I was thinking about the Mathematical universe hypothesis and a natural question popped into my mind:
Assuming that the universe (by universe I mean the complete physical reality) is really isomorphic to some conceivable, mathematically constructible structure, how would one begin to narrow down the possibilities? How would one identify its properties without necessarily finding the structure itself?
My first guess is that one should look at QFT and GR and assume that the mathematical structure we want would have to be consistent with those two theories at least in the appropriate approximations/limits and that we could somehow find all the familiar symmetries in some form on that structure.
But those are just words, I don't understand what would one have to do rigorously to rule out some of the structures, I would really appreciate some non-handwavy guidelines if it's possible.
P.S. I'll gladly elaborate further and edit my question if something is particularly unclear.
EDIT 1: I'm not asking how theoretical physics should proceed in general and how should the scientific method be used in order to understand the world. I'm interested in how much can we say about the "final, true and complete" theory (assuming it exists) without actually having it. I'm interested in the mathematical structure associated with that theory, what are the most general statements about it that are almost certainly true?