In fact, the definition of these concepts, with a realization that interesting and ingenious considerations could be applied to them, is the first demonstration of the ingeniousness of the mathematician who defines them. The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used.
Within computer programming alone, topics as diverse as decidability and Turing completeness, computational complexity, discrete probability, number theory for cryptography, calculus for almost any optimization problem, geometry not only for graphics but also for information theory, which is necessary for compression and coding - show that math is the heart and soul of all of these concepts! Beyond that, so many of the operations that computers are actually useful for carrying out are inherently mathematical. I get why so many people are dismissive of "higher" math - there is no shortage of lousy teachers or rote arithmetic in early education, boring classes and an overall negative reinforcement that can leave people jaded and scornful. But I've learned from experience that it IS possible to get young kids interested in real math, mostly by knowing some of the relationships to fascinating phenomena. Regardless, I think it is tragic to see such disparaging opinions of mathematics.