Higher dimmensions and parallel universes.

[Rota]

On the other hand, can you say definitively that we possess no dimension higher than three or four ourselves? I saw a discussn of Flatland somewhere here earlier. In that book, it was said that A. Square did in fact have a third dimension to him; it was just extremely thin, so that it appeared nonexistent. Could the same be true for ourselves relative to other dimensions?

Again, you can't say this is true. You can't say it's not true. But, it does render your use of 'impossible" earlier more or less incorrect.

(By extension, and using the same book's logic, we can exist in lesser dimensions. The only issue is, only a cross-section of ourselves would exist on that plane/line/whatever, while the rest of us is outside of it.)

Here is the discussion you were looking for...

Here: I'd like to introduce you to my good friend, A. Square. He lives in Flatland, a wonderful, two-dimensional country with his two-dimensional family and friends. One day, A. Square was approached by a visitor. However, this visitor didn't come in through the front door, he appeared in Square's house, first as a point, then as an expanding circle. This would be the Sphere, a denizen of a new place called Spaceland. Square simply could not comprehend it; how could there be space "up" or "down" from his current location? However, the Sphere eventually took A. Square out of Flatland, so that he could see that Flatland was indeed just tw parts of a three-dimensional world...

Am I making sense, here?

Your friend square is still a three dimensional object. Let's say you know him as a pretty green square with only height and width. You merely cannot make out his "depth," but I can assure you it is there. If he is on paper, then his depth is measurement of the crayon's wax, the pencil's graphite, the pen's ink, or the marker's dye. If he is on TV, then his depth is just a few electrons deep of the excited electrons producing the green color.

If there was no depth, then you could not perceive him. You cannot measure a truly 2 dimensional object. You can walk right through any spacial area that you would like to imagine a 2 dimensional figure exists. It MUST have the third dimension to be comprehended in the glory of it's 2 large dimensions. We all laugh and mock it's pathetic, measly, visually inadequate third dimension.

Mr. Square cannot deny his third dimension, no matter how small. Without it, he would not exist.