*INTRODUCTION*
Data is that what we see by using the eyes. Information is that what we do
not see by using the eyes, but we see by using the brain; because it is the
background to that what we see by using the eyes.
Data are the foreground, the text, which are put into a context by the
information, which is the background. In Wittgenstein terms: Sachverhalt
and Zusammenhang (which I translate – unofficially – as facts /data/ and
context /relationships/).
The “context” component shows the equivalent alternatives to the data
object. This we find by using the commutative symbols which generate the
categories in which the data object is included. The “data” component is
then the individuating sequential symbol which points out one specific from
among the members of the communicative category.
The formal definition of the term “information” is as follows:
*“Let x = ak. This is a statement, no information contained. Let x = ak and
k **Î** {1,2,...,k,...,n}**. This statement contains the information k **Ï**
{1,2,...,k-1,k+1,...,n}”**.*
A numeric approach uses the concept of counting in terms of consolidation
of displacements, and points out the data as a specific element of a cycle,
the information part being the communication about which cycle the element
is part of /= data about the remaining elements /. In the following
paragraphs we show the technique, by which the term “information” is to be
explained, can be established
*METHODOLOGY*
*Material we work with:* We begin by creating elements of a set. The set we
construct to demonstrate how to establish information content consists of
realisations of the logical sentence *a+b=c. *That is, if we use *d
*distinguishing
categories of elements, we shall have a set that contains the elements *{(1,1),
(1,2), (2,2), (1,3), (2,3), (3,3), (1,4), (2,4), …., (d,d)}. *These
elements we refer to as *(a,b), a <=b.*
The number *n *of the elements of the set is of course dependent of *d*, *n
= f(d). * *n = d(d+1)/2.*
While the *principle *of information management is valid over a wide range
of values of *d*, it can be shown (OEIS A242615) that for reasons of
numerical facts, the *efficiency *of information management is the highest
when using *d=16, *which yields *n=136. *Nature also appears to use the
mathematically optimal method of information transmission.
*What we look into the material 1:* properties of the elements. We use the
set of elements created such as a kind of Rorschach cards, looking aspects
into them.
We use, next to the traditional aspects *{a, b, c=a+b}* some additional
aspects of *a+b=c *also, namely *{u=b-a, k=2b-a, t=3b-2a, q=2a-b, s=(d+1) -
(a+b), w=3a-2b},* that is, altogether *9 aspects of a+b=c. *
Users are of course free and invited to introduce additional or different
aspects to categorise logical sentences with. The *number* of aspects needs
not to be higher of 8 if they are used in combination (we refer here again
to the facts discussed in OEIS A242615), and as to the *kinds* of aspects:
one is always open to improvements.
*What we look into the material 2:* properties of the set. We impose
sequential orders on the elements of the set, using combinations of aspects.
We generate *sequencing aspects * by using always 2 of the 9 *primary
aspects, *by creating sequential orders within the set such that each of
the primary aspects is once the *first* and once the *second *ordering
aspect. That is, we sequence the set on the criteria *{ab, ac, ak, au, …,
as, aw, ba, bc, bu, …, bw, ca, cb, …, cw, ka, kb, …, …, wt, ws}. * This
brings forth 72 sequential enumerations of the elements of the set. Of
these, about 20 are actually different. (The inexactitude regarding the
number of identical sequential enumerations has to do with the *sequence*
of the primary aspects and will be of fundamental importance in the course
of the applications of the model.)
The 72 different sequences the elements – of which some are different in
name only - of the set have been brought into are called the *catalogued
sequences. *These are by no means random but are as closely related to
each other as aspects of *a+b=c *can be closely related to each other. Each
of the catalogued sequences is equally legitimate and each is an implicated
corollary of *a+b=c, *now having been made explicit (=realised).
*What we observe within the material 1:* logical conflicts. We will not
ignore conflicts between place and inhabitant, inhabitant and place.
