Measuring Fractal Dimension ?

[Steven D'Aprano]

On Thu, 25 Jun 2009 12:23:07 +0100, Robin Becker wrote:

> Paul Rubin wrote:
[...]
>> No really, it is just set theory, which is a pretty bogus subject in
>> some sense.  There aren't many discontinuous functions in nature. 

Depends on how you define "discontinuous". Catastrophe theory is full of 
discontinuous changes in state. Animal (by which I include human) 
behaviour often displays discontinuous changes. So does chemistry: one 
minute the grenade is sitting there, stable as can be, the next it's an 
expanding cloud of gas and metal fragments. Then there's spontaneous 
symmetry breaking. At an atomic level, it's difficult to think of things 
which *aren't* discontinuous.

And of course, if quantum mechanics is right, nature is *nothing but* 
discontinuous functions.


>> 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, there's also discrete mathematics, including discrete 
versions of calculus. 


> 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?

That's a question for scientific investigation, not mathematics or 
philosophy.

It may be that the universe is fundamentally discontinuous, and the 
continuous functions we see are only because of our insufficiently high 
resolution senses and instruments. Or it may be that the discontinuities 
we see are only because we're not capable of looking closely enough to 
see the smooth function passing between the two ends of the discontinuity.

My money is on the universe being fundamentally discontinuous. We can 
explain the continuous behaviour of classical-scale phenomenon in terms 
of discontinuous quantum behaviour, but it doesn't seem possible to 
explain discontinuous quantum behaviour in terms of lower-level 
continuous behaviour.


-- 
Steven