[Fis] general theory of order and disorder

Dear FIS,
it is a pleasure to contribute to a discussion about order and disorder. You
will (not) be surprised that this again boils down to linearisation of
structured sets.

Definition: order means that objects that could be other places are on one
of specific places that carry symbols that point to those objects that are
this moment in fact there. (This means: things have properties. Properties
of things regulate which place they belong to. Places appear to have
properties that match - in our perception - those of the objects. E.g. in a
"clean" place one puts "clean" things but not "dirty" ones.) If everything
is maximally matched to where it belongs, we have maximal order.
Disorder therefore means that objects are on a place among possible
differing places that does not suit the match between descriptor_object and
descriptor_place maximally.
Now we start to count:
a) we count how many differences are there among n places;
b) we count how many differences can there be among n objects;
c) we count if the proportion between possible charactersitics of objects
and possible characteristics of places changes as n grows.
To the results:
ad a): Places do not have properties of their own. We can assign an
individual attribute to each place. (Like in a new house, it is not that
important, where the place for clean and for dirty objects is, but we must
foresee it, that it is somewhere.) For the individual attributes we take the
numbers 1,2,3... etc. and we number the places. As the places are the
background on which order (or disorder) shall be diagnosed, we shall take
into consideration each and every way the places can be enumerated. There
are n! ways of assigning a number from among 1..n to an object from among n.
ad b): Objects have those properties as symbols attached to objects show. On
a finite set of objects only a finite number of nonredundant, distinct
symbols can be distinguished. The distinction happens by contrasting objects
that have a symbol (of a specific kind) with objects that have not this
(specific) symbol. The meaning of the symbol is derived from the differences
between those objects that have this symbol and those that have this symbol
not. (The terms "red", "hot" and "continuous" are learnt by pointing out
those that are so and contrasting these to such objects that have this
symbol not.) Therefore, parts of a set can have only a limited number of
properties (as there is on a finite set only a limited number of contrast
backgrounds available).
A set of n objects can possess E(n)**ln(E(n)) ways of inner differentiation.
Beyond this number of distinct arrangements of symbols, only repetitions of
properties that already have been pointed out can happen. About the parts of
this set everything will have been said (with respect to the inner
differentiation of its objects).
Ad c): Of course, as n grows (as we have more objects that can be in an
orderly or a disorderly arrangement) we have more places these objects can
be in and also more differences these objects can have among themselves.
(The more differences the objects have, the more they can be ordered. If all
objects were alike, they could be neither in order nor in disorder, because
no one would register that you have changed the position of one object with
that of another.)
What is surprising (to me) is
that the concept of a maximally structured set has so far been neglected by
mathematics, information theory and combinatorics;
that it explains very clearly how places and characteristics of objects
interdepend;
that it helps understanding why nature uses a long sequence (the DNA) that
interacts with a broad presence (the cell).

A more detailed elaboration is to be found in the article published in
Entropy - thanks to Pedro and Su-Kun.

To the terminology: order can be defined by pointing out one from among
several possible alternative states of property/place matches. If a thing
with <this> property is on a place with <that> property then we speak of
order. If either a thing with <this> property is on a place that has not the
 property <that>; or, on a place with <that> property an object with not
<this> property, then we speak of disorder.
The extent of disorder can be measured by using measures of similarity
between properties and between places and making a matrix.

Looking forward your suggestions on this.
Karl

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Received on Thu Apr 29 12:15:06 2004