> That said, I think when people speak of math, they speak of its application to the real-world, not the proofs.
I think it very much depends on who the people are. I'm a math teacher, and, when I speak of math, I definitely don't just mean its applications to the real world. I definitely think programming—in the sense of thinking about the craft, not just cudgelling the computer into doing what one wants—is good preparation for learning the proof-theoretic arts of mathematics.
One could have a valid proof to a theorem, but there's the human element of having people understand and accept the result.
That said, I think when people speak of math, they speak of its application to the real-world, not the proofs.
[1] - From the Wikipedia page for the axiom, Jerry Bona has an amusing quote: "The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"