I think that the problem physics is having (the clashes of ideas that space has 3 dimensions, spacetime is 4-D and that string theory predicts extra dimensions) are down to a philosophical mistake in our understanding. I believe that it is matter which is three dimensional (hence all matter is made up from 3 fundamental particles.... 2 up quarks and a down, or vice versa) and that space simply curves around the matter to form a 3 dimensional veil if you like. Using this perspective, it can be seen that space has infinite dimensions, but is restricted to being measured as three dimensional because of the 3-D nature of matter. Thus... if matter existed in 4 spatial dimensions then space would be seen to be 4-D and spacetime 5-D etc etc.
I should have a paper on this is the research library very soon.
DG