Correct but unhelpful answer: only through inordinate amount of practice with solving problems. A possibly helpful answer to the question "Why is it probably a

Correct but unhelpful answer: only through inordinate amount of practice with solving problems. A possibly helpful answer to the question "Why is it probably a good idea for most people to give up on this?": A dimension R^n is only a collection of points each of which is specified by an n-tuple over R. This isn't hard to hold in your head, but it tells you very little about how objects behave in any given space. By "think in", presumably the inquirer wants to be able to predict the behaviors of objects. The trouble is that there are many more different kinds of objects and behaviors than one is naturally inclined to assume, because one doesn't normally think of his ordinary 3D intuition as an insanely complex piece of specialized hardware that it is. You can certainly learn to do in software small parts of it, one by one. You would take those n-tuples, and do some particular bit of math on them to get you where you want to go. This will be difficult, like all math. Gather a big pile of these small pieces, train them until they're fast enough, and eventually you've got something like a crude emulator. I would hazard that unless you are interested in deep abstract (as in, non-visual) problems of the relevant math branches, you won't have the discipline to carry all this out.