Although my mathematician friends would probably yell at me if they heard this, as far as I see things it's hardly any different from physics. Just a model we create to describe observed phenomenon.
Oh, math only works this way in intuitive fields, like elementary calculus, elementary probability and staticstis, graph theory or Euclidean geometry. However, there are fields in math that are different -- for instance, there are topological spaces that exhibit phenomenons unseen anywhere else and that are very hard to grasp intuitively (I had very hard time trying to imagine what Cech-Stone compactification construction based on ultrafilters look like, and eventually I gave up after I understood that this space is only supposed to satisfy an universal property, and not have any intuitive shape). Deep results in algebraic methods in topology say interesting things about the models, while being completely indescribable outside of them, apart from some trivial examples (for instance, what _exactly_ are cohomology classes?).
The bottom line is, if the only things you've seen are geometry and calculus, with intuitive concepts like speed of change, area, length, then yeah, it's only making our intuitions more formal. Otherwise, it's something completely different.
I'm exactly the intuitive thinker you describe, with only an undergrad degree in math. I have not a clue what a Cech-Stone compactification construction based on ultrafilters is.
But... I think that whatever it is, you decide on some fairly simple properties you want to satisfy, and then go off discovering what they lead to and what the consequences are. Sometimes (mostly all the time?) you get something trivial or that reduces to being isomorphic to something else, but sometimes you get out a lot more than you put in, in surprising ways. I call that a lot more like "discovery" than "invention" though both are strained as analogies.
Again, I would argue this is no different than physics. The strength and maturity of a model is not only based on its ability to describe what we can verify, but predict what we have never (and may never) seen, and sometimes what we have never even conceived of- and I think both physics and mathematics are doing that.
My point is, many concepts in math do not exist outside of the models describing them, while if we throw away physical models, the reality does not go away.
Bingo. Mathematics is a consistent model which can be applied to do useful things and make useful predictions, but like any logical system it springs from axioms that must simply be taken as given. Most of these axioms (Modus Ponens[1], for example) are so simple and obvious that it seems bizarre to construct a way of thinking without them, but that does not intrinsically make them "true".
