Measuring Fractal Dimension ?

Steven D'Aprano steve at
Sun Jun 28 06:07:38 EDT 2009

On Sat, 27 Jun 2009 23:52:02 -0700, Paul Rubin wrote:

> Steven D'Aprano <steve at> writes:
>> Depends on how you define "discontinuous".
> The mathematical way, of course.  For any epsilon > 0, etc.

I thought we were talking about discontinuities in *nature*, not in 
mathematics. There's no "of course" about it.

>> 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.
> If that transition from grenade to gas cloud takes a minute (or even a
> femtosecond), it's not a mathematical discontinuity.  

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.

Mathematics is an abstraction. It doesn't necessarily correspond to 
reality. Assuming that reality "really is" the mathematical abstraction 
underneath is just an assumption, and not one supported by any evidence.

> The other examples work out about the same way.

Quantum phenomenon are actual mathematical discontinuities, or at least 
they can be, e.g. electron levels in an atom. Even when they are 
continuous, they're continuous because they consist of an infinity of 
discontinuous levels infinitesimally far apart.


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