First of all, I want to thank everyone for their contributions and comments, there's some really great stuff here, and I've (and I feel anyone who has viewed this question would agree) definitely gained a lot from all that has been said. I've given my question a lot of thought, and I feel compelled to give an answer that encompasses some of the more elegant ideas expressed by other users. Here goes:
Mathematics, in its most essential form, is simply a set of ideas about quantity. These ideas are ways to view the world through the lens of measurement and of dimension. They have a unique "flavor", and in an exactly analogous manner, every other science (or field of study) is a group of ideas that help us view the world in a "flavored" way. Mathematicians discover these ideas, invent them, use them to find other ideas, clarify them and bring them to precise formulation, work with them, and this list may go on. We have a body of knowledge, and that is the key.
Thus far I've said nothing of deduction and for good reason: Deduction is the tool we use in order to ground our ideas in a firm, unshakable system. We strive to find axioms that will properly serve as the "seeds" for all of the mathematical ideas we'd like to have in our system. We wish to build up a logical fortress of implications and equivalences that will house our mathematical ideas. The deductive system is the means to derive and discuss the ideas of mathematics. Let me put this in different words: In order to further our quantitative knowledge, it becomes necessary to clearly define our abstract structures; it also becomes necessary to clearly perceive the properties of these structures, to manipulate them, and ultimately use them to arrive at other structures. This is the nature of the mathematical-deductive system, and the benefit is twofold: Firstly, we have a precise and logical way to house our intuitive quantitative ideas which form the main body of math. Secondly, once the dust has settled and we have worked out our precise deductive system to the last jot and tittle, it becomes possible to create new ideas, to reach novelties! That is the magic of the deductive system; the precision is a way to formalize our intuition, and the new-found rigor may even result in new ideas hitherto unsuspected. Once you have a logical system, novelties might be reached by utilizing the system, playing by its logic rules of derivation. So the true mathematics is a combination of both intuition and rigor.
It doesn't matter that, in some technical sense, everything might be traced back to axiomatic roots. That is just how deduction works, you can't fault a logical system for being logical. But at each derivative step, we have a new idea, that is the essential point! Even in an equality which seems to be the biggest triviality, the LHS and the RHS represent two different ideas, and their being equal a third. To reiterate, the deduction is a rigorous, "tautological" (not in the strict sense) system; but in each each step that we take, that brings us further and further away from the axioms, a new idea is birthed or represented. It is the concepts contained in each theorem or identity that are at the heart of math, and not the logical steps between them. Those logical steps are just due to the nature of deduction, and are of necessity if we are to employ the tool of deduction.
I hope I have added something new to this (in my opinion) intriguing discussion, or at least synthesized what's already here. Thanks for reading.