Re: [Fis] Group Theory, Quantum Mechanics, and Music

Dear Colleagues:

This message responds to recent posts by Karl, Pedro, and Scott. I
suspect that most of the issues come from the difficulty of trans
disciplinary communication.

Karl chooses a narrative that explores some facets of biological
diversity and related combinatorics. Most biologists and physicians
would, I think, concur with Karl's narrative concerning biological
diversity. (Pedro appears to concur with this facet of Karl's post.)

In addition, Karl raises points which I do not understand. For
example, when he writes:

"There is such an approach. It starts from mechanical principles and
generates chemical structures. Please see the usage of multidimensional
partitions, specifically the clusters of subsets with predictable properties
that are alqways present if the set size is above a minimum. These
constructs can well be used as representing the chemical elements. The
constructs obey the ordering principle n=2i**2, as observed in the periodic
table of elements."

I do not understand how to go from a particular (multidimensional)
subset of the chemical elements to a specific chemical structure in
terms of Karl's mathematical hypothesis. Most (nearly all)
biological molecules are unique isomers (including optical isomers) -
one particular multidimensional subset organized into a **specific
chemical bonding** pattern. The concept of an optical isomer is a
property of the organization of the **whole** molecule and not a
attribute of any particular chemical element.

Karl, my favorite molecule is known as NAD (Nicotinamide Adenine Dinucleotide).
It structural formula is H(27)C(21)N(7)O(14)P(2). Six of the 21
carbon atoms are optically active. The NAD molecule includes five
covalent rings, three aromatic and two aliphatic. (See an elementary
biochemistry textbook for the exact structure.)

Karl, can you demonstrate how your approach would generate the
particular biological isomer of NAD that is common to all living
organisms?

Pedro's response suggests that my paragraph concerning the generation
of chemical structures was mis - understood. I was referring to a
GLOBAL mechanical process for generating the particular organization
of atoms (and ions) into molecules. That is, given a set of atoms
(ie, a multidimensional set of atoms in the sense of Karl), which
particular structures (graphs) will exist? This includes the
question of which optical isomers are possible.

Quantum chemistry seeks to answer a much simpler (LOCAL) question.
Given a set of atoms composed into a particular chemical structure,
what are the motions of the particles relative to one another. This
is a traditional physical question concerning time, space, and
energy. As different chemical isomers have the same molecular
formula but contain different chemical structures, the motions in
time and space are different. It is well known that these
differences are studied by analysis of the various classes of spectra
(Ultra violet, visible, Infa red, NMR, microwave, etc). On this
point, I think that we are addressing two different concepts.

The issue of group theory and its relation to the biological sciences
is another matter. The languaging of meaning is difficult because
group theory is essential to many mathematical structures of far
greater complexity than a mathematical group.
Let us restrict the discussion to the category of finite groups with
the usual set of four axioms.
What systems can be described with this structure?

Joe Rosen, in Symmetry in Science (1995), writes exclusively from a
physical view point. He makes no claims about the intrinsic metabolic
dynamics and communications within living systems.

Robert Rosen has two books -- "Anticipatory Systems" and "Life
Itself" which include extensive critical analysis of the
relationships between group structures and living systems. He
concludes from his systematic analysis that life is not a mechanism.
Many of his arguments are persuasive to me (and I hasten to add, some
of his arguments are not consistent with chemical theory!).

My evaluation of applied mathematics is based on an empirical
correspondence principle - that is, the operations of the mathematics
must correspond with the operations of the system. Imaginative
narratives are simply imaginative narratives. In biological and
chemical research, the empirical correspondences are often partitions
- multisets or multidimensional partitions in the sense that Karl
uses these terms. The experimental correspondences often support a
reflexive relation without supporting a symmetric relation in space /
time or a transitive relation with respect to matter.

Scott writes about about many issues that are irrelevant to the
mathematics of the "simplest possible" mathematical representation of
a chemical structure. It appears that the assumption is made that a
bipartite graph defined in terms of mathematics (nodes and edges) is
the same graph as one defined in terms of chemical symbols. It is
not. It is well known that chemical operations are different from
mathematical operations and such an assumption is false. Scott and I
are not communicating about the same topic.

  I am now preparing this work for publication where explicit examples
(including cyclic molecules) will detail the correspondence relations

Underlying these three communications are varying philosophies of the
relations between pure mathematics, applied mathematics and our
knowledge of various scientific disciplines. These concerns were
amply motivated and underscore the difficulty of trans disciplinary
communication when the individuals have varying perspectives of the
structures of mathematics and the structures of natural systems. For
me, the systematic categorization of extraordinarily richness of
mathematical structures within category theory has eased the task of
creating correspondence relations between mathematical systems and
natural systems.

Cheers

Jerry LR Chandler

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Received on Sun Jun 15 07:14:09 2003