It is obvious that 2 different catalogued orders unveil logical conflicts.
If in order *αβ* element *e *is to be found on place *p1* and in order *γδ*
element *e *is to be found on place *p2*, there is apparently a conflict.
The same conflict can also be stated by using the formulation: If in order
*αβ* on place *p *element *e1* is to be found and in order *γδ *on place *p
*element *e2* is to be found, there is apparently a conflict.
We observe potentially or actually conflicting assignments of a sequential
number *{1..n}* to one and the same element of the set, in dependence of
which of the catalogued orders we deem to be actually the case. As we
decline to entertain an epistemological attitude of human decisions
creating and ceasing logical conflicts, we look into methods of solving
these potential and realised conflicts. As the two orders *αβ* and *γδ*, if
they are different, create dislocations of the elements relative to their
own conceptions of which is the correct place for an individual element to
be in, we speak of *a consolidation of dislocations* that we aim at.
*What we look into the material 3:* series of place changes. We transform
linear arrangement *αβ* into linear arrangement *γδ*.
We have observed that if we once order the set sequentially according to
sorting order *αβ* (say, e.g. on arguments: *a,b*), and then again on a
different sorting order *γδ* (say, e.g. on arguments: *b,a*), then we have
conflicting assignments of places to most of the elements. (This example is
worked through in great detail in OEIS A235647. The sequence: *{(1,1),
(1,2), (2,2), (1,3), (2,3), (3,3), (1,4), (2,4), …., (d,d)} *shows the set
to be sorted on arguments *b,a*, while the sequence *{(1,1), (1,2), (1,3),
…, (1,d), (2,2), (2,3), …, (2,d), (3,3), (3,4), (3,d), …., (d,d)} *shows
the set to be sorted on arguments *a,b.*)
The position of an element *(a,b) *in the linear sequence that has been
created by sorting the set on arguments *b,a*
is given by (1),
* x = position ( (a,b), d, (b,a) ) = b* (b-1) / 2 +
a
(1)*
The position of an element *(a,b) *in the linear sequence that has been
created by sorting the set on arguments *a,b*
is given by (2),
* x = position ( (a,b), d, (a,b) ) = d*(d + 1) / 2 – ( d – a + 1 ) * ( d –
a + 2 ) / 2 + ( b – a + 1) (2)*
where the SYNTAX is:
x resulting position
of element
position ( *(a,b)*, d, (a,b) ) elements with numerical values
a,b
position ( (a,b), *d, *(a,b) ) number of distinct categories,
resulting for number of elements in s
set *n = d*(d+1)/2*
position ( (a,b), d, (*a,b*) ) sorting order resulting from
first criterium: a, second criterium: b
We now visualise a series of *travels - bumps – pushes – travels – bumps –
pushes – travels - … - travels. *In this vivid imagination about how the
reordering, from order according to *αβ* into order according to *γδ*,
actually takes place, one starts off with the first element one encounters
in the present order *αβ *and moves it to its new place in order *γδ. *There,
the element *bumps* into that element which occupies that place in this
moment. This element is then *pushed* out of its place and *travels* to its
new place, where it *bumps *into that element which occupies its place, *pushes
*that element out of its place, which then *travels *etc. etc. etc. until
the Last element of the cycle *travels *to that place which has been
vacated by the first element one has started the reorder with. On that
place a void has been the case since the first element of the cycle had
begun its *travel.*
*What we observe within the material 2:* a web of paths. We see
tautologies, compromises and discontinuities.
Extending the concept demonstrated above on a rearrangement from sorting
order *(a,b)* into sorting order *(b,a) *onto rearrangements from any of
the 72 catalogued orders into any different of the catalogued orders, we
arrive at 72*71/2 = 2.556 *transitions, *where each transition
contains 136 *steps.
