The Candidate Laws for CoEvolving Systems

Or, as Stuart Kauffman puts it:

Four Candidate Laws for the CoConstruction of a Biosphere

• <i>Candidate Law #1: The Dynamical Edge of Chaos</i> -- Communities of autonomous agents will evolve to the dynamical "edge of chaos" within and between members of the community, thereby simultaneously achieving an optimal coarse graining of each agent's world that maximizes the capacity of each agent to discriminate and act without trembling hands.
• <i>Candidate Law #2: Community Assembly Reaches a Self-Organized Critical State</i> -- A coassembling community of agents, on a short timescale with respect to coevolution, will assemble to a self-organized critical state with some maximum number of species per community. In the vicinity of that maximum, a power law distribution of avalanches of local extinction events will occur. As the maximum is approached the net rate of entry of new species slows, then halts.
• <i>Candidate Law #3: Coevolutionary Tuning of Fitness Landscapes and Organisms to a Self-Organized Critical State</i> -- On a coevolutionary timescale, coevolving autonomous agents as a community attain a self-organized critical state by tuning landscape structure (ways of making a living) and coupling between landscapes, yielding a global power law distribution of extinction and speciation events and a power law distribution of species lifetimes.
• <i>Candidate Law #4: Expanding the Adjacent Possible in a Self-Organized Critical Way</i> -- Autonomous agents will evolve such that causally local communities are on a generalized "subcritical-supracritical boundary" exhibiting a generalized self-organized critical average for the sustained expansion of the adjacent possible of the effective phase space of the community.

If these four Laws can be expressed in metamathematical terms, it would be interesting to see if they can be proven as theorems in Goedel's metamath.

The Gottman Hypothesis

In <i>The Mathematics of Marriage</i> Gottman et al. argue convincingly that the necessity of non-linear systems is demonstrated by the instability of linear systems, specifically the inability of linear systems to resolve into stable set points. It would be interesting to see if such arguments could be extended to these candidate laws.

See also Math and The Goedel Connection.

— Scotus - 26 May 2002