Velocity Reviews > Re: Measuring Fractal Dimension ?

# Re: Measuring Fractal Dimension ?

Paul Rubin
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Posts: n/a

 06-28-2009
Steven D'Aprano <(E-Mail Removed)> 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.

Steven D'Aprano
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Posts: n/a

 06-28-2009
On Sun, 28 Jun 2009 03:28:51 -0700, Paul Rubin wrote:

> Steven D'Aprano <(E-Mail Removed)> 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).

I would argue that it's the other way around: mathematical *continuity*
doesn't occur in nature. If things look continuous, it's only because
we're not looking close enough.

But that depends on what you call "things"... if electron shells are real
(and they seem to be) and discontinuous, and the shells are predicted/
specified by eigenvalues of some continuous function, is the continuous
function part of nature or just a theoretical abstraction?

>> 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?

The point is that you can't safely draw conclusions about *nature* from
*mathematics*. The existence or non-existence of discontinuities/
continuities in nature is an empirical question that can't be settled by
any amount of armchair theorising, even very intelligent theorising, by
theorists, philosophers or mathematicians. You have to go out and look.

By the way, the reason you can't do to a pea in reality what you can do
with a mathematical abstraction of a pea is because peas are made of
discontinuous atoms.

--
Steven

Paul Rubin
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Posts: n/a

 06-28-2009
Steven D'Aprano <(E-Mail Removed)> writes:
> But that depends on what you call "things"... if electron shells are real
> (and they seem to be) and discontinuous, and the shells are predicted/
> specified by eigenvalues of some continuous function, is the continuous
> function part of nature or just a theoretical abstraction?

Again, electron shells came up in the context of a question about
quantum theory, which is a mathematical theory involving continuous
operators. That theory appears to very accurately model and predict
observable natural phenomena. Is the real physical mechanism
underneath observable nature actually some kind of discrete "checkers
game" to which quantum theory is merely a close approximation? Maybe,
but there's not a predictive mathematical theory like that right now,
and even if there was, we'd be back to the question of just how it is
that the checkers get from one place to another.

> By the way, the reason you can't do to a pea in reality what you can do
> with a mathematical abstraction of a pea is because peas are made of
> discontinuous atoms.

Not so much discontinuity, as the physical unreality of non-measurable
sets.

greg
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Posts: n/a

 06-29-2009
Steven D'Aprano wrote:
> one
> minute the grenade is sitting there, stable as can be, the next it's an
> expanding cloud of gas and metal fragments.

I'm not sure that counts as "discontinuous" in the mathematical
sense. If you were to film the grenade exploding and play it
back slowly enough, the process would actually look fairly
smooth.

Mathematically, it's possible for a system to exhibit chaotic
behaviour (so that you can't tell exactly when the grenade is
going to go off) even though all the equations describing its
behaviour are smooth and continuous.

> My money is on the universe being fundamentally discontinuous.

That's quite likely true. Quantum mechanics doesn't actually
predict discrete behaviour -- the mathematics deals with
continuously-changing state functions. It's only the interpretation
of those functions (as determining the probabilities of finding
the system in one of a discrete set of states) that introduces
discontinuities.

So it seems quite plausible that the continuous functions are
just approximations of some underlying discrete process.

The trick will be figuring out how such a process can work
without running afoul of the various theorems concerning the
non-existince of hidden variable theories...

--
Greg

greg
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Posts: n/a

 06-29-2009
Paul Rubin wrote:
> Steven D'Aprano <(E-Mail Removed)> writes:
>
>>But that depends on what you call "things"... if electron shells are real
>>(and they seem to be) and discontinuous, and the shells are predicted/
>>specified by eigenvalues of some continuous function, is the continuous
>>function part of nature or just a theoretical abstraction?

Another thing to think about: If you put the atom in a
magnetic field, the energy levels of the electrons get
shifted slightly. To the extent that you can vary the
magnetic field continuously, you can continuously

This of course raises the question of whether it's
really possible to continuously adjust a magnetic field.
But it's at least possible to do so with much finer
granularity than the differences between energy levels
in an atom.

So if there is a fundamentally discrete model
underlying everything, it must be at a much finer
granularity than anything we've so far observed, and
the discrete things that we have observed probably
aren't direct reflections of it.

--
Greg

Terry Reedy
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Posts: n/a

 06-29-2009
greg wrote:
> Steven D'Aprano wrote:
>> one minute the grenade is sitting there, stable as can be, the next
>> it's an expanding cloud of gas and metal fragments.

>
> I'm not sure that counts as "discontinuous" in the mathematical
> sense. If you were to film the grenade exploding and play it
> back slowly enough, the process would actually look fairly
> smooth.

radioactive emission might be a better example then.
I do not believe there is any acceleration like you see with grenade
fragments. Certainly, none with em radiation. Nothing....emission at
light speed.

pdpi
Guest
Posts: n/a

 06-30-2009
On Jun 29, 3:17*am, greg <(E-Mail Removed)> wrote:
> Paul Rubin wrote:
> > Steven D'Aprano <(E-Mail Removed)> writes:

>
> >>But that depends on what you call "things"... if electron shells are real
> >>(and they seem to be) and discontinuous, and the shells are predicted/
> >>specified by eigenvalues of some continuous function, is the continuous
> >>function part of nature or just a theoretical abstraction?

>
> Another thing to think about: If you put the atom in a
> magnetic field, the energy levels of the electrons get
> shifted slightly. To the extent that you can vary the
> magnetic field continuously, you can continuously
>
> This of course raises the question of whether it's
> really possible to continuously adjust a magnetic field.
> But it's at least possible to do so with much finer
> granularity than the differences between energy levels
> in an atom.
>
> So if there is a fundamentally discrete model
> underlying everything, it must be at a much finer
> granularity than anything we've so far observed, and
> the discrete things that we have observed probably
> aren't direct reflections of it.
>
> --
> Greg

Electron shells and isolated electrons stuck in a magnetic field are
different phenomena that can't be directly compared. Or, at least,
such a comparison requires you to explain why it's proper.