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From: Robert L. Oldershaw on 19 Jul 2010 19:12
I would like to “introduce into evidence” two quotations that appear
to be highly relevant to the discussions of this thread.


“One sees that both dimensionality and sense derive from the fact
that
affine geometry holds in the infinitely small. While topology has
succeeded fairly well in mastering continuity, we do not yet
understand the inner meaning of the restriction to differentiable
manifolds. Perhaps one day physics will be able to discard it.”
Hermann Weyl, 1963, Philosophy of Mathematics and Natural Science


“Never in the annals of science and engineering has there been a
phenomenon so ubiquitous, a paradigm so universal, or a discipline so
multidisciplinary as that of chaos. Yet chaos represents only the
tip
of an awesome iceberg, for beneath it lies a much finer structure of
immense complexity, a geometric labyrinth of endless convolutions,
and
a surreal landscape of enchanting beauty. The bedrock which anchors
these local and global bifurcation terrains is the omnipresent
nonlinearity that was once wantonly linearized by the engineers and
applied scientists of yore, thereby forfeiting their only chance to
grapple with reality.” Leon O. Chua, 1991, Int. J. Bifurcation and
Chaos, Vol. 1, No. 1, 1-2.


Perhaps we should be considering a new unified paradigm for physics
based primarily upon a revised foundational geometry for nature.


The envisioned progression of universal geometries could be outlined
as follows.


Euclidean Geometry (flat, continuous, differentiable) --> Non-
Euclidean Geometry (curved, continuous, differentiable) --> Non-
Differentiable Fractal Geometry (curved, continuous, non-
differentiable).


Quite possibly it is a fundamentally nonlinear and non-differentiable
world.


To be sure, differentiable approximations would be useful, and even
necessary, in modeling limited and subjectively chosen segments of
nature’s hierarchy. However, it would be important to remember that
these restricted differentiable models are only approximations to the
actual physical structure of nature, which would be non-
differentiable
when viewed without the subjective restrictions in scale and
resolution.


RLO
www.amherst.edu/~rloldershaw

From: TKeating on 19 Jul 2010 21:40
On Jul 19, 4:12 pm, "Robert L. Oldershaw" <rlolders...(a)amherst.edu>
wrote:
> I would like to “introduce into evidence” two quotations that appear
> to be highly relevant to the discussions of this thread.
>
> “One sees that both dimensionality and sense derive from the fact
> that
> affine geometry holds in the infinitely small.  While topology has
> succeeded fairly well in mastering continuity, we do not yet
> understand the inner meaning of the restriction to differentiable
> manifolds.  Perhaps one day physics will be able to discard it.”
> Hermann Weyl, 1963, Philosophy of Mathematics and Natural Science
>
> “Never in the annals of science and engineering has there been a
> phenomenon so ubiquitous, a paradigm so universal, or a discipline so
> multidisciplinary as that of chaos.  Yet chaos represents only the
> tip
> of an awesome iceberg, for beneath it lies a much finer structure of
> immense complexity, a geometric labyrinth of endless convolutions,
> and
> a surreal landscape of enchanting beauty.  The bedrock which anchors
> these local and global bifurcation terrains is the omnipresent
> nonlinearity that was once wantonly linearized by the engineers and
> applied scientists of yore, thereby forfeiting their only chance to
> grapple with reality.” Leon O. Chua, 1991, Int. J. Bifurcation and
> Chaos, Vol. 1, No. 1, 1-2.
>
> Perhaps we should be considering a new unified paradigm for physics
> based primarily upon a revised foundational geometry for nature.
>
> The envisioned progression of universal geometries could be outlined
> as follows.
>
> Euclidean Geometry (flat, continuous, differentiable)  --> Non-
> Euclidean Geometry (curved, continuous, differentiable) --> Non-
> Differentiable Fractal Geometry (curved, continuous, non-
> differentiable).
>
> Quite possibly it is a fundamentally nonlinear and non-differentiable
> world.
>
> To be sure, differentiable approximations would be useful, and even
> necessary, in modeling limited and subjectively chosen segments of
> nature’s hierarchy. However, it would be important to remember that
> these restricted differentiable models are only approximations to the
> actual physical structure of nature, which would be non-
> differentiable
> when viewed without the subjective restrictions in scale and
> resolution.
>
> RLOwww.amherst.edu/~rloldershaw

You Xposted to sci.chem and I am replying from there.

Your narrative is not quite consistent. Narratives that lack
mathematical formulae tend to be inconsistent.

The simplest Euclidean geometry that can be concieved is that of the
2D plane or if you prefer, the complex plane. Yet the
Mandelbrot set exists in the complex plane. See:

http://en.wikipedia.org/wiki/Mandelbrot_set

Your narratives are like those of Archimedes Plutonium. How is your
thesis research progressing young man?

Moral: If you do not like your narratives to be challenged by
realists, do not Xpost to sci.chem

PS Have a look into:

http://en.wikipedia.org/wiki/Autism_spectrum

There is a lot of ASD going around. Look around you.
From: Robert L. Oldershaw on 19 Jul 2010 23:34
On Jul 19, 1:57 pm, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
>
> I rather much suspect you will not grasp or even visibly respond to this
> point, because you are incapable of responding to my analysis with anything
> other than one liners.
------------------------------------------------------

I'm genuinely concerned about you, Woofy.

There's spittle on your chin.

Are you up-to-date on all your shots?
From: eric gisse on 20 Jul 2010 00:13
Robert L. Oldershaw wrote:

> On Jul 19, 1:57 pm, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
>>
>> I rather much suspect you will not grasp or even visibly respond to this
>> point, because you are incapable of responding to my analysis with
>> anything other than one liners.
> ------------------------------------------------------
>
> I'm genuinely concerned about you, Woofy.
>
> There's spittle on your chin.
>
> Are you up-to-date on all your shots?

Dance, puppet. Keep doing exactly as I predict.

From: Jacko on 20 Jul 2010 03:33
The sci.chem man below knows something.

> Perhaps we should be considering a new unified paradigm for physics
> based primarily upon a revised foundational geometry for nature.
>
> The envisioned progression of universal geometries could be outlined
> as follows.


> Euclidean Geometry (flat, continuous, differentiable)

Is easily calculable compared to other geometries.

> Non-Euclidean Geometry (curved, continuous, differentiable)

This can be mapped to a euclidian geometry with parametric variation
of relativly apparent constants known as invariants.

> Non-Differentiable Fractal Geometry (curved, continuous, non-
> differentiable).

Fractal does not imply non-differentialble, it implies non-
computability of outcome due to constrains of input accuracy. Many,
but not all fractals are non-differentiable. x(i+1)=L.x(i).[1-x(i)] is
in some ways. It does have singularities though.

> Quite possibly it is a fundamentally nonlinear and non-differentiable
> world.

Fundementally non-linear is for sure, while integration exists so does
non-linearity.

> To be sure, differentiable approximations would be useful, and even
> necessary, in modeling limited and subjectively chosen segments of
> nature’s hierarchy. However, it would be important to remember that
> these restricted differentiable models are only approximations to the
> actual physical structure of nature

> which would be non-differentiable
> when viewed without the subjective restrictions in scale and
> resolution.

I really think you miss-understood your 'we do not yet
understand the inner meaning of the restriction to differentiable
manifolds' quote. It is to do with discontinousness. A singularity is
discontinuous in some function, all continous smooth functions are
differentiable, and discountinueties create points of non-
differentiability, or indeterminate values. Non-smooth functions
attatch no meaning to rates of change. Any system having a rate of
change due to 'inertia' interaction differentials therefore must be
smooth. Any smooth curve can occupy a space of any dimension,
including fractional dimension. The smooth projection of any fractal
may be difficult to calculate without quantization approximation
causeing potential sources of error in model outcome. And that's about
it for now.

Cheers Jacko
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