Is mathematics one big tautology? Is mathematics one big tautology? Let me put the question in clearer terms:

Mathematics is a deductive system; it works by starting with arbitrary axioms, and deriving therefrom "new" properties through the process of deduction. As such, it would seem that we are simply creating a string of equivalences; each property can be traced back logically to the axioms. It must be that way, that's how deductive systems work!

If that be the case, then in what sense are we introducing novel or new ideas? It would seem that everything is simply equivalent to the fundamental set of axioms that we choose to start with. Is there a precise step in mathematical derivation that we can isolate as going beyond pure logic? If so, then how does this fit in with the fact that mathematics is deductive? Must we change our view of mathematics as purely deductive? And if not, then is there a way to reconcile the feeling of creativity in mathematics with the fact that it boils down to pure logic?

I'm trying to figure out the true nature of what's going on here.
Disclaimer: different people view this differently. I side with Lakatos: Logic is a tool. Proofs are a way to verify one's intuition (and in many cases to improve one's intuition) and it is a tool to check the consistency of theories in a process of refining the axioms. The fact that every proof boils down to a tautology is true but irrelevant to mathematics. 

Here is an isomorphic question to the question you posed: A painting is just blobs of paint of different colour on canvas. So, are we to deduce from this fact that the art of painting is reduced to just placing paint on canvas? Technically, the answer is yes. But the painter does much more than that. In fact, it is clear that while the painter must possess quite a large amount of skill in placing paint on canvas, this skill is the least relevant (while absolutely necessary) for the creative process of painting.

So it is with mathematics. Being able to prove is essential, but is the least relevant skill for doing mathematics. In mathematics we don't deduce things from axioms. Rather we try to capture a certain idea by introducing axioms, check which theorems follow from the axioms and compare these results against the idea we are trying to capture. If the results agree we are happy. If the results disagree, we change the axioms. The ideas we try to capture transcend the deductive system. The deductive system is there to help us find consequences from the axioms, but it does not tell us how to gauge the validity of results against the idea we try to capture, nor how to adjust the axioms. 

This is my personal point of view of what mathematics is (or at least what a sizable portion of it is). It is very close to what physics is. Physics is not just some theories about matter and its interactions with stuff. Rather it is trying to model reality. So does mathematics, it's just not entirely clear which reality it is trying to model.
Mathematicians seem to be persons that:


study mathematics;
pick up new ideas and formulate consistent theories that capture the essence of these ideas;
formulate and examine statements and try to prove them or find counterexamples;
publish their discoveries and results to be reviewed and used by other mathematicians;
teach mathematics to others.


I conclude that mathematics is a process of accumulation of knowledge about mathematical theories on an increasing number of topics. 

About the relation between mathematics and tautologies, Kurt Gödel said (Jean van Heijenoort, 1967, From Frege to Gödel s. 601):


  "The class of provable formulas is defined to be the smallest class of
  formulas that contains the axioms and is closed under the relation
  'immediate consequence', i.e. formula $c$ of $a$ and $b$ is defined as
  an immediate consequence in terms of modus ponens or substitution."


Which in my opinion makes it clear why proved theorems doesn't have counterexamples.

Logic is the trivial part of a mathematical proof and that's the tricky part which is mathematics.