Measuring Fractal Dimension ?

[pdpi]

On Jun 25, 10:38 am, Paul Rubin <http://phr...@NOSPAM.invalid> wrote:
> Robin Becker <ro... at reportlab.com> writes:
> > someone once explained to me that the set of systems that are
> > continuous in the calculus sense was of measure zero in the set of all
> > systems I think it was a fairly formal discussion, but my
> > understanding was of the hand waving sort.
>
> That is very straightforward if you don't mind a handwave.  Let S be
> some arbitrary subset of the reals, and let f(x)=0 if x is in S, and 1
> otherwise (this is a discontinuous function if S is nonempty).  How
> many different such f's can there be?  Obviously one for every
> possible subset of the reals.  The cardinality of such f's is the
> power set of the reals, i.e. much larger than the set of reals.
>
> On the other hand, let g be some arbitrary continuous function on the
> reals.  Let H be the image of Q (the set of rationals) under g.  That
> is, H = {g(x) such that x is rational}.  Since g is continuous, it is
> completely determined by H, which is a countable set.  So the
> cardinality is RxN which is the same as the cardinality of R.  
>
> > If true that makes calculus (and hence all of our understanding of
> > such "natural" concepts) pretty small and perhaps non-applicable.
>
> No really, it is just set theory, which is a pretty bogus subject in
> some sense.  There aren't many discontinuous functions in nature.
> There is a philosophy of mathematics (intuitionism) that says
> classical set theory is wrong and in fact there are NO discontinuous
> functions.  They have their own mathematical axioms which allow
> developing calculus in a way that all functions are continuous.
>
> > On the other hand R Kalman (of Bucy and Kalman filter fame) likened
> > study of continuous linear dynamical systems to "a man searching for
> > a lost ring under the only light in a dark street" ie we search
> > where we can see. Because such systems are tractable doesn't make
> > them natural or essential or applicable in a generic sense.
>
> Really, I think the alternative he was thinking of may have been
> something like nonlinear PDE's, a horribly messy subject from a
> practical point of view, but still basically free of set-theoretic
> monstrosities.  The Banach-Tarski paradox has nothing to do with nature.

I'll take the Banach-Tarski construct (it's not a paradox, damn it!)
over non-linear PDEs any day of the week, thankyouverymuch. :)