Absolutely spot on Bill.
It is very well covered in http://en.wikipedia.org/wiki/Dimension_(mathematics_and_physics)
You will find the key point
The state-space of quantum mechanics is an infinite-dimensional function space.
As per Bill's post our "classic" world has a small set of dimensions of freedom being 3+1 ... those degrees of freedom are 3 of space and 1 of time.
If you think about electromagnetism it clearly shows this problem in that it is definitely 4 dimensional. Almost all humans are also well versed in turning 3D images into 2D pictures and drawings and we can even do 4D. When we draw the classic electromagnetic wave we are actually bringing a 4D waveform back to a 2D drawing and we usually draw it like this
Try drawing the above image moving in 3D space ... see the problem time is already occupying the 3rd dimension and you can't. The dimensionality issue is not just restricted to QM even a basic EM wave has the problem
So extra dimensions don't have to be real to us and whether we can represent them depends on our imagination rather that any physical reality.
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it.
So perhaps the real issue is in classic physics we tend to think of dimensions in a certain physical way because of the way we are taught and experience the world via our eyes and so the dimensions in the two discussions don't really refer to the same thing .... dimensional space versus abstract dimension.
