TITLE: How do you explain the use of studying higher mathematics to non-mathematicians?
QUESTION [13 upvotes]: Currently, I am an undergraduate studying math and will soon be taking my first courses at a graduate level. I'm often approached by my friends, my parents, or others, with an innocuous question: "What do you learn by studying math?" Often, I respond with something I'm sure will appeal to the pragmatic nature of their question:

"It develops strong deductive skills."
"It treats every problem by its unseen assumptions."

While these answers are ostensibly true, they don't really portray the beauty of an elegant solution. I feel that by studying mathematics, I can really know something. Unlike English, or even History, Mathematics' acknowledgements of its extremely basic assumptions makes me feel like I'm discovering something universally true. Of course, this hand-wavy, feeling-based argument doesn't really sate my questioners' curiosity. Moreover, sometimes my audience doesn't know what lies beyond their high school Calculus class, that math can be more than arithmetic; for these folk, it is especially hard to explain the uses of math.  
I guess what I'm asking is: How might I respond to these questions in a way that doesn't make me trivialize or compromise the deeper aspects of math?

REPLY [2 votes]: I will try to answer the hidden presumption that this question takes for granted rather than the popular answer or list of quotes from great mathematicians. 
Mathematics is a tool. One of many in your arsenal. It is based on deductive logic but it is not a synonym of deductive logic, rather a broad subset. 
As for the question of elegance or beauty, remember that form follows the function. Beauty and elegance is derived by the purpose. Nothing is elegant in  a vacuum nor does a tool automatically gain elegance by majority vote.
Other than that, Maths is great. Even greater is that you enjoy it. Just keep the automagical presumptions of elegance at bay and prove it's elegance by application if you can.