From: Jerry LR Chandler <Jerry.LR.Chandler@Cox.net>
Date: June 7, 2004 2:25:57 PM EDT
To: fis@listas.unizar.es
Subject: Definitions of order
Dear Colleagues:
This note responds to comments by Michel, Rafael, Loet and Michael D..
I seek to explore once again the relations between mathematics, chemistry, philosophy and the necessity for biosemiotics as part of a structure for the concept of information.
The communication closes with two questions designed to challenge the status quo.
First, Michael brings up the question of the role of partially ordered sets (in mathematical jargon, posets) and the relation to transitive relations.
Transitive relations can be defined as:
If x is greater than y and
if y is greater than z, then
x is greater than z.
This is a simple re-arrangement of the general structure of a Quine syllogism:
If every y is a z and
if x is a y, then
x is a z.
The transitive relation per se, like the syllogism, does not address the question of the beginning or the end of the ordering relation. Thus, a transitive relation not not address the least element criteria of set theory. Multiple beginning elements and ending elements are consistent with a transitive relation on a set. Uniqueness is not addresses in either the transitive relation nor the syllogism.
For example, the transitive relation could be restricted to any three three components of an entropy generating string.
Another mathematical theory, known as lattice theory, addresses the issue of least element and greatest element. (Note that the Latin root of order, as described in my last post) requires a beginning, not merely a question of internal ranking of components.
However, lattice theory is not based on a transitive relation, rather it presupposes a anti-symmetric relation.
An anti-symmetric relation is defined in terms of
x is greater than y or
y is greater than x.
No syllogism emerges from the anti-symmetric relation as only two indeterminate symbols are used.
Michel asks about relations to disorder.
Chaos, a term mathematically defined only for continuous dynamic systems, is not related to the concept of disorder when disorder is defined as the negation of the order of time. In other words, the points in time follow one another in a simple transitive relation, quite consistent with Koichio's language of time.
By definition, chaotic systems are iterative systems, the relationships cycle around and around and around and ...
The critical distinction that separates chaotic orbits from classical physical orbits is that these are not regular cycles.
Indeed, a chaotic system may never repeat the same cycle twice.
(The calculations are in terms of real numbers, values that can NOT be calculated EXACTLY.)
Michael's conclusion, " So, in this sense, there is no ambiguity about what is order."
seems a bit over simplified for me.
Loet's proposition
The four-dimensional probability distribution representing a hyperspace can encompass the various geometrical and temporal subdynamics plus their interaction terms.
is quite confusing to me. I really do not understand the mathematical characterization of the system which he describes. In particular, without more information about the relations among the variables in the system, I do not see the basis of the conclusion. Generally speaking, the richness of the classes of behaviors of such systems is so great that...
What is the importance of these distinctions about order for information theory in the chemical sciences?
Consider the labeling of a path between atomic numbers representing atoms within a molecule.
Transitive relations among atomic numbers within one path of molecule may not be possible yet may exist in another path. Thus, thus a specific order of atoms in the molecule may be true in one path and false in another path!
The concept of "order" within the chemical sciences (molecular structures) is remote from the concept of "order" in physical sciences and mathematics. We use a different approach to creating relations among atomic numbers. Each atomic number serves as a source of local order. And, each atomic number within the molecule contributes to the global properties of the molecule. These are deterministic in the sense that the structure is determined by the specific relations between the atomic numbers. Different ordering of the same set of atomic numbers gives different molecules (different individuals, isomers).
The fact that local order is created by each nucleus presents a deep problem for any generalization about chemical systems. For example, a simple protein may contain 1000 atoms, each a source of local order. A social metaphor for a protein would be a village of 1000 people, each tugging and pulling at one another to create the economic, social and political fabric of the village.
Why would a "probability mass function" be useful in describing these relations?
(In other words, in what sense is the concept of a vector space meaningful in this context?)
In particular, why should scalar relations apply to compositions of atoms?
Rafeal's comment about being, point and place requires further examination.
"There is a famous definition of points by Aristotle in his Physics (V, 3) in
which he distinguishes between natural beings (physei onta), points
(stigme), and unity (monas).
- physei onta are charakterized by unity, place and position (hen,
topos,thetos)
- points (stigme) are characterized by lack of place (atopos), position
(thetos) and touch (syneches, continuum)
- unity (monas) is charakterized by lack of place (atopos) and lack of
position (athetos)
There is a progressive detachment (or separation: chorismos) of points and
unity from natural beings, i.e. a process of "abs-traction")"
Within English usage, the concept of "place" and "position" are intimately related to one another.
For example, one "places" an object in a "position". And, a position is a specific place.
Is this distinction captured by the distinction between " topos" and " thetos"?
It is useful to point out that the concept of " monas" as "unity lacking place and position" is remote from mathematical usage. For example, the unit "0" gives the basis for identity of the arithmetic operation of addition while the unit "1" gives the basis of the identity of the arithmetic operation of multiplication.
Thus, in terms of ordering relations, the concept of "unity / identity" in these two arithmetic operations
specifies both place (at the beginning) and position with respect to all other units of the system.
This appears almost the opposite of the Greek usage. Is this merely a matter of a irregular translations or is it a substantial issue with respect to our view of "order"?
Finally, Rafeal's comment seems closely tied to the concept of language as if the words themselves were really the objects and not symbols representing our representation of the language. Rota ( a mathematical philosopher) points out that Heidigger created a universal "as" to distinguish "Being" from "being".
Rafeal, in your view, what is the distinction between the following two phrases:
"Natural Beings as characterized by place, position and touch, are objects."
"Natural beings are characterized by place, position and touch as objects."
Michael D.'s excellent post gets to the heart of the problems of transdisciplinary communication.
The conditions for the usage of the term "entropy" are well established in physical sciences (including physical chemistry) and engineering. In these disciplines, the binding conceptual relations are the system of units -- meters, kilograms, seconds -- and the arithmetic operations of addition, multiplication and exponentiation.
In other sciences, particularly particularly chemical structures, biology and medicine, the binding conceptual relations are different from the physical system of units and the arithmetic operations.
The identity of the individual plays a critical role for two reasons:
1. In general, neither the MKS system of units nor the arithmetic operations function on these individual units in a manner analogous to the physical sciences. (For example, if one multiples the atomic number of helium (2) by the abstract number (4), one does not generate the atomic number (8) for oxygen.)
2. The function role of time is fundamentally different. Time is individualized to become an interior property of the particular individual. (For example. each individual biological organism occupies a unique position in the evolutionary tree of inheritance. Each molecule has a property of a specific spectra, a time dependent property of the relations.)
Primary conceptual bases of the chemical sciences include the concept of the table of chemical elements, the concept of the chemical bond, the concept of historical unity of life and the biological concept of reproduction.
Mathematically, every calculation of a chemical structure related to the functional dynamics of these four binding concepts is based on integer values of atomic numbers. The calculations produce exact multisets of the numbers of atoms present without reliance on the MKS system of units. If one desires to express communication in terms of the MKS system, one switches from the binding concept of number to the binding concept of mass and introduces artificial physical constants of non-integer value that can only be approximated
An open question to the physical and biological scientists:
What are the correspondence relations between the conceptual binding relationships of the chemical sciences and the conceptual binding relationships of the physical sciences?
A specific question to Rafeal:
Are the philosophical stances of Husserl and Heidigger deep enough to bridge the fundamentally different referential conceptual binding systems that generate the logics of the physical and chemical sciences?
Cheers
Jerry LR Chandler
Research Professor
Krasnow Institute for Advanced Study
President
WESS
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Received on Mon Jun 7 20:44:35 2004
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