This reminds me of the distinction between complicated and complex systems.  A complicated system can be predictably modeled, even if it requires a large model or many steps to do so.  A complex system, on the other hand, is inherently unpredictable and surprising.

Mathematics and other axiomatic systems are very complicated, but still fundamentally predictable.  Arguably physics is complicated rather than complex (though with the observer effect in quantum mechanics, I'm not totally sure about that).  The weird thing is that, even though, say, biology is "just" physics applied to living organisms, it is incredibly complex, unpredictable, and surprising.

Your point about humans being able to shift between systems is particularly salient.  If reality is fundamentally ordered and knowable, we would expect a coherence between various axiomatic systems insofar as they reflect reality, even though no one system has the whole picture.  So we have to remain humble and recognize the extent of our ability to know on our own.

All that said, if we develop and test various axiomatic systems to show they have some bearing on reality, then we ought to be able to triangulate some knowledge about the fundamental reality that those systems describe.  Catholicism recognizes this as "natural revelation," by which, with the aid of reason, we can observe the created world to come to know some things about the Creator, who is ultimately outside of any systems we can devise.

The medieval theologians had this idea of "divine simplicity," which says that God is identical with all of His attributes.  So we might say "God is omnipotent," but in reality, God is not a being who possesses omnipotence, he *is* omnipotence.  And also omnipresence, omniscience, goodness, love, and so on.  Thus God, who is the fundament of reality, is fundamentally simple; He is one thing.  Everything we can say about God is just another way of triangulating on that ultimately simple Being.

Which brings me to a question, based on the end of your post: 

If you are appreciative of and wistful towards religion, what is keeping you from making the plunge and pursuing it further?