Dimension really doesn't make sense to me. I'm not talking about mathematically. As an abstract, it is perfectly sensible. It's just that philosophical problem of existing in three or maybe four dimensions and trying to grasp something we cannot experience. It takes the kind of faith usually not reserved for the sciences.

"Nowadays the notion of multidimensional space has become so widespread in the
mathematical sciences that it is taken almost for granted. The hersy would be to
deny its existence, not to assert it." --Ian Stewart in "Does God Play Dice"

Of course, we're not asserting that multidimensional space just yet. We're asserting a fractional dimension space. This is too baffling for me to even come up with good questions about. However, the reason I imagine, is because I am thinking of dimensions spatially as opposed to in the abstract. In a strictly mathematical sense, this is sort of o.k. If your unknown is defined in terms of the result of another dependent function, it seems reasonable that you may get a fractional number of unknowns - which seems to mesh with the simpler dimension definition that the dimension is the number of unknowns.

But of course, it's not that simple. What really surprised me, and I don't know why, is to find that there is no uniformly agreed upon method for determining the dimension. Most of the math professors I have come across have the view that math is the language of god, perfect and indisputable. Leading me to believe that mathematicians should at least agree upon the method of speech, should they take this view.

At least I have found through the modern dance metaphor that finally I sit in a forum where I will not be shot down for asking critical questions of the process or questioning the methodology or result. (I don't intend to do this often, but sometimes, like today, it matters.)