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Beyond Incompletenss

Paul Grobstein's picture

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: Demarcation, Mind, Chance

The place I would like to get to and why ... (updated in italics)

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. 

Generalizing Gödel

All humans use formal systems whether they are aware of it or not and to the extent they do are accordingly constrained in what they can explore

there is no absolutely justifiable ("fair") grading system nor any absolutely defensible ("virtuous") way of behaving

there is no "infinity" meaning everything/all possibiliities; there are countable infinities and successively larger ones without end

"The more philosophers I read, the clearer it seemed to me that each of them could carry their views back to first principles which were incompatible with the first principles of their opponents, and that none of them every got to that fabled place 'beyond hypotheses'. There seemed to be nothing like a neutral standpoint from which these alternative first principles could be evaluated." ... Rorty

"Of that which we cannot speak we must remain silent" ... Wittgenstein

the "inexpressible" = that which cannot be generated by a formal system

All formal systems capable of self-referentiality can be shown, within the systems itself to be "incomplete."  The limitation relates in principle to the fact that the inferences from formal systems are countably infinite, and so there are "inexpressibles" outside that set of inferences.  An interest in consistency as opposed to completeness may also contributes to the limitations.

"Gödel's proof should not be construed as an invitation to despair or as an excuse for mystery-mongering ... does not mean that there are truths which are forever incapable of becoming known, or that a "mystic" intuition (radically different in kind and authority from what is generally operative in intellectual advances) must replace cogent truth.  It does not mean ... that there are "ineluctable limits to human reason."  It does mean that the resources of the human intellect have not been, and cannot be fully formalized, and that new principles of demonstration foreverawait invention and discovery ...

Nor do the inherent limitations of calculating machines imply that we cannot hope to explain living matter and human reason in physical and chemical terms ... The theorm does indicate that the structure and power of the human mind are far more complex and subtle than any non-living machine yet envisaged." 

Ernest Nagel and James R. Newman, Gödel's Proof


One can, however, use an awareness of Gödel's limitations to transcend them in any given case (see Forms of Inquiry)

"The answer to the question 'What is the way the world is? What are the ways the world is?' is not a shush, but a chatter." ... Nelson Goodman

Further exploration of ways to conceive/deal with the "inexpressible"? 

To do so or not to do so ...

"More and more, the 'demarcation problem' comes to mind...what are the bounds of science, nonscience, and pseudoscience?...what is it and is it not that science can inform us about the natural world? Discussions of 'mindless' or 'soulful' seem to be attaching values to actions ... My intuition tells me that such analogies of 'mindlessness' and 'soulfulness' are falling into the realm of pseudoscience"  ... Mike

"why is demarcation the thing to do?  I've been thinking ... about how so often in institutions, and academia, people set up categories, departments, programs and then spend so much time defining what is inside and what is outside of them.  In this case, demarcation is a problem, not a solution. " .... Alice

"one place my mind went wandering to this morning, when Alan asked Paul what he meant by "mindless," and Mike asked what alternatively "soulful" activities he had in mind" .... Anne

"My intuition tells me that such analogies of 'mindlessness' and 'soulfulness' are falling into the realm of pseudoscience, if we are trying to use formal systems to justify them" ... Mike

Is there a way to speak of "mind" and "soul"  that brings them into the realm of "formal systems," of the expressible, and so into the realm of legitimate and productive inquiry?  

Computers and their limitations - Turing

Beyond the computable - Chaitin

Liebnitz's principle of sufficient reason challenged: randomness

Irreducible complexity

The human bipartite brain