Re: Creates more problems than it solves?
> leads to the concept of infinite _dimensions_
Erm, no. That's a pretty fundamental misunderstanding of what fractional dimensions are.
To take the example you mentioned, of coastlines not being circles, the length we measure depends on the length of the ruler we pick to measure it. The fractal part is due to self-similarity at various scales and the overall "crinkiliness" of the thing being measured.
The thing is/things are:
* physical law determines that things have to bottom-out at the Planck scale, so any weirdnesses observed with your set of rulers is merely an epiphenomenon when compared with c/Planck-based metrics
* Mandelbrot's "nature" is not the same "nature" as in the "nature of reality" (whether it be relativistic, string-theoretic or multiversal or whatever); Mandelbrot's "nature" is stochastic and has underlying power laws
* using relativistic rulers is by definition the "wrong thing" when dealing with the fundamental nature of things; it's like measuring how "plaid" the universe is
* something like the fractal/Hausdorff dimension is a mathematical abstraction, not a real "dimension" (again, see power laws)
Besides, just because there are fractions doesn't mean that there have to be an infinite number of numerators and denominators (and associated explanations for them as separate things) in the universe. Unless you want to try to argue that, your argument falls apart.