I really do mean representation theory, in the widest sense of the word. Wide because I am trying to make connections here, not distinctions.
So for instance, of course group representations. On the one hand there is an abstract group multiplication, and on the other side we are representing this as a geometric action.
The same thing goes for "bigger" gadgets, lie algebras, hall algebras, etc. All kind of algebraic objects that we can define abstractly (syntactically) but then lo! it is actually represented by some kind of (geometric?) action.
And then in a more logical vein you have the lattice type theories. Boolean algebras, heyting algebras. These things have topological representations (or is it functions on topological spaces). And then to denotational semantics of lambda calculus. And it's all kind of a syntax on one side, and a semantics on the other.
I don't know much about models of set theory, but I'm assuming these also involve a kind of "active" representation of syntax. I would also include things like algebraic topology, topological quantum field theory. These are all "functors" which act to represent an algebraic gadget in a more dynamic way (i'm possibly over-generalizing here.)
Anyway, there's no concrete definition for what I mean by a "representation" so that's why i gave some examples. It's quite mysterious imho.
> So anyway -- yeah, physicists.
Well, it's easy to criticize physicists, but I also think mathematicians are guilty of this. Ok, so they define their calculations rigorously (not always), but then it's gloves off and away we go! I've come to realize that the whole program of categorification is an attempt to bring more context to these calculations (this is what I would call type theory for mathematics btw.) And most mathematics is written down as calculations, without this context. Probably because it's much harder to categorify everything. The simplest example of what I mean is this: "why keep track of a (finite) set when you can just say how many elements are in the set?" And so on. I really do think that alot of mathematicians are just as guilty as physicists of the "shut up and calculate" attitude. But, they are certainly much more humble :-)