Someone recently asked this on Math.SE, and it’s something I think about a lot, so here is what I think about it currently:
This is a really old question, but one I ask myself all the time. (Warning: I’m going to stretch a metaphor so far that the Bard would be worried)
I once heard a description of Grothendieck which said the following: Many mathematicians saw the peak of the mountain and tried to climb it, coming up with the most ingenious ways to summit it, and yet all of them fell short. Grothendieck came along, saw the mountain and instead of trying to climb it he stood back and tried to build the airplane, and when he had succeeded he flew above the mountain from high above, seeing what all the other mathematicians wished, and all without ever trying to climb the mountain. Eventually his airplane revealed so much more about the geography than any mountain climber could have.
Why do people build airplanes? I would argue that planes come not out of seeing the heights of mountains, but out of an infatuation with air itself, with the notion of flight, with the freedom it provides.
Often I think mathematicians as a whole do what Grothendieck did in this metaphor, they see the mountains which humans wish to summit, and they do not climb them (or at least pure mathematicians do not wish to climb them), they simply try to build better airplanes to fulfill their love of flight, and those same planes reveal more than anyone could ever predict.
Mathematics can not reveal everything: Sometimes humans wish to explore under the earth, where we see the natural philosophical caves of the earth and sometimes go spelunking in them, and often times those mavericks among us write literature to dig deeper caves, and we must explore those so that one day we might understand the core of humanity and the world, and in this case airplanes and mountain climbers can only get us so far.