Physicists and mathematicians who think about higher-dimensional spaces are, if they allow their interest to somehow become public knowledge, inevitably asked: "How can you visualize more than three dimensions of space?" There are at least three correct answers: (1) You can't. (2) You don't have to; manipulating abstract symbols is enough to help you figure things out. (3) There are tricks to help you pseudo-visualize higher-dimensional objects by cleverly projecting them into three dimensions; see here and here.
But really, why can't we visualize things in more than three dimensions of space? Could a Flatlander, living in a world with only two spatial dimensions, learn to visualize our three-dimensional world? Could we somehow, through practice or direct intervention in the brain, train ourselves to truly visualize more dimensions?
I can think of a couple of explanations why it's so hard, with different ramifications. One would be simply that our imaginations ...
