[Fis] Problems with a math definition of information

Dear Fis,

I. structured presentations

let me pick up the suggestion of some contributions ago that each of us presents his/her concept of what is info. Maybe the following proposal for a structured questionnaire could be of help:

1) origin of concept (what other concepts does it link to/root in?)

2) explanational advantages (which phaenomena can be specifically well explained by it)

3) static or dynamic (will the info content remain the same or does it grow/shrink)

4) organised or self-evolving (is there an external cause, reason, purpose, design; is it made or has it grown)

5) is there an ultimate state in the process (entropy, paradise, well-ordered homeostasis, etc.)

6) cyclical or linear (does it have recurring states or no/a few/only recurring states)

7) other remarkable properties of the concept.

If the Fis peer-group agrees to it, I shall present my concept of information according to these main lines in the next contribution.

II. definitorial difficulties with the concept of information within mathematical logic

In “classical”, traditional mathematical logic, there is no place for the concept of information. This because, as Wittgenstein has put it, “mathematics is in itself a system of tautologies”. The idea of information (“pointing out of one alternative among several”) presupposes, however, that there are alternatives, thence that it could be otherwise. In a tautological system there can be no “otherwise”.

The information is subjectively experienced as the release of a tension and technically understood to close a phase of ambiguity. During the phase of the tension (during the period of ambiguity) the future can “turn out” / “become” reality. In dependence of which of the alternatives gets pointed out, different realities (subsequent states) will evolve. But a tautological system cannot have any varieties, because in such a case it would not be tautological but indeed have an informational content.

The ground, basic belief in mathematical logic is that it is eternal (not time-dependent), ubiquitous (valid everywhere) and all-pervading (not application-bound, true by and in itself). So the system is tautologic, meaning it contains no news and cannot contain anything new (because all is in it ever since and shall remain in it for all times).

III. Solution: make math intrinsically wrong

Only after some quite serious hair-splitting has it been possible to make mathematical logic more amenable to reality, specifically biology. It is necessary to allow – within mathematics – for something to be possibly otherwise. There must be an intrinsic contradiction within the system so that it can meaningfully state <it> is <so> where this sentence is not a tautology. (if <it> could not be possibly otherwise but <so>, it would remain in a system in which each and all alternatives are known and fit together free of contradiction. That approach allows depicting mathematics as an in itself closed, thoroughly true system, but by this tool we could not picture something that grows and evolves. A system in which all is included cannot grow or change.)

Only in that case if we allow mathematics to be intrinsically inconsistent can we hope to model a Nature in which we see that it is in quite many fashions inconsistent.

Therefore we have to find a way to present mathematics as a system of references which consist of subsystems wherein each subsystem is contradiction-free and true within itself, but the subsystems do not fit well together. (Like the law systems of national states in Europe: each legal system is within itself consistent but they do not fit together in some aspects.)

IV. Looking at a set of objects with symbols on them

The hair-splitting referred to above brings forth this welcome inner inconsistency within mathematics. The hair-splitting is done by putting a dozen or so objects on a table before you and wondering whether these objects have more in common or more un-common (diversity). How diverse and how similar are the objects? is the question by which one finds basic contradictions within mathematics. I have mentioned that the differences are not that obvious, they come to about 1/4-th of 1/10**160. This is a number with 0,00…here come 160 zeroes …03 or so.

But, this difference allows a mathematical approach to the observation from physics that objects and logical relations can translate into each other. (The physicists call objects “matter” and logical relations “energy”.) Also, this difference makes it meaningful to ask: “is the spatial neighbourhood distribution more improbable than the qualitative distribution?” while regarding a collection of objects with symbols on them.

Due to the surprisingly complex system of sur-, in- and bijectivity relations of logical relations to be found on a collection of objects with symbols on them, in dependence of whether one sees the objects one-after-the-other or all-at-the-same-time (laying more emphasis on differences or similarities in the process), one finds a quite satisfying mbols on them. ical relations can translate into each other. e objects have more number of possible explanational approaches to a wide variety of applications, including theoretical genetics and artificial intelligence.

Conclusion:

For the idea of information to be integrated into the mathematical system of thoughts, it is necessary that the mathematical system of thoughts allows for internal contradictions. The internal contradictions can come about by applying both methods of evaluating an assembly: one after the other or several groups together.

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Received on Tue Sep 6 11:14:44 2005