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International Conference on Complex Systems (ICCS2006)

Complexity in the Paradox of Simplicity

Jonathan Post
Computer Futures, Inc.

Philip Fellman
University of Southern New Hampshire

     Full text: Not available
     Last modified: March 31, 2006

Complexity in the Paradox of Simplicity
Jonathan Vos Post
Computer Futures, Inc.
3225 North Marengo Avenue
Altadena, CA 91001
(626) 398-1673
Draft 6.0; 34,700 words; 100 pages, of noon 11 Oct 2005


David Hilbert's long lost “24th problem” [Thiele] was intended to clarify the notion that for every theorem, there is a “simplest” proof. His grand program was demolished by Godel, but his problems, upon solution, bestow instant success and even immortality to the solver. His notion of “simplest” raises key questions for the 21st century (when computerized automated theorem proving has solved some famous problems but created debate as to what constitutes proof), these questions capable of analysis in the domains of Complexity and the Philosophy of Science. In particular, given multiple definitions of “simplest,” involving differing definitions of, usage of, and justifications for elegance and (qualitative and quantitative) parsimony, by what meta-criterion do we choose the simplest of those? And how are our hands tied by neurological and psychological limitations on our ability to introspect on how we choose (i.e. “choice blindness), and how automated theorem provers operate? Are we, in Zeilberger’s phrase, “slaves of Occam's razor?”

This paper surveys the literature of Philosophy as it pertains to Simplicity and Occam's Razor in the abstract, in Mathematics, and in Science. This survey is essentially an annotation of a survey by Alan Baker. Several new branches of Proof Theory and the Theory of Complexity, particularly from Chaitin, Hutter, Kolmogorov, Martin-Lof, Matiyasevich, Rissanen, Solomonoff, and involving the “Minimum Description Length” (MDL) formalism are explored in this context, yielding some new axiomatic and computational approaches to Simplicity which avoid decidability and computability limits of the general problems of Complexity.

[end abstract]

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