*This gives us a *Basic Table of Transitions *that has 347.616 rows. Each
row contains the data *from_order, into_order, value_a, value_b, cycle_nr,
step_in_cycle, place_from, place_to. *
Of these data, one may construct many planes and some spaces. The *place_from,
place_to *values are coordinates on a plane, of which the axes are
named *from_order,
into_order, *and the x,y values are the integers that represent the
sequential place of the element in the respective sequential orders.
In some cases, one can observe that spaces can be constructed with 3
rectangular axes. For more details on this observation, please generate the
tables on your own computer, or see the discussion in *Learn To Count In
Twelve Easy Steps [1], *or *Natural Orders [2].*
The cycles connect points of planes, in some cases: points that have
coordinates in 3 dimensions.
*What we observe within the material 3:* logical relations. The entries in
our databases are connected among each other in manifold ways, both in
commutative and also in sequential enumerations. We make use of the
algorithmic tool known as “cyclic permutations”. This tool assigns symbols
to the elements which are *both commutative and sequential. *
One has to distance oneself from the quasi-objective impression which the
sensual apparatus confers: Our senses lead us to erroneously believe that
the property of an assembly to be sequenced or to be commutative are two
mutually exclusive categories.
*DISCUSSION OF INFORMATION CONTENT*
We now come to the information content of a communication. The concept we
now put forward is an improvement on the Shannon concept. In the Shannon
understanding of data transmission, the communication points out 1 place
within a limited range and transmits the relevant value by using symbols
which can be {0,1}. In the Shannon concept, the individuality of the
element is negated: the element is understood to be identical in meaning
with the place it occupies. In the Shannon understanding, any element *i **Î
N *receives its properties by the place *i *it occupies on the number line.
Here, we also use a limited range – namely the number of elements of the
corpus of a cycle – and transmit descriptions which allow us to point out
the value of 1 element. In a visualisation, we bend the section of the
number line which Shannon imagines straight, albeit limited, until it
becomes a closed cycle. In this loop, we point out, *which* of the elements
of the loop we mean, and also, *which loops *this element is included in.
*Example: *In order to clarify the differences between the Shannon method
and the method apparently used by the central nervous system of animals, we
discuss an example with *n *pupils of a class.
When identifying a pupil in a class of *n *pupils, we can proceed with
Shannon and:
1) Line up the pupils,
2) Give each pupil 1 sequential number
3) Transmit repeatedly, which half of the line-up the pupil is
included in, until we point out the place the pupil whom we wish to point
out stands on. The identification of the inhabitant happens by means of the
place it inhabits.
One will note, hat the Shannon approach also uses a *limited length *subsection
of the number line *N*, otherwise the transmission of symbols would not be
interpretable.
Our approach would be:
1) Do not line up the pupils, but rather sort them on
a) hair colour
b) age
c) weight
d) height
e) voice pitch
f) etc. (One can read off OEIS 242615, how many properties can maximally be
used with n pupils.)
2) Establish, who would change places with whom if a linear reorder
would take place from property “x” into property “y” (in this example,
there are 5*4/2=10 reorders); which cycles the individual pupils would
belong to;
3) Transmit the cycles the element is included in, using parallel
channels;
4) The background to the element sought are those elements which are
not included in all cycles.
In the former method, one knows the number of bits needed to identify 1 of
n elements in a set. In the latter method, one can learn and one can adapt.
In the former method, the elements have no relationships among each other:
the method is well suited to transmit data that can be random. The latter
method uses the immanent facts of belonging-to among elements that exist
prior to human knowledge. This method is well-suited to transmit
observations that relate to a world, in which the rule of *a+b=c* is valid.
Shannon does not distinguish between the properties of any inhabitant of a
place on *N *and the place.
In our method, we have a clear distinction between the object, its
properties, its tendency to be place-bound and any place it might occupy.
This approach mirrors the *level of intelligence evidenced by insects*
which recognise food, irrespective of its place, and places, irrespective
of whether a food is there. In a discussion in FIS a few months ago, the
thesis was brought forward, that the human brain employs older, more
archaic regions and methods to create an inner chart of its surroundings
than the regions or methods which are used to distinguish objects.
The individuality of the object one can point out by the proposed
(“tautomatic”) method of information transmission relies not on the place
the object had occupied or is supposed to continuously occupy, but on the
relative differences it possesses to the other objects of the set.
In the example:
Object Mausi is that object which is changing places with object Hansi when
compared no more on hair colour but on voice pitch; object Lili is that
object which is changing places with objects Mausi and Clarita when being
reordered from weight into hair colour;
object *j *is included (i.),
while the set is being reordered from property *α *(induced sequential
linear enumeration) into property *β * (ii.),
together with objects * p,r,q,s *into cycle *(**α,**β,n,Cz)* (iii.),
and concurrently is included (iv.),
while the set is being reordered from property *γ* (induced sequential
linear enumeration) into property *δ * (v.),
together with objects * k,l,p,f,w,s *into cycle *(**γ,**δ,n,Cq)* (vi.),
and concurrently is included (vii.),
while the set is being reordered from property *κ *(induced sequential
linear enumeration) into property *λ * (viii.),
together with objects * b,r,t,s *into cycle *(**κ,**λ,n,Cs)* (ix.),
and so forth.
SYNTAX:
Objects *i,I,p,r,q,s,k,l,p,f,w,s,b,r,t,s *are pupils, of whom *l, p, r *and*
s *appear more than once, the object which appears in all of the corporis
is object *i*;
Properties *α,β,γ,δ,κ,λ *are categories like hair colour, weight, age,
etc.: on these properties linear enumerations are possible and are
considered to be existent;
cycle *(**κ,**λ,n,Cs) *symbolises the corpus of the cycle numbered *s *which
is to be observed during a reorder of a set’s *n *elements from a linear
enumeration according to property *κ* into a linear enumeration according
to property *λ.*
In plain speech explained, the principle is that
An individual is characterised by its rank in a number of properties (the
maximal number of relevant properties describing a collection of *n
*individuals
can be read off the function presented in OEIS A242615); its ranks in the
properties are made observable by the elements which are co-included in the
respective corpus of the differing cycles which are generated during
reorders among the properties.
In actual fact, this method of screening and filtering out individuals from
among a multitude is widely used by industry and military organisations.
There, they may filter out people who, say, are good in French, can draw
freehand, can swim well, but need not shoot precisely or have absolute
pitch hearing. In these examples of factual usage of the principle, there
exist rankings outside of the data which are applied unto the data: in the
method shown here, the rankings are implicit corollaries of the collection
which the human intellect objectifies as independent of the objects,
erroneously in the same fashion as optical illusions are erroneous.
*CONCLUSION*
We have looked properties into a collection of logical objects. We have
observed logical conflicts. The logical conflicts can be – in some cases –
consolidated by means of compromise-building. We point out that the normal,
usual state of the world as depicted by this imagination is that of a
collection in which dislocations and differences, under- and
over-densities, agglomerations and voids exist.
In case one is ready to rely on one’s abilities to think conclusively, one
will arrive at the conclusion that a world which is depicted by the logical
sentence *a+b=c *is a world in which a disorder *a priori *exists.
Furthermore, one would be inclined to assume that the property of the world
to appear disordered to us would be a constituent property of that world
which we imagine while observing rules of logic: therefore, one can assume
that a *usual, average, medium* degree of disorder exists; while the *form
and appearance* of that disorder will be changing, its *substance and
extent* remains stable. (That is: there should be assumed a global
coefficient of being disordered.)
We propose to invite a name-giving convention, which can assign descriptive
labels to the variety of quantitative and qualitative dislocations one
observes. It may turn out, that “energy”, “information”, “force”, “mass”
are as closely related to each other as readings of the statement *a+b=c*
are related to each other.
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