Dear Colleagues:
I will seek to address, from a very narrow focus, Michel concerns.
( I will approach the topic from a mathematical and biosemiotic perspective.)
Michel Petitjean Wrote:
>More than 70 messages were emitted since the beginning of the
>session, discussing about entropy and information, and sometimes
>about symmetry.
>Nevertheless, the relations of these concepts with order and disorder
>has been little evoked.
>There are definitions for information, entropy, and symmetry.
>I am convinced that order (or disorder) merits its own definition,
>and that, once this definition is exhibited, we could work to
>establish relations with other concepts. Existence of order and
>disorder only through intuitive assertions is not satisfactory.
>Does somebody could exhibit physical or mathematical definitions
>of order or disorder, apart from "entropy is disorder" ?
I will start my ad hoc comments with linguistic concepts, move on to the
intertwining of linguistic and mathematical concepts and then to the
intertwining of mathematical and chemical concepts and close with
philosophical concepts.
The English word "order" according to OED is derived from Latin,
ordin, ordo, -- row, series, course, array, rank, class, degree;
related to ordiri, begin
"disorder" -- lack of order or regular arrangement.
The concept of "order" plays a central role in mathematics.
Note that English synonyms of order are the names of advanced concepts in
mathematics, particularly algebra. For example, the group structure of
polynomials is displayed as an array.
Note that the terms "order" and "disorder" are not opposites in the
mathematical sense.
If one wishes to express the opposite of the concept of "order" then, in
English, the term is the negation, "not ordered". The concept of "not
ordered" is synonymous with chaos in the mathematical sense of J. Yorke.
The concept of order is very deep, both mathematically and
philosophically. Literally hundreds of definitions and theorems of
mathematics relate to the concept of order. In the context of usage, order
could be viewed as any system that meets the definition or axiom. Disorder
could be failure to agree with one or more of the definitions or
axioms. Thus, what is "order" in one mathematical context is "disorder"
in another context because of the structure of mathematical logic.
Order is central to the foundations of mathematics.
Set theory is one foundation of mathematics. An abstract concept of order
enters set theory via the "well - ordered axiom." One expression of this
axiom ( Stoll, Set Theory and Logic, 1961) states:
"a well ordered set is a partially ordered set such that each non-empty
subset has a least or first element."
[[ Contrast this statement of a mathematical concept with the simplicity of
"time's arrow"!]]
The concept of order also lies at the heart of practical mathematics in the
concepts of arithmetic.
Arithmetic calculations with integers use ordinal numbers (the term ordinal
is also derived from the L. ordo.)
Thus, an essential feature of counting is to organize a set of symbols in
an ordinal manner.
Abstractly, the symbols are merely 'place holders' for the concept of order
or sequence.
But the structure of mathematical order is not restricted to simple regular
counting.
For example, the prime numbers form an increasing sequence.
But, no simple ordering relations predicts the next prime number!
Summary view: The mathematical concept of order has a very elaborate fine
structure that permeates virtually all aspects of mathematical discourse.
Critical notion:
In mathematics, order has the relation of being anti-symmetric.
Anti-symmetric, as applied to order, means simply "is greater than" , a
comparison of a pair.
In other words, if one has an ordering relation, then either one term is
first and the other is second or vice versa. (The possibility of
"twins" is not permitted; no ties)
Note that this corresponds readily with the set theory definition of the
existence of a first element of a set.
Note also that the mathematical concept of anti-symmetry is crisply defined
and one can decide based solely on the concept of size, when a relation is
anti-symmetric. ( From the definition above, the concept of size does not
enter into the concept of axiom of well ordered sets.)
Note that the concept of "order" is the parent of a parent of the concept
of "anti-symmetry"
The opposite relation does not hold, that is, anti-symmetry is not the
parent of the concept of order
Note the difference between anti - symmetry and the generic concept of
order. An array is viewed as a relation of order. For example, a matrix
of 3 by 5 is an ordered set because the relation (position) of each of the
15 elements of the set of the matrix (of the arithmetic operation of
multiplication of 3 times 5) is placed in a regular pattern.
Confusing? Of course it is!!! Logically, several syllogisms derived from
the root concept of "order" are related in various chains of sorites,
forming a branching, tree like structure of semantic relations.
Part of the confusion arises from the distinction between the very
difficult concept of a "relation" and the simpler (arithmetic) concept of
"number".
Why are these concepts important to FIS discussions of entropy and
information?
Clearly, the concept of "order" and "size" play a critical role in both
theories (Shannon and Boltzmann).
in the mathematical sense of order defined above, chemical semiotics is not
based on "order" or "size".
Indeed, a molecular structure is not made up from antisymmetric relations
of atomic numbers (nor are anti-symmetric sequences of atomic numbers
excluded!)
What is the relation between "order" and "symmetry" in chemistry and biology?
Since the mid- 19 th Century discovery of Pasteur, the the concept of
symmetry and asymmetry (not "anti-symmetry")
have been defined in terms of properties of the whole molecule. The
symmetry or asymmetry of a molecule is a wholistic function of the totality
of the organization of the atoms. The particular species of atoms play
only a subsidiary role in the decision logic of chemical symmetry and
asymmetry. In simple terms, atomic numbers (as an ordering relation of
abstract numbers) is not the basis of the logical basis of sequences of
atoms in molecules.
My conclusions are both strong and direct:
The information content of chemical molecules does not correspond with the
mathematical concepts of order or size.
