The immediate occasion of the question was a comment posted at the n-Category Cafe:
Interview with Manin → The Wand Chooses The Wizard
I must explain to you how I imagine mathematics. I am an emotional Platonist (not a rational one: there are no rational arguments in favor of Platonism). Somehow or other, for me mathematical research is a discovery, not an invention. I imagine for myself a great castle, or something like that, and you gradually start seeing its contours through the deep mist, and begin to investigate something. How you formulate what it is you've seen depends on your type of thinking and on the scale of what you have seen, and on the social circumstances around you, and so on.
Yuri Manin, in Mikhail Gelfand (interviewer), Mark Saul (translator), "We Do Not Choose Our Profession, It Chooses Us : Interview with Yuri Manin", Notices of the AMS 56 (November 2009), 1268–1274.
Dialogging of comments at the n-Category Cafe is rather highly selective, so I've made a practice of backing up my comments on that blog at the Inquiry List. In particular, my comments on this question and the context out of which it arose will be logged on this thread.
I see an important question here, but maybe it's a question that lurks beneath the surface appearance of an opinon poll. At any rate, I am reading the second person address of the question as more rhetorical than essential.
But I'm still getting used to this space, so I may spend some time fiddling with this entry until I see what can be done here.