I have some similar thoughts. For example, when people learn about 2-space and 3-space (or 2-d and 3-d arrays), the most common thing to do is to draw out the

I have some similar thoughts. For example, when people learn about 2-space and 3-space (or 2-d and 3-d arrays), the most common thing to do is to draw out the spatial analogue: a WxH table, a WxHxD cube. And so the examples always stop at 3. But in math you don't stop anywhere. I have wondered if people who have extreme finesse with math at arbitrary dimensions change strategies when dealing with higher dimensions, or if they never did this spatial conversion all along and deal with it completely symbolically. Then I tried to think of what certain things mean symbolically. In integration it's always the area. But in symbols it's just a cold, unfriendly product. I'm sure there are merits to doing it this way, but I have trouble seeing intuitive meaning in it. Perhaps the better mathematicians simply calculate and don't care about what sense it makes?