Besides, the exact fields and their delineations are largely arbitrary. They're just historical artifacts. Why is there a line between math and statistics? Where did "applied math" as a separate field come from?
We could just go all European and call CS "informatics". After all, that's probably the best description of the field: it's the study of information. If not for historical and social reasons, I'd say that CS was a superset of math, not the other way around. "Computer" is just the name we give to the whole class of physical tools we use to study information.
Economics is about modelling behaviour (be it macro or micro), it has nothing to do with making money.
Then you'd be ignoring that a whole mess of math is not, in fact, computable (axiom of choice, law of excluded middle, etc). I think the problem is way too many "computer scientists" are pretty much software engineers.
The things you listed are extremely close to one of the foundational fields of CS, namely programming language theory. It's funny that you should bring up the law of the excluded middle because I learned a lot about it by studying PL theory: did you know that adding callCC to your typed lambda calculus turns it from a constructive logic system to a classical one? And that double negation translation is just a CPS transform which, at least for me, is a heck of a lot easier to reason about than proof theory directly?
Everything in math is, at least in theory, provable. Provable in a discrete system (ie a formal logic) even if it's reasoning about something that isn't discrete. So there's nothing fundamental stopping us from doing exactly the same proofs in a formal system that a computer can understand. Oh, say, a programming language.
Of course, with current technology, it's difficult from a practical standpoint, but there's nothing fundamental stopping us. And ideally it's something we should do to ward off errors and also to create proof artifacts that can be analyzed and manipulated systematically.
Wouldn't static analysis for mathematical proofs be great? Or papers written as libraries so that relying on a lemma literally links against the proof it was first defined in.
Seriously, if you dive into PL theory you'll see that the line separating CS from math doesn't really exist except as a social and technological phenomenon. (Lots of things are fundamentally possible but too difficult to do in reality, at least right now.)
That the software industry is anti-scientific does not mean there is not a computer science.
I don't formally write them down (unless it's a particularly complex case), but during debugging and performance tweaking I find myself making hypotheses and experiments all the time (according to the lay definitions of those words, at least)