I believe that Tarski's truth undefinability theorem has uses in philosophy, as the mentioned incompleteness theorems, and that's besides cranks attributing it to proofs that god exists, that mathematics is just a bunch of crap etc etc.
I have a friend who's studying cognitive science (as well as a math degree) and he says that there's some good math modeling there, and we all saw the financial Nobel prizes that were given to mathematicians.
However, I have a quarrel with the idea of using mathematics outside the ideal realm of mathematical objects. This is because mathematics is inherently precise and perfect, you can define notions that capture exactly what you want them to capture, while our "real" world is inherently imperfect and imprecise - and we can never judge what's true in the real world, as Tarski's theorem tells us.
The above argument means that taking mathematics into the real world is to allow imperfect and imprecise definitions and "latitude" for things to change beyond our original meaning. This is not mathematics anymore, in my eyes anyway.
In the very first math class I had in the academia, the teacher came in and said "Mathematics is the science of deducing from certain assumptions." and three and a half years later, I only grow to understand deeper and deeper how true this is. In the real world, i.e. outside of mathematical idealism, you can't prove anything - only find evidence to supposedly support a claim, or disprove it. So the ability to assume things and deduce things with absolute certainty becomes problematic. Which is non-mathematical in my eyes anyway.