Beyond Incompleteness II

Notes for an Evolving Systems conversation related to

Chance: Its meaning and significance
Paul Grobstein
2 June 2010

(on line forum at /exchange/evolsys/chance10)

Beyond Incompleteness II

The place I would like to get to and why ... (updated with emphasis)

It is provably NOT the case that a full understanding of the universe, including the participation of humans trying to understand it, can be achieved by a any description in terms of an underlying set of initial principles and deterministic rules of interaction (contra Wolfram and the agendas, conscious or otherwise, of many disciplines).  The problem has to do not solely with complexity or numbers of variables or time or human proclivities but reflects as well inherent characteristics of the explanatory capabilities of logic, computability, and formal systems.  Such systems by their nature are limited in the range of understandings they can elaborate and explore. 

The significance of this limitation is not at all restricted to logic or mathematics or science.  All human thought expressible symbolically (eg all language) is subject to this limitation insofar as it reflects an "underlying set of initial principles and deterministic rules of interaction."  Conversely, formal systems themselves provide the wherewithal to move beyond their own limitations, not only for logic/mathematics/science but for human thought generally. 

It follows from this that there is a need for a less constrained approach to characterizing the objectives and methods of inquiry.  Evolving systems, with their fundamental dependence on some degree of randomness, seem to provide an example of such a less constrained approach, one that if clarified might provide a reasonable alternative to existing conceptions of the nature of inquiry.

Moving beyond a concern about "science" and "demarcation" (in order to "loop back"?)

"It seems like we are trying to "move beyond science" based on the limitations of formal systems" ... Doug

"I realize, though, from recent discussions that my optimism (that specialization and conflict across disciplinary boundaries are good things) is not necessarily shared by all. In fact, there seems to be suspicion amongst disciplines for primacy in some perceived (if not in many respects real) power structure ... a consequence of what it means to be a human individual vying for position within an evolving social system. Hopefully, the conversation can move beyond this awkward aspect of human nature, and maybe acknowledge where demarcation can be a good thing and something that we can all take advantage of to become better learners and teachers." ... Mike

" I’d like to draw on, and offer others, what is useful in this story, but not be limited to or by it. I can imagine that for others, there may be parallel stories -- perhaps about religion as authoritarian/suffocating, or about belief as a limit on intellectual growth -- that could orient and re-orient -- and be re-oriented by -- the investigation we are sharing." ... Alice

"We should be open-minded to the anomalies in our worldview and the successes of alternative viewpoints.” ... Joel Brown, via Mike

"How many of us became (and continue to practice as) academics, in search of the (illusive?) security that formal systems offer? How many of us come to recognize it that search as illusory, and make adjustments....?" ... Anne

"science does try to restrict the definitions [of words]. To try to frame this in the conversation of formal systems, a system cannot be built if the terms on which the system is constructed have multiple meanings ... exploring the different meanings of words might be very important in other disciplines ..." Mike ("Against ambiguity")

Progress to date (PG's story)

Any general theory of inquiry must both include and go beyond an understanding/appreciation of both the values and the limitations of formal systems (systems in which conclusions can be reached "justifiably" and "mindlessly" by a fixed set of starting conditions and rules of deduction from them).  All inquiry uses formal systems, whether inquirers are aware of it or not.  And all formal systems are "incomplete" whether inquirers are aware of it or not.  One way to move beyond particular formal systems is to appreciate their existence and value, use one's awareness of a fixed set of starting conditions/rules as the starting point for creating alternative formal systems/understandings by changing one or more of the starting conditions/rules. 

Is there a way to speak of values and ambiguity, and of "mind" and "soul,"  that brings them into the realm of the expressible, into the realm of legitimate and productive inquiry?  Might a recognition of the limitations of any given formal system help us to do? 

"Computers" as formal systems and their limitations - Turing

• The Turing machine as a formal system
• algorithms as deterministic and  "mindless"
• universal Turing machines
• The Church-Turing hypothesis
• Non-computable: the halting problem
• The brain as a Turing machine? (Warren McCulloch and Walter Pitts, "A Logical Calculus of the Ideas Immanent in Nervous Activity," 1943; Roger Penrose, The Emperor's New Mind, 1989)

Beyond the computable - Chaitin

• Liebnitz's principle of sufficient reason challenged: randomness
• Irreducible complexity
• Randomness and "pseudo" randomness

The human bipartite brain: beyond Turing machine limitations?

• randomness (pseudo?)
• ability to generate alternative outputs for the same input
• ability to challenge first principles and methods, including "principle of sufficient reason"
• ability to make productive use of inconsistencies - multiple encapsulated form systems

Inquiry is the effort to achieve "understanding" defined as

• mirroring of nature
• achieving oneness with the ... sublime, supernatural, etc
• a transient state of agreement among multiple unconscious, conscious, and collective understandings, from which it follows that both the products and methods of inquiry are always local, context-dependent, and subject to revision, that skepticism/interest in the as yet to be conceived is the only consistent feature of method, and that randomness, the aleatory, stochasticity is a desireable feature rather than a bug.