You can't fly with your eyes closed - at all

I thought I new most everything, but I didn't know this ...

James Fallows (August 21, 2007) - Aerodynamics 101 (following JFK Jr crash discussion)

...No one who has received an instrument rating in an airplane doubts any of the above, because as part of the training everyone has been through this exercise: When the plane is safely several thousand feet in the air, the instructor will tell the trainee pilot to close his eyes and keep flying the plane. He is supposed to keep it level just by feel.

Within sixty seconds or so, the instructor crisply says "my airplane" and takes control again. The trainee pilot opens his eyes and is startled to see the ground coming up toward him, since the plane is spiraling* down. This always happens when the plane is flown blind, it usually happens very quickly, and generally the eyes-closed pilot had no physical sense that anything was wrong. This process is what tragically kills most pilots without instrument training when they unexpectedly enter a cloud; having no visual cue to what is up and down is the same as wearing a blindfold. It is also what is assumed to have happened to poor John Kennedy Jr and his passengers, in the dark and fog over the water.

It's the same reason you can pour tea more or less correctly in a plane flying upside down -- they're designed to keep the "floor" in place (meaning, of course, that you're losing altitude pretty fast ...)