The information content in chemical molecules does not correspond with the
mathematical concept of order because the atomic numbers are the basis of
ordering relations in the table of elements but the structure of a molecule
is not defined by the either the atomic number of the component elements of
the molecule nor the multiset of atomic elements composing the
molecule. (This is simply a fancy way of stating that the concept of
"isomers" is a critical logical distinction between the organization of
mathematical thought and the organization of chemical thought. In still
other words, Pasteur was right!)
Also, the information content of chemical molecules does not correspond
with the mathematical concepts of size. If one presupposes that "size" in
the mathematical sense corresponds with "information", then knowledge of
the atomic numbers and the multiset of atomic numbers of a molecule would
determine the chemical properties and the information content. No such
empirical correspondence exists. Indeed, the opposite empirical information
exists. Chemical molecules of exactly the same (mathematical) size have
different chemical properties. Compounds of the same size (same multiset
of atomic numbers) and with different properties are called isomers. In
simple language, a particular molecular formula may represent a very large
number of different isomers, all of the same "size".
(In order to avoid a more technical discussion, the two preceding
paragraphs neglect to recognize the critical role of ions in
chemistry. Inclusion of the concept of ions would not change the
conclusions but would require finer distinctions of meanings and usage with
respect to the nature of relationships between atomic number and molecular
properties.)
The concept of biological semiotics can be viewed as an holistic extension
of chemical communications.
From the developments in molecular genetics over the past 50 years,
biological semiotics is without doubt, grounded in chemical
structures. Indeed, in many cases, a single molecule with a highly complex
structure ( a multiset of atoms organized into one particular isomer) is
the critical source of sexual communications between the male and female of
a species. Natural history has chosen one specific molecule from the set
of all molecules to act as the specific communicator between sexes.
By extension, then, I conclude that if the mathematical concepts of "order"
and "size" are not the basis of chemical isomers, then the mathematical
concepts of "order" and "size" are not the basis of biological semiotics.
If the the mathematical concepts of "order" and "size" fail to provide the
critical concepts essential to biological and biomedical information, what
is the alternative?
A simple logical explication is available.
This logical explication is the organization of independent physical
particles into dependent relationships by "chemical bonds".
Each chemical bond is an encoding of relationships between atoms as objects,
chemical objects with properties that include the abstract concept of
number as well as physical properties. (Again, a fancy way of expressing
a simple idea that an atomic number is a number.)
Thus, each molecule is an specific encoding of informational relations.
The specific organization of a multiset of bonds among a multiset of atoms
creates a specific code word, a code word that conveys information .
Living systems have elaborated an extensive network of related code words
that specify relations among molecules.
It goes without saying that the genetic matter, one of the sources of
biological motion, is a critical set of code words that is an obvious
source of specifying biochemical and biological relations.
Some months ago, I posted a comment about the philosophical concept of
organization, a mathematical sequence of composing relations (propositions,
sorites) leading to emergence.
I would refer to this conceptualization as the organity of natural
systems. By extension, biosemiotics concerns itself with the organity of
natural codes.
A general conclusion is possible:
Organic mathematics and mechanical mathematics both use symbols to
communicate information.
A communications systems grounded in either organic or mechanical
mathematics uses specific encoding and decoding processes to transmit
messages. In mechanical mathematics, the encoding and decoding systems
are abstract concepts based on concepts of order and size. In organic
mathematics, a natural encoding and decoding process provides the
structural basis (molecular structures) which generate biosemiotics which
in turn is the basis for a wide range of phenomena.
If anyone detects any tensions between the mathematics of the present
comment and the earlier comment, I would be delighted to learn of potential
flaws in either line of reasoning.
In closing, I offer three comments to earlier posts.
1. An alert reader will note the frequent usage of the term, "concept" in
this comment. As described in earlier writings, I use the term "concept"
in the sense of "to give birth to", separate and distinct from other terms.
2. The question of evolution or emergence of living systems is not a
question of selection, it is one of choice, choices of encodings that
persist long after the putative "selective conditions" have vanished.
In living systems, the durable choices are encoded into the chemical memory
of the genetic system.
3. Rafael writes:
"It is well known that definitions are not true or false, but more
or less fruitful. In a way, people are free to define terms as they like,
but in reality their definitions may encounter problems. In children's
play, a chair can be defined as a table and vice versa. This works as long
as the children remember and obey their own decisions and do not apply
their own conventions with outsiders. However, when somebody defines a term
in such an idiosyncratic way, that definition will be neglected and will
not contribute to understanding, communication, or the advance of practice.".
Rafeal's paragraph is a perfectly reasonable working hypothesis for
philosophy. Within the communities of chemistry and mathematics, a
different, wholistic (not-relativistic) standard is held.
We start with definitions and propositions.
In order to develop a consistent set of propositions,
we seek definitions for terms in propositions that fit into pairs of
implications.
We seek implicational pairs that generate syllogisms.
We seek syllogisms that we can concatenate into sorites.
In the chemical sciences, a further standard is essential to chemical
communication and information.
Idealistically, we seek propositions, implications, syllogisms, and
sorites that are in harmony with the natural order and size of species
existing in the material world.
Cheers
Jerry LR Chandler
Research Professor
Krasnow Institute for Advanced Study
George Mason University
Received on Mon May 24 09:33:51 2004
This archive was generated by hypermail 2.1.8 : Mon 07 Mar 2005 - 10:24:46 CET