Recently I watched this presentation by Brian Greene on string theory. In it he describes how the reason we don't observe the extra dimensions required by string theory could be because they are very small and "hidden" from us.
He uses an analogy of looking across a parking lot at a telephone wire. From our vantage point the wire seems flat. However, to an ant crawling on the wire, it would not seem flat at all. The ant can see this clearly, though the fact is "hidden" from us in our vantage point. I can see how this crude analogy can provide a conceptualization of why possible other dimensions cannot be seen, but I don't see how this can in any way hint toward the extra dimension being "small." It seems to me that it is the object (telephone wire) that seems small to us because we are so far away. This creates the illusion that the wire exists in two dimensions and not three. This does not mean that the third dimension itself is small.
My question is, what does it mean to call a dimension "small?" Is this a physical size or is this a mathematical quantity/expression that is associated with size for the benefit of the layman? Also, if a dimension can indeed have a physical size, then can we assign a size to the three dimensions that we can see?
I found a similar Phys.SE question here. However, the answers on this question deal more with how dimensions can be experimentally measured, and did not completely answer my question.