faul_sname comments on How to avoid dying in a car crash - Less Wrong
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Your brain gives the illusion that you can, because it can switch quite quickly. But this is just like the illusion that you can see the whole world around you - it's not actually so. The proof is straightforward, and needs a friend.
One person holds up two fingers, one on each hand, and holds them up about a foot apart in front of them. The other person looks rapidly back and forth between the two fingers, switching their gaze from finger to finger twice a second in a regular rhythm. It's not that hard to do this.
The person holding up the fingers watches the eyes of the other person, and once they've established a rhythm they ask them a visual memory question. They will be unable to answer it without breaking rhythm on their eye movements, which the friend can observe.
Corollary - you at some level only have one internal screen which can either view external images, or internal ones. Not both at the same time.
Alternatively, read the rest of this comment while you visualize slowly spinning a rubik's cube on the axis that cuts through opposite corners. If you don't have any trouble doing so, you know that you can see while visualizing. As for myself, I find that I can't do both tasks simultaneously.
I kind-of can, though the cube image is not that vivid. (It's still something I wouldn't do while driving, though.)
EDIT: BTW, I have several reasons to think that in my case reading mostly involves a part of my brain also used for processing spoken language and different from that used to process non-linguistic visual information, which may be unusual.