Beyond Reversibility and Computability and Consistency and Skepticism: Information?

Paul Grobstein
Center for Science Information Working Group
15, 22 July 2004

THREE WEEK'S WORK:
DISCOVERING THE LIMITS OF THE BOX

Physics/physicists are not in fact committed to time irreversibility - 22 June

They have and may continue to find it a useful tool in many situations but acknowledge problems with it in thermodynamics and possibly in quantum mechanics as well.

Hence, investigators of other phenomena ought not to presume that time reversibility is necessarily applicable in, or should be taken as a model for, their own inquiries.

There are important non-computable numbers - 1,8 July "Non-computable" = not "Turing-computable", ie not derivable using formal axiomatic systems as a basis assumption for exploration

"Important" in the sense that such numbers are a consequence of any formal axiomatic system adequate to support arithmetic. Perhaps also important in that such numbers can be generated in other ways and so may play a role in "reality".

Arithmetic is not "complete". Hence, investigators of other phenomena ought not to presume that either arithmetic (and, by inference, formal mathematics generally) or "Turing-computability" is necessarily applicable in, or should be taken as a model for, their own inquiries.

Formal axiomatic systems sufficiently rich to support arithmetic (and anything more sophisticated) cannot be both consistent and complete - 1 July If they are consistent, they will always be capable of producing outputs that follow from the starting conditions but cannot be inferred from the starting conditions using any given set of inference rules.

Hence, investigators of other phenomena ought not to presume that formal axiomatic systems are necessarily applicable, or should be taken as a model for, their own inquiries.

"Reality" may or may not be the playing out of a formal axiomatic system but, regardless, trying to understand it in terms of formal axiomatic systems will inevitably fail in the long run.

If "reality" were a formal axiomatic system, no set of inference rules would suffice to account for all observations even if the starting conditions were known. (Additional conjecture: no finite set of observations of the outputs of a formal axiomatic system can yield its starting conditions, and there is no assurance whatsoever of "convergence" to the starting conditions)

If "reality" is not a formal axiomatic system, then the conclusion follows directly.

Ergo ..... it is time to stop trying to "ground" inquiry/exploration on some foundation of "certainty", ie on any principle or set of principles which are not themselves subject to inquiry and change

Known tools that work are fine when and where they work but it should not be presumed they are the only tools that one will ever need.

The best known tools derive from one or another form of the presumption that there is a solid starting point or set of starting points from which everything else derives/can be understood.

The best known tools clearly generate things they are unable to handle, at least in some cases because of self-referencing.

"Maybe its time to seriously entertain the possibility that looking for a single solid starting point just isn't the right way to go, that one has to find another, different way to proceed." ... Letter to René

Computability beyond Turing-computability?

Logics/mathematics beyond formal axiomatic systems?

POSSIBLE WAYS OUT OF THE BOX:
IRREVERSIBILITY, INCONSISTENCY, INDETERMINACY, SELF-REFERENTIALITY (RECURSION) AND THE BRAIN

One Trick: Look at your starting points and see what you can change ...

"A foolish consistency is the hobgoblin ..."
(Ralph Waldo Emerson, Self-Reliance, 1841) Could one relax the demand for consistency? Is it useful? Would it yield greater completeness?

"To be, or not to be: that is the question"
Shakespeare, Hamlet

Another Trick: Take a fresh look at what evolution has created as an effective inquirer into/explorer of "reality" ....

Self-referentiality, recursion (more on the brain trick)


22 July
Click to go back to beginning on 15 July

How to proceed in face of "profound skepticism"?, ie with no fixed starting point that one can treat as unquestionable ...

Trying it out in practice

BAYESIAN INFERENCE:
WORKING WITHOUT A FOUNDATION?

AN UPDATED THEORY OF INFORMATION ?
(From the top down, hoping to meet from the bottom up in the middle)

What IS "information"?

At the top ("story teller") level

"I have two children. one of whom is a girl. What sex is my other child?"

contrast with

"I have two children, one of whom is a girl. She is the taller of the two. What sex is my other child?"

von Bayer, Information, Chapter 9

Information, to a human story-teller, is something that changes in a story the degree of uncertainty about something (alters Batesian "probability", critically dependent on ability to appreciate counter-factuals). There is no "information" unless there is such a change (information is not an intrinsic property of anything; it is fundamentally relational)

At the bottom level

Information "may perhaps be defined as that which is is transformed by a decoder and used by a model to make predictions; a "decoder" is any organized matter that associates with some predictability elements of one set of organized matter with elements of another set of organized matter .... (where organized=not randomly distributed)".

"Information" can exist without humans, without counter-factuals. It is any organization of matter/energy that affects, in an at least somewhat non-random way, the organization of other organized matter/energy.

Between the "active inanimate" and "story tellers"

models =? non random distributions of decoders that to some degree "represent" and "predict" their surroundings (without any counter-factuals, representations of alternatives to their surroundings), ie are "adapted" - have "shapes" that fit other "shapes" (antibodies, DNA) - can be self-referential

model makers =? as above except with added capability to become further modified by their own activity as decoders

humans (other model makers?) =? as above except with added capability to represent/use counterfactuals (Batesian inference can exist in model makers but doesn't there represent to the entity "uncertainty"). Implies (working from bottom up) that "information" is something that "fits", hence that in human thought there is something analogous to "shape", something that defines "spaces". Can be origin of "exploration"?

To continue:




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