Measuring Fractal Dimension ?

[Paul Rubin]

Steven D'Aprano <steve at REMOVE-THIS-cybersource.com.au> writes:
> But that depends on what you call "things"... if electron shells are real 
> (and they seem to be) and discontinuous, and the shells are predicted/
> specified by eigenvalues of some continuous function, is the continuous 
> function part of nature or just a theoretical abstraction?

Again, electron shells came up in the context of a question about
quantum theory, which is a mathematical theory involving continuous
operators.  That theory appears to very accurately model and predict
observable natural phenomena.  Is the real physical mechanism
underneath observable nature actually some kind of discrete "checkers
game" to which quantum theory is merely a close approximation?  Maybe,
but there's not a predictive mathematical theory like that right now,
and even if there was, we'd be back to the question of just how it is
that the checkers get from one place to another.

> By the way, the reason you can't do to a pea in reality what you can do 
> with a mathematical abstraction of a pea is because peas are made of 
> discontinuous atoms.

Not so much discontinuity, as the physical unreality of non-measurable
sets.