Why do Gödel's incompleteness theorems make mathematicians so sad? There are complete, decidable and consistent fragments of mathematics like the arithmetic of real numbers, complex numbers, quaternions, etc., Euclidean geometry and the non-Euclidean geometries (as they are interpretable inside Euclidean geometry). Is this big portion of mathematics not enough or not worthy of studying?
One possible answer is this: Hilbert's program was to prove the consistency of math (including e.g. set theory) using a very modest set of assumptions. Gödel's second incompleteness theorem shows this to be impossible. In fact, it shows that the method should be reversed -- we must use stronger theories to prove consistency, rather than weaker ones.
Peter Smith has a clear written passage about this in his great Introduction to Gödel's Theorems:
The real impact of the Second Theorem isn't in the limitations it places on a theory's proving its own consistency. The key point is this. If a nice arithmetical theory T can't even prove itself to be consistent, it certainly can't prove that a richer theory T+ is consistent. Hence we can't use 'safe' reasoning to prove that other more 'risky' mathematical theories are in good shape. For example, we can't use unproblematic mathematical reasoning to convince ourselves of the consistency of set theory (with its postulation of a universe of wildly infinite sets).
And that is a very interesting result, for it seems to sabotage what is called Hilbert's Programme, which is precisely the project of defending the wilder reaches of infinitistic mathematics by giving consistency proofs which use only 'safe' methods.
Does this actually trouble mathematicians? Probably not in practice.
As for the first incompleteness theorem, again I don't think it troubles mathematicians. Even if confronted with an unprovable sentence, they can try and prove it (or its negation) in a stronger theory. See, for example, the case of Goodstein's Theorem.