Measuring Fractal Dimension ?

[Robin Becker]

Paul Rubin wrote:
.........
> 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.  
> 

ok so probably true then
> 
>> 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.
> 

so does this render all the discreteness implied by quantum theory unreliable? 
or is it that we just cannot see(measure) the continuity that really happens? 
Certainly there are people like Wolfram who seem to think we're in some kind of 
giant calculating engine where state transitions are discontinuous.


>> 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.

My memory of his seminar was that he was concerned about our failure to model 
even the simplest of systems with non-linearity and/or discreteness. I seem to 
recall that was about the time that chaotic behaviours were starting to appear 
in the control literature and they certainly depend on non-linearity.
-- 
Robin Becker