[Fis] A rational definition of information
Louis Kauffman
loukau at gmail.com
Mon Mar 9 07:19:18 CET 2020
Dear Folks,
It may help to recall two modes of mathematical definition.
In Euclid we find definitions in the form:
Def. 1.1. A point is that which has no part.
Def. 1.2. A line is a breadthless length.
Def. 1.3. The extremities of lines are points.
Def. 1.4. A straight line lies equally with respect to the points on itself.
In present day mathematics we find definitions such as
A group G is a set equipped with a binary operation that is associative and has an identity and inverses.
Any set satisfying these requirements is a group.
To one who already knows some geometric ideas, Euclid’s definitions are quite informative of his point of view about the idealization of experience that is the geometry.
From the modern point of view Euclid’s definitions are not regarded as definitions at all, but rather as motivating statements. All of Euclid’s terms point, line, etc are seen as “undefined terms” whose relations are
the only subject matter of the geometry. This abstract point of view is different from the way the Greeks thought about geometry, but once you have the idea of geometry, the abstract point of view is useful for keeping track of the logic of things. Also, all mathematical consequences in modern geometry are just logical tautologies based on the axioms. It is all seen in a context of emptiness of everything but the given relations such as “two distinct points determine a line” and logical consequences. All the “information” in the geometry is contained in its axioms, but must be unfolded by logical discourse (the proofs) in order to be seen, or comprehended.
There is genuine information in the geometry nevertheless. For example, the Pythagorean Theorem tells us something about right triangles that we would not see directly just looking at the axioms for the geometry, or empirically by examining right triangles. The theorem gives us understanding that the Theorem will hold for any right triangle and this is information of a high order.
I point this out to reinforce Mark’s statement that
"Defining is a process of seeking abstract principles which are generative not only of phenomena themselves, but of our narrative capacities for explaining them and our empirical faculties for exploring them.”
I suspect that he is thinking about mathematics when he makes this statement.
Nevertheless, we find something different in the empirical domain.
We do not demand that our abstract principles generate the phenomena there.
In fact we find that concept and percept arise together in the examination of phenomena
and that it is in this arising, with the help of thinking and the fundamental circularity of thought knowing thought,
that we come to agree that information is present.
Very best,
Lou Kauffman
I
> On Mar 8, 2020, at 12:25 PM, Mark Johnson <johnsonmwj1 at GMAIL.COM> wrote:
>
> Dear Karl, all,
>
> Do we need a definition of "defining" first?
>
> This is my effort:
> "
> Defining is a process of seeking abstract principles which are generative not only of phenomena themselves, but of our narrative capacities for explaining them and our empirical faculties for exploring them."
>
> Perhaps others can do better.
>
> Is "Information" a phenomenon to be explained - for which a set of generative abstract principles are sought?
>
> Or is "Information" the process we are in as we seek generative abstract principles?
>
> I find myself thinking the latter. I think we can (and will) get better generative abstract principles. But not from where we are standing at the moment.
>
> The distinction between "explanandum" and "explanans" must at least in the first instance be clarified. But it is worth noting that neither of those ancient terms addresses the process of performing them. We need a different kind of language.
>
> Best wishes,
>
> Mark
>
>
>
>
>
> On Sun, 8 Mar 2020 at 11:52, Karl Javorszky <karl.javorszky at gmail.com <mailto:karl.javorszky at gmail.com>> wrote:
> A rational definition of information (20200307)
>
> Semantic description of the meaning of the term information
> Information is understood to relate to the framework, background of the assertion. That, what is the case, is imbedded in such surroundings, to which that what is the case, contrasts. Information relates to that, what we do not focus on, such parts of the stage, which are not highlighted. In this sense, information is the description of the remaining alternatives, of that what is not the case. This property of the mental construct information makes it so complicated and tricky to get a clear definition of the concept: as we want to investigate that, what remains in the half-shade or shade, we cannot avoid directing the torch of our intellect towards the subject to be investigated, therefore removing it from the half-shade or shade.
>
> Information, definition of
> a.: semantic
> The sentence
> It was Peter who did it
> is a statement and contains no information
>
> The sentence
> It was Peter, from among {Peter, Paul, Susan, Mary}, who did it
> contains the information {Paul, Susan, Mary} did not do it
>
> b: formal
> The sentence
> Let x = ak
> is a statement and contains no information
>
> The sentence
> Let x = ak and k Î {1,2,...,k,...,n}
> contains the information k Ï {1,2,...,k-1,k+1,...,n}
>
> Information, semantic examples
> We watch a group consisting of Alma, Bella, Cesar, Daniel.
> x = ak
> k Ï {1,2,...,k-1,k+1,...,n}
> Cesar does the cooking today
> The other three can go out;
> It is undecided, who among A,B,D will do the dishes
> Bella got arrested
> That will be the talk among A,C,D
> Daniel deceived us all
> We shall build an alliance; we are a natural alliance
> Alma can’t leave poets and composers alone
> If you need someone with moral rectitude, ask for any of us.
>
> We observe the assignment of a symbol of assertion to 1 of the elements in a group of 4, which leaves the remaining 3 elements being assigned a symbol of negation. One will find using the notation of (1,3) practical for the above examples.
>
> (Assertion, negation) : (inclusion, exclusion) : (is the case, remaining alternatives)
> The notation (n1,n2) (where n1 + n2 = n) is a description of the informational value of an assertion. It has two values. The concept is well known in test theory, where we speak of item discrimination characteristics.
>
> Extracting information by using double negation
> The systematics presented here allows integrating the so-called Shannon school of information, by showing that the Shannon concept is but a simple, degenerated, trivial special case of the general principle expressed here. If we possess a symbol set of {0,1}, then 0 is “not among those which are different to 0 /are not 0/” and 1 is “not among those which are not 1 /are different to 1/)”
> The logical operators are {same, different} and {included, excluded}. For organisms that possess a memory, the {included, excluded} property is but a variant of the {same, different} property of constituents of perception.
>
> Relevance for calculation of properties of informational processes in Nature
> The approach of using the “remaining alternatives” method for modelling processes of Nature stands and falls with the availability of a catalogue of alternatives, of which always some remain. This is no problem, as we consider that processes of Nature are periodic, at least those which we experience on this here planet we inhabit.
> Periodic processes consist of a limited number of distinct states, which follow each other in a strict sequence. In the following example, we shall speak of the periodic changes caused by the rotation of the Earth, using the notation D/N for day/night changes, and of the periodic changes caused by the movements of the Moon, using the notation H/L for high and low tide.
>
> To be well ordered
> Let us imagine a collection of some proto-somethings that are dissolved in a small niche in the sea. The things will become adapted to their surroundings by being ordered according to the ordering principle D/N. Concurrently, the things will become adapted to their surroundings by being ordered according to the ordering principle H/L, also. For the argument’s sake, we shall assume that being optimally ordered for the changes coming from day/night will be different to that order, which is optimal for the changes coming from the tides.
>
> Concurrent ordering requirements
> Some of the learned friends here will remember that the term cycles had already been mentioned a few times in this excellent chat room. Cycles appear as movement patterns and as constitutors of groups of elements (members of the corpus of the cycle) which move together during a resequencing. The corpus of the cycle contains elements which share the common symbol of the cycle, and also symbols that are sequential, depicting the strict sequence in which the members of the corpus follow each other during a resequencing.
> The proto-somethings that are subject to both D/N and H/L periodic changes will inevitably fall into cycles during the reorders that are the consequences of the periodic processes: day/night and high/low tide. The paths created by the cycles remain unchanged for millions and billions of repetitions of identical rearrangements. One can well speak of a stable collection of possible states the collection of the proto-somethings are in. Here we have the collection of remaining alternatives, that is: an environment with information content.
>
> Variations on a theme by Nature
> The re-arrangements of the elements of the collection of proto-somethings are in principle all alike. Yet, their appearance may have a wide range of varieties. This comes from two main causes:
> a. differing numbers of elements in the corpora: this causes appearances of faster /slower changes within sub-systems of one and the same system;
> b. differing offsets causing differing collections of elements to be contemporaneous: this causes appearances of types and individuals among the realisations.
>
> Closing remarks
> One would wish for the mechanics behind the Laws of Nature to be more in accordance with our imaginations about, how a self-evident, self-explanatory set of interdependences of rules will look like. Unfortunately, sequencing has to do with sequences, and these deal with order and changes of places. The pictures arising from re-ordering mental concepts can cause some inner disturbances, the more so for learned friends who are used to the idea that they have their mental concepts in the best order possible, and any re-arrangements can only be irritating. Let me end on an encouraging note: once one has accepted the thought, that properties and order determine places, the underlying mechanism of wheels, cogs, belts, valves and pistons creates a highly fascinating and educative inner entertainment. Good luck!
>
> Karl
>
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> --
> Dr. Mark William Johnson
> Institute of Learning and Teaching
> Faculty of Health and Life Sciences
> University of Liverpool
>
> Phone: 07786 064505
> Email: johnsonmwj1 at gmail.com <mailto:johnsonmwj1 at gmail.com>
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