It's a Self-Similar World

I am splitting off this topic from the previous thread to give Nick and
R... a chance to abuse their cats without interference or superposition
of topics.

1. To my comment: "Maybe we can retain the continuous S-T paradigm if
we are willing to consider continuous, but non-differentiable
geometries", Charles Francis replied:

>Why would we want to. After all ther is no empirical evidence ... that
>there is such a thing as a continuum.

Einstein's vision of dynamics based on geometry rather than forces(or
even more dubious constructs) still appeals to me and I don't want to
give up on such a beautiful vision until all possibilities have been
explored.


2. Regarding my purposely provocative statement: "After all, it is a
fractal world", Charles Francis wondered:

>Is that a statement of faith? How would you go about demonstrating scientifically, bearing in mind the infinite >process you would have to undertake to examine the proposition.

Firstly, many mathematical and physical structures are recognized as
fractals but do not have an infinite number of levels of structure.
The concept of infinite substructure is an extremely interesting case,
but finite fractals are equally worthy of study. To avoid unproductive
semantic squabbling, I'm going to change my manifesto from "It's a
fractal world" to "It's a self-similar world", and propose that a
self-similar structure or process manifests physically meaningful
self-similarity on at least three distinct scales.

If you want some justification for making self-similarity such a
fundamental concept, I would ask you to go to:
www.amherst.edu/~rloldershaw
and click on "Selected Papers", then click on a paper entitled
"Self-Similarity Everywhere" near the end of the list. I cite about 80
examples of self-similarity from various realms of nature. The website
is devoted to a cosmological paradigm based on discrete
self-similarity.


3. Now let me try to tie 1. and 2. together. In his last scientific
writing (for an Italian conference entitled "Fifty Years of
Relativity") Einstein noted the intrinsic self-similarity of his field
equations. At that point he regarded this property as potentially
problematic. If he had lived long enough to see the evidence for
ubiquitous self-similarity in nature, from subatomic to metagalactic
scales, he might have had renewed confidence in his vision of a unified
field theory, based on that very property: self-similarity.

.

Relevant Pages

  • Re: Black holes, dark matter
    ... It seems to me that your use of fractals is largely qualitative. ... really not important in your system, only the idea of self-similarity. ... Are you aware of work of mathematicians such as Stephen Semmes, ... I guess I'm saying that the expository style makes me ...
    (sci.astro)
  • Self-Similarity In Nature
    ... of the term self-similarity because it is a key ingredient of fractals ... perhaps excusable to anthropomorphize a bit and say that nature truly ... were it not for this elegant design ... distinction between fractals and self-similarity. ...
    (sci.fractals)
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