I think you've got an interesting point, and I'd like to add to it a bit and refine/correct it a bit.

First off, minor thing, but as Jeremy has pointed out, your terminology is wrong; "representation theory" means something else. You seem to mean something more like "model theory".

But anyway -- I'm a mathematician, and I've noticed something like this too when I've dealt with physicists or their writings. But before I discuss physicists, let's discuss mathematicians, because I think you've mischaracterized them a little.

(Note, all claims here are just based on my experience.)

You talk about considering syntax vs. considering the different models of the syntax. But most mathematicians, unless they are logicians or set theorists or something, don't take either of those views. They take the Platonic view -- they're discussing a world of mathematical objects; the statements just describe it. They're not considering the statements as primary and then considering multiple models of it, they're considering the one true world of mathematical objects and describing it with statements (which might have other models, but who cares). This point of view breaks down somewhat when you have to deal with things like the continuum hypothesis and such, but that sort of thing doesn't really come up in ordinary mathematics.

(Note though that taking the Platonic view does not necessarily mean taking the "theory of programming" perspective the article discusses. I mean, I wouldn't take Aaronson's view, that computation is prior to everything else; that's pretty incompatible with it. But to my mind, computation is fundamentally about finite strings over a finite alphabet. Talking about computation with higher types -- where the Church-Turing thesis can fail -- as though it were, well, computation, feels really weird to me.)

Also, another terminological correction: Most mathematicans don't care about type theory. They care about types, at least implicitly, though they might not use that term; but they are not going to consider types as an object of study in and of themself. But yes -- when you want to understand a mathematical object, one of the first questions is, "What sort of object is that?" or "What space does that live in?"

So anyway -- yeah, physicists. I've noticed this too, and I don't know what to make of it. Where physicists don't tend to fully specify their, let's call it an ontology, and have a hard time answering questions like "What sort of object is that?" or "So that's a function from what to what?" Where they don't seem to properly distinguish between a mathematical model or description, and a method of calculation that's useful when working with that model or description. (I have so often failed to get an answer to, "So are virtual particles an actual physical phenomenon predicted by the quantum field theory, or are they just part of a useful method of calculation for working with quantum field theory?") Where they seem to care primarily about methods for getting numbers out, rather than building full, coherent models.

It's all weirdly instrumentalist, which to me seems backwards from what you'd expect -- like, mathematical Platonism is a philosophical question, but the physical world is definitely real! And sure, maybe the bits we can't measure are more of a philosophical question too -- especially since one can quite possibly come up with multiple isomorphic usable models for the laws of physics, whatever they turn out to be -- but still, mathematicians have no trouble just treating their mathematical objects as real things to be reasoned about, so this kind of instrumentalist view among physicists seems weird to me.

I hadn't even considered the possibility that they were thinking "syntax-first", so to speak! But I guess that does kind of sum it up pretty nicely. Bugs the hell out of me, though. Makes it pretty difficult to talk about the math of their theories with them, too.