Actually, it's the opposite. Here's a simplistic overview of why: Suppose you do a finite amount of mathematics (prove a finite number of theorems from a finite axiom set). But by the incompleteness theorems, there are some theorems that simply cannot be proved. Thus, no finite amount of mathematics suffices to encompass all of mathematics.
It's instructive to look at a specific case study of what situations like these actually look like in the course mathematical research. The first and still most famous case is the ancient question of whether or not Euclid's parallel postulate (PP) can be derived as a theorem from the first four axioms (the postulate states that There is at most one line that can be drawn parallel to another given one through an external point).
One way geometer's posed this problem was to attempt to re-prove all the old Euclidean results from scratch without invoking the axiom of parallels, and see how far they could get. This system of geometry, Euclidean minus parallels, is called Neutral Geometry because it is neutral about whether or not the parallel axiom is true. The question is then, does Neutral Geometry end up being equivalent to Euclidean Geometry anyway?
And of course, they failed time and again to prove PP within Neutral Geometry, and so they eventually started focusing their research on more indirect approaches, like considering what taking "PP is false" as an axiom would imply and presumably something absurd would result. One way to falsify PP is to say that we can find at least two parallel lines through a point, in contrast to at most one. Counter-intuitive results followed, but it this system was surprisingly proved to be consistent regardless, and it came to be called hyperbolic geometry. Whereas before geometry was a single unified system codified by Euclid, both complete and consistent, was shown to be but a stem of an ever branching family geometr*ies*.
This is what really makes the implications Godel's incompleteness theorems so profound. It guarantees the openendedness of mathematics. In the beginning there was geometry and arithmetic, and then algebra and eventually calculus/analysis. The subject matter of mathematics could be understood in completely taxonomic terms: space, quantity, structure and change. But today we can see that mathematics exceeds any definition in terms of subject matter. On might say that mathematics is more of an art: the art of being creatively logical through the media of abstraction.