Yes, I think we do have a deep, philosophical disagreement here. You seem to be taking a more "coherentist" approach (very roughly, if theories X, Y, and Z all

Yes, I think we do have a deep, philosophical disagreement here. You seem to be taking a more "coherentist" approach (very roughly, if theories X, Y, and Z all explain observed phenomena P_1, P_2, ... P_n, then there's probably something to all of them.") OTOH, I would counter that if theories Y and Z offer no testable predictions beyond that which established theory X does, then they are of limited value. In this way, I would follow Lakatos and Kuhn most closely. The irony here is that, although I am a professional software engineer, the entirety of my post-secondary education is in mathematics. I did some research in graph theory when I was in grad school, so I am actually quite familiar with this notion of "phase transition" that you're referring to. But, I also don't consider mathematics to be a science, because I define "science" in the same way as Wikipedia: "...a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe." There are many things mathematics can describe that are inherently unphysical, which, IMO, makes mathematics itself not a science, however excellent a tool for science it is. And, it's on this basis that I'd come at all of your points 1-4. Regarding QG theories and their explanatory power, I would say that the fact that they have a handful of universally agreed upon predictions does not make any one of them correct in the least. The value in any one theory of QG is not in the areas where it agrees with other theories of QG; it's in where that theory disagrees with established theory in a way that we can see in a lab, a telescope, or an accelerator. What we're trusting here is the physical behavior of the universe, and we're hoping that it's correctly predicted by the mathematical construct that is the theory.