AW: [Fis] Number theory and chemical syntax

Dear Jerry and Fis,

thank you for entering a dialogue. This is the art insights may (or may not)
evolve, according to Plato and the other demi-gods of fundamental
consilience.
In your question, you distinguish between "number" and "operation". E.g. in
"3+3=6" you distinguish between the "3" resp. "6" and "+" resp. "=". Me not.
For me, "3+3=6" translates into "a set of a total extent of 6 is now in 2
identical components; the probability for a set of this size to be in two
parts is p1=...; the probability for two parts to be of identical extent is
with a set of this size p2=...; m-class=0; trutch class=.2., ..."
In fact, I propose to use a kind of (multi-dimensional) table the crude
model of which is as follows:
 Let us pick for a demonstration the following properties of a {description
of a set|partition|addition}:
Property list:
(1) size,
(2) no of fragments,
(3) max no of identical fragm,
(4) number of different fragm,
(5) diff between max and min fragm,
(6) disj class,
(7) truth class,
(8) no of distinct sequences,
(9) overlap depth.
Now we tabulate the describing aspects and classify the observed according
to its probability (which we derive from dividing the frequencies of this
class by the marginal sums for this property). This should come like a
triangle with identical rows' and columns' entry receiving a "n.a." (like
in cities distance matrix or currency translation tables, where €, $, �, �
can be translated into each other). We throw together the usual, the left-
and the right-improbables. (~). (The distinction between left-improbable,
probable and right-improbable appears to explain quite many questions
connected to what the triplets are good for. To understand left, middle and
right probabilities, please look at Fig 3 of "Possible Uses".
(http://hal.ccsd.cnrs.fr/ccsd-00002878))
An example with 64 = 4+4+3+5+4+1+7+3+4+3+5+4+2+5+3+2+5
Compared to all partitions of 64, this has 17 summands, slightly more than
the most probable of 13, so we assign a "+" for "right-improbable class" for
property "(2) no of fragments" as predicted by "(1) size".
So we go thru all properties, always comparing this observation to the
statistics of all partitions. E.g. we can say "(4) no of different
fragments" is 6, about ~. (I cannot publish the relevant statistics here.
Maybe you will want to make a thoro statistics of additions on N so you can
compare any individual addition to that universe and establish its
/im-/probability.)
Then, and this is the revolutionary innovation, we re-extrapolate from
property i unto property j (like we did from (1) unto (2) and (4) in the
paragraph above.
This reads like:
Compared to all partitions with 17 summands, this one points on the property
"(1) size" to 64. This is slightly lower than the most probable value of
about 71, so we assign a "-" for "left-improbable class" for property "(1)
size" as predicted by "(2) no of fragments".
We redo this and redo again and find that size is one of the least stable
properties of a set. The pointers to the most probable associated property
appear to make a screwdriver. This is an inherently instable (or
quasi-stable) system of references where some constellations reappear in a
different size environment. It is the same just bigger. In a probability
numbering space some additions appear to be (and now, I am stuck for words,
because this observation has no name) {"linear translations",
"similarities", "form identities", "relatives", "<you give it a name>"} like
the betting chances of picking any one from 8 out of 26 as compared to
picking any one from 16 out of 52.
 Do you get the picture? In my model of a counting system, we do not deal
too much with "size" but watch whether RELATIVE to size there is a) about
adequate (expected) no of fragments, or left-improb (too few fragments) or
right-improb (too finely ground, too many fragments), b) about adequate
(expected) no of identical fragments, or left-improb (too few identical
fragments) or right-improb (too uniformly ground, too many identical
fragments), c) about adequate (expected) no of distinct fragments, or
left-improb (too few distinct fragments) or right-improb (too differently
ground, too many distinct fragments), ... etc., and then
RELATIVE to no of rragments there is a) about adequate (expected) size, or
left-improb (too small) or right-improb (too big), b) about adequate
(expected) no of identical fragments, or left-improb (too few identical
fragments) or right-improb (too uniformly ground, too many identical
fragments), c) about adequate (expected) no of distinct fragments, or
left-improb (too few distinct fragments) or right-improb (too differently
ground, too many distinct fragments), ... etc., and so on.
It may help to know that there is maximally ln(E(n)) distinct aspects
(properties) to a set of size n (E(n) being the number of partitions of n).
After having categorised the set according to lnE(n) dimensions, one will
certainly run into redundancies. (The probability that an additional
classification will repeat one that has already been made reaches 1.0; the
extent of nonredundant information transported by a classification after
lnEn classification falls to 0.0)
It may also be useful to know that the collection is NOT infinite, so it is
useful to build up the statistics of all partitions of each natural number
(below ca. 140). (Please, relase me from explaining why 140. "Possible Uses"
gives a short discussion on that.) The probability that a sentence detailing
congruent relations on a set n > 139 is redundant is 1.0.
Of course, to do the grunt work, one should have the cooperation of and an
interest by a competent center for high-power computing. Maybe, you know
anyone who is interested in basic science. If so, please, do drop me a line.
So, Jerry et al, in this model you do not distinguish "operator" and
"argument" when describing a collection in its present (and most probable
next and next and next and next ...) fragmentational state.
Sorry for the inappropriate polish of this contribution. As I understand,
FIS does not support tables and graphics. Is there a way to upload files or
tables or statistics?
Looking forward your next question:
Karl
PS.: as to entries into the periodic table of elements, these appear to be
logical archetypes. Please look up the chi-square distribution and make the
frequencies tables of partitions, according to their formal properties (like
outlined above, that should be sufficient for a starter) before I can
explain this to you. But please do, because the concept is worth unfolding
and explaining and understanding. It allows for superb combinatorics
(building molecules, etc.) and I believe this is what you are after with
your theoretical chemistry. Yell after you got the tables.

_______________________________________________
fis mailing list
fis@listas.unizar.es
http://webmail.unizar.es/mailman/listinfo/fis
Received on Fri Nov 5 11:41:42 2004