Measuring Fractal Dimension ?

[Steven D'Aprano]

On Sun, 28 Jun 2009 03:28:51 -0700, Paul Rubin wrote:

> Steven D'Aprano <steve at REMOVE-THIS-cybersource.com.au> writes:
>> I thought we were talking about discontinuities in *nature*, not in
>> mathematics. There's no "of course" about it.
> 
> IIRC we were talking about fractals, which are a topic in mathematics.
> This led to some discussion of mathematical continuity, and the claim
> that mathematical discontinuity doesn't appear to occur in nature (and
> according to some, it shouldn't occur in mathematics either).

I would argue that it's the other way around: mathematical *continuity* 
doesn't occur in nature. If things look continuous, it's only because 
we're not looking close enough.

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?


>> In mathematics, you can cut up a pea and reassemble it into a solid
>> sphere the size of the Earth. Try doing that with a real pea.
> 
> That's another example of a mathematical phenomenon that doesn't occur
> in nature.  What are you getting at?

The point is that you can't safely draw conclusions about *nature* from 
*mathematics*. The existence or non-existence of discontinuities/
continuities in nature is an empirical question that can't be settled by 
any amount of armchair theorising, even very intelligent theorising, by 
theorists, philosophers or mathematicians. You have to go out and look.

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.



-- 
Steven