In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining --- they depend very heavily on the community of mathematicians.
Bill Thurston's response to "What's a mathematician to do?"
I thank you for your friendly letters. You sure know how to cheer up a really old geezer (84) in his sunset years. I don't make public appearances any more because I now resemble nothing so much as an iguana.
What I had to say to you, moreover, would not take long, to wit: Practice any art, music, singing, dancing, acting, drawing, painting, sculpting, poetry, fiction, essays, reportage, no matter how well or badly, not to get money and fame, but to experience becoming, to find out what's inside you, to make your soul grow.
Seriously! I mean starting right now, do art and do it for the rest of your lives. Draw a funny or nice picture of Ms. Lockwood, and give it to her. Dance home after school, and sing in the shower and on and on. Make a face in your mashed potatoes. Pretend you're Count Dracula.
Here's an assignment for tonight, and I hope Ms. Lockwood will flunk you if you don't do it: Write a six line poem, about anything, but rhymed. No fair tennis without a net. Make it as good as you possibly can. But don't tell anybody what you're doing. Don't show it or recite it to anybody, not even your girlfriend or parents or whatever, or Ms. Lockwood. OK?
Tear it up into teeny-weeny pieces, and discard them into widely separated trash recepticals. You will find that you have already been gloriously rewarded for your poem. You have experienced becoming, learned a lot more about what's inside you, and you have made your soul grow.
Kurt Vonnegut's letter to a bunch of kids, asking him to visit their school.
Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it's all illuminated and you can see exactly where you were. Then you enter the next dark room…
Andrew Wiles, on doing math