Dimensions

I don't see how we can conceive of a 2D world? Our piece of paper we talk of drawing on has 3 dimensions. Anything we use to draw on it with has 3 dimensions, the marks of lead, ink, paint or what not have 3 dimensions. At least it seems that way according to our senses and what we have learned to perceive by them, but I suppose that does not make it necessarily true. How many dimensions are there in our dreams?