, which express/measure some kind of "strength" that theories have. The "taller" the proof number, the stronger the theory. The classical basic example is how, since Peano arithmetic (PA) reaches towards the totalization of exponentiation, then the proof-theoretic ordinal of PA is ωωω... (where the tower goes on for ω-many steps upward).
But finding the proof numbers of much stronger theories proves beyond-exponentially more difficult. For reasons of informativeness, it's not good enough to just make up an ω-long list of increasingly stronger theories and just generically declare that entry #2 on some list, say, has a proof number = β, say, but we want to know more about the "inner character" of these β-terms.
It seems like, the greater the proof number you end up with for some theory, the harder it would be to justify that theory, though, then. For example, calculus (when known as analysis/second-order arithmetic) has an unknown, and presumably very, very, very tall, β to its name. So there is this thing called "Friedman's grand conjecture," from a philosopher-of-mathematics Harvey Friedman, which says that a remarkably short-but-countable ordinal goes with a relatively "weak" theory that can ground much of so-called "normal mathematics." Now, as an inclusivist about pluralities in logic/mathematics, I take greater-and-greater theories to be justifiable, but in a supererogatory manner, i.e. it is optional whether we wish to work in them, and only some lesser threshold marks the boundary of what mathematics we ought to agree with. And I would expect second-order arithmetic to be within view of that threshold, perhaps.
However, given how difficult it is to extract a proof number from such a specified theory, why not bypass the issue altogether, and instead of trying to go from some theory T to some proof number β, one would go from β to T? That is, if we could identify some β, in detail, on its own terms first, and then reflect on its possible abstract justification per se, then if this β seemed (intuitively?) justifiable as such, we would then try to construct some T (by coming up with axioms, etc.) that fit to this β, and we wouldn't have to trouble ourselves with the extra effort of tailoring β to some prior T.
Would this be more intellectually economical a theoretical practice, in this context? For example, while considering structural analogies between PA's known β and the known β of a small fragment of analysis, I discovered that a structurally similar, but intrinsically much taller, ordinal would be the proof number of some theory, but I don't know which. It seems relevant to that for analysis, granted, but I'm not competent to judge how relevant it is, how much it would help in discovering the β of analysis. Still, it seems like whatever T it would apply to would be much closer to full second-order arithmetic than the fragment in the analogy, and might count as ismorphically justifiable (i.e. its justification, such as it is, would be isomorphic to the justifiability of PA and the minimal fragment of analysis).
Or, then, if we could generate an extremely tall, justified β "for its own sake" (first), then we could generically move to a justified T that went beyond mere analysis or even ZFC? Is it preferable, for reasons of mathematical pragmatism, to stick with the traditional direction-of-fit, here, or to reverse our perspective and focus on the justifiability of proof-theoretical ordinals before worrying about which theories these ordinals pertain to?