"We should believe in our best [scientific] theories as they are, even if we realize that we could have done things completely differently. Approaches that show that we could have done science or mathematics differently shouldn't undermine our belief in science and mathematics as it is, or at least they shouldn't undermine it by much."Let's leave aside the wrinkle in the last clause. I want to focus on how the authority of physical theory is being used to make claims about the ontological status of numbers (a familiar strategy following Quine-Putnam). I think there is a slide in the argument where Thomas conflates claims about what the theory is about and the way the theory is formulated.
To be clear: for the sake of argument I grant that notational variants, even excessive ones, do not undermine our belief in the truth of a scientific theory. Even conflicting theories [a la Duhem-Quine] that can account for the same current and future data need not undermine our belief in our current ruling theory. (I am very fond of Newton's fourth rule of reasoning.) [I also grant, for the sake of argument, that there is such a thing a standard ruling theory.]
But...scientific-practitioners do not take physical theories as making claims about the nature of number nor is there a reasonable way to interpret most of these theories as such. [I am open to the idea that string theory and parts of quantum mechanics may be interpreted at odds with this claim!]. As Jody Azzouni has argued, scientists don't create measuring tools to interact with numbers. And the particular math one uses is often introduced pragmatic purposes. Not to mention that the standard of "proof" in science is often very different from the standard of proof in mathematics--so here's a case where science's authority is very much questioned).
So, one cannot appeal to the *general* authority of physics to settle the question about the nature of number or mathematical objects. Rather than accepting a theory without further analysis (as a whole package), I believe one needs to do piece-meal investigation of what mathematical objects are required for (indispensable to) the empirical content of particular physical theories. (After all, the physical theory comes as a package for physics purposes, but it wasn't designed to meet all philosophical or mathematical purposes at once--note, for example, that standards of proof in physics and math can differ.)
There is more to be said about this (and I haven't reported Thomas' responses), but this has grown long enough.