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From: Robert L. Oldershaw on 19 Jul 2010 19:10
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: eric gisse on 19 Jul 2010 22:03
Robert L. Oldershaw wrote:
[snip all]

Good job on the multi posting.
From: Robert L. Oldershaw on 20 Jul 2010 11:56
On Jul 19, 10:03 pm, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
>
> Good job on the multi posting.
-------------------------------

Morning Woofy,

There is an unpleasant surprise waiting for you at
sci.physics.research.

Oh, that's right. It's a site for physicists and you are a "barker".

Back to sleep, then. Perchance to dream of chasing squirrels.

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