Dear Sungchul and Colleagues,
Let me join the harmonious and musical – as Francesco has put it – concert
among us, trying to understand and formalize our diverging concepts of what
is information.
Sungchul writes: “… DNA viewed as a set of linear sequences of genes …”.
This we all agree on. We will please also agree on the view “…. The living
organism is a set of commutative elements…”, as this cannot be denied (the
liver not being before or after the kidneys or the brain, but concurrently
with them). This is the small aspect I wish to draw your attention to.
The following points merit consideration:
a) Copying to and fro: size questions
The DNA and the organism exchange information (whatever that is in detail)
with each other. The information content of sequences is copied from the
products of the testicles and follicles to the organism while expanding
(unfolding, de-packaging). The information content of the organism is
copied unto the sperm or the egg, respectively while containing
(registering, storing, packaging). Therefore, two processes of copying take
place between source and target, where once the sequence is the source and
the commutative assembly the target, and once the commutative organism is
the source and the sequence is the target.
Now, a copying process implies by its nature that the target must have at
least such capacity (enumerable distinct states) as the source, otherwise
something would get lost. As the process goes both directions, we have a
copying problem, where the two targets must be at least as differentiated
(possessing distinguishable, enumerable states) as the other. Following
example may be used: we need two of such sacks into which the other sack’s
contents plus the sack must fit. There must obviously be some slack: maybe
this slack is a candidate to be called information.
b) Maximal efficiency, some accounting kind of solution needed
Assuming that Nature works at top efficiency and abhors vacui, one has to
find the back-door Nature uses to fit a bigger set into a smaller set, and
then back. This she does by using an interplay between the respective sizes
(capacities, number of enumerable distinguishable distinct states) of two
functions (for details, please look up OEIS/A242615).
The accounting trick is, that information lies in the eyes of the beholder.
If we see 66 objects as one commutative collection, the answering
information is related to “where are subgroups of these things?”, while
seeing the same objects as 11 sequences of 6 objects each, the answering
information is related to “what kind of thing happens /is contemporary/,
where and when?”. One may not like the numeric facts, but they are simply
there, and Nature uses them.
While being in full conformity of Sungchul’s point: “… communications …
must be mediated by messages (or signs) (i) … (ii) obeying a set of
syntactic rules…”, let me suggest that his idea of a wide range of
alternatives regarding the number of symbol-carrying tokens and
articulation pattern among the words of the logical language, that this
grammatical openness has been found to be restricted to 3 tokens in one
message and 4 varieties of tokes, in the ideal case, as dictated by the
numbers. Nature appears to follow the rules of elementary arithmetic. The 3
planes in succession, that generate a space, and the 4 possible readings of
a linear-sequential fact, on each of the planes represented in space, are
facts coming from the rules of counting. One needs 3 planes to construct a
space: this one can do, based on a sequence, but one arrives – as the cost
of the exercise – at 4 alternatives on each of 3 readings of the space.
Another candidate for the term “information”.
c) Forever changing, always the same
The interplay between what is in store and what is actually taking place
now can well be studied if one takes the time and sits down and reorders
the content of his office (in thoughts). One will find that the task is
rather complicated and will try to avoid the eye-opening work of figuring
out what the term “order” means. Would it be possible to make your office
maximally disordered? What would be where, in that case?
Yet, one will not be able to avoid (will surely be seduced into listening
to the sweet voice of adventure and discovery as the rosy-fingered maiden
of Reason will rise in the East of his/her brain) seeing cycles and
associations everywhere.
The great change in logic FIS is approximating, cycling and circling
around, refers to a concept of the “atomos”, “object”, “element”,
“thing-as-such”, “number as such”. In the new logic, these are never alone,
always together with others, in movement, being subsystems among other
subsystems.
Belonging to cycles confers powers of prediction to elements. (The DNA is a
prediction on the sequence of components’ assembly.) If **
elements have passed, surely * *will come next, because the
process is apparently **. The observation: “*{i,j,k} *have