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comment     share|improve this answer add comment  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 Man