Measuring Fractal Dimension ?

[Paul Rubin]

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).

> 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?

> Quantum phenomenon are actual mathematical discontinuities, or at
> least they can be, e.g. electron levels in an atom.

I'm sure you know more physics than I do, but I was always taught
that observables (like electron levels) were eigenvalues of underlying
continuous operators.  That the eigenvalues are discrete just means
some continuous function has multiple roots that are discrete.

There is a theorem (I don't know the proof or even the precise
statement) that if quantum mechanics has the slightest amount of
linearity, then it's possible in principle to solve NP-hard problems
in polynomial time with quantum computers.  So I think it is treated
as perfectly linear.