Re: [Fis] Re: miscellanea / temperature / symmetry

Dear Michel et al.,

On Apr 22, 2004, at 1:56 AM, Michel Petitjean wrote:

> About symmetry and its relations to entropy: symmetry theory and
> entropy (information theory) are both related to distributions.
> But, symmetry theory has to work for distributions, discarding whether
> they are finite discrete, or infinite, or continuous. Informational
> entropy is mainly connected to finite discrete distributions, despite
> some extensions in the continuous case.
> Much more problematic is that the entropy associated to a discrete
> distribution does not care of the numerical values to which non null
> probabilities are attached, and in fact, entropy exists even if the
> probabilities are defined on a non numerical space: e.g.:
> P(red)=1/3, P(green)=1/3, P(blue)=1/3 is a distribution for which
> entropy can be calculated. This is false for symmetry calculations,
> which deals mainly with euclidean multivariate distributions.
> So, entropy and symmetry are quite different.

It seems that you have staked out a position that is entirely at odds
with the view advocated earlier by Shu-Kun. My own view is very much
aligned with that of Shu-Kun, so I would like to consider how your
rational argument leads to such a contradictory conclusion. It seems
to me that the source of the difference lies in the distinction between
the concept of entropy and our tools for estimating its value. You
might argue that the concept of entropy was originally derived as a
statistical calculation, so that subsequent development of the concept
distorted its correct meaning; but I would argue that its conceptual
development represents an ordinary and heuristically valuable aspect of
science. This process has arguably led to a more useful concept of
entropy (e.g., one that includes meaning related to symmetry), despite
an increase in conceptual ambiguity and divergence between the concept
and its original measure. I think that the relationship between
concept and measure has converged on the kind of relationship that we
often encounter when theory provides a concept first. In this case, a
variety of measures may later be advocated to represent the concept,
but it is recognized that each measure is only an error-prone
approximation of the concept. As an evolutionary biologist I would
point to the concept of fitness and its measures as an example.

In sum, I think that the growth and development of the concept of
entropy beyond the scope of its traditional measures to include
concepts like symmetry has been heuristically positive, although I also
recognize the difficulties that this introduces.

Cheers,

Guy Hoelzer

Department of Biology
University of Nevada Reno
Reno, NV 89557

Phone: 775-784-4860
Fax: 775-784-1302

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Received on Thu Apr 22 18:15:42